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\section{Introduction}
Spallation reactions, i.e. proton-induced reactions on heavy targets at a few hundred MeV, have been the subject of many studies since 1950. They are known to be a valuable tool for the study of the de-excitation of hot nuclei because, contrarily to reactions between heavy ions, they lead to the formation of hot prefragments with only a limited excitation of the collective degrees of freedom such as rotation or compression. Their study has also been motivated by astrophysics, as cosmic rays undergo spallation reactions with the hydrogen and helium nuclei of the interstellar medium.
Recently, progresses in high-power accelerator technologies have made possible the realisation of intense neutron sources based on spallation reactions. Such sources are needed for Accelerator-Driven Systems~\cite{Bowmann,Rubbia}, and also find applications in nuclear physics~\cite{NToF} and for material physics and biology~\cite{ESS}. Furthermore, spallation reactions can also be used to produce exotic nuclei and, hence, secondary beams~\cite{ISOLDE,EURISOL}. Those new applications have motivated a large number of experimental works and created a strong demand for high-precision calculation codes.
In order to bring new data and therefore new constraints for the codes, measurements of reaction residues have been undertaken at GSI by an international collaboration. These experiments are based on the inverse-kinematics method. Fragments are identified in-flight using the FRS spectrometer~\cite{FRS}, making possible the first measurements of complete nuclide distributions. In the frame of those studies, production cross sections have already been published for several systems: Au+p at 800A~MeV~\cite{Au_Fanny,Au_Pepe}, Pb+p at 1A~GeV~\cite{Wlazlo,Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, U+p at 1A~GeV~\cite{Taieb,Monique} and U+d at 1A~GeV\cite{U_Enrique,U_Jorge}.
These results have helped to partially discriminate between the respective influence of the two main steps of the spallation process, the intranuclear cascade and the fission/evaporation process. The behavior of several codes (the ISABEL~\cite{ISABEL}, INCL4~\cite{INCL4} or BRIC~\cite{BRIC} intranuclear cascades, the ABLA fission/evaporation code~\cite{ABLA}) has proved to be now overall satisfactory for proton energies around 1 GeV. On the other hand, important failures in the description of the emission of charged particles in the Dresner evaporation code~\cite{Dresner} have been put in evidence~\cite{Au_Fanny}. In the 1 GeV energy region, the only serious, remaining deficiency is the underestimation of the lightest evaporation products, which are related to the most violent collisions. Despite large differences in the description of the spallation process, all the codes mentioned above present this weakness. This indicates that some phenomena have not been taken into account. In recent experiments conducted at GSI on lighter nuclei ($^{56}$Fe, $^{136}$Xe), our collaboration explored a range of nuclear temperatures higher than in the systems mentioned in the previous paragraph. Indications were found that fast break-up decay may play an important role in high-energy spallation reactions~\cite{Paolo}.
The question of understanding of the evolution of the reaction mechanisms with decreasing energy also remains open. This is an important point in the perspective of technical applications, because nuclear reactions in thick targets happen in a broad energy range: beam particles are subject to electronic slowing down, and also fast particles emitted in the first stage of the reactions are likely to produce additional nuclear reactions, giving rise to an {\it internuclear} cascade. To address the question of the dependance of the reaction on the energy of the incident particle, an experiment has been performed at GSI aiming at measuring production cross sections of residues in the spallation of lead by protons at 500A~MeV.
The present paper deals with the experimental results on the fragmentation-evaporation residues obtained in this experiment. It completes the results already published obtained during the same experiment for the fission products~\cite{NPA_Bea}. Detailed confrontations between the results from codes dedicated to the description of the spallation process and these data as well as other data on evaporation residues obtained by our group and other related measurements on spallation reactions like light particle production are postponed to a forthcoming paper.
The energy chosen for this experiment, which is low in comparison to the FRS standards for experiments involving nuclei as heavy as lead~\cite{achromat}, was a source of difficulties for the identification of the fragments in the spectrometer. The modified setting and the analysis methods developed for this experiment have been presented in a dedicated paper~\cite{NIM_loa}. We will briefly recall their main features in section~\ref{chap:setup}. We will then discuss in section~\ref{chap:secondary} the influence of multiple reactions taking place in the liquid hydrogen target and modifying the observed fragment production, and the method which has been employed to remove their contribution. In section~\ref{chap:kinematics} we will present the results on the reaction kinematics. Finally, in section~\ref{chap:results} we will present the production cross sections.
\section{Experimental setup and analysis process} \label{chap:setup}
The GSI synchrotron (SIS) was used to produce a 500A~MeV $^{208}$Pb pulsed beam with a pulse duration of 4 seconds and a repetition time of 8 seconds. The beam was sent onto a 87.3~mg/cm$^2$~liquid hydrogen target~\cite{target} located at the entrance of the FRagment Separator (FRS). The target window consisted of two 9~mg/cm$^2$~Ti foils on each side. The beam intensity was monitored during all the experiment by a beam-intensity monitor~\cite{SEETRAM}. In order to maximise the proportion of fully stripped fragments in the spectrometer, a 60~mg/cm$^2$~Nb foil was placed after the target.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{frs_2.eps}
\caption{Schematic view of the FRagment Separator. Each magnetic section between focal planes (the target location, $S_2$ and $S_4$) consists of two dipoles plus several quadrupoles and sextupoles (the latter are not represented here as they were not used during this experiment).}
\label{fig:FRS}
\end{center}
\end{figure}
Fragments were identified in-flight using the FRS spectrometer (see figure~\ref{fig:FRS}). The rigidity of the fragments in each of the two magnetic sections is given by:
\begin{equation}
B \rho = \frac{m_0 c}{e} \frac{A}{q} ( \beta \gamma )
\label{eqn:brho}
\end{equation}
where $B$ is the magnetic field, $\rho$ the curvature radius of the fragment trajectory, $A$ the mass number, $q$ the ionic charge of the fragment, and $\beta$ and $\gamma$ are the Lorentz relativistic coefficients. The presence of $q$ in equation~\ref{eqn:brho} is a critical point, as a large part of the fragments produced at the energy of 500A~MeV chosen for this experiment were not fully stripped.
In order to measure the nuclear charge ($Z$) of the fragment, 4 MUlti-Sampling Ionisation Chambers (MUSIC; see~\cite{MUSIC} for complete description) were placed at the exit of the spectrometer, each one filled with 2 bar of P10 gas mixture (90\% Ar, 10\% CH$_4$). The total gas thickness was 800~mg/cm$^2$, with the measurement of the energy loss effectively performed only in some 2/3 of the length of each chamber. Using such a large gas thickness was necessary in order to maximise the charge exchanges (electron pick-up and stripping) of the fragment in the gas, thus washing out the influence of the incoming ionic charge state of the fragment on the energy loss, and ensuring that the latter represents the nuclear charge of the fragment with sufficient resolution~\cite{NIM_loa}. The obtained resolution, $\Delta$Z{\it (FWHM)}/Z, ranged between 0.6\% for the lightest fragments and 0.9\% for the heaviest.
The horizontal position of the fragments at the intermediate and final focal planes (respectively $S_2$~and $S_4$; see figure~\ref{fig:FRS}) was measured using 3~mm thick plastic scintillators. The signals of these detectors were also used to measure the time of flight of the fragments in the second part of the spectrometer. The $A/q$~ratio of the fragment was deduced from the combination of these measurements, according to equation~\ref{eqn:brho}.
The thick aluminium degrader (1700~mg/cm$^2$) located at the intermediate focal plane ($S_2$) was used as a passive energy-loss measurement device. As the energy loss can be related to the variation of the magnetic rigidity and the ionic charge states in the two magnetic sections, the latter may thus be deduced from the energy loss as obtained from the nuclear-charge identification and the velocity measurement~\cite{NIM_loa}.
The resolution obtained in this measurement was not sufficient to discriminate the ionic charge states on an event-by-event basis. The only information obtained was the charge-state changing between the first and the second part of the spectrometer, an integer value that we will note $\Delta q$. Besides, the evaluation of the variation of magnetic rigidity was also used to reject fragments that underwent a nuclear reaction at $S_2$.
The mass of each fragment was determined assuming that its number of electrons in the FRS was the minimum required by its $\Delta q$ value~\cite{NIM_loa}. The production rate for each nuclide was then calculated by constructing its full velocity distribution in the first part of the FRS, using formula~\ref{eqn:brho}. For many nuclides, the momentum width was larger than the momentum acceptance of the FRS ($\pm1.5\%$); in this case several settings of the magnets were used in order to cover the full momentum distribution of the fragment. Due to the hypothesis made on the ionic charge state, some fragments were misidentified; the corresponding correction factor for the production rates was deduced from ionic charge-state probabilities calculated using the code GLOBAL~\cite{GLOBAL}.
The above procedure was performed separately for each group of fragments characterised by a given $\Delta q$ value. The probability of each $\Delta q$ value was then deduced from the scaling factor necessary for all isotopic distributions obtained for a given element to match with each other. The obtained values were found to be in good agreement with the GLOBAL calculations (discrepancies lower than 10\% for the most abundant ionic charge-state combinations, and less than 20\% for other combinations)~\cite{NIM_loa}.
The production rates were corrected for the losses in the different layers of matter located in the path of the fragments after the target area (degrader, plastic scintillators, MUSIC chambers). The total reaction cross sections were calculated using the Karol optical-model-based code~\cite{Karol}. Losses were found to be of the order of 30\%, including reactions in the MUSIC chambers (the latter being characterised by signals of first and last chambers being improperly correlated). The reaction rates in the different layers of matter are presented in table~\ref{table:reac_prob}. The dead time of the acquisition and the detector efficiencies were also taken into account. Fragment losses due to limited angular acceptance of the FRS were found to be negligible. All the measurements and the analysis procedure above were repeated with an empty target, and the resulting production rates were subtracted from the total production rates. Finally, the production cross sections were obtained by normalising the production rates to the number of atoms in the liquid hydrogen target, which had been measured in a previous experiment, and to the beam intensity.
\renewcommand{\arraystretch}{0.4}
\begin{table}[ht]
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& & & & & \\
Layer & Target windows & Stripper & Scintillator & Degrader & MUSICs \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Focal plane & \multicolumn{2}{c|}{$S_0$}&\multicolumn{2}{c|}{$S_2$}& $S_4$ \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& & & & & \\
Reaction & 2.1\% & 2.1\% & 8.4\% & 16.6\% & 4.9\% \\
probability & & & & & \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Method of & \multicolumn{2}{c|}{{\it none}} & \multicolumn{2}{c|}{$B\rho$ change} & $\Delta E$ change\\
rejection & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
Method of & \multicolumn{2}{c|}{Dedicated measurement} & \multicolumn{3}{c|}{Calculation} \\
correction & \multicolumn{2}{c|}{(empty target)} & \multicolumn{3}{c|}{(Karol model)} \\
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
\hline
\end{tabular}
}
\caption{Reaction probabilities of the beam ($^{208}Pb$ at $500A~MeV$ at the entrance of the FRS) in the various layers of matter of the FRS beam line. Rejection method of the formed nuclei and correction method of the production rates are also mentioned. See text for details.}
\label{table:reac_prob}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
\section{Secondary reactions in the target} \label{chap:secondary}
Any fragment formed in a collision with a proton of the target may undergo additional nuclear collisions, in the target as well as in surrounding material (target window, stripper foil). These secondary reactions are expected to play an important role in the production of nuclides far from the projectile: at relativistic energies, proton-induced reactions mainly produce nuclei lighter than the heavy partner of the reaction, therefore in most cases multiple reactions will remove more nucleons than a single reaction.
There is no way to identify such multiple reactions during the analysis process. Therefore, one has to unfold their contribution by using calculated reaction cross sections or by performing a self-consistent calculation. This section is dedicated to the presentation of a new method developed for this experiment aiming at estimating the contribution of the multiple reactions with a high precision while minimising the input from codes.
\subsection{Unfolding method}
The production cross section of a nuclide $f$ from projectile (indexed as $0$ in the following) is written as:
\begin{equation}
\sigma_{0\rightarrow f} = \frac{e^{\frac{\sigma_0+\sigma_f}{2}x}}{x} \;
\left(T_f(x) - \frac{x^2}{2} \sum_{A_0 < A_i < A_f}^{} \;
\sigma_{0\rightarrow i} \; \sigma_{i\rightarrow f} \;
e^{-\frac{\sigma_0+\sigma_i+\sigma_f}{3}x} \right)
\label{eqn:secondary}
\end{equation}
where $\sigma_i$ is the total reaction cross sections of a nuclide $i$ on a nuclide of the target, $\sigma(i,j)$ is the production cross section of a nuclide $(A_j,Z_j)$ from a nuclide $(A_i,Z_i)$, $T$ is the observed production rate, and $x$ is the thickness of the target. A derivation of this equation is presented in appendix~\ref{chap:sec_calc}. This corresponds to a first-order approximation (i.e. only double reactions in the hydrogen are taken into account), but it is easily extended to higher orders. In our calculations, we actually accounted for the second order reactions: triple reactions in hydrogen, and reactions involving one reaction in a target window and one in the hydrogen (or the reverse). We found that those second order terms actually accounted for less than 20\% of the multiple reactions.
Solving this system of equations (one equation for each observed nuclide) requires the calculation of all the $\sigma$ terms. For the total cross sections, several reliable codes exist; we used the optical-model-based code of Karol~\cite{Karol}, the same one we used for the probability of nuclear reactions at the intermediate focal plane of the FRS. For the partial cross sections involving heavy target nuclei (target windows and stripper foil), the EPAX parametrisation~\cite{EPAX} offers reliable results with minimal calculation time. In the case of proton-induced reactions, the Monte-Carlo cascade-evaporation codes would seem an obvious choice, but they could hardly be used here for two reasons. First, the calculation required to evaluate the hundreds of possible reactions would have been very time consuming. Second, as one of the goals of this experiment was to produce data to constrain these codes in this poorly-known energy region, the use of those codes might have introduced an artificial consistency between the data and the codes.
In order to calculate the isotopic cross sections, we can decompose each cross section of proton-induced reactions in a product of 3 factors:
\begin{equation}
\sigma(x,y) = \sigma_x \; P_A((A_x,Z_x) \rightarrow A_y )
\; P_Z((A_x,Z_x,A_y)\rightarrow Z_y)
\end{equation}
Here, the first term is the total reaction cross section of a nuclide $x$ (we have already stated that it could be calculated using the Karol formula~\cite{Karol}), the second term is the probability to form a nuclide of mass $A_y$ from a nuclide of mass $A_x$, and the third term is the probability that the nuclide formed with a mass $A_y$ has an atomic number $Z_y$.
In order to estimate the second term, we used a property of proton-induced spallation reactions: nearly all nuclides $b$ formed from a nuclide $a$ have a mass strictly smaller than the one from $a$. For example, in the 500A~MeV experiment on $^{208}$Pb we observed no formation of any nuclide of mass 209, and nuclides of mass 208 ($^{208}$Bi) are formed in less than 0.1\% of the reactions. Furthermore, we assumed that, as far as only the probability of mass loss is concerned, the influence of the isospin of the target nuclide is weak enough to be neglected. Using these assumptions, one can solve the system of equations~\ref{eqn:secondary} isobar by isobar, in the decreasing masses order, because the term $P_A(A_x \rightarrow A_y )$ required by each equation is immediately obtained from the previously corrected data as $P_A(A_0-(A_x-A_y))$ (where $A_0$ is the projectile mass).
For the third term, this kind of simple scaling law cannot be applied because, although $P_Z$ depends only on $A_y$ for large values of $A_x-A_y$. Indeed, this well-known property defines the so-called residue corridor~\cite{Dufour}: the statistical nature of the evaporation process favors its ending close to nuclei for which the probability to evaporate a neutron and a proton (in other words, the neutron and proton separation energies), are the closest, a property which is completely independent of the entrance channel. But, in the case of short evaporation chains, the influence of the entrance channel is not suppressed; in other words, for low values of $A_x-A_y$, $P_Z$ depends not only on $A_y$ but also on $Z_x$ . This memory effect is fully taken into account in the EPAX parametrisation~\cite{EPAX}. We will describe hereafter how the EPAX formula can be used even though it is out of its energy domain applicability. Please note that, as EPAX does not take into account the fission process, this method would not be appropriate for reactions involving highly fissile nuclei such as uranium.
\subsection{Calculation of isobaric distributions using EPAX}
Some characteristics of the EPAX parametrisation~\cite{EPAX} correspond to the requirements for a multiple reactions calculation: it needs very little computing time, and it proved to be reliable, not only for reactions involving nuclides close to the stability valley, but also for proton-rich nuclides~\cite{112Sn}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{epax_a.eps}
\caption{Comparison of production rates obtained in the Pb+p at 500A~MeV experiment (dots) with calculations performed with both the standard (discontinuous lines) and a version we modified (continuous lines) of the EPAX parametrisation~\cite{EPAX}. For each isobaric spectrum, calculations have been renormalized to the data.}
\label{fig:EPAX}
\end{center}
\end{figure}
EPAX has been written in order to reproduce residues from reactions in the limiting-fragmentation regime~\cite{limit_frag}, which is reached in spallation only for projectile energies of several GeV. The mass distribution of residues formed in 1A~GeV proton-induced spallation reactions exhibits a very different shape from the one formed in the limiting-fragmentation regime~\cite{Pb_p}. Therefore, the residues from the same reaction with half the incident energy may certainly not be reproduced by the mass-loss formula of EPAX.
On the other hand, the shape of the isobaric spectra is mainly a consequence of the sequential evaporation mechanism. Therefore, there is no reason why its validity should be limited to high incident energies. To check this assumption we extracted the isobaric component of the EPAX formula and compared it to our data after proper renormalization for each isobaric spectrum. Only minor adjustments were necessary to obtain a very satisfactory reproduction of the measured production rates, as it can be seen in figure~\ref{fig:EPAX}. Such a comparison makes sense as the secondary reactions are not expected to play an important role in the production of nuclides with a mass loss of less than, roughly, 30 mass units with respect to the projectile. Unexpectedly, we observed that the charge-pickup reactions were also reasonably well reproduced by the parametrisation, despite the fact that the EPAX authors didn't take this phenomena into account during the development process.
\subsection{Contribution of the multiple reactions in the target}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{secondary72.eps}
\includegraphics[width=0.45\textwidth]{secondary78.eps}
\caption{Production cross sections in hydrogen before and after the subtraction of the multiple reactions (empty and full dots, respectively), and contribution of the multiple reactions in the target (continuous lines), for Hf (left) and Pt (right) isotopes.}
\label{fig:secondary}
\end{center}
\end{figure}
The results of multiple-reaction calculations are presented in figure~\ref{fig:secondary} for 2 isotopic distributions, each one corresponding to an extreme situation regarding the contribution of the multiple reactions. For $Z$ around 78, the multiple reactions are an important contributor for very proton-rich nuclides only. Their contribution increases and spreads towards neutron-rich nuclides with decreasing $Z$. The very proton-rich part of the isotopic distributions of the light elements such as Hf is reproduced by our calculation with differences being less than 20\% in most cases. This demonstrates the validity of our approach. As the uncertainty on this calculation could not be estimated in a systematic way, we quoted the value of 20\% mentioned above.
We chose to consider as results of the experiment only the cross sections deduced from production rates for which multiple reactions contributed for less than 50\%. This discards nearly all nuclides with $Z<70$, which represent only a very small fraction of the fragmentation residues.
\section{Kinematics of the reaction} \label{chap:kinematics}
Once nuclei are identified, their velocity in the first part of the FRS can be calculated using the equation~\ref{eqn:brho}:
\begin{equation}
( \beta \gamma )_1 = \frac{(B\rho)_1}{A/q_1}
\end{equation}
Here the index 1 stands for the first part of the FRS (before $S_2$). Using this technique, the resolution is expected to be of the same order as the one obtained for the magnetic rigidity, roughly $5.10^{-4}$. At the energy used in this experiment, this is by a factor of 3 better than what can be achieved by a time-of-flight measurement in the second part of the FRS. This high resolution makes the FRS a remarkable tool to study the kinematics of nuclear reactions.
We have already pointed out that, because of the limited momentum acceptance of the FRS, the reconstruction of the full velocity spectra of each nuclide is a necessary step in order to evaluate the production rates (section~\ref{chap:setup}). The measured velocity spectra are Lorentz transformed into the reference frame of the beam, and corrected for the contribution of the beam width and for the velocity scattering due to the passage through the target and the surrounding materials. The resulting spectra give direct access to the longitudinal momentum transfer and to the longitudinal momentum spread caused by the nuclear reactions.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{p_parrallel_mean.eps}
\includegraphics[width=0.48\textwidth]{sigma_mean.eps}
\caption{Momentum transfer (left) and momentum width (right) measured in the reactions Pb+p at 500A~MeV (triangles), Pb+p at 1A~GeV (squares) and Au+p at 800A~MeV (circles). Data are compared to Morrissey systematics~\cite{Morrissey} (continuous lines) and Goldhaber formula~\cite{Goldhaber}, the later being computed with a Fermi momentum of 118~MeV.c$^{-1}$ (dashed line) and 95~MeV.c$^{-1}$ (dashed-dotted line). All data have been normalised following the Morrissey prescription.}
\label{fig:kin}
\end{center}
\end{figure}
In figure~\ref{fig:kin} these quantities are compared to results of previous spallation experiments as well as to the well-known Morrissey systematics~\cite{Morrissey} and to the Goldhaber formula~\cite{Goldhaber}. The data were averaged over each isobaric distribution using the production cross sections as weighting factor.
A very similar tendency is obtained for all the experimental data regarding the momentum transfer. The simple linear dependence proposed by the Morrissey systematics is not fulfilled by the experiment. The longitudinal momentum transfer is underestimated for fragments corresponding to a mass loss of 10 to 45 units with respect to the projectile. This underestimation vanishes with increasing mass losses.
The momentum width exhibit a linear dependance to the square root of the mass loss. The Morrissey systematics offers a fair reproduction of the data. Using the result of a direct measurement of the Fermi momentum (118~MeV/c)~\cite{fermi_mes}, the Goldhaber formula overestimates the momentum width. This is not unexpected as this formula only takes into account the nucleons removed during the cascade stage, which lead to larger momentum fluctuations with respect to the nucleons emitted in the evaporation phase~\cite{Hanelt}. Nevertheless, a better agreement with data can be obtained by using an arbitrary Fermi momentum value of 95~MeV/c, as often done in heavy-ion calculations. The dispersion between the different data sets is probably related to the delicate corrections applied to the data, namely the beam width in momentum and position, which are difficult to estimate.
\section{Production cross sections} \label{chap:results}
\subsection{Isotopic cross sections}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{isotopic.eps}
\caption{Isotopic production cross-section distributions of residues in the reaction Pb+p at 500A~MeV.}
\label{fig:cs}
\end{center}
\end{figure}
Figure~\ref{fig:cs} shows the measured distributions of isotopic production cross-sections for elements between erbium and bismuth (see appendix~\ref{chap:annexe_xs} for the full list of cross sections). Some 250 spallation-evaporation cross sections have been measured. One observes that the cross sections of isotopic chains vary smoothly. As no even-odd fluctuations are expected~\cite{Valentina}, and considering the statistical nature of the evaporation process, this corroborates that the measured production rates do not hide any other source of fluctuations. The upper limit of 50\% production of multiple reactions in the target, that we have decided to set, removed from the distributions a growing part of the lightest isotopes when $Z$ is decreasing. The most neutron-rich Pt and Ir isotopes have not been measured because of missing settings of the magnetic fields during the experiment.
\subsection{Total cross sections}
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c}
\hline
Reaction & $^{208}$Pb+p & $^{208}$Pb+p & $^{208}$Pb+d \\
& (500A MeV) & (1A GeV) & (1A GeV) \\
\hline
\hline
Spallation-evaporation (measured) & 1.44 (0.21) & 1.68 (0.22) & 1.91 (0.24)\\
Total (measured) & 1.67 (0.23) & 1.84 (0.23) & 2.08 (0.24) \\
Total (calculated) & 1.70 & 1.80 & 2.32 \\
\hline
\end{tabular}
\caption{Spallation cross sections (in barns) for reactions $^{208}$Pb+p at 500A~MeV and 1A~GeV, and $^{208}$Pb+d at 1A~GeV. The measured spallation-evaporation and total cross sections (adding fission) are compared to a Glauber calculation performed using updated density distributions. Values in parenthesis are the total uncertainty of the measurements.}
\label{tab:total_cs}
\end{center}
\end{table}
We have estimated the total production cross-section of evaporation residues by summing all the measured residue cross sections, obtaining a value of (1.44$\pm$0.21)~b. Our measurement does not strictly cover all the range of the possible residues. However, the nuclides for which we have no measurement are mostly the lightest fragmentation products. Considering the steepness of the mass curve in this region (see figure~\ref{fig:cs_mass}), we can assume that the contribution of these nuclides is small, and probably much smaller than the error bars.
Adding the fission cross-section for this reaction~\cite{NPA_Bea} we estimated the total cross section to (1.67 $\pm$ 0.23) b.
This value is very close to the 1.70~b found by a Glauber-type calculation performed with updated density distributions. On table~\ref{tab:total_cs} we compare those values with the ones obtained in the reactions Pb+p and Pb+d at 1A~GeV. Although the slight decrease of the total cross section with respect to 1A~GeV measurements is in agreement with the expected trend, the 500A~MeV fission cross section is higher than previous measurements conducted at this energy, and also higher than the systematics of Prokofiev~\cite{Prokofiev}. This question has been discussed in detail in the corresponding paper~\cite{NPA_Bea}. Let us only underline that the agreement of a well-established model with our experimental total cross section comes in support of our measurement.
\subsection{Comparison to radiochemical measurements}
In recent years, a large number of measurements of spallation residues have been performed by the team of R. Michel. Of special relevance to our work is the measurement of residues of spallation of natural lead by protons at 550~MeV published by Gloris \etal~\cite{Gloris}. Cross sections with independent yields ({\it i.e.} nuclides that are not produced by $\beta$ decay) can be compared directly, while cross sections corresponding to accumulation of $\beta$-decaying nuclei require a summation of our data along the decay chain.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.9\textwidth]{gloris.eps}
\caption{Ratio between production cross-sections of residues measured in the reaction $^{208}$Pb+p at 500A~MeV (this work) and in the reaction p+$^{nat}$Pb at 550~MeV as a function of the mass of the residue for the isotopes measured in~\cite{Gloris}. Filled and empty circles represent nuclides with independent yields and cumulated yields, respectively. Calculations of the ratio between production cross sections at 550 and 500 MeV have been performed in two systems: with the same $^{208}$Pb target nuclei (continuous line) and with a different nucleus, $^{207}$Pb (dashed line), in order to study the effect of the use of natural lead in the Gloris experiment (see text).}
\label{fig:chemistry}
\end{center}
\end{figure}
The ratio between our data and those of Gloris \etal\ are presented in figure~\ref{fig:chemistry}. In the case of the cumulated yields, cross sections measured at GSI have been summed along the decay chain in order to be comparable with the radiochemical measurements. The agreement between the two data sets is overall fair for heavy residues, although a systematic shift of roughly 10\% may be guessed. Considering the error bars, all the measurements seem to be compatible, with the exception of $^{203}$Pb and $^{202}$Tl. As we have already underlined, our data points are very consistent with respect to one another. This makes such a large error in our measurement for these two nuclides rather unlikely, since it should have been clearly visible on our isotopic distributions.
For lighter nuclei the ratio decreases rapidly with increasing mass loss with respect to the projectile. This effect is the direct consequence of the differences between the measured systems: the 10\% higher energy strongly favors the production of lighter residues, up to a factor of 3 for mass losses around 35 nucleons.
This statement can be checked by using Monte-Carlo calculations. For this purpose we used the ISABEL~\cite{ISABEL} intranuclear cascade and the ABLA~\cite{ABLA} evaporation code. Although one of the purposes of these measurements is precisely to check the validity of those codes in the few hundreds of MeV region, we can assume that they are reliable enough if one only wants to calculate variations in a very limited energy and mass range, as it is the case here.
Results of the calculation of the ratio between isobaric cross sections for the two systems are also represented in figure~\ref{fig:chemistry}. A calculation that only takes into account the different incident energies offers a satisfactory reproduction of the decrease of the ratio for the light fragments. Replacing $^{208}$Pb by $^{207}$Pb (in order to mock the isotopic mixing of natural lead of which the targets of Gloris experiment were made) leads to a slight reduction of the calculated ratio for light nuclides, which improves the agreement with the data in this mass range. For heavy nuclides the calculations indicate that results at 500A~MeV should be larger than at 550A~MeV, which is not what we observed for most of the points. However, only 3 points are not compatible with the calculations when one considers the error bars. This leads us to conclude that, taking into account the differences between the systems measured in the Gloris experiment and our experiment, the agreement between these data sets is satisfactory.
\subsection{Mass spectra and comparison to previous GSI experiments}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.95\textwidth]{mass_loss.eps}
\caption{Production cross-sections of the residues of the reaction Pb+p at 500A~MeV as a function of the mass loss with respect to the projectile (full circles). Data obtained in previously mentioned experiments are also represented: Au+p at 800A~MeV (triangles), Pb+p (squares) and Pb+d (diamonds) at 1A~GeV. The isolated points at $\Delta A=0$ correspond to a single nuclide, $^{208}$Bi.}
\label{fig:cs_mass}
\end{center}
\end{figure}
Figure~\ref{fig:cs_mass} presents the production cross sections, summed for all isobars, as a function of the mass loss with respect to the projectile. The data obtained from several experiments performed at the FRS are presented here: Pb+p at 500A~MeV, Pb+p at 1A~GeV~\cite{Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, and Au+p at 800A~MeV~\cite{Au_Fanny}.
For small mass losses, each spectrum has a nearly constant value. In this mass range, the lower the incident energy, the higher the cross sections. With increasing mass losses, the cross sections start to decrease. Here, the lower the energy, the earlier and the steeper the fall. This is easily understood as the direct consequence of the exploration by the prefragment of all the possible range of excitation energy available in each system. In this respect, the measurement with deuterons gives insights about what would be obtained in a measurement conducted with protons at twice the energy. The shape of the mass-loss curve obtained from the measurement on gold is fully compatible with the tendencies observed for lead.
In the 500A~MeV experiment, a clear separation exists between the group of the evaporation products (which does not extend beyond mass losses of 40 mass units) and the group of the fission product (which starts around mass losses of 70~\cite{NPA_Bea}). This absence of mixing could also be demonstrated by studying the velocity spectra of the light fragments, which all exhibit a quasi-perfect Gaussian shape, while the presence of fission products would have introduced a characteristic double-bumped shape due to the forward-backward selection of the fission fragments by the FRS~\cite{Pb_p}.
\subsection{Charge pick-up}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.45\textwidth]{charge_pickup.eps}
\includegraphics*[width=0.45\textwidth]{iso_pickup.eps}
\caption{Left figure: charge-pickup cross sections measured on lead (full symbols) and gold (empty symbols) as a function of the incident energy. The sum of the partial cross sections measured at the FRS (this work, full dots; Keli\'c \etal~\cite{Kelic}, full squares; Rejmund \etal~\cite{Au_Fanny}, upward triangles) is compared to elemental cross sections from Waddington \etal~\cite{Waddington} (downward triangles) and Binns \etal~\cite{Binns} (diamonds), which were both extracted from CH$_2$ and C measurements. Right figure: isotopic charge-pickup cross sections at 3 energies as a function of the mass loss with respect to the heavy partner of the reaction.}
\label{fig:pickup}
\end{center}
\end{figure}
An especially interesting result of this experiment is the measurement of the production cross section of 15 isotopes of Bi (see figure~\ref{fig:pickup}). Those nuclides are formed by charge-pickup reactions. In the energy range considered here, the capture of the incident proton is not initially possible, as the incident proton energy is well above the Fermi energy of the target nuclide. Therefore the formation of $^{209}$Bi is improbable, and the formation of $^{208}$Bi is only possible via, either the formation of a resonant state ($\Delta$ and pions), or a quasi-elastic collision between the incident proton and a neutron from the target nuclide, in which the neutrons leaves with an energy very close to the initial energy of the proton. The cross section for the charge-pickup is one of the few data that bring direct constraints for the intranuclear-cascade codes.
In the left part of figure~\ref{fig:pickup} we compare our measurement of the total charge-pickup cross section to previous measurements performed by our collaboration~\cite{Au_Fanny,Kelic}, Waddington \etal~\cite{Waddington} and Binns \etal~\cite{Binns}. For a qualitative discussion we do not need to discriminate between gold and lead as those nuclides are close to one another, both in atomic number and mass. Our measurement confirms the trend of a strong increase of the total cross section of the charge-pickup with decreasing energy.
This increase of the Bi production in Pb+p experiments concerns all Bi isotopes, as it can be seen in the right part of figure~\ref{fig:pickup}. The shapes of the 500A~MeV and 1A~GeV distributions are overall similar, but the overproduction at 500A~MeV increases slowly from a factor of 2 for the heaviest isotopes to a factor of 4 for the lightest. Problems in the separation of the ionic charge states~\cite{NIM_loa} prevented us to use the kinematic spectra to distinguish between the respective contribution of the $\Delta$ resonance and the quasi-elastic reactions in the formation of the heaviest Bi isotopes, as it was done by Keli\'c \etal~\cite{Kelic}.
The shape of the Hg spectrum (obtained in the Au+p at 800A~MeV measurement) is slightly different from the lead spectra. On one hand, for mass losses up to 7 mass units, the shape of the isotopic distribution is nearly identical to the Pb spectra, with values in-between the two Pb experiments, which is consistent with a smooth evolution as a function of the projectile energy. On the other hand the production of the lightest isotopes decreases faster than in the Pb+p experiments. This difference in shape can be explained by the lower Coulomb barrier and the shorter distance from the residue corridor~\cite{Dufour} for Hg nuclides with respect to Bi nuclides, which favor the emission of protons by the excited prefragments~\cite{Summerer_pickup}.
\subsection{Isobaric cross sections}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.95\textwidth]{isobaric3.eps}
\caption{Isobaric spectra of production cross-sections (in mb) in the reactions Pb+p and Pb+Ti at 500A~MeV (full circles and crosses, respectively). The data obtained in the experiments Au+p at 800A~MeV \cite{Au_Fanny}, Pb+p and Pb+d at 1A~GeV \cite{Pb_p,Pb_d} are also plotted (triangles, squares and diamonds, respectively). }
\label{fig:cs_isobaric}
\end{center}
\end{figure}
In figure~\ref{fig:cs_isobaric} the data from the same experiments as in previous sections are plotted as isobaric spectra.
For heavy fragments, all distributions issued from reactions of Pb with p or d are very similar, both in shape and in magnitude. Low-energy reactions slightly dominate the cross sections for masses down to 185. With decreasing masses, the isobaric spectra behave in accordance with the mass distributions: the spectra for the highest-energy reaction scale down very slowly, whereas this scaling is steeper and steeper when one considers reactions at decreasing incident energy. However, for each isobar, the shapes of the different spectra remains extremely similar, as does its centroid (this can be seen in the left part of figure~\ref{fig:cs_means}).
Data from the reactions of Pb on the dummy target (which consists mainly of Ti in the target itself and Nb for the stripper foil placed after the target) at 500A~MeV have been added to the figure~\ref{fig:cs_isobaric} in order to illustrate the so-called limiting-fragmentation regime~\cite{limit_frag}. We observe no difference of shape or centroid between the spectra issued from the reaction on heavy ions and from the reaction on hydrogen isotopes.
The gold data offer an interesting point of comparison with the lead data. The gold fragments with mass close to 197 are associated with rather cold reactions and have therefore kept a $A/Z$ ratio very close to the initial system, while Pb fragments close to the same mass have lost roughly 10 nucleons, mostly neutrons because of the hindrance of charged-particle emission due to the Coulomb barrier. Therefore the gold and lead residue spectra are strongly shifted with respect to one another. This shift slowly vanishes with the increasing mass loss, which is easily understood as the slow move of the gold fragment distributions towards the residue corridor~\cite{Dufour}. This corridor is clearly visible on the left part of figure~\ref{fig:cs_means}: the barycenter of the isobaric distributions of the residues of all reactions converge on the same line.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.45\textwidth]{mean_z.eps}
\includegraphics*[width=0.45\textwidth]{mean_a.eps}
\caption{Average atomic number as a function of the mass of the residue (left figure) and average mass of the residue as a function of the atomic number (right figure) in the reactions Pb+p at 500A~MeV (full circles, this work), Au+p at 800A~MeV (triangles, \cite{Au_Fanny}) and Pb+p at 1A~GeV (squares, \cite{Pb_p}). In the calculation of the average values, points have been weighted according to their cross section.}
\label{fig:cs_means}
\end{center}
\end{figure}
This universal behavior, well known for reactions between heavy ions, is here demonstrated to be valid in a very broad energy range, even for fragments which are at the very end of the mass distribution. In other words, the isobaric distributions are independent of the incident energy in the system studied if properly renormalized. This is an experimental proof that the factorisation hypothesis is valid at energies as low as a few hundreds of MeV. This further strengthens the discussion regarding the agreement between EPAX and the data obtained from proton-induced reactions in systems in which fission does not play a major role (section~\ref{chap:secondary}).
Conversely, various projectile energies lead to variations of the center of the residues isotopic distributions, as it can be seen on the right part of the figure~\ref{fig:cs_means} as a deviation of the average mass value of the residues produced at 500A MeV. If this effect is washed out by the slow variations of the mass curve for higher energy reactions, it becomes noticeable at lower energies when the fall of the mass distribution becomes so steep that the production of the most neutron-deficient isotopes is strongly inhibited. Therefore, the reproduction of the isotopic and elemental cross-sections using a scaling factor between different systems is not appropriate at energies below the fragmentation limit.
\section{Conclusion}
The production cross sections and the momentum distributions have been measured for about 250 nuclei formed in the reaction of $^{208}$Pb on protons at 500A~MeV, covering most of the nuclides created down to a mass loss of 40 units with respect to the projectile, and with cross sections as low as 5~$\mu$b. The reaction products were identified in-flight in atomic number and mass-over-ionic-charge using the FRS spectrometer. The large proportion of non-fully stripped ions in the spectrometer was accounted for in detail, thus allowing to calculate the production cross section for each nuclide. The contribution of multiple reactions in the target to the residue production was carefully subtracted.
The production cross sections are in good agreement with previous radiochemical measurements. The isobaric distributions of the production cross sections are found to be very close to the ones measured at higher energies, thus extending the validity range of the factorisation hypothesis to energies of a few hundreds of MeV. The large variations observed on the isotopic cross sections can be nearly fully ascribed to the variations of the residue distributions with mass-loss at decreasing energy. Kinematical data are consistent with previous measurements.
The data obtained in this experiment, combined with previous measurements performed with the same technique (especially in the same system at 1A~GeV), constitute a set of information that is highly relevant for the development of reliable nuclear-reaction codes and, thus, the design of ADS.
|
{
"timestamp": "2005-12-09T12:08:54",
"yymm": "0503",
"arxiv_id": "nucl-ex/0503021",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0503021"
}
|
\section{Introduction}
The purpose of this paper is to study the structure of the bounded derived
category $\Dbcoh(\boldsymbol{E})$ of coherent sheaves on a singular irreducible
projective curve $\boldsymbol{E}$ of arithmetic genus one.
In the smooth case, such structure results are easily obtained from Atiyah's
description \cite{Atiyah} of indecomposable vector bundles over elliptic
curves. However, if $\boldsymbol{E}$ has a node or a cusp, some crucial
properties fail to hold. This is illustrated by the following table.
\begin{center}
\begin{tabular}[t]{p{6cm}|c|c}
&smooth&singular\\ \hline
homological dimension of $\Coh_{\boldsymbol{E}}$
&$1$&$\infty$\\ \hline
Serre duality holds&in general&\multicolumn{1}{p{3cm}}{
with one object being perfect}\\ \hline
torsion free implies locally free&yes&no\\ \hline
indecomposable coherent sheaves are semi-stable&yes&no\\ \hline
any indecomposable complex is isomorphic to a shift of a sheaf&yes&no\\
\hline
\end{tabular}
\end{center}
Despite these difficulties, the main goal of this article is to find the common
features between the smooth and the singular case. A list of such can be
found in Remark \ref{rem:common}.
In Section \ref{sec:background}, we review the smooth case and highlight where
the properties mentioned above are used. Our approach was inspired by
\cite{LenzingMeltzer}.
Atiyah's algorithm to construct indecomposable vector bundles of any slope can
be understood as an application of a sequence of twist functors with spherical
objects. From this point of view, Atiyah shows that any indecomposable object
of $\Dbcoh(\boldsymbol{E})$ is the image of an indecomposable torsion sheaf
under an exact auto-equivalence of $\Dbcoh(\boldsymbol{E})$.
In the case of a singular Weierstra{\ss} curve $\boldsymbol{E}$, as our main
technical tool we use Harder-Narasimhan filtrations in
$\Dbcoh(\boldsymbol{E})$, which were introduced by Bridgeland
\cite{Stability}. Their general properties are studied in Section
\ref{sec:HNF}.
The key result of Section \ref{sec:dercat} is the preservation of stability
under Seidel-Thomas twists \cite{SeidelThomas} with spherical objects. This
allows us to show that, like in the smooth case, any category of semi-stable
sheaves with fixed slope is equivalent to the category of coherent torsion
sheaves on $\boldsymbol{E}$.
In the case of slope zero, this was shown in our previous work
\cite{BurbanKreussler}. For the nodal case, an explicit description of
semi-stable sheaves of degree zero via \'etale coverings was given there as
well.
A combinatorial description of semi-stable sheaves of arbitrary slope over
a nodal cubic curve was found by Mozgovoy \cite{Mozgovoy}.
On the other hand, a classification of all indecomposable objects of
$\Dbcoh(\boldsymbol{E})$ was presented in \cite{BurbanDrozd}. A description of
the Harder-Narasimhan filtrations in terms of this classification is a task for
future work.
However, if the singular point of $\boldsymbol{E}$ is a cusp, the description
of all indecomposable coherent torsion sheaves is a wild problem in the sense
of representation theory, see for example \cite{Drozd72}. Nevertheless,
stable vector bundles on a cuspidal cubic have been classified by Bodnarchuk
and Drozd \cite{Lesya}.
It turns out that semi-stable sheaves of infinite homological dimension are
particularly important, because only such sheaves appear as Harder-Narasimhan
factors of indecomposable objects in $\Dbcoh(\boldsymbol{E})$ which are not
semi-stable (Proposition \ref{prop:extreme}).
The main result (Proposition \ref{prop:spherical}) of Section \ref{sec:dercat}
is the answer to a question of Polishchuk, who asked in \cite{YangBaxter},
Section 1.4, for a description of all spherical objects on $\boldsymbol{E}$.
We also prove that the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of spherical objects.
In Section \ref{sec:tstruc} we study $t$-structures on $\Dbcoh(\boldsymbol{E})$
and stability conditions in the sense of \cite{Stability}.
We completely classify all $t$-structures on this category (Theorem
\ref{thm:tstruc}). This allows us to
deduce a description of the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ (Corollary \ref{cor:auto}).
As a second application, we calculate Bridgeland's moduli space of
stability conditions on $\boldsymbol{E}$ (Proposition \ref{prop:stabmod}).
The hearts $\mathsf{D}(\theta,\theta+1)$ of the $t$-structures constructed in
Section \ref{sec:tstruc} are finite-dimensional non-Noetherian Abelian
categories of infinite global dimension.
In the case of a smooth elliptic curve, this category is equivalent to the
category of holomorphic vector bundles on a non-commutative torus in the
sense of Polishchuk and Schwarz \cite{PolSchw}, Proposition 3.9.
It is an interesting problem to find such a differential-geometric
interpretation of these Abelian categories in the case of singular
Weierstra{\ss} curves.
Using the technique of Harder-Narasimhan filtrations, we gain new insight into
the classification of indecomposable complexes, which was obtained in
\cite{BurbanDrozd}.
It seems plausible that similar methods can be applied to study the derived
category of representations of certain derived tame associative algebras, such
as gentle algebras, skew-gentle algebras or degenerated tubular algebras, see
for example \cite{BuDro}.
The study of Harder-Narasimhan filtrations in conjunction with the action of
the group of exact auto-equivalences of the derived category should provide new
insight into the combinatorics of indecomposable objects in these derived
categories.
\textbf{Notation.} We fix an algebraically closed field $\boldsymbol{k}$ of
characteristic zero. By $\boldsymbol{E}$ we always denote a Weierstra{\ss}
curve. This is a reduced irreducible curve of arithmetic genus one, isomorphic
to a cubic curve in the projective plane. If not smooth, it has precisely one
singular point $s\in\boldsymbol{E}$, which can be a node or a cusp.
If $x\in\boldsymbol{E}$ is arbitrary, we denote by $\boldsymbol{k}(x)$ the
residue field of $x$ and consider it as a sky-scraper sheaf supported at $x$.
By $\Dbcoh(\boldsymbol{E})$ we denote the derived category of complexes of
$\mathcal{O}_{\boldsymbol{E}}$-modules whose cohomology sheaves are coherent
and which are non-zero only in finitely many degrees.
\textbf{Acknowledgement.} The first-named author would like to thank
Max-Planck-Institut f\"ur Mathematik in Bonn for financial support.
Both authors would like to thank Yuriy Drozd, Daniel Huybrechts, Bernhard
Keller, Rapha\"el Rouquier and Olivier Schiffmann for helpful discussions,
and the referee for his or her constructive comments.
\section{Background: the smooth case}\label{sec:background}
The purpose of this section is to recall well-known results about the structure
of the bounded derived category of coherent sheaves over a smooth elliptic
curve. Proofs of most of these results can be found in \cite{Atiyah},
\cite{Oda}, \cite{LenzingMeltzer} and \cite{Tu}.
The focus of our presentation is on the features and techniques
which are essential in the singular case as well.
At the end of this section we highlight the main differences between the smooth
and the singular case. It becomes clear that the failure of Serre duality is
the main reason why the proofs and even the formulation of some of the main
results do not carry over to the singular case.
The aim of the subsequent sections will then be to overcome these difficulties,
to find correct formulations which generalise to the singular case and to
highlight the common features of the bounded derived category in the smooth and
singular case.
With the exception of subsection \ref{subsec:diff}, throughout this section
$\boldsymbol{E}$ denotes a smooth elliptic curve over $\boldsymbol{k}$.
\subsection{Homological dimension}
For any two coherent sheaves $\mathcal{F}, \mathcal{G}$ on $\boldsymbol{E}$,
Serre duality provides an isomorphism $$\Ext^{\nu}(\mathcal{F},\mathcal{G})
\cong \Ext^{1-\nu}(\mathcal{G},\mathcal{F})^{\ast}.$$
This follows from the usual formulation of Serre duality and the fact that any
coherent sheaf has a finite locally free resolution. As a consequence,
$\Ext^{\nu}(\mathcal{F},\mathcal{G})=0$ for any $\nu \ge 2$, which means that
$\Coh_{\boldsymbol{E}}$ has homological dimension one. This implies that any
object $X\in\Dbcoh(\boldsymbol{E})$ splits into the direct sum of appropriate
shifts of its cohomology sheaves. To see this, start with a complex
$X=(\mathcal{F}^{-1} \stackrel{f}{\longrightarrow} \mathcal{F}^{0})$
and consider the distinguished triangle in $\Dbcoh(\boldsymbol{E})$
$$\ker(f)[1] \rightarrow X \rightarrow \coker(f) \stackrel{\xi}{\rightarrow}
\ker(f)[2].$$
Because $\xi\in\Hom(\coker(f),\ker(f)[2]) = \Ext^{2}(\coker(f), \ker(f)) =0$,
we obtain $X\cong \ker(f)[1] \oplus \coker(f)$. Using the same idea we can
proceed by induction to get the claim.
\subsection{Indecomposable sheaves are semi-stable}
It is well-known that any coherent sheaf $\mathcal{F}\in\Coh_{\boldsymbol{E}}$
has a Harder-Narasimhan filtration
$$0\subset \mathcal{F}_{n} \subset \ldots \subset \mathcal{F}_{1} \subset
\mathcal{F}_{0} = \mathcal{F}$$
whose factors $\mathcal{A}_{\nu} := \mathcal{F}_{\nu}/\mathcal{F}_{\nu+1}$ are
semi-stable with decreasing slopes $\mu(\mathcal{A}_{n})>
\mu(\mathcal{A}_{n-1}) > \ldots > \mu(\mathcal{A}_{0})$.
Using the definition of semi-stability, this implies
$\Hom(\mathcal{A}_{\nu+i}, \mathcal{A}_{\nu})
= 0$ for all $\nu\ge 0$ and $i>0$. Therefore,
$\Ext^{1}(\mathcal{A}_{0},\mathcal{F}_{1}) \cong
\Hom(\mathcal{F}_{1}, \mathcal{A}_{0})^{\ast} =0$, and the exact sequence
$0\rightarrow \mathcal{F}_{1} \rightarrow \mathcal{F} \rightarrow
\mathcal{A}_{0} \rightarrow 0$ must split.
In particular, if $\mathcal{F}$ is indecomposable, we have $\mathcal{F}_{1}=0$
and $\mathcal{F}\cong \mathcal{A}_{0}$ and $\mathcal{F}$ is semi-stable.
\subsection{Jordan-H\"older factors}
The full sub-category of $\Coh_{\boldsymbol{E}}$ whose objects are the
semi-stable sheaves of a fixed slope is an Abelian category in which any object
has a Jordan-H\"older filtration with stable factors.
If $\mathcal{F}$ and $\mathcal{G}$ are non-isomorphic stable sheaves which
have the same slope, we have $\Hom(\mathcal{F},\mathcal{G})=0$. Based on this
fact, in the same way as before, we can deduce that an indecomposable
semi-stable sheaf has all its Jordan-H\"older factors isomorphic to each
other.
\subsection{Simple is stable}
It is well-known that any stable sheaf $\mathcal{F}$ is simple,
i.e. $\Hom(\mathcal{F},\mathcal{F}) \cong \boldsymbol{k}$. On a smooth
elliptic curve, the converse is true as well, which equips us with a useful
homological characterisation of stability.
To see that simple implies stable, we suppose for a contradiction that
$\mathcal{F}$ is simple but not stable. This implies the existence of an
epimorphism $\mathcal{F}\rightarrow \mathcal{G}$ with $\mathcal{G}$ stable and
$\mu(\mathcal{F})\ge \mu(\mathcal{G})$. Serre duality implies
$\dim \Ext^{1}(\mathcal{G},\mathcal{F}) = \dim \Hom(\mathcal{F},\mathcal{G}) >
0$, hence, $\chi(\mathcal{G},\mathcal{F}) := \dim
\Hom(\mathcal{G},\mathcal{F}) - \dim \Ext^{1}(\mathcal{G},\mathcal{F}) < \dim
\Hom(\mathcal{G},\mathcal{F})$. Riemann-Roch gives
$\chi(\mathcal{G},\mathcal{F}) = (\mu(\mathcal{F}) -
\mu(\mathcal{G}))/\rk(\mathcal{F})\rk(\mathcal{G}) > 0$, hence
$\Hom(\mathcal{G},\mathcal{F})\ne 0$. But this produces a
non-zero composition $\mathcal{F}\rightarrow \mathcal{G} \rightarrow
\mathcal{F}$ which is not an isomorphism, in contradiction to the assumption
that $\mathcal{F}$ was simple.
\subsection{Classification}
Atiyah \cite{Atiyah} gave a description of all stable sheaves with a fixed
slope in the form $\mathcal{E}(r,d)\otimes \mathcal{L}$, where $\mathcal{L}$
is a line bundle of degree zero and $\mathcal{E}(r,d)$ is a particular stable
bundle of the fixed slope. The bundle $\mathcal{E}(r,d)$ depends on the choice
of a base point $p_{0}\in\boldsymbol{E}$ and
its construction reflects the Euclidean algorithm on the pair $(\rk,\deg)$.
We look at this description from a slightly different perspective. We use the
twist functors $T_{\mathcal{O}}$ and $T_{\boldsymbol{k}(p_{0})}$, which were
constructed by Seidel and Thomas \cite{SeidelThomas} (see also \cite{Meltzer}).
They act as equivalences on $\Dbcoh(\boldsymbol{E})$ and, hence, preserve
stability.
A stable sheaf of rank $r$ and degree $d$ is sent by $T_{\mathcal{O}}$ to one
with $(\rk,\deg)$ equal to $(r-d,d)$. If $r<d$ this is a shift of a stable
sheaf. The functor $T_{\boldsymbol{k}(p_{0})}$ sends the pair $(r,d)$ to
$(r,r+d)$ and its inverse sends it to $(r,d-r)$. Therefore, if we follow the
Euclidean algorithm, we find a composition of such functors which provides an
equivalence between the category of stable sheaves with slope $d/r$ and the
category of simple torsion sheaves. Such sheaves are precisely the structure
sheaves of closed points $\boldsymbol{k}(x)$, $x\in\boldsymbol{E}$. They are
considered to be stable with slope $\infty$.
More generally, this procedure provides an equivalence between the category of
semi-stable sheaves of rank $r$ and degree $d$ with the category of torsion
sheaves of length equal to $\gcd(r,d)$. This shows, in particular, that the
Abelian category of semi-stable sheaves with fixed slope is equivalent to the
category of coherent torsion sheaves.
\subsection{Auto-equivalences}
By $\Aut(\Dbcoh(\boldsymbol{E}))$ we denote the group of all exact
auto-equivalences of the triangulated category $\Dbcoh(\boldsymbol{E})$. This
group acts on the Grothendieck group $\mathsf{K}(\boldsymbol{E}) \cong
\mathsf{K}(\Dbcoh(\boldsymbol{E}))$.
As the kernel of the Chern character is the radical of the Euler-form
$\langle X,Y \rangle = \dim(\Hom(X,Y)) - \dim(\Hom(X,Y[1])$
which is invariant under this action, it induces an action on the even
cohomology $H^{2\ast}(\boldsymbol{E}, \mathbb{Z}) \cong \mathbb{Z}^{2}$.
Because $\dim(\Hom(\mathcal{F},\mathcal{G}))>0$ if and only if $\langle
\mathcal{F},\mathcal{G} \rangle >0$, provided $\mathcal{F}\not\cong
\mathcal{G}$ are stable sheaves, the induced action on $\mathbb{Z}^{2}$ is
orientation preserving. So, we obtain a homomorphism of groups $\varphi:
\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z})$, which is
surjective because $T_{\mathcal{O}}$ and $T_{\boldsymbol{k}(p_{0})}$ are
mapped to a pair of generators of $\SL(2,\mathbb{Z})$. Explicitly, if
$\mathbb{G}$ is an auto-equivalence, $\varphi(\mathbb{G})$ describes its
action on the pair $(\rk,\deg)$.
To understand $\ker(\varphi)$, we observe that $\varphi(\mathbb{G}) =
\boldsymbol{1}$ implies that $\mathbb{G}$ sends a simple torsion sheaf
$\boldsymbol{k}(x)$ to some $\boldsymbol{k}(y)[2k]$, because indecomposability
is retained. By the same reason, $\mathbb{G}(\mathcal{O})$ is a shifted line
bundle of degree zero.
However, $\Hom(\mathcal{L},\boldsymbol{k}(y)[l]) = 0$, if $\mathcal{L}$ is a
line bundle and $l\ne 0$. Hence, after composing $\mathbb{G}$ with a shift, it
sends all simple torsion sheaves to simple torsion sheaves, without a shift.
Because $\boldsymbol{E}$ is smooth, we can apply a result of Orlov \cite{Orlov}
which says that any auto-equivalence $\mathbb{G}$ is a Fourier-Mukai transform
\cite{Mukai}.
However, any such functor, which sends the sheaves $\boldsymbol{k}(x)$ to
torsion sheaves of length one is of the form
$\mathbb{G}(X)=f^{\ast}(\mathcal{L}\otimes X)$,
where $f:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is an automorphism and
$\mathcal{L}\in\Pic(\boldsymbol{E})$ a line bundle. Hence, $\ker(\varphi)$ is
generated by $\Aut(\boldsymbol{E}), \Pic^{0}(\boldsymbol{E})$ and even
shifts. This gives a complete description of the group
$\Aut(\Dbcoh(\boldsymbol{E}))$. A similar approach was used by Lenzing and
Meltzer to describe the group of exact auto-equivalences of tubular weighted
projective lines
\cite{LenzingMeltzerAuto}.
\subsection{Difficulties in the singular case}\label{subsec:diff}
Let now $\boldsymbol{E}$ be an irreducible but singular curve of arithmetic
genus one. The technical cornerstones of the theory as described in this
section fail to be true in this case. More precisely:
\begin{itemize}
\item the category of coherent sheaves $\Coh_{\boldsymbol{E}}$ has infinite
homological dimension;
\item there exist indecomposable complexes in $\Dbcoh(\boldsymbol{E})$ which
are not just shifted sheaves, see \cite{BurbanDrozd}, section 3;
\item Serre duality fails to be true in general;
\item not all indecomposable vector bundles are semi-stable;
\item there exist indecomposable coherent sheaves which are neither torsion
sheaves nor torsion free sheaves, see \cite{BurbanDrozd}.
\end{itemize}
Most of the trouble is caused by the failure of Serre duality.
The basic example is the following. Suppose, $s\in\boldsymbol{E}$ is a node,
then
$$\Hom(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong \boldsymbol{k}\quad
\text{ and }\quad
\Ext^{1}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong \boldsymbol{k}^{2}.$$
Serre duality is available, only if at least one of the two sheaves involved
has finite homological dimension. This might suggest that replacing
$\Dbcoh(\boldsymbol{E})$ by the sub-category of perfect complexes would solve
most of the problems. But see Remark \ref{rem:notperfect}.
In the subsequent sections we overcome these difficulties and point out the
similarities between the smooth and the singular case.
\section{Harder-Narasimhan filtrations}\label{sec:HNF}
Throughout this section, $\boldsymbol{E}$ denotes an irreducible reduced
projective curve over $\boldsymbol{k}$ of arithmetic genus one.
The notion of stability of coherent torsion free sheaves on an irreducible
curve is usually defined with the aid of the slope function
$\mu(\,\cdot\,)=\deg(\,\cdot\,)/\rk(\,\cdot\,)$. To use the phase function instead is
equivalent, but better adapted for the generalisation to derived categories
described below.
By definition, the \emph{phase} $\varphi(\mathcal{F})$ of a non-zero
coherent sheaf $\mathcal{F}$ is the unique number which satisfies $0 <
\varphi(\mathcal{F})\le 1$ and $m(\mathcal{F}) \exp(\pi
i\varphi(\mathcal{F})) = -\deg(\mathcal{F}) + i \rk(\mathcal{F})$, where
$m(\mathcal{F})$ is a positive real number, called the \emph{mass} of the
sheaf $\mathcal{F}$.
In particular, $\varphi(\mathcal{O}) = 1/2$ and all non-zero torsion sheaves
have phase one.
A torsion free coherent sheaf $\mathcal{F}$ is called semi-stable if for any
exact sequence of torsion free coherent sheaves
$$0 \rightarrow \mathcal{E} \rightarrow \mathcal{F}
\rightarrow \mathcal{G} \rightarrow 0$$
the inequality $\varphi(\mathcal{E}) \le \varphi(\mathcal{F})$, or
equivalently, $\varphi(\mathcal{F}) \le \varphi(\mathcal{G})$, holds.
It is well-known \cite{Rudakov} that any torsion free coherent sheaf
$\mathcal{F}$ on a projective variety has a Harder-Narasimhan filtration
$$0 \subset \mathcal{F}_{n} \subset \mathcal{F}_{n-1} \cdots \subset
\mathcal{F}_{1} \subset \mathcal{F}_{0} = \mathcal{F},$$
which is uniquely characterised by the property that all factors
$\mathcal{A}_{i} = \mathcal{F}_{i}/\mathcal{F}_{i+1}$ are semi-stable and
satisfy
$$\varphi(\mathcal{A}_{n}) > \varphi(\mathcal{A}_{n-1}) > \cdots >
\varphi(\mathcal{A}_{0}).$$
Originally, this concept of stability was introduced in the 1960s in order to
construct moduli spaces using geometric invariant theory. It could also be
seen as a method to understand the structure of the category of coherent
sheaves on a projective variety.
By Simpson, the notion of stability was extended to coherent sheaves of
pure dimension.
A very general approach was taken by Rudakov \cite{Rudakov}, who introduced
the notion of stability on Abelian categories. Under some finiteness
assumptions on the category, he shows the existence and uniqueness of a
Harder-Narasimhan filtration for any object of the category in question. As an
application of his work, the usual slope stability extends to the whole
category $\Coh_{\boldsymbol{E}}$ of coherent sheaves on $\boldsymbol{E}$. In
particular, any non-zero coherent sheaf has a Harder-Narasimhan filtration and
any non-zero coherent torsion sheaf on the curve $\boldsymbol{E}$ is
semi-stable.
Inspired by work of Douglas on $\Pi$-Stability for D-branes, see for example
\cite{Douglas}, it was shown by Bridgeland \cite{Stability} how to extend the
concept of stability and Harder-Narasimhan filtration to the derived category
of coherent sheaves, or more generally, to a triangulated category. These new
ideas were merged with the ideas from \cite{Rudakov} in the paper
\cite{GRK}.
We shall follow here the approach of Bridgeland \cite{Stability}. In Section
\ref{sec:tstruc} we give a description of Bridgeland's moduli space of
stability conditions on the derived category of irreducible singular curves
of arithmetic genus one. However, throughout the present chapter we stick to
the classical notion of stability on the category of coherent sheaves and the
stability structure it induces on the triangulated category.
In order to generalise the concept of a Harder-Narasimhan filtration to the
category $\Dbcoh(\boldsymbol{E})$, Bridgeland \cite{Stability} extends the
definition of the phase of a sheaf to shifts of coherent sheaves by:
$$\varphi(\mathcal{F}[n]) := \varphi(\mathcal{F})+n,$$
where $\mathcal{F}\ne 0$ is a coherent sheaf on $\boldsymbol{E}$ and
$n\in\mathbb{Z}$. A complex which is non-zero at position $m$ only has,
according to this definition, phase in the interval $(-m,-m+1]$. If
$\mathcal{F}$ and $\mathcal{F}'$ are non-zero coherent sheaves and $a,b$
integers, we have the implication:
$$\varphi(\mathcal{F}[-a]) > \varphi(\mathcal{F}'[-b])
\quad\Rightarrow\quad a\le b.$$
For any $\varphi\in\mathbb{R}$ we denote by $\mathsf{P}(\varphi)$ the Abelian
category of shifted semi-stable sheaves with phase $\varphi$. Of course,
$0\in\mathsf{P}(\varphi)$ for all $\varphi$.
If $\varphi\in(0,1]$, this is a full Abelian subcategory of
$\Coh_{\boldsymbol{E}}$. For any $\varphi\in\mathbb{R}$ we have
$\mathsf{P}(\varphi+n) = \mathsf{P}(\varphi)[n]$. A non-zero object of
$\Dbcoh(\boldsymbol{E})$ will be called \emph{semi-stable}, if it is an
element of one of the categories $\mathsf{P}(\varphi)$, $\varphi\in\mathbb{R}$.
Bridgeland's stability conditions \cite{Stability} involve so-called
central charges.
In order to define the central charge of the standard stability condition, we
need a definition of degree and rank for arbitrary objects in
$\Dbcoh(\boldsymbol{E})$.
Let $K =\mathcal{O}_{\boldsymbol{E},\eta}$ be the field of rational
functions on the irreducible curve $\boldsymbol{E}$ with generic point
$\eta\in\boldsymbol{E}$. The base change $\eta:\Spec(K)\rightarrow
\boldsymbol{E}$ is flat, so that $\eta^{\ast}(F)$,
taken in the non-derived sense, is correctly defined for any
$F\in\Dbcoh(\boldsymbol{E})$.
We define $\rk(F):=\chi(\eta^{\ast}(F))$, which is the
alternating sum of the dimensions of the cohomology spaces of the complex
$\eta^{\ast}(F)$ which are vector spaces over $K$.
In order to define the degree, we use the functor
$$\boldsymbol{R}\Hom(\mathcal{O}_{\boldsymbol{E}},\,\cdot\,): \Dbcoh(\boldsymbol{E})
\rightarrow \Dbcoh(\boldsymbol{k}),$$
and set $\deg(F):= \chi(\boldsymbol{R}\Hom(\mathcal{O}_{\boldsymbol{E}},F))$.
Here, we denoted by $\Dbcoh(\boldsymbol{k})$ the bounded derived category of
finite dimensional vector spaces over $\boldsymbol{k}$.
For coherent sheaves, these definitions coincide with the usual
definitions of rank and degree. In particular, a torsion sheaf of length $m$
which is supported at a single point of $\boldsymbol{E}$ has rank $0$ and
degree $m$.
These definitions imply that rank and degree are additive on distinguished
triangles in $\Dbcoh(\boldsymbol{E})$. Hence, they induce homomorphisms on the
Grothendieck group $\mathsf{K}(\Dbcoh(\boldsymbol{E}))$ of the triangulated
category $\Dbcoh(\boldsymbol{E})$, which is by definition the quotient of the
free Abelian group generated by the objects of $\Dbcoh(\boldsymbol{E})$ modulo
expressions coming from distinguished triangles.
Recall that $\mathsf{K}_{0}(\Coh(\boldsymbol{E})) \cong
\mathsf{K}(\Dbcoh(\boldsymbol{E}))$,
see \cite{Groth}. We denote this group by $\mathsf{K}(\boldsymbol{E})$
\begin{lemma}\label{lem:GrothGrp}
If $\boldsymbol{E}$ is an irreducible singular curve of arithmetic genus
one, we have $\mathsf{K}(\boldsymbol{E}) \cong \mathbb{Z}^{2}$ with
generators $[\boldsymbol{k}(x)]$ and $[\mathcal{O}_{\boldsymbol{E}}]$.
\end{lemma}
\begin{proof}
Recall that the Grothendieck-Riemann-Roch Theorem, see \cite{BFM} or
\cite{Fulton}, provides a homomorphism
$$\tau_{\boldsymbol{E}}:\mathsf{K}(\boldsymbol{E}) \rightarrow
A_{\ast}(\boldsymbol{E})\otimes \mathbb{Q},$$
which depends functorially on $\boldsymbol{E}$ with respect to proper direct
images.
Moreover,
$(\tau_{\boldsymbol{E}})_{\mathbb{Q}}:\mathsf{K}(\boldsymbol{E})\otimes
\mathbb{Q} \rightarrow A_{\ast}(\boldsymbol{E})\otimes \mathbb{Q}$
is an isomorphism, see \cite{Fulton}, Cor.\/ 18.3.2.
If $\boldsymbol{E}$ is an irreducible singular projective curve of
arithmetic genus one, we easily see that the Chow group
$A_{\ast}(\boldsymbol{E})$ is isomorphic to $\mathbb{Z}^{2}$.
The two generators are $[x]\in A_{0}(\boldsymbol{E})$ with
$x\in\boldsymbol{E}$ and $[\boldsymbol{E}]\in A_{1}(\boldsymbol{E})$.
Note that $[x]=[y]\in A_{0}(\boldsymbol{E})$ for any two closed points
$x,y\in\boldsymbol{E}$, because the normalisation of $\boldsymbol{E}$ is
$\mathbb{P}^{1}$.
Using \cite{Fulton}, Thm.\/ 18.3 (5), we obtain
$\tau_{\boldsymbol{E}}(\boldsymbol{k}(x))= [x] \in A_{0}(\boldsymbol{E})$
for any $x\in\boldsymbol{E}$. On the other hand, from \cite{Fulton}, Expl.\/
18.3.4 (a), we obtain $\tau_{\boldsymbol{E}}(\mathcal{O}_{\boldsymbol{E}}) =
[\boldsymbol{E}] \in A_{1}(\boldsymbol{E})$.
Therefore, the classes of $\boldsymbol{k}(x)$ and
$\mathcal{O}_{\boldsymbol{E}}$ define a basis of
$\mathsf{K}(\boldsymbol{E})\otimes \mathbb{Q}$.
However, these two classes generate the group $\mathsf{K}(\boldsymbol{E})$,
so that it must be a free Abelian group.
\end{proof}
The \emph{central charge} of the standard stability structure on
$\Dbcoh(\boldsymbol{E})$ is the homomorphism of Abelian groups
$$
Z: \mathsf{K}(\boldsymbol{E}) \rightarrow \mathbb{Z}\oplus i\mathbb{Z}
\subset \mathbb{C},
$$
which is given by
$$
Z(F) := -\deg(F) + i \rk(F).
$$
If $F$ is a non-zero coherent sheaf, $Z(F)$ is a point on the ray from the
origin through $\exp(\pi i \varphi(F))$ in $\mathbb{C}$. Its distance from the
origin was called the mass of $F$.
Although the phase $\varphi(F)$ is defined for sheaves and their shifts only,
we are able to define the slope $\mu(F)$ for any object in
$\Dbcoh(\boldsymbol{E})$ which is not equal to zero in the Grothendieck
group. Namely, the usual definition $\mu(F):=\deg(F)/\rk(F)$ gives us now a
mapping $$\mu:\mathsf{K}(\boldsymbol{E})\setminus\{0\} \rightarrow
\mathbb{Q} \cup \{\infty\},$$ which extends the usual definition of the slope
of a sheaf.
Because $Z(\mathcal{O}_{\boldsymbol{E}})=i$ and $Z(\boldsymbol{k}(x))=-1$,
Lemma \ref{lem:GrothGrp} implies that $Z$ is injective. Therefore, $\mu$ is
defined for any non-zero element of the Grothendieck group.
For arbitrary objects $X\in\Dbcoh(\boldsymbol{E})$ we have $Z(X[1]) = -Z(X)$,
hence $\mu(X[1]) = \mu(X)$ when defined.
In case of shifted sheaves, in contrast to the slope $\mu$, the
phase $\varphi$ keeps track of the position of this sheaf in the complex.
As an illustration, we include an example of an indecomposable object in
$\Dbcoh(\boldsymbol{E})$ which has a zero image in the Grothendieck group.
\begin{example}
Let $s\in\boldsymbol{E}$ be the singular point and denote, as usual, by
$\boldsymbol{k}(s)$ the torsion sheaf of length one which is supported at
$s$. This sheaf does not have finite homological dimension. To see this, we
observe first that $\Ext^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong
H^{0}(\mathcal{E}xt^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s)))$. Moreover,
as an
$\mathcal{O}_{\boldsymbol{E},s}$-module,
$\boldsymbol{k}(s)$ has an infinite periodic locally free resolution of the
form
$$
\cdots \stackrel{A}{\longrightarrow} \mathcal{O}_{\boldsymbol{E},s}^{2}
\stackrel{B}{\longrightarrow} \mathcal{O}_{\boldsymbol{E},s}^{2}
\stackrel{A}{\longrightarrow}\mathcal{O}_{\boldsymbol{E},s}^{2}
\longrightarrow \mathcal{O}_{\boldsymbol{E},s}
\longrightarrow \boldsymbol{k}(s) \longrightarrow 0
$$
where $AB=BA=f\cdot I_{2}$ is a reduced matrix factorisation of an equation
$f$ of $\boldsymbol{E} \subset \mathbb{P}^{2}$. For example, if $s$ is a
node, so that $\boldsymbol{E}$ is locally given by the polynomial $f = y^{2}
- x^{3} -x^{2}\in\boldsymbol{k}[x,y]$, we can choose
$A=\bigl(\begin{smallmatrix}
y&x^{2}+x\\x&y
\end{smallmatrix}\bigr)$ and
$B=\bigl(\begin{smallmatrix}
y&-x^{2}-x\\-x&y
\end{smallmatrix}\bigr)$ considered modulo $f$. More generally, any
singular Weierstra{\ss} cubic $f$ can be written as $y\cdot y - R\cdot S$
with $y, R,S$ all vanishing at the singular point. The off-diagonal elements
of $A$ and $B$ are then formed by $\pm R,\pm S$. Therefore, all entries of
the matrices $A$ and $B$ are elements of the maximal ideal of the local ring
$\mathcal{O}_{\boldsymbol{E},s}$. Hence, the application of $\Hom(\,\cdot\,,
\boldsymbol{k}(s))$ produces a complex with zero differential, which implies
that $\Ext^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s))$ is
two-dimensional for all $k\ge 1$.
In particular, $\Ext^{2}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong
\boldsymbol{k}^{2}$, and we can pick a non-zero element
$w\in\Hom(\boldsymbol{k}(s), \boldsymbol{k}(s)[2])$. There exists a complex
$X\in\Dbcoh(\boldsymbol{E})$ which sits in a distinguished triangle
$$X\rightarrow \boldsymbol{k}(s) \stackrel{w}{\longrightarrow}
\boldsymbol{k}(s)[2] \stackrel{+}{\longrightarrow}.$$
Because the shift by one corresponds to multiplication by $-1$ in the
Grothendieck group, this object $X$ is equal to zero in
$\mathsf{K}(\boldsymbol{E})$. On the other hand, $X$ is
indecomposable. Indeed, if $X$ would split, it must be $X\cong
\boldsymbol{k}(s) \oplus \boldsymbol{k}(s)[1]$, because the only non-zero
cohomology of $X$ is $H^{-1}(X) \cong \boldsymbol{k}(s)$ and $H^{0}(X) \cong
\boldsymbol{k}(s)$. But, because $\Hom(\boldsymbol{k}(s)[1],
\boldsymbol{k}(s)) = 0$, Lemma \ref{lem:PengXiao}, applied to the
distinguished triangle
\[\begin{CD}
\boldsymbol{k}(s)[1] @>>> X @>>> \boldsymbol{k}(s) @>{+}>{w}>
\end{CD}\]
with $X\cong \boldsymbol{k}(s) \oplus \boldsymbol{k}(s)[1]$, implies $w=0$.
\end{example}
\begin{definition}[\cite{Stability}]
A Harder-Narasimhan filtration (HNF) of an object $X \in
\Dbcoh(\boldsymbol{E})$ is a finite collection of distinguished triangles
\[
\xymatrix@C=.5em
{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
\\
}
\]
with $A_j\in\mathsf{P}(\varphi_j)$ and $A_{j}\ne 0$ for all $j$, such that
$\varphi_{n} > \varphi_{n-1} > \cdots > \varphi_{0}.$
\end{definition}
If all ingredients of a HNF are shifted by one, we obtain a HNF of $X[1]$.
The shifted sheaves $A_{j}$ are called \emph{the semi-stable HN-factors} of
$X$ and we define $\varphi_{+}(X):=\varphi_{n}$ and
$\varphi_{-}(X):=\varphi_{0}$. Later, Theorem \ref{thm:uniqueHNF}, we show that
the HNF of an object $X$ is unique up to isomorphism. This justifies this
notation. For the moment, we keep in mind that $\varphi_{+}(X)$ and
$\varphi_{-}(X)$ might depend on the HNF and not only on the object $X$.
Before we proceed, we include a few remarks about the notation we use.
Distinguished triangles in a triangulated category are either displayed in the
form
$X\rightarrow Y\rightarrow Z \stackrel{+}{\longrightarrow}
\quad\text{ or as }\quad
\xymatrix@C=.5em{
X \ar[rr] && Y, \ar[dl]\\ & Z \ar[ul]^{+}
}
$
where the arrow which is marked with $+$ is in fact a morphism $Z\rightarrow
X[1]$.
We shall use the octahedron axiom, the axiom (TR4) in Verdier's list, in the
following convenient form: if two morphisms $X\stackrel{f}{\longrightarrow} Y
\stackrel{g}{\longrightarrow} Z$ are given, for any three distinguished
triangles with bases $f, g$ and $g\circ f$ there exists a fourth distinguished
triangle which is indicated below by dashed arrows, such that we obtain the
following commutative diagram:
\begin{center}
\mbox{\begin{xy} 0;<10mm,0mm>:0,
(0,3) *+{Z'} ="top" ,
(-3,0) *+{X} ="left" ,
(3,0) *+{X'} ="right" ,
(-1.5,1.5) *+{Y} ="midleft" ,
(1.5,1.5) *+{Y'} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{Z}="center",
{"left" \ar@{->}^{f} "midleft" \ar@{->}_{g\circ f} "center"},
{"center" \ar@{->} "right" \ar@{->} "midright"},
{"midleft" \ar@{->}^{g} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
The remainder of this section is devoted to the proofs of the crucial
properties of Harder-Narasimhan filtrations in triangulated categories.
These properties can be found in \cite{Stability, GRK}, where most of
them appear to be either implicit or without a detailed proof.
\begin{lemma}\label{lem:connect}
Let
$\xymatrix@C=.4em{U \ar[rr]^{f} && X \ar[dl]\\ & V \ar[ul]^{+}}$
and
$A \longrightarrow V \longrightarrow V' \stackrel{+}{\longrightarrow}$
be distinguished triangles. Then there exists a factorisation
$U\longrightarrow W \stackrel{f'}{\longrightarrow} X$
of $f$ and two distinguished triangles
$$\xymatrix@C=.5em{U \ar[rr] && W \ar[dl]\ar[rr]^{f'} && X\ar[dl]\\
& A \ar[ul]^{+} && V'.\ar[ul]^{+}}$$
\end{lemma}
\begin{proof}
If we apply the octahedron axiom to the composition $A\rightarrow V
\rightarrow U[1]$ we obtain the following commutative diagram, which gives
the claim.
\begin{center}
\mbox{\begin{xy} 0;<10mm,0mm>:0,
(0,3) *+{V'} ="top" ,
(-3,0) *+{A} ="left" ,
(3,0) *+{X[1]} ="right" ,
(-1.5,1.5) *+{V} ="midleft" ,
(1.5,1.5) *+{W[1]} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{U[1]}="center",
{"left" \ar@{->} "midleft" \ar@{->} "center"},
{"center" \ar@{->}_{f[1]} "right" \ar@{->} "midright"},
{"midleft" \ar@{->} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->}^{f'[1]} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
\end{proof}
\begin{lemma}\label{lem:split}
Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}V \ar[rr] \ar[dl]_{\cong}&& F_{n-1}V \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}V \ar[rr] && F_{0}V \ar@{=}[r]\ar[dl] & V
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $V\in\Dbcoh(\boldsymbol{E})$ and $F_{k}V \longrightarrow V
\longrightarrow V' \stackrel{+}{\longrightarrow}$ a distinguished triangle
with $1\le k \le n$. Then, $F_{k}V$ has a HNF with HN-factors $A_{n},
A_{n-1}, \ldots, A_{k}$ and $V'$ one with HN-factors $A_{k-1},
A_{k-2}, \ldots, A_{0}$.
\end{lemma}
\begin{proof}
The first statement is clear, because we can cut off the HNF of $V$ at
$F_{k}V$ to obtain a HNF of $F_{k}V$. Let us define objects $F_{i}V'$ by
exact triangles $F_{k}V \longrightarrow F_{i}V \longrightarrow F_{i}V'
\stackrel{+}{\longrightarrow}$, where the first arrow is the composition of
the morphisms in the HNF of $V$. Using the octahedron axiom, we obtain for
any $i\le k$ a commutative diagram
\begin{center}
\mbox{\begin{xy} 0;<12mm,0mm>:0,
(0,3) *+{F_{i}V'} ="top" ,
(-3,0) *+{F_{k}V} ="left" ,
(3,0) *+{A_{i-1}} ="right" ,
(-1.5,1.5) *+{F_{i}V} ="midleft" ,
(1.5,1.5) *+{F_{i-1}V'} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{F_{i-1}V}="center",
{"left" \ar@{->} "midleft" \ar@{->} "center"},
{"center" \ar@{->} "right" \ar@{->} "midright"},
{"midleft" \ar@{->} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
which implies the second claim.
\end{proof}
\begin{remark}\label{rem:split}
The statement of Lemma \ref{lem:split} is true with identical proof if we
relax the assumption of being a HNF by allowing $\varphi(A_{k}) =
\varphi(A_{k-1})$ for the chosen value of $k$.
\end{remark}
\begin{lemma}\label{lem:bounds}
If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
is a HNF of $X\in\Dbcoh(\boldsymbol{E})$ such that $A_{0}[k]$ is a sheaf,
then $H^{k}(X)\ne 0$. In particular, the following implication is true:
$$X\in \mathsf{D}^{\le m} \quad\Longrightarrow\quad
\forall i\ge0: A_{i}\in \mathsf{D}^{\le m}.$$
\end{lemma}
\begin{proof}
The assumption $A_{0}[k]\in \Coh_{\boldsymbol{E}}$ means
$H^{k}(A_{0})=A_{0}[k]\ne 0$ and $\varphi(A_{0}) \in (-k,-k+1]$. Because for
all $i>0$ we have $\varphi(A_{i}) > \varphi(A_{0})$, we obtain
$\varphi(A_{i})>-k$. This implies $H^{k+1}(A_{i})=0$ for all $i\ge 0$. The
cohomology sequences of the distinguished triangles
$F_{i+1}\longrightarrow F_{i}\longrightarrow A_{i}
\stackrel{+}{\longrightarrow}$ imply $H^{k+1}(F_{i}X)=0$ for all $i>0$ and
an exact sequence $H^{k}(X) \rightarrow H^{k}(A_{0}) \rightarrow
H^{k+1}(F_{1}X)$, hence $H^{k}(X)\ne 0$. The statement about the other
HN-factors $A_{i}$ follows now from $\varphi(A_{i})\ge \varphi(A_{0})$.
\end{proof}
\begin{proposition}
Any non-zero object $X\in\Dbcoh(\boldsymbol{E})$ has a HNF.
\end{proposition}
\begin{proof}
The existence of a HNF for objects of $\Coh_{\boldsymbol{E}}$ is
classically known, see \cite{HarderNarasimhan, Rudakov}. Therefore, we can
proceed by induction on the number of non-zero cohomology sheaves of
$X\in\Dbcoh(\boldsymbol{E})$. If $n$ is the largest integer with
$H^{n}(X)\ne 0$, we have a distinguished triangle
\begin{equation}
\tau^{\le n-1} X \longrightarrow X \longrightarrow H^{n}(X)[-n]
\stackrel{+}{\longrightarrow}
\end{equation}
By inductive hypothesis, there exists a HNF of $\tau^{\le n-1} X$. From
Lemma \ref{lem:bounds} we conclude that all HN-factors of $\tau^{\le n-1} X$
are in $\mathsf{D}^{\le n-1}$ and so $\varphi_{-}(\tau^{\le n-1} X)>-n+1$.
Because $H^{n}(X)$ is a sheaf, we have $\varphi_{+}(H^{n}(X)[-n])
\in (-n,-n+1]$, hence $\varphi_{-}(\tau^{\le n-1} X) >
\varphi_{+}(H^{n}(X)[-n])$.
We prove now for any distinguished triangle
\begin{equation}
\label{eq:induction}
U \longrightarrow X \longrightarrow V \stackrel{+}{\longrightarrow}
\end{equation}
in which $V[n]$ is a coherent sheaf that the existence of a HNF for $U$ with
$\varphi_{-}(U)> \varphi_{+}(V)$ implies the existence of a HNF of $X$.
Because $V[n]$ is a sheaf, $V$ has a HNF and we proceed by induction on the
number of HN-factors of $V$. Let $A$ be the leftmost object in a HNF of $V$,
i.e.\/ $A\in\mathsf{P}(\varphi_{+}(V))$. By Lemma \ref{lem:connect} applied
to the distinguished triangles (\ref{eq:induction}) and $A \longrightarrow V
\longrightarrow V' \stackrel{+}{\longrightarrow}$, there exist two
distinguished triangles in which $V'[n]$ is a coherent sheaf with a smaller
number of HN-factors as $V$:
$$\xymatrix@C=.5em{U \ar[rr] && W \ar[dl]\ar[rr] && X.\ar[dl]\\
& A \ar[ul]^{+} && V'\ar[ul]^{+}}$$
Because $\varphi_{-}(U)\ge \varphi(A) =\varphi_{+}(V)$, the left triangle can
be concatenated to the given HNF of $U$ in order to provide a HNF for
$W$. The start of the induction is covered as well: it is the case $V'=0$.
\end{proof}
\begin{lemma}\label{wesPT:ii}
If $X,Y\in \Dbcoh(\boldsymbol{E})$ with $\varphi_{-}(X) > \varphi_{+}(Y)$,
then $$\Hom(X,Y)=0.$$
\end{lemma}
\begin{proof}
If $X,Y$ are semi-stable sheaves, this is well-known and follows easily from
the definition of semi-stability. Because $\Hom(X,Y[k])=0$, if $X,Y$ are
sheaves and $k<0$, the claim follows if $X\in \mathsf{P}(\varphi)$ and $Y\in
\mathsf{P}(\psi)$ with $\varphi>\psi$. Let now $X\in\mathsf{P}(\varphi)$ and
$Y\in \Dbcoh(\boldsymbol{E})$ with $\varphi > \varphi_{+}(Y)$. Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{m}Y \ar[rr] \ar[dl]_{\cong}&& F_{m-1}Y \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}Y \ar[rr] && F_{0}Y \ar@{=}[r]\ar[dl] & Y
\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
be a HNF of $Y$. We have $\varphi(B_{j})\le \varphi(B_{m}) =
\varphi_{+}(Y)$, hence $\varphi(X)>\varphi(B_{j})$ and $\Hom(X,B_{j})=0$ for
all $j$. If we apply the functor $\Hom(X,\,\cdot\,)$ to the distinguished
triangles $F_{j+1}Y \longrightarrow F_{j}Y \longrightarrow B_{j}
\stackrel{+}{\longrightarrow}$, we obtain surjections $\Hom(X,F_{j+1}Y)
\twoheadrightarrow \Hom(X,F_{j}Y)$. From $\Hom(X,F_{m}Y)=
\Hom(X,B_{m})=0$, we obtain $\Hom(X,Y)=\Hom(X,F_{0}Y)=0$.
Let now $X,Y$ be arbitrary non-zero objects of $\Dbcoh(\boldsymbol{E})$
which satisfy $\varphi_{-}(X) > \varphi_{+}(Y)$. If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
is a HNF of $X$, we have $\varphi(A_{i})\ge \varphi(A_{0})=\varphi_{-}(X) >
\varphi_{+}(Y)$. We know already $\Hom(A_{i},Y)=0$ for all $i\ge 0$. If we
apply the functor $\Hom(\,\cdot\,,Y)$ to the distinguished triangles
$F_{i+1}X \longrightarrow F_{i}X \longrightarrow A_{i}
\stackrel{+}{\longrightarrow}$, we obtain injections $\Hom(F_{i}X,Y)
\hookrightarrow \Hom(F_{i+1}X,Y)$. Again, this implies $\Hom(X,Y)=0$.
\end{proof}
\begin{theorem}[\cite{Stability,GRK}]\label{thm:uniqueHNF}
The HNF of any non-zero object $X\in\Dbcoh(\boldsymbol{E})$ is unique up to
unique isomorphism.
\end{theorem}
\begin{proof}
If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
and
\[\xymatrix@C=.5em{
0\; \ar[rr] && G_{m}X \ar[rr] \ar[dl]_{\cong}&& G_{m-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&G_{1}X \ar[rr] && G_{0}X \ar@{=}[r]\ar[dl] &
X\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
are HNFs of $X$, we have to show that there exist unique isomorphisms of
distinguished triangles for any $k\ge 0$
\[\begin{CD}
F_{k+1}X @>>> F_{k}X @>>> A_{k} @>{+}>>\\
@VV{f_{k+1}}V @VV{f_{k}}V @VV{g_{k}}V \\
G_{k+1}X @>>> G_{k}X @>>> B_{k} @>{+}>>
\end{CD}\]
with $f_{0}=\mathsf{Id}_{X}$. This is obtained by induction on $k\ge0$ from
the following claim: if an isomorphism $f:F\rightarrow G$ and two
distinguished triangles
$F' \longrightarrow F \longrightarrow A \stackrel{+}{\longrightarrow }$ and
$G' \longrightarrow G \longrightarrow B \stackrel{+}{\longrightarrow }$ are
given such that $A\in\mathsf{P}(\varphi), B\in\mathsf{P}(\psi)$ and $F',G'$
have HNFs with $\varphi_{-}(F')>\varphi$ and $\varphi_{-}(G')>\psi$, then
there exist unique isomorphisms $f':F'\rightarrow G'$ and $g:A\rightarrow B$
such that $(f',f,g)$ is a morphism of triangles. In particular,
$\varphi=\psi$.
Without loss of generality, we may assume $\varphi\ge \psi$. This implies
$\varphi_{-}(F'[1]) > \varphi_{-}(F') > \psi$. Lemma \ref{wesPT:ii} implies
therefore $\Hom(F',B) = \Hom(F'[1],B) = 0$. From \cite{Asterisque100},
Proposition 1.1.9, we obtain the existence and uniqueness of the morphisms
$f',g$. It remains to show that they are isomorphisms. If $g$ were zero, the
second morphism in the triangle $G' \longrightarrow G
\stackrel{0}{\longrightarrow} B \stackrel{+}{\longrightarrow }$ would be
zero. Hence, $B$ were a direct summand of $G'[1]$ which implies
$\Hom(G'[1],B)\ne 0$. This contradicts Lemma \ref{wesPT:ii}, because
$\varphi_{-}(G'[1]) > \varphi(G') > \psi=\varphi(B)$. Hence, $g\ne 0$ and
Lemma \ref{wesPT:ii} implies $\varphi(A)\le \varphi(B)$, i.e.\/
$\varphi=\psi$. So, the same reasoning as before gives a unique morphism of
distinguished triangles in the other direction. The composition of both are
the respective identities of $F' \longrightarrow F \longrightarrow A
\stackrel{+}{\longrightarrow }$ and $G' \longrightarrow G \longrightarrow B
\stackrel{+}{\longrightarrow }$ respectively, which follows again from the
uniqueness part of \cite{Asterisque100}, Proposition 1.1.9. This proves the
claim.
\end{proof}
We need the following useful lemma.
\begin{lemma}(\cite{PengXiao}, Lemma 2.5)\label{lem:PengXiao}
Let $\mathsf{D}$ be a triangulated category and
\[\begin{CD}
F @>>> G @>>> H_1 \oplus H_2 @>{+}>{(0,w)}>
\end{CD}\]
be a distinguished triangle in $\mathsf{D}$. Then $G\cong H_{1}\oplus G'$
splits and the given triangle is isomorphic to
\[\begin{CD}
F
@>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
H_1 \oplus G'
@>{\bigl(\begin{smallmatrix}1&0\\0&f' \end{smallmatrix}\bigr)}>>
H_1 \oplus H_2
@>{+}>{(0,w)}>
\end{CD}\]
Dually, if
\[\begin{CD}
F @>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
G_{1}\oplus G_{2} @>>> H @>{+}>>
\end{CD}\]
is a distinguished triangle then $H\cong G_{1}\oplus H'$ and the given
triangle is isomorphic to
\[\begin{CD}
F
@>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
G_1 \oplus G_{2}
@>{\bigl(\begin{smallmatrix}1&0\\0&f' \end{smallmatrix}\bigr)}>>
G_1 \oplus H'
@>{+}>{(0,w)}>
\end{CD}\]
\end{lemma}
The results in this section are true for more general triangulated
categories than $\Dbcoh(\boldsymbol{E})$. Without changes, the proofs apply if
we replace $\Dbcoh(\boldsymbol{E})$ by the bounded derived category of an
Abelian category which is equipped with the notion of stability in the sense
of \cite{Rudakov}. In particular, these results hold for polynomial stability
on the triangulated categories $\Dbcoh(X)$ where $X$ is a projective
variety over $\boldsymbol{k}$.
\section{The structure of the bounded derived category of coherent sheaves on
a singular Weiersta{\ss} curve}\label{sec:dercat}
In this section, we prove the main results on which our understanding of
$\Dbcoh(\boldsymbol{E})$ is based. Again, $\boldsymbol{E}$ denotes a
Weierstra{\ss} curve. Our main focus is on the singular case, however all the
results remain true in the smooth case as well.
A speciality of this category is the
non-vanishing result Proposition \ref{wesPT}. Unlike the smooth case, there
exist indecomposable objects in $\Dbcoh(\boldsymbol{E})$, which are not
semi-stable. Their Harder-Narasimhan factors are characterised in Proposition
\ref{prop:extreme}. We propose to visualise indecomposable objects by their
``shadows''. As an application of our results, we give a complete
characterisation of all spherical objects in $\Dbcoh(\boldsymbol{E})$. As a
consequence, we show that the group of exact auto-equivalences acts
transitively on the set of spherical objects. This answers a question which was
posed by Polishchuk \cite{YangBaxter}.
Let us set up some notation.
For any $\varphi\in(0,1]$ we denote by $\mathsf{P}(\varphi)^{s} \subset
\mathsf{P}(\varphi)$ the full subcategory of stable sheaves with phase
$\varphi$. We extend this definition to all $\varphi\in\mathbb{R}$ by
requiring $\mathsf{P}(\varphi +n)^{s} = \mathsf{P}(\varphi)^{s}[n]$ for all
$n\in\mathbb{Z}$ and all $\varphi\in\mathbb{R}$.
We already know the structure of $\mathsf{P}(1)^{s}$. Because $\mathsf{P}(1)$
is the category of coherent torsion sheaves on $\boldsymbol{E}$, the objects
of $\mathsf{P}(1)^{s}$ are precisely the structure sheaves $\boldsymbol{k}(x)$
of closed points $x\in\boldsymbol{E}$. In order to understand the structure of
all the other categories $\mathsf{P}(\varphi)^{s}$, we use Fourier-Mukai
transforms. Our main technical tool will be the transform $\mathbb{F}$ which
was studied in \cite{BurbanKreussler}. It depends on the choice of a regular
point $p_{0}\in\boldsymbol{E}$.
Let us briefly recall its definition and main properties. It was defined
with the aid of Seidel-Thomas twists \cite{SeidelThomas}, which are functors
$T_{E}: \Dbcoh(\boldsymbol{E}) \rightarrow \Dbcoh(\boldsymbol{E})$
depending on a spherical object $E\in\Dbcoh(\boldsymbol{E})$. On objects, these
functors are characterised by the existence of a distinguished triangle
$$\boldsymbol{R}\Hom(E,F) \otimes E \rightarrow F \rightarrow T_{E}(F)
\stackrel{+}{\longrightarrow}.$$
If $p_{0}\in\boldsymbol{E}$ is a smooth point, the functor
$T_{\boldsymbol{k}(p_{0})}$ is isomorphic to the tensor product with the
locally free sheaf $\mathcal{O}_{\boldsymbol{E}}(p_{0})$, see
\cite{SeidelThomas}, 3.11. We defined
$$\mathbb{F} :=
T_{\boldsymbol{k}(p_{0})}T_{\mathcal{O}}T_{\boldsymbol{k}(p_{0})}.$$
In \cite{SeidelThomas} is was shown that twist functors can be described as
integral transforms and that $\mathbb{F}$ is isomorphic to the functor
$\FM^{\mathcal{P}}$, which is given by
$$\FM^{\mathcal{P}}(\,\cdot\,) :=
\boldsymbol{R}\pi_{2\ast}(\mathcal{P}\dtens \pi_{1}^{\ast}(\,\cdot\,)),$$
where $\mathcal{P}=\mathcal{I}_{\Delta}\otimes
\pi_{1}^{\ast}\mathcal{O}(p_{0}) \otimes
\pi_{2}^{\ast}\mathcal{O}(p_{0})[1]$. This is a shift of a coherent sheaf on
$\boldsymbol{E}\times \boldsymbol{E}$, on which we denote the ideal of the
diagonal by $\mathcal{I}_{\Delta} \subset
\mathcal{O}_{\boldsymbol{E}\times\boldsymbol{E}}$ and the two projections by
$\pi_{1}, \pi_{2}$.
In order to understand the effect of $\mathbb{F}$ on rank and degree, we look
at the distinguished triangle
$$\boldsymbol{R}\Hom(\mathcal{O},F) \otimes \mathcal{O} \rightarrow F
\rightarrow T_{\mathcal{O}}(F) \stackrel{+}{\longrightarrow}.$$
The additivity of rank and degree implies $\rk(T_{\mathcal{O}}(F))= \rk(F) -
\deg(F)$ and
$\deg(T_{\mathcal{O}}(F))= \deg(F)$. On the other hand, it is well-known that
$\deg(T_{\boldsymbol{k}(p_{0})}(F)) = \deg(F)+\rk(F)$ and
$\rk(T_{\boldsymbol{k}(p_{0})}(F)) = \rk(F)$.
So, if we use $[\mathcal{O}_{\boldsymbol{E}}], -[\boldsymbol{k}(p_{0})]$ as a
basis of $\mathsf{K}(\boldsymbol{E})$, which means that we use coordinates
$(\rk,-\deg)$, then the action of $T_{\mathcal{O}}, T_{\boldsymbol{k}(p_{0})}$
and of $\mathbb{F}$ on $\mathsf{K}(\boldsymbol{E})$ is given by the matrices
$$
\begin{pmatrix}
1&1\\0&1
\end{pmatrix},
\begin{pmatrix}
1&0\\-1&1
\end{pmatrix} \quad \;\text{and}\quad
\begin{pmatrix}
0&1\\-1&0
\end{pmatrix}\;\text{respectively.}
$$
In particular, for any object $F\in\Dbcoh(\boldsymbol{E})$ which has a slope,
we have $\mu(T_{\boldsymbol{k}(p_{0})}(F)) = \mu(F)+1$ and
$\mu(\mathbb{F}(F))=-\frac{1}{\mu(F)}$ using the usual conventions in dealing
with $\infty$.
If $F$ is a sheaf or a twist thereof, we defined the phase
$\varphi(F)$. In order to understand the effect of $\mathbb{F}$ on phases, it
is not sufficient to know its effect on the slope. This is because the slope
determines the phase modulo $2\mathbb{Z}$ only.
However, if $F$ is a coherent sheaf, the description of $\mathbb{F}$ as
$\FM^{\mathcal{P}}$ shows that $\mathbb{F}(F)$ can have non-vanishing
cohomology in degrees $-1$ and $0$ only. If, in addition, $\mathbb{F}(F)$ is a
shifted sheaf, this implies $\varphi(\mathbb{F}(F))\in (0,2]$.
From the formula for the slope it is now clear that $\varphi(\mathbb{F}(F)) =
\varphi(F)+\frac{1}{2}$ for any shifted coherent sheaf $F$.
The following result was first shown in \cite{Nachr}. We give an independent
proof here, which was inspired by \cite{Nachr}, Lemma 3.1.
\begin{theorem}\label{thm:mother}
$\mathbb{F}$ sends semi-stable sheaves to semi-stable sheaves.
\end{theorem}
\begin{proof}
Note that, by definition, a semi-stable sheaf of positive rank is
automatically torsion free. The only sheaf with degree and rank equal to
zero is the zero sheaf. Throughout this proof, we let $\mathcal{F}$ be a
semi-stable sheaf on $\boldsymbol{E}$.
If $\deg(\mathcal{F})=0$ this sheaf is torsion free and the claim
was shown in \cite{BurbanKreussler}, Thm.\/ 2.21, see also \cite{FMmin}.
For the sake of clarity we would like to stress here the fact that
\cite{BurbanKreussler}, Section 2, deals with nodal as well as cuspidal
Weierstra{\ss} curves.
Next, suppose $\deg(\mathcal{F})>0$. If
$\rk(\mathcal{F})=0$, $\mathcal{F}$ is a coherent torsion sheaf. Again, the
claim follows from \cite{BurbanKreussler}, Thm.\/ 2.21 and Thm.\/ 2.18,
where it was shown that $\mathbb{F}\circ\mathbb{F}= i^{\ast}[1]$, for any
Weierstra{\ss} curve. Here, $i:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is
the involution which fixes the singularity and which corresponds to taking
the inverse on the smooth part of $\boldsymbol{E}$ with its group structure
in which $p_{0}$ is the neutral element.
Therefore, we may suppose $\mathcal{F}$ is torsion free. As observed before,
the complex $\mathbb{F}(\mathcal{F})\in\Dbcoh(\boldsymbol{E})$ can have
non-vanishing cohomology in degrees $-1$ and $0$ only. We are going to show
that $\mathbb{F}(\mathcal{F})[-1]$ is a sheaf, which is equivalent to the
vanishing of the cohomology object
$\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))\in\Coh_{\boldsymbol{E}}$.
Recall from \cite {BurbanKreussler}, Lemma 2.13, that for any smooth point
$x\in\boldsymbol{E}$ the sheaf of degree zero $\mathcal{O}(x-p_{0})$
satisfies
$\mathbb{F}(\mathcal{O}(x-p_{0})) \cong T_{\mathcal{O}}(\mathcal{O}(x))
\cong \boldsymbol{k}(x)$. Moreover, if $s\in\boldsymbol{E}$ denotes the
singular point, $n:\mathbb{P}^{1}\rightarrow \boldsymbol{E}$
the normalisation and
$\widetilde{\mathcal{O}}:=n_{\ast}(\mathcal{O}_{\mathbb{P}^{1}})$, then
$\mathbb{F}(\widetilde{\mathcal{O}}(-p_{0})) \cong
T_{\mathcal{O}}(\widetilde{\mathcal{O}}) \cong \boldsymbol{k}(s)$. The sheaf
$\widetilde{\mathcal{O}}(-p_{0})$ has degree zero on $\boldsymbol{E}$.
Because $\mathbb{F}$ is an equivalence, we obtain isomorphisms
\begin{align*}
\Hom(\mathbb{F}(\mathcal{F}),\boldsymbol{k}(x)) &\cong
\Hom(\mathcal{F}, \mathcal{O}(x-p_{0}))\\
\intertext{and}
\Hom(\mathbb{F}(\mathcal{F}),\boldsymbol{k}(s)) &\cong
\Hom(\mathcal{F}, \widetilde{\mathcal{O}}(-p_{0}))
\end{align*}
where $x\in\boldsymbol{E}$ is an arbitrary smooth point. These vector
spaces vanish as $\mathcal{F}$ was assumed to be semi-stable and of positive
degree.
Because cohomology of the complex $\mathbb{F}(\mathcal{F})$ vanishes in
positive degree, there is a canonical
morphism $\mathbb{F}(\mathcal{F})\rightarrow
\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))$ in $\Dbcoh(\boldsymbol{E})$, which
induces an injection of functors
$\Hom(\mathcal{H}^{0}(\mathbb{F}(\mathcal{F})), \,\cdot\,) \hookrightarrow
\Hom(\mathbb{F}(\mathcal{F}), \,\cdot\,)$. Therefore, the vanishing which was
obtained above, shows
$$\Hom(\mathcal{H}^{0}(\mathbb{F}(\mathcal{F})), \boldsymbol{k}(y)) = 0$$
for any point $y\in\boldsymbol{E}$. This implies the vanishing of the sheaf
$\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))$. Hence,
$\widehat{\mathcal{F}}:=\mathbb{F}(\mathcal{F})[-1]$ is a coherent sheaf and
the definition of $\mathbb{F}$ implies that there is an exact sequence of
coherent sheaves
$$0\rightarrow \widehat{\mathcal{F}}(-p_{0}) \rightarrow
H^{0}(\mathcal{F}(p_{0})) \otimes \mathcal{O}_{\boldsymbol{E}} \rightarrow
\mathcal{F}(p_{0}) \rightarrow 0.$$
This sequence implies, in particular, that $\widehat{\mathcal{F}}$ is
torsion free.
Before we proceed to show that $\widehat{\mathcal{F}}$ is semi-stable, we
apply duality to prove that $\mathbb{F}(\mathcal{F})$ is a sheaf if
$\deg(\mathcal{F})<0$. Let us denote the dualising functor by $\mathbb{D}:=
\boldsymbol{R}\mathcal{H}om(\,\cdot\,, \mathcal{O}_{\boldsymbol{E}})$. This
functor satisfies $\mathbb{D}\mathbb{D}\cong \boldsymbol{1}$. In
\cite{BurbanKreusslerRel}, Cor.\/ 3.4, we have shown that there exists an
isomorphism
$$\mathbb{D}\mathbb{F} [-1] \cong i^{\ast} \mathbb{F} \mathbb{D}.$$
Using $\mathbb{D}\circ[1] \cong [-1]\circ \mathbb{D}$, this implies
$$\mathbb{F} \cong \mathbb{D}i^{\ast}[-1]\mathbb{F}\mathbb{D}.$$
Because $\mathcal{F}$ is a torsion free sheaf on a curve, it is
Cohen-Macaulay and since $\boldsymbol{E}$ is Gorenstein, this implies
$\mathcal{E}xt^{i}(\mathcal{F},\mathcal{O}) = 0$ for any $i>0$.
Therefore, we have $\mathbb{D}(\mathcal{F})\cong \mathcal{F}^{\vee}$ and
this is a semi-stable coherent sheaf of positive degree. Thus, $[-1]\circ
\mathbb{F}$ sends $\mathcal{F}^{\vee}$ to a torsion free sheaf, on which
$\mathbb{D}$ is just the usual dual. Now, we see that
$\mathbb{F}(\mathcal{F})$ is a torsion free sheaf if $\mathcal{F}$ was
semi-stable and of negative degree.
It remains to prove that $\mathbb{F}$ preserves semi-stability. If
$\deg(\mathcal{F})=0$ or $\mathcal{F}$ is a torsion sheaf, this was shown
for any Weierstra{\ss} curve in \cite{BurbanKreussler}.
If $\deg(\mathcal{F})\ne 0$ the proof is based upon
$\mathbb{F}\mathbb{F}[-1]\cong i^{\ast}$, see \cite{BurbanKreussler}, Thm.\/
2.18. Suppose $\deg(\mathcal{F})>0$, then
$\mathbb{F}(\widehat{\mathcal{F}})\cong i^{\ast}(\mathcal{F})$ and this is a
coherent sheaf.
If $\widehat{\mathcal{F}}$ were not semi-stable, there would exist a
semi-stable sheaf $\mathcal{G}$ with $\mu(\widehat{\mathcal{F}}) >
\mu(\mathcal{G})$ and a non-zero morphism $\widehat{\mathcal{F}} \rightarrow
\mathcal{G}$. Because $\mu(\widehat{\mathcal{F}}) = -1/\mu(\mathcal{F})<0$,
$\mathbb{F}(\mathcal{G})$ is a coherent sheaf and application of
$\mathbb{F}$ produces a non-zero morphism $i^{\ast}(\mathcal{F}) \cong
\mathbb{F}(\widehat{\mathcal{F}}) \rightarrow
\mathbb{F}(\mathcal{G})$. However, $\mu(i^{\ast}(\mathcal{F})) =
\mu(\mathcal{F}) > -1/\mu(\mathcal{G}) = \mu(\mathbb{F}(\mathcal{G}))$
contradicts semi-stability of $i^{\ast}(\mathcal{F})$. Hence,
$\widehat{\mathcal{F}}$ is semi-stable. The proof in the case
$\deg(\mathcal{F})<0$ starts with a non-zero morphism
$\mathcal{U}\rightarrow \mathbb{F}(\mathcal{F})$ and proceeds similarly.
\end{proof}
It was shown in \cite{BurbanKreussler} that we obtain an action of the group
$\widetilde{\SL}(2,\mathbb{Z})$ on $\Dbcoh(\boldsymbol{E})$ by sending
generators of this group to $T_{\mathcal{O}}$, $T_{\boldsymbol{k}(p_{0})}$ and
the translation functor $[1]$ respectively.
Let us denote $$\mathsf{Q}:=\{\varphi\in\mathbb{R}\mid
\mathsf{P}(\varphi) \text{ contains a non-zero object}\}.$$
The action of a group $G$ on $\mathsf{Q}$ is called \emph{monotone}, if
$\varphi\le\psi$ implies $g\cdot\varphi\le g\cdot\psi$ for every $g\in G$ and
$\varphi,\psi\in \mathsf{Q}$.
\begin{proposition}\label{prop:transit}
The $\widetilde{\SL}(2,\mathbb{Z})$-action on $\Dbcoh(\boldsymbol{E})$
induces a monotone and transitive action on the set $\mathsf{Q}$. All
isotropy groups of this action are isomorphic to $\mathbb{Z}$.
\end{proposition}
\begin{proof}
As seen above, for any $\psi\in\mathsf{Q}$ and $0\ne
A\in\mathsf{P}(\psi)$, we have $\varphi(\mathbb{F}(A)) =
\varphi(A)+\frac{1}{2}$ and $\mu(T_{\boldsymbol{k}(p_{0})}(A)) = \mu(A)+1$.
Therefore, by Theorem \ref{thm:mother} it is clear that we obtain an induced
monotone action of $\widetilde{\SL}(2,\mathbb{Z})$ on $\mathsf{Q}$.
The group $\SL(2,\mathbb{Z})$ acts transitively on the set of all pairs of
co-prime integers which we interpret as primitive vectors of the lattice
$\mathbb{Z}\oplus i\mathbb{Z}\subset\mathbb{C}$. Hence, the action
of $\widetilde{\SL}(2,\mathbb{Z})$ on $\mathsf{Q}$ is transitive as
well. So, all isotropy groups are isomorphic. Finally, it is easy
to see that the isotropy group of $1\in\mathsf{Q}$ is generated by
$T_{\boldsymbol{k}(p_{0})}$.
\end{proof}
As an important consequence we obtain the following clear structure result
for the slices $\mathsf{P}(\varphi)$.
\begin{corollary}\label{cor:equiv}
The category $\mathsf{P}(\varphi)$ of semi-stable objects of phase
$\varphi\in\mathsf{Q}$ is equivalent to the category $\mathsf{P}(1)$ of
torsion sheaves. Any such equivalence restricts to an equivalence between
$\mathsf{P}(\varphi)^{s}$ and $\mathsf{P}(1)^{s}$. Under such an
equivalence, stable vector bundles correspond to structure sheaves of
smooth points. Moreover, if $\varphi\in(0,1)\cap \mathsf{Q}$,
$\mathsf{P}(\varphi)^{s}$ contains a unique torsion free sheaf, which is not
locally free. It correspond to the structure sheaf
$\boldsymbol{k}(s)\in\mathsf{P}(1)^{s}$ of the singular point.
\end{corollary}
Recall that an object $E\in\Dbcoh(\boldsymbol{E})$ is called \emph{perfect},
if it is isomorphic in the derived category to a bounded complex of locally
free sheaves of finite rank. Thus, a sheaf or shift thereof is called perfect,
if it is perfect as an object in $\Dbcoh(\boldsymbol{E})$.
If $\boldsymbol{E}$ is smooth, any object in $\Dbcoh(\boldsymbol{E})$ is
perfect. However, if $s\in\boldsymbol{E}$ is a singular point, the torsion
sheaf $\boldsymbol{k}(s)$ is not perfect.
If $\boldsymbol{E}$ is singular with one singularity $s\in\boldsymbol{E}$, the
category $\mathsf{P}(1)^{s}$ contains precisely one object which is not
perfect, the object $\boldsymbol{k}(s)$.
Hence, by Proposition \ref{prop:transit}, for any $\varphi\in\mathsf{Q}$ there
is precisely one element in $\mathsf{P}(\varphi)^{s}$ which is not perfect. We
shall refer to it as the \emph{extreme} stable element with phase
$\varphi$. So, the sheaf $\boldsymbol{k}(s)$ is the extreme stable element
with phase $1$. The extreme stable element is never locally free. A stable
object is either perfect or extreme.
We shall need the following version of Serre duality, which can be deduced
easily from standard versions:
If $E,F\in\Dbcoh(\boldsymbol{E})$ and at least one of them is perfect, then
there is a bi-functorial isomorphism
\begin{equation}
\label{wesPT:i}\Hom(E,F) \cong \Hom(F,E[1])^{\ast}.
\end{equation}
If neither of the objects is perfect, this is no longer true. For example,
$\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s))\cong \boldsymbol{k}$, but
$\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s)[1]) \cong
\Ext^{1}(\boldsymbol{k}(s),\boldsymbol{k}(s)) \cong \boldsymbol{k}^{2}$.
Any object $X$ in the Abelian category $\mathsf{P}(\varphi)$ has a
Jordan-H\"older filtration (JHF)
$$0\subset F_{n}X \subset \ldots \subset F_{1}X \subset F_{0}X = X$$
with stable JH-factors $J_{i}=F_{i}X/F_{i+1}X \in
\mathsf{P}(\varphi)^{s}$. The graded object $\oplus_{i=0}^{n}J_{i}$ is
determined by $X$. Observe that for any two objects $J\not\cong
J'\in\mathsf{P}(\varphi)^{s}$ we can apply Serre duality because at most one
of them is non-perfect.
\begin{corollary}\label{cor:sheaves}
\begin{enumerate}
\item\label{cor:i} If $\varphi,\psi \in \mathsf{Q}$ with $\varphi -1 < \psi
\le \varphi$ there exists $\Phi\in \widetilde{\SL}(2,\mathbb{Z})$ such that
$\Phi(\varphi)=1$ and $\Phi(\psi)\in(0,1]$.
\item\label{cor:ii} If $A,B\in\mathsf{P}(\varphi)^{s}$, then $A\cong B \iff
\Hom(A,B)\ne 0.$
\item\label{cor:iii} If $0\ne X\in\mathsf{P}(\varphi)$ and $0\ne
Y\in\mathsf{P}(\psi)$ with $\varphi < \psi < \varphi+1$, then $\Hom(X,Y)\ne
0$.
\item \label{cor:iv} If $J\in\mathsf{P}(\varphi)^{s}$ is not a JH-factor of
$X\in \mathsf{P}(\varphi)$, for all $i\in\mathbb{Z}$ we have
$\Hom(J,X[i])=0$.
\item \label{cor:v} If $X\in\mathsf{P}(\varphi)$ is indecomposable, all its
JH-factors are isomorphic to each other.
\item \label{cor:vi} If $X,Y\in\mathsf{P}(\varphi)$ are non-zero
indecomposable objects, both with the same JH-factor, then $\Hom(X,Y) \ne
0$.
\end{enumerate}
\end{corollary}
\begin{proof}
(\ref{cor:i}) This follows from Proposition \ref{prop:transit} because the
shift functor corresponds to an element in the centre of
$\widetilde{\SL}(2,\mathbb{Z})$ and therefore $\Phi(\mathsf{P}(\varphi)) =
\mathsf{P}(1)$ implies $\Phi(\mathsf{P}(\varphi-1)) = \mathsf{P}(0)$.
(\ref{cor:ii}) The statement is clear in case $\varphi=1$ and follows from
(\ref{cor:i}) in the general case.
(\ref{cor:iii}) Using (\ref{cor:i}) we can assume $\psi=1$, which means
that $Y$ is a coherent torsion sheaf. By Proposition \ref{prop:transit} this
implies $\varphi\in(0,1)$ and $X$ is a torsion free coherent sheaf. If
$Y\in\mathsf{P}(1)^{s}$ the statement is clear, because any torsion free
sheaf has a non-zero morphism to any $Y=\boldsymbol{k}(x)$,
$x\in\boldsymbol{E}$. If $Y\in\mathsf{P}(1)$ is arbitrary, there exists a
point $x\in\boldsymbol{E}$ and a non-zero morphism $\boldsymbol{k}(x)
\rightarrow Y$. The claim follows now from left-exactness of the functor
$\Hom(X,\,\cdot\,)$.
(\ref{cor:iv}) If $J'\in\mathsf{P}(\varphi)^{s}$ is a JH-factor of $X$, we
have $J\not\cong J'$. From (\ref{cor:ii}) and Serre duality together with
Lemma \ref{wesPT:ii} we obtain $\Hom(J,J'[i])=0$ for any
$i\in\mathbb{Z}$. Using the JHF of $X$, the claim now follows.
(\ref{cor:v}) It is easy to prove by induction that any
$X\in\mathsf{P}(\varphi)$ can be split as a finite direct sum $X\cong\oplus
X_{k}$, where each $X_{k}$ has all JH-factors isomorphic to a single element
$J_{k}\in\mathsf{P}(\varphi)^{s}$. This implies, the claim.
(\ref{cor:vi}) By (\ref{cor:i}) we may assume
$\varphi(X)=\varphi(Y)=1$. This means, both objects are indecomposable
torsion sheaves with support at the singular point $s\in\boldsymbol{E}$.
Such sheaves always have an epimorphism to and a monomorphism from the
extreme object $\boldsymbol{k}(s)$, hence the claim.
\end{proof}
It is interesting and important to note that an indecomposable semi-stable
object can be perfect even though all its JH-factors are extreme. This is
made explicit in \cite{BurbanKreussler}, Section 4, in the case of the
category $\mathsf{P}(1)$ of coherent torsion sheaves. If $\boldsymbol{E}$ is
nodal, there are two kinds of indecomposable torsion sheaves with support at
the node $s\in\boldsymbol{E}$: the so-called \emph{bands} and
\emph{strings}. The bands are perfect, whereas the strings are not
perfect. Using the action of $\widetilde{\SL}(2,\mathbb{Z})$ this carries over
to all other categories $\mathsf{P}(\varphi)$ with $\varphi\in\mathsf{Q}$.
An object $X\in\mathsf{P}(\varphi)$ will be called \emph{extreme} if it
does not have a direct summand which is perfect. This implies that, but is not
equivalent to the property that all its JH-factors are extreme. An example can
be found below, see Ex.~\ref{ex:extremefactors}.
From the above we deduce that any $X\in\mathsf{P}(\varphi)$ can be split as a
direct sum $X\cong X^{e}\oplus X^{p}$ with $X^{e}$ extreme and $X^{p}$
perfect. All direct summands of the extreme part have the unique
extreme stable element with phase $\varphi$ as its JH-factors. On the
other hand, all the direct summands of $X^{p}$ are perfect and they can have
any object of $\mathsf{P}(\varphi)^{s}$ as JH-factor.
\begin{corollary}
Any coherent sheaf $\mathcal{F}$ with $\End(\mathcal{F}) = \boldsymbol{k}$
is stable.
\end{corollary}
\begin{proof}
The assumption implies that $\mathcal{F}$ is indecomposable.
If $\mathcal{F}$ were not even semi-stable, it would have at least two
HN-factors. Using Corollary \ref{cor:sheaves}, we may assume that
$\varphi_{+}(\mathcal{F})=1$. Thus, $\mathcal{F}$ is a coherent sheaf which
is neither torsion nor torsion free. This implies that there is a
non-invertible endomorphism
$\mathcal{F} \rightarrow \boldsymbol{k}(s) \rightarrow \tors(\mathcal{F})
\rightarrow \mathcal{F}$, in contradiction to the assumption. Hence,
$\mathcal{F}\in\mathsf{P}(\varphi)$ is semi-stable. Let
$\mathcal{J}\in\mathsf{P}(\varphi)$ be its JH-factor.
From Corollary \ref{cor:sheaves} (\ref{cor:vi}) we obtain a non-zero
endomorphism $\mathcal{F}\rightarrow \mathcal{J}\rightarrow \mathcal{F}$,
which can only be an isomorphism, if $\mathcal{F}\cong \mathcal{J}$, so
$\mathcal{F}$ is indeed stable.
\end{proof}
The following method can be used to visualise the structure of the category
$\Dbcoh(\boldsymbol{E})$: the vertical slices in Figure \ref{fig:slices} are
thought to correspond to the categories $\mathsf{P}(t)^{s}$ of stable objects.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1,1){\line(0,1){3}}\put(1,0.8){\makebox(0,0)[t]{$2$}}
\put(4,1){\line(0,1){3}}\put(4,0.8){\makebox(0,0)[t]{$1$}}
\put(7,1){\line(0,1){3}}\put(7,0.8){\makebox(0,0)[t]{$0$}}
\put(10,1){\line(0,1){3}}\put(10,0.8){\makebox(0,0)[t]{$-1$}}
\thinlines
\put(4.9,1){\line(0,1){3}}\put(4.9,0.8){\makebox(0,0)[t]{$t$}}
\put(1.8,2.5){$\Coh_{\boldsymbol{E}}[1]$}
\put(5.3,2.5){$\Coh_{\boldsymbol{E}}$}
\put(7.6,2.5){$\Coh_{\boldsymbol{E}}[-1]$}
\end{picture}
\end{center}
\caption{slices}\label{fig:slices}
\end{figure}
They are non-empty if and only if $t\in\mathsf{Q}$, i.e.\/
$\mathbb{R}\exp(\pi it) \cap \mathbb{Z}^{2} \ne \{(0,0)\}$. A point on such a
slice represents a stable object. The extreme stable objects are those which
lie on the dashed upper horizontal line. The labelling below the picture
reflects the phases of the slices. We have chosen to let it decrease from the
left to right in order to have objects with cohomology in negative degrees on
the left and with positive degrees on the right.
By Proposition \ref{prop:transit}, the group $\widetilde{\SL}(2,\mathbb{Z})$
acts on the set of all stable objects, hence it acts on such pictures.
This action sends slices to slices and acts transitively on the set of
slices with phase $t\in\mathsf{Q}$. The dashed line of extreme stable objects
is invariant under this action.
Any indecomposable object $0\ne X\in\Dbcoh(\boldsymbol{E})$ has a
\emph{shadow} in such a picture: it is the set of all stable objects which
occur as JH-factors in the HN-factors of $X$. If this set consists of more
than one point, the shadow is obtained by connecting these points by line
segments.
The following proposition shows that the shadow of an indecomposable object
which consists of more than one point is completely contained in the
extreme line.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1,1){\line(0,1){3}}\put(1,0.8){\makebox(0,0)[t]{$2$}}
\put(4,1){\line(0,1){3}}\put(4,0.8){\makebox(0,0)[t]{$1$}}
\put(7,1){\line(0,1){3}}\put(7,0.8){\makebox(0,0)[t]{$0$}}
\put(10,1){\line(0,1){3}}\put(10,0.8){\makebox(0,0)[t]{$-1$}}
\thinlines
\put(1.8,2.5){$\Coh_{\boldsymbol{E}}[1]$}
\put(5.3,2.5){$\Coh_{\boldsymbol{E}}$}
\put(7.6,2.5){$\Coh_{\boldsymbol{E}}[-1]$}
\put(4,2){\circle*{0.2}}\put(4.2,2){\makebox(0,0)[l]{$X_{1}$}}
\put(8.2,3.4){\circle*{0.2}}\put(8.4,3.4){\makebox(0,0)[l]{$X_{2}$}}
\put(0.3,4){\circle*{0.2}}
\thicklines\put(0.3,4){\line(1,0){1.5}}
\put(1.8,4){\circle*{0.2}}
\thicklines\put(1.8,4){\line(1,0){0.8}}
\put(2.6,4){\circle*{0.2}}\put(1.3,4.2){\makebox(0,0)[b]{$X_{3}$}}
\put(4.3,4){\circle*{0.2}}
\thicklines\put(4.3,4){\line(1,0){1}}
\put(5.3,4){\circle*{0.2}}\put(4.8,4.2){\makebox(0,0)[b]{$X_{4}$}}
\put(6.3,4){\circle*{0.2}}\put(6.3,4.2){\makebox(0,0)[b]{$X_{5}$}}
\end{picture}
\end{center}
\caption{shadows}\label{fig:example}
\end{figure}
Figure \ref{fig:example} shows the shadows of five different
indecomposable objects:
\begin{itemize}
\item $X_{1}\in\Coh_{\boldsymbol{E}}$ an indecomposable torsion sheaf,
\item $X_{2}\in \Coh_{\boldsymbol{E}}[-1]$ the shift of an indecomposable
semi-stable locally free sheaf,
\item $X_{3}$ a genuine complex with three extreme HN-factors, one in
$\Coh_{\boldsymbol{E}}[2]$ and the other two in $\Coh_{\boldsymbol{E}}[1]$,
\item $X_{4}$ an indecomposable torsion free sheaf which is not semi-stable,
\item $X_{5}\in\Coh_{\boldsymbol{E}}$ an indecomposable and semi-stable
torsion free sheaf which could be perfect or not (a band or a string in the
language of representation theory).
\end{itemize}
The shadow of an indecomposable object is a single point if and only if this
object is semi-stable.
\begin{proposition}\label{prop:extreme}
Let $X \in \Dbcoh(\boldsymbol{E})$ be an indecomposable object which is not
semi-stable. Then, all HN-factors of $X$ are extreme.
\end{proposition}
\begin{proof}
Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $X$. If the HN-factor $A_{i}$ were not
extreme, it could be split into a direct sum $A_{i} \cong A_{i}'
\oplus A_{i}''$ with $0\ne A_{i}'$ perfect and $A_{i}',
A_{i}''\in\mathsf{P}(\varphi_{i})$.
Because $\varphi_{-}(F_{i+1}X) > \varphi_{i}=\varphi(A_{i}')$, Lemma
\ref{wesPT:ii} and Serre duality imply
$$\Hom(A_{i}', F_{i+1}X[1]) \cong \Hom(F_{i+1}X, A_{i}')^{\ast} = 0.$$
Hence, we can apply Lemma \ref{lem:PengXiao} to the distinguished triangle
$$F_{i+1}X \rightarrow F_{i}X \rightarrow A_{i}
\stackrel{+}{\longrightarrow}$$
and obtain a decomposition $F_{i}X \cong F_{i}'X \oplus A_{i}'$.
We proceed by descending induction on $j\le i$ to show that there exist
decompositions $F_{j}X \cong F_{j}'X \oplus A_{i}'$. This is
obtained from Lemma \ref{lem:PengXiao} applied to the distinguished triangle
$$F_{j}'X\oplus A_{i}' \rightarrow F_{j-1}X \rightarrow A_{j-1}
\stackrel{+}{\longrightarrow}$$
and using Lemma \ref{wesPT:ii}, Serre duality and
$\varphi(A_{i}') > \varphi(A_{j-1})$ to get
$$\Hom(A_{j-1}, A_{i}'[1]) \cong \Hom(A_{i}', A_{j-1})^{\ast} = 0.$$
We obtain a decomposition $X=F_{0}X \cong F_{0}'X \oplus A_{i}'$ in which
we have $A_{i}'\ne 0$. Because $X$ was assumed to be indecomposable, we
should have $X\cong A_{i}'$, but this was excluded by assumption. This
contradiction shows that all HN-factors $A_{i}$ are necessarily
extreme.
\end{proof}
\begin{corollary}\label{cor:types}
There exist four types of indecomposable objects in the category
$\Coh_{\boldsymbol{E}}$:
\begin{enumerate}
\item \label{type:i} semi-stable with perfect JH-factor;
\item \label{type:ii} semi-stable, perfect but its JH-factor extreme;
\item \label{type:iii} semi-stable and extreme;
\item \label{type:iv} not semi-stable, with all its HN-factors extreme.
\end{enumerate}
\end{corollary}
A similar statement is true for $\Dbcoh(\boldsymbol{E})$. In this case, the
objects of types (\ref{type:i}), (\ref{type:ii}) and (\ref{type:iii}) are
shifts of coherent sheaves, whereas genuine complexes are possible for objects
of type (\ref{type:iv}).
Types (\ref{type:ii}), (\ref{type:iii}) and (\ref{type:iv}) were not available
in the smooth case.
Examples of type (\ref{type:i}) are simple vector bundles and structure sheaves
$\boldsymbol{k}(x)$ of smooth points $x\in\boldsymbol{E}$. All indecomposable
objects with a shadow not on the extreme line fall into type (\ref{type:i}).
Under the equivalences of Corollary \ref{cor:equiv}, indecomposable semi-stable
locally free sheaves with extreme JH-factor correspond, in the nodal case,
precisely to those torsion sheaves with support at the node $s$, which are
called bands, see \cite{BurbanKreussler}.
Examples of type (\ref{type:iii}) are the stable coherent sheaves which are not
locally free and the structure sheaf $\boldsymbol{k}(s)$ of the singular point
$s\in\boldsymbol{E}$. Moreover, in the nodal case, the torsion sheaves with
support at $s$, which are called strings in \cite{BurbanKreussler}, are of
type (\ref{type:iii}) as well. Examples of objects of type (\ref{type:iv}) are
given below.
\begin{example}
We shall construct torsion free sheaves on nodal $\boldsymbol{E}$ with an
arbitrary finite number of HN-factors. This implies that the number of points
in a shadow of an indecomposable object in $\Dbcoh(\boldsymbol{E})$ is not
bounded.
Recall from \cite{DrozdGreuel} that any indecomposable torsion free sheaf
which is not locally free, is isomorphic to a sheaf
$\mathcal{S}(\boldsymbol{d}) = p_{n\ast} \mathcal{L}(\boldsymbol{d})$. We use
here the notation of \cite{BurbanKreussler}, Section 3.5, so that $p_{n}:
\boldsymbol{I_{n}} \rightarrow \boldsymbol{E}$ denotes a certain morphism
from the chain $\boldsymbol{I_{n}}$ of $n$ smooth rational curves to the
nodal curve $\boldsymbol{E}$. If $\boldsymbol{d}=(d_{1},\ldots,d_{n})
\in\mathbb{Z}^{n}$, we denote by $\mathcal{L}(\boldsymbol{d})$ the line
bundle on $\boldsymbol{I_{n}}$ which has degree $d_{\nu}$ on the $\nu$-th
component of $\boldsymbol{I_{n}}$. We know $\rk(\mathcal{S}(\boldsymbol{d}))
= n$ and $\deg(\mathcal{S}(\boldsymbol{d})) = 1+\sum d_{\nu}$. We obtain,
in particular, that for any $\varphi\in\mathsf{Q} \cap (0,1)$ there exist
$n\in\mathbb{Z}$ and $\boldsymbol{d}(\varphi)\in\mathbb{Z}^{n}$ such that
$\mathcal{S}(\boldsymbol{d}(\varphi))$ is the unique extreme element in
$\mathsf{P}(\varphi)^{s}$. On the other hand, if $\boldsymbol{d}'\in
\mathbb{Z}^{n'}, \boldsymbol{d}''\in \mathbb{Z}^{n''}$ and
$\boldsymbol{d} = (\boldsymbol{d}_{+}', \boldsymbol{d}'')\in
\mathbb{Z}^{n'+n''}$, where $\boldsymbol{d}_{+}'$ is obtained from
$\boldsymbol{d}'$ by adding $1$ to the last component, we have an exact
sequence
$$0\rightarrow \mathcal{S}(\boldsymbol{d}') \rightarrow
\mathcal{S}(\boldsymbol{d}) \rightarrow
\mathcal{S}(\boldsymbol{d}'') \rightarrow 0$$
see for example \cite{Mozgovoy}. Hence, if we start with a sequence
$0<\varphi_{0} <\varphi_{1}< \ldots <\varphi_{m} <1$ where
$\varphi_{\nu}\in\mathsf{Q}$ and define
$$\boldsymbol{d}^{(m)} = \boldsymbol{d}(\varphi_{m})\quad\text{ and }\quad
\boldsymbol{d}^{(\nu)} =
(\boldsymbol{d}_{+}^{(\nu+1)},\boldsymbol{d}(\varphi_{\nu})) \text{ for }
m > \nu \ge 0,$$
we obtain an indecomposable torsion free sheaf
$\mathcal{S}(\boldsymbol{d}^{(0)})$ whose HN-factors are the extreme stable
sheaves $\mathcal{S}(\boldsymbol{d}(\varphi_{\nu})) \in
\mathsf{P}(\varphi_{\nu}), 0\le \nu \le m$. The HNF of this sheaf is given by
$$\mathcal{S}(\boldsymbol{d}^{(m)}) \subset
\mathcal{S}(\boldsymbol{d}^{(m-1)}) \subset \ldots \subset
\mathcal{S}(\boldsymbol{d}^{(0)}).$$
The sheaf $\mathcal{S}(\boldsymbol{d}^{(0)})$ is of type (\ref{type:iv}) and
not perfect.
\end{example}
\begin{example}\label{ex:extremefactors}
Suppose $\boldsymbol{E}$ is nodal and let
$\pi:C_{2}\rightarrow\boldsymbol{E}$ be an \'etale morphism of degree
two, where $C_{2}$ denotes a reducible curve which has two components, both
isomorphic to $\mathbb{P}^{1}$ and which intersect transversally at two
distinct points.
By $i_{\nu}:\mathbb{P}^{1}\rightarrow \boldsymbol{E},\;\nu=1,2$ we denote the
morphisms which are induced by the embeddings of the two components of
$C_{2}$.
There is a $\boldsymbol{k}^{\times}$-family of
line bundles on $C_{2}$, whose restriction to one component is
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$ and to the other is
$\mathcal{O}_{\mathbb{P}^{1}}(2)$. The element in $\boldsymbol{k}^{\times}$
corresponds to a gluing parameter over one of the two singularities of
$C_{2}$. If $\mathcal{L}$ denotes one such line bundle,
$\mathcal{E}:=\pi_{\ast}\mathcal{L}$ is an
indecomposable vector bundle of rank two and degree zero on $\boldsymbol{E}$.
Let us fix notation so that $i_{1}^{\ast}\mathcal{E}
\cong \mathcal{O}_{\mathbb{P}^{1}}(-2)$ and $i_{2}^{\ast}\mathcal{E} \cong
\mathcal{O}_{\mathbb{P}^{1}}(2)$. There is an exact sequence of coherent
sheaves on $\boldsymbol{E}$
\begin{equation}\label{eq:nonssvb}
0\rightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow
\mathcal{E}
\rightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2) \rightarrow 0.
\end{equation}
Because the torsion free sheaves $i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}$ and
$i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2)$ have rank one and
$\boldsymbol{E}$ is irreducible, they are stable. Because $\varphi(i_{2\ast}
\mathcal{O}_{\mathbb{P}^{1}}) = 3/4$ and $\varphi(i_{1\ast}
\mathcal{O}_{\mathbb{P}^{1}}(-2)) = 1/4$, Theorem \ref{thm:uniqueHNF}
implies that the HNF of $\mathcal{E}$ is given by the
exact sequence (\ref{eq:nonssvb}). The HN-factors are the two torsion
free sheaves of rank one $i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}$ and
$i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2)$, which are not locally
free. These are the extreme stable elements with phases $3/4$ and $1/4$
respectively. Therefore, the indecomposable vector bundle $\mathcal{E}$ is a
perfect object of type (\ref{type:iv}) which satisfies
$\varphi_{-}(\mathcal{E})=1/4$ and $\varphi_{+}(\mathcal{E})=3/4$.
\end{example}
\begin{remark}\label{rem:notperfect}
This example shows that the full sub-category of perfect complexes in the
category $\Dbcoh(\boldsymbol{E})$ is not closed under taking
Harder-Narasimhan factors. We interpret this to be an indication that the
derived category of perfect complexes is not an appropriate object for
homological mirror symmetry on singular Calabi-Yau varieties.
\end{remark}
\begin{remark}
It seems plausible that methods similar to those of this section could be
applied to study the derived category of representations of certain derived
tame associative algebras. Such may include gentle algebras, skew-gentle
algebras and degenerated tubular algebras.
The study of Harder-Narasimhan filtrations in conjunction with the action of
the group of exact auto-equivalences of the derived category may provide new
insight into the combinatorics of indecomposable objects in these derived
categories.
\end{remark}
\begin{proposition}\label{wesPT}
Suppose $X,Y\in \Dbcoh(\boldsymbol{E})$ are non-zero.
\begin{enumerate}
\item \label{wesPT:iii} If $\varphi_{-}(X) <
\varphi_{+}(Y) < \varphi_{-}(X)+1$, then $\Hom(X,Y)\ne 0$.
\item \label{wesPT:iv} If $X$ and $Y$ are indecomposable objects which are
not of type (\ref{type:i}) in Corollary \ref{cor:types} and which satisfy
$\varphi_{-}(X) = \varphi_{+}(Y)$, then $\Hom(X,Y)\ne 0$.
\end{enumerate}
\end{proposition}
\begin{proof}
If $X$ and $Y$ are semi-stable objects, the claim
(\ref{wesPT:iii}) was proved in Corollary \ref{cor:sheaves} (\ref{cor:iii}).
Similarly, (\ref{wesPT:iv}) for two semi-stable objects follows from
Corollary \ref{cor:sheaves} (\ref{cor:vi}), because there is only one
non-perfect object in $\mathsf{P}(\varphi)^{s}$.
For the rest of the proof we treat both cases, (\ref{wesPT:iii}) and
(\ref{wesPT:iv}) simultaneously.
For the proof of (\ref{wesPT:iv}) we keep in mind that Proposition
\ref{prop:extreme} implies that no HN-factor has a perfect summand, if
the object is indecomposable but not semi-stable.
If $X\in\mathsf{P(\varphi)}$ is semi-stable but $Y\in\Dbcoh(\boldsymbol{E})$
is arbitrary, we let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{m}Y \ar[rr] \ar[dl]_{\cong}&& F_{m-1}Y \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}Y \ar[rr] && F_{0}Y \ar@{=}[r]\ar[dl] &
Y\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
be a HNF of $Y$. As $\varphi(B_{m})=\varphi_{+}(Y)$ we know already
$\Hom(X,B_{m})\ne 0$.
By assumption, we have $\varphi(B_{i}[-1]) = \varphi(B_{i})
-1 \le \varphi_{+}(Y)-1 < \varphi(X)$. Hence, by Lemma \ref{wesPT:ii},
$\Hom(X, B_{i}[-1]) =0$ and the cohomology sequence of the distinguished
triangle $F_{i+1}Y\rightarrow F_{i}Y\rightarrow B_{i}
\stackrel{+}{\rightarrow}$ provides an inclusion $\Hom(X, F_{i+1}Y) \subset
\Hom(X, F_{i}Y)$. This implies $0\ne \Hom(X,B_{m})\subset \Hom(X,Y)$.
Finally, in the general case, we let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $X$. As $\varphi(A_{0})=\varphi_{-}(X)$ we have
$\Hom(A_{0},Y)\ne 0$.
Because $\varphi_{-}(F_{1}X[1]) = \varphi_{-}(F_{1}X) +1 =
\varphi(A_{1}) +1 > \varphi_{-}(X)+1 > \varphi_{+}(Y)$, Lemma
\ref{wesPT:ii} implies $\Hom(F_{1}X[1], Y)=0$. The distinguished triangle
$F_{1}X \rightarrow X \rightarrow A_{0} \stackrel{+}{\rightarrow}$ gives us
now an inclusion $0\ne \Hom(A_{0},Y) \subset \Hom(X,Y)$ and so the claim.
\end{proof}
In \cite{YangBaxter}, Polishchuk asked for the classification of all
spherical objects in the bounded derived category of a singular projective
curve of arithmetic genus one. Below, we shall solve this problem for
irreducible curves.
Let $\boldsymbol{E}$ be an irreducible projective curve of arithmetic genus
one over our base field $\boldsymbol{k}$.
Recall that in this case an object $X\in\Dbcoh(\boldsymbol{E})$ is
\emph{spherical} if
$$X \text{ is perfect and }\quad
\Hom(X,X[i]) \cong
\begin{cases}
\boldsymbol{k} & \text{if }\; i \in \{0,1\} \\
0 & \text{if }\; i \not\in \{0,1\}
\end{cases}
$$
\begin{proposition}\label{prop:spherical}
Let $\boldsymbol{E}$ be an irreducible projective curve of arithmetic genus
one and $X\in\Dbcoh(\boldsymbol{E})$. Then the following are equivalent:
\begin{enumerate}
\item\label{spher:i} $X$ is spherical;
\item \label{spher:ii}$\Hom(X,X[i]) \cong
\begin{cases}
\boldsymbol{k} & \text{if }\; i = 0 \\
0 & \text{if }\; i = 2 \;\text{ or }\; i<0;
\end{cases}$
\item\label{spher:iii} $X$ is perfect and stable;
\item\label{spher:iv} there exists $n\in\mathbb{Z}$ such that $X[n]$ is
isomorphic to a simple vector bundle or to a torsion sheaf of length one
which is supported at a smooth point of $\boldsymbol{E}$.
\end{enumerate}
In particular, the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of all spherical
objects.
\end{proposition}
\begin{proof}
The implication (\ref{spher:i})$\Rightarrow$(\ref{spher:ii}) is obvious.
Let us prove (\ref{spher:ii})$\Rightarrow$(\ref{spher:iii}).
First, we observe that $\Hom(X,X) \cong \boldsymbol{k}$ implies that
$X$ is indecomposable. Suppose, $X$ is not semi-stable. This is equivalent
to $\varphi_{+}(X)>\varphi_{-}(X)$. By Proposition \ref{prop:extreme}
we know that all HN-factors of $X$ are extreme.
Let $M\ge 0$ be the unique integer with $M\le \varphi_{+}(X) -
\varphi_{-}(X) < M+1$.
If $M< \varphi_{+}(X) - \varphi_{-}(X) <M+1$,
Proposition \ref{wesPT} (\ref{wesPT:iii}) implies $\Hom(X,X[-M])\ne
0$. Under the assumption (\ref{spher:ii}), this is possible only if $M=0$.
On the other hand, if $M=\varphi_{+}(X) - \varphi_{-}(X)$, we obtain from
Proposition \ref{wesPT} (\ref{wesPT:iv}) $\Hom(X,X[-M]) \ne 0$.
Again, this implies $M=0$.
So, we have $0< \varphi_{+}(X) - \varphi_{-}(X) <1$.
If we apply the functor $\Hom(\,\cdot\,,X)$ to
$F_{1}X\stackrel{u}{\rightarrow} X \rightarrow A_{0}
\stackrel{+}{\longrightarrow}$,
the rightmost distinguished triangle of the HNF of $X$, we obtain the exact
sequence
$$\Hom(F_{1}X[1],X) \rightarrow \Hom(A_{0},X) \rightarrow \Hom(X,X)
\rightarrow \Hom(F_{1}X,X),$$
in which the leftmost term $\Hom(F_{1}X[1],X)=0$ by Lemma \ref{wesPT:ii},
because $\varphi_{-}(F_{1}X[1]) > \varphi_{-}(X)+1>\varphi_{+}(X)$. The third
morphism in this sequence is not the zero map, as it sends $\mathsf{Id}_{X}$
to $u\ne 0$. Because $\Hom(X,X)$ is one dimensional, this is only possible
if $\Hom(A_{0},X)=0$. But Proposition \ref{wesPT} (\ref{wesPT:iii}) and
$\varphi(A_{0})<\varphi_{+}(X)< \varphi(A_{0})+1$ imply $\Hom(A_{0},X)\ne 0$.
This contradiction shows that $X$ must be semi-stable.
We observed earlier that all the
JH-factors of an indecomposable semi-stable object are isomorphic to each
other. Therefore, any indecomposable semi-stable object which is not stable
has a space of endomorphisms of dimension at least two. So, we conclude
$X\in\mathsf{P}(\varphi)^{s}$ for some $\varphi\in\mathbb{R}$.
Because $\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s)[2]) \cong
\Ext^{2}(\boldsymbol{k}(s),\boldsymbol{k}(s))\ne 0$, the transitivity of the
action of $\widetilde{\SL}(2,\mathbb{Z})$ on the set $\mathsf{Q}$ implies
that none of the extreme stable objects satisfies the condition
(\ref{spher:ii}). Hence, $X$ is perfect and stable.
To prove (\ref{spher:iii})$\Rightarrow$(\ref{spher:i}), we observe that
the group of automorphisms of the curve $\boldsymbol{E}$ acts
transitively on the regular locus $\boldsymbol{E}\setminus\{s\}$. Hence, by
Proposition \ref{prop:transit}, the group of auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of all perfect stable
objects. Because, for example, the structure sheaf
$\mathcal{O}_{\boldsymbol{E}}$ is spherical, it is now clear that all
perfect stable objects are indeed spherical and that the group of
exact auto-equivalences of $\Dbcoh(\boldsymbol{E})$ acts transitively on the
set of all spherical objects.
To show the equivalence with (\ref{spher:iv}), it remains to recall that any
perfect coherent torsion free sheaf on $\boldsymbol{E}$ is locally free. This
follows easily from the Auslander-Buchsbaum formula because we are working in
dimension one.
\end{proof}
\section{Description of $t$-structures in the case of a
singular Weierstra\ss{} curve}\label{sec:tstruc}
The main result of this section is a description of all $t$-structures on the
derived category of a singular Weierstra\ss{} curve $\boldsymbol{E}$. This
generalises results of \cite{GRK} and \cite{Pol1}, where the smooth case was
studied. As an application, we obtain a description of the group
$\Aut(\Dbcoh(\boldsymbol{E}))$ of all exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$. A second application is a description of Bridgeland's
space of stability conditions on $\boldsymbol{E}$.
Recall that a $t$-structure on a triangulated category $\mathsf{D}$ is a pair
of full subcategories $(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 0})$ such that,
with the notation $\mathsf{D}^{\ge n} := \mathsf{D}^{\ge
0}[-n]$ and $\mathsf{D}^{\le n} := \mathsf{D}^{\le 0}[-n]$ for any $n\in
\mathbb{Z}$, the following holds:
\begin{enumerate}
\item $\mathsf{D}^{\le 0} \subset \mathsf{D}^{\le 1}$ and $\mathsf{D}^{\ge 1}
\subset \mathsf{D}^{\ge 0}$;
\item $\Hom(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 1}) = 0$;
\item\label{def:tiii} for any object $X \in \mathsf{D}$ there exists a
distinguished triangle
$$A \rightarrow X \rightarrow B \stackrel{+}{\longrightarrow}$$
with $A \in \mathsf{D}^{\le 0}$ and $B \in \mathsf{D}^{\ge 1}.$
\end{enumerate}
If $(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 0})$ is a $t$-structure then ${\sf
A} = \mathsf{D}^{\le 0} \cap \mathsf{D}^{\ge 0}$ has a structure of an
Abelian category. It is called the \emph{heart} of the $t$-structure.
In this way, $t$-structures on the derived category
$\Dbcoh(\boldsymbol{E})$ lead to interesting Abelian categories embedded into
it. The natural $t$-structure on $\Dbcoh(\boldsymbol{E})$ has $\mathsf{D}^{\le
n}$ equal to the full subcategory formed by all complexes with non-zero
cohomology in degree less or equal to $n$ only. Similarly, the full subcategory
$\mathsf{D}^{\ge n}$ consists of all complexes $X$ with $H^{i}(X)=0$ for all
$i<n$. The heart of the natural $t$-structure is the Abelian category
$\Coh_{\boldsymbol{E}}$.
In addition to the natural $t$-structure we also have many interesting
$t$-structures on $\Dbcoh(\boldsymbol{E})$.
In order to describe them, we introduce the following notation. We continue to
work with the notion of stability and the notation introduced in the previous
section.
If $\mathsf{P}\subset\mathsf{P}(\theta)^{s}$ is a subset, we denote by
$\mathsf{D}[\mathsf{P}, \infty)$ the full subcategory of
$\Dbcoh(\boldsymbol{E})$ which is defined as follows:
$X\in\Dbcoh(\boldsymbol{E})$ is in $\mathsf{D}[\mathsf{P},
\infty)$ if and only if $X=0$ or all its HN-factors, which have at least one
JH-factor which is not in $\mathsf{P}$, have phase $\varphi>\theta$.
Similarly, $\mathsf{D}(-\infty,\mathsf{P}]$ denotes the category which is
generated by $\mathsf{P}$ and all $\mathsf{P}(\varphi)$ with
$\varphi<\theta$.
If $\mathsf{P}=\mathsf{P}(\theta)^{s}$ we may abbreviate
$\mathsf{D}[\theta,\infty) = \mathsf{D}[\mathsf{P}, \infty)$ and
$\mathsf{D}(-\infty,\theta] = \mathsf{D}(-\infty,\mathsf{P}]$.
Similarly, if $\mathsf{P}=\emptyset$ we use
the abbreviations $\mathsf{D}(\theta,\infty)$ and $\mathsf{D}(-\infty,\theta)$.
For any open, closed or half-closed interval $I\subset\mathbb{R}$ we define
the full subcategories $\mathsf{D}I$ precisely in the same way. Thus, an
object $0\ne X\in\Dbcoh(\boldsymbol{E})$ is in $\mathsf{D}I$ if and only if
$\varphi_{-}(X)\in I$ and $\varphi_{+}(X)\in I$.
\begin{proposition}\label{prop:texpl}
Let $\theta\in\mathbb{R}$ and $\mathsf{P}(\theta)^{-} \subset
\mathsf{P}(\theta)^{s}$ be arbitrary. Denote by $\mathsf{P}(\theta)^{+} =
\mathsf{P}(\theta)^{s} \setminus \mathsf{P}(\theta)^{-}$ the complement of
$\mathsf{P}(\theta)^{-}$. Then,
$$\mathsf{D}^{\le0} := \mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)$$
defines a $t$-structure on $\Dbcoh(\boldsymbol{E})$ with
$$\mathsf{D}^{\ge1} := \mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}].$$
The heart $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ of it is
the category $\mathsf{D}[\mathsf{P}(\theta)^{-},
\mathsf{P}(\theta)^{+}[1]]$, which consists of those objects
$X\in\Dbcoh(\boldsymbol{E})$ whose HN-factors either have
phase $\varphi\in(\theta,\theta+1)$ or have all its JH-factors in
$\mathsf{P}(\theta)^{-}$ or $\mathsf{P}(\theta)^{+}[1]$.
\end{proposition}
\begin{proof}
The only non-trivial property which deserves a proof is (\ref{def:tiii}) in
the definition of $t$-structure. Given $X\in \Dbcoh(\boldsymbol{E})$, we
have to show that there exists a distinguished triangle $A\rightarrow X
\rightarrow B \stackrel{+}{\rightarrow}$ with $A\in \mathsf{D}^{\le0}$ and
$B\in \mathsf{D}^{\ge1}$. In order to construct it, let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be the HNF of $X$. Because $\varphi(A_{i+1})>\varphi(A_{i})$ for all $i$,
there exists an integer $k$, $0\le k \le n+1$ such that $\varphi(A_{k})\ge
\theta >\varphi(A_{k-1})$. If $\varphi(A_{k})>\theta$, this implies
$A_{i}\in \mathsf{D}^{\le0}$, if $i\ge k$ and $A_{i}\in \mathsf{D}^{\ge1}$,
if $i<k$. In particular, $F_{k}X\in \mathsf{D}^{\le0}$. In this case, we
define $A:=F_{k}X$ and let $A=F_{k}X \rightarrow X$ be the composition of
the morphisms in the HNF.
If, however, $\varphi(A_{k})=\theta$, there is a splitting $A_{k}\cong
A_{k}^{-} \oplus A_{k}^{+}$ such that all JH-factors of $A_{k}^{-}$ (resp.\/
$A_{k}^{+}$) are in $\mathsf{P}(\theta)^{-}$ (resp.\/
$\mathsf{P}(\theta)^{+}$).
Now, we apply Lemma \ref{lem:connect} to the distinguished triangles
$F_{k+1}X \stackrel{f}{\longrightarrow} F_{k}X \longrightarrow A_{k}
\stackrel{+}{\longrightarrow}$
and
$A_{k}^{-} \longrightarrow A_{k} \longrightarrow A_{k}^{+}
\stackrel{+}{\longrightarrow}$, given by the splitting of $A_{k}$,
to obtain a factorisation $F_{k+1}X \rightarrow A \rightarrow F_{k}X$ of
$f$ and two distinguished triangles
$$\xymatrix@C=.5em{F_{k+1}X \ar[rr] && A \ar[dl]\ar[rr] && F_{k}X.\ar[dl]\\
& A_{k}^{-} \ar[ul]^{+} && A_{k}^{+}\ar[ul]^{+}}$$
Part of the given HNF of $X$ together with the left one of these two
triangles form a HNF of $A$, whence $A\in \mathsf{D}^{\le0}$.
Again, we let $A\rightarrow X$
be obtained by composition with the morphisms in the HNF of $X$.
In any case, we choose a distinguished triangle $A\rightarrow X \rightarrow B
\stackrel{+}{\rightarrow}$, where $A\rightarrow X$ is the morphism chosen
before. From Lemma \ref{lem:split} or Remark \ref{rem:split} we obtain $B\in
\mathsf{D}^{\ge1}$. This proves the proposition.
\end{proof}
We shall also need the following standard result.
\begin{lemma}\label{lem:tsummands}
Let $(\mathsf{D}^{\le 0}, \sf{D}^{\ge 0})$ be a $t$-structure on a
triangulated category. If $X \oplus Y \in \sf{D}^{\le 0}$ then
$X \in \sf{D}^{\le 0}$ and $Y \in \sf{D}^{\le 0}$.
The corresponding statement holds for $\sf{D}^{\ge 0}$.
\end{lemma}
\begin{proof}
Let $A \stackrel{f}{\longrightarrow} X \stackrel{g}{\longrightarrow} B
\stackrel{+}{\longrightarrow}$ be a distinguished triangle with
$A\in\mathsf{D}^{\le 0}$ and $B\in\mathsf{D}^{\ge 1}$, which exists due to
the definition of a $t$-structure. If $X\not\in \mathsf{D}^{\le 0}$, we
necessarily have $g\ne 0$ and $B \ne 0$.
Because $\Hom(\sf{D}^{\le 0}, \sf{D}^{\ge 1}) = 0$, the composition $X
\oplus Y \stackrel{p}{\longrightarrow} X \stackrel{g}{\longrightarrow} B$,
in which $p$ denotes the natural projection, must be zero. If
$i:X\rightarrow X\oplus Y$ denotes the canonical morphism, we
obtain $g=g\circ p\circ i=0$, a contradiction. In the same way it follows
that $Y\in\mathsf{D}^{\le 0}$.
\end{proof}
Recall that an Abelian category is called \emph{Noetherian}, if any sequence of
epimorphisms stabilises, this means that for any sequence of epimorphisms
$f_{k}:A_{k}\rightarrow A_{k+1}$ there exists an integer $k_{0}$ such that
$f_{k}$ is an isomorphism for all $k\ge k_{0}$.
\begin{lemma}\label{lem:heart}
The heart $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ of the $t$-structure,
which was described in Proposition \ref{prop:texpl}, is Noetherian if and
only if $\mathsf{P}(\theta) \ne \{0\}$ and $\mathsf{P}(\theta)^{-} =
\emptyset$. In this case, ${\sf A}(\theta, \emptyset) = \mathsf{D}(\theta,
\theta +1]$.
\end{lemma}
\begin{proof}
If $\mathsf{P}(\theta) = \{0\}$ then
$\mathsf{A}(\theta,\mathsf{P}(\theta)^{-}) =
\mathsf{D}(\theta,\theta+1)$. This category is not Noetherian.
To prove this, we follow the proof of Polishchuk in the smooth case
\cite{Pol1}, Proposition 3.1.
We are going to show for any non-zero locally free shifted sheaf $E \in
\mathsf{D}(\theta, \theta +1)$, the
existence of a locally free shifted sheaf $F$ and an epimorphism
$E\twoheadrightarrow F$ in $\mathsf{D}(\theta, \theta +1)$, which is not an
isomorphism. This will be sufficient to show that $\mathsf{D}(\theta, \theta
+1)$ is not Noetherian.
By applying an appropriate shift, we may assume $0<\theta<1$. Under this
assumption, for every stable coherent sheaf $G$ we have
\begin{align*}
G\in\mathsf{D}(\theta, \theta +1) &\iff \theta<\varphi(G)\le 1\\
G[1]\in\mathsf{D}(\theta, \theta +1) &\iff 0<\varphi(G)< \theta.
\end{align*}
For any two objects $X,Y\in\Dbcoh(\boldsymbol{E})$ we define the Euler form
to be
$$\langle X,Y \rangle = \rk(X)\deg(Y) - \deg(X)\rk(Y)$$
which is the imaginary part of $\overline{Z(X)}Z(Y)$. If $X$ and $Y$ are
coherent sheaves and one of them is perfect, we have
$$\langle X,Y \rangle = \chi(X,Y) := \dim\Hom(X,Y) - \dim \Ext^{1}(X,Y).$$
This remains true, if we apply arbitrary shifts to the sheaves $X,Y$, where
we understand $\chi(X,Y)=\sum_{\nu} (-1)^{\nu} \dim \Hom(X,Y[\nu]).$
Let $E\in\mathsf{D}(\theta, \theta +1)$ be an arbitrary non-zero locally
free shifted sheaf. We look at the strip in the plane between the lines
$L(0):= \mathbb{R}\exp(i\pi\theta)$ and $L(E):=L(0)+Z(E)$. This strip must
contain lattice points in its interior.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,6)
\put(1.5,2){\vector(1,0){9.5}}\put(11,1.9){\makebox(0,0)[t]{$-\deg$}}
\put(6,0){\vector(0,1){6}}\put(5.8,6){\makebox(0,0)[r]{$\rk$}}
\put(2,0){\line(2,1){9}}
\put(11,4.3){\makebox(0,0)[t]{$\theta$}}
\put(2.8,0){\makebox(0,0)[l]{$\theta+1$}}
\put(1,1.5){\line(2,1){9}}
\put(6,2){\vector(-1,1){1}}\put(4.8,2.9){\makebox(0,0)[t]{$F$}}
\put(6,2){\vector(2,3){2}}\put(8.2,4.9){\makebox(0,0)[t]{$E$}}
\put(5,3){\vector(3,2){3}}
\put(10.5,4.5){\makebox(0,0)[b]{$L(0)$}}
\put(10.5,6){\makebox(0,0)[t]{$L(E)$}}
\end{picture}
\end{center}
\caption{}\label{fig:strip}\end{figure}
Therefore, there exists a lattice point $Z_{F}$ in this strip which enjoys
the following properties:
\begin{enumerate}
\item\label{nopoint} the only lattice points on the closed triangle whose
vertices are $0, Z(E), Z_{F}$, are its vertices;
\item\label{phase} $\varphi_{F} > \varphi(E)$.
\end{enumerate}
By $\varphi_{F}$ we denote here the unique number which satisfies
$\theta <\varphi_{F} < \theta+1$ and $Z_{F}\in \mathbb{R}\exp(i\pi
\varphi_{F})$.
Because $\SL(2,\mathbb{Z})$ acts transitively on $\mathsf{Q}$, there exists
a stable non-zero locally free shifted sheaf $F\in\mathsf{D}(\theta, \theta
+1)$ with $Z(F)=Z_{F}$ and $\varphi(F)=\varphi_{F}$.
The assumption $\mathsf{P}(\theta)=\{0\}$ implies
$\mathbb{R}\exp(i\pi\theta)\cap\mathbb{Z}^{2} = \{0\}$, hence, $Z(E)$ is the
only lattice point on the line $L(E)$. This implies that $Z(F)$ is not on
the boundary of the stripe between $L(0)$ and $L(E)$. In particular,
$Z(E)-Z(F)$ is contained in the same half-plane of $L(0)$ as $Z(E)$ and
$Z(F)$, see Figure \ref{fig:strip}.
Condition (\ref{nopoint}) implies $\langle E,F \rangle = 1$. Because $E$ is
locally free, condition (\ref{phase}) implies
$$\Ext^{1}(E,F) = \Hom(F,E) = 0.$$
Hence, $\Hom(E,F)\cong \boldsymbol{k}$. The evaluation map gives, therefore,
a distinguished triangle
$$\Hom(E,F)\otimes E \rightarrow F \rightarrow T_{E}(F)
\stackrel{+}{\longrightarrow}$$
with $T_{E}(F)\in \Dbcoh(\boldsymbol{E})$.
If $C:=T_{E}(F)[-1]$ we obtain a distinguished triangle
\begin{equation}
\label{eq:mutation}
C\rightarrow E\rightarrow F \stackrel{+}{\longrightarrow}
\end{equation}
with $Z(C)=Z(E)-Z(F)$.
Because $E$ is a stable non-zero shifted locally free sheaf, it is spherical
by Proposition \ref{prop:spherical} and so $T_{E}$ is an equivalence. This
implies that $T_{E}(F)$ is spherical and, by Proposition
\ref{prop:spherical} again, $C$ is a stable non-zero shifted locally free
sheaf. All morphisms in the distinguished triangle (\ref{eq:mutation}) are
non-zero because $C, E, F$ are indecomposable, see Lemma \ref{lem:PengXiao}.
Using Lemma \ref{wesPT:ii}, this implies $\theta-1<\varphi(C)<\theta+1$.
However, we have seen in which half-plane $Z(C)$ is contained, so that we
must have $\theta<\varphi(C)<\theta+1$, which implies
$C\in\mathsf{D}(\theta,\theta+1)$. The distinguished triangle
(\ref{eq:mutation}) and the definition of the structure of Abelian category
on the heart $\mathsf{D}(\theta,\theta+1)$ imply now that the
morphism $E\rightarrow F$ in (\ref{eq:mutation}) is an epimorphism in
$\mathsf{D}(\theta,\theta+1)$. This gives an infinite chain of epimorphisms
which are not isomorphisms, so that the category
$\mathsf{D}(\theta,\theta+1)$ is indeed not Noetherian.
In order to show that $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ is not
Noetherian for $\mathsf{P}(\theta)^{-} \ne \emptyset$ we may assume $\theta
= 0$. If there exists a stable element $\boldsymbol{k}(x) \in
\mathsf{P}(0)^{-}[1]\subset \mathsf{P}(1)$, where $x\in\boldsymbol{E}$ is a
smooth point, we have exact sequences
\begin{equation}
\label{eq:sequence}
0 \rightarrow \mathcal{O}(mx)
\rightarrow \mathcal{O}((m+1)x)
\rightarrow \boldsymbol{k}(x)
\rightarrow 0
\end{equation}
in $\Coh_{\boldsymbol{E}}$ with arbitrary
$m\in\mathbb{Z}$.
Hence the cone of the morphism $\mathcal{O}(mx) \rightarrow
\mathcal{O}((m+1)x)$ is isomorphic to $\boldsymbol{k}(x)[0]$. Because
$\boldsymbol{k}(x)[0]$ is an object of $\mathsf{D}^{\le-1}$, with regard to
the $t$-structure which is defined by $\mathsf{P}({0})^{-}$, we obtain
$\tau_{\ge0}(\boldsymbol{k}(x)[0])=0$, which is the cokernel of
$\mathcal{O}(mx) \rightarrow \mathcal{O}((m+1)x)$
in the Abelian category $\mathsf{A}(0,\mathsf{P}(0)^{-})$, see
\cite{Asterisque100}, 1.3.
Hence, there is an exact sequence
$$0 \rightarrow \boldsymbol{k}(x)[-1] \rightarrow \mathcal{O}(mx)
\rightarrow \mathcal{O}((m+1)x) \rightarrow 0$$
in $\mathsf{A}(0,\mathsf{P}(0)^{-})$
and we obtain an infinite chain of epimorphisms
$$ \mathcal{O}(x) \rightarrow \mathcal{O}(2x) \rightarrow \mathcal{O}(3x)
\rightarrow \cdots$$ in the category $\mathsf{A}(0,\mathsf{P}(0)^{-})$,
which, therefore, is not Noetherian.
If $\mathsf{P}(0)^{-}[1]$ contains $\boldsymbol{k}(s)$ only, where
$s\in\boldsymbol{E}$ is the singular point, we proceed as follows. First,
recall that there exist coherent torsion modules with support at $s$
which have finite injective dimension, see for example
\cite{BurbanKreussler}, Section 4. To describe examples of them, we can
choose a line bundle $\mathcal{L}$ on $\boldsymbol{E}$ and a section
$\sigma\in H^{0}(\mathcal{L})$, such that the cokernel of
$\sigma:\mathcal{O}\rightarrow \mathcal{L}$ is a coherent torsion module
$\mathcal{B}$ of length two with support at $s$. If we embed
$\boldsymbol{E}$ into $\mathbb{P}^{2}$, such a line bundle $\mathcal{L}$ is
obtained as the tensor product of the restriction of
$\mathcal{O}_{\mathbb{P}^{2}}(1)$ with $\mathcal{O}_{\boldsymbol{E}}(-x)$,
where $x\in\boldsymbol{E}$ is a smooth point. The section $\sigma$
corresponds to the line in the plane through $x$ and $s$. By twisting with
$\mathcal{L}^{\otimes m}$ we obtain exact sequences
$$ 0 \rightarrow \mathcal{L}^{\otimes m} \rightarrow \mathcal{L}^{\otimes
(m+1)} \rightarrow \mathcal{B} \rightarrow 0$$ in $\Coh_{\boldsymbol{E}}$.
Because $\mathcal{B}$ is a semi-stable torsion sheaf with support at $s$,
all its JH-factors are isomorphic to $\boldsymbol{k}(s)$ and we conclude as
above.
\end{proof}
\begin{proposition}\label{prop:eitheror}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a $t$-structure on
$\Dbcoh(\boldsymbol{E})$ and $B$ a semi-stable indecomposable object
in $\Dbcoh(\boldsymbol{E})$. Then either $B\in \mathsf{D}^{\le0}$ or $B\in
\mathsf{D}^{\ge1}$.
\end{proposition}
\begin{proof}
Let $X\stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} Y
\stackrel{+}{\longrightarrow}$ be a distinguished triangle with $X\in
\mathsf{D}^{\le0}$ and $Y\in \mathsf{D}^{\ge1}$. Suppose $X\ne 0$ and $Y\ne
0$ in $\Dbcoh(\boldsymbol{E})$. We decompose both objects into
indecomposables $X=\bigoplus X_{i}$ and $Y=\bigoplus Y_{j}$. By Lemma
\ref{lem:tsummands} we have $X_{i}\in \mathsf{D}^{\le0}$ and $Y_{j}\in
\mathsf{D}^{\ge1}$. If one of the components of the morphisms
$Y[-1]\rightarrow X=\bigoplus X_{i}$ or $\bigoplus Y_{j}=Y\rightarrow X[1]$
were zero, by Lemma \ref{lem:PengXiao} we would obtain a direct summand
$X_{i}$ or $Y_{j}$ in $B$. Because $B$ was assumed to be indecomposable,
this implies the claim of the proposition.
For the rest of the proof we suppose that all components of these two
morphisms are non-zero. This implies that $X_{i}$ and $Y_{j}$ are
non-perfect for all $i,j$. Indeed, if $X_{i}$ were perfect, we could
apply Serre duality (\ref{wesPT:i}) to obtain $\Hom(Y,X_{i}[1]) =
\Hom(X_{i},Y)^{\ast}$, which is zero because $X_{i}\in \mathsf{D}^{\le0}$
and $Y\in \mathsf{D}^{\ge1}$. The case with perfect $Y_{j}$ can be dealt
with similarly.
Using Lemma \ref{lem:PengXiao} again, it follows that none of the components
of $f:\bigoplus X_{i} \rightarrow B$ or $g:B\rightarrow \bigoplus Y_{j}$ is
zero, because none of the $X_{i}$ could be a direct summand of $Y[-1]$ and
none of the $Y_{j}$ could be a summand of $X[1]$.
Using Lemma \ref{wesPT:ii}, this implies
$\varphi_{-}(X_{i}) \le \varphi(B) \le \varphi_{+}(Y_{j})$ for all $i,j$.
If there exist $i,j$ such that $\varphi_{-}(X_{i}) -
\varphi_{+}(Y_{j})\not\in \mathbb{Z}$, there exists an integer $k\ge 0$ such
that $\varphi_{-}(X_{i}[k]) < \varphi_{+}(Y_{j}) < \varphi_{-}(X_{i}[k])
+1$. Using Proposition \ref{wesPT} (\ref{wesPT:iii}) this implies
$\Hom(X_{i}[k], Y_{j}) \ne 0$. But, for any
integer $k\ge 0$ we have $X_{i}[k]\in \mathsf{D}^{\le0}$ and because
$Y_{j}\in \mathsf{D}^{\ge1}$, we should have $\Hom(X_{i}[k], Y_{j}) =
0$. This contradiction implies $\varphi_{-}(X_{i}) - \varphi_{+}(Y_{j}) \in
\mathbb{Z}$ for all $i,j$. But, if $k=\varphi_{+}(Y_{j}) -
\varphi_{-}(X_{i})$, we still have $\Hom(X_{i}[k], Y_{j}) \ne 0$, which
follows from Proposition \ref{wesPT} (\ref{wesPT:iv}) because $X_{i}$ and
$Y_{j}$ are not perfect. The conclusion is now that we must have $X=0$ or
$Y=0$, which implies the claim.
\end{proof}
\begin{lemma}\label{lem:inequ}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a $t$-structure on
$\Dbcoh(\boldsymbol{E})$. If $F\in \mathsf{D}^{\le0}$ and $G\in
\mathsf{D}^{\ge1}$, then $\varphi_{-}(F)\ge \varphi_{+}(G)$.
\end{lemma}
\begin{proof}
Suppose $\varphi_{-}(F)< \varphi_{+}(G)$. It is sufficient to derive a
contradiction for indecomposable objects $F$ and $G$.
Because, for any $k\ge0$, $F[k]\in \mathsf{D}^{\le0}$, we may replace $F$
by $F[k]$ and can assume $0< \varphi_{+}(G) - \varphi_{-}(F)\le 1$. Now,
there exists a stable vector bundle $\mathcal{B}$ on $\boldsymbol{E}$ and an
integer $r$ such that
$$\varphi_{-}(F) < \varphi(\mathcal{B}[r]) < \varphi_{+}(G) \le
\varphi_{-}(F) + 1.$$
By Proposition \ref{prop:eitheror},
$\mathcal{B}[r]$ is in $\mathsf{D}^{\le0}$ or in
$\mathsf{D}^{\ge1}$. But, from Proposition \ref{wesPT} (\ref{wesPT:iii}) we
deduce $\Hom(F, \mathcal{B}[r])\ne 0$ and $\Hom(\mathcal{B}[r],G)\ne 0$. If
$\mathcal{B}[r]\in \mathsf{D}^{\ge1}$, the first inequality contradicts $F\in
\mathsf{D}^{\le0}$ and if $\mathcal{B}[r]\in \mathsf{D}^{\le0}$, the second
one contradicts $G\in \mathsf{D}^{\ge1}$.
\end{proof}
\begin{theorem}\label{thm:tstruc}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a t-structure on
$\Dbcoh(\boldsymbol{E})$. Then there exists a number $\theta\in \mathbb{R}$
and a subset
$\mathsf{P}(\theta)^{-}\subset \mathsf{P}(\theta)^{s}$, such that
$${\sf D}^{\le 0} = \mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)
\quad\text{ and }\quad
{\sf D}^{\ge 1} = \mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}].$$
\end{theorem}
\begin{proof}
From Lemma \ref{lem:inequ} we deduce the existence of $\theta \in
\mathbb{R}$ such that $\mathsf{D}(\theta,\infty)\subset\mathsf{D}^{\le 0}$
and $\mathsf{D}(-\infty, \theta)\subset\mathsf{D}^{\ge 1}.$
If we define
$\mathsf{P}(\theta)^{-}=\mathsf{P}(\theta)^{s}\cap \mathsf{D}^{\le 0}$
and
$\mathsf{P}(\theta)^{+}=\mathsf{P}(\theta)^{s}\cap \mathsf{D}^{\ge1}$,
Proposition \ref{prop:eitheror} implies
$\mathsf{P}(\theta)^{s} = \mathsf{P}(\theta)^{-}\cup\mathsf{P}(\theta)^{+}$.
Hence, $\mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)\subset \mathsf{D}^{\le0}$
and $\mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}]\subset \mathsf{D}^{\ge 1}$.
From Proposition \ref{prop:texpl} we know that
$(\mathsf{D}[\mathsf{P}(\theta)^{-}, \infty),
\mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}[1]])$ defines a
$t$-structure. Now, the statement of the theorem follows.
\end{proof}
\begin{remark}
In the case of a smooth elliptic curve Theorem \ref{thm:tstruc} was proved in
\cite{GRK}. If $\theta\not\in\mathsf{Q}$ the heart
$\mathsf{D}(\theta,\theta+1)$ of the corresponding $t$-structure is a
finite-dimensional non-Noetherian Abelian category of infinite global
dimension. In the smooth case, they correspond to the category of holomorphic
vector bundles on a non-commutative torus in the sense of Polishchuk and
Schwarz \cite{PolSchw}. It is an interesting problem to find a similar
interpretation of these Abelian categories in the case of a singular
Weierstra{\ss} curve $\boldsymbol{E}$.
\end{remark}
To complete this section we give two applications of Theorem
\ref{thm:tstruc}. The first is a description of the group of exact
auto-equivalences of the triangulated category $\Dbcoh(\boldsymbol{E})$. The
second application is a description of Bridgeland's space of all stability
structures on $\Dbcoh(\boldsymbol{E})$. In both cases, $\boldsymbol{E}$ is an
irreducible curve of arithmetic genus one over $\boldsymbol{k}$.
\begin{corollary}\label{cor:auto}
There exists an exact sequence of groups
$$
\boldsymbol{1} \longrightarrow \Aut^0(\Dbcoh(\boldsymbol{E}))
\longrightarrow \Aut(\Dbcoh(\boldsymbol{E}))
\longrightarrow \SL(2,\mathbb{Z}) \longrightarrow \boldsymbol{1}
$$
in which $\Aut^0(\Dbcoh(\boldsymbol{E}))$ is generated by tensor products
with line bundles of degree zero, automorphisms of the curve and the shift
by $2$.
\end{corollary}
\begin{proof}
The homomorphism $\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z})$
is defined by describing the action of an auto-equivalence on
$\mathsf{K}(\boldsymbol{E})$ in terms of the coordinate functions $(\deg,
\rk)$.
That this is indeed in $\SL(2,\mathbb{Z})$ follows, for example, because
$\Aut(\Dbcoh(\boldsymbol{E}))$ preserves stability and the Euler-form
\begin{align*}
\langle \mathcal{F},\mathcal{G}\rangle &=
\dim\Hom(\mathcal{F},\mathcal{G}) -
\dim\Hom(\mathcal{G},\mathcal{F})\\
&= \rk(\mathcal{F}) \deg(\mathcal{G}) -
\deg(\mathcal{F})\rk(\mathcal{G})
\end{align*}
for stable and perfect sheaves $\mathcal{F},\mathcal{G}$.
Clearly, tensor products with line bundles of degree
zero, automorphisms of the curve and the shift by $2$ are contained in the
kernel of this homomorphism. In order to show that the kernel coincides with
$\Aut^0(\Dbcoh(\boldsymbol{E}))$, we let $\mathbb{G}$ be an arbitrary exact
auto-equivalence of $\Dbcoh(\boldsymbol{E})$.
Then, $\mathbb{G}(\Coh_{\boldsymbol{E}})$ is still Noetherian and it is the
heart
of the $t$-structure $(\mathbb{G}(\mathsf{D}^{\le0}),
\mathbb{G}(\mathsf{D}^{\ge0}))$.
From Theorem \ref{thm:tstruc} and Lemma \ref{lem:heart} we know all
Noetherian hearts of $t$-structures. We obtain
$\mathbb{G}(\Coh_{\boldsymbol{E}}) = \mathsf{D}(\theta,\theta+1]$ with
$\mathsf{P}(\theta)\ne \{0\}$.
Now, by Corollary \ref{cor:sheaves} there exists
$\Phi\in\widetilde{\SL}(2,\mathbb{Z})$ which maps
$\mathsf{D}(\theta,\theta+1]$ to $\mathsf{D}(0, 1]=\Coh_{\boldsymbol{E}}$.
This implies that the auto-equivalence $\Phi\circ\mathbb{G}$ induces an
auto-equivalence of the category $\Coh_{\boldsymbol{E}}$.
It is well-known that such an auto-equivalence has the form $f^*(\mathcal{L}
\otimes \,\cdot\,)$, where $f:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is an
isomorphism and $\mathcal{L}$ is a line bundle.
Note that $f^*(\mathcal{L} \otimes \,\cdot\,)$ is sent to the identity in
$\SL(2,\mathbb{Z})$, if and only if $\mathcal{L}$ is of degree zero.
The composition of $\Phi\circ\mathbb{G}$ with the inverse of $f^*(\mathcal{L}
\otimes \,\cdot\,)$ satisfies the assumptions of \cite{BondalOrlov}, Prop.~A.3,
hence is isomorphic to the identity.
Because the kernel of the homomorphism $\widetilde{\SL}(2,\mathbb{Z})
\rightarrow \SL(2,\mathbb{Z})$, which is induced by the action of
$\widetilde{\SL}(2,\mathbb{Z})$ on $\Dbcoh(\boldsymbol{E})$ and the above
homomorphism $\Aut(\Dbcoh(\boldsymbol{E})) \longrightarrow
\SL(2,\mathbb{Z})$, is generated by the element of
$\widetilde{\SL}(2,\mathbb{Z})$ which acts as the shift by $2$, the claim
now follows.
\end{proof}
For our second application, we recall Bridgeland's definition of stability
condition on a triangulated category \cite{Stability}.
Recall that we set $\mathsf{K}(\boldsymbol{E}) =
\mathsf{K}_{0}(\Coh(\boldsymbol{E})) \cong
\mathsf{K}_{0}(\Dbcoh(\boldsymbol{E}))$.
Following Bridgeland \cite{Stability}, we call a pair $(W,\mathsf{R})$ a
\emph{stability condition} on $\Dbcoh(\boldsymbol{E})$, if
$$W:\mathsf{K}(\boldsymbol{E})\rightarrow\mathbb{C}$$
is a group homomorphism and $\mathsf{R}$ is a compatible slicing of
$\Dbcoh(\boldsymbol{E})$. A \emph{slicing} $\mathsf{R}$ consists of a
collection of full additive subcategories $\mathsf{R}(t) \subset
\Dbcoh(\boldsymbol{E})$, $t\in\mathbb{R}$, such that
\begin{enumerate}
\item $\forall t\in\mathbb{R}\quad \mathsf{R}(t+1) = \mathsf{R}(t)[1]$;
\item If $t_{1}>t_{2}$ and $A_{\nu}\in\mathsf{R}(t_{\nu})$, then
$\Hom(A_{1},A_{2}) =0$;
\item each non-zero object $X\in\Dbcoh(\boldsymbol{E})$ has a HNF
\[\xymatrix@C=.4em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
in which $0\ne A_{\nu}\in\mathsf{R}(\varphi_{\nu})$ and
$\varphi_{n}>\varphi_{n-1}> \ldots > \varphi_{1}>\varphi_{0}$.
\end{enumerate}
Compatibility means for all non-zero $A\in\mathsf{R}(t)$
$$W(A)\in \mathbb{R}_{>0}\exp(i\pi t).$$
By $\varphi^{\mathsf{R}}$ we denote the phase function on
$\mathsf{R}$-semi-stable objects. Similarly, we denote by
$\varphi^{\mathsf{R}}_{+}(X)$ and
$\varphi^{\mathsf{R}}_{-}(X)$ the largest, respectively smallest, phase of an
$\mathsf{R}$-HN factor of $X$.
The standard stability condition, which was studied in the previous section,
will always be denoted by $(Z, \mathsf{P})$. This stability condition has a
slicing which is \emph{locally finite}, see \cite{Stability}, Def.\/ 5.7.
A slicing $\mathsf{R}$ is called locally finite, iff there exists $\eta>0$
such that for any $t\in\mathbb{R}$ the quasi-Abelian category
$\mathsf{D}^{\mathsf{R}}(t-\eta, t+\eta)$ is of finite length, i.e. Artinian
and Noetherian.
This category consists of those objects $X\in\Dbcoh(\boldsymbol{E})$ which
satisfy $t-\eta<\varphi^{\mathsf{R}}_{-}(X) \le \varphi^{\mathsf{R}}_{+}(X) <
t+\eta$.
In order to obtain a good moduli space of stability conditions, Bridgeland
\cite{Stability} requires the stability conditions to be
\emph{numerical}. This means that the central charge $W$ factors
through the numerical Grothendieck group. This makes sense if for any two
objects $E,F$ of the triangulated category in question, the vector spaces
$\bigoplus_{i} \Hom(E,F[i])$ are finite-dimensional. This condition is not
satisfied for $\Dbcoh(\boldsymbol{E})$, if $\boldsymbol{E}$ is
singular. However, in the case of our interest, we do not need such an extra
condition, because the Grothendieck group $\mathsf{K}(\boldsymbol{E})$ is
sufficiently small. From Lemma \ref{lem:GrothGrp} we know
$\mathsf{K}(\boldsymbol{E}) \cong \mathbb{Z}^{2}$ with generators
$[\mathcal{O}_{\boldsymbol{E}}]$ and $[\boldsymbol{k}(x)]$,
$x\in\boldsymbol{E}$ arbitrary.
Because $Z(\boldsymbol{k}(x))=-1$ and $Z(\mathcal{O}_{\boldsymbol{E}})=i$, it
is now clear that any homomorphism $W:\mathsf{K}(\boldsymbol{E}) \rightarrow
\mathbb{C}$ can be written as $W(E)=w_{1}\deg(E) + w_{2}\rk(E)$ with $w_{1},
w_{2}\in\mathbb{C}$. Equivalently, if we identify $\mathbb{C}$ with
$\mathbb{R}^{2}$, there exists a $2\times 2$-matrix $A$ such that $W=A\circ Z$.
\begin{definition}
By $\Stab{\boldsymbol{E}}$ we denote the set of all stability conditions $(W,
\mathsf{R})$ on $\Dbcoh(\boldsymbol{E})$ for which $\mathsf{R}$ is a locally
finite slicing.
\end{definition}
\begin{lemma}\label{lem:notaline}
For any $(W, \mathsf{R}) \in \Stab(\boldsymbol{E})$ there exists a matrix
$A\in\GL(2,\mathbb{R})$, such that $W=A\circ Z$.
\end{lemma}
\begin{proof}
As seen above, there exists a not necessarily invertible matrix $A$ such
that $W=A\circ Z$. If $A$ were not invertible, there would exist a number
$t_{0}\in\mathbb{R}$ such that $W(\mathsf{K}(\boldsymbol{E})) \subset
\mathbb{R}\exp(i\pi t_{0})$. This implies that there may exist a non-zero
object in $\mathsf{R}(t)$ only if $t-t_{0}\in\mathbb{Z}$. The assumption
that the slicing $\mathsf{R}$ is locally finite implies now that
$\mathsf{R}(t)$ is of finite length for any $t\in\mathbb{R}$. On the other
hand, the heart of the $t$-structure, which is defined by $(W,\mathsf{R})$
is $\mathsf{R}(t_{0})$ up to a shift. However, in Lemma \ref{lem:heart} we
determined all Noetherian hearts of $t$-structures on
$\Dbcoh(\boldsymbol{E})$ and none of them is Artinian. This contradiction
shows that $A$ is invertible.
\end{proof}
\begin{lemma}\label{lem:function}
If $(W,\mathsf{R}) \in \Stab(\boldsymbol{E})$, there exists a unique
strictly increasing function $f:\mathbb{R} \rightarrow \mathbb{R}$ with
$f(t+1) = f(t)+1$ and $\mathsf{R}(t) = \mathsf{P}(f(t))$.
\end{lemma}
\begin{proof}
By definition, $W(\mathsf{R}(t)) \subset \mathbb{R}_{>0} \exp(i\pi
t)$. By Lemma \ref{lem:notaline}, there exists a linear isomorphism $A$ such
that $W=A^{-1}\circ Z$. This implies that there is a function
$f:\mathbb{R}\rightarrow \mathbb{R}$ such that $Z(\mathsf{R}(t))
\subset \mathbb{R}_{>0} \exp(i\pi f(t))$.
On the other hand, $\mathsf{R}(t)$ is the intersection of two hearts of
$t$-structures. By Proposition \ref{prop:texpl} these hearts are of the form
$\mathsf{D}[\mathsf{P}(\theta_{1})^{-}, \mathsf{P}(\theta_{1})^{+}[1]]$ and
$\mathsf{D}[\mathsf{P}(\theta_{2})^{-}, \mathsf{P}(\theta_{2})^{+}[1]]$ with
$\theta_{1}\le \theta_{2}$. These have non-empty intersection only if
$\theta_{2} \le \theta_{1}+1$. Their intersection is contained in
$\mathsf{D}[\theta_{2},\theta_{1}+1]$, see Figure \ref{fig:intersection}.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1.5,2){\line(0,1){2}}\put(1.5,0.8){\makebox(0,0)[t]{$\theta_{2}+1$}}
\put(1.4,3){\makebox(0,0)[r]{$\mathsf{P}(\theta_{2})^{+}[1]$}}
\put(5.5,1){\line(0,1){1}}\put(5.5,0.8){\makebox(0,0)[t]{$\theta_{2}$}}
\put(5.6,1.4){\makebox(0,0)[l]{$\mathsf{P}(\theta_{2})^{-}$}}
\put(4.5,2.3){\line(0,1){1}}\put(4.5,0.8){\makebox(0,0)[t]{$\theta_{1}+1$}}
\put(4.4,3){\makebox(0,0)[r]{$\mathsf{P}(\theta_{1})^{+}[1]$}}
\put(8.5,1){\line(0,1){1.3}}\put(8.5,0.8){\makebox(0,0)[t]{$\theta_{1}$}}
\put(8.5,3.3){\line(0,1){0.7}}
\put(8.6,1.5){\makebox(0,0)[l]{$\mathsf{P}(\theta_{1})^{-}$}}
\thinlines
\multiput(1.5,1)(0,0.2){5}{\line(0,1){0.1}}
\multiput(5.5,2)(0,0.2){10}{\line(0,1){0.1}}
\multiput(4.5,1)(0,0.2){7}{\line(0,1){0.1}}
\multiput(4.5,3.3)(0,0.2){4}{\line(0,1){0.1}}
\multiput(8.5,2.3)(0,0.2){5}{\line(0,1){0.1}}
\put(1.9,4){\line(-2,-3){0.4}}
\put(2.7,4){\line(-2,-3){1.2}}
\multiput(1.5,1)(0.8,0){3}{\line(2,3){2}}
\put(3.9,1){\line(2,3){1.6}}
\put(4.7,1){\line(2,3){0.8}}
\put(8.1,4){\line(2,-3){0.4}}
\put(7.3,4){\line(2,-3){1.2}}
\multiput(8.5,1)(-0.8,0){3}{\line(-2,3){2}}
\put(6.1,1){\line(-2,3){1.6}}
\put(5.3,1){\line(-2,3){0.8}}
\end{picture}
\end{center}
\caption{}\label{fig:intersection}
\end{figure}
Moreover, if $\theta_{2}<\theta_{1}+1$, there exist $\alpha,
\beta\in\mathsf{Q}$ with $\theta_{2}< \alpha < \beta < \theta_{1}+1\le
\theta_{2}+1$. In this case we have two non-trivial subcategories
$\mathsf{P}(\alpha)\subset \mathsf{R}(t)$ and $\mathsf{P}(\beta)\subset
\mathsf{R}(t)$. However, because $0<\beta-\alpha<1$ and
$Z(\mathsf{R}(t)) \subset \mathbb{R}_{>0} \exp(i\pi f(t))$, we cannot have
$Z(\mathsf{P}(\alpha)) \subset \mathbb{R}_{>0} \exp(i\pi\alpha)$ and
$Z(\mathsf{P}(\beta)) \subset \mathbb{R}_{>0} \exp(i\pi\beta)$.
Hence, $\theta_{2}=\theta_{1}+1=f(t)$ and we obtain $\mathsf{R}(t) \subset
\mathsf{P}(f(t))$.
From $\mathsf{R}(t+m)=\mathsf{R}(t)[m]$ we easily obtain
$f(t+m)=f(t)+m$. Moreover, $f(t_{2})=f(t_{1})+m$ with $m\in\mathbb{Z}$
implies $t_{2}-t_{1}\in\mathbb{Z}$, because the image of $W$ is not
contained in a line by Lemma \ref{lem:notaline}.
Next, we show that $f$ is strictly increasing. Suppose $t_{1}<t_{2}$,
$t_{2}-t_{1}\not\in\mathbb{Z}$ and both $\mathsf{R}(t_{i})$ contain non-zero
objects $X_{i}$. For any $m\ge0$ we have $\Hom(X_{2}, X_{1}[-m]) = 0$. If
$f(t_{2}) < f(t_{1})$, we choose $m\ge0$ such that $f(t_{2}) < f(t_{1}) -m <
f(t_{2}) +1$ and obtain $X_{2}\in\mathsf{P}(f(t_{2}))$ and $X_{1}[-m] \in
\mathsf{P}(f(t_{1})-m)$. But this implies, by Corollary \ref{cor:sheaves}
(\ref{cor:iii}), $\Hom(X_{2}, X_{1}[-m]) \ne 0$, a contradiction. Hence, we
have shown that $f$ is strictly increasing with $f(t+1)=f(t)+1$ and
$\mathsf{R}(t)\subset \mathsf{P}(f(t))$. In particular, any $\mathsf{R}$-HNF
is a $\mathsf{P}$-HNF as well. Therefore, all $\mathsf{P}$-semi-stable
objects are $\mathsf{R}$-semi-stable and we obtain $\mathsf{R}(t) =
\mathsf{P}(f(t))$.
\end{proof}
It was shown in \cite{Stability} that the group
$\widetilde{\GL}^{+}(2,\mathbb{R})$ acts naturally on the moduli space of
stability conditions $\Stab(\boldsymbol{E})$.
This group is the universal cover of $\GL^{+}(2, \mathbb{R})$ and has the
following description:
$$\widetilde{\GL}^{+}(2,\mathbb{R}) =
\{(A,f) \mid A\in\GL^{+}(2,\mathbb{R}), f:\mathbb{R}\rightarrow \mathbb{R}
\text{ compatible}\},$$
where compatibility means that $f$ is strictly increasing, satisfies
$f(t+1)=f(t)+1$ and induces the same map on $S^{1}\cong\mathbb{R}/2\mathbb{Z}$
as $A$ does on $S^{1}\cong\mathbb{R}^{2}\setminus\{0\}/\mathbb{R}^{\ast}$.
The action is simply $(A,f)\cdot (W,\mathsf{Q})=(A^{-1}\circ W,\mathsf{Q}\circ
f)$. So, this action basically is a relabelling of the slices.
The following result generalises \cite{Stability}, Thm.\/ 9.1, to the singular
case.
\begin{proposition}\label{prop:stabmod}
The action of $\widetilde{\GL}^{+}(2,\mathbb{R})$ on $\Stab(\boldsymbol{E})$
is simply transitive.
\end{proposition}
\begin{proof}
If $(W,\mathsf{R})\in\Stab(\boldsymbol{E})$, the two values
$W(\mathcal{O}_{\boldsymbol{E}})$ and $W(\boldsymbol{k}(p_{0}))$ determine a
linear transformation $A^{-1}\in\GL(2, \mathbb{R})$ such that $W=A^{-1}\circ
Z$, see Lemma \ref{lem:notaline}.
By construction, the function $f:\mathbb{R}\rightarrow \mathbb{R}$ of Lemma
\ref{lem:function} induces the same mapping on
$S^{1}\cong\mathbb{R}/2\mathbb{Z}$ as $A^{-1}$ does on
$S^{1}\cong\mathbb{R}^{2}\setminus\{0\}/\mathbb{R}^{\ast}$. Therefore,
$A\in\GL^{+}(2, \mathbb{R})$ and we obtain $(A,f)\in\widetilde{\GL}^{+}(2,
\mathbb{R})$ which satisfies $(W,\mathsf{R}) = (A,f)\cdot
(Z,\mathsf{P})$. Finally, if $(A,f)\cdot (Z,\mathsf{P}) =(Z,\mathsf{P})$ for
some $(A,f)\in\widetilde{\GL}^{+}(2, \mathbb{R})$, we obtain $f(t)=t$ for
all $t\in\mathbb{R}$. This implies easily $A=\boldsymbol{1}$.
\end{proof}
The group $\Aut(\Dbcoh(\boldsymbol{E}))$ acts on $\Stab(\boldsymbol{E})$ by
the rule $$\mathbb{G} \cdot (W, \mathsf{R}) := (\overline{\mathbb{G}}\circ W,
\mathbb{G}(\mathsf{R})).$$
Here, $\overline{\mathbb{G}}\in\SL(2,\mathbb{Z})$ is the image of
$\mathbb{G}\in\Aut(\Dbcoh(\boldsymbol{E}))$ under the homomorphism of Corollary
\ref{cor:auto} and $\mathbb{G}(\mathsf{R})(t):= \mathbb{G}(\mathsf{R}(t))$.
Because automorphisms of $\boldsymbol{E}$ and twists by line
bundles act trivially on $\Stab(\boldsymbol{E})$, we obtain
$$\Stab(\boldsymbol{E})/\Aut(\Dbcoh(\boldsymbol{E})) \cong \GL^{+}(2,
\mathbb{R})/\SL(2,\mathbb{Z}),$$
which is a $\mathbb{C}^{\times}$-bundle over the coarse moduli space of
elliptic curves. This result coincides with Bridgeland's result in the smooth
case. The main reason for this coincidence seems to be the irreducibility of
the curve. Example \ref{ex:marginalst} below shows that the situation is
significantly more difficult in the case of reducible degenerations of
elliptic curves.
\begin{remark}\label{rem:common}
Our results show that singular and smooth Weierstra{\ss} curves
$\boldsymbol{E}$ share the following properties:
\begin{enumerate}
\item A coherent sheaf $\mathcal{F}$ is stable if and only if
$\End(\mathcal{F}) \cong \boldsymbol{k}$.
\item Any spherical object is a shift of a stable vector bundle or of a
structure sheaf $\boldsymbol{k}(x)$ of a smooth point $x\in\boldsymbol{E}$.
\item The category of semi-stable sheaves of a fixed slope is
equivalent to the category of coherent torsion sheaves.
Such an equivalence is induced by an auto-equivalence of
$\Dbcoh(\boldsymbol{E})$.
\item There is an exact sequence of groups\\
$\boldsymbol{1} \rightarrow \langle \Aut(\boldsymbol{E}),
\Pic^{0}(\boldsymbol{E}),[2]\rangle \rightarrow
\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z}) \rightarrow
\boldsymbol{1}.$
\item $\widetilde{\GL}^{+}(2,\mathbb{R})$ acts transitively on
$\Stab(\boldsymbol{E})$.
\item $\Stab(\boldsymbol{E})/\Aut(\Dbcoh(\boldsymbol{E})) \cong \GL^{+}(2,
\mathbb{R})/\SL(2,\mathbb{Z}).$
\end{enumerate}
\end{remark}
\begin{example}\label{ex:marginalst}
Let $C_{2}$ denote a reducible curve which has two components, both
isomorphic to $\mathbb{P}^{1}$ and which intersect transversally at two
distinct points. This curve has arithmetic genus one and appears as a
degeneration of a smooth elliptic curve.
On this curve, there exists a line bundle $\mathcal{L}$ which
fails to be stable with respect to some stability conditions. To construct
an explicit example, denote by $\pi:\widetilde{C}_{2}\rightarrow C_{2}$ the
normalisation, so that $\widetilde{C}_{2}$ is the disjoint union of two
copies of $\mathbb{P}^{1}$. There is a $\boldsymbol{k}^{\times}$-family of
line bundles whose pull-back to $\widetilde{C}_{2}$ is
$\mathcal{O}_{\mathbb{P}^{1}}$ on one component and
$\mathcal{O}_{\mathbb{P}^{1}}(2)$ on the other. The element in
$\boldsymbol{k}^{\times}$
corresponds to a gluing parameter over one of the two singularities. Let
$\mathcal{L}$ denote one such line bundle.
If $i_{\nu}:\mathbb{P}^{1}\rightarrow C_{2},\;\nu=1,2$ denote the embeddings
of the two components, we fix notation so that $i_{1}^{\ast}\mathcal{L}
\cong \mathcal{O}_{\mathbb{P}^{1}}$ and $i_{2}^{\ast}\mathcal{L} \cong
\mathcal{O}_{\mathbb{P}^{1}}(2)$. There is an exact sequence of coherent
sheaves on $C_{2}$
\begin{equation}\label{eq:linebundle}
0\rightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow \mathcal{L}
\rightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow 0.
\end{equation}
Moreover, the only non-trivial quotients of $\mathcal{L}$ are
$\mathcal{L}\twoheadrightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}$
and
$\mathcal{L}\twoheadrightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}(2)$.
For arbitrary positive real numbers $a,b$ we define a centred
slope-function $W_{a,b}$ on the category $\Coh_{C_{2}}$ by
$$W_{a,b}(F):= -\deg(F) + i(a\cdot \rk(i_{1}^{\ast}F) + b\cdot
\rk(i_{2}^{\ast} F)),$$
where $\deg(F)=h^{0}(F) - h^{1}(F)$. For example,
\begin{align*}
W_{a,b}(i_{1\ast}\mathcal{O}_{\mathbb{P}^{1}}(d)) &= -d-1+ia
\quad\text{ and }\\
W_{a,b}(i_{2\ast}\mathcal{O}_{\mathbb{P}^{1}}(d)) &= -d-1+ib.
\end{align*}
Using the exact sequence (\ref{eq:linebundle}), we obtain
$W_{a,b}(\mathcal{L}) = -2+i(a+b)$.
Using results of \cite{Rudakov}, it is easy to see that $W_{a,b}$ has the
Harder-Narasimhan property in the sense of \cite{Stability}. Hence, by
\cite{Stability}, Prop.\/ 5.3, $W_{a,b}$ defines a stability condition on
$\Dbcoh(C_{2})$. With respect to this stability condition, the line bundle
$\mathcal{L}$ is stable precisely when $2/(a+b) < 1/a$, which is
equivalent to $a<b$. It is semi-stable, but not stable, if $b=a$. If $a>b$,
$\mathcal{L}$ is not even semi-stable.
\end{example}
This example illustrates an interesting effect, which was not available on an
irreducible curve of arithmetic genus one. It is an interesting problem to
describe the subset in $\Stab(\boldsymbol{E})$ for which a given line
bundle $\mathcal{L}$ is semi-stable. This is a closed subset, see
\cite{Stability}. Physicists call the boundary of this set the line of
marginal stability, see e.g. \cite{AspinwallDouglas}. The example above
describes the intersection of this set with a two-parameter family of
stability conditions in $\Stab(\boldsymbol{E})$.
\begin{remark}
In the case of an irreducible curve of arithmetic genus one, we have shown
in Proposition \ref{prop:spherical} that $\Aut(\Dbcoh(\boldsymbol{E}))$ acts
transitively on the set of all spherical objects on
$\boldsymbol{E}$. Polishchuk \cite{YangBaxter} conjectured that this is
likewise true in the case of reducible curves with trivial dualising sheaf.
However, on the curve $C_{2}$ there exists a
spherical complex which has non-zero cohomology in two different degrees, see
\cite{BuBu}. This indicates that the reducible case is more difficult and
involves new features.
\end{remark}
|
{
"timestamp": "2006-02-14T17:10:40",
"yymm": "0503",
"arxiv_id": "math/0503496",
"language": "en",
"url": "https://arxiv.org/abs/math/0503496"
}
|
\section{Introduction}
\label{sec:intro}
Kaonic atoms and kaonic nuclei
carry important information concerning the $K^-$-nucleon interaction
in nuclear medium. This information is very important to determine the
constraints on kaon condensation in high density matter. The properties
of kaons in nuclei are strongly influenced by the change undergone by
$\Lambda(1405)$ in nuclear medium, because $\Lambda(1405)$ is a resonance
state just below the kaon-nucleon threshold. In fact, there are studies
of kaonic atoms carried out by modifying the properties of $\Lambda(1405)$ in nuclear medium.
\cite{alberg76,wei,miz} These works reproduce the properties of kaonic atoms very well,
which come out to be as good as the
phenomenological study of Batty.\cite{bat}
Recently, there have been significant developments in the description
of hadron properties in terms of the $SU(3)$ chiral Lagrangian.
The unitarization of the chiral Lagrangian allows the interpretation of the
$\Lambda(1405)$ resonance state as a baryon-meson coupled
system.\cite{kai,ose} Subsequently, the properties of $\Lambda(1405)$
in nuclear medium
using the $SU(3)$ chiral unitary model were also investigated by
Waas et al.,\cite{waa} \ Lutz,\cite{lut} Ramos and Oset,\cite{ram} and Ciepl$\acute{\rm y}$ et al.\cite{ciep01} \
All of these works considered the Pauli effect on the
intermediate nucleons. In addition, in Ref.~\citen{lut}, the self-energy of the
kaon in the intermediate states is considered, and in Ref.~\citen{ram}, the
self-energies of the pions and baryons are also taken into account. These
approaches lead to a kaon self-energy in nuclear medium that can be
tested with kaonic atoms and kaonic nuclei. There are also $\bar{K}$ potential studies based on meson-exchange J$\ddot{\rm u}$lich $\bar{K}N$ interaction.\cite{tolo01,tolo02}
In a previous work,\cite{hirenzaki00} \ we adopted the scattering amplitude in nuclear
medium calculated by Ramos and Oset\cite{ram} for studies of
kaonic atoms, and demonstrated the ability to
reproduce the existing kaonic atom data as accurately as the optical potential
studied by Batty.\cite{bat} \ We then calculated the deeply bound kaonic
atoms for $^{16}$O and $^{40}$Ca, which have narrow widths
and are believed to be observable with well-suited experimental methods.\cite{16.5_Friedman99,Friedman99} \
We also obtained
very deep kaonic nuclear states, which have large decay widths,
of the order of several tens of MeV. The $(K^-, \gamma)$ reaction
was studied for the formation of the deeply bound kaonic atomic
states,\cite{hirenzaki00} \ which could
not be observed with kaonic X-ray spectroscopy, using the formulation
developed in Ref.~\citen{nie} for the formation of
deeply bound
pionic atoms with the $(\pi^-, \gamma)$ reaction.
Another very important development in recent years is that in the study
of kaonic nuclear states, which are kaon-nucleus bound systems determined mainly
by the strong interaction. Experimental studies of the kaonic nuclear
states using in-flight ($K,N$) reactions were proposed and performed by Kishimoto and his collaborators.\cite{Kishimoto99,Kishimoto03} \ Experiments employing stopped ($K,N$) reactions were carried out
by Iwasaki, T. Suzuki and their collaborators and reported in Refs.~\citen{Iwasaki03} and \citen{Suzuki04}.\
In these experiments, they found some indications of the existence of kaonic nuclear states.
There are also theoretical studies of the structure and formation of kaonic nuclear states related to
these experimental activities.\cite{Akaishi02} \ It should be noted that these theoretical studies
predict the possible existence of ultra-high density states in kaonic nuclear systems. \cite{Akaishi02,dote04}
In this paper, we study in-flight ($K^-, p$) reactions systematically with regard to their role in populating
deeply bound kaonic states and the observation of their properties in experiments.
We have found the usefulness of
direct reactions in the formation of deeply bound pionic atoms
using the ($d,^3$He) reactions.\cite{tok,hirenzaki91,gilg00,itahashi00} \
However, in the present case, $K^+$ must be produced in
addition in this ($d,^3$He) reaction,
and there would be a large momentum mismatch. For this reason,
the $(K^-, \gamma)$ reaction was considered first in Ref.~\citen{hirenzaki00}. \
Here we theoretically study another reaction, ($K^-, p$), and present systematic results that elucidate
the experimental feasibility of the reaction. The ($K^-, p$) reaction
was proposed in Refs.~\citen{Friedman99} and \citen{Kishimoto99}. \ However,
realistic spectra have not yet been calculated. We calculate the
spectra theoretically using the approach of Ref.~\citen{hirenzaki91}
for the deeply bound pionic atom formation reaction.
We believe that this theoretical evaluation will be interesting and important for
studies of kaon properties in nuclear medium.
In $\S$\ref{sec:structure}, we describe the theoretical model of the structure of kaon-nucleus
bound systems and present the numerical results. The theoretical formalism and numerical results for the
($K^-,p$) reactions are discussed in $\S$\ref{sec:formation}. We give summary in $\S$\ref{sec:conclusion}.
\section{Structure of kaonic atoms and kaonic nuclei}
\label{sec:structure}
\subsection{Formalism}
\label{S_Form}
We study the properties of kaonic bound systems by solving the Klein-Gordon equation
\begin{equation}
[-{\bf {\nabla}}^2+\mu^2+2\mu V_{\rm opt}(r)]\phi(\mbox{\boldmath $r$})=[E-V_{\rm coul}(r)]^2 \phi(\mbox{\boldmath $r$})~.
\label{KGeq}
\end{equation}
\noindent
Here, $\mu$ is the kaon-nucleus reduced mass and $V_{\rm coul}(r)$ is the Coulomb potential with a
finite nuclear size:
\begin{equation}
V_{\rm coul}(r)=-e^2 \int \frac{\rho_p(r')}{|\mbox{\boldmath $r$-$r'$}|}d^3 r'~,
\label{V_coul}
\end{equation}
\noindent
where $\rho_p(r)$ is the proton density distribution. We employ the empirical Woods-Saxon form
for the density and keep the shapes of the neutron and proton density distributions fixed as
\begin{equation}
\rho (r) = \rho_n(r)+\rho_p(r) = \frac{\rho_{\rm 0}}{1+\exp[(r-R)/a]}~,
\label{rho}
\end{equation}
\noindent
where we use $R=1.18A^{1/3}-0.48~[{\rm fm}]$ and $a=0.5~[{\rm fm}]$ with $A$, the nuclear mass number. It is noticed that the point nucleon density distributions are
deduced from $\rho$ in Eq. (\ref{rho}) by using the same prescription described in Sect. 4 in Ref.~\citen{nieves93} and are used to evaluate the kaon-nucleus optical potential.
The kaon-nucleus optical potential $V_{\rm opt}$ is given by
\begin{equation}
2\mu V_{\rm opt} (r) = -4 \pi \eta a_{\rm eff}(\rho)\rho(r) ,
\label{V_opt}
\end{equation}
\noindent
where $a_{\rm eff}$($\rho$) is a density dependent effective scattering length and $\eta=1+m_K/
M_N$.
In this paper, we use two kinds of effective scattering lengths, that obtained with
the chiral unitary approach\cite{ram} and that obtained with a phenomenological fit.\cite{batty97}~\ Here, we do not introduce any energy dependence for the
effective scattering lengths, and we use the scattering lengths at the $KN$
threshold energy.
The effective scattering length $a_{\rm eff}$ of the chiral unitary approach is described in Ref.~\citen{hirenzaki00} \ in detail. It
is defined by the kaon self-energy in nuclear matter, with the local density approximation. The form of
$a_{\rm eff}$ obtained in a phenomenological fit is one of the results reported in Ref.~\citen{batty97}, \ and it
is parameterized as
\begin{equation}
a_{\rm eff}(\rho)=(-0.15+0.62i)+(1.66-0.04i)(\rho/\rho_{\rm 0})^{0.24} [{\rm fm}] .
\label{batty_a}
\end{equation}
\noindent
The reason we consider these two potentials is that they provide
equivalently good descriptions of the observed kaonic atom data, even though they have very
different potential depths, as we will see in next subsection.
Thus, it should be extremely interesting to compare the results obtained with these potentials
in the ($K^-,p$) reaction spectra, including the kaonic nuclear region.
We solve the Klein-Gordon equation numerically, following the method of Oset and Salcedo.\cite{oset85} \
The application of this method to pionic atom studies are reported in detail in Ref.~\citen{nieves93}.
\subsection{Numerical results}
\label{S_results}
We show the kaon nucleus potential for the $^{39}$K case in Fig.~\ref{fig_V_opt} as an example.
Because the real part of $a_{\rm eff}$($\rho$) changes sign at a certain nuclear density in
both the chiral unitary and phenomenological models, the kaon nucleus optical potential is
attractive, while keeping the repulsive sign for the kaon-nucleon scattering length in free space.
The real part of the scattering length for the phenomenological fit depends on the density much
more strongly than the results of the chiral unitary model and yields Re $a_{\rm eff}$($\rho_0)=1.51 [{\rm fm}]$.
Hence, as we can see in Fig.~\ref{fig_V_opt}, \ the depths of the real
optical potentials of these models differ significantly.
On the other hand, the density
dependence of the imaginary part of the phenomenological
scattering length is rather flat, and its strength is similar to that of
the chiral unitary model.
\begin{figure}[htbp]
\epsfysize=5cm
\centerline{\epsfbox{fig1.eps}}
\caption{The kaon-nucleus optical potential for $^{39}$K as a function of the radial coordinate $r$.
The left and right panels show the real and imaginary part, respectively.
The solid line indicates the potential strength of the chiral unitary approach and the dashed line
of the phenomenological fit.}
\label{fig_V_opt}
\end{figure}
The calculated energy levels for the atomic states and nuclear states
in $^{39}$K are shown in Fig.~\ref{fig:39K_Energy}, where the results of the chiral
model and the phenomenological model, Eq.~(\ref{batty_a}), are compared.
We see that the results obtained with the two potentials are very similar for the atomic states.
We find that the deep atomic states, such as atomic 1$s$ in $^{39}$K (still unobserved),
appear with narrower widths than the separation between levels and are predicted
to be quasi-stable states. Similar results are reported in previous works.
\cite{hirenzaki00,16.5_Friedman99} \ Because several model potentials predict
the existence of quasi-stable deep atomic states, it would be interesting to
observe the states experimentally. On the other hand, the predicted binding
energies and widths are very stable and almost identical for all of the potential models
considered here.
Hence, it is very difficult to distinguish the theoretical potentials from only
the observation of atomic levels.
In the lower panels of Fig.~\ref{fig:39K_Energy}, we also show the energy levels
of the deep nuclear kaonic states of $^{39}$K using the chiral unitary model
potential and the phenomenological model potential. The deep nuclear states
are represented by the solid bars with numbers, which indicate their widths in units of MeV.
These nuclear states have extremely large widths in all cases and would not be observed
as peak structures in experiments if they indeed do have such large widths. We should,
however, mention here that the level structures of these potential models differ
significantly. In the chiral unitary potential, only two nuclear
states are predicted, while eight states are predicted with the phenomenological
model. This difference presents the opportunity to distinguish the theoretical potentials in
observations of kaonic nuclear states.
\begin{figure}[htbp]
\epsfxsize=10cm
\centerline{\epsfbox{fig2.eps}}
\caption{(Upper panel) Energy levels of kaonic atoms of $^{39}$K obtained with the optical
potentials of the chiral unitary model (left) and the phenomenological fit (right).
The hatched areas indicate the level widths.
(Lower panel) Energy levels of kaonic nuclear states of $^{39}$K obtained with the optical
potentials of the chiral unitary model (left) and the phenomenological fit (right).
The level width is indicated by the number appearing at each level, in units of MeV.}
\label{fig:39K_Energy}
\end{figure}
Calculated density distributions of nuclear 1$s$ and 2$s$ and atomic 1$s$ kaonic
states in $^{39}$K are shown in Fig.~\ref{fig:39K_wf} for the case of the
phenomenological optical potential. It is seen that the wavefunctions of the deep nuclear kaonic
states remain almost entirely inside the nuclear radius, which is about 3.5 fm for
$^{39}$K. Hence, the widths become extremely large, of the order of 100 MeV.
The wavefunctions of the atomic states are pushed outward by the imaginary part of
the strong interaction. It should be noted that the atomic 1$s$ state
corresponds to the 4-th $s$ state in the solutions of the Klein-Gordon
(KG) equation, Eq.~(\ref{KGeq}). We divided the series of KG solutions into two categories,
'atomic states' and 'nuclear states', since the properties of these
states are very different, and there are no ambiguities in this
classification, as can be seen in Figs. \ref{fig:39K_Energy} and \ref{fig:39K_wf}.
\begin{figure}[htpd]
\epsfxsize=6cm
\centerline{\epsfbox{fig3.eps}}
\caption{The kaonic bound state density distributions
$|r\phi(r)|^2$ in coordinate space for $^{39}$K obtained with
the phenomenological optical potential. The
solid and dotted curves indicate the distributions of the 1$s$ and 2$s$ states. The dashed
curve represents the density of the 4$s$ state and is regarded as a kaonic atom 1$s$ state. The
half-density radius of $^{39}$K is also shown.}
\label{fig:39K_wf}
\end{figure}
We have also calculated the kaon-nucleus binding energies and widths for both
atomic and nuclear states in other nuclei.
The obtained results are compiled in Tables~\ref{tab:ph} and \ref{tab:chi}
for the phenomenological optical potential and for the chiral unitary potential
cases, respectively.
We selected $^{11}$B,~$^{15}$N,~$^{27}$Al and $^{39}$K nuclei, which appear
in the final states of the formation reactions for $^{12}$C,~$^{16}$O,~$^{28}$Si
and $^{40}$Ca targets, as described in the next section.
In all cases, we found kaonic atom states and kaonic nuclear states. The
results for the atomic states are similar for the two potentials and
are known to reproduce existing data reasonably well. On the other hand,
certain differences are found in the energy spectra of kaonic nuclear states,
as expected, and they should be investigated experimentally.
\begin{table}[htbp]
\begin{center}
\caption{Calculated binding energies and widths of kaon-$^{11}$B, -$^{15}$N,
-$^{27}$Al and -$^{39}$K systems with the phenomenological optical potential
in units of MeV for kaonic nuclear states and in units of keV for kaonic atom states.}
\begin{tabular}{c|cc|cc|cc|cc}
\hline
\hline
&&&&&&&&\\
Nuclear State& \multicolumn{2}{|c|}{$^{11}$B}&\multicolumn{2}{|c|}{$^{15}$N}
&\multicolumn{2}{|c|}{$^{27}$Al}&\multicolumn{2}{|c}{$^{39}$K}\\
(MeV)&B.E.& $\Gamma$&B.E.& $\Gamma$&B.E.& $\Gamma$&B.E.& $\Gamma$ \\
\hline
1$s$ & 132.5 & 183.0 & 155.7 & 205.5&190.8&239.5&206.7&253.8\\
2$s$&-&-&19.0&96.2&69.8&142.3&101.3&165.6\\
3$s$&-&-&-&-&-&-&2.1$\times$10$^{-1}$&88.8\\
2$p$&58.4&127.0&86.9&151.1&136.3&191.2&161.7&211.3\\
3$p$&-&-&-&-&16.5&103.3&50.2&131.6\\
3$d$&-&-&22.9&108.3&80.4&152.2&113.3&175.2\\
4$d$&-&-&-&-&-&-&3.9&98.7\\
4$f$&-&-&-&-&26.2&119.5&64.3&144.9\\
\hline
Atomic State&&&&&&&&\\
(keV)&&&&&&&&\\
\hline
1$s$&192.4&40.6&338.1&68.3&844.9&243.1&1408.0&355.4\\
2$s$&60.5&7.3&111.8&13.3&319.0&58.3&580.6&97.6\\
3$s$&29.2&2.5&55.1&4.6&166.8&22.3&316.9&39.9\\
4$s$&17.1&1.1&21.7&1.1&102.3&10.8&199.4&20.0\\
2$p$&78.5&6.2$\times$10$^{-1}$&154.6&4.0&507.5&37.4&988.7&132.7\\
3$p$&34.9&2.2$\times$10$^{-1}$&68.8&1.4&229.4&12.7&458.4&45.8\\
4$p$&19.6&9.6$\times$10$^{-2}$&38.7&6.1$\times$10$^{-1}$&130.5&5.7&265.1&20.8\\
3$d$&34.9&2.3$\times$10$^{-4}$&69.3&2.9$\times$10$^{-3}$&243.0&4.2$\times$10$^{-1}$&520.9&5.0\\
4$d$&19.6&1.4$\times$10$^{-4}$&39.0&1.7$\times$10$^{-3}$&136.6&2.5$\times$10$^{-1}$&292.6&2.9\\
4$f$&19.6&1.0$\times$10$^{-8}$&38.9&4.8$\times$10$^{-7}$&136.5&3.2$\times$10$^{-4}$&293.7&1.7$\times$10$^{-2}$\\
\hline
\end{tabular}
\label{tab:ph}
\end{center}
\end{table}
\begin{table}[htpb]
\begin{center}
\caption{Calculated binding energies and widths of kaon-$^{11}$B, -$^{15}$N,
-$^{27}$Al and -$^{39}$K systems with the optical potential of the chiral unitary model,
in units of MeV for kaonic nuclear states and in units of keV for kaonic atom states.}
\begin{tabular}{c|cc|cc|cc|cc}
\hline
\hline
&&&&&&&&\\
Nuclear State& \multicolumn{2}{|c|}{$^{11}$B}&\multicolumn{2}{|c|}{$^{15}$N}
&\multicolumn{2}{|c|}{$^{27}$Al}&\multicolumn{2}{|c}{$^{39}$K}\\
(MeV)&B.E.& $\Gamma$&B.E.& $\Gamma$&B.E.& $\Gamma$&B.E.& $\Gamma$ \\
\hline
1$s$&4.6&81.6&11.0&87.9&22.5&96.6&29.3&100.3\\
2$p$&-&-&-&-&-1.1&79.5&9.0&87.4\\
\hline
Atomic State&&&&&&&&\\
(keV)&&&&&&&&\\
\hline
1$s$&197.5&35.0&340.5&67.8&852.1&199.1&1422.7&337.3\\
2$s$&61.3&6.2&112.3&13.2&320.4&47.6&584.3&92.5\\
3$s$&29.5&2.1&55.3&1.6&167.2&18.2&318.3&37.8\\
4$s$&17.3&9.3$\times$10$^{-1}$&21.7&1.1&102.5&8.8&200.1&19.0\\
2$p$&78.4&6.3$\times$10$^{-1}$&154.4&3.1&507.3&34.8&986.8&109.3\\
3$p$&34.8&2.2$\times$10$^{-1}$&68.7&1.1&229.3&11.8&457.7&37.7\\
4$p$&19.6&9.7$\times$10$^{-2}$&38.7&4.8$\times$10$^{-1}$&130.5&5.3&264.8&17.1\\
3$d$&34.9&1.6$\times$10$^{-4}$&69.3&2.6$\times$10$^{-3}$&243.0&3.4$\times$10$^{-1}$&520.8&4.5\\
4$d$&19.6&9.5$\times$10$^{-5}$&39.0&1.6$\times$10$^{-3}$&136.6&2.0$\times$10$^{-1}$&292.6&2.6\\
4$f$&19.6&1.0$\times$10$^{-8}$&38.9&3.7$\times$10$^{-7}$&136.5&2.8$\times$10$^{-4}$&293.7&1.5$\times$10$^{-2}$\\
\hline
\end{tabular}
\label{tab:chi}
\end{center}
\end{table}
We should mention here that the kaonic nuclear 2$p$ state in $^{27}$Al described in
Table \ref{tab:chi} provides a negative value for the binding energy.
This state, however, is interpreted as a bound state, since the sign of the corresponding eigenenergy
in the non-relativistic Schr$\ddot{\rm o}$dinger equation is opposite
to that of the Klein-Gordon solution, due to the large widths of the nuclear states, as shown below.
The binding energies $B_{\rm KG}$ and widths $\Gamma_{\rm KG}$ of the
Klein-Gordon equation, which are tabulated in Tables \ref{tab:ph} and
\ref{tab:chi}, are defined as $E=(\mu - B_{\rm KG})-\frac{i}{2}\Gamma_{\rm KG}$
by the eigenenergy $E$ in Eq.~(\ref{KGeq}). The non-relativistic binding energy
$B_{\rm S}$ and width $\Gamma_{\rm S}$ of the Schr$\ddot{\rm o}$dinger
equation are related to $B_{\rm KG}$ and $\Gamma_{\rm KG}$ as
\begin{equation}
B_{\rm S}=B_{\rm KG}-\frac{B_{\rm KG}^2}{2\mu}+\frac{\Gamma_{\rm KG}^2}{8\mu} ,
\label{eq:E_s-KG}
\end{equation}
\begin{equation}
\Gamma_{\rm S}=\Gamma_{\rm KG}-\frac{B_{\rm KG}}{\mu}\Gamma_{\rm KG}.
\label{eq:G_s-KG}
\end{equation}
\noindent
Thus, in the case of the kaonic 2$p$ nuclear state in $^{27}$Al described in Table
\ref{tab:chi}, the non-relativistic binding energies and widths
are $B_{\rm S}\sim 0.5$MeV
and $\Gamma_{\rm S}\sim \Gamma_{\rm KG}$, which indicate that the state is
bound. It should be noted that the asymptotic behavior of the wavefunction
is determined by $B_S$.
\section{Kaonic atoms and kaonic nuclei formation in ($K^-,p$) reactions}
\label{sec:formation}
\subsection{Formalism}
We adopt the theoretical model presented in Ref.~\citen{hirenzaki91}
to calculate the formation cross sections of the kaonic atoms and kaonic nuclei
in the ($K^-, p$) reaction.
In this model, the emitted proton energy
spectra can be written as
\begin{equation}
\frac{d^2\sigma}{dE_ p d\Omega_ p} =
\left( \frac{d \sigma}{d \Omega} \right)^{\rm lab}_{K^- p \rightarrow p K^-}
\sum_f \frac{ \Gamma_ K}{2 \pi}
\frac{1}{\Delta E^2 + \Gamma_ K^2/4} N_{\rm eff} ,
\label{eqn:cross-section}
\end{equation}
\noindent
where the sum is over all (kaon-particle) $\otimes$ (proton-hole) configurations in the
final kaonic bound states.
The differential cross section for the elementary process of the reaction
$K^- + p $ $\rightarrow$ $p + K^-$ in the laboratory frame,
$( d \sigma/d \Omega)^{\rm lab}_{K^- p \rightarrow p K^-} $
is evaluated using the $K^-p$ total elastic cross section data in
Ref.~\citen{PDG2004} \ by assuming a flat angular distribution in the
center-of-mass frame at each energy.
The resonance peak energy is determined by $\Delta E$ appearing
in Eq.~(\ref{eqn:cross-section}),
which is defined as
\begin{equation}
\Delta E = T_ p - (T_ K - S_ p (j_ p^{-1}) + B_ K) ,
\label{eq:DE}
\end{equation}
\noindent
where $T_ K$ is the incident kaon kinetic energy, $T_ p$ the emitted
proton kinetic energy, and $B_ K$ the kaon binding energy in the final state.
The proton separation energy $S_ p$ from each single particle
level listed in Table~\ref{tb:sp} is obtained from the data in
Refs.~\citen{Belo85,yosoi04,ajz91,mougey76,amaldi64,nakamura74}. We use the data in
Ref.~\citen{TOI96} for the separation energies of the proton-hole
levels corresponding to the ground states of the daughter nuclei. The widths of the hole states $\Gamma_ p$ are
also listed in Table~\ref{tb:sp}. These were obtained from the same data
sets by assuming the widths of the ground states of the daughter nuclei to
be zero because of their stabilities.
\begin{center}
\begin{table}[hpbt]
\begin{center}
\caption{One proton separation energies $S_ p$ and widths $\Gamma_ p$
of the hole states
of $^{12}$C, $^{16}$O, $^{28}$Si and $^{40}$Ca
deduced from the data given in Ref.~\protect \citen{Belo85} \ for $^{12}$C, those given
in Refs.~\protect \citen{yosoi04} and~\protect \citen{ajz91} \ for $^{16}$O,
those given in Refs.~\protect \citen{mougey76} and~\protect \citen{amaldi64} \ for $^{28}$Si,
and those given in Ref~\protect \citen{nakamura74} \ for $^{40}$Ca.
The separation energies corresponding to the ground states of the daughter
nuclei are taken from Ref.~\protect \citen{TOI96}.
The widths $\Gamma_ p$ indicate FWHM of the Lorentz distribution for
$^{16}$O and of Gaussian distributions for other nuclei.
The widths of the ground states of the daughter nuclei are fixed
to zero, because of their stabilities.
For the 1$p$ states in $^{28}$Si and $^{40}$Ca, two levels, 1$p_{1/2}$ and 1$p_{3/2}$,
have not been observed separately, and therefore $S_ p$ and $\Gamma_ p$
are set to the same values for both levels.
}
\vspace{3mm}
\begin{tabular}{|c|cc|cc|cc|cc|}
\hline
&&&&&&&&\\
single particle & \multicolumn{2}{|c|}{$^{12}$C}&\multicolumn{2}{|c|}{$^{16}$O}&
\multicolumn{2}{|c|}{$^{28}$Si}&\multicolumn{2}{|c|}{$^{40}$Ca}\\
states [MeV]& $S_ p$ & $\Gamma_ p$ & $S_ p$ & $\Gamma_ p$
&$S_ p$ & $\Gamma_ p$ &$S_ p$ & $\Gamma_p$\\ \hline
1$d_{3/2}$ & &&&&&&8.3 & 0 \\
2$s_{1/2}$ & &&&&&&11.5 & 7.7 \\
1$d_{5/2}$ & &&&&11.6&0&16.3 & 3.7 \\
1$p_{1/2}$ & &&12.1&0&27.5&17.0&33.2 & 21.6 \\
1$p_{3/2}$ &16.0&0&18.4&3.1$\times$10$^{-6}$&27.5&17.0&33.2& 21.6 \\
1$s_{1/2}$ &33.9&12.1&41.1&19.0&46.5&21.0& 56.3 & 30.6 \\ \hline
\end{tabular}
\label{tb:sp}
\end{center}
\end{table}
\end{center}
The effective number $N_{\rm eff}$ is defined as
\begin{eqnarray}
N_{\rm eff} = \sum_{J M m_s}
\Bigl|\int d^3r
\chi^{\ast}_f(\mbox{\boldmath $r$}) \xi^{\ast}_{1/2,m_s}
[\phi^{\ast}_{l_K}(\mbox{\boldmath $r$}) \otimes \psi_{j_p}(\mbox{\boldmath $r$})]_{JM}
\chi_i(\mbox{\boldmath $r$})\Bigr|^2.
\end{eqnarray}
\noindent
The proton and the kaon wavefunctions are denoted by $\psi_{j_ p}$
and $\phi_{l_ K}$. We adopt the
harmonic oscillator wavefunction for $\psi_{j_ p}$. The spin wave
function is denoted by
$\xi_{1/2,m_s}$,
and we take the spin average with respect to $m_s$, so as to
take into account the possible spin directions of
the protons in the target nucleus. The functions $\chi_i$ and $\chi_f$ are the initial and final distorted waves of the
projectile and ejectile, respectively. We use the eikonal
approximation and replace $\chi_f$ and
$\chi_i$ by employing the relation
\begin{eqnarray}
\chi^{\ast}_f(\mbox{\boldmath $r$}) \chi_i(\mbox{\boldmath $r$}) = \exp (i \mbox{\boldmath $q$} \cdot \mbox{\boldmath $r$})D(z, \mbox{\boldmath $b$}),
\end{eqnarray}
where $\mbox{\boldmath $q$}$ is the momentum transfer between the projectile and
ejectile, and the distortion factor $D(z,\mbox{\boldmath $b$})$ is defined as
\begin{eqnarray}
D(z, \mbox{\boldmath $b$}) = \exp \left[
-\frac{1}{2} \sigma_ {KN} \int^{z}_{-\infty}d z^{\prime}
\rho_A (z^{\prime},\mbox{\boldmath $b$})
-\frac{1}{2} \sigma_{pN} \int^{\infty}_{z}d z^{\prime}
\rho_{A-1} (z^{\prime},\mbox{\boldmath $b$})
\right].
\end{eqnarray}
Here, the kaon-nucleon and proton-nucleon total cross sections
are denoted by $\sigma_{KN}$ and
$\sigma_{pN}$. The functions $\rho_A(z,\mbox{\boldmath $b$})$ and $\rho_{A-1}(z,\mbox{\boldmath $b$})$ are
the density distributions of the target and daughter nuclei
in the beam direction coordinate $z$ with impact parameter $\mbox{\boldmath $b$}$,
respectively.
We calculated the kaonic bound state wavefunctions using the optical potentials
obtained with the chiral unitary model \cite{hirenzaki00} \
and the phenomenological fit, \cite{batty97} as
described in \S\ref{sec:structure}.
In the chiral unitary model, the depth of the attractive potential is
only approximately 50 MeV at the center of the nucleus, which is much weaker than
the phenomenological potential used in Ref.~\citen{batty97}.
For the case of the phenomenological potential, there exist kaonic
nuclear bound states with very large binding energies, for example, 100 -- 200 MeV. For these
bound states, the ${\bar K}N$ system cannot decay into $\pi\Sigma$, because of the
threshold, and hence, the widths of these states are expected to be narrower.
On the other hand, for chiral unitary potential cases, we do not have
narrow nuclear states, like those in Refs.~\citen{Kishimoto99} and~\citen{Akaishi02},
because the decay phase space for the ${\bar K}N$ system to the $\pi\Sigma$ channel
is sufficiently large,
owing to the smaller binding energies.
In order to include 'narrowing effects' for the widths due to the
phase space suppression, we introduce a phase space factor $f^{\rm MFG}$, defined
in Ref.~\citen{mares04} \ by Mare$\check{\rm s}$, Friedman, and Gal as,
\begin{equation}
f^{\rm MFG}(E)=0.8f^{\rm MFG}_1(E)+0.2f^{\rm MFG}_2(E) ,
\label{eq:mfg}
\end{equation}
\noindent
where $f^{\rm MFG}_1$ and $f^{\rm MFG}_2$ are the phase space factors for
$\bar{K} N \rightarrow \pi \Sigma$ decay and
$\bar{K} NN \rightarrow \Sigma N$, respectively. These factors are defined as
\begin{equation}
f^{\rm MFG}_1(E)=\frac{M^3_{01}}{M^3_1}\sqrt{\frac{[M^2_1-(m_{\pi}+m_{\Sigma})^2][M^2_1-(m_{\Sigma}-m_{\pi})^2]}
{[M^2_{01}-(m_{\pi}+m_{\Sigma})^2][M^2_{01}-(m_{\Sigma}-m_{\pi})^2]}}\theta(M_1-m_{\pi}-m_{\Sigma}) ,
\label{eq:mfg1}
\end{equation}
and
\begin{equation}
f^{\rm MFG}_2(E)=\frac{M^3_{02}}{M^3_2}\sqrt{\frac{[M^2_2-(m_ N+m_{\Sigma})^2][M^2_2-(m_{\Sigma}-m_ N)^2]}
{[M^2_{02}-(m_ N+m_{\Sigma})^2][M^2_{02}
-(m_{\Sigma}-m_ N)^2]}}\theta(M_2-m_{\Sigma}-m_ N) .
\label{eq:mfg2}
\end{equation}
Here, the branching ratios of mesic decay and non-mesic decay are
assumed to be 80$\%$ and 20$\%$. The masses are defined as
$M_{01}=m_{\bar K}+m_{ N}$,
$M_1=M_{01}+E$, $M_{02}=m_{\bar K}+2m_{ N}$,
$M_2=M_{02}+E$, and $E$ is the kaon energy defined as
$E=T_{K}-T_{p}-S_{p}$, using the same kinematical variables as in Eq. (\ref{eq:DE}).
We multiply the energy independent kaonic widths $\Gamma_K$ by the phase space factor $f^{\rm MFG}$ in order to introduce the energy dependence due to the
phase space suppression as
\begin{equation}
\Gamma_{K} \rightarrow \Gamma _{K}(E)=\Gamma_{K} \times f^{\rm MFG}(E) .
\label{eq:mfg_G}
\end{equation}
\subsection{Numerical results}
We present the numerical results for the kaon bound state formation
spectra in this subsection.
First, we consider the momentum transfer of the ($K^-,p$)
reactions as a function of the incident kaon energy,
which is an important guide to determine suitable incident
energies in order to obtain a large production rate of the bound states.
Because we consider both atomic and nuclear kaon states,
we assume four different binding energies to calculate the
momentum transfer. We consider the forward reactions and
the momentum transfer in the laboratory frame, as shown in Fig.~\ref{fig:momentum}.
As can be seen in the figure, the condition of zero recoil can be
satisfied only for atomic states with $T_{K} = $10 -- 20 MeV.
For deeply bound nuclear states, the reaction always requires
a certain momentum transfer. However, the incident
energy dependence of the momentum transfer is not strong for
kaonic nuclear states, as shown in Fig.~\ref{fig:momentum}.
\begin{figure}[htpb]
\epsfxsize=6cm
\centerline{\epsfbox{fig4.eps}}
\caption{Momentum transfers in the ($K^-,p$) reactions with $^{40}$Ca for targets the
formation of kaon-$^{39}$K bound systems. The proton separation energy $S_{p}$ is
fixed at 8.3 MeV, and the kaon binding energies are assumed to be 0 MeV (solid curve),
50 MeV (dotted curve), 100 MeV (dashed cureve), and 150 MeV (long-dashed curve).}
\label{fig:momentum}
\end{figure}
We consider formation of kaonic atoms and kaonic nuclei
separately, because their properties, and hence,
the optimal kinematical conditions for their formation are expected to be different.
We first consider the formation of atomic states. Because the binding energies of
atomic states are sufficiently small, we can safely ignore the phase space effect for
the decay widths considered in Eqs. (\ref{eq:mfg}) -- (\ref{eq:mfg_G}).
For atomic states, the obtained wavefunctions and energy spectra are almost
identical for the chiral unitary and phenomenological potentials.
For this reason, we show the results only for chiral unitary potential.
We first study the energy dependence of the atomic
1$s$ state formation rate in order to
determine the optimal incident energy for the deepest atomic 1$s$ state
observation in the ($K^-, p$) reactions. For this purpose, we show in Fig.~\ref{fig:1} \
the energy dependence of the
ratio of the calculated effective numbers of the kaonic atom 1$s$ and 2$p$ states
coupled to the [$d_{3/2}^{-1}$] proton-hole state in $^{39}$K.
We find that the contribution of the
1$s$ state is significantly larger than that of the 2$p$ state at $T_{K}$ = 20 MeV, and
we therefore hypothesize that the 1$s$ state can be observed clearly without a large
background due to the 2$p$ kaonic state at this energy, where the
momentum transfer is reasonably small for atomic state formation.
Next, we consider the
energy dependence of the peak height of the 1$s$ kaonic atom state coupled to
the [$d_{3/2}^{-1}$] proton-hole state in $^{39}$K to determine the
suitable incident energy to have a large cross section. As we can see
in Fig.~\ref{fig:2}, the cross section is maximal somewhere in the range $T_{K}$ = 30 -- 40
MeV, and we find that the cross section has a local maximum value
near $T_{K}$ = 400 MeV. From these observations, we take $T_{K}$ = 20
and 400 MeV as the incident kaon energies to calculate the energy
spectrum of the emitted proton. We also consider $T_{K}$ = 100 MeV as an
energy between 20 and 400 MeV. We mention here that the eikonal approximation is
known to be valid only for high energies. Thus, the results for low energies, i.e.
$T_{K}$ $\leq$ 100 MeV, should be regarded as rough estimations.
\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}{0.5\hsize}
\epsfxsize=6cm
\centerline{\epsfbox{fig5.eps}}
\caption{Ratio of the effective numbers of 1$s$ and 2$p$ kaonic atom states formed
with the [$d_{3/2}^{-1}$] proton-hole state in $^{39}$K plotted as a function of the
incident kaon energy $T_{K}$.}
\label{fig:1}
\end{minipage}
\begin{minipage}{0.5\hsize}
\epsfxsize=6cm
\centerline{\epsfbox{fig6.eps}}
\caption{Double differential cross section at the resonance peak energy of the
kaonic atom 1$s$ state formation with the [$d^{-1}_{3/2}$] proton-hole state in $^{39}$K
at $\theta^{\rm Lab}_{p}$ = 0 [degrees] plotted as a function of the
incident kaon kinetic energy.}
\label{fig:2}
\end{minipage}
\end{tabular}
\end{figure}
In Fig.~\ref{fig:3} we show the calculated spectra for a
$^{40}$Ca target, including contributions from
the kaonic atom states up to 4$f$ for [1$d_{3/2}^{-1}$], [1$d_{5/2}^{-1}$] and
[2$s_{1/2}^{-1}$] proton states in $^{39}$K at $T_{K}$ = 20 MeV.
The spectra
without the widths of the proton-hole states $\Gamma_{p}$
are represented by the dashed curve.
We find that the contributions coupled to different
proton-hole states are localized in different energy regions and are
well separated from each other. This feature of the spectra
is different from that of the pionic atom formation in the ($d,^3$He) reaction,
\cite{hirenzaki91} \ where the contributions from different neutron-hole
states overlap.
We find the same features of the spectra for other incident energies.
\cite{okumura00} \
The realistic spectra including $\Gamma_{p}$ are plotted by the solid
lines in the same figure. As can be seen in the figure, $\Gamma_{p}$ is
too large to identify each kaonic bound state,
except for the [$d_{3/2}^{-1}$] hole state, corresponding to the
ground state of the final daughter nucleus $^{39}$K.
\begin{figure}[htbp]
\epsfxsize=6cm
\centerline{\epsfbox{fig7.eps}}
\caption{Kaonic atom formation cross sections for each proton-hole state
in the ($K^-, p$) reactions
plotted as functions of the emitted proton energy at the incident kaon
energy $T_{K}$ = 20 MeV and $\theta^{\rm Lab}_{p}$ = 0 [degrees].
The solid and dashed curves are results with and without
the widths of the proton-hole states, respectively.}
\label{fig:3}
\end{figure}
We show in Fig.~\ref{fig:4} the detailed structure of the kaonic atom formation cross
section coupled to the ground state of the daughter nucleus $^{39}$K.
Contributions from deeper proton-hole states provide the smooth
background of the spectra in this energy region because of their large
widths.
We find that the contributions from deeply bound 1$s$ and 2$p$ kaonic atom states
are well separated. At $T_{K}$ = 20 MeV, the peak due to the 1$s$ state
is significantly larger than that due to the 2$p$ state, as expected from the
result in Fig.~\ref{fig:1}. The peak height of the atomic 1$s$ contribution is approximately 7
[$\mu b$/($sr$ MeV)] at $T_{K}$ = 20 and 100 MeV. At $T_{K}$ = 100 MeV, the 2$p$ peak
is enhanced and is approximately 24 [$\mu b$/($sr$ MeV)]. The overall shapes of the
cross sections at 100 MeV and 400 MeV are similar, while the
absolute strength at 400 MeV is approximately 1/3 -- 1/4 of that at 100 MeV.
We should mention here that the energy resolution of experiments must be
good enough so as to observe the separate peak stracture in the spectrum
for the atomic states formation.
\begin{figure}[htbp]
\epsfxsize=14cm
\centerline{\epsfbox{fig8.eps}}
\caption{Kaonic atom formation cross sections in $^{40}$Ca($K^-, p$) reactions
coupled to the [$d_{3/2}^{-1}$] proton-hole state in $^{39}$K
plotted as functions of the emitted proton energy at
$\theta^{\rm Lab}_{p}$ = 0 [degrees] at the incident kaon
energies $T_{K}$ = 20 MeV, 100 MeV and 400 MeV,
respectively. }
\label{fig:4}
\end{figure}
We next consider the formation spectra of the kaonic nuclear states in
the ($K^-,p$) reactions. We choose the incident kaon energy to be
$T_{K} = 600$ MeV, for which there exist experimental data. \cite{Kishimoto03}
\ We show in Fig.~\ref{fig:40Ca} the formation spectra of kaonic nuclei
together with those of kaonic atoms as functions of the emitted proton kinetic
energies. Here, the widths of the kaonic states are fixed to the values
listed in Tables \ref{tab:ph} and \ref{tab:chi}. The effects of
the proton-hole widths $\Gamma_{p}$ are included. We found that
the spectra do not exhibit any peak-like structure due to the nuclear
state formation but have only a smooth slope in both the chiral unitary
and phenomenological optical potential cases. The contributions from the atomic
state formation appear as two very narrow
peaks around, the threshold energies
for both potentials. Each large peak contains several smaller peaks, due to the
formation of several atomic states, as in the spectrum shown in Fig. \ref{fig:4}.
\begin{figure}[htpb]
\epsfxsize=12cm
\centerline{\epsfbox{fig9.eps}}
\caption{Kaonic nucleus formation cross sections in $^{40}$Ca($K^-,p$) reactions
plotted as functions of the emitted proton energies at $\theta^{\rm Lab}_{p} = 0$
[degrees] and $T_{K} = 600$ MeV for (left) the chiral unitary model and (right)
the phenomenological $K$-nucleus optical potential. The vertical dashed lines
indicate the threshold energies, and the sharp peaks around the threshold
are due to the atomic state formations.}
\label{fig:40Ca}
\end{figure}
In order to include the phase space effects on the decay widths for the final kaon
system, we multiply the widths of kaonic states $\Gamma_K$ by the phase space factor defined in
Eqs. (\ref{eq:mfg})--(\ref{eq:mfg_G})
and calculate the ($K^-,p$) spectra.
We present the results in Fig. \ref{fig:39K_phase} for both potentials.
We find that the spectrum for the chiral unitary potential is not
affected significantly by the phase space effect. However, the ($K^-,p$)
spectrum shape for the phenomenological potential is distorted by including
the phase space factor and is expected to possess a bumpy structure, as
reported in Ref.~\citen{Kishimoto03}.
\begin{figure}[htpb]
\epsfxsize=12cm
\centerline{\epsfbox{fig10.eps}}
\caption{ Kaonic nucleus formation cross sections in $^{40}$Ca($K^-,p$) reactions
plotted as functions of the emitted proton energies at $\theta^{\rm Lab}_{p} = 0$
[degrees] and $T_{K} = 600$ MeV for (left) the chiral unitary model and (right)
the phenomenological $K$-nucleus optical potential. The vertical dashed lines
indicate the threshold energies, and the sharp peaks around the threshold
are due to the atomic state formations. The energy dependent decay
widths for kaonic states are used. (See the main text in details.)}
\label{fig:39K_phase}
\end{figure}
We performed systematic calculations for other target nuclei, $^{12}$C,
$^{16}$O and $^{28}$Si at $T_{K} = 600$ MeV and present the results in
Fig.~\ref{fig:spectrum_other}. In Ref.~\citen{Kishimoto03}, experimental
data for a $^{16}$O target are reported and data for $^{12}$C and $^{28}$Si
targets could be obtained in the future.~\cite{kishimoto_p}
As shown in Fig.~\ref{fig:spectrum_other}, we have found that
some bumpy structures in the ($K^-,p$) spectra may appear, due to the formation of
the kaon nucleus states, especially in the case of the $^{12}$C target, if the
kaon-nucleus optical potential is as deep as 200 MeV, as reported in Ref.~\citen{batty97}.
On the other hand, if the depth of the optical potential is as shallow as
50 MeV, as predicted by the chiral unitary model, the spectrum may not possess any bumpy structures,
but only exhibit a smooth slope for all targets considered here.
Finally, we present energy integrated cross sections $\frac{d\sigma}{d\Omega}$ for kaonic nuclear $1s$ state formation for $^{12}$C and $^{28}$Si targets with the results obtained in Refs.~\citen{ciep01} and~\citen{Kishimoto99}.
We found that our results with the phenomenological optical potential qualitatively
agree with those in Ref.~\citen{ciep01} and are smaller than those in Ref.~\citen{Kishimoto99}. The present results with the chiral unitary potential are significantly larger than those with the phenomenological potential because of the smaller momentum transfer in the ($K^-,p$) reactions due to the smaller binding energies of kaonic nuclear $1s$ states.
\begin{table}[htpd]
\begin{center}
\caption{Energy integrated cross sections for the formation of kaonic $1s$ nuclear states in units of [$\mu b/sr$]. The results in Refs.~\protect\citen{ciep01} and~\protect\citen{Kishimoto99} are also shown for comparison. Proton hole states are [$1p_{3/2}$]$^{-1}$ and [$1d_{5/2}$]$^{-1}$ for $^{12}$C and $^{28}$Si targets, respectively.}
\begin{tabular}{c|cccc}
\hline
\hline
&\multicolumn{4}{c}{($d\sigma/d\Omega$)$_{(K^-,p)}$ [$\mu b/sr$]}\\
Target nucleus&Chiral Unitary&Phenomenology&Ref.~\protect\citen{ciep01}&Ref.~\protect\citen{Kishimoto99} \\ \hline
$^{12}$C&425&65&47&100--490\\
$^{28}$Si&92.6&2.7&6.0&35--180\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[htpb]
\epsfxsize=10cm
\centerline{\epsfbox{fig11.eps}}
\caption{Same as Fig.~\protect\ref{fig:39K_phase}, except that here the target nuclei
are (top) $^{12}$C, (middle) $^{16}$O, and (bottom) $^{28}$Si.}
\label{fig:spectrum_other}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We have studied the structure and formation of kaonic atoms and kaonic nuclei in this
paper. We used two different kaon-nucleus optical potentials, which are obtained
from the chiral unitary model and a phenomenological fit of existing
kaonic atom data. We theoretically studied the structure of kaonic atoms and
kaonic nuclei using these potentials and determined the differences between the obtained level
schemes of the kaonic nuclear states.
We also studied the formation cross sections of deeply bound kaonic atoms
and kaonic nuclei which cannot be observed with standard X-ray spectroscopy. All the
atomic states are theoretically predicted to be quasi-stable.
We
investigate the ($K^-, p$) reaction theoretically and evaluate the cross
section of the $^{40}$Ca($K^-, p$) reaction in detail. For deep atomic state formation, the cross sections
are predicted to be approximately 7
[$\mu b$/($sr$ MeV)] at $T_{K}$ = 20 and 100 MeV
and 2 [$\mu b$/($sr$ MeV)] at 400 MeV
for kaonic atom 1$s$ state formation. For the atomic 2$p$ state, the cross section is predicted to be approximately
24 [$\mu b$/($sr$ MeV)] at $T_{K}$ = 100 MeV.
We also systematically studied the formation cross sections of kaonic nuclear states in
($K^-,p$) reactions for various targets.
In order to take into account the phase space suppression effects of the
decay widths, we introduced a phase space factor to obtain the ($K^-,p$)
spectra. We found in our theoretically studies that in the ($K^-,p$) reactions,
a certain bumpy structure due to kaonic nucleus formation can be seen only
for the case of a deep ($\sim200$ MeV) phenomenological kaon nucleus potential.
Due to the phase space suppression, the decay widths of kaonic states
become so narrow that we can see certain bumpy structure in the reaction spectrum,
which could be seen in experiments. For the case of the chiral unitary potential,
the binding energies are too small to reduce the decay widths and to see
the bumpy structure in the spectra of the ($K^-,p$) reactions. However, we should properly include the
energy dependence of the chiral unitary potential in future studies of kaonic
nuclear states to evaluate more realistic formation spectra.
In order to obtain more conclusive theoretically results, we need to apply
Green function methods for states with large widths\cite{morimatsu85} and to consider
the energy dependence of the optical potential properly.
Furthermore, we should consider the changes and/or deformations of the nucleus due
to the existence of the kaon inside and solve the problem in a self-consistent manner
for kaonic nucleus states. However, we believe that the present theoretical effort
to evaluate the absolute cross sections for the kaonic bound state formation
are relevant for determining a suitable method to observe them and helpful for
developing the physics of kaon-nuclear bound systems and kaon behavior in
nuclear medium. Further investigations both theoretically
and experimentally are needed to understand kaon behavior in nuclear medium more
precisely.
\section*{Acknowledgements}
We acknowledge E. Oset and A. Ramos for stimulating
discussions on kaon bound systems and the chiral unitary model.
We would like to thank
M. Iwasaki and T. Kishimoto for stimulating discussions on the latest experimental data
of kaonic nucleus formation. We also would like to thank T. Yamazaki,
Y. Akaishi, and A. Dot$\acute{\rm e}$ for useful discussions on theoretical
aspects of kaonic nucleus systems. We are grateful to H. Toki and E. Hiyama
for many suggestions and discussions regarding the kaon-nucleus systems.
We also thank A. Gal for his careful reading of our preprint and useful comments.
This work is partly supported by Grants-in-Aid for scientific research of MonbuKagakusho and Japan Society for the Promotion of Science (No. 16540254).
|
{
"timestamp": "2005-08-25T03:10:23",
"yymm": "0503",
"arxiv_id": "nucl-th/0503039",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503039"
}
|
\section{Introduction}
\label{intro}
There is the evident nigh affinity between the classical probability
function and the Boolean function of the classical propositional logic \cite{LYN66}. These functions are differed by the range of value, only. That is if
the range of values of the Boolean function shall be expanded from the
two-elements set $\left\{ 0;1\right\} $ to the segment $\left[ 0;1\right] $
of the real numeric axis then the logical analog of the Bernoulli Large
Number Law \cite{BER13} can be deduced from the logical axioms. These topics
is considered in this article.
\section{The classical logic}
\label{sec:1}
{\bf Definition 2.1} Sentence $\ll \Theta \gg $ is a true
sentence if and only if $\Theta $ \cite{TAR44}.
For example: sentence $\ll $it rains$\gg $ is the true sentence if and
only if it rains.
{\bf Definition 2.2} Sentence $\ll \Theta \gg $ is a false
sentence if and only if it is not that $\Theta $.
{\bf Definition 2.3} Sentences $A$ and $B$ are equal ($A=B$%
) if $A $ is true if and only if $B$ is true.
Hereinafter we use the usual notions of the classical propositional logic
\cite{MEN63}.
{\bf Definition 2.4} Sentence $C$ is a conjunction of the
sentences $A$ and $B$ \\($C=\left( A\wedge B\right) $) if $C$ is true if and
only if $A$ is true and $B$ is true.
{\bf Definition 2.5} Sentence $C$ is a negation of the
sentence $A$ ( $C=\overline{A}$), if $C$ is true if and only if $A$ is false.
{\bf Theorem 2.1}
1) $(A\wedge A)=A$;
2) $(A\wedge B)=(B\wedge A)$;
3) $(A\wedge (B\wedge C))=((A\wedge B)\wedge C)$;
4) if $T$ is the true sentence then for every sentence $A$: $(A\wedge T)=A$;
5) if $F$ is false sentence then $\overline{F}$ is true sentence.
{\bf Proof of the Theorem {2.1}: }From Definitions {2.1},
{2.2}, {2.3}, {2.4}.
{\bf Definition 2.6} Each function $\rm{g}$ with domain in
the set of the sentences and with the range of values on the two-elements set
$\left\{ 0;1\right\} $ is a Boolean function if:
1) $\rm{g}\left( \overline{A}\right) =1-$ ${%
\rm{g}}\left( A\right) $ for every sentence $A$;
2) $\rm{g}\left( A\wedge B\right) ={%
\rm{g}}\left( A\right) \cdot \rm{g}\left( B\right) $ for all
sentences $A$ and $B$.
{\bf Definition 2.7} Set $\Im $ of the sentences is a basic set if for every element
$A$ of this set there exist Boolean functions $\rm{g}_1$ and $\rm{g}_2$ such
that the following conditions fulfill:
1) $\rm{g}_1\left( A\right) \neq \rm{g}_2\left( A\right) $;
2) $\rm{g}_1\left( B\right) =\rm{g}_2\left( B\right) $ for
each element $B$ of $\Im $ such that $B\neq A$.
{\bf Definition 2.8} Set $\left[ \Im \right] $ of the sentences is
a propositional closure of the set $\Im $ if the following
conditions fulfill:
1) if $A\in \Im $ then $A\in \left[ \Im \right ] $;
2) if $A\in \left[\Im \right]$ then $\overline{A}\in \left[ \Im \right ] $;
3) if $A\in \left[ \Im \right ]$ and $B\in \left[ \Im \right ] $ then $%
\left( A\wedge B\right) \in \left[ \Im \right ] $;
4) there do not exist other elements of $\left[ \Im \right ] $ except the
listed by 1), 2), 3) points of this definition.
In the following text the elements of $\left[ \Im \right] $ are called as
the $\mathit{\Im }$-sentences.
{\bf Definition 2.9} $\Im $-sentence $A$ is a tautology
if for all Boolean functions $\rm{g}$:
\[
\rm{g}(A)=1\mbox{.}
\]
{\bf Definition 2.10} A disjunction and an
implication are defined by the usual way:
\[
\begin{array}{c}
\left( A\vee B\right) =\overline{\left( \overline{A}\wedge \overline{B}%
\right) }\mbox{,} \\
\left( A\Rightarrow B\right) =\overline{\left( A\wedge \overline{B}\right) }%
\mbox{.}
\end{array}
\]
By this definition and the Definitions {2.4} and {2.5}:
$\left( A\vee B\right) $ is the false sentence if and only if $A$ is the
false sentence and $B$ is the false sentence.
$\left( A\Rightarrow B\right) $ is the false sentence if and only if $A$
is the true sentence and $B$ is the false sentence.
{\bf Definition 2.11} A $\Im $-sentence is a propositional axiom
\cite{MEN63} if this sentence has got one some amongst the following forms:
\textbf{A1}. $\left( A\Rightarrow \left( B\Rightarrow A\right) \right) $;
\textbf{A2. }$\left( \left( A\Rightarrow \left( B\Rightarrow C\right)
\right) \Rightarrow \left( \left( A\Rightarrow B\right) \Rightarrow \left(
A\Rightarrow C\right) \right) \right) $;
\textbf{A3}. $\left( \left( \overline{B}\Rightarrow \overline{A}\right)
\Rightarrow \left( \left( \overline{B}\Rightarrow A\right) \Rightarrow
B\right) \right) $.
Let $\Im $ be some basic set. In the following text I consider $\Im $-sentences,
only.
{\bf Definition 2.12} Sentence $B$ is obtained from the sentences $\left( A\Rightarrow
B\right) $ and $A$ by the logic rule "modus ponens".
{\bf Definition 2.13} \cite{MEN63} Array $A_1,A_2,\ldots ,A_n$ of the
sentences is a propositional deduction of the sentence $A$ from
the hypothesis list $\Gamma $ (denote: $\Gamma \vdash A$) if $A_n=A$ and
for all numbers $l$ ($1\leq l\leq n$): $A_l$ is either the propositional
axiom or $A_l$ is obtained from some sentences $A_{l-k}$ and $A_{l-s}$ by
the modus ponens or $A_l\in \Gamma $.
{\bf Definition 2.14} A sentence is a propositional proved
sentence if this sentence is the propositional axiom or this sentence is
obtained from the propositional proved sentences by the modus ponens.
Hence, if $A$ is the propositional proved sentence then the propositional
deduction
\begin{center}
$\vdash A$
\end{center}
exists.
{\bf Theorem: 2.2} \cite{MEN63} If sentence $A$ is the propositional
proved sentence then for all Boolean function $\rm{g}$: $\rm{g}\left( A\right) =1$.
{\bf Proof of the Theorem {2.2}: }\cite{MEN63}.
{\bf Theorem: 2.3} {\bf (The completeness Theorem). }\cite{MEN63} All tautologies
are the propositional proved sentences.
{\bf Proof of the Theorem {2.3}: }\cite{MEN63}.
\section{B-functions}
\label{sec:2}
{\bf Definition 3.1} Each function $\rm{b}\left( x\right)$
with domain in the sentences set and with the range of values on the numeric
axis segment $\left[ 0;1\right] $ is called as a B-function if
\[
\rm{b}\left( C\right) =1
\]
for some sentence $C$ and
\[
\rm{b}\left( A\wedge B\right) +\rm{b}\left( A\wedge
\overline{B}\right) =\rm{b}\left( A\right)
\]
for every sentences $A$ and $B$.
{\bf Theorem: 3.1} For each B-function $\rm{b}$:
1) for every sentences $A$ and $B$: $\rm{b}\left( A\wedge B\right)
\leq \rm{b}\left( A\right) $;
2) for every sentence $A$: if $T$ is the true sentence, then \\$\rm{b}%
\left( A\right) +\rm{b}\overline{\left( A \right )}=\rm{b}%
\left( T\right) $
3) for every sentence $A$: if $T$ is the true sentence, then $\rm{b}%
\left( A\right) \leq \rm{b}\left( T\right) $;
{\bf Proof of the Theorem {3.1}:}
1)From Definitions {3.1}.
2) From the points 4 and 2 of the Theorem {2.1}:
\[
\rm{b}\left( T\wedge A\right) +\rm{b}\left( T\wedge
\overline{A}\right) =\rm{b}\left( A\right) +\rm{b}\left(
\overline{A}\right) .
\]
3) From previous point of that Theorem.
Therefore, if $T$ is the true sentence, then
\begin{equation}
\rm{b}\left( T\right) =1\mbox{.} \label{b2}
\end{equation}
Hence, for every sentence $A$:
\begin{equation}
\rm{b}\left( A\right) +\rm{b}\left( \overline{A}\right) =1%
\mbox{.} \label{b3}
\end{equation}
{\bf Theorem: 3.2} If sentence $D$ is the propositional proved
sentence then for all B-functions $\rm{b}$: $\rm{b}\left(
D\right) =1 $.
{\bf Proof of the Theorem {3.2}: }
If $D$ is A1 then by Definition {2.10}:
\[
\rm{b}\left( D\right) =\rm{b}\left( \overline{\left( A\wedge
\overline{\overline{\left( B\wedge \overline{A}\right) }}\right) }\right) %
\mbox{.}
\]
By (\ref{b3}):
\[
\rm{b}\left( D\right) =1-\rm{b}\left( A\wedge \overline{%
\overline{\left( B\wedge \overline{A}\right) }}\right) \mbox{.}
\]
By the Definition {3.1} and the Theorem {2.1}:
\[
\begin{array}{c}
\rm{b}\left( D\right) =1-\rm{b}\left( A\right) +\rm{b}\left( A\wedge \overline{\left( B\wedge \overline{A}\right) }\right) %
\mbox{,} \\
\rm{b}\left( D\right) =1-\rm{b}\left( A\right) +\rm{b}\left( A\right) -\rm{b}\left( A\wedge \left( B\wedge \overline{A}%
\right) \right) \mbox{,} \\
\rm{b}\left( D\right) =1-\rm{b}\left( \left( A\wedge
B\right) \wedge \overline{A}\right) \mbox{,} \\
\rm{b}\left( D\right) =1-\rm{b}\left( A\wedge B\right) +%
\rm{b}\left( \left( A\wedge B\right) \wedge A\right) \mbox{,} \\
\rm{b}\left( D\right) =1-\rm{b}\left( A\wedge B\right) +%
\rm{b}\left( \left( A\wedge A\right) \wedge B\right) \mbox{,} \\
\rm{b}\left( D\right) =1-\rm{b}\left( A\wedge B\right) +%
\rm{b}\left( A\wedge B\right) \mbox{.}
\end{array}
\]
The proof is similar for the rest propositional axioms .
Let for all B-function $\rm{b}$: $\rm{b}(A)=1$ and $%
\rm{b}(A\Rightarrow D)=1$.
By Definition {2.10}:
\[
\rm{b}\left( A\Rightarrow D\right) =\rm{b}\left( \overline{%
A\wedge \overline{D}}\right) \mbox{.}
\]
By (\ref{b3}):
\[
\rm{b}\left( A\Rightarrow D\right) =1-\rm{b}\left( A\wedge
\overline{D}\right) \mbox{.}
\]
Hence,
\[
\rm{b}\left( A\wedge \overline{D}\right) =0\mbox{.}
\]
By Definition {3.1}:
\[
\rm{b}\left( A\wedge \overline{D}\right) =\rm{b}\left(
A\right) -\rm{b}\left( A\wedge D\right) \mbox{.}
\]
Hence,
\[
\rm{b}\left( A\wedge D\right) =\rm{b}\left( A\right) =1%
\mbox{.}
\]
By Definition {3.1} and the Theorem {2.1}:
\[
\rm{b}\left( A\wedge D\right) =\rm{b}\left( D\right) -{%
\rm{b}}\left( D\wedge \overline{A}\right) =1\mbox{.}
\]
Therefore, for all B-function $\rm{b}$:
\[
\rm{b}\left( D\right) =1\mbox{.}
\]
{\bf Theorem: 3.3}
1) If for all Boolean functions $\rm{g}$:
\[
\rm{g}\left( A\right) =1
\]
then for all B-functions $\rm{b}$:
\[
\rm{b}\left( A\right) =1\mbox{.}
\]
2) If for all Boolean functions $\rm{g}$:
\[
\rm{g}\left( A\right) =0
\]
then for all B-functions $\rm{b}$:
\[
\rm{b}\left( A\right) =0\mbox{.}
\]
{\bf Proof of the Theorem {3.3}: }
1) This just follows from the preceding Theorem and from the Theorem {2.3}.
2) If for all Boolean functions $\rm{g}$: $\rm{g}\left(
A\right) =0$, then by the Definition {2.6}: $\rm{g}\left( \overline{A}%
\right) =1$. Hence, by the point 1 of this Theorem: for all B-function ${%
\rm{b}}$: $\rm{b}\left( \overline{A}\right) =1$. By (\ref{b3}%
): $\rm{b}\left( A\right) =0$.
{\bf Theorem: 3.4} All Boolean functions are the B-functions.
Hence, the B-function is the generalization of the logic Boolean
function. Therefore, the B-function is the logic function, too.
{\bf Proof of the Theorem {3.4}: } If $C$ is {\bf A1} then
$\rm{g}\left( C\right) =1$.
By Definition {2.6}: for all Boolean functions $\rm{g}$:
$\rm{g}\left( A\wedge B\right) +\rm{g}\left( A\wedge
\overline{B}\right) =\rm{g}\left( A\right) \cdot \rm{g}%
\left( B\right) +{\rm{g}}\left( A\right) \cdot \left( 1-\rm{g}%
\left( B\right) \right) =\rm{g}\left( A\right) $.
{\bf Theorem: 3.5}
\[
\rm{b}\left( A\vee B\right) =\rm{b}\left( A\right) +%
\rm{b}\left( B\right) -\rm{b}\left( A\wedge B\right) \mbox{.}
\]
{\bf Definition 3.2} Sentences $A$ and $B$ are
inconsistent sentences for the B-function $\rm{b}$ if
\[
\rm{b}\left( A\wedge B\right) =0\mbox{.}
\]
{\bf Proof of the Theorem {3.5}: }By the Definition {2.10} and
(\ref{b3}):
\[
\rm{b}\left( A\vee B\right) =1-\rm{b}\left( \overline{A}%
\wedge \overline{B}\right) .
\]
By Definition {3.1}:
\[
\rm{b}\left( A\vee B\right) =1-\rm{b}\left( \overline{A}%
\right) +\rm{b}\left( \overline{A}\wedge B\right) =\rm{b}%
\left( A\right) +\rm{b}\left( B\right) -\rm{b}\left( A\wedge
B\right) \mbox{.}
\]
{\bf Theorem: 3.6} If sentences $A$ and $B$ are the inconsistent
sentences for the B-function $\rm{b}$ then
\[
\rm{b}\left( A\vee B\right) =\rm{b}\left( A\right) +%
\rm{b}\left( B\right) \mbox{.}
\]
{\bf Proof of the Theorem {3.6}: }This just follows from the preceding
Theorem and Definition {3.2}.
{\bf Theorem: 3.7} If $\rm{b}\left( A\wedge B\right) =\rm{b}\left( A\right
\cdot \rm{b}\left( B\right) $ then $\rm{b}%
\left( A\wedge \overline{B}\right) =\rm{b}\left( A\right) \cdot {%
\rm{b}}\left( \overline{B}\right) $.
{\bf Proof of the Theorem {3.7}: }By the Definition {3.1}:
\[
\rm{b}\left( A\wedge \overline{B}\right) =\rm{b}\left(
A\right) -\rm{b}\left( A\wedge B\right) \mbox{.}
\]
Hence,
\[
\rm{b}\left( A\wedge \overline{B}\right) =\rm{b}\left(
A\right) -\rm{b}\left( A\right) \cdot \rm{b}\left( B\right) =%
{\rm{b}}\left( A\right) \cdot \left( 1-\rm{b}\left( B\right)
\right) \mbox{.}
\]
Hence, by (\ref{b3}):
\[
\rm{b}\left( A\wedge \overline{B}\right) =\rm{b}\left(
A\right) \cdot \rm{b}\left( \overline{B}\right) \mbox{.}
\]
{\bf Theorem: 3.8} $\rm{b}\left( A\wedge \overline{A}\wedge
B\right) =0$.
{\bf Proof of the Theorem {3.8}: }By the Definition {3.1} and by
the points 2 and 3 of the Theorem {2.1}:
\[
\rm{b}\left( A\wedge \overline{A}\wedge B\right) =\rm{b}%
\left( A\wedge B\right) -\rm{b}\left( A\wedge A\wedge B\right) ,
\]
hence, by the point 1 of the Theorem {2.1}:
\[
\rm{b}\left( A\wedge \overline{A}\wedge B\right) =\rm{b}%
\left( A\wedge B\right) -\rm{b}\left( A\wedge B\right) \mbox{.}
\]
{\bf Theorem: 3.9}
\[
\mathrm{P}\left( A\wedge \left( B\vee C\right) \right) = %
\mathrm{P}\left( A\wedge B\right) +\mathrm{P}\left( A\wedge C\right) -%
\mathrm{P}\left( A\wedge B\wedge C\right) \mbox{.}
\]
{\bf Proof of the Theorem {3.9}}:
By Definition 3.1:
$\mathrm{P}\left( A\wedge \left( B\vee C\right) \right) =\mathrm{P}\left(
A\wedge \overline{\left( \overline{B}\wedge \overline{C}\right) }\right) =%
\mathrm{P}\left( A\right) -\mathrm{P}\left( A\wedge \overline{B}\wedge
\overline{C}\right) =\mathrm{P}\left( A\right) -\mathrm{P}\left( A\wedge
\overline{B}\right) +\mathrm{P}\left( A\wedge \overline{B}\wedge C\right) =
\mathrm{P}\left( A\wedge B\right) +\mathrm{P}\left( A\wedge C\right) -%
\mathrm{P}\left( A\wedge B\wedge C\right) $
\section{The independent tests}
\label{sec:3}
{\bf Definition 4.1} Let $st(n)$ be a function such that $st(n)$
has got the domain on the set of natural numbers and has got the range of
values in the set of the $\Im $-sentences.
In this case $\Im $-sentence $A$ is a [st]-series of range $r$ with
V- number $k$ if $A$, $r$ and $k$ fulfill to some one amongst the following
conditions:
1) $r=1$ and $k=1$, $A=st\left( 1\right) $ or $k=0$, $A=\overline{st\left(
1\right) }$;
2) $B$ is [st]-series of range $r-1$ with V-number $k-1$ and
\[
A=\left( B\wedge st\left( r\right) \right) \mbox{,}
\]
or $B$ is [st]-series of range $r-1$ with V-number $k$ and
\[
A=\left( B\wedge \overline{st\left( r\right) }\right) \mbox{.}
\]
Let us denote a set of [st]-series of range $r$ with V-number $%
k$ as $[st](r,k)$.
For example, if $st\left( n\right) $ is a sentence $B_n$ then the
sentences:
$\left( B_1\wedge B_2\wedge \overline{B_3}\right) $, $\left( B_1\wedge
\overline{B_2}\wedge B_3\right) $, $\left( \overline{B_1}\wedge B_2\wedge
B_3\right) $
are the elements of $[st](3,2)$, and \\$\left( B_1\wedge B_2\wedge \overline{%
B_3}\wedge B_4\wedge \overline{B_5}\right) \in [st](5,3)$.
{\bf Definition 4.2} Function $st(n)$ is independent for B-function $\rm{b}$ if for
$A$: if \\$A\in $ $[st](r,r)$ then:
\[
\rm{b}\left( A\right) =\prod\limits_{n=1}^r\rm{b}\left(
st\left( n\right) \right) \mbox{.}
\]
{\bf Definition 4.3} Let $st(n)$ be a function such that $st(n)$
has got the domain on the set of natural numbers and has got the range of
values in the set of the $\Im $-sentences.
In this case sentence $A$ is [st]-disjunction of range $r$ with V-number $k$
(denote: $\rm{t}[st](r,k)$) if $A$ is the disjunction of all elements
of $[st](r,k)$.
For example, if $st\left( n\right) $ is the sentence $C_n$ then:
$\left( \overline{C_1}\wedge \overline{C_2}\wedge \overline{C_3}\right) ={%
\rm{t}}[st]\left( 3,0\right) $,
$\rm{t}[st]\left( 3,1\right) =\left( \left( C_1\wedge \overline{C_2}%
\wedge \overline{C_3}\right) \vee \left( \overline{C_1}\wedge C_2\wedge
\overline{C_3}\right) \vee \left( \overline{C_1}\wedge \overline{C_2}\wedge
C_3\right) \right) $,
$\rm{t}[st]\left( 3,2\right) =\left( \left( C_1\wedge C_2\wedge
\overline{C_3}\right) \vee \left( \overline{C_1}\wedge C_2\wedge C_3\right)
\vee \left( C_1\wedge \overline{C_2}\wedge C_3\right) \right) $,
$\left( C_1\wedge C_2\wedge C_3\right) =\rm{t}[st]\left( 3,3\right) $.
{\bf Definition 4.4} A rational number $\omega$ is called as a
frequency of sentence $A$ in the [st]-series of $r$ independent for B-function
$\rm{b}$ tests (designate: $\omega=\nu _r\left[ st\right] \left( A\right)$) if
1) $st(n)$ is independent for B-function $\rm{b}$,
2) for all $n$: $\rm{b}\left( st\left( n\right) \right) =\rm{b}\left( A\right) $,
3) $\rm{t}[st](r,k)$ is true and $\omega=k/r$.
{\bf Theorem: 4.1} {\bf (the J.Bernoulli formula }\cite{BER13}\textbf{)}
If $st(n) $ is independent for B-function $\rm{b}$ and there exists
a real number $p$ such that for all $n$: $\rm{b}\left( st\left( n\right)
\right) =p$ then
\[
\rm{b}\left( \rm{t}\left[ st\right] \left( r,k\right)
\right) =\frac{r!}{k!\cdot \left( r-k\right) !}\cdot p^k\cdot \left(
1-p\right) ^{r-k}\mbox{.}
\]
{\bf Proof of the Theorem {4.1}: }By the Definition {4.2} and
the Theorem {3.7}: if $B\in \left[ st\right] \left( r,k\right) $ then:
\[
\rm{b}\left( B\right) =p^k\cdot \left( 1-p\right) ^{r-k}\mbox{.}
\]
Since $\left[ st\right] \left( r,k\right) $ contains ${r!}/\left({k!\cdot
\left( r-k\right) !}\right)$ elements then by the Theorems {3.7}, {3.8}
and {3.6} this Theorem is fulfilled.
{\bf Definition 4.5} Let function $st(n)$ has got the domain on
the set of the natural numbers and has got the range of values in the set of
the $\Im $-sentences.
Let function $f(r,k,l)$ has got the domain in the set of threes of the natural
numbers and has got the range of values in the set of the $\Im $-sentences.
In this case $f(r,k,l)=\rm{T}[st](r,k,l)$ if
1) $f(r,k,k)=\rm{t}[st](r,k)$,
2) $f(r,k,l+1)=(f(r,k,l)\vee \rm{t}[st](r,l+1)) $.
{\bf Definition 4.6} If $a$ and $b$ are real numbers and $k-1<a\leq k$
and $l\leq b<l+1$ then $\rm{T}[st](r,a,b)=\rm{T}[st](r,k,l)$.
{\bf Theorem: 4.2}
\[
\rm{T}[st](r,a,b)=\ll \frac ar \leq \nu _r\left[ st\right] \left(
A\right) \leq \frac br\gg \mbox{.}
\]
{\bf Proof of the Theorem {4.2}: }By the Definition {4.6}: there
exist natural numbers $r $ and $k$ such that $k-1<a\leq k$ and $l\leq b<l+1$.
The recursion on $l$:
1. Let $l=k$.
In this case by the Definition {4.4}:
\[
\rm{T}[st](r,k,k)=\rm{t}[st](r,k)=\ll \nu _r\left[ st\right]
\left( A\right) =\frac kr\gg \mbox{.}
\]
2. Let $n$ be any natural number.
The recursive assumption: Let
\[
\rm{T}[st](r,k,k+n)=\ll \frac kr\leq \nu _r\left[ st\right] \left(
A\right) \leq \frac {k+n}r\gg \mbox{.}
\]
By the Definition {4.5}:
\[
\rm{T}[st](r,k,k+n+1)=(\rm{T}[st](r,k,k+n)\vee \rm{t}%
[st](r,k+n+1))\mbox{.}
\]
By the recursive assumption and by the Definition {4.4}:
\[
\rm{T}[st](r,k,k+n+1)=
\]
\[
=(\ll \frac kr\leq \nu _r\left[ st\right] \left( A\right) \leq \frac
{k+n}r\gg \vee \ll \nu _r\left[ st\right] \left( A\right) =\frac {k+n+1}r\gg)%
\mbox{.}
\]
Hence, by the Definition {2.10}:
\[
\rm{T}[st](r,k,k+n+1)=\ll \frac kr\leq \nu _r\left[ st\right] \left(
A\right) \leq \frac {k+n+1}r\gg \mbox{.}
\]
{\bf Theorem: 4.3} If $st(n)$ is independent for B-function ${%
\rm{b}}$ and there exists a real number $p$ such that ${%
\rm{b}}\left( st\left( n\right) \right) =p$ for all $n$ then
\[
\rm{b}\left( \rm{T}[st](r,a,b)\right) =\sum_{a\leq k\leq
b}\frac {r!}{k!\cdot \left( r-k\right) !}\cdot p^k\cdot \left( 1-p\right)
^{r-k}\mbox{.}
\]
{\bf Proof of the Theorem {4.3}: }This is the consequence from the
Theorem {4.1} by the Theorem {3.6}.
{\bf Theorem: 4.4} If $st(n)$ is independent for the B-function ${%
\rm{b}}$ and there exists a real number $p$ such that ${%
\rm{b}}\left( st\left( n\right) \right) =p$ for all $n$ then
\[
\rm{b}\left( \rm{T}[st](r,r\cdot \left( p-\varepsilon
\right) ,r\cdot \left( p+\varepsilon \right) )\right) \geq 1-\frac{p\cdot
\left( 1-p\right) }{r\cdot \varepsilon ^2}
\]
for every positive real number $\varepsilon $.
{\bf Proof of the Theorem {4.4}:} Because
\[
\sum_{k=0}^r\left( k-r\cdot p\right) ^2\cdot \frac{r!}{k!\cdot \left(
r-k\right) !}\cdot p^k\cdot \left( 1-p\right) ^{r-k}=r\cdot p\cdot \left(
1-p\right)
\]
then if
\[
J=\left\{ k\in \mathbf{N}|0\leq k\leq r\cdot \left( p-\varepsilon \right)
\right\} \cap \left\{ k\in \mathbf{N}|r\cdot \left( p+\varepsilon \right)
\leq k\leq r\right\}
\]
then
\[
\sum_{k\in J}\frac {r!}{k!\cdot \left( r-k\right) !}\cdot p^k\cdot \left(
1-p\right) ^{r-k}\leq \frac {p\cdot \left( 1-p\right) }{r\cdot \varepsilon
^2}\mbox{.}
\]
Hence, by (\ref{b3}) this Theorem is fulfilled.
Hence
\begin{equation}
\lim\limits_{r\rightarrow \infty }\rm{b}\left( \rm{T}[st](r,r%
\cdot \left( p-\varepsilon \right) ,r\cdot \left( p+\varepsilon \right) )%
\right) =1 \label{uh}
\end{equation}
for all tiny positive numbers $\varepsilon $.
\section{The logic probability function}
\label{sec:4}
{\bf Definition 5.1} B-function $\mathrm{P}$ is $P$-function
if for every $\Im $-sentence $\ll \Theta \gg$: \\If $\mathrm{P \left( \ll
\Theta \gg \right) = 1}$ then $\ll \Theta \gg$ is true sentence.
Hence from Theorem {4.2} and (\ref{uh}): if $\rm{b}$ is a $P$-function
then the sentence
\[
\ll \left( p-\varepsilon \right) \leq \nu _r\left[ st\right] \left( A\right)
\leq \left( p+\varepsilon \right) \gg
\]
is almost true sentence for large $r$ and for all tiny $\varepsilon $.
Therefore, it is almost truely that
\[
\nu _r\left[ st\right] \left( A\right) =p
\]
for large $r$.
Therefore, it is almost true that
\[
\rm{b}\left( A\right) =\nu _r\left[ st\right] \left( A\right)
\]
for large $r$.
Therefore, the function, defined by the Definition {5.1} has got the
statistical meaning. That is why I'm call such function as the
logic probability function.
\section{Conditional probability}
{\bf Definition 6.1:} {\it Conditional probability} $B$ for $C$ is the
following function:
\begin{equation}
\mathfrak{b}\left( B/C\right) \stackrel{def}{=}\frac{\mathfrak{b}\left( C\wedge B\right)
}{\mathfrak{b}\left( C\right) }\mbox{.}\label{CP}
\end{equation}
{\bf Theorem 6.1} The conditional probability function is a B-function.
{\bf Proof of Theorem 6.1} From Definition 6.1:
\begin{center}
$\mathfrak{b}\left( C/C\right) =\frac{\mathfrak{b}\left( C\wedge C\right) }{\mathfrak{b}%
\left( C\right) }$.
\end{center}
Hence by point 1 of Theorem 2.1:
\begin{center}
$\mathfrak{b}\left( C/C\right) =\frac{\mathfrak{b}\left( C\right) }{\mathfrak{b}\left(
C\right) }=1$.
\end{center}
Form Definition 6.1:
\begin{center}
$\mathfrak{b}\left( \left( A\wedge B\right) /C\right) +\mathfrak{b}\left( \left(
A\wedge \left( \neg B\right) \right) /C\right) =\frac{\mathfrak{b}\left( C\wedge \left(
A\wedge B\right) \right) }{\mathfrak{b}\left( C\right) }+\frac{\mathfrak{b}\left(
C\wedge \left( A\wedge \left( \neg B\right) \right) \right) }{\mathfrak{b}\left( C\right) }
$.
\end{center}
Hence:
\begin{center}
$\mathfrak{b}\left( \left( A\wedge B\right) /C\right) +\mathfrak{b}\left( \left(
A\wedge \left( \neg B\right) \right) /C\right) =\frac{\mathfrak{b}\left( C\wedge \left(
A\wedge B\right) \right) +\mathfrak{b}\left( C\wedge \left( A\wedge \left( \neg B\right)
\right) \right) }{\mathfrak{b}\left( C\right) }$.
\end{center}
By point 3 of Theorem 2.1:
\begin{center}
$\mathfrak{b}\left( \left( A\wedge B\right) /C\right) +\mathfrak{b}\left( \left(
A\wedge \left( \neg B\right) \right) /C\right) =\frac{\mathfrak{b}\left( \left(
C\wedge A\right) \wedge B\right) +\mathfrak{b}\left( \left( C\wedge A\right) \wedge \left( \neg
B\right) \right) }{\mathfrak{b}\left( C\right) }$.
\end{center}
Hence by Definition 3.1:
\begin{center}
$\mathfrak{b}\left( \left( A\wedge B\right) /C\right) +\mathfrak{b}\left( \left(
A\wedge \left( \neg B\right) \right) /C\right) =\frac{\mathfrak{b}\left( C\wedge A\right)
}{\mathfrak{b}\left( C\right) }$.
\end{center}
Hence by Definition 6.1:
\begin{center}
$\mathfrak{b}\left( \left( A\wedge B\right) /C\right) +\mathfrak{b}\left( \left(
A\wedge \left( \neg B\right) \right) /C\right) =\mathfrak{b}\left( A/C\right) $
$_{\bf \Box }$
\end{center}
\section{Classical probability}
Let $\mathrm{P}$ be $P$-function.
{\bf Definition 7.1} $\left\{ B_1,B_2,\ldots ,B_n\right\} $ is called as {\it
complete set} if the following conditions are fulfilled:
1. if $k\neq s$ then $\left( B_k\wedge B_s\right)$ is a false sentence;
2. $\left( B_1\vee B_2\vee \ldots \vee B_n\right)$ is a true sentence.
{\bf Definition 7.2} $B$ is favorable for $A$ if $\left(B\wedge\overline{A}%
\right)$ is a false sentence, and $B$ is unfavorable for $A$ if $\left(B\wedg
A\right)$ is a false sentence.
Let
1. $\left\{ B_1,B_2,\ldots ,B_n\right\}$ be complete set;
2. for $k\in \left\{ 1,2,\ldots ,n\right\} $ and $s\in \left\{ 1,2,\ldots ,n%
\right\} $: $\mathrm{P}\left( B_k\right) =\mathrm{P}\left( B_s\right) $;
3. if $1\leq k\leq m$ then $B_k$ is favorable for $A$, and if $m+1\leq s\leq n$
then $B_s$ is unfavorable for $A$.
In that case from point 5 of Theorem 2.1 and from (\ref{b2}) and (\ref{b3}):
\[
\mathrm{P}\left( \overline{A}\wedge B_k\right) = 0
\]
for $k\in \left\{ 1,2,\ldots ,m\right\} $ and
\[
\mathrm{P}\left( A\wedge B_s\right) =0
\]
for $s\in \left\{ m+1,m+2,\ldots ,n\right\} $.
Hence from Definition 3.1:
\[
\mathrm{P}\left( A\wedge B_k\right) =\mathrm{P}\left( B_k\right)
\]
for $k\in \left\{ 1,2,\ldots ,n\right\} $.
By point 4 of Theorem 2.1:
\[
A=\left( A\wedge \left( B_1\vee B_2\vee \ldots \vee B_m\vee B_{m+1}\ldots
\vee B_n\right) \right) \mbox{.}
\]
Hence by Theorem 3.9:
$\mathrm{P}\left( A\right) =\mathrm{P}\left( A\wedge B_1\right) +\mathrm{P}%
\left( A\wedge B_2\right) +\ldots +$
$+\mathrm{P}\left( A\wedge B_m\right) +\mathrm{P}\left( A\wedge
B_{m+1}\right) +\ldots +\mathrm{P}\left( A\wedge B_n\right) =$
$=\mathrm{P}\left( B_1\right) +\mathrm{P}\left( B_2\right) +\ldots +\mathrm{P%
}\left( B_m\right) $.
Therefore
\[
\mathrm{P}\left( A\right) =\frac mn\mbox{.}
\]
\section{Conclusion}
\label{sect:concl}
The logic probability function is the extension of the logic B-function.
Therefore, \textbf{the probability is some generalization of the classic
propositional logic.} That is the probability is the logic of events such
that these events do not happen, yet.
\section{Appendix. Consistency}
\subsection{THE NONSTANDARD NUMBERS}
Let us consider the set ${\bf N}$ of natural numbers.
{\bf Definition A.1:} The $n${\it -part-set} ${\bf S}$ of ${\bf N}$ is
defined recursively as follows:
1) ${\bf S}_1=\left\{ 1\right\} $;
2) ${\bf S}_{\left( n+1\right) }={\bf S}_n\cup \left\{ n+1\right\} $.
{\bf Definition A.2: }If ${\bf S}_n$ is the $n$-part-set of ${\bf N}$ and
{\bf A}\subseteq {\bf N}$ then $\left\| {\bf A}\cap {\bf S}_n\right\| $ is
the quantity elements of the set ${\bf A}\cap {\bf S}_n$, and if
\[
\varpi _n\left( {\bf A}\right) =\frac{\left\| {\bf A}\cap {\bf S}_n\right\|
n\mbox{,}
\]
then $\varpi _n\left( {\bf A}\right) $ is {\it the frequency} of the set
{\bf A}$ on the $n$-part-set ${\bf S}_n$.
{\bf Theorem A.1:}
1) $\varpi _n({\bf N})=1$;
2) $\varpi _n(\emptyset )=0$;
3) $\varpi _n({\bf A})+\varpi _n({\bf N}-{\bf A})=1$;
4) $\varpi _n({\bf A}\cap {\bf B})+\varpi _n({\bf A}\cap ({\bf N}-{\bf B
))=\varpi _n({\bf A})$.
{\bf Proof of the Theorem A.1:} From Definitions A.1 and A.2.
{\bf Definition A.3: }If ''$\lim $'' is the Cauchy-Weierstrass ''limit''
then let us denote:
\[
{\bf \Phi ix=}\left\{ {\bf A}\subseteq {\bf N}|\lim_{n\rightarrow \infty
}\varpi _n({\bf A})=1\right\} \mbox{.}
\]
{\bf Theorem A.2: }${\bf \Phi ix}$ is the filter \cite{DVS}, i.e.:
1) ${\bf N}\in {\bf \Phi ix}$,
2) $\emptyset \notin {\bf \Phi ix}$,
3) if ${\bf A}\in {\bf \Phi ix}$ and ${\bf B}\in {\bf \Phi ix}$ then $({\bf
}\cap {\bf B})\in {\bf \Phi ix}$ ;
4) if ${\bf A}\in {\bf \Phi ix}$ and ${\bf A}\subseteq {\bf B}$ then ${\bf B
\in {\bf \Phi ix}$.
{\bf Proof of the Theorem A.2:} From the point 3 of Theorem A.1:
\[
\lim_{n\rightarrow \infty }\varpi _n({\bf N}-{\bf B})=0\mbox{.}
\]
From the point 4 of Theorem A.1:
\[
\varpi _n({\bf A}\cap ({\bf N}-{\bf B}))\leq \varpi _n({\bf N}-{\bf B}
\mbox{.}
\]
Hence,
\[
\lim_{n\rightarrow \infty }\varpi _n\left( {\bf A}\cap ({\bf N}-{\bf B
)\right) =0\mbox{.}
\]
Hence,
\[
\lim_{n\rightarrow \infty }\varpi _n\left( {\bf A}\cap {\bf B}\right)
=\lim_{n\rightarrow \infty }\varpi _n({\bf A})\mbox{.}
\]
In the following text we shall adopt to our topics the definitions and the
proofs of the Robinson Nonstandard Analysis \cite{DVS2}:
{\bf Definition A.4:} The sequences of the real numbers $\left\langle
r_n\right\rangle $ and $\left\langle s_n\right\rangle $ are {\it Q-equivalen
} (denote: $\left\langle r_n\right\rangle \sim \left\langle s_n\right\rangle
$) if
\[
\left\{ n\in {\bf N}|r_n=s_n\right\} \in {\bf \Phi ix}\mbox{.}
\]
{\bf Theorem A.3:} If ${\bf r}$,${\bf s}$,${\bf u}$ are the sequences of the
real numbers then
1) ${\bf r}\sim {\bf r}$,
2) if ${\bf r}\sim {\bf s}$ then ${\bf s}\sim {\bf r}$;
3) if ${\bf r}\sim {\bf s}$ and ${\bf s}\sim {\bf u}$ then ${\bf r}\sim {\bf
u}$.
{\bf Proof of the Theorem A.3:} By Definition A.4 from the Theorem A.2 is
obvious.
{\bf Definition A.5:} {\it The Q-number} is the set of the Q-equivalent
sequences of the real numbers, i.e. if $\widetilde{a}$ is the Q-number and
{\bf r}\in \widetilde{a}$ and ${\bf s}\in \widetilde{a}$, then ${\bf r}\sim
{\bf s};$ and if ${\bf r}\in \widetilde{a}$ and ${\bf r}\sim {\bf s}$ then
{\bf s}\in \widetilde{a}$.
{\bf Definition A.6:} The Q-number $\widetilde{a}$ is {\it the standard
Q-number} $a$ if $a$ is some real number and the sequence $\left\langle
r_n\right\rangle $ exists, for which: $\left\langle r_n\right\rangle \in
\widetilde{a}$ and
\[
\left\{ n\in {\bf N}|r_n=a\right\} \in {\bf \Phi ix}\mbox{.}
\]
{\bf Definition A.7:} The Q-numbers $\widetilde{a}$ and $\widetilde{b}$ are
{\it the equal Q-numbers} (denote: $\widetilde{a}=\widetilde{b}$) if a
\widetilde{a}\subseteq \widetilde{b}$ and $\widetilde{b}\subseteq \widetilde
a}$.
{\bf Theorem A.4: }Let $\mathfrak{f}(x,y,z)$ be a function, which has got the
domain in ${\bf R}\times {\bf R}\times {\bf R}$, has got the range of values
in ${\bf R}$ (${\bf R}$ is the real numbers set).
Let $\left\langle y_{1,n}\right\rangle $ , $\left\langle
y_{2,n}\right\rangle $ , $\left\langle y_{3,n}\right\rangle $ ,
\left\langle z_{1,n}\right\rangle $ , $\left\langle z_{2,n}\right\rangle $ ,
$\left\langle z_{3,n}\right\rangle $ be any sequences of real numbers.
In this case if $\left\langle z_{i,n}\right\rangle \sim \left\langle
y_{i,n}\right\rangle $ then $\left\langle \mathfrak{f}(y_{1,n},y_{2,n},y_{3,n}
\right\rangle \sim \left\langle \mathfrak{f}(z_{1,n},z_{2,n},z_{3,n})\righ
\rangle $.
{\bf Proof of the Theorem A.4:} Let us denote:
if $k=1$ or $k=2$ or $k=3$ then
\[
{\bf A}_k=\left\{ n\in {\bf N}|y_{k,n}=z_{k,n}\right\} \mbox{.}
\]
In this case by Definition A.4 for all $k$:
\[
{\bf A}_k\in {\bf \Phi ix}\mbox{.}
\]
Because
\[
\left( {\bf A}_1\cap {\bf A}_2\cap {\bf A}_3\right) \subseteq \left\{ n\in
{\bf N}|{\mathfrak f}(y_{1,n},y_{2,n},y_{3,n})={\mathfrak f
(z_{1,n},z_{2,n},z_{3,n})\right\} \mbox{,}
\]
then by Theorem A.2:
\[
\left\{ n\in {\bf N}|{\mathfrak f}(y_{1,n},y_{2,n},y_{3,n})={\mathfrak f
(z_{1,n},z_{2,n},z_{3,n})\right\} \in {\bf \Phi ix}\mbox{.}
\]
{\bf Definition A.8:} Let us denote: $Q{\bf R}$ is the set of the Q-numbers.
~
{\bf Definition A.9: }The function $\widetilde{\mathfrak{f}}$, which has got
the domain in $Q{\bf R}\times Q{\bf R}\times Q{\bf R}$, has got the range of
values in $Q{\bf R}$, is {\it the Q-extension of the function} $\mathfrak{f}$,
which has got the domain in ${\bf R}\times {\bf R}\times {\bf R}$, has got
the range of values in ${\bf R}$, if the following condition is accomplished:
Let $\left\langle x_n\right\rangle $ ,$\left\langle y_n\right\rangle $ ,
\left\langle z_n\right\rangle $ be any sequences of real numbers. In this
case: if
$\left\langle x_n\right\rangle \in \widetilde{x}$, $\left\langle
y_n\right\rangle \in \widetilde{y}$, $\left\langle z_n\right\rangle \in
\widetilde{z}$, $\widetilde{u}=\widetilde{\mathfrak{f}}\left( \widetilde{x}
\widetilde{y},\widetilde{z}\right) $,
then
$\left\langle \mathfrak{f}\left( x_n,y_n,z_n\right) \right\rangle \in
\widetilde{u}$.
{\bf Theorem A.5:} For all functions $\mathfrak{f}$, which have the domain in
{\bf R}\times {\bf R}\times {\bf R}$, have the range of values in ${\bf R}$,
and for all real numbers $a$, $b$, $c$, $d$: if $\widetilde{\mathfrak{f}}$ is
the Q-extension of $\mathfrak{f}$; $\widetilde{a}$, $\widetilde{b}$,
\widetilde{c}$, $\widetilde{d}$ are standard Q-numbers $a$, $b$, $c$, $d$,
then:
if $d=\mathfrak{f}(a,b,c)$ then $\widetilde{d}=\widetilde{\mathfrak{f}}(\widetilde
a},\widetilde{b},\widetilde{c})$ and vice versa.
{\bf Proof of the Theorem A.5:} If $\left\langle r_n\right\rangle \in
\widetilde{a}$, $\left\langle s_n\right\rangle \in \widetilde{b}$,
\left\langle u_n\right\rangle \in \widetilde{c}$, $\left\langle {\mathfrak t
_n\right\rangle \in \widetilde{d}$ then by Definition A.6:
\[
\begin{array}{c}
\left\{ n\in {\bf N}|r_n=a\right\} \in {\bf \Phi ix}\mbox{,} \\
\left\{ n\in {\bf N}|s_n=b\right\} \in {\bf \Phi ix}\mbox{,} \\
\left\{ n\in {\bf N}|u_n=c\right\} \in {\bf \Phi ix}\mbox{,} \\
\left\{ n\in {\bf N}|t_n=d\right\} \in {\bf \Phi ix}\mbox{.}
\end{array}
\]
1) Let $d={\mathfrak f}(a,b,c)$.
In this case by Theorem A.2:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(r_n,s_n,u_n)\right\} \in {\bf \Phi ix
\mbox{.}
\]
Hence, by Definition A.4:
\[
\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f}(r_n,s_n,u_n)\righ
\rangle \mbox{.}
\]
Therefore by Definition A.5:
\[
\left\langle {\mathfrak f}(r_n,s_n,u_n)\right\rangle \in \widetilde{d}\mbox{.}
\]
Hence, by Definition A.9:
\[
\widetilde{d}=\widetilde{{\mathfrak f}}(\widetilde{a},\widetilde{b},\widetilde{c
)\mbox{.}
\]
2) Let $\widetilde{d}=\widetilde{{\mathfrak f}}(\widetilde{a},\widetilde{b}
\widetilde{c})$.
In this case by Definition A.9:
\[
\left\langle {\mathfrak f}(r_n,s_n,u_n)\right\rangle \in \widetilde{d}\mbox{.}
\]
Hence, by Definition A.5:
\[
\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f}(r_n,s_n,u_n)\righ
\rangle \mbox{.}
\]
Therefore, by Definition A.4:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(r_n,s_n,u_n)\right\} \in {\bf \Phi ix
\mbox{.}
\]
Hence, by the Theorem A.2:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(r_n,s_n,u_n),r_n=a,s_n=b,u_n=c,t_n=
\right\} \in {\bf \Phi ix}\mbox{.}
\]
Hence, since this set does not empty, then
\[
d={\mathfrak f}(a,b,c)\mbox{.}
\]
By this Theorem: if $\widetilde{\mathfrak{f}}$ is the Q-extension of the
function $\mathfrak{f}$ then the expression ''$\widetilde{\mathfrak{f}}(\widetilde
x},\widetilde{y},\widetilde{z})$'' will be denoted as ''$\mathfrak{f}
\widetilde{x},\widetilde{y},\widetilde{z})$'' and if $\widetilde{u}$ is the
standard Q-number then the expression ''$\widetilde{u}$'' will be denoted as
''$u$''.
{\bf Theorem A.6:} If for all real numbers $a$, $b$, $c$:
\[
\varphi (a,b,c)=\psi (a,b,c)
\]
then for all Q-numbers $\widetilde{x}$, $\widetilde{y}$, $\widetilde{z}$:
\[
\varphi (\widetilde{x},\widetilde{y},\widetilde{z})=\psi (\widetilde{x}
\widetilde{y},\widetilde{z})\mbox{.}
\]
{\bf Proof of the Theorem A.6:} If $\left\langle x_n\right\rangle \in
\widetilde{x}$, $\left\langle y_n\right\rangle \in \widetilde{y}$,
\left\langle z_n\right\rangle \in \widetilde{z}$, $\widetilde{u}=\varphi
\widetilde{x},\widetilde{y},\widetilde{z})$, then by Definition A.9:
\left\langle \varphi (x_n,y_n,z_n)\right\rangle \in \widetilde{u}$.
Because $\varphi (x_n,y_n,z_n)=\psi (x_n,y_n,z_n)$ then $\left\langle \psi
(x_n,y_n,z_n)\right\rangle \in \widetilde{u}$.
If $\widetilde{v}=\psi (\widetilde{x},\widetilde{y},\widetilde{z})$ then by
Definition A.9: $\left\langle \psi (x_n,y_n,z_n)\right\rangle \in \widetilde
v}$, too.
Therefore, for all sequences $\left\langle t_n\right\rangle $ of real
numbers: if $\left\langle t_n\right\rangle \in \widetilde{u}$ then by
Definition A.5: $\left\langle t_n\right\rangle \sim \left\langle \psi
(x_n,y_n,z_n)\right\rangle $.
Hence, $\left\langle t_n\right\rangle \in \widetilde{v}$; and if
\left\langle t_n\right\rangle \in \widetilde{v}$ then $\left\langle
t_n\right\rangle \sim \left\langle \varphi (x_n,y_n,z_n)\right\rangle $;
hence, $\left\langle t_n\right\rangle \in \widetilde{u}$.
Therefore, $\widetilde{u}=\widetilde{v}$.
{\bf Theorem A.7:} If for all real numbers $a$, $b$, $c$:
\[
\mathfrak{f}\left( a,\varphi (b,c)\right) =\psi (a,b,c)
\]
then for all Q-numbers $\widetilde{x}$, $\widetilde{y}$, $\widetilde{z}$:
\[
\mathfrak{f}\left( \widetilde{x},\varphi (\widetilde{y},\widetilde{z})\right)
=\psi (\widetilde{x},\widetilde{y},\widetilde{z})\mbox{.}
\]
{\bf Consequences from Theorems A.6 and A.7:} \cite{DVS3}: For all Q-numbers
$\widetilde{x}$, $\widetilde{y}$, $\widetilde{z}$:
${\bf \Phi }${\bf 1:} $(\widetilde{x}+\widetilde{y})=(\widetilde{y}
\widetilde{x})$,
${\bf \Phi }${\bf 2:} $(\widetilde{x}+(\widetilde{y}+\widetilde{z}))=(
\widetilde{x}+\widetilde{y})+\widetilde{z})$,
${\bf \Phi }${\bf 3:} $(\widetilde{x}+0)=\widetilde{x}$,
${\bf \Phi }${\bf 5:} $(\widetilde{x}\cdot \widetilde{y})=(\widetilde{y
\cdot \widetilde{x})$,
${\bf \Phi }${\bf 6:} $(\widetilde{x}\cdot (\widetilde{y}\cdot \widetilde{z
))=((\widetilde{x}\cdot \widetilde{y})\cdot \widetilde{z})$,
${\bf \Phi 7}${\bf : }$(\widetilde{x}\cdot 1)=\widetilde{x}$,
${\bf \Phi }${\bf 10:} $(\widetilde{x}\cdot (\widetilde{y}+\widetilde{z}))=(
\widetilde{x}\cdot \widetilde{y})+(\widetilde{x}\cdot \widetilde{z}))$.
{\bf Proof of the Theorem A.7:} Let $\left\langle w_n\right\rangle \in
\widetilde{w}$, ${\mathfrak f}(\widetilde{x},\widetilde{w})=\widetilde{u}$,
\left\langle x_n\right\rangle \in \widetilde{x}$, $\left\langle
y_n\right\rangle \in \widetilde{y}$, $\left\langle z_n\right\rangle \in
\widetilde{z}$, $\varphi (\widetilde{y},\widetilde{z})=\widetilde{w}$, $\psi
(\widetilde{x},\widetilde{y},\widetilde{z})=\widetilde{v}$.
By the condition of this Theorem: ${\mathfrak f}(x_n,\varphi (y_n,z_n))=\psi
(x_n,y_n,z_n)$.
By Definition A.9: $\left\langle \psi (x_n,y_n,z_n)\right\rangle \in
\widetilde{v}$, $\left\langle \varphi (x_n,y_n)\right\rangle \in \widetilde{
}$, $\left\langle {\mathfrak f}(x_n,w_n)\right\rangle \in \widetilde{u}$.
For all sequences $\left\langle t_n\right\rangle $ of real numbers:
1) If $\left\langle t_n\right\rangle \in \widetilde{v}$ then by Definition
A.5: $\left\langle t_n\right\rangle \sim \left\langle \psi
(x_n,y_n,z_n)\right\rangle $.
Hence $\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f}(x_n,\varphi
(y_n,z_n))\right\rangle $.
Therefore, by Definition A.4:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(x_n,\varphi \left( y_n,z_n\right)
)\right\} \in {\bf \Phi ix}
\]
and
\[
\left\{ n\in {\bf N}|w_n=\varphi \left( y_n,z_n\right) \right\} \in {\bf
\Phi ix}\mbox{.}
\]
Hence, by Theorem A.2:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(x_n,w_n)\right\} \in {\bf \Phi ix}\mbox{.}
\]
Hence, by Definition A.4:
\[
\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f}(x_n,w_n)\righ
\rangle \mbox{.}
\]
Therefore, by Definition A.5: $\left\langle t_n\right\rangle \in \widetilde{
}$.
2) If $\left\langle t_n\right\rangle \in \widetilde{u}$ then by Definition
A.5: $\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f
(x_n,w_n)\right\rangle $.
Because $\left\langle w_n\right\rangle \sim \left\langle \varphi
(y_n,z_n)\right\rangle $ then by Definition A.4:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(x_n,w_n)\right\} \in {\bf \Phi ix}\mbox{,}
\]
\[
\left\{ n\in {\bf N}|w_n=\varphi \left( y_n,z_n\right) \right\} \in {\bf
\Phi ix}\mbox{.}
\]
Therefore, by Theorem A.2:
\[
\left\{ n\in {\bf N}|t_n={\mathfrak f}(x_n,\varphi \left( y_n,z_n\right)
)\right\} \in {\bf \Phi ix}\mbox{.}
\]
Hence, by Definition A.4:
\[
\left\langle t_n\right\rangle \sim \left\langle {\mathfrak f}(x_n,\varphi
(y_n,z_n))\right\rangle \mbox{.}
\]
Therefore,
\[
\left\langle t_n\right\rangle \sim \left\langle \psi
(x_n,y_n,z_n)\right\rangle \mbox{.}
\]
Hence, by Definition A.5: $\left\langle t_n\right\rangle \in \widetilde{v}$.
From above and from 1) by Definition A.7: $\widetilde{u}=\widetilde{v}$.
{\bf Theorem A.8: }${\bf \Phi }${\bf 4:} For every Q-number $\widetilde{x}$
the Q-number $\widetilde{y}$ exists, for which:
$(\widetilde{x}+\widetilde{y})=0$.
{\bf Proof of the Theorem A.8: }If $\left\langle x_n\right\rangle \in
\widetilde{x}$ then $\widetilde{y}$ is the Q-number, which contains
\left\langle -x_n\right\rangle $.
{\bf Theorem A.9: }${\bf \Phi 9}${\bf :} There is not that $0=1$.
{\bf Proof of the Theorem A.9:} is obvious from Definition A.6 and
Definition A.7.
{\bf Definition A.10:} The Q-number $\widetilde{x}$ is {\it Q-less} than the
Q-number $\widetilde{y}$ (denote: $\widetilde{x}<\widetilde{y}$) if the
sequences $\left\langle x_n\right\rangle $ and $\left\langle
y_n\right\rangle $ of real numbers exist, for which: $\left\langle
x_n\right\rangle \in \widetilde{x}$, $\left\langle y_n\right\rangle \in
\widetilde{y}$ and
\[
\left\{ n\in {\bf N}|x_n<y_n\right\} \in {\bf \Phi ix}\mbox{.}
\]
{\bf Theorem A.10:} For all Q-numbers $\widetilde{x}$, $\widetilde{y}$,
\widetilde{z}$: \cite{DVS4}
${\bf \Omega 1}$: there is not that $\widetilde{x}<\widetilde{x}$;
${\bf \Omega 2}$: if $\widetilde{x}<\widetilde{y}$ and $\widetilde{y}
\widetilde{z}$ then $\widetilde{x}<\widetilde{z}$;
${\bf \Omega 4}$: if $\widetilde{x}<\widetilde{y}$ then $(\widetilde{x}
\widetilde{z})<(\widetilde{y}+\widetilde{z})$;
${\bf \Omega 5}$: if $0<\widetilde{z}$ and $\widetilde{x}<\widetilde{y}$,
then $(\widetilde{x}\cdot \widetilde{z})<(\widetilde{y}\cdot \widetilde{z})$;
${\bf \Omega 3}^{\prime }$: if $\widetilde{x}<\widetilde{y}$ then there is
not, that $\widetilde{y}<\widetilde{x}$ or $\widetilde{x}=\widetilde{y}$ and
vice versa;
${\bf \Omega 3}^{\prime \prime }$: for all standard Q-numbers $x$, $y$, $z$:
$x<y$ or $y<x$ or $x=y$.
{\bf Proof of the Theorem A.10:} is obvious from Definition A.10 by the
Theorem A.2.
{\bf Theorem A.11: }${\bf \Phi }${\bf 8:} If $0<|\widetilde{x}|$ then the
Q-number $\widetilde{y}$ exists, for which $(\widetilde{x}\cdot \widetilde{y
)=1$.
{\bf Proof of the Theorem A.11:} If $\left\langle x_n\right\rangle \in
\widetilde{x}$ then by Definition A.10: if
\[
{\bf A}=\left\{ n\in {\bf N}|0<\left| x_n\right| \right\}
\]
then ${\bf A}\in {\bf \Phi ix}$.
In this case: if for the sequence $\left\langle y_n\right\rangle $ : if
n\in {\bf A}$ then $y_n=1/x_n$
- then
\[
\left\{ n\in {\bf N}|x_n\cdot y_n=1\right\} \in {\bf \Phi ix}\mbox{.}
\]
Thus, Q-numbers are fulfilled to all properties of real numbers, except
\Omega $3 \cite{DVS5}. The property $\Omega $3 is accomplished by some weak
meaning ($\Omega $3' and $\Omega $3'').
{\bf Definition A.11:} The Q-number $\widetilde{x}$ is {\it the
infinitesimal Q-number} if the sequence of real numbers $\left\langle
x_n\right\rangle $ exists, for which: $\left\langle x_n\right\rangle \in
\widetilde{x}$ and for all positive real numbers $\varepsilon $:
\[
\left\{ n\in {\bf N}||x_n|<\varepsilon \right\} \in {\bf \Phi ix}\mbox{.}
\]
Let the set of all infinitesimal Q-numbers be denoted as $I$.
{\bf Definition A.12:} The Q-numbers $\widetilde{x}$ and $\widetilde{y}$ are
t{\it he infinite closed Q-numbers} (denote: $\widetilde{x}\approx
\widetilde{y}$) if $|\widetilde{x}-\widetilde{y}|=0$ or $|\widetilde{x}
\widetilde{y}|$ is infinitesimal.
{\bf Definition A.13}: The Q-number $\widetilde{x}$ is {\it the infinite
Q-number} if the sequence $\left\langle r_n\right\rangle $ of real numbers
exists, for which $\left\langle r_n\right\rangle \in \widetilde{x}$ and for
every natural number $m$:
\[
\left\{ n\in {\bf N}|m<r_n\right\} \in {\bf \Phi ix}\mbox{.}
\]
\subsection{Model}
Let us define the propositional calculus like to (\cite{MEN63}), but the
propositional forms shall be marked by the script greek letters.
{\bf Definition C1: }A set $\Re $ of the propositional forms is{\it \ a
U-world} if:
1) if $\alpha _1,\alpha _2,\ldots ,\alpha _n\in \Re $ and $\alpha _1,\alpha
_2,\ldots ,\alpha _n\vdash \beta $ then $\beta \in \Re $,
2) for all propositional forms $\alpha $: it is not that $(\alpha \& \left(
\neg \alpha \right) )\in \Re $,
3) for every propositional form $\alpha $: $\alpha \in \Re $ or $(\neg
\alpha )\in \Re $.
{\bf Definition C2: }The sequences of the propositional forms $\left\langle
\alpha _n\right\rangle $ and $\left\langle \beta _n\right\rangle $ are {\it
Q-equivalent} (denote: $\left\langle \alpha _n\right\rangle \sim
\left\langle \beta _n\right\rangle $) if
\[
\left\{ n\in {\bf N}|\alpha _n\equiv \beta _n\right\} \in {\bf \Phi ix
\mbox{.}
\]
Let us define the notions of {\it the Q-extension of the functions} for like as
in the Definitions A.5, A.2, A.9, A.5, A.6.
{\bf Definition C3:} The Q-form $\widetilde{\alpha }$ is {\it Q-real} in the
U-world $\Re $ if the sequence $\left\langle \alpha _n\right\rangle $ of the
propositional forms exists, for which: $\left\langle \alpha _n\right\rangle
\in \widetilde{\alpha }$ and
\[
\left\{ n\in {\bf N}|\alpha _n\in \Re \right\} \in {\bf \Phi ix}\mbox{.}
\]
{\bf Definition C4: }The set $\widetilde{\Re }$ of the Q-forms is t{\it he
Q-extension of the U-world }$\Re $ if $\widetilde{\Re }$ is the set of
Q-forms $\widetilde{\alpha }$, which are Q-real in $\Re $.
{\bf Definition C5:} The sequence $\left\langle \widetilde{\Re
_k\right\rangle $ of the Q-extensions is {\it the S-world}.
{\bf Definition C6: }The Q-form $\widetilde{\alpha }$ is {\it S-real in the
S-world }$\left\langle \widetilde{\Re }_k\right\rangle $ if
\[
\left\{ k\in {\bf N}|\widetilde{\alpha }\in \widetilde{\Re }_k\right\} \in
{\bf \Phi ix}\mbox{.}
\]
{\bf Definition C7:} The set ${\bf A}$ (${\bf A}\subseteq {\bf N}$) is {\it
the regular set} if for every real positive number $\varepsilon $ the
natural number $n_0$ exists, for which: for all natural numbers $n$ and $m$,
which are more or equal to $n_0$:
\[
|w_n({\bf A})-w_m({\bf A})|<\varepsilon \mbox{.}
\]
{\bf Theorem C1:} If ${\bf A}$ is the regular set and for all real positive
\varepsilon $:
\[
\left\{ k\in {\bf N}|w_k({\bf A})<\varepsilon \right\} \in {\bf \Phi ix
\mbox{.}
\]
then
\[
\lim_{k\rightarrow \infty }w_k({\bf A})=0\mbox{.}
\]
{\bf Proof of theTheorem C1:} Let be
\[
\lim_{k\rightarrow \infty }w_k({\bf A})\neq 0\mbox{.}
\]
That is the real number $\varepsilon _0$ exists, for which: for every
natural number $n^{\prime }$ the natural number $n$ exists, for which:
\[
n>n^{\prime }\mbox{ and }w_n({\bf A})>\varepsilon _0.
\]
Let $\delta _0$ be some positive real number, for which: $\varepsilon
_0-\delta _0>0$. Because ${\bf A}$ is the regular set then for $\delta _0$
the natural number $n_0$ exists, for which: for all natural numbers $n$ and
m$, which are more or equal to $n_0$:
\[
|w_m({\bf A})-w_n({\bf A})|<\delta _0\mbox{.}
\]
That is
\[
w_m({\bf A})>w_n({\bf A})-\delta _0\mbox{.}
\]
Since $w_n({\bf A})\geq \varepsilon _0$ then $w_m({\bf A})\geq \varepsilon
_0-\delta _0$.
Hence, the natural number $n_0$ exists, for which: for all natural numbers
m $: if $m\geq n_0$ then $w_m({\bf A})\geq \varepsilon _0-\delta _0$.
Therefore,
\[
\left\{ m\in {\bf N}|w_m({\bf A})\geq \varepsilon _0-\delta _0\right\} \in
{\bf \Phi ix}\mbox{.}
\]
and by this Theorem condition:
\[
\left\{ k\in {\bf N}|w_k({\bf A})<\varepsilon _0-\delta _0\right\} \in {\bf
\Phi ix}\mbox{.}
\]
Hence,
\[
\left\{ k\in {\bf N}|\varepsilon _0-\delta _0<\varepsilon _0-\delta
_0\right\} \in {\bf \Phi ix}\mbox{.}
\]
That is $\emptyset \notin {\bf \Phi ix}$. It is the contradiction for the
Theorem 2.2.
{\bf Definition C8:} Let $\left\langle \widetilde{\Re }_k\right\rangle $ be
a S-world.
In this case the function ${\mathfrak W}(\widetilde{\beta })$, which has got the
domain in the set of the Q-forms, has got the range of values in $Q{\bf R}$,
is defined as the following:
If ${\mathfrak W}(\widetilde{\beta })=\widetilde{p}$ then the sequence
\left\langle p_n\right\rangle $ of the real numbers exists, for which:
\left\langle p_n\right\rangle \in \widetilde{p}$ and
\[
p_n=w_n\left( \left\{ k\in {\bf N}|\widetilde{\beta }\in \widetilde{\Re
_k\right\} \right) \mbox{.}
\]
{\bf Theorem C2:} If $\left\{ k\in {\bf N}|\widetilde{\beta }\in \widetilde
\Re }_k\right\} $ is the regular set and ${\mathfrak W}(\widetilde{\beta
)\approx 1$ then $\widetilde{\beta }$ is S-resl in $\left\langle \widetilde
\Re }_k\right\rangle $.
{\bf Proof of the Theorem C2: }Since ${\mathfrak W}(\widetilde{\beta })\approx 1$
then by Definitions.2.12 and 2.11: for all positive real $\varepsilon $:
\[
\left\{ n\in {\bf N}|w_n\left( \left\{ k\in {\bf N}|\widetilde{\beta }\in
\widetilde{\Re }_k\right\} \right) >1-\varepsilon \right\} \in {\bf \Phi ix
\mbox{.}
\]
Hence, by the point 3 of the Theorem 2.1: for all positive real $\varepsilon
$:
\[
\left\{ n\in {\bf N}|\left( {\bf N}-w_n\left( \left\{ k\in {\bf N}
\widetilde{\beta }\in \widetilde{\Re }_k\right\} \right) \right)
<\varepsilon \right\} \in {\bf \Phi ix}\mbox{.}
\]
Therefore, by the Theorem C1:
\[
\lim_{n\rightarrow \infty }\left( {\bf N}-w_n\left( \left\{ k\in {\bf N}
\widetilde{\beta }\in \widetilde{\Re }_k\right\} \right) \right) =0\mbox{.}
\]
That is:
\[
\lim_{n\rightarrow \infty }w_n\left( \left\{ k\in {\bf N}|\widetilde{\beta
\in \widetilde{\Re }_k\right\} \right) =1\mbox{.}
\]
Hence, by Definition.2.3:
\[
\left\{ k\in {\bf N}|\widetilde{\beta }\in \widetilde{\Re }_k\right\} \in
{\bf \Phi ix}\mbox{.}
\]
And by Definition C6: $\widetilde{\beta }$ is S-real in $\left\langle
\widetilde{\Re }_k\right\rangle $.
{\bf Theorem C3: }The P-function exists.
{\bf Proof of the Theorem C3:} By the Theorems C2 and 2.1: ${\mathfrak W}
\widetilde{\beta })$ is the P-function in $\left\langle \widetilde{\Re
_k\right\rangle $.
|
{
"timestamp": "2005-05-20T06:21:32",
"yymm": "0503",
"arxiv_id": "math/0503624",
"language": "en",
"url": "https://arxiv.org/abs/math/0503624"
}
|
\section{Introduction}
\mlabel{sec:intro}
It is well-known that the natural functor from the category of associative
algebras to that of Lie algebras and the adjoint functor
play a fundamental role in the study of
these algebraic structures and their applications.
This paper establishes a similar relationship between Rota-Baxter algebras
and dendriform dialgebras and dendriform trialgebras by using
free Rota-Baxter algebras.
\medskip
A Rota-Baxter algebra is an algebra $A$ with a linear endomorphism $R$ satisfying
the {\bf Rota-Baxter equation}:
\begin{equation}
R(x)R(y) = R\big(R(x)y + xR(y) + \lambda xy\big),\ \forall x,y \in A.
\mlabel{eq:RB}
\end{equation}
Here $\lambda$ is a fixed element in the base ring and is sometimes denoted by
$-\theta$. This equation was introduced by the mathematician Glen E.
Baxter~\mcite{Ba} in 1960 in his probability study, and was popularized
mainly by the work of Gian-Carlo Rota~\mcite{Ro1, Ro2, Ro3} and his school.
Linear operators satisfying equation
(\mref{eq:RB}) in the context of Lie algebras were introduced
independently by Belavin and Drinfeld \mcite{B-D}, and
Semenov-Tian-Shansky~\mcite{STS1} in the 1980s and were related to
solutions, called $r$-matrices, of the (modified) classical
Yang-Baxter equation, named after the physicists Chen-ning Yang
and Rodney Baxter.
Recently, there have been several interesting developments of Rota-Baxter
algebras in theoretical physics and
mathematics, including quantum field
theory~\mcite{C-K1,C-K2}, Yang-Baxter
equations~\mcite{Ag1,Ag2,Ag3}, shuffle
products~\mcite{shuf,G-K1,G-K2},
operads~\mcite{A-L,EF1,prod,Le1,Le2,Le3}, Hopf
algebras~\mcite{A-G-K-O,shuf,EMP07}, combinatorics~\mcite{Gu2} and
number theory~\mcite{shuf,mzv,Gu5,G-Z,MP1,MP2,zhao}. The most prominent of
these is the work~\mcite{C-K1,C-K2} of Connes and Kreimer in their Hopf algebraic
approach to renormalization theory in perturbative quantum field
theory, continued in a series
of papers
\mcite{E-G-G-V,mat,EGfields06,E-G-K2,E-G-K3,egm2006,ek2005,
em2006,EMP07}.
\smallskip
A dendriform dialgebra is a module $D$ with two binary operations
$\prec$ and $\succ$ that satisfy three relations between them (see Eq.~(\mref{eq:dia})).
This concept was introduced by Loday~\mcite{Lo1} in 1995
with motivation from algebraic $K$-theory, and was further
studied in connection with several areas in mathematics and
physics, including operads~\mcite{Lo2}, homology~\mcite{Fra1,Fra2},
Hopf algebras~\mcite{Ch,Hol2,L-R2,Ron,KDF2007}, Lie and Leibniz algebras~\mcite{Fra2},
combinatorics~\mcite{A-S1,A-S2,Fo,L-R1}, arithmetic~\mcite{Lo3} and
quantum field theory~\mcite{Fo,Hol1}.
A few years later Loday and Ronco defined dendriform trialgebras
in their study~\mcite{L-R2} of polytopes and Koszul duality.
Such a structure is a module $T$ equipped with binary operations
$\prec,\succ$ and
$\spr$ that satisfy seven relations that will be recalled in Eq.~(\mref{eq:tri}).
The dendriform dialgebra and trialgebra share the property that the sum of
the binary operations $\prec+\succ$ (for dialgebra) or $\prec+\succ+\, \spr$
(for trialgebra) is associative. Other dendriform algebra structures
have the similar property of ``splitting associativity" in the sense that
an associative
product decomposes into a linear combination of several binary operations.
Many such structures have been obtained lately, such as
the quadri-algebra of Loday and Aguiar~\mcite{A-L} and
the ennea- and NS-algebra of Leroux~\mcite{Le1,Le2}. In \mcite{prod}
(see also \mcite{Lo4}),
we showed how these more complex structures, equipped with large
numbers of compositions and relations, can be derived from an
operadic point of view in terms of products.
Further examples and developments can be found in~\mcite{unit,Lo2}.
\smallskip
The first link between Rota-Baxter algebras and dendriform
algebras was given by Aguiar~\mcite{Ag1} who showed that a
Rota-Baxter algebra of weight $\lambda=0$ carries a dendriform
dialgebra structure, resembling the Lie algebra structure on an associative
algebra. This has been extended to
further connections between linear operators and dendriform type
algebras~\mcite{EF1,Le2,A-L,prod}, in particular to dendriform trialgebras by
the first named author. See Theorem~\mref{thm:EFs} for details.
Consequently, there are natural functors from the category of
Rota-Baxter algebras of weight $\lambda$ to the categories of
dendriform dialgebras and trialgebras. We study the adjoint functors in this paper.
\smallskip
As a preparation, we first construct in Section~\mref{sec:nonua}
free Rota-Baxter algebras (Theorem~\mref{thm:freeao}) which play a central role in the study
of the adjoint functors. This is in analogy to the central role played by
the free associative algebras in the study of the adjoint functor
from the category of Lie algebras to the category of associative
algebras.
As we will see, free Rota-Baxter algebras can be defined in various
generalities, such as over a set or over another algebra, in various
contexts, such as unitary or nonunitary algebras, and they can be
constructed in various terms, such as by words or by trees, either explicitly
or recursively. For the purpose of our application to adjoint functors,
we only consider a special case of free Rota-Baxter algebras, namely
free nonunitary Rota-Baxter algebras $\ncshao(A)$ generated by another algebra $A$ that
possesses a basis over the base ring. Further studies of free Rota-Baxter algebras can be found in~\mcite{A-M,free,EMP07,Gu6,GK3,G-S}.
Then in Section \mref{sec:adj},
we use these free Rota-Baxter algebras to
obtain adjoint functors of the functors from Rota-Baxter algebras
to dendriform dialgebras (Theorem~\mref{thm:envdend}) and trialgebras (Theorem~\mref{thm:env}) by proving the existence of the corresponding universal enveloping Rota-Baxter algebras. In the case of dendriform trialgebras,
let $D=(D,\prec,\succ,\spr)$ be a dendriform trialgebra. Let $\ncshao(D)$ be the free nonunitary
Rota-Baxter algebra over the nonunitary algebra $(D,\spr)$ constructed in
Theorem~\mref{thm:freeao}. Let $I_R$ be
a suitable Rota-Baxter ideal of $\ncshao(D)$ generated by relations from $\prec$ and $\succ$. Theorem~\mref{thm:envdend} shows that
the quotient Rota-Baxter algebra $\ncshao(D)/I_R$ is the universal enveloping Rota-Baxter algebra of $D$ in the sense of Definition~\mref{de:env}.
The special case of free dendriform algebras is considered in
Section~\mref{sec:dfree} where we realize the free dendriform
dialgebra and trialgebra of Loday and Loday-Ronco
in terms of decorated planar rooted trees as canonical subalgebras of free Rota-Baxter algebras.
\medskip
\noindent
{\bf Notations:}
In this paper, $\bfk$ is a commutative unitary ring which will be
further assumed to be a field in Sections~\mref{sec:adj} and
\mref{sec:dfree}. Let $\Alg$ be the category of unitary
$\bfk$-algebras $A$ whose unit is identified with the unit
$\bfone$ of $\bfk$ by the structure homomorphism $\bfk\to A$. Let
$\Algo$ be the category of nonunitary $\bfk$-algebras. Similarly
let $\RB_\lambda$ (resp. $\RBo_\lambda$) be the category of
unitary (resp. nonunitary) Rota-Baxter $\bfk$-algebras of weight
$\lambda$. The subscript $\lambda$ will be suppressed if there is
no danger of confusion.
\medskip
\noindent
{\bf Acknowledgements:}
We thank M. Aguiar, J.-L. Loday and M. Ronco for helpful
discussions. The first named author was supported by a Ph.D. grant
from the Ev. Studienwerk e.V., and would like to thank the people
at the Theory Department of the Physics Institute at Bonn
University for encouragement and help. The second named author
acknowledges support from NSF grant DMS 0505643 and
a Research Council grant from the
Rutgers University. Both authors acknowledge the warm hospitality
of I.H.\'E.S. (LG) and L.P.T.H.E. (KEF) where this work was
completed.
\section{Free nonunitary Rota-Baxter algebras on an algebra}
\mlabel{sec:nonua}
We now construct free nonunitary \rbas over another nonunitary
algebra. Other than its theoretical significance, our main purpose
is for the application in later sections to
study universal enveloping \rbas of dendriform dialgebras and
trialgebras.
The reader can regard such free \rbas over another algebra as
the Rota-Baxter analog of the tensor algebra over a module.
It is well-known that such tensor algebras are essential in the
study of enveloping algebras of Lie algebras~\mcite{Reu}. Because of the
nonunitariness of Lie algebras, it is the free nonunitary, instead of
unitary, associative algebras
that are used in the study of the adjoint functor from Lie algebra to
associative algebras. For the similar reason, free nonunitary Rota-Baxter
algebras are convenient in the study of the adjoint functor from dendriform
algebras to Rota-Baxter algebras.
As remarked earlier, other cases of free Rota-Baxter algebras are considered
elsewhere~\mcite{free}.
Let $B$ be a nonunitary $\bfk$-algebra. Recall~\mcite{G-K1,G-K2} that
a free nonunitary \rba over $B$ is a nonunitary \rba $\ncshao(B)$ with a Rota-Baxter
operator $R_B$ and a nonunitary algebra homomorphism $j_B: B\to \ncshao(B)$ such that,
for any nonunitary \rba $A$ and any nonunitary algebra
homomorphism $f:B\to A$, there is a unique nonunitary \rba homomorphism
$\free{f}: \ncshao(B)\to A$ such that $\free{f}\circ j_B=f$.
$$ \xymatrix{ B \ar[rr]^{j_B}\ar[drr]^{f} && \ncshao(B) \ar[d]_{\free{f}} \\
&& A}
$$
We assume that the nonunitary algebra $B$ possesses a basis over the
base ring $\bfk$. This is no restriction if the base ring is a field
as is customarily taken to be the case in the study of dendriform
algebras/operads and therefore in our later sections.
We first display a $\bfk$-basis of the free \rba in terms of words
in \S~\mref{ss:base}. The product on the free
\rba is given in~\mref{ss:prodao} and the universal property of
the free \rba is proved in~\mref{ss:proof}.
\subsection{A basis of a free Rota-Baxter algebra as words}
\mlabel{ss:base}
Let $B$ be a nonunitary $\bfk$-algebra with a $\bfk$-basis $X$.
We first display a $\bfk$-basis $\frakX_\infty$ of $\ncshao(B)$ in terms of
words from the alphabet set $X$.
Let $\lc$ and $\rc$ be symbols, called brackets,
and let $X'=X\cup \{\lc,\rc\}$.
Let $M(X')$ be the free semigroup generated by $X'$.
\begin{defn}
Let $Y,Z$ be two subsets of $M(X')$. Define the {\bf alternating product}
of $Y$ and $Z$ to be
\allowdisplaybreaks{
\begin{eqnarray}
\altx(Y,Z)&=&\Big( \bigcup_{r\geq 1} \big (Y\lc Z\rc \big)^r \Big) \bigcup
\Big(\bigcup_{r\geq 0} \big (Y\lc Z\rc \big)^r Y\Big) \notag \\
&& \bigcup \Big( \bigcup_{r\geq 1} \big( \lc Z\rc Y \big )^r \Big)
\bigcup \Big( \bigcup_{r\geq 0} \big (\lc Z\rc Y\big )^r \lc Z\rc \Big).
\mlabel{eq:wordsao}
\end{eqnarray}}
\mlabel{de:alt}
\end{defn}
We construct a sequence $\frakX_n$ of subsets of $M(X')$ by the following
recursion. Let $\frakX_0=X$ and, for $n\geq 0$, define
\allowdisplaybreaks{
\begin{equation}
\frakX_{n+1}=\altx(X,\frakX_n). \notag
\end{equation}
More precisely,
\begin{eqnarray}
\frakX_{n+1}&=& \Big( \bigcup_{r\geq 1} \big (X\lc \frakX_{n}\rc\big )^r \Big) \bigcup
\Big(\bigcup_{r\geq 0} \big (X\lc \frakX_{n}\rc\big )^r X\Big)
\notag \\
&& \bigcup \Big( \bigcup_{r\geq 1} \big (\lc \frakX_{n}\rc X\big )^r \Big)
\bigcup \Big( \bigcup_{r\geq 0} \big( \lc \frakX_{n}\rc X \big)^r
\lc \frakX_{n-1}\rc \Big). \mlabel{eq:x1ao}
\end{eqnarray}
Further, define
\begin{eqnarray}
\frakX_\infty &=& \bigcup_{n\geq 0} \frakX_n = \dirlim \frakX_n. \mlabel{eq:x3ao}
\end{eqnarray}}
Here the second equation in Eq. (\mref{eq:x3ao}) follows since
$\frakX_1\supseteq \frakX_0$ and, assuming $\frakX_n\supseteq \frakX_{n-1}$,
we have
$$\frakX_{n+1}=\altx(X,\frakX_n) \supseteq \altx(X,\frakX_{n-1})
\supseteq \frakX_n.$$
\begin{defn}
A word in $\frakX_\infty$ is called a {\bf (strict) Rota-Baxter (bracketed)
word (RBWs)}.
\mlabel{de:rbw}
\end{defn}
A similar concept of parenthesized words has appeared in the work of
Kreimer~\mcite{Kr1} to represent Hopf algebra structure on Feynman
diagrams in pQFT, with a different set of restrictions on the
words. We use the brackets $\lc$ and $\rc$ instead of $($ and $)$ to avoid
confusion with the
usual meaning of parentheses.
The verification of the following properties of RBWs are quite easy and is left to the reader.
\begin{lemma}
\begin{enumerate}
\item
For each $n\geq 1$, the union of
$\frakX_n=\altx(X,\frakX_{n-1})$ expressed in Eq.(\mref{eq:x1ao}) is
disjoint:
\allowdisplaybreaks{
\begin{eqnarray}
\frakX_n & =&
\Big( \dbigcup_{r\geq 1} \big (X\lc \frakX_{n-1}\rc\big )^r \Big) \dbigcup
\Big(\dbigcup_{r\geq 0} \big (X\lc \frakX_{n-1}\rc\big )^r X\Big) \notag\\
&& \dbigcup \Big( \dbigcup_{r\geq 1} \big (\lc \frakX_{n-1}\rc X\big )^r \Big)
\dbigcup \Big( \dbigcup_{r\geq 0} \big( \lc \frakX_{n-1}\rc X \big)^r
\lc \frakX_{n-1}\rc \Big).
\mlabel{eq:words2}
\end{eqnarray}}
\mlabel{it:disjoint}
\item
We further have the disjoint union
\allowdisplaybreaks{
\begin{eqnarray}
\frakX_\infty & =&
\Big( \dbigcup_{r\geq 1} \big (X\lc \frakX_{\infty}\rc\big )^r \Big) \dbigcup
\Big(\dbigcup_{r\geq 0} \big (X\lc \frakX_{\infty}\rc\big )^r X\Big) \notag\\
&& \dbigcup \Big( \dbigcup_{r\geq 1} \big (\lc \frakX_{\infty}\rc X\big )^r \Big)
\dbigcup \Big( \dbigcup_{r\geq 0} \big( \lc \frakX_{\infty}\rc X \big)^r
\lc \frakX_{\infty}\rc \Big).
\mlabel{eq:words3}
\end{eqnarray}}
\mlabel{it:disjointi}
\item
Every RBW $\frakx\neq \bfone$ has a unique decomposition
\begin{equation}
\frakx=\frakx_1 \cdots \frakx_b,
\mlabel{eq:st}
\end{equation}
where $\frakx_i$, $1\leq i\leq b$, is alternatively in $X$ or in $\lc \frakX_\infty\rc$.
This decomposition will be called the {\bf standard decomposition}
of $\frakx$.
\mlabel{it:st}
\end{enumerate}
\mlabel{lem:ex}
\end{lemma}
For a \rbw $\frakx$ in ${\frakX}_\infty$ with standard decomposition
$\frakx_1 \cdots \frakx_b$,
we define $b$ to be the {\bf breadth} $b(\frakx)$ of $\frakx$, we define
the {\bf head}
$h(\frakx)$ of $\frakx$ to be 0 (resp. 1) if $\frakx_1$ is in $X$
(resp. in $\lc \frakX_\infty \rc$). Similarly define the {\bf tail}
$t(\frakx)$ of $\frakx$ to be 0 (resp. 1) if $\frakx_b$ is in $X$
(resp. in $\lc \frakX_\infty \rc$).
In terms of the decomposition~(\mref{eq:words2}),
the head, tail and breadth of a word $\frakx$
are given in the following table.
\begin{center}
\begin{tabular}{c|c|c|c|c}
$\frakx$ & $(X\lc \frakX_{n-1}\rc)^{r}$
&$(X \lc \frakX_{n-1}\rc)^{ r} X$ &
$(\lc \frakX_{n-1}\rc X)^{ r} $ &
$(\lc \frakX_{n-1}\rc X)^{ r} \lc \frakX_{n-1}\rc$ \\ \hline
$h(\frakx)$& $0$ & 0& 1& 1 \\
$t(\frakx)$ & $1$ & 0 & 0 & 1\\
$b(\frakx)$& $2r$ &$2r+1$& $2r$& $2r+1$
\end{tabular}
\end{center}
Finally, define the {\bf depth} $d(\frakx)$ to be
$$ d(\frakx)=\min \{n\ \big |\ \frakx\in \frakX_n \}.$$
So, in particular, the depth of elements in $X$ is 0 and depth of elements
in $\lc X\rc$ is one.
\begin{exam} For
$x_1,x_2, x_3\in X$, the word $\lc\lc x_1\rc x_2\rc x_3$ has
head 1, tail 0, breadth 2 and depth 2.
\end{exam}
\subsection{The product in a free Rota-Baxter algebra}
\mlabel{ss:prodao}
Let
$$\ncshao(B)=\bigoplus_{\frakx\in \frakX_\infty} \bfk \frakx.$$
We now define a product $\shpr$ on $\ncshao(B)$ by defining
$\frakx\shpr \frakx'\in \ncshao(B)$ for $\frakx,\frakx'\in \frakX_\infty$ and then
extending bilinearly.
Roughly speaking, the product of $\frakx$ and $\frakx'$ is defined
to be the concatenation whenever $t(\frakx)\neq h(\frakx')$. When
$t(\frakx)=h(\frakx')$, the product is defined by the product in
$B$ or by the Rota-Baxter relation in Eq.~(\mref{eq:shprod0}).
To be precise, we use induction on the sum $n:=d(\frakx)+d(\frakx')$
of the depths of $\frakx$ and $\frakx'$.
Then $n\geq 0$.
If $n=0$, then $\frakx,\frakx'$ are in $X$ and so are in $B$ and we
define $\frakx\shpr \frakx'=\frakx \spr \frakx'\in B \subseteq \ncshao(B)$.
Here $\spr$ is the product in $B$.
Suppose $\frakx\shpr \frakx'$ have been defined for all $\frakx,\frakx'\in
\frakX_\infty$ with $n\geq k\geq 0$ and let
$\frakx, \frakx'\in \frakX_\infty$ with $n=k+1$.
First assume the breadth $b(\frakx)=b(\frakx')=1$. Then
$\frakx$ and $\frakx'$ are in $X$ or $\lc \frakX_\infty\rc$. We accordingly
define
\begin{equation}
\frakx\shpr \frakx'=\left \{ \begin{array}{ll}
\frakx \spr \frakx', & {\rm if\ } \frakx,\frakx'\in X,\\
\frakx \frakx', & {\rm if\ } \frakx\in X, \frakx'\in \lc \frakX_\infty\rc,\\
\frakx \frakx', & {\rm if\ } \frakx\in \lc \frakX_\infty\rc, \frakx'\in X,\\
\lc \lc \ox\rc \shpr \ox'\rc +\lc \ox \shpr \lc \ox'\rc \rc
+\lambda \lc \ox \shpr \ox' \rc, & {\rm if\ } \frakx=\lc \ox\rc,
\frakx'=\lc \ox'\rc \in \lc \frakX_\infty \rc.
\end{array} \right .
\mlabel{eq:shprod0}
\end{equation}
Here the product in the first case is the product in $B$, in the second and
third case are by concatenation and in the fourth case is by the induction
hypothesis since for the three products on the right hand side we have
\begin{eqnarray*}
d(\lc\ox \rc)+ d(\ox')
&=& d(\lc \ox \rc)+d(\lc \ox' \rc)-1
= d(\frakx)+d(\frakx')-1,\\
d(\ox)+d(\lc \ox'\rc) &=& d(\lc \ox \rc)+d(\lc \ox'\rc)-1
= d(\frakx)+ d(\frakx')-1,\\
d(\ox)+ d(\ox') &=& d(\lc \ox \rc)-1+ d(\lc \ox' \rc)-1
= d(\frakx)+d(\frakx')-2
\end{eqnarray*}
which are all less than or equal to $k$.
Now assume $b(\frakx)>1$ or $b(\frakx')>1$. Let
$\frakx=\frakx_1\cdots\frakx_b$ and $\frakx'=\frakx'_1\cdots\frakx'_{b'}$
be the standard decompositions from Lemma~\mref{lem:ex}. We then define
\begin{equation}
\frakx \shpr \frakx'= \frakx_1\cdots \frakx_{b-1}(\frakx_b\shpr \frakx'_1)\,
\frakx'_{2}\cdots \frakx'_{b'}
\end{equation}
where $\frakx_b\shpr \frakx'_1$ is defined by Eq.~(\mref{eq:shprod0}) and
the rest is given by concatenation. The concatenation is well-defined since by
Eq.~(\mref{eq:shprod0}), we have $h(\frakx_b)=h(\frakx_b\shpr \frakx'_1)$
and $t(\frakx'_1)=t(\frakx_b\shpr \frakx'_1)$. Therefore,
$t(\frakx_{b-1})\neq h(\frakx_b\shpr \frakx'_1)$ and
$h(\frakx'_2)\neq t(\frakx_b\shpr \frakx'_1)$.
\medskip
We record the following simple properties of $\shpr$ for later applications.
\begin{lemma} Let $\frakx,\frakx'\in \frakX_\infty$. We have the following
statements.
\begin{enumerate}
\item $h(\frakx)=h(\frakx\shpr \frakx')$ and $t(\frakx')=t(\frakx\shpr \frakx')$.
\mlabel{it:mat0}
\item If $t(\frakx)\neq h(\frakx')$, then
$\frakx \shpr \frakx' =\frakx \frakx'$ (concatenation).
\mlabel{it:mat1}
\item If $t(\frakx)\neq h(\frakx')$, then for any $\frakx''\in \frakX_\infty$,
$$(\frakx\frakx')\shpr \frakx'' =\frakx(\frakx' \shpr \frakx''), \quad
\frakx''\shpr (\frakx \frakx') =(\frakx'' \shpr \frakx) \frakx'.$$
\mlabel{it:mat2}
\end{enumerate}
\mlabel{lem:match}
\end{lemma}
Extending $\shpr$ bilinearly, we obtain
a binary operation
$$ \ncshao (B)\otimes \ncshao(B) \to \ncshao(B).$$
For $\frakx\in \frakX_\infty$, define
\begin{equation}
R_B(\frakx)=\lc \frakx \rc.
\mlabel{eq:RBop}
\end{equation}
Obviously $\lc \frakx \rc$ is again in $\frakX_\infty$. Thus $R_B$ extends to
a linear operator $R_B$ on $\ncshao(B)$.
Let
$$j_X:X\to \frakX_\infty \to \ncshao(B)$$
be the natural injection which extends to an algebra injection
\begin{equation}
j_B: B \to \ncshao(B).
\mlabel{eq:jo}
\end{equation}
The following is our first main result which will be proved in the next subsection.
\begin{theorem}
Let $B$ be a nonunitary $\bfk$-algebra with a $\bfk$-basis $X$.
\begin{enumerate}
\item
The pair $(\ncshao(B),\shpr)$ is a nonunitary associative algebra.
\mlabel{it:alg}
\item
The triple $(\ncshao(B),\shpr,R_B)$ is a nonunitary \rba of weight $\lambda$.
\mlabel{it:RB}
\item
The quadruple $(\ncshao(B),\shpr,R_B,j_B)$ is the free nonunitary \rba of
weight $\lambda$ on
the algebra $B$.
\mlabel{it:free}
\end{enumerate}
\mlabel{thm:freeao}
\end{theorem}
The following corollary of the theorem will be used later in the paper.
\begin{coro}
Let $V$ be a $\bfk$-module and let $T(V)=\bigoplus_{n\geq 1} V^{\ot n}$
be the tensor algebra over $V$. Then $\ncshao(T(V))$, together with the
natural injection $i_V: V\to T(V) \xrightarrow{j_{T(V)}} \ncshao(T(V))$,
is a free nonunitary Rota-Baxter algebra over $V$, in the sense that,
for any nonunitary Rota-Baxter algebra $A$ and $\bfk$-module map
$f: V\to A$ there is a unique nonunitary Rota-Baxter algebra homomorphism
$\freev{f}: \ncshao(T(V)) \to A$ such that $k_V \circ \free{f} = f$.
\mlabel{co:vecfree}
\end{coro}
\begin{proof}
The maps in the corollary and in this proof are organized in the following
diagram
$$ \xymatrix{ T \ar[rr]^{k_V} \ar[dd]_{f} \ar[rrdd]^(.4){i_V}
&& T(V) \ar[dd]^{j_{T(V)}} \ar[lldd]_(.7){\free{f}} \\
&& \\
A && \ncshao(T(V)) \ar[ll]_{\freev{f}}}$$
For the given $\bfk$-module $V$, note that $T(V)$, together with the natural
injection $k_V: V\to T(V)$, is the free nonunitary
$\bfk$-algebra over $V$. So for the given $\bfk$-algebra $A$ and $\bfk$-module
map $f: V\to A$, there is a unique nonunitary $\bfk$-algebra homomorphism
$\freea{f}: T(V) \to A$ such that $\freea{f} \circ k_V=f$. Then by the
universal property of the free Rota-Baxter algebra $\ncshao(T(V))$, there is
a unique $\free{\freea{f}}: \ncshao(T(V)) \to A$ such that
$ \free{\freea{f}}\circ j_{T(V)}=\freea{f}$. Since $i_V=j_{T(V)}\circ k_V$,
we have
$ \free{\freea{f}} i_V = \freea{f} \circ k_V=f$.
So we have proved the existence of $\freev{f}=\free{\freea{f}}.$
For the uniqueness of $\freev{f}$. Suppose there is another
$\freev{f}':\ncshao(T(V)) \to A$ such that $ \freev{f}'\circ i_V =f$.
Then we have
$$ \freev{f}' \circ j_{T(V)} \circ k_V = \freev{f}'\circ i_V=f
= \freev{f} \circ i_V = \freev{f} \circ j_{T(V)} \circ k_V.$$
By the universal property of the free algebra $T(V)$, we have
$\freev{f}'\circ j_{T(V)} = \freev{f}\circ j_{T(V)}$.
Then by the universal property of the free Rota-Baxter algebra
$\ncshao(T(V))$, we have
$\freev{f}'=\freev{f}$, as needed.
\end{proof}
\subsection{The proof of Theorem~\mref{thm:freeao}}
\mlabel{ss:proof}
\begin{proof}
\mref{it:alg}. We just need to verify the associativity. For this we only need to verify
\begin{equation}
(\frakx'\shpr \frakx'')\shpr \frakx''' =\frakx'\shpr(\frakx'' \shpr \frakx''')
\mlabel{eq:assx}
\end{equation}
for $\frakx',\frakx'',\frakx'''\in \frakX_\infty$.
We will do this by induction on the sum of the depths
$n:=d(\frakx')+d(\frakx'')+d(\frakx''')$. If $n=0$, then
all of $\frakx',\frakx'',\frakx'''$ have depth zero and so are
in $X$. In this case the product $\shpr$ is given by the product $\spr$
in $B$ and so is associative.
Assume the associativity holds for $n\leq k$ and assume that
$\frakx',\frakx'',\frakx'''\in \frakX_\infty$ have
$n=d(\frakx')+d(\frakx'')+d(\frakx''')=k+1.$
If $t(\frakx')\neq h(\frakx'')$, then by Lemma~\mref{lem:match},
$$ (\frakx' \shpr \frakx'') \shpr \frakx'''=(\frakx'\frakx'')\shpr \frakx'''
= \frakx' (\frakx'' \shpr \frakx''') =\frakx'\shpr (\frakx''\shpr \frakx''').$$
Similarly if $t(\frakx'')\neq h(\frakx''')$.
Thus we only need to verify the associativity when
$t(\frakx')=h(\frakx'')$ and $t(\frakx'')=h(\frakx''')$.
We next reduce the breadths of the words.
\begin{lemma}
If the associativity
$$(\frakx' \shpr \frakx'')\shpr \frakx'''=
\frakx'\shpr (\frakx'' \shpr \frakx''') $$
holds for all $\frakx', \frakx''$ and $\frakx'''$ in $\frakX_\infty$ of breadth one, then
it holds for all $\frakx', \frakx''$ and $\frakx'''$ in $\frakX_\infty$.
\mlabel{lem:ell}
\end{lemma}
\begin{proof}
We use induction on the sum of breadths
$m:=b(\frakx')+b(\frakx'')+b(\frakx''')$.
Then $m\geq 3$. The case when $m=3$ is the assumption of the lemma.
Assume the associativity holds for
$3\leq m \leq j$ and take $\frakx',
\frakx'',\frakx'''\in \frakX_\infty$ with
$m = j+1.$
Then $j+1\geq 4$. So at least one of $\frakx',\frakx'',\frakx'''$ have
breadth greater than or equal to 2.
First assume $b(\frakx')\geq 2$. Then $\frakx'=\frakx'_1\frakx'_2$
with $\frakx'_1,\, \frakx'_2\in \frakX_\infty$ and
$t(\frakx'_1)\neq h(\frakx'_2)$.
Thus
\allowdisplaybreaks{\begin{eqnarray*}
(\frakx'\shpr \frakx'') \shpr \frakx'''&=&
((\frakx'_1\frakx'_2)\shpr \frakx'')\shpr \frakx'''\\
&=& (\frakx'_1 (\frakx'_2 \shpr \frakx''))\shpr \frakx'''
\quad {\rm by\ Lemma~\mref{lem:match}.\mref{it:mat2}}\\
& =& \frakx'_1 ((\frakx'_2 \shpr \frakx'') \shpr \frakx''')
\quad {\rm by\ Lemma~\mref{lem:match}.\mref{it:mat0}\ and\ \mref{it:mat2}}.
\end{eqnarray*}}
Similarly, \allowdisplaybreaks{
\begin{eqnarray*}
\frakx'\shpr (\frakx'' \shpr \frakx''')&=&
(\frakx'_1\frakx'_2)\shpr (\frakx''\shpr \frakx''')\\
&=& \frakx'_1 (\frakx'_2 \shpr (\frakx''\shpr \frakx''')).
\end{eqnarray*}}
Thus $$ (\frakx'\shpr \frakx'') \shpr \frakx'''=
\frakx'\shpr (\frakx'' \shpr \frakx''')$$
whenever
$$ (\frakx'_2 \shpr \frakx'') \shpr \frakx'''=
\frakx'_2 \shpr (\frakx''\shpr \frakx''')$$
which follows from the induction hypothesis.
A similar proof works if $b(\frakx''')\geq 2.$
Finally if $b(\frakx'')\geq 2$, then $\frakx''=\frakx''_1\frakx''_2$
with $\frakx''_1,\,\frakx''_2\in \frakX_\infty$ and $t(\frakx''_1)\neq
h(\frakx''_2)$. So using Lemma~\mref{lem:match} repeatedly, we
have \allowdisplaybreaks{
\begin{eqnarray*}
(\frakx' \shpr \frakx'')\shpr \frakx'''&=&
(\frakx' \shpr (\frakx''_1 \frakx''_2)) \shpr \frakx''' \\
&=& ((\frakx' \shpr \frakx''_1)\frakx''_2)\shpr \frakx'''
\quad {\rm by\ Lemma~\mref{lem:match}.\mref{it:mat0}\ and\ \mref{it:mat2}}\\
&=& (\frakx'\shpr \frakx''_1)(\frakx''_2 \shpr \frakx''')
\quad {\rm by\ Lemma~\mref{lem:match}.\mref{it:mat0}\ and\ \mref{it:mat2}}
\end{eqnarray*}}
In the same way, we have
$$(\frakx'\shpr \frakx''_1)(\frakx''_2 \shpr \frakx''')
= \frakx'\shpr (\frakx'' \shpr \frakx''').$$
This again proves the associativity.
\end{proof}
To summarize, our proof of the associativity
has been reduced to the special case when
$\frakx',\frakx'',\frakx'''\in \frakX_\infty$ are chosen so that
\begin{enumerate}
\item
$n:= d(\frakx')+d(\frakx'')+d(\frakx''')=k+1\geq 1$ with the assumption that
the associativity holds when $n\leq k$.
\mlabel{it:sp1}
\item
the elements are of breadth one and
\mlabel{it:sp2}
\item
$t(\frakx')=h(\frakx'')$ and $t(\frakx'')=h(\frakx''')$.
\mlabel{it:sp3}
\end{enumerate}
By item \mref{it:sp2}, the head and tail of each of the elements are the same.
Therefore by item \mref{it:sp3}, either all the three elements are in $X$
or they are all in $\lc \frakX_\infty \rc$.
If all of $\frakx',\frakx'',\frakx'''$ are in $X$,
then as already shown, the associativity follows from the associativity in $B$.
So it remains to consider $\frakx',\frakx'',\frakx'''$ all in $\lc
\frakX_\infty \rc$.
Then $\frakx'=\lc
\ox'\rc, \frakx''=\lc \ox'' \rc, \frakx'''=\lc \ox'''\rc$ with
$\ox',\ox'',\ox'''\in \frakX_\infty$. Using Eq.~(\mref{eq:shprod0}) and bilinearity
of the product $\shpr$,
we have
\allowdisplaybreaks{\begin{eqnarray*} (\frakx'\shpr \frakx'')\shpr
\frakx''&=& \big \lc \lc \ox'\rc \shpr \ox '' +\ox'\shpr
\lc\ox''\rc
+\lambda\ox'\shpr \ox'' \big \rc \shpr \lc \ox'''\rc \\
&=& \lc\lc \ox'\rc \shpr \ox''\rc \shpr \lc\ox'''\rc
+ \lc\ox'\shpr \lc \ox''\rc \rc\shpr \lc \ox'''\rc
+\lambda \lc \ox'\shpr \ox''\rc \shpr \lc\ox'''\rc \\
&=& \lc\lc\lc \ox'\rc\shpr \ox''\rc\shpr \ox''' \rc
+ \lc\big(\lc\ox'\rc \shpr \ox''\big) \shpr \lc\ox'''\rc\rc
+\lambda \lc\big(\lc\ox'\rc \shpr\ox''\big)\shpr \ox'''\rc\\
&& + \lc\lc\ox'\shpr\lc\ox''\rc\rc \shpr \ox'''\rc
+ \lc\big(\ox'\shpr\lc \ox''\rc\big) \shpr\lc \ox'''\rc\rc
+\lambda \lc\big(\ox'\shpr \lc \ox''\rc \big) \shpr \ox'''\rc \\
&& + \lambda \lc \lc \ox'\shpr \ox''\rc\shpr \ox'''\rc
+\lambda \lc \big(\ox'\shpr \ox''\big)\shpr \lc \ox'''\rc \rc
+ \lambda^2 \lc \big(\ox'\shpr \ox''\big) \shpr \ox'''\rc.
\end{eqnarray*}}
Applying the induction hypothesis in $n$ to the fifth term $\big
(\ox'\shpr\lc \ox''\rc\big) \shpr\lc \ox'''\rc$ and then use
Eq.~(\mref{eq:shprod0}) again, we have \allowdisplaybreaks{
\begin{eqnarray*}
(\frakx'\shpr \frakx'')\shpr \frakx''
&=& \lc\lc\lc \ox'\rc\shpr \ox''\rc\shpr \ox''' \rc
+ \lc\big(\lc\ox'\rc \shpr \ox''\big) \shpr \lc\ox'''\rc\rc
+\lambda \lc\big(\lc\ox'\rc \shpr\ox''\big)\shpr \ox'''\rc\\
&& + \lc\lc\ox'\shpr\lc\ox''\rc\rc \shpr \ox'''\rc
+ \lc\ox'\shpr\lc\lc\ox''\rc\shpr\ox'''\rc\rc
+ \lc\ox' \shpr \lc\ox''\shpr \lc \ox'''\rc\rc\rc \\
&& +\lambda \lc \ox' \shpr \lc\ox''\shpr \ox'''\rc\rc
+\lambda \lc\big(\ox'\shpr \lc \ox''\rc \big) \shpr \ox'''\rc \\
&& + \lambda \lc \lc \ox'\shpr \ox''\rc\shpr \ox'''\rc
+\lambda \lc \big(\ox'\shpr \ox''\big)\shpr \lc \ox'''\rc \rc
+ \lambda^2 \lc \big(\ox'\shpr \ox''\big) \shpr \ox'''\rc.
\end{eqnarray*}}
Similarly we obtain \allowdisplaybreaks{
\begin{eqnarray*} \frakx'
\shpr \big(\frakx''\shpr \frakx'''\big) &=& \lc\ox'\rc \shpr
\Big(\lc\lc\ox''\rc \shpr \ox'''\rc
+ \lc\ox''\shpr \lc \ox'''\rc\rc +\lambda\lc \ox''\shpr\ox'''\rc \Big)\\
&=& \lc\lc\ox'\rc\shpr \big(\lc\ox''\rc\shpr\ox'''\big)\rc
+\lc \ox'\shpr \lc \lc \ox''\rc \shpr \ox'''\rc\rc
+ \lambda \lc \ox'\shpr \big(\lc\ox''\rc\shpr \ox'''\big)\rc\\
&& + \lc\lc \ox'\rc\shpr \big(\ox''\shpr \lc \ox'''\rc \big) \rc
+ \lc \ox' \shpr \lc \ox'' \shpr \lc \ox'''\rc\rc\rc
+ \lambda \lc \ox' \shpr \big(\ox''\shpr \lc \ox'''\rc \big)\rc\\
&& + \lambda\lc\lc\ox'\rc\shpr \big( \ox''\shpr \ox'''\big) \rc
+ \lambda \lc \ox'\shpr \lc \ox''\shpr \ox'''\rc\rc
+ \lambda^2 \lc \ox' \shpr \big( \ox''\shpr\ox'''\big) \rc \Big)\\
&=& \lc\lc\lc\ox'\rc\shpr \ox''\rc\shpr \ox'''\rc
+ \lc \lc \ox'\shpr \lc\ox''\rc\rc \shpr \ox'''\rc
+ \lambda \lc\lc\ox'\shpr\ox''\rc\shpr\ox'''\rc\\
&& +\lc \ox'\shpr \lc \lc \ox''\rc \shpr \ox'''\rc\rc
+ \lambda \lc \ox'\shpr \big(\lc\ox''\rc\shpr \ox'''\big)\rc\\
&& + \lc\lc \ox'\rc\shpr \big(\ox''\shpr \lc \ox'''\rc \big) \rc
+ \lc \ox' \shpr \lc \ox'' \shpr \lc \ox'''\rc\rc\rc
+ \lambda \lc \ox' \shpr \big(\ox''\shpr \lc \ox'''\rc \big)\rc\\
&& + \lambda\lc\lc\ox'\rc\shpr \big( \ox''\shpr \ox'''\big) \rc
+ \lambda \lc \ox'\shpr \lc \ox''\shpr \ox'''\rc\rc
+ \lambda^2 \lc \ox' \shpr \big( \ox''\shpr\ox'''\big) \rc.
\end{eqnarray*}}
Now by induction, the $i$-th term in the expansion of
$(\frakx'\shpr \frakx'')\shpr \frakx'''$ matches with the $\sigma(i)$-th
term in the expansion of $\frakx'\shpr(\frakx'' \shpr \frakx''')$.
Here the permutation $\sigma\in \Sigma_{11}$ is
\begin{equation}
\left ( \begin{array}{c} i\\\sigma(i)\end{array}\right)
= \left ( \begin{array}{ccccccccccc} 1&2&3&4&5&6&7&8&9&10&11\\
1&6&9&2&4&7&10&5&3&8&11\end{array} \right ).
\mlabel{eq:sigma}
\end{equation}
This completes the proof of the first part of Theorem~\mref{thm:freeao}.
\mref{it:RB}. The proof is immediate from the definition
$R_B(\frakx)=\lc \frakx\rc$ and Eq. (\mref{eq:shprod0}).
\mref{it:free}. Let $(A,R)$ be a unitary \rba of weight
$\lambda$. Let $f: B\to A$ be
a nonunitary $\bfk$-algebra morphism. We will construct a $\bfk$-linear map
$\free{f}:\ncsha(B)\to A$ by defining $\free{f}(\frakx)$ for
$\frakx\in \frakX_\infty$.
We achieve this by defining $\free{f}(\frakx)$ for $\frakx\in \frakX_n,\ n\geq 0$,
using induction on $n$.
For $\frakx\in \frakX_0:=X$, define
$\free{f}(\frakx)=f(\frakx).$ Suppose $\free{f}(\frakx)$ has been
defined for $\frakx\in \frakX_n$ and consider $\frakx$ in $\frakX_{n+1}$ which
is, by definition and Eq.~(\mref{eq:words2}), \allowdisplaybreaks{
\begin{eqnarray*} \altx(X,\frakX_{n})& =&
\Big( \dbigcup_{r\geq 1} (X\lc \frakX_{n}\rc)^r \Big) \dbigcup
\Big(\dbigcup_{r\geq 0} (X\lc \frakX_{n}\rc)^r X\Big) \\
&& \dbigcup \Big( \dbigcup_{r\geq 0} \lc \frakX_{n}\rc (X\lc \frakX_{n}\rc)^r \Big)
\dbigcup \Big( \dbigcup_{r\geq 0} \lc \frakX_{n}\rc (X\lc \frakX_{n}\rc)^r X\Big).
\end{eqnarray*}}
Let $\frakx$ be in the first union component
$\dbigcup_{r\geq 1} (X\lc \frakX_{n}\rc)^r$ above.
Then
$$\frakx = \prod_{i=1}^r(\frakx_{2i-1} \lc \frakx_{2i} \rc)$$
for
$\frakx_{2i-1}\in X$ and $\frakx_{2i}\in \frakX_n$, $1\leq i\leq r$.
By the construction of the multiplication $\shpr$ and the Rota-Baxter operator
$R_B$, we have
$$\frakx= \shpr_{i=1}^r(\frakx_{2i-1} \shpr \lc \frakx_{2i}\rc)
= \shpr_{i=1}^r(\frakx_{2i-1} \shpr R_B(\frakx_{2i})).$$
Define
\begin{equation}
\free{f}(\frakx) = \ast_{i=1}^r \big(\free{f}(\frakx_{2i-1})
\ast R\big (\free{f}(\frakx_{2i})) \big).
\mlabel{eq:hom}
\end{equation}
where the right hand side is well-defined by the induction hypothesis.
Similarly define $\free{f}(\frakx)$ if $\frakx$ is in the other union
components.
For any $\frakx\in \frakX_\infty$, we have
$R_B(\frakx)=\lc \frakx\rc\in \frakX_\infty$, and
by definition (Eq. (\mref{eq:hom})) of $\free{f}$, we have
\begin{equation}
\free{f}(\lc \frakx \rc)=R(\free{f}(\frakx)).
\mlabel{eq:hom1-2}
\end{equation}
So $\free{f}$ commutes with
the Rota-Baxter operators.
Combining this equation with Eq.~(\mref{eq:hom}) we see that if
$\frakx=\frakx_1\cdots \frakx_b$ is the standard decomposition of $\frakx$,
then
\begin{equation}
\free{f}(\frakx)=\free{f}(\frakx_1)*\cdots * \free{f}(\frakx_b).
\mlabel{eq:staohom}
\end{equation}
Note that this is the only possible way to define $\free{f}(\frakx)$ in order
for $\free{f}$ to be a Rota-Baxter algebra homomorphism extending $f$.
We remain to prove that the map $\free{f}$ defined in Eq.~(\mref{eq:hom}) is
indeed an algebra homomorphism.
For this we only need to check the multiplicity
\begin{equation}
\free{f} (\frakx \shpr \frakx')=\free{f}(\frakx) \ast \free{f}(\frakx')
\mlabel{eq:hom2}
\end{equation}
for all $\frakx,\frakx'\in \frakX_\infty$.
For this we use induction on the sum of depths
$n:=d(\frakx)+d(\frakx')$. Then $n\geq 0$.
When $n=0$, we have $\frakx,\frakx'\in X$. Then Eq.~(\mref{eq:hom2}) follows from
the multiplicity of $f$. Assume the multiplicity holds for $\frakx,\frakx'
\in \frakX_\infty$ with $n\geq k$ and take $\frakx,\frakx'\in \frakX_\infty$ with
$n=k+1$.
Let $\frakx=\frakx_1\cdots \frakx_b$ and $\frakx'=\frakx'_1\cdots\frakx'_{b'}$
be the standard decompositions.
By Eq.~(\mref{eq:shprod0}),
\begin{align*}
\free{f}(\frakx_b\shpr \frakx'_1)&=
\left \{\begin{array}{ll}
\free{f}(\frakx_b \spr \frakx'_1), & {\rm if\ } \frakx_b,\frakx'_1\in X,\\
\free{f}(\frakx_b \frakx'_1), & {\rm if\ } \frakx_b\in X, \frakx'_1\in \lc \frakX_\infty\rc,\\
\free{f}(\frakx_b \frakx'_1), & {\rm if\ } \frakx_b\in \lc \frakX_\infty\rc,
\frakx'_1\in X,\\
\free{f}\big( \lc \lc \ox_b\rc \shpr \ox'_1\rc +\lc \ox_b \shpr \lc \ox'_1\rc \rc
+\lambda \lc \ox_b \shpr \ox'_1 \rc\big), & {\rm if\ } \frakx_b=\lc \ox_b\rc,
\frakx'_1=\lc \ox'_1\rc \in \lc \frakX_\infty \rc.
\end{array} \right .
\end{align*}
In the first three cases, the right hand side is
$\free{f}(\frakx_b)*\free{f}(\frakx'_1)$ by the definition of $\free{f}$.
In the fourth case, we have, by Eq.~(\mref{eq:hom1-2}), the induction
hypothesis and the Rota-Baxter relation of $R$,
\begin{align*}
&\free{f}\big( \lc \lc \ox_b\rc \shpr \ox'_1\rc
+\lc \ox_b \shpr \lc \ox'_1\rc \rc
+\lambda \lc \ox_b \shpr \ox'_1 \rc\big)\\
=&\free{f}(\lc \lc \ox_b\rc \shpr \ox'_1\rc)
+ \free{f}(\lc \ox_b \shpr \lc \ox'_1\rc \rc)
+\free{f}(\lambda \lc \ox_b \shpr \ox'_1 \rc)\\
=&R(\free{f}(\lc \ox_b\rc \shpr \ox'_1))
+ R(\free{f}(\ox_b \shpr \lc \ox'_1\rc ))
+ \lambda R(\free{f}(\ox_b \shpr \ox'_1 ))\\
=&R(\free{f}(\lc \ox_b\rc)*\free{f}(\ox'_1))
+ R(\free{f}(\ox_b) *\free{f}( \lc \ox'_1\rc ))
+ \lambda R(\free{f}(\ox_b) * \free{f}(\ox'_1) )\\
=&R(R(\free{f}(\ox_b))*\free{f}(\ox'_1))
+ R(\free{f}(\ox_b) *R(\free{f}(\ox'_1)))
+ \lambda R(\free{f}(\ox_b) * \free{f}(\ox'_1) )\\
=& R(\free{f}(\ox_b))*R(\free{f}(\ox'_1))\\
=& \free{f}(\lc \ox_b\rc) * \free{f}(\lc\ox'_1\rc)\\
=& \free{f} (\frakx_b) *\free{f}(\frakx'_1).
\end{align*}
Therefore $\free{f}(\frakx_b\shpr \frakx'_1)=\free{f}(\frakx_b)*\free{f}(\frakx'_1)$.
Then
\begin{align*}
\free{f}(\frakx\shpr \frakx')&=
\free{f}\big(\frakx_1\cdots\frakx_{b-1}(\frakx_b\shpr \frakx'_1)\frakx'_2\cdots
\frakx'_{b'}\big) \\
&= \free{f}(\frakx_1)*\cdots *\free{f}(\frakx_{b-1})*
\free{f}(\frakx_b\shpr \frakx'_1)*\free{f}(\frakx'_2)\cdots
\free{f}(\frakx'_{b'})\\
&= \free{f}(\frakx_1)*\cdots *\free{f}(\frakx_{b-1})*
\free{f}(\frakx_b)* \free{f} (\frakx'_1)*\free{f}(\frakx'_2)\cdots
\free{f}(\frakx'_{b'})\\
&= \free{f}(\frakx)*\free{f}(\frakx').
\end{align*}
This is what we need.
\end{proof}
\section{Universal enveloping algebras of dendriform trialgebras}
\mlabel{sec:adj}
\subsection{Dendriform dialgebras and trialgebras}
\mlabel{sec:dend}
We recall the following definitions.
A dendriform dialgebra~\mcite{Lo1} is a module $D$ with two binary operations
$\prec$ and $\succ$ such that
\begin{eqnarray}
&& (x \prec y) \prec z= x \prec (y\prec z +y \succ z),
(x \succ y ) \prec z= x \succ (y\prec z), \notag \\
&& (x \prec y +x\succ y)\succ z = x \succ (y\succ z)
\mlabel{eq:dia}
\end{eqnarray}
for $x,y,z\in D$.
A dendriform trialgebra~\mcite{L-R2} is
a module $T$ equipped with binary operations $\prec,\succ$ and
$\spr$ that satisfy the relations \allowdisplaybreaks{
\begin{eqnarray}
&&(x\prec y)\prec z=x\prec (y\star z),
(x\succ y)\prec z=x\succ (y\prec z),\notag \\
&&(x\star y)\succ z=x\succ (y\succ z),
(x\succ y)\spr z=x\succ (y\spr z),
\mlabel{eq:tri}\\
&&(x\prec y)\spr z=x\spr (y\succ z),
(x\spr y)\prec z=x\spr (y\prec z),
(x\spr y)\spr z=x\spr (y\spr z). \notag
\end{eqnarray}
Here
$\star=\prec+\succ+\spr.$
The category of dendriform trialgebras $(D,\prec,\succ,\spr)$ is denoted by $\DT$.
Recall that $\spr$, as well as $\star$, is an associative product.
The category $\Dend$ of dendriform dialgebras can be identified with the subcategory
of $\DT$ of objects with $\spr=0$.
These algebras are related to Rota-Baxter algebras by the following theorem.
\begin{theorem} {\bf (Aguiar~\mcite{Ag2}, Ebrahimi-Fard~\mcite{EF1})}
\begin{enumerate}
\item A Rota-Baxter algebra $(A,R)$ of weight zero defines
a dendriform dialgebra $(A,\prec_R,\succ_R)$, where
\begin{equation}
x\prec_R y=xR(y),\ x\succ_R y=R(x)y.
\mlabel{it:ags}
\end{equation}
\item A Rota-Baxter algebra $(A,R)$ of weight $\lambda$ defines a
dendriform trialgebra $(A,\prec_R,\succ_R,\spr_R)$, where
\begin{equation} x\prec_R y=xR(y),\ x\succ_R y=R(x)y, x\spr_R y=\lambda xy.
\mlabel{it:ef1s}
\end{equation}
\item A Rota-Baxter
algebra $(A,R)$ of weight $\lambda$ defines a dendriform dialgebra
$(A,\prec'_R,\succ'_R)$, where
\begin{equation} x\prec'_R y=xR(y) + \lambda xy,\ x\succ'_R y=R(x)y.
\mlabel{it:ef2s}
\end{equation}
\end{enumerate}
\mlabel{thm:EFs}
\end{theorem}
We note that (\mref{it:ef2s}) specializes to (\mref{it:ags}) when $\lambda=0$.
The same can be said of (\mref{it:ef1s}) since when $\lambda=0$, the product
$\spr_R$ is zero and the relations of the trialgebra reduces to the relations
of a dialgebra.
It is easy to see that the maps between objects in the categories $\RBo_\lambda$,
$\Dend$ and $\DT$ in Theorem~\mref{thm:EFs} are
compatible with the morphisms. Thus we obtain functors
$$\cale: \RBo_\lambda \to \DT,\ \calf: \RBo_\lambda \to \Dend.$$
We will study their adjoint functors. The two functors $\cale$ and
$\calf$ are related by the following simple observation.
\begin{prop}
\begin{enumerate}
\item
Let $(D,\prec,\succ,\spr)$ be in $\DT$. Then $(D,\prec',\succ')$ is in $\Dend$.
Here $\prec'=\prec+\spr$ and $\succ'=\succ$.
\mlabel{it:dte}
\item
Let $\calg:\DT \to \Dend$ be the functor obtained from \mref{it:dte}. Then
we have $\calf=\calg\circ \cale$.
\mlabel{it:dtf}
\item Fix a $\lambda\in \bfk$.
If the adjoint functors $\cale': \DT\to \RBo_\lambda$ and
$\calg':\Dend \to \DT$ exist, then the adjoint functor
$\calf':\Dend\to \RBo_\lambda$ exists and
$\calf'=\cale' \circ \calg'$.
\mlabel{it:dtc}
\end{enumerate}
\mlabel{pp:DtT}
\end{prop}
\begin{proof}
\mref{it:dte}
Let $\star' = \prec'+\succ$. Then we have $\star'=\star$. We have
\allowdisplaybreaks{
\begin{eqnarray*}
(a\prec' b)\prec' c &=& (a\spr b+a \prec b)\prec' c \\
&=& (a\spr b+a \prec b)\spr c + (a\spr b+a \prec b)\prec c \\
&=& (a\spr b) \spr c +(a \prec b)\spr c + (a\spr b)\prec c + (a \prec b)\prec c \\
&=& a\spr (b \spr c) +a \spr (b\succ c) + a \spr (b\prec c) + a \prec (b\star c)
\ \ {\rm (by\ Eq.~(\mref{eq:tri}))}\\
&=& a\prec' (b\star' c).
\end{eqnarray*}}
This verifies the first relation for the dendriform dialgebra.
The other two relations are also easy to verify:
$$ (a\succ' b)\succ' c = (a\succ b) \succ c= a \succ (b\star c)=a \succ' (b\star' c).$$
$$ (a\succ' b)\prec' c = (a\succ b)\spr c+(a\succ b)\prec c
=a\succ (b\spr c)+a \succ (b\prec c)=a \succ'(b\prec' c).$$
\mref{it:dtf}
For $(A,R)\in \RBo_\lambda$, by Theorem~\mref{thm:EFs} and item \mref{it:dte},
we have
\begin{eqnarray*}
\calg(\cale((A,R)))&=& \calg((A,\prec_R,\succ_R,\cdot_R))\\
&=& (A,\prec_R+\cdot_R, \succ_R)\\
&=& \calf((A,R)).
\end{eqnarray*}
It is easy to check that the composition is also compatible with the morphisms.
So we get the equality of functors.
\mref{it:dtc}
is standard: for any $C\in \Dend$ and $A\in \RBo_\lambda$, we have
\begin{eqnarray*}
\Hom(C,\calg(\calf(A))) &\cong & \Hom(\calg'(C),\calf(A))\\
&\cong& \Hom(\calf'(\calg'(C)),A).
\end{eqnarray*}
So $\calf'(\calg'(C))=\cale'(C)$.
\end{proof}
\subsection{Universal enveloping Rota-Baxter algebras}
\mlabel{ss:envel}
Motivated by the enveloping algebra of a Lie algebra, we are naturally led
to the following definition.
\begin{defn}
Let $D\in \DT$ (resp. $\Dend$) and let $\lambda\in \bfk$.
A {\bf universal enveloping Rota-Baxter algebra} of weight $\lambda$ of $D$ is
a Rota-Baxter algebra $\rbadj(D):=\rbadj_\lambda(D)\in \RBo_\lambda$ with a morphism
$\rho: D\to \rbadj(D)$
in $\DT$ (resp. $\Dend$) such that for any $A\in \RBo_\lambda$ and morphism
$f:D\to A$ in $\DT$ (resp. $\Dend$), there is a unique
$\den{f}: \rbadj(D)\to A$ in $\RBo_\lambda$ such that $\den{f} \circ \rho =f$.
\mlabel{de:env}
\end{defn}
By the universal property of $\rbadj(D)$, it is unique up to isomorphisms in $\RBo_\lambda$.
\subsection{The existence of enveloping algebras}
We will separately consider the enveloping algebras for dialgebras
and trialgebras.
\subsubsection{The trialgebra case}
Let $D=(D,\prec,\succ,\spr)\in \DT$. Then $(D,\spr)$ is a
nonunitary $\bfk$-algebra. Let $\lambda\in \bfk$
be given. Let $\ncshao(D):=\ncshao_\lambda(D)$ be the free nonunitary
Rota-Baxter algebra over $D$ of weight $\lambda$ constructed in
\S\mref{ss:prodao}. Identify $D$ as a subalgebra of $\ncshao(D)$
by the natural injection $j_D$ in Eq.(\mref{eq:jo}). Let $I_R$ be
the Rota-Baxter ideal of $\ncshao(D)$ generated
by the set
\begin{equation}
\big \{ x\prec y - x\lc y\rc,\; x\succ y - \lc x\rc y\ \big|\
x,y\in D \big\}. \mlabel{eq:gen}
\end{equation}
Here a Rota-Baxter ideal of $\ncshao(D)$ is an ideal $I$ of $\ncshao(D)$ such that
$R_B(I)\subseteq I$, and the Rota-Baxter ideal of $\ncshao(D)$ generated by
a subset of $\ncshao(D)$ is the intersection of all Rota-Baxter ideals of $\ncshao(D)$ that
contain the subset.
Let $\pi: \ncshao(D)\to \ncshao(D)/I_R$ be the quotient map.
\begin{theorem}
The quotient Rota-Baxter algebra $\ncshao(D)/I_R$, together with
$\rho:=\pi \circ j_D$, is the universal enveloping Rota-Baxter algebra of $D$.
\mlabel{thm:env}
\end{theorem}
The theorem provides the adjoint functor $\cale':\DT \to \RBo$ of the
functor $\cale: \RBo\to \DT$.
\begin{proof}
Let $(A,R)\in \RBo_\lambda$. It gives an object in $\DT$ by Theorem~\mref{thm:EFs}
which we still denote by $A$. Let $f:D\to A$ be a morphism
in $\DT$. We will complete the following commutative diagram
\begin{equation}
\xymatrix{
D \ar[rr]^{j_D} \ar[d]_f && \ncshao(D) \ar[d]^\pi \ar@{.>}[dll]_{\free{f}} \\
A && \ncshao(D)/I_R \ar@{.>}[ll]_{\den{f}}
}
\end{equation}
By the freeness of $\ncshao(D)$, there is a morphism
$\free{f}:\ncshao(D) \to A$ in $\RB^0$ such that the upper left
triangle commutes. So for any $x,y\in D$, by Eq. (\mref{eq:hom}), we have
\allowdisplaybreaks{
\begin{eqnarray*}
\free{f}(x\prec y - x\lc y \rc)
&=& \free{f}(x\prec y) - \free{f}(x\lc y\rc) \\
&=&\free{f}(x\prec y)-\free{f}(x)R(\free{f}(y))\\
&=& f(x\prec y) -f(x)R(f(y))\\
&=& f(x\prec y)-f(x)\prec_R f(y)\\
&=& f(x\prec y)-f(x\prec y)=0.
\end{eqnarray*}}
Therefore, $x\prec y - x\lc y\rc$ is in $\ker(\free{f})$. Similarly,
$x\succ y -\lc x\rc y$ is in $\ker(\free{f})$. Thus $I_R$ is in $\ker(\free{f})$
and there is a morphism $\den{f}: \ncshao(D)/I_R\to A$ in $\RBo$ such that
$\free{f}=\den{f} \circ \pi$.
Then
$$ \den{f}\circ \rho = \den{f} \circ \pi \circ j_D=\free{f}\circ j_D=f.$$
This proves the existence of $\den{f}$.
Suppose $\den{f}':\ncshao(D)/I_R \to A$ is a morphism in $\RBo$ such that
$\den{f}'\circ \rho=f$. Then
$$ (\den{f}' \circ \pi)\circ j_D = f = (\den{f}\circ \pi)\circ j_D.$$
By the universal property of the free Rota-Baxter algebra $\ncshao(D)$ over $D$,
we have $\den{f}'\circ \pi = \den{f} \circ \pi$ in $\RBo$. Since $\pi$ is
surjective, we have $\den{f}'=\den{f}$. This proves the uniqueness of $\den{f}$.
\end{proof}
\subsubsection{The dialgebra case}
Now let $D=(D,\prec,\succ)\in \Dend$.
Let $T(D)=\bigoplus_{n\geq 1} D^{\ot n}$ be the tensor product algebra over $D$.
Then $T(D)$ is the free nonunitary algebra generated by the $\bfk$-module $D$~\cite[Prop. II.5.1]{Ka}.
By Corollary~\mref{co:vecfree},
$\ncshao(T(D))$, with the natural injection
$i_D: D\to T(D) \to \ncshao(T(D))$, is the free Rota-Baxter
algebra over the vector space $D$.
Let $J_R$ be the Rota-Baxter ideal of $\ncshao(T(D))$ generated
by the set
\begin{equation}
\big \{ x\prec y - x\lc y\rc-\lambda x\ot y,\;
x\succ y - \lc x\rc y\ \big|\ x,y\in D \big\}
\mlabel{eq:gendend}
\end{equation}
Let $\pi: \ncshao(T(D))\to \ncshao(T(D))/J_R$ be the quotient map.
\begin{theorem}
The quotient Rota-Baxter algebra $\ncshao(T(D))/J_R$, together
with $\rho:= \pi \circ i_D$, is the universal
enveloping Rota-Baxter algebra of $D$ of weight $\lambda$.
\mlabel{thm:envdend}
\end{theorem}
\begin{proof}
Let $(A,R)$ be a Rota-Baxter algebra of weight $\lambda$ and let $f:D\to A$ be
a morphism in $\Dend$. More precisely, we have $f:D\to \calg A$ where
$\calg A=(A,\prec_R',\succ_R')$ is the dendriform dialgebra in Theorem~\mref{thm:EFs}.
We will complete the following commutative diagram, using notations from
Corollary~\mref{co:vecfree}.
\begin{equation}
\xymatrix{ & T(D) \ar[rd]^{j_{T(D)}} \ar@{.>}[lddd]^{\freea{f}} & \\
D \ar[rr]^{i_D} \ar[dd]_f \ar[ru]^{k_D} && \ncshao(T(D)) \ar[dd]^\pi \ar@{.>}[ddll]_{\freev{f}} \\
&& \\
A && \ncshao(T(D))/J_R \ar@{.>}[ll]_{\den{f}}
}
\end{equation}
By the universal property of the free algebra $T(D)$ over $D$, there is a
unique morphism $\freea{f}:T(D)\to A$ in $\Algo$ such that
$\freea{f}\circ k_D =f$ and so
$\freea{f}(x_1\ot \cdots \ot x_n)=f(x_1) * \cdots * f(x_n)$.
Here $*$ is the product in $A$.
Then by the universal property of the free Rota-Baxter algebra
$\ncshao(T(D))$ over $T(D)$, there is a unique morphism
$\free{\freea{f}}:\ncshao(T(D)) \to A$ in $\RBo$
such that $\free{\freea{f}}\circ j_{T(D)} =\freea{f}$.
By Corollary~\mref{co:vecfree}, $\free{\freea{f}}=\freev{f}.$
Then
\begin{equation} \freev{f}\circ i_D =\freev{f} \circ j_{T(D)} \circ k_D
= \freea{f} \circ k_D = f.
\mlabel{eq:free2}
\end{equation}
So
for any $x,y\in D$, we have
\begin{eqnarray*}
\freev{f}(x\prec y - x\lc y \rc-\lambda x\ot y)
&=&\freev{f}(x\prec y)-\freev{f}(x)*R(\freev{f}(y))-\lambda \freev{f}(x\ot y)\\
&=&\freev{f}(x\prec y)-\freev{f}(x)*R(\freev{f}(y))-\lambda \freea{f}(x\ot y)\\
&=& f(x\prec y) -f(x)*R(f(y)) -\lambda f(x)* f(y)\\
&=& f(x\prec y)-f(x)\prec_R' f(y)\\
&=& f(x\prec y)-f(x\prec y)=0.
\end{eqnarray*}
Therefore, $x\prec y - x\lc y\rc-\lambda x\ot y$ is in $\ker(\freev{f})$. Similarly,
$x\succ y -\lc x\rc y$ is in $\ker(\freev{f})$. Thus $J_R$ is in $\ker(\freev{f})$
and there is a morphism $\den{f}: \ncshao(T(D))/J_R\to A$ in $\RBo$ such that
$\freev{f}=\den{f} \circ \pi$.
Then by the definition of $\rho=\pi \circ i_D$ in the theorem and
Eq. (\mref{eq:free2}), we have
$$ \den{f}\circ \rho = \den{f} \circ \pi \circ i_D=\freev{f}\circ i_D=f.$$
This proves the existence of $\den{f}$.
Suppose $\den{f}':\ncshao(T(D))/J_R \to A$ is also a morphism in $\RBo$
such that $\den{f}'\circ \rho=f$. Then
$$ (\den{f}' \circ \pi)\circ i_D = f = (\den{f}\circ \pi)\circ i_D.$$
By Corollary~\mref{co:vecfree}, the free Rota-Baxter algebra $\ncshao(T(D))$
over the algebra $T(D)$ is also the free Rota-Baxter algebra over the
vector space $D$ with respect the natural injection $i_D$.
So we have $\den{f}'\circ \pi = \den{f} \circ \pi$ in $\RBo$. Since $\pi$ is
surjective, we have $\den{f}'=\den{f}$. This proves the uniqueness of $\den{f}$.
\end{proof}
\section{Free dendriform di- and trialgebras and free Rota-Baxter algebras}
\mlabel{sec:dfree}
The results in this section can be regarded as more precise forms of results
in \S\mref{sec:adj} in special cases.
Our emphasis here is to interpret free dendriform dialgebras
and free dendriform trialgebras as natural subalgebras of free Rota-Baxter
algebras. This interpretation also suggests a planar tree structure on free Rota-Baxter algebras which will be made precise in~\mcite{free}.
\subsection{The dialgebra case}
\subsubsection{Free dendriform dialgebras}
Let $\bfk$ be a field.
We briefly recall the construction of free dendriform dialgebra $\Dend(V)$
over a $\bfk$-vector space $V$ as colored planar binary trees. For details, see
\mcite{Lo1,Ron}.
Let $X$ be a basis of $V$. For $n\geq 0$, let $Y_n$ be the set of
planar binary trees with $n+1$ leaves and one root such that the
valence of each internal vertex is exactly two. Let $Y_{n,X}$ be
the set of planar binary trees with $n+1$ leaves and with vertices
decorated by elements of $X$. The unique tree with one leave is
denoted by $|$. So we have $Y_0=Y_{0,X}=\{|\}$. Let
$\bfk[Y_{n,X}]$ be the $\bfk$-vector space generated by $Y_{n,X}$.
Here are the first few of them without decoration.
$$Y_0 = \{ \ \vert\ \} ,\qquad \ Y_1 = \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{1tree.eps} \\ \end{array}\Big\} ,
\qquad
Y_2 = \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{2tree.eps}\ , \!\!\includegraphics[scale=0.51]{3tree.eps}\ \\ \end{array} \Big\}
$$
\allowdisplaybreaks{\begin{eqnarray*}
Y_3 &=& \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{5tree.eps},\
\!\!\includegraphics[scale=0.51]{6tree.eps},\ \!\!\includegraphics[scale=0.51]{12tree.eps},\ \ldots
\\ \end{array} \Big\}.
\end{eqnarray*}}
For $T\in Y_{m,X},U\in Y_{n,X}$ and $x\in X$, the grafting of $T$ and $U$ over
$x$ is $T\vee_x U\in Y_{m+n+1,X}$.
Let $\Dend(V)$ be the graded vector space $\bigoplus_{n\geq 1} \bfk[Y_{n,X}]$.
Define binary operations $\prec$ and $\succ$ on $\Dend(V)$ recursively by
\begin{enumerate}
\item
$|\succ T = T\prec |=T$ and $|\prec T = T\succ |=0$ for $T\in Y_{n,X}, n\geq 1$;
\item
For $T=T^\ell\vee_x T^r$ and $U=U^\ell\vee_y U^r$, define
$$ T\prec U= T^\ell \vee_x (T^r\prec U+T^r \succ U),\quad
T\succ U = (T\prec U^\ell+T\succ U^\ell) \vee_y U^r.$$
\end{enumerate}
Since $|\prec |$ and $|\succ |$ is not defined, the binary operations $\prec$ and
$\succ$ are only defined on $\Dend(V)$ though the operation $\star:=\prec +\succ$
can be extended to $H_\LR:=\bfk[Y_0]\oplus \Dend(V)$ by defining $|\star T=T\star |=T.$
By~\mcite{Lo1}} $(\Dend(V),\prec,\succ)$ is the free
dendriform dialgebra over $V$.
\begin{theorem}
Let $V$ be a $\bfk$-vector space. The free dendriform dialgebra over
$V$ is a sub dendriform dialgebra of the free Rota-Baxter algebra
$\ncshao(V)$ of weight zero.
\mlabel{thm:dial}
\end{theorem}
The proof will be given in the next subsection.
\subsubsection{Proof of Theorem~\mref{thm:dial}}
For the given vector space $V$, make $V$ into a $\bfk$-algebra without identity
by given
$V$ the zero product. Let $\ncshao(V)$ be the free nonunitary Rota-Baxter algebra
of weight zero over
$V$ constructed in Theorem~\ref{thm:freeao}. Since $\ncshao(V)$
is a dendriform dialgebra, the natural map $j_V: V\to \ncshao(V)$ extends
uniquely to
a dendriform dialgebra morphism $D(j): \Dend(V)\to \ncshao(V)$.
We will prove that this map is injective and identifies $\Dend(V)$ as a subalgebra
of $\ncshao(V)$ in the category of dendriform dialgebras.
We first define a map
$$\phi: \Dend(V)\to \ncshao(V)$$
and then show in Theorem~\mref{thm:freedend} below that it agrees with
$D(j)$.
We construct $\phi$ by defining $\phi(T)$ for $T\in Y_{n,X}, n\geq 1,$ inductively
on $n$. Any $T\in Y_{n,X}, n\geq 1$ can be uniquely written as
$T=T^\ell \vee_x T^r$ with $x\in X$ and $T^\ell,T^r\in \cup_{0\leq i<n} Y_{i,X}$.
We then define
\begin{equation}
\phi(T)=\left \{\begin{array}{ll}
\lc \phi(T^\ell)\rc x \lc \phi(T^r) \rc, & T^\ell\neq 1, T^r \neq 1,\\
x \lc \phi(T^r) \rc, & T^\ell =1, T^r\neq 1,\\
\lc \phi(T^\ell) \rc x, & T^\ell\neq 1, T^r =1,\\
x, & T^\ell=1,T^r=1.
\end{array} \right .
\end{equation}
For example,
$$
\phi \Big(\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.41]{xtree.eps} \\ \end{array} \Big)= x,
\qquad \phi \Bigg(\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.41]{xyztree.eps} \\ \end{array}\Bigg)
= \lc x \rc z \lc y \rc.
$$ We recall~\mcite{Lo1} that $\Dend(V)$ with the operation
$\star:=\,\prec+\succ$ is an associative algebra.
We now describe a submodule of $\ncshao(V)$ to be identified with the
image of $\phi$ in Theorem~\mref{thm:freedend}.
\begin{defn} {\rm
A $y\in \frakX_\infty$ is called a
{\bf dendriform diword (DW)} if it satisfies the following {\em
additional} properties.
\begin{enumerate}
\item $y$ is not in $\lc \frakX_\infty \rc$;
\item There is no subword $\lc \lc \frakx \rc \rc$ with $\frakx\in \frakX_\infty$
in the word;
\item There is no subword of the form
$\frakx_1\lc \frakx_2 \rc \frakx_3$ with $\frakx_1,\frakx_3\in X$ and
$\frakx_2\in \frakX_\infty$.
\end{enumerate}
We let $DW(V)$ be the subspace of $\ncshao(V)$ generated by the
dendriform diwords. }
\end{defn}
For example
$$ x_0\lc x_1\lc x_2 \rc \rc,
\lc x_0\rc x_1\lc x_2 \rc$$
are dendriform diwords while
$$
\lc \lc x_1\rc \rc, \lc \lc x_1\rc x_2 \lc x_3\rc\rc,
x_1 \lc x_2 \rc x_3 $$
are in $\frakX_\infty$ but not dendriform diwords.
Equivalently, $DW(V)$ can be characterized in terms of the
decomposition (\mref{eq:words3}). For subsets $Y,Z$ of $\frakX_\infty$, define
$$ D(Y,Z)=(Y\lc Z \rc ) \bigcup (\lc Z\rc Y) \bigcup \lc Z\rc Y \lc Z\rc.$$
Then define $D_0(V)=X$ and, for $n\geq 0$, inductively define
\begin{equation}
D_{n+1}(V)=D(X,D_n(V))
= (X\lc D_n(V) \rc ) \bigcup (\lc D_n(V) \rc X) \bigcup \lc D_n(V)\rc X \lc D_n(V)\rc.
\mlabel{eq:dend}
\end{equation}
Then $D_\infty: = \cup_{n\geq 0} D_n(V)$ is the set of dendriform
diwords and $DW(V)=\oplus_{\frakx\in D_\infty} \bfk \frakx.$
Theorem~\mref{thm:dial} follows from the following theorem.
\begin{theorem}
\begin{enumerate}
\item $\phi: \Dend(V)\to \ncshao(V)$ is a homomorphism of
dendriform dialgebras. \mlabel{it:hom}
\item $\phi=D(j)$, the morphism of dendriform dialgebras induced
by $j:V\to \ncshao(V)$. \mlabel{it:agree}
\item $\phi(\Dend(V))=DW(V)$.
\mlabel{it:image}
\item $\phi$ is injective. \mlabel{it:injd}
\end{enumerate}
\mlabel{thm:freedend}
\end{theorem}
\begin{proof}
\mref{it:hom}
we first note that the operations $\prec$ and $\succ$ can be equivalently defined
as follows.
Let $T\in Y_{m,X}, U\in Y_{n,X}$ with $m\geq 1, n\geq 1$. Then
$T=T^\ell \vee_x T^r, U=U^\ell \vee_y U^r$ with $x,y\in X$ and
$T^\ell,T^r,U^\ell,U^r\in \cup_{i\geq 0} Y_{i,X}.$ Define
\begin{eqnarray}
T\prec U: &=& \left \{ \begin{array}{ll}
T^\ell \vee_x (T^r \prec U + T^r \succ U), &{\rm if\ } T^r\neq |,\\
T^\ell \vee_x U,& {\rm if\ } T^r=|.
\end{array} \right .
\\
T\succ U: &=&\left \{ \begin{array}{ll}
(T\prec U^\ell+T\succ U^\ell) \vee_y U^r, & {\rm if\ } U^\ell \neq |,\\
T \vee_y U^r, & {\rm if\ } U^\ell = |.
\end{array} \right .
\end{eqnarray}
Thus we have
\begin{eqnarray*}
\phi(T\prec U) &=& \left \{ \begin{array}{ll}
\phi( T^\ell \vee_x (T^r \prec U + T^r \succ U)), &{\rm if\ } T^r\neq |,\\
\phi(T^\ell \vee_x U), & {\rm if\ } T^r=|.
\end{array} \right . \\
&=& \left \{\begin{array}{ll}
\lc \phi(T^\ell)\rc x \lc \phi(T^r \prec U + T^r \succ U)\rc,
&{\rm if\ } T^r\neq |, T^\ell\neq |,\\
x \lc \phi(T^r \prec U + T^r \succ U)\rc,
&{\rm if\ } T^r\neq |, T^\ell = |,\\
\lc \phi(T^\ell)\rc x \lc \phi(U) \rc, &{\rm if\ } T^r = |, T^\ell\neq |,\\
x \lc \phi(U)\rc, &{\rm if\ } T^r = |, T^\ell = |.
\end{array} \right . \\
&&{\rm(by\ definition\ of\ }\phi{)} \\
&=& \left \{\begin{array}{ll}
\lc \phi(T^\ell)\rc x \lc \phi(T^r) \prec_R \phi(U) + \phi(T^r) \succ_R \phi(U)\rc,
&{\rm if\ } T^r\neq |, T^\ell\neq |,\\
x \lc (\phi(T^r) \prec_R \phi(U) + \phi(T^r) \succ_R \phi(U))\rc,
&{\rm if\ } T^r\neq |, T^\ell = |,\\
\lc \phi(T^\ell)\rc x \lc \phi(U)\rc , &{\rm if\ } T^r = |, T^\ell\neq |,\\
x \lc \phi(U)\rc, &{\rm if\ } T^r = |, T^\ell = |.
\end{array} \right . \\
&&{\rm(by\ induction\ hypothesis)}
\end{eqnarray*}
On the other hand, we have
\begin{eqnarray*}
\phi(T) \prec_R \phi(U)&=& \phi(T^\ell\vee_x T^r) \lc \phi(U) \rc \\
&=& \left \{\begin{array}{ll}
\lc \phi(T^\ell) \rc x \lc \phi(T^r)\rc\lc \phi(U)\rc,
&{\rm\ if\ }T^r\neq |, T^\ell \neq |, \\
x \lc \phi(T^r)\rc \lc \phi(U)\rc, &{\rm if\ } T^r\neq |, T^\ell=|, \\
\lc \phi(T^\ell) \rc x \lc \phi(U) \rc ,&{\rm if\ } T^r=|, T^\ell\neq |,\\
x\lc \phi(U) \rc, & {\rm if\ } T^r=|,T^\ell=|.
\end{array} \right . \\
&& {\rm (by\ definition\ of\ }\phi{)} \\
&=& \left \{\begin{array}{ll}
\lc \phi(T^\ell) \rc x
\big \lc \phi(T^r)\lc \phi(U)\rc +\lc \phi(T^r) \rc \phi(U) \big\rc,
&{\rm\ if\ }T^r\neq |, T^\ell \neq |, \\
x \lc \phi(T^r) \big \lc \phi(U)\rc +\lc \phi(T^r) \rc \phi(U) \big \rc,
&{\rm if\ } T^r\neq |, T^\ell=|, \\
\lc \phi(T^\ell) \rc x \lc \phi(U) \rc ,&{\rm if\ } T^r=|, T^\ell\neq |,\\
x\lc \phi(U) \rc, & {\rm if\ } T^r=|,T^\ell=|.
\end{array} \right . \\
&& {\rm (by\ Rota-Baxter\ relation\ of\ }R(T)=\lc T\rc{)}. \\
\end{eqnarray*}
This proves $\phi(T\prec U)=\phi(T)\prec_R \phi(U)$. We similarly prove
$\phi(T\succ U) =\phi(T)\succ_R \phi(U).$
Thus $\phi$ is a homomorphism in $\Dend$.
\mref{it:agree} follows from the uniqueness of the dendriform dialgebra
morphism $\Dend(V)\to \ncshao(V)$ extending the map
$j_V:V\to \ncshao(V)$.
\mref{it:image} We only need to prove $DW(V)\subseteq \phi(\Dend(V))$
and $\phi(\Dend(V)) \subseteq DW(V)$. To prove the former, we prove
$D_n\subseteq \phi(\Dend(V))$ by induction on $n$.
When $n=0$, $D_n=X$ so the inclusion is clear.
Suppose the inclusion holds for $n$. Then by the definition of $D_{n+1}(V)$
in Eq.~(\mref{eq:dend}), an element of $D_{n+1}(V)$ is of the following three
forms:
i) It is $\frakx \lc \frakx'\rc$ with $\frakx\in X$, $\frakx'\in D_n(V)$.
Then it is $\frakx \prec_R \frakx'$ which is in $\phi(\Dend(V))$ by the induction
hypothesis and the fact that $\phi(\Dend(V))$ is a sub dendriform algebra.
ii) It is $\lc \frakx \rc \frakx'$ with $\frakx\in D_n(V)$ and $\frakx'\in X$.
Then the same proof works.
iii) It is $\lc \frakx\rc \frakx' \lc \frakx''\rc$ with $\frakx,\frakx''\in D_n(V)$
and $\frakx'\in X$. Then it is
$$(\frakx \succ_R \frakx') \prec_R \frakx''=\frakx' \succ_R (\frakx' \prec_R \frakx'').$$
By induction, $\frakx$ and $\frakx''$ are in the sub dendriform dialgebra
$\phi(\Dend(V))$. So the element itself is in $\phi(\Dend(V))$.
The second inclusion follows easily by induction on degrees of trees in
$\Dend(V)$.
\mref{it:injd} By the definition of $\phi$ and part \mref{it:image}, $\phi$ gives
a one-one correspondence between
$\cup_{n\geq 0} Y_{n,X}$ as a basis of $\Dend(V)$ and $DW(V)$ as a basis of
$\phi(\Dend(V))$. Therefore $\phi$ is injective.
\end{proof}
\subsection{The trialgebra case}
\subsubsection{Free dendriform trialgebras}
We describe the construction of free dendriform trialgebra $\DT(V)$
over a vector space $V$ as colored planar trees. For details when $V$ is
of rank one over $\bfk$, see~\mcite{L-R1}.
Let $\Omega$ be a basis of $V$. For $n\geq 0$, let $T_n$ be the set of
planar trees with $n+1$ leaves and one root such that the valence
of each internal vertex is at least two. Let $T_{n,\Omega}$ be the set
of planar trees with $n+1$ leaves and
with vertices {\bf valently decorated} by elements of $\Omega$, in the
sense that if a vertex has valence $k$, then the vertex is
decorated by a vector in $\Omega^{k-1}$. For example the vertex of
\!\!\includegraphics[scale=0.41]{1tree.eps} is decorated by $x\in \Omega$ while the vertex of
\!\!\includegraphics[scale=0.41]{4tree.eps} is decorated by $(x,y)\in \Omega^2.$ The unique tree with one
leaf is denoted by $|$. So we have $T_0=T_{0,\Omega}=\{|\}$. Let
$\bfk[T_{n,\Omega}]$ be the $\bfk$-vector space generated by $T_{n,\Omega}$.
Here are the first few of them without decoration.
$$T_0 = \{ \ \vert\ \} ,\qquad \ T_1 = \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{1tree.eps} \\ \end{array}\Big\} ,
\qquad
T_2 = \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{2tree.eps}\ , \!\!\includegraphics[scale=0.51]{3tree.eps}\ , \!\!\includegraphics[scale=0.51]{4tree.eps}\ \\ \end{array} \Big\}
$$
\allowdisplaybreaks{\begin{eqnarray*}
T_3 &=& \Big\{\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{5tree.eps},\
\!\!\includegraphics[scale=0.51]{6tree.eps},\ \!\!\includegraphics[scale=0.51]{7tree.eps},\ \!\!\includegraphics[scale=0.51]{8tree.eps},\ \!\!\includegraphics[scale=0.51]{9tree.eps},\ \cdots,\ \!\!\includegraphics[scale=0.51]{11tree.eps},\ \!\!\includegraphics[scale=0.51]{12tree.eps},\
\!\!\includegraphics[scale=0.51]{13tree.eps},\ \!\!\includegraphics[scale=0.51]{14tree.eps},\ \ldots \\ \end{array} \Big\}.
\end{eqnarray*}}
For $T^{(i)}\in T_{n_i,\Omega},\, 0\leq i\leq k,$ and $x_i\in \Omega,\, 1\leq i\leq k$,
the grafting of $T^{(i)}$ over $(x_1,\cdots,x_k)$ is
$$T^{(0)}\vee_{x_1} T^{(1)}\vee_{x_2} \cdots \vee_{x_k} T^{(k)}.$$
Any tree can be uniquely expressed as such a grafting of lower degree trees.
For example
$$ \!\!\includegraphics[scale=0.51]{dectree1.eps} \ \ \ \ = | \vee_x | \vee_y |.$$
Let $\DT(V)$ be the graded vector space $\bigoplus_{n\geq 1} \bfk[T_{n,\Omega}]$.
Define binary operations $\prec$, $\succ$ and $\spr$ on $\DT(V)$ recursively by
\begin{enumerate}
\item
$|\succ T = T\prec |=T$, $|\prec T = T\succ |=0$ and
$| \spr T = T\spr |=0$ for $T\in T_{n,\Omega}, n\geq 1$;
\item
For $T=T^{(0)}\vee_{x_1}\cdots \vee_{x_m} T^{(m)}$ and
$U=U^{(0)}\vee_{y_1}\cdots \vee_{y_n} U^{(n)}$, define
\begin{eqnarray*}
T\prec U &=& T^{(0)}\vee_{x_1} \cdots \vee_{x_m} (T^{(m)} \star U),\\
T\succ U &=& (T \star U^{(0)}) \vee_{y_1}\cdots \vee_{y_n} U^{(n)},\\
T\spr U &=& T^{(0)}\vee_{x_1} \cdots \vee_{x_m} (T^{(m)} \star U^{(0)})
\vee_{y_1}\cdots \vee_{y_n} U^{(n)}.
\end{eqnarray*}
\end{enumerate}
Here $\star:=\prec +\succ+\, \spr$
Since $|\prec |$, $|\succ |$ and $|\spr |$ are not defined, the binary operations
$\prec$, $\succ$ and $\spr$ are only defined on $\DT(V)$ though the operation $\star$
can be extended to $H_\DT:=\bfk[T_0]\oplus \DT(V)$ by defining $|\star T=T\star |=T.$
\begin{theorem} $(\DT(V),\prec,\succ,\spr)$ is the free dendriform trialgebra
over $V$.
\mlabel{thm:LR}
\end{theorem}
\begin{proof}
The proof is given by Loday and Ronco in~\mcite{L-R1} when $V$ is of dimension one.
The proof for the general case is the same.
\end{proof}
Our goal is to prove
\begin{theorem}
Let $V$ be a $\bfk$-vector space. The free dendriform trialgebra over
$V$ is a canonical sub-dendriform trialgebra of the free Rota-Baxter algebra
$\ncshao(T(V))$ of weight one.
\mlabel{thm:tri}
\end{theorem}
We restrict the weight of the Rota-Baxter algebra to one to ease the notations.
The proof will be given in the next subsection.
\subsubsection{Proof of Theorem~\mref{thm:tri}}
Let $V$ be the given $\bfk$-vector space with basis $\Omega$.
Let $T(V)=\bigoplus_{n\geq 1} V^{\ot n}$ be the tensor product algebra over $V$.
Then $T(V)$ is the free nonunitary algebra generated by the $\bfk$-space $V$.
A basis of $T(V)$ is $X:=M(\Omega)$, the free semigroup generated
by $\Omega$.
By Theorem~\mref{thm:freeao}, $\ncshao(T(V)):=\ncshao_\bfone
(T(V))$ is the free nonunitary Rota-Baxter algebra over $T(V)$ of
weight $\bfone$ constructed in \S\mref{ss:prodao}.
Since $\ncshao(T(V))$
is a dendriform trialgebra, the natural map $j_V: V\to \ncshao(T(V))$ extends
uniquely to
a dendriform trialgebra morphism $T(j): \DT(V)\to \ncshao(T(V))$.
We will prove that this map is injective and identifies $\DT(V)$ as a subalgebra
of $\ncshao(T(V))$ in the category of dendriform trialgebras.
We first define a map
$$\psi: \DT(V)\to \ncshao(T(V))$$
and then show in Theorem~\mref{thm:freetri} below that it agrees with
$T(j)$.
We construct $\psi$ by defining $\psi(T)$ for $T\in T_{n,\Omega}, n\geq 1,$ inductively
on $n$.
Any $T\in T_{n,\Omega}, n\geq 1$, can be uniquely written as
$T=T^{(0)} \vee_{x_1} \cdots \vee_{x_k} T^{(k)}$ with $x_i\in \Omega$ and
$T^{(i)}\in \cup_{0\leq i<n} T_{i,\Omega}$.
We then define
\begin{equation}
\psi(T)=
\overline{\lc\psi(T^{(0)})\rc} x_1 \overline{\lc\psi(T^{(1)})\rc}
\cdots \overline{\lc\psi(T^{(k-1)})\rc} x_k \overline{\lc \psi(T^{(k)})\rc},
\mlabel{eq:psi}
\end{equation}
where $\overline{\lc\psi(T^{(i)})\rc}=\lc\psi(T^{(i)})\rc$ if
$\psi(T^{(i)}) \neq |$. If $\psi(T^{(i)}) = |$, then the factor
$\lc\psi(T^{(i)})\rc$ is dropped when $i=0$ or $k$, and is replaced by
$\otimes$ when $0<i<k$. For example,
$$
\overline{\lc\psi(|)\rc} x_1 \overline{\lc\psi(T^{(1)})\rc} x_2
\cdots x_k \overline{\lc \psi(T^{(k)})\rc}
= x_1 \overline{\lc\psi(T^{(1)})\rc} x_2
\cdots x_k \overline{\lc \psi(T^{(k)})\rc}$$
and
$$
\overline{\lc\psi(T^{(0)})\rc} x_1 \overline{\lc\psi(|)\rc} x_2
\overline{\lc\psi(T^{(2)})\rc} \cdots x_k \overline{\lc \psi(T^{(k)})\rc}
= \overline{\lc\psi(T^{(0)})\rc} (x_1 \otimes x_2)
\overline{\lc\psi(T^{(2)})\rc}\cdots x_k \overline{\lc \psi(T^{(k)})\rc}.$$
In particular,
$$ \psi \bigg(\ \begin{array}{c} \\[-0.4cm] \!\!\includegraphics[scale=0.51]{dectree1.eps}\ \\ \end{array}\bigg)
=\psi( | \vee_x | \vee_y |) = \overline{\lc\psi(|)\rc} \vee_x
\overline{\lc \psi(|)\rc} \vee_y \overline{\lc \psi(|)\rc} = x
\otimes y.$$
We now describe a submodule of $\ncshao(T(V))$ to be identified with the
image of $\psi$ in Theorem~\mref{thm:freetri}.
\begin{defn} {\rm Let $X=M(\Omega)$.
A $y\in \frakX_\infty$ is called a
{\bf dendriform triword (TW)} if it satisfies the following {\em
additional} properties.
\begin{enumerate}
\item $y$ is not in $\lc \frakX_\infty \rc$;
\item There is no subword $\lc \lc \frakx \rc \rc$ with $\frakx\in \frakX_\infty$
in the word;
\end{enumerate}
We let $TW(V)$ be the subspace of $\ncshao(T(V))$ generated by the
dendriform triwords. }
\end{defn}
For example
$$ x_0\lc x_1\lc x_2 \rc \rc,
\lc x_0\rc x_1\lc x_2 \rc, \lc x_0\rc x_1\lc x_2 \rc x_3 \lc x_4\rc,
x_0\otimes x_1$$
are dendriform triwords while
$$
\lc \lc x_1\rc \rc,
\lc x_1 \lc x_2 \rc x_3\rc $$
are in $\frakX_\infty$ but not dendriform triwords.
Equivalently, TWs can be characterized in terms of the
decomposition (\mref{eq:words3}). For subsets $Y,Z$ of
$\frakX_\infty$, define \allowdisplaybreaks{
\begin{eqnarray}
S(Y,Z)&=&\Big( \bigcup_{r\geq 1} (Y\lc Z\rc)^r \Big) \bigcup
\Big(\bigcup_{r\geq 0} (Y\lc Z\rc)^r Y\Big) \notag \\
&& \bigcup \Big( \bigcup_{r\geq 1} \lc Z\rc (Y\lc Z\rc)^r \Big)
\bigcup \Big( \bigcup_{r\geq 0} \lc Z\rc (Y\lc Z\rc)^r Y\Big).
\mlabel{eq:twords}
\end{eqnarray}}
Then define $S_0(V)=M(X)$. For $n\geq 0$, inductively define
\begin{equation}
S_{n+1}(V)=S(M(X),S_n(V)).
\mlabel{eq:tris}
\end{equation}
Then $S_\infty: = \cup_{n\geq 0} S_n(V)$ is the set of dendriform triwords and
$TW(V)=\oplus_{\frakx\in S_\infty} \bfk \frakx.$
Theorem~\mref{thm:tri} follows from the following theorem.
\begin{theorem}
\begin{enumerate}
\item $\psi: \DT(V)\to \ncshao(T(V))$ is a homomorphism of
dendriform trialgebras. \mlabel{it:homt}
\item $\psi=T(j)$, the
morphism of dendriform trialgebras induced by $j:V\to
\ncshao(T(V))$. \mlabel{it:agreet}
\item $\psi(\DT)=DT(V)$.
\mlabel{it:imaget}
\item $\psi$ is injective. \mlabel{it:injt}
\end{enumerate}
\mlabel{thm:freetri}
\end{theorem}
\begin{proof} The proof is similar to Theorem~\mref{thm:freedend}.
For the lack of a uniform approach for both cases, we give
some details.
\mref{it:homt}
we first note that the operations $\prec$ and $\succ$ can be equivalently defined
as follows without using $|\prec T$, etc.
Let $T\in T_{i,X}, U\in T_{j,X}$ with $i\geq 1, j\geq 1$. Then
$T=T^{(0)}\vee_{x_1}\cdots \vee_{x_m} T^{(m)}$ and
$U=U^{(0)}\vee_{y_1}\cdots \vee_{y_n} U^{(n)}$, define
\begin{eqnarray*}
T\prec U &=& \left \{ \begin{array}{ll}
T^{(0)}\vee_{x_1} \cdots \vee_{x_m} (T^{(m)} \star U), & {\rm\ if\ } T^{(m)}\neq |,\\
T^{(0)}\vee_{x_1} \cdots \vee_{x_m} U, & {\rm\ if\ } T^{(m)}= |
\end{array} \right . \\
T\succ U &=& \left \{ \begin{array}{ll}
(T \star U^{(0)}) \vee_{y_1} \cdots \vee_{y_n} U^{(n)},& {\rm if\ } U^{(0)}\neq |,\\
T \vee_{y_1} \cdots \vee_{y_n} U^{(n)},& {\rm if\ } U^{(0)}= |
\end{array} \right . \\
T\spr U &=& \left \{ \begin{array}{ll}
T^{(0)}\vee_{x_1} \cdots \vee_{x_m} (T^{(m)} \star U^{(0)})
\vee_{y_1} \cdots \vee_{y_n} U^{(n)},& {\rm if\ } T^{(m)}\neq |, U^{(0)}\neq |,\\
T^{(0)}\vee_{x_1} \cdots \vee_{x_m} U^{(0)}
\vee_{y_1} \cdots \vee_{y_n} U^{(n)},& {\rm if\ } T^{(m)}= |, U^{(0)}\neq |,\\
T^{(0)}\vee_{x_1} \cdots \vee_{x_m} T^{(m)}
\vee_{y_1} \cdots \vee_{y_n} U^{(n)},& {\rm if\ } T^{(m)}\neq |, U^{(0)}= |
\end{array} \right .
\end{eqnarray*}
Now we use induction on $i+j$ to prove
\begin{eqnarray}
&&\psi(T\prec U) = \psi(T) \prec_{R} \psi(U), \
\psi(T\succ U) = \psi(T) \succ_{R} \psi(U), \ \\
&&\psi(T\spr U) = \psi(T) \spr_{R} \psi(U).
\mlabel{eq:homt}
\end{eqnarray}
Here $R:=R_{T(V)}$ is the Rota-Baxter operator on $\ncshao(T(V))$.
Since $i+j\geq 2$, we can first take $i+j=2$. Then
$T=|\vee_x |$, $U=|\vee_y |$. So by Eq. (\mref{eq:psi}),
$$ \psi(T\prec U) = \psi( (|\vee_x |) \prec U)
= \psi( | \vee_x U) = x \lc \psi(U)\rc = x \lc y \rc
=x \prec_{R} y.$$
We similarly have $\psi(T\succ U)= x\succ_{R} y$ and
$$ \psi(T\spr U)=\psi((|\vee_x |) \spr (| \vee_y |))
= \psi( |\vee_x | \vee_ y |) = x\ot y = x \spr_{R} y.$$
Assume Equations (\mref{eq:homt}) hold for $T\in T_{i,X},\ U\in T_{j,X}$ with
$i+j\geq k\geq 2$. Then we also have
\begin{eqnarray}
\psi(T\star U)&=&\psi (T\prec U+T\succ U +T\spr U) \notag \\
&=& \psi(T) \prec_{R} \psi(U)+
\psi(T) \succ_{R} \psi(U)+
\psi(T) \spr_{R} \psi(U) \mlabel{eq:start}\\
&=& \psi(T)\star_{R} \psi(U). \notag
\end{eqnarray}
Here $\star_{R}=\prec_{R} +\succ_{R}+\spr_{R}.$ Consider $T,U$
with $m+n=k+1$. We consider two cases of
$T=T^{(0)}\vee_{x_1}\cdots \vee_{x_m} T^{(m)}$. Since $U\neq |$,
we have $\overline{\lc T^{(m)} \star U\rc}=\lc T^{(m)} \star U\rc$
if $T^{(m)}\neq |$, and $\overline{\lc U\rc}=\lc U\rc$ if
$T^{(m)}= |$.
{\bf Case 1.} If $T^{(m)}\neq |$, then
\begin{eqnarray*}
\psi(T\prec U) &=& \psi (T^{(0)}\vee_{x_1} \cdots
\vee_{x_m} (T^{(m)}\star U))
\ \ {\rm (definition\ of\ } \prec {\rm )} \\
&=& \overline{\lc \psi(T^{(0)}) \rc} x_1 \cdots x_m \lc \psi(T^{(m)} \star U)\rc
\ \ {\rm (definition\ of\ } \psi {\rm )} \\
&=& \overline{\lc \psi(T^{(0)}) \rc} x_1 \cdots x_m
\lc \psi(T^{(m)}) \star_R \psi(U)\rc
\ \ {\rm (induction\ hypothesis\ (\mref{eq:start}))} \\
&=& \overline{\lc \psi(T^{(0)}) \rc} x_1 \cdots x_m
\lc \psi(T^{(m)})\rc \lc \psi(U)\rc
\ \ {\rm (relation~(\mref{eq:RB}))} \\
&=& \psi (T^{(0)}\vee_{x_1} \cdots \vee_{x_m}
T^{(m)})\prec_R \psi(U)
\ \ {\rm (defintion\ of\ } \psi {\rm )}\\
&=& \psi(T)\prec_R \psi(U).
\end{eqnarray*}
{\bf Case 2.} If $T^{(m)}=|$, then
\begin{eqnarray*}
\psi(T\prec U) &=& \psi (T^{(0)}\vee_{x_1} \cdots
\vee_{x_m} U)
\ \ {\rm (definition\ of\ } \prec {\rm )} \\
&=& \overline{\lc \psi(T^{(0)}) \rc} x_1 \cdots x_m \lc \psi(U)\rc
\ \ {\rm (definition\ of\ } \psi {\rm )} \\
&=& \psi (T^{(0)}\vee_{x_1} \cdots \vee_{x_m} T^{(m)}) \lc \psi(U)\rc
\ \ {\rm (defintion\ of\ } \psi {\rm )}\\
&=& \psi(T)\prec_R \psi(U).
\end{eqnarray*}
This proves $\psi(T\prec U)=\psi(T)\prec_R \psi(U)$. We similarly prove
$\psi(T\succ U) =\psi(T)\succ_R \psi(U)$ and
$\psi(T\spr U) =\psi(T)\spr_R \psi(U)$.
Thus $\psi$ is a homomorphism in $\DT$.
\mref{it:agreet} follows from the uniqueness of the
morphism $\DT(V)\to \ncshao(T(V))$ of dendriform trialgebra extending the map
$i:V\to \ncshao(T(V))$.
\mref{it:imaget} We only need to prove $TW(V)\subseteq \psi(\DT(V))$
and $\psi(\DT(V)) \subseteq TW(V)$. To prove the former, we prove
$S_n(V)\subseteq \psi(\DT(V))$ by induction on $n$.
When $n=0$, $S_n(V)=X$ so the inclusion is clear.
Suppose the inclusion holds for $1\leq n\leq k$. Then by the definition of
$S_{k+1}(V)$ in Eq.~(\mref{eq:tris}), an element of $S_{k+1}(V)$ has length
greater or equal to 2. We apply induction on its length.
If the length is 2, then it is one of the following two cases.
i) It is $\frakx \lc \frakx'\rc$ with $\frakx\in X$, $\frakx'\in
S_{k}(V)$. Then it is $\frakx \prec_R \frakx'$
which is in $\psi(\DT(V))$ by the induction hypothesis and the
consequence from part \mref{it:homt} that $\psi(\DT(V))$ is a
sub dendriform algebra.
ii) It is $\lc \frakx \rc \frakx'$ with
$\frakx\in S_{k}(V)$ and
$\frakx'\in X$. Then the same proof works.
Suppose all elements of $S_{k+1}$ with length $\leq q$ and $\geq 2$ are in
$\psi(\DT(V))$. Consider an element $\frakx$ of $S_{k+1}$ with length $q+1$.
Then $q+1\geq 3$. If $q+1=3$, we again have two cases.
i) $\frakx=\lc \ox_1\rc \frakx_2 \lc \ox_3\rc$ with $\ox_1,\ox_2\in S_n(V)$
and $\frakx_1\in X$. Then it is
$(\ox_1 \succ_R \frakx_2) \prec_R \ox_3.$
By induction hypothesis on $n$, $\ox_1$ and $\ox_3$ are in the sub dendriform
dialgebra $\psi(\DT(V))$. So the element itself is in $\psi(\DT(V))$.
ii) $\frakx=\frakx_1 \lc \ox_2 \rc \frakx_3$ with $\frakx_1,\frakx_3\in X$
and $\ox_2\in S_n(V)$. Then
$\frakx=\frakx_1 \cdot_R (\ox_2 \succ \frakx_3)$ which is in $\psi(\DT(V))$.
If $q+1\geq 4$, then $\frakx$ can be expressed as the concatenation of
$\frakx_1$ and $\frakx_2$ of lengths at least two and hence are in
$TW(V)$. By induction hypotheses, $\frakx_1$ and $\frakx_2$ are in $\psi(\DT(V))$.
Therefore $\frakx = \frakx_1 \spr_R \frakx_2$ is in $\psi(\DT(V))$.
This completes the proof of the first inclusion. The proof of the second
inclusion follows from a similar induction on the degree of trees in $\DT(V)$.
\mref{it:injt} By the definition of $\psi$ and part \mref{it:image}, $\psi$ gives
a one-one correspondence between
$\cup_{n\geq 0} T_{n,X}$ as a basis of $\DT(V)$ and $TW(V)$ as a basis of
$\psi(\DT(V))$. Therefore $\psi$ is injective.
\end{proof}
|
{
"timestamp": "2007-05-31T19:49:29",
"yymm": "0503",
"arxiv_id": "math/0503647",
"language": "en",
"url": "https://arxiv.org/abs/math/0503647"
}
|
\section{Introduction}
\label{Introduction}
In this article we prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. In the first section we will give the statements of Siegel's and Picard's theorems, and we will recall why these two theorems from such seemingly different areas of mathematics are related. We will then proceed to give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. These include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions. These conjectures appear to be fundamentally new. However, in some special cases we will be able to relate our conjectures to Vojta's conjectures. In this respect, we are also led to formulate a new conjecture relating the absolute discriminant and height of an algebraic point on a projective variety over a number field (Conjecture \ref{conj4}).
We will then summarize our progress on these conjectures. We have been able to get results in all dimensions, with best-possible results in many cases for surfaces. Our technique is based on the new proof of Siegel's theorem given by Corvaja and Zannier in \cite{Co}. They showed how one may use the Schmidt Subspace Theorem to obtain a very simple and elegant proof of Siegel's theorem. More recently, they have used this technique to obtain other results on integral points (see \cite{Co5}, \cite{Co3}, and \cite{Co2}) and Ru has translated the approach to Nevanlinna theory \cite{Ru3}. We will use the Schmidt Subspace Theorem approach to get results on integral points on higher-dimensional varieties, and analogously, we will use a version of Cartan's Second Main Theorem due to Vojta to obtain results on holomorphic curves in higher-dimensional complex varieties, generalizing Picard's theorem.
As an application of our results, we show how to improve a result of Faltings on integral points on the complements of certain singular plane curves, proving a statement about hyperbolicity as well. We end with a discussion of our conjectures, relating them to previously known results and conjectures, and giving examples limiting any improvement to their hypotheses and conclusions.
\section{Theorems of Siegel and Picard}
\label{sclassical}
It has been observed by Osgood, Vojta, Lang, and others that there is a striking correspondence between statements in Nevanlinna theory and in Diophantine approximation (see \cite{Ru} and \cite{Vo2}). This correspondence has been extremely lucrative, influencing results and conjectures in both subjects considerably. The correspondence can be formulated in both a qualitative and quantitative way. In this section, we will concentrate on the simplest case of the qualitative correspondence, Siegel's and Picard's theorems.
Let $V\subset \mathbb{A}^n$ be an affine variety defined over a number field $k$. We will also view $V$ as a complex analytic space. Then it has been noticed that $V(\mathcal{O}_{L,S})$ (the set of points with all coordinates in $\mathcal{O}_{L,S}$, the $S$-integers of $L$) seems to be infinite for sufficiently large number fields $L$ and sets of places $S$ if and only if there exists a non-constant holomorphic map $f:\mathbb{C}\to V$. When $V=C$ is a curve (i.e. one-dimensional variety), this correspondence has been proven to hold exactly, and it is known precisely for which curves $C$ the two statements hold. On the number theory side, Siegel's theorem is the fundamental theorem on integral points on curves. On the analytic side the analogue is a theorem of Picard. We now give the following formulations of these two theorems.
\begin{theorema}[Siegel]
\label{Siegel2}
Let $k$ be a number field. Let $S$ be a finite set of places of $k$ containing the archimedean places. Let $C$ be an affine curve defined over $k$ embedded in affine space $\mathbb{A}^m$. Let $\tilde{C}$ be a projective closure of $C$. If $\# \tilde{C}\backslash C >2$ (over $\overline{k}$) then $C$ has finitely many points in $\mathbb{A}^m(\mathcal{O}_{k,S})$.
\end{theorema}
\begin{theoremb}[Picard]
\label{Picard}
Let $\tilde{C}$ be a compact Riemann surface. Let $C \subset \tilde{C}$. If $\# \tilde{C}\backslash C > 2$, then all holomorphic maps $f:\mathbb{C} \to C$ are constant.
\end{theoremb}
In other words, Siegel's and Picard's theorems state that if $D$ consists of many distinct points on a curve $X$, then any set of integral points on $X\backslash D$ is finite and any holomorphic map $f:\mathbb{C}\to X\backslash D$ is constant. We will thus view as generalizing Siegel's or Picard's theorem any theorem that asserts that if $D$ has ``enough components" then there is some limitation on the integral points on $X\backslash D$ or on the holomorphic maps $f:\mathbb{C}\to X\backslash D$. In Picard's theorem it may also be shown that the curves $C$ in question satisfy the stronger condition of being Kobayashi hyperbolic. We will frequently be able to generalize this fact to higher dimensions as well.
Siegel's theorem is usually stated with the extra information that the $\# \tilde{C}\backslash C >2$ hypothesis is unnecessary for nonrational affine curves $C$. However, it may be shown that this stronger version of Siegel's theorem may be derived from Siegel's theorem as we have stated it by using \'etale coverings of the curve $C$ (see \cite{Co}). A similar statement holds for Picard's theorem. It is Siegel's and Picard's theorems in the form we have given above that we will generalize.
We note that when the geometric genus of $C$ is greater than one, Siegel's theorem follows from the much stronger theorem of Faltings that $C$ has only finitely many $k$-rational points. Similarly, it is a theorem of Picard that there are no nonconstant holomorphic maps $f:\mathbb{C}\to \tilde{C}$ when $\tilde{C}$ is a projective curve of geometric genus greater than one.
\section{Some Preliminary Definitions}
In order to state our conjectures and results we will need a few definitions. In Vojta's Nevanlinna-Diophantine dictionary, the Diophantine object corresponding to a holomorphic map $f:\mathbb{C}\to X\backslash D$ is a set of $(D,S)$-integral points on $X$. We'll now sketch the definition of a set of $(D,S)$-integral points on $X$ in terms of Weil functions.
Let $D$ be a Cartier divisor on a projective variety $X$, both defined over a number field $k$. Let $M_k$ denote the set of places of $k$ (see Section \ref{sDio}). Let $v\in M_k$. Extend $|\cdot|_v$ to an absolute value on $\overline{k}_v$. We define a local Weil function for $D$ relative to $v$ to be a function $\lambda_{D,v}:X(\overline{k}_v)\backslash D \to \mathbb{R}$ such that if $D$ is represented locally by $(f)$ on an open set $U$ then
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{equation*}
\lambda_{D,v}(P)=-\log{|f(P)|_v}+\alpha_v(P)
\end{equation*}
where $\alpha_v$ is a continuous function on $U(\overline{k}_v)$ (in the $v$-topology).
By choosing embeddings $k\to \overline{k}_v$ and $\overline{k}\to \overline{k}_v$, we may also think of $\lambda_{D,v}$ as a function on $X(k)\backslash D$ or $X(\overline{k})\backslash D$. A global Weil functions consists of a collection of local Weil functions, $\lambda_{D,v}$, for $v\in M_k$, where the $\alpha_v$ above satisfy certain reasonable boundedness conditions as $v$ varies. We refer the reader to \cite{La} and \cite{Vo2} for a further discussion of this.
\begin{definition}
Let $D$ be an effective Cartier divisor on a projective variety $X$, both defined over a number field $k$. Let $S$ be a finite set of places in $M_k$ containing the archimedean places. Let $R \subset X(\overline{k})\backslash D$. Then $R$ is defined to be a $(D,S)$-integral set of points if there exists a global Weil function $\lambda_{D,v}$ and constants $c_v$, with $c_v=0$ for all but finitely many $v$, such that for all $v\in M_k\backslash S$ and all embeddings $\overline{k} \to \overline{k}_v$\\
\begin{equation*}
\lambda_{D,v}(P) \leq c_v
\end{equation*}
for all $P$ in $R$.
\end{definition}
We will frequently just say $D$-integral, omitting the reference to $S$, when $S$ has been fixed or when the statement is true for all possible $S$. Except where explicitly stated, we will also require from now on that a set of $D$-integral points be $k$-rational, i.e. $R\subset X(k)$.
For us, the key property of a set of $(D,S)$-integral points is given by the following theorem.
\begin{theorem}
\label{reg}
Let $R\subset X(\overline{k})\backslash D$ be a set of $(D,S)$-integral points on $X$. Then for any regular function $f$ on $X\backslash D$ (defined over $\overline{k}$) there exists a constant $a\in k$ such that $af(P)$ is $S$-integral for all $P$ in $R$, that is $af(P)$ lies in the integral closure of $\mathcal{O}_{k,S}$ in $\overline{k}$ for all $P\in R$.
\end{theorem}
In fact, in what follows, most of our results hold, and our conjectures should hold, for any set $R$ satisfying the conclusion of Theorem \ref{reg}. We will prefer to work with sets of $D$-integral points because they are better geometrically behaved (e.g. under pullbacks) and because they are the right objects to use so that the Diophantine exceptional set we are about to define matches (conjecturally) the holomorphic exceptional set we will define. We note that sets of $D$-integral points are also essentially the same as the sets of scheme-theoretic integral points one would get from working with models of $X\backslash D$ over $\mathcal{O}_{k,S}$ (see \cite[Prop. 1.4.1]{Vo2}).
It will be necessary to define various exceptional sets of a variety.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{definitiona}
Let $X$ be a projective variety and $D$ an effective Cartier divisor on $X$, both defined over a number field $k$. Let $L$ be a number field, $L\supset k$, and $S$ a finite set of places of $L$ containing the archimedean places. We define the Diophantine exceptional set of $X\backslash D$ with respect to $L$ and $S$ to be
\begin{equation*}
\ExcL(X\backslash D)=\bigcup_R \dim_{>0}(\overline{R})
\end{equation*}
where the union runs over all sets $R$ of $L$-rational $(D,S)$-integral points on $X$ and $\dim_{>0}(\overline{R})$ denotes the union of the positive dimensional irreducible components of the Zariski-closure of $R$. We define the absolute Diophantine exceptional set of $X\backslash D$ to be
\begin{equation*}
\Excd(X\backslash D)=\bigcup_{L \supset k,S} \ExcL(X\backslash D),
\end{equation*}
with $L$ ranging over all number fields and $S$ ranging over all sets of places of $L$ as above.
\end{definitiona}
These definitions depend only on $X\backslash D$ and not on the choices of $X$ and $D$.
\begin{definitionb}
Let $X$ be a complex variety. We define the holomorphic exceptional set of $X$, $\Exch(X)$, to be the union of all images of non-constant holomorphic maps $f:\mathbb{C}\to X$.
\end{definitionb}
Conjecturally, it is expected that $\Excd(X\backslash D)=\Exch(X\backslash D)$ (it may also be necessary to take the Zariski-closures of both sides first).
\begin{definitiona}
Let $X$ be a projective variety defined over a number field $k$. Let $D$ be an effective Cartier divisor on $X$. Then we define $X\backslash D$ to be Mordellic if $\Excd(X\backslash D)$ is empty. We define $X\backslash D$ to be quasi-Mordellic if $\Excd(X\backslash D)$ is not Zariski-dense in $X$.
\end{definitiona}
\begin{definitionb}
Let $X$ be a complex variety. We define $X$ to be Brody hyperbolic if $\Exch(X)$ is empty. We define $X$ to be quasi-Brody hyperbolic if $\Exch(X)$ is not Zariski-dense in $X$.
\end{definitionb}
Note that $X$ being quasi-Brody hyperbolic is a stronger condition than the non-existence of holomorphic maps $f:\mathbb{C}\to X$ with Zariski-dense image. Similarly, $X\backslash D$ being quasi-Mordellic is stronger than the non-existence of dense sets of $D$-integral points on $X$.
We will also need a convenient measure of the size of a divisor. We will use $\mathcal{O}_X(D)$, or simply $\mathcal{O}(D)$ when there is no ambiguity, to denote the invertible sheaf associated to a Cartier divisor $D$ on $X$, and $h^i(D)$ to denote the dimension of the vector space $H^i(X,\mathcal{O}(D))$. When $h^0(D)>0$, we will also frequently use the notation $\Phi_D$ to denote the rational map (unique up to projective automorphisms) from $X$ to $\mathbb{P}^{h^0(D)-1}$ corresponding to a basis of $H^0(X,\mathcal{O}(D))$.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{definition}
\label{defk}
Let $D$ be a divisor on a nonsingular projective variety $X$. We define the dimension of $D$ to be the integer $\kappa(D)$ such that there exists positive constants $c_1$ and $c_2$ such that
\begin{equation*}
c_1 n^{\kappa(D)} \leq h^0(nD)\leq c_2 n^{\kappa(D)}
\end{equation*}
for all sufficiently divisible $n>0$. If $h^0(nD)=0$ for all $n>0$ then we let $\kappa(D)=-\infty$.
\end{definition}
Alternatively, if $\kappa(D)\geq 0$, one can show that
\begin{equation*}
\kappa(D)=\max \{\dim \Phi_{nD}(X)|n>0,h^0(nD)>0\}.
\end{equation*}
If $D$ is a Cartier divisor on a singular complex projective variety, we define $\kappa(D)=\kappa(\pi^*D)$ where $\pi:X'\to X$ is a desingularization of $X$. It is easy to show that this is independent of the chosen desingularization. For more properties of $\kappa(D)$ we refer the reader to \cite[Ch. 10]{Ii}.
\begin{definition}
\label{defbig}
We define a Cartier divisor $D$ on $X$ to be quasi-ample (or big) if $\kappa(D)=\dim X$.
\end{definition}
If $D$ is quasi-ample then there exists an $n>0$ such that $\Phi_{nD}$ is birational, justifying the name.
\section{General Setup and Notation}
\label{gsetup}
Throughout this paper we will use the following general setup and notation.\\\\
\textbf {General setup}: Let $X$ be a complex projective variety. Let $D=\sum_{i=1}^r D_i$ be a divisor on $X$ with the $D_i$'s effective Cartier divisors for all $i$. Suppose that at most $m$ $D_i$'s meet at a point, so that the intersection of any $m+1$ distinct $D_i$'s is empty.\\\\
In the Diophantine setting, we will also assume that $X$ and $D$ are defined over a number field $k$ and we let $S$ be a finite set of places of $k$ containing the archimedean places.
From now on, we will freely use the notation $X$, $D$, $D_i$, $r$, $m$, $k$, and $S$ as above without further explanation.
\section{Siegel and Picard-type Conjectures}
In this section we give conjectures generalizing Siegel's theorem and Picard's theorem in various directions.
\subsection{Main Conjectures}
Some special cases of the conjectures given in this section are related to Vojta's Main Conjecture. Later, we will also give conjectures related to Vojta's General Conjecture, hence our terminology in this section and the next. We remind the reader that throughout we are using the general setup of the last section.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{conjecturea}[Main Siegel-type Conjecture]
\label{conjmaina}
Suppose that $\kappa(D_i)\geq \kappa_0>0$ for all $i$. If $r>m+\frac{m}{\kappa_0}$ then there does not exist a Zariski-dense set of $k$-rational $(D,S)$-integral points on $X$.
\end{conjecturea}
\begin{conjectureb}[Main Picard-type Conjecture]
\label{conjmainb}
Suppose that $\kappa(D_i)\geq \kappa_0>0$ for all $i$. If $r>m+\frac{m}{\kappa_0}$ then there does not exist a holomorphic map $f:\mathbb{C} \to X \backslash D$ with Zariski-dense image.
\end{conjectureb}
As mentioned earlier, we will usually just say $D$-integral, omitting $k$ and $S$ from the notation.
Siegel's theorem (resp. Picard's theorem) is the case $m=\kappa_0=\dim X=1$ of Conjecture \ref{conjmaina} (resp. Conjecture \ref{conjmainb}). We note that the dimension of $X$ does not appear in the conjectures, but $\kappa(D_i)$ is bounded by $\dim X$. We will now discuss some consequences and special cases of these conjectures which seem important enough in their own right to be listed separately as new conjectures, and which will sometimes contain extra conjectures (e.g. on the exceptional sets) which do not follow from the main conjectures above. At the two extremes of $\kappa_0$ we have
\begin{conjecturea}
\label{conj1a}
If $\kappa(D_i)>0$ for all $i$ and $r> 2m$ then there does not exist a Zariski-dense set of $D$-integral points on $X$.
\end{conjecturea}
\begin{conjectureb}
\label{conj1b}
If $\kappa(D_i)>0$ for all $i$ and $r>2m$ then there does not exist a holomorphic map $f:\mathbb{C} \to X \backslash D$ with Zariski-dense image.
\end{conjectureb}
\begin{conjecturea}
\label{conj1ab}
If $D_i$ is quasi-ample for all $i$ and $r>m+\frac{m}{\dim X}$ then $X\backslash D$ is quasi-Mordellic.
\end{conjecturea}
\begin{conjectureb}
\label{conj1bb}
If $D_i$ is quasi-ample for all $i$ and $r>m+\frac{m}{\dim X}$ then $X\backslash D$ is quasi-Brody hyperbolic.
\end{conjectureb}
We note that when the $D_i$'s are in some sort of general position, so that $m=\dim X$, the inequalities in the last two conjectures above take the nicer form $r>\dim X +1$. The statements on quasi-Mordellicity and quasi-Brody hyperbolicity do not follow (directly at least) from the Main Conjectures.
Of particular interest is the case where $D_i$ is ample for all $i$. In this case we conjecture very precise bounds on the dimensions of the exceptional sets (see Remark \ref{rbig} for a possible generalization to quasi-ample divisors).
\begin{conjecturea}[Main Siegel-type Conjecture for Ample Divisors]
\label{conj2a}
Suppose that $D_i$ is ample for all $i$.\\\\
(a). If $r>m+\frac{m}{\dim X}$ then $\dim \Excd(X) \leq \frac{m}{r-m}$.\\
(b). In particular, if $r>2m$ then $X\backslash D$ is Mordellic.
\end{conjecturea}
\begin{conjectureb}[Main Picard-type Conjecture for Ample Divisors]
\label{conj2b}
Suppose that $D_i$ is ample for all $i$.\\\\
(a). If $r>m+\frac{m}{\dim X}$ then $\dim \Exch(X) \leq \frac{m}{r-m}$.\\
(b). If $r>2m$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{conjectureb}
It is not hard to show that the Main Conjectures for ample divisors follow from Conjectures \ref{conj1ab} and \ref{conj1bb}.
\subsection{General Conjectures}
We will also consider the situation where the field that the integral points are defined over is allowed to vary over all fields of degree less than or equal to $d$ over some fixed field $k$. So in this section we do not require that the integral points be $k$-rational.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{definition}
Let $R\subset X(\overline{k})$. We define the degree of $R$ over $k$ to be $\deg_k R=\sup_{P\in R} [k(P):k]$.
\end{definition}
Generalizing the Main Siegel-type Conjecture of last section, we conjecture
\begin{conjecture}[General Siegel-type Conjecture]
\label{congen}
Suppose that $\kappa(D_i)\geq \kappa_0>0$ for all $i$. Let $d$ be a positive integer. If $r>m+\frac{m(2d-1)}{\kappa_0}$ then there does not exist a Zariski-dense set of $D$-integral points on $X$ of degree $d$ over $k$.
\end{conjecture}
\noindent We will see later that this conjecture and others in this section are related to Vojta's General Conjecture.
We will also want to define a degree $d$ Diophantine exceptional set for a variety $V$. With the notation from our earlier definition for $\Excd$ we define
\begin{definition}
Let $X$ be a projective variety and $D$ an effective Cartier divisor on $X$, both defined over a number field $k$. Let $L$ be a number field, $L\supset k$, and $S$ a finite set of places of $L$ containing the archimedean places. We define the degree $d$ Diophantine exceptional set of $X\backslash D$ with respect to $L$ and $S$ to be
\begin{equation*}
\dExcL(X\backslash D)=\bigcup_R \dim_{>0}(\overline{R})
\end{equation*}
where the union runs over all sets $R$ of $(D,S)$-integral points on $X$ of degree $d$ over $L$. We define the degree $d$ absolute Diophantine exceptional set of $X\backslash D$ to be
\begin{equation*}
\dExcd(X\backslash D)=\bigcup_{L \supset k,S} \dExcL(X\backslash D),
\end{equation*}
with $L$ ranging over all number fields and $S$ ranging over all sets of places of $L$ as above.
\end{definition}
Similarly we define $X\backslash D$ to be degree $d$ Mordellic (resp. degree $d$ quasi-Mordellic) if $\dExcd(X\backslash D)$ is empty (resp. not Zariski-dense in X). At the two extremes of $\kappa_0$ we have
\begin{conjecture}
Let $d$ be a positive integer. If $\kappa(D_i)>0$ for all $i$ and $r>2dm$ then there does not exist a Zariski-dense set of $D$-integral points on $X$ of degree $d$ over $k$.
\end{conjecture}
\begin{conjecture}
Let $d$ be a positive integer. If $D_i$ is quasi-ample for all $i$ and $r>m+\frac{m(2d-1)}{\dim X}$ then $X\backslash D$ is degree $d$ quasi-Mordellic.
\end{conjecture}
We can also give a conjecture for ample divisors giving bounds on the degree $d$ Diophantine exceptional set.
\begin{conjecture}[General Siegel-type Conjecture for Ample Divisors]
Suppose that $D_i$ is ample for all $i$.\\\\
(a). If $r>m+\frac{m(2d-1)}{\dim X}$ then $\dim \dExcd(X\backslash D)\leq \frac{m(2d-1)}{r-m}$.\\
(b). In particular, if $r>2dm$ then $X\backslash D$ is degree $d$ Mordellic.
\end{conjecture}
\subsection{Conjectures over $\mathbb{Z}$ and Complex Quadratic Rings of Integers}
When $\#S=1$, or equivalently, when $\mathcal{O}_{k,S}$ is $\mathbb{Z}$ or the ring of integers of a complex quadratic field, and $D_i$ is defined over $k$ for all $i$, we conjecture improvements to our previous conjectures. We will refer to these conjectures as ``over $\mathbb{Z}$", though they apply equally well to rings of integers of complex quadratic fields.
\begin{conjecture}[Main Siegel-type Conjecture over $\mathbb{Z}$]
Let $k=\mathbb{Q}$ or a complex quadratic field and let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. Suppose that $D_i$ is defined over $k$ for all $i$ and that $\kappa(D_i)>0$ for all $i$. If $r>m$ then there does not exist a Zariski-dense set of $(D,S)$-integral points on $X$.
\end{conjecture}
We emphasize that in contrast to our previous conjectures, each $D_i$ must be defined over $k$. We also conjecture that in the above if each $D_i$ is quasi-ample, then $\Exck(X)$ is not Zariski-dense in $X$. For ample divisors, as usual, we conjecture something more.
\begin{conjecture}[Main Siegel-type Conjecture over $\mathbb{Z}$ for Ample Divisors]
\label{conjS}
Let $k=\mathbb{Q}$ or a complex quadratic field and let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. Suppose that $D_i$ is ample and defined over $k$ for all $i$.\\\\
(a). All sets $R$ of $(D,S)$-integral points on $X$ have $\dim R \leq 1+\dim (\bigcap_i D_i)$.\\
(b). In particular, if $D=D_1+D_2$ is a sum of two ample effective Cartier divisors on $X$, both defined over $k$, with no irreducible components in common, then there does not exist a Zariski-dense set of $(D,S)$-integral points on $X$.
\end{conjecture}
\begin{conjecture}[General Siegel-type Conjecture over $\mathbb{Z}$]
\label{GZ}
Let $k=\mathbb{Q}$ or a complex quadratic field and let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. Suppose that $D_i$ is defined over $k$ for all $i$ and that $\kappa(D_i)\geq \kappa_0>0$ for all $i$. Let $d$ be a positive integer. If $r>m+\frac{m(d-1)}{\kappa_0}$ then there does not exist a Zariski-dense set of $(D,S)$-integral points on $X$ of degree $d$ over $k$.
\end{conjecture}
If $D_i$ is quasi-ample for all $i$ in the above conjecture, then we also conjecture that $\dExck(X\backslash D)$ is not Zariski-dense in $X$. For ample divisors we have
\begin{conjecture}[General Siegel-type Conjecture over $\mathbb{Z}$ for Ample Divisors]
Let $k=\mathbb{Q}$ or a complex quadratic field and let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. Suppose that $D_i$ is ample and defined over $k$ for all $i$. Let $d$ be a positive integer. If $r>m+\frac{m(d-1)}{\dim X}$ then $\dim \dExck(X\backslash D)\leq \frac{m(d-1)}{r-m}$.
\end{conjecture}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
We will discuss the conjectures in greater detail in Section \ref{Remarks}.
\section{Overview of Results}
Sections \ref{smain}-\ref{SVgeneral} will be concerned with proving special cases of the above conjectures. In this section we highlight some of our results.
Along the lines of the Main Conjectures we have
\begin{theorema}
Suppose $r>2m\dim X$.\\\\
(a). If $D_i$ is quasi-ample for all $i$ then $X\backslash D$ is quasi-Mordellic.\\
(b). If $D_i$ is ample for all $i$ then $X\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
Suppose $r>2m\dim X$.\\\\
(a). If $D_i$ is quasi-ample for all $i$ then $X\backslash D$ is quasi-Brody hyperbolic.\\
(b). If $D_i$ is ample for all $i$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{theoremb}
If in addition the $D_i$'s have no irreducible components in common, then in the part (a)'s above we only need $r>2[\frac{m+1}{2}]\dim X$ where $[x]$ denotes the greatest integer in $x$.
When $X$ is a surface, $m\leq 2$, and the $D_i$'s have no irreducible components in common, we are able to prove the Main Conjectures, Conjectures \ref{conjmaina},B through \ref{conj2a},B.
\begin{theorema}
Suppose $X$ is a surface and the $D_i$'s have no irreducible components in common.\\\\
(a). If $m=1$, $\kappa(D_i)>0$ for all $i$, and $r>2$ then there does not exist a Zariski-dense set of $D$-integral points on $X$.\\
(b). If $m=2$, $\kappa(D_i)>0$ for all $i$, and $r>4$ then there does not exist a Zariski-dense set of $D$-integral points on $X$.\\
(c). If $m=2$, $D_i$ is quasi-ample for all $i$, and $r>3$ then $X\backslash D$ is quasi-Mordellic.\\
(d). If $m=2$, $D_i$ is ample for all $i$, and $r>4$ then $X\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
Suppose $X$ is a surface and the $D_i$'s have no irreducible components in common.\\\\
(a). If $m=1$, $\kappa(D_i)>0$ for all $i$, and $r>2$ then there does not exist a holomorphic map $f:\mathbb{C}\to X\backslash D$ with Zariski-dense image.\\
(b). If $m=2$, $\kappa(D_i)>0$ for all $i$, and $r>4$ then there does not exist a holomorphic map $f:\mathbb{C}\to X\backslash D$ with Zariski-dense image.\\
(c). If $m=2$, $D_i$ is quasi-ample for all $i$, and $r>3$ then $X\backslash D$ is quasi-Brody hyperbolic.\\
(d). If $m=2$, $D_i$ is ample for all $i$, and $r>4$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{theoremb}
We will see later that if $m=1, r>1,$ and $\kappa(D_i)>0$ for all $i$, then we must necessarily have $\kappa(D_i)=1$ for all $i$.
As to the General Conjectures, when the integral points are allowed to vary over fields of a bounded degree, we prove
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{theorem}
Let $d$ be a postive integer. If $D_i$ is ample for all $i$ and $r>2d^2m\dim X$ then $X\backslash D$ is degree $d$ Mordellic (all sets of $D$-integral points on $X$ of degree $d$ over $k$ are finite).
\end{theorem}
\begin{theorem}
Let $k=\mathbb{Q}$ or a complex quadratic field. Let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. Let $d$ be a positive integer. If $D_i$ is ample and defined over $k$ for all $i$ and $r>dm$ then all sets of $(D,S)$-integral points on $X$ of degree $d$ over $k$ are finite.
\end{theorem}
As an application of our results, we will discuss an improvement to a result of Faltings.
Faltings \cite{Fa} has recently shown how theorems on integral points on the complements of divisors with many components may occasionally be used to prove theorems on the complements of irreducible divisors. He shows how to do this with certain very singular curves on $\mathbb{P}^2$ by reducing the problem to a covering surface and applying the method of \cite{Fa2}. In \cite{Co4}, Zannier uses the subspace theorem approach instead of \cite{Fa2} to prove a result similar to Faltings. In Section \ref{Faltings} we will prove a theorem which generalizes both results. As an added bonus, we also prove the theorem in the case of holomorphic curves.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\section{Preliminaries}
\subsection{Diophantine Approximation}
\label{sDio}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
Let $k$ be a number field. Let $\mathcal{O}_k$ be the ring of integers of $k$. As usual, we have a set $M_k$ of absolute values (or places) of $k$ consisting of one place for each prime ideal $\mathfrak{p}$ of $\mathcal{O}_k$, one place for each real embedding $\sigma:k \to \mathbb{R}$, and one place for each pair of conjugate embeddings $\sigma,\overline{\sigma}:k \to \mathbb{C}$. Let $k_v$ denote the completion of $k$ with respect to $v$. We normalize our absolute values so that $|p|_v=p^{-[k_v:\mathbb{Q}_p]/[k:\mathbb{Q}]}$ if $v$ corresponds to $\mathfrak{p}$ and $\mathfrak{p}|p$, and $|x|_v=|\sigma(x)|^{[k_v:\mathbb{R}]/[k:\mathbb{Q}]}$ if $v$ corresponds to an embedding $\sigma$ (in which case we say that $v$ is archimedean). If $v$ is a place of $k$ and $w$ is a place of a field extension $L$ of $k$, then we say that $w$ lies above $v$, or $w|v$, if $w$ and $v$ define the same topology on $k$.
With the above definitions we have the product formula
\begin{equation*}
\prod_{v \in M_k}|x|_v=1 \quad \text{for all } x\in k^*.
\end{equation*}
For a point $P=(x_0,\ldots,x_n)\in \mathbb{P}^n(k)$ we define the height to be
\begin{equation*}
H(P)=\prod_{v\in M_k} \max(|x_0|_v,\ldots,|x_n|_v).
\end{equation*}
It follows from the product formula that $H(P)$ is independent of the choice of homogeneous coordinates for $P$. It is also easy to see that the height is independent of $k$. We define the logarithmic height to be
\begin{equation*}
h(P)=\log H(P).
\end{equation*}
At the core of our Diophantine results is the following version of Schmidt's Subspace Theorem due to Vojta \cite{Vo6}.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
Let $k$ be a number field. Let $S$ be a finite set of places in $M_k$ containing the archimedean places. Let $H_1,\ldots,H_m$ be hyperplanes in $\mathbb{P}^n$ defined over $\overline{k}$ with corresponding Weil functions $\lambda_{H_1},\ldots,\lambda_{H_m}$. Then there exists a finite union of hyperplanes $Z$, depending only on $H_1,\ldots,H_m$ (and not $k$ or $S$), such that for any $\epsilon>0$,
\begin{equation}
\sum_{v\in S}\max_I \sum_{i \in I} \lambda_{H_i,v}(P) \leq (n+1+\epsilon)h(P)
\end{equation}
holds for all but finitely many $P$ in $\mathbb{P}^n(k)\backslash Z$, where the max is taken over subsets $I \subset \{1,\ldots,m\}$ such that the linear forms defining $H_i,i \in I$ are linearly independent.
\end{theorema}
Explicitly, if $H$ is a hyperplane on $\mathbb{P}^n$ defined by the linear form $L(x_0,\ldots,x_n)$ then a Weil function for $H$ is given by
\begin{equation}
\label{Weila}
\lambda_{H,v}(P)=\log \max_i \frac{|x_i|_v}{|L(P)|_v}.
\end{equation}
where $P=(x_0,\cdots,x_n)$.
We will also need the close relative of Schmidt's theorem, the $S$-unit lemma.
\begin{theorema}
Let $k$ be a number field and let $n>1$ be an integer. Let $\Gamma$ be a finitely generated subgroup of $k^*$. Then all but finitely many solutions of the equation
\begin{equation}
u_0+u_1+\cdots+u_n=1, u_i\in \Gamma
\end{equation}
lie in one of the diagonal hyperplanes $H_I$ defined by the equation $\sum_{x\in I}x_i=0$, where $I$ is a proper subset of $\{0,\ldots,n\}$ with at least two elements.
\end{theorema}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
For the convenience of the reader, we have collected various properties of $D$-integral points that we will use (sometimes implicitly) throughout the paper (see \cite{Vo2}).
\begin{lemma}
\label{Integral}
Let $k$ be a number field and $S$ a finite set of places in $M_k$ containing the archimedean places. Let $D$ be an effective Cartier divisor on a projective variety $X$, both defined over $k$.\\\\
(a). Let $L$ be a finite extension of $k$ and let $T$ be the set of places of $L$ lying over places in $S$. If $R$ is a set of $(D,S)$-integral points then it is a set of $(D,T)$-integral points.\\
(b). Let $E$ be an effective Cartier divisor on $X$. If $R$ is a set of $(D+E)$-integral points then $R$ is a set of $D$-integral points.\\
(c). The $D$-integrality of a set is independent of the multiplicities of the components of $D$.\\
(d). Let $Y$ be a projective variety defined over $k$. Let $\pi:Y\to X$ be a morphism defined over $k$ with $\pi(Y)\not\subset D$. If $R$ is a set of $(D,S)$-integral points on $X$ then $\pi^{-1}(R)$ is a set of $(\pi^*D,S)$-integral points on $Y$.
\end{lemma}
Note also in (d), that if in addition $\pi:Y\backslash \pi^*D \to X \backslash D$ is a finite \'etale map, then by the Chevally-Weil theorem there exists a number field $L$ such that $\pi^{-1}(R)\subset Y(L)$ \cite[Th. 1.4.11]{Vo2}.
\subsection{Nevanlinna Theory and Kobayashi Hyperbolicity}
We will be interested in Nevanlinna theory as it applies to holomorphic maps $f:\mathbb{C} \to \mathbb{P}^n$ and hyperplanes on $\mathbb{P}^n$.
Let $f:\mathbb{C} \to \mathbb{P}^n$ be a holomorphic map. Then we may choose a representation of $f$, $\mathbf{f}=(f_0,\ldots,f_n)$ where $f_0,\ldots,f_n$ are entire functions without common zeros. Let us define $\|\mathbf{f}\|=(|f_0|^2+\cdots +|f_n|^2)^{\frac{1}{2}}$. Then we define a characteristic function $T_f(r)$ of $f$ to be
\begin{equation*}
T_f(r)=\int_{0}^{2\pi} \log \|\mathbf{f}(re^{i\theta})\|\frac{d\theta}{2\pi}.
\end{equation*}
Note that by Jensen's formula this function is well-defined up to a constant. Let $H$ be a hyperplane in $\mathbb{P}^n$ defined by a linear form $L$. Then we define a Weil function $\lambda_H(f(z))$ of $f$ with respect to $H$ by
\begin{equation}
\label{Weilb}
\lambda_H(f(z))=-\log \frac{|L(\mathbf{f}(z))|}{\|\mathbf{f}(z)\|}.
\end{equation}
We note that this is independent of the choice of $\mathbf{f}$ and depends on the choice of $L$ only up to a constant. The analogue of Schmidt's Subspace Theorem that we will need is the following version of Cartan's Second Main Theorem, due to Vojta \cite{Vo3}.
\begin{theoremb}
Let $H_1,\ldots H_m$ be hyperplanes in $\mathbb{P}^n$ with corresponding Weil functions $\lambda_{H_1},\ldots,\lambda_{H_m}$. Then there exists a finite union of hyperplanes $Z$ such that for any $\epsilon >0$ and any holomorphic map $f:\mathbb{C}\to \mathbb{P}^n\backslash Z$
\begin{equation}
\int_{0}^{2\pi} \max_I \sum_{i \in I} \lambda_{H_i}(f(re^{i\theta}))\frac{d\theta}{2\pi} \leq (n+1+\epsilon)T_f(r)
\end{equation}
holds for all $r$ outside a set of finite Lebesgue measure, where the max is taken over subsets $I \subset \{1,\ldots,m\}$ such that the linear forms defining $H_i,i \in I$, are linearly independent.
\end{theoremb}
The analogue of the $S$-unit lemma is the Borel lemma.
\begin{theoremb}
Let $f_1,\ldots,f_n$ be entire functions. Suppose that
\begin{equation}
e^{f_1}+\cdots+e^{f_n}=1.
\end{equation}
Then $f_i$ is constant for some $i$.
\end{theoremb}
Closely connected to questions about holomorphic curves is the Kobayashi pseudo-distance and Kobayashi hyperbolicity. We refer the reader to \cite{La2} for the definitions of the Kobayashi pseudo-distance, Kobayashi hyperbolic, complete hyperbolic, and hyperbolically imbedded. It is trivial that Kobayashi hyperbolic implies Brody hyperbolic. We will want a criterion for proving the converse in special cases. On projective varieties, this is given by Brody's theorem. More generally, we will use the following theorem of Green (see \cite{Gr2} and \cite{La2}).
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{theorem}[Green]
\label{hyperbolic}
Let $X$ be a complex projective variety. Let $Y=\bigcup_{i \in I} D_i$ be a finite union of Cartier divisors $D_i$ on $X$. Suppose that for every subset $\emptyset \subset J \subset I$,
\begin{equation*}
\bigcap_{j\in J}D_j\backslash \bigcup_{i\in I\backslash J}D_i
\end{equation*}
is Brody hyperbolic, where $\bigcap_{j\in \emptyset}D_j=X$. Then $X\backslash Y$ is complete hyperbolic and hyperbolically imbedded in $X$.
\end{theorem}
\subsection{Nef and Quasi-ample Divisors}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
We now recall some basic definitions and facts regarding nef and quasi-ample divisors. We will use the theory of intersection numbers on projective varieties as presented in, for instance, \cite{Kl}. We will use the notation $D^n$ to denote the intersection number of the $n$-fold intersection of $D$ with itself. In what follows $X$ will be a projective variety over an algebraically closed field of characteristic $0$.
\begin{definition}
A Cartier divisor $D$ (or invertible sheaf $\mathcal{O}(D)$) on $X$ is said to be numerically effective, or nef, if $D.C\geq 0$ for any closed integral curve $C$ on $X$.
\end{definition}
The next lemma summarizes some basic properties of nef divisors (see \cite{Kl}).
\begin{lemma}
Nef divisors satisfy the following:\\\\
(a). Let $n=\dim X$. If $D_1,\ldots,D_n$ are nef divisors on $X$ then $D_1.D_2.\ldots.D_n\geq 0$.\\
(b). Let $D$ be a nef divisor and $A$ an ample divisor on $X$. Then $A+D$ is ample.\\
(c). Let $f:X \to Y$ be a morphism and let $D$ be a nef divisor on $Y$. Then $f^*\mathcal{O}(D)$ is nef on $X$.
\end{lemma}
Recall that we have defined $\kappa(D)$ and quasi-ampleness for a Cartier divisor (Definitions \ref{defk} and \ref{defbig}). It is always true that $\kappa(D) \leq \dim X$, so $D$ is quasi-ample (or big) if and only if it has the largest possible dimension for a divisor on $X$. For nef divisors it is possible to give a more numerical criterion for a divisor to be quasi-ample. It is also possible in this case to get an asymptotic formula for $h^0(nD)$. We have the following lemma, due to Sommese, as it appears in \cite{Ka}.
\begin{lemma}
\label{nefbig}
Let $D$ be a nef divisor on a nonsingular projective variety $X$. Let $q=\dim X$. Then $h^0(nD)=\frac{D^q}{q!}n^q+O(n^{q-1})$. In particular, $D^q>0$ if and only if $D$ is quasi-ample.
\end{lemma}
\begin{proof}
Let $K_X$ denote the canonical divisor on $X$. Let $L$ be an ample divisor on $X$ such that $L+K_X$ is very ample. Since $D$ is nef, $nD+L$ is ample, and so by Kodaira's vanishing theorem we have
\begin{equation*}
H^i(X,\mathcal{O}(nD+L+K_X))=0 \text{ for } i>0.
\end{equation*}
Therefore,
\begin{equation*}
h^0(nD+L+K_X)=\chi(\mathcal{O}(nD+L+K_X))=\frac{D^q}{q!}n^q+O(n^{q-1})
\end{equation*}
by Riemann-Roch. Let $Y$ be a general member of the linear system $|L+K_X|$, so that $Y$ is nonsingular and irreducible. Then we have an exact sequence
\begin{equation*}
0 \to H^0(X,\mathcal{O}(nD)) \to H^0(X,\mathcal{O}(nD+L+K_X))\to H^0(Y,i^*\mathcal{O}(nD+L+K_X))
\end{equation*}
where $i:Y\to X$ is the inclusion map. But since $\dim Y=q-1$, we have $\dim H^0(Y,i^*\mathcal{O}(nD+L+K_X))\leq O(n^{q-1})$. It follows that $h^0(nD)=\frac{D^q}{q!}n^q+O(n^{q-1})$.
\end{proof}
Since we will use it multiple times, we state the exact sequence used above as a lemma.
\begin{lemma}
\label{exact}
Let $D$ be an effective Cartier divisor on $X$ with inclusion map $i:D \to X$. Let $E$ be any Cartier divisor on $X$. Then we have exact sequences
\begin{align}
&0 \to \mathcal{O}(E-D) \to \mathcal{O}(E) \to i_{*}(i^*\mathcal{O}(E)) \to 0\\
&0 \to H^0(X,\mathcal{O}(E-D))\to H^0(X,\mathcal{O}(E)) \to H^0(D,i^*(\mathcal{O}(E)).
\end{align}
\end{lemma}
\begin{proof}
If $D$ is an effective Cartier divisor, then a fundamental exact sequence is
\begin{equation*}
0 \to \mathcal{O}(-D) \to \mathcal{O}_X \to i_{*} \mathcal{O}_D \to 0.
\end{equation*}
Tensoring with $\mathcal{O}(E)$ and using the projection formula, we get the first exact sequence. Taking global sections then gives the second exact sequence.
\end{proof}
We can prove a little more for surfaces.
\begin{lemma}
\label{surfbig}
Let $D$ be an effective divisor on a nonsingular projective surface $X$. If $D^2>0$ then $h^0(nD)\geq \frac{n^2D^2}{2}+O(n)$ and $D$ is quasi-ample.
\end{lemma}
\begin{proof}
By Riemann-Roch,
\begin{equation}
h^0(nD)-h^1(nD)+h^0(K-nD)=\frac{n^2D^2}{2}-\frac{nD.K}{2}+1+p_a.
\end{equation}
Since $D$ is effective, $D \neq 0$, $h^0(K-nD)=0$ for $n\gg 0$ (for example, choose $n>K.H$ where $H$ is an ample divisor). We also have $h^1(nD)\geq 0$, so $h^0(nD)\geq \frac{n^2D^2}{2}+O(n)$ and $D$ is quasi-ample.
\end{proof}
It is not always true that if $D$ is nef then $h^0(E-D) \leq h^0(E)$. If $h^0(D)=0$ (for example if $D$ corresponds to a non-zero torsion element of Pic $X$) then when $E=D$ we have $h^0(E-D)=h^0(D-D)=h^0(0)=1 > h^0(E)=0$. We will want some control over $h^0(E-D)$ when $D$ is nef, and so we prove the following weak lemma.
\begin{lemma}
\label{nef}
Let $X$ be a nonsingular projective variety of dimension $q$. Let $D$ be a nef divisor on $X$. Let $E$ be any divisor on $X$. Then
\begin{equation*}
h^0(nE-mD) \leq h^0(nE)+O(n^{q-1})
\end{equation*}
for all $m,n\geq 0$, where the implied constant is independent of $m$.
\end{lemma}
\begin{proof}
We first claim that if $F$ is any nef divisor then there exists a divisor $C$, independent of $F$, such that $h^0(C+F)>0$. Explicitly, we may take $C=(q+2)A+K_X$, where $A$ is a very ample divisor on $X$. We prove this by induction on the dimension $q$. The case $q=1$ is easy. For the inductive step, we have an exact sequence
\begin{multline*}
0\to H^0(X,\mathcal{O}((q+1)A+K_X+F))\to H^0(X,\mathcal{O}((q+2)A+K_X+F)) \\
\to H^0(Y,i^*(\mathcal{O}((q+2)A+K_X+F)))\to H^1(X,\mathcal{O}((q+1)A+K_X+F))
\end{multline*}
where $Y$ is an irreducible nonsingular element of $|A|$ with inclusion map $i:Y\to X$. Since $(q+1)A+F$ is ample, by Kodaira vanishing, the last term above is $0$. Since $\omega_Y\cong i^*(\mathcal{O}(A+K_X))$, by induction we get that $\dim H^0(Y,i^*(\mathcal{O}((q+2)A+K_X+F)))>0$. Since the penultimate map in the exact sequence above is surjective, we therefore also have $h^0((q+2)A+k_X+F)=h^0(C+F)>0$, proving our claim.
Then we have
\begin{equation*}
h^0(nE-mD)\leq h^0(nE-mD+(C+mD))= h^0(nE+C) \leq h^0(nE) +O(n^{q-1})
\end{equation*}
independently of $m$, where the last inequality follows from Lemma \ref{exact} as in the proof of Lemma \ref{nefbig}.
\end{proof}
\section{Fundamental Theorems on Large Divisors}
\label{smain}
In this section we prove a slightly expanded version of a theorem of Corvaja and Zannier and its analogue for holomorphic curves. These theorems will be fundamental to our future results.
Let $D$ be a divisor on a nonsingular projective variety $X$ defined over a field $k$. Let $\overline{k}(X)$ denote the function field of $X$ over $\overline{k}$. We will write $D\geq E$ if $D-E$ is effective. Let div$(f)$ denote the principal divisor associated to $f$. Let $L(D)$ be the $\overline{k}$-vector space $L(D)=\{f \in \overline{k}(X)|\text{div}(f)\geq -D\}$, and let $l(D)=\dim L(D)=h^0(D)$. If $E$ is a prime divisor we let $\text{ord}_E f$ denote the coefficient of $E$ in div$(f)$. We make the following definition.
\begin{definition}
Let $D$ be an effective divisor on a nonsingular projective variety $X$ defined over a field $k$. Then we define $D$ to be a very large divisor on $X$ if for every $P\in D(\overline{k})$ there exists a basis $B$ of $L(D)$ such that $\text{ord}_E\prod_{f \in B}f>0$ for every irreducible component $E$ of $D$ such that $P\in E$. We define $D$ to be a large divisor if some nonnegative integral linear combination of its irreducible components is very large on $X$.
\end{definition}
\begin{remark}
\label{remlarge}
Suppose $D$ is very large. Let $P\in D$ and let $\mathcal E$ be the set of irreducible components $E$ of $D$ such that $P\in E$. If $B$ is a basis of $L(D)$ that has the property in the definition of very large with respect to $P$, then $B$ also works as a basis with respect to any $Q\in \bigcap_{E \in \mathcal{E}}E$. Thus, it is easily seen that in the definition of very large one only needs to use bases $B\in \mathcal{B}$ for some finite set of bases $\mathcal{B}$ for any very large divisor $D$.
\end{remark}
We will see (Theorem \ref{cor3}) for example that on any nonsingular projective variety $X$ the sum of sufficiently many ample effective divisors in general position is large. On the other hand, it is obvious from the definition that if $D$ is an irreducible effective divisor on $X$ then $D$ cannot be large. Roughly speaking, large divisors have a lot of irreducible components of high $D$-dimension.
With this definition we have the following theorems.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}[Corvaja-Zannier]
\label{maina}
Let $X$ be a nonsingular projective variety defined over a number field $k$. Let $S\subset M_k$ be a finite set of places of $k$ containing the archimedean places. Let $D$ be a large divisor on $X$ defined over $k$. Then there does not exist a Zariski-dense set of $D$-integral points on $X$. Furthermore, if $D$ is very large and $\Phi_D$ is a rational map to projective space corresponding to $D$, then there exists a proper closed subset $Z\subset X$ depending only on $D$ (and not $k$ or $S$) such that $\Phi_D(R\backslash Z)$ is finite for any set $R$ of $D$-integral points on $X$. In particular, if $\Phi_D$ is birational, $X\backslash D$ is quasi-Mordellic.
\end{theorema}
\begin{theoremb}
\label{mainb}
Let $X$ be a nonsingular complex projective variety. Let $D$ be a large divisor on $X$. Then there does not exist a holomorphic map $f:\mathbb{C} \to X \backslash D$ with Zariski-dense image. Furthermore, if $D$ is very large and $\Phi_D$ is a rational map to projective space corresponding to $D$, then there exists a proper closed subset $Z\subset X$ depending only on $D$ such that for all holomorphic maps $f:\mathbb{C} \to X \backslash D$, either $f(\mathbb{C})\subset Z$ or $\Phi_D\circ f$ is constant. In particular, if $\Phi_D$ is birational, $X\backslash D$ is quasi-Brody hyperbolic.
\end{theoremb}
Theorem \ref{maina} appears, essentially, in the proof of the Main Theorem in \cite{Co2}, and for curves in \cite{Co}. We have added the last two statements to the theorem by using Vojta's result on the exceptional hyperplanes in the Schmidt Subspace Theorem.
Given these theorems, many of our results mentioned in the introduction reduce to showing that certain divisors are large.
Let us prove Theorem \ref{maina} first. Before proving this theorem, we need a lemma.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{lemma}
\label{seq}
Let $X$ be a projective variety defined over a number field $k$. Let $R\subset X(k)$ be a Zariski-dense subset of $X$. Let $v\in M_k$. Then there exists a point $P$ in $X(k_v)$ and a sequence $\{P_i\}$ in $R$ such that $\{P_i\}\to P$ in the $v$-topology on $X(k_v)$ and $\bigcup \{P_i\}$ is Zariski-dense in $X$.
\end{lemma}
\begin{proof}
We will always be working in the $v$-topology on $X(k_v)$. First we claim that there exists a $P$ in $\overline{R}\subset X(k_v)$ such that for every neighborhood $U$ of $P$ in $X(k_v)$, $U\cap R$ is Zariski-dense in $X$. Indeed, suppose there is no such $P$. Then for each $P$ in $R$, let $U_P$ be a neighborhood of $P$ such that $U_P \cap R$ is not Zariski-dense in $X$. Since $X(k_v)$ is compact because $X$ is projective, $\overline{R}$ is compact, so we may cover $\overline{R}$ by finitely many open sets $U_{P_1},\ldots,U_{P_n}$. But then $R=(U_{P_1}\cap R)\cup\cdots \cup (U_{P_n}\cap R)$ is not Zariski-dense in $X$, a contradiction.
Now pick some $P$ as in the claim above. Embed $X$ in $\mathbb{P}^n_k$ for some $n$. Since $k$ is countable, the set of hypersurfaces in $\mathbb{P}^n_k$ not containing $X$ is countable. Let $\{H_i\}$ be an enumeration of these. There also exists a countable collection of neighborhoods $\{U_i\}$ of $P$ in $X(k_v)$ such that $U_i \subset U_j$ for $i>j$ and $\bigcap U_i=\{P\}$. Since $U_i\cap R$ is Zariski-dense in $X$, for all $i$ there exists a $P_i \in U_i \cap R$ such that $P_i \notin H_i$. Then $\{P_i\}\to P$ in $X(k_v)$ and $\bigcup \{P_i\}$ is Zariski-dense in $X$ since it is not contained in any hypersurface.
\end{proof}
\begin{proof}[Proof of Theorem \ref{maina}]
Let $D$ be a large divisor and $S$ and $X$ as in Theorem \ref{maina}. Clearly, we may reduce to the case where $D$ is very large. Extending $k$ if necessary and enlarging $S$, we may assume without loss of generality that every irreducible component of $D$ is defined over $k$ and that all of the finitely many functions in $L(D)$ we use (see Remark \ref{remlarge}) are defined over $k$. Let $\{\phi_1,\ldots,\phi_{l(D)}\}$ be a basis of $L(D)$ over $k$. Let $R$ be a $(D,S)$-integral set of points on $X$. It suffices to prove the theorem in the case that $\overline{R}$ is irreducible. By repeatedly applying Lemma~\ref{seq}, we see that there exists a sequence $P_i$ in $R$ such that for each $v$ in $S$, $\{P_i\}$ converges to a point $P_v\in X(k_v)$ and $\bigcup \{P_i\}$ is Zariski-dense in $\overline{R}$.
Let $S'$ be the set of places $v\in S$ such that $P_v\in D(k_v)$, and let $S''=S\backslash S'$. Since $D$ is very large, for each $v\in S'$ let $L_{iv},i=1,\ldots,l(D)$ be a basis for $L(D)$ such that $\text{ord}_E\prod_{i=1}^{l(D)}L_{iv}>0$ for all irreducible components $E$ of $D$ such that $P_v\in E(k_v)$. Of course each $L_{iv}$ is a linear form in the $\phi_j$'s over $k$.
For $v\in S''$, we set $L_{jv}=\phi_j$ for $j=1,\ldots,l(D)$. Let $\phi(P)=(\phi_1(P),\ldots,\phi_{l(D)}(P))$ for $P\in X\backslash D$. Let $H_{jv}$ denote the hyperplane in $\mathbb{P}^{l(D)-1}$ determined by $L_{jv}$ with respect to the basis $\phi_1,\ldots,\phi_{l(D)}$. Let $\lambda_{H_{jv},v}$ be the Weil function for $H_{jv}$ given in Equation (\ref{Weila}).
We will now show that there exists $\epsilon>0$ and a constant $C$ such that
\begin{equation}
\label{Schmidt}
\sum_{v\in S}\sum_{j =1}^{l(D)} \lambda_{H_{jv},v}(\phi(P_i)) > (l(D)+\epsilon)h(\phi(P_i))+C.
\end{equation}
Since $R$ is a set of $(D,S)$-integral points, we have
\begin{equation*}
h(\phi(P_i))<\sum_{v \in S}\log{\max_j|\phi_j(P_i)|_v}+O(1).
\end{equation*}
Using this it suffices to prove that
\begin{equation*}
\sum_{v\in S}\sum_{j =1}^{l(D)}\log \max_{j'} \frac{|\phi_{j'}(P_i)|_v}{|L_{jv}(P_i)|_v}>(l(D)+\epsilon)\sum_{v\in S}\log{\max_{j'}|\phi_{j'}(P_i)|_v}+C'
\end{equation*}
for some $C'$ or rearranging things, simplifying, and exponentiating
\begin{equation*}
\prod_{v\in S} |\max_{j'}(\phi_{j'}(P_i))^{\epsilon} \prod_{j=1}^{l(D)}L_{jv}(P_i)|_v
\end{equation*}
is bounded for some $\epsilon>0$. Let
\begin{equation*}
M=\max\{-\text{ord}_E \phi_j|E \text{ is an irreducible component of } D, j=1,\ldots,l(D)\}.
\end{equation*}
Let $\epsilon=\frac{1}{M}$. For $v\in S''$ both $|\phi_{j'}(P_i)|_v$ and $|L_{jv}(P_i)|_v$ are bounded for all $i$ since $P_v \notin D(k_v)$ and $\phi_{j'}$ and $L_{jv}$ have poles lying only in the support of $D$. Let $v\in S'$. So $P_v\in D(k_v)$. It follows from the definition of $M$ and the fact that $\text{ord}_E \prod_{i=1}^{l(D)}L_{iv}>0$ for any irreducible component $E$ of $D$ such that $P_v \in E(k_v)$ that $\text{ord}_E\phi_{j'} (\prod_{i=1}^{l(D)}L_{iv})^M\geq -M+ M \geq 0$ for any irreducible component $E$ of $D$ such that $P_v \in E(k_v)$. Since the $\phi_{j'}$ and $L_{iv}$ have poles only in the support of $D$, it follows from the previous order computation that $|\max_{j'}(\phi_{j'}(P_i))^{\epsilon} \prod_{j=1}^{l(D)}L_{jv}(P_i)|_v$ is bounded for all $i$ and all $v\in S$ when $\epsilon=\frac{1}{M}>0$. So we have proved Equation (\ref{Schmidt}).
Note that either $h(\phi(P_i))\to \infty$ as $i\to \infty$ or $\phi(P_i)=\phi(\overline{R})$, and $\phi(P_i)$ is constant for all $i$. In the latter case the theorem is proved, so we may assume the former. Therefore, making $\epsilon$ smaller, we see that Equation (\ref{Schmidt}) holds with $C=0$ for all but finitely many $i$. So by Schmidt's Subspace Theorem, there exists a finite union of hyperplanes $Z\subset \mathbb{P}^{l(D)-1}$ such that all but finitely many of the points in the set $\{\phi(P_i)=(\phi_1(P_i),\ldots,\phi_{l(D)}(P_i))|i\in \mathbb{N}\}$ lie in $Z$. Using Remark \ref{remlarge} we see that we may choose the hyperplanes $H_{iv}$ used above from a finite set of hyperplanes independent of $R$. Therefore, using the statement on the exceptional hyperplanes in the Schmidt Subspace Theorem, we see that $Z$ may be chosen to depend only on $D$ and not $R$, $k$, or $S$. Since it was assumed that $\overline{R}$ is irreducible and $\phi(\overline{R})$ is not a point, it follows that $\phi(R)\subset Z$. Since $\phi_1,\ldots,\phi_{d}$ are linearly independent functions in $K(X)$ and $Z$ is a finite union of hyperplanes, it follows that $\phi^{-1}(Z)$ is a finite union of proper closed subvarieties of $X$. So $R\subset \phi^{-1}(Z)$ and the theorem is proved.
\end{proof}
The proof of Theorem \ref{mainb} is very similar.
\begin{proof}[Proof of Theorem \ref{mainb}]
Since our assertion depends only on the support of $D$ we may assume without loss of generality that $D$ is very large on $X$. Let $f:\mathbb{C} \to X \backslash D$ be a holomorphic map. By Remark \ref{remlarge} there exists a finite set $J$ of elements in $L(D)$ such that for any $P\in D$ there exists a subset $I \subset J$ that is a basis of $L(D)$ such that $\text{ord}_E \prod_{g\in I}g>0$ for every irreducible component $E$ of $D$ such that $P\in E$. Let $\phi_1,\ldots,\phi_{l(D)}$ be a basis for $L(D)$. Let $\phi=(\phi_1,\ldots,\phi_{l(D)}):X\backslash D \to \mathbb{P}^{l(D)-1}$. Let $J'$ be the set of linear forms $L$ in $l(D)$ variables over $\mathbb{C}$ such that $L\circ \phi \in J$. If $L$ is a linear form, let $H_L$ be the corresponding hyperplane. We will now show that there exists $\epsilon>0$ and a constant $C$ such that
\begin{equation}
\label{Cartan}
\int_{0}^{2\pi} \max_I \sum_{L \in I} \lambda_{H_L}(\phi \circ f(re^{i\theta}))\frac{d\theta}{2\pi} > (l(D)+\epsilon)T_{\phi \circ f}(r)-C
\end{equation}
for all $r>0$, where the max is taken over subsets $I \subset J'$ such that $I$ consists of exactly $l(D)$ linearly independent linear forms. Substituting the definition of the Weil function in Equation (\ref{Weilb}) and the definition of $T_{\phi \circ f}$, after some manipulation the inequality in Equation (\ref{Cartan}) becomes
\begin{equation*}
\int_{0}^{2\pi} \epsilon \log|\phi\circ f(re^{i\theta})|+\min_I \sum_{L \in I} \log |L\circ \phi \circ f(re^{i\theta})| \frac{d\theta}{2\pi}<C
\end{equation*}
with $I$ as before. Since $|\phi\circ f(re^{i\theta})|\leq \sqrt{l(D)} \max_j |\phi_j \circ f(re^{i\theta})|$ it clearly suffices to show that
\begin{equation}
\label{Cartan2}
\max_j|\phi_j \circ f(re^{i\theta})|^{\epsilon} \min_I \prod_{L \in I} |L\circ \phi \circ f(re^{i\theta})|
\end{equation}
is bounded independently of $r$ and $\theta$ for some $\epsilon>0$.
Let $D_1,\ldots,D_m$ be the irreducible components of $D$. Let
\begin{equation*}
M=\max\{-\text{ord}_{D_i} \phi_j|i=1,\ldots,m, j=1,\ldots,l(D)\}.
\end{equation*}
We will work in the classical topology. Let $P\in D$. Then there exists a neighborhood $U$ of $P$ such that for all $Q\in \overline{U}$ if $Q \in D_i$ for some $i$ then $P \in D_i$. Let $I\subset J'$ be a subset of $J'$ such that $\text{ord}_{D_i} \prod_{L\in I} L \circ \phi>0$ for all $i$ such that $P\in D_i$. If $P \in D_i$, then by the definition of $M$ we have $\text{ord}_{D_i}\phi_j (\prod_{L\in I} L \circ \phi)^M\geq 0$ for all $j$. By the definition of $U$ we see that $\phi_j (\prod_{L\in I} L \circ \phi)^M$ is bounded for all $j$ on the compact set $\overline{U}$. Since $D$ is compact and may be covered by such sets we see that $\max_j|\phi_j|\min_I \prod_{L \in I} |L\circ \phi|^M$ is bounded on $X\backslash D$ (using also that away from $D$ everything is obviously bounded since the $\phi_j$'s have poles only in $D$). Therefore Equation (\ref{Cartan2}) is bounded independently of $r$ and $\theta$ for $\epsilon=\frac{1}{M}$.
If $\phi \circ f$ is constant then there is nothing to prove, so assume otherwise. Then $T_{\phi \circ f}(r)\to \infty$ as $r\to \infty$, and so making $\epsilon$ smaller, we see that we have proven the inequality (\ref{Cartan}) with $C=0$ for all sufficiently large $r$. Therefore by Cartan's Second Main Theorem, there exists a finite union of hyperplanes $Z\subset \mathbb{P}^{l(D)-1}$ depending only on $D$ (the $H_L$'s depended only on $D$) such that $\phi(f(\mathbb{C}))\subset Z$. Since the $\phi_j$'s are linearly independent and $Z$ is a finite union of hyperplanes, $\phi^{-1}(Z)$ is a finite union of closed subvarieties of $X$ and $f(\mathbb{C})\subset \phi^{-1}(Z)$.
\end{proof}
\begin{remark}
If $D$ is very large and one can explicitly compute the map $\phi$ and the hyperplanes used in the above proofs, then one can explicitly compute the closed set $Z$ in the theorems above. This follows from the explicit description of the exceptional hyperplanes in \cite{Vo6} and \cite{Vo3}.
\end{remark}
\section{Large Divisors}
For an effective divisor $D=\sum_{i=1}^r D_i$ on $X$ and $P\in D(\overline{k})$, we let $D_P=\sum_{i:P\in D_i}D_i$.
\begin{lemma}
\label{large}
Let $D=\sum_{i=1}^r D_i$ be a divisor on a nonsingular projective variety $X$ with $D_i$ effective for each $i$. Let $P\in D$. Let $f_P(m,n)=l(nD-mD_P)-l(nD-(m+1)D_P)$. If there exists $n>0$ such that $\sum_{m=0}^{\infty}(m-n)f_P(m,n)>0$ for all $P\in D$ then $nD$ is very large.
\end{lemma}
\begin{proof}
Let $n>0$ be such that $\sum_{m=0}^{\infty}(m-n)f_P(m,n)>0$ for all $P\in D$. This sum is clearly finite for all $P\in D$ and we let $M_P(n)$ be the largest integer such that $f_P(M_P(n),n)>0$. Let $P\in D$. Let $M=M_P(n)$. Let $V_j=L(nD- jD_P)$. So $\dim V_j/V_{j+1}=f_P(j,n)$. We have $L(nD)=V_0 \supset V_1 \supset \ldots \supset V_M\neq 0$. Choose a basis of $V_M$ and successively complete it to bases of $V_{M-1},V_{M-2},\ldots,V_0$, to obtain a basis $f_1,\ldots,f_{l(nD)}$. Let $E$ be an irreducible component of $D$ such that $P \in E$. If $f_j \in V_m$ then $\text{ord}_E f_j\geq (m-n)\ord_ED$. So we get that $\text{ord}_E\prod_{i=1}^{l(nD)}f_i\geq (\ord_ED)\sum_{m=0}^{M}(m-n)f_P(m,n)>0$. So $nD$ is very large.
\end{proof}
\begin{theorem}
\label{cor2}
Let $X$ be a nonsingular projective variety. Let $q= \dim X$. Let $D=\sum_{i=1}^{r}D_i$ be a divisor on $X$ such that $D_i$ is effective and nef for each $i$. Suppose also that every irreducible component of $D$ is nonsingular. If
\begin{equation*}
D^q>2q D^{q-1}.D_P, \qquad \forall P\in D
\end{equation*}
then $nD$ is very large for $n\gg 0$. In particular, $D$ is large.
\end{theorem}
\begin{proof}
Let $P \in D$. Let $D_P=\sum_{j=1}^{k}a_j E_j$, where each $E_j$ is a distinct prime divisor. Repeatedly applying Lemma~\ref{exact}, we obtain
\begin{multline*}
\dim H^0(X,\mathcal{O}(nD-m D_P))-\dim H^0(X,\mathcal{O}(nD-(m+1)D_P)) \\
\leq \sum_{j=1}^k \sum_{l=0}^{a_{j}-1}\dim H^0(E_{j},i^*_{E_{j}}\mathcal{O}(nD-mD_P -\sum_{j'=1}^{j-1}a_{j'}E_{j'}-lE_{j}))
\end{multline*}
It follows from the fact that $D_P$ is nef, Lemma \ref{exact}, and Lemma \ref{nef} that
\begin{multline*}
\dim H^0(E_{j},i^*_{E_{j}}\mathcal{O}(nD-mD_P -\sum_{j'=1}^{j-1}a_{j'}E_{j'}-lE_{j}))\\
\leq \dim H^0(E_{j},i^*_{E_{j}}\mathcal{O}(nD))+O(n^{q-2}).
\end{multline*}
Therefore,
\begin{multline*}
\dim H^0(X,\mathcal{O}(nD-m D_P))-\dim H^0(X,\mathcal{O}(nD-(m+1)D_P))\\
\leq \sum_{j=1}^k a_j \dim H^0(E_j,i_{E_j}^*\mathcal{O}(nD))+O(n^{q-2}).
\end{multline*}
Since $D$ is nef, $l(nD)=\frac{n^q}{q!}D^q + O(n^{q-1})$. Since $i_{E_j}^*\mathcal{O}(D)$ is also nef, we have $\dim H^0(E_j,i_{E_j}^* \mathcal{O}(nD))= \frac{n^{q-1}}{(q-1)!} D^{q-1}.E_j+ O(n^{q-2})$. So
\begin{equation*}
f_P(m,n)\leq \frac{n^{q-1}}{(q-1)!} \sum_{j=1}^k a_j D^{q-1}.E_j+ O(n^{q-2}) =\frac{n^{q-1}}{(q-1)!} D^{q-1}.D_P+ O(n^{q-2}).
\end{equation*}
To use this estimate, we borrow a lemma from \cite{Co2}.
\begin{lemma}
\label{CZlemma}
Let $h$ and $R$ be integers with $R\leq h$ and let $x_1,\ldots,x_h,U_1,\ldots,U_R$ be real numbers. If $0\leq x_i\leq U_i$ for $i=1,\ldots, R$ and $\sum_{j=1}^RU_j\leq \sum_{j=1}^hx_j$ then $\sum_{j=1}^hjx_j\geq \sum_{j=1}^RjU_j$.
\end{lemma}
\begin{proof}
We have
\begin{align*}
\sum_{j=1}^RjU_j+\sum_{j=1}^h(R+1-j)x_j&\leq \sum_{j=1}^RjU_j+\sum_{j=1}^R(R+1-j)x_j\\
&\leq \sum_{j=1}^RjU_j+\sum_{j=1}^R(R+1-j)U_j=(R+1)\sum_{j=1}^RU_j
\end{align*}
So, rearranging things.
\begin{equation*}
\sum_{j=1}^h jx_j\geq \sum_{j=1}^RjU_j+(R+1)\left(\sum_{j=1}^hx_j-\sum_{j=1}^RU_j\right)
\end{equation*}
and the last term is positive by assumption.
\end{proof}
Let $R_n=\frac{n^q}{q!}D^q$ and $S_n= \frac{n^{q-1}}{(q-1)!} D^{q-1}.D_P$. In the notation of Lemma \ref{large}, we have
\begin{equation*}
\sum_{m=0}^{M_P(n)}f_P(m,n)=l(nD)=R_n+O(n^{q-1}).
\end{equation*}
and $f_P(m,n)\leq S_n+O(n^{q-2})$. We will assume from now on that $S_n\neq 0$ (the case $S_n=0$ is similar).
Then using our estimate, we have $M_P(n) \geq \frac{R_n}{S_n}+O(1)$ and $\sum_{m=0}^{\frac{R_n}{S_n}+O(1)}(S_n+ O(n^{q-2}))\leq \sum_{m=0}^{M_P(n)}f_P(m,n)$. So using Lemma \ref{CZlemma}, for $n \gg 0$ we get the estimate
\begin{align*}
\sum_{m=0}^{M_P(n)}(m-n)f_P(m,n) & \geq \sum_{m=0}^{\frac{R_n}{S_n}+O(1)}m(S_n+O(n^{q-2}))-n\sum_{m=0}^{M_P(n)}f_P(m,n)\\
&\geq \frac{R_n^2}{2S_n}-nR_n+O(n^q)\\
&\geq \frac{R_n}{S_n}\left[\frac{n^q}{2q!}\left(D^q-2q D^{q-1}.D_P\right)+O(n^{q-1})\right].
\end{align*}
So for $n \gg 0$, $ \sum_{m=0}^{M_P}(m-n)f_P(m,n)>0$ if $D^q>2q D^{q-1}.D_P$. Then we are done by Lemma \ref{large}.
\end{proof}
When $q=1$ we obtain
\begin{corollary}
Let $D$ be an effective divisor on a nonsingular projective curve $X$. If $D$ is a sum of more than 2 distinct points on $X$ then $D$ is large.
\end{corollary}
By Theorems \ref{maina} and \ref{mainb} we then recover
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{corollarya}
Siegel's theorem (Theorem~\ref{Siegel2})
\end{corollarya}
\begin{corollaryb}
Picard's theorem (Theorem \ref{Picard})
\end{corollaryb}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
Actually we have only proved these theorems for nonsingular curves $\tilde{C}$. However, the general case follows from this case by looking at the normalization of $\tilde{C}$.
Suppose that we have a divisor $D=\sum_{i=1}^{r}D_i$ satisfying the hypotheses of Theorem~\ref{cor2}. We would like to modify $D$ to a divisor $D'=\sum_{i=1}^{r}a_iD_i$ so that we may optimally apply the theorem. When each $D_i$ is ample, this amounts to choosing the $a_i$'s so that in the embedding given by $nD'$ for $n\gg 0$ the degree of each $a_iD_i$ is the same. In terms of intersection theory, we would like $a_iD_i.(D')^{q-1}$ to be the same for each $i$. We make the following definition:
\begin{definition}
Let $X$ be a nonsingular projective variety. Let $q=\dim X$. Let $D=\sum_{i=1}^rD_i$ be a divisor on $X$ with $D_1,\ldots,D_r$ effective.. Then $D$ is said to have equidegree with respect to $D_1,\ldots,D_r$ if $D_i.D^{q-1}=\frac{D^q}{r}$ for $i=1,\ldots,r$. We will say that $D$ is equidegreelizable (with respect to $D_1,\ldots,D_r$) if there exist real numbers $a_i>0$ such that if $D'=\sum_{i=1}^ra_iD_i$ then $D'$ has equidegree with respect to $a_1D_1,\ldots,a_r D_r$. (extending intersections to $\Div X\otimes \mathbb{R}$ in the canonical way).
\end{definition}
We will frequently omit the reference to the $D_i$'s when it is clear what we mean.
\begin{lemma}
\label{equi}
Let $X$ be a nonsingular projective variety. Let $q=\dim X$. Let $D_1,\ldots,D_r$ be divisors on $X$ with $D_i^q>0$ for all $i$. Suppose that all $q$-fold intersections of the $D_i$'s are nonnegative. Then $\sum_{i=1}^r D_i$ is equidegreelizable with respect to $D_1,\ldots,D_r$.
\end{lemma}
\begin{proof}
Consider the function $f(a_1,\ldots,a_r)=(\sum_{i=1}^r e^{a_i}D_i)^q$ subject to the constraint $g(a_1,\ldots,a_r)=\sum_{i=1}^r a_i=0$. Since all $q$-fold intersections of the $D_i$'s are nonnegative, $f(a_1,\ldots,a_r)\geq e^{q a_i}D_i^q$ for any $i$. Since $D_i^q>0$ for all $i$, as $\max \{a_i\}\to \infty$ we have $f(a_1,\ldots,a_r) \to \infty$. It follows that $f$ attains a minimum on the plane $\sum_{i=1}^r a_i=0$. Therefore there exists a solution $\lambda,a_1,\ldots,a_r$ to the Lagrange multiplier equations $g=0,\frac{\partial f}{\partial a_i}=e^{a_i}D_i.(\sum_{i=1}^r e^{a_i} D_i)^{q-1}=\lambda \frac{\partial g}{\partial a_i}=\lambda, i=1,\ldots,r$. So $D'=\sum_{i=1}^r e^{a_i}D_i$ has equidegree with respect to $D_1,\ldots,D_r$ and trivially $e^{a_i}>0$.
\end{proof}
We give an example to show that not all divisor sums are equidegreelizable.
\begin{example}
Let $X=\mathbb{P}^1 \times \mathbb{P}^1$. Let $D_1=P_1 \times \mathbb{P}^1,D_2=P_2 \times \mathbb{P}^1$, and $D_3=\mathbb{P}^1 \times Q$, where $P_1,P_2$, and $Q$ are points in the various $\mathbb{P}^1$'s. So $D_1.D_2=D_1^2=D_2^2=D_3^2=0$ and $D_1.D_3=D_2.D_3=1$. Let $D=a_1D_1+a_2D_2+a_3D_3$. Since $a_3D_3.D=a_1D_1.D+a_2D_2.D$, it is clear that there do not exist $a_1,a_2,a_3>0$ such that $a_iD_i.D=\frac{D^2}{3}$ for $i=1,2,3$. So $D=D_1+D_2+D_3$ is not equidegreelizable with respect to $D_1, D_2$, and $D_3$.
\end{example}
With the above definition, we have the following theorem.
\begin{theorem}
\label{cor3}
Let $X$ be a nonsingular projective variety. Let $q=\dim X$. Let $D=\sum_{i=1}^r D_i$ be a quasi-ample divisor on $X$ equidegreelizable with respect to $D_1,\ldots, D_r$, with $D_1,\ldots, D_r$ nef and effective. Suppose that every irreducible component of $D$ is nonsingular. Suppose that the intersection of any $m+1$ distinct $D_i$'s is empty. If $r>2mq$ then $D$ is large. Furthermore, there exists a very large divisor $E$ with the same support as $D$ such that $\Phi_E$ is birational.
\end{theorem}
\begin{proof}
Since $D$ is equidegreelizable, we may find positive integers $a_i$ such that if $D'=\sum_{i=1}^{r}a_iD_i$ then $\frac{a_iD_i.(D')^{q-1}}{(D')^q}$ is arbitrarily close to $\frac{1}{r}$ for each $i$. Note that $D'$ is again quasi-ample. Since for any $P\in D(\overline{k})$, $P$ belongs to at most $m$ divisors $D_i$, and $r>2mq$, we have that
\begin{equation*}
2q(D')^{q-1}.(D')_P=2q \sum_{i: P\in D_i(\overline{k})}a_i D_i.(D')^{q-1}<(D')^q.
\end{equation*}
So the hypotheses of Theorem~\ref{cor2} are satisfied and so $nD'$ is very large for $n \gg 0$. The last statement then follows from the fact that $D'$ is quasi-ample.
\end{proof}
\begin{lemma}
\label{reduce}
Let $X$ be a complex projective variety. Let $D=\sum_{i=1}^rD_i$ be a sum of effective Cartier divisors on $X$. Then there exists a nonsingular projective variety $X'$, a birational morphism $\pi:X'\to X$, and a divisor $D'=\sum_{i=1}^rD_i'$ on $X'$ such that $\supp D_i'\subset \supp \pi^*D_i$ for all $i$, every irreducible component of $D'$ is nonsingular, $|D_i'|$ is base-point free for all $i$ (in particular $D_i'$ is nef), and $\kappa(D_i')=\kappa(D_i)=\dim \Phi_{D_i'}(X')$ for all $i$.
\end{lemma}
\begin{proof}
Taking a resolution of the singularities of $X$ and of the embedded singularities of the irreducible components of $D$ we may assume that $X$ and every irreducible component of $D$ are nonsingular. For each $i$, let $m_i>0$ be such that $\dim \Phi_{m_iD_i}(X)=\kappa(D_i)$. Let $\pi:X'\to X$ be the map obtained by blowing up the base-points of all the linear systems $|m_iD_i|$. Then $\pi^*(m_iD_i)=D_i'+F_i$ for each $i$, where $|D_i'|$ is base-point free and $F_i$ is the fixed part of $|\pi^*(m_iD_i)|$. We have, trivially from the definition, $\kappa(D_i)=\kappa(m_iD_i)$. Further, $\kappa(m_iD_i)=\kappa(\pi^*(m_iD_i))$ (in fact $l(mD_i)=l(\pi^*(mD_i))$ for all $m$ follows easily from $\pi_*\mathcal{O}_{X'}=\mathcal{O}_X$ and the projection formula). Finally, $\kappa(\pi^*(m_iD_i))=\kappa(D_i')$ since by construction $\kappa(D_i')=\max_{n>0}\dim \Phi_{nD_i'}(X')\geq \kappa(D_i)=\kappa(\pi^*(m_iD_i))$ (the other inequality being trivial). So $\kappa(D_i')=\kappa(D_i)$ for all $i$ and therefore $X',\pi$, and $D'=\sum_{i=1}^rD_i'$ satisfy the requirements of the lemma.
\end{proof}
We now obtain one of our main results.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
\label{cor4}
Let $X$ be a projective variety defined over a number field $k$. Let $q=\dim X$. Let $D=\sum_{i=1}^{r} D_i$ be a divisor on $X$ defined over $k$ such that the $D_i$'s are effective Cartier divisors and the intersection of any $m+1$ distinct $D_i$'s is empty.\\\\
(a). If $D_i$ is quasi-ample for each $i$ and $r> 2mq$ then $X\backslash D$ is quasi-Mordellic.\\
(b). If $D_i$ is ample for each $i$ and $r> 2mq$ then $X\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
\label{cor4b}
Let $X$ be a complex projective variety. Let $q=\dim X$. Let $D=\sum_{i=1}^{r} D_i$ be a divisor on $X$ such that the $D_i$'s are effective Cartier divisors and the intersection of any $m+1$ distinct $D_i$'s is empty.\\\\
(a). If $D_i$ is quasi-ample for each $i$ and $r>2mq$ then $X\backslash D$ is quasi-Brody hyperbolic.\\
(b). If $D_i$ is ample for each $i$ and $r> 2mq$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{theoremb}
Aside from the statement about being complete hyperbolic and hyperbolically imbedded, the same proof works for both Theorems \ref{cor4} and \ref{cor4b}.
\begin{proof}
We'll prove part (a) first. Note that if $\pi:X'\to X$ is a birational morphism and the conclusions of part (a) of the theorems hold for $\pi^*D$ on $X'$ then they hold for $D$ on $X$. Therefore, by Lemma \ref{reduce}, we may assume (extending $k$ in the Diophantine case if necessary) that $X$ is nonsingular, every irreducible component of $D$ is nonsingular, and $D_i$ is nef for all $i$. The statement then follows from Lemma \ref{equi}, Theorem \ref{cor3}, and Theorems \ref{maina} and \ref{mainb}.
For part (b), we note that by (a) any set of $D$-integral points (resp. the image of any holomorphic map $f:\mathbb{C} \to X \backslash D$) is not Zariski-dense. Let $R$ be a set of $D$-integral points (resp. the image of a holomorphic map $f:\mathbb{C} \to X \backslash D$). Let $Y$ be an irreducible component of the Zariski-closure of $R$. Suppose $\dim Y>0$. Then $D$ pulls back to a sum of $r$ ample (hence quasi-ample) divisors on $Y$ such that the intersection of any $m+1$ of them is empty. But $R\cap Y$ is a dense set of $D|_Y$-integral points on $Y$ (resp. the image of a holomorphic map $f:\mathbb{C} \to Y \backslash D$), contradicting part (a) proven above since $r>2mq>2m\dim Y$. Therefore $\dim Y=0$.
To prove the extra hyperbolicity statements in (b) in the analytic case, we use Theorem \ref{hyperbolic}. Let $\emptyset \subset J \subset \{1,\ldots,r\}$. Let $s=\#J$. Let $X'=\bigcap_{j\in J}D_j$. We may clearly assume that $X'\not\subset D_i$ for any $i\in I\backslash J$ and that $\dim X'>0$. Let $D'=\sum_{i\in I\backslash J}D_i|_{X'}$. Then $D'$ is a sum of $r-s$ ample divisors on $X'$ and the intersection of any $m-s+1$ of the ample divisors is empty since $X'$ is already an intersection of $s$ of the $D_i$'s. But $r>2mq$ implies that $r-s>2(m-s)\dim X'$. Therefore by what we have proven above, $X'\backslash D'$ is Brody hyperbolic. So by Theorem~\ref{hyperbolic}, $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$.
\end{proof}
We can prove our Main Conjectures in the simple case $m=1$ by reducing to Siegel's and Picard's theorems. We will need the following Bertini theorem (see \cite[Th. 7.19]{Ii}).
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{theorem}
\label{Bertini}
Let $|D|$ be a base-point free linear system on a nonsingular projective variety $X$ with $\dim \Phi_{D}(X)\geq 2$. Then every member of $|D|$ is connected and a general member of $|D|$ is nonsingular and irreducible.
\end{theorem}
\begin{lemma}
\label{m1}
Suppose $D=D_1+D_2$ is a Cartier divisor on a projective variety $X$ with $\kappa(D_1)>0,\kappa(D_2)>0$ and $D_1\cap D_2=\emptyset$. Then $\kappa(D)=\kappa(D_1)=\kappa(D_2)=1$.
\begin{proof}
By Lemma \ref{reduce}, we may assume that $X$ is nonsingular and $|D|$ is base-point free. If $\kappa(D)\geq 2$ then $\dim \Phi_{nD}(X)\geq 2$ for some $n>0$. But by Theorem \ref{Bertini}, every divisor in $|nD|$ is connected, contradicting $D_1\cap D_2=\emptyset$.
\end{proof}
\end{lemma}
\begin{theorem}
\label{tm1}
The Main Conjectures, Conjectures \ref{conjmaina},B through \ref{conj2a},B, are true if $m=1$ (i.e. $D_i\cap D_j=\emptyset$ for all $i\neq j$).
\end{theorem}
\begin{proof}
By the above lemma, it suffices to prove the conjectures when $D=\sum_{i=1}^rD_i$ with $r>2$, and $\kappa(D)=1$. By Lemma \ref{reduce}, we may assume that $X$ is nonsingular and $D$ is base-point free. For $n\gg 0$, $\Phi_{nD}(X)$ is a nonsingular curve $C$ and $\Phi_{nD}$ has connected fibers. Therefore, since $D_i\cap D_j=\emptyset$ for $i\neq j$, we have $\Phi_{nD}(X\backslash D)=C\backslash\{r\text{ points}\}$. Since $r>2$, we are done by Siegel's and Picard's theorems.
\end{proof}
\section{A Filtration Lemma}
We'll now show how some of the results in the last section may be improved by use of a linear algebra lemma on filtrations. The idea of using this lemma, as well as its statement and proof, are taken from the paper \cite{Co2}. Corvaja and Zannier used it to prove a result on integral points on surfaces, and it will be essential for our results on surfaces in the next section also.
\begin{lemma}
Let $V$ be a vector space of finite dimension $d$ over a field $k$. Let $V=W_1\supset W_2\supset \cdots \supset W_h,V=W_1^*\supset W_2^*\supset \cdots \supset W_{h^*}^*$ be two filtrations on $V$. There exists a basis $v_1,\dots, v_d$ of $V$ which contains a basis of each $W_j$ and $W_j^*$.
\end{lemma}
\begin{proof}
The proof will be by induction on $d$. The case $d=1$ is trivial. By refining the first filtration, we may assume without loss of generality that $W_2$ is a hyperplane in $V$. Let $W_i'=W_i^*\cap W_2$. By the inductive hypothesis there exists a basis $v_1,\ldots,v_{d-1}$ of $W_2$ containing a basis of each of $W_3,\ldots, W_h$ and $W_1',\ldots,W_h'$. If $W_i^*\subset W_2$ for $i>1$ then $W_i'=W_i^*$ for $i>1$. So in this case if we complete $v_1,\ldots,v_{d-1}$ to any basis of $V$ we are done. Otherwise, let $l$ be the maximal index with $W_l^* \not\subset W_2$ and let $v_d\in W_l^*\backslash W_l'$. We claim that $B=\{v_1,\ldots,v_d\}$ is a basis of $V$ with the required property. It clearly contains a basis of $W_i$ for each $i$. Let $i\in \{1,\ldots,h^*\}$. If $i>l$ then $W_i^*=W_i'$ and so by construction $B$ contains a basis of $W_i^*$. If $i\leq l$ then $v_d\in W_l^*\backslash W_l' \subset W_i^*\backslash W_i'$. Since $B$ contains a basis $B_i'$ of $W_i'$ and $W_i'$ is a hyperplane in $W_i^*$, we see that $B_i'\cup \{v_d\}$ is a basis of $W_i^*$.
\end{proof}
Using our notation from the last section, suppose that for $P\in D$ we have $D_P=D_{P,1}+D_{P,2}$ where $D_{P,1}$ and $D_{P,2}$ are effective divisors with no irreducible components in common. We may then prove the following versions of Lemma \ref{large} and Theorem \ref{cor2}.
\begin{lemma}
\label{large2}
Let $D=\sum_{i=1}^r D_i$ be a nonzero divisor on a nonsingular variety $X$ with $D_i$ effective for each $i$. Let $P\in D$. Let $f_{P,j}(m,n)=l(nD-mD_{P,j})-l(nD-(m+1)D_{P,j})$ for $j=1,2$. If there exists $n>0$ such that either $\sum_{m=0}^{\infty}(m-n)f_{P,j}(m,n)>0$ or $D_{P,j}=0$ for all $P\in D$ and $j=1,2$ then $nD$ is very large.
\end{lemma}
\begin{theorem}
\label{cor22}
Let $X$ be a nonsingular variety. Let $q= \dim X$. Let $D=\sum_{i=1}^{r}D_i$ be a divisor on $X$ such that $D_{P,j}$ is nef for all $P\in D$ and $j=1,2$. Suppose also that every irreducible component of $D$ is nonsingular. If
\begin{equation*}
D^q>2q D^{q-1}.D_{P,j}, \qquad \forall P\in D, j=1,2
\end{equation*}
then $nD$ is very large for $n\gg 0$.
\end{theorem}
The proofs are similar to the proofs of Lemma \ref{large} and Theorem \ref{cor2}. The only difference is that in the proof of Lemma \ref{large2}, we look at the two filtrations of $L(nD)$ given by $W_j=L(nD-jD_{P,1})$ and $W_j^*=L(nD-jD_{P,2})$ and we use the filtration lemma to construct a basis $f_1,\ldots,f_{l(nD)}$ that contains a basis for each $W_j$ and $W_j^*$.
Suppose now that $D=\sum_{i=1}^rD_i$ where the $D_i$'s are effective divisors and the intersection of any $m+1$ distinct $D_i$'s is empty. We may then write $D_P=D_{P,1}+D_{P,2}$ where $D_{P,1}$ and $D_{P,2}$ are each not a sum of more than $[\frac{m+1}{2}]$ $D_i$'s, where $[x]$ denotes the greatest integer in $x$. Using this, we get the following improvements to the part (a)'s of Theorems \ref{cor4} and \ref{cor4b}.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
\label{cor42}
Let $X$ be a projective variety defined over a number field $k$. Let $q=\dim X$. Let $D=\sum_{i=1}^{r} D_i$ be a divisor on $X$ defined over $k$ such that the $D_i$'s are effective Cartier divisors with no irreducible components in common and the intersection of any $m+1$ distinct $D_i$'s is empty. If $D_i$ is quasi-ample for each $i$ and $r> 2[\frac{m+1}{2}]q$ then $X\backslash D$ is quasi-Mordellic.
\end{theorema}
\begin{theoremb}
\label{cor4b2}
Let $X$ be a complex projective variety. Let $q=\dim X$. Let $D=\sum_{i=1}^{r} D_i$ be a divisor on $X$ such that the $D_i$'s are effective Cartier divisors with no irreducible components in common and the intersection of any $m+1$ distinct $D_i$'s is empty. If $D_i$ is quasi-ample for each $i$ and $r>2[\frac{m+1}{2}]q$ then $X \backslash D$ is quasi-Brody hyperbolic.
\end{theoremb}
Unfortunately, we need the requirement that the $D_i$'s have no irreducible components in common so that we may have $D_{P,1}$ and $D_{P,2}$ with no irreducible components in common (which is necessary in proving Lemma \ref{large2}). Because of this, we cannot prove a finiteness result about ample divisors as we did in the last section, since the restrictions of the $D_i$'s to a subvariety of $X$ may have irreducible components in common.
\section{Surfaces}
\label{ssurf}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
We will now see that we may make the results of the last two sections more precise if we restrict to the case where $X$ is a surface. With regards to integral points, this section builds on some of the work in \cite{Co2}. Corvaja and Zannier prove, essentially, Theorem \ref{surf} \cite[Main Theorem]{Co2} and they prove Theorem \ref{surf3a} when $m=2$ and the $D_i$'s have multiples which are all numerically equivalent. The Nevanlinna-theoretic analogues of the results in \cite{Co2} were proved by Ru and Liu in \cite{Ru4}. Our results overlap with their results as well.
We first prove a consequence of the Hodge Index theorem.
\begin{lemma}
\label{Hodge}
Let $D$ be a divisor on a nonsingular surface $X$ with $D^2>0$. Then $(D^2)(E^2)\leq (D.E)^2$ for any divisor $E$ on $X$.
\end{lemma}
\begin{proof}
By the Hodge index theorem, the intersection pairing on Num $X\bigotimes\mathbb{R}$ can be diagonalized with one $+1$ on the diagonal and all other diagonal entries $-1$. We will identify elements of Pic $X$ as elements of Num $X\bigotimes\mathbb{R}$ in the canonical way. Extend $D$ to an orthogonal basis $B$ of Num $X\bigotimes\mathbb{R}$. Let $E$ be any divisor on $X$. Writing $E$ in the basis $B$, it is apparent from the Hodge index theorem that $(D^2)(E^2)\leq (D.E)^2$.
\end{proof}
For surfaces, the more precise version of Theorem~\ref{cor22} is
\begin{theorem}[Corvaja-Zannier]
\label{surf}
Let $X$ be a nonsingular projective surface. Let $D=\sum_{i=1}^{r}D_i$ be a nef divisor on $X$ with the $D_i$'s effective divisors and $D^2>0$. For $P\in D$, let $D_P=\sum_{i:P\in D_i}D_i=D_{P,1}+D_{P,2}$ where $D_{P,1}$ and $D_{P,2}$ are effective divisors with no irreducible components in common. Suppose that for all $P\in D$, $j=1,2$ and $m,n>0$ we have either $l(nD-mD_{P,j})=0$ or
\begin{equation*}
l(nD-mD_{P,j})-l(nD-(m+1)D_{P,j})\leq (nD-mD_{P,j}).mD_{P,j}+O(1)
\end{equation*}
where the constant does not depend on $m$ or $n$. Let $A_{P,j}=D_{P,j}.D_{P,j},B_{P,j}=D.D_{P,j}$, and $C=D.D$ for $j=1,2$. If for all $P \in D$ and $j=1,2$ either we have $D_{P,j}=0$ or we have
\begin{align*}
&A_{P,j}>0 \Longrightarrow B_{P,j}^2-2A_{P,j}C+3A_{P,j}B_{P,j}+(3A_{P,j}-B_{P,j})\sqrt{B_{P,j}^2-A_{P,j}C}<0\\
&A_{P,j}=0 \Longrightarrow C>4B_{P,j}\\
&A_{P,j}<0 \Longrightarrow B_{P,j}^2-2A_{P,j}C+3A_{P,j}B_{P,j}+(3A_{P,j}-B_{P,j})\sqrt{B_{P,j}^2-A_{P,j}C}>0
\end{align*}
then $nD$ is very large for $n\gg 0$ (note that by Lemma \ref{Hodge} $B_{P,j}^2-A_{P,j}C>0$).
\end{theorem}
\begin{proof}
Let $P \in D$ and $j\in \{1,2\}$ with $D_{P,j}\neq 0$. Let $A=A_{P,j}$ and $B=B_{P,j}$. By assumption, we have
\begin{align*}
f_{P,j}(m,n)&=\dim H^0(X,\mathcal{O}(nD-m D_{P,j}))-\dim H^0(X,\mathcal{O}(nD-(m+1)D_{P,j}))\\
&\leq nB-mA+O(1)
\end{align*}
where the constant in the $O(1)$ does not depend on $m$ or $n$. We have
\begin{equation*}
l(nD)=\frac{D^2}{2}n^2+O(n)=\frac{C}{2}n^2+O(n).
\end{equation*}
Solving
\begin{equation*}
\sum_{m=0}^{M(n)} nB-mA+O(1)=\frac{C}{2}n^2+O(n)= l(nD)
\end{equation*}
for $M(n)$, we get
\begin{align*}
&M(n)=\frac{B \pm \sqrt{B^2-AC}}{A}n+O(1), &A \neq 0\\
&M(n)=\frac{C}{2B}n+O(1), &A=0,B\neq 0\\
&M(n)=O(n^2), &A=0,B=0.
\end{align*}
From now on, we will always choose the minus sign in the first expression above. We also have $\sum_{m=0}^{\infty}f_{P,j}(m,n)=l(nD)$. Therefore by Lemma \ref{CZlemma},
\begin{equation}
\label{MP}
\sum_{m=0}^{\infty}(m-n)f_{P,j}(m,n) \geq \sum_{m=0}^{M(n)}m(nB-mA+O(1))-nl(nD).
\end{equation}
Let $K=\frac{B - \sqrt{B^2-AC}}{A}$. If $A \neq 0$ then substituting $K$ into (\ref{MP}) we get
\begin{equation*}
\sum_{m=0}^{\infty}(m-n)f_{P,j}(m,n)\geq(-\frac{A}{3}K^3+\frac{B}{2}K^2-\frac{C}{2})n^3+O(n^2)
\end{equation*}
So if $-\frac{A}{3}K^3+\frac{B}{2}K^2-\frac{C}{2}>0$ then by Lemma \ref{large2}, $nD$ will be very large for $n\gg 0$. Algebraic simplification then gives the theorem in the case $A \neq 0$. The other cases are similar.
\end{proof}
\begin{lemma}
\label{clemma}
Let $X$ be a nonsingular projective surface. Let $C$ be an irreducible curve on $X$ and $D$ any divisor on $X$. Then
\begin{equation*}
h^0(D)-h^0(D-C)\leq \max\{0,1+C.D\}.
\end{equation*}
\end{lemma}
\begin{proof}
The statement depends only on the linear equivalence class of $D$, so replacing $D$ by an appropriate divisor linearly equivalent to $D$, we may assume that the support of $D$ does not contain any possible singularity of $C$. By Lemma \ref{exact} we have
\begin{equation*}
h^0(D)-h^0(D-C)\leq \dim H^0(C,\mathcal{O}(D)|_C)
\end{equation*}
Since the support of $D$ does not contain any singularity of $C$, $\mathcal{O}(D)|_C$ has degree $C.D$ on $C$ and $\dim H^0(C,\mathcal{O}(D)|_C)\leq \max\{0,1+C.D\}$.
\end{proof}
\begin{lemma}
\label{slemma}
Let $X$ be a nonsingular projective surface. Let $D$ be a nef divisor on $X$. Let $E$ be an effective divisor on $X$ such that either $E$ is linearly equivalent to an irreducible curve or for every irreducible component $C$ of $E$, $C.E\leq 0$. Then for all $m,n>0$ either $l(nD-mE)=0$ or
\begin{equation}
\label{ineq}
l(nD-mE)-l(nD-(m+1)E)\leq (nD-mE).E+O(1)
\end{equation}
where the constant is independent of $m$ and $n$.
\end{lemma}
\begin{proof}
In the first case, suppose $E$ is linearly equivalent to an irreducible curve $C$. If $(nD-mE).E\geq 0$ then (\ref{ineq}) holds by Lemma \ref{clemma}. If $(nD-mE).E=nD.C-mC.C<0$ then since $D$ is nef, we must have $C.C>0$. But if $l(nD-mE)>0$ then $nD-mE$ is linearly equivalent to an effective divisor $F=G+mC$ where $m\geq 0$ and $G$ is an effective divisor not containing $C$. Since clearly $G.C\geq 0$, $F.C=(nD-mE).E<0$ implies $C.C<0$, a contradiction. So either $l(nD-mE)=0$ or (\ref{ineq}) holds in this case.
Now suppose we are in the second case, where for every irreducible component $C$ of $E$, $C.E\leq 0$. Let $E=\sum_{j=1}^{k}a_j C_j$, where each $C_j$ is a distinct prime divisor. Then as in the proof of Theorem~\ref{cor2} we have
\begin{multline*}
l(nD-mE)-l(nD-(m+1)E) \leq \\
\sum_{j=1}^k \sum_{l=0}^{a_{j}-1}\dim H^0(C_{j},i^*_{C_{j}}\mathcal{O}(nD-mE -\sum_{j'=1}^{j-1}a_{j'}C_{j'}-lC_{j}))
\end{multline*}
But
\begin{align*}
\dim H^0(C_{j},i^*_{C_{j}}\mathcal{O}(nD-mE -\sum_{j'=1}^{j-1}a_{j'}C_{j'}-lC_{j}))&\leq \dim H^0(C_{j},i^*_{C_{j}}\mathcal{O}(nD-mE))+O(1)\\
&\leq (nD-mE).C_j+O(1),
\end{align*}
where the constant is independent of $m$ and $n$. The second inequality follows since $(nD-mE).C_j\geq nD.C_j\geq 0$ as $D$ is nef and $E.C_j\leq 0$. Combining the above inequalities, we then see that (\ref{ineq}) always holds in this case.
\end{proof}
Going back to the General Setup of Section \ref{gsetup}, we have
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
\label{surf3a}
Let $X$ be a projective surface. Suppose the $D_i$'s have no irreducible components in common.\\\\
(a). If $D_i$ is quasi-ample for all $i$ and $r\geq 4[\frac{m+1}{2}]$ then $X\backslash D$ is quasi-Mordellic.\\
(b). If $D_i$ is ample for all $i$ and either $m$ is even and $r>2m$ or $m$ is odd and $r>2m+1$ then $X\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
\label{surf3b}
Let $X$ be a projective surface. Suppose the $D_i$'s have no irreducible components in common.\\\\
(a). If $D_i$ is quasi-ample for all $i$ and $r\geq 4[\frac{m+1}{2}]$ then $X\backslash D$ is quasi-Brody hyperbolic.\\
(b). If $D_i$ is ample for all $i$ and either $m$ is even and $r>2m$ or $m$ is odd and $r>2m+1$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{theoremb}
\begin{proof}
We'll prove the part (a)'s first. It suffices to prove these in the case $r=4[\frac{m+1}{2}]$. As in the proofs of Theorems \ref{cor4} and \ref{cor4b}, we may use Lemma \ref{reduce} to reduce to the case where $X$ is nonsingular, $|D_i|$ is base-point free for all $i$, and $\dim \Phi_{D_i}(X)=2$ for all $i$. Therefore $D_i^2>0$ and $D_i$ is nef for each $i$. By Lemma \ref{equi}, $D$ is equidegreelizable. So we may find positive integers $a_1,\ldots,a_r$ such that if $D'=\sum_{i=1}^{r}a_iD_i$ then $\frac{a_iD_i.D'}{(D')^2}$ is arbitrarily close to $\frac{1}{r}$ for all $i$. Since at most $m$ $D_i$'s meet at any given point, $D'_P$ is a sum of at most $m$ $a_iD_i$'s for any $P\in D'$. Therefore we may write $D'_P=D'_{P,1}+D'_{P,2}$ where each $D'_{P,j}$ is a sum of at most $[\frac{m+1}{2}]$ $a_iD_i$'s, and $D'_{P,1}$ and $D'_{P,2}$ have no irreducible components in common. Note that when $D'_{P,j}\neq 0$, we have, from our assumptions on the $D_i$'s, that $|D'_{P,j}|$ is base-point free and $\dim \Phi_{D'_{P,j}}(X)=2$. So by Theorem \ref{Bertini}, $D'_{P,j}$ is linearly equivalent to an irreducible curve. Therefore, by Lemma \ref{slemma}, we will be able to apply Theorem \ref{surf} to $D'$.
The hardest case is clearly when $D'_{P,j}$ is a sum of the maximum $[\frac{m+1}{2}]$ $a_iD_i$'s. For simplicity, we will now restrict to this case. It follows that in the notation of Theorem \ref{surf} we may take, for all such $P$ and $j$,
\begin{equation*}
\left|\frac{C}{B_{P,j}}-\frac{r}{[\frac{m+1}{2}]}\right|=\left|\frac{C}{B_{P,j}}-4\right|<\epsilon
\end{equation*}
where by adjusting the $a_i$'s in $D'$, $\epsilon$ may be made arbitrary close to $0$ while at the same time $\frac{A_{P,j}}{B_{P,j}}$ is positive and bounded away from $0$. Furthermore, by Lemma \ref{Hodge}, $\frac{A_{P,j}}{B_{P,j}}\leq \frac{B_{P,j}}{C}$. Let $a=\frac{A_{P,j}}{B_{P,j}}$ and $c=\frac{C}{B_{P,j}}$. Then by Theorem \ref{surf}, we must show that
\begin{equation}
\label{sineq}
1-2ac+3a+(3a-1)\sqrt{1-ac}<0
\end{equation}
where $0<a\leq\frac{1}{c}$. When $c=4$, we get $1-5a+(3a-1)\sqrt{1-4a}$, which is easily seen to have a root only at $a=0$ for $0\leq a\leq\frac{1}{4}$, and is negative for $0<a\leq\frac{1}{4}$ since putting $a=\frac{1}{4}$ gives $-\frac{1}{4}$. So when $c=4+\epsilon$, since $a$ is bounded away from zero as $\epsilon \to 0$, we see that (\ref{sineq}) is negative for small enough $\epsilon$. Therefore by Theorem \ref{surf}, $nD'$ is very large for $n\gg 0$. Since $D'$ is quasi-ample, $\Phi_{nD'}$ is a birational map to projective space for some arbitrarily large $n$. Therefore by Theorems \ref{maina} and \ref{mainb} we are done, as $D$ and $D'$ have the same support.
Assume the hypotheses in the part (b)'s. Let $Y$ be the Zariski-closure of a set of $D$-integral points (resp. $f(\mathbb{C})$). By what we have proven above, $\dim Y\leq 1$. If $\dim Y=1$, let $C$ be an irreducible component of this curve with $\dim C>0$. Since each $D_i$ is ample, $D_i$ must intersect $C$ in a point. Since at most $m$ $D_i$'s meet at a point and $r>2m$, we see that $D|_C$ contains at least $3$ distinct points. Therefore by Siegel's (resp. Picard's) theorem we get a contradiction as the above gives a dense set of $D|_C$-integral points (resp. a dense holomorphic map $\mathbb{C} \to C\backslash D|_C$). This same argument and Theorem \ref{hyperbolic} show that in the analytic case $X\backslash D$ is hyperbolic and hyperbolically embedded in $X$.
\end{proof}
It is possible to make minor improvements to this theorem. For example,
\begin{theorema}
\label{surf4a}
Let $X$ be a nonsingular projective surface. Suppose $m=2$, $D=\sum_{i=1}^4D_i$, $D_i.D_j>0$ for $i\neq j$, $D_1^2>0$, $D_i$ is nef for all $i$, and the $D_i$'s have no irreducible components in common. Suppose also that the conclusion of Lemma \ref{slemma} holds with $D$ any positive integral linear combination of the $D_i$'s and $E=D_i$, for $i=1,2,3,4$. Then $X\backslash D$ is quasi-Mordellic.
\end{theorema}
\begin{theoremb}
\label{surf4b}
With the same hypotheses as above, in the analytic setting, $X\backslash D$ is quasi-Brody hyperbolic.
\end{theoremb}
\begin{proof}
We first show that for any $\epsilon>0$, $(\sum_{i=1}^4e^{a_i}D_i)^2\geq e^{\frac{2}{3}\max_i\{a_i\}}$ on the plane $(1+\epsilon)a_1+\sum_{i=2}^4a_i=0$. If $\max_i\{a_i\}=a_1$ then $(\sum_{i=1}^4e^{a_i}D_i)^2\geq e^{2a_1}D_1^2\geq e^{2a_1}$. Otherwise, if $\max_i\{a_i\}=a_j$, $j>1$, then clearly we must have $a_k\geq -\frac{a_j}{3}$ for some $j\neq k$. Then $(\sum_{i=1}^4e^{a_i}D_i)^2\geq e^{a_j+a_k}D_j.D_k\geq e^{\frac{2}{3}a_j}$ since $D_j.D_k\geq 1$. Therefore $(\sum_{i=1}^4e^{a_i}D_i)^2$ takes a minimum on the plane $(1+\epsilon)a_1+\sum_{i=2}^4a_i=0$. So looking at the Lagrange multiplier equations as in Lemma \ref{equi}, there exist real numbers $b_i>0,\lambda>0$ (depending on $\epsilon$) such that if $D'=\sum_{i=1}^4b_iD_i$ then $b_1D_1.D'=(1+\epsilon)\lambda$ and $b_iD_i.D'=\lambda$ for $i=2,3,4$, or written differently, $\frac{(D')^2}{b_1D_1.D'}=\frac{4+\epsilon}{1+\epsilon}$ and $\frac{(D')^2}{b_iD_i.D'}=4+\epsilon>4$ for $i=2,3,4$. Note also that it follows from the inequality we proved above that in terms of $a_1,\ldots,a_4$, the region where $(\sum_{i=1}^4e^{a_i}D_i)^2$ takes a minimum may be bounded independently of $\epsilon$. Therefore there exist positive constants $K,K'$ independent of $\epsilon$, such that we may choose $K<b_i<K'$ for all $i$, and in particular, as $\epsilon\to 0$, $\frac{(b_1D_1)^2}{b_1D_1.D'}$ is bounded away from zero.
We now choose positive integers $c_i$ such that $\frac{c_i}{c_j}$ closely approximates $\frac{b_i}{b_j}$, and let $E=\sum_{i=1}^4c_iD_i$. Having chosen $\epsilon$ small enough and the integers $c_i$ correctly, we will then have $E^2>4c_iD_i.E$ for $i=2,3,4$ and we will have $\frac{E^2}{c_1D_1.E}$ close enough to $4$ (see the proof of Theorem \ref{surf3a}, B) so that the inequalities in $\ref{surf}$ hold for $E_{P,j}=c_iD_i$ for any $i$. Since $m=2$, we may always take $E_{P,j}=0$ or $E_{P,j}=c_iD_i$ for some $i$. By our hypotheses, we may apply Theorem \ref{surf}, so $nE$ is very large for $n\gg 0$. Since $D_1^2>0$, $E$ is quasi-ample. So we are done by Theorems \ref{maina} and \ref{mainb}, as $D$ and $E$ have the same support.
\end{proof}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{example}
Let $X=\mathbb{P}^1\times \mathbb{P}^1$. Let $D_1=\{0\}\times \mathbb{P}^1, D_2=\mathbb{P}^1\times \{0\}$, and let $D_3$ and $D_4$ be ample effective divisors on $X$. Suppose also that the intersection of any three of the $D_i$'s is empty. Let $D=\sum_{i=1}^4D_i$. Then the hypotheses of Theorems \ref{surf4a}, B are satisfied and $X\backslash D$ is quasi-Mordellic and quasi-Brody hyperbolic. Note also that $X\backslash D_1\cup D_2\cong \mathbb{A}^2\cong \mathbb{P}^2\backslash \{\text{a line}\}$. Therefore, we can also prove many theorems for $\mathbb{P}^2\backslash D$, where $D$ is a sum of three effective divisors on $\mathbb{P}^2$.
\end{example}
Recently, Corvaja and Zannier \cite{Co6} have shown another way their methods may get results on $\mathbb{P}^2\backslash D$ where $D$ is a sum of three effective divisors satisfying certain hypotheses.
We have the following general corollary to the above theorems.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{corollarya}
Let $X$ be a projective surface. Suppose $m=2$, $D=\sum_{i=1}^4D_i$, $D_1,D_2,D_3$ are quasi-ample, $\kappa(D_4)>0$, and the $D_i$'s have no irreducible components in common. Then $X\backslash D$ is quasi-Mordellic.
\end{corollarya}
\begin{corollaryb}
Let $X$ be a projective surface. Suppose $m=2$, $D=\sum_{i=1}^4D_i$, $D_1,D_2,D_3$ are quasi-ample, $\kappa(D_4)>0$, and the $D_i$'s have no irreducible components in common. Then $X\backslash D$ is quasi-Brody hyperbolic.
\end{corollaryb}
\begin{proof}
We first reduce to the situation of Lemma \ref{reduce}. So $X$ is nonsingular, each $D_i$ is nef, and $D_1^2,D_2^2,D_3^2>0, D_4^2\geq 0$ By Lemma \ref{m1}, $D_i.D_j>0$ for $i\neq j$. For $i=1,2,3$ and $n>0$ $nD_i$ is linearly equivalent to an irreducible curve by Theorem \ref{Bertini}, since by our reductions $|nD_i|$ is base-point free and $\dim \Phi_{nD_i}(X)=2$. The same holds for $nD_4$ if $D_4^2>0$. If $D_4^2=0$, then for every irreducible component $C$ of $D_4$ we must have $C.D_4=0$ since $D_4$ is nef. This verifies the hypotheses of Lemma \ref{slemma} with $E=D_i$ for $i=1,2,3,4$. Therefore, we may apply Theorems \ref{surf4a}, B.
\end{proof}
We note that one can construct examples where $m=2$, $D_1$ and $D_2$ are quasi-ample, $\kappa(D_3)=\kappa(D_4)=1$, the $D_i$'s have no irreducible components in common, and there exist dense sets of $D$-integral points. We now prove a theorem in the case where we only have $\kappa(D_i)>0$ for all $i$.
\begin{theorema}
Let $X$ be a projective surface. Suppose the $D_i$'s have no irreducible components in common. If $\kappa(D_i)>0$ for all $i$ and $r> 4[\frac{m+1}{2}]$ then there does not exist a Zariski-dense set of $D$-integral points on $X$.
\end{theorema}
\begin{theoremb}
Let $X$ be a projective surface. Suppose the $D_i$'s have no irreducible components in common. If $\kappa(D_i)>0$ for all $i$ and $r> 4[\frac{m+1}{2}]$ then there does not exist a holomorphic map $f:\mathbb{C}\to X\backslash D$ with Zariski-dense image.
\end{theoremb}
\begin{proof}
We first reduce to the situation of Lemma \ref{reduce}, so in particular $|D_i|$ is base-point free for all $i$. In this case, for any subset $I\subset \{1,\ldots,r\}$, if $D_I=\sum_{i\in I}D_i$ is quasi-ample, there exists $n_I>0$ such that $\dim \Phi_{n_ID_I}(X)=2$. Since the $D_i$'s are nef, this happens if and only if $D_i.D_j>0$ for some $i,j\in I$. Let $N=\prod_{I}n_I$ where the $I$ ranges over subsets such that $D_I$ is quasi-ample. Let $D'=ND$ and $D_i'=ND_i$. Then we see that for any nonnegative integral linear combination, $E$, of the $D_i'$'s, if $E$ is quasi-ample, then $E$ is linearly equivalent to an irreducible divisor since $|E|$ is base-point free and $\dim \Phi_E(X)=2$, and otherwise, for every irreducible component $C$ of $E$ we have $C.E=0$. Therefore, by Lemma \ref{slemma}, replacing $D$ by $D'$, we may assume that we may apply Theorem \ref{surf} to any nonnegative linear combination of the $D_i$'s.
By Theorem \ref{tm1}, we are done if any three of the $D_i$'s have pairwise empty intersection. So suppose that this is not the case. Then we have $m\geq 2$ and $r\geq 5$. We now show that $D$ is equidegreelizable. As in the proof of Lemma \ref{equi}, it suffices to show that $(\sum_{i=1}^re^{a_i}D_i)^2$ attains a minimum on the plane $\sum_{i=1}^ra_i=0$. For this, it will suffice to show that $(\sum_{i=1}^re^{a_i}D_i)^2\geq e^{\frac{1}{3}\max_i\{a_i\}}$. Suppose $\max_i\{a_i\}=a_j$ for $j\in \{1,\ldots,r\}$. Let $a_k$ and $a_l$ be some choice of the next largest $a_i$'s. Clearly, since $\sum_{i=1}^ra_i=0$, we must have $a_k,a_l\geq -\frac{2a_j}{r-2}\geq -\frac{2}{3}a_j$ since $r\geq 5$. We now show that either $D_j.D_k\geq 1$ or $D_j.D_l\geq 1$. Suppose otherwise. Then by our assumption, we must have $D_k.D_l\geq 1$. But then $D_k+D_l$ is quasi-ample, and so we must have $(D_k+D_l).D_j\geq 1$ by Lemma \ref{m1}, a contradiction. So if, say, $D_j.D_k\geq 1$ then $(\sum_{i=1}^re^{a_i}D_i)^2\geq e^{a_j+a_k}D_j.D_k\geq e^{\frac{1}{3}\max_i\{a_i\}}$, as was to be shown. Since $D$ is equidegreelizable, there exist positive integers $c_i$ such that $D'=\sum_{i=1}^rc_iD_i$, and $\frac{c_iD_i.D'}{(D')^2}$ is as close as we like to $\frac{1}{r}$. Since we may choose $D'_{P,j}$ to consist of a sum of at most $[\frac{m+1}{2}]$ $c_iD_i$'s and $r>4[\frac{m+1}{2}]$, we may choose the $c_i$'s so that we always have $C>4B_{P,j}$. We also have $A_{P,j}\geq 0$. But then, as we have seen previously, the inequalities of Theorem \ref{surf} will be satisfied.
\end{proof}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
To summarize some of the results in this section:
\begin{theorem}
Let $X$ be a projective surface. Suppose that $m\leq 2$ and the $D_i$'s have no irreducible components in common. Then all of the Main Conjectures (Conjectures \ref{conjmaina},B-\ref{conj2a},B) are true.
\end{theorem}
\section{Small S}
\label{ssmall}
We now prove some theorems in the special case that $\#S$ is small relative to the number of components of $D$. Throughout we use the general Diophantine setup of Section \ref{gsetup}.
\begin{theorem}
\label{sthm}
Suppose that $D_i$ is defined over $k$ for all $i$. Let $s=\#S$.\\\\
(a). If $D_i$ is quasi-ample for all $i$ and $r>ms$ then there exists a proper closed subset $Z\subset X$ such that for any set $R$ of $(D,S)$-integral points on $X$, $R\backslash Z$ if finite.\\
(b). If $D_i$ is ample for all $i$ and $r>ms$ then all sets of $(D,S)$-integral points on $X$ are finite.
\end{theorem}
\begin{proof}
We reduce to the case where $X$ is nonsingular. We prove part (a) first. Our proof is a modification of the proof of Theorem~\ref{maina}. Suppose $R$ is a Zariski-dense set of $(D,S)$-integral points on $X$. Then as in the proof of Theorem \ref{maina}, there exists a sequence $P_i$ in $R$ such that for each $v$ in $S$, $\{P_i\}$ converges to a point $P_v\in X(k_v)$ and $\bigcup\{P_i\}$ is Zariski-dense in $X$. Since $r>ms$, there exists an index $i$ such that $P_v\notin D_i(k_v)$ for all $v\in S$. Since $D_i$ is quasi-ample, it follows from Lemma \ref{exact} and the argument in Lemma \ref{nefbig} that for some $n>0$, $l(nD_i-\sum_{j\neq i}D_i)>0$. Then $(n+1)D_i-\sum_{j\neq i}D_i$ is quasi-ample, and so for some $n'>0$, and $E=n'(n+1)D_i-n'\sum_{j\neq i}D_i$, $\Phi_{E}$ is birational. Now let $S'$ be the set of places $v\in S$ such that $P_v \in D(k_v)$. Let $\phi_1,\ldots,\phi_{l(E)}$ be a basis for $L(E)$ over $k$. Then for any $v\in S'$, $\text{ord}_F\prod_{i=1}^{l(E)}\phi_i>0$ for every irreducible component $F$ of $D$ such that $P_v \in F(k_v)$. This is precisely what we used the largeness hypothesis for in the proof of Theorem \ref{maina}. Let $\phi=(\phi_1,\ldots,\phi_{l(E)})$. Let $L_{jv}=\phi_j$ for $j=1,\ldots,l(E)$ and $v\in S$. Then the same proof as in Theorem \ref{maina} (replacing $D$ by $E$ in appropriate places) proves part (a).
For (b), let $R$ be a set of $(D,S)$-integral points on $X$. Let $Y$ be an irreducible component of the Zariski-closure, $\overline{R}$, of $R$. Suppose $\dim Y>0$. Then $D$ pulls back to a sum of $r$ ample effective divisors on $Y$ such that at most $m$ of them meet at a point. But then part (a) applied to $D|_Y$ contraticts the fact that $R\cap Y$ is a dense set of $(D|_Y,S)$-integral points. Therefore $\dim Y=0$.
\end{proof}
When $\#S=1$ this theorem gives a particularly strong result.
\begin{corollary}
\label{qs1}
Suppose $\#S=1$. If $D_i$ is ample for all $i$ and $r>m$ then all sets of $(D,S)$-integral points on $X$ are finite.
\end{corollary}
It follows from the Dirichlet unit theorem that $\#S=1$ if and only if $\mathcal{O}_{k,S}^*$ is finite if and only if $\mathcal{O}_{k,S}=\mathbb{Z}$ or the ring of integers of a complex quadratic field. The inequality in Corollary \ref{qs1} is sharp as the next example shows.
\begin{example}
Let $X=\mathbb{P}^n$. Let $k=\mathbb{Q}$ and let $S$ consist only of the prime at infinity. Let $D_i$ be the divisor on $\mathbb{P}^n$ defined by $x_i=0$, where $x_0,\ldots, x_n$ are homogeneous coordinates on $\mathbb{P}^n$. Let $D=\sum_{i=1}^n a_iD_i$. Let $m=\sum_{i=1}^na_i$. Then the set of points with $x_0\in \mathbb{Z}$ and $x_i=1$, $i=1,\ldots,n$, is an infinite set of $(D,S)$-integral points on $X$ and $D$ is a sum of $m$ ample divisors defined over $\mathbb{Q}$.
\end{example}
\begin{theorem}
\label{Pic}
Let $X$ be a nonsingular projective variety. Suppose $\#S=1$. Let $\rho$ denote the Picard number of $X$ and let $n$ be the rank of the group of $k$-rational points of $\text{Pic}^0(X)$. Suppose that the $D_i$'s are defined over $k$ for all $i$ and have no irreducible components in common. If $r>\rho+n$ then there does not exist a dense set of $(D,S)$-integral points on $X$.
\end{theorem}
Our proof is essentially the first half of the proof of Theorem 2.4.1 in \cite{Vo2}.
\begin{proof}
It follows from the definitions that the group of divisor classes with a representative defined over $k$ has rank at most $\rho+n$. Since $r>\rho+n$, there exists a linear combination of the $D_i$'s that is principal, equal to $(f)$ for some nonconstant rational function $f$ on $X$. Let $R$ be a set of $(D,S)$-integral points on $X$. Since all of the poles of $f$ lie in $D$ there exists an $a\in k$ such that $af$ takes on integral values on $R$. Since the poles of $\frac{1}{f}$ also lie in $D$, the same reasoning applies to $\frac{1}{f}$. Therefore $f(R)$ lies in only finitely many cosets of the group of units $\mathcal{O}_{k,S}^*$. But since $\#S=1$, $\mathcal{O}_{k,S}^*$ is finite. Therefore $R$ lies in the finite union of proper subvarieties of $X$ of the form $f=a$ for a finite number of $a\in k$.
\end{proof}
We note that the requirement in all of these results that not only $D$ be defined over $k$, but that the $D_i$'s be defined over $k$ is absolutely necessary. For example, if $X=\mathbb{P}^1$, $k=\mathbb{Q}$, $S=\{\infty\}$, and $D=P+Q$ where $P$ and $Q$ are conjugate over a real quadratic field, then from Pell's equation there do exist dense sets of $(D,S)$-integral points on $X$.
\section{Results on the General Conjectures}
\label{SVgeneral}
We will now consider the case where the integral points are allowed to vary over number fields of a bounded degree over some number field $k$. As an application of their results on surfaces in \cite{Co2}, Corvaja and Zannier prove
\begin{theorem}
Let $X$ be a projective curve defined over a number field $k$. Let $S$ be a finite set of places of $k$ containing the archimedean places. Let $D=\sum_{i=1}^r P_i$ be a divisor on $X$ defined over $k$ such that the $P_i$'s are distinct points. If $r>4$ then all sets of $D$-integral points on $X$ quadratic over $k$ are finite.
\end{theorem}
This theorem can also be obtained as a consequence of a result of Vojta (see Section \ref{sVo}). Using the same technique Corvaja and Zannier used, looking at symmetric powers of $X$, our higher-dimensional results give
\begin{theorem}
Let $n=\dim X$. If $D_i$ is ample for all $i$ and $r>2d^2mn$ then all sets of $D$-integral points on $X$ of degree $d$ over $k$ are finite.
\end{theorem}
\begin{proof}
Suppose $r>2d^2mn$ and let $R\subset X(\overline{k})$ be a set of $D$-integral points on $X$ of degree $d$ over $k$. It suffices to prove the finiteness of $R$ in the case where for every $P\in R$ we have $[k(P):k]=d$. Let $X^d$ be the $d$-fold product of $X$ with itself, and let $\pi_i:X^d\to X$ be the $i$-th projection map for $i=1,\ldots, d$. Let $\Sym^d X$ denote the $d$-fold symmetric product of $X$ with itself and let $\phi:X^d\to \Sym^d X$ be the natural map. Let $E_i=\phi(\pi_1^*D_i)$ and $E=\sum_{i=1}^rE_i$. We have that $\phi^*E_i=\sum_{j=1}^d \pi_j^*D_i$ which is ample on $X^d$. Since $\phi$ is a finite surjective morphism, it follows that $E_i$ is ample. By looking at the corresponding statement on $X^d$ we see that the intersection of any $dm+1$ distinct $E_i$'s is empty. We also have $\dim \Sym X^d=dn$. Since $r>2(dm)(dn)$, by Theorem \ref{cor4}(b) we have that all sets of $k$-rational $E$-integral points on $\Sym^d X$ are finite. For $P\in R$ let $P^{(1)},\ldots,P^{(d)}$ denote the $d$ conjugates of $P$ over $k$. Then $R'=\{(P^{(1)},\ldots,P^{(d)})\in X^d|P\in R\}$ is a set of $\sum_{i=1}^d\pi_i^*D$-integral points on $X^d$. So $\phi(R')$ is a set of $E$-integral points on $\Sym^d X$. Note that $\phi(R')$ is actually a set of $k$-rational points on $\Sym^d X$. Therefore, from above, $\phi(R')$ must be finite, and so clearly $R$ must be finite.
\end{proof}
When $\#S=1$ we have the stronger theorem
\begin{theorem}
Let $X$ be a projective variety defined over $k=\mathbb{Q}$ or a complex quadratic field $k$. Let $S=\{v_\infty\}$ consist of the unique archimedean place of $k$. If $D_i$ is ample and defined over $k$ for all $i$ and $r>dm$ then all sets of $D$-integral points on $X$ of degree $d$ over $k$ are finite.
\end{theorem}
\begin{proof}
The proof is identical with the proof of the previous theorem, except that instead of using Theorem \ref{cor4}(b) we use Corollary \ref{qs1}.
\end{proof}
\section{A Result of Faltings}
\label{Faltings}
In \cite{Fa}, Faltings proves the finiteness of integral points on the complements of certain irreducible singular curves in $\mathbb{P}^2$. Recently a similar result has also been obtained by Zannier in \cite{Co4}. We show, as simple corollaries of our work on surfaces, how we may improve both results on integral points, and at the same time we will prove the analogous statement for holomorphic curves.
Let $X$ be an irreducible nonsingular projective surface over an algebraically closed field $k$ of characteristic $0$. Let $\mathcal{L}=\mathcal{O}_X(L)$ be an ample line bundle on $X$ with $K_X+3L$ ample.
Assume that the global sections $\Gamma(X,\mathcal{L})$ generate\\\\
(a). $\mathcal{L}_x/\mathfrak{m}_x^4\mathcal{L}_x$ for all points $x\in X$\\
(b). $\mathcal{L}_x/\mathfrak{m}_x^3\mathcal{L}_x \bigoplus \mathcal{L}_y/\mathfrak{m}_y^3\mathcal{L}_y$ for all pairs $\{x,y\}$ of distinct points\\
(c). $\mathcal{L}_x/\mathfrak{m}_x^2\mathcal{L}_x \bigoplus \mathcal{L}_y/\mathfrak{m}_y^2\mathcal{L}_y \bigoplus \mathcal{L}_z/\mathfrak{m}_z^2\mathcal{L}_z$ for all triples $\{x,y,z\}$ of distinct points.\\
A three-dimensional subspace $E\subset \Gamma(X,\mathcal{L})$ that generates $\mathcal{L}$ gives a morphism $f_E:X \to \mathbb{P}^2$. Faltings studies this map when $E$ is suitably generic.
\begin{definition}
\label{Fgeneric}
Let $E\subset \Gamma(X,\mathcal{L})$ be a three-dimensional subspace. We call $E$ generic if:\\\\
(a). E generates $\mathcal{L}$.\\
(b). The discriminant locus $Z\subset X$ of $f_E$ is nonsingular.\\
(c). The restriction of $f_E$ to $Z$ is birational onto its image $D\subset \mathbb{P}^2$.\\
(d). $D$ has only cusps and nodes as singularities.
\end{definition}
Three-dimensional subspaces $E\subset \Gamma(X,\mathcal{L})$ are naturally parametrized by a Grassmannian $G$. Let $n=L^2$. It is then proven that
\begin{theorem}
\label{F2}
With notation as above\\
(a). Generic $E$'s form a dense open subset $G'$ of $G$.\\
(b). For generic $E$ let $\pi:Y\to X \to \mathbb{P}^2$ denote the associated normal Galois covering. Then $Y$ is smooth, $Z$ is irreducible, and the covering group $Aut(Y/\mathbb{P}^2)$ is the full symmetric group $S_n$.
\end{theorem}
Faltings also proves
\begin{theorem}
\label{DP}
Let $\pi^*D$ be the pullback of $D$ to $Y$. Then $\pi^*D=2\sum_{1\leq i<j \leq n}Z_{ij}=\sum_{i=1}^nA_i$ where $Z_{ij}$ is effective and nonsingular for every $i$ and $j$, and $A_i=\sum_{j\neq i}Z_{ij}$ is the pullback of $Z$ under the $i$th projection map $Y\to X$. Furthermore, let $P\in \pi^*D$. Then one of the following holds:\\\\
(a). $\pi(P)$ is a smooth point of $D$ and $P\in Z_{ij}$ for exactly one $\{ij\}$.\\
(b). $\pi(P)$ is a node of $D$ and exactly two components $Z_{ij}$ and $Z_{kl}$ of $\pi^*D$ for disjoint $\{ij\}$ and $\{kl\}$ intersect at $P$.\\
(c). $\pi(P)$ is a cusp of $D$ and exactly three components $Z_{ij},Z_{ik},Z_{jk}$ intersect at $P$ for some $i,j,k$.
\end{theorem}
Let $d=\deg D$ and assume that everything above is defined over a number field. The main result of \cite{Fa} is
\begin{theorem}[Faltings]
\label{Famain}
If $dL-\alpha Z$ is ample on $X$ for some $\alpha>12$ then $\mathbb{P}^2\backslash D$ is Mordellic.
\end{theorem}
Zannier proves this unconditionally if the Kodaira number of $X$ is nonnegative, and more generally he gives a numerical condition replacing the condition on $L$ and $Z$ above. We will prove Theorem \ref{Famain} unconditionally, i.e. without the ampleness condition. We also prove the analogue for holomorphic curves. Under the assumptions discussed above, we prove
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
$\mathbb{P}^2\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
$\mathbb{P}^2\backslash D$ is complete hyperbolic. In particular, $\mathbb{P}^2\backslash D$ is Brody hyperbolic.
\end{theoremb}
\begin{proof}
Since $\pi:Y\backslash \pi^*D\to \mathbb{P}^2\backslash D$ is a finite \'etale covering, the problem is reduced to proving the theorems for $Y\backslash \pi^*D$. The assumption (a) on $L$ given at the beginning of the section implies that $n=L^2\geq 9$. We have $\pi^*D=\sum_{i=1}^n A_i$ and that $A_i$ is the pullback of $Z$ under the $i$th projection map $Y\to X$. Therefore $A_i$ is ample as the projection is a finite map (recall that we assumed $Z\sim K_X+3L$ was ample). It follows from Theorem \ref{DP} that at most four $A_i$'s meet at a point. Therefore we're done by Theorems \ref{surf3a}(b) and \ref{surf3b}(b) with $r\geq 9$ and $m=4$. That $\mathbb{P}^2\backslash D$ is complete hyperbolic follows from the fact that $Y\backslash \pi^*D$ is complete hyperbolic (see \cite{La2}).
\end{proof}
\section{Remarks on the Siegel and Picard-type Conjectures}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\label{Remarks}
In this section we will show the sharpness of the inequalities and the necessity of certain hypotheses in many of the conjectures, how our conjectures relate to other conjectures that have been made, and what special cases of the conjectures are known by previous work.
\subsection{Main Conjectures}
\subsubsection{Examples Limiting Improvements to the Conjectures}
Our main goal here is to show that the inequalities in all of the main conjectures cannot be improved. We'll start with two fundamental examples on $\mathbb{P}^n$.
\begin{examplea}
\label{NHypera}
Let $X=\mathbb{P}^n$. Let $D=\sum_{i=0}^nD_i$, where $D_i$ is the hyperplane defined by $x_i=0,i=0,\ldots,n$. Let $k$ be a number field with an infinite number of units. Let $S$ be the set of archimedean places. Let $R$ be the set of points in $\mathbb{P}^n$ which have a representation where the coordinates are all units. Then $R$ is a set of $D$-integral points on $X$. It follows from the $S$-unit lemma that $R$ is Zariski-dense in $X$.
\end{examplea}
\begin{exampleb}
\label{NHyper}
Let $X$ and $D$ be as above. Let $f_i,i=0,\ldots,n$ be linearly independent entire functions. Let $f:\mathbb{C}\to X$ be defined by $f=(e^{f_0},\ldots,e^{f_n})$. Clearly the image of $f$ does not intersect $D$. It follows from Borel's lemma that the image of $f$ is Zariski-dense in $X$.
\end{exampleb}
We will give two variants of these examples which show that the inequalities in the Main Siegel and Picard-type Conjectures, Conjectures~\ref{conjmaina} and \ref{conjmainb}, are sharp for all values of $m$ and $\kappa_0$.
\begin{examplea}
\label{var1a}
Let $X$, $k$, $S$, $D$, $D_i$, and $R$ be as in Example \ref{NHypera}. Let $Y=X^q$ and let $\pi_j$ be the $j$th projection map from $Y$ to $X$ for $j=1,\ldots,q$. Let $R'=R^q\subset Y$. Let $E_{i,j}=\pi_j^*D_i$ for $0\leq i \leq n,1\leq j \leq q$. Let $1\leq m\leq nq$. Let $r=\left[m+\frac{m}{n}\right]$ and $r'=\left[\frac{r}{n+1}\right]=\left[\frac{m}{n}\right]$. Let
\begin{equation*}
E=\sum_{j=1}^{r'}\sum_{i=0}^n E_{i,j}+\sum_{i=1}^{r-r'(n+1)}E_{i,r'+1}.
\end{equation*}
Then $R'$ is a set of $E$-integral points on $Y$ and it follows, again, from the $S$-unit lemma that $R'$ is Zariski-dense in $Y$. Furthermore, there are at most $nr'+r-r'(n+1)=r-r'=m$ of the $E_{i,j}$'s in $E$ meeting at a given point, and $E$ is a sum of $r=\left[m+\frac{m}{n}\right]$ of the $E_{i,j}$'s with $\kappa(E_{i,j})=n$ for all $i$ and $j$.
\end{examplea}
\begin{exampleb}
\label{var1b}
Same as the above example, except that instead of $R'$, we use a holomorphic map $f:X\to Y\backslash E$ given by $f=(e^{f_{0,1}},\ldots,e^{f_{n,1}})\times\cdots \times(e^{f_{0,t}},\ldots,e^{f_{n,t}})$ where the $f_{i,j}$'s are linearly independent entire functions. It follows from Borel's lemma that $f$ has Zariski-dense image in $Y$.
\end{exampleb}
The second variants of Examples \ref{NHypera} and \ref{NHyper} are
\begin{examplea}
\label{var2a}
Let $m$ and $n$ be positive integers. Let $X$, $k$, $S$, $D_i$, $r$, and $r'$ be as in Example \ref{var1a}. Let $D=\sum_{i=0}^{n}a_iD_i$ where $a_i=r'+1$ for $i=0,\ldots,r-(n+1)r'-1$ and $a_i=r'$ for $i=r-(n+1)r',\ldots,n$. Then counting the $D_i$'s with their multiplicity in $D$, $D$ is a sum of $\sum_{i=0}^{n}a_i=r$ effective divisors such that the intersection of any $m+1$ of them is empty. We have $\kappa(D_i)=n$ for all $i$. By Example \ref{NHypera} there exist dense sets of $D$-integral points on $X$.
\end{examplea}
\begin{exampleb}
\label{var2b}
The same example as above, except we use the holomorphic map from Example \ref{NHyper}.
\end{exampleb}
The above four examples also show that one cannot improve the inequalities in Conjectures \ref{conj1a},B and \ref{conj1ab},B.
We have not yet discussed the $\kappa_0=0$ case. If $D$ is a divisor on a projective variety $X$, then by blowing up subvarieties of $D$ on $X$ we may get a divisor $D'$ on $X'$ with arbitrarily many components and $X\backslash D \cong X'\backslash D'$. In this case, the new components $C$ have $\kappa(C)=0$. So, as is suggested by the $\kappa_0$ in the denominators of the inequalities, there is no possible result of the type in the Main Siegel and Picard-type Conjectures if one allows divisors $D_i$ with $\kappa(D_i)=0$. However, all is not lost in this case. If we are willing to include in the inequalities numerical invariants of the variety such as the Picard number, then it is possible to give theorems for arbitrary effective divisors. We will discuss this in Section \ref{mainknown}.
There are also examples showing that the exceptional sets may be dense, even if the hypotheses of the Main Siegel and Picard-type Conjectures are satisfied. For example, let $X=\mathbb{P}^1\times \mathbb{P}^1$ and let $D=\sum_{i\in I} P_i\times \mathbb{P}^1$ be a finite sum with $P_i\in \mathbb{P}^1(k), i \in I$, for some number field $k$. Then it is easy to show that $\Excd(X\backslash D)=\Exch(X\backslash D)=X\backslash D$.
For the Main Conjectures for Ample Divisors we have
\begin{examplea}
Let $D$ be the sum of any $r$ hyperplanes in general position (i.e. the intersection of any $n+1$ of them is empty) in $\mathbb{P}^n$ with $n<r\leq 2n$. Assume also that $D$ is defined over a number field. Then one may show that there exists a linear subspace $L\subset \mathbb{P}^n$ with $\dim L=\left[\frac{n}{r-n}\right]$ such that $L$ contains a dense set of $D|_L$-integral points (for some $k$ and $S$) (see \cite{Fu2}, \cite{Gr}, and \cite{No} for the constructions).
\end{examplea}
\begin{exampleb}
In the same situation as above, one may also show that there exists a holomorphic map $f:\mathbb{C}\to L\backslash D$ with Zariski-dense image.
\end{exampleb}
In the simplest case, where $r=2m=2n$, we may simply take $L$ to be a line that passes through points $P$ and $Q$ where $P$ is the intersection of, say, the first $n$ hyperplanes and $Q$ is the intersection of the last $n$ hyperplanes. Then $L\cap D$ is a $\mathbb{P}^1$ minus two points, and so we see that we cannot have finiteness or constancy for the objects in question.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\begin{remark}
\label{rbig}
It is quite possible that our Main Conjectures for Ample Divisors may be extended to quasi-ample divisors. Let $D$ be a quasi-ample divisor on a projective variety $X$. Let $n>0$ be large enough such that the map $\Phi=\Phi_{nD}$, corresponding to $nD$, is birational. It is then quite plausible that all of our conclusions that held for ample divisors generalize to quasi-ample divisors if we state things in terms of $\Phi$, that is, replace $\dim \Excd(X\backslash D)$ and $\dim \Exch(X\backslash D)$ by $\dim \Phi(\Excd(X\backslash D))$ and $\dim \Phi(\Excd(X\backslash D))$ in the conjectures.
\end{remark}
\subsubsection{Relation to Vojta's Main Conjecture}
\label{mainrelation}
We now show how some special cases of the Main Conjectures are related to Vojta's Main Conjecture. If $D$ is a divisor on a nonsingular complex variety $X$, we say that $D$ has normal crossings if every point $P\in D$ has an analytic open neighborhood in $X$ with analytic local coordinates $z_1,\ldots,z_n$ such that $D$ is locally defined by $z_1\cdot z_2\cdots z_i=0$ for some $i$. Inspired by results in equi-dimensional Nevanlinna theory, Vojta made the following conjecture in \cite{Vo2}.
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{conjecturea}[Vojta's Main Conjecture]
\label{Vmain}
Let $X$ be a nonsingular projective variety with canonical divisor $K$. Let $D$ be a normal crossings divisor on $X$, and let $k$ be a number field over which $X$ and $D$ are defined. Let $A$ be a quasi-ample divisor on $X$. Let $\epsilon>0$. Then there exists a proper Zariski-closed subset $Z=Z(X,D,\epsilon,A)$ such that
\begin{equation*}
m(D,P)+h_K(P)\leq \epsilon h_A(P)+O(1)
\end{equation*}
for all points $P\in X\backslash Z$.
\end{conjecturea}
Similarly, the analogue is conjectured for holomorphic curves
\begin{conjectureb}
\label{Vmainb}
Let $X$ be a nonsingular complex projective variety with canonical divisor $K$. Let $D$ be a normal crossings divisor on $X$. Let $A$ be a quasi-ample divisor on $X$. Let $\epsilon>0$. Then there exists a proper Zariski-closed subset $Z=Z(X,D,\epsilon,A)$ such that for all holomorphic maps $f:\mathbb{C}\to X$ whose image is not contained in $Z$,
\begin{equation*}
m(D,r)+T_K(r)\leq \epsilon T_A(r)+O(1)
\end{equation*}
holds for all $r$ outside a set of finite Lebesgue measure.
\end{conjectureb}
Qualitatively, these conjectures have the following simple consequences.
\begin{conjecturea}
\label{conj3}
Let $X$ be a nonsingular projective variety, defined over a number field $k$. Let $K$ be the canonical divisor of $X$, and $D$ a normal crossings divisor on $X$, defined over $k$. Suppose that $K+D$ is quasi-ample. Then $X\backslash D$ is quasi-Mordellic.
\end{conjecturea}
\begin{conjectureb}
\label{conj3b}
Let $X$ be a nonsingular complex projective variety. Let $K$ be the canonical divisor of $X$, and $D$ a normal crossings divisor on $X$. Suppose that $K+D$ is quasi-ample. Then $X\backslash D$ is quasi-Brody hyperbolic.
\end{conjectureb}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
To relate these conjectures to our conjectures we recall the following theorem, which is a consequence of Mori theory \cite[Lemma 1.7]{Mo}.
\begin{theorem}
Let $X$ be a nonsingular complex projective variety of dimension $n$. If $D_1,\ldots,D_{n+2}$ are ample divisors on $X$ then $K+\sum_{i=1}^{n+2}D_i$ is ample.
\end{theorem}
So when $X$ is nonsingular, the $D_i$'s are ample, and $D$ has normal crossings, we see that Conjectures \ref{conj1ab} and \ref{conj1bb} are consequences of Conjectures \ref{conj3} and \ref{conj3b}.
\subsubsection{Previously Known Results Related to the Conjectures}
\label{mainknown}
As was discussed earlier, our work builds on previous work of Corvaja and Zannier, who obtained results on surfaces in \cite{Co2}, and initiated the general method we have used in \cite{Co}. The Nevanlinna theoretic analogues of \cite{Co2} were proved by Liu and Ru in \cite{Ru4}. We briefly discussed these previous results in Section \ref{ssurf}.
We now discuss what is known for arbitrary divisors. As a consequence of his work on integral points on subvarieties of semi-abelian varieties, Vojta \cite{Vo1} proved
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}A}
\setcounter{theoremb}{\value{theorem}}
\begin{theorema}
\label{Vojtaa}
Let $X$ be a projective variety defined over a number field $k$. Let $\rho$ denote the Picard number of $X$. Let $D$ be an effective divisor on $X$ defined over $k$ which has more than $\dim X - h^1(X,\mathcal{O}_X)+\rho$ (geometrically) irreducible components. Then $X\backslash D$ is quasi-Mordellic.
\end{theorema}
Similarly, a special case of work of Noguchi \cite{No2} gives
\begin{theoremb}
\label{Vojtab}
Let $X$ be a complex projective variety. Let $\rho$ denote the Picard number of $X$. Let $D$ be an effective divisor on $X$ which has more than $\dim X - h^1(X,\mathcal{O}_X)+\rho$ irreducible components. Then $X\backslash D$ is quasi-Brody hyperbolic.
\end{theoremb}
We note that it is easily shown that both theorems are sharp in that there are divisors with $\dim X - h^1(X,\mathcal{O}_X)+\rho$ irreducible components for which the conclusions of the theorems are false. For a weaker, but more elementary theorem along these lines, see also Th. 2.4.1 in~\cite{Vo2}. As consequences of Theorems \ref{Vojtaa},B we see that Conjectures \ref{conj1ab},B are true for $X=\mathbb{P}^n$, and more generally for any projective variety $X$ with Picard number one.
From the work of Noguchi and Winkelmann \cite{No} we have the following theorems related to our Main Conjectures for Ample Divisors (some special cases of these results had been obtained previously by various people; see \cite{No} for the history).
\begin{theorema}
Let $X$ be a projective variety of dimension $n$ defined over a number field $k$. Let $S$ be a finite set of places of $k$ containing the archimedean places. Let $\rho$ be the Picard number of $X$. Let $D=\sum_{i=1}^rD_i$ be a divisor on $X$ defined over $k$ with the $D_i$'s effective reduced ample Cartier divisors such that the intersection of any $n+1$ of them is empty.\\\\
(a). If $r>n+1$ then all sets of $D$-integral points $R$ have $\dim R\leq \frac{n}{r-n}\rho$.\\
(b). If $r>n(\rho+1)$ then $X\backslash D$ is Mordellic.\\
(c). If $X\subset \mathbb{P}^N$, all $D_i$'s are hypersurface cuts of $X$, and $r>2n$ then $X\backslash D$ is Mordellic.
\end{theorema}
\begin{theoremb}
Let $X$ be a complex projective variety of dimension $n$. Let $\rho$ be the Picard number of $X$. Let $D=\sum_{i=1}^rD_i$ be a divisor on $X$ with the $D_i$'s effective reduced ample Cartier divisors such that the intersection of any $n+1$ of them is empty.\\\\
(a). If $r>n+1$ then all holomorphic maps $f:\mathbb{C}\to X\backslash D$ have $\dim f(\mathbb{C})\leq \frac{n}{r-n}\rho$.\\
(b). If $r>n(\rho+1)$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.\\
(c). If $X\subset \mathbb{P}^N$, all $D_i$'s are hypersurface cuts of $X$, and $r>2n$ then $X\backslash D$ is complete hyperbolic and hyperbolically imbedded in $X$. In particular, $X\backslash D$ is Brody hyperbolic.
\end{theoremb}
Consequently, when $m=\dim X$, the $D_i$'s are reduced divisors, and $\rho(X)=1$, we have that the Main Conjectures for Ample Divisors, Conjectures \ref{conj2a},B, are true modulo the statements on the exceptional sets (i.e. replace $\Excd(X\backslash D)$ by any particular set of integral points $R$ in Conjecture \ref{conj2a}, etc.) Similarly, the part (c)'s of the above theorems give special cases of the part (b)'s of Conjectures \ref{conj2a},B.
\subsection{General Conjectures}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\subsubsection{Examples Limiting Improvements to the Conjectures}
We start off with an example showing that the inequalities in the General Conjectures are best possible when $X$ is a curve.
\begin{example}
\label{genex}
Let $X$ be a projective curve defined over a number field $k$ with $\mathcal{O}_k^*$ infinite. Let $f:X\to \mathbb{P}^1$ be a morphism of degree $d$ defined over $k$. Let $P,Q\in \mathbb{P}^1(k)$ be two distinct points over which $f$ is unramified, and let $D=P+Q$. Then there exists an infinite set $R$ of $k$-rational $D$-integral points on $\mathbb{P}^1\backslash D$. Since $f$ has degree $d$, $f^{-1}(R)$ is a set of $f^*D$-integral points on $X\backslash f^*D$ of degree $d$ over $k$ and $f^*D$ is a sum of $2d$ distinct points on $X$.
\end{example}
Taking products of curves, we then get examples in all dimensions showing that the inequality in the General Siegel-type Conjecture cannot be improved in the case $\kappa_0=1$.
\begin{example}
\label{genex2}
Let $D=\sum_{i=1}^{2md}H_i$ be a sum of hyperplanes on $\mathbb{P}^n$ defined over a number field $k$ such that the intersection of any $m+1$ of the $H_i$'s is empty. Suppose also that $\bigcap_{i=(j-1)m+1}^{jm}H_i=\{P_j\}$ consists of a single point for $j=1,\ldots,2d$ and the $P_j$'s are collinear. Then there exist infinite sets of $D$-integral points of degree $d$ on $\mathbb{P}^n\backslash D$ over large enough number fields. Indeed, the line $L$ through the $P_j$'s intersects $D$ in $2d$ points, and we see from Example \ref{genex} that $L\backslash L\cap D$ contains infinite sets of integral points over large enough number fields.
\end{example}
This shows that the inequality in the finiteness part of the General Siegel-type Conjecture for Ample Divisors cannot be improved. We expect that using only divisors that are sums of hyperplanes on projective space, one may show that the other inequalities in the General Conjectures may not be improved for any set of parameters. For example, it should be true that if $D$ is a sum of $2d+n-1$ hyperplanes in general position on $\mathbb{P}^n$, then for some number field $k$ there exist dense sets of $D$-integral points on $\mathbb{P}^n$ of degree $d$ over $k$. In any case, it is easy to show that $\Excd(\mathbb{P}^n\backslash D)=\mathbb{P}^n\backslash D$. If $P$ is a point where $n$ of the hyperplanes intersect, then any line through $P$ will intersect $D$ in $2d$ points. But as we have seen, over some number field $k$, such lines will contain infinitely many integral points of degree $d$ over $k$. To show the existence of a Zarisk-dense set of $D$-integral points, one needs to show that if the lines and their sets of integral points are chosen correctly, then the infinite union of the sets of integral points will still be a set of $D$-integral points (there is no problem for finite unions).
\subsubsection{Vojta's General Conjecture and a Conjectural Discriminant-Height Inequality}
We will now investigate how the General Siegel-type Conjecture, Conjecture \ref{congen}, is related to Vojta's General Conjecture. In order to make a connection between the two conjectures, we will need to formulate a new conjecture bounding the absolute logarithmic discriminant in terms of heights. We will digress briefly to discuss this new conjecture. Let $X$ be a variety defined over a number field $k$ and let $P\in X(\overline{k})$. Let $d(P)=\frac{1}{[k(P):\mathbb{Q}]}\log |D_{k(P)/\mathbb{Q}}|$ where $D_{k(P)/\mathbb{Q}}$ is the discriminant of $k(P)$ over $\mathbb{Q}$. We call $d(P)$ the absolute logarithmic discriminant of $P$. Let $m(D,P)=\sum_{v\in S}\lambda_{D,v}(P)$. Then Vojta's General Conjecture states
\begin{conjecture}[Vojta's General Conjecture]
\label{Vgeneral}
Let $X$ be a complete nonsingular variety with canonical divisor $K$. Let $D$ be a normal crossings divisor on $X$, and let $k$ be a number field over which $X$ and $D$ are defined. Let $A$ be a quasi-ample divisor on $X$. Let $\epsilon>0$. If $\nu$ is a positive integer then there exists a Zariski-closed subset $Z=Z(\nu,X,D,\epsilon,A)$ such that
\begin{equation*}
m(D,P)+h_K(P)\leq d(P)+\epsilon h_A(P)+O(1)
\end{equation*}
for all points $P\in X(\overline{k})\backslash Z$ such that $[k(P):k]\leq \nu$.
\end{conjecture}
Actually, Vojta's General Conjecture as it appears in \cite{Vo2} has the discriminant term as $(\dim X)d(P)$, but it is now believed that the $\dim X$ term is unecessary (see \cite[Conjecture 8.7]{Vo5} or the discussion at the end of \cite{Vo6}).
Vojta's General Conjecture, with $D=0$, can be seen as giving a lower bound on the absolute logarithmic discriminant in terms of heights (outside some Zariski-closed subset). As a companion to this, we give the following conjectural upper bound on the logarithmic discriminant in terms of heights.
\begin{conjecture}
\label{conj4}
Let $X$ be a nonsingular projective variety of dimension $n$ defined over a number field $k$ with canonical divisor $K$. Let $A$ be an ample divisor on $X$. Let $\nu$ be a positive integer. Let $\epsilon>0$. Then
\begin{equation*}
d(P)\leq h_K(P)+(2[k(P):k]+n-1+\epsilon)h_A(P)+O(1)
\end{equation*}
for all $P\in X(\overline{k})$ with $[k(P):k]\leq \nu$.
\end{conjecture}
\begin{remark}
It is possible that with the hypothesis $A$ ample weakened to $A$ quasi-ample that the inequality holds outside of some Zariski-closed subset of $X$ (it is not hard to see the necessity of the Zariski-closed subset in this case). It is also possible that the conjecture is true with $\epsilon=0$. As with Vojta's General Conjecture, it is quite plausible that one may take $\nu=\infty$, i.e. the inequality holds for all $P\in X(\overline{k})$.
\end{remark}
It is a result of Silverman \cite{Si2} that Conjecture \ref{conj4} is true for $X=\mathbb{P}^n$ with $\epsilon=0$ and $\nu=\infty$. For curves, Conjecture \ref{conj4} is true by a result of Song and Tucker \cite[Eq. 2.0.3]{Tu}. They proved the stronger statement
\begin{theorem}
\label{thTu}
Let $X$ be a nonsingular projective curve defined over a number field $k$ with canonical divisor $K$. Let $A$ be an ample divisor on $X$. Let $\nu$ be a positive integer. Let $\epsilon>0$. Then
\begin{equation*}
d(P)\leq d_a(P)\leq h_K(P)+(2[k(P):k]+\epsilon)h_A(P)+O(1)
\end{equation*}
for all $P\in X(\overline{k})$ with $[k(P):k]\leq \nu$, where $d_a(P)$ is the arithmetic discriminant of $P$ (see \cite{Vo8} for the definition and properties).
\end{theorem}
We now show how Vojta's General Conjecture, combined with our conjectural upper bound on the discrimant, imply a special case of the General Siegel-type Conjecture.
\begin{theorem}
Assume Vojta's General Conjecture, Conjecture \ref{Vgeneral}, and the conjectural upper bound on the absolute logarithmic discriminant, Conjecture \ref{conj4}. Let $X$ be a nonsingular projective variety defined over a number field $k$. Let $n=\dim X$. Let $D=\sum_{i=1}^r D_i$ be a normal crossings divisor defined over $k$ with $D_i$ ample and effective for all $i$. If $r>2\nu+n-1$ then $X\backslash D$ is degree $\nu$ quasi-Mordellic. In particular, there do not exist Zariski-dense sets of $D$-integral points on $X$ of degree $\nu$ over $k$.
\end{theorem}
\begin{proof}
Let $R$ be a set of $D$-integral points on $X$ of degree $\nu$ over $k$. Then $m(D,P)+h_K(P)=h_D(P)+h_K(P)+O(1)$ for $P\in R$. By Conjecture \ref{conj4}, for any $\epsilon>0$, $h_{D_i}(P)\geq \frac{d(P)-h_K(P)}{2\nu+n-1+\epsilon}+O(1)$. So since $r>2\nu+n-1$, we have $h_D(P)\geq d(P)-h_K(P)+(1-\epsilon)h_{D_1}(P)+O(1)$. Therefore
\begin{equation*}
m(D,P)+h_K(P)>d(P)+\epsilon h_A(P)+O(1)
\end{equation*}
for all $P\in R$, for any ample divisor $A$ on $X$ and small enough $\epsilon$. So we're done by Vojta's General Conjecture.
\end{proof}
So assuming Vojta's General Conjecture and Conjecture \ref{conj4}, we see that the General Siegel-type Conjecture is true if $D_i$ is ample for all $i$ and $D$ has normal crossings.
\subsubsection{Previously Known Results Related to the Conjectures}
\label{sVo}
In \cite{Vo7}, Vojta proved the following generalization of Falting's theorem on rational points on curves and the Thue-Siegel-Roth-Wirsing theorem.
\begin{theorem}
Let $X$ be a nonsingular projective curve defined over a number field $k$ with canonical divisor $K$. Let $D$ be an effective divisor on $X$ defined over $k$ with no multiple components and $A$ an ample divisor on $X$. Let $\nu$ be a positive integer and let $\epsilon>0$. Then
\begin{equation*}
m(D,P)+h_K(P)\leq d_a(P)+\epsilon h_A(P)+O(1)
\end{equation*}
for all $P\in X(\overline{k})\backslash D$ with $[k(P):k]\leq \nu$, where the constant in $O(1)$ depends on $X,D,\nu,A$, and $\epsilon$.
\end{theorem}
Using Theorem \ref{thTu} we then easily obtain the following theorem.
\begin{corollary}
Let $X$ be a nonsingular projective curve defined over a number field $k$. Let $D$ be an effective divisor on $X$ that is a sum of more than $2\nu$ distinct points. Then $X\backslash D$ is degree $\nu$ Mordellic.
\end{corollary}
Therefore our General Siegel-type Conjectures are true for curves. Of course for $\mathbb{P}^1$ this was already known from the Thue-Siegel-Roth-Wirsing theorem. As mentioned earlier, the special case $\nu=2$ was also proven by Corvaja and Zannier using the Schmidt Subspace Theorem technique \cite{Co2}.
\subsection{Conjectures over $\mathbb{Z}$ and Complex Quadratic Rings of Integers}
I am not aware of any previous results that pertain to these conjectures, or any way to relate them to other known conjectures. An open problem then is to formulate quantitative conjectures explaining the qualitative conjectures I have made over $\mathbb{Z}$ and complex quadratic rings of integers. We now briefly discuss some examples showing that in many cases the inequalities in these conjectures may not be improved.
For the Main Siegel-type Conjecture over $\mathbb{Z}$, to show that the inequality in the conjecture may not be improved we may simple take $D=mH$ where $H$ is a hyperplane on $\mathbb{P}^n$ defined over $\mathbb{Q}$. Examples where the $m$ divisors have no components in common are easily obtained from products of projective spaces.
For the Main Conjecture on Ample Divisors over $\mathbb{Z}$, if $D=\sum_{i=1}^mH_i$ is a sum of $m<n$ distinct hyperplanes on $\mathbb{P}^n$ defined over $\mathbb{Q}$ then $\dim \cap_{i=1}^mH_i=n-m$ and there is a $Y=\mathbb{P}^{n-m+1}\subset \mathbb{P}^n$ with $D|_Y$ a hyperplane on $Y$ defined over $\mathbb{Q}$. So there are sets of $D$-integral points on $\mathbb{P}^n$ with dimension $n-m+1$.
Examples for the General Conjectures over $\mathbb{Z}$ are nearly identical to Examples \ref{genex} and \ref{genex2}, except that we must replace $2d$ by $d$ everywhere, since we are using $\mathbb{A}^1$ as our starting point. Again, we expect that using only divisors that are sums of hyperplanes on projective space, one may show that the inequalities in the General Conjectures over $\mathbb{Z}$ may not be improved for any set of parameters.
|
{
"timestamp": "2005-03-30T10:19:28",
"yymm": "0503",
"arxiv_id": "math/0503699",
"language": "en",
"url": "https://arxiv.org/abs/math/0503699"
}
|
\section{Introduction}
It is common knowledge that Lifshitz formula \cite{19}
describes the van der Waals and Casimir force acting between
two thick plane parallel material plates separated by a gap
of width $a$. According to this formula, the free energy
of the van der Waals and Casimir interaction can be
represented in terms of reflection coefficients
\begin{equation}
{\cal{F}_R}=\frac{k_BT}{2\pi}
\int_0^{\infty}\!\!\!{k_{\!\bot}\,dk_{\!\bot}}
\sum\limits_{l=0}^{\infty}{\vphantom{\sum}}^{\prime}\left\{
\ln\left[1-r_{\|}^{2}(\xi_l,k_{\!\bot})e^{-2aq_l}\right]
+
\ln\left[
\vphantom{r_{\|}^{2}(\xi_l,k_{\!\bot})e^{-2aq_l}}
1-r_{\bot}^{2}(\xi_l,k_{\!\bot})e^{-2aq_l}\right]
\right\}.
\label{e1}
\end{equation}
\noindent
Here prime means the addition of a multiple 1/2 near the term with
$l=0$, and
the Lifshitz reflection
coefficients take the form
\begin{eqnarray}
&& r_{\|}^{2}(\xi_l,k_{\!\bot})\equiv
r_{\|,L}^{2}(\xi_l,k_{\!\bot})=
\left(\frac{\varepsilon_lq_l-k_l}{\varepsilon_lq_l+k_l}\right)^2,
\nonumber \\
&& r_{\bot}^{2}(\xi_l,k_{\!\bot})\equiv
r_{\bot,L}^{2}(\xi_l,k_{\!\bot})=
\left(\frac{q_l-k_l}{q_l+k_l}\right)^2,
\label{e2}
\end{eqnarray} \noindent
where
$\varepsilon_l\equiv\varepsilon(i\xi_l)$, $\varepsilon(\omega)$ is
the dielectric permittivity of the plate material,
$\xi_l=2\pi k_B Tl/\hbar$ ($l=0,1,2,\ldots$) are the Matsubara
frequencies, and
$k_l^2\equiv k_{\!\bot}^2+\varepsilon_l\xi_l^2/c^2$,
{\ }$q_l^2\equiv k_{\!\bot}^2+\xi_l^2/c^2$.
Beginning in 2000, the behavior of the thermal correction to the
Casimir force between real metals has been hotly debated. It was shown
that Lifshitz formula
leads to different results depending on the model of metal
conductivity used. For real metals at low frequencies $\omega$,
the dielectric permittivity $\varepsilon$ varies as $\omega^{-1}$. After
substituting $\varepsilon\sim\omega^{-1}$
into the Lifshitz formula, the result is a thermal
correction which is several hundred times greater than for
ideal metals at separations of a few tenths of a micrometer \cite{3,8}
The attempt \cite{6} to modify the zero-frequency term of the Lifshitz
formula for real metals, assuming that it behaves as in the case of
ideal metals, also leads
to a large thermal correction to the Casimir force at short separations.
It is important to note that in the
approaches of both \cite{3,8} and also of
\cite{6} a thermodymanic puzzle
arises, i.e., the Nernst heat theorem is violated for a perfect
lattice \cite{12,17}. (See also \cite{16} where it is shown that for
the preservation of the Nernst heat theorem in the approach of
\cite{3,8} it is necessary to have metals with
defects or impurities; it is common knowledge, however, that thermodynamics
must be valid for both perfect and imperfect lattices.)
This puzzle casts doubt on the many applications of the Lifshitz theory
of dispersion forces, and thus represents a potentially serious challenge to
both experimental and theoretical physics. By contrast, the use of
$\varepsilon\sim\omega^{-2}$, as holds in a free electron plasma
model neglecting relaxation, leads \cite{5,4} to a small thermal
correction to the Casimir force at short separations.
This is in qualitative agreement with
the case of an ideal metal and is consistent with the Nernst heat
theorem. It should be borne in mind, however, that the plasma model
is not universal, and is applicable only in the case when the characteristic
frequency is in the domain of infrared optics.
The present paper demonstrates that the main
reason why the Drude model in combination with the Lifshitz theory
had failed to describe the thermal Casimir force is the
inadequacy of the standard concept of a fluctuating electromagnetic
field on the background of $\varepsilon$ depending only on
frequency inside a lossy real metal. To avoid a contradiction with
thermodynamics, one should use the reflection coefficients expressed
in terms of the surface impedance.
\section{The fluctuating field and
the surface impedance}
The concept of a fluctuating electromagnetic
field works well for the description of zero-point oscillations
in media with a frequency-dependent dielectric permittivity
where no real electric current does arise. We will now consider a
conductor in an external electric
field, which varies with some
frequency $\omega$ satisfying the conditions
\begin{equation}
l\ll\delta_n(\omega),\qquad l\ll\frac{v_F}{\omega}, \label{e3}
\end{equation}
\noindent where $l$ is the mean free path of a conduction electron,
$\delta_n(\omega)=c/\sqrt{2\pi\sigma\omega}$ is the penetration
depth of the field inside a metal, $\sigma$ is the conductivity,
and $v_F$ is the Fermi velocity. Eqs.~(\ref{e3}) determine the
domain of the normal skin effect \cite{21}. In this frequency
region the external field leads to the initiation of a real
current of the conduction electrons and the dielectric permittivity
is modelled by the Drude function $\varepsilon\sim\omega^{-1}$
leading to the difficulties with the Lifshitz formula mentioned
in Introduction.
The physical reason for these difficulties becomes clear
when one observes that the alternating electric field with
frequencies characteristic for the normal skin effect inevitably
leads to heating of a metal when it penetrates through the skin
layer. By contrast, the thermal photons in thermal equilibrium
with a metal plate or the virtual photons (giving rise
to the Casimir effect) can not lead to
the initiation of a real current and heating of the metal
(this is prohibited by thermodynamics). Hence the
concept of a fluctuating electromagnetic field penetrating inside
a metal cannot describe virtual and thermal photons in the
frequency region (\ref{e3}). As a consequence, the Lifshitz formula
can not be applied in combination with the Drude dielectric
function in the domain of the normal skin effect.
As is evident from the foregoing, another theoretical basis is needed
to find the thermal Casimir force between real metals different
from the approach used in the case of dielectrics. Here we show that
this basis is given by the surface impedance boundary conditions
introduced by M.~A.~Leontovich \cite{19,24}.
The fundamental difference of the surface impedance
boundary conditions from
the other approaches is that they permit not to consider the
electromagnetic fluctuations inside a metal. Instead, the
following boundary conditions are imposed taking into account
the properties of real metal
\begin{equation}
{{\mathbf{E}}}_t=Z(\omega)
\left[{{\mathbf{B}}}_t\times{{\mathbf{n}}}\right],
\label{e9}
\end{equation} \noindent
where $Z(\omega)=1/\sqrt{\varepsilon(\omega)}$ is
the Leontovich surface impedance of
the conductor, ${\mathbf{E}}_t$ and ${\mathbf{B}}_t$
are the tangential components of electric and magnetic fields, and
${\mathbf{n}}$ is the unit normal vector to the surface
(pointed inside a metal). The boundary condition (\ref{e9}) can
be used to determine the electromagnetic field outside a metal.
Note, that the impedance $Z(\omega)$ and the condition (\ref{e9})
suggest a more universal description than the one by means of
$\varepsilon$. They still hold in the domain of the anomalous skin
effect where a description in terms of the dielectric permittivity
$\varepsilon(\omega)$ is impossible. For ideal metals it holds $Z\equiv 0$.
The use of the Leontovich impedance in Eq.~(\ref{e9}) which does not depend
on the polarization state and transverse momentum, is of prime importance.
Note that in \cite{12n} the exact impedances
depending on a transverse momentum were used. This has led to the
same conclusions as were obtained previously from the Lifshitz formula
combined with the dielectric permittivity $\varepsilon\sim\omega^{-1}$.
As was already mentioned above, these conclusions are in violation
of the Nernst heat theorem for a perfect
lattice \cite{12,17,16}.
Although a recent review \cite{12n} claims agreement
with the Nernst heat theorem
in \cite{3,8}, no specific objections
against the rigorous analytical proof of the opposite statement in
\cite{17} are presented.
The fallacy in the calculations of \cite{12n}
concerning the type of the impedance
is that it disregards the requirement that the reflection properties
for virtual photons on a classical boundary should be the same as
for real photons. Paper \cite{17} demonstrates
in detail that by enforcing this requirement the exact and Leontovich
impedances coincide at zero frequency and lead to the conclusions
of \cite{15} which are in perfect agreement with the Nernst
heat theorem.
\section{Lifshitz formula in terms of surface impedance}
Let us consider the case of real eigenfrequencies
$\omega_{k_{\bot},n}^{\|}$, $\omega_{k_{\bot},n}^{\bot}$ (i.e.,
the pure imaginary im\-pe\-dan\-ce). The total free energy
of the electromagnetic oscillations is given by the sum of the
free energies of oscillators over all possible values of
their quantum numbers,
\begin{equation} {\cal{F}}= \sum\limits_{\alpha}\left[
\frac{\hbar\omega_{\alpha}}{2}+k_BT\ln
\left(1-e^{-\frac{\hbar\omega_{\alpha}}{k_BT}}\right)\right]=
k_BT \sum\limits_{\alpha}
\ln\left(2\sinh{\frac{\hbar\omega_{\alpha}}{2k_BT}}\right).
\label{e14}
\end{equation} \noindent
At $T\to 0$, the value of
$\cal{F}$ from Eq.~(\ref{e14}) coincides with the sum
of the zero-point energies which is usually considered
at zero temperature.
Applying this to the electromagnetic oscillations between metal
plates, where $\alpha=\{p,{\mbox{$k$}}_{\!\bot},n\}$,
and $p=\bot$ or $\|$ labels the polarization states,
we obtain
\begin{equation}
{\cal{F}}=k_BT
\int_0^{\infty}\frac{k_{\!\bot}\,dk_{\!\bot}}{2\pi}
\sum\limits_{n}\left[
\ln\left(2\sinh{\frac{\hbar\omega_{k_{\bot},n}^{\|}}{2k_BT}}\right)
+\ln\left(2\sinh{\frac{\hbar\omega_{k_{\bot},n}^{\bot}}{2k_BT}}\right)
\right]. \label{e15}
\end{equation} \noindent
Using the impedance boundary conditions, it can be easily shown that
the eigenfrequencies of
the electromagnetic field between plates with parallel and
perpendicular polarizations are determined by the equations
\begin{equation}
\Delta_{\|}(\omega,k_{\!\bot})\equiv
\frac{1}{2}e^{-aq}\left(1-\eta^2\right)\left(\sinh aq-
\frac{2i\eta}{1-\eta^2}\cosh aq\right)=0,
\label{e16}
\end{equation}
\noindent
\begin{equation}
\Delta_{\bot}(\omega,k_{\!\bot})\equiv
\frac{1}{2}e^{-aq}\left(1-\kappa^2\right)\left(\sinh aq+
\frac{2i\kappa}{1-\kappa^2}\cosh aq\right)=0,
\label{e17}
\end{equation}
\noindent
where $\eta=\eta(\omega)=Z\omega/(cq)$,
$\kappa=\kappa(\omega)=Zcq/\omega$,
and $q^2=k_{\bot}^2-\omega^2/c^2$.
The expression in the right-hand side of Eq.~(\ref{e15}) is
evidently divergent. Before performing a renormalization, let us
equivalently represent the sum over the eigenfrequencies
$\omega_{k_{\bot},n}^{\|,\bot}$ by the use of the argument theorem
\cite{2}.
Then Eq.~(\ref{e15}) can be rewritten as
\begin{equation}
{\cal{F}}=k_BT
\int_0^{\infty}\frac{k_{\!\bot}\,dk_{\!\bot}}{2\pi} \frac{1}{2\pi
i}\oint_{C_1} \ln\left(2\sinh{\frac{\hbar\omega}{2k_BT}}\right)
d\left[\ln\Delta_{\|}(\omega,k_{\!\bot})+
\ln\Delta_{\bot}(\omega,k_{\!\bot})\right].
\label{e18}
\end{equation}
\noindent
Here, the closed contour $C_1$ is bypassed counterclockwise. It
consists of two arcs, one having an infinitely small radius
$\varepsilon$ and the other one an infinitely large radius $R$,
and two straight lines $L_1,\,L_2$ inclined at the angles $\pm 45$
degrees to the real axis. The quantities
$\Delta_{\|,\bot}(\omega,k_{\!\bot})$ have their roots at the
photon eigenfrequencies and are defined in Eqs.~(\ref{e16}) and
(\ref{e17}). Unlike the usual derivation of the Lifshitz
formula at $T\neq 0$ \cite{15n} the function under the
integral in (\ref{e18}) has branch points rather than poles at the
imaginary frequencies $\omega_l=i\xi_l$.
The contour $C_1$ is
chosen so as to avoid all these branch points and enclose all the
photon eigenfrequencies.
After the integration by parts and some rearrangement \cite{15},
we find the equivalent but more simple
expression for the Casimir free energy
\begin{equation}
{\cal{F}}=\frac{k_BT}{2\pi}
\int_0^{\infty}{k_{\!\bot}\,dk_{\!\bot}}
\sum\limits_{l=0}^{\infty}{\vphantom{\sum}}^{\prime}\left[
\ln\Delta_{\|}(\xi_l,k_{\!\bot})+
\ln\Delta_{\bot}(\xi_l,k_{\!\bot})\right].
\label{e19}
\end{equation}
Expression (\ref{e19}) is still infinite. To remove the
divergences, we subtract from the right-hand side of
Eq.~(\ref{e19}) the free energy in the case of infinitely separated
interacting bodies ($a\to\infty$). Then the physical,
renormalized, free energy vanishes for infinitely remote plates.
From Eqs.~(\ref{e16}) and (\ref{e17}) after the substitution $\omega\to
i\xi_l$ in the limit $a\to\infty$ it follows
\begin{eqnarray} &&
\Delta_{\|}^{\!\infty}(\xi_l,k_{\!\bot})=\frac{1}{4}\left(1+\eta_l^2\right)
\left(1+\frac{2\eta_l}{1+\eta_l^2}\right),
\nonumber \\
&&
\Delta_{\bot}^{\!\infty}(\xi_l,k_{\!\bot})=\frac{1}{4}
\left(1+\kappa_l^2\right)
\left(1+\frac{2\kappa_l}{1+\kappa_l^2}\right).
\label{e20}
\end{eqnarray}
\noindent
The renormalization prescription is equivalent to the change of
$\Delta_{\|,\bot}(\xi_l,k_{\!\bot})$ in Eq.~(\ref{e19}) for
\begin{equation}
\Delta_{\|,\bot}^{\! R}(\xi_l,k_{\!\bot})\equiv
\frac{\Delta_{\|,\bot}(\xi_l,k_{\!\bot})}{\Delta_{\|,\bot}^{\!\infty}
(\xi_l,k_{\!\bot})}=
1-r_{\|,\bot}^{2}(\xi_l,k_{\!\bot})e^{-2aq_l},
\label{e21}
\end{equation}
\noindent
where the quantities $r_{\|,\bot}(\xi_l,k_{\!\bot})$ have the
meaning of reflection coefficients and are given by
\begin{equation}
r_{\|}^{2}(\xi_l,k_{\!\bot})=
\left(\frac{cq_l-Z_l\xi_l}{cq_l+Z_l\xi_l}\right)^2,
\quad
r_{\bot}^{2}(\xi_l,k_{\!\bot})=
\left(\frac{\xi_l-Z_lcq_l}{\xi_l+Z_lcq_l}\right)^2.
\label{e22}
\end{equation}
\noindent
Here $Z_l\equiv Z(i\xi_l)$.
The reflection coefficients (\ref{e22}) are in accordance with
\cite{24} where the reflection of a plane electromagnetic
wave incident from vacuum onto the plane surface of a metal was
described in terms of the Leontovich surface impedance.
Thus, the final renormalized expression for the Casimir
free energy in the surface impedance approach is given once more
by the Lifshitz formula (\ref{e1}),
where the reflection coefficients are expressed in terms of
impedance according Eq.~(\ref{e22}).
The above derivation was performed under the assumption that the
photon eigenfrequencies are real. This is, however, not the case
for arbitrary complex impedance. If the photon eigenfrequencies
are complex, the free energy is not given by Eq.~(\ref{e14}) (which
is already clear from the complexity of the right-hand side of
this equation). For arbitrary complex impedance the correct
expression for the free energy should be determined from the
solution of an auxiliary electrodynamic problem \cite{31}.
In fact the Casimir free energy is the
functional of the impedance even when the impedance has a nonzero
real part taking absorption into account. The solution of the
auxiliary electrodynamic problem leads to conclusion \cite{31}
that the correct free energy is obtained from
Eqs.~(\ref{e1}), (\ref{e22}) by analytic continuation to arbitrary
complex impedances, i.e., to arbitrary oscillation spectra. The
qualitative reason for the validity of this statement is that the
free energy depends only on the behavior of $Z(\omega)$ at the
imaginary frequency axis where $Z(\omega)$ is always real.
It must be emphasized that the
surface impedance approach is in perfect agreement with
thermodynamics. In the impedance approach the entropy, defined as
\begin{equation}
S(a,T)=-\frac{\partial{\cal{F}}_R(a,T)}{\partial T},
\label{e24}
\end{equation}
\noindent
is positive and equal to zero at zero temperature in accordance
with the Nernst heat theorem (remind that this is not the case in
the approaches based on the use of a frequency dependent
dielectric permittivity).
\section{Conclusions and discussion}
In the above we demonstrated that the Lifshitz formula for
the Casimir free energy can be derived starting from the boundary
condition on the surface of real metal containing the Leontovich
impedance. In doing so there is no need to use the concept of
a fluctuating electromagnetic field inside a metal. We argued
that the standard concept of a fluctuating field inside a metal,
described by the dielectric permittivity depending only on
frequency cannot serve as an adequate model for the
zero-point oscillations and thermal photons.
It follows from the
fact that the vacuum oscillations and thermal photons in
equilibrium can not lead to a heating of a
metal as do the electromagnetic fluctuations on the background
of $\varepsilon(\omega)$.
If this fact is overlooked, contradictions with the
thermodynamics arise when one substitutes into the Lifshitz
formula the Drude dielectric
function taking into account the volume relaxation and,
consequently, the Joule heating.
In the impedance approach the
entropy is in all cases nonnegative and takes zero value at zero
temperature. Thus, Eqs.~(\ref{e1}) and (\ref{e22}) lay down
the theoretical foundation for the calculation of the thermal Casimir
effect. In fact the approaches of \cite{3,8,6} and the
impedance approach of \cite{17,15} predict quite different
magnitudes of the thermal corrections to the Casimir force.
Up to separation distances of a few hundred nanometers, the thermal
correction predicted by the impedance approach is negligibly small.
As to the thermal corrections of \cite{3,8,6}, they may
achieve several percent of force magnitude. Recent experiment
\cite{32} on measuring the Casimir force by means of a
micromechanical torsional oscillator is consistent with the
theoretical predictions of the impedance approach. At the same
time, the experimental data of this experiment is in drastic
contradiction with the approaches of \cite{3,8,6}.
The experiments on measuring the Casimir and van der Waals
forces by means of an atomic force microscope also suggest
good opportunity to distinguish between the two approaches.
To conclude, different derivations of the Lifshitz formula
for the Casimir free energy in the case of real metals
lead to one and the same mathematical expression containing,
however, two different pairs of reflection coefficients.
Recent results demonstrate quite clear that for real metals
the use of reflection coefficients, expressed in terms of the
frequency-dependent dielectric permittivity, is not only
in violation of thermodynamics but is also in contradiction
with experiment. By this reason the reflection coefficients in
terms of the Leontovich impedance are evidently preferable.
\section*{Acknowledgments}
V.M.M.\ is grateful to the participants of the Seminar at Centro Brasileiro
de Pesquisas F\'{\i}sicas (CCP) for discussion.
The authors acknowledge FAPERJ for financial support.
|
{
"timestamp": "2005-03-06T19:39:25",
"yymm": "0503",
"arxiv_id": "quant-ph/0503064",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503064"
}
|
\section{Introduction}
Communities of ecologically similar species that compete with each other solely for resources
are often described by neutral community models (NCM) \cite{Hubbel,Bell,Chave,Norris}. These models proved to be successful and useful in describing many of the basic patterns of biodiversity such as the distribution of abundance, distribution of range, the range-abundance relation and the species-area relation \cite{Bell,Chave}. The neutral theory is considered by many ecologists as a radical shift from
established niche theories and generated considerable controversy \cite{Levine,Enquist,Abrams,Clark,Landau}. The relevance of NCM for describing the dynamics and statistics of real communities is still much debated and criticized \cite{Nee}.
Nowadays NCM are studied mostly by Monte Carlo type computer simulations \cite{Chave,Bell,McGill}, and
apparently there are no analytical results. One of the key issues that macro-ecologist are often investigating is the species abundances distribution (SAD), introduced for characterizing the frequency of species with a given abundance \cite{Preston,May,Pielou}. In case of NCM, SAD is generated numerically and it is called the Zero Sum Multinomial (ZSM) distribution \cite{McGill,Condit}.
The aim of the present paper is to give an analytical mean-field type approximation for ZSM. By using the
invariance of the system against the intrinsic fluctuations characteristic for NCM, we derive an
analytic solution that describes the results of computer simulations. The derived
analytical form of SAD leads also an interesting relation between the total number of individuals, total
number of species and the size of the most abundant species of the considered meta-community. This
novel scaling relation is confirmed by computer simulations on neutral models.
\section{Neutral Community Models}
NCM are usually defined on lattice sites, on which a given number of $S_{max}$ species can coexist
\cite{Hubbel,Bell,McGill} and compete for resources. Each lattice site can be occupied by many individuals belonging to different species, however the total number of individuals for each lattice site is limited to a fixed $N_{max}$ value. This limiting value models the finite amount of available resources in a given territory. As time passes individuals in the system can give birth to individuals belonging to the same species, can die or can migrate to a nearby site. The neutrality of the model implies that all individuals (regardless of the species they belong) are considered to be equally fit for the given ecosystem, and have thus the same $b$ multiplication, $d$ death and $q$ diffusion rate. The system is considered also in contact with a reservoir, from where with a small $w<<1$ probability per unit time an individual from a randomly chosen species can be assigned to a randomly chosen lattice site. This effect models the random fluctuations that can happen in the abundances of species. The dynamics of the considered community is than as follows:
\begin{itemize}
\item A given number of individuals from randomly chosen species are assigned on randomly chosen lattice sites.
\item With the initially fixed probabilities we allow each individual to give birth to another individual of the same species, to die or to migrate on a nearby site.
\item We constantly verify the saturation condition on each site. Once the number of individuals on a site exceeds the $N_{max}$ value, a randomly chosen individual is removed from that site.
\item We apply the random fluctuations resulting from the reservoir.
\end{itemize}
After on each lattice site saturation is achieved a dynamical equilibrium sets in, and one can study
the statistical properties of several relevant quantities. Computer simulations usually focus on
generating SAD and on studying several scaling relations like species-area and range-abundance relations.
\section{Analytical approximation for SAD}
Let us consider a fixed area in a NCM (a delimited region in the lattice) on which we study SAD.
In the selected area we denote by $S(x,t)$ the number of species with size $x$ at the time-moment $t$.
($x$ is a discrete variable $x=1,2, ....k...$). $S(x,t)$ divided by the total number of species yields the mathematically rigorously defined SAD (Species Abundances Distribution). We mention here that in most of the papers dealing with SAD, instead of this rigorously defined distribution function a histogram on
intervals increasing as a power of 2 is constructed \cite{Preston,May,Pielou}. On a log-normal scale this
histogram usually has a Gaussian shape, and thus SAD is called a log-normal distribution.
Without arguing on the relevance of this histogram (a nice treatment on this subject is
given by May \cite{May}) for the sake of mathematical simplicity we will not use this representation, and calculate instead the mathematically rigorous distribution function. It is of course anytime
possible to re-plot the obtained distribution function in the form that is usually used
by ecologists, using instead of the $x$ variable the $z=log_2(x)$ variable.
In the framework of the considered model the time evolution of $S(x,t)$ for an infinitesimally short $dt$
time can be approximated by the following master-equation:
\begin{eqnarray}
S(x,t+dt)=S(x,t)+[W_+(x-1) S(x-1,t)+W_-(x+1)S(x+1,t)-\nonumber \\
-W_+(x) S(x,t)-W_-(x) S(x,t)]dt
\label{master}
\end{eqnarray}
In the equation from above $W_+(x)$ denotes the probability that one species with size $x$ increases its size to $x+1$ in unit time and $W_-(x)$ denotes the probability that one species with size $x$ decreases it's size to $x-1$ in unit time. We neglected here the possibility that in the small $dt$ time-interval one species increases or decreases it's size by more than one individual. The value of $dt$ can be
always taken as small, as needed so that this starting assumption should hold. It worth also mentioning
that this master equation is not applicable in the neighborhood of the limiting values of $x$ since here either $x-1$ or $x+1$ is not existing. We expect thus that the shape of SAD determined from (\ref{master}) can have problems for very low and very high values of $x$.
We assume now that SAD reaches a steady-state in time. All computer simulations on
neutral models shows that this is true. This means that $S(x,t)$ should be time invariant in respect of the fluctuation governed by equation (\ref{master}).
Under this stationarity assumption we get the equation:
\begin{equation}
W_+(x-1) S(x-1,t)+W_-(x+1)S(x+1,t)-W_+(x) S(x,t) - W_-(x) S(x,t)=0
\label{form}
\end{equation}
We have to approximate now the $W_+(x)$ and $W_-(x)$ probabilities. We will work with
the assumptions of the NCM, and consider all species having the
same birth, death and migration rate. Let us denote by
$P_+$ the probability that one individual multiplies itself in unit time (we assume $P_+$ is the
same for all individuals and species). Let us denote by $P_-$ the probability that one
individual disappears from the considered territory in unit time (again the same for all individuals and species). Further we assume that:
\begin{eqnarray}
P_+<<1 \nonumber\\
P_-<<1
\label{assum}
\end{eqnarray}
By simple probability theory we get:
\begin{equation}
W_+(x)= x P_+ [1-(P_++P_-)]^{x-1}
\label{w+}
\end{equation}
The above equation tells us, that the increase by unity of the size of one species can be
realized if any of the $x$ individual from a selected species
multiplies itself, while the other individuals remain unchanged. (Of course there
are many other possibilities involving the birth and death of more than one individual.
However, since we considered the (\ref{assum}) assumption
all other possibilities will be with orders of magnitude smaller).
It is also worth mentioning that for the selected local community the effect of migration and the
stochastic contribution from the reservoir can be
taken into account through the birth and death processes, changing slightly the values of this
probabilities. Migration inside the considered area is equivalent with a birth process, while migration
outside from the considered territory is equivalent with death of individuals.
Using the assumptions (\ref{assum}) we can make now the following approximations:
\begin{eqnarray}
W_+(x)= xP_+[1-(P_++P_-)]^{x-1} = x P_+ [1-(P_++P_-)]^{[1/(P_++P_-)] \cdot (P_++P_-)(x-1)} \approx
\nonumber \\
\approx xP_+e^{-(P_++P_-)(x-1)}
\label{am}
\end{eqnarray}
In the same manner, one can write:
\begin{equation}
W_-(x)= x P_- [1-(P_++P_-)]^{x-1} \approx x P_-e^{-(P_++P_-)(x-1)}
\label{ap}
\end{equation}
Instead of $P_+$ and $P_-$ we introduce now two new notations:
\begin{eqnarray}
s=P_++P_- \\
q=P_+-P_-
\end{eqnarray}
from where:
\begin{eqnarray}
P_+=\frac{s+q}{2} \\
P_-=\frac{s-q}{2}
\end{eqnarray}
From the assumptions (\ref{assum}) it is clear that it also holds:
\begin{eqnarray}
s<<1 \\
q<<1
\end{eqnarray}
Let us assume now that
\begin{equation}
|P_+|=|P_-| \longrightarrow q=0,
\end{equation}
which would mean that the probability of multiplication and death is the same, so
there is a constant number of individuals in the considered local community. In other words this means
that the territory is saturated, and although the size of different species fluctuates, the total
number $N_t$ of population is constant.
The probability density for the species abundances distribution (SAD) is given than as:
\begin{equation}
\rho(x,t)=\frac{S(x,t)}{S_t}.
\end{equation}
Instead of $x$ let us introduce now a new variable $y=x/N_t<<1$ ($N_t>>1$ is
the total number of individuals in the system)
For $\rho(y,t)$ we have the (\ref{form}) equation:
\begin{equation}
W_+(yN_t-1) \rho(y-\frac{1}{N_t},t)+W_-(yN_t+1) \rho(y+\frac{1}{N_t},t)-W_+(yN_t) \rho(y,t) - W_-(yN_t) \rho(y,t)=0
\end{equation}
Since $\rho(y,t)$ is a limiting distribution (not depending on $t$ anymore) we will simply
denote is as $\rho(y)$.
\begin{equation}
W_+(yN_t-1) \rho(y-\frac{1}{N_t})+W_-(yN_t+1) \rho(y+\frac{1}{N_t})-W_+(yN_t) \rho(y) - W_-(yN_t) \rho(y)=0
\label{start}
\end{equation}
We can use now Taylor series expansion to get $\rho(y-\frac{1}{N_t})$ and $\rho(y+\frac{1}{N_t})$:
\begin{eqnarray}
\rho(y-\frac{1}{N_t})=\rho(y)-\frac{1}{N_t}\rho'(y)+\frac{1}{2{N_t}^2} \rho''(y) \\
\rho(y+\frac{1}{N_t})=\rho(y)+\frac{1}{N_t}\rho'(y)+\frac{1}{2{N_t}^2} \rho''(y)
\end{eqnarray}
We denoted here by $\rho'(y)$ and $\rho''(y)$ the first and second order derivatives
of the $\rho(y)$ function, respectively. Taking account of $q=0$, the values of $W_{\pm}(y)$ are given by (\ref{am}, \ref{ap}) as follows:
\begin{eqnarray}
W_+(yN_t-1)=\frac{(yN_t-1) s}{2} exp[-(yN_t-2)s] \\
W_-(yN_t+1)= \frac{(yN_t+1) s}{2} exp[-yN_ts] \\
W_+(yN_t)= \frac{yN_ts}{2} exp[-(yN_t-1)s]\\
W_-(yN_t)= \frac{yN_ts}{2} exp[-(yN_t-1)s]
\end{eqnarray}
Plugging all these in equation (\ref{start}):
\begin{eqnarray}
\frac{(yN_t-1) s}{2} exp[-(yN_t-2)s] [\rho(y)-\frac{1}{N_t}\rho'(y)+\frac{1}{2{N_t}^2} \rho''(y) ]+
\nonumber \\ \frac{(yN_t+1) s}{2} exp[-yN_ts]
[\rho(y)+\frac{1}{N_t}\rho'(y)+\frac{1}{2N_t^2} \rho''(y) ] =
\frac{yN_ts}{2} exp[-(yN_t-1)s] \rho(y)
\end{eqnarray}
Simplifying both sides with $s \cdot exp[-yN_ts]$, some immediate algebra yields the following second order differential equation for $\rho(y)$:
\begin{eqnarray}
\rho(y)[\frac{yN_t}{2}(e^{2s}-2e^s+1)-\frac{1}{2}(e^{2s}-1)] + \rho'(y) \frac{1}{N_t} [ \frac{yN_t}{2}(1-e^{2s}) + \frac{1}{2}(1-e^{2s})] + \nonumber \\
+\rho''(y) \frac{1}{2N_t^2} [(\frac{yN_t}{2} (e^{2s}+1)+\frac{1}{2}(1-e^{2s})]=0
\end{eqnarray}
Since $s<<1$ the following approximations are justified
\begin{eqnarray}
e^{2s} \approx 1+2s \\
e^{s} \approx 1+s ,
\end{eqnarray}
and the differential equation becomes:
\begin{equation}
-\rho(y) s + \rho'(y) \frac{1}{N_t} [s+1-yN_ts] + \rho''(y) \frac{1}{2N_t^2} [yN_t+yN_ts-s] = 0
\end{equation}
For solving this differential equation, in the {\bf first approximation} we neglect
all term that are proportional with the $1/N_t \rightarrow 0$ quantity. This yields
a first order differential equation:
\begin{equation}
s \rho(y) = -y s \rho'(y)
\end{equation}
This equation has the immediate solution
\begin{equation}
\rho_I(y)=C_1/y,
\label{app1}
\end{equation}
with $C_1$ an integration constant.
The histogram $\sigma (z)$ that is usually used for SAD can be immediately determined from (\ref{app1}),
writing the $\rho_I(y)$ distribution as a function of the $z=log_2(x)=log_2(yN_t)$ variable. It is immediate to realize that this would yield a constant distribution ($\sigma_{I}(z)=C$).
A {\bf better approximation} can be achieved by keeping the terms proportional with $1/N_t$ and
neglecting the second orderly small $1/N_t^2$ and $s/N_t$ terms. This yields the
\begin{equation}
-s \rho(y) + \rho'(y) \frac{1}{N_t} [1-yN_ts] + \rho''(y)\frac{1}{2N_t} y =0
\end{equation}
differential equation. Going back now to the $x=yN_t$ variable
\begin{equation}
-s \rho(x) + \rho'(x) [1-xs] + \rho ''(x) \frac{x}{2} =0,
\end{equation}
we get the general solution
\begin{equation}
\rho_{II}(x)=\frac{C_1}{x}+\frac{e^{2sx} C_2}{x},
\end{equation}
where $C_1$ and $C_2$ are two integration constants.
By visually comparing with the experimental and simulated SAD curves we can conclude that
we need $C_1>0$ and $C_2<0$ to get the right shape. The general solution for SAD, should write thus
\begin{equation}
\rho_{II}(x)= \frac{K_1}{x}(K_2-e^{2sx}),
\end{equation}
with $K_1$ and $K_2$ two real, positive constants.
It is immediate to observe that the obtained distribution for SAD, has a cutoff, i.e. there
is a maximum value of $x$ until $\rho(x)$ is acceptable (remains positive). This results, is not surprising, since due to the finite number of individuals in the system and the finite value of
the number of species one would naturally expect a cutoff in the distribution.
There are three fitting parameters in the mathematical expression of $\rho_{II}(x)$ ($K_1$, $K_2$ and $s$). Since $\rho_{II}(x)$ has to be normalized, we can
determine $K_1$ as a function of $K_2$ and $s$. The normalization of this distribution function
is not easy and cannot be done analytically, since there is no primitive function for
$exp(\alpha x)/x$.
However, if we can use the $sx<<1$ assumption and consider a Taylor expansion
in the exponential we obtain the more simple
\begin{equation}
\rho_{II}(x) \approx F_n \frac{F_1-x}{x},
\label{form2}
\end{equation}
($F_n$ and $F_1$ are again two positive real constants) distribution, which has a cutoff
for $x=F_1$.
This distribution function is exactly the same as the one proposed by Dewdney using
totally different arguments \cite{Dewdney},
and named {\em logistic-J distribution}. As argued in \cite{Dewdney} it describes well
the SAD for many real communities.
The normalization condition for this distribution function is:
\begin{equation}
\int_{1}^{F_1} F_n \frac{F_1-x}{x} dx =1,
\label{normal}
\end{equation}
and an immediate calculus gives:
\begin{equation}
F_n=\frac{1}{F_1 ln(F_1)-F_1+1},
\end{equation}
The approximated normalized distribution function for SAD is then:
\begin{equation}
\rho_{II}(x) \approx \frac{1}{F_1 ln(F_1)-F_1+1} \frac{F_1-x}{x}
\label{result}
\end{equation}
We can consider thus the above simple one-parameter fit to approximate the results for SAD on NCM.
The shape of $\sigma(z)$ can be again quickly obtained
from $\rho_{II}(x)$, by changing the variable in this distribution function to $z=log_2(x)$.
A simple calculation yields the form
\begin{equation}
\sigma_{II}(z)=C*(F_1-2^z),
\label{app2}
\end{equation}
where C is another normalization constant. It is important to realize, that $\sigma(z)$
given by the above approximation does not show the generally observed bell shaped curve,
and for small values of $z$ it is a constant. We must remember however that the shape of SAD given by our approximation can not be trusted for small $z$ values, since in this limit the starting master equation (\ref{master}) is not valid.
\section{SAD from computer simulations}
In order to check the validity of our analytical approximation for SAD we performed computer simulations
on the model presented in Section 2. We considered a lattice of size $20 \times 20$, $S_t=400$ species, and $N_{max}=1000$ for each lattice site. We studied a local community on a square of
$9\times 9$ lattice sites, and we fixed several values for the dynamical parameters $d/b$ and $q/b$.
We used periodic boundary conditions, and the efficient kinetic or resident time Monte Carlo
algorithm was implemented. The simulations were made on a $Pentium^{(TM)} 4$ cluster. As a general results, we obtained that the analytical form given by (\ref{result}) describes well the simulation data for SAD. On Figure 1 we present a characteristic fit for the simulation data. The parameters
used in the simulation were $d/b=0.3$ and $q/b=0.2$. The obtained best fit parameter for
equation (\ref{result}) was $F_1=14500$. The rigorously
defined $\rho(x)$ distribution function suggest that in NCM SAD has a scale-invariant nature. The finite size of the system introduces a natural cutoff in this
scale-invariant behavior.
Computer simulations on NCM proves thus the applicability of our analytical approximations for the
form of the ZSM distribution.
\section{Scaling laws resulting from SAD}
Starting from the analytical approximation (\ref{result}) for the form of SAD, we can derive
an interesting relation between the size of the most abundant species ($N_s$), the total number
of individuals ($N_t$) and the number of detected species ($S_t$) in the considered meta-community.
The distribution function (\ref{result}) has a cutoff at $x=F_1$, from where it results that
$F_1 \approx N_s$. It is also immediate to realize that from the definition of $\rho(x)$ it results
\begin{eqnarray}
N_t=\int_{1}^{N_s} C x \frac{\rho(x)}{F_n} dx = C \int_{1}^{N_s} x \frac{N_s-x}{x} dx \\
S_t=\int_{1}^{N_s} C \frac{\rho(x)}{F_n} dx = C \int_{1}^{N_s} \frac{N_s-x}{x} dx,
\end{eqnarray}
where $C$ is a normalization constant, which normalizes $\rho(x)$ to the total number of
species in the local community. The above two integrals are easily calculated and
leads to the following two coupled differential equations:
\begin{eqnarray}
N_t=C N_s (N_s-1) - \frac{C}{2} (N_s^2-1) \\
S_t=C N_s ln(N_s) - C (N_s-1)
\end{eqnarray}
Working on relatively large habitats, one can use the $N_s>>1$ assumption, and the coupled equation system from above can be simplified:
\begin{eqnarray}
N_t \approx \frac{C}{2} N_s^2 \\
S_t \approx C N_s [ln(N_s)-1]
\end{eqnarray}
Eliminating from this system the normalization constant $C$ we obtain the important relation:
\begin{equation}
\frac{S_t N_s}{N_t [ln(N_s)-1]}=2
\label{magic}
\end{equation}
Computer simulation results on NCM supports again the validity of the magic formula from above.
(The simulations were made on a $20 \times 20$ lattice, and we choose $S_t=400$, $N_{max}=1000$, $d/b=0.3$ and $q/b=0.2$).
On Figure 2 we plotted the simulation results for different local community sizes, and
the plot shows that equation (\ref{magic}) works well, however the constant on the right
side of the equation seems to be slightly different from 2.
We think that this slight difference is the result of our crude approximation: $F_1 \approx N_s$, and in reality we should have $F_1$ slightly bigger than $N_s$. The simulation data from Figure 2
was obtained after averaging on several local communities of size $A$.
Increasing the size $A$ of the considered habitat one would naturally expect $N_t \sim A$. Using
equation (\ref{magic}) one would immediately get thus the interesting scaling-law:
\begin{equation}
\frac{S_t N_s}{ln(N_s)-1} \sim A
\label{scaling}
\end{equation}
The (\ref{scaling}) scaling relation can be also immediately verified in computer simulations
on NCM. Results for a $20 \times 20$ lattice, $S_t=400$, $N_{max}=1000$, $d/b=0.3$ and $q/b=0.2$ are shown on Figure 3. On the figure with a dashed line we indicated the power-law with exponent $1$.
As seen from the figure, the simulation data supports the scaling-law given by our analytical
approach.
\section{Conclusions}
We have given here an mean-field type analytical approximation for the species abundances distribution function for neutral community models. By using the invariance of this distribution regarding the internal
fluctuations characteristic for the model, we derived an analytical approximation for the
distribution function which describes well the simulation data obtained on NCM. The derived distribution
function has a natural cutoff, governed by the finite extent of the system, and leads to
an interesting relation between the total number of individuals, total number of species and the size of the most abundant species, found in the considered habitat. Computer simulations on neutral models
confirms the validity of this scaling relation.
\section{Acknowledgments}
The present study was supported by the Sapientia KPI foundation for interdisciplinary research.
We are grateful for Dr. N. Stollenwerk for helpful suggestions and discussions. We also
thank Dr. A. Balogh and Dr. V. Mark\'o for introducing us in this fascinating interdisciplinary field,
and for providing us a lot of interesting bibliography on the subject.
|
{
"timestamp": "2005-03-17T10:51:17",
"yymm": "0503",
"arxiv_id": "q-bio/0503026",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503026"
}
|
\section{From ASDYM equations to Einstein--Weyl structures}
\setcounter{equation}{0}
The idea of allowing
infinite--dimensional groups of diffeomorphisms of some manifold $\Sigma$
as gauge groups provides a link between
the Yang--Mills--Higgs theories
on $\mathbb{R}^n$ and conformal gravity theories on
$\mathbb{R}^{n}\times\Sigma$. The gauge--theoretic covariant derivatives
and Higgs fields are reinterpreted as a frame of vector fields
thus leading to a conformal structure \cite{Wa90}.
This program has lead, among other things, to a dual description of certain
two--dimensional integrable systems: as symmetry reductions of
anti--self--dual Yang--Mills (ASDYM), or as special curved anti--self--dual
conformal structures \cite{Wa92, MW96, DMW98, D02}.
In this paper we shall give the first example of a dispersionless
integrable system in 2+1 dimension which fits into this framework
(Theorem \ref{asdred}).
As a spin--off we shall obtain a gauge--theoretic
characterisation of hyperCR Einstein--Weyl spaces in 2+1 dimensions
(Theorem \ref{asdth}). We shall also construct two explicit new
classes of solutions to the system (\ref{PMA}) out of solutions to
the nonlinear Schr\"{o}dinger equation, and
the Korteweg de Vries equation (formulae (\ref{EWnls}) and (\ref{EWkdv})).
Consider a pair of quasi-linear PDEs
\begin{equation}
\label{PMA}
u_t+w_y+uw_x-wu_x=0,\qquad u_y+w_x=0,
\end{equation}
for two real functions $u=u(x, y, t), w=w(x, y, t)$.
This integrable system has recently been used to characterise a class
of Einstein--Weyl structures in 2+1 dimensions \cite{D04}. It has also
appeared in other contexts \cite{Pa03, MSh03, MSh04, FK04} as an example of 2+1 dimensional dispersionless integrable models.
The equations (\ref{PMA}) arise as compatibility conditions
$[L, M]=0$ of an overdetermined system of linear equations
$L\Psi=M\Psi=0$, where $\Psi=\Psi(x, y, t, \lambda)$ is a function,
$\lambda$ is a spectral parameter, and the Lax pair is given by
\begin{equation}
\label{Lax11}
L=\partial_t-w\partial_x-\lambda\partial_y,\qquad
M=\partial_y+u\partial_x-\lambda\partial_x.
\end{equation}
This should be contrasted with Lax pairs for other
dispersionless integrable systems \cite{Ta90, Z94, GMM, Kon3, BK04}
which contain derivatives w.r.t the spectral parameter.
The first equation in (\ref{PMA}) resembles a flatness
condition for a connection with the underlying Lie algebra diff$(\Sigma)$,
where $\Sigma=S^1$ or $\mathbb{R}$. The following result makes this
interpretation precise
\begin{theo}
\label{asdred}
The system {\em(\ref{PMA})} arises as a symmetry reduction
of the anti--self--dual Yang Mills equations in signature $(2, 2)$
with the infinite--dimensional gauge
group Diff$(\Sigma)$ and two commuting translational symmetries
exactly one of which is
null. Any such symmetry reduction is gauge equivalent to
(\ref{PMA}).
\end{theo}
{\bf Proof.}
Consider the flat metric of signature $(2, 2)$ on $\mathbb{R}^4$ which
in double null coordinates $y^{\mu}=(t, z, \tilde{t}, \tilde{z})$ takes the
form
\[
\d s^2=\d t\d \tilde{t}-\d z\d \tilde{z},
\]
and choose the volume element
$\d t\wedge\d \tilde{t}\wedge\d z\wedge\d \tilde{z}$.
Let $A\in T^*\mathbb{R}^4\otimes\mathfrak{g}$ be a connection one--form, and
let $F$ be its curvature two--form. Here $\mathfrak{g}$ is the Lie algebra of some
(possibly infinite dimensional) gauge group $G$.
In a local trivialisation $A=A_\mu\d y^\mu$
and $F=(1/2)F_{\mu\nu}\d y^{\mu}\wedge\d y^{\nu}$, where
$F_{\mu\nu}=[D_{\mu}, D_{\nu}]$
takes its values in $\mathfrak{g}$. Here
$D_\mu=\partial_\mu-A_\mu$ is the covariant derivative.
The connection is defined up to gauge transformations
$A\rightarrow b^{-1}Ab-b^{-1}\d b$, where $b\in \mbox{Map}(\mathbb{R}^4, G)$.
The ASDYM equations on $A_\mu$ are $F=-\ast F$, or
\[
F_{tz}=0, \qquad F_{t\tilde{t}}-F_{z\tilde{z}}=0,\qquad
F_{\tilde{t}\tilde{z}}=0.
\]
These equations are equivalent to the commutativity of the Lax pair
\[
L=D_t-\lambda D_{\tilde{z}}, \qquad M=D_z-\lambda D_{\tilde{t}}
\]
for every value of the parameter $\lambda$.
We shall require that the connection possesses two commuting
translational symmetries, one null and one non--null
which in our coordinates are in $\partial_{\tilde{t}}$
and $\partial_{\tilde{y}}$ directions,
where $z=y+\tilde{y}, \tilde{z}=y-\tilde{y}$.
Choose a gauge such that $A_{\tilde{z}}=0$ and one of the
Higgs fields $\Phi=A_{\tilde{t}}$ is constant. The Lax pair has so
far been reduced to
\begin{equation}
\label{2Dlax}
L=\partial_t-W-\lambda\partial_y, \qquad M=\partial_y-U-\lambda\Phi,
\end{equation}
where $W=A_t$ and $U=A_z$ are
functions of $(y, t)$ with values in the Lie algebra
$\mathfrak{g}$, and $\Phi$ is an element of $\mathfrak{g}$ which doesn't depend on $(y, t)$.
The reduced ASDYM equations are
\[
\partial_yW-\partial_tU+[W, U]=0, \qquad \partial_y U+[W, \Phi]=0.
\]
Now choose $G=\mbox{Diff}(\Sigma)$, where $\Sigma$ is some one--dimensional
manifold, so that $(U, W, \Phi)$ become vector fields
on $\Sigma$. We can choose a local
coordinate $x$ on $\Sigma$ such that
\begin{equation}
\label{from1}
\Phi=\partial_x, \qquad W=w(x, y, t)\partial_x, \qquad U=-u(x, y, t)\partial_x
\end{equation}
where $u, w$ are smooth functions on $\mathbb{R}^3$. The reduced
Lax pair (\ref{2Dlax}) is identical to (\ref{Lax11}) and the ASDYM equations
reduce to the pair of PDEs (\ref{PMA}). \hfill $\Box $\medskip
Recall that a Weyl structure on an $n$ dimensional manifold ${\cal W}$ consists
of a torsion-free connection $D$ and a conformal structure
$[h]$ which is compatible with $D$ in a sense
that $Dh=\omega\otimes h$ for some one-form $\omega$ and $h\in[h]$.
We say that a Weyl structure
is Einstein--Weyl if the traceless part of the symmetrised Ricci tensor of
$D$ vanishes. The three--dimensional Einstein--Weyl structure is called hyperCR \cite{CP99,
GT98, DT01, D04} if
its mini-twistor space \cite{H82} is a holomorphic bundle over $\mathbb{CP}^1$.
In \cite{D04} it was demonstrated that if $n=3$, and $[h]$ has signature
$(++-)$ then
all Lorentzian hyperCR Einstein--Weyl structures are locally of the form
\begin{equation}
\label{PMAEW}
h=(\d y+u\d t)^2-4(\d x+w\d t)\d t,\qquad
\omega =u_x\d y+(uu_x+2u_y)\d t,
\end{equation}
where $u, w$ satisfy (\ref{PMA}).
This result combined with Theorem \ref{asdred} yields the following
coordinate independent characterisation of the hyperCR Einstein--Weyl
condition
\begin{theo}
\label{asdth}
The ASDYM equations
in $2+2$ dimensions with two commuting translational symmetries
one null and one non--null, and the gauge group Diff$(\Sigma)$
are gauge equivalent to the hyperCR Einstein--Weyl equations in
\em{2+1} dimensions.
\end{theo}
This is a Lorentzian analogue of a Theorem
proved in \cite{C01} in the Euclidean case. The readers should note
that in \cite{C01} the result is formulated in terms of the Hitchin system,
and not reductions of the ASDYM system.
\section{Reductions to KdV and NLS}
\setcounter{equation}{0}
Reductions of the ASDYM equations with $G=SU(1, 1)$
by two translations (one of which is null)
lead to well--known integrable systems
KdV, and NLS \cite{MS89}. The group $SU(1, 1)$ is a subgroup
of Diff$(S^1)$ which can be seen by considering the Mobius
action of $SU(1, 1)$
\[
\zeta\longrightarrow M(\zeta)=\frac{\alpha\zeta+\beta}{\overline{\beta}\zeta+\overline{\alpha}}, \qquad
|\alpha|^2-|\beta|^2=1
\]
on the unit disc. This restricts to the action on the circle as
$|M(\zeta)|=1$ if $|\zeta|=1$.
We should therefore expect that equation (\ref{PMA}) contains
KdV and NLS as its special cases (but not necessarily symmetry
reduction). To find explicit classes of solutions to (\ref{PMA}) out of solutions
to KdV and NLS we proceed as follows.
Consider the matrices
\[
\tau_+=\left(
\begin{array}{cc}
0&1\\
0&0
\end{array}
\right ), \qquad
\tau_-=\left(
\begin{array}{cc}
0&0\\
1&0
\end{array}
\right ),\qquad
\tau_0=\left(
\begin{array}{cc}
1&0\\
0&-1
\end{array}
\right )
\]
with the commutation relations
\[
[\tau_+, \tau_-]=\tau_0, \qquad [\tau_0, \tau_+]=2\tau_+, \qquad
[\tau_0, \tau_-]=-2\tau_-.
\]
The NLS equation
\begin{equation}
\label{nls}
i\phi_t=-\frac{1}{2}\phi_{yy}+\phi|\phi|^2, \qquad \phi=\phi(y, t)
\end{equation}
arises from the reduced Lax pair (\ref{2Dlax}) with
\[
W=\frac{1}{2i}(-|\phi|^2\tau_0+\phi_y\tau_--\overline{\phi}_y\tau_+),
\qquad
U=-\phi\tau_--\overline{\phi}\tau_+,
\qquad
\Phi=i\tau_0.
\]
Now we replace the matrices by vector fields on $\Sigma$ corresponding
to the embedding of $su(1, 1)$ in diff$(\Sigma)$
\[
\tau_+\longrightarrow \frac{1}{2i}e^{2ix}\frac{\partial}{\partial x}, \qquad
\tau_-\longrightarrow -\frac{1}{2i}e^{-2ix}\frac{\partial}{\partial x}, \qquad
\tau_0\longrightarrow \frac{1}{i}\frac{\partial}{\partial x},
\]
and read off the solution to (\ref{PMA}) from (\ref{from1})
\begin{equation}
\label{EWnls}
u=\frac{1}{2i}(\overline{\phi}e^{2ix}-\phi e^{-2ix}), \qquad
w=\frac{1}{2}|\phi|^2+\frac{1}{4}(e^{2ix}\overline{\phi}_y+e^{-2ix}\phi_y).
\end{equation}
The second equation in (\ref{PMA}) is satisfied identically, and the
first is satisfied if $\phi(y, t)$
is a solution to the NLS equation (\ref{nls}).
Analogous procedure can be applied to the KdV
equation
\begin{equation}
\label{KdV}
4v_t-v_{yyy}-6vv_y=0, \qquad v=v(y, t).
\end{equation}
The Lax pair for this equation
is given by (\ref{2Dlax}) with
\[
W=\Big(q_y\tau_+
-\kappa\tau_- -\Big(\frac{1}{2}q_{yy}+qq_y\Big)\tau_0\Big),
\qquad
U=\tau_+-q\tau_0 -(q_y+q^2)\tau_-,
\qquad \Phi=\tau_-,
\]
where
\[
\kappa=\frac{1}{4}q_{yyy}+qq_{yy}+\frac{1}{2}{q_y}^2+q^2q_y,\qquad
\mbox{and}\; v=2q_y.
\]
Now we choose $x$ such that
\[
\tau_+\longrightarrow -x^2\frac{\partial}{\partial x}, \qquad
\tau_-\longrightarrow \frac{\partial}{\partial x}, \qquad
\tau_0\longrightarrow 2x\frac{\partial}{\partial x},
\]
and read off the
expressions for $u$ and $w$
\begin{equation}
\label{EWkdv}
u=x^2+2xq+q_y+q^2, \qquad w=-x^2q_y-x(q_{yy}+2qq_y)-\kappa.
\label{KdVsol}
\end{equation}
The second equation in (\ref{PMA}) holds identically, and the
first is satisfied if $v$ is a solution to (\ref{KdV}).
In references \cite{MSh03, MSh04} the so called `universal hierarchy' was
studied and a general procedure of constructing its
differential reductions was proposed.
The system (\ref{PMA}) arises from the first two flows of this hierarchy,
but it is not clear how the differential constraints
imposed in \cite{MSh03, MSh04} can be understood from the Diff$(S^1)$ point of
view. It would be interesting to see whether our reductions to NLS and KdV
are `differential' in the sense of the above references.
One remark is in place: There is a standard procedure \cite{JT85} of
constructing
anti--self--dual conformal structures with symmetries out of EW
structures in 3 or 2+1 dimensions. The procedure is based on
solving a linear
generalised monopole equation on the EW background.
Moreover, the hyperCR EW structures
always lead to hyper--complex conformal structures with a
tri--holomorphic Killing vector, and it is possible to
choose a monopole such that there exist a Ricci--flat metric in the conformal
class \cite{GT98}. Any hyperCR EW (\ref{PMAEW})
structure given in terms of KdV, or NlS potential by (\ref{EWkdv}) or
(\ref{EWnls}) will therefore lead to a $(++--)$ ASD Ricci--flat metric
with a tri--holomorphic homothety.
The explicit formulae for the metric in terms
of solutions to (\ref{PMA}) can be found in \cite{D04}.
Another class of ASD Ricci--flat metrics has been constructed from
KdV and NLS, by embedding $SU(1,1)$ in a Lie algebra of volume preserving
transformations of the Poincare disc \cite{DMW98}. These metrics generically
do not admit any symmetries, and therefore are different from ours.
\section*{Acknowledgements}
Both authors
were partly supported by NATO grant PST.CLG.978984. MD is a member of the
European Network in Geometry,
Mathematical Physics and Applications. We wish to thank
the anonymous referee for valuable comments.
|
{
"timestamp": "2005-06-10T19:44:19",
"yymm": "0503",
"arxiv_id": "nlin/0503030",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503030"
}
|
\section{Introduction}
Models of non-linear spatially extended systems exhibit a variety of spatial
and temporal pattern forming phenomena. A subclass of these patterns are
spatially localized structures \cite{rev1} that include pulses, solitons,
fronts, and domain walls. The standard analysis of these localized structures
assumes that, on large length and time scales, they can be
treated as ``coherent objects'' \cite{rev1}, with effective parameters like
position, and velocity attributed to them. A perturbative expansion about
this isolated coherent object profile is then used to understand
its response to external forces, interaction with
other localized structures \cite{ephlick1,ephlick}, noise, or internal
instabilities \cite{meron4,skyrabin}. Perturbative calculations reduce the
original non-linear problem to a series of linear problems that require
consistency criteria known as solvability conditions for their solution.
Typically, the solution of a linear equation $L \phi= \psi$, requires the
orthogonality of $\psi$ to the zero modes $\chi$, ie., $(\psi,\chi)=0$, in the
null space of the adjoint homogeneous problem $L^{\dag} \chi=0$.
Often, the symmetries in a particular system are responsible for the
zero modes of the operators obtained after a perturbative expansion.
For instance, since a localized structure profile and the same
profile translated infinitesimally are both solutions of the underlying
non-linear equation, the difference of the two profiles provides a zero
(neutral or Goldstone) mode. Strictly, the zero mode is the derivative of the
localized structure profile, and the underlying symmetry is translation
invariance. Zero modes extracted from symmetry arguments may then be employed
straightforwardly into solvability integrals.
The argument above, based on translational invariance, works if the system
size is infinite. For a localized
structure near a system boundary, due to the relevant
boundary conditions that have to be imposed there, the localized structure
solution and its infinitesimally translated counterpart are no longer
solutions of the same equation. Hence, translational invariance is broken.
Therefore, in this case, one has to contend, not only
with the incorporation of the boundary data into the solvability condition, but
also the appropriate treatment of broken translation invariance.
Most treatments of localized structures follow analytical techniques that
fall in the realm of moving boundary approximations \cite{fife}. A common
feature to these approximations, for instance, in excitable waves
\cite{meronrp}, or bistable fronts \cite{meron5, siam1, meron2}, is the
separation of the description of the localized structure into an
``inner region'' and ``outer region''. The inner region, characterized
by short spatial scales
and fast time scales, captures the internal dynamics of the
localized structure. In contrast, the dynamics of the localized structure
as a whole is captured by
the long spatial and time scales comprising the outer problem. The solvability
integrals in moving boundary type approximations occur in the inner
problem. Since it is the fields in the outer region that mediate the
interaction with the boundary \cite{meron1,yadav}, the boundary data is not
incorporated into solvability conditions arising in the inner problem.
There are ample situations however, where it may not be possible to have
separate inner and outer regions of a localized structure by manipulating
relevant system parameters \cite{Coullet}. In such cases, the boundary
data must be directly incorporated into the solvability condition.
In this paper, through an appropriately chosen adjoint operator
$L^{\dag}$ defined for the semi-infinite system (localized structure near
a boundary), we develop techniques that not only include the boundary data
into the solvability condition, but also directly incorporate the effects
of broken translational invariance into it. We accomplish this by extending
the definition of the Goldstone mode to include the possibility that the
corresponding eigenvalue be non-zero, with its magnitude dependent on how
close the localized structure is to the boundary. This leads further to a
modified solvability criteria.
As a case study, we develop our techniques in the context of reaction-diffusion
systems and apply it to non-equilibrium domain walls (fronts) found in
bistable regimes. In bistable reaction-diffusion systems, fronts connecting
the two homogeneous steady states can undergo a bifurcation, called a front
bifurcation, where a stationary Ising front loses stability to two
counter-propagating Bloch fronts\cite{Coullet}. This bifurcation can be
regarded as an internal instability of the Ising front, the localized
structure about which a perturbative expansion is carried out to obtain the
propagating Bloch wall solution. This bifurcation, also known as the
Ising-Bloch bifurcation, has been observed in several systems, like chemical
reactions \cite{meron4, haas, Li} and also in liquid crystals \cite{Frisch,kai}.
In a recent work \cite{yadav}, we examined the influence of boundaries on
Ising-Bloch fronts in a FitzHugh-Nagumo (FHN) reaction diffusion model. We
were able to derive order parameter equations (OPE) for front dynamics, where
the fronts were perturbed by the imposition of Dirichlet and possibly other
boundary conditions at the boundaries. This derivation for the two component
FHN model required restrictive assumptions about the relative size of the
fronts for the two concentration fields, allowing for the use of moving boundary
approximation like singular perturbation
methods detailed in \cite{siam1,meron2}. These singular perturbation
techniques are quite versatile, predicting exotic phenomena like front
reversal, trapping, and oscillation at the boundary. However, as observed
earlier, we wish to examine the effects of boundary data on localized
structures, where moving boundary type approximations are not applicable,
and the explicit incorporation of boundary data in a solvability condition is
required.
In the next section we discuss the extension of the solvability condition to
incorporate boundary data and broken translational invariance
via the extension of the Goldstone mode in a generic system exhibiting a
localized structure. In Sec. III, we describe the modification of the
slow manifold of a generic Ising-Bloch front due to boundary effects.
In Sec. IV, we
apply our method of solvability condition extension to study the effects of
finite size and Dirichlet boundary conditions on the dynamics of Ising-Bloch
fronts in a parametrically forced complex Ginzburg Landau equation (CGLE)
\cite{Coullet,skyrabin}. An important reason behind this choice is its
experimental context, modeling Ising-Bloch fronts in Liquid crystals subjected
to rotating magnetic fields \cite{Frisch,kai}. Liquid crystal systems are
ideal candidates to study boundary effects, as lateral boundary conditions
may be imposed in a controlled manner by appropriate electric fields
\cite{book}. Another experimental test bed is presented Ref.~\cite{bode2},
in the form of coupled non-linear electrical oscillators, where the application
of boundary conditions requires a minor and straightforward variation of the
original circuit. In Sec. V we discuss in detail the implications of the
derived order parameter equations for the parametrically forced CGLE. In
Sec. VI we present our conclusions.
\section{Goldstone modes and solvability criteria}
Consider a general non-linear PDE,
\begin{eqnarray}
\partial_t U= {\cal L} U +N(U),
\label{eq:gone}
\end{eqnarray}
where $U(x,t)$ is the solution vector, ${\cal L}$ are the linear terms,
and $N(U)$ are the non-linear terms. Let $U_0(x)$ be a stationary
localized solution of Eq.~(\ref{eq:gone}), with the asymptotic behavior
$U(x)\rightarrow 0; x\rightarrow \pm \infty$. In principle, $U_0(x)$ also
encompasses uniformly translating localized structures, which are stationary
in a co-moving frame.
Due to translational invariance in the system, one has $A(x)=U_{0x}$, the
derivative with respect to $x$ of the localized structure profile, as the
zero eigenvalue (neutral or Goldstone) mode of the operator
$\pounds={\cal L}+N^{\prime}(U_0)$. Also, it is reasonable to expect that
due to translational invariance $\pounds^{\dag}$ has a
corresponding zero eigenvector, given by the solution of
$\pounds^{\dag} A^{\dag}=0$. A detailed discussion of this issue may be found
in \cite{sarlos} and the references therein.
While examining the stability of $A=U_{0x}$ to perturbations, which may
include a small external perturbation $p(U,x)$ added onto Eq.~(\ref{eq:gone}),
one obtains,
\begin{eqnarray}
& &[{\cal L}+N^{\prime}(U_0)]\delta U =f;~~~~~~\nonumber\\
f&=&\partial_t (\delta U)-\{N^{\prime \prime}(U_0){(\delta U)}^2/2+p(U_0,x)
\nonumber\\&+&p^{\prime}(U_0,x)\delta U
+p^{\prime \prime}(U_0,x){(\delta U)}^2/2 +{\cal O}[{(\delta U)}^3]\},
\label{eq:gfive}
\end{eqnarray}
where $\delta U$ is the small deviation from the localized structure profile.
Realizing that the operator $\pounds={\cal L}+N^{\prime}(U_0)$ has a
Goldstone mode, the solvability of Eq.~(\ref{eq:gfive}) requires,
\begin{eqnarray}
(f,A^{\dag})=0.
\label{eq:gsix}
\end{eqnarray}
The brackets indicate an inner product or the projection of the dynamical
terms $f$ onto the Goldstone mode (its corresponding adjoint) $A^{\dag}$.
Equation.~(\ref{eq:gsix}) represents the generic response of a localized
structure to a wide variety of perturbations, both internal and external.
From an informal and intuitively appealing point of view, the Goldstone mode
with its associated zero eigenvalue is a slow (relevant) mode, which coupled
with other slow modes in the system, should dominate the dynamics. The
projection in Eq.~(\ref{eq:gsix}) is a formal prescription to capture
this slow dynamics.
Let a localized structure be located near a boundary at $x=-l$, with the
origin fixed at the position of the localized structure.
Although, $A^{\dag}$ is still a solution of $\pounds^{\dag} A^{\dag}=0$
in this case, it does not
assume the homogeneous boundary value $A^{\dag}(-l)=0$.
Consequently, $A^{\dag}$ is no longer the zero eigenvector of the adjoint
homogeneous problem in the semi-finite interval $[-l,\infty]$.
However, we still expect $A^{\dag}$ to play a central role in the dynamics of
the localized structure, all be it in a slightly modified form
$A^{\dag}_l=A^{\dag}+\delta A^{\dag}_l$. The subscript $l$ denotes the
proximity of
the localized structure to the boundary, and $\delta A^{\dag}_l$ is a
proximity dependent correction to $A^{\dag}$. We require that in the
limit $l\rightarrow \infty$, $A^{\dag}_l\rightarrow A^{\dag}$, and
$\delta A^{\dag}_l\rightarrow 0$.
This requirement is reasonable on physical grounds.
The slow dynamics of the localized structure far away from the boundary
involves $A^{\dag}$ as a relevant constituent by virtue of it being a slow
mode. As the localized structure gradually nears the boundary, we still expect
$A^{\dag}$, in its modified form $A^{\dag}_l$, to be the relevant (slow)
constituent of the dynamics.
$A^{\dag}_l$ may be determined in two possible ways. Firstly, we may extract
$A^{\dag}_l$ as the solution of
\begin{eqnarray}
\pounds^{\dag} A^{\dag}_l=0,~~A^{\dag}_l(-l)=0,~~A^{\dag}_l(\infty)=0,
\label{eq:gseven}
\end{eqnarray}
with the implication that $A^{\dag}_l=A^{\dag}+\delta A^{\dag}_l$ is still a
zero eigenvector in the finite system. Or we may
extract $A^{\dag}_l$ as a solution of
\begin{eqnarray}
\pounds^{\dag} A^{\dag}_l=\lambda_l A^{\dag}_l,~~A^{\dag}_l(-l)=0.~~
A^{\dag}_l(\infty)=0.
\label{eq:geight}
\end{eqnarray}
Thus, as the localized structure gradually closes in on a
boundary, the zero eigenvector $A^{\dag}$ is modified to $A^{\dag}_l$, and
the zero eigenvalue gradually migrates away from zero, assuming the
value $\lambda_l$. Hence, as $l \rightarrow \infty$,
$\lambda_l \rightarrow 0$, and $A^{\dag}_l \rightarrow A$.
The first scenario is easily discarded using uniqueness arguments. If
Eq.~(\ref{eq:gseven}) is obeyed, then $\delta A^{\dag}_l$
should obey, $\pounds^{\dag} \delta A^{\dag}_l = 0,
\delta A^{\dag}_l(-l)=-A^{\dag}(-l), \delta A^{\dag}_l(\infty)=0$, with the
unique solution $\delta A^{\dag}_l=-A^{\dag}$. Therefore, since
$A^{\dag}_l=A^{\dag}+\delta A^{\dag}_l$,
Eq.~(\ref{eq:gseven}) only has the trivial solution $A^{\dag}_l=0$
(the uniqueness of homogeneous and inhomogeneous problems involving linear
differential operators on semi-infinite intervals can be proved by a
transformation that takes the semi-infinite interval into a finite interval,
followed by the utilization of theorems on uniqueness available for finite
intervals. We provide a proof in Appendix A for the CGLE that is studied in
detail in later sections. Moreover, such a transformation may also be
applied to operators with an asymptotic structure similar to that of
the CGLE). This leads us to conclude that the modification of $A^{\dag}$
in a finite system is appropriately represented by Eq.~(\ref{eq:geight}).
For arbitrary functions $u$ (not the field $U$ in Eq.~(\ref{eq:gone}))
and $v$, and using integration by parts, we have,
\begin{eqnarray}
(\pounds u,v)&=&(u,\pounds^{\dag}v)+v(b)u_x(b)-v(a)u_x(a)\nonumber\\
&+&v_x(a)u(a)-v_x(b)u(b),
\label{eq:gten}
\end{eqnarray}
where we assume for simplicity that $\pounds$ is a reaction-diffusion
type operator comprised of second order differential terms only.
$x=a$ and $x=b$ are arbitrary boundary points. If needed, one may evaluate
surface terms for more general operators using integration by parts.
For the localized structure $a=-l$ and $b=\infty$.
We invoke Eq.~(\ref{eq:geight}) and substitute
$v=A_l^{\dag}$, $u=\delta U_l$ (the subscript $l$ denotes that $\delta U$ is
now considered in a finite system) in Eq.~(\ref{eq:gten}), to obtain,
\begin{eqnarray}
(\pounds \delta U_l,A_l^{\dag})&=&(f,A_l^{\dag})=
(\delta U_l,\lambda_l A_l^{\dag})+A_{lx}^{\dag}(-l)\delta U_l(-l)
\nonumber\\&-&A_{lx}^{\dag}(\infty)\delta U_l(\infty).
\label{eq:geleven}
\end{eqnarray}
This is the sought after finite system extension of the solvability criteria
Eq.~(\ref{eq:gsix}). Also, as $l \rightarrow \infty$, Eq.~(\ref{eq:geleven})
reduces to $(f,A^{\dag})=0$. Since $\pounds$ is obtained by linearizing about
the localized structure $U_0(x)$, $\delta U_l(-l)$ is simply the
difference $U(-l)-U_0(-l)$, where $U(-l)$ is the Dirichlet boundary value
imposed on field $U$, the solution of Eq.~(\ref{eq:gone}).
The extension Eq.~(\ref{eq:geleven}), tailored to incorporate non-homogeneous
Dirichlet boundary conditions on the field $U$, is not unique. For instance,
one may consider the effects of non-homogeneous Neumann boundary conditions on
the field $U$ by requiring that $A_l^{\dag}$ obeys
\begin{eqnarray}
\pounds^{\dag} A_l^{\dag}=\lambda_l A_l^{\dag},
~~A_{lx}^{\dag}(-l)=0,~~A_{lx}^{\dag}(\infty)=0.
\label{eq:gtwe}
\end{eqnarray}
Here, the derivatives, rather than $A_l^{\dag}$ itself, assume zero values at
the boundary. Furthermore, an
extension $A_l^{\dag}$ for a general set of homogeneous boundary conditions,
with homogeneous Dirichlet and Neumann boundary conditions as special cases,
may also be developed. Next, we apply the techniques and criteria developed
so far to analyze non-equilibrium Ising-Bloch fronts, as the fronts interact
with the system boundary.
\section{Boundary effects in a generic Ising-Bloch system}
Ising-Bloch fronts provide an interesting arena to apply the methods developed
in the last section. Along with the usual Goldstone mode associated with
translational invariance, the slow manifold for Ising-Bloch fronts also
includes a spatially localized slow mode responsible for the Ising-Bloch
bifurcation \cite{skyrabin,michaelis,bode}. Chirality preserving stationary
Ising fronts \cite{Coullet}, bifurcate into a pair of chirality broken,
counter-propagating Bloch fronts. The slow manifold for Ising-Bloch fronts
comprised of the Goldstone and chirality breaking modes, manifests itself in
the form of order parameter equations (OPE) \cite{meron4,skyrabin,bode} for
the order parameters, front velocity and front position. The front velocity is
a measure of the effects of the chirality breaking mode. The Goldstone mode
captures front translations by infinitesimal changes in the front position,
the other order parameter. We seek the coupling between these order parameters
induced by the boundary data and broken translational invariance.
A generic Ising front denoted by $U_0(x)$, gives the Goldstone mode $U_{0x}$.
Close to the Ising-Bloch bifurcation threshold, propagating Bloch wall
solutions are regarded as perturbations of the stationary Ising wall
solution \cite{Coullet}. The front velocity $c$ controls the strength of these
perturbations. Therefore, expanding the deviation $\delta U$ in powers
of $c$, we have,
\begin{eqnarray}
U_b&=&U_0 +\delta U\nonumber\\
&=& U_0 +c\delta U_1 +c^2 \delta U_2 +c^3 \delta U_3 + ..,
\label{eq:h1}
\end{eqnarray}
with the perturbed Bloch wall solution $U_b$.
For convenience we transform into a frame of reference moving along with
the Bloch wall. This transformation amounts to $\partial_t (\delta U)
\rightarrow \partial_t (\delta U)-c(U_{0x}+\delta U_x)$.
Invoking Eq.~(\ref{eq:gfive}) and substituting into it the expansion of
$\delta U$, while at the same time disregarding the influence of any external
perturbation $p(U,x)$, we obtain,
\begin{eqnarray}
&&\pounds[c\delta U_1+ c^2 \delta U_2+c^3 \delta U_3]=\partial_t(c \delta U_1)
\nonumber\\
&-&c[U_{0x} +c\delta U_{1x} +c^2 \delta U_{2x}]-c^2 N_2 -c^3 N_3+\cdots
\label{eq:h2}
\end{eqnarray}
$N_2$ and $N_3$ represent the coefficients of second order and third order
velocity terms respectively.
Equating terms which are first order in velocity $c$ in Eq.~(\ref{eq:h2}), we
obtain,
\begin{eqnarray}
\pounds \delta U_1+U_{0x}=0.
\label{eq:h3}
\end{eqnarray}
This means that $\pounds$ has a double zero eigenvalue at the Ising-Bloch
bifurcation threshold \cite{skyrabin,bode}. Therefore, along with the zero
Goldstone mode, we have another eigenvalue that passes through zero at the
bifurcation. The Goldstone mode $U_{0x}$ and the
generalized eigenvector $\delta U_1$ obtained from Eq.~(\ref{eq:h3}), span
the slow manifold. The chirality breaking mode is then constructed as a
linear combination of these two modes \cite{skyrabin}.
Employing the projection criteria Eq.~(\ref{eq:gsix}) for an Ising-Bloch front
close to the bifurcation threshold, ie., the solvability of Eq.~(\ref{eq:h2}),
results in,
\begin{eqnarray}
(\delta U_1,A^\dag)\partial_t c&=&c(U_{0x},A^\dag)
+c^2 (\delta U_{1x}+N_2,A^\dag)\nonumber\\
&+& c^3 (\delta U_{2x}+ N_3, A^\dag) +\cdots
\label{eq:h4}
\end{eqnarray}
This is the generic OPE for the velocity
of Ising-Bloch fronts close to the bifurcation threshold. The particular form
of the inner products in Eq.~(\ref{eq:h4}) is system specific. If one
assumes further symmetries in the system, for example
$U\rightarrow-U$, inner products that are coefficients of even powers of the
velocity in Eq.~(\ref{eq:h4}) vanish, resulting in the normal form of a
pitchfork bifurcation. The inner product $(U_{0x},A^\dag)$ in
Eq.~(\ref{eq:h4}) controls the distance from the Ising-Bloch bifurcation
threshold, where for consistency (Ising-Bloch bifurcation is a pitchfork) it
is further required that $(U_{0x},A^\dag) \sim c^2$, $\partial_t c \sim c^3$
\cite{skyrabin,bode}. Hence, all the terms in Eq.~(\ref{eq:h4}) are of size
$c^3$.
We invoke the extended solvability criteria Eq.~(\ref{eq:geleven}) to
evaluate the effects of boundary data on the dynamics of Ising-Bloch fronts.
For generic Ising-Bloch fronts interacting with boundaries where Dirichlet
data is present, the extended solvability criteria assumes the form,
\begin{eqnarray}
(\delta U_{1l},A_l^{\dag})\partial_t c &=&c(U_{0x},A_l^\dag)
+c^2 (\delta U_{1lx}+N_2,A_l^\dag)\nonumber\\
&+& c^3 (\delta U_{2lx}+ N_3, A_l^\dag)\nonumber\\
&+& \lambda_l(c\delta U_{1l}
+c^2\delta U_{2l}+c^3\delta U_{3l}+\cdots,A_l^{\dag})\nonumber\\
&+& A_{lx}^{\dag}(-l)\delta U_{l}(-l)-A_{lx}^{\dag}(\infty)\delta U_{l}(\infty).
\label{eq:h5}
\end{eqnarray}
In contrast to earlier works \cite{meron4,skyrabin,bode} focused on the
effects of external perturbations, $p(U,x)$, on the slow manifold, the
constituent modes of the slow manifold require appropriate modifications in
order to capture the effects arising due to confinement by boundaries.
While, the modification of the adjoint Goldstone
mode $A^\dag$ to $A_l^{\dag}$ is generic to any confined localized
structure, or alternatively, a localized structure being considered in the
vicinity of system boundaries, the modification
of the generalized eigenvector $\delta U_1$ to $\delta U_{1l}$ is a unique
characteristic of Ising-Bloch fronts.
Simplifications to the slow manifold Eq.~(\ref{eq:h5}) are made by the
following observations. Consider the term,
$f_0=\lambda_l(c\delta U_{1l}+c^2\delta U_{2l}
+c^3\delta U_{3l}+\cdots,A_l^{\dag})$, on the right hand side of
Eq.~(\ref{eq:h5}). The inner product
$f_1=\lambda_l(c\delta U_{1l},A_l^{\dag})$ has the largest contribution since
it involves the first power of the velocity $c$. Now, as mentioned
before, all terms should be of size $c^3$, a requirement imposed for the
Ising-Bloch bifurcation to be a pitchfork.
Therefore, $f_1 \sim \lambda_l c \sim c^3$, implying
$\lambda_l \sim c^2$. Moreover, the size of $\lambda_l$ is controlled by the
distance of the Bloch fronts from the boundary. If the front is far away from
the boundary, that is, if $\lambda_l \sim {\cal O}(c^3)$, then
$f_1 \sim {\cal O}(c^4)$, and its contribution to Eq.~(\ref{eq:h5}) can be
neglected. As the front moves towards the boundary, so that
$\lambda_l \sim c^2$, then $f_1 \sim c^3$ contributes to Eq.~(\ref{eq:h5}),
and the ensuing front dynamics. If the front gets too close to the boundary,
ie., $\lambda_l \sim c$, then $f_1 \sim c^2$, and the scaling requiring that
all the terms be of size $c^3$ breaks down. In other words, if
$\lambda_l \sim c$, the effects of the boundary are too strong for them to be
accurately considered as small perturbations on the dynamics of Ising-Bloch
fronts. Consequently, the size of $\lambda_l$ serves as a measure of the
strength of the boundary perturbation. In light of the present discussion,
Eq.~(\ref{eq:h5}) simplifies to
\begin{eqnarray}
(\delta U_{1l},A_l^{\dag})\partial_t c &=&c(U_{0x},A_l^\dag)
+c^2 (\delta U_{1lx}+N_2,A_l^\dag)\nonumber\\
&+& c^3 (\delta U_{2lx}+ N_3, A_l^\dag)\nonumber\\
&+& \lambda_l(c\delta U_{1l},A_l^{\dag})\nonumber\\
&+& A_{lx}^{\dag}(-l)\delta U_{l}(-l).
\label{eq:h6}
\end{eqnarray}
The surface terms at infinity contribute zero, since by construction
$A_l^{\dag}(\infty)=0$.
\section{boundary effects in the parametrically forced CGLE}
The CGLE reads,
\begin{eqnarray}
\partial_\tau F=(\gamma+i\nu)F-|F|^2 F + \mu F^* + \partial{^2}_{X}F+ \alpha.
\label{eq:PCGLE}
\end{eqnarray}
Equation.~(\ref{eq:PCGLE}) and its generalizations \cite{Coullet,skyrabin,
ephlick} have been thoroughly analyzed in the context of the Ising-Bloch
bifurcation. The field $F$ may be regarded as the amplitude of diffusively
coupled auto oscillators that oscillate above the Hopf bifurcation threshold
determined by the parameter $\gamma$. $\mu$ represents the strength of
parametric forcing at twice the natural frequency, and $\nu$ is the detuning.
The parameter $\alpha$, which models forcing at the natural frequency of the
system, breaks the $(F\rightarrow -F)$ symmetry. As a result, the pitchfork
normal form of the Ising-Bloch bifurcation for $\alpha=0$ unfolds into a
saddle node for a non-zero $\alpha$.
We briefly recount the results of \cite{skyrabin} concerning
the dynamics of Ising-Bloch fronts in the parametrically forced CGLE valid
for an infinite system. This lays down the framework for the subsequent
consideration of finite system sizes and boundary effects.
For $\alpha=0$ and in the bistable regime determined by the constraints,
$|\nu|<\mu$, $\gamma>-\sqrt{\mu^2-\nu^2}$, Eq.~(\ref{eq:PCGLE}) possesses
a stationary Ising wall solution
$F_I=\sqrt{\kappa}\tanh(\sqrt{\kappa/2X})e^{i\phi}$. Here $\kappa=\gamma
+\sqrt{\mu^2-\nu^2}$ and $\phi$ is obtained by solving $\sin(2\phi)=\nu
/\mu$. Bloch wall solutions of Eq.~(\ref{eq:PCGLE}) are then obtained as a
perturbation to the Ising wall,
\begin{equation}
F_b(x,t)=\sqrt{\kappa}[\tanh(x)+u(x,t)+iw(x,t)]e^{i\phi},
\label{perturb}
\end{equation}
where the space-time scaling $t=\kappa \tau/2$, $x=\sqrt{\kappa/2}X$ is
introduced by the authors, resulting in,
\begin{eqnarray}
\pounds =
\left[
\begin{array}{c}
D_1~~~~~~-4\nu/\kappa\\
\\
0~~~~~D_2-3+4\gamma/\kappa
\end{array}\right]\nonumber,\\ \nonumber \\ \nonumber \\
D_1=\partial^2_x+2-6\tanh^2(x),\nonumber \\ \nonumber \\
D_2=\partial^2_x+1-2\tanh^2(x),\nonumber\\ \nonumber\\
\widetilde{N}=-2\tanh(x)
~\left[
\begin{array}{c}
3u^2+w^2\\ \\
2uw
\end{array}\right]
&-2&\left[
\begin{array}{c}
u^3+uw^2\\ \\
w^3+wu^2
\end{array}\right]\nonumber.
\label{eq:donga}
\end{eqnarray}
For clarity and continuation of the conventions used in the previous
sections, we stress the following points.
Firstly, we recognize that $\delta U=\{u,w\}^T$. Secondly, $\delta U$ obeys
\begin{eqnarray}
\partial_t \delta U=\pounds \delta U+\widetilde{N},
\end{eqnarray}
which when compared with Eq.~(\ref{eq:gfive}), leads to the realization that
$\widetilde{N}=N^{\prime \prime}(U_0){(\delta U)}^2/2+{\cal O}[{(\delta U)}^3$.
Thirdly, $\pounds$ is obtained by linearizing about the solution $U_0(x)$.
In the present case the stationary solution is the
Ising wall $F_I(x)=\sqrt{\kappa}\tanh(x)e^{i\phi}$, and $U_0(x)=\tanh(x)$,
where the constant factor $\sqrt{\kappa}e^{i\phi}$ should be dropped if the
perturbation $\delta U=\{u,w\}^T$ is defined through Eq.~(\ref{perturb}).
For the specific case of the parametrically forced CGLE, one has
\cite{skyrabin},
\begin{eqnarray}
\delta U_1&=&\left[
\begin{array}{c}
\frac{8}{3\pi}I_{11}(x)-I_{12}(x)\\ \\
\frac{8\gamma}{9\pi\nu}$sech(x)$
\end{array}\right],~~~\nonumber
U_{0x}=\left[
\begin{array}{c}
$sech$^{2}(x)\\ \\
0
\end{array}\right],\nonumber\\ \\
\text{and}&~&~
A^\dag=\left[
\begin{array}{c}
\frac{9(\mu_c-\mu)\mu_c}{\pi\gamma\nu}$sech$^{2}(x)\\ \\
$sech(x)$
\end{array}\right].\nonumber
\label{eq:thevecs}
\end{eqnarray}
Substituting these vectors into Equation.~(\ref{eq:h4}) gives \cite{skyrabin},
\begin{eqnarray}
\partial_t c &=& \frac{27(\mu_c-\mu)\mu_c}{4\gamma^2} c
-\left({\left[\frac{8\gamma}{9\pi\nu}\right]}^2+0.36 \right) c^3.
\label{eq:opecgl}
\end{eqnarray}
Eq.~(\ref{eq:opecgl}) possesses three stationary
states, two counter-propagating Bloch walls and a stationary Ising wall. These
steady states exchange stability via the Ising-Bloch bifurcation at the
critical bifurcation parameter $3\mu_c=\sqrt{9\nu^2+\gamma^2}$.
The components of the vectors $\delta U=c \delta U_1+c^2 \delta U_2+..$, $U_0$
and $A^\dag$, in an infinite system, exponentially decay to zero as one
moves away from the front both to the left and to the right. This
signifies that Ising and Bloch walls are localized structures that are not
influenced by boundary conditions imposed on either boundary sufficiently
far away. Furthermore, no explicit dependence on $x$ in Eq.~(\ref{eq:opecgl})
indicates translational invariance, a residue of infinite system size.
We now calculate $A^\dag_l$ and the associated value
of $\lambda_l$. $A^\dag_l$ satisfies the boundary conditions
$A^{\dag}_l(-l)=0$, $A^{\dag}_l(\infty)=0$ (homogeneous problem), since we
wish to examine the influence of Dirichlet boundary conditions on $U$
(non-homogeneous problem). Close to the bifurcation threshold determined by
the magnitude of $\mu_c-\mu$, the operator $\pounds^\dag$ has the form
\begin{eqnarray}
\pounds^\dag&=&\left[
\begin{array}{c}
D_1~~~~~~~0\\ \\
-\nu/\gamma~~~~~D_2
\end{array}\right]\nonumber
+
\frac{27\mu_c(\mu-\mu_c)}{4\gamma^2}\left[
\begin{array}{c}
0~~~~~~~0\\ \\
\nu/\gamma~~~~-1
\end{array}\right]\nonumber\\ \nonumber\\
&=& \pounds^\dag_1+(\mu-\mu_c)\pounds^\dag_2.
\label{eq:matsum}
\end{eqnarray}
The operator $\pounds^\dag_2$ is a perturbative correction to the
operator $\pounds^\dag_1$, since $\mu-\mu_c \sim c^2$. Hence, we first
examine $\pounds^\dag_1$ the dominant term in $\pounds^\dag$.
The operators $D_1$ and $D_2$ populate the diagonals of $\pounds^\dag_1$, and
possess zero eigenvectors given by $Z_1=\text{sech}^2(x)$ and
$Z_2=\text{sech}(x)$ respectively, in an infinite system. These
eigenvectors satisfy the constraint of being zero at positive and negative
infinity. Imagine a traveling Bloch front sufficiently distant from the
boundary, where Dirichlet boundary conditions are imposed. The front does not
sense the boundary and the condition $D_1Z_1=D_2Z_2=0$ holds. This is because
the solutions $Z_1$ and $Z_2$ exponentially approach zero on either side of
the front. As the front closes in on the boundary,
such that it is barely able to sense it ($Z_1$ and $Z_2$ have small finite
values at the boundary), the eigenvectors $Z_1~\text{and}~Z_2$
are modified to $Z_{1l}~\text{and}~Z_{2l}$ by constraining them to
have zero values at the boundary. Meanwhile, in a
semi-infinite or finite domain, the only solutions to $D_1Z_{1l}=D_2Z_{2l}=0$
which have a zero value at both boundaries are the trivial solutions
$Z_{1l}=Z_{2l}=0$ (uniqueness arguments). Hence, requiring that the
solutions $Z_{1l}$($Z_{2l}$) are only slight modifications of $Z_1$($Z_2$) and
are not trivial zero solutions demands that these solutions
obey $D_1Z_{1l}=\lambda_{1l}Z_{1l}$ and $D_2Z_{2l}=\lambda_{2l}Z_{2l}$.
Figure.~\ref{fig:eigvec}(a) shows the plot of $Z_1$ in grey, where the left
boundary is at a finite distance $l$ from the peak. $Z_1$ has a finite nonzero
value at the boundary. We require that the modified eigenvector $Z_{1l}$ have a
zero value at the boundary and not be all that different from $Z_1$ elsewhere.
We make the ansatz that this can be accomplished by subtracting from $Z_1$
its image to the left of the boundary. Therefore, we have, $Z_{1l}=
\text{sech}^2(x)-\text{sech}^2(x+2l)$.
Figure.~\ref{fig:eigvec}(b) shows a good agreement between our guess and the
actual numerically evaluated $Z_{1l}$. This is so because in the asymptotic
limit $\exp{2x}>>1$, $D_1=\partial_x^2-4$, and the image is approximately a
zero eigenvector of this operator in the same limit.
Introducing images into a semi-infinite problem is by no means
a coincidence. Images are a common occurrence whenever boundary data is
involved. For the extension $A^{\dag}_{l}$ (correspondingly
$Z_{1l}$ and $Z_{2l}$) to assume a zero value at the boundary, the introduction
of the image becomes a natural necessity. Furthermore, we wish to stress that
the concept of images is quite general in its utility. Extensions of Goldstone
modes can be readily obtained for other systems, with linear operators having
similar properties of exponential decay asymptotics.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{aeigvecD_1.ps}} \\
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{neigvecD_1.ps}}
\end{tabular}
\caption{(a) Shows the plot of $Z_1$. The peak is at a distance of $l=2$
from the boundary. (b) The squares represent the numerically obtained
$Z_{1l}$. The analytical guess $Z_{1l}=\text{sech}^2(x)-\text{sech}^2(x+2l)$
is the solid line.}
\label{fig:eigvec}
\end{center}
\end{figure}
An upper bound, $\lambda_{1l}^\uparrow$, on the eigenvalue $\lambda_{1l}$,
is easily obtained by a variational principle, given by,
\begin{eqnarray}
|\lambda_{1l}|<|\lambda_{1l}^\uparrow|=(Z_{1l},D_1Z_{1l})/(Z_{1l},Z_{1l}).
\label{eq:varia}
\end{eqnarray}
A more refined variational guess of $Z_{1l}$ may be made
by introducing an extra parameter $a_1$. Consequently, we
have $Z_{1l}=\exp{(a_1x)}[\text{sech}^2(x)-\text{sech}^2(x+2l)]$. Manipulation
of this parameter provides a better guess of the change in shape of the peak
in the actual modified eigenvector $Z_{1l}$. Figure.~\ref{fig:eigenva}(a)
compares the numerical and variationally calculated eigenvalues as a function
of the distance $l$ of the front from the boundary. The dashed curve
represents the numerically calculated eigenvalues of $D_1$. The thin curve
depicts the variationally calculated eigenvalues with $Z_{1l}=\text{sech}^2(x)-
\text{sech}^2(x+2l)$. The squares signify a better variational calculation
of the eigenvalues using $Z_{1l}=\exp{(a_1x)}[\text{sech}^2(x)-
\text{sech}^2(x+2l)]$. An improved guess of $Z_{2l}$, and eigenvalue
$\lambda_{2l}$ for the operator $D_2$, similarly involves taking
$Z_{l2}=\exp{(a_2x)}[\text{sech}(x)-\text{sech}(x+2l)]$. Depicted in
Fig.~\ref{fig:eigenva}(b) are the eigenvalues $\lambda_{2l}$, numerically
calculated (dashed curve), variationally calculated with respective guesses
$Z_{2l}=\text{sech}(x)-\text{sech}(x+2l)$ (thin line), and
$Z_{2l}=\exp{(a_2x)}[\text{sech}(x)- \text{sech}(x+2l)]$ (squares).
The numerical calculation of the eigenvalues $\lambda_{1l}$
and $\lambda_{2l}$ involved using a standard QR algorithm on the matrix
obtained by a finite difference approximation to the operators $D_1$ and
$D_2$. The grid spacing was adjusted until we obtained convergence. The
eigenvectors were calculated using inverse iterations, with the number of
iterations optimized for convergence.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{eigenD_1.ps}} \\
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{eigenD_2.ps}}
\end{tabular}
\caption{(a) Comparison of variational and numerical calculations of
$\lambda_{1l}$
(b) Similar comparison of $\lambda_{2l}$ calculated using numerical and
variational techniques.}
\label{fig:eigenva}
\end{center}
\end{figure}
The first row in the matrix representation of the adjoint operator
Eq.~(\ref{eq:matsum}) consists only of the operator $D_1$. Therefore,
since $\pounds^\dag A^\dag_l=\lambda_l A^\dag_l$, we immediately
obtain $\lambda_l=\lambda_{1l}$. We recall
that in the limit of infinite front distance from the boundary
$l\rightarrow\infty$, we have $A^\dag_l\rightarrow A^\dag$. Combining this
asymptotic limit constraint with the requirement that the sought after
eigenvector has zero values at both boundaries, we obtain,
\begin{eqnarray}
A^\dag_l=\left[
\begin{array}{c}
\frac{(\mu_c-\mu)\mu_c}{\pi\gamma\nu}
Z_{1l} \\ \\
Z_{2l}
\end{array}\right].\nonumber\\
\label{eq:advec}
\end{eqnarray}
A more rigorous derivation involving a step by step consideration of the
operators $L^\dag_1$ and $L^\dag_2$ in a perturbative scheme also yields
Eq.~(\ref{eq:advec}).
We now focus on incorporating the effects of the Dirichlet boundary values
$X_b$ and $Y_b$, the values of the real and imaginary components of the
field $F$ in Eq.~(\ref{eq:PCGLE}), into the dynamics of fronts close to the
boundary. Bloch walls are perturbed Ising walls, with the
perturbation $\delta U_l$. The boundary value of this perturbation
$\delta U_l(-l)$ is obtained by fixing $F(-l)=X_b+iY_b$ and subtracting from
it the value that the Ising wall
assumes $F_I(-l)=\sqrt{\kappa}\tanh(-l)e^{i\phi}$. Recalling
Eq.~(\ref{perturb}), and $\delta U=\{u,w\}^T$, we obtain,
\begin{eqnarray}
\delta U_l(-l)=\left[
\begin{array}{c}
( X_b $cos$(\phi)+Y_b $sin$(\phi))/\sqrt{\kappa}+$tanh$(l) \\ \\
( Y_b $cos$(\phi)-X_b $sin$(\phi))/\sqrt{\kappa}
\end{array}\right].\nonumber \\
\label{eq:bdev}
\end{eqnarray}
\section{OPE}
To extract a reduced description of the influence of Dirichlet boundary
conditions on the motion of Ising-Bloch fronts, we invoke Eq.~(\ref{eq:h6}),
and substitute into it the explicit forms of $A^\dag_l$ and $\lambda_l$
derived in the previous section. Consider the term
$f_1=\lambda_l(c\delta U_{1l},A_l^{\dag})$ on the right hand
side (RHS) of Eq.~(\ref{eq:h6}). For the CGLE, as seen in
Eq.~(\ref{eq:advec}), the first component of $A_l^{\dag}$, denoted by,
$A_{l1}^{\dag }$, is smaller by a factor of $c^2$ than the second component
$A_{l2}^{\dag }$. This is so because $\mu_c -\mu \sim c^2$. Hence, while
evaluating $f_1$, we need only consider the inner product of the second
component of the generalized eigenvector, $\delta U_{1l}$, denoted by
$\delta U_{1l2}$, and $A_{l2}^{\dag }$. The generalized eigenvector
$\delta U_1$ is known Eq.~(\ref{eq:thevecs}), and its finite system
modification $\delta U_{1l}$
needs to be evaluated (only the second component $\delta U_{1l2}$) to
evaluate the inner product in $f_1$.
To evaluate $\delta U_{1l2}$ we recall that $Z_2=\text{sech}(x)$,
with $D_2Z_{2}=0$. The second component of
$\delta U_1$, is given by $\delta U_{12}=[8\gamma/9\pi\nu]\text{sech}(x)$.
Hence, $D_2 \delta U_{12}=0$.
In a confined system with the left boundary at $x=-l$, $Z_2$ is modified to
$Z_{2l}=\text{sech}(x)-\text{sech}(x+2l)$, requiring that the
homogeneous boundary condition, $Z_{2l}(-l)=0$, holds good. In the confined
system $\delta U_{12}$ is modified to $\delta U_{1l2}$. However, to obtain
$\delta U_{1l2}$, the requirement that it obeys the inhomogeneous boundary
condition $c\delta U_{1l2}(-l)=\delta U_{l2}(-l)$, since $\delta U_l=
c\delta U_1 +{\cal O}(c^2)$, needs to be imposed. Therefore we construct
$\delta U_{1l2}(x)=c\delta U_{12}-\beta \text{sech}(x+2l)$, followed by
imposing the inhomogeneous boundary condition
$c\delta U_{1l2}(-l)=\delta U_{l2}(-l)$, to evaluate $\beta$. After doing so,
we have,
\begin{eqnarray}
c \delta U_{1l2}=\frac{c8\gamma}{9 \pi \nu} Z_{2l}
-\frac{\delta U_{l2}(-l)}{\text{sech}(l)}\text{sech} (x+2l).
\label{eq:modi}
\end{eqnarray}
We, finally have the ingredients to calculate all the inner products in
Eq.~(\ref{eq:h6}). The bulk of the boundary
influence, we contend, is captured by the interplay of the terms,
$c(U_{0x},A_l^\dag)$, $\lambda_l(c\delta U_{1l},A_l^{\dag})$, and
the surface term $A_{lx}^{\dag}(-l)\delta U_{l}(-l)$ in Eq.~(\ref{eq:h6}).
Therefore, although, strictly speaking, the inner products containing higher
order terms $c^2 (\delta U_{1lx}+N_2,A_l^\dag)$, and
$c^3 (\delta U_{2lx}+ N_3, A_l^\dag)$, in Eq.~(\ref{eq:h6}), should be
evaluated in the finite domain $[-l,\infty]$, we approximate them by taking
the inner product in the infinite interval $[-\infty,\infty]$.
Performing all the inner products in Eq.~(\ref{eq:h6}) and rearranging the
terms, we obtain
\begin{eqnarray}
\partial_t c &=& \frac{27(\mu_c-\mu)\mu_c}{4\gamma^2} c+\lambda_l c
-\left({\left[\frac{8\gamma}{9\pi\nu}\right]}^2+p\right) c^3\nonumber\\
&-&\left[\frac{9\pi\nu}{16\gamma}\right]\text{tanh}(l)
\text{sech}(l) \delta U_{l2}(-l)\nonumber\\
&+&\left[\frac{81(\mu_c-\mu)\mu_c}{4\gamma^2}\right]
\text{tanh}(l) \delta U_{l1}(-l)\nonumber\\
&-&\left[\lambda_l\frac{9\pi\nu}{16\gamma}\right]2l\text{cosech}(2l).
\label{eq:dopecgl}
\end{eqnarray}
In deriving Eq.~(\ref{eq:dopecgl})
we have used $Z_{1l}=\text{sech}^2(x)-\text{sech}^2(x+2l)$ and
$Z_{2l}=\text{sech}(x)-\text{sech}(x+2l)$, where $\lambda_l=\lambda_{1l}$ is
given by Eq.~(\ref{eq:varia}), and $p=0.36$ Eq.~(\ref{eq:opecgl}).
Equation.~(\ref{eq:dopecgl}) along with
$\partial_t l=-c$ represents the coupling of the two degrees of freedom,
front velocity $c$ and position $l$, by the influence of Dirichlet boundary
conditions imposed at the boundary. As required, in the limit of infinite front distance from the boundary Eq.~(\ref{eq:dopecgl}) reduces to
Eq.~(\ref{eq:opecgl}).
We now examine the consequences of the coupling of the front velocity and
position close to the boundary. Firstly, we report the findings of our
numerical simulations of Eq.~(\ref{eq:PCGLE}), which is a system with infinite
degrees of freedom. Secondly, we corroborate these findings by solving the
reduced, two degree of freedom OPE we have derived.
We performed numerical simulations of Eq.~(\ref{eq:PCGLE}), where Bloch fronts
were created at infinity (far from the boundaries) and launched towards a
boundary. The velocity of these Bloch fronts was chosen to be one of the
steady states of Eq.~(\ref{eq:opecgl}) resulting in uniform front translation
with this velocity until the fronts closed in on the boundary. Near the
boundary, contingent upon
the Dirichlet boundary value imposed, the incoming Bloch fronts were either
trapped or bounced back. Bloch fronts that bounce evolve into the
counter-propagating Bloch front near the boundary and move away. Trapped Bloch
fronts, as opposed to bouncing Bloch fronts, evolve into non-trivial steady
state solutions (See Ref.\cite{yadav}) of the CGLE Eq.~(\ref{eq:PCGLE}).
We summarize our numerical observations of Bloch front behavior as a function
of the boundary conditions $X_b$ and $Y_b$ in Figure.~\ref{fig:trans} .
This phase diagram in the plane of
boundary values reveals a curve separating regions of bouncing and trapped
fronts represented by diamonds. We compare these results with the
transition curve predicted by the reduced model Eq.~(\ref{eq:dopecgl}),
plotted as the dashed curve in Figure.~\ref{fig:trans}.
The plots show a good agreement (within $0.5\%$)
between the two transition curves. This is a striking result considering the
fact that in calculating $A^\dag_l$ and $\lambda_l$ we have employed
approximate vectors $Z_{1l}$ and $Z_{2l}$.
\begin{figure}
\includegraphics[width=8cm,height=8cm,angle=0]{transpcgle.ps}
\caption{The transition curve for the full model Eq.~(\ref{eq:PCGLE}) plotted
using squares, the same curve obtained from the reduced OPE
Eq.~(\ref{eq:dopecgl}), plotted as a dashed line. Here,
$\nu=0.3$, $\gamma=1.0$, $\mu=0.448$.}
\label{fig:trans}
\end{figure}
Bouncing fronts gradually slow down as they near the boundary, attain zero
velocity at a certain critical distance from it, and finally move away as
the sign of the velocity flips.
As we change the boundary values and get closer to
the transition curve, bouncing fronts attain zero velocity at a much smaller
critical distance from the boundary, until eventually right at the transition
curve they reach the point of closest approach to the boundary. As we
cross the transition curve and move into the trapping region, approaching
fronts no longer attain zero velocity close to the boundary, their velocity
never flips sign, and hence they never bounce. The distance from the boundary
of the point of closest approach depends on where exactly on the phase
diagram the transition curve is crossed.
The agreement between the transition curves obtained from the full model
Eq.~(\ref{eq:PCGLE}) and the reduced model Eq.~(\ref{eq:dopecgl}) is better
when the point of closest approach is further away from the boundary. This is
because, as detailed earlier, the vectors $Z_{1l}$ and $Z_{2l}$ are better
approximations to the actual solutions of $D_1Z_{1l}=\lambda_1 Z_{1l}$ and
$D_2Z_{2l}=\lambda_2 Z_{2l}$, further away from the boundary. Consequently,
a better guess of these vectors, valid close to the boundary, should improve
the agreement between the transition curves, even if, the point of closest
approach is closer to the boundary. However, the approximate vectors we
use are sufficient for the purpose of establishing the usefulness of our
general method that accounts for the broken translational invariance in a
spatially finite system through the extension of solvability conditions. Our
method incorporates into it the eigenvalue $\lambda_l$, the most direct
measure of broken translational invariance, which can be obtained accurately
via a variational principle using relatively crude guesses for the
eigenvectors.
We now, by examining Eq.~(\ref{eq:dopecgl}) in more detail, extract the
mechanism behind the transition from bouncing to trapped fronts as Dirichlet
boundary conditions are changed. Figure.~\ref{fig:bounce1}(a)
shows the nullclines, invariant
manifold, and trajectories of Eq.~(\ref{eq:dopecgl}) inside the bouncing
region of the phase diagram. A saddle, present at the point of intersection of
the nullclines, controls the flows in this bouncing regime. Far away from the
boundary, situated at $x=0$ in the plot, the nullclines are three parallel
straight lines that represent two counter-propagating Bloch wall steady state
solutions, and a stationary Ising wall solution of Eq.~(\ref{eq:opecgl}).
The bouncing involves the Bloch front initially flowing towards the saddle.
Thereupon, influenced by the unstable manifold, the front flows away.
Figure.~\ref{fig:bounce1}(b)
still depicts flows inside the bouncing region, but much closer to
the transition curve. In this regime bouncing and trapped fronts can coexist.
The invariant manifolds demarcate two basins, one of attraction towards the
boundary, and the other of repulsion away from it. Inside the repulsion basin
all incoming Bloch fronts bounce with the same mechanism as in
Fig.~\ref{fig:bounce1}(a).
All the flows in the attraction basin are directed towards the system boundary,
with no possibility of a bounce. Figure.~\ref{fig:bounce1}(b)
shows both bouncing and trapped
Bloch front trajectories in their respective basins. We reported on the
the coexistence region in our numerical study of Eq.~(\ref{eq:PCGLE}) in
Ref.\cite{yadav}. Here, we have provided an analytical explanation of this
phenomena.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{1.116n.ps}} \\
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{1.112n.ps}}
\end{tabular}
\caption{(a) The plot deep inside the bouncing region, the nullclines are
thin black curves, the thick curves correspond to the trajectories in the
phase plane, and the invariant manifolds are plotted as dashed lines. Here,
$\nu=0.3$, $\gamma=1.0$, $\mu=0.448$, $X_b=-1.116$, and $Y_b=-0.4262$.
(b) Plot still in the bouncing region, but close to the transition curve. The
same plotting scheme and parameters used, with boundary values
$X_b=-1.112$, $Y_b=-0.4262$.}
\label{fig:bounce1}
\end{center}
\end{figure}
The flows in the trapping region close to the transition curve are shown in
Figure.~\ref{fig:trap1}(a) .
Trapped Bloch fronts, created at infinity and on the upper branch
of the nullcline (corresponding to one of the steady states of
Eq.~(\ref{eq:opecgl})), lie inside the basin of attraction towards the
boundary. Consequently, the transition from bouncing to trapped fronts is
marked by the initial front velocity and position moving from the basin of
repulsion (Fig.~\ref{fig:bounce1}(b)) to the
basin of attraction (Figure.~\ref{fig:trap1}(a)) as the boundary values
are varied. Deep inside the trapping region the saddle no longer exists,
and we have a sink instead (Fig.~\ref{fig:trap1}(b)). All incoming Bloch
front trajectories end up at this sink.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{1.11n.ps}} \\
\resizebox{80mm}{!}{\includegraphics[width=7cm,height=6cm,angle=0]
{0.98n.ps}}
\end{tabular}
\caption{(a) Plot in the trapping
region close to the transition curve. The same plotting scheme and
parameters used, with boundary values $X_b=-1.11$, $Y_b=-0.4262$.
(b) The plot deep inside the trapping region, the nullclines are
thin black curves, the trajectory is the thick curve. Here,
$\nu=0.3$, $\gamma=1.0$, $\mu=0.448$, $X_b=-1.09$, and $Y_b=-0.4262$.}
\label{fig:trap1}
\end{center}
\end{figure}
Summarizing, the nonuniform motion of Bloch fronts close to the boundary
is governed by the fixed point of Eq.~(\ref{eq:dopecgl}), giving rise to
bouncing, trapping, and coexistence of the two. Well inside the
bouncing region this fixed point is a saddle. Deep into the trapping region
the fixed point changes into a sink.
\section{Conclusion}
We have developed a general method of analyzing the influence of broken
translational invariance due to finite size and boundary effects on the
dynamics of localized
solutions of generic non-linear spatially extended systems. We apply our
method to the special case of a bistable reaction-diffusion system, where
the localized solutions are fronts Eq.~(\ref{eq:dopecgl}). The implementation
of this method involves the extension of the infinite system size limit
solvability conditions, used to extract a reduced description of the infinite
dimensional system, into solvability conditions that account for finite system
size and boundary effects. The extended solvability criteria works by naturally
incorporating into it the concept of images. As a result, the method affords
a direct grasp of the broken translational invariance in a confined system
through the calculation of relevant eigenvalues.
In the special case of Dirichlet boundary conditions imposed on the CGLE,
we were able to provide mechanisms for Bloch front trapping, bouncing and
coexistence of the two at the boundary. This nonuniform front motion
is a result of the coupling of the two degrees of freedom, front velocity and
position, by the influence of boundary conditions. We have explicitly derived
this coupling by using our method of solvability condition extension. The
role of other types of boundary conditions, either Neumann or mixed can be
explored in a similar fashion by constructing a suitable extension of the
modified Goldstone mode. For example, exploring Neumann boundary conditions
requires the extension to always have zero derivatives at the boundary.
This can be accomplished in the CGLE or other systems by adding, rather than
subtracting, the image.
Finally, we comment on the generality of solvability condition extension.
In any system, whenever it is possible to derive reduced dynamical equations
through projections on the Goldstone mode, our method can be applied to
obtain the finite size and boundary effects in terms of the modifications of
these reduced dynamical equations.
\begin{acknowledgements}
This work was supported in part by NSF Grant No. DMR-9710608 and by a Faculty
Research Grant from the Louisiana State University office of Sponsered
Research.
\end{acknowledgements}
|
{
"timestamp": "2005-08-24T23:54:38",
"yymm": "0503",
"arxiv_id": "nlin/0503039",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503039"
}
|
\section{Introduction}
Let $K,K'$ be closed convex pointed cones with non-empty interior, residing in finite-dimensional real
vector spaces $E,E'$. Then an element $w \in E \otimes E'$ of the tensor product space is called {\sl separable}
if it can be represented as a convex combination of product elements $v \otimes v'$, where $v \in K$, $v' \in K'$.
It is not hard to show that the set of separable elements is itself a closed convex pointed cone with non-empty
interior. This cone is called the $K \otimes K'$-separable cone. Cones of elements that are separable with respect
to more than two initial cones are defined similarly.
Separable cones have many applications in Mathematical Programming. So, the dual cones to the cones of
positive polynomials, which frequently appear in optimization problems \cite{PosPolsinControl},\cite{NesterovSOS}, can be
represented as separable cones. In Quantum Information Theory the set of unnormalized unentangled mixed
states of a multi-partite quantum system also forms a separable cone. In this case the underlying cones are
cones of positive semidefinite matrices. These separable cones and the corresponding positive maps have been
subject of intense study in the recent Quantum Information Theory literature \cite{Gurvits0302102},\cite{Horodeckis96},\cite{Peres96},
but drew attention of the mathematical community also before the emergence of this field \cite{Choi74},\cite{Stormer63},\cite{Terpstra},\cite{Woronowicz}.
In this paper we treat ball-ball separable cones, i.e.\ cones of $K \otimes K'$-separable elements where the
underlying cones $K,K'$ are conic hulls of closed Euclidean balls or solid ellipsoids not containing the
origin. Such cones have a relatively simple structure. Thus they are suitable for the approximation of more
complex separable cones. This can be done by approximating the underlying cones by appropriate ball-generated
cones. The idea of replacing an underlying cone by a ball-generated cone was put forward by Leonid Gurvits and Howard Barnum, who
successfully used it to obtain lower bounds on the largest ball of unnormalized separable elements around
the identity matrix for multipartite systems \cite{Gurvits0302102},\cite{Gurvits0409095}. In this contribution we compute several characteristics of
ball-ball separable cones exactly, which allows for a more efficient application of such approximations.
One such application makes use of the fact that the cone of positive semidefinite hermitian $2 \times 2$
matrices is isomorphic to the 4-dimensional Lorentz cone $L_4$ and hence is also a ball-generated cone. This
enables us to refine Gurvits' bounds for the case of multi-qubit systems. We prove that for a 3-qubit system
a ball of radius $\sqrt{4/5}$ around the identity matrix consists only of separable elements, as opposed to the best bound
$\sqrt{8/11}$ known previously \cite{Gurvits0409095}. For systems consisting of more than 3 qubits we obtain an
improvement of roughly 12\% with respect to the best bounds known before \cite{Gurvits0409095}. Namely, we prove
that for an $m$-qubit system, a ball of radius $\frac{2^{m/2}}{\sqrt{3^{m-1}+1}}$ around the identity matrix consists only
of separable elements. The exponent in the asymptotics (as $m \to \infty$) of this bound is the same as the
one obtained in \cite{Gurvits0409095}. Recently Stanislaw Szarek showed that this exponent delivers the exact asymptotics
in the multi-qubit case (an earlier result is published in \cite{Szarek0310061}).
\smallskip
The paper is organized as follows.
In the next section we characterize the extreme rays of the cone dual to the cone of ball-ball separable
elements, namely the cone of linear maps that take the Lorentz cone to the Lorentz cone in the respective
source and target spaces (Lorentz-to-Lorentz positive maps). It is well-known that the extreme rays of the
dual cones characterize the largest faces in the primal cones \cite{Rockafellar}. These faces are of interest to
us because they determine the radii of the largest separable balls around chosen elements in the separable
cones. Note that describing the extreme rays of cones of Lorentz-to-Lorentz positive maps yields also a
description of cones dual to ellipsoid-ellipsoid separable cones, because extreme rays are taken to extreme
rays by invertible linear mappings.
In Section 3 we describe the largest faces of ball-ball separable cones using the obtained families of
extreme rays in the dual cones. This allows to get some insight into the structure of ball-ball separable
cones.
In Section 4 we compute the radius of the largest ball of ellipsoid-ellipsoid separable elements around the tensor
product of points defining the central rays of the initial ellipsoid-generated cones.
In Section 5 we apply this result to study the cone of separable unnormalized states of a multi-qubit system. We compute the above-mentioned lower bounds
on the radii of largest separable balls around the unnormalized uniformly mixed state.
In the last section we summarize our results.
\section{Extreme rays of cones of Lorentz-to-Lorentz positive maps}
In this section we compute the extreme rays of the cone of positive maps, i.e.\ those linear maps which take the Lorentz cone in the Lorentz cone
in the respective source and target spaces.
\smallskip
Let $L_m \subset {\bf R}^m$, $L_n \subset {\bf R}^n$ be standard Lorentz cones of dimensions $m$ and $n$, i.e.
\begin{eqnarray*}
L_m &=& \{(x_0,x_1,\dots,x_{m-1})^T \,|\, x_0 \geq
|(x_1,\dots,x_{m-1})^T|\}, \\ L_n &=& \{(y_0,y_1,\dots,y_{n-1})^T
\,|\, y_0 \geq |(y_1,\dots,y_{n-1})^T|\}.
\end{eqnarray*}
We assume throughout the paper that $\min(n,m) \geq 2$. Since $L_1$ is isomorphic to the ray ${\bf R}_+$, the
case $\min(n,m) = 1$ is trivial. We call a linear map $M: {\bf R}^m \to {\bf R}^n$ {\sl $L_m$-to-$L_n$
positive} or just {\sl positive} if $M[L_m] \subset L_n$. Since the Lorentz cones are self-dual, the cone
${\cal P}$ of such maps is dual to the cone of $L_m \otimes L_n$-separable elements. Moreover, as a
consequence of this self-duality $M$ is positive if and only if the adjoint map $M^T$ is positive in the
sense that $M^T[L_n] \subset L_m$.
In this section we determine the extreme rays of the cone ${\cal
P}$ of positive maps. We represent maps from ${\bf R}^m$ to ${\bf
R}^n$ by $n \times m$ matrices partitioned as
\begin{equation} \label{partition}
M = \left( \begin{array}{cc} s & h \\ v & A \end{array}
\right),
\end{equation}
where $s$ is a scalar, $h$ is a row vector, $v$ is a column vector
and $A$ is a $(n-1)\times(m-1)$-matrix. Note that if $M$ is a
non-zero positive map, then the scalar $s$ is strictly positive.
Define two diagonal matrices $J_n = diag(1,I_{n-1})$, $J_m =
diag(1,I_{m-1})$, where $I_k$ denotes the $k \times k$ identity matrix. Note that if $x \in {\bf R}^m$ is in the
interior of $L_m$, then $x^TJ_mx > 0$. If $x \in \partial L_m$,
then $x^TJ_mx = 0$.
{\lemma \label{poscond} A map
\[ M = \left( \begin{array}{cc} 1 & h \\ v & A \end{array} \right) \in {\bf R}^{n \times m}
\]
is $L_m$-to-$L_n$ positive if and only if
\[ \exists\, \lambda \geq 0: \quad M^T J_n M \succeq \lambda J_m, \quad |h| \leq 1,
\]
or equivalently,
\[ \exists\, \lambda' \geq 0: \quad M J_m M^T \succeq \lambda J_n, \quad |v| \leq 1.
\] }
{\it Proof.} By definition, $M$ is positive if for any $x_0 \geq
0$, $x \in {\bf R}^{m-1}$ such that $x_0 \geq |x|$ we have $y_0
\geq |y|$, where
\[ \left( \begin{array}{c} y_0 \\ y \end{array} \right) = M \left( \begin{array}{c} x_0 \\ x \end{array}
\right).
\]
We can rewrite this equivalently as
\[ \forall\,x_0,x\ |\ x_0 \geq 0,\ ( x_0\ x^T) J_m \left( \begin{array}{c} x_0 \\ x \end{array} \right) \geq 0: \quad
x_0 + hx \geq 0, \quad ( x_0\ x^T) M^T J_n M \left( \begin{array}{c} x_0 \\ x \end{array} \right) \geq 0.
\]
By the ${\cal S}$-lemma \cite{Dines43},\cite{Yakubovich71} this is equivalent to the conditions
\[ (1\ h)^T \in L_m, \quad \exists\, \lambda \geq 0: \quad M^T J_n M \succeq \lambda J_m,
\]
which gives the first set of conditions claimed by the lemma. The
second set is obtained by considering the adjoint maps $M^T$ in
the $L_n$-to-$L_m$ positive cone. $\Box$
\smallskip
Let us establish necessary and sufficient conditions for a
positive map to generate an extreme ray of the cone ${\cal P}$. Note also that $M$ generates an extreme ray if and only if $M^T$ generates an extreme ray.
First we show that if a non-zero positive map $M$ does not take
the interior of $L_m$ in the interior of $L_n$, then $M$ is of
rank 1.
{\lemma \label{rk1} Let $M \not= 0$ be a positive map. Suppose there exists $x
\in int\,L_m$ such that $Mx \in \partial L_n$. Then the rank of
$M$ is equal to 1. }
{\it Proof.} Let the conditions of the lemma hold. Denote the point $Mx \in \partial L_n$ by $y$. Then $y$ is
contained in the linear subspace $M[{\bf R}^m] = Im\,M$. Since $M \not= 0$, this subspace is non-zero.
Moreover, since $x \in int\,L_m$, there exists a neighbourhood $U_m \subset {\bf R}^m$ of $x$ which is
entirely contained in $L_m$. Its image $M[U_m]$ will be a neighbourhood $U_n$ of $y$ relative to $Im\,M$. By
the positivity of $M$ the set $U_n$ is contained in $L_n$ and therefore in the face of $y$ with respect to
the cone $L_n$. But $y \in \partial L_n$, hence this face equals the intersection of $L_n$ with the linear
subspace generated by $y$. Therefore $y \not= 0$ and $Im\,M = \{ \alpha y\,|\, \alpha \in {\bf R}\}$. Thus
$M$ has rank 1. $\Box$
{\lemma \label{rk1extr} A positive map $M$ of rank 1, partitioned as in (\ref{partition}), generates an
extreme ray if and only if $|h| = |v| = s$. }
{\it Proof.} Let $M$ be a rank 1 map, partitioned as in
(\ref{partition}), and let $s > 0$. Then we have
\[ M = \frac{1}{s} \left( \begin{array}{c} s \\ v \end{array}
\right) \left( s \ h \right).
\]
The condition of positivity provided by Lemma \ref{poscond} then
takes the form
\[ (s\ h)^T \in L_m, \quad \exists\, \lambda \geq 0: \quad \frac{1}{s^2} \left( \begin{array}{c} s \\ h^T \end{array}
\right) \left( s \ v^T \right) J_n \left( \begin{array}{c} s \\
v \end{array} \right) \left( s \ h \right) \succeq \lambda J_m
\]
\[ \Leftrightarrow\ s \geq |h|,\ \exists\lambda \geq 0:\ \frac{1}{s^2} (s^2 -
|v|^2) \left( \begin{array}{c} s \\ h^T \end{array} \right) \left(
s \ h \right) \succeq \lambda J_m
\]
\begin{equation} \label{poscrk1}
\Leftrightarrow\ s \geq |h|,\ s \geq |v|.
\end{equation}
Hence a rank 1 map is positive if and only if $s \geq |h|$, $s
\geq |v|$, $s > 0$.
Let now $M$ be a positive map of rank 1 and suppose that $s >
|h| \not= 0$. Then the maps
\[ M(\lambda) = \frac{1}{s} \left( \begin{array}{c} s \\ v \end{array}
\right) \left( s \ \lambda h \right)
\]
are positive for all $\lambda \in [-s/|h|,s/|h|]$ and $M = M(1)$. The point $\lambda = 1$ lies in the interior of the interval $[-s/|h|,s/|h|]$,
and the matrices $M(\lambda)$ are not multiples
of each other for different $\lambda$. Hence $M$ does not generate an extreme ray of ${\cal P}$.
A slightly modified argument can be applied if $h = 0$. Choose any
vector $h'$ with $|h'| = s$ and consider the maps
\[ M(\lambda) = \frac{1}{s} \left( \begin{array}{c} s \\ v \end{array}
\right) \left( s \ \lambda h' \right).
\]
Then $M(\lambda)$ is positive for all $\lambda \in [-1,1]$ and $M = M(0)$. Hence $M$ cannot
generate an extreme ray neither.
The same reasoning applies if $s > |v|$. Thus if $M$ generates an extreme ray, then $|h| = |v| = s$.
It rests to show that any rank 1 matrix with $|h| = |v| = s > 0$
generates an extreme ray. Let $M$ be such a matrix. Suppose there
exists a matrix $\delta M$ such that $M(\lambda) = M + \lambda
\delta M \in {\cal P}$ for all $\lambda$ in a neighbourhood $U$ of
zero. Let $V$ be a neighbourhood of the unit vector $e_0 \in {\bf
R}^m$ that lies entirely in the interior of $L_m$. Then for all
$\lambda \in U$ and for all $w \in V$ we have
\[ M(\lambda) w = Mw + \lambda\,\delta M\,w \in L_n,
\quad Mw = \frac{1}{s} (s\ h)w \cdot \left( \begin{array}{c} s \\ v \end{array} \right) \in \partial L_n.
\]
Since $\lambda$ can vary in a neighbourhood of zero, the vectors $Mw + \lambda\,\delta M\,w$ have to lie in
the face of $Mw$ with respect to the cone $L_n$. Then $\delta M\,w$ lies in the tangent space to that face.
This tangent space is the linear subspace generated by $Mw$. Hence $\delta M\,w$ has to be a multiple of the
vector $Mw$. Since this holds for all $w \in V$, the image of $\delta M$ must be contained in the linear
subspace of ${\bf R}^n$ generated by the set $\{Mw\,|\,w \in V\}$. Note that $\frac{1}{s} (s\ h)w > 0$,
because $w \in int\,L_m$. Therefore this subspace is one-dimensional and generated by $(s\ v^T)^T$. It
follows that $\delta M$ is of the form $(s\ v^T)^T u$ for some vector $u \in {\bf R}^m$. If we apply the same
line of reasoning for the positive map $M^T$, we conclude that $\delta M^T$ is of the form $(s\ h)^T u'$ for
some vector $u' \in {\bf R}^n$. Thus $\delta M$ is proportional to $M$ and $M$ generates an extreme ray of
${\cal P}$.
This completes the proof of the lemma. $\Box$
\smallskip
It rests to consider the positive maps of rank strictly greater than 1. Let $M$ be such a map, partitioned as in (\ref{partition}). By Lemma \ref{rk1} $M$ takes the interior of $L_n$ to the interior of $L_m$.
Let $Aut(L_m)$, $Aut(L_n)$ be the automorphism groups of the cones $L_m$, $L_n$, respectively.
We shall now show that if $M$ generates an extreme ray, then there exist automorphisms $U_n \in Aut(L_n)$, $U_m \in Aut(L_m)$ such that
$U_nMU_m$ is {\sl doubly stochastic}. (A positive map $M$ is called doubly stochastic if
$M$ and $M^T$ take the central elements $e_0$ of the cones $L_m,L_n$ into each other. Otherwise spoken, $M$ is
doubly stochastic if $s = 1$ and $h = v = 0$.)
Define two functions $p,q: {\bf R}^m \to {\bf R}$ by $p(x) = x^TJ_mx$, $q(x) = x^TM^TJ_nMx$. Then the set $N
= \{ (p(x),q(x)) \in {\bf R}^2 \,|\, x \in {\bf R}^m \}$ is called the {\sl joint numerical range} of the
matrices $J_m$, $M^TJ_nM$ underlying the quadratic forms $p,q$. It is known \cite{Dines43} that the set $N$
is a convex cone. Lemma \ref{poscond} states the existence of a number $\lambda \geq 0$ such that $q(x) \geq
\lambda p(x)$ for all $x \in {\bf R}^m$. Let $\lambda^*$ be the maximal such $\lambda$. Since $M$ takes the
interior of $L_m$ to the interior of $L_n$, the set $N$ has a non-empty intersection with the open first
orthant. Therefore $\lambda^*$ exists. Moreover, $\lambda^* > 0$, otherwise the matrix $M^TJ_nM$ would be
positive semidefinite, which is not possible if the rank of $M$ is strictly greater than 1. We have $M^TJ_nM
- \lambda^*J_m \succeq 0$ and $M^TJ_nM - \lambda^*J_m - \delta J_m \not\succeq 0$ for any $\delta > 0$. Hence
there exists $x \not= 0$ such that $q(x)-\lambda^*p(x) = x^T(M^TJ_nM - \lambda^*J_m)x = 0$ and $p(x) =
x^TJ_mx \geq 0$. Let us distinguish two cases.
\smallskip
1. There exists a point $x^* \in {\bf R}^m$ such that $q(x^*) - \lambda^*p(x^*) = 0$ and $p(x^*) > 0$.
Without restriction of generality we can choose $x^*$ such that $x^* \in int\,L_m$ and $p(x^*) = 1$. Denote
$Mx^*$ by $y^*$. Since $M$ takes $int\,L_m$ to $int\,L_n$, we have $y^* \in int\,L_n$. In fact,
$(y^*)^TJ_ny^* = q(x^*) = \lambda^* > 0$. Let $U_m$, $U_n$ be automorphisms of the cones $L_m$, $L_n$,
respectively, preserving the quadratic forms $J_m$, $J_n$, respectively, such that $U_m x^* = e_0^m \in {\bf
R}^m$ and $U_n y^* = \sqrt{\lambda^*} e_0^n \in {\bf R}^n$. (Here $e_0^m,e_0^n$ are the unit vectors in the
direction of the coordinate $x_0$ in the respective spaces.) Such automorphisms exist since the Lorentz cones
are homogeneous \cite{Vinberg63}.
Define a map $\tilde M = (\lambda^*)^{-1/2}U_nMU_m^{-1}$. By the positivity of $M$ this map is also positive.
We have $\tilde M e_0^m = (\lambda^*)^{-1/2}U_nMU_m^{-1}U_m x^* = (\lambda^*)^{-1/2}U_ny^* = e_0^n$. Since
$x^*$ is contained in the nullspace of the positive semidefinite matrix $M^TJ_nM - \lambda^*J_m$, we have
$M^TJ_nMx^* = M^TJ_ny^*= \lambda^* J_m x^*$. It follows that
\begin{eqnarray*}
\tilde M^T e_0^n &=& \tilde M^T (J_n e_0^n) = (\lambda^*)^{-1/2}U_m^{-T}M^TU_n^T (U_n^{-T}J_nU_n^{-1}) e_0^n
= (\lambda^*)^{-1/2}U_m^{-T}M^T J_n [(\lambda^*)^{-1/2} y^*] \\
&=& (\lambda^*)^{-1} U_m^{-T} [\lambda^* J_m x^*] = U_m^{-T} J_m x^* = J_m U_m x^* = e_0^m.
\end{eqnarray*}
Hence $\tilde M$ is doubly stochastic.
{\it Remark:} A similar statement for cones of maps that take the positive semidefinite cone to the positive semidefinite
cone was proven by Leonid Gurvits \cite{Gurvits0303055}.
\smallskip
2. For any point $x \in int\,L_m$ we have $q(x) > \lambda^* p(x)$.
We noted above that there exists $x^* \not= 0$ such that $q(x^*) = \lambda^* p(x^*)$ and $p(x^*) \geq 0$.
Since $p(x) > 0$ yields $q(x) > \lambda^* p(x)$, we have $p(x^*) = 0$. Without restriction of generality we
can choose $x^*$ such that $x^* \in \partial L_m$. Denote by $L_N$ the nullspace of the positive semidefinite
matrix $M^TJ_nM - \lambda^*J_m$. This linear subspace contains $x^* \in \partial L_m$ and does not intersect
the interior of $L_m$. Hence it lies in the orthogonal complement to the element $J_mx^* \in \partial L_m$.
On the other hand, $(M^TJ_nM - \lambda^*J_m)x^* = 0$ yields for any vector $v \in Ker\,M$ that $v^T(M^TJ_nM - \lambda^*J_m)x^* =
-\lambda^* v^TJ_mx^* = 0$. Hence the kernel of $M$ lies also in the orthogonal complement of $J_mx^*$. Equivalently, $J_mx^*$ lies in the
image of the matrix $M^T$ and there exists a vector $v \in {\bf R}^n$ such that $J_mx^* = M^Tv$.
Let now $\Delta = J_nv(J_mx^*)^T$ and consider the family of maps $M(\alpha) = M + \alpha \Delta$. For any vector $w \in L_N$ and
for any $\alpha$ we have
\[ [M(\alpha)^TJ_nM(\alpha) - \lambda^* J_m]w = [\alpha \Delta^T J_n M + \alpha M^T J_n \Delta + \alpha^2 \Delta^T J_n \Delta]w = 0,
\]
because $\Delta w = J_nv \langle J_mx^*, w \rangle = 0$ and $\Delta^T J_n M w = J_mx^* v^T J_n^TJ_n M w =
J_mx^* \langle J_mx^*, w \rangle = 0$. Hence there exists $\delta > 0$ such that for all $\alpha \in
(-\delta,+\delta)$ the matrix $M(\alpha)^TJ_nM(\alpha) - \lambda^* J_m$ lies in the face of the
positive semidefinite cone generated by the matrix $M^TJ_nM - \lambda^* J_m$.
Let the matrix $M(\alpha)$ be partitioned as
\[ M(\alpha) = \left( \begin{array}{cc} s(\alpha) & h(\alpha) \\ v(\alpha) & A(\alpha) \end{array} \right).
\]
We have $(s(0)\ h(0)) = (s\ h) \in int L_m$, because otherwise the positive map $M^T$ would take the vector $e_0^n \in int\,L_n$ to the
vector $(s\ h)^T \in \partial L_m$, and $M^T$ would have rank 1 by Lemma \ref{rk1}. Therefore there exists $\delta' > 0$ such that
$(s(\alpha)\ h(\alpha))^T \in int\,L_m$ for all $\alpha \in (-\delta',+\delta')$.
Then by Lemma \ref{poscond} the map $M(\alpha)$ is positive for all $\alpha$ with $|\alpha| < \min(\delta,\delta')$ and hence contained
in the cone ${\cal P}$. Since the rank of $\Delta$ equals 1, but the rank of $M$ is strictly greater than 1, the matrices $M,\Delta$ cannot be collinear. It follows that
$M$ does not generate an extreme ray of ${\cal P}$.
\smallskip
We have proven the following
{\corollary Let $M$ be a positive map of rank strictly greater than 1 and let $M$ generate an extreme ray of ${\cal P}$.
Then there exist automorphisms $U_n \in Aut(L_n)$, $U_m \in Aut(L_m)$ such that
$U_nMU_m$ is doubly stochastic. $\Box$ }
Note that for any automorphisms $U_n \in Aut(L_n)$, $U_m \in Aut(L_m)$ the matrix $U_nMU_m$ generates an extreme ray of ${\cal P}$
if and only if $M$ generates an extreme ray of ${\cal P}$. Let us characterize the extreme rays that are generated by doubly stochastic
matrices. From Lemma \ref{poscond} it follows that a doubly stochastic matrix, partitioned as in (\ref{partition}), is positive if and only if $\sigma_{\max}(A) \leq 1$, where
$\sigma_{\max}$ denotes the maximal singular value.
{\lemma \label{dblstochlem} Let $M$ be a doubly stochastic positive map, partitioned as in (\ref{partition}),
and let $M$ generate an extreme ray of ${\cal P}$. Then all singular values of $A$ equal 1. }
{\it Proof.} Let us assume the contrary. Suppose $M$ is doubly stochastic and positive, partitioned as in (\ref {partition}), with $\sigma_{\min}(A) = \hat\sigma < 1$.
Let $A = UDV$ be the singular value decomposition of $A$ and $\sigma_1,\dots,\sigma_{\min(m-1,n-1)}$ its singular values in decreasing order.
Here $U,V$ are orthogonal matrices of appropriate size and $D = diag(\sigma_1,\sigma_2,\dots,\sigma_{\min(m-1,n-1)})$ is a $(n-1)\times(m-1)$ matrix with the singular values of $A$ on its main diagonal,
all other elements being zero. Note that $\sigma_{\min(m-1,n-1)} = \hat\sigma < 1$. Let us define an affine one-parametric family of diagonal $(n-1)\times(m-1)$ matrices
by $D(\alpha) = diag(\sigma_1,\sigma_2,\dots,\sigma_{\min(n,m)-2},\alpha)$. Then the maps
\[ M(\alpha) = \left( \begin{array}{cc} 1 & 0 \\ 0 & UD(\alpha)V \end{array} \right)
\]
are positive and hence belong to ${\cal P}$ for all $\alpha \in [-1,+1]$. Note that $M = M(\hat\sigma)$. Since these matrices are not proportional for different values of $\alpha$, and $\hat\sigma \in (-1,1)$,
the map $M$ does not generate an extreme ray of the cone ${\cal P}$. This proves the lemma. $\Box$
{\lemma Let $M$ be a doubly stochastic positive map, partitioned as in (\ref{partition}), and let all singular values of $A$ equal 1.
Then $M$ generates an extreme ray of ${\cal P}$ if and only if $\min(n,m) > 2$. }
{\it Proof.} Let $M$ be a map satisfying the assumptions of the lemma. Assume also without restriction of generality that
$n \geq m$. Then we have $A^TA = I_{m-1}$. Let us first show that $M$ does not generate an extreme ray if $\min(n,m) = 2$.
If $m = 2$, then the matrix $A$ is a unit length column vector. Consider the two maps
\[ M_1 = \left( \begin{array}{c} 1 \\ A \end{array} \right) ( 1\ 1 ), \qquad M_2 = \left( \begin{array}{c} 1 \\ -A \end{array} \right) ( 1\ -1 ).
\]
These maps are positive by condition (\ref{poscrk1}) and not proportional. Moreover, we have $M = \frac{1}{2}(M_1+M_2)$. Hence $M$ does
not generate an extreme ray of ${\cal P}$.
Suppose now that $n \geq m \geq 3$. Assume there exists an $n \times m$ matrix
\[ M_{\delta} = \left( \begin{array}{cc} 0 & h_{\delta} \\ v_{\delta} & A_{\delta} \end{array} \right)
\]
and a number $\varepsilon > 0$ such that the map $M(\alpha) = M + \alpha M_{\delta}$ is positive for all $\alpha \in (-\varepsilon,+\varepsilon)$.
The assumption that the upper left element of $M_{\delta}$ is zero does not restrict the generality, because this element can be made zero by
adding to $M_{\delta}$ an appropriate multiple of $M$. Let us develop the positivity condition of Lemma \ref{poscond}. We have that $M(\alpha)$ is positive
if and only if $|\alpha||h_{\delta}| \leq 1$ and there exists $\lambda \geq 0$ such that
\begin{eqnarray}
&& M(\alpha)^TJ_nM(\alpha) - \lambda J_m = \left( \begin{array}{cc} 1 & \alpha v_{\delta}^T \\ \alpha h_{\delta}^T & A^T + \alpha A_{\delta}^T \end{array} \right) J_n
\left( \begin{array}{cc} 1 & \alpha h_{\delta} \\ \alpha v_{\delta} & A + \alpha A_{\delta} \end{array} \right) - \lambda J_m \nonumber\\
&=& \left( \begin{array}{cc} 1-\lambda-\alpha^2|v_{\delta}|^2 & \alpha (h_{\delta}-v_{\delta}^TA) - \alpha^2v_{\delta}^TA_{\delta} \\
\alpha (h_{\delta}^T-A^Tv_{\delta}) - \alpha^2A_{\delta}^Tv_{\delta} &
(\lambda - 1)I_{m-1} - \alpha (A_{\delta}^TA + A^TA_{\delta}) + \alpha^2(h_{\delta}^Th_{\delta} - A_{\delta}^TA_{\delta}) \end{array} \right) \succeq 0. \label{commat}
\end{eqnarray}
Here $\lambda$ may depend on $\alpha$. We obtain in particular $-\alpha (A_{\delta}^TA + A^TA_{\delta}) +
\alpha^2(h_{\delta}^Th_{\delta} - A_{\delta}^TA_{\delta}) \succeq (1-\lambda)I_{m-1} \succeq \alpha^2|v_{\delta}|^2 I_{m-1}
\succeq 0$. A necessary condition for this inequality to hold for all $\alpha \in (-\varepsilon,+\varepsilon)$ is that
$A_{\delta}^TA + A^TA_{\delta} = 0$. It follows that $h_{\delta}^Th_{\delta} \succeq A_{\delta}^TA_{\delta} +
|v_{\delta}|^2 I_{m-1}$. The left-hand side of this inequality is a matrix of rank not exceeding 1, while the
right-hand side is positive semidefinite. Hence the rank of the right-hand side cannot exceed 1 too. Since $m-1 \geq
2$, it follows that $v_{\delta} = 0$ and $A_{\delta}$ is of the form $wh_{\delta}$, where $w$ is a column vector of
appropriate size. This yields the inequality $\alpha^2h_{\delta}^Th_{\delta}(1 - |w|^2) \succeq (1-\lambda)I_{m-1}$,
which implies $\lambda \equiv 1$ for a similar reason. But then the upper left element of matrix (\ref{commat}) is
zero. Therefore $\alpha (h_{\delta}-v_{\delta}^TA) - \alpha^2v_{\delta}^TA_{\delta} = \alpha h_{\delta} = 0$ for all
$\alpha \in (-\varepsilon,+\varepsilon)$ and $h_{\delta} = 0$, $A_{\delta} = wh_{\delta} = 0$. This proves that $M$
generates an extreme ray of ${\cal P}$. $\Box$
\smallskip
Combining the results obtained so far, we can characterize the extreme rays of the cone ${\cal P}$ as follows.
{\lemma Let the positive map $M$ be partitioned as in (\ref{partition}) and suppose that it generates an
extreme ray of the cone ${\cal P}$. Then either $M$ is of rank 1, with $|h| = |v| = s$, or there exist
automorphisms $U_m \in Aut(L_m)$, $U_n \in Aut(L_n)$ such that
\[ U_n M U_m = \left( \begin{array}{cc} 1 & 0 \\ 0 & A' \end{array}
\right)
\]
with all singular values of $A'$ equal to $1$.
If $\min(m,n) \geq 3$, then all matrices of the above types generate extreme
rays. If $\min(m,n) = 2$, then only those of them which are of
rank 1 generate extreme rays. $\Box$}
\smallskip
Note that for any pair of non-zero elements $x,y \in \partial L_m$ in the boundary of the Lorentz cone $L_m$ there exists an
automorphism $U_m$ of $L_m$ that takes $x$ to $y$. Further, for any orthogonal matrix $U_{m-1}$ of dimension $(m-1) \times (m-1)$
the $m \times m$ matrix $diag(1,U_{m-1})$ represents an automorphism of $L_m$. This allows us to reduce the extreme rays of ${\cal P}$
to two canonical forms. Define the two positive maps
\[ M_1 = diag\left( \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right), 0, \dots, 0 \right) = \left( \begin{array}{cc} {\bf 1}_{2 \times 2} & {\bf 0}_{2 \times (m-2)} \\
{\bf 0}_{(n-2) \times 2} & {\bf 0}_{(n-2) \times (m-2)} \end{array} \right),
\]
\begin{equation} \label{standpos}
M_2 = diag(1,\dots,1) = \left\{ \begin{array}{ccl}
\left( \begin{array}{c} I_m \\ {\bf 0}_{(n-m) \times m} \end{array} \right), & \ & n \geq m, \\
( I_n\ {\bf 0}_{n \times (m-n)}), && n < m. \end{array} \right.
\end{equation}
Here ${\bf 1}_{k \times l}, {\bf 0}_{k \times l}$ denote $k \times l$ matrices filled with ones and zeros, respectively.
{\definition We call a positive map $M$ of {\sl Type I} if there exist automorphisms $U_m \in Aut(L_m)$, $U_n \in Aut(L_n)$
such that $U_n M U_m = M_1$. We call $M$ of {\sl Type II} if there exist automorphisms $U_m \in Aut(L_m)$, $U_n \in Aut(L_n)$
such that $U_n M U_m = M_2$. }
\smallskip
We have the following theorem.
{\theorem \label{extreme}
Let the positive map $M$ generate an extreme ray of the cone ${\cal P}$. Then $M$ is either of Type I or of Type II.
If $\min(m,n) \geq 3$, then all matrices of Types I and II generate extreme rays.
If $\min(m,n) = 2$, then only the matrices of Type I generate extreme rays. $\Box$ }
\medskip
The theorem shows that the structure of the cone ${\cal P}$ of positive maps is more complex than the
structure of the Lorentz cones $L_k$, but is still relatively simple. While the Lorentz cone has only one
kind of extreme rays (which are equivalent with respect to the action of the automorphism group), the cone of
positive maps has two kinds. An exception are the cones of $L_2$-to-$L_n$ positive maps. In this case the
extreme rays form two copies of the boundary $\partial L_n$ of the cone $L_n$ which are located in mutually
orthogonal subspaces.
\section{Largest faces of ball-ball separable cones}
In this section we give a description of the largest faces of ball-ball separable cones, departing from the
two families of extreme rays of the cone of positive maps obtained in the previous section.
\smallskip
We call an element $B$ of the space ${\bf R}^m \otimes {\bf R}^n$ {\sl $L_m \otimes L_n$-separable} or just
{\sl separable} if $B$ can be expressed as a finite sum $\sum_{k=1}^N x_k \otimes y_k$ of product elements
such that $x_k \in L_m, y_k \in L_n$ for all $k = 1,\dots,N$. The separable elements form a convex cone in
${\bf R}^m \otimes {\bf R}^n$, the {\sl separable cone}, which will be denoted by $K_{sep}$. This cone is
dual to the cone ${\cal P}$ of positive maps considered in the previous section. For convenience we will
represent the elements of ${\bf R}^m \otimes {\bf R}^n$ as $n \times m$ matrices such that the scalar product
of a linear map $M: {\bf R}^m \to {\bf R}^n$ with an element $B \in {\bf R}^m \otimes {\bf R}^n$ is given by
$\langle B,M \rangle = tr\,(M^TB) = tr\,(B^TM)$. In this representation a product element $x \otimes y$ is
given by the rank 1 matrix $yx^T$.
It is well-known that the largest faces (i.e.\ non-trivial faces that are not an intersection of other, strictly larger faces)
of a convex cone $K$ have the form $\{x \in K \,|\, \langle x, y
\rangle = 0\}$, where $y$ generates an extreme ray of the dual cone $K^*$ \cite{Rockafellar}. Let us compute the
faces corresponding to the extreme rays of the cone of positive maps ${\cal P}$ described by Theorem
\ref{extreme}. By this theorem, there are two kinds of extreme rays. These generate two kinds of largest
faces of the separable cone.
Let us define two standard faces of the separable cone $K_{sep}$ by
\[ F_1 = \{ B \in K_{sep} \,|\, \langle B, M_1 \rangle = 0 \}, \quad
F_2 = \{ B \in K_{sep} \,|\, \langle B, M_2 \rangle = 0 \},
\]
where $M_1,M_2$ are the positive maps (\ref{standpos}).
{\definition We call a face $F$ of $K_{sep}$ of {\sl Type I} if there exist automorphisms $U_m \in
Aut(L_m)$, $U_n \in Aut(L_n)$ such that $\{ U_n B U_m \,|\, B \in F \} = F_1$. We call a face $F$ of
$K_{sep}$ of {\sl Type II} if there exist automorphisms $U_m \in Aut(L_m)$, $U_n \in Aut(L_n)$ such that $\{
U_n B U_m \,|\, B \in F \} = F_2$ (or $U_nFU_m = F_2$ for short). }
\smallskip
Hence all faces of Type I are affinely isomorphic to $F_1$, while all faces of Type II are affinely
isomorphic to $F_2$. Let us determine the structure of these two sets.
{\prop \label{F1} $F_1$ is affinely isomorphic to the convex conic hull of the union
\[ Z_1 = \left\{ z = (z_0,z_1,\dots,z_{n+m-2})^T \,\Big|\, z_0 = 1,(z_1-1)^2+\sum_{k=2}^{m-1}z_k^2=1,z_m=\cdots=z_{n+m-2}=0
\right\} \cup
\]
\[ \cup \left\{ z = (z_0,z_1,\dots,z_{n+m-2})^T \,\Big|\, z_0 =
1,z_1=\cdots=z_{m-1}=0,(z_m-1)^2+\sum_{k=m+1}^{n+m-2}z_k^2=1 \right\} \subset {\bf R}^{n+m-1}.
\] }
{\it Remark:} Thus a section of the cone $F_1$ is affinely isomorphic to the convex hull of two intersecting spheres $S^{m-1},S^{n-1}$ which are located in orthogonal subspaces.
{\it Proof.} The set $F_1 = \{ B \in K_{sep} \,|\, \langle B, M_1 \rangle = 0 \}$ is given by the convex hull of
those extreme rays of $K_{sep}$ that are orthogonal to $M_1$. The extreme rays of $K_{sep}$ are tensor
products of the extreme rays generating the individual Lorentz cones $L_m,L_n$, i.e.\ generated by elements
of the form
\begin{equation} \label{tensor}
B = \left( \begin{array}{c} 1 \\ h^T \end{array} \right) \otimes \left( \begin{array}{c} 1 \\ v \end{array} \right) = \left( \begin{array}{cc} 1 & h \\ v & vh \end{array} \right),
\end{equation}
where $h \in {\bf R}^{m-1}$ is a row vector, $v \in {\bf R}^{n-1}$ is a column vector with $|h| = |v| = 1$. We have
\[ \langle B, M_1 \rangle = tr \left( \begin{array}{cc} 1 & h \\ v & vh \end{array} \right)^T \left( \begin{array}{cc} 1 & (e_1^{m-1})^T \\
e_1^{n-1} & e_1^{n-1}(e_1^{m-1})^T \end{array} \right) = (1 + \langle v, e_1^{n-1} \rangle)(1 + \langle h,
e_1^{m-1} \rangle).
\]
Here $e_1^k$ is the unit vector in the direction of the first coordinate in the space ${\bf R}^k$. Therefore
$\langle B, M_1 \rangle = 0$ if and only if $v = -e_1^{n-1}$ or $h = -e_1^{m-1}$. Thus we obtain
\begin{equation} \label{charF1}
F_1 = conv \left\{ \left( \begin{array}{c} 1 \\ -e_1^{n-1} \end{array} \right) x^T + y \,(1\
-(e_1^{m-1})^T) \,\Big|\, x \in \partial L_m, y \in \partial L_n \right\}
\end{equation}
\[ = \left\{ \left(
\begin{array}{ccccc} x_0+y_0 & x_1-y_0 & x_2 & \cdots & x_{m-1} \\ -x_0+y_1 & -x_1-y_1 & -x_2 & \cdots & -x_{m-1} \\
y_2 & -y_2 & \\ \vdots & \vdots & & {\bf 0}_{(n-2) \times (m-2)} & \\ y_{n-1} & -y_{n-1} & \end{array}
\right) \,\Big|\, \begin{array}{c} x_0 \geq |(x_1,\dots,x_{m-1})^T|, \\ y_0 \geq |(y_1,\dots,y_{n-1})^T|
\end{array} \right\}.
\]
It is now easily seen that the affine map $f: {\bf R}^{n+m-1} \to {\bf R}^m \otimes {\bf R}^n$ given by
\[ z = \left( \begin{array}{c} z_0 \\ \vdots \\ z_{n+m-2} \end{array} \right) \mapsto \left( \begin{array}{ccccc} z_0 & z_1-1 & z_2 & \cdots & z_{m-1} \\
z_m-1 & 2-z_0-z_1-z_m & -z_2 & \cdots & -z_{m-1} \\
z_{m+1} & -z_{m+1} & \\
\vdots & \vdots & & {\bf 0}_{(n-2) \times (m-2)} & \\ z_{n+m-2} & -z_{n+m-2} & \end{array} \right)
\]
is an affine bijection between the union $Z_1$ and a set of generators of the cone $F_1$. $\Box$
\smallskip
Let us now consider the second kind of largest faces. Denote by $Sym(k)$ the space of real symmetric $k
\times k$ matrices.
{\prop \label{F2} The face $F_2$ is affinely isomorphic to the set
\[ Z_2 = \{ A \in Sym(\min(n,m)) \,|\, A \succeq 0, A_{00} = tr\, A/2 \},
\]
where $A_{00}$ is the upper left element of the matrix $A$. }
{\it Proof.} Let $n \geq m$ without restriction of generality. Then the positive map
\[ \tilde M_2 = \left( \begin{array}{cc} 1 & {\bf 0}_{1 \times (m-1)} \\ {\bf 0}_{(m-1) \times 1} & -I_{m-1} \\ {\bf 0}_{(n-m) \times 1} & {\bf 0}_{(n-m) \times (m-1)} \end{array} \right)
= M_2 \left( \begin{array}{cc} 1 & {\bf 0}_{1 \times (m-1)} \\ {\bf 0}_{(m-1) \times 1} & -I_{m-1} \end{array} \right)
\]
generates an extreme ray of ${\cal P}$ and is of Type II. Instead of the face $F_2$ we will consider the
isomorphic face $\tilde F_2 = \{ B \in K_{sep} \,|\, \langle B, \tilde M_2 \rangle = 0 \}$.
This face is given by the convex hull
of those extreme rays of $K_{sep}$ that are orthogonal to $\tilde M_2$. Let such an extreme ray be generated by the tensor product (\ref{tensor}).
Let the vector $v$ be partitioned in a subvector $v_a$ of dimension $m-1$ and a subvector $v_b$ of dimension $n-m$. We have
\[ \langle B, \tilde M_2 \rangle = tr \left[ \left( \begin{array}{cc} 1 & h \\ v & vh \end{array} \right)^T \tilde M_2 \right] =
tr \left( \begin{array}{cc} 1 & -v_a^T \\ h^T & -h^Tv_a^T \end{array} \right) = 1 - hv_a.
\]
Note that $|h| = 1$, $|v_a| \leq 1$. Therefore $\langle B, \tilde M_2 \rangle = 0$ if and only if $v_a = h^T$ and $v_b = 0$. Thus $\tilde F_2$ is given by the
convex conic hull of the set
\[ \left\{ \left( \begin{array}{cc}
1 & h \\ h^T & h^Th \\
{\bf 0}_{(n-m) \times 1} & {\bf 0}_{(n-m) \times (m-1)}
\end{array} \right) \,\Big|\, |h| = 1 \right\}.
\]
This hull is equal to the set
\[ conv \left\{ \left( \begin{array}{c} A \\ {\bf 0}_{(n-m) \times m} \end{array} \right) \,\Big|\, A \in Sym(m),\, A \succeq 0,\, rk\, A \leq 1,\, A_{00} = tr\,A/2 \right\}
\]
\[ = \left\{ \left( \begin{array}{c} A \\ {\bf 0}_{(n-m) \times m} \end{array} \right) \,\Big|\, A \in Sym(m),\, A \succeq 0,\, A_{00} = tr\,A/2 \right\}.
\]
The last relation is a consequence of the following fact.
{\it If $L$ is an linear subspace of $Sym(k)$ of codimension 1,
then the intersection of $L$ with the cone $S_+(k)$ of PSD matrices in $Sym(k)$ equals the convex conic hull of all rank 1 PSD matrices
which are contained in $L$. }
Indeed, if these two sets do not coincide, then there exists a linear functional $L'$ on $Sym(k)$ that strictly separates some point in $L \cap S_+(k)$ from
the convex conic hull of all rank 1 PSD matrices in $L$. But this contradicts the convexity of the joint numerical range \cite{Dines43} of the quadratic forms on
${\bf R}^k$ induced by $L$ and $L'$.
This completes the proof. $\Box$
\smallskip
Above characterizations of the standard faces $F_1,F_2$ allow us to characterize all faces of Types I and II.
{\lemma The faces of Type I are parameterized by a pair of vectors $(h,v)$, where $h \in S^{m-2} \subset
{\bf R}^{m-1}$ is a row vector of length 1 and $v \in S^{n-2} \subset {\bf R}^{n-1}$ is a column vector of
length 1. The face $F_I(h,v)$ corresponding to such a pair $(h,v)$ is given by
\[ F_I(h,v) = conv \left\{ \left( \begin{array}{c} 1 \\ v \end{array} \right) x^T
+ y \,(1\ h) \,\Big|\, x \in \partial L_m, y \in \partial L_n \right\}.
\] }
{\it Proof.} Let $F$ be a face of Type I. Then there exist automorphisms $U_n,U_m$ of $L_n,L_m$,
respectively, such that $F = U_n F_1 U_m$. Define the vectors
\[ h' = U_n \left( \begin{array}{c} 1 \\ -e_1^{n-1} \end{array} \right) \in \partial L_n, \quad v' = U_m^T \left( \begin{array}{c} 1 \\ -e_1^{m-1} \end{array} \right) \in \partial L_m.
\]
Let $h = (h')^T/h'_0$, $v = v'/v'_0$ be normalized multiples of $h',v'$. Then description (\ref{charF1}) of
the standard face $F_1$ shows that $F$ has the form $F_I(h,v)$ defined in the theorem.
On the other hand, the generator sets of $F_I(h,v)$, $F_I(h',v')$ are different whenever $(h,v) \not= (h',v')$. Hence $F_I(h,v) \not= F_I(h',v')$ for $(h,v) \not= (h',v')$. $\Box$
\smallskip
{\corollary \label{corr1} Any two faces of Type I have a non-trivial intersection. }
{\it Proof.} Let $F_I(h,v)$, $F_I(h',v')$ be two faces of Type I. Then the elements
\[ \left( \begin{array}{cc} 1 & h \\ v' & v'h \end{array} \right), \left( \begin{array}{cc} 1 & h' \\ v & vh' \end{array} \right) \in K_{sep}
\]
are contained in both $F_I(h,v)$ and $F_I(h',v')$. $\Box$
{\corollary \label{corr2} Any face of Type I has a non-trivial intersection with any face of Type II. }
{\it Proof.} Let $F_I(h,v)$ be a face of Type I, and let $F_{II}$ be a face of Type II. Then there exist
automorphisms $U_n,U_m$ such that $F_{II} = U_n \tilde F_2 U_m$. We assume $n \geq m$ without loss of
generality. Choose $v' \in \partial L_n$ such that $U_n^{-1}(1\ (v')^T)^T(1\ h)U_m^{-1}$ is in
$\tilde F_2$. Then the element $(1\ (v')^T)^T(1\ h)$ is shared by the faces $F_{II}$ and $F_I(h,v)$.
$\Box$
\smallskip
On the other hand, faces of Type II do not necessarily have a non-trivial intersection.
\smallskip
In this section we have described the largest faces of the separable cone $K_{sep}$. There are two types of
such faces. All faces of one type are equivalent with respect to the action of the automorphism groups of
the underlying Lorentz cones. Any other non-trivial face is an intersection of some largest faces.
The faces of Type I are affinely isomorphic to the convex conic hull of two
spheres $S^{n-2}$,$S^{m-2}$ which intersect each other in one point, but lie in orthogonal subspaces. The
faces of Type II are intersections of the cone of positive semidefinite $\min(n,m) \times \min(n,m)$-matrices
with a linear subspace of codimension 1. Note that the manifold formed by the union of relative interiors of Type I faces has
$(n+m-1) + (n-2) + (m-2) = 2(n+m)-5$ dimensions, whereas the boundary of $K_{sep}$ is $nm-1$-dimensional. But
$2(n+m)-5 < nm-1$ if $\min(n,m) \geq 3$. Hence the boundary of the $L_2 \otimes L_n$-separable cones is
formed by Type I faces, while the boundary of $K_{sep}$ for $\min(n,m) \geq 3$ is formed by Type II faces.
\section{Radii of largest separable balls}
Extreme rays and largest faces remain invariant under linear bijections. Therefore the results obtained in
the last two sections are extendible to cones that are separable with respect to linear images of standard
Lorentz cones. In this section we compute radii of largest separable balls. These radii are naturally
invariant only under orthogonal mappings, therefore results obtained for $L_m \otimes L_m$-separable cones
will not extend to arbitrary linear images of the Lorentz cones. In order to cover this more general case, we
will consider general ellipsoid-generated cones. We shall compute the radius of the maximal
ellipsoid-ellipsoid separable ball around the tensor product of elements generating the central rays of the
two individual ellipsoid-generated cones. Let us first give a precise definition of an ellipsoid-generated
cone and its central ray.
Let $B \subset E$ be a closed solid ellipsoid with nonempty interior in some $n$-dimensional real vector
space. Suppose that the origin of the space is not contained in $B$. Then the conic hull of $B$ is the image
of the standard Lorentz cone $L_n$ under a regular linear mapping. Moreover, by a rotation it can be
transformed to some standardized ellipsoidal cone
\[
K_{st}(P) = \left\{ (x_0,x_1,\dots,x_{n-1})^T \,\big|\, x_0 \geq \sqrt{x^T P x},\ x = (x_1,\dots,x_{n-1})^T
\right\},
\]
where $P$ is a positive definite symmetric $(n-1)\times(n-1)$-matrix. The set of positive definite symmetric
$(n-1)\times(n-1)$-matrices parameterizes the set of standardized ellipsoidal cones in ${\bf R}^n$. We define
the central ray of $K_{st}(P)$ as the ray generated by the unit vector $e_0 = (1,0,\dots,0)^T$.
Let now $K_1 \subset {\bf R}^m$, $K_2 \subset {\bf R}^n$ be standardized ellipsoidal cones given by positive
definite matrices $P_1,P_2$ of dimensions $(m-1)\times(m-1)$ and $(n-1)\times(n-1)$, respectively:
\begin{eqnarray*}
K_1 &=& K_{st}(P_1) = \left\{ (x_0,x_1,\dots,x_{m-1})^T \in {\bf R}^m \,\big|\, x_0 \geq \sqrt{x^T P_1 x},\ x = (x_1,\dots,x_{m-1})^T \right\}, \\
K_2 &=& K_{st}(P_2) = \left\{ (y_0,y_1,\dots,y_{n-1})^T \in {\bf R}^n \,\big|\, y_0 \geq \sqrt{y^T P_2 y},\ y
= (y_1,\dots,y_{n-1})^T \right\}.
\end{eqnarray*}
Denote by $e_0^N,e_1^N,\dots,e_{N-1}^N$ the unit vectors along the coordinate axes of the space ${\bf R}^N$.
Then the unit vectors $e_0^m,e_0^n$ define the central rays of the cones $K_1,K_2$.
As in the previous section, we shall call an element $B \in {\bf R}^m \otimes {\bf R}^n$ {\sl $K_1 \otimes
K_2$-separable} or just {\sl separable} if $B$ can be expressed as a finite sum $\sum_{k=1}^N x_k \otimes
y_k$ of product elements such that $x_k \in K_1, y_k \in K_2$ for all $k = 1,\dots,N$. The cone of $K_1
\otimes K_2$-separable elements, the {\sl separable cone}, will be denoted by $K_{sep}$. We will represent
the elements of ${\bf R}^m \otimes {\bf R}^n$ as $n \times m$ matrices such that a product element $x \otimes
y$ is given by the rank 1 matrix $yx^T$. Then the product $e_0^m \otimes e_0^n$ is given by a matrix that has
zero elements everywhere except a 1 in the upper left corner.
We shall compute the radius of the largest ball around the unit length vector $e_0^m \otimes e_0^n \in {\bf
R}^m\otimes {\bf R}^n$ consisting of $K_1 \otimes K_2$-separable elements. A ball $B \subset {\bf R}^m\otimes
{\bf R}^n$ consists of separable elements if and only if the conical hull of $B$ is contained in the
separable cone $K_{sep}$. This conical hull is a ball-generated cone, and as such an ellipsoid-generated
cone.
{\lemma \label{radii_rel} Consider the real vector space ${\bf R}^N$. The cone generated by a ball of radius
$\rho < 1$ around the unit vector $e_0$ equals the standardized ellipsoidal cone $K_{st}(r^{-2}I_{N-1})$ with
$r = \frac{\rho}{\sqrt{1-\rho^2}}$. }
{\it Proof.} By the rotational symmetry of the ball-generated cone it must equal a standardized ellipsoidal
cone $K_{st}(P)$ with the matrix $P$ being proportional to the identity matrix. From the definition of
$K_{st}(r^{-2}I_{N-1})$ it follows that $r$ is the radius of the ball created by the intersection of the cone
$K_{st}(r^{-2}I_{N-1})$ with the hyperplane given by the equation $x_0 = 1$. The relation $r =
\frac{\rho}{\sqrt{1-\rho^2}}$ is a consequence of the similarity of appropriate rectangular triangles formed
in the $e_0$-$e_1$ plane of ${\bf R}^N$. $\Box$
Note that $r = \frac{\rho}{\sqrt{1-\rho^2}}$ is a monotonous function of $\rho \in [0,1)$.
\smallskip
Let us denote the ball-generated cone $K_{st}(r^{-2}I_{nm-1})$ by $K_{ball}(r)$. Identify ${\bf R}^{nm}$ with
${\bf R}^m\otimes {\bf R}^n$ by identifying the basis vectors $e_{kn+l}^{mn}$, $k = 0,\dots,m-1$, $l =
0,\dots,n-1$, with the orthonormal basis of tensor products $e_k^m \otimes e_l^n$. Then the cone
$K_{ball}(r)$ is generated by a ball centered on $e_0^m \otimes e_0^n$.
Let $K_{sep}^*,K_{ball}(r)^*$ denote the cones dual to $K_{sep},K_{ball}(r)$. These cones reside in the space
$({\bf R}^{nm})^* = {\bf R}^{nm}$, whose elements will likewise be represented by $n \times m$ matrices. The
scalar product of a matrix $B \in {\bf R}^m\otimes {\bf R}^n$ with a matrix $M \in ({\bf R}^{nm})^*$ will be
defined as $\langle B,M \rangle = tr\,(M^TB) = tr\,(B^TM)$.
{\lemma Let $r$ be the largest number such that the inclusion $K_{ball}(r) \subset K_{sep}$ holds. Then
\begin{equation} \label{maxi}
r^{-1} = \sqrt{ \max \left\{ hP_1h^T + v^TP_2v + tr\,(A^TP_2AP_1) \,\left\vert\, \tilde M = \left(
\begin{array}{cc} 1 & h \\ v & A \end{array} \right) \mbox{ is }L_m\mbox{-to-}L_n\mbox{ positive} \right. \right\} }.
\end{equation} }
{\it Proof.} We have $K_{ball}(r) \subset K_{sep}$ if and only if $K_{sep}^* \subset K_{ball}(r)^*$. By means
of standard linear algebra one establishes that $K_{ball}(r)^* = K_{ball}(r^{-1})$, $K_{st}(P)^* =
K_{st}(P^{-1})$ and $K_{sep}^*$ is the cone of $K_1$-to-$K_2^*$, or $K_{st}(P_1)$-to-$K_{st}(P_2^{-1})$
positive maps. It follows that the largest $r$ satisfying the inclusion $K_{ball}(r) \subset K_{sep}$ equals
the inverse of the smallest $R$ such that any $K_{st}(P_1)$-to-$K_{st}(P_2^{-1})$ positive map lies in the
cone $K_{ball}(R)$.
Let us characterize the cone of $K_{st}(P_1)$-to-$K_{st}(P_2^{-1})$ positive maps and the cone $K_{ball}(R)$.
Let $M: {\bf R}^m \to {\bf R}^n$ be a linear map, partitioned as
\begin{equation} \label{part2}
M = \left( \begin{array}{cc} 1 & h \\ v & A \end{array} \right),
\end{equation}
where $h$ is a row vector of length $m-1$, $v$ is a column vector of length $n-1$ and $A$ is a $(n-1) \times
(m-1)$ matrix.
Since for a positive definite matrix $P$ and for any vector $x$ we have $\sqrt{x^TPx} = |P^{1/2}x|$, we can
characterize the cones $K_1,K_2^*$ as follows:
\begin{eqnarray*}
K_1 &=& \left\{ (x_0,x_1,\dots,x_{m-1})^T \,\big|\, (x_0, (x_1,\dots,x_{m-1})P_1^{1/2})^T \in L_m \right\}, \\
K_2^* &=& \left\{ (y_0,y_1,\dots,y_{n-1})^T \,\big|\, (y_0, (y_1,\dots,y_{n-1})P_2^{-1/2})^T \in L_n
\right\}.
\end{eqnarray*}
It follows that the map $diag(1,P_1^{-1/2})$ is an isomorphism between $L_m$ and $K_1$ and the map
$diag(1,P_2^{-1/2})$ an isomorphism between $K_2^*$ and $L_n$. Hence $M$ is $K_1$-to-$K_2^*$ positive if and
only if the map
\[
\tilde M = diag(1,P_2^{-1/2})\cdot M\cdot diag(1,P_1^{-1/2}) = \left( \begin{array}{cc} 1 & \tilde h \\
\tilde v & \tilde A
\end{array} \right) = \left( \begin{array}{cc} 1 & hP_1^{-1/2} \\ P_2^{-1/2}v & P_2^{-1/2}AP_1^{-1/2}
\end{array} \right)
\]
is $L_m$-to-$L_n$ positive.
Let us examine the cone $K_{ball}(R)$. By definition $M$ is in $K_{ball}(R)$ if $M_{00} \geq
\sqrt{\sum_{(k,l) \not= (0,0)} M_{kl}^2}/R$ (here $M_{kl}$ denotes the elements of $M$). If $M$ is
partitioned as in (\ref{part2}), then $M_{00} = 1$ and $M \in K_{ball}(R)$ if and only if $R \geq
\sqrt{\sum_{(k,l) \not= (0,0)} M_{kl}^2} = \sqrt{|h|^2 + |v|^2 + ||A||_2^2}$.
Hence we obtain the following characterization of the largest number $r$ such that any $K_1$-to-$K_2^*$
positive map is contained in $K_{ball}(r^{-1})$:
\begin{eqnarray*}
r^{-1} &=& \max \left\{ \sqrt{|h|^2 + |v|^2 + ||A||_2^2} \,\left\vert\, \tilde M = \left( \begin{array}{cc} 1
& hP_1^{-1/2} \\ P_2^{-1/2}v & P_2^{-1/2}AP_1^{-1/2} \end{array} \right) \mbox{ is }L_m\mbox{-to-}L_n\mbox{
positive} \right. \right\} \\ &=& \max \left\{ \sqrt{|hP_1^{1/2}|^2 + |P_2^{1/2}v|^2 +
||P_2^{1/2}AP_1^{1/2}||_2^2} \,\left\vert\, \tilde M = \left(
\begin{array}{cc} 1 & h \\ v & A \end{array} \right) \mbox{ is }L_m\mbox{-to-}L_n\mbox{ positive} \right. \right\} \\
&=& \sqrt{ \max \left\{ hP_1h^T + v^TP_2v + tr\,(A^TP_2AP_1) \,\left\vert\, \tilde M = \left(
\begin{array}{cc} 1 & h \\ v & A \end{array} \right) \mbox{ is }L_m\mbox{-to-}L_n\mbox{ positive} \right. \right\}
}.\quad \Box
\end{eqnarray*}
We shall now calculate expression (\ref{maxi}). We have to compute the maximum of the function $F(M) =
hP_1h^T + v^TP_2v + tr\,(A^TP_2AP_1)$ over the set $S$ of $L_m$-to-$L_n$ positive maps $M$ which are
partitioned as in (\ref{part2}), i.e.\ with the upper left element being equal to 1. We shall show that $F$
achieves its maximum either at a rank 1 map or at a doubly stochastic map.
The following lemma is verified by direct calculation.
{\lemma For any integer $n \geq 2$ and any row vector $b \in {\bf R}^{n-1}$ the linear transformation
\[ U_n(b) = \left( \begin{array}{cc} 1 + |b|^2 & \frac{b\left( I_{n-1} - \frac{b^Tb}{(1 + |b|^2)^2} \right)^{-1/2}}{1 + |b|^2} \\
b^T & \left( I_{n-1} - \frac{b^Tb}{(1 + |b|^2)^2} \right)^{-1/2} \end{array} \right)
\]
preserves the quadratic form $J_n$, i.e.\ $J_n = U_n(b)^T J_n U_n(b)$, and is hence an automorphism of the cone $L_n$.
$\Box$ }
\smallskip
Let $M$ be an $L_m$-to-$L_n$ positive map, partitioned as in (\ref{part2}). By the preceding lemma the maps
$U_n(b_n)M$, $MU_m(b_m)$ are also positive for all $b_n \in {\bf R}^{n-1}$, $b_m \in {\bf R}^{m-1}$. The
upper left elements of these products are given by
\[ m_l(b_n) = 1 + |b_n|^2 + \frac{b_n\left( I_{n-1} - \frac{b_n^Tb_n}{(1 + |b_n|^2)^2} \right)^{-1/2}}{1 + |b_n|^2}v > 0, \quad m_r(b_m) = 1 + |b_m|^2 + hb_m^T > 0.
\]
Consider the families of positive maps $M_l(b_n) = U_n(b_n)M/m_l(b_n)$, $M_r(b_m) = MU_m(b_m)/m_r(b_m)$,
parameterized by row vectors $b_n \in {\bf R}^{n-1}$, $b_m \in {\bf R}^{m-1}$. The upper left element of the
corresponding matrices equals 1. Hence $M_l(b_n)$, $M_r(b_m)$ can be partitioned as in (\ref{part2}):
\begin{equation} \label{Ms}
M_l(b_n) = \left( \begin{array}{cc} 1 & h_l(b_n) \\ v_l(b_n) & A_l(b_n) \end{array} \right), \quad M_r(b_m) = \left( \begin{array}{cc} 1 & h_r(b_m) \\ v_r(b_m) & A_r(b_m) \end{array} \right),
\end{equation}
where $h_l(b_n)$, $v_l(b_n)$, $A_l(b_n)$, $h_r(b_m)$, $v_r(b_m)$, $A_r(b_m)$ are vectors and matrices
depending accordingly on the parameter vectors $b_n,b_m$. Define two scalar functions
\begin{eqnarray*}
F_l(b_n) &=& F(M_l(b_n)) = h_l(b_n)P_1h_l(b_n)^T + v_l(b_n)^TP_2v_l(b_n) + tr\,(A_l(b_n)^TP_2A_l(b_n)P_1), \\
F_r(b_m) &=& F(M_r(b_m)) = h_r(b_m)P_1h_r(b_m)^T + v_r(b_m)^TP_2v_r(b_m) + tr\,(A_r(b_m)^TP_2A_r(b_m)P_1).
\end{eqnarray*}
{\lemma Let a $L_m$-to-$L_n$ positive map $M$, partitioned as in (\ref{part2}), realize the maximum of the
function $F$. Then $M$ generates an extreme ray of the cone ${\cal P}$ of $L_m$-to-$L_n$ positive maps. The
corresponding functions $F_l(b_n)$, $F_r(b_m)$ have global maxima at $b_n = 0$, $b_m = 0$, respectively. As a
consequence, their gradients at $b_n = 0$ and $b_m = 0$ vanish. }
{\it Proof.} The function $F(M) = hP_1h^T + v^TP_2v + tr\,(A^TP_2AP_1)$ is strictly convex on the convex set
$S$. Hence its maximum is achieved at an extreme point of this set. Equivalently, the map realizing the
maximum of $F$ generates an extreme ray of ${\cal P}$.
Let the map $M$ realize the maximum of $F$. Define the families of maps (\ref{Ms}). Now note that
$U_n(0),U_m(0)$ are the identity maps, hence $M_l(0) = M_r(0) = M$. Since the maps $M_l(b_n)$, $M_r(b_m)$ are
in $S$ for all $b_n,b_m$, the functions $F_l(b_n)$, $F_r(b_m)$ attain their global maxima at the origin.
$\Box$
{\lemma \label{maxrk1} Let an $L_m$-to-$L_n$ positive map $M$, partitioned as in (\ref{part2}), realize the
maximum of $F$. If $M$ has rank 1, then this maximum is given by $F_{\max} = -1 + (1 + \lambda_{\max}(P_1))(1
+ \lambda_{\max}(P_2))$ (here $\lambda_{\max}$ denotes the maximal eigenvalue). }
{\it Proof.} Let $M$ satisfy the assumptions of the lemma. If $M$ is of rank 1, then $A = vh$. By the
previous lemma $M$ generates an extreme ray of ${\cal P}$. By Lemma \ref{rk1extr} we then have $|h| = |v| =
1$. On the other hand, any pair of unit length vectors $(h',v')$ defines a positive map of rank 1 via
\[ M(h',v') = \left( \begin{array}{cc} 1 & h' \\ v' & v'h' \end{array} \right).
\]
We have
\begin{eqnarray*}
F(M(h',v')) &=& h'P_1(h')^T + (v')^TP_2v' + (v')^TP_2v' + (v')^TP_2v'h'P_1(h')^T \\
&=& -1 + (1+h'P_1(h')^T)(1+(v')^TP_2v').
\end{eqnarray*}
It follows that
\[
F_{\max} = \max_{|h'|=|v'|=1} F(M(h',v')) = -1 + (1 + \lambda_{\max}(P_1))(1 + \lambda_{\max}(P_2)). \quad
\Box
\]
{\lemma \label{sig1} Let $M$ be an $L_m$-to-$L_n$ positive map, partitioned as in (\ref{part2}). Then
$\sigma_{\max}(A) \leq 1$. }
{\it Proof.} Let $M$ be a map satisfying the assumptions of the lemma and suppose that $\sigma =
\sigma_{\max}(A) > 1$. Then there exist unit length column vectors $u,w$ of appropriate dimensions such that
$\sigma u = Aw$ and hence $\sigma = u^TAw$. Without restriction of generality we can assume that $hw-u^Tv
\leq 0$ (otherwise we multiply $u,w$ by $-1$). Then we have
\[ \left( \begin{array}{c} 1 \\ -u \end{array} \right)^T \left( \begin{array}{cc} 1 & h \\ v & A \end{array}
\right) \left( \begin{array}{c} 1 \\ w \end{array} \right) = 1 + hw - u^Tv - \sigma \leq 1 - \sigma < 0.
\]
But
\[ \left( \begin{array}{c} 1 \\ -u \end{array} \right) \in L_n, \quad \left( \begin{array}{c} 1 \\ w \end{array}
\right)\in L_m, \quad M \left( \begin{array}{c} 1 \\ w \end{array} \right) \in L_n,
\]
the last inclusion being due to the positivity of $M$. Hence the scalar product of two vectors in $L_n$ is
negative, which leads to a contradiction with the self-duality of $L_n$. Thus the assumption
$\sigma_{\max}(A) > 1$ was false, which completes the proof. $\Box$
{\lemma Let an $L_m$-to-$L_n$ positive map $M$, partitioned as in (\ref{part2}), realize the maximum of $F$.
Suppose further that this maximum is strictly greater than the maximum over the rank 1 maps established in
Lemma \ref{maxrk1}. Then $h=v=0$. }
{\it Proof.} Let $M$ satisfy the assumptions of the lemma. We shall now compute the gradients of the functions $F_l(b_n)$, $F_r(b_m)$ at $b_n = 0$, $b_m = 0$. We have
\[ U_n'(b)|_{b = 0} = \left( \begin{array}{cc} 0 & b' \\ (b^T)' & 0 \end{array} \right), \quad m_l'(b)|_{b = 0} = b'v, \quad m_r'(b)|_{b = 0} = h(b^T)'.
\]
Hence we obtain
\[ h_l'(b_n)|_{b_n = 0} = b_n'(A - vh), \quad v_l'(b_n)|_{b_n = 0} = (I_{n-1} - vv^T)(b_n^T)', \quad A_l(b_n)|_{b_n = 0} = (b_n^T)'h - (v^T(b_n^T)')A,
\]
\[ h_r'(b_m)|_{b_m = 0} = b_m'(I_{m-1} - h^Th), \quad v_r'(b_m)|_{b_m = 0} = (A-vh)(b_m^T)', \quad A_r(b_m)|_{b_m = 0} =
vb_m' - A(b_m'h^T).
\]
It follows that
\begin{eqnarray*}
F_l'(b_n)|_{b_n = 0} &=& 2\left( h_l'P_1h^T + (v_l^T)'P_2v + tr\,((A_l^T)'P_2AP_1) \right) \\ &=& 2b_n'\left[
(I_{n-1}+P_2)AP_1h^T + (-(hP_1h^T+v^TP_2v+tr(A^TP_2AP_1))I_{n-1} + P_2)v \right], \\
F_r'(b_m)|_{b_m = 0} &=& 2\left( h_r'P_1h^T + (v_r^T)'P_2v + tr\,((A_r^T)'P_2AP_1) \right) \\ &=& 2b_m'\left[
(I_{m-1}+P_1)A^TP_2v + (-(hP_1h^T+v^TP_2v+tr(A^TP_2AP_1))I_{m-1} + P_1)h^T \right].
\end{eqnarray*}
Since the vanishing of the gradient is a necessary condition of maximality of the functions $F_l,F_r$, we obtain the
equations
\begin{eqnarray*}
(I_{n-1}+P_2)AP_1h^T &=& [(1+hP_1h^T+v^TP_2v+tr(A^TP_2AP_1))I_{n-1} - (I_{n-1}+P_2)]v, \\
(I_{m-1}+P_1)A^TP_2v &=& [(1+hP_1h^T+v^TP_2v+tr(A^TP_2AP_1))I_{m-1} - (I_{m-1}+P_1)]h^T.
\end{eqnarray*}
The maximum of $F$ is given by $F_{\max} = 1+hP_1h^T+v^TP_2v+tr(A^TP_2AP_1)$. It follows that
\begin{equation} \label{Aeqs}
AP_1h^T = [(I_{n-1}+P_2)^{-1}F_{\max} - I_{n-1}]v, \quad A^TP_2v = [(I_{m-1}+P_1)^{-1}F_{\max} - I_{m-1}]h^T.
\end{equation}
By the assumptions of the lemma we have $F_{\max} > (1+\lambda_{\max}(P_1))(1+\lambda_{\max}(P_2))$.
Therefore $(I_{n-1}+P_2)^{-1}F_{\max} - I_{n-1} \succ \lambda_{\max}(P_1)I_{n-1}$ and
$(I_{m-1}+P_1)^{-1}F_{\max} - I_{m-1} \succ \lambda_{\max}(P_2)I_{m-1}$. Hence the matrices on the right-hand
sides of (\ref{Aeqs}) are invertible and $h = 0$ implies $v = 0$ and vice versa. Let us assume that $h \not=
0$ and $v \not= 0$. Taking the norms on both sides of equations (\ref{Aeqs}), we get
\[ ||A||_{\infty} \lambda_{\max}(P_1) |h| \geq |AP_1h^T| = |[(I_{n-1}+P_2)^{-1}F_{\max} - I_{n-1}]v| >
\lambda_{\max}(P_1) |v|, \] \[ ||A||_{\infty} \lambda_{\max}(P_2) |v| \geq |A^TP_2v| =
|[(I_{m-1}+P_1)^{-1}F_{\max} - I_{m-1}]h^T| > \lambda_{\max}(P_2) |h|.
\]
Combining, we obtain $||A||_{\infty} = \sigma_{\max}(A) > 1$, which by Lemma \ref{sig1} leads to a
contradiction with the positivity of $M$. Hence $h = v = 0$, which completes the proof. $\Box$
\smallskip
The lemma implies that if $M$ realizes the maximum of $F$ and has a rank greater than 1, then it must be
doubly stochastic.
{\lemma Let $\lambda_1(P_1),\lambda_2(P_1),\dots,\lambda_{m-1}(P_1)$ and
$\lambda_1(P_2),\lambda_2(P_2),\dots,\lambda_{n-1}(P_2)$ be the eigenvalues of the matrices $P_1,P_2$,
respectively, in decreasing order. Then the maximum of $F$ is given by the expression $F_{\max} = \max\{ -1 +
(1 + \lambda_1(P_1))(1 + \lambda_1(P_2)), \sum_{k=1}^{\min(n,m)-1} \lambda_k(P_1) \lambda_k(P_2) \}$. }
{\it Proof.} We have shown above that the maximum of $F$ is achieved either at a rank 1 map, in which case it
equals $F_{\max} = -1 + (1 + \lambda_1(P_1))(1 + \lambda_1(P_2))$, or at a doubly stochastic map. Suppose we
are in the second case, and the map realizing the maximum of $F$ is partitioned as in (\ref{part2}) with
$h=v=0$.
Since the maximum is achieved at a map generating an extreme ray of the cone ${\cal P}$, all singular values
of the matrix $A$ equal 1 by Lemma \ref{dblstochlem}. Assume without restriction of generality that $n \geq
m$. Then the singular value decomposition of $A$ is given by
\[ A = UDV = U \left( \begin{array}{c} I_{m-1} \\ {\bf 0}_{(n-m)\times(m-1)} \end{array} \right) V,
\]
where $U,V$ are orthogonal matrices of appropriate dimensions.
On the other hand, by Lemma \ref{poscond} any pair $(U',V')$ of orthogonal matrices of appropriate size
defines a doubly stochastic positive map
\[ M(U',V') = \left( \begin{array}{cc} 1 & 0 \\ 0 & U'DV' \end{array} \right).
\]
Therefore
\begin{eqnarray*}
F_{\max} &=& \max_{U',V'} F(M(U',V')) = \max_{U',V'} tr(V^TD^TU^TP_2UDVP_1) \\
&=& \max_{U',V'} tr(D^T(U^TP_2U)D(VP_1V^T)).
\end{eqnarray*}
The pair $(U,V)$ of orthogonal matrices maximizes the function $F(M(U',V'))$.
Denote $VP_1V^T$ by $\tilde P_1$ and $U^TP_2U$ by $\tilde P_2$. Then the first order maximality condition is
given by the commutation relations $[D^T\tilde P_2D,\tilde P_1] = [D\tilde P_1 D^T, \tilde P_2] = 0$.
Partition the matrix $\tilde P_2$ as
\[ \tilde P_2 = \left( \begin{array}{cc} \tilde P_2^{11} & \tilde P_2^{12} \\ \tilde P_2^{21} & \tilde
P_2^{22} \end{array} \right),
\]
where $\tilde P_2^{11}$ is of size $(m-1) \times (m-1)$. Then above commutation relations imply $[\tilde
P_1,\tilde P_2^{11}] = 0$, $\tilde P_2^{12} = \tilde P_2^{21} = 0$. Let now $W_{m-1}$ be an orthogonal matrix
that simultaneously block-diagonalizes $\tilde P_1$ and $\tilde P_2^{11}$, and let $W_{n-m}$ be an orthogonal
matrix that diagonalizes $\tilde P_2^{22}$. Now note that $F_{\max} = tr(D^T \tilde P_2 D \tilde P_1) =
tr(D^T [diag(W_{m-1},W_{n-m})\,\tilde P_2\,diag(W_{m-1},W_{n-m})^T] D [W_{m-1}\tilde P_1W_{m-1}^T])$. The
products in brackets are diagonal and have the form $(U')^TP_2U'$, $V'P_1(V')^T$ for some orthogonal matrices
$U',V'$. Hence we can assume without loss of generality that $\tilde P_1,\tilde P_2$ are both diagonal.
Therefore there exist pairwise distinct indices $j_1,\dots,j_{m-1} \in \{1,\dots,n-1\}$ such that $F_{\max} =
\sum_{k=1}^{m-1} \lambda_k(P_1)\lambda_{j_k}(P_2)$. Obviously this sum is maximal if $j_k = k$ for all $k$,
and we arrive at the inequality $F_{\max} \leq \sum_{k=1}^{m-1} \lambda_k(P_1)\lambda_k(P_2)$.
On the other hand, there exist orthogonal matrices $U',V'$ such that \\ $(U')^TP_2U' =
diag(\lambda_1(P_2),\lambda_2(P_2),\dots,\lambda_{n-1}(P_2))$, $V'P_1(V')^T =
diag(\lambda_1(P_1),\dots,\lambda_{m-1}(P_1))$. Then we have $F(M(U',V')) = \sum_{k=1}^{m-1}
\lambda_k(P_1)\lambda_k(P_2)$ and $F_{\max} \geq \sum_{k=1}^{m-1} \lambda_k(P_1)\lambda_k(P_2)$. The proof is
complete. $\Box$
\smallskip
We have proven the following
{\corollary Let $r$ be the largest number such that the inclusion $K_{ball}(r) \subset K_{sep}$ holds. Then
\[ r = \left[ \max\left\{ -1 +
(1 + \lambda_1(P_1))(1 + \lambda_1(P_2)), \sum_{k=1}^{\min(n,m)-1} \lambda_k(P_1) \lambda_k(P_2) \right\}
\right]^{-1/2}. \quad \Box
\] }
\smallskip
By Lemma \ref{radii_rel} we now have the following theorem.
{\theorem \label{thrad} The radius of the largest $K_1 \otimes K_2$-separable ball around $e_0^m \otimes
e_0^n$ is given by
\[ \rho = \left[ \max\left\{
(1 + \lambda_1(P_1))(1 + \lambda_1(P_2)), 1 + \sum_{k=1}^{\min(n,m)-1} \lambda_k(P_1) \lambda_k(P_2) \right\}
\right]^{-1/2}. \quad \Box
\] }
{\corollary \label{ballcor} Let $B_1 \subset {\bf R}^m$, $B_2 \subset {\bf R}^n$ be balls of radii
$\rho_1,\rho_2 < 1$ around the unit vectors $e_0^m,e_0^n$, respectively. Let $K_1,K_2$ be the cones generated
by these balls. Then the radius of the largest $K_1 \otimes K_2$-separable ball around the unit vector $e_0^m
\otimes e_0^n \in {\bf R}^{mn}$ equals
\[ \left[ \max\left\{ \rho_1^{-2}\rho_2^{-2}, 1+ (\min(n,m)-1) (\rho_1^{-2}-1)(\rho_2^{-2}-1) \right\} \right]^{-1/2}.
\] }
The corollary is a direct consequence of the preceding theorem and Lemma \ref{radii_rel}.
\section{Application to multi-qubit systems}
In this section we apply the obtained results to compute largest $K_1\otimes K_2$-separable balls of
bipartite matrices around the identity, where the cones $K_1,K_2$ are generated by balls around the
identities in the factor spaces. We provide the exact value of the radius of such largest balls in dependence
on the radii of the original balls and the dimensions of the matrices. These results will be used to compute
lower bounds on the largest separable ball of unnormalized mixed states for multi-qubit systems.
Denote the space of $k \times k$ hermitian matrices by ${\cal H}(k)$. Let $B_{r_1} \subset {\cal H}(m)$,
$B_{r_2} \subset {\cal H}(n)$ be balls of radii $r_1 < \sqrt{m}$, $r_2 < \sqrt{n}$ around the corresponding
identities $I_m,I_n$ and let $K_1,K_2$ be the conic hulls of these balls. We look for the largest ball around
the identity $I_{nm} \in {\cal H}(mn) = {\cal H}(m)\otimes {\cal H}(n)$ which is contained in the cone of
$K_1 \otimes K_2$-separable matrices.
The following corollary is a consequence of Corollary \ref{ballcor} and the fact that the identity in ${\cal
H}(n) \cong {\bf R}^{n^2}$ has norm $\sqrt{n}$.
{\corollary \label{matrixballs} The largest ball around $I_{nm} \in {\cal H}(mn) = {\cal H}(m)\otimes {\cal
H}(n)$ which is contained in the cone of $K_1 \otimes K_2$-separable matrices has radius
\[ r = \min\left( r_1r_2,
\frac{\sqrt{mn}r_1r_2}{\sqrt{(\min(m^2,n^2)-1)(m-r_1^2)(n-r_2^2) + r_1^2r_2^2}} \right). \ \Box
\] }
\bigskip
We see that for large dimensions and small $r_1,r_2$ $r$ is asymptotically equal to
$\frac{r_1r_2}{\min(m,n)}$. This asymptotics was independently found by Leonid Gurvits\footnote{Leonid Gurvits, personal communication}.
\smallskip
Let us use this result to obtain a bound on the radius of the largest separable ball of unnormalized density
matrices for multi-qubit systems. Let $m = 2$, $r_1 = 1$ and set $n(k) = 2^{k-1}$. Define a sequence $\rho_k$
recursively by $\rho_1 = 1$ and
\begin{eqnarray} \label{thhilf}
\rho_k &=& \min\left( r_1\rho_{k-1},
\frac{\sqrt{mn(k)}r_1\rho_{k-1}}{\sqrt{(\min(m^2,n(k)^2)-1)(m-r_1^2)(n(k)-\rho_{k-1}^2) + r_1^2\rho_{k-1}^2}}
\right) \\ && = \min\left( \rho_{k-1}, \frac{\sqrt{2^k}\rho_{k-1}}{\sqrt{3(2^{k-1}-\rho_{k-1}^2) +
\rho_{k-1}^2}} \right) = \frac{\sqrt{2^k}\rho_{k-1}}{\sqrt{3\cdot 2^{k-1} - 2\rho_{k-1}^2}} \nonumber
\end{eqnarray}
for $k \geq 2$. It follows that
\[ \rho_1^{-2} = 1,\quad \rho_k^{-2} = \frac{3}{2}\rho_{k-1}^{-2} - 2^{-k+1}
\]
and we get the explicit expression
\[ \rho_k^{-2} = \frac{1}{3}\left( \frac{3}{2} \right)^k + 2^{-k},\quad \rho_k = \frac{2^{k/2}}{\sqrt{3^{k-1}+1}}.
\]
{\theorem \label{multiqubit} $\rho_k = \frac{2^{k/2}}{\sqrt{3^{k-1}+1}}$ is a lower bound on the radius of the largest separable
ball of unnormalized multi-partite mixed states of a $k$-qubit system around the identity matrix in the space
${\cal H}(2)^{\otimes k}$. }
{\it Proof.} We prove the theorem by induction.
For a one-qubit system $\rho_1 = 1$ is the radius of the largest ball around $I_2$ in the cone ${\cal
H}_+(2)$ of positive semidefinite hermitian $2 \times 2$ matrices. Hence for $k=1$ the bound $\rho_k$ is
exact.
Assume now that the ball $B_{k-1} \subset {\cal H}(2)^{\otimes(k-1)}$ of radius $\rho_{k-1}$ around the
identity matrix $I_{2^{k-1}} \in {\cal H}(2)^{\otimes(k-1)}$ consists of unnormalized separable states of a
$(k-1)$-qubit system. Let us apply Corollary \ref{matrixballs} with $m = 2$ and $r_1 = 1$. Since the cone
${\cal H}_+(2)$ is isometric to the standard Lorentz cone $L_4 = K_{st}(I_3)$, it will be generated by a ball
of radius 1 around $I_2$ and we get $K_1 = {\cal H}_+(2)$. Let further $n = n(k) = 2^{k-1}$ and $r_2 =
\rho_{k-1}$. If we identify the space ${\cal H}(n)$ with the space ${\cal H}(2)^{\otimes(k-1)}$, then the
cone $K_2$ will be generated by $B_{k-1}$.
But then the ball $B_k \subset {\cal H}(2)^{\otimes k}$ of radius $\rho_k$ around the identity matrix
$I_{2^k}$ is ${\cal H}_+(2) \otimes B_{k-1}$-separable by (\ref{thhilf}) and Corollary \ref{matrixballs}.
Thus it is also ${\cal H}_+(2)^{\otimes k}$-separable by the assumption on $B_{k-1}$. $\Box$
\smallskip
{\it Remark:} $\rho_k$ is the best bound one can obtain by tensoring in the spaces ${\cal H}(2)$ successively
and approximating each time the separable cone by the largest ball-generated cone contained therein. This
general approach was proposed and successfully applied by Gurvits and Barnum in \cite{Gurvits0302102}.
{\it Remark:} Since both factor cones in the ${\cal H}_+(2)^{\otimes 2}$-separable cone are isometric to
$L_4$, Corollary \ref{matrixballs} provides the exact result also for $k = 2$. Gurvits and Barnum obtained the exact
result for a general bipartite space in \cite{Gurvits0204159}.
For a 3-qubit system we get a radius of $\sqrt{4/5}$ instead of $\sqrt{8/11}$ and for $n$-qubit systems with
$n \geq 4$ an improvement of over $12.3\%$ with respect to Gurvits' result in \cite{Gurvits0409095}. The new
bounds imply that with standard NMR preparation technique one needs at least 36 qubits to obtain
entanglement, which is a slightly stronger restriction than the one proven by Gurvits and Barnum \cite{Gurvits0409095}.
\section{Conclusion}
In this contribution we dealt with cones consisiting of elements separable with respect to two Lorentz cones.
Such cones are prospective candidates for the approximation of more complex separable cones, such as the
cones of unnormalized separable states of a multi-partite quantum system. The idea of using a Lorentz cone to
approximate one of the factor cones in a bipartite setting and recursively in a multi-partite setting was
introduced by Leonid Gurvits and Howard Barnum in \cite{Gurvits0302102}. Later they obtained asymptotically exact results on the size of largest
separable balls in \cite{Gurvits0409095}.
We considered different aspects of ball-ball separable cones. Theorem \ref{extreme} describes the extreme
rays generating the cone dual to a ball-ball separable cone, i.e.\ a cone of Lorentz-to-Lorentz positive
maps. There are two kinds of such rays, and all rays of one kind are equivalent under the action induced by
the automorphism groups of the individual Lorentz cones. Correspondingly, the ball-ball separable cones
possess two kinds of largest faces. Here by a "largest" face we mean a non-trivial face that is not
the intersection of other, strictly larger faces. The shape of these faces is described in Propositions
\ref{F1} and \ref{F2}. In Corollaries \ref{corr1} and \ref{corr2} we established that the
largest faces are highly intersecting each other, unlike the largest faces of a single Lorentz cone. In
Theorem \ref{thrad} and Corollary \ref{ballcor} we compute the radius of the largest ball around an element
on the central ray of a ball-ball separable cone that is contained in this cone. This result extends to the
case of balls in ellipsoid-ellipsoid separable cones. Such cones are affinely isomorphic, but not isometric
to a ball-ball separable cone. The extension to ellipsoid-ellipsoid separable cones allows to use more
flexible approximations of individual factor cones by ellipsoidal cones, which in may be more appropriate
than Lorentz cones in some situations.
Finally, we applied the developed theory to the case of a multi-qubit quantum system. Due to the exactness of
our estimates we were able to sharpen previously available bounds on the radii of maximal separable balls
around the uniformly mixed state. Our bounds in Theorem \ref{multiqubit} are about 12\% tighter than the best bounds obtained so far
\cite{Gurvits0409095}.
|
{
"timestamp": "2005-03-24T16:46:37",
"yymm": "0503",
"arxiv_id": "quant-ph/0503194",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503194"
}
|
\section{Introdu\c{c}\~ao}
\hspace{.5cm} Nos dias atuais a Teoria Qu\^antica de Campos \'e largamente
empregada em diversas \'areas da f\'{\i}sica, tais como, altas energias,
mec\^anica estat\'{\i}stica, mat\'eria condensada, etc. Sendo a Teoria
Qu\^antica de Campos fundamentalmente de aspectos perturbativos, ela sofre
de pesados problemas de diverg\^encias. O tratamento destas diverg\^encias
tem sido um enorme desafio para os f\'{\i}sicos. A natureza matem\'atica do
problema \'e bem conhecida. Diverg\^encias ocorrem nos c\'alculos
perturbativos porque duas distribui\c{c}\~oes n\~ao podem ser multiplicadas
em um mesmo ponto. V\'arios m\'etodos tem sido propostos para solucionar
este problema. Entretanto somente \'e poss\'{\i}vel eliminar estes infinitos
de uma maneira f\'{\i}sica e consistente por absorv\^e-los nos par\^ametros
livres da teoria (massa e constante de acoplamento).
O procedimento usual para sanar o problema das diverg\^encias \'e empregar
um m\'etodo de regulariza\c{c}\~ao (cut-off, dimensional, zeta, etc ),
tornando a teoria finita atrav\'es do uso de um regulador (par\^ametro de
regulariza\c{c}\~ao) a fim de isolar as diverg\^encias e, ent\~ao,
restabelecer a teoria original com a elimina\c{c}\~ao do regulador usando
uma prescri\c{c}\~ao de renormaliza\c{c}\~ao, subtra\c{c}\~ao dos p\'olos ou
adi\c{c}\~ao de contra-termos.
De maneira geral o entendimento do procedimento de renormaliza\c{c}\~ao
empregado fica prejudicado devido a complexidade da Teoria Qu\^antica de
Campos. A fim de contornar esta dificuldade, vamos tratar aqui de dois
problemas simples e bem conhecidos por qualquer aluno de gradua\c{c}\~ao em
f\'{\i}sica e possivelmente dos demais cursos da \'area de Ci\^encias Exatas.
Os problemas aos quais nos referimos \'e o da determina\c{c}\~ao do
potencial escalar el\'etrico e do potencial vetor magn\'etico de um fio
infinito de carga e de corrente, respectivamente. Tais problemas, de um modo
geral parecem amb\'{\i}guos para os alunos, pois escondido neles existe um
procedimento de renormaliza\c{c}\~ao, como apontou Hans em seu artigo [1].
Uma maneira encontrada para se evitar diretamente as diverg\^encias nos
c\'alculos dos potenciais, \'e primeiramente determinar os campos
el\'etricos e magn\'etico e em seguida calcular os potenciais escalar
el\'etrico e vetorial magn\'etico do fio infinito.
O artigo est\'a organizado com segue. Na se\c{c}\~ao-2 tratamos do c\'alculo
do potencial escalar el\'etrico de um fio infinito com densidade linear de
carga $\lambda$ e do potencial vetor magn\'etico de um fio infinito de
corrente constante, que nos conduzir\'a a uma integral divergente. Nas se\c{c%
}\~oes 3, 4 e 5 n\'os regularizamos a integral divergente obtida na se\c{c}%
\~ao anterior usando os m\'etodos, cut-off [3], dimensional [4] e fun\c{c}%
\~ao zeta [5] respectivamente. Na se\c{c}\~ao-6 usando as prescri\c{c}\~oes
de renormaliza\c{c}\~ao, determinamos os potenciais renormalizados,
discutimos o par\^ametro de escala e apresentamos as id\'eias b\'asicas da
Teoria de Renormaliza\c{c}\~ao em Teoria Qu\^antica de Campos.
\section{Potencial Escalar El\'etrico e Potencial Vetor Magn\'etico}
\hspace{.5cm} O potencial escalar el\'etrico $\Phi(\vec{r})$ gerado por um
fio infinito com densidade linear de carga $\lambda$ em um ponto qualquer do
espa\c{c}o exceto no fio \'e dado por [2-3]
\begin{eqnarray}
\Phi(\vec{r})=\frac{\lambda}{4\pi\varepsilon_{0}}\int_{-\infty}^{\infty}%
\frac{dz} {\sqrt{z^{2}+\rho^{2}}},
\end{eqnarray}
onde temos colocado o fio sobre o eixo z e $\rho$ \'e a dist\^ancia do ponto
ao fio, coordenada radial cil\'{\i}ndrica.
O potencial vetor magn\'etico $\vec{A}(\vec{r})$ produzido por um fio
infinito de corrente el\'etrica constante $i$, \'e dado por [3]
\begin{eqnarray}
\vec{A}(\vec{r})=\frac{\mu_{0}i}{4\pi}\int_{-\infty}^{\infty}\frac{dz}{\sqrt{%
z^{2}+\rho^{2}}}\hat{k},
\end{eqnarray}
onde temos usando a mesma geometria anterior.
Uma an\'alise dimensional da integral
\begin{eqnarray}
I=\int_{-\infty}^{\infty}\frac{dz}{\sqrt{z^{2}+\rho^{2}}},
\end{eqnarray}
que aparece nas equa\c{c}\~oes dos potenciais, mostra que ela \'e
adimensional e portanto sofre de uma diverg\^encia logar\'{\i}tmica.
Assim, vemos que para estes dois problemas simples devemos empregar um
procedimento de renormaliza\c{c}\~ao a fim de obtermos os potenciais
renormalizados, isto \'e, "observados" (a difer\^en\c{c}a de potencial entre
dois pontos, pois ele \'e uma grandeza relativa e n\~ao absoluta).
A fim de tornar a teoria finita e assim manuze\'avel, devemos empregar um
m\'etodo de regulariza\c{c}\~ao. Isto vai nos permitir separarmos a parte
finita da divergente. Por\'em, a teoria fica dependente de um par\^ametro de
regulariza\c{c}\~ao e uma prescri\c{c}\~ao de renormaliza\c{c}\~ao dever\'a
se empregada para restabelecermos a teoria original. Vamos utilizar
diferentes m\'etodos de regulariza\c{c}\~ao e mostrar que, embora cada um
forne\c{c}a um resultado diferente, a teoria final, isto \'e, renormalizada
(f\'{\i}sica) \'e independente do m\'etodo de regulariza\c{c}\~ao usado.
\section{Cut-off}
\hspace{.5cm} Esse m\'etodo de regulariza\c{c}\~ao se baseia no emprego de
um corte nos limites da integral, isto \'e, trocamos o limite infinito por
um valor finito $\Lambda$ (par\^ametro regularizador).
Com a inclus\~ao do corte tornamos a teoria finita, por\'em dependente de $%
\Lambda$. Portanto, para restabelecermos a teoria original, devemos ao final
tomar o limite com $\Lambda$ tendendo a infinito.
Na integral da eq.(3) vamos introduzir um corte
\begin{eqnarray}
I_{\Lambda}=\int_{0}^{\Lambda}\frac{dz}{\sqrt{z^{2}+\rho^{2}}}.
\end{eqnarray}
Uma vez que tomaremos o limite, \'e conveniente obtermos o resultado da
integral da eq.(4) em pot\^encias de $\Lambda$ e de $\frac{1}{\Lambda}$ de
forma a permitir a separa\c{c}\~ao do(s) p\'olo(s) da parte finita. Vamos
dividir a integral da eq.(4) em duas partes
\begin{eqnarray}
I_{\Lambda}=\int_{0}^{\rho}\frac{dz}{\rho\sqrt{\frac{z^{2}}{\rho^{2}}+1}} +
\int_{\rho}^{\Lambda}\frac{dz}{z\sqrt{\frac{\rho^{2}}{z^{2}}+1}},
\end{eqnarray}
para considerarmos os casos em que $z<\rho$ e $z>\rho$. Realizando as
expans\~oes em s\'erie de Taylor dos integrandos da eq. (5) e depois
integrando termo a termo obtemos
\[
I_{\Lambda }=C+\ln \left( \frac{\Lambda }{\rho }\right) +O\left( \frac{1}{%
\Lambda ^{2}}\right) ,
\]
onde $C$ \'{e} uma constante.
Podemos observar que quando tentamos restabelecer a teoria original, ou
seja, tomamos o limite de $\Lambda$ tendento a infinito, presenciamos uma
diverg\^encia logar\'{\i}tmica, como j\'a esperavamos.
\section{Regulariza\c{c}\~ao Dimensional}
\hspace{.5cm} Este m\'etodo de regulariza\c{c}\~ao consiste em modificar a
dimens\~ao da integral atrav\'es de uma continua\c{c}\~ao anal\'{\i}tica de
forma a torn\'a-la finita. Consegue-se isto trocando a dimens\~ao do
diferenciando por uma outra complexa, atrav\'es da inclus\~ao de um
par\^ametro regularizador complexo, $\omega$
\begin{eqnarray}
I(\rho,\omega)=\int_{-\infty}^{\infty}\frac{d^{1-\omega}z}{\sqrt{z^{2}+
\rho^{2}}}.
\end{eqnarray}
A integral (7) agora \'e finita e pode ser realizada usando a rela\c{c}\~ao
[4]
\[
\int_{-\infty }^{\infty }\left( k^{2}+a^{2}\right) ^{-\alpha }d^{m}k=\pi ^{%
\frac{m}{2}}\frac{\Gamma (\alpha -\frac{m}{2})}{\Gamma (\alpha )}\left(
a^{2}\right) ^{\frac{m}{2}-\alpha },
\]
obtendo
\[
I(\rho ,\omega )=\pi ^{\frac{-\omega }{2}}\Gamma \left( \frac{\omega }{2}%
\right) (\rho )^{-\omega }.
\]
Para separarmos a parte finita da diverg\^{e}nte quando $\omega $ vai a
zero, vamos fazer uma expans\~{a}o em pot\^{e}ncias de $\omega $ da eq.(9),
para isto usamos para $|\omega |\ll 1$ as seguintes rela\c{c}\~{o}es
\[
\Gamma \left( \frac{\omega }{2}\right) =\frac{2}{\omega }-\gamma +O(\omega )
\]
e
\[
\rho ^{-\omega }=1-\frac{\omega }{2}\ln (\rho ^{2})+O(\omega ^{2}),
\]
onde $\gamma $ \'{e} o n\'{u}mero de Euler. Ent\~{a}o temos
\[
I(\rho ,\omega )=\pi ^{-\frac{\omega }{2}}\left[ \frac{2}{\omega }-\gamma
-\ln \left( \frac{\rho ^{2}}{\mu ^{2}}\right) +O(\omega )\right] ,
\]
onde temos inclu\'{i}do um par\^{a}metro de escala $\mu $ com dimens\~{a}o
de comprimento, a fim de tornar o logaritmando adimensional.
\section{Regulariza\c{c}\~ao por Fun\c{c}\~ao Zeta}
\hspace{.5cm} A fun\c{c}\~ao zeta generalizada associada a um operador $M$,
\'e definida como
\begin{eqnarray}
\zeta_{M}(s)=\sum_{i}\lambda_{i}^{-s},
\end{eqnarray}
onde $\lambda_i$, s\~ao os auto-valores do operador $M$ e $s$ um par\^ametro
complexo
Definimos, para o nosso caso, a fun\c{c}\~ao zeta como
\[
\zeta (s+1/2)=\int_{-\infty }^{\infty }\left( \frac{z^{2}}{\mu ^{2}}+\frac{%
\rho ^{2}}{\mu ^{2}}\right) ^{-s-1/2}d\left( \frac{z}{\mu }\right)
\]
e a integral (3) fica
\[
I(\rho ,s)=\zeta (s+1/2).
\]
O par\^{a}metro de escala $\mu $, com dimens\~{a}o de comprimento foi
inclu\'{i}do para tornar a fun\c{c}\~{a}o zeta admensional para todo $s$.%
\newline
Usando a rela\c{c}\~{a}o (8) obtemos
\[
\zeta (s+1/2)=\sqrt{\pi }\frac{\Gamma (s)}{\Gamma (s+1/2)}\left( \frac{\rho
^{2}}{\mu ^{2}}\right) ^{-s}
\]
que com a aproxima\c{c}\~{a}o
\[
2\sqrt{\pi }\frac{\Gamma (s)}{\Gamma (s-1/2)}\approx -\frac{1}{s},
\]
para $|s|\ll 1$, temos
\[
\zeta (s+1/2)=-\frac{\left( \frac{\rho ^{2}}{\mu ^{2}}\right) ^{-s}}{%
2s(s-1/2)}.
\]
A continua\c{c}\~{a}o anal\'{i}tica para s igual a zero da eq.(18) \'{e}
obtida multiplicando a equa\c{c}\~{a}o por s e em seguida derivando em $s=0$
[5]. Assim
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}-\frac{\lambda }{2\pi
\varepsilon _{0}}\ln \left( \frac{\rho }{\mu }\right) ,
\]
\[
\vec{A}(\vec{r})=\frac{\mu _{0}i}{2\pi }\hat{k}-\frac{\mu _{0}i}{2\pi }\ln
\left( \frac{\rho }{\mu }\right) \hat{k}.
\]
\section{Condi\c{c}\~oes de Renormaliza\c{c}\~ao}
\hspace{.5cm} Como podemos observar os potenciais obtidos atrav\'es dos
resultados dados pelas eq.(6) e (12) s\~ao ainda divergentes, portanto,
devemos lan\c{c}ar m\~ao de uma prescri\c{c}\~ao de renormaliza\c{c}\~ao a
fim de eliminar a parte divergente (p\'olo).
Como prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o usaremos a condi\c{c}%
\~{a}o f\'{i}sica, de que os potenciais n\~{a}o s\~{a}o grandezas absolutas
e sim relativas, isto \'{e}, somente diferen\c{c}as de potenciais podem ser
observadas. Assim, usando as eq.(6) e (12) obtemos
\[
\Phi (\vec{r})-\Phi (\vec{r_{0}})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln
\left( \frac{\rho _{0}}{\rho }\right)
\]
e
\[
\vec{A}(\vec{r})-\vec{A}(\vec{r_{0}})=\frac{\mu _{0}i}{2\pi }\ln \left(
\frac{\rho _{0}}{\rho }\right) \hat{k}
\]
Agora tomando o potencial nulo no ponto de refer\^{e}ncia $\vec{r_{0}}$,
temos
\[
\Phi _{R}(\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{%
\rho _{0}}{\rho }\right)
\]
e
\[
\vec{A}_{R}(\vec{r})=\frac{\mu _{0}i}{2\pi }\ln \left( \frac{\rho _{0}}{\rho
}\right) \hat{k}.
\]
Note que o ponto de refer\^encia $\vec{r_{0}}$ \'e completamente
arbitr\'ario.
Embora os resultados obtidos nas eq.(19) e (20) sejam finitos, eles ainda
n\~ao representam os resultados f\'{\i}sicos, pois n\~ao sabemos se o que
retiramos da parte divergente foi mais que o necess\'ario. Uma renormaliza\c{%
c}\~ao finita deve ser realizada para que os potenciais obtidos sejam
aqueles que representem a f\'{\i}sica do problema.
Novamente usando a diferen\c{c}a de potencial como condi\c{c}\~ao de
renormaliza\c{c}\~ao, obtemos das eq.(19) e (20) os mesmos resultados
obtidos nas eq.(23) e (24)
\'E importante comentarmos a presen\c{c}a do par\^ametro de escala $\mu$ nas
eq.(12), (19) e (20).
A prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o usada aqui fornece
imediatamente o resultado f\'{i}sico, isto \'{e}, o potencial no ponto $\vec{%
r}$ medido em rela\c{c}\~{a}o aquele medido no ponto de refer\^{e}ncia $\vec{%
r_{0}}$. Se desejassemos como primeira etapa obter um resultado finito para
as eq.(6) e (12) poder\'{i}amos usar como prescri\c{c}\~{a}o a subtra\c{c}%
\~{a}o do termo divergente (p\'{o}lo). Na eq.(6) a fim de separarmos a parte
divergente da finita devemos multiplicar e dividir o logaritimando por um
par\^{a}metro arbitr\'{a}rio finito, o par\^{a}metro de escala $\mu $.
\[
I(\rho ,\mu ,\Lambda )=C-\left[ \ln \left( \frac{\rho }{\mu }\right) -\ln
\left( \frac{\Lambda }{\mu }\right) \right] +O\left( \frac{1}{\Lambda ^{2}}%
\right) .
\]
Agora usando como prescri\c{c}\~{a}o a subtra\c{c}\~{a}o do p\'{o}lo,
obtemos, para o cut-off
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{\rho }{%
\mu }\right) +\frac{\lambda }{2\pi \varepsilon _{0}}C,
\]
e para a dimensional
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{\rho }{%
\mu }\right) +\frac{\gamma }{2\pi \varepsilon _{0}}.
\]
Ent\~ao, notamos que no caso da regulariza\c{c}\~ao dimensional e zeta, esta
separa\c{c}\~ao j\'a foi realizada de alguma forma escondida dentro dos
procedimento usados.
Uma maneira mais elegante e formal de introduzimos o par\^{a}metro de
escalar \'{e} fazendo com que a integral inicial (3) seja adimensional, isto
\'{e},
\[
I=\int_{-\infty }^{\infty }\frac{d\left( \frac{z}{\mu }\right) }{\sqrt{\frac{%
z^{2}}{\mu ^{2}}+\frac{\rho ^{2}}{\mu ^{2}}}}.
\]
E desta forma tornando a eq.(7) adimensional para qualquer $\omega $.
\'{E} claro que a continua\c{c}\~{a}o anal\'{i}tica usada no m\'{e}todo da
fun\c{c}\~{a}o zeta \'{e} a prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o
necess\'{a}ria para se obter o resultado finito e \'{e} equivalente a subtra%
\c{c}\~{a}o do p\'{o}lo. Isso fica claro se tivessemos realizado a
expans\~{a}o em s\'{e}rie de Laurent da eq.(18)
\[
I(\rho ,s)=\frac{a_{-1}}{s}+\ln \left( \frac{\rho }{\mu }\right) +O(s),
\]
onde $a_{-1}$ \'{e} o res\'{i}duo.
Note que os resultados das eq.(19),(26) e (27) diferem por uma constante e
s\~ao dependentes do par\^ametro de escala. Como j\'a dissemos, embora os
resultados destas equa\c{c}\~oes sejam finitos eles ainda n\~ao representam
a f\'{\i}sica da teoria. Isto \'e obvio, pois, n\~ao podemos ter os
resultados f\'{\i}sicos (observados) dependentes do m\'etodo de regulariza\c{%
c}\~ao. Uma renormaliza\c{c}\~ao finita deve ser feita para ajustar os
potenciais obtidos aqueles observados (diferen\c{c}as). Esta condi\c{c}\~ao
de renormaliza\c{c}\~ao nos permite escrever os potenciais em fun\c{c}\~ao
daqueles observados em um determinado ponto. Ela tamb\'em permite que o
par\^ametro de escala seja escrito em fun\c{c}\~ao do ponto de refer\^encia $%
\rho_{0}$. \'E claro que o ponto de refer\^encia \'e arbitr\'ario e portanto
tamb\'em o par\^ametro de escala.
Agora estamos aptos a sintetizar como funciona a renormaliza\c{c}\~ao. Os
potenciais dados pelas eq.(6), (12) e (19), n\~ao s\~ao aqueles f\'{\i}sicos
(observ\'aveis) sendo at\'e mesmo divergentes. Para torn\'a-los aqueles
observados devemos ajust\'a-los. Assim, medimos (na verdade aqui definimos
um valor qualquer, em geral zero) o potencial em um ponto de refer\^encial
qualquer $\vec{r_{0}}$ que no caso da Teoria Qu\^antica de Campos \'e
chamado ponto de renormaliza\c{c}\~ao ou subtra\c{c}\~ao. Por fim escrevemos
o potencial f\'{\i}sico (observado) como fun\c{c}\~ao daquele medido no
ponto de refer\^encia (ponto de renormaliza\c{c}\~ao). Este procedimento
ent\~ao absorve a diverg\^encia do potencial original n\~ao f\'{\i}sico.
Em resumo:
i) Potencial original n\~ao f\'{\i}sico
\begin{eqnarray}
\Phi_d(\vec{r})=D + C + \Phi_F(\vec{r}),
\end{eqnarray}
onde D \'e o termo divergente separado por um m\'etodo qualquer de regulariza%
\c{c}\~ao, e C \'e uma constante que depende do m\'etodo de regulariza\c{c}%
\~ao e $\Phi_F(\vec{r})$ \'e o potencial.
ii) Potencial medido no ponto de refer\^encia (renormaliza\c{c}\~ao)
\begin{eqnarray}
\Phi_{0}=D + C + \Phi_F(\vec{r_{0}}).
\end{eqnarray}
Neste caso para $\Phi_{0}$ \'e determinado um valor arbitr\'ario e n\~ao
realmente medido
Agora escrevemos
\begin{eqnarray}
D + C = \Phi_{0}-\Phi_F(\vec{r_{0}})
\end{eqnarray}
e substituindo na eq.(31), fica
\begin{eqnarray}
\Phi_R(\vec{r})=\Phi(\vec{r})-\Phi(\vec{r_{0}})+\Phi_{0},
\end{eqnarray}
onde $\Phi_R(\vec{r})$ \'e o potencial renormalizado.
Note que mesmo no caso de um m\'etodo de regulariza\c{c}\~ao que forne\c{c}a
um resultado finito, ainda temos de ajustar este resultado aquele
f\'{\i}sico.
Finalmente, podemos analizar como funciona a renormaliza\c{c}\~ao na Teoria
Qu\^antica de Campos. A teoria original depende de alguns par\^ametros em
geral divergentes, tais como $m$ e $\lambda$. Tais par\^ametro n\~ao
representam a massa $(m)$ e a constante de acoplamento $\lambda$ observados
da teoria e sim s\~ao ajustando atrav\'es das condi\c{c}\~oes de renormaliza%
\c{c}\~ao a estas quantidades f\'{\i}sicas renormalizadas, medidas em caso
de teorias realistas, ou definidas no caso de teorias n\~ao realistas, em um
determinado ponto, chamado ponto de renormaliza\c{c}\~ao ou subtra\c{c}\~ao.
Este ponto, pode ser o quadri-momento da teoria ou um determinado estado do
sistema, em geral o de menor energia, ou estado de v\'acuo, embora qualquer
ponto seja t\~ao bom quanto outro, isto \'e, o ponto de renormaliza\c{c}\~ao
\'e arbitr\'ario.
Escrevendo agora a teoria original em fun\c{c}\~ao n\~ao mais dos
par\^ametros originais $m$ e $\lambda$ e sim das quantidades f\'{\i}sicas
renormalizadas ("observadas") $m_{R}$ e $\lambda_{R}$, as diverg\^encias
s\~ao absorvidas de forma semelhante ao que ocorreu com o potencial.
Uma maneira alternativa usada \'e tomar os par\^ametros $m$ e $\lambda$ da
teoria original como sendo realmente aquele observados (renormalizados) e
absorver as diverg\^encias da teoria em contra-termos $\delta$$m$ e $%
\delta\lambda$ inclu\'{\i}dos na teoria. Tais contra-termos, \'e claro,
devem ser de termos de mesma pot\^encia nos campos que aqueles de $m$ e $%
\lambda$. Ent\~ao, usando as condi\c{c}\~oes de renormaliza\c{c}\~ao os
contra-termos s\~ao determinados de forma a anular as diverg\^encias e
fornecer a f\'{\i}sica da teoria.
\section{Conclus\~ao}
\hspace{.5cm} Atrav\'es de um exemplo simples do c\'alculo dos potenciais
escalar e vetorial de um fio infinito de carga e de corrente,
respectivamente, podemos apresentar as diverg\^encias que sofrem algumas
teorias, os m\'etodos usados para lidar com estas diverg\^encias
(separ\'a-los da parte finita) e o procedimento usado para tornar tais
teorias em teorias f\'{\i}sicas (renormaliza\c{c}\~ao).
|
{
"timestamp": "2005-03-13T17:39:26",
"yymm": "0503",
"arxiv_id": "physics/0503107",
"language": "pt",
"url": "https://arxiv.org/abs/physics/0503107"
}
|
\section{Introduction}
An important part of the theory of vector bundles over homogeneous spaces $G/P$ is
the study of {\em homogeneous} vector bundles. This class of vector bundles has first
been investigated by Kostant and Bott in the 50's, who clarified the relation
between the representation theory of $G$ and homogeneous bundles. This relation
in many cases allows to determine properties of homogeneous vector bundles very
explicitly, and so homogeneous bundles have played a great role in the field of
studying general vector bundles, notably over projective spaces.
In a more general situation, one considers a {\em quasi-homogeneous} space, i.e. a
space $X$ together with the action of an algebraic group $G$ such that this action
has a dense open orbit in $X$. In this context it is customary to speak about {\em
equivariant} rather than homogeneous vector bundles; denote $\sigma, p_2 : G \times
X \longrightarrow X$ the group action and the projection onto the second factor,
respectively, then a vector bundle (or a more general sheaf) \msh{E} on $X$ is {\em
equivariant} if there exists an isomorphism
\begin{equation*}
\Phi: \sigma^* \sh{E} \overset{\cong}{\longrightarrow} p_2^* \sh{E}
\end{equation*}
such that
\begin{equation*}
(\mu \times 1_X)^* \Phi = p_{23}^*\phi \circ (1_G \times \sigma)^* \Phi,
\end{equation*}
where $\mu$ is the group multiplication morphism and $p_{23}$ the projection onto the
second and third factor of $G \times G \times X$ (see also \cite{GIT}).
This situation in general is considerably more difficult than the case of homogeneous
spaces, as (at least) the following two things can happen: in general, $X$ has a
rather complicated
orbit structure, such that there are lower-dimensional invariant loci which allow
equivariant
vector bundles to degenerate to more general equivariant sheaves, if considered in
families in a suitable sense; moreover, the representation theory of $G$ contributes
only marginal information. So the conclusion is that one has to study the
complete category of equivariant sheaves over $X$, which in particular means:
\begin{enumerate}[(i)]
\item construct good invariants for equivariant sheaves over $X$,
\item study moduli spaces with respect to these invariants.
\end{enumerate}
In this work, we attempt to carry out part of such a program for equivariant sheaves
over toric varieties, which are probably the easiest examples of quasi-homogeneous
spaces.
\paragraph{Reflexive Sheaves.}
Our approach is based on the framework of $\Delta$-families which we have
developed in earlier work (\cite{perling1}), which in turn
generalizes the characterization of Klyachko (\cite{Kly90}, \cite{Kly91}) of
equivariant reflexive sheaves. Klyachko's observation was that every such sheaf
\msh{E} is equivalent to a finite dimensional vector space $\mathbf{E}$ together
with a finite set of full filtrations
\begin{equation*}
\cdots \subset E^\rho(i) \subset E^\rho(i + 1) \subset \cdots \subset \mathbf{E}
\end{equation*}
for $i \in \mathbb{Z}$ and every torus invariant divisor $\rho \in {\Delta(1)}$ (see section
\ref{toricvarieties} for notation). Naively, one can seperate two kinds of data from
such a set of filtrations: first, the indices $i$, preferably those where the
dimension of the filtration jumps $E^\rho(i) \subsetneq E^\rho(i + 1)$, and second,
the flags underlying the filtrations, when we forget about the indices. One could
think of the indices as a discrete invariant for \msh{E}, and the flags as {\em
moduli} for the sheaf. However, it turns out that the indices essentially only
determine the first equivariant Chern class of \msh{E}, and the moduli of flags do
not behave very well in sheaf theoretic sense. This has been investigated in detail
in \cite{perling2} for case of equivariant vector bundles of rank two over toric
surfaces.
One could proceed now and declare the equivariant Chern classes as invariants for
equivariant sheaves and construct moduli with respect to these (this has been done in
\cite{perling2}), but we are interested in a more direct approach and want to analyze
the flags underlying the filtrations. These flags and their intersections determine
a subvector space arrangement of $\mathbf{E}$, and as there is no more data left to
describe \msh{E}, one intuitively assumes that all further properties of \msh{E} are
somehow encoded in this arrangement.
Our approach is to construct a global resolution
for any given equivariant sheaf \msh{E} over $X$. From the point of view of
homogeneous coordinate rings (see \cite{Cox}) it has been observed (\cite{ERR}) that
every such sheaf has a finite global resolution
\begin{equation*}
\label{resolutionsequence}
0 \longrightarrow \sh{F}_s \longrightarrow \cdots \longrightarrow \sh{F}_0
\longrightarrow \sh{E} \longrightarrow 0
\end{equation*}
where $\sh{F}_i \cong \bigoplus_j \sh{O}(D_{ij})$ for every $i$. Here, the $D_{ij}$
are torus invariant Weil divisors, and the sheaves $\sh{O}(D_{ij})$ are equivariant
reflexive sheaves of rank one; in the case where $X$ is smooth, these sheaves always
are invertible.
We will give an explicit construction for such resolutions, which for the case of
reflexive sheaves will only depend on the underlying vector space arrangements.
Our results generalize a result of Klyachko, who in \cite{Kly90} constructed a
{\em canonical} resolution in the case where \msh{E} is locally free and $X$ is
smooth and complete.
\paragraph{Vector space arrangements.}
An interesting aspect of our construction is the solution of the following problem;
consider any subvector space arrangement in some vector space $\mathbf{E}$, and its
underlying poset $\mathcal{P}$ which is given by the set of subvector spaces in the
arrangement together with the partial order which is given by inclusion. Then, does
there exist a vector space $\mathbf{F}$ together with a {\em coordinate} vector space
arrangement such that the underlying poset is isomorphic to $\mathcal{P}$? The answer is yes,
and it is rather straightforward to see that one just needs to choose $\mathbf{F}$
large enough, such that the combinatorics of $\mathcal{P}$ can be modelled by coordinate
spaces of $\mathbf{F}$. As a byproduct, we obtain a surjection $\mathbf{F}
\twoheadrightarrow \mathbf{E}$ such that for every element $V \in \mathcal{P}$ and its
corresponding subvector space $F_V$ of $\mathbf{F}$, we have a commutative
exact diagram
\begin{equation*}
\xymatrix{
0 \ar[r] & \mathbf{K} \ar[r] & \mathbf{F} \ar[r] & \mathbf{E} \ar[r] & 0 \\
0 \ar[r] & K_V \ar[r] \ar@{^{(}->}[u] & F_V \ar[r] \ar@{^{(}->}[u] & V \ar[r]
\ar@{^{(}->}[u] & 0.
}
\end{equation*}
The vector spaces $K_V$ again form a vector space arrangement in $\mathbf{K}$ whose
underlying poset is a subset of the original poset $\mathcal{P}$. We call the arrangement
$K_V$ the {\em first syzygy arrangement} of $\mathcal{P}$.
By iterating this procedure, we obtain an exact sequence of vector spaces
\begin{equation*}
0 \longrightarrow \mathbf{F}_s \longrightarrow \cdots \longrightarrow \mathbf{F}_0
\longrightarrow \mathbf{E} \longrightarrow 0
\end{equation*}
where every $\mathbf{F}_i$ contains a coordinate vector space arrangement whose
underlying poset coincides with the poset underlying the $i$-th
syzygy arrangement. In the case where the arrangement in $\mathbf{E}$ is closed
under performing intersections, we have even a good notion of {\em minimal}
resolutions; we obtain a unique representation of such an arrangement in terms of
the purely combinatorial information encoded in the successive coordinate space
arrangements. In a sense, we can think of the resolution as providing a
``K-theory''-class in a suitable category of vector space arrangements. We formulate
the following
\
\noindent
{\bf Conjecture:} Let \msh{E} be a reflexive equivariant sheaf over a toric variety
$X$, then every property of \msh{E} depends only on the indices of the filtrations
$E^\rho(i)$ and the $K$-theory class of the underlying vector space arrangement.
\
\noindent
One can read this conjecture also the way that the ``K-theory''-class of a
vector space arrangement is its finest possible invariant. The class of coordinate
vector space arrangements is a well-studied subject (see \cite{BuchstaberPanov1}),
and it would be interesting to see whether properties of general arrangements can
be studied through free resolutions.
\paragraph{Poset representations.}
The construction of some global resolution for an arbitrary equivariant coherent sheaf
over $X$ is not necessarily a difficult task, but in general the organization of all
the needed data is rather elaborate. Any nuts and bolts approach, starting from
scratch, would probably be rather cumbersome for the reader to follow; therefore we
adopt in
this paper a more formal approach, by developing a certain amount of
framework in the context of poset representations. Such representations, as a subtopic
of quiver representations \cite{Gabriel72}, have been studied since long (see
\cite{Nazarova80}). Any poset $\mathcal{P}$ with a partial order $\leq$ in a natural way is
equivalent to some category. In this category the objects are the elements of $\mathcal{P}$,
and the morphisms are the relations $x \leq y$, i.e. there exists at most one
morphism between two objects $x, y \in \mathcal{P}$. A {\em representation} of $\mathcal{P}$ is a
functor $F: \mathcal{P} \longrightarrow k\operatorname{-\bf Vect}$, $x \mapsto F_x$, the category
of vector spaces over some field $k$. The representations
themselves form an abelian category whose morphisms are the natural transformations.
On a poset $\mathcal{P}$ there exists a natural topology, which is generated by the basis
$U(x) = \{x \leq y \in \mathcal{P}\}$. Using this topology, every representation $F$ of $\mathcal{P}$
induces a sheaf over $\mathcal{P}$ by setting $\sh{F}\big(U(x)\big) := F_x$, and conversely,
any sheaf over $\mathcal{P}$ with values in $k\operatorname{-\bf Vect}$ induces a representation of $\mathcal{P}$. In fact,
the categories of representations of $\mathcal{P}$ and of sheaves over $\mathcal{P}$ with values in
$k\operatorname{-\bf Vect}$ are equivalent. However, it will be more comfortable for us to have both
points of view in mind and to switch the picture freely. The additional bonus of
sheaves over $\mathcal{P}$ is that by the continuation to the whole topology of $\mathcal{P}$, they
automatically incorporate inverse limits via $\sh{F}(U) =
\underset{\leftarrow}{\lim} \sh{F}\big(U(x)\big)$, where the limit runs over all
$x \in U$. For
us, this is a very natural way to encode all possible pullback diagrams over the poset
$\mathcal{P}$. Sheaves over posets have been in the literature before, the first reference
we are aware of being \cite{Baclawski}. More recently, this kind of sheaves has been
used in similar contexts like ours, for the study of certain modules over semigroup
rings \cite{Yanagawa01}, and for vector space arrangements
\cite{DeligneGoreskyMacPherson}.
\paragraph{Posets and graded modules.}
Our general principle will be to start with local constructions and to globalize these
by some gluing procedure, where 'local' and 'global' means over affine and general
toric varieties, respectively. Recall that an affine toric variety $U_\sigma$ over
some algebraically
closed field $k$ on which the torus $T$ acts, is equivalent to the spectrum of a
normal semigroup ring $k[\sigma_M]$. The semigroup $\sigma_M$ is a subsemigroup of
the character group $M \cong \mathbb{Z}^r$ of $T$, which is given by the intersection of a
convex rational polyhedral cone $\check{\sigma}$ in $M \otimes_\mathbb{Z} \mathbb{R}$ with $M$.
Any equivariant sheaf \msh{E} over an affine toric variety $U_\sigma$ is equivalent
to an $M$-graded $k[\sigma_M]$-module $E^\sigma = \Gamma(U_\sigma, \sh{E})$, i.e.
\begin{equation*}
E^\sigma = \bigoplus_{m \in M} E^\sigma_m.
\end{equation*}
A fundamental observation is that this grading is the reason that equivariant sheaves
over toric varieties have still a {\em semi-combinatorial} nature, in contrast to the
completely combinatorial description of the toric varieties themselves. To see this,
note that $\sigma_M$ endows $M$ with the structure of a poset by setting $m
\leq_\sigma m'$ iff $m' - m \in \sigma_m$ (we simplify here, as in fact this in
general only defines a preorder). This way, $E^\sigma$ is equivalent to a
representation of $M$ which maps every $m$ to the vector space $E^\sigma_m$, and every
relation $m \leq_\sigma m'$ is mapped to the vector space homomorphism $E^\sigma_m
\longrightarrow E^\sigma_{m'}$, which is given by multiplication with the monomial
$\chi(m' - m)$. It turns out that the category of representations of the {\em poset}
$M$ is equivalent to the category of equivariant quasicoherent sheaves over
$U_\sigma$.
To be able to work truly with a finitely generated module, one needs an expedient
finite representation for it. For this, we introduce the notion of a {\em polyhedral
decomposition} of $M$. For any $\rho \in \sigma(1)$ and any integer $n_\rho$ we have
the shifted halfspace $\{m \in M_\mathbb{R} \mid \langle m, n(\rho) \rangle \geq n_\rho\}$ in
$M_\mathbb{R}$, and
for any tuple ${\underline{n}} = \big(n_\rho \mid \rho \in \sigma(1)\big) \in \mathbb{Z}^{\sigma(1)}$ the
intersection of half spaces $P_{\underline{n}} = \{m \mid \langle m, n(\rho) \rangle \geq n_\rho
\text{ for all } \rho \in \sigma(1)\}$. We call such an unbounded domain $P_{\underline{n}}$ a
{\em polyhedron}. Note that the dual cone $\hat{\sigma}$ itself is a polyhedron which
has the zero face as its unique compact face. Figure \ref{f-introexample2} shows an
example of a cone where $\sigma(1)$ consists of four rays, and a polyhedron defined
with respect to these four rays.
\begin{figure}[htb]
\begin{center}
\includegraphics[height=4cm]{introexample2b.eps}\qquad\qquad
\includegraphics[height=4cm]{introexample2.eps}
\end{center}
\caption{Example of a cone with four maximal faces and a polyhedron}\label{f-introexample2}
\end{figure}
The intersection of two polyhedra $P_{{\underline{n}}_1}, P_{{\underline{n}}_2}$ is again a polyhedron, $P_{\underline{n}}$,
where ${\underline{n}} = (\max\{(n_{1, \rho}, n_{2, \rho}\} \mid \rho \in \sigma(1))$. This way,
any collection of polyhedra $P_{{\underline{n}}_1}, \dots, P_{{\underline{n}}_s}$ gives rise to a partition
of $M$ as follows. Define the 'least common multiple' ${\underline{n}}$ of any collection
${\underline{n}}_{i_1}, \dots, {\underline{n}}_{i_k}$, by the componentwise maximum of the ${\underline{n}}_{i_j}$. Then the
equivalence classes $T_{\underline{n}}$ contain all $m \in M$ with $\langle m, n(\rho) \rangle \geq
n_{i, \rho}$ for all $\rho \in \sigma(1)$, for which there is no bigger least common
multiple ${\underline{n}}'$ satisfying these inequalities. Figure \ref{f-introexample1} shows a
partition of $\mathbb{Z}^2$ generated by three polyhedra.
\begin{figure}[htb]
\begin{center}
\includegraphics[height=4cm,width=4cm]{introexample1.eps}
\end{center}
\caption{A polyhedral decomposition of $\mathbb{Z}^2$ into seven regions}\label{f-introexample1}
\end{figure}
The set of $\operatorname{lcm}$'s of the ${\underline{n}}_1, \dots, {\underline{n}}_s$ in a natural way becomes a poset, as a
subposet of $\mathbb{Z}^{\sigma(1}$ with partial order induced by the componentwise order. The
$\operatorname{lcm}$'s are a special case of a polyhedral decomposition which is induced by an
{\em admissible poset}. A finite poset $\mathcal{P}^\sigma \subset \mathbb{Z}^{\sigma(1)}$ is
admissible if for any $m \in M$ there exists a unique maximal element ${\underline{n}} \in
\mathcal{P}^\sigma$ such that $\langle m, n(\rho) \rangle \geq n_\rho$ for all $\rho \in
\sigma(1)$. $\mathcal{P}^\sigma$ is admissible {\em with respect to $E^\sigma$} if moreover
for every ${\underline{n}} \in \mathcal{P}^\sigma$ there exist a vector space $E_{\underline{n}}$ such that
$E_{\underline{n}} \cong E_m^\sigma$ for all $m \in T_{\underline{n}}$. The vector spaces $E_{\underline{n}}$ together with
appropriate morphisms $E_{\underline{n}} \longrightarrow E_{{\underline{n}}'}$ (whose existence is part of our
definition \ref{admissibledef} for admissible posets),
yield a representation of $\mathcal{P}^\sigma$, which encodes the
complete structure of $E^\sigma$. We can think of it, euphemistically, as a
compression of $E^\sigma$.
The most important feature of our constructions is that the compression of $E^\sigma$
is functorial, because we systematically exploit the
formalism of sheaves on posets; we finally arrive at an equivalence of categories
between sheaves over $\mathcal{P}^\sigma$ and $k[\sigma_M]$-modules with respect to which
$\mathcal{P}^\sigma$ is admissible (theorem \ref{admissibleequivalence}).
This in particular enables us to construct resolutions of $E^\sigma$ in terms of
free resolutions of the sheaf $E_{\underline{n}}$ over $\mathcal{P}^\sigma$. The resolutions obtained this
way are not free resolutions, but rather resolutions by reflexive modules of rank one.
Any ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$ gives rise to a $T$-invariant Weil divisor $D_{\underline{n}} = -
\sum_{\rho \in \sigma(1)} n_\rho D_\rho$ on $U_\sigma$, and thus to a reflexive sheaf
of rank one $\sh{O}_{U_\sigma}(D_{\underline{n}})$. Write $S_{({\underline{n}})}$ for the associated reflexive
$k[\sigma_M]$-module, then its $M$-graded decomposition is given by
\begin{equation*}
S_{(n)} \cong \bigoplus_{m \in P_{\underline{n}} \cap M} k \cdot \chi(m).
\end{equation*}
Every equivalence class $T_{\underline{n}}$ has the shape of the forepart of the polyhedron $P_{\underline{n}}$,
and thus provides a 'slot' by which we can define a map $S_{({\underline{n}})} \rightarrow
E^\sigma_m$ without missing any $M$-degree in $T_{\underline{n}}$. This leads to a somewhat
different philosophy of resolutions than the usual one --- instead of a generating set
of $E^\sigma$ as basic input for our resolutions, we use a polyhedral decomposition.
This at least leads to finite resolutions and reduces in many cases the problem to
understanding the modules $S_{({\underline{n}})}$ (see theorem \ref{CMresolution} for such an
application). We want to remark that our notions of admissible posets and polyhedral
decompositions are very close, though not entirely identical, to the sector
partitions in \cite{helmmiller}.
\paragraph{Gluing of posets and globalization.}
A sheaf \msh{E} is equivalent to a collection of $k[\sigma_M]$-modules $E^\sigma$,
where $\sigma$ runs over the fan associated to $X$, which glues in an appropriate
sense over the $U_\sigma$. On the other hand, \msh{E} can be represented by a
collection of sheaves over some admissible posets $\mathcal{P}^\sigma$, which we have to glue
--- in an appropriate sense. The problem of gluing posets might be interesting in a
somewhat broader mathematical context, so that we decided to define it slightly
more general than necessary.
We remark that the naive idea of gluing posets like topological spaces,
which of course can be done, probably does not lead to anything interesting. For
instance, one can easily show that a topological space which is covered by two open
sets, each of which is homeomorphic to a poset, can globally be given the structure of
a poset. By induction, one concludes that every set which has a finite cover by
posets is a poset again.
Our notion of gluing is different from this, and indeed it is a derived
concept which comes very naturally from toric geometry, suitable for us to
construct global resolutions.
Our idea is to realize gluing by passing from posets to {\em preordered} sets.
In contrast to our statements above, a semigroup $\sigma_M$ in general induces only
a preorder on $M$, rather than a partial order. For any two $m, m' \in M$ we have
$m \leq_\sigma m'$ and $m' \leq_\sigma m$ iff $m - m' \in \sigma_M^\bot$, the
maximal subgroup of $\sigma_M$. We can turn $\leq_\sigma$ into a proper partial order
if we pass to the induced order on the quotient $M / \sigma_M^\bot$. For any
$M$-graded module $E$ and any pair $m, m'$ with $m \leq_\sigma m'$ and
$m' \leq_\sigma m$, the multiplication homomorphisms by $\chi(m' - m)$ and
$\chi(m - m')$ necessarily are isomorphisms, and in fact, the categories of
$M$-graded $k[\sigma_M]$-modules and of $M / \sigma_M^\bot$-graded
$k[\sigma_M / \sigma_M^\bot]$-modules are equivalent. Now for simplicity assume that
$\leq_\sigma$ is a partial order and let $\tau < \sigma$ be a proper face, such that
$\leq_\tau$ is a proper preorder. $\tau_M$ is of the form $\sigma_M + \mathbb{Z}_{\geq 0}
\cdot (-m_\tau)$ for some $m_\tau \in \sigma_M$ such that $\tau^\bot \cap
\check{\sigma}$ is a proper face of $\check{\sigma}$ and $\tau^\bot_M =
(\sigma_M \cap \tau_M^\bot) + \mathbb{Z}_{\geq 0} \cdot (-m_\tau)$ is a nontrivial subgroup.
The set $\tau_M^\bot \cap \sigma_M$ is a subsemigroup of $\tau_M^\bot$, giving rise
to a partial order on $\tau_M^\bot$. For any $m \in M$, we can think of the affine
subset $m + \tau_M^\bot$ as a {\em slice} in $M$, and every such slice has its own
partial order. With respect to such a slice, we can consider the directed system
$E^\sigma_{m'}$ with $m' \in m + \tau_M^\bot$, and the directed limit of this system:
\begin{equation*}
E^\tau_m := \underset{\rightarrow}{\lim} E^\sigma_{m'}, \qquad m' \in m + \tau^\bot_M.
\end{equation*}
It turns out that $\bigoplus_{m \in M} E^\tau_m \cong E^\sigma_{\chi(m_\tau)}$, i.e.
the localization of
$E^\sigma$ by the character $\chi(m_\tau)$, which we now can interpret as some kind
of limit figure of $E^\sigma$ along the direction $m_\tau$.
This example is our prototype for defining gluing of partially ordered sets and
sheaves over them. Let $\mathcal{P}$ be an abstract poset with some partial order $\leq$. Then
a {\em localization} of $\leq$ is a preorder $\leq'$, such that $x \leq y$ implies
$x \leq' y$, and $x \leq' y$ implies $x \leq w$ for some element $w$ with $w
\leq' y$ and $y \leq' w$. This is the abstract analogoue of the slicing above, where
the preorder $\leq'$ groups together certain subsets of $\mathcal{P}$. By this definition,
if we pass to the quotient $\mathcal{P} / \sim$, where $x \sim y$ iff $x \leq' y$ and $y
\leq' x$, every representation $F$ of $\mathcal{P}$ induces a representation $\bar{F}$ of
$\mathcal{P} / \sim$, where
\begin{equation*}
\bar{F}_{[x]} = \underset{\rightarrow}{\lim} F_y,
\end{equation*}
the limit is taken over all $y \in x$ {\em with respect to the partial order
$\leq$}. This way, we obtain a quite canonical procedure for gluing sheaves $F_1,
F_2$ over two posets $\mathcal{P}_1, \mathcal{P}_2$; we simply require that the posets have
localizations $\leq'_1, \leq'_2$ such that there exists isomorphisms $l : \mathcal{P}_1 / \sim
\longrightarrow \mathcal{P}_2 / \sim$ and $\phi : l^*\bar{F}_2 \longrightarrow \bar{F}_1$.
We refer to subsections \ref{posetgluing} and \ref{sheafgluing} for the precise
definitions.
\paragraph{Overview of the paper.}
This paper tries to be self-contained in the sense that all required notation related
to toric geometry are introduced. However, we refrain from
giving any account on these subjects; we refer to \cite{perling1} for a more details.
The paper consists of four principal parts. In section \ref{posheaves}, we present
our principal technical framework from the theory of poset representations; in
addition to well-known material, in this section gluing of posets and of sheaves
over posets are introduced.
In section \ref{toricvarieties}, we recall general notions from toric geometry and
the formalism of $\Delta$-families as developed in \cite{perling1}. We present a
partial reformulation of the material in view of the formalism of section
\ref{posheaves}. We show that the Krull-Schmidt theorem holds in the category of
equivariant coherent sheaves over {\em any} toric variety.
Section \ref{resolutions} contains the biggest part of the work;
starting from polynomial rings (subsection \ref{polynomialrings}), and then
generalizing to normal semigroup rings (subsection \ref{semigrouprings}), we construct
resolutions for finitely generated modules over affine toric varieties. In subsections
\ref{homext} and \ref{deltaglobres} we construct global resolutions, both from the
point of view of gluing over posets, and homogeneous coordinate rings.
In section \ref{reflexivesheaves} we analyze the special case of reflexive modules,
and in particular we amplify their close relationship to vector space arrangements.
As an application, in subsection \ref{cmmodules} we show that our resolutions in the
case of reflexive modules behave well in sense of homological algebra. In subsection
\ref{reflexivemodels} we discuss how resolutions of vector space arrangements can
effectively be computed in terms of associated modules over polynomial rings.
\
This work extends results of my thesis \cite{perlingdiss}. Most of this paper
has been written during my stay at the Abdus Salam ICTP, Trieste for whose
hospitality I am deeply grateful.
\section{Preliminaries on Preordered Sets}
\label{posheaves}
In this section let $\mathcal{P}$ be a countable set on which a preorder
$\leq$ is defined. Recall that a preorder is defined by the same axioms as a partial
order, except for the reflexivity axiom, i.e. there may exist elements $x, y \in \mathcal{P}$
such that $x \leq y$ and $y \leq x$, but $x \neq y$. For such pairs we write $x
\lessgtr y$; in the sequel we will frequently put indices on the symbol $\leq$, such
as $\leq', \leq_\sigma$, etc.; then these indices also apply to $\lessgtr$.
If there is no ambiguity in the preorder chosen, we just write $\mathcal{P}$, else we write
$(\mathcal{P}, \leq)$.
\subsection{Representations of preordered sets}
Any preordered set $\mathcal{P}$ in a natural way forms
a category; its objects are given by the set underlying
$\mathcal{P}$ and the morphisms for $x, y \in \operatorname{Ob}(\mathcal{P})$ are:
\begin{equation*}
\operatorname{Mor}(x, y) =
\begin{cases}
\text{the pair } (x, y) & \text{ if } x \leq y \\
\emptyset & \text{ else.}
\end{cases}
\end{equation*}
Here the pair $(x, x)$ represents the identity morphism for all $x \in \mathcal{P}$.
\begin{definition}
A functor from $\mathcal{P}$ to $k\operatorname{-\bf Vect}$, the category of vector spaces over the
field $k$, is called a {\em $k$-linear representation} of $\mathcal{P}$.
\end{definition}
As a general notation, if $E$ denotes a $k$-linear representation of a preordered set
$\mathcal{P}$, an element $x \in \mathcal{P}$ is mapped to the vector space denoted $E_x$, and the
relation $x \leq y$ is mapped to a vector space homomorphism $E(x, y)$.
The $k$-linear representations of $\mathcal{P}$ form an abelian category whose
morphisms are natural transformations.
Representations of $\mathcal{P}$ are equivalent to {\em sheaves} over $\mathcal{P}$. On $\mathcal{P}$ there is
defined a topology which is generated by the basis
\begin{equation*}
U(x) := \{y \geq x\}
\end{equation*}
for all $x \in \mathcal{P}$. The continuous maps between posets then are precisely the
order preserving maps.
A sheaf \msh{E} on $\mathcal{P}$ with respect to this topology with values in $k\operatorname{-\bf Vect}$
automatically induces a representation of $\mathcal{P}$. On the other hand, for any
representation $E$, following \cite{EGAI} \S 0.3.2,
we obtain a presheaf \msh{E} on $\mathcal{P}$ by setting $\sh{E}\big((U(x)\big)
:= E_x$ for all $x \in \mathcal{P}$ and $\sh{E}(U) := \underset{\leftarrow}{\lim}
\ \sh{E}\big(U(x)\big)$ for some open set $U$, where
the limit runs over all $x \in U$. Note that the stalk $\sh{E}_x$ is isomorphic
to $\sh{E}\big(U(x)\big)$. By observing that for some $U(x)$ every open cover
necessarily contains $U(x)$, and applying the criterion of \S 0.3.2.2 in \cite{EGAI},
it follows that every presheaf automatically is a sheaf.
A distinguished class of representations are the representations $F^x$ associated to
some element $x \in \mathcal{P}$, which are given by:
\begin{equation*}
y \mapsto
\begin{cases}
k & \text{ if } x \leq y \\
0 & \text{ else},
\end{cases}
\end{equation*}
and relations $y \leq z$ mapped to identity if $x \leq y$, and to the zero map
else. In terms of sheaves over $\mathcal{P}$, one can alternatively define $F^x$ as follows.
Denote $j_x$ the canonical inclusion $U(x) \hookrightarrow \mathcal{P}$, and let $\mathbf{k}$
be the constant sheaf with $\mathbf{k}(U(y)) = k$ for all $y \in \mathcal{P}$.
Then $F^x$ corresponds to $j_{x!}j_x^* \mathbf{k}$. We say that a representation
of $\mathcal{P}$ is {\em free} if it is isomorphic to a direct sum of objects of the form
$F^x$. We have:
\begin{proposition}
The representations $F^x$ are projective objects in the category of $k$-linear
representations of $\mathcal{P}$.
\end{proposition}
Using the notion of free objects, we can introduce {\em free resolutions}.
\begin{definition}
Let $\mathcal{P}$ be any preordered set and $E$ a $k$-linear representation.
Then a {\em free resolution} of $E$ is an exact sequence
\begin{equation*}
\dots \longrightarrow F_i \longrightarrow \dots \longrightarrow F_0 \longrightarrow
E \longrightarrow 0
\end{equation*}
where for every $i$ : $F_i \cong \bigoplus_{j} F^{x_{ij}}$ for some $x_{ij} \in \mathcal{P}$.
\end{definition}
Let $x \in \mathcal{P}$, then we consider the subvector space of $E_x$ which is generated
by the image of all $E_y$, $y < x$, by the morphisms $E(x, y)$,
$E_{<x} := \sum_{y < x} E(y, x) E_y$, where we set $E_{<x} := 0$ if the
set $\{y < x\}$ is empty. $\operatorname{codim}_{E_x} E_{< x}$ is
the {\em free dimension} of $E_x$.
\begin{proposition}
\label{repres}
Let $\mathcal{P}$ be a finite preordered set. Then for every $k$-linear representation of
$\mathcal{P}$ there exists a {\em finite} free resolution, that
is, there exists a free resolution as above and some $n \geq 0$ such that $F_i = 0$
for all $i > n$.
\end{proposition}
\begin{proof}
Let $\mathcal{X} \subset \mathcal{P}$ be the set of elements such that $E_x$ has positive free
dimension. For every $x \in \mathcal{X}$ we consider the short exact
sequence of vector spaces
\begin{equation*}
\xymatrix{
0 \ar[r] & E_{<x} \ar[r] & E_x \ar[r] & E_x / E_{<x} \ar[r]
\ar@/^1pc/@{.>}[l]^{\mu_x} & 0,
}
\end{equation*}
where we have chosen some section $\mu_x$. For every such $x$, we can consider
the constant sheaf $E_x / E_{<x}$ on $\mathcal{P}$ and its restriction $E^x := j_{x!}j^*_x (
E_x / E_{<x})$. Using the section $\mu_x$, there exists a natural homomorphism
$\phi_x : E^x \longrightarrow E$ by setting $\phi_x = E(x, y) \circ \mu_x :
E^x_y \longrightarrow E_y$
for every pair $x \leq y$ and the zero map for all $x \nleq y$.
The sheaf $E^x$ is isomorphic to $(F^x)^{f_x}$, where $f_x$ is the free dimension of
$E_x$. Thus we define $F_0 = \bigoplus_{x \in \mathcal{X}} E^x \cong
\bigoplus_{x \in \mathcal{X}} (F^x)^{f_x}$ and a homomorphism
$\phi_0: F_0 \longrightarrow E$ by setting $\phi_0 := \sum_{x \in \mathcal{X}}
\phi_x$. By construction, $\phi_0$ is a surjective map, and we obtain thus
a short exact sequence of representations of $\mathcal{P}$:
\begin{equation*}
0 \longrightarrow K_0 \longrightarrow F_0 \overset{\phi_0}{\longrightarrow} E
\longrightarrow 0.
\end{equation*}
Now we can repeat this construction with $K_0$, and by iterating we obtain a free
resolution of $E$ which is concatenated of short exact sequences $0 \longrightarrow
K_{i + 1} \longrightarrow F_{i + 1} .\overset{\phi_{i + 1}}{\longrightarrow} K_i
\longrightarrow 0$.
Now observe that for $K_{i + 1, x} = 0$ whenever the free dimension of $K_{i, x}$ is
equal to $\dim K_{i, x}$,
and $K_{i, x} = 0$ implies that $K_{i + 1, x} = 0$. The set of such
$K_{i, x}$ whose free dimension is equal to $\dim K_{i, x}$ is always nonempty as long
as $K_i$ is nontrivial, because the set contains at least the minimal elements
$x \in \mathcal{P}$ which have nontrivial $K_{i, x}$. So, as $\mathcal{P}$ is finite, it follows that
there exists some $r > 0$ for which $K_{i, x} = 0$ for all $i > r$.
\end{proof}
\begin{definition}
Let $0 \longrightarrow F_r \overset{\phi_r}{\longrightarrow} \dots,
\overset{\phi_1}{\longrightarrow} F_0 \overset{\phi_0}{\longrightarrow} E
\longrightarrow 0$ be a free resolution of a $k$-linear representation $E$, then we
call the kernel of $\phi_i$ the {\em $i$th syzygy representation} of $E$.
\end{definition}
\subsection{Direct and inverse limits}
Now we recall some basic facts about direct and inverse limits in the
category of vector spaces. This is only intended as a reminder to the reader,
as we will be using limits extensively during the rest of this paper.
As we have seen in the previous subsection, every preordered set $\mathcal{P}$ in a natural
way is a directed family. Thus, a representation $E$ of $\mathcal{P}$ becomes a directed
family of vector spaces. Recall, that the {\em inverse limit} of $E$ is a vector
space
\begin{equation*}
\underset{\leftarrow}{\lim} E =: \mathbf{E}^i
\end{equation*}
which has the following universal properties:
\begin{enumerate}[(i)]
\item for every element $x \in \mathcal{P}$ there exists a unique homomorphism $\phi_x :
\mathbf{E}^i \longrightarrow E_x$ such that $E(x, y) \circ \phi_x = \phi_y$ for
every $x \leq y$;
\item for every vector space $\mathbf{F}$ with homomorphisms $\psi_x: \mathbf{F}
\longrightarrow E_x$, where $\psi_y = E(x, y) \circ \psi_x$ for every $x \leq y$,
there exists a unique homomorphism $\delta: \mathbf{F} \longrightarrow \mathbf{E}^i$
with $\psi_x = \phi_x \circ \delta$ for all $x \in \mathcal{P}$.
\end{enumerate}
\begin{definition}
We denote the vector space homomorphism $\delta: \mathbf{F} \longrightarrow
\mathbf{E}^i$ {\em diagonal homomorphism} from $\mathbf{F}$ to $\mathbf{E}^i$.
\end{definition}
Explicitly, such a limit can be constructed as the subvector space of the direct
product $\prod_{x \in \mathcal{P}} E_x$ consisting of sequences $(e_x \mid x \in \mathcal{P})$ such
that $E(x, y)(e_x) = e_y$ for every pair $x \leq y$. If $\mathcal{P}$ has a unique minimal
element $x_{\min}$, then $\phi_{x_{\min}} : \mathbf{E}^i \rightarrow E_{x_{\min}}$
becomes an isomorphism. This construction is a straightforward
generalization of the
pullback in the category of vector spaces; the pullback is the special case where
the poset consists of three elements $x, y, z$ with $x < z$ and $y < z$.
Dually, there exists the {\em direct limit}
\begin{equation*}
\underset{\rightarrow}{\lim} E =: \mathbf{E}^d
\end{equation*}
which generalizes pushout. It can explicitly be constructed as the quotient of
the vector space $\prod_{x \in \mathcal{P}} E_x$ by the subvector space generated by vectors
$E(x, y)(e_x) - e_x$. For every $x \in \mathcal{P}$ there exists a homomorphism $\phi^x :
E_x \longrightarrow \mathbf{E}^d$ such that universal properties analogously to the
inverse limit are fulfilled. Note that in case that there exists a unique maximal
element $x_{\max}$, the homomorphism $\phi^{x_{\max}}$ is an isomorphism.
Both limits behave covariantly; consider two preordered sets $\mathcal{P}$, $\mathcal{Q}$,
and any two representations $E$, $F$ of $\mathcal{P}$ and $\mathcal{Q}$, respectively, and
a order preserving map $f : \mathcal{P} \longrightarrow \mathcal{Q}$. Then any natural
transformation $r : E \longrightarrow f^* F$ induces a homomorphism of limits:
\begin{equation*}
\underset{\leftarrow}{\lim}\ r : \underset{\leftarrow}{\lim}\ E \longrightarrow
\underset{\leftarrow}{\lim}\ f^*F \text{\quad resp. \quad }
\underset{\rightarrow}{\lim} \ r : \underset{\rightarrow}{\lim}\ E \longrightarrow
\underset{\rightarrow}{\lim}\ f^*F.
\end{equation*}
In particular, if $\mathcal{P}$ and $\mathcal{Q}$ have unique minimal elements $x_{\min}$ and
$y_{\min}$, respectively, and $f(x_{\min}) = y_{\min}$, we obtain
\begin{equation*}
\underset{\leftarrow}{\lim}\ r : \underset{\leftarrow}{\lim}\ E \longrightarrow
\underset{\leftarrow}{\lim}\ F,
\end{equation*}
and analogously for the direct limit with respect to maximal elements.
\subsection{Gluing of preordered sets}
\label{posetgluing}
\begin{definition}
Let $(\mathcal{P}, \leq)$ be a preordered set. Then we denote $\mathcal{P}_\lessgtr$ the quotient
of $\mathcal{P}$ by the equivalence relation which is given by $x \sim y$ iff $x \lessgtr y$.
\end{definition}
Clearly, $\leq$ induces a partial order on the set $\mathcal{P}_\lessgtr$.
\begin{definition}
A {\em localization} of $\mathcal{P}$ is a preorder $\leq'$ on $\mathcal{P}$ such that the
following conditions are fulfilled:
\begin{enumerate}[(i)]
\item for all $x, y \in \mathcal{P}$, $x \leq y$ implies $x \leq' y$,
\item for all $x \leq' y$ there exists some $w \lessgtr' y$ such that
$x \leq w$.
\end{enumerate}
\end{definition}
Let $\leq'$ be a localization of $(\mathcal{P}, \leq)$, then $\leq$ induces a relation on
$\mathcal{P}_{\lessgtr'}$ by setting $[x] \leq [y]$ iff there exist $u \in [x]$, $v \in [y]$
with $u \leq v$.
\begin{proposition}
Let $\leq'$ a localization of $(\mathcal{P}, \leq)$, then the relation on $\mathcal{P}_{\lessgtr'}$
induced by $\leq$ coincides with the partial order induced by $\leq'$.
\end{proposition}
\begin{proof}
We check the poset axioms for $\leq$: {\em 1)} $[x] \leq [x]$ follows because
$u \leq' u$ implies that there exists $v \lessgtr' u$ such that $u \leq v$. {\em 2)}
Let $[x] \leq [y]$ and $[y] \leq [x]$; then there exist $u, p \in [x]$, $v, q \in
[y]$ such that $u \leq v$ and $p \leq q$; then $u \leq'v \leq' p \leq' u$, and thus
$v \lessgtr' u$, hence $[x] = [y]$. {\em 3)} Let $[x] \leq [y]$ and $[y] \leq [z]$;
then there exist $u \in [x]$, $v, p \in [y]$ and $q \in [z]$ such that $u \leq v$ and
$p \leq q$; thus $u \leq' v \leq' p \leq' q$ and there exists $w \lessgtr'q$ such
that $u \leq w$, and thus $[x] \leq [z]$.
Now the equivalence of the partial orders $\leq$ and $\leq'$ on $\mathcal{P}_{\lessgtr'}$ is
trival.
\end{proof}
Let $(\mathcal{P}_\alpha, \leq_\alpha)$ be a finite family of preordered sets, together
with a family of representations $F_\alpha$, where $\alpha$ runs over some index set
$A$. Our aim is to {\em glue} these representations when certain conditions on the
$\mathcal{P}_\alpha$ are fulfilled. For this, we need the following notion:
\begin{definition}
Let $f : \mathcal{P} \longrightarrow \mathcal{Q}$ be an order preserving
map between preordered sets. Then $f$ is a {\em contraction} if
\begin{enumerate}[(i)]
\item for every $x \in \mathcal{P}$ there exists some $y \in \mathcal{Q}$ such that
$f\big(U(x)\big) = U(y)$,
\item for every $y \in \mathcal{Q}$: $f^{-1}\big(U(y)\big) = U(x)$ for some $x \in
\mathcal{P}$.
\end{enumerate}
\end{definition}
These conditions imply that $f$ is surjective and that for every $x \in
\mathcal{P}$ with $f\big(U(x)\big) = U(y)$ there exists $z \in \mathcal{P}$ with $U(x) \subset U(z) =
f^{-1}\big(U(y)\big)$. By this we can define a map $h : \mathcal{Q} \longrightarrow
\mathcal{P}$ by mapping $y \mapsto z$. This map is an order preserving injection of
$\mathcal{Q}$ into $\mathcal{P}$.
\begin{definition}
Let $f : \mathcal{P} \longrightarrow \mathcal{Q}$ be a contraction. Then the unique map $h :
\mathcal{Q} \longrightarrow \mathcal{P}$ mapping $y \in \mathcal{Q}$ to $z \in \mathcal{P}$ such that
$f^{-1}(U(y)) = U(z)$ is called {\em hooking} of $\mathcal{Q}$ into $\mathcal{P}$.
\end{definition}
Using this definition, one can think of our gluing of posets as a process of
{\em hooking} different posets along common contractions.
Let $\mathcal{P}, \mathcal{Q}$ be two finite preordered sets, $f : \mathcal{P} \longrightarrow
\mathcal{Q}$ a contraction, and $E$ a representation of $\mathcal{Q}$. Then the
pullback $f^* F^x$ for any free representation for some $x \in \mathcal{Q}$ then
is isomorphic to the free representation $F^{h(x)}$ of $\mathcal{P}$. For any free
resolution $0 \rightarrow F_r \rightarrow \cdots \rightarrow F_0 \rightarrow E
\rightarrow 0$, one can consider the pullback sequence $0 \rightarrow f^*F_r
\rightarrow \cdots \rightarrow f^*F_0 \rightarrow f^*E \rightarrow 0$. We observe:
\begin{lemma}
\label{contractionliftres}
The sequence $0 \rightarrow f^*F_r \rightarrow \cdots \rightarrow f^*F_0 \rightarrow
f^*E \rightarrow 0$ is isomorphic to the free resolution of $f^*E$ in the sense of
proposition \ref{repres}.
\end{lemma}
\begin{proof}
It suffices to check the first step of the resolution $0 \rightarrow K_0 \rightarrow
f^* F_0 \rightarrow f^* E \rightarrow 0$ and to show that $K_0$ and $f^* F_0$
coincide with the representations obtained by the procedure of proposition
\ref{repres}. But this follows directly from the fact that for every $y \in
\mathcal{Q}$, the homomorphisms $(f^*E)_{h(x)} \rightarrow (f^*E)_y$ are isomorphisms
for all $h(x) \leq y \in f^{-1}(x)$.
\end{proof}
\begin{definition}
Let $(A, \preceq)$ be a finite poset and $\mathcal{P}_\alpha$, $\alpha \in A$ be a
family of preordered sets. We say that the posets $\mathcal{P}_\alpha$ {\em glue over $A$} if
\begin{enumerate}[(i)]
\item for every $\beta < \alpha \in A$ there exists a localization
$\leq_\alpha^\beta$ of $\leq_\alpha$ and a contraction
$l_{\alpha\beta}: (\mathcal{P}_\alpha)_{\lessgtr_\alpha^\beta} \rightarrow
(\mathcal{P}_\beta)_{\lessgtr_\beta}$;
\item for every triple $\gamma
\preceq \beta \preceq \alpha \in A$, the composition of maps $\mathcal{P}_\alpha
\rightarrow (\mathcal{P}_\alpha)_{\lessgtr^\beta_\alpha}
\overset{l_{\alpha\beta}}{\rightarrow} (\mathcal{P}_\beta)_{\lessgtr_\beta}
\rightarrow (\mathcal{P}_\beta)_{\lessgtr_\beta^\gamma}\overset{l_{\beta\gamma}}{\rightarrow}
(\mathcal{P}_\gamma)_{\lessgtr_\gamma}$ coincides with $\mathcal{P}_\alpha \rightarrow
(\mathcal{P}_\alpha)_{\lessgtr^\gamma_\alpha}
\overset{l_{\alpha\gamma}}{\rightarrow} (\mathcal{P}_\gamma)_{\lessgtr_\gamma}$.
\end{enumerate}
\end{definition}
Our principal example, where the maps $l_{\alpha\beta}$ actually are isomorphisms,
will be the preorderings associated to a fan $\Delta$ in section
\ref{toricvarieties}.
\subsection{Gluing of sheaves over preordered sets}
\label{sheafgluing}
Let $(\mathcal{P}, \leq)$ be some preordered set; if $E$ is some representation of $\mathcal{P}$, then
for any pair $x \lessgtr y$, the map $E(x,y) : E_x \longrightarrow E_y$ is an
isomorphism whose inverse is $E(y,x)$. Thus $E$ descends to a representation of
$\mathcal{P}_\lessgtr$ by setting $E_{[x]} := \underset{\rightarrow}{\lim} E_y$, where
the direct limit is taken over all elements $y \leq x$.
For any $y \leq x$ there is the canonical inclusion of directed systems $\{E_z \mid
z \leq y\} \hookrightarrow \{E_z \mid z \leq x\}$, which induces a functorial
homomorphism
$E_{[y]} \longrightarrow E_{[x]}$. On the other hand, every representation $F$ of
$\mathcal{P}_\lessgtr$ lifts to a representation of $\mathcal{P}$ by setting $E_x := E_{[x]}$ and
$E(x,y) := E([x],[y])$. By descend and lift, we have:
\begin{lemma}
\label{popreequivrep}
Let $(\mathcal{P}, \leq)$ be a preordered set. The category of representations of $\mathcal{P}$ is
equivalent to the category of representations of $\mathcal{P}_\lessgtr$.
\end{lemma}
\comment{
Let $(\mathcal{P}, \leq_1)$ be a preorderet set and let $\leq_2$ be another preorder on
$\mathcal{P}$ such that $x \leq_1 y$ implies $x \leq_2 y$ for all $x, y \in \mathcal{P}$. Denote
$\sim_i$ the equivalence relation with respect to $\leq_i$, $i = 1, 2$; the
preorder $\leq_2$ induces a preorder on $\mathcal{P} / \sim_1$, and forming equivalence
classes of $\mathcal{P} / \sim_1$ with respect to the induced preorder coincides with
$\mathcal{P} / \sim_2$.
}
Let $\leq'$ be a localization of $\leq$. For any $x \in \mathcal{P}$, denote
$\mathcal{P}_x = \{z \in \mathcal{P} \mid z \leq' x\}$. We construct a representation on
$\mathcal{P}_{\lessgtr'}$ by mapping $[x]' \in (\mathcal{P})_{\lessgtr'}$ to the vector space $E_{[x]'}
:= \underset{\rightarrow}{\lim} E_z$, the direct limit taken over
$\mathcal{P}_x$ {\em with respect to the partial order $\leq$}.
The inclusion $\mathcal{P}_x \hookrightarrow \mathcal{P}_y$ induces an inclusion of directed sets
with respect to $\leq$, and thus we obtain a morphism $E_{[x]'} \longrightarrow
E_{[y]'}$. By lemma \ref{popreequivrep}, this representation lifts to
a representation of $(\mathcal{P}, \leq')$.
\begin{definition}
Let $(\mathcal{P}, \leq)$ be a preordered set, $\leq'$ a localization of $\leq$, and $E$ a
representation of $(\mathcal{P}, \leq)$. Consider the poset $\mathcal{P}_{\lessgtr'}$. Then we call
the induced sheaf on $(\mathcal{P}, \leq')$ a {\em localization} of $F$.
\end{definition}
Now we assume that we are given some partially ordered set $(A, \preceq)$, a
collection of preordered sets $\mathcal{P}_\alpha$ which glues over $A$, and a collection of
sheaves $E^\alpha$ over $\mathcal{P}_\alpha$ for every $\alpha \in A$. We want {\em glue} this
collection of sheaves to give some kind of global object over the glued preordered
sets.
\begin{definition}
We say that the collection $E^\alpha$ {\em glues} over the collection $\mathcal{P}_\alpha$, if
\begin{enumerate}[(i)]
\item for every $\beta \preceq \alpha$, and morphism of posets $l_{\alpha\beta}:
(\mathcal{P}_\alpha)_{\lessgtr^\beta_\alpha} \longrightarrow (\mathcal{P}_\beta)_{\lessgtr_\beta}$
there is an isomorphism of sheaves $\phi^{\alpha\beta} : l_{\alpha\beta}^*E^\beta
\overset{\cong}{\longrightarrow} E^\alpha$
\item for every triple $\gamma \preceq \beta \preceq \alpha$: $\phi^{\alpha\gamma} =
\phi^{\alpha\beta} \circ l^*_{\alpha\beta}\phi^{\beta\gamma}$.
\end{enumerate}
We call a such a collection a {\em sheaf} over $\mathcal{P}^\alpha$.
\end{definition}
Let $E^\alpha$, $F^\alpha$ be sheaves over $\mathcal{P}^\alpha$, where we denote the gluing
homomorphisms $\phi^{\alpha\beta}$ and $\psi^{\alpha\beta}$, respectively.
A homomorphism from $E^\alpha$ to $F^\alpha$ is given by a collection of
homomorphisms $f_\alpha : E^\alpha \longrightarrow F^\alpha$ such that $f_\alpha
\circ \phi^{\alpha\beta} = \psi^{\alpha\beta} \circ l^*_{\alpha\beta}f_\alpha$
for every pair $\beta \preceq \alpha$. One checks straightforwardly that this is
compatible with the cocycle conditions on $\phi^{\alpha\beta}$ and
$\psi^{\alpha\beta}$, and moreover that the corresponding families of kernels and
cokernels of $f^\alpha$ glues over $A$:
\begin{proposition}
The category of sheaves over $\mathcal{P}^\alpha$ is abelian.
\end{proposition}
\comment{
Now we define an anologon for invertible sheaves:
\begin{definition}
Let $\mathcal{P}^\alpha$ be a collection of posets which glues over $A$ and consider some
collection $\{x_\alpha \in \mathcal{P}_\alpha\}$ which has the property that for every
$\beta < \alpha$ $\bar{x}_\alpha = h_{\alpha\beta}(x_\beta)$. Then a {\em locally
free} representation of the system $x_\alpha$ is given by a collection $F^{x_\alpha}$
which glues over the collection $\mathcal{P}_\alpha$.
\end{definition}
}
\paragraph{Compression of sheaves over preordered sets.}
Let $A$ be a finite poset and denote $\mathbf{P}^f_A$ the category of collections of
{\em finite} preordered sets $\{\mathcal{P}^\alpha \mid \alpha \in A\}$ which glue over $A$.
Let $\{\mathcal{Q}^\alpha\}$ be any collection of not necessarily finite preordered
sets which glues over $A$. Denote $\mathbf{C}$ any subcategory of the category of
sheaves which glue over the collection $\mathcal{Q}^\alpha$. A {\em compression} of
$\mathbf{C}$ is any object $\{\mathcal{P}^\alpha\}$ of $\mathbf{P}^f$ together with a pair of
functors
\begin{align*}
\operatorname{zip} & : \mathbf{C} \longrightarrow \mathbf{Sheaves}(\mathcal{P}^\alpha) \\
\operatorname{unzip} & : \mathbf{Sheaves}(\mathcal{P}^\alpha) \longrightarrow \mathbf{C}.
\end{align*}
which induce an equivalence of categories between $\mathbf{C}$ and
$\mathbf{Sheaves}(\mathcal{P}^\alpha)$.
\comment{
Denote $\mathbf{SP}$ the category of {\em sheaves on posets}. Objects in this
category are pairs $(\mathcal{P}, \sh{F})$, where $\mathcal{P}$ is a poset and $\sh{F}$ is a sheaf on
$\mathcal{P}$. For
any two posets $\mathcal{P}$, $\mathcal{Q}$ with sheaves $\sh{F}$, and $\sh{G}$, respectively,
we define the morphisms to be pairs $(f, h)$, where $f : \mathcal{P} \longrightarrow
\mathcal{Q}$ is an order preserving map and $h$ is a sheaf homomorphism in
$\operatorname{Hom}(\sh{F}, f^*\sh{G})$; the composition $(f_2, g_2) \circ (f_1, g_1)$ is given
by $\big(f_2 \circ f_1, (f_1^* g_2) \circ g_1\big)$.
We denote $\mathbf{SP}^f$ the full subcategory of $\mathbf{SP}$ whose objects are the
{\em finite} posets. Analogously, if $(A, \preceq)$ is a poset, then we denote
$\mathbf{SP}_A$, $\mathbf{SP}_A^f$ the categories of families of sheaves on
preordered sets, respectively on finite preordered sets, which glue over $A$.
\begin{definition}
Let $\mathbf{C}$ and $\mathbf{C}^f$ be subcategories of $\mathbf{SP}_A$ and
$\mathbf{SP}_A^f$, respectively. A {\em compression} of $\mathbf{C}$ is an
equivalence of categories given by a pair of functors
\end{definition}
Below, we will be considering the also the case of sheaves over a single poset, in
which case we use the notion of $\operatorname{zip}$ and $\operatorname{unzip}$ without reference to and poset
$A$.
}
\comment{are concerned with a family of subcategories of $\mathbf{SP}$, indexed
by a poset $A$, from which we want to construct a compression with respect to
$\mathbf{SP}_A$, respectively $\mathbf{Sp}_A^f$.
\begin{definition}
For any $\alpha \in A$ let $\operatorname{zip}^\alpha$, $\operatorname{unzip}^\alpha$ be compressions with
respect to subcategories $\mathbf{C}^\alpha$, $\mathbf{C}^{\alpha, f}$ of
$\mathbf{SP}$ and $\mathbf{SP}^f$, respectively. Denote $\mathbf{C} \subset
\coprod_{\alpha \in A} \mathbf{C}^\alpha$ and $\mathbf{C}^f \subset \coprod_{\alpha
\in A} \mathbf{C}^{\alpha, f}$ the subcategories of objects which glue over $A$.
Then we say that the collection $\operatorname{zip}^\alpha$, $\operatorname{unzip}^\alpha$ {\em glues} over $A$
if:
\begin{enumerate}[(i)]
\item for any collection $(F^\alpha \mid \alpha \in A) \in \operatorname{Ob}(\mathbf{C})$ the
collection $\operatorname{zip}^\alpha F^\alpha$ glues over $A$,
\item for any collection $(F^\alpha \mid \alpha \in A) \in \operatorname{Ob}(\mathbf{C}^f)$ the
collection $\operatorname{unzip}^\alpha F^\alpha$ glues over $A$,
\item the collection $\operatorname{unzip}^\alpha \circ \operatorname{zip}^\alpha$, $\alpha \in A$, is an
equivalence of $\mathbf{C}$ with itself.
\end{enumerate}
\end{definition}
If there is no ambiguity, we will denote the collections $\operatorname{zip}^\alpha$,
$\operatorname{unzip}^\alpha$ simply by $\operatorname{zip}$ and $\operatorname{unzip}$.
}
\section{Toric Varieties and $\Delta$-Families}
\label{toricvarieties}
In this section we briefly recall basic facts for toric varieties and our results
from \cite{perling1} on equivariant sheaves over toric varieties.
For general information about toric varieties we refer
to \cite{Oda} and \cite{Fulton}. In this work $X$ will always denote an
$r$-dimensional
toric variety over a fixed algebraically closed field $k$, and $T$ the open dense
torus contained in $X$. Moreover, we use the following notation:
\begin{itemize}
\setlength{\itemsep}{-4pt}
\item $M \cong \mathbb{Z}^n$ is the character group of $T$, and $N$ the
$\mathbb{Z}$-module dual to $M$,\\
$M_\mathbb{R} := M \otimes_Z \mathbb{R}$, $N_\mathbb{R} := N \otimes_Z \mathbb{R}$;
\item elements of $M$ are denoted $m, m'$ etc. if written additively and $\chi(m),
\chi(m')$ etc. if written multiplicatively, i.e. $\chi(m + m') = \chi(m)\chi(m')$;
\item $\Delta$ denotes the fan associated to $X$, and cones in $\Delta$ are denoted
by small Greek letters $\rho$, $\sigma$, $\tau$, etc.;
the natural order among cones is denoted by $\tau < \sigma$, \\
$\Delta(i) := \{\sigma \in \Delta \mid \dim \sigma = i\}$ the set of
all cones of fixed dimension $i$,\\
$\sigma(i) := \{\tau \in \Delta(i) \mid \tau < \sigma\}$;
\item $\check{\sigma} := \{m \in M_\mathbb{R} \mid \langle m, n \rangle \geq 0
\text{ for all $n \in \sigma$}\}$ is the cone {\it dual} to $\sigma$,\\
$\sigma^\bot = \{m \in M_\mathbb{R} \mid \langle m, n \rangle = 0 \text{ for all } n \in \sigma \}$, \\
$\sigma_M := \check{\sigma} \cap M$ is the subsemigroup of $M$ associated to
$\sigma$. \\
$\sigma_M^\bot := \sigma^\bot \cap M$ is the maximal subgroup of $\sigma_M$;
\item the affine toric variety associate to a cone $\sigma$ is denoted $U_\sigma$,\\
$U_\sigma \cong \spec{k[\sigma_M]}$, where $k[\sigma_M]$ is the semigroup ring over
$\sigma_M$;
\item elements of $\Delta(1)$ are called {\it rays}, and the torus invariant Weil
divisor associated to some ray $\rho \in {\Delta(1)}$ is denoted $D_\rho$.
\end{itemize}
\subsection{Equivariant sheaves and $\Delta$-families}
\label{deltafamilies}
Consider any rational polyhedral convex cone $\sigma$, then the subsemigroup
$\sigma_M$ induces a {\em directed preorder} $\leq_\sigma$ on $M$ by setting
$m \leq_\sigma m'$ iff $m' - m \in \sigma_M$. The following properties of
$\leq_\sigma$ are easy to see:
\begin{enumerate}[(i)]
\setlength{\itemsep}{-5pt}
\item $m \leq_\sigma m'$ and $m' \leq_\sigma m$ iff $m - m' \in \sigma_M^\bot$.
\item If $\tau \leq \sigma$, then $m \leq_\sigma m'$ implies $m \leq_\tau m'$.
\item If $\sigma$ is of maximal dimension in $N_\mathbb{R}$, then $\leq_\sigma$ is a partial
order.
\end{enumerate}
Let $\sh{E}$ be an equivariant sheaf over $X$ and denote $E^\sigma :=
\Gamma(U_\sigma, \sh{E})$ for every affine open $T$-invariant subvariety $U_\sigma$
of $X$. The dual action of $T$ on $E^\sigma$ induces an isotypical decomposition
\begin{equation*}
E^\sigma = \bigoplus_{m \in M}E^\sigma_m
\end{equation*}
For any two $m \leq_\sigma m'$, there exists a distinguished $k$-linear map
spaces $\chi^\sigma_{m, m'} : E_m \longrightarrow E_{m'}$ which is given by
multiplication by the monomial
$\chi(m' - m) \in k[\sigma_M]$. These distinguished maps completely specifiy the
module structure of $E^\sigma$ over $k[\sigma_M]$. Observing that $\chi(m'' - m')
\chi(m' - m) = \chi(m'' - m)$ and $\chi(m - m) = 1$, we even obtain a
{\em functorial} description of $E^\sigma$. By mapping $m \mapsto E^\sigma_m$ and
$(m, m') \mapsto \chi^\sigma_{m, m'}$ for $m \leq_\sigma m'$, every $M$-graded
$k[\sigma_M]$-module
$E^\sigma$ defines a functor from the preordered set $(M, \leq_\sigma)$ to the
category $k\operatorname{-\bf Vect}$ of $k$-vector spaces.
\begin{proposition}[\cite{perling1}, Proposition 5.5]
Let $U_\sigma = \spec{k[\sigma_M]}$ be an affine toric variety. Then the following
categories are equivalent:
\begin{enumerate}[(i)]
\setlength{\itemsep}{-4pt}
\item equivariant quasicoherent sheaves over $U_\sigma$,
\item $M$-graded $k[\sigma_M]$-modules,
\item $k$-linear representations of the preordered set $(M, \leq_\sigma)$.
\end{enumerate}
\end{proposition}
\begin{definition}
We call a representation of $(M, \leq_\sigma)$ a {\em $\sigma$-family}.
\end{definition}
In the sequel, we will use the notation $E^\sigma$ exchangeably for the
$k[\sigma_M]$-module and for the $\sigma$-family.
Now for any pair $\tau < \sigma$, there exists some $m_\tau \in \sigma_M^\bot$ such
that $\tau_M = \sigma_M + \mathbb{Z}_{\geq 0} (-m_\tau)$ and $\tau_M^\bot = (\tau_M^\bot \cap
\sigma_M) + \mathbb{Z}_{\geq 0} (-m_\tau)$. In terms of preordered sets, this translates the
way that we can consider $(M, \leq_\tau)$ as a localization of $(M, \leq_\sigma)$ in
the sense of subsection \ref{posetgluing}. Moreover, the localization of $(M,
\leq_\sigma)$ by $\leq_\tau$ coincides with $(M, \leq_\tau)$, and thus the
contractions
$l_{\sigma\tau} : M_{\lessgtr_\sigma^\tau} \longrightarrow M_{\lessgtr_\tau}$ are
isomorphisms. We have:
\begin{proposition}
The family of preordered sets $(M, \leq_\sigma)$, $\sigma \in \Delta$, glues over
$\Delta$.
\end{proposition}
The restriction of $\sh{E}\vert_{U_\sigma}$ to $U_\tau$ corresponds to the
localization $E^\sigma_{\chi(m_\tau)}$. To
understand this in terms of $\sigma$-families, we first observe that the canonical
map $E^\sigma \longrightarrow E^\sigma_{\chi(m_\tau)}$ at the same time is a
homomorphism of directed systems.
\begin{proposition}
For every $m \in M$ there exists a natural isomorphism $E^\tau_m \cong
\underset{\rightarrow}{\lim} E^\sigma_{m'}$, where the limit is taken over the
directed
system of all $E^\sigma_{m'}$ with $m' \leq_\tau m$ {\em with respect to the preorder
$\leq_\sigma$}.
\end{proposition}
\begin{proof}
By definition of localization, the vector space $E^\tau_m$ is the set of equivalence
classes $\{[\frac{e}{\chi(m')}] \mid \deg_M e = m + m'\}$, where
$\frac{e_1}{\chi(m_1)}
\sim \frac{e_2}{\chi(m_2)}$ if and only if $\chi(m_1) \cdot e_2 = \chi(m_2) \cdot
e_1$ in $E^\sigma$, where without loss of generality, $m_1$ and $m_2$ can be chosen
from $\sigma_M$. In other notation, this reads $\chi^\sigma_{m + m_1, m + m_1 +
m_2} e_2 = \chi^\sigma_{m + m_2, m + m_1 + m_2} e_1$. So, in a natural way, we can
identify $E^\tau_m$ with the direct limit $\underset{\rightarrow}{\lim}
E^\sigma_{m'}$.
\end{proof}
\comment{ Now for any two $m_1 \lessgtr_\tau
m_2$, the map $\chi^\tau_{m_1, m_2}$ is an isomorphism, and moreover there exist
$m'_1, m'_2 \in \sigma^\bot_M$ such that $\chi(m'_1) \cdot E_{m_1} = \chi(m'_2)
\cdot E_{m_2}$. This in particular implies that there is an isomorphism
\begin{equation*}
E^\tau_m \cong \underset{\rightarrow}{\lim} E^\sigma_{m'}
\end{equation*}
for every $m \in M$, where the limit is taken over the directed system of all
$E^\sigma_{m'}$ with $m' \leq_\tau m$ {\em with respect to the preorder
$\leq_\sigma$}. The canonical map
\begin{equation*}
E^\sigma \longrightarrow E^\sigma_{\chi(m_\tau)}
\end{equation*}
in terms of the $\sigma$- and $\tau$-families translates into a localization of
the representation $E^\sigma$ of $(M, \leq_\sigma)$ to $\leq_\tau$.
}
By this proposition, we see that the localization of $E^\sigma$ by $\chi(m_\tau)$
translates into the localization of $E^\sigma$, considered as {\em sheaf over
$(M, \leq_\sigma)$}, to $(M, \leq_\tau)$. We get:
\begin{definition}[see also \cite{perling1}, Definition 5.8]
A $\Delta$-family is a collection $\{E^\sigma \mid \sigma \in \Delta\}$ of
$\sigma$-families which glues over $\Delta$.
\end{definition}
\begin{theorem}[\cite{perling1}, Theorem 5.9]
The category of equivariant sheaves over $X$ is equivalent to the category of
$\Delta$-families.
\end{theorem}
\subsection{The Krull-Schmidt property}
\label{krullschmidt}
Let $\mathfrak{C}$ be any category in which direct
sums exist. We say that the Krull-Schmidt theorem holds in $\mathfrak{C}$ if
for every object $A$ in $\mathcal{C}$ and for every two decompositions into
indecomposable objects
\begin{equation*}
A \cong X_1 \oplus X_2 \oplus \dots \oplus X_n \cong Y_1 \oplus Y_2 \oplus \dots
\oplus Y_m
\end{equation*}
we have $m = n$, and there exists a permutation $\pi$ of $\{1, \dots, n\}$ such that
$X_i \cong Y_{\pi(i)}$ for every $i$.
It is well known that the Krull-Schmidt theorem holds in the category of coherent
sheaves over a complete variety. For the category of equivariant coherent sheaves
over a toric variety, we can drop the completeness condition:
\begin{theorem}
\label{krullschmidttheorem}
Let $X$ be any toric variety, then the Krull-Schmidt theorem holds for the category
of equivariant coherent sheaves over $X$.
\end{theorem}
\begin{proof}
According to a classical result of Atiyah (\cite{Atiyah1}),
it suffices to show that for every
two equivariant sheaves \msh{E} and $\sh{F}$, the vector space
$\operatorname{Hom}(\sh{E}, \sh{F})^T$ of $T$-equivariant sheaf homomorphisms is
finite-dimensional. As we are dealing only with finite fans, it is enough to consider
the case where $X = U_\sigma$ is an affine toric variety such that $\sh{E}$ and
$\sh{F}$ correspond to finitely generated $k[\sigma_M]$-modules $E^\sigma$ and
$F^\sigma$. In this case the statement follows because every generator of $E^\sigma$
of degree $m$ must be mapped to some element $f \in F^\sigma_m$ and every vector
space $F^\sigma_m$ is finite dimensional (\cite{perling1}, Proposition 5.11).
\end{proof}
\subsection{The quotient representation of a toric variety}
Every toric variety can be represented as a good quotient of a quasi-affine toric
variety (see \cite{Cox}). This representation starts with the exact sequence
\begin{equation*}
0 \longrightarrow M_0 \longrightarrow M \longrightarrow {\mathbb{Z}^\rays}
\longrightarrow A \longrightarrow 0
\end{equation*}
where the map from $M$ to $\weildivisors$\ is given by $m \mapsto (\langle m, n(\rho)
\rangle \mid \rho \in {\Delta(1)})$. In the sequel we will assume that the fan $\Delta$ is
not contained in a proper subvector space of $N_\mathbb{R}$. In this case $M_0$ is the zero
module.
We consider the polynomial ring $S = k[x_\rho \mid \rho \in {\Delta(1)}]$; this
ring is endowed with a natural $\weildivisors$-grading by setting $\deg x^{\underline{n}} = {\underline{n}}$ for
every monomial $x_\rho$. Via the surjection of $\weildivisors$\ onto $A$, the ring $S$
automatically acquires an $A$-grading,
\begin{equation*}
S \cong \bigoplus_{\alpha \in A} S_\alpha.
\end{equation*}
We define the {\em irrelevant} ideal $B = \langle x^{\hat{\sigma}} \mid \sigma \in
\Delta \rangle$, where $x^{\hat{\sigma}} = \prod_{\rho \in {\Delta(1)} \setminus \sigma(1)}
x_\rho$ for every $\sigma \in \Delta$.
The variety $\mathbf{V}(B)$ defined by $B$ is a finite union of linear subspaces of
$\spec{S} \cong k^{\Delta(1)}$, which has codimension at least two.
The complement of $\mathbf{V}(B)$, which we denote $\hat{X}$, is a quasi-affine toric
variety, on which the torus $\hat{T} \cong (k^*)^{\Delta(1)}$ acts. Denote $e_\rho$ the
standard basis vectors of $\mathbb{R}^{\Delta(1)}$, then the fan of $\hat{X}$ is generated by
the cones $\hat{\sigma} = \sum_{\rho \in \sigma(1)} \mathbb{R}_{\geq 0} e_\rho$, for every
$\sigma \in \Delta$. The affine open subsets
$U_{\hat{\sigma}}$ form a cover of $\hat{X}$, and we will call $\hat{\Delta}
= \{\hat{\sigma} \mid \sigma \in \Delta\}$
the fan of $\hat{X}$, although in general $\hat{\Delta}$ is not a proper fan, unless
$X$ is a simplicial toric variety.
There is a canonical morphism $\pi: \hat{X} \longrightarrow X$ which is described by
the map of fans induced by the linear map given by $e_\rho \mapsto n(\rho)$. By this
morphism, $X$ becomes a {\em good quotient} of $\hat{X}$ by the diagonalizable
group $G = \operatorname{Hom}(A, k^*)$.
The coordinates $x_\rho$ then serve as global coordinates for
$X$, and $S$ is denoted the {\em homogeneous coordinate ring}
of $X$.
\comment{
As every $U_\sigma$ is a toric variety, every such $U_\sigma$ has its own homogeneous
coordinate ring $S_\sigma$. Below it will be useful for us to observe that we can
obtain $S$ as a limit ring of the $S_\sigma$.
This quotient
has locally the description of $U_\sigma = U_{\hat{\sigma}} // G$, and $k[\sigma_M]
= S_{x^{\hat{\sigma}}}^G = \big(S_{x^{\hat{\sigma}}}\big)_0$ with respect to the
$A$-grading.
Note that for any $\sigma \in \Delta$, the localization $S_{x^{\hat{\sigma}}}$
automatically becomes a homogeneous coordinate ring for $U_\sigma$, and the
restriction $\pi \vert_{U_{x^{\hat{\sigma}}}} \longrightarrow U_\sigma$ becomes a
quotient representation for $U_\sigma$.
}
\paragraph{$A$-graded $S$-modules.}
Any $A$-graded $S$-module $F$ defines a $G$-equivariant sheaf over
$k^{\Delta(1)}$ and thus over $\hat{X}$, and it has been shown (see \cite{Mustata1}) that
every quasicoherent sheaf over $X$ can be represented as a descend of an $A$-graded
$S$-module $F$ of the
form $\big(\pi_* (\tilde{F}\vert_{\hat{X}})\big)^G$, where $\tilde{\ }$ denotes the
usual sheafification functor over the affine space $k^{\Delta(1)}$. We abbreviate the
descend of a module $F$ by $\breve{F}$. In the other direction, every coherent sheaf
\msh{F} over $X$ gives rise to an $A$-graded $S$-module $\Gamma(\hat{X},
\pi^*\sh{F})$. There is always an isomorphism $\Gamma(\hat{X}, \pi^*\sh{F})\breve{\ }
\cong \sh{F}$, but in general there is no isomorphism between any $A$-graded module
$F$ and $\Gamma(\hat{X}, \pi^* \breve{F})$.
\paragraph{Fine-graded $S$-modules.}
For the study of equivariant sheaves, we have to
consider {\em fine graded} modules, i.e. $\weildivisors$-graded $S$-modules. Such a
module $F$ is equivalent to $\hat{T}$-equivariant sheaf over $k^{\Delta(1)}$, and its
descend $\breve{F}$ then in a natural way is a $T$-equivariant sheaf over $X$.
On the other hand, the pullback $\pi^*\sh{E}$ of some $T$-equivariant sheaf over $X$
has a natural $\hat{T}$-equivariant structure, and thus $\hat{E} := \Gamma(\hat{X},
\pi^* \sh{E})$ is fine graded. The most important examples for us are
the modules which are defined as the descend of free $S$-modules of rank one. These
are the modules of the form $S({\underline{n}})$, the degree shift of $S$ by some element
${\underline{n}} \in {\mathbb{Z}^\rays}$, where $S({\underline{n}})_{{\underline{n}}'} = S_{{\underline{n}} + {\underline{n}}'}$. The descend
$\breve{S}({\underline{n}})$ is isomorphic to $\sh{O}_X(D_{\underline{n}})$, the reflexive sheaf of rank one
which is associated to the Weil-divisor $D_{\underline{n}} := \sum_{\rho \in {\Delta(1)}} -n_\rho
D_\rho$. As a general notation, we write $S_{({\underline{n}})}$ instead of $S({\underline{n}})_0$; note that
this shift is in the $\weildivisors$-grading, not in the $A$-grading and therefore
fixes a unique equivariant structure on $\breve{S}({\underline{n}})$.
\paragraph{Global and local quotient representations.}
For any $\sigma \in \Delta$ there is an exact sequence
\begin{equation*}
0 \longrightarrow \sigma_M^\bot \longrightarrow M \longrightarrow \mathbb{Z}^{\sigma(1)}
\longrightarrow A^\sigma \longrightarrow 0,
\end{equation*}
by which we have a splitting $M \cong \sigma_M^\bot \oplus M / \sigma_M^\bot$, where
we identify $M / \sigma_M^\bot \cong M_{\lessgtr_\sigma}$ with the image of $M$ in
$\mathbb{Z}^{\sigma(1)}$. This
induces a splitting $U_\sigma \cong T_\sigma \times U_{\sigma'}$, where $T_\sigma
\cong \spec{k[\sigma^\bot_M]}$ is the minimal orbit of $U_\sigma$, and $U_{\sigma'}$
is the affine toric variety associated to the subsemigroup $\sigma_M' = \sigma_M /
\sigma^\bot_M$ of $M / \sigma^\bot_M$. Below, every construction with respect to
$(M, \leq_\sigma)$ will up to natural equivalence only depend on
$M_{\lessgtr_\sigma}$, and so for clearer presentation we will always neglect the
factor $\sigma_M^\bot$ and identify any $m \in M$ with its image in $\mathbb{Z}^{\sigma(1)}$.
The embedding of $M$ in $\mathbb{Z}^{\sigma(1)}$ is in a natural way compatible with the
partial order $\leq$ on $\mathbb{Z}^{\sigma(1)}$ induced by the subsemigroup
$\mathbb{N}^{\sigma(1)}$, i.e. $m \leq_\sigma m'$ iff $m \leq m'$. We consider the order
$\leq$ as an {\em extension} of $\leq_\sigma$ to $\mathbb{Z}^{\sigma(1)}$. For any $\tau <
\sigma$, the localization of $\leq_\sigma$ by $\tau_\sigma$ extends to a localization
of $\leq$ by the preorder $\leq'$ induced by the subsemigroup $\mathbb{N}^{\tau(1)} \oplus
\mathbb{Z}^{\sigma(1) \setminus \tau(1)}$, and we have a natural identification
$(\mathbb{Z}^{\sigma(1)})_{\lessgtr'} = \mathbb{Z}^{\tau(1)}$. This localization is naturally
compatible with the localization of $M$ by $\leq_\sigma^\tau$ and we have the
following commutative exact diagram:
\begin{equation*}
\xymatrix{
0 \ar[r] & \sigma^\bot_M \ar@{ >->}[d] \ar[r] & M \ar@{=}[d] \ar[r] & \mathbb{Z}^{\sigma(1)}
\ar@{-{>>}}[d]^{\pi} \ar[r] & A^\sigma \ar[r] \ar[d] & 0 \\
0 \ar[r] & \tau^\bot_M \ar[r] & M \ar[r] & \mathbb{Z}^{\tau(1)} \ar[r] & A^\tau \ar[r] & 0,
}
\end{equation*}
where $\pi$ is the canonical projection from $\mathbb{Z}^{\sigma(1)}$ onto $\mathbb{Z}^{\tau(1)}$.
Having these natural compatibilities in mind, in the sequel we will use the notation
$\leq_\sigma$ for both preorders on $M$ and on $\mathbb{Z}^{\sigma(1)}$; we will write
${\underline{n}} \leq_\sigma m$ and the like for ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$ and $m \in M$.
We now describe more precisely the relation between the $\Delta$-family of \msh{E}
and the
$\hat{\Delta}$-family of $\pi^*\sh{E}$. For any $\sigma \in \Delta$ consider the
quotient representation $\pi_\sigma : U_{\hat{\sigma}} \twoheadrightarrow U_\sigma$,
where $U_{\hat{\sigma}} \cong k^{\sigma(1)}$. Here without loss of generality we
assume for the moment that $\sigma$ has full dimension in $N_\mathbb{R}$.
Denote $E^\sigma := \Gamma(U_\sigma, \sh{E})$ and $\hat{E}^\sigma := \Gamma(
U_{\hat{\sigma}}, \pi_\sigma^* \sh{E})$. The homogeneous coordinate ring has a
natural $A^\sigma$-grading $S^\sigma = \bigoplus_{\alpha \in A^\sigma}
S_\alpha^\sigma$, with $S_0 \cong k[\sigma_M]$, so that we can write:
\begin{equation*}
\hat{E}^\sigma \cong E^\sigma \otimes_{S^\sigma_0} S^\sigma \cong E^\sigma
\otimes_{S^\sigma_0} \big(\bigoplus_{\alpha \in A^\sigma} S_\alpha^\sigma\big) \cong
\bigoplus_{\alpha \in A^\sigma} \big(E^\sigma \otimes_{S^\sigma_0} S^\sigma_\alpha
\big).
\end{equation*}
By $E^\sigma \otimes_{S^\sigma_0} S^\sigma_0 \cong E^\sigma$, we find that
$E^\sigma$ is naturally embedded in $\hat{E}^\sigma$, and thus the $\sigma$-family
of $E^\sigma$ is a subfamily of the $\hat{\sigma}$-family of $\hat{E}^\sigma$.
Denote $\leq_\Delta$ the preorder on $\weildivisors$, then every $(\mathbb{Z}^{\sigma(1)},
\leq_\sigma)$ is isomorphic to the localization of $(\mathbb{Z}^{\Delta(1)}, \leq_\Delta)$ by the
preorder $\leq'_\sigma$ where ${\underline{n}} \lessgtr'_\sigma {\underline{n}}'$ iff ${\underline{n}} - {\underline{n}}' \in
\mathbb{Z}^{{\Delta(1)} \setminus \sigma(1)}$. By the natural projection ${\mathbb{Z}^\rays}
\longrightarrow \mathbb{Z}^{\sigma(1)}$, every fine graded module over the localization
$S_{x^{\hat{\sigma}}} \cong k[\mathbb{Z}^{{\Delta(1)} \setminus \sigma(1)} \times \mathbb{N}^{\sigma(1)}]$
is equivalent to a fine graded module over $S^\sigma \cong k[\mathbb{N}^{\sigma(1)}]$. The
localization
$\hat{E}_{x^{\hat{\sigma}}}$ of $\hat{E}$ then is equivalent to a representation
of $(\mathbb{Z}^{\sigma(1)}, \leq_\sigma)$, and by naturality,
the $\Delta$-family $E^\Delta$ glues as a subfamily of the $\hat{\Delta}$-family
$\hat{E}^{\hat{\Delta}}$.
\paragraph{Resolutions of $\hat{E}$.}
As for every equivariant coherent sheaf \msh{E} we can consider its associated fine
graded module $\hat{E}$, in principle there is nothing which prevents us from doing
this and to compute some finite free resolution of $\hat{E}$ over $S$, which then
descends to a resolution of \msh{E} of the desired type:
\begin{equation*}
0 \longrightarrow \breve{F}_s \longrightarrow \cdots \longrightarrow \breve{F}_0
\longrightarrow \sh{E} \longrightarrow 0,
\end{equation*}
where $\breve{F}_i \cong \bigoplus_{j = i}^{k_j} \sh{O}_X(D_{{\underline{n}}_{ij}})$ and the
length $s$ by Hilbert's syzygy theorem being bounded by the numbers of rays in
$\Delta$. At this point we could stop with this paper and leave the problem as an
application of traditional methods. However, there are some drawbacks of this
point of view, which motivate our further investigations. One problem is that the
pullback of a coherent sheaves along good quotients so far seems not to be
understood very well, not even for the toric case --- and as we will see in example
\ref{pullbacktorsionexample} below, such pullbacks might show pathological behaviour,
such as acquiring additional torsion. Another problem is that due to its global
nature, the module $\hat{\sh{E}}$ contains much more relations which might be
irrelevant to consider for getting a resolution.
\begin{example}
\label{pullbacktorsionexample}
Consider the subsemigroup $\sigma_M$ of $M \cong \mathbb{Z}^2$ which is generated by the
elements $(1, 0)$, $(1, 1)$, $(1, 2)$ and its associated semigroup ring $S_0 =
k[\sigma_M]$. Its fan is spanned by the primitive vectors ${\underline{n}}_1 = (2, -1)$ and
${\underline{n}}_2 = (0, 1)$ in
$N_\mathbb{R}$, and the homogeneous coordinate ring $S = S^\sigma$ is $\mathbb{Z}_2$-graded.
Denote ${\underline{n}} := (-1, 0)$ and consider the reflexive $S_0$-module $S_{({\underline{n}})} \cong S_1$.
For the pullback we have:
\begin{equation*}
S_{({\underline{n}})} \otimes_{S_0} (S_0 \otimes S_1) \cong (S_1 \otimes_{S_0} S_0) \oplus
(S_1 \otimes_{S_0} S_1) \cong S_1 \otimes (S_1 \otimes_{S_0} S_1).
\end{equation*}
To compute $(S_1 \otimes_{S_0} S_1)$, we directly evaluate it as an $M$-graded tensor
product. The module $S_{({\underline{n}})}$ is a $M$-graded, where
\begin{equation*}
S_{({\underline{n}}), m} =
\begin{cases}
k & \text{ if } \langle m, {\underline{n}}_i \rangle \geq 0 \text{ for } i = 1, 2 \\
0 & \text{ else}.
\end{cases}
\end{equation*}
In degree $m$, $S_{({\underline{n}})} \otimes_{k[\sigma_M]} S_{({\underline{n}})}$ is generated by all elements
$\chi(m_1) \otimes \chi(m_2)$ such that $m_1 + m_2 = m$ modulo the relation that
$\chi(m_1) \otimes \chi(m_2)$ is equivalent to $\chi(m_1 - m') \otimes \chi(m_2)$
and $\chi(m_1) \otimes \chi(m_2 - m'')$, respectively, whenever there exist some
$\chi(m')$ or $\chi(m'')$ such that $\chi(m_1 - m')$ and $\chi(m_2 - m'')$,
respectively, are in $S_{({\underline{n}})}$. It turns out that these relations cancel most of
the generators in every degree, so that for all nonzero degrees:
\begin{equation*}
\dim(S_{({\underline{n}})} \otimes_{S_0} S_{({\underline{n}})})_m =
\begin{cases}
2 & \text{ if } m = (1, 1) \\
1 & \text{ else}.
\end{cases}
\end{equation*}
Note that the nonzero degrees are precisely thos contained in the intersection of the
half spaces $\langle m, {\underline{n}}_2 \rangle
\geq 0$, $\langle m, 2{\underline{n}}_1 \rangle \geq 0$ and $\langle m, (1, 0) \rangle \geq 0$.
So, in degree $(1,1)$, our module has dimension two, whereas in all other degrees
it has at dimension one, which implies that it has torsion in degree $(1, 1)$, as
for any character $(1, 1) \leq_\sigma m$ the homomorphism $\chi_{(1, 1), m}$ can not
be injective. This is indeed an example where pullback of a torsion free, and even
reflexive, module along a geometric quotient aqcuires some new torsion.
\end{example}
Another phenomenon which we want to mention is that there are also other relevant
effects which one has to consider if one tries to choose some alternative module
instead of $\hat{E}$ whose descend coincides with that of $\hat{E}$.
For instance, for any affine toric variety $U_\sigma$ for which $A^\sigma$ is
nontrivial, there exist nonzero $S$-modules $F$ whose zero component vanishes;
the most easiest example is the one-dimensional module $S^\sigma / \langle x_\rho \mid
\rho \in \sigma(1) \rangle$ whose degree gets shifted by some nonzero $\alpha \in
A^\sigma$.
\section{Compression and Resolutions}
\label{resolutions}
\subsection{$\operatorname{lcm}$-lattices in $\mathbb{Z}^r$}
The partial order on $\mathbb{Z}^r$ induced by $\mathbb{N}^r$ coincides with the partial order given
by componentwise ordering, i.e. if we write ${\underline{n}} = (n_1, \dots, n_r)$, ${\underline{n}}' = (n'_1,
\dots, n'_r)$, then ${\underline{n}} \leq {\underline{n}}'$ iff $n_i \leq n'_i$ for every $1 \leq i \leq r$.
We set $\bar{\mathbb{Z}} := \{-\infty\} \cup \mathbb{Z}$ which is totally ordered by $-\infty < n$
for all ${\underline{n}} \in \mathbb{Z}$. Like $\mathbb{Z}^r$, the set $\bar{\mathbb{Z}}^r$ is partially ordered by the
componentwise total order, and the the canonical inclusion $\mathbb{Z}^r \hookrightarrow
\bar{\mathbb{Z}}^r$ is order preserving. We call any element
in $\bar{\mathbb{Z}}^r \setminus \mathbb{Z}^r$ {\em infinitary}.
For any element ${\underline{n}} \in \mathbb{Z}^r$ we can consider the subset ${\underline{n}} + \mathbb{N}^r$, which is
the intersection of the shifted cone ${\underline{n}} + \mathbb{R}_{\geq 0}^r$ with $\mathbb{Z}^r$. It is easy to
see that for any finite set of elements ${\underline{n}}_1, \dots {\underline{n}}_s$ in $\mathbb{Z}^r$, the intersection
$\bigcap_{i = 1}^s \big({\underline{n}}_i + \mathbb{N}^r\big)$ is again of the form ${\underline{n}} + \mathbb{N}^r$. The
element ${\underline{n}}$ is called the {\em least common multiple} of ${\underline{n}}_1, \dots, {\underline{n}}_r$, denoted
$\operatorname{lcm}\{{\underline{n}}_1, \dots, {\underline{n}}_s\}$, and it is given by componentwise maximum of the ${\underline{n}}_i$.
The $\operatorname{lcm}$ extends canonically to $\bar{\mathbb{Z}}^r$. In the geometric picture, for some
infinitary element ${\underline{n}} = (n_1, \dots, n_r)$ with $n_{i_j} = -\infty$ for some
$\{i_1, \dots, i_r\} \subset \{1, \dots, r\}$, we write ${\underline{n}} + C$ for the cone, where
$C = \{ \underline{c} \in \mathbb{R}^r \mid c_i \geq 0 \text{ if } i \notin \{i_1, \dots,
i_k\}\}$. One can think of the cone $C$ of the standard orthant moved to minus
infinity in the directions $i_1, \dots, i_k$.
In our actual definition of the $\operatorname{lcm}$-lattice, we will need inifinitary elements
to generate the lattice, but after generation, we throw away all these elements.
Instead, we close every $\operatorname{lcm}$-lattice from below by adding the unique minimal
element $(-\infty, \dots, -\infty) =: \hat{0}$.
\begin{definition}
Let $\mathcal{P} \subset \bar{\mathbb{Z}}^r$ be some poset and $\operatorname{lcm}(\mathcal{P})$ the lattice generated by
the $\operatorname{lcm}$'s of elements in $\mathcal{P}$. Then we denote the set $(\operatorname{lcm}(\mathcal{P}) \cap \mathbb{Z}^r)
\cup \hat{0}$ the {\em $\operatorname{lcm}$-lattice} of $\mathcal{P}$.
\end{definition}
Every $\operatorname{lcm}$-lattice $\mathcal{L}$ gives rise to a partition of $\mathbb{Z}^r$, respectively to an
equivalence relation,
on $\mathbb{Z}^r$. Namely, for every element $\underline{n} \in \mathbb{Z}^r$, there exists a unique
maximal element ${\underline{n}}' \in \mathcal{L}$ with ${\underline{n}}' \leq {\underline{n}}$.
\begin{definition}
Let ${\underline{n}} \in \mathbb{Z}^r$ and ${\underline{n}}' \in \mathcal{L}$ maximal such that ${\underline{n}}' \leq {\underline{n}}$. Then we call
${\underline{n}}'$ the {\em anchor element} $A({\underline{n}})$ of ${\underline{n}}$ in $\mathcal{L}$. Any two elements ${\underline{n}}_1, {\underline{n}}_2
\in \mathbb{Z}^r$ are equivalent iff $A({\underline{n}}_1) = A({\underline{n}}_2)$. We denote $T_{\underline{n}}$ the equivalence
class associated to ${\underline{n}} \in \mathcal{L}$.
\end{definition}
\subsection{Polynomial rings}
\label{polynomialrings}
In this subsection we consider the special case where $X$ is an affine toric variety
isomorphic to the affine space $k^r$, so that we can assume without loss of
generality that $\sigma$ and $\sigma_M$ coincide with the standard orthant
$\mathbb{R}^r_{\geq 0}$ in $\mathbb{R}^r$, and the subsemigroup $\mathbb{N}^r$ of $\mathbb{Z}^r$, respectively.
We denote $S \cong k[\mathbb{N}^r]$ the coordinate ring of $X$ and $E$ a nonzero finitely
generated $S$-module. We formally extend the representation of $(\mathbb{Z}^r, \leq)$ by $E$
to a representation of $(\bar{\mathbb{Z}}^r, \leq)$ by setting $E_{\underline{n}} = 0$ for all infinitary
${\underline{n}}$. In order
to construct a compression functor for $E$, we have to extract all nontrivial maps
(i.e. the nonisomorphisms) of the corresponding $\sigma$-family, as well as all
possible relations among them.
\begin{definition}
Let ${\underline{n}} \in \mathbb{Z}^r$, then we define the set $I_E({\underline{n}})$ to contain those elements ${\underline{n}}'$
in $\bar{\mathbb{Z}}^r$ which are minimal with the property that for all ${\underline{n}}'' \in
\bar{Z}^r$ with ${\underline{n}}' \leq {\underline{n}}'' \leq {\underline{n}}$ the morphisms $\chi_{{\underline{n}}', {\underline{n}}''}: E_{{\underline{n}}'}
\rightarrow E_{{\underline{n}}''}$ and $\chi_{{\underline{n}}'', {\underline{n}}}: E_{{\underline{n}}''} \rightarrow E_{{\underline{n}}}$ are
isomorphisms. We denote $\mathcal{I}_E := \bigcup_{{\underline{n}} \in \mathbb{Z}^r} I_E({\underline{n}})$.
\end{definition}
Note that the case where $I_E({\underline{n}})$ contains an infinitary elemement can only
(but not necessarily has to) occur when $E_{\underline{n}}$ is zero. Moreover, note that it
follows immediately from the finitely generatedness of $E$ that
$\mathcal{I}_E$ and the $I_E({\underline{n}})$ are finite sets.
\begin{definition}
We denote $\mathcal{L}_E$ the $\operatorname{lcm}$-lattice generated by $\mathcal{I}_E$.
For any ${\underline{n}} \in \mathbb{Z}^r$, we denote the corresponding anchor element by $A_E({\underline{n}})$.
\end{definition}
We can depict the set of equivalence classes as a tiling of $\mathbb{R}^r$ by cubic, possibly
non-compact blocks, where the anchor elements are precisely those elements sitting on
the smallest vertex with respect to $\leq$.
Observe that $\operatorname{lcm}\{{\underline{n}}_1, \dots, {\underline{n}}_s\} \in \mathbb{Z}^r$ as soon as at least one of the
${\underline{n}}_i$ is non-infinitary. Moreover, if $I_E({\underline{n}})$ contains an infinitary element, this
implies that $E_{\underline{n}} = 0$. In general, the set $I_E({\underline{n}})$ will contain
infinitary elements only if there exists no ${\underline{n}}' < {\underline{n}}$ such that $E_{{\underline{n}}'} \neq 0$.
In that case, $I_E({\underline{n}})$ will contain $\hat{0}$ as its only element. An exception
are those modules $E$, which are of rank zero, and thus are torsion modules.
The infinitary elements $I_E({\underline{n}})$ for all ${\underline{n}} \in \mathbb{Z}^r$ in that case describe the
support of $E$.
\begin{example}
Let $J \subset S$ be a monomial ideal, generated by monomials $x^{{\underline{n}}_1}, \dots,
x^{{\underline{n}}_s}$. Then we have
\begin{equation*}
I_J({\underline{n}}) =
\begin{cases}
\{{\underline{n}}_i \leq {\underline{n}} \} & \text{ if } x^{\underline{n}} \in J \\
\hat{0} & \text{ else},
\end{cases}
\end{equation*}
and the anchor element $A_J({\underline{n}})$ being $\operatorname{lcm}\{{\underline{n}}_i \leq {\underline{n}}\}$. The lattice $\mathcal{L}_J$
then coincides with the $\operatorname{lcm}$-lattice introduced in \cite{GPW99}.
\end{example}
\begin{example}
\label{lcmexample1}
Consider the torsion module $T= k[x, y] / \langle x^2, xy, y^2\rangle$. We have
\begin{equation*}
I_T({\underline{n}}) =
\begin{cases}
\{(0, 0)\} & \text{ for } {\underline{n}} \in \{(0, 0), (1, 0), (0, 1)\} \\
\{(1, 1)\} & \text{ for } {\underline{n}} = (1, 1) \\
\{(2, -\infty)\} & \text{ for } {\underline{n}} = (k, 0), k > 1 \\
\{(-\infty, 2)\} & \text{ for } {\underline{n}} = (0, k), k > 1 \\
\{(1, 1), (2, -\infty)\} & \text{ for } {\underline{n}} = (k, 1), k > 1 \\
\{(1, 1), (-\infty, 2)\} & \text{ for } {\underline{n}} = (1, k), k > 1 \\
\{(1, 1), (2, -\infty), (-\infty, 2)\} & \text{ for } (2, 2) \leq {\underline{n}} \\
\{\hat{0}\} & \text{ else}.
\end{cases}
\end{equation*}
The corresponding $\operatorname{lcm}$-lattice then is the set $\{\hat{0}, (0, 0), (2, 0),
(0, 2), (1, 1), (1, 2),$ $(2, 1),$ $(2, 2)\}$. Figure \ref{f-lcmexample1} shows
the partitioning of $\mathbb{Z}^2$ by the $\operatorname{lcm}$-lattice. The rectangular figure indicates
the degrees $(0, 0), (1, 0), (0, 1)$, where $T$ is nonzero; the light grey
triangles indicate all the initial elements $I_T({\underline{n}})$, and the darker grey triangles
denote the additional elements of the $\operatorname{lcm}$-lattice. The infinitary
elements become merged to $\hat{0}$ in $\mathcal{L}_T$.
\end{example}
\insfig{lcmexample1}{$\operatorname{lcm}$-lattice for example \ref{lcmexample1}}
Denote $\mathcal{L}_E$-Rep the category of finite-dimensional $k$-linear representations
of $\mathcal{L}_E$; denote $\mathcal{M}_E$ the full subcategory of the category of
fine-graded $S$-modules whose objects are those modules $F$ whose associated
$\operatorname{lcm}$-lattice
$\mathcal{L}_F$ is a sublattice of $\mathcal{L}_E$. Let $\iota_E : \mathcal{L}_E \hookrightarrow \mathbb{Z}^r$ be
the canonical inclusion. Then we define the functor $\operatorname{zip}^E$ from $\mathcal{M}_E$
into $\mathcal{L}_E$-Rep by
\begin{equation*}
\operatorname{zip}^E(F) := \iota_E^* F
\end{equation*}
where $\iota_E^*$ denotes the sheaf pullback.
To define the $\operatorname{unzip}$ functor, we have to do a little bit more. Let $F$ be some
representation of $\mathcal{L}_E$, mapping ${\underline{n}}$ to $F({\underline{n}})$, and ${\underline{n}} \leq {\underline{n}}'$ to $F({\underline{n}}, {\underline{n}}')$.
Then we define a representation of $\mathbb{Z}^r$ by setting $F_{\underline{n}} := F\big(A_E({\underline{n}})\big)$ and
$\chi_{{\underline{n}}, {\underline{n}}'} := F\big(A_E({\underline{n}}), A_E({\underline{n}}')\big)$ for every pair ${\underline{n}}, {\underline{n}}' \in \mathbb{Z}^r$.
This indeed establishes a well defined functor, where $F\big(A_E({\underline{n}}),
A_E({\underline{n}}')\big) = \operatorname{id}$ whenever $A_E({\underline{n}}) = A_E({\underline{n}}')$ and $F\big(A_E({\underline{n}}),
A_E({\underline{n}}'')\big) = F\big(A_E({\underline{n}}'), A_E({\underline{n}}'')\big) \circ F\big(A_E({\underline{n}}), A_E({\underline{n}}')\big)$
whenever ${\underline{n}} \leq {\underline{n}}' \leq {\underline{n}}''$.
\begin{theorem}
The pair of functors $\operatorname{zip}$ and $\operatorname{unzip}$ establishes an equivalence of categories
between $\mathcal{M}_E$ and $\mathcal{L}_E$-Rep.
\end{theorem}
\begin{proof}
We show that $\operatorname{unzip} \circ \operatorname{zip} \cong 1_{\mathcal{M}_E}$ and $\operatorname{zip} \circ \operatorname{unzip}
\cong 1_{\mathcal{L}_E\operatorname{-Rep}}$. In the first case, let $F$ be some
representation of $(\mathbb{Z}^r, \leq)$. Denote $F'_{\underline{n}} := \operatorname{unzip}(\iota_E^*F)({\underline{n}})$ for
every ${\underline{n}} \in \mathbb{Z}^r$ and define $h : F'_{\underline{n}} \longrightarrow F_{\underline{n}}$ by setting
$h := \chi_{A_E({\underline{n}}), {\underline{n}}}$. Now $h$ is an isomorphism for every ${\underline{n}} \in \mathbb{Z}^r$, and
moreover, for every pair ${\underline{n}} \leq {\underline{n}}'$, we have $\chi_{A_E({\underline{n}}'), {\underline{n}}'} \circ
\chi_{A_E({\underline{n}}), A_E({\underline{n}}')} = \chi_{{\underline{n}}, {\underline{n}}'} \circ \chi_{A_E({\underline{n}}), {\underline{n}}} = \chi_{A_E({\underline{n}}),
{\underline{n}}'}$. So we obtain $\operatorname{unzip} \circ \operatorname{zip} \cong 1_{\mathcal{M}_E}$.
The other direction is immediate, and we even obtain $\operatorname{zip} \circ \operatorname{unzip} =
1_{\mathcal{L}_E\operatorname{-Rep}}$
\end{proof}
\begin{corollary}
$\mathcal{M}_E$ is an abelian category.
\end{corollary}
Let ${\underline{n}}$ be any element in $\mathcal{L}_E$, then we can consider the free representation
$F^{\underline{n}}$ of $\mathcal{L}_E$. Its unzipping has a particularly easy structure, namely
$\operatorname{unzip}(F^{\underline{n}}) \cong S(-{\underline{n}})$, i.e. the free fine-graded $S$-module with degree
shifted by $-{\underline{n}}$. $\operatorname{unzip}(F^{\underline{n}})$ is the unique $S$-module which has the property
that its ${\underline{n}}'$-th degree is one-dimensional if ${\underline{n}} \leq {\underline{n}}'$ and zero else.
Now we can consider a free resolution of $\operatorname{zip}(E)$ in terms of free representations
of $\mathcal{L}_E$:
\begin{equation*}
0 \longrightarrow F_s \longrightarrow \cdots \longrightarrow F_0 \longrightarrow
\operatorname{zip}(E) \longrightarrow 0
\end{equation*}
where for every $1 \leq i \leq s$:
\begin{equation*}
F_i \cong \bigoplus_{{\underline{n}} \in \mathcal{L}_E} (F^{\underline{n}})^{f^i_{\underline{n}}}
\end{equation*}
where $f^i_{\underline{n}}$ is the free dimension of the vector space associated to ${\underline{n}}$ in the
$(i - 1)$-th syzygy representation.
By unzipping, we obtain an exact sequence of fine-graded $S$-modules:
\begin{equation}
\label{Sres}
0 \longrightarrow \operatorname{unzip}(F_s) \longrightarrow \cdots \longrightarrow \operatorname{unzip}(F_0)
\longrightarrow E \longrightarrow 0
\end{equation}
where for every $1 \leq i \leq s$:
\begin{equation*}
\operatorname{unzip}(F_i) \cong \bigoplus_{{\underline{n}} \in \mathcal{L}_E} S(-{\underline{n}})^{f^i_{\underline{n}}}.
\end{equation*}
In order to show, that this is a minimal free resolution of $E$ over $S$, we consider
the first step of the resolution $0 \rightarrow K_0 \rightarrow \operatorname{unzip} F_0
\rightarrow E
\rightarrow 0$. We define a map $\phi : \mathcal{L}_E \longrightarrow \mathcal{L}_{K_0}$ by mapping
every ${\underline{n}} \in \mathcal{L}_E$ to its anchor element in $\mathcal{L}_{K_0}$:
\begin{equation*}
\phi({\underline{n}}) := A_{K_0}({\underline{n}}).
\end{equation*}
We have the following:
\begin{proposition}
\label{syzcontraction}
The map $\phi$ is a contraction.
\end{proposition}
\begin{proof}
We first show that $\phi(U_E({\underline{n}})) = U_{K_0}(A_{K_0}({\underline{n}}))$ for all ${\underline{n}} \in \mathcal{L}_E$,
where we write $U_E$ and $U_{K_0}$ for open subsets in $\mathcal{L}_E$ and $\mathcal{L}_{K_0}$,
respectively. Clearly, $\phi(U_E({\underline{n}})) \subset U(A_{K_0}(n))$; by construction of
$K_0$, the lattice $\mathcal{L}_{K_0}$ is a sublattice of $\mathcal{L}_E$, so that for any ${\underline{n}}' \in
U_{K_0}(A_{K_0})$ there is ${\underline{n}}'' \in U_E({\underline{n}})$ with $\phi({\underline{n}}'') = {\underline{n}}'$.
Now let ${\underline{n}} \in \mathcal{L}_{K_0}$ and consider the set $\phi^{-1}\big(U_{K_0}({\underline{n}})\big)$,
which consists of all ${\underline{n}}' \in \mathcal{L}_E$ such that ${\underline{n}} \leq A_{K_0}({\underline{n}}')$. ${\underline{n}} \leq {\underline{n}}'$
implies ${\underline{n}} = A_{K_0}({\underline{n}}) \leq A_{K_0}({\underline{n}}')$, and thus $U_E({\underline{n}}) \subset
\phi^{-1}(U_{K_0}({\underline{n}}))$. Moreover, $\phi^{-1}\big(U_{K_0}({\underline{n}})\big) = \{{\underline{n}}' \in \mathcal{L}_E
\mid {\underline{n}} \leq A_{K_0}({\underline{n}}')\}$, and thus $\phi^{-1}\big(U_{K_0}({\underline{n}})\big) \subset
U_E({\underline{n}})$. Hence, $\phi^{-1}\big(U_{K_0}({\underline{n}})\big) = U_E({\underline{n}})$, and $\phi$ is a
contraction.
\end{proof}
\begin{theorem}
Sequence (\ref{Sres}) is a minimal free resolution of $E$ over $S$.
\end{theorem}
\begin{proof}
Observe that the number of $k$-linear independent generators of the module $E$
degree ${\underline{n}}$ is the codimension of the subvector space $\sum_{{\underline{n}}' < {\underline{n}}} x^{{\underline{n}} - {\underline{n}}'}
\cdot E_{{\underline{n}}'}$ of $E_{\underline{n}}$, which coincides with the free dimension of $E_{\underline{n}}$. Thus
$\operatorname{unzip} F_0$ is the minimal free module which surjects onto $E$.
Using proposition \ref{syzcontraction} and lemma \ref{contractionliftres}, we see
that a resolution of $K_0$ over $\mathcal{L}_E$ is a lift of some resolution of $K_0$
restricted to $\mathcal{L}_{K_0}$. Hence, the theorem follows by induction.
\end{proof}
\subsection{Admissible posets and normal semigroup rings}
\label{semigrouprings}
To extend our considerations to the case of normal semigroup rings, consider the
map $M \rightarrow \mathbb{Z}^{\sigma(1)}$,
which without loss of generality we assume to be injective. This corresponds to a
quotient representation $\pi : k^{\sigma(1)} \twoheadrightarrow U_\sigma$ together
with an $A$-graded homogeneous coordinate ring $S := k[x_\rho \mid \rho \in
\sigma(1)]$. For any coherent sheaf $\sh{E}$ over $U_\sigma$, we can consider its
pullback $\pi^* \sh{E}$ over $k^{\sigma(1)}$.
Applying the machinery from subsection \ref{polynomialrings}, we can obtain a
reflexive resolution for $\sh{E}$ by sheafification of the resolution of $\hat{E}$
with respect to the $\operatorname{lcm}$-lattice $\mathcal{L}_{\hat{E}}$:
\begin{equation*}
0 \longrightarrow \operatorname{unzip}(F_r)\breve{\ } \longrightarrow \cdots \longrightarrow
\operatorname{unzip}(F_0)\breve{\ } \longrightarrow \sh{E} \longrightarrow 0,
\end{equation*}
where $\sh{E} \cong \big(\operatorname{unzip} \iota_{\hat{E}}^* \pi^* E\big)\breve{\ }$.
For any anchor element ${\underline{n}} \in \mathcal{L}_{\hat{E}}$, the unzipping of the associated free
representation of $\mathcal{L}_{\hat{E}}$ is isomorphic to $S(-{\underline{n}})$.
Unlike the case of smooth toric varieties, in the general case such a
resolution is not uniquely defined, and it can be possible to obtain shorter
resolutions which are of this type.
\begin{definition}
\label{admissibledef}
Let $E$ be a $M$-graded $k[\sigma_M]$-module. A finite subposet $\mathcal{P} \subset \mathbb{Z}^r \cup
\hat{0}$ is {\em admissible with respect to $E$} if
\begin{enumerate}[(i)]
\item\label{admissibledefi} for all $m \in M$ there exists a {\em unique} ${\underline{n}} \in
\mathcal{P}$ with ${\underline{n}} \leq m$, such that ${\underline{n}}' \leq m$ implies ${\underline{n}}' \leq {\underline{n}}$ for all ${\underline{n}}'
\in \mathcal{P}$;
\item\label{admissibledefii} consider the open set $U_{\underline{n}} = \bigcup_{{\underline{n}} \leq m} U(m)$
in $M$ and the vector space $E(U_{\underline{n}}) = \underset{\leftarrow}{\lim} E_m$, there exists
a vector space $E_{\underline{n}}$ and a diagonal homomorphism $E_{\underline{n}} \longrightarrow E(U_{\underline{n}})$ such
that every induced homomorphism $E_{\underline{n}} \longrightarrow E_m$ is an isomorphism for all
$m \in T_{\underline{n}}$.
\end{enumerate}
We call $E_{\underline{n}}$ the {\em anchor completion} of $E$ at ${\underline{n}}$ and we denote $A_E(m)$ the
unique maximal element ${\underline{n}} \in \mathcal{P}$ with ${\underline{n}} \leq m$.
\end{definition}
Note that in the definition we have identified the elements $m \in M$ with their
image in $\mathbb{Z}^{\sigma(1)}$. For any ${\underline{n}} \in \mathcal{P}$, the homomorphism $E_{\underline{n}} \rightarrow
E(U_{\underline{n}})$ necessarily is injective, and for every ${\underline{n}} \leq {\underline{n}}'$, the composition
\begin{equation*}
E_{\underline{n}} \longrightarrow E(U_{\underline{n}}) \longrightarrow E(U_{{\underline{n}}'})
\end{equation*}
is a diagonal morphism, which factors through the image of $E_{{\underline{n}}'}$, such that we
obtain a morphism between the anchor completions $E_{\underline{n}} \longrightarrow E_{{\underline{n}}'}$.
\begin{lemma}
Assume that $U_\sigma$ is smooth and thus $M \cong \mathbb{Z}^{\sigma(1)}$ and let $\mathcal{P}$ be
some admissible poset with respect to $E$. Then for every subset $m_1, \dots, m_s$ of
$\mathcal{P}$, $\operatorname{lcm}\{m_1, \dots, m_s\}$ is also contained in $\mathcal{P}$. In particular, $\mathcal{P}$
contains the $\operatorname{lcm}$-lattice $\mathcal{L}_E$.
\end{lemma}
\begin{proof}
Denote $m_l := \operatorname{lcm}\{m_1, \dots, m_s\}$. There exists a unique $m \in \mathcal{P}$ such that
$m \geq_\sigma m_l$; but such an $m$ must coincide with $m_l$.
\end{proof}
\comment{
In the case where $\sigma$ has not full dimension, we can consider a splitting of
$M \cong M^\bot_\sigma \oplus M_\sigma$, where $M_\sigma$ is the image of $M$ in
$\mathbb{Z}^{\sigma(1)}$ by the map $M \rightarrow \mathbb{Z}^{\sigma(1)}$ and $M_\sigma^\bot$ its
kernel. $(M_\sigma) \otimes_\mathbb{Z} \mathbb{R}$ the can naturally be identified with the dual
space of the minimal subspace $N_\sigma \subset N_\mathbb{R}$ containing $\sigma$. Moreover,
as partially ordered set, $M_\sigma$ is isomorphic to $M_{\lessgtr_\sigma}$, and
any representation of $M_\sigma$ with respect to the induced partial order is
equivalent to a representation of the preordered set $(M, \leq_\sigma)$.
}
From the observation that $\mathcal{L}_{\hat{E}}$ is admissible, we conclude:
\begin{proposition}
Every finitely generated $k[\sigma_M]$-module $E$ has an admissible poset.
\end{proposition}
\begin{proof}
We take the poset of all ${\underline{n}} \in \mathcal{L}_{\hat{E}}$ such that $\{m \in M \mid A_E(m)
= {\underline{n}}\} \neq \emptyset$.
\end{proof}
Let $\mathcal{P} \subset \mathbb{Z}^{\sigma(1)}$ be an admissible poset, and denote $\mathcal{M}_\mathcal{P}$
the category of finitely generated, $M$-graded $k[\sigma_M]$-modules for which $\mathcal{P}$
is admissible.
Then we define the functor $\operatorname{zip}^\mathcal{P}$ from $\mathcal{M}_\mathcal{P}$ to the category of
$k$-linear representations of $\mathcal{P}$ by:
\begin{equation*}
\operatorname{zip}^\mathcal{P} (E)_{\underline{n}} := E_{\underline{n}},
\end{equation*}
where $E_{\underline{n}}$ is the anchor completion at ${\underline{n}}$.
\begin{remark}
\label{bigadmissible}
Our definition also allows to add anchor elements ${\underline{n}}$ such that the corresponding
set $T_{\underline{n}}$ is empty. In that case we set $E_{\underline{n}} = \underset{\leftarrow}{\lim} E_{{\underline{n}}'}$
for all ${\underline{n}} < {\underline{n}}' \in \mathcal{P}$ such that $T_{{\underline{n}}'} \neq \emptyset$.
\end{remark}
In the opposite direction, from every representation $E$ of an admissible poset $\mathcal{P}$
one can construct a representation of $M$. We define $\operatorname{unzip}^\mathcal{P} (E)$ by setting:
\begin{enumerate}[(i)]
\item $\operatorname{unzip}^\mathcal{P} (E)_m := E_{A(m)}$,
\item $\chi_{m, m'} := E\big(A(m), A(m')\big)$.
\end{enumerate}
\begin{theorem}
\label{admissibleequivalence}
The pair $\operatorname{zip}^\mathcal{P}$ and $\operatorname{unzip}^\mathcal{P}$ is a compression of $\mathcal{M}_\mathcal{P}$, i.e.
$\operatorname{zip}^\mathcal{P}$ and $\operatorname{unzip}^\mathcal{P}$ are functors which establish an equivalence of categories.
\end{theorem}
\begin{proof}
By construction, $\operatorname{unzip}^\mathcal{P} \circ \operatorname{zip}^\mathcal{P} (E) \cong E$ for every $k[\sigma_M]$-module
for which $\mathcal{P}$ is admissible. To obtain functors, we show that any morphism $E
\longrightarrow F$ of objects in $\mathcal{M}_\mathcal{P}$ induces a morphism of the
corresponding representations of $\mathcal{P}$ and vice versa. First, any homomorphism $E
\rightarrow F$ is a homomorphism of sheaves over $(M, \leq_\sigma)$, and thus there
is an induced homomorphism $E_{\underline{n}} \rightarrow E(U_{\underline{n}}) \rightarrow F(U_{\underline{n}})$ for every
${\underline{n}} \in \mathcal{P}$, which factors through the diagonal $F_{\underline{n}}$, hence we obtain a
homomorphism $E_{\underline{n}} \rightarrow F_{\underline{n}}$; the family of such morphisms for every ${\underline{n}} \in
\mathcal{P}$ in a natural way represents a homomorphism of representations of $\mathcal{P}$. In the
other direction, a homomorphism $f: \operatorname{zip}^\mathcal{P} (E) \rightarrow \operatorname{zip}^\mathcal{P}(F)$ unzips
componentwise as $f_m := f_{A(m)} : E_{A(m)} \rightarrow F_{A(m)}$.
\end{proof}
\begin{proposition}
Let ${\underline{n}} \in \mathcal{P}$, then $\mathcal{P}$ is admissible with respect to the reflexive module
$S_{({\underline{n}})}$, and moreover, $S_{{\underline{n}}} \cong \operatorname{unzip}^\mathcal{P} F^{\underline{n}}$.
\end{proposition}
\begin{proof}
Let $m \in M$, then ${\underline{n}} \leq A_E(m)$ iff ${\underline{n}} \leq m$: the first implication is clear,
because $m \leq A_E(m)$; for the second, observe that $A_E(m) \geq \operatorname{lcm} \{A_E(m),
{\underline{n}}\}$, and thus ${\underline{n}} \leq A_E(m)$. So $\mathcal{P}$ is admissible with respect to $S_{({\underline{n}})}$
and $\operatorname{unzip}^\mathcal{P} F^{\underline{n}} \cong S_{({\underline{n}})}$.
\end{proof}
As in the case for polynomial rings, we obtain a reflexive resolution for $E$:
\begin{equation*}
0 \longrightarrow \operatorname{unzip}^\mathcal{P}(F_s) \longrightarrow \cdots \longrightarrow
\operatorname{unzip}^\mathcal{P}(F_0) \longrightarrow E \longrightarrow 0.
\end{equation*}
\begin{example}
\label{admissibleexample1}
Consider the semigroup $\sigma_M$ from \ref{pullbacktorsionexample} and the torsion
sheaf $T$ which is given by:
\begin{equation*}
T_m =
\begin{cases}
k & m = (p, 0), p \geq 0\\
k & m = (0, 1) + p \cdot (1, 2), p \geq 0\\
0 & \text{ else},
\end{cases}
\end{equation*}
and $\chi_{m, m'} = \operatorname{id}$ whenever $T_m, T_{m'} \neq 0$.
We can compare the following two admissible posets,
\begin{equation*}
\mathcal{P}_1 = \{\hat{0}, (0, 0), (-1, 1), (0, 1)\}
\end{equation*}
and
\begin{equation*}
\mathcal{P}_2 = \{\hat{0}, (-1, 0), (0, 1)\}.
\end{equation*}
We have $(\operatorname{zip}^{\mathcal{P}_1}T)_{(0, 0)} = k$, $(\operatorname{zip}^{\mathcal{P}_1}T)_{(-1, 1)} = k$,
$(\operatorname{zip}^{\mathcal{P}_1}T)_{(0, 1)} = 0$, and $\operatorname{zip}^{\mathcal{P}_2}T_{(-1, 0)} = k$,
$\operatorname{zip}^{\mathcal{P}_2}T_{(0, 1)} = 0$. But in the latter case, we have that $T(U_{(-1, 0)})$ is
is the fiber product $k \times_0 k \cong k^2$, such that $\operatorname{zip}^{\mathcal{P}_2}T_{(-1, 0)}$
corresponds to a proper
diagonal homomorphism $k \rightarrow k^2$. These compressions give rise to two
somewhat different resolutions. Via resolving over $\mathcal{P}_1$ and by $\operatorname{unzip}^{\mathcal{P}_1}$, we
obtain:
\begin{equation*}
0 \longrightarrow S_{(0, -1)} \longrightarrow S_{(1, -1)} \oplus S_{(0, 0)}
\longrightarrow T \longrightarrow 0
\end{equation*}
and for $\mathcal{P}_2$:
\begin{equation*}
0 \longrightarrow S_{(0, -1)} \longrightarrow S_{(1, 0)} \longrightarrow T
\longrightarrow 0.
\end{equation*}
In a sense, the module $T$ is like the module $k[x, y] / \langle xy \rangle$
over the polynomial ring $k[x, y]$, whose $\operatorname{lcm}$-lattice is isomorphic to $\mathcal{P}_2$.
However, there exists no unique minimal element in $m \in M$ with $T_m \neq 0$,
so that the consideration of the diagonal morphism indeed is necessary to obtain
a resolution which is like the minimal resolution of $k[x, y] / \langle xy \rangle$.
The left part of figure \ref{f-admissibleexample1} shows a part of the lattice
$\mathbb{Z}^2$; the light grey areas indicate the degrees, where $T_m$ is nonzero. The right
part of figure \ref{f-admissibleexample1} shows the partitioning of $\mathbb{Z}^2$ according
to the two admissible posets $\mathcal{P}_1$ and $\mathcal{P}_2$.
\end{example}
\begin{figure}[ht]
\includegraphics[height=5cm,width=5cm]{admissibleexample1a.eps}\quad\quad\quad\quad
\includegraphics[height=5cm, width=8cm]{admissibleexample1b.eps}
\caption{The module from example \ref{admissibleexample1} and the partitions of
$\mathbb{Z}^2$ with respect to $\mathcal{P}_1$ and $\mathcal{P}_2$}\label{f-admissibleexample1}
\end{figure}
\subsection{Extension of a module to the homogeneous coordinate ring}
\label{homext}
Consider $E$ any $M$-graded $k[\sigma_M]$-module and $S$ the homogeneous coordinate
ring for $U_\sigma$. As we have seen in the previous subsection, one can in a natural
way associate the $S$-module $\hat{E} = E \otimes_{S_0} S$ to $E$. In this subsection
we want to discuss another way to associate a module, denoted $EE$, to $E$ which also
has the property that $E\breve{E} \cong E$, but which behaves better, for instance it
preserves
the property of torsion freeness. For this, for every ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$ we
denote $U_{\underline{n}} := \bigcup_{{\underline{n}} \leq m} U(m)$ an open subset of $(M, \leq_\sigma)$. To
see that the set $\{{\underline{n}} \leq m\}$ is always nonempty, just choose some $m \in
\sigma_M$ with $\langle m, n(\rho) \rangle > 0$ for every $\rho \in \sigma(1)$, then
for every ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$ we can choose an integer $c > 0$ such that ${\underline{n}} \leq
c \cdot m$. Thus for every ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$ the vector space $E(U_{\underline{n}})$ exists,
and can be identified with $\underset{\leftarrow}{\lim} E_m$. If $E$ is finitely
generated, then the $E(U_{\underline{n}})$ are finite dimensional.
\begin{definition}
We define a representation of $(\mathbb{Z}^{\sigma(1)}, \leq)$ by:
\begin{equation*}
EE_{\underline{n}} := E(U_{\underline{n}}).
\end{equation*}
\end{definition}
For every ${\underline{n}} \leq {\underline{n}}'$, the set $U_{\underline{n}}'$ is contained in $U_{\underline{n}}$, and thus we have
a functorial homomorphism $EE_{\underline{n}} \rightarrow EE_{{\underline{n}}'}$, and indeed we obtain a
well-defined representation of $\mathbb{Z}^{\sigma(1)}$.
\begin{proposition}
\label{EEprop}
$EE$ has the following properties:
\begin{enumerate}[(i)]
\item\label{EEpropi} $E\breve{E} = E$.
\item\label{EEpropii} If $E$ is finitely generated, then also $EE$ is finitely
generated.
\item\label{EEpropiii} If $E$ is torsion free, then also $EE$ is torsion free.
\end{enumerate}
\end{proposition}
\begin{proof}
(\ref{EEpropi}): By definition, if ${\underline{n}} = m \in M$, then $EE_m = E(U(m)) = E_m$, thus
$EE_0 = E$.
(\ref{EEpropii}): We apply the criteria of \cite{perling1}, \S 5.3. We have already
stated that the $EE_{\underline{n}}$ are finite dimensional. For all infinite chains $\cdots <
{\underline{n}}_i < {\underline{n}}_{i + 1} < \cdots$, we know that there exists an index $i_0$ such that the
$E_m$ vanish for $m \leq {\underline{n}}_{i_0}$, and thus the $EE_{\underline{n}}$ are zero. To see that there
are only finitely many ${\underline{n}}$ such that $\bigoplus_{{\underline{n}}' < {\underline{n}}} EE_{{\underline{n}}'} \rightarrow
EE_{\underline{n}}$ is not surjective, we choose some finite poset in $\mathbb{Z}^{\sigma(1)}$ which is
admissible with respect to $E$; using this, we find that there are only finitely many
isomorphism classes of vector spaces $EE_{\underline{n}}$.
(\ref{EEpropiii}): As the morphisms $\chi_{m, m'}$ are injective for every $m
\leq_\sigma m'$, the induced morphisms of the limits $EE_{\underline{n}} \rightarrow EE_{{\underline{n}}'}$
are also injective for every ${\underline{n}} \leq {\underline{n}}'$.
\end{proof}
Note that in general $EE$ is not just $\hat{E}$ modulo torsion. For instance, the
module $\check{E}$ of example \ref{pullbacktorsionexample} modulo torsion is not
reflexive, whereas $EE$ is reflexive (see subsection \ref{reflext}).
\begin{proposition}
The $\operatorname{lcm}$-lattice of $EE$ is admissible with respect to $E$.
\end{proposition}
\begin{proof}
Denote $\mathcal{L}$ the $\operatorname{lcm}$-lattice of $E$.
Let $m \in M$ and ${\underline{n}} \in \mathcal{L}$ its anchor element. By definition, the
map $EE_{\underline{n}} \longrightarrow E_m$ is an isomorphism, and thus $\mathcal{L}$ is
admissible.
\end{proof}
So we can use $EE$ as alternative module by which we can construct resolutions of
\msh{E}. In subsection \ref{reflext}, we will do a more explicit analysis of $EE$
for the case where \msh{E} is reflexive.
\subsection{Global resolutions for $\Delta$-families}
\label{deltaglobres}
Now let \msh{E} be an equivariant coherent sheaf over $X$ and $E^\Delta$ its
associated $\Delta$-family. To obtain global resolution of \msh{E}, we want to
extend the techniques considered in the previous two subsections. Denoting
$E^\sigma := \Gamma(U_\sigma, \sh{E})$, we assume that we have a family
$\mathfrak{P} = \{\mathcal{P}^\sigma \mid \sigma \in \Delta\}$ of posets and
compressions $\operatorname{zip}^{\mathcal{P}^\sigma}, \operatorname{unzip}^{\mathcal{P}^\sigma}$ with respect to these posets.
For nicer notation, we write $\operatorname{zip}^\sigma$
and $\operatorname{unzip}^\sigma$ instead of $\operatorname{zip}^{\mathcal{P}^\sigma}$ and $\operatorname{unzip}^{\mathcal{P}^\sigma}$.
For any $m \in M$ we write $A_E^\sigma(m)$ for the anchor element of $m$ in
$\mathcal{P}^\sigma$. We denote $l_{\sigma\tau}$ and $k_{\sigma\tau}$ the gluing maps
for the families $(M, \leq_\sigma)$ and $\mathcal{P}^\sigma$.
\begin{definition}
The collection $\mathfrak{P} = \{\mathcal{P}^\sigma \mid \sigma \in \Delta\}$ is called
{\em admissible} with respect to \msh{E} if it glues over $\Delta$ and for every
$\sigma \in \Delta$, the poset $\mathcal{P}^\sigma$ is admissible with respect to $E^\sigma$.
\end{definition}
\comment{
For any $\tau < \sigma$
and the corresponding admissible posets $\mathcal{P}^\tau$ and
$\mathcal{P}^\sigma$, by abuse of notation we denote $\leq_\tau$ and $\leq_\sigma$ the
partial orders on $\mathcal{P}^\tau$ and $\mathcal{P}^\sigma$, respectively, which in a natural way
are compatible with the corresponding preorders on $M$. The localization
in a natural way is compatible the localization of $\leq_\sigma$ by
$\leq_\tau$ on $M$. This way, via the canonical projection of $\mathbb{Z}^{\sigma(1)}$ onto
$\mathbb{Z}^{\tau(1)}$ we identify the set
$(\mathcal{P}^\sigma)_{\lessgtr_\sigma^\tau}$ with its image in $\mathbb{Z}^{\tau(1)}$.
}
\paragraph{Compressions of $\Delta$-families.}
\begin{proposition}
Let $\mathfrak{P} = \{\mathcal{P}^\sigma \mid \sigma \in \Delta\}$ be a collection of posets
which is admissible with respect to \msh{E} and assume that we have a family of
sheaves $F^\sigma$ which glues over the collection $\mathcal{P}^\sigma$, then the family
$\operatorname{unzip}^\sigma F^\sigma$ is a $\Delta$-family.
\end{proposition}
\begin{proof}
We show that $l^*_{\sigma\tau} \operatorname{unzip}^\tau F^\tau \cong \operatorname{unzip}^\sigma
k_{\sigma\tau}^* F^\tau$ for every $\tau < \sigma$. This follows componentwise from
$(l^*_{\sigma\tau} \operatorname{unzip}^\tau F^\tau)_m = (\operatorname{unzip}^\tau F^\tau)_{l_{\sigma\tau}(m)}
= (\operatorname{unzip}^\tau F^\tau)_m = F^\tau_{A^\tau(m)}$ and
$(\operatorname{unzip}^\sigma k_{\sigma\tau}^* F^\tau)_m = (k^*_{\sigma\tau} F^\tau)_{A^\sigma(m)}
= F^\tau_{k_{\sigma\tau}(A^\sigma(m))} \cong F^\tau_{A^\tau(m)}$, where the last
isomorphism follows from the fact that $k_{\sigma\tau}$ is a contraction.
Denote $\Psi^{\sigma\tau} : k_{\sigma\tau}^* F^\tau \overset{\cong}{\longrightarrow}
F^\sigma$ the gluing maps over the family $\mathcal{P}^\sigma$, then we set $\Phi^{\sigma\tau}
:= \operatorname{unzip}^\sigma \Psi^{\sigma_\tau}$.
By the isomorphisms $l^*_{\sigma\tau} \operatorname{unzip}^\tau F^\tau \overset{\cong}{\rightarrow}
\operatorname{unzip}^\sigma k_{\sigma\tau}^* F^\tau$ for all $\tau < \sigma$ and the functoriality
of $l_{\sigma\tau}^*$, we have for any triple $\rho < \tau < \sigma$ the natural
identification $\Phi^{\sigma\rho} = \Phi^{\sigma \tau} \circ l_{\sigma\tau}^*
\Phi^{\tau\rho}$, and the proposition follows.
\end{proof}
Denote $\mathbf{S}^{\mathfrak{P}}$ the category of coherent equivariant sheaves over
$X$ with respect to which the collection $\mathcal{P}^\sigma$ is admissible.
The operations $\operatorname{zip}^\Delta$ and
$\operatorname{unzip}^\Delta$ are, up to natural isomorphism, mutually inverse functors from
$\mathbf{S}^{\mathfrak{P}}$ to the category sheaves over $\mathfrak{P}$. Thus, we
have:
\begin{theorem}
$\operatorname{zip}^\Delta$ and $\operatorname{unzip}^\Delta$ are a compression of $\mathbf{S}^{\mathfrak{P}}$.
\end{theorem}
In general, there is no canonical choice for admissible posets which automatically
glues over $\Delta$. However, below we will give a gluing procedure starting from
a family of admissible posets over $\Delta_{\max}$, which yields a set of admissible
posets together with a compression for any coherent $\Delta$-family.
For smooth toric varieties, the $\operatorname{lcm}$-lattices already will do the job:
\begin{proposition}
Assume that $X$ is a smooth toric variety, then the family $\operatorname{zip}^\sigma E^\sigma$,
with respect to the $\operatorname{lcm}$-lattices of the modules $E^\sigma$, glues over $\Delta$.
\end{proposition}
\begin{proof}
Without loss of generality assume that $\sigma$ has full dimension in $N_\mathbb{R}$. Let
$\tau < \sigma$ and for $m \in M$ denote $\bar{m}$ its class in $M / \tau^\bot_M$.
For any $m' \leq_\sigma m \in M$, an isomorphism $\chi^\sigma_{m', m} : E^\sigma_{m'}
\longrightarrow E^\sigma_m$ implies an isomorphism $\chi^\tau_{m', m} :
E^\tau_{m'} \longrightarrow E^\tau_m$, and for any $m \in M$, we have that $\bar{m}'
\in I^\tau(\bar{m})$ implies that there exists some $m'' \in \bar{m}$ with $m'' \in
I^\sigma(m)$. Therefore, if we denote $\mathcal{P}^\sigma$, $\mathcal{P}^\tau$ the $\operatorname{lcm}$-lattices of
$E^\sigma$ and $E^\tau$, respectively, we have a canonical contraction
$(\mathcal{P}^\sigma)_{\lessgtr_\sigma^\tau} \longrightarrow (\mathcal{P}^\tau)_{\lessgtr_\tau}$ given
by $\mathcal{P}^\sigma \ni m \mapsto A^\tau_E(\bar{m})$.
\end{proof}
\paragraph{Refining compressions.}
For some arbitrary choice of $\mathfrak{P}$, the category $\mathbf{S}^\mathfrak{P}$
in general contains not enough reflexive sheaves of rank one to construct global
resolutions. This is true even for the collection of
$\operatorname{lcm}$-lattices of \msh{E} over a smooth toric variety. The reason for this is that
relations of the module $E^\sigma$ which are encoded in the lattice $\mathcal{P}^\sigma$,
must no longer be present in the localization $E^\tau$ for $\tau < \sigma$, and thus
are ``contracted'' over $\mathcal{P}^\tau$. But if we construct a resolution over $U_\sigma$,
these relations still are present after we restrict to $U_\tau$, and thus are also
felt by neighbouring cones $\sigma'$ with $\tau \subset \sigma \cap \sigma'$. So,
in order to construct a resolution with respect to any admissible collection of
posets which glues over $\Delta$, we have to refine these posets in a way which
allows that any locally given reflexive sheaf $\sh{O}_{U_\sigma}(D)$ can be extended
to a suitable reflexive sheaf of rank one over $X$
\comment{
To construct a resolution of \msh{E} starting from the local data given by
compression with respect to admissible posets, one has to take into account that
every free resolution over some poset $\mathcal{P}^\sigma$ will interfer with the free
resolutions
over all the other $\mathcal{P}^\tau$'s; this is meant in the sense that for any ${\underline{n}} \in
\mathcal{P}^\sigma$ we have to extend the associated sheaf $\sh{O}_{U_\sigma}(\sum_{\rho}
n_\rho D_\rho)$, and correspondingly the map $\sh{O}_{U_\sigma} (\sum_{\rho} n_\rho
D_\rho) \longrightarrow \sh{E}\mid_{U_\sigma}$, to the whole of $X$. This corresponds
to extending in a suitable sense the map $F^{\underline{n}} \longrightarrow \operatorname{zip}^\sigma E^\sigma$
to maps over the other $\mathcal{P}^\sigma$. For this, the $\mathcal{P}^\sigma$ a priori are not fine
enough, as they do not take into account relations of the $E^\tau$ coming from
neighbouring $\mathcal{P}^\tau$, which are killed by restricting to $\mathcal{P}^{\sigma \cap \tau}$,
as these will be felt by globalized resolutions. So the first step to construct
global resolutions is to construct a refinements of the $\mathcal{P}^\sigma$, i.e. admissible
posets $\tilde{\mathcal{P}}^\sigma$ with $\mathcal{P}^\sigma \subset \tilde{\mathcal{P}}^\sigma$, such that
$\big(\tilde{\mathcal{P}}^{\sigma_1}\big)_{\lessgtr_{\sigma_1}^\tau} =
\big(\tilde{\mathcal{P}}^{\sigma_1}\big)_{\lessgtr_{\sigma_2}^\tau}$
for every $\tau, \sigma_1, \sigma_2 \in \Delta$, where $\tau = \sigma_1 \cap
\sigma_2$.
}
\begin{example}
\label{p1p1example}
Consider the toric surface $\mathbb{P}^1 \times \mathbb{P}^1$. The associated fan has
four rays $\rho^1, \dots, \rho^4$ and four maximal cones $\sigma^{12}, \sigma^{23},
\sigma^{34}, \sigma^{41}$, where $\sigma_{ij}$ is spanned by the rays $\rho^i$,
$\rho^j$. The associated semigroups $\sigma_M^{ij}$ are generated in $M \cong \mathbb{Z}^2$
by $\{(1, 0), (0, 1)\}$, $\{(0, 1), (-1, 0)\}$, $\{(-1, 0), (0, -1)\}$, $\{(0, -1),
(1, 0)\}$, respectively. We consider the sky\-scraper sheaf \msh{S} which has two
stalks at the orbits $\orb{\sigma^{12}}$ and $\orb{\sigma^{23}}$, respectively,
which are, as $k[\sigma_M^{ij}]$-modules, given by:
\begin{equation*}
\Gamma(U_{12}, \sh{S}) = k \cdot \chi\big((0, 0)\big), \quad
\Gamma(U_{23}, \sh{S}) = k \cdot \chi\big((-2, 2)\big).
\end{equation*}
Figure \ref{f-p1p1example} shows the four dual cones describing $\mathbb{P}^1 \times
\mathbb{P}^1$, slightly moved away from each other, and an indication of the
associated $\operatorname{lcm}$-lattices. The squares indicate the degrees $(0, 0)$ and $(-2, -2)$
where the stalks of \msh{S} sit, the light grey triangles indicate the anchor
elements of the two $\sigma$-families.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{p1p1example.eps}
\end{center}
\caption{Skyscraper sheaf over $\mathbb{P}^1 \times \mathbb{P}^1$.}\label{f-p1p1example}
\end{figure}
The dark grey triangles show the additional anchor elements which come from the
transition from one $\sigma$-family into another some of which have to enter a global
resolution. One possible resolution would be:
\begin{gather*}
0 \longrightarrow \sh{O}(-D_1 - D_2 - 2D_3) \oplus \sh{O}(-3D_2 - 3 D_3)
\longrightarrow \\
\sh{O}(-D_1 - 2 D_3) \oplus \sh{O}(-D_2-2D_3) \oplus \sh{O}(-2D_2 - 3 D_3) \oplus
\sh{O}(-3D_2 - 2 D_3) \longrightarrow \\ \sh{O}(-2D_3) \oplus
\sh{O}(-2 D_2 - 2 D_3) \longrightarrow
\sh{S} \longrightarrow 0
\end{gather*}
where we write $D_i$ instead of $D_{\rho_i}$. Note that the choice of other admissible
posets instead of the $\operatorname{lcm}$-lattices can lead to more convenient resolutions.
\end{example}
We consider any family of posets $\mathfrak{P} = \{\mathcal{P}^\sigma \mid \sigma
\in \Delta\}$, which glues over $\Delta$ and which is admissible with respect to
\msh{E}. We are going to construct a family of posets $\tilde{\mathfrak{P}} =
\{\tilde{\mathcal{P}}^\sigma \mid \sigma \in \Delta\}$ which glues over $\Delta$, is
admissible with respect to \msh{E}, and whose associated category
$\mathbf{S}^{\tilde{\mathfrak{P}}}$ has enough reflexive sheaves.
We start bottom-up and we consider $\mathcal{P}^\rho \subset \mathbb{Z}^{\rho(1)} \cong \mathbb{Z}$ for some
$\rho \in {\Delta(1)}$. In fact, $\mathcal{P}^\rho$ is a linear chain, i.e. a totally orderd
subset of $\mathbb{Z}$. For every $\sigma > \rho$, we consider
$\big(\mathcal{P}^\sigma\big)_{\lessgtr_\sigma^\rho}$ as a subset of $\mathbb{Z}^{\rho(1)}$, such that
the hooking $h_{\sigma\rho}$ becomes the natural inclusion $\mathcal{P}^\rho \subset
\big(\mathcal{P}^\sigma\big)_{\lessgtr_\sigma^\rho}$ in $\mathbb{Z}^{\rho(1)}$. We set
\begin{equation*}
\tilde{\mathcal{P}}^\rho := \bigcup_{\rho < \sigma}\big(\mathcal{P}^\sigma\big)_{\lessgtr_\sigma^\rho}.
\end{equation*}
Now fix some $\sigma \in \Delta$ together with its admissible lattice
$\mathcal{P}^\sigma \subset \mathbb{Z}^{\sigma(1)}$. For every $\tau < \sigma$, we consider the
natural embedding $\mathbb{Z}^{\tau(1)} \hookrightarrow \mathbb{Z}^{\sigma(1)}$ which is induced
by the inclusion $\tau(1) \subset \sigma(1)$. In particular, every element $i \in
\tilde{\mathcal{P}}^\rho$ becomes an element of $\mathbb{Z}^{\sigma(1)}$ which is nonzero only at the
$\rho$th position. For every $m \in M$ there exists a unique anchor
element $A(m) \in \mathcal{P}^\sigma$. We refine now by setting:
\begin{equation*}
\tilde{A}(m) = \operatorname{lcm} \{i \in \tilde{\mathcal{P}}^\rho \mid i \leq \langle m,
n(\rho) \rangle\}_{\rho \in \sigma(1)}
\end{equation*}
and
\begin{equation*}
\tilde{\mathcal{P}}^\sigma := \{\tilde{A}(m) \mid m \in M\} \cup \mathcal{P}^\sigma,
\end{equation*}
where we observe that $A(m)$ is of the form
\begin{equation*}
A(m) = \big(\max\{i \in \tilde{\mathcal{P}}^\rho \mid i \leq A(m)_{\lessgtr_\sigma^\rho}\}
\mid \rho \in \sigma(1)\big)
\end{equation*}
and thus $\mathcal{P}^\sigma \subset \tilde{\mathcal{P}}^\sigma$
Clearly, $\tilde{\mathcal{P}}^\sigma$ is admissible with respect to $E^\sigma$, and we can
consider the compressions $\operatorname{zip}^{\tilde{P}^\sigma}$, $\operatorname{unzip}^{\tilde{P}^\sigma}$.
Using the identification of $(\mathcal{P}^\sigma)_{\lessgtr^\tau_\sigma}$ with its image
in $\mathbb{Z}^{\tau(1)}$, we have:
\begin{proposition}
For any $\tau \in \Delta$, $\tilde{\mathcal{P}}^\tau = \bigcup_{\tau < \sigma}
(\mathcal{P}^\sigma)_{\lessgtr^\tau_\sigma}$, where the union runs over all $\sigma
\in \Delta_{\max}$ with $\tau < \sigma$.
\end{proposition}
\begin{proof}
This follows because for any $\eta < \tau$, $\mathcal{P}^\eta = (\mathcal{P}^\eta)_{\lessgtr_\eta} \in
\mathbb{Z}^\tau$ is a subset of the image of $(\mathcal{P}^\sigma)_{\lessgtr_\sigma^\eta}$ in
$\mathbb{Z}^{\eta(1)}$. Thus $\tilde{\mathcal{P}}^\rho = \bigcup_{\rho < \sigma}
(\mathcal{P}^\sigma)_{\lessgtr_\sigma^\rho}$ where the union runs over all maximal cones. Now
the proposition follows from $\mathcal{P}^\tau \subset (\mathcal{P}^\sigma)_{\lessgtr^\tau_\sigma}$
and by the generatedness of $\tilde{\mathcal{P}}^\sigma$ by $\mathcal{P}^\sigma$ and the
$\tilde{\mathcal{P}}^\rho$.
\end{proof}
By this proposition, we can conclude that the choice of any collection of admissible
posets leads to a collection of admissible posets which glue over $\Delta$:
\begin{corollary}
The family $\tilde{\mathcal{P}}^\tau$ is generated by the $\mathcal{P}^\sigma$, where $\tau$ runs over
$\Delta_{\max}$.
\end{corollary}
\begin{corollary}
$(\tilde{\mathcal{P}}^\sigma)_{\lessgtr_\sigma^\tau} = \tilde{\mathcal{P}}^\tau$ for all
$\tau < \sigma \in \Delta$.
\end{corollary}
By combining these two corollaries, we obtain:
\begin{proposition}
The family of sheaves $\operatorname{zip}^{\tilde{\mathcal{P}}^\sigma} E^\sigma$ glues over $\Delta$.
\end{proposition}
\paragraph{Global resolutions.}
Recall from section \ref{homext} that for every $\sigma \in \Delta$ we can construct
the extension module $EE^\sigma$ of $E^\sigma$ over the ring $S^\sigma$. In the
equivariant setting, the category of modules over $S^\sigma$ is equivalent to that
of the
ring $S_{x^{\hat{\sigma}}}$, and we can extend $EE^\sigma$ to a module over this
ring. By naturality of the construction, the $EE^\sigma$ glue to a sheaf $E\sh{E}$
over $\hat{X}$, and we obtain the $S$-module
$E\hat{E} := \Gamma(k^{\Delta(1)}, E\sh{E})$. We have the following properties for
$E\hat{E}$, which immediately follow from the corresponding properties of
proposition \ref{EEprop}:
\begin{proposition}
$E\hat{E}$ has the following properties:
\begin{enumerate}[(i)]
\item $E\hat{E}\breve{\ } \cong \sh{E}$.
\item If \msh{E} is coherent, then $E\hat{E}$ is finitely generated.
\item if \msh{E} is torsion free, then $E\hat{E}$ is torsion free.
\end{enumerate}
\end{proposition}
This way, a global resolution can be constructed as the descend of a
resolution of the $S$-module $E\hat{E}$ with respect to its $\operatorname{lcm}$-lattice. However,
there are more possibilites to resolve \msh{E} which use $E\hat{E}$ but do not
require the cost of computing the whole $\operatorname{lcm}$-lattice of $E\hat{E}$.
For this, we give a more precise picture of $E\hat{E}$.
For every $\sigma \in \Delta$ and every $\tau < \sigma$, denote $\pi_\sigma :
\mathbb{Z}^{\Delta(1)} \longrightarrow \mathbb{Z}^{\sigma(1)}$ and $\pi^\sigma_\tau: \mathbb{Z}^{\sigma(1)}
\longrightarrow \mathbb{Z}^{\tau(1)}$ the canonical projections. For any
${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$, $EE^\sigma_{\underline{n}}$ is defined to be the inverse limit
$E^\sigma(U_{\underline{n}}) := \underset{\leftarrow}{\lim} E^\sigma(U_m)$.
For any $\tau < \sigma \in \Delta$, there is the canonical map induced by
localization: $E^\sigma(U_{\underline{n}}) \longrightarrow E^\tau(U_{\pi^\sigma_\tau({\underline{n}})})$, and
for any ${\underline{n}} \in {\mathbb{Z}^\rays}$, we obtain the directed system
\begin{equation*}
\xymatrix{
E\hat{E}_{\underline{n}} \ar[r]^{\pi_\sigma} \ar[rd]_{\pi_\tau} & EE^\sigma_{\pi_\sigma({\underline{n}})}
\ar[d]^{\pi^\sigma_\tau} \\
& EE^\tau_{\pi_\tau({\underline{n}})}
}
\end{equation*}
whose final object is the vector space $EE^0_{\pi_0({\underline{n}})}$.
The component $E\hat{E}_{\underline{n}}$ then has the universal property of the inverse limit of
this system:
\begin{equation*}
E\hat{E}_{\underline{n}} = \underset{\leftarrow}{\lim} E^\sigma_{\underline{n}} = \underset{\leftarrow}{\lim}
E^\sigma_m
\end{equation*}
where the latter limit runs over all $\sigma \in \Delta$ and the system of all $m
\in M$ such that ${\underline{n}} \leq_\sigma m$.
\begin{definition}
A {\em lift} $\tilde{\mathcal{P}}^\lambda$ of the collection $\tilde{\mathcal{P}}^\sigma$ is a
collection of injective,
order preserving maps $\lambda_\sigma: \tilde{\mathcal{P}}^\sigma \hookrightarrow \mathbb{Z}^{\Delta(1)}
\cup \hat{0}$ such that
\begin{enumerate}[(i)]
\item $\pi_\sigma\big(\lambda_\sigma({\underline{n}})\big) = {\underline{n}}$ for all ${\underline{n}} \in
\tilde {\mathcal{P}}^\sigma$;
\item $\lambda_\tau({\underline{n}}) = \operatorname{lcm}\Big\{\lambda_\sigma \big((\pi^\tau_\sigma)^{-1}({\underline{n}})
\cap \tilde{\mathcal{P}}^\sigma\big) \mid \tau < \sigma\Big\}$
for every ${\underline{n}} \in \tilde{\mathcal{P}}^\tau$;
\item for all ${\underline{n}} \in \mathcal{P}^\sigma$ the composition $EE_{\lambda_\sigma({\underline{n}})}
\longrightarrow (EE_{\lambda_\sigma({\underline{n}})})_{\lessgtr^\sigma_\Delta} \longrightarrow
EE^\sigma_{\underline{n}}$ is surjective.
\end{enumerate}
We identify $\tilde{\mathcal{P}}^\lambda$ with the poset given by the image of the maps
$\lambda_\sigma$.
\end{definition}
There is, of course, no most natural choice for a lift $\lambda$, but a general
choice which always works, is:
\begin{equation*}
\lambda_\sigma({\underline{n}})_\rho =
\begin{cases}
n_\rho & \text{ if } \rho \in \sigma(1), \\
\max\{i \in \tilde{\mathcal{P}}^\rho\} & \text{ if } \tilde{\mathcal{P}}^\rho \neq \hat{0}, \\
0 & \text{ else}.
\end{cases}
\end{equation*}
In the case where \msh{E} is reflexive, it is possible to do a more efficient
general choice, as we will see in subsection \ref{reflext}.
With respect to a lift $\lambda$, we can define the submodule $E\hat{E}_\lambda
\subset E\hat{E}$ as follows. For every $\sigma \in \Delta$ and all ${\underline{n}} \in
\tilde{\mathcal{P}}^\sigma$ we choose a subvector space $E'_{\lambda_\sigma({\underline{n}})} \subset
E\hat{E}_{\lambda_\sigma({\underline{n}})}$ such that the induced morphism
$E'_{\lambda_\sigma({\underline{n}})} \longrightarrow EE^\sigma_{\underline{n}}$ is surjective. Then we define
$E\hat{E}_\lambda$ to be the module generated by the $E'_{\lambda_\sigma({\underline{n}})}$.
Note that in spite we do not make explicit this choice in the notation, we always
assume it implicitly.
\begin{proposition}
$(E\hat{E}_\lambda)\breve{\ } \cong \sh{E}$.
\end{proposition}
\begin{proof}
As the homomorphism $EE_{\lambda_\sigma({\underline{n}})} \longrightarrow EE^\sigma_{\underline{n}}$ is
surjective, the map $(EE_{\lambda_\sigma({\underline{n}})})_{\lessgtr^\sigma_\Delta}
\longrightarrow EE^\sigma_{\underline{n}}$ becomes an isomorphism. Moreover, $\pi_\sigma \big(
\tilde{\mathcal{P}}^\sigma\big) = \tilde{\mathcal{P}}^\sigma$, so that the induced representation
on $\tilde{\mathcal{P}}^\sigma$ is the same as for $E\hat{E}$.
\end{proof}
So, we can also use $E\hat{E}_\lambda$ to construct resolutions:
\begin{corollary}
Every lift $\tilde{\mathcal{P}}^\lambda$ gives rise to a resolution of \msh{E}.
\end{corollary}
Instead of taking $E\hat{E}_\lambda$, we can also take directly the poset
$\tilde{\mathcal{P}}^\lambda$. Denote $i : \tilde{\mathcal{P}}^\lambda \hookrightarrow \mathbb{Z}^{\Delta(1)}$ the
canonical inclusion. We define:
\begin{equation*}
\operatorname{zip}^\lambda\sh{E} := i^* E\hat{E}_\lambda.
\end{equation*}
For any given representation $E$ of $\tilde{\mathcal{P}}^\lambda$, by the canonical projection
we obtain back the admissible poset $\tilde{\mathcal{P}}^\sigma$ as the image of
$\tilde{\mathcal{P}}^\lambda$ in $\mathbb{Z}^{\sigma(1)}$, together with the localization of the
representation $E$. These representations glue naturally over $\tilde{\mathcal{P}}^\sigma$,
and we can use them reconstruct the
module $E\hat{E}$. By taking the submodule generated in the degrees given by
$\tilde{\mathcal{P}}^\lambda$, we obtain $E\hat{E}_\lambda$. Using either $E\hat{E}$ or
$E\hat{E}_\lambda$, by sheafification we get back the sheaf \msh{E}. We denote this
procedure $\operatorname{unzip}^\lambda E$.
\begin{proposition}
The category of representations of $\tilde{\mathcal{P}}^\lambda$ is a full subcategory of
the category of sheaves which glue over the collection $\tilde{\mathcal{P}}^\sigma$.
\end{proposition}
\begin{proof}
We only remark that these categories in general can not be equivalent, as the lift
$\lambda^\sigma$ for every ${\underline{n}} \in \tilde{\mathcal{P}}^\sigma$ fixes the choice of free
representations of $\tilde{\mathcal{P}}^{\sigma'}$, $\sigma' \in \Delta$, which glue together
with the free representation of ${\underline{n}}$ over $\tilde{\mathcal{P}}^\sigma$.
\end{proof}
Using this correspondence, we obtain finally the finest class of global resolutions
for \msh{E}. For the representation $E^\lambda$ of $\tilde{\mathcal{P}}^\lambda$ for some
lift $\lambda$, we construct the free resolution in the category of
$\tilde{\mathcal{P}}^\lambda$-representations, and by unzipping we obtain:
\begin{equation*}
0 \longrightarrow \operatorname{unzip}^\lambda F_s \longrightarrow \cdots \longrightarrow
\operatorname{unzip}^\lambda F_0 \longrightarrow \sh{E} \longrightarrow 0.
\end{equation*}
However, nothing prevents us from taking a different lift $\lambda$ for every
syzygy of \msh{E}, and as we will see below, this will be quite natural for doing so
in the case of reflexive sheaves. So we can consider a sequence of lifts
$\lambda_0, \dots \lambda_s$ and a corresponding resolution:
\begin{equation*}
0 \longrightarrow \operatorname{unzip}^{\lambda_s} F_s \longrightarrow \cdots \longrightarrow
\operatorname{unzip}^{\lambda_0} F_0 \longrightarrow \sh{E} \longrightarrow 0.
\end{equation*}
Here, the finiteness of the sequence follows that in every step we eliminate
minimal elements of the induced representations of the admissible posets
$\tilde{\mathcal{P}}^\sigma$, but the length $s$ finally may depend on the successive choice
of the lifts.
\begin{example}
\label{p1p1example2}
Consider the variety $\mathbb{P}^1 \times \mathbb{P}^1$ and skyscraper sheaf \msh{S}
similar to that of example \ref{p1p1example}, but this time with slightly different
gradings:
\begin{equation*}
\Gamma(U_{12}, \sh{S}) = k \cdot \chi(1, 1), \quad \Gamma(U_{23}, \sh{S}) = k \cdot
\chi(-1, 1).
\end{equation*}
Figure \ref{f-p1p1example2} shows the corresponding posets. Explicitly, we have:
\begin{align*}
\tilde{\mathcal{P}}^{12} & = \{\hat{0}, (1, 1), (2, 1), (1, 2), (2, 2)\} \\
\tilde{\mathcal{P}}^{23} & = \{\hat{0}, (1, 1), (2, 1), (1, 2), (2, 2)\} \\
\tilde{\mathcal{P}}^2 & = \{\hat{0}, 1, 2\}
\end{align*}
(for brevity, we suppress the ray $\rho_4$). We consider the lifts:
\begin{align*}
\lambda_{12}\big(\tilde{\mathcal{P}}^{12}\big) & = \{\hat{0}, (1, 1, 1), (2, 1, 2), (1, 2, 1),
(2, 2, 2)\} \\
\lambda_{12}\big(\tilde{\mathcal{P}}^{23}\big) & = \{\hat{0}, (1, 1, 1), (1, 2, 1), (2, 1, 2),
(2, 2, 2)\} \\
\lambda_{12}\big(\tilde{\mathcal{P}}^2\big) & = \{\hat{0}, (2, 1, 2), (2, 2, 2)\}.
\end{align*}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{p1p1example2.eps}
\end{center}
\caption{Another skyscraper sheaf over $\mathbb{P}^1 \times \mathbb{P}^1$.}\label{f-p1p1example2}
\end{figure}
The sheaf $E\hat{E}$ is given by
\begin{equation*}
E\hat{E}_{\underline{n}} \cong
\begin{cases}
k^2 & \text{ if } {\underline{n}} = (1, 1, 1) \\
0 & \text{ else}.
\end{cases}
\end{equation*}
Choosing the one-dimensional diagonal $k \subset E \hat{E}_{(1,1,1)}$, we obtain as
resolution:
\begin{gather*}
0 \longrightarrow \sh{O}(- 2 D_1 - 2 D_2 - 2 D_3) \longrightarrow
\sh{O}(- 2 D_1 - D_2 - 2 D_3) \oplus \sh{O}(- D_1 - 2 D_2 - D_3) \\ \longrightarrow
\sh{O}(-D_1 - D_2 - D_3)
\longrightarrow \sh{S} \longrightarrow 0
\end{gather*}
\end{example}
\comment{
X smooth => p^lambda \subset lcm-lattice of EE_\lambda
\begin{proposition}
\end{proposition}
\begin{proof}
\end{proof}
}
\comment{
Now consider $\sh{O} := \sh{O}_X(\sum_{\rho \in {\Delta(1)}} i_\rho D_\rho)$ any reflexive
sheaf of rank one over $X$. For every $\sigma \in \Delta$, the module $O^\sigma :=
\Gamma(U_\sigma, \sh{O})$ can
irredundantly be described by the admissible poset consisting of two elements,
$\{\hat{0}, {\underline{n}}_\sigma\}$, where ${\underline{n}}_\sigma = (-i_\rho \mid \rho \in \sigma(1)) \in
\mathbb{Z}^{\sigma(1)}$, and the free representation $F^{{\underline{n}}_\sigma}$. Every admissible
subposet of $\mathbb{Z}^{\sigma(1)}$ which contains
this poset is also admissible for $O^\sigma$. In turn, consider any collection
of admissible posets $\tilde{\mathfrak{P}} = \{\tilde{P}^\sigma \mid \sigma \in
\Delta\}$ as constructed above. If $-i_\rho \in \tilde{\mathcal{P}}^\sigma$ for every $\rho
\in {\Delta(1)}$, then ${\underline{n}}_\sigma$ is contained in $\tilde{\mathcal{P}}^\sigma$ for every $\sigma$,
and the category $\mathbf{S}^{\tilde{\mathfrak{P}}}$
contains \msh{O}.
We begin by taking for any $\sigma \in
\Delta$ the first step of a resolution of $E^\sigma$ with respect to the
admissible poset $\tilde{\mathcal{P}}^\sigma$, and extend it a sheaf homomorphism to all of
$X$. Let this be
\begin{equation*}
0 \longrightarrow K^\sigma_0 \longrightarrow F^\sigma_0 \longrightarrow E^\sigma
\longrightarrow 0,
\end{equation*}
where $F_0 \cong \bigoplus_{{\underline{n}} \in \tilde{P}^\sigma} S_{({\underline{n}})}^{f^\sigma_{\underline{n}}}$. Now,
for any ${\underline{n}} \in \tilde{\mathcal{P}}^\sigma$ with $f^\sigma_{\underline{n}} \neq 0$, consider the
map $\sh{O}_{U_\sigma}(D_{\underline{n}}) \longrightarrow \sh{E}\vert_{U_\sigma}$.
\begin{lemma}
The homomorphism $\sh{O}_{U_\sigma}(D_{\underline{n}}) \longrightarrow \sh{E}\vert_{U_\sigma}$
extends to $\sh{O}_X(D_{{\underline{n}}'}) \longrightarrow \sh{E}$.
\end{lemma}
\begin{proof}
For every $\eta \in \Delta$, the sheaf $\sh{O}_{U_\eta}(D_{{\underline{n}}'})$ corresponds to
a representation $F^{{\underline{n}}'_\eta}$ of $\tilde{\mathcal{P}}^\eta$, where ${\underline{n}}'_\eta$ is the image
of ${\underline{n}}'$ in $\mathbb{Z}^{\eta(1)}$ by the projection $\mathbb{Z}^{\Delta(1)} \twoheadrightarrow
\mathbb{Z}^{\eta(1)}$. Let $\tau = \sigma \cap \eta$, then we choose a homomorphism of the
free representation $F^{{\underline{n}}'_\eta}$ to $E^\eta$ such that the morphism induced over
$\tilde{P}^\tau$ coincides with the homomorphism $F^{{\underline{n}}'_\tau} \rightarrow E^\tau$
induced by the homomorphism $F^{\underline{n}} \rightarrow E^\sigma$. Note that we always can
do this because $E^\tau_{{\underline{n}}_\tau} \cong E^{\eta}_{{\underline{n}}'_\eta}$. So we have a system
of homomorphisms which glue over $\Delta$, corresponding to a sheaf homomorphism
$\sh{O}_X(D_{{\underline{n}}'}) \longrightarrow \sh{E}$.
\end{proof}
As a first step of a global resolution we define:
\begin{equation*}
\sh{F}_0 := \bigoplus_{\sigma \in \Delta_{\max}} \bigoplus_{{\underline{n}} \in \tilde{\mathcal{P}}^\sigma}
\sh{O}_X(D_{{\underline{n}}'})^{f^\sigma_{\underline{n}}},
\end{equation*}
the map $\sh{F}_0 \twoheadrightarrow \sh{E}$ being given by the summation of all the
maps $\sh{O}_X(D_{{\underline{n}}'}) \rightarrow \sh{E}$. We then have a short exact sequence
$0 \rightarrow \sh{K}_0 \rightarrow \sh{F}_0 \rightarrow \sh{E} \rightarrow 0$ where
$\sh{K}_0$ is some torsion free sheaf for which $\tilde{\mathfrak{P}}$ is a
collection of admissible posets. By iterating, we obtain a global resolution of
\msh{E}. In fact, this resolution is finite:
}
\comment{
\begin{theorem}
Above construction leads to a finite global resolution of \msh{E}
\begin{equation*}
0 \longrightarrow \sh{F}_s \longrightarrow \cdots \longrightarrow
\sh{F}_0 \longrightarrow \sh{E} \longrightarrow 0.
\end{equation*}
where $\sh{F}_i \cong \bigoplus_{\sigma \in \Delta_{\max}}
\bigoplus_{{\underline{n}} \in \tilde{\mathcal{P}}^\sigma} \sh{O}_X(D_{{\underline{n}}'})^{f^\sigma_{{\underline{n}}, i}}$ for
some integers $f^\sigma_{{\underline{n}}, i} \geq 0$.
\end{theorem}
\begin{proof}
It remains only to show that the above resolution is finite. The choice of ${\underline{n}}' \in
{\mathbb{Z}^\rays}$ for every ${\underline{n}} \in \tilde{\mathcal{P}}^\sigma$ amounts to an embedding of
$\tilde{\mathcal{P}}^\sigma$ in ${\mathbb{Z}^\rays}$. Therefore we can consider the poset
consisting of the union of all embeddings of $\tilde{\mathcal{P}}^\sigma$ in $\weildivisors$.
As this set is finite and thus has minimal elements, by analogous arguments as in
proposition \ref{repres}, it follows that the
number of those ${\underline{n}}'$ whose $i$-th syzygy representation is nonzero, decreases in
every step of the resolution and at the end must become zero.
\end{proof}
}
\comment{
\begin{theorem}
Let \msh{E} be a coherent equivariant sheaf over a toric variety $X$. Then there
exists a compression $\{\operatorname{zip}^\sigma, \operatorname{unzip}^\sigma \mid \sigma \in \Delta\}$ for the
full subcategory of equivariant coherent sheaves whose objects are sheaves \msh{F}
such that $\mathcal{L}^{\sigma, M}_\sh{F} \subset \mathcal{L}^{\sigma, M}_\sh{E}$ for every $\sigma
\in \Delta$.
\end{theorem}
\begin{proof}
\end{proof}
}
\comment{
\subsection{Reduction to smaller global resolutions}
The prescription given in the previous section in general is far from being optimal
in the sense that requires too many summands in each step. Consider any coherent
equivariant sheaf \msh{E} and the family of admissible posets $\tilde{\mathcal{P}}^\sigma$.
Let $\{{\underline{n}}^\sigma \mid {\underline{n}}^\sigma \in \mathcal{P}^\sigma\}$ be any collection of anchor elements
such that $\pi_1({\underline{n}}^{\sigma_1}) = \pi_2({\underline{n}}^{\sigma_2})$ for all $\sigma_1, \sigma_2
\in \Delta$ and $\pi_i : \mathbb{Z}^{\sigma_i(1)} \longrightarrow
\mathbb{Z}^{(\sigma_1 \cap \sigma_2)(1)}$. For any ${\underline{n}}^\sigma$, we have a homomorphism to
the limit vector space, $E^\sigma_{{\underline{n}}^\sigma} \longrightarrow \mathbf{E}$, and with
respect to this system of homomorphisms, we can consider the inverse limit
$\mathbf{E}^i := \underset{\leftarrow}{\lim} E^\sigma_{{\underline{n}}_\sigma}$. Likewise, for
the reflexive sheaf $\sh{O}_X(D_{\underline{n}})$ we have the sheaf over $\tilde{\mathcal{P}}^\sigma$
given by free representations $O^{{\underline{n}}_\sigma}$, and the associated limits
$\mathbf{O}$, $\mathbf{O}^i$. The homomorphisms $F^{{\underline{n}}_\sigma} \longrightarrow
E^\sigma_{{\underline{n}}_\sigma}$ induce the homomorphisms $\mathbf{O} \longrightarrow
\mathbf{E}$ and $\mathbf{O}^i \longrightarrow \mathbf{E}^i$ where the latter is a
{\em diagonal homomorphism}. Consider the subsystem of $E^\sigma_{{\underline{n}}_\sigma}$ which
is given by the vector spaces
\begin{equation*}
E^\sigma_{< {\underline{n}}_\sigma} := \sum_{{\underline{n}}' < {\underline{n}}} E^\sigma({\underline{n}}', {\underline{n}}) E_{{\underline{n}}'}
\end{equation*}
for every $\sigma \in \Delta$. We denote $\mathbf{E}^i_<$ the inverse limit of this
system.
}
\section{Reflexive Sheaves and Vector Space Arrangements}
\label{reflexivesheaves}
\subsection{Reflexive sheaves and their canonical admissible posets}
For an equivariant reflexive sheaf over a toric variety, i.e. a sheaf \msh{E} which is
isomorphic to its bidual, $\sh{E} \cong \sh{E}\check{\ }\check{\ }$, the associated
$\Delta$-family has a quite efficient representation. To every equivariant
coherent sheaf \msh{E} over $U_\sigma$, one can associate a limit vector space
$\mathbf{E}^\sigma := \underset{\rightarrow}{\lim} E_m^\sigma$, and by the gluing
of the $E^\sigma$ over the collection of posets $(M, \leq_\sigma)$, there is a
functorial isomorphism $\mathbf{E}^\sigma \rightarrow \mathbf{E}^0 =: \mathbf{E}$,
where $0$ denotes the zero cone in $\Delta$, and moreover, $\dim \mathbf{E} = \operatorname{rk}
\sh{E}$. As explained in detail in
\cite{perling1}, section 5 (see also \cite{Kly90}, \cite{Kly91}), every
equivariant reflexive sheaf \msh{E} is determined by a set of filtrations
\begin{equation*}
\cdots \subset E^\rho(i) \subset E^\rho(i + 1) \subset \cdots \subset \mathbf{E}
\end{equation*}
for every ray $\rho \in {\Delta(1)}$. These filtrations must be {\em full}, i.e.
$E^\rho(i) = 0$ for very small $i$, and $E^\rho(i) = \mathbf{E}$
for $i$ very large. The corresponding $\sigma$-families then can be constructed from
these filtrations by setting
\begin{equation*}
E^\sigma_m = \bigcap_{\rho \in \sigma(1)} E^\rho\big(\langle m, n(\rho) \rangle\big).
\end{equation*}
In fact, this construction establishes an equivalence of categories between
equivariant reflexive sheaves and vector spaces with full filtrations.
The morphisms in the latter category are vector space homomorphisms which are
compatible with the filtrations in the $\Delta$-family sense (\cite{perling1},
Theorem 5.29).
Consider a reflexive module $E^\sigma$ over the ring $k[\sigma_M]$, where without
loss of generality we assume that $\sigma$ has full dimension in $N_\mathbb{R}$. To any
such module there is associated the {\em subvector space arrangement} $\{E^\sigma_m
\mid m \in M\}$ in the limit vector space $\mathbf{E}$, where $E_m^\sigma =
\bigcup_{\rho \in \sigma(1)} E^\rho\big(\langle m, n(\rho) \rangle\big)$. This
arrangement in a natural way is a poset, where the partial order is given by
inclusion. We will show that we can embed this poset into $\mathbb{Z}^{\sigma(1)}$ such
that it becomes an admissible poset for $E^\sigma$.
\begin{definition}
For every $m \in M$, we define $\kappa_\rho(m) = \min \{i \in \mathbb{Z} \mid E_m^\sigma
\subset E^\rho(i)\}$ and the {\em anchor} of $m$ by:
\begin{equation*}
A(m) = \big(\kappa^\rho_m \mid \rho \in \sigma(1)\big) \in \mathbb{Z}^{\sigma(1)}.
\end{equation*}
We denote $\mathcal{P}_{E^\sigma}$ the subposet $\{A(m) \mid m \in M\}$ of
$\mathbb{Z}^{\sigma(1)}$.
\end{definition}
\begin{proposition}
\label{admissibleproof}
$\mathcal{P}_{E^\sigma}$ is admissible with respect to $E^\sigma$.
\end{proposition}
\begin{proof}
First, clearly, $A(m) \leq m$ for all $m \in M$. Now assume that $A(m') \leq m$ for
some $m' \in M$. $A(m') \leq m$ implies that $E^\sigma_{m'} \subset E^\sigma_m$,
and thus $A(m') \leq A(m)$. Now, by definition $\bigcap_{\rho \in \sigma(1)} E^\rho
\big(n_\rho) = E^\sigma_m$ for all $m \in T_{\underline{n}}$ for some ${\underline{n}} \in \mathcal{P}_{E^\sigma}$.
\end{proof}
\begin{definition}
We call $\mathcal{P}_{E^\sigma}$ the {\em canonical admissible poset} of $E^\sigma$.
\end{definition}
An important fact for understanding the structure of reflexive modules is the
following
\begin{lemma}
\label{reflkeylemma}
Let $\mathcal{P}_{E^\sigma}$ be the canonical admissible poset of $E^\sigma$. Then $E^\sigma_m
\subset E^\sigma_{m'}$ iff $A(m) \leq_\sigma A(m')$. Moreover, $E^\sigma_m =
E^\sigma_{m'}$ iff $A(m) = A(m')$.
\end{lemma}
\begin{proof}
Assume first that $E^\sigma_m \subset E^\sigma_{m'}$. Then for every $\rho \in
\sigma(1)$ it follows that $\min\{i \mid E^\sigma_m \subset E^\rho(i)\} \leq \min\{i
\mid E^\sigma_{m'} \subset E^\rho(i)\}$, and thus $A(m) \leq_\sigma A(m')$. In the
other direction, denote ${\underline{n}} := A(m)$, ${\underline{n}}' := A(m')$, then $n_\rho \leq n'_\rho$ for
every $\rho \in \sigma(1)$ and $E^\rho(n_\rho) \subseteq E^\rho(n'_\rho)$, and thus
$E^\sigma_m \subset E^\sigma_{m'}$.
\end{proof}
\begin{proposition}
If $U_\sigma$ is smooth, then, as a poset, the vector space arrangement associated
to $E^\sigma$ is isomorphic to its $\operatorname{lcm}$-lattice.
\end{proposition}
\begin{proof}
Because $U_\sigma$ is smooth, for every $m \in M$, the anchor element $A(m)$ is an
element of $M$, and we conclude from the proof of proposition \ref{admissibleproof},
that $A(m)$ is the unique member of $I(m)$. For any two $A(m) \neq A(m')$, the vector
spaces $E^\sigma_m$ and $E^\sigma_{m'}$ do not coincide, and thus the vector space
$E^\sigma_{m''}$, where $m'' = \operatorname{lcm} \{m, m'\}$, contains at least the sum $E^\sigma_m
+ E^\sigma_{m'}$. Moreover, we have that $E^\sigma_{m''} = \bigcap_{\rho \in
\sigma(1)} E^\rho\big(\langle m'', n(\rho) \rangle\big)$, where for every $\rho \in
\sigma(1)$ $E^\rho\big(\langle m'', n(\rho) \rangle\big)$ contains $E^\sigma_m$ and
$E^\sigma_{m'}$, and thus $\langle m'', n(\rho) \rangle \geq \max\{\langle m, n(\rho)
\rangle, \langle m', n(\rho) \rangle\}$. So $m''$ is the minimal element of $M$
with respect to the partial order $\sigma_M$, such that $E^\sigma_{m''}$ contains
both, $E^\sigma_m$ and $E^\sigma_{m'}$.
\end{proof}
\begin{example}
\label{intersectionimprove}
We give an example which shows that the choice of another admissible poset instead of
the canonical one can improve the resolution.
Consider the subsemigroup $\sigma_M$ of $\mathbb{Z}^2$ which is generated by $(1, 0)$,
$(1, 1)$ and $(1, 2)$; the corresponding cone $\sigma$ has two rays $\rho_1, \rho_2$
with primitive elements $n(\rho_1) = (2, 1)$, $n(\rho_2) = (0, 1)$. Let $\mathbf{E}
\cong k^3$ and consider the filtrations
\begin{equation*}
E^{\rho_1}(i) =
\begin{cases}
0 & \text{ for } i < 0 \\
E_1 & \text{ for } i = 0 \\
\mathbf{E} & \text{ for } i > 0
\end{cases}\qquad
E^{\rho_2}(i) =
\begin{cases}
0 & \text{ for } i < 1 \\
E_2 & \text{ for } i = 1 \\
\mathbf{E} & \text{ for } i > 1.
\end{cases}
\end{equation*}
with $\dim E_i = 2$ and the $E_i$ in general position.
The corresponding canonical admissible poset is
$\mathcal{P} = \{\hat{0}, (0, 2), (1, 1), (1, 2)\}$ and it leads to the resolution
\begin{equation*}
0 \longrightarrow S_{(2, 2)} \longrightarrow S_{(1, 1)}^2 \oplus S_{(0, 2)}^2
\longrightarrow E \longrightarrow 0
\end{equation*}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{intersectionimprove.eps}
\end{center}
\caption{Canonical admissible poset and the poset generated by its intersections}\label{f-intersectionimprove}
\end{figure}
If we choose instead the poset $\mathcal{P}'= \{\hat{0}, (0, 1), (0, 2), (1, 1), (1, 2)\}$,
the associated representation of $\mathcal{P}'$ maps $(0, 1)$ to the the subvector space
$E_1 \cap E_2$ of $\mathbf{E}$. The corresponding vector space arrangements are shown
as linear configurations in $\mathbb{P}\mathbf{E} \cong \mathbb{P}^2$ in figure
\ref{f-intersectionimprove}. The grey dot in the right figure denotes the intersection
$E_1 \cap E_2$. The corresponding resolution becomes:
\begin{equation*}
0 \longrightarrow S_{(0, 1)} \oplus S_{(0, 2)} \oplus S_{(1, 1)} \longrightarrow E
\longrightarrow 0,
\end{equation*}
i.e. $E$ splits into a direct sum of reflexive sheaves of rank one.
\end{example}
\subsection{Extensions to the homogeneous coordinate ring}
\label{reflext}
We first investigate the structure of the module $E\hat{E}$ where \msh{E} is
reflexive.
For this, we first consider the module $EE^\sigma$ for any $\sigma \in \Delta$. Its
determination is a straightforward computation:
\begin{proposition}
Let $E^\sigma$ be a reflexive $k[\sigma_M]$-module given by filtrations $E^\rho(i)$.
Then its extension is given by:
\begin{equation*}
EE^\sigma_{\underline{n}} = \bigcap_{\rho \in \sigma(1)} E^\rho(n_\rho).
\end{equation*}
\end{proposition}
\begin{proof}
We have $EE^\sigma_{\underline{n}} = \underset{\leftarrow}{\lim} E_m^\sigma$, where the limit runs
over all ${\underline{n}} \leq m$. As all morphisms $\chi^\sigma_{m, m'}$ are injective, this
direct limit immediately translates into an intersection in $\mathbf{E}^\sigma$:
\begin{align*}
\underset{\leftarrow}{\lim} E_m^\sigma & = \bigcap_{{\underline{n}} \leq m} E_m^\sigma \\
& = \bigcap_{{\underline{n}} \leq m} \bigcap_{\rho \in \sigma(1)} E^\rho\big(\langle m, n(\rho)
\rangle \big).
\end{align*}
It is always possible to find $m \in M$ for some $\tau \in \sigma(1)$ such that
$\langle m, n(\tau) \rangle = n_\tau$ and $\langle m, n(\rho) \rangle >> 0$ for any
$\tau \neq \rho$, such that $\bigcap_{\rho \in \sigma(1)} E^\rho\big(\langle m,
n(\rho) \rangle\big) = E^\tau(n_\tau)$. Thus we obtain $\bigcap_{\rho \in \sigma(1)}
E^\rho(n_\rho) \subset EE^\sigma_{\underline{n}} \subset \bigcap_{\rho \in \sigma(1)}
E^\rho(n_\rho)$ and the proposition follows.
\end{proof}
So the module $E\hat{E}^\sigma$ can explicitly be described by the filtrations for
\msh{E} and in fact, it is a reflexive module. To describe its filtrations more
explicitly, we use the quotient representation $\pi : k^{\sigma(1)} \longrightarrow
U_\sigma$.
For each $\rho \in \sigma(1)$, the restriction of \msh{E} to $U_\rho$ is a locally
free sheaf and thus if we restrict $\pi$ to $U_{\hat{\rho}}$, the pullback
\begin{equation*}
\hat{\sh{E}}^{\hat{\rho}} :=
(\pi\vert_{U_{\hat{\rho}}})^*\sh{E}\vert_{U_{\rho}}
\end{equation*}
is locally free over $U_{\hat{\rho}}$. To determine the filtration associated to
$\hat{\sh{E}}^{\hat{\rho}} $, consider the injective map
\begin{equation*}
\alpha_\rho: M / \rho^\bot_M \longrightarrow \mathbb{Z}^{\rho(1)}.
\end{equation*}
Then every element $i \in {\mathbb{Z}^\rays} / \hat{\rho}^\bot_{\hat{M}}$ lies in a unique
intervall $\alpha_\rho(j) \leq i < \alpha_\rho(j + 1)$ for some $j \in M /
\rho^\bot_M \cong \mathbb{Z}$. $\hat{\sh{E}}^{\hat{\rho}}$ then can be described
by a filtration of $\mathbf{E}$, which is given by
\begin{equation*}
E\hat{E}^{\hat{\rho}}(i) = E^\rho(j) \text{ for } \alpha_\rho(j) \leq i
< \alpha_\rho(j + 1).
\end{equation*}
The reflexive $S$-module defined by set of filtrations $E\hat{E}^{\hat{\rho}}(i)$
for every $\rho \in {\Delta(1)}$ then can be identified with $E\hat{E}$.
\begin{proposition}
Let \msh{E} be a reflexive sheaf, then there is an isomorphism
$\hat{E}\check{\ }\check{\ } \cong E\hat{E}$.
\end{proposition}
\subsection{Resolutions for vector space arrangements and reflexive equivariant
sheaves}
\paragraph{The affine case.}
First we consider resolutions for a reflexive $M$-graded module $E^\sigma$ over
$k[\sigma_M]$ with filtrations $E^\rho(i)$ for $\rho \in \sigma(1)$. Revisiting the
resolution process of proposition \ref{repres} for the corresponding representation
of the canonical admissible poset $\mathcal{P}_{E^\sigma}$, we find by lemma
\ref{reflkeylemma} that for any ${\underline{n}} \in \mathcal{P}_{E^\sigma}$, the vector space
$E^\sigma_{< {\underline{n}}}$ is the subvector space of $E^\sigma_{\underline{n}}$ which is spanned by all its
{\em sub}vector spaces in the arrangement $\mathcal{P}_{E^\sigma}$. We have the first step of
its resolution
\begin{equation*}
0 \longrightarrow K_0 \longrightarrow F_0 \longrightarrow E^\sigma \longrightarrow 0
\end{equation*}
such that $F_0$ is a reflexive module $F_0 \cong \bigoplus_{{\underline{n}} \in \mathcal{P}^\sigma}
S_{({\underline{n}})}^{f_{\underline{n}}}$
which is defined by filtrations $F^\rho(i)$ in a limit vector space $\mathbf{F}$,
defining a vector space arrangement $\mathcal{Q} := \{F_m \mid m \in M\}$.
\begin{proposition}
The poset underlying the vector space arrangement $\mathcal{Q}$ is isomorphic to
$\mathcal{P}_{E^\sigma}$.
\end{proposition}
\begin{proof}
The dimension of the vector space $F_{0, {\underline{n}}}$ is given by the number of ${\underline{n}}'
\leq_\sigma {\underline{n}}$; by lemma \ref{reflkeylemma} we have that $E^\sigma_m \subsetneq
E^\sigma_{m'}$ iff $A(m) < A(m')$, and thus the number of ${\underline{n}}'' \in \mathcal{P}$ for which
$E^\sigma_m$ has positive free dimension and which ${\underline{n}}'' \leq A(m)$ is smaller than
the number of such elements with ${\underline{n}}'' \leq A(m')$.
\end{proof}
The kernel $K_0$ is a reflexive module, given by filtrations $K^\rho(i) =
\operatorname{ker}(F^\rho(i)\rightarrow E\rho(i))$ of the kernel vector space
$\mathbf{K} = \operatorname{ker}(\mathbf{F} \rightarrow \mathbf{E})$. However,
the canonical admissible poset of $K_0$ is no longer isomorphic to $\mathcal{P}_{E^\sigma}$,
but we have the following:
\begin{proposition}
The canonical admissible poset of $K_0$ is a contraction of $\mathcal{P}_{E^\sigma}$.
\end{proposition}
\begin{proof}
We define the retraction morphism $r : \mathcal{P}_{E^\sigma} \longrightarrow \mathcal{P}_{K_0}$ by
mapping $A_{E^\sigma}(m)$ to $A_{K_0}(m)$ for all $m \in M$. For any $E^\sigma_m
\subset E^\sigma_{m'}$ we have $K_{0, m} \subset K_{0, m'}$, and thus
$r\big(U(A_{E^\sigma}(m))\big) \subset U(A_{K_0}(m))$. The other inclusion follows
because $\mathcal{P}_{K_0}$ is admissible for $K_0$. On the other hand, let ${\underline{n}} \in
\mathcal{P}_{K_0}$, then ${\underline{n}} \leq {\underline{n}}'$ for every ${\underline{n}}' \in r^{-1}\big(U({\underline{n}})\big)$, and
${\underline{n}} \in r^{-1}\big(U({\underline{n}})\big)$, thus $r^{-1}\big(U({\underline{n}})\big) = U({\underline{n}})$ in
$\mathcal{P}_{E^\sigma}$.
\end{proof}
By \ref{contractionliftres} this in particular implies that we can iterate and
the resolution of the vector space arrangement $\mathcal{P}_{E^\sigma}$ is equivalent to
a resolution of $E^\sigma$. We have:
\begin{equation*}
0 \longrightarrow F_s \longrightarrow \cdots \longrightarrow F_0 \longrightarrow
E^\sigma \longrightarrow 0
\end{equation*}
where $F_i \cong \bigoplus_{{\underline{n}} \in \mathcal{P}_{E^\sigma}} S_{({\underline{n}})}^{f^i_{\underline{n}}}$.
The shape of the resolution can be changed by chosing another admissible poset for
$E^\sigma$. This in turn is equivalent to adding {\em any} set of intersections of
vector spaces in $\mathcal{P}_{E^\sigma}$. To see this, we pass to the module $EE^\sigma$.
The arrangement of this module is complete with respect to intersections, and
every anchor element of the canonical admissible poset of $E^\sigma$ is by definition
an anchor element of the $\operatorname{lcm}$-lattice of $EE^\sigma$. In particular, for every
${\underline{n}} \in \mathcal{L}_{EE^\sigma}$ with ${\underline{n}} \leq m$, we have ${\underline{n}} \leq A_{E^\sigma}({\underline{n}})$
by lemma \ref{reflkeylemma}, so that condition (\ref{admissibledefi}) of definition
\ref{admissibledef} is fulfilled. Moreover, as $T_{\underline{n}}$ is empty if ${\underline{n}}$ is not from
$\mathcal{P}_{E^\sigma}$, condition (\ref{admissibledefii}) is trivially fulfilled.
\paragraph{The global case.}
Now we assume that \msh{E} is a reflexive sheaf over an arbitrary toric variety $X$,
represented by filtrations $E^\rho(i)$ of some vector space $\mathbf{E}$ for every
$\rho \in {\Delta(1)}$. We denote
$\mathcal{P}^\sigma$ the canonical admissible posets for every $E^\sigma$. To make contact
with the formalism of section \ref{deltaglobres}, we first consider the refinements
$\tilde{\mathcal{P}}^\sigma$.
\begin{lemma}
$\mathcal{P}^\sigma$ is a contraction of $\tilde{\mathcal{P}}^\sigma$ for every $\sigma \in \Delta$.
\end{lemma}
\begin{proof}
For every $\rho \in {\Delta(1)}$, the canonical admissible poset $\mathcal{P}^\rho$ is given by
$\hat{0}$ and some sequence $i^\rho_1 < \dots < i^\rho_{k_\rho}$ in $\mathbb{Z}$, where
$k_\rho < \operatorname{rk} \sh{E}$, such that $E^\rho(i) = E^\rho(i + j)$ for $j \geq 0$ if and
only if there exists no $i_{p}^\rho$ for some $p \in \{1, \dots, k_\rho\}$
such that $i < i^\rho_p \leq i + j$.
For every $\rho \in {\Delta(1)}$ and every $\rho < \sigma$, we have
$(\mathcal{P}^\sigma)_{\lessgtr_\sigma^\rho} = \mathcal{P}^\rho$, and thus $\mathcal{P}^\rho = \tilde{\mathcal{P}}^\rho$.
Recall that $\tilde{A}^\sigma$ was defined as the least common multiple of the
elements $\max\{i \in \tilde{\mathcal{P}}^\rho \mid i \leq \langle m, n(\rho) \rangle\}$,
where $\tilde{\mathcal{P}}$ is considered as subset of $\mathbb{Z}^{\sigma(1)}$ via the canonical
embedding $\mathbb{Z}^\rho \hookrightarrow \mathbb{Z}^{\sigma(1)}$. Denote $r : \tilde{\mathcal{P}}^\sigma
\longrightarrow \mathcal{P}^\sigma$, mapping the anchor $\tilde{A}^\sigma(m)$ to
$A^\sigma(m)$. Clearly, $r$ is surjective. Then for any $\tilde{A}^\sigma(m) \in
\tilde{\mathcal{P}}^\sigma$, the image of $U\big(\tilde{A}^\sigma(m)\big)$ is
$U\big(A^\sigma(m)\big)$. For any ${\underline{n}} \in \mathcal{P}^\sigma$, $r^{-1}({\underline{n}}) = {\underline{n}}$, so
$r^{-1}\big(U({\underline{n}})\big) = U({\underline{n}})$ (the latter as an open subset of $\tilde{\mathcal{P}}^\sigma$,
and the lemma follows.
\end{proof}
For resolving \msh{E}, we now must define a lift $\lambda$ of the collection
$\tilde{\mathcal{P}}^\sigma$ to $\weildivisors$. For every $\sigma \in \Delta$, we define
$\lambda_\sigma : \tilde{\mathcal{P}}^\sigma \longrightarrow {\mathbb{Z}^\rays}$ by
\begin{equation*}
\big(\lambda_\sigma({\underline{n}})\big)_\rho =
\begin{cases}
\min\{i \mid E^\sigma_{\underline{n}} \subset E^\rho(i)\} & \text{ for } \rho \in {\Delta(1)} \setminus
\sigma(1) \\
n_\rho & \text{ for } \rho \in \sigma(1).
\end{cases}
\end{equation*}
\begin{proposition}
The collection $\lambda_\sigma$ is a lift of $\tilde{\mathcal{P}}^\sigma$.
\end{proposition}
\begin{proof}
By definition, $(\pi_\sigma \circ \lambda_\sigma)({\underline{n}}) = {\underline{n}}$ for every ${\underline{n}} \in
\tilde{\mathcal{P}}^\sigma$. We show that $\lambda_\tau({\underline{n}}) = \operatorname{lcm}\Big\{\lambda_\sigma
\big((\pi^\tau_\sigma)^{-1}({\underline{n}}) \cap \tilde{\mathcal{P}}^\sigma\big) \mid \tau < \sigma\Big\}$
for every $\tilde{\mathcal{P}}^\tau$.
For this, observe that $E\hat{E}_{\lambda_\sigma({\underline{n}})} = E^\sigma_{\underline{n}}$, because
\begin{align*}
E^\sigma_{\underline{n}} & = \bigcap_{\rho \in \sigma(1)} E^\rho(n_\rho)
\subset E\hat{E}_{\lambda_\sigma({\underline{n}})}
= \bigcap_{\rho \in {\Delta(1)}} E^\rho( \lambda_\sigma({\underline{n}})_\rho) \\
& = E^\sigma_{\underline{n}} \cap \big(\bigcap_{\rho \in {\Delta(1)} \setminus \sigma(1)}
E^\rho(\lambda_\sigma({\underline{n}})_\rho)\big) \subset E^\sigma_{\underline{n}}.
\end{align*}
\end{proof}
Now, the lift $\lambda$ gives rise to a subarrangement of the subvector space
arrangement of the arrangement associated to $E\hat{E}$, which is given by the
union of arrangements in $\mathbf{E}$:
\begin{equation*}
\mathcal{P}^\Delta := \bigcup_{\sigma \in \Delta} \mathcal{P}^\sigma = \Big\{\bigcap_{\rho \in
\sigma(1)} E^\rho\big(\langle m, n(\rho) \rangle\big) \mid \sigma \in \Delta, m \in
M \Big\} = \bigcup_{\sigma \in \Delta} \{\lambda_\sigma({\underline{n}}) \mid {\underline{n}} \in \mathcal{P}^\sigma\}
\end{equation*}
The first step $0 \rightarrow \sh{K}_0 \rightarrow \sh{F}_0 \rightarrow \sh{E}
\rightarrow 0$ of the global resolution of $\mathbf{E}$ then is given by the sheaf
\begin{equation*}
\sh{F}_0 \cong \bigoplus_{{\underline{n}} \in \mathcal{P}^\Delta} \sh{O}\big(D_{\lambda({\underline{n}})}\big)^{f^0_{\underline{n}}},
\end{equation*}
where $f^0_{\underline{n}}$ is the free dimension of the vector space $E_{\underline{n}}$. By iteration, we
get a free resolution, which at the same time is a resolution of the vector
space arrangement $\mathcal{P}^\Delta$.
Note that this resolution coincides with the resolution of the module
$E\hat{E}_\lambda$ over $S$. The global resolution of \msh{E}
constructed using $E\hat{E}$ is given by the
minimal resolution given by the vector space arrangement in $\mathbf{E}$ which is
generated by {\em all} intersections of the vector spaces $E^\rho(i)$.
\subsection{Resolutions of Cohen-Macaulay modules}
\label{cmmodules}
Let $E$ be a (maximal) Cohen-Macaulay module over $k[\sigma_M]$, where $\sigma$
has full dimension in $N_\mathbb{R}$. We show that our resolutions behave well in the sense
that the maximal length of regular sequences does not decrease. We follow
\cite{brunsherzog} \S 1.5, and say that the graded module $E$ is Cohen-Macaulay if
$\operatorname{grade}_\mathfrak{m} E = \dim k[\sigma_M]$, where $\mathfrak{m}$ is the
maximal homogeneous ideal of $k[\sigma_M]$ which is generated by all non-unit
monomials.
\begin{theorem}
\label{CMresolution}
Let $E$ be an $M$-graded Cohen-Macaulay module over $k[\sigma_M]$ and consider the
resolution
\begin{equation*}
0 \longrightarrow F_s \longrightarrow \cdots \longrightarrow F_0 \longrightarrow E
\longrightarrow 0
\end{equation*}
corresponding to the canonical admissible poset of $E$. Then every $F_i$ is a
direct sum of Cohen-Macaulay modules of rank one.
\end{theorem}
\begin{proof}
We need only to consider the first step of the resolution $0 \rightarrow K_0
\rightarrow F_0 \rightarrow E \rightarrow 0$, as $K_0$ will be Cohen-Macaulay if
$F_0$ and $E$ are Cohen-Macaulay; the result then follows by induction. If we
restrict the surjection from $F_0$ to $E$ to a direct summand of rank one $R$ of
$F_0$, we necessarily obtain an injection $0 \rightarrow R \rightarrow E$.
We show that any $E$-regular sequence by construction also is a $R$-regular sequence.
Let $x_1, \dots, x_r$ be a $E$-regular sequence and denote $\mathbf{x}_i$ the ideal
generated by $x_1, \dots, x_i$, for $1 \leq i \leq r$. We consider the diagram
\begin{equation*}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & & \\
0 \ar[r] & \mathbf{x}_i R \ar[r] \ar[d] & \mathbf{x}_i E \ar[r] \ar[d] & \mathbf{x}_i
E / \mathbf{x}_i R \ar[r] \ar[d]^\alpha & 0 \\
0 \ar[r] & R \ar[r] \ar[d] & E \ar[r] \ar[d] & E / R \ar[r] & 0 \\
& R / \mathbf{x}_i R \ar[r]^\beta \ar[d] & E / \mathbf{x}_i E \ar[d] & & \\
& 0 & 0 & &
}
\end{equation*}
If $\alpha$ is injective, then also $\beta$ is injective, and the element $x_{i + 1}$
is a nonzero divisor of $R / \mathbf{x}_i R$, as it is a nonzero divisor of $E /
\mathbf{x}_i E$. To show that $\alpha$ is injective, we show that there exists no
$e_1, \dots, e_i \in E$ such that $y := \sum_{j = 1}^i x_j e_j$ is in $R$ but not in
$\mathbf{x}_i R$. This sum decomposes into homogeneous summands $y = \sum_{m \in M}
y_m$ where $y_m = \sum_{j = 1}^i \sum_{m' \in M} x_{j, m'} \cdot e_{j, m - m'}$.
If we write $x_{j, m'} = a_{j, m'} \chi(m')$, this sum can be written as
$\sum_{j = 1}^i \sum_{m' \in M} a_{j, m'} \chi(m') \cdot e_{j, m - m'}$. Now we
split the set $\{m' \in M \mid e_{j, m - m'} \neq 0\} = U_j \coprod V_j$, where
$U_j = \{m' \mid R_{m - m'} \neq 0\}$. By construction of the inclusion of $R$ in
$E$, there does not exist any $m'' \in V_j$ such that $E_{m''}$ contains a one
dimensional subvector space whose image in $\mathbf{E}$ coincides with the image of
$R$. For any $m' \in V_j$, the elements
$\chi(m') \cdot e_{j, m - m'}$ must be contained in the subvector space $F_m$ spanned
by all $E_{m''}$ with $m'' < m$, and writing the equations modulo $F_m$, we can
replace every $e_j$ by some $f_j$ such that $f_{j, m - m'} = 0$ if $m' \in V_j$
and $\sum_j x_j f_j = y$. Thus we have for every $m$ the equation $x_m =
\sum_{j = 1}^i \sum_{m' \in U_j} a_{j, m'} \chi(m') \cdot f_{j, m - m'}$.
For $m' \in U_j$, we can project every
$f_{j, m - m'}$ to some appropriate $r_{j, m - m'} \in R_{m - m'}$, such that
$x_m = \sum_{j = 1}^i \sum_{m' \in U_j} a_{j, m'} \chi(m') \cdot r_{j, m - m'}
$.
Therefore, we have $x_m \in \mathbf{x}_i R$, from which follows that $\alpha$ is
injective.
\comment{
By construction
of the inclusion of $R$ in $E$, there does not exist any $m'' \leq_\sigma m$ such
that $E_{m''}$ contains a one dimensional subvector space whose image in $\mathbf{E}$
coincides with the image of $R$. Thus the summands in the sum over $V_j$ are all
contained in a proper subvector space of $E_m$ which does not contain $R_m$. Hence
we have a sum of two vectors in $E_m$ which lies in $R_m$, where one of the summands
lies in $R_m$, and the other outside, and so the second summand must vanish.
Therefore, we have $e_{j, m - m'} = r_{j, m - m'}$ for every nonzero $e_{j, m - m'}$,
and $e_1, \dots, e_i \in R$, from which follows that $\alpha$ is injective.}
\comment{
$x = \sum_{j = 1}^t \chi(m_j) f_j$, where the $f_j$ are homogeneous elements of $E$.
The image of $R$ in $\mathbf{E}$ spans a one-dimensional subvector space $\langle
R \rangle$ of $\mathbf{E}$, and for $x$ to be in $R$, it is a necessary condition
that the image of every summand $\chi(m_j) f_j$ is contained in $\langle R \rangle$.
The $\chi(m_j)$ act as identity homomorphism on the vector space $\mathbf{E}$ and
thus on its subvector spaces. So, it follows that every $f_j$ must be in $\langle R
\rangle$. By construction, the submodule $R$ of $E$ is the reflexive module
associated to an anchor element ${\underline{n}} \in \mathbb{Z}^{\sigma(1)}$, and by this it lies outside
of the span of all subvector spaces of $\bigcap_{\rho \in \sigma(1)} E^\rho(n_\rho)$,
hence the $f_j$ must be contained in $R$, and thus $x \in \mathbf{x}_i R$,
}
\end{proof}
\begin{corollary}[from proof of theorem \ref{CMresolution}]
Let $E$ be any reflexive $k[\sigma_M]$-module and $F_i$ as in the theorem, then
$\operatorname{grade}_\mathfrak{m} F_i \geq \operatorname{grade}_\mathfrak{m} E$ for
all $0 \leq i \leq s$.
\end{corollary}
\subsection{Reflexive models for vector space arrangements}
\label{reflexivemodels}
In this subsection we want to make a few remarks on how resolutions of vector space
arrangements can efficiently be constructed by passing to appropriate reflexive
modules over the polynomial ring. The point here is resolutions of such modules are
a standard task for many computer algebra systems. However, to make use of such
systems, one has to construct appropriate input data from the arrangement.
Let $\mathcal{V}$ be a subvector space arrangement of some vector space $\mathbf{V}$.
We make two assumptions on $\mathcal{V}$; the first is that $\mathcal{V}$ is complete
with respect to intersections, that is, for any subset $W_1, \dots, W_r \in
\mathcal{V}$, the intersection $W_1 \cap \dots \cap W_n$ is also in $\mathcal{V}$.
The second assumption is that the input data for $\mathcal{V}$ is given by a set of
vectors $v^W_1, \dots, v^W_{i_W}$ such that $W$ is the span over $k$ of all $v^V_i$
where $V \subset W$ and $i = 1, \dots, i_V$. Moreover, we assume that this set is
irredundant, i.e. $i_W = \operatorname{codim}_W \sum_{V \subsetneq W} V$.
Using this input data, the first step
\begin{equation*}
0 \longrightarrow K_0 \longrightarrow F_0 \overset{M}{\longrightarrow} \mathbf{V}
\longrightarrow 0
\end{equation*}
of the resolution of $\mathcal{V}$ is nearly tautological. Assume that we have chosen
a basis for $\mathbf{V}$, then $F_0$ is given by a basis $e^W_i$, $i = 1, \dots, i_W$
in one-to-one correspondence to the vectors $v^W_i$, and the matrix $M$ then can
simply be chosen as having the vectors $v^W_i$ as its columns, i.e. $M = (v^W_{ij})$.
By associating to
$\mathcal{V}$ the structure of some appropriate fine-graded module, the matrix $M$
becomes a monomial matrix for which syzygies can be computed.
\begin{definition}
\begin{enumerate}[(i)]
\item A {\em reflexive model} for $\mathcal{V}$ is an inclusion $\mathcal{V}
\hookrightarrow (\mathbb{Z}^r, \leq)$ for some $r > 0$ such that that its image in $\mathbb{Z}^r$ is
an $\operatorname{lcm}$-lattice.
\item A set of {\em generating flags} of $\mathcal{V}$ is a set of tuples
$\{E^1_1 \subsetneq \cdots \subsetneq E^1_{n_1}\}, \dots, \{E^r_1 \subsetneq \dots
\subsetneq E^r_{n_r}\} \subset \mathcal{V}$ such that $E^i_{n_i} = \mathbf{V}$ for
every $i$ and $\mathcal{V}$ is the set of all intersections among the $E^i_j$.
\end{enumerate}
\end{definition}
Let $E^i_j$ be any set of generating flags, then we associate to each of the flags
a tuple of integers $\underline{k}^i := (k^i_1 < \dots < k^i_{n_i})$.
This data defines a reflexive model, where we map every $W \in \mathcal{V}$ to the
tuple $\underline{k}_V := (\min\{k^i_j \mid W \subset E^i_{k^i_j}\} \mid i = 1,
\dots, r)$. As easily
can be seen, this reflexive model gives rise to a reflexive fine-graded module $E$
over the polynomial ring $S = k[x_1, \dots, x_r]$ which is given by filtrations
\begin{equation*}
E^i(j) =
\begin{cases}
0 & \text{ if } j < k^i_1, \\
E^i_l & \text{ if } k^i_l \leq j < k^i_{l + 1}, l < n_i, \\
\mathbf{V} & \text{ if } n_i \leq j.
\end{cases}
\end{equation*}
$E$ has an embedding into the free module $S(-\underline{k}_{\min})^{\dim
\mathbf{V}}$, where
$\underline{k}_{\min} = (k^1_1, \dots, k^r_1)$, and the module $F_0$ is given by
the direct sum $\bigoplus_{W \in \mathcal{V}} S(-\underline{k}_V)^{f_V}$, where
$f_V$ is the free dimension of $V$. So, we have
\begin{equation*}
0 \longrightarrow K_0 \longrightarrow \bigoplus_{W \in \mathcal{V}}
S(-\underline{k}_V)^{f_V} \overset{\bar{M}}{\longrightarrow}
S(-\underline{k}_{\min})^{\dim \mathbf{V}},
\end{equation*}
where $\bar{M}$ is a monomial matrix whose entries are of the form
$(v^W_{ij} x^{\underline{k}_V - \underline{k}_{\min}})$, where the $v^W_{ij}$ are the
corresponding entries of the matrix $M$. The image of $M$ then is the module $E$.
So the only effect seen by the choice of the reflexive model for $\mathcal{V}$ are
the number of variables in the ring $S$ and the degrees of the monomials in
$\bar{M}$ and the subsequent matrices in the resolution, whereas the coefficients in
$\bar{M}$ are precisely the entries of the matrix $M$.
\addtocontents{toc}{\vskip4mm}
\addtocontents{toc}{\bf References \hfill \thepage}
|
{
"timestamp": "2005-03-23T19:06:05",
"yymm": "0503",
"arxiv_id": "math/0503501",
"language": "en",
"url": "https://arxiv.org/abs/math/0503501"
}
|
\section{Introduction}
\label{sec:intro}
Spectral properties of the Laplacian on a compact manifold is a
well-established and still active field of research. Much less is
known on the spectrum of \emph{non-compact} manifolds. We restrict
ourselves here to the class of non-compact \emph{covering} manifolds
$X \to M$ with compact quotient $M$, in which the covering group
$\Gamma$ plays an important role. In the open problem section of
\cite[Ch.~IX, Problem~37]{schoen-yau:94}, Yau posed the question about
the nature and the stability of the (purely essential) spectrum of
such a covering $X \to M$.
The aim of this paper is to provide a large class of examples of
Riemannian coverings $X \to M$ having spectral gaps in the essential
spectrum of its Laplacian~$\laplacian X$. Here, a spectral gap is a
non-void open interval $(\alpha, \beta)$ with $(\alpha, \beta) \cap
\spec {\laplacian X } = \emptyset$ and $\alpha, \beta \in \spec
{\laplacian X}$. The manifolds $X$ and $M$ are $d$-dimensional,
$d\geq 2$, and we denote by $D$ a fundamental domain associated to
this covering. The main idea for producing spectral gaps is to
construct a family of Riemannian metrics $(g_\eps)_{\eps>0}$ on $X$
such that the length scale w.r.t.\ the metric $g_\eps$ is of order
$\eps$ at the boundary of a fundamental domain $D$ and unchanged
elsewhere (cf.~Figure~\ref{fig:per-mfd}). If such a fundamental domain
exists, we say that the family of metrics $(g_\eps)$ \emph{decouples}
the manifold $X$. The covering $X \to M$ with a decoupling family of
metrics $(g_\eps)$ ``converges'' in a sense to be specified below to a
limit covering consisting of the infinite disjoint (``decoupled'') union of the
limit quotient manifold $N$ which are again $d$-dimensional (see
Subsection~\ref{ssec:outline} and Section~\ref{sec:construye} for
details). We stress that the curvature does not remain bounded as
$\eps \to 0$; in contrast to degeneration of Riemannian metrics under
curvature bounds developed~e.g.\ in~\cite{cheeger:01}. All groups
$\Gamma$ are assumed to be discrete and finitely generated throughout
the present article.
\subsection{Statement of the main results}
\begin{maintheorem}[cf.~Theorem~\ref{thm:gaps.res.fin}]
\label{mthm:1}
Suppose that $X \to M$ is a Riemannian covering with residually
finite covering group $\Gamma$ and metric $g$. Then by a local
deformation of $g$ we
construct a family of metrics $(g_\eps)$ decoupling $X$, such
that for each $n \in \N$ there exists $\eps_n>0$ where $\spec
{\laplacian {(X,g_{\eps_n})}}$ has at least $n$ gaps, i.e.\ $n+1$
components as subset of $[0,\infty)$.
\end{maintheorem}
Basically, we will give two different constructions for the family of
manifolds $(X,g_\eps)$: first, ``adding small handles'' to a given
manifold $(N,g)$ and second, a conformal perturbation of $g$. As a
set, $(X,g_\eps)$ converges to a limit manifold consisting of
infinitely many disjoint copies of the limit quotient manifold $N$ as
$\eps \to 0$.
A \emph{residually finite} group is a countable
discrete group such that the intersection of all its normal subgroups
of finite index is trivial. Roughly speaking, a residually
finite group has many normal subgroups of finite index.
Geometrically, a covering with a residually finite covering group can
be approximated by a sequence of finite coverings $M_i \to M$ (a
\emph{tower of coverings}). The class of residually finite groups is
very large, containing e.g.~finitely generated abelian groups, type~I
groups (i.e.\ finite extensions of $\Z^r$), free groups or finitely
generated subgroups of the isometries of the $d$-dimensional
hyperbolic space $\Hyp^d$ (cf.\ Section~\ref{sec:res.fin}).
Denote by $\mathcal N(g,\lambda)$ the number of components of
$\spec\laplacian {(X,g)}$ which intersect the interval $[0, \lambda]$.
Our result gives a \emph{lower} bound on $\mathcal N(g,\lambda)$, in
particular, we can reformulate the Main~Theorem~\ref{mthm:1} as
follows: \emph{For each $n \in \N$ there exists
$g=g_{\eps_n}$ such that $\mathcal N(g,\lambda) \ge n+1$.}
Using the Weyl eigenvalue asymptotic on the limit $d$-dimensional
manifold $(N,g)$ associated to the decoupling family $(g_\eps)$ on $X
\to M$, we obtain the following asymptotic lower bound on the number
of gaps (where $\omega_d$ denotes the volume of the $d$-dimensional
Euclidean unit ball):
\begin{maintheorem}[cf.~Theorem~\ref{thm:band}]
Assume that the covering group is residually finite and that the
spectrum of the Laplacian on the limit manifold $(N,g)$ is simple,
i.e.~ all eigenvalues have multiplicity one. Then for each $\lambda
\ge 0$ there exists $\eps(\lambda)>0$ such that
\begin{equation*}
\liminf_{\lambda\to\infty}
\frac{\mathcal N (g_{\eps(\lambda)}, \lambda)}
{(2\pi)^{-d} \omega_d \vol (N,g) \lambda^{d/2}}
\ge 1.
\end{equation*}
\end{maintheorem}
The assumption on the spectrum of $(N,g)$ is natural since $\mathcal
N(g,\lambda)$ counts components in the spectrum \emph{without}
multiplicity.
A priori, the number of gaps $\mathcal N(g,\lambda)$ could be
infinite, e.g.~if $\spec {\laplacian {(X,g)}}$ contains a Cantor set.
But Br\"uning and Sunada showed in~\cite{bruening-sunada:92} that for
covering groups $\Gamma$ with positive \emph{Kadison constant}
$C(\Gamma)>0$ (cf.~Section~\ref{sec:kadison}) asymptotic upper bound
\begin{equation*}
\limsup_{\lambda\to\infty} \frac{\mathcal N(g, \lambda)}
{(2\pi)^{-d} \omega_d \vol (M,g) \lambda^{d/2}}
\le \frac 1 {C(\Gamma)}
\end{equation*}
holds. In particular, $\mathcal N(g,\lambda)$ is finite, and the
spectrum of $\laplacian {(X,g)}$ does not contain Cantor-like subsets.
Applying these results to our situation we give a partial answer on
the question of Yau of the nature of the spectrum:
\begin{maintheorem}[cf.~Theorem~\ref{thm:band}]
\sloppy Suppose that $X \to M$ is a Riemannian $\Gamma$-covering
with decoupling family of metrics $(g_\eps)$, where $\Gamma$ is a
residually finite group that has positive Kadison constant
$C(\Gamma)>0$. Then $\spec \laplacian {(X,g_\eps)}$ has
band-structure, i.e.~$\mathcal N(g_\eps,\lambda)~<~\infty$ for all
$\lambda \ge 0$ and $\mathcal N(g_\eps,\lambda)$ can be made
arbitrary large provided $\eps$ is small and $\lambda$ is large
enough.
\end{maintheorem}
Some examples of groups with positive Kadison
constant and which are residually finite are finitely generated,
abelian groups, the free (non-abelian) group in $r \ge 2$ generators
or fundamental groups of compact, orientable surfaces (see also
Section~\ref{sec:examples}).
\subsection{Motivation and related work}
A main motivation for our work comes from the spectral theory of
Schr\"odinger operators $H=-\Delta+V$ on $\R^d$, $d \ge 2$, with $V$
periodic w.r.t.~the action of a discrete abelian
group~$\Gamma_{\mathrm{ab}}=\Z^d$ on $\R^d$. For such operators, it is
a well known fact that if $V$ has high barriers near the boundary of a
fundamental domain $D$, then gaps appear in the spectrum of $H$. In
this way, the potential $V$ essentially decouples the fundamental
domain $D$ from its neighbouring domains (see \cite{hempel-post:03}
for an overview on this subject).
A natural generalisation into a geometric context is to replace the
periodic structure $(\R^d, \Z^d)$ by a Riemannian covering $X \to M$
with a discrete (in general non-abelian) group $\Gamma$. Our work
shows that the decoupling effect of the potential $V$ can be replaced
purely by geometry, in particular by the decoupling family of metrics
$(g_\eps)$ on $X \to M$. From a quantum mechanical or probabilistic
point of view, the correspondence seems to be natural: One has a small
probability to find a particle (with low energy) in a region with a
high potential barrier or where the manifold $(X,g_\eps)$ is very thin
and the absolute value of the curvature is very large.
It was already observed by, e.g., Br\"uning, Gruber, Kobayashi, Ono
and Sunada \cite{bruening-sunada:92,gruber:01, sunada:90,kos:89} that
many properties of the spectrum of a periodic Schr\"odinger operator
(e.g.~band-structure, Bloch's property etc.) generalise to the
context of Riemannian coverings. An important difference is the
existence of $\Lsymb_2$-eigenvalues in the context of manifolds
(cf.~\cite{kos:89}). Such eigenvalues cannot occur in the spectrum of
a periodic Schr\"odinger operator on $\R^d$ (cf.~\cite{sunada:90}).
The existence of (covering) manifolds with spectral gaps has also been
established by Br\"uning, Exner, Geyler and Lobanov
in~\cite{beg:03,bgl:05}. They couple compact manifolds by points or
line-segments with certain boundary condition at the coupling points;
the point coupling corresponds to the case $\eps=0$ in our situation
(with decoupled boundary condition). The case of abelian \emph{smooth}
coverings has been established in \cite{post:03a} (cf.~also the
references therein). Spectral gaps of Schr\"odinger operators on the
hyperbolic space have been analysed in~\cite{karp-peyerimhoff:00}. For
other manifolds with spectral gaps (not necessarily periodic), we
refer to \cite{exner-post:05, post:06}. Under certain topological
restrictions on the middle degree homology group one can show the
existence of spectral gaps also for the differential form Laplacian on
a $\Z$-covering (see~\cite{acp:pre07}).
Some further interesting results on the group $\Gamma$ and spectral
properties of a Riemannian $\Gamma$-covering were shown by
Brooks~\cite{brooks:81}, e.g.\ that $\Gamma$ is amenable iff $0 \in
\spec \laplacian X$. Moreover, Brooks~\cite{brooks:86} provided a
combinatorial criterion whether the second eigenvalue of $\laplacian
{M_i}$ is bounded from below as $i \to \infty$, where $M_i \to M$ is a
tower of coverings.
For physical applications of our results we refer to
Section~\ref{sec:outlook}. Let us finish with two consequences of our
result giving partial answers to the question of Yau on the nature and
stability of the spectrum of $\laplacian X$:
\begin{consequence}[Manifold with given spectrum]
First, we can solve the following inverse spectral problem: Given a
compact (connected) manifold $N$ of dimension $d \ge 3$ and a
sequence of numbers $0=\lambda_1(0)<\ldots<\lambda_n(0)$ it is
possible to construct a metric $g$ on $N$ having exactly the numbers
$\lambda_k(0)$ as first $n$ eigenvalues with multiplicity $1$
(cf.~\cite{colin:87}). Then, applying our Main Theorem~3 and using
the relation between $\spec \laplacian {(X,g_\eps)}$ and $\spec
\laplacian {(N,g)}$ we can construct a covering $X \to M$ with
decoupling family $(g_\eps)$ having band spectrum close to the given
points $\{\lambda_k(0)\}$, $k=1,\dots, n$. The covering $(X,g_\eps)
\to (M,g_\eps)$ is obtained roughly by joining copies of $N$ through
small, thin cylinders (see first construction mentioned below). In
particular, we have constructed a covering manifold with
approximatively given spectrum in a finite spectral interval
$[0,\lambda]$, \emph{independently} of the covering group!
\end{consequence}
\begin{consequence}[Instability of gaps]
Suppose $X=\Hyp^d$ is the $d$-dimensional ($d \ge 3$) hyperbolic
space (or more generally, a simply connected, complete, symmetric
space of non-compact type) with its natural metric $g$. It is known,
that $\laplacian {(X,g)}$ has no spectral gaps, in particular $\spec
{\laplacian {(X,g)}} = [\lambda_0, \infty)$ for some constant
$\lambda_0 \ge 0$ (see e.g.~\cite{donnelly:79}). Let $\Gamma$ be a
finitely generated subgroup of the isometries of $X$ such that $M =
X/\Gamma$ is compact. Note that such groups are residually finite.
The second construction described below allows us to find a
decoupling family $(g_\eps)$ on $X$ where $g_\eps = \rho_\eps^2 g$
is conformally equivalent to $g$. We then apply Main~Theorem~1 and
obtain for each $n \in \N$ a metric $g_{\eps_n}$ such that the
corresponding Laplacian has at least $n$ gaps. In particular, the
number of gaps is \emph{not} stable, even under uniform conformal
changes of the metric. Note that the conformal factor $\rho_\eps$
can be chosen in such a way that $\rho_\eps \to \rho_{\eps_0}$
uniformly as $\eps \to \eps_0$ provided $\eps_0>0$. Nevertheless,
the band-gap structure remains invariant due to Main Theorem~3, once
$\Gamma$ has a positive Kadison constant.
\end{consequence}
\begin{figure}
\begin{center}
\begin{picture}(0,0)
\includegraphics{nc-floquet-fig1.eps}
\end{picture}%
\setlength{\unitlength}{4144sp}
\begin{picture}(5244,1501)(259,-695)
\put(811,-556){$X$}
\put(4501,-556){$D$}
\put(2791,-646){$\eps$}
\end{picture}
\caption{A covering manifold $X$ with fundamental domain $D$. The
junctions between different translates of $D$ are of order
$\eps$.}
\label{fig:per-mfd}
\end{center}
\end{figure}
\subsection{An outline of the argument}
\label{ssec:outline}
In the rest of the introduction we will present the main ideas of
the construction of the decoupling metrics and mention the strategy
for showing the existence of spectral gaps.
The first construction starts from a compact Riemannian manifold $N$
of dimension $d \ge 2$ (for simplicity without boundary) and a group
$\Gamma$ with generators $\gamma_1, \dots, \gamma_r$. We choose $2r$
different points $x_1,y_1, \dots, x_r, y_r$. For each generator, we
endow $x_i$ and $y_i$ with a cylindrical end of radius and length of
order $\eps>0$ (by changing the metric appropriately on $D:=N
\setminus \{x_1, y_1, \dots, x_r, y_r\}$). If we join $\Gamma$ copies
of these decorated manifolds $(D,g_\eps)$ according to the Cayley
graph of $\Gamma$ associated to $\gamma_1, \dots, \gamma_r$, we obtain a
$\Gamma$-covering $X \to M$ with a decoupling family of metrics
$(g_\eps)$ (cf.~Figure~\ref{fig:per-mfd}).
The second construction starts with an arbitrary covering $(X,g) \to
(M,g)$ (with compact quotient) of dimension $d\ge 3$ and changes the
metric conformally, i.e.\ $g_\eps := \rho_\eps^2 g$, in such a way,
that $\rho_\eps$ is still periodic and of order $\eps$ close to the
boundary of a fundamental domain $D$; more details can be found in
Section~\ref{sec:construye}. In the case of abelian coverings these
constructions have already been used in~\cite{post:03a}.
Once the construction of the family of decoupling metrics $(g_\eps)$
has been done, the strategy to show the existence of spectral gaps
goes as follows. We consider first the Dirichlet $(+)$ and Neumann
$(-)$ eigenvalues $\EWDN k (\eps)$ of the Laplacian on the fundamental
domain $(D,g_\eps)$. One can show that $\EWDN k (\eps)$ converges to
the eigenvalues $\EW k(0)$ of the Laplacian on the limit manifold
$(N,g)$ (see \cite{post:03a} and references therein). In other words,
the Dirichlet-Neumann intervals
\begin{equation*}
I_k(\eps) := [\EWN k (\eps), \EWD k (\eps)]
\end{equation*}
converge to a point as $\eps\to 0$. Therefore, if $\eps$ is small
enough, the union
\begin{equation*}
I(\eps) := \bigcup_{k \in \N} I_k(\eps)
\end{equation*}
is a closed set having at least $n$ gaps, i.e.\ $n+1$ components as a
subset of $[0, \infty)$.
The rest of the argument depends on the properties of the covering
group $\Gamma$:
\begin{enumerate}
\item For abelian groups $\Gamma_{\mathrm{ab}}$, the inclusion $\spec
\laplacian {(X, g_\eps)} \subset I(\eps)$ is given by the Floquet
theory (cf.~Section~\ref{sec:floquet} or \cite{kuchment:93,
sunada:88}). Basically, one shows that $\laplacian {(X,g_\eps)}$
is unitary equivalent to a direct integral of operators on
$(D,g_\eps)$ acting on $\rho$-equivariant functions, where $\rho$
runs through the set of irreducible unitary representations
$\widehat \Gamma_{\mathrm{ab}}$ (characters). Note that in the
abelian case all $\rho$ are one-dimensional and $\widehat
\Gamma_{\mathrm{ab}}$ is homeomorphic to (disjoint copies of) the
torus $\Torus^r$. The Min-max principle ensures that the $k$-th
eigenvalue of the equivariant operator lies in $I_k(\eps)$.
\item If the group is non-abelian but still has only
finite-dimensional irreducible representations, then one can show
that the spectrum of the $\rho$-equivariant Laplacian is still
included in $I(\eps)$. In this case the (non-abelian) Floquet
theory guarantees again that $\spec\laplacian {(X, g_\eps)} \subset
I(\eps)$. The class of groups which satisfy the previous condition
are type~I groups, i.e finite extensions of abelian groups. These
groups have a dual object $\widehat \Gamma$ which is a nice measure
space (\emph{smooth} in the terminology
of~\cite[Chapter~2]{mackey:76}).
\item If the group is \emph{residually finite} (a much wider class of
groups including type I groups), then one can construct a so-called
\emph{tower of coverings} consisting of finite coverings $M_i \to M$
``converging'' to the original covering $X \to M$. The inclusion of
the spectrum of $\laplacian {(X,g_\eps)}$ in the closure of the
union over all spectra of $\laplacian {(M_i,g_\eps)}$ was shown
in~\cite{ass:94,adachi:95}. For the \emph{finite} coverings $M_i
\to M$ we again have the inclusion $\spec {\laplacian{(M_i,g_\eps)}}
\subset I(\eps)$.
\item For non-amenable groups (i.e.\ groups, for which $\spec
\laplacian {(M,g_\eps)}$ is not included in $\spec \laplacian
{(X,g_\eps)})$, cf.~Remark~\ref{rem:amenable}, we have to assure
that any of the intervals $I_k(\eps)$ intersects $\spec \laplacian
X$ non-trivially. This will be done in~Theorem~\ref{thm:spectrum}.
\end{enumerate}
\subsection*{Organisation of the paper}
In the following section we set up the problem, present the
geometrical context and state some results and conventions that will
be needed later. In Section~\ref{sec:construye} we present in detail
the two procedures for constructing covering manifolds with a
decoupling family of metrics. In this case the set $I(\eps)$ defined
above will have at least a prescribed finite number of spectral gaps.
Each procedure is well adapted to a given initial geometrical context
(cf.~Remark~\ref{ExplainMethods} as well as
Examples~\ref{ex:fund.group} and \ref{ex:heisenberg}). In
Section~\ref{sec:floquet} we show the inclusion of the spectrum of
equivariant Laplacians into the union of the Dirichlet-Neumann
intervals $I_k(\eps)$ and review briefly the Floquet theory for
non-abelian groups. The Floquet theory is applied in
Section~\ref{sec:type.I} for coverings with type~I groups. In
Section~\ref{sec:res.fin} we study a class of covering manifolds with
residually finite groups. In Section~\ref{sec:kadison} we consider
residually finite groups $\Gamma$ that in addition have a positive
Kadison constant. In Section~\ref{sec:examples} we illustrate the
results obtained with some classes of examples and point out their
mutual relations. Subsection~\ref{sec:OpenQuestion} contains an
interesting example of a covering with an amenable, \emph{not}
residually finite group which cannot be treated with our methods. We
expect though that in this case one can still generate spectral gaps
by the construction presented in Section~\ref{sec:construye}. Finally,
we conclude mentioning several possible applications for our results.
\section{Geometrical preliminaries: covering manifolds and Laplacians}
\label{sec:prelim}
We begin fixing our geometrical context and recalling some results
that will be useful later on. We denote by $X$ a \emph{non-compact}
Riemannian manifold of dimension $d \ge 2$ with a metric $g$. We also
assume the existence of a finitely generated (infinite) discrete group
$\Gamma$ of isometries acting \emph{properly discontinuously} and
\emph{cocompactly} on $X$, i.e.\ for each $x \in X$ there is a
neighbourhood $U$ of $x$ such that the sets $\gamma U$ and $\gamma'U$ are
disjoint if $\gamma \ne \gamma'$ and $M:=X/\Gamma$ is compact. Moreover, the
quotient $M$ is a Riemannian manifold which also has dimension $d$ and
is locally isometric to $X$. In other words, $\map \pi X M$ is a
\emph{Riemannian covering space} with covering group $\Gamma$. We
call such a manifold \emph{$\Gamma$-periodic} or simply \emph{periodic}.
All groups $\Gamma$ appearing in this paper will satisfy the preceding
properties.
We also fix a \emph{fundamental domain} $D$, i.e.\ an open set $D
\subset X$ such that $\gamma D$ and $\gamma' D$ are disjoint for all $\gamma \ne
\gamma'$ and $\bigcup_{\gamma \in \Gamma} \gamma \overline D = X$. We always
assume that $\overline D$ is compact and that $\bd D$ is piecewise
smooth. If not otherwise stated we also assume that $D$ is connected.
Note that we can embed $D\subset X$ isometrically into the quotient
$M$. In the sequel, we will not always distinguish between $D$ as a
subset of $X$ or $M$ since they are isometric. For details we refer
to~\cite[\S6.5]{ratcliffe:94}.
As a prototype for an elliptic operator we consider the Laplacian
$\laplacian X$ on a Riemannian manifold $(X,g)$ acting on a dense
subspace of the Hilbert space $\Lsqr X$ with norm $\norm[X] \cdot$.
For the formulation of the Theorems~\ref{thm:gaps.type.I} and
\ref{thm:gaps.res.fin} and at other places, it is useful to denote
explicitly the dependence on the metric, since we deform the manifold
by changing the metric. In this case we will write $\laplacian
{(X,g)}$ for $\laplacian X$ or $\Lsqr {X,g}$ for $\Lsqr X$.
The positive self-adjoint operator
$\laplacian X$ can be defined in terms of a
suitable qua\-dra\-tic form $q_X$ (see e.g.~\cite[Chapter~VI]{kato:95},
\cite{reed-simon-1} or \cite{davies:96}). Concretely we have
\begin{equation}
\label{def:quad.form}
q_X(u):=\normsqr[X] {d u} = \int_X {|d u|^2},\quad
u \in \Cci X
\end{equation}
where the integral is taken with respect to the volume density measure
of $(X,g)$. In coordinates we write the pointwise norm of the
$1$-form $d u$ as
\begin{displaymath}
|d u |^2(x)= \sum_{i,j} g^{ij}(x) \partial_i u(x) \,
\partial_j\overline {u(x)} ,
\end{displaymath}
where $(g^{ij})$ is the inverse of the metric tensor $(g_{ij})$ in a
chart. Taking the closure of the quadratic form we can extend $q_X$ onto
the Sobolev space
\begin{displaymath}
\Sob X = \Sob {X,g} = \set{ u \in \Lsqr X}{ q_X(u) < \infty}.
\end{displaymath}
As usual the operator $\laplacian X$ is related with the quadratic form by the
formula $\iprod {\laplacian X u} u = q_X(u)$, $u \in \Cci X$. Since the metric
on $X$ is $\Gamma$-invariant, the Laplacian $\laplacian X$ (i.e.\ its
resolvent) commutes with the translation on $X$ given by
\begin{equation}
\label{eq:transl}
(T_\gamma u )(x) := u(\gamma^{-1}x), \quad u \in \Lsqr X, \gamma \in \Gamma.
\end{equation}
Operators with this property are called \emph{periodic}.
For an open, relatively compact subset $D \subset X$ with sufficiently smooth
boundary $\bd D$ (e.g.~Lipschitz) we define the Dirichlet (respectively,
Neumann) Laplacian $\laplacianD D$ (resp., $\laplacianN D$) via its quadratic
form $q_D^+$ (resp., $q_D^-$) associated to the closure of $q_D$ on $\Cci D$,
the space of smooth functions with compact support, (resp., $\Ci {\overline
D}$, the space of smooth functions with continuous derivatives up to the
boundary). We also use the notation $\Sobn D = \dom q_D^+$ (resp., $\Sob D =
\dom q_D^-$). Note that the usual boundary condition of the Neumann Laplacian
occurs only in the \emph{operator} domain via the Gau{\ss}-Green formula.
Since $\overline D$ is compact, $\laplacianD D$ has purely discrete spectrum
$\EWD k$, $k \in \N$. It is written in ascending order and repeated according
to multiplicity. The same is true for the Neumann Laplacian and we denote the
corresponding purely discrete spectrum by $\EWN k$, $k\in\N$.
One of the advantages of the quadratic form approach is that
one can easily read off from the inclusion of domains an order relation
for the eigenvalues. In fact, by the
the \emph{min-max principle} we have
\begin{equation}
\label{eq:min.max}
\EWDN k =
\inf_{L_k} \sup_{u \in L_k \setminus \{0\} }
\frac {q_D^\pm(u)}{\normsqr u},
\end{equation}
where the infimum is taken over all $k$-dimensional subspaces $L_k$ of the
corresponding \emph{quadratic} form domain $\dom q_D^\pm$,
cf.~e.g.~\cite{davies:96}. Then the inclusion
\begin{equation}
\label{eq:dom.mono}
\dom q_D^+ = \Sobn D \subset \Sob D = \dom q_D^-
\end{equation}
implies the following important relation between the corresponding
eigenvalues
\begin{equation}
\label{eq:ew.mono}
\EWD k \ge \EWN k .
\end{equation}
This means, that the Dirichlet $k$-th eigenvalue is in general
larger than the $k$-th Neumann eigenvalue and this justifies
the choice of the labels $+$, respectively, $-$.
\section{Construction of periodic manifolds}
\label{sec:construye}
In the present section we will give two different construction
procedures (labelled by the letters `A' and `B') for covering manifolds,
such that the corresponding Laplacian will have a prescribed finite
number of spectral gaps. In contrast with \cite{post:03a} (where only
abelian groups were considered) we will base the construction on the
specification of the quotient space $M=X/\Gamma$. By doing this, the
spectral convergence result in Theorem~\ref{thm:mfd.conv} becomes
manifestly independent of the fact whether $\Gamma$ is abelian or not.
Both constructions are done in two steps: first, we specify in two
ways the quotient $M$ together with a family of metrics $g_\eps$.
Second, we construct in either case the covering manifold with
covering group $\Gamma$ which has $r$ generators. In the last section
we will localise the spectrum of the covering Laplacian in certain
intervals given by an associated Dirichlet, respectively, Neumann
eigenvalue problem. Some reasons for presenting two different
methods~(A) and~(B) are formulated in a final remark of this section.
\subsection{Construction of the quotient}
\label{ssec:quotient}
In the following two methods we define a family of Riemannian
manifolds $(M,g_\eps)$ that converge to a Riemannian manifold $(N,g)$
of the same dimension (cf.~Figure~\ref{fig:constr-mfd}). In each case
we will also specify a domain $D\subset M$ (in the following section
$D$ will become a fundamental domain of the corresponding covering):
\begin{enumerate}
\item[(1A)] \textbf{Attaching $r$ handles:} We construct the manifold
$M$ by attaching $r$ handles diffeomorphic with $C := (0,1) \times
\Sphere^{d-1}$ to a given $d$-dimensional compact orientable
manifold $N$ with metric $g$. For simplicity we assume that $N$ has
no boundary. Concretely, for each handle we remove two small discs
of radius $\eps>0$ from $N$, denote the remaining set by $R_\eps$
and identify $\{0\} \times \Sphere^{d-1}$ with the boundary of the
first hole and $\{1\} \times \Sphere^{d-1}$ with the boundary of the
second hole. We denote by $D$ the open subset of $M$ where the mid
section $\{1/2\} \times \Sphere^{d-1}$ of each handle is removed.
One can finally define a family of metrics $(g_\eps)_\eps$,
$\eps>0$, on $M$ such that the diameter and length of the handle is
of order $\eps$ (see e.g.~\cite{post:03a,chavel-feldman:81}). In
this situation the handles shrink to a point as $\eps\to 0$. Note
that $(R_\eps, g)$ can be embedded isometrically into $(N,g)$, resp.,
$(M, g_\eps)$. This fact will we useful for proving
Theorem~\ref{thm:spectrum}.
\item[(1B)] \textbf{Conformal change of metric:} In the second
construction, we start with an arbitrary compact $d$-dimensional
Riemannian manifold $M$ with metric $g$. We consider only the case
$d \ge 3$ (for a discussion of some two-dimensional examples
see~\cite{post:03a}). Moreover, we assume that $N$ and $D$ are two
open subsets of $M$ such that (i)~$\bd N$ is smooth, (ii)~$\overline
N \subset D$, (iii)~$\overline D = M$ and (iv)~$D \setminus N$ can
completely be described by Fermi coordinates (i.e.\ coordinates
$(r,y)$, $r$ being the distance from $N$ and $y \in \bd N$) up to a
set of measure $0$ (cf.~Figure~\ref{fig:constr-mfd}~(B)). The last
assumption assures that $N$ is in some sense large in $D$.
Suppose in addition, that $\map {\rho_\eps} M {(0,1]}$, $\eps>0$, is
a family of smooth functions such that $\rho_\eps(x)=1$ if $x \in N$
and $\rho_\eps(x)=\eps$ if $x \in M \setminus N$ and $\dist(x,\bd N)
\ge \eps^d$. Then $\rho_\eps$ converges pointwise to the
characteristic function of $N$. Furthermore, the Riemannian manifold
$(M, g_\eps)$ with $g_\eps:= \rho_\eps^2 g$ converges to $(N,g)$ in
the sense that $M \setminus N$ shrinks to a point in the metric
$g_\eps$.
\end{enumerate}
\begin{figure}[h]
\begin{center}
\begin{picture}(0,0)
\includegraphics{nc-floquet-fig2.eps}
\end{picture}
\setlength{\unitlength}{4144sp}
\begin{picture}(5335,2429)(361,-1861)
\put(3601,-1800){(B)}
\put(360,-1800){(A)}
\put(3550,460){$r$}
\put(4394,-44){$y$}
\put(1391, 59){$R_\eps$}
\put(360,-1550){$C=(0,1)\times\Sphere^{d-1}$}
\put(360,250){$(N,g)$}
\put(2000,-1650){$\beta_1$ (mid section)}
\put(860,-400){$\alpha_1$}
\put(4494,-854){$N$}
\put(4970,178){$D \setminus N$}
\put(4000,-1489){$\rho_\eps(x)=O(\eps)$}
\put(4150,-1100){$\rho_\eps(x)=1$}
\end{picture}%
\caption{Two constructions of a family of manifold $(M, g_\eps$), $\eps >
0$: In both cases, the grey area has a length scale of order
$\eps$ in all directions. (A)~We attach $r$ handles (here $r=1$)
of diameter and length of order $\eps$ to the manifold $(N,g)$.
We also denoted the two cycles $\alpha_1$ and $\beta_1$. (B)~We
change the metric conformally to $g_\eps = \rho_\eps^2 g$. The
grey area $D \setminus N$ (with Fermi coordinates in the upper
left corner) shrinks conformally to a point as $\eps \to 0$
whereas $N$ remains fixed. Note that the opposite sides of the
square are identified (to obtain a torus as manifold $M$).}
\label{fig:constr-mfd}
\end{center}
\end{figure}
Now we can formulate the following spectral convergence result which was
proven in~\cite{post:03a}:
\begin{theorem}
\label{thm:mfd.conv}
Suppose $(M,g_\eps)$ and $D \subset M$ are constructed as in
parts~(1A) or~(1B) above. In Case~(1B) we assume in addition that $d
\ge 3$. Then
\begin{displaymath}
\EWDN k (\eps) \to \EW k (0)
\end{displaymath}
as $\eps \to 0$ for each $k$. Here, $\EWDN k (\eps)$ denotes the
$k$-th Dirichlet, resp., Neumann eigenvalue of the Laplacian on
$(D,g_\eps)$ whereas $\EW k (0)$ is the $k$-th eigenvalue of $(N,g)$
(with Neumann boundary conditions at $\bd N$ in Case~(1B)).
\end{theorem}
\subsection{Construction of the covering spaces}
\label{ssec:constr.cov.sp}
Given $(M,g_\eps)$ and $D$ as in the previous subsection, we will
associate a Riemannian covering $\map \pi {(X,g_\eps)} {(M, g_\eps)}$
with covering group $\Gamma$ such that $D$ is a fundamental
domain. Note that we identify $D \subset M$ with a component of the
lift $\widetilde D := \pi^{-1} (D)$. Moreover, $\Gamma$ is isomorphic
to a normal subgroup of the fundamental group $\pi_1(M)$.
\begin{enumerate}
\item[(2A)] Suppose that $\Gamma$ is a discrete group with $r$ generators
$\gamma_1, \dots, \gamma_r$. We will construct a $\Gamma$-covering
$(X,g_\eps) \to (M, g_\eps)$ with fundamental domain $D$ where $D$
and $(M,g_\eps)$ are given as in Part~(1A) of the previous
subsection. Roughly speaking, we glue together $\Gamma$ copies of
$D$ along the handles according to the Cayley graph of $\Gamma$
w.r.t.\ the generators $\gamma_1, \dots, \gamma_r$. For convenience
of the reader, we specify the construction:
The fundamental group of $M$ is given by $\pi_1(M) = \pi_1(N) *
\Z^{*r}$ in the case $d \ge 3$. Here, $G_1*G_2$ denotes the free
product of $G_1$ and $G_2$, and $\Z^{*r}$ is the free group in $r$
generators $\alpha_1, \dots, \alpha_r$. If $d=2$ we know from the
classification result for $2$-dimensional orientable manifolds that
$N$ is diffeomorphic to an $s$-holed torus. In this case the
fundamental group is given by
\begin{equation}
\label{eq:fund.group}
\pi_1(M) = \langle \alpha_1, \beta_1, \dots,
\alpha_{r+s}, \beta_{r+s} \mid [\alpha_1, \beta_1] \cdot \ldots \cdot
[\alpha_{r+s},\beta_{r+s}] = e \rangle,
\end{equation}
where $[\alpha,\beta]:=\alpha \beta \alpha^{-1}\beta^{-1}$ is the usual
commutator. We may assume that $\alpha_i$
represents the homotopy class of the cycle \emph{transversal} to the section
of the $i$-th handle and that $\beta_i$ represents the section itself ($i=1,
\dots, r$) (cf.~Figure~\ref{fig:constr-mfd}~(A)).
One easily sees that there exists an epimorphism $\map \phi {\pi_1(M)} \Gamma$
which maps $\alpha_i \in \pi_1(M)$ to $\gamma_i \in \Gamma$ ($i=1, \dots, r$) and
all other generators to the unit element $e \in \Gamma$.
Note that this map is also well-defined in the case $d=2$,
since the relation in~\eqref{eq:fund.group} is trivially satisfied in the
case when the $\beta_i$'s are mapped to $e$.
Finally, $\Gamma\cong \pi_1(M) / \ker \phi$, and
$X \to M$ is the associated covering with respect to the universal
covering $\widetilde M \to M$
(considered as a principal bundle with discrete fibre $\Gamma$) and
the natural action of $\Gamma$ on $\pi_1(M)$.
Then $X \to M$ is a normal $\Gamma$-covering with fundamental domain $D$
constructed as in~(1A) of the preceding subsection. Here we use the
fact that $\alpha_i$ is \emph{transversal} to the section of the
handle in dimension~$2$.
\item[(2B)] Suppose $(X,g) \to (M,g)$ is a Riemannian covering
with fundamental
domain $D$ such that $\bd D$ is piecewise smooth. Then $\overline D = M$,
where we have embedded $D$ into the quotient,
cf.~\cite[Theorem~6.5.8]{ratcliffe:94}.
According to~(1B) we can conformally change the metric on $M$, to produce
a new covering $(X,g_\eps) \to (M,g_\eps)$ that satisfies the required
properties.
\end{enumerate}
In both cases, we lift for each $\eps >0$ the metric $g_\eps$ from $M$
to $X$ and obtain a Riemannian covering $(X,g_\eps) \to (M, g_\eps)$.
Note that the set $D$ specified in the first step of the previous
construction becomes a fundamental domain after the specification of
the covering in the second step.
The following statement is a direct consequence of the spectral convergence
result in Theorem~\ref{thm:mfd.conv}:
\begin{theorem}
\label{thm:gaps}
Suppose $(X,g_\eps) \to (M, g_\eps)$ ($\eps>0$) is a family of Riemannian
coverings with fundamental domain $D$ constructed as in the
previous parts~(2A) or~(2B).
Then for each $n \in \N$ there exists $\eps = \eps_n > 0$ such that
\begin{equation}
\label{eq:gaps}
I(\eps):=\bigcup_{k \in \N} I_k(\eps), \qquad \text{with} \qquad
I_k(\eps) := [\EWN k (\eps), \EWD k (\eps)],
\end{equation}
is a closed set having at least $n$ gaps, i.e.\ $n+1$ components as subset
of $[0,\infty)$. Here, $\EWDN k (\eps)$ denotes the $k$-th Dirichlet,
resp., Neumann eigenvalue of the Laplacian on $(D,g_\eps)$.
\end{theorem}
\begin{proof}
First, note that $\set{\EWDN k (\eps)}{ k \in \N}$, $\eps\geq 0$,
has no finite accumulation point, since the spectrum is discrete.
Second, Theorem~\ref{thm:mfd.conv}
shows that the intervals $I_k(\eps)$ reduce
to the point $\{\EW k (0) \}$ as $\eps \to 0$.
Therefore, $I(\eps)$ is a locally finite
union of compact intervals, hence closed.
\end{proof}
\subsection{Existence of spectrum outside the gaps}
\label{ssec:ex.spec}
In the following subsection we will assure that each Neumann-Dirichlet
interval $I_k(\eps)$ contains at least one point of $\spec \laplacian
{(X,g_\eps)}$ provided $\eps$ is small enough. In our general setting
described below (cf.\ Theorems~\ref{thm:gaps.type.I}
and~\ref{thm:gaps.res.fin}) we will show the inclusion
\begin{equation}
\label{eq:spec.incl}
\spec \laplacian {(X,g_\eps)} \subset \bigcup_{k \in \N} I_k(\eps).
\end{equation}
It is a priori not clear that each $I_k(\eps)$ intersects the spectrum
of the Laplacian on $(X,g_\eps)$, i.e.\ that gaps in $\bigcup_{k \in
\N} I_k(\eps)$ are also gaps in $\spec \laplacian {(X,g_\eps)}$. If
the covering group is amenable, the $k$-th eigenvalue of the Laplacian
on the quotient $(M,g_\eps)$ is always an element of $I_k(\eps) \cap
\spec (\laplacian X, g_\eps)$ (cf.~the argument in the proof of
Theorem~\ref{thm:gaps.type.I}). In general, this need not to be true.
Therefore, we need the following theorem which will be used in
Theorems~\ref{thm:gaps.res.fin} and~\ref{thm:band}:
\begin{theorem}
\label{thm:spectrum}
With the notation of the previous theorem, we have
\begin{equation}
\label{eq:spectrum}
I_k(\eps) \cap \spec \laplacian {(X,g_\eps)} \ne \emptyset
\end{equation}
for all $k \in \N$.
\end{theorem}
We begin with a general criterion which will be useful to detect
points in the spectra of a parameter-dependent family of operators
using only its sesquilinear form. A similar result is also stated
in~\cite[Lemma~5.1]{krejcirik-kriz:05}.
Suppose that $H_\eps$ is a self-adjoint, non-negative, unbounded
operator in a Hilbert space $\HS_\eps$ for each $\eps>0$. Denote by
$\HS_\eps^1 := \dom h_\eps$ the Hilbert space of the corresponding
quadratic form $h_\eps$ associated to $H_\eps$ with norm $\norm[1] u
:= (h_\eps(u) + \norm[\HS_\eps] u)^{1/2}$ and by $\HS_\eps^{-1}$ the
dual of $\HS_\eps^1$. Note that
$\map{H_\eps}{\HS_\eps^1}{\HS_\eps^{-1}}$ is continuous. In the next
lemma we characterise for each $\eps$ certain spectral points of
$H_\eps$.
\begin{lemma}
\label{lem:char.spec}
Suppose there exist a family $(u_\eps) \subset \HS_\eps^1$ and
constants $\lambda \ge 0$, $c>0$ such that
\begin{equation}
\label{eq:char.spec}
\norm[-1] {(H_\eps-\lambda)u_\eps} \to 0 \qquad \text{as} \qquad
\eps \to 0
\end{equation}
and $\norm {u_\eps} \ge c > 0$ for all $\eps>0$, then there exists
$\delta=\delta(\eps) \to 0$ as $\eps \to 0$ such that
\begin{displaymath}
\lambda + \delta(\eps) \in \spec H_\eps.
\end{displaymath}
\end{lemma}
\begin{proof}
Suppose that the conclusion is false. Then there exist a sequence
$\eps_n \to 0$ and a constant $\delta_0>0$ such that
\begin{displaymath}
I_\lambda \cap \spec H_{\eps_n} = \emptyset \qquad \text{with} \qquad
I_\lambda := (\lambda - \delta_0, \lambda + \delta_0)
\end{displaymath}
for all $n \in \N$. Denote by $E_t$ the spectral resolution of
$H_\eps$. Then
\begin{multline*}
\normsqr[-1] {(H_\eps - \lambda)u_\eps} =
\Dint {\R_+ \setminus I_\lambda} {\frac{(t-\lambda)^2} {(t+1)}}
{\iprod {E_tu_\eps} {u_\eps}} \\ \ge
\frac {\delta_0^2} {\lambda + \delta_0 + 1}
\Dint {\R_+ \setminus I_\lambda}
{} {\iprod {E_tu_\eps} {u_\eps}} \ge
\frac {c \delta_0^2} {\lambda + \delta_0 + 1}
\end{multline*}
since $I_\lambda$ does not lie in the support of the spectral
measure. But this inequality contradicts~\eqref{eq:char.spec}.
\end{proof}
\begin{remark}
Eq.~\eqref{eq:char.spec} is equivalent to the inequality
\begin{equation}
\label{eq:char.spec2}
|h_\eps(u_\eps, v_\eps) - \lambda \iprod {u_\eps} {v_\eps} | \le
o(1) \norm[1] {v_\eps} \qquad \text{for all $v_\eps \in \HS_\eps^1$}
\end{equation}
as $\eps \to 0$. Note that $o(1)$ could depend on $u_\eps$. The
advantage of the criterion in the previous lemma is that one only
needs to find a family $(u_\eps)$ in the domain of the quadratic
form $h_\eps$.
\end{remark}
We will need the following lemma in order to define a cut-off function
with convergent $L_2$-integral of its derivative. Its proof is
straightforward.
\begin{lemma}
\label{lem:cut-off}
Denote by $h(r):= r^{-d+2}$ if $d \ge 3$ and $h(r) = \ln r$ if
$d=2$. For $\eps \in (0,1)$ define
\begin{equation}
\label{eq:cut-off}
\chi_\eps(r):=
\begin{cases}
0, & 0 < r \le \eps\\
\frac{h(r)-h(\eps)}{h(\sqrt \eps) - h(\eps)}, &
\eps \le r \le \sqrt \eps\\
1, & \sqrt \eps \le r
\end{cases}
\end{equation}
then $\chi_\eps \in \Sob{(0,1)}$ and
\begin{displaymath}
\normsqr {\chi_\eps'} := \Dint[1] 0 {|\chi_\eps'(r)|^2 r^{d-1}} r = o(1)
\end{displaymath}
as $\eps \to 0$.
\end{lemma}
Remember that $(N,g)$ is the unperturbed manifold as in
Figure~\ref{fig:constr-mfd}. In Case~A of
Subsection~\ref{ssec:quotient}, we denoted by $R_\eps$ the manifold
$N$ with a closed ball of radius $\eps$ removed around each point
where the handles have been attached (note that $R_\eps$ is also
contained in $D$) and denote by $(r,y)$ the polar coordinates around
such a point ($r=\eps$ corresponds to a component of $\bd R_\eps$).
\begin{proof}[Proof of Theorem~\ref{thm:spectrum}]
Let $\phi$ be the $k$-th eigenfunction of the limit operator
$\laplacian N$ with eigenvalue $\lambda=\lambda_k(0)$. We will treat
Cases~A and~B of Subsection~\ref{ssec:quotient} separately.
\noindent
(3A) Set $u_\eps(r,y) := \chi_\eps(r) \phi(r,y)$ in the polar
coordinates described above and $u_\eps := \phi$ on $R_{\sqrt
\eps}$. Now, $\normsqr[R_{\sqrt \eps}] {\phi} \ge c$ since
$\normsqr[R_{\sqrt \eps}] \phi \to \normsqr[N] \phi>0$ as $\eps \to
0$. In addition, $u_\eps \in \Sobn {R_\eps} \subset \Sob {X,g_\eps}$
and
\begin{multline*}
|\iprod {du_\eps} {dv_\eps} - \lambda \iprod {u_\eps} {v_\eps}| \\ =
\Bigl| \int_{R_\eps} \bigl[\iprod {d\phi} {d(\chi_\eps v_\eps)}
-\lambda \phi \overline{\chi_\eps v} \bigr] +
\int_{R_\eps} \phi \iprod {d\chi_\eps}{dv_\eps} -
\int_{R_\eps} \overline v \iprod {d\phi}{d\chi_\eps}
\Bigr|
\end{multline*}
for all $v_\eps \in \Sob {D_\eps}$. Now the first integral
vanishes since $\phi$ is the eigenfunction with eigenvalue
$\lambda$ on $N$. Note that $\chi_\eps v \in \Sobn {R_\eps}$ can
be interpreted as function in $\Sob N$. The second and third
integral can be estimated from above by
\begin{displaymath}
\sup_{x \in N} \bigl[|\phi(x)| + |d\phi(x)| \bigr]
\norm {\chi_\eps'} \norm[1] {v_\eps} = o(1) \norm[1] {v_\eps}
\end{displaymath}
since $\phi$ is a smooth function on an $\eps$-independent space
and due to Lemma~\ref{lem:cut-off}.
\noindent
(3B) Set $u_\eps := \phi$ on $N$ and $u_\eps(r,y):=\widetilde
\chi_\eps(r) \phi(0,y)$, $r>0$, i.e.\ on $D \setminus N$ with
$\widetilde \chi_\eps(r):=\chi_\eps(\sqrt \eps + \eps^d - r)$,
where $\chi_\eps$ is defined in~\eqref{eq:cut-off} with $d=2$. Note
that $\widetilde \chi_\eps'(r) \ne 0$ only for those $r=\dist(x,\bd
N)$ where the conformal factor $\rho_\eps(x)=\eps$. Now, $u_\eps
\in \Sobn {D,g_\eps} \subset \Sob {X,g_\eps}$. Furthermore, for
$v_\eps \in \Sob {D,g_\eps}$ we have
\begin{multline*}
|\iprod {du_\eps} {dv_\eps} - \lambda \iprod {u_\eps} {v_\eps}| \le
\int_{D \setminus N}
\Bigl[
\bigl|
\widetilde \chi_\eps'(r)\phi(0,y) \partial_r v_\eps
\bigr| \rho_\eps^{d-2} \\ +
\bigl|
\widetilde \chi_\eps (r) \iprod {d_y\phi(0,y)} {d_y v_\eps}
\bigr| \rho_\eps^{d-2} +
\lambda \widetilde \chi_\eps(r) |\phi(0,y) v_\eps| \rho_\eps^d
\Bigr] \, \mathrm dr \, \mathrm dy \\ \le
C
\Bigl[
\Bigl(
\int\limits_{\eps^d}^{\sqrt \eps + \eps^d - \eps}
|\widetilde \chi_\eps'(r)|^2 \eps^{d-2} \,\mathrm dr
\Bigr)^{\frac 12} \\+
\Bigl(
\int\limits_0^{\sqrt \eps}
|\widetilde \chi_\eps(r)|^2 \rho_\eps^{d-2} \, \mathrm dr
\Bigr)^{\frac 12} +
\Bigl(
\int\limits_0^{\sqrt \eps}
|\widetilde \chi_\eps(r)|^2 \rho_\eps^d \, \mathrm dr
\Bigr)^{\frac 12}
\Bigr] \norm[1]{v_\eps}
\end{multline*}
where we have used that $\phi$ is the Neumann eigenfunction on
$N$. Furthermore, $C$ depends on the supremum of $\phi$ and
$d\phi$ and on $\lambda$. Note that the conformal factor
$\rho_\eps$ equals $\eps$ on the support of $\widetilde
\chi_\eps'$, therefore, the first integral converges to $0$ since
$d \ge 3$. Finally, estimating $\widetilde \chi_\eps$ and
$\rho_\eps$ by $1$, the second and third integral are bounded by
$\eps^{1/4}$.
\end{proof}
We finally can define formally the meaning of ``decoupling'':
\begin{definition}
We call a family of metrics $(g_\eps)_\eps$ on $X \to M$
\emph{decoupling}, if the conclusions of Theorems~\ref{thm:gaps}
and~\ref{thm:spectrum} hold, i.e., if there exists a fundamental
domain $D$ such that for each $n$ there exists $\eps_n>0$ such that
$I(\eps_n)$ in \eqref{eq:gaps} has at least $n+1$ components and
if~\eqref{eq:spectrum} holds for all $k \in \N$.
\end{definition}
\begin{remark}
\label{ExplainMethods}
In the present section we have specified two constructions of
decoupling families of metrics on covering manifolds, such that the
corresponding Laplacians will have at least a prescribed number of
spectral gaps (cf.~Sections~\ref{sec:type.I} and~\ref{sec:res.fin}).
The construction specified in method~(A) is feasible for every given
covering group $\Gamma$ with $r$ generators. Note that this method
produces fundamental domains that have smooth boundaries (see
e.g.~Example~\ref{ex:fund.group} below).
The construction in~(B) applies for every given Riemannian
covering $(X,g) \to (M,g)$, since, by the procedure described,
one can modify conformally this covering in order
to satisfy the spectral convergence
result of Theorem~\ref{thm:mfd.conv}
(cf.~Example~\ref{ex:heisenberg}).
\end{remark}
\section{Floquet theory for non-abelian groups}
\label{sec:floquet}
The aim of the present section is to state a spectral inclusion result
(cf.~Theorem~\ref{thm:spec.incl}) and the direct integral
decomposition of $\laplacian X$ (cf.~Theorem~\ref{thm:floquet}) for
certain \emph{non-abelian} discrete groups $\Gamma$. These results will
be used to prove the existence of spectral gaps in the situations
analysed in the next two sections. A more detailed presentation of
the results in this section may be found in \cite{lledo-post:07}.
\subsection{Equivariant Laplacians}
\label{ssec:equiv.lapl}
We will introduce next a new operator that lies ``between'' the
Dirichlet and Neumann Laplacians and that will play an important role
in the following. Suppose $\rho$ is a unitary representation of the
discrete group $\Gamma$ on the Hilbert space $\HS$, i.e.\ $\map \rho
\Gamma{\Unitary \HS}$ is a homomorphism. We fix a fundamental domain
$D$ for the $\Gamma$-covering $X \to M$.
We now introduce the space of smooth $\rho$-equivariant functions
\begin{equation}
\label{def:equiv.fct}
\CiR {D,\HS} := \set {h \restr D}
{h \in \Ci {X, \HS}, \quad h(\gamma x) = \rho_\gamma h(x), \quad
\gamma \in \Gamma, x \in X}.
\end{equation}
This definition coincides with the usual one for abelian groups,
cf.~\cite{lledo-post:07}. Note that we need \emph{vector-valued}
functions $\map h X \HS$ since the representation $\rho$ acts on the
Hilbert space $\HS$, which, in general, has dimension greater than
$1$.
We define next the so-called \emph{equivariant Laplacian} (w.r.t.\ the
representation $\rho$) on $\Lsqr {D,\HS} \cong \Lsqr D \otimes \HS$:
Let a quadratic form be defined by
\begin{equation}
\label{eq:quad.form}
\normsqr[D] {dh} := \Dint D {\normsqr[\HS] {dh(x)}} {X(x)}
\end{equation}
for $h \in \CiR {D,\HS}$, where the integrand is locally given by
\begin{displaymath}
\normsqr[\HS] {dh(x)} =
\sum_{i,j} g^{ij}(x) \, \iprod[\HS] {\partial_i h(x)} {\partial_j h(x)},
\qquad x \in D.
\end{displaymath}
This generalises Eq.~\eqref{def:quad.form} to the case of vector-valued
functions. We denote the domain of the closure of the quadratic form by
$\SobR{D,\HS}$. The corresponding non-negative,
self-adjoint operator on $\Lsqr {D,\HS}$, the
\emph{$\rho$-equivariant Laplacian}, will be denoted by $\laplacianR {D,\HS}$
(cf.~\cite[Chapter~VI]{kato:95}).
\subsection{Dirichlet-Neumann bracketing}
\label{ssec:dir-neu}
We study in this section the spectrum of a $\rho$-equivariant
Laplacian $\Delta^\rho$ associated with a finite-dimensional
representation $\rho$. In particular, we show that $\spec
\Delta^\rho$ is contained in a suitable set determined by the spectrum
of the Dirichlet and Neumann Laplacians on $D$. The key ingredient in
dealing with non-abelian groups is the observation that this set is
\emph{independent} of $\rho$.
We begin with the definition of certain operators acting in $\Lsqr
{D,\HS}$ and its eigenvalues. We denote by $\EWN m (\HS)$, $\EWR m
(\HS)$, resp., $\EWD m (\HS)$ the $m$-th eigenvalue of the operator
$\laplacianN {D,\HS}$, $\laplacianR {D,\HS}$, resp., $\laplacianD {D,
\HS}$ corresponding to the quadratic form~\eqref{eq:quad.form} on
$\Sobn {D,\HS}$, $\SobR {D,\HS}$, resp., $\Sob {D,\HS}$. Recall that
$\Sobn {D,\HS}$ is the $\Sobsymb^1$-closure of the space of smooth
functions $\map h D \HS$ with support away from $\bd D$ and $\Sob
{D,\HS}$ is the closure of the space of smooth functions with
derivatives continuous up to the boundary.
The proof of the next lemma follows, as in the abelian case
(cf.~Eqs.~\eqref{eq:dom.mono} and~\eqref{eq:ew.mono}), from the
reverse inclusions of the quadratic form domains
\begin{equation}
\label{eq:dom.mono.2}
\Sob {D,\HS} \supset \SobR {D,\HS} \supset \Sobn {D,\HS}
\end{equation}
and the min-max principle~\eqref{eq:min.max}.
\begin{lemma}
\label{lem:bracketing}
We have
\begin{displaymath}
\EWN m (\HS) \le \EWR m (\HS) \le \EWD m (\HS)
\end{displaymath}
for all $m \in \N$.
\end{lemma}
>From the definition of the quadratic form in the Dirichlet, resp., Neumann
case we have that the corresponding
vector-valued Laplacians are a direct sum of the scalar operators. Therefore
the eigenvalues of the corresponding vector-valued Laplace operators consist
of repeated eigenvalues of the scalar Laplacian. We can
arrange the former in the following way:
\begin{lemma}
\label{lem:dn.scalar}
If $n:=\dim \HS < \infty$ then
\begin{displaymath}
\EWDN m (\HS) = \EWDN k, \qquad
m=(k-1)n+1, \dots, kn,
\end{displaymath}
where $\EWDN k$ denotes the (scalar) $k$-th Dirichet/Neumann eigenvalue on
$D$.
\end{lemma}
\begin{proof}
Note that $\laplacianDN {D, \HS}$ is unitarily equivalent to an $n$-fold
direct sum of the scalar operator $\laplacianDN D$ on $\Lsqr D$ since there
is no coupling between the components on the boundary.
\end{proof}
Recall the definition of the intervals $I_k := [\EWN k, \EWD k]$ in
Eq.~\eqref{eq:gaps} (for simplicity, we omit in the following the
index $\eps$). From the preceding two lemmas we may collect the $n$
eigenvalues of $\laplacianR {D,\HS}$ which lie in $I_k$:
\begin{equation}
\label{eq:band.rho}
B_k(\rho) := \set {\EWR m (\HS)}
{m=(k-1)n+1, \dots, kn}
\subset I_k, \qquad n := \dim \HS.
\end{equation}
Therefore, we obtain the following spectral inclusion for equivariant
Laplacians. This result will be applied in
Theorems~\ref{thm:gaps.type.I} and \ref{thm:gaps.res.fin}
below.
\begin{theorem}
\label{thm:spec.incl}
If $\rho$ is a unitary representation on a finite-dimensional
Hilbert space $\HS$ then
\begin{displaymath}
\spec \laplacianR {D, \HS} =
\bigcup_{k \in \N} B_k(\rho) \subseteq
\bigcup_{k \in \N} I_k
\end{displaymath}
where $\laplacianR {D, \HS}$ denotes the $\rho$-equivariant Laplacian.
\end{theorem}
\subsection{Non-abelian Floquet transformation}
\label{ssec:floquet}
Consider first the right, respectively, left regular representation $R$,
resp., $L$ on the Hilbert space $\lsqr \Gamma$:
\begin{equation}
\label{def:reg.rep}
(R_\gamma a)_\tg = a_{\tg \gamma}, \qquad
(L_\gamma a)_\tg = a_{\gamma^{-1}\tg}, \quad\qquad
a = (a_\gamma)_\gamma \in \lsqr \Gamma,
\quad \gamma,\tg \in \Gamma.
\end{equation}
Using standard results we introduce the following unitary map (see
e.g., \cite[Section~3 and the appendix]{lledo-post:07} and
references cited therein)
\begin{equation}
\label{eq:fourier}
\map F {\lsqr \Gamma}{\OintZ {\HS(z)}}
\end{equation}
for a suitable measure space $(Z, \mathrm dz)$. The map $F$ is a
generalisation of the Fourier transformation in the abelian case.
Moreover, it transforms the right regular representation $R$ into the
following direct integral representation
\begin{equation}
\label{eq:reg.rep.trafo}
\widehat R_\gamma = F R_\gamma F^* = \OintZ {R_\gamma(z)}, \qquad \gamma \in \Gamma.
\end{equation}
\begin{remark}
\label{rem:meas.space}
Let $\al R$ be the von Neumann algebra generated by all unitaries
$R_\gamma$, $\gamma \in \Gamma$, i.e.
\begin{equation}
\label{eq:gen.vn.algebra}
\al R = \set {R_\gamma} {\gamma \in \Gamma}'',
\end{equation}
where $\al R'$ denotes the commutant of $\al R$ in $\End {\lsqr
\Gamma}$. Then we decompose $\al R$ with respect to a maximal abelian
von Neumann subalgebra $\al A\subset \al R'$ (for a concrete example
see Example~\ref{ex:dir.int}). The space $Z$ is the compact
Hausdorff space associated, by Gelfand's isomorphism, to a
\emph{separable} $C^*$-algebra $\al C$, which is strongly dense in
$\al A$. Furthermore, $\mathrm d z$ is a regular Borel measure on
$Z$. We may identify the algebra $\al A$ with $\Linfty {Z,\mathrm
dz}$ and since it is maximal abelian, the fibre representations
$R(z)$ are irreducible a.e.\ (see
\cite[Section~14.8~ff.]{wallach:92}).
\end{remark}
The generalised Fourier transformation introduced in
Eq.~\eqref{eq:fourier} can be used to decompose $\Lsqr X$ into a
direct integral. In particular, we define for a.e.~$z \in Z$:
\begin{equation}
\label{eq:floquet.short}
(Uu)(z)(x) := \sum_{\gamma \in \Gamma} \,u(\gamma x) R_{\gamma^{-1}}(z) v(z) ,
\end{equation}
where $v:=F \delta_e \in \lsqr \Gamma$, $u \in \Cci X$ and $x \in D$.
The map $U$ extends to a unitary map
\begin{displaymath}
\map U {\Lsqr X} {\OintZ
{\Lsqr {D,\HS(z)}} \cong \OintZ {\HS(z)} \otimes \Lsqr D},
\end{displaymath}
the so-called \emph{Floquet} or \emph{partial Fourier transformation}.
Moreover, operators commuting with the translation $T$ on $\Lsqr X$
are decomposable, in particular, we can decompose $\laplacian X$ since
its resolvent commutes with all translations~\eqref{eq:transl}.
We denote by $\CiEq {D, \HS(z)}$ the set of smooth $R(z)$-equivariant
functions defined in~\eqref{def:equiv.fct} and $\laplacianZ D$ is the
$R(z)$-equivariant Laplacian in $\Lsqr {D,\HS(z)}$. One can show in
this context (cf.~\cite{sunada:88,lledo-post:07}):
\begin{theorem}
\label{thm:floquet}
The operator $U$ maps $\Cci X$ into
$\OintZ {\CiEq {D,\HS(z)}}$. Moreover, $\laplacian X$ is unitary
equivalent to $\OintZ {\laplacianZ D}$ and
\begin{equation}
\label{eq:spec.dir.int}
\spec \laplacian X \subseteq
\overline {\bigcup_{z \in Z} \spec \laplacianZ D}.
\end{equation}
If $\Gamma$ is amenable (cf.\ Remark~\ref{rem:amenable}), then we
have equality in \eqref{eq:spec.dir.int}.
\end{theorem}
\begin{example}
\label{ex:dir.int}
Let us illustrate the above direct integral decomposition in the
case of the free group $\Gamma = \Z * \Z$ generated by $\alpha$ and
$\beta$. Let $A \cong \Z$ be the cyclic subgroup generated by
$\alpha$. We can decompose the algebra $\mathcal R$ given
in~\eqref{eq:gen.vn.algebra} w.r.t.\ the abelian algebra $\mathcal A
:= \set{L_a \in \mathcal L(\lsqr \Gamma)}{a \in A} \subset \mathcal
R'$, and, in this case, we have $Z = \Sphere^1$. Since the set
$\set{a \gamma a^{-1}}{a \in A}$ is infinite provided $\gamma \notin
A$, the algebra is \emph{maximal} abelian in $\mathcal R'$ (i.e.
$\mathcal A = \mathcal A' \cap \mathcal R'$), and therefore, each
fibre representation $R(z)$ is irreducible in $\HS(z)$.
Moreover, since $L_a \in \mathcal A'$ ($a \in A$) we can also
decompose these operators w.r.t\ the previous direct integral.
We can give a more concrete realisation of the abstract Fourier
transformation $F=F_\Gamma$ (see e.g.~\cite[Section~19]{robert:83}):
We interprete $\Gamma \to A \setminus \Gamma$ as covering space with
abelian covering group $A$ acting on $\Gamma$ from the left; the
corresponding translation action $T_a$ on $\lsqr \Gamma$ coincides
with the left regular representation $L_a$ ($a \in A$). The
(abelian) Floquet transformation $U=U_A$ gives a direct integral
decomposition
\begin{equation*}
\map{F_\Gamma = U_A}
{\lsqr \Gamma} {\Oint {\widehat A} {\HS(\chi)} \chi},
\end{equation*}
where $\HS(\chi) \cong \lsqr {A \setminus \Gamma}$ is the space of
$\chi$-equivariant sequences in $\lsqr \Gamma$. Note that $\HS(\chi)$
is infinite dimensional. A straightforward calculation shows that
\begin{equation*}
R_\gamma \cong \Oint {\widehat A} {R_\gamma(\chi)} \chi
\quad \text{and} \quad
L_a \cong \Oint {\widehat A} {L_a (\chi)} \chi,
\end{equation*}
where $R_\gamma(\chi) u(\tg) = u(\tg \gamma)$ and $L_a(\chi) u(\tg)=
\overline \chi(a) u(\tg)$ for $u \in \HS(\chi)$. Note that
$L_\gamma$, $\gamma \notin A$, does not decompose into a direct
integral over $Z$ since it mixes the fibres. Furthermore, one sees
that $v = (U \delta_e)(\chi)$ is the \emph{unique} normalised
eigenvector of $R_a(\chi)$ with eigenvalue $\chi(a)$. This follows
from the fact that the set of cosets $\set{A\gamma a}{a \in A}
\subset A\setminus \Gamma$ is infinite provided $\gamma \notin A$.
From the previous facts one can directly check that each $R(\chi)$
is an irreducible representation of $\Gamma$ in $\HS(\chi)$ and that
these representations are mutually inequivalent. Finally, $R(\chi)$
is also inequivalent to any irreducible component of the direct
integral decomposition obtained from a different maximal abelian
subgroup $B \ne A$.
\end{example}
\section{Spectral gaps for type~I groups}
\label{sec:type.I}
We will present in this section the first method to show that the
Laplacian of the manifolds constructed in Section~\ref{sec:construye}
with (in general \emph{non-abelian}) type~I covering groups have an
arbitrary finite number of spectral gaps. We begin recalling the
definition of type~I groups in the context of discrete groups.
\begin{definition}
\label{def:type.I}
A discrete group $\Gamma$ is of \emph{type~I} if $\Gamma$ is a finite
extension of an abelian group, i.e.\ if there is an exact sequence
\begin{displaymath}
0 \longrightarrow
A \longrightarrow
\Gamma\longrightarrow
\Gamma_0 \longrightarrow
0 ,
\end{displaymath}
where $A \lhd \Gamma$ is abelian and $\Gamma_0 \cong \Gamma/ A$ is a finite
group.
\end{definition}
\begin{remark}
\label{rem:type.I}
\begin{enumerate}
\item
\label{rem:type.I.i}
In the previous definition we have used a simple characterisation
of countable, \emph{discrete} groups of type~I due to Thoma,
cf.~\cite{thoma:64}. Moreover, all irreducible representations of
a type~I group $\Gamma$ are finite-dimensional and have a uniform
bound on the dimension (see~\cite{thoma:64,moore:72}). Therefore,
the following properties are all equivalent: (a)~there is a
uniform bound on the dimensions of irreducible representations of
$\Gamma$, (b)~all irreducible representations of $\Gamma$ are
finite-dimensional, (c) $\Gamma$ is a finite extension of an
abelian group, (d)~$\Gamma$ is CCR (completely continuous
representation, cf.~\cite[Ch.~14]{wallach:92}), (e)~$\Gamma$ is of
type~I. Recall also that $\Gamma$ is of type~I iff the von
Neumann algebra $\al R$ generated by $\Gamma$
(cf.~Eq.~\eqref{eq:gen.vn.algebra}) is of \emph{type~I}
(cf.~\cite{kaniuth:69}).
Note that for our application it would be enough if $\Gamma$ has a
decomposition over a measure space $(Z,\mathrm d z)$ as in
Remark~\ref{rem:meas.space} such that \emph{almost} every
representation $\rho(z)$ is finite-dimensional. But such a group
is already of type~I: indeed, if the set $\set {z \in Z} {\dim
\HS(z) = \infty}$ has measure $0$, then it follows from
\cite[Section~II.3.5]{dixmier:81} that the von Neumann Algebra
$\mathcal R$ (cf.~Eq.~\eqref{eq:gen.vn.algebra}) is of type~I. By
the above equivalent characterisation this implies that $\Gamma$
is of type~I.
\item
\label{rem:type.I.ii}
The following criterion (cf.~\cite{kaniuth:69,kallman:70}) will
be used in Examples~\ref{ex:heisenberg} and \ref{ex:free.group} to
decide that a group is not of type~I: The von Neumann algebra $\al
R$ is of type~II$_1$ iff $\Gamma_{\mathrm{fcc}}$ has infinite
index in $\Gamma$. Here,
\begin{equation}
\label{eq:fcc}
\Gamma_{\mathrm{fcc}} :=
\set{\gamma \in \Gamma} {C_\gamma \text{ is finite}}
\end{equation}
is the set of elements $\gamma\in\Gamma$ having finite conjugacy
class $C_\gamma$. In particular such a group is not of type~I.
Even worse: Almost all representations in the direct integral
decomposition~\eqref{eq:reg.rep.trafo} are of type~II$_1$
(\cite[Section~II.3.5]{dixmier:81}) and therefore
infinite-dimensional (see e.g.~Example~\ref{ex:dir.int}).
\end{enumerate}
\end{remark}
\begin{remark}
\label{rem:amenable}
The notion of amenable discrete groups will be useful at different
stages of our approach. For a definition of \emph{amenability} of
a discrete group $\Gamma$ see e.g.~\cite{day:57} or
\cite{brooks:81}. We will only need the following equivalent
characterisations: (a)~$\Gamma$ is amenable. (b)~$0 \in \spec
\laplacian X$~\cite{brooks:81}. (c)~$\spec \laplacian M \subset
\spec \laplacian X$~\cite[Propositions~7--8]{sunada:88}. Here, $X
\to M$ is a covering with covering group $\Gamma$. Note that
discrete type~I groups are amenable since they are finite extensions
of abelian groups (extensions of amenable groups are again amenable,
cf.~\cite[Section~4]{day:57}).
We want to stress that Theorem~\ref{thm:spectrum} is no
contradiction to the fact that $\Gamma$ is amenable iff $0 \in \spec
\laplacian {(X,g_\eps)}$ although the first interval
$I_1(g_\eps)=[0,\EWD k(g_\eps)]$ tends to $0$ as $\eps \to 0$. Note
that we have only shown that $I_1(g_\eps) \cap \spec \laplacian
{(X,g_\eps)} \ne \emptyset$ and \emph{not} $0=\EW 1(M,g_\eps) \in
\spec \laplacian {(X,g_\eps)}$ which is only true in the amenable
case.
\end{remark}
The \emph{dual of $\Gamma$}, which we denote by $\hG$, is the set of
equivalence classes of unitary irreducible representations of $\Gamma$. We
denote by $[\rho]$ the (unitary) equivalence class of a unitary
representation $\rho$ on $\HS$. Note that the spectrum of a
$\rho$-equivariant Laplacian and $\dim \HS$ only depend on the
\emph{equivalence class} of $\rho$.
If $\Gamma$ is of type~I, then the dual $\hG$ becomes a nice measure space
(``smooth'' in the terminology of \cite[Chapter~2]{mackey:76}).
Furthermore, we can use $\hG$ as measure space in the direct integral
decomposition defined in Subsection~\ref{ssec:floquet}. In particular,
combining the results of Section~\ref{sec:prelim}
and~\ref{sec:floquet} we obtain the main result for type~I
groups:
\begin{theorem}
\label{thm:gaps.type.I}
Suppose $X \to M$ is a Riemannian $\Gamma$-covering with fundamental
domain $D$, where $\Gamma$ is a type~I group and denote by $g$ the
Riemannian metric on $X$. Then
\begin{displaymath}
\spec \laplacian {(X,g)} \subset \bigcup_{k \in \N} I_k(g),
\qquad\mathrm{and}\qquad
I_k(g) \cap \spec \laplacian{(X,g)} \ne \emptyset, \quad k \in \N,
\end{displaymath}
where $I_k(g):=[\EWN k (D,g), \EWD k (D,g)]$ is the
Neumann-Dirichlet interval defined as in~\eqref{eq:gaps}. In
particular, for each $n \in \N$ there exists a metric $g=g_{\eps_n}$
constructed as in Subsection~\ref{ssec:constr.cov.sp} such that
$\spec \laplacian {(X,g)}$ has at least $n$ gaps, i.e.\ $n+1$
components as subset of $[0, \infty)$.
\end{theorem}
\begin{proof}
We have
\begin{displaymath}
\spec \laplacian X =
\overline {\bigcup_{[\rho] \in \hG} \spec \laplacianR {D, \HS}}
\subseteq \overline {\bigcup_{k \in \N} I_k(g)}
= \bigcup_{k \in \N} I_k(g),
\end{displaymath}
where we used the Theorem~\ref{thm:floquet} with $Z=\hG$ for the
first equality and Theorem~\ref{thm:spec.incl} for the inclusion.
Note that $\Gamma$ is amenable and that the latter theorem applies
since all (equivalence classes of) irreducible representations of a
type~I group are finite-dimensional
(cf.~Remark~\ref{rem:type.I}~(\ref{rem:type.I.i})). The existence of
gaps in $\bigcup_k I_k(g)$ follows from Theorem~\ref{thm:gaps}.
Since $\Gamma$ is amenable, $\spec \laplacian M \subset \spec
\laplacian X$ (cf.~(c) in Remark~\ref{rem:amenable}). Moreover,
from Eq.~\eqref{eq:band.rho} with $\rho$ the trivial representation
on $\HS=\C$, we have that $\lambda_k(M) \in I_k$. Note that
functions on $M$ correspond to functions on $D$ with periodic
boundary conditions. Therefore, we have shown that every gap of the
union $\bigcup_k I_k(g)$ is also a gap of $\spec \laplacian X$.
\end{proof}
\section{Spectral gaps for residually finite groups}
\label{sec:res.fin}
In this section, we present a new method to prove the existence of a
finite number of spectral gaps of $\laplacian X$. The present
approach is applicable to so-called residually finite groups $\Gamma$,
which is a much larger class of groups containing type~I groups
(cf.~Section~\ref{sec:examples}). Roughly speaking, residually finite
means that $\Gamma$ has a lot of normal subgroups with finite index.
Geometrically, this implies that one can approximate the covering
$\map \pi X M$ with covering group $\Gamma$ by \emph{finite} coverings
$\map {p_i} {M_i} M$, where the $M_i$'s are compact.
Since the present section is central to the paper we will give for
completeness proofs of known results, namely for
Theorem~\ref{thm:res.fin} (see~\cite{ass:94,adachi:95}).
\subsection{Subcoverings and residually finite groups}
\label{ssec:sub.cov}
Suppose that $\map \pi X M$ is a covering with covering group $\Gamma$ (as
in Section~\ref{sec:prelim}). Corresponding to a normal subgroup $\Gamma_i
\lhd \Gamma$ we associate a covering $\map {\pi_i} X {M_i}$ such that
\begin{equation}
\label{eq:sub.cov}
\begin{diagram}
& & X & &\\
& \ldTo(2,2)^{\pi_i}_{\Gamma_i} & & \rdTo(2,2)^\pi_\Gamma& \\
M_i & & \rTo^{p_i}_{\Gamma/\Gamma_i}
& & M
\end{diagram}
\end{equation}
is a commutative diagram. The groups under the arrows denote the
corresponding covering groups.
\begin{definition}
\label{def:res.fin}
A (countable, infinite) discrete group $\Gamma$ is residually finite if
there exists a monotonous decreasing sequence of normal subgroups
$\Gamma_i \lhd \Gamma$ such that
\begin{equation}
\label{eq:sub.groups}
\Gamma=\Gamma_0 \rhd \Gamma_1 \rhd \dots \rhd \Gamma_i \rhd \cdots, \quad
\bigcap_{i \in \N} \Gamma_i = \{e\} \quad \text{and} \quad
\text{$\Gamma/\Gamma_i$ is finite.}
\end{equation}
Denote by $\mathfrak R \mathcal F$ the class of residually finite
groups.
\end{definition}
Suppose now that $\Gamma$ is residually finite. Then there exists a
corresponding sequence of coverings $\map {\pi_i} X {M_i}$ such that
$\map {p_i} {M_i} M$ is a \emph{finite} covering
(cf.~Diagram~\eqref{eq:sub.cov}). Such a sequence of covering maps is
also called \emph{tower of coverings}.
\begin{remark}
We recall also the following equivalent definitions of residually
finite groups (see e.g.~\cite{magnus:69} or~\cite[Section~2.3]{robinson:82}).
\begin{enumerate}
\item A group $\Gamma$ is called \emph{residually finite} if for
all $\gamma \in \Gamma\setminus \{e\}$ there is a group homomorphism
$\map \Psi \Gamma G$ such that $\Psi(\gamma) \ne e$ and
$\Psi(\Gamma)$ is a \emph{finite} group.
\item Let $\al F$ denote the class of finite groups. Then $\Gamma$ is
residually finite, iff the so-called \emph{$\al F$-residual}
\begin{equation}
\label{eq:residual}
\mathfrak{R}_{\al F}(\Gamma) :=
\bigcap_{ \substack{N \lhd \Gamma\\\Gamma/N \in \al F}} N
\end{equation}
is trivial, i.e.~$\mathfrak{R}_{\al F}(\Gamma)=\{e\}$.
\end{enumerate}
\end{remark}
Next we give some examples for residually finite groups (cf.~the
survey article~\cite{magnus:69}):
\begin{example}
\label{ex:res.fin}
(i)~Abelian and finite groups are residually finite. (ii)~Free
products of residually finite groups are residually finite, in
particular, the free group in $r$ generators $\Z^{*r}$ is residually
finite. (iii)~Finitely generated linear groups are residually finite
(for a simple proof of this fact cf.~\cite{alperin:87}; a group is
called \emph{linear} iff it is isomorphic to a subgroup of
$\mathrm{GL}_n(\C)$ for some $n \in \N$.) In particular,
$\mathrm{SL}_n(\Z)$, fundamental groups of closed, orientable
surfaces of genus $g$ or, more generally, finitely generated
subgroups of the isometry group on the hyperbolic space $\Hyp^d$ are
residually finite.
\end{example}
Next we need to introduce a metric on the discrete space $\Gamma$:
\begin{definition}
\label{def:word.met}
Let $G$ be a set which generates $\Gamma$. The \emph{word metric}
$d=d_G$ on $\Gamma$ is defined as follows: $d(\gamma,e)$ is the
minimal number of elements in $G$ needed to express $\gamma$ as a word
in the alphabet $G$; $d(e,e):=0$ and $d(\gamma,\tg) := d(\gamma
\tg^{-1}, e)$.
\end{definition}
Geometrically, residually finiteness means that, given any compact set
$K \subset X$, there exists a finite covering $\map {p_i} {M_i} M$ and
a covering $\map {\pi_i} X {M_i}$ which is injective on $K$
(cf.~\cite{brooks:86}). This idea is used in the following lemma:
\begin{lemma}
\label{lem:seq.fund.dom}
Fix a fundamental domain $D$ for the covering $\map \pi X M$ and
suppose that $\map {\pi_i} X {M_i}$ ($i \in \N$) is a tower of
coverings as above. Then for each covering $\map {\pi_i} X {M_i}$
there is a fundamental domain $D_i$ (not necessarily connected) such
that
\begin{displaymath}
D_0 := D \subset D_1 \subset \dots \subset D_i \subset \cdots
\qquad \text{and} \qquad
\bigcup_{i \in \N} D_i = X.
\end{displaymath}
\end{lemma}
\begin{proof}
It is enough to show the existence of a family of representants $R_i
\subset \Gamma$ of $\Gamma/\Gamma_i$, $i\in\N$, satisfying
\begin{displaymath}
R_0 := \{e\} \subset R_1 \subset \dots \subset R_i \subset \cdots
\qquad \text{and} \qquad
\bigcup_{i \in \N} R_i = \Gamma.
\end{displaymath}
In this case the fundamental domains are given explicitly by
\begin{displaymath}
D_i := \intr \bigcup_{r \in R_i} r^{-1} \overline D,
\end{displaymath}
where $\intr$ denotes the topological interior.
Let $d$ be the word metric on $\Gamma$ with respect to the set of
generators $G := \set{\gamma \in \Gamma}{\gamma\overline D \cap
\overline D \ne \emptyset}$, which is naturally adapted to the
fundamental domain $D$. Note that $G$ is finite and generates $\Gamma$
since $\overline D$ is compact (cf.~\cite[Theorems~6.5.10
and~6.5.11]{ratcliffe:94}).
We choose a set of representants $R_i$ of $\Gamma/\Gamma_i$ that
have minimal distance in the word metric to the neutral element,
i.e.~if $r\in R_i$, then $d(r,e)\leq d(r\Gamma_i, e)$. Note that
since $\Gamma_{i+1}\subset\Gamma_i$ we have $R_{i+1}\supset R_i$.
To conclude the proof we have to show that every $\gamma \in \Gamma$ is
contained in some $R_i$, $i \in \N$. Since $\Gamma$ is finitely
generated, there exists $n \in \N$ such that $\gamma\in B_{n} :=
\{\gamma \in \Gamma\mid d(\gamma,e)\le n\}$. Moreover, since $B_{2n}$ is
finite and $\Gamma$ residually finite we also have
$B_{2n}\cap\Gamma_i=\{e\}$ for $i$ large enough. Therefore, any
other element $\tg=\gamma \gamma_i^{-1}$ in the class $\gamma
\Gamma_i$ with $\gamma_i \in \Gamma_i \setminus \{e\}$ has a distance
greater than $n$, since
\begin{displaymath}
d(\tg, e) = d(\gamma\gamma_i^{-1},e) = d(\gamma, \gamma_i) \ge
d(e,\gamma_i) - d(\gamma,e) > 2n - n = n.
\end{displaymath}
This implies that $\gamma\in R_i$
by the minimality condition in the choice of the representants.
\end{proof}
\begin{theorem}
\label{thm:res.fin}
Suppose $\Gamma$ is residually finite with the associated sequence of coverings
$\map {\pi_i} X {M_i}$ and $\map {p_i} {M_i} M$ as in~\eqref{eq:sub.cov}.
Then
\begin{displaymath}
\spec \laplacian X \subseteq \overline {\bigcup_{i \in \N} \spec \laplacian
{M_i}},
\end{displaymath}
and the Laplacian $\laplacian {M_i}$ w.r.t.\ the finite covering
$\map {p_i} {M_i} M$ has discrete spectrum. Equality holds iff
$\Gamma$ is amenable.
\end{theorem}
\begin{proof}
(Cf.~\cite{adachi:95}) If $\lambda \in \spec \laplacian X$, then for each
$\eps>0$ there exists $u \in \Cci X$ such that
\begin{displaymath}
\frac{\normsqr[X] {(\laplacian X - \lambda) u}} {\normsqr[X] u} < \eps.
\end{displaymath}
Applying Lemma~\ref{lem:seq.fund.dom} there is an $i=i(\eps)$ such that
$\supp u \subset D_i$. Furthermore, since
$D_i \hookrightarrow M_i=X/\Gamma_i$ is an
isometry, $u$ can be written as the lift of a smooth $f$ on $M_i$,
i.e.\ $f \circ \pi_i = u$. Therefore,
\begin{displaymath}
\frac{\normsqr[M_i] {(\laplacian {M_i} - \lambda) f}} {\normsqr[M_i] f} =
\frac{\normsqr[X] {(\laplacian X - \lambda) u}} {\normsqr[X] u} < \eps,
\end{displaymath}
which implies $\lambda \in \overline {\bigcup_{i \in \N} \spec
\laplacian {M_i}}$. Finally, since $M_i \to M$ is a finite
covering and $M$ is compact, $\spec\laplacian {M_i}$ is discrete.
For the second assertion cf.~\cite{adachi:95} or~\cite{ass:94}. One
basically uses the characterisation due to \cite{brooks:81} that
$\Gamma$ is amenable iff $0 \in \spec \laplacian X$ (cf.\
Remark~\ref{rem:amenable}).
\end{proof}
Next we analyse the spectrum of the finite covering $M_i \to M$. Note that
$D$ is also isometric
to a fundamental domain for \emph{each} finite covering $M_i \to M$,
$i \in \N$.
\begin{lemma}
\label{lem:fin.group}
We have
\begin{displaymath}
\spec \laplacian {M_i} =
\bigcup_{[\rho] \in \widehat{G_i}}
\spec \laplacianR {D,\HS(\rho)},
\end{displaymath}
where $\Delta^\rho$ is the equivariant Laplacian introduced in
Subsection~\ref{ssec:equiv.lapl} and $G_i := \Gamma/\Gamma_i$ is a finite
group and $\widehat{G_i}$ its dual.
\end{lemma}
\begin{proof}
Applying the results of Subsection~\ref{ssec:floquet} to the finite group
$G_i$ and the finite measure space $Z:=\widehat{G_i}$ with the counting
measure all direct integrals become direct sums. By Peter-Weyl's theorem
(see e.g.~\cite[\S27.49]{hewitt-ross-2}) we also have
\begin{displaymath}
\map F {\lsqr {G_i}}
{\bigoplus_{[\rho] \in \widehat{G_i}} n(\rho) \HS(\rho)},
\end{displaymath}
where each multiplicity satisfies $n(\rho)=\dim \HS(\rho)<\infty$.
Finally,
\begin{displaymath}
\laplacian {M_i} \cong
\bigoplus_{[\rho] \in \widehat{G_i}} \laplacianR {D, \HS(\rho)}
\end{displaymath}
and the result follows.
\end{proof}
We now can formulate the main result of this section:
\begin{theorem}
\label{thm:gaps.res.fin}
Suppose $X \to M$ is a Riemannian $\Gamma$-covering with fundamental
domain $D$, where $\Gamma$ is a residually finite group and denote by
$g$ the Riemannian metric on $X$. Then
\begin{displaymath}
\spec \laplacian {(X,g)} \subset \bigcup_{k \in \N} I_k(g), \qquad
I_k(g) \cap \spec \laplacian{(X,g)} \ne \emptyset, \quad k \in \N,
\end{displaymath}
where $I_k(g):=[\EWN k (D,g), \EWD k (D,g)]$ is defined as
in~\eqref{eq:gaps}. In particular, for each $n \in \N$ there exists
a metric $g=g_{\eps_n}$, constructed as in
Subsection~\ref{ssec:constr.cov.sp}, such that $\spec \laplacian
{(X,g)}$ has at least $n$ gaps, i.e.\ $n+1$ components as subset of
$[0, \infty)$.
\end{theorem}
\begin{proof}
We have
\begin{displaymath}
\spec \laplacian X \subseteq
\overline {\bigcup_{i \in \N} \spec \laplacian {M_i}} =
\overline {\bigcup_{\substack{i \in \N\\ [\rho] \in \widehat {G_i}}}
\spec \laplacianR {D, \HS(\rho)}}
\subseteq
\overline {\bigcup_{k \in \N} I_k(g) } = \bigcup_{k \in \N} I_k(g),
\end{displaymath}
where we used Theorem~\ref{thm:res.fin}, Lemma~\ref{lem:fin.group}
and Theorem~\ref{thm:spec.incl}. Note that the latter theorem
applies since all (equivalence classes of) irreducible
representations of the finite groups $G_i$, $i\in\N$, are
finite-dimensional. The existence of gaps in $\bigcup_k I_k(g)$
follows from Theorem~\ref{thm:gaps}. Finally, by
Theorem~\ref{thm:spectrum}, a gap of $\bigcup_k I_k(g)$ is in fact a
gap of $\spec \laplacian X$.
\end{proof}
\section{Kadison constant and asymptotic behaviour}
\label{sec:kadison}
In the present section we will combine our main result stated in
Theorem~\ref{thm:gaps.res.fin} with some results by Sunada and
Br\"uning (cf.~\cite[Theorem~1]{sunada:92} or
\cite{bruening-sunada:92}), to give a more complete description of the
spectrum of the Laplacian $\laplacian X$, where $X \to M$ is the
$\Gamma$-covering constructed in Section~\ref{sec:construye}. For
this, we need a further definition:
\begin{definition}
\label{def:kadison}
Let $\Gamma$ be a finitely generated discrete group.
The \emph{Kadison constant} of $\Gamma$ is defined as
\begin{displaymath}
C(\Gamma) := \inf \set{ \tr_\Gamma (P)}
{\text{$P$ non-trivial projection in
$C^*_{\mathrm{red}}(\Gamma,\al K)$}},
\end{displaymath}
where $\tr_\Gamma (\cdot)$ is the canonical trace on
$C^*_{\mathrm{red}}(\Gamma,\al K)$ , the tensor product of the
reduced group $C^*$-algebra of $\Gamma$ and the algebra $\al K$ of
compact operators on a separable Hilbert space of infinite dimension
(see \cite[Section~1]{sunada:92} for more details.)
\end{definition}
In this section, we assume that $\Gamma$ is is residually finite and
has a strictly positive Kadison constant, i.e.~$C(\Gamma)>0$. For
example, the free product $\Z^{*r} * \Gamma_1 * \dots * \Gamma_a$ with
finite groups $\Gamma_i$ satisfies both properties
(cf.~e.g.~\cite{magnus:69}, \cite[Appendix]{sunada:92}). Another such
group is the fundamental group (cf.~Eq.~\eqref{eq:fund.group}) of a
(compact, orientable) surface of genus $g$
(see~\cite{marcolli-mathai:99}).
\begin{remark}
Suppose that $K$ is an integral operator on $\Lsqr X$ commuting
with the group action, having smooth kernel $k(x,y)$ and satisfying
\begin{displaymath}
k(x,y) = 0 \qquad \text{for all $x,y \in X$ with $d(x,y) \ge c$}
\end{displaymath}
for some constant $c>0$. Then $K$ can be interpreted as an element
of $C_{\mathrm {red}}^*(\Gamma, \al K)$ and one can write the
$\Gamma$-trace as
\begin{displaymath}
\tr_\Gamma K = \int_D k(x,x)\, dx
\end{displaymath}
(see \cite[Section~1]{sunada:92} as well as \cite{atiyah:76} for
further details), where $D$ is a fundamental domain of $X \to M$.
If we consider the spectral
resolution of the Laplacian $\laplacian X \cong \Oint{} \lambda
{E(\lambda)}$, then it follows that
\begin{displaymath}
E(\lambda_2) - E(\lambda_1) \in C_{\mathrm {red}}^*(\Gamma, \al K)
\end{displaymath}
if $\lambda_1 < \lambda_2$ and $\lambda_1, \lambda_2 \not\in \spec
\laplacian X$ (cf.~\cite[Section~2]{sunada:92}).
\end{remark}
Denote by $\mathcal N(g,\lambda)$ the number of components of
$\spec\laplacian {(X,g)} \cap [0, \lambda]$.
From~\cite{bruening-sunada:92,sunada:92} we obtain the following
asymptotic estimate on $\mathcal N(g,\lambda)$:
\begin{theorem}
\label{thm:lower.asym}
Suppose $(X,g) \to (M,g)$ is a Riemannian $\Gamma$-covering where
$\Gamma$ has a positive Kadison
constant, i.e.\ $C(\Gamma)>0$ then
\begin{equation}
\limsup_{\lambda\to\infty}
\frac{\mathcal N(g, \lambda)}
{(2\pi)^{-d} \omega_d \vol (M,g) \lambda^{d/2}}
\le \frac 1 {C(\Gamma)}.
\end{equation}
In particular, the spectrum of $\laplacian X$ has band-structure,
i.e.\ $\mathcal N(g,\lambda)<\infty$ for all $\lambda \ge 0$.
\end{theorem}
\begin{remark}
\label{rem:bethe.sommer}
Note that Theorem~\ref{thm:lower.asym} only gives an \emph{asymptotic}
upper bound on the number of components of $\spec \laplacian X \cap
[0,\lambda]$, not on the \emph{whole} spectrum itself. Therefore,
we have no assertion about the so-called \emph{Bethe-Sommerfeld
conjecture} stating that the number of spectral gaps for a
periodic operator in dimensions $d \ge 2$ remains \emph{finite}.
\end{remark}
Combining Theorem~\ref{thm:lower.asym} with our result on spectral gaps
we obtain more information on the spectrum and a \emph{lower}
asymptotic bound on the number of components:
\begin{theorem}
\label{thm:band}
Suppose $(X,g) \to (M,g)$ is a Riemannian $\Gamma$-covering where
$\Gamma$ is a residually finite group and where $g=g_\eps$ is the
family of decoupling metrics constructed in
Section~\ref{sec:construye}. Then we have:
\begin{enumerate}
\item For each $n \in \N$ there exists $g=g_{\eps_n}$ such that $\spec
\laplacian {(X,g)}$ has at least $n$ gaps. If in addition
$C(\Gamma)>0$ then there exists $\lambda_0>0$ such that
\begin{equation*}
n +1 \le \mathcal N(g,\lambda) < \infty
\end{equation*}
for all $\lambda \ge \lambda_0$, i.e.\ $\spec {\laplacian{(X,g)}}$
has band-structure.
\item Suppose in addition that the limit manifold $(N,g)$ has simple
spectrum, i.e.\ all eigenvalues $\lambda_k(0)$ have multiplicity $1$
(cf.~Theorem~\ref{thm:mfd.conv}). Then for each $\lambda \ge 0$
there exists $\eps(\lambda)>0$ such that
\begin{equation*}
\liminf_{\lambda\to\infty}
\frac{\al N (g_{\eps(\lambda)}, \lambda)}
{(2\pi)^{-d} \omega_d \vol (N,g) \lambda^{d/2}}
\ge 1.
\end{equation*}
Here, $g_\eps$ denotes the metric constructed in
Section~\ref{sec:construye}.
\end{enumerate}
\end{theorem}
\begin{proof}
(i)~follows immediately from Theorems~\ref{thm:gaps.res.fin}
and~\ref{thm:lower.asym}. (ii)~Suppose $\lambda \notin \spec
\laplacian N$, then $\lambda_k(0) < \lambda < \lambda_{k+1}(0)$ for
some $k\in\N$. Let $\eps=\eps(\lambda) \in (0,1]$ be the largest
number such that $\al N(\lambda,g_\eps)$ is (at least) $k$, in other
words, $\al N(\lambda, g_\eps) \ge k = \al N(\lambda, \laplacian N)$
where the latter number denotes the number of eigenvalues of
$\laplacian N$ below $\lambda$. We conclude with the Weyl theorem,
\begin{displaymath}
\lim_{\lambda\to\infty}
\frac{\al N (\lambda, \laplacian N)}
{(2\pi)^{-d} \omega_d \vol (N,g) \lambda^{d/2}} = 1,
\end{displaymath}
where $\omega_d$ denotes the volume of the $d$-dimensional Euclidean
unit ball.
\end{proof}
\sloppy To conclude the section we remark that generically,
$\laplacian {(N,g)}$ has simple spectrum (cf.~\cite{uhlenbeck:76}).
The assumption on the spectrum of $(N,g)$ is natural since $\mathcal
N(g,\lambda)$ counts the components without multiplicity.
\section{Examples}
\label{sec:examples}
\subsection{Relation between the approaches presented in
Sections~\ref{sec:type.I} and \ref{sec:res.fin}}
\label{Relation5.6}
We begin comparing the two main approaches presented in this
paper which assure the existence of spectral gaps
(cf.~Sections~\ref{sec:type.I} and \ref{sec:res.fin}).
One easily sees from Definition~\ref{def:res.fin} that a \emph{finite}
extension of a residually finite group is again residually finite. In
particular, type~I groups are residually finite as finite extensions
of abelian groups (cf.\ Definition~\ref{def:type.I}). Therefore, for
type~I groups one can also produce spectral gaps by the approximation
method with finite coverings introduced in Section~\ref{sec:res.fin}.
Nevertheless we believe that the direct integral method will be useful
when analysing further spectral properties:
\begin{example}
\label{rem:ev.dep.cont}
One of the advantages of the method described in
Section~\ref{sec:type.I} is that one has more information about the
bands. Suppose $\Gamma$ is finitely generated and \emph{abelian},
i.e.\ $\Gamma \cong \Z^r \oplus \Gamma_0$, where $\Gamma_0$ is the
torsion subgroup of $\Gamma$. Then $\hG$ is the disjoint union of
finitely many copies of $\Torus^r$. From the continuity of the map
$\rho \to \EWR k$ (cf.~\cite{bjr:99} or~\cite{sunada:90}), we can
simplify the characterisation of the spectrum in
Theorem~\ref{thm:floquet} and obtain
\begin{equation}
\label{eq:char.spec.ab}
\spec \laplacian X = \bigcup_{k \in \N} B_k, \quad \text{where} \quad
B_k := \set{\EWR k} {\rho \in \hG} \subseteq I_k,
\end{equation}
the $k$-th \emph{band}. Since $\hG$ is compact, $B_k$ is also
compact, but in general, $B_k$ need not to be connected (recall that
$\hG$ is connected iff $\Gamma$ is torsion free, i.e.\ $\Gamma= \Z^r$).
Note also that $B_k$ has only finitely many components. For
non-abelian groups this approach may be generalised in the direction
of Hilbert C*-modules (cf.~\cite{gruber:01}).
\end{example}
In principle one could also consider a combination of the methods of
Section~\ref{sec:type.I} and~\ref{sec:res.fin}: denote by $\al T_1$
the class of type~I groups and by $\mathfrak{R}\mathcal{T}_1$ the
class of \emph{residually type~I} groups, i.e.
$\Gamma\in\mathfrak{R}\mathcal{T}_1$ iff the $\al T_1$-residual
$\mathfrak{R}_{\mathcal{T}_1}(\Gamma)$ is trivial
(cf.~Eq.~\eqref{eq:residual}). Similarly we denote by
$\mathfrak{R}\mathcal{F}$ the class of residually finite groups
(cf.~Definition~\ref{def:res.fin}). If we consider a covering with a
group $\Gamma\in\mathfrak{R}\mathcal{T}_1$, then instead of the
\emph{finite} covering $\map{p_i} {M_i} M$ considered in
Eq.~\eqref{eq:sub.cov} we would have a covering with a type~I group.
For these groups, we can replace Lemma~\ref{lem:fin.group} by the
direct integral decomposition of Theorem~\ref{thm:floquet}.
Nevertheless the following lemma shows that the class of residually
finite and residually type~I groups coincide.
\begin{lemma}
\label{lem:res.class}
From the inclusion
$\mathcal{F}\subset\mathcal{T}_1\subset\mathfrak{R}\mathcal{F}$ it
follows that the corresponding residuals for the group $\Gamma$
coincide, i.e.\ $\mathfrak R_{\al F}(\Gamma)=\mathfrak R_{\al T_1}
(\Gamma)$. Moreover,
$\mathfrak{R}\mathcal{F}=\mathfrak{R}\mathcal{T}_1$.
\end{lemma}
\begin{proof}
From the inclusion $\mathcal{F}\subset\mathcal{T}_1$ it follows
immediately that $\mathfrak R_{\al F}(\Gamma)\supset \mathfrak
R_{\al T_1} (\Gamma)$. To show the reverse inclusion one uses the
following characterisation: a group is residually $\al F$ iff it is
a subcartesian product of finite groups
(cf.~\cite[\S~2.3.3]{robinson:82}). Finally, from the equality of
the residuals it follows that
$\mathfrak{R}\mathcal{F}=\mathfrak{R}\mathcal{T}_1$.
\end{proof}
\subsection{Examples with residually finite groups}
\label{ResFinGroups}
In the rest of this subsection we present several examples
of residually finite groups which are not type~I. They
show different aspects of our analysis.
For the next example recall the construction~(A) described in
Section~\ref{sec:construye}.
\begin{example}[Fundamental groups of oriented, closed surfaces]
\label{ex:fund.group}
Suppose that $N:=\Sphere^2$ is the two-dimensional sphere with a
metric such that $\laplacian N$ has simple spectrum
(cf.~\cite{uhlenbeck:76} for the existence of such metrics).
Suppose, in addition, that $M$ is obtained by adding $r$ handles to
$N$ as described in Section~\ref{sec:construye}, Case~A. The
fundamental group $\Gamma$ of $M$ (cf.~Eq.~\eqref{eq:fund.group}
with $s=0$) is residually finite (recall
Example~\ref{ex:res.fin}~(iii)). Moreover, from the proof of
Proposition~2.16 in \cite{marcolli-mathai:99}, it follows that
$\Gamma$ has a positive Kadison constant. Therefore,
Theorem~\ref{thm:band} applies to the the universal cover $X
:=\widetilde M \to M$ with the metric $g_\eps$ specified in
Section~\ref{sec:construye}.
\end{example}
The following example uses the construction~(B) in
Section~\ref{sec:construye}.
\begin{example}[Heisenberg group]
\label{ex:heisenberg}
Let $\Gamma := H_3(\Z)$ be the \emph{discrete Heisenberg group},
where $H_3(R)$ denotes the set of matrices
\begin{equation}
\label{eq:heisenberg}
A_{x,y,z} := \begin{pmatrix}
1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1
\end{pmatrix}
\end{equation}
with coefficients $x,y,z$ in the ring $R$. A covering with group
$\Gamma$ is given e.g.~by $X:=H_3(\R)$ with compact quotient
$M:=H_3(\R)/H_3(\Z)$. Note that $X$ is diffeomorphic to $\R^3$.
Clearly, $\Gamma$ is a finitely generated linear group and therefore
residually finite (cf.\ Example~\ref{ex:res.fin}~(iii)). Now, by
Theorem~\ref{thm:gaps.res.fin} one can deform conformally a
$\Gamma$-invariant metric $g$ as in Case~(B) of
Section~\ref{sec:construye}, such that $\spec \laplacian X$ has at
least $n$ spectral gaps, $n\in\N$.
In this case, $\Gamma$ is also amenable as an extension of amenable groups
(cf.\ Remark~\ref{rem:amenable}). In fact,
$\Gamma$ is isomorphic to the semi-direct product
$\Z\ltimes \Z^2$, where $1 \in \Z$ acts on $\Z^2$ by the matrix
\begin{displaymath}
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix}.
\end{displaymath}
Therefore, we have equality in the
characterisation of $\spec \laplacian X$ in
Theorems~\ref{thm:floquet} and~\ref{thm:res.fin}.
Note finally that the group $\Gamma$ is not of type~I since
$\Gamma_{\mathrm {fcc}} = \set{A_{0,y,0}}{y \in \Z}$ has infinite
index in $\Gamma$ (cf.\
Remark~\ref{rem:type.I}~(\ref{rem:type.I.ii})). Thus, our method in
Section~\ref{sec:type.I} does not apply since the measure $\mathrm
dz$ in~\eqref{eq:fourier} is supported only on infinite-dimensional
Hilbert spaces. Curiously, one can construct a \emph{finitely}
additive measure on the group dual $\hG$ supported by the set of
finite-dimensional representations of $\hG$ (cf.~\cite{pytlik:79}).
The group dual $\hG$ is calculated
e.g.~in~\cite[Beispiel~1]{kaniuth:68}.
\end{example}
\begin{example}[Free groups]
\label{ex:free.group}
Let $\Gamma = \Z^{*r}$ be the free group with $r>1$
generators. Then $\Gamma$ is residually finite
(recall Example~\ref{ex:res.fin}~(ii))
and has positive Kadison constant (cf.~\cite[Appendix]{sunada:92}).
Therefore, Theorem~\ref{thm:band} applies to the $\Gamma$-coverings
$X \to M$ specified in Section~\ref{sec:construye}.
Note that $\Gamma$ is not of type~I since $\Gamma_{\mathrm
{fcc}}=\{e\}$ (cf.~Remark~\ref{rem:type.I}~(\ref{rem:type.I.ii})).
Such groups are called \emph{ICC (infinite conjugacy class) groups}.
Again, for any direct integral decomposition~\eqref{eq:fourier},
almost all Hilbert spaces $\HS(z)$ are infinite-dimensional. Finally,
$\Gamma$ is not amenable.
\end{example}
\subsection{An example with an amenable, non-residually finite group}
\label{sec:OpenQuestion}
Kirchberg mentioned in \cite[Section~5]{kirchberg:94} an interesting
example of a finitely generated \emph{amenable} group which is not
residually finite: Denote by $S_0$ the group of permutations of $\Z$
which leave unpermuted all but a finite number of integers. We call
$A_0$ the normal subgroup of even permutations in $S_0$. Let $\Z$ act
on $S_0$ as shift operator. Then the semi-direct product $\Gamma: =\Z
\ltimes S_0$ is (finitely) generated by the shift $n \mapsto n+1$ and
the transposition interchanging $0$ and $1$. Note that $\Gamma$ and
$S_0$ are ICC groups.
\begin{lemma}
\label{S0amenable}
The group $\Gamma$ is amenable. Moreover, $\mathfrak{R}_{\al
F}(\Gamma)=A_0$, hence $\Gamma$ is not residually finite.
\end{lemma}
\begin{proof}
The group $S_0$ is amenable as inductive limit of amenable groups;
therefore, $\Gamma$ is amenable as semi-direct product of amenable
groups (cf.~\cite[Section~4]{day:57}).
The equality $\mathfrak R_{\mathcal F} (\Gamma) = A_0$ follows from
the fact that $A_0$ is simple.
\end{proof}
\begin{proposition}
\label{prop:OpenFinRep}
Every finite-dimensional unitary representation $\rho$ of $\Gamma$
leaves $A_0$ elementwise invariant, i.e.\ $\rho(\gamma)=\1$ for all
$\gamma \in A_0$.
\end{proposition}
\begin{proof}
Let $\al E$ be the class of countable subgroups of $\mathrm U(n)$,
$n \in \N$, and $\al {FG}$ the class of finitely generated groups.
Note that $\al F \subset \al E \cap \al {FG}$ and that finitely
generated linear groups are residually finite
(cf.~Example~\ref{ex:res.fin}~(iii)), i.e.~$\al E \cap \al {FG}
\subset\mathfrak R \al F$. Arguing as in the proof of
Lemma~\ref{lem:res.class} we obtain from the inclusions $\al F
\subset \al E \cap \al {FG} \subset\mathfrak R \al F$ that
$\mathfrak R_{\al E \cap \al{FG}}(\Gamma) = \mathfrak R_{\al
F}(\Gamma)$. Now by Lemma~\ref{S0amenable} the $\al F$-residual
of $\Gamma$ is $A_0$. Finally, since $\Gamma$ itself is finitely
generated (i.e.\ $\Gamma \in \al {FG}$), we have
\begin{displaymath}
\mathfrak R_{\al E}(\Gamma) =
\mathfrak R_{\al E \cap \al{FG}}(\Gamma)=A_0.
\end{displaymath}
This concludes the proof since $\rho$ is a finite-dimensional
unitary representation iff $\mathrm {im} (\rho) \cong \Gamma/\ker
\rho \in \al E$, i.e.\ $\mathfrak R_{\al E}(\Gamma)$ is the
intersection of all $\ker \rho$, where $\rho$ are the
finite-dimensional, unitary representations of $\Gamma$.
\end{proof}
In conclusion, we cannot analyse the spectrum of $\laplacian X$ by
none of the above methods since $\Gamma$ is not residually finite (and
therefore neither of type~I). Nevertheless, equality holds
in~\eqref{eq:spec.dir.int}, but we would need infinite-dimensional
Hilbert spaces $\HS(z)$ in the direct integral decomposition in order to
describe the spectrum of the whole covering $X \to M$ and not only of
the subcovering $X/A_0 \to M$ (with covering group $\Z \times \Z_2$,
cf.\ Diagram~\eqref{eq:sub.cov}).
\begin{remark}
Coverings with transformation groups as in the present subsection
cannot be treated with the methods developed in this paper. It
seems though reasonable that even for non-residually finite groups
the construction specified in Section~\ref{sec:construye} still
produces at least $n$ spectral gaps, $n\in\N$. To show this one
needs to replace the techniques of Section~\ref{sec:floquet} that
use the min-max principle in order to prove the existence of
spectral gaps for these types of covering manifolds.
\end{remark}
\section{Conclusions and applications}
\label{sec:outlook}
Given a Riemannian covering $(X,g)\to (M,g)$ with a residually finite
transformation group $\Gamma$ we constructed a deformed
$\Gamma$-covering $(X,g_\eps)\to (M,g_\eps)$ such that
$\spec\Delta_{(X,g_\eps)}$ has $n$ spectral gaps, $n\in\N$.
Intuitively one decouples neighbouring fundamental domains by
deforming the metric $g\to g_\eps$ in such a way that the junctions of
the fundamental domains are scaled down (cf.~Figure~\ref{fig:per-mfd}).
Therefore, our construction may serve as a model of how to use
geometry to remove unwanted frequencies or energies in certain
situations which may be relevant for technological applications.
For instance, the Laplacian on $(X,g_\eps)$ may serve to give an
approximate description of the energy operator of a quantum mechanical
particle moving along the periodic space $X$. Usually, the energy
operator contains additional potential terms coming form the curvature
of the embedding in some ambient space, cf.~\cite{froese-herbst:00},
but, nevertheless, $\laplacian {(X,g_\eps)}$ is still a good
approximation for describing properties of the particle. A spectral
gap in this context is related to the transport properties of the
particle in the periodic medium, e.g., an insulator has a large first
spectral gap.
Another application are photonic crystals, i.e.\ optical materials
that allow only certain frequencies to propagate. Usually, one has to
consider differential forms in order to describe the propagation of
classical electromagnetic waves in a medium. Nevertheless, if we
assume that the Riemannian density is related to the dielectric
constant of the material, one can use the scalar Laplacian on a
manifold as a simplified model. For more details, we refer
to~\cite{kuchment:01,figotin-kuchment:98} and the references therein.
A further interesting line of research would be to consider the
opposite situation as in the present paper; that means the use of
geometry to prevent the appearance of spectral gaps
(cf.~\cite{friedlander:91,mazzeo:91}). In fact, these authors proved
that $\EWN {k+1}(D) \le \EWD k(D)$ for all $k \in \N$, i.e, that $I_k
\cap I_{k+1} \ne \emptyset$ for all $k \in \N$ provided $D$ is an open
subset of $\R^n$ or a Riemannian symmetric space of non-compact type.
On such a space, we have a priory no information on the existence of
gaps.
It would also be interesting to connect the number of gaps with
geometric quantities, e.g., isoperimetric constants or the curvature.
We want to stress that the curvature of $(X,g_\eps)$ is \emph{not}
bounded as $\eps \to 0$ (cf.~\cite{post:03a}) in contrast to the
degeneration of Riemannian metrics under curvature bounds
(cf.~e.g.~\cite{cheeger:01}).
In the present paper we have considered $\laplacian X$ as a prototype
of an elliptic operator and have avoided the use of a potential $V$.
In this way we isolate the effect of geometry on $\spec {\laplacian X}$.
Of course, our methods and results may also be extended to more
general periodic structures that have a ``reasonable'' Neumann
Laplacian as a lower bound and satisfy the spectral ``localisation''
result in Theorem~\ref{thm:spec.incl}. For example, one can also study
periodic operators like $\laplacian X + V$, operators on quantum wave
guides, more general periodic elliptic operators or operators on
metric graphs (cf.~e.g.~\cite{exner-post:05} for examples of periodic
metric graphs with spectral gaps).
Finally, we conclude mentioning that we can not apply directly our
result to disprove the Bethe-Sommerfeld conjecture on manifolds, which
says that the number of spectral gaps for a periodic operator in
dimensions $d \ge 2$ remains \emph{finite}. Even if we know that the
spectrum of the Laplacian on $(X,g_\eps)$ converges to the discrete
set $\set{\lambda_k}{k \in \N}$ as $\eps \to 0$, we cannot expect a
\emph{uniform} control of the spectral convergence on the whole
interval $[0,\infty)$ since there are topological obstructions
(cf.~\cite{chavel-feldman:81}). Note that a uniform convergence would
immediately imply that $\spec {\laplacian{(X,g_\eps)}}$ would have an
\emph{infinite} number of spectral gaps. Nevertheless, we hope that
our construction will contribute to the clarification of the status of
this conjecture.
\section*{Acknowledgements}
It is a pleasure to thank Mohamed Barakat for helpful discussions on
residually finite groups. We are also grateful to David
Krej{\v{c}}i{\v{r}}{\'\i}k and Norbert Peyerimhoff for useful
comments. Finally, we would like to thank Volker En{\ss},
Christopher Fewster, Luka Grubi\v{s}i\'c and Vadim Kostrykin for
valuable remarks and suggestions on the manuscript.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
{
"timestamp": "2007-12-10T16:59:19",
"yymm": "0503",
"arxiv_id": "math-ph/0503005",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503005"
}
|
\section{Introduction}
We study the discrete random Schr\"odinger operator
\begin{eqnarray}
H_\omega=\Delta+\lambda V_\omega
\end{eqnarray}
on $\ell^2({\Bbb Z}^2)$,
where $\Delta$ is the (centered) nearest neighbor Laplacian, with
spectrum $[-4,4]$, and $\lambda$ is a small parameter
(the disorder strength).
The random potential is given by $V_\omega(x)=v_{\sigma}(x)\omega_x$,
where $v_\sigma(x)\sim|x|^{-{\sigma}}$
and $\{\omega_x\}_{x\in{\Bbb Z}^2}$ are Gaussian i.i.d. random variables.
The restriction to Gaussian randomness has expository advantages,
but is not essential for our techniques to apply.
Extension of our methods to non-Gaussian random potentials can be
accessed along the lines demonstrated in \cite{ch}.
The purpose of this paper is to derive lower bounds on the localization
lengths of eigenfunctions of $H_\omega$.
In the supercritical case ${\sigma}>\frac12$,
it was proven by Bourgain in \cite{bo1} that with large probability,
$H_\omega$ (with Bernoulli or Gaussian randomness)
has, for small $\lambda$, pure a.c. spectrum in $(-4+\tau,-\tau)\cup(\tau,4-\tau)$
($\tau>0$ arbitrary, but fixed); moreover,
the wave operators were constructed, and asymptotic completeness was established.
The (generalized) eigenfunctions are therefore delocalized.
Certain other classes of lattice Schr\"odinger operators
with decaying random potentials have been proven to exhibit a.c. spectrum,
scattering, and asymptotic completeness by Bourgain in \cite{bo2},
and by Rodnianski and Schlag in \cite{rosc}. We also note the contextually
related work of Denissov in \cite{de}.
In the case ${\sigma}=0$, Schlag, Shubin and Wolff have proven lower bounds
on the localization length of eigenfunctions of the form
$\lambda^{-2+\eta}$, for any $\eta>0$, \cite{shscwo}.
For ${\sigma}=0$ and $d=3$, lower bounds of the form
$\lambda^{-2}|\log\lambda|^{-1}$ were derived in \cite{ch}.
We shall here address the case $0<{\sigma}\leq\frac12$ in dimension two.
Our main results are as follows.
For the critical decay exponent ${\sigma}=\frac12$, the problem is
{\em marginal} in the language of renormalization group theory.
Accordingly, we obtain a comparison of the {\em logarithm}
of the localization length to powers of $\lambda$,
yielding lower bounds on the localization length
that are {\em exponential} in $\frac1\lambda$, of the form
$2^{\lambda^{-\frac14+\eta}}$ ($\eta>0$ arbitrary).
In the subcritical case $0<{\sigma}<\frac12$,
it is suspected that the model exhibits a significant component
of point spectrum. In the language of renormalization group theory,
the potential scales like a {\em relevant} perturbation,
whereby we obtain a comparison of
the localization length to powers of $\lambda$.
Consequently, our lower bounds on the localization lengths are
{\em polynomial} in $\frac1\lambda$ for $0<{\sigma}<\frac12$,
of the form $\lambda^{-\frac{2-\eta}{1-2{\sigma}}}$ ($\eta>0$ arbitrary).
On the one hand, our strategy employs graph expansion methods
due to Erd\"os and Yau \cite{erd, erdyau}, and further elaborated on
by the author \cite{ch, ch1}. On the other hand, we use a
smoothing of resolvent multipliers by dyadic restriction,
inspired by Bourgain's approach in \cite{bo1}.
Our methods can be extended to higher dimensions,
but we will here only focus on the case $d=2$.
The following works, which determine macroscopic hydrodynamic limits of the quantum
dynamics in the Anderson model at small disorders (without spatial decay, i.e. ${\sigma}=0$),
are closely related to the topics discussed here.
In an important early work, Spohn proved in \cite{sp} that the kinetic macroscopic
scaling and low coupling limit is determined by a linear Boltzmann equation,
locally in macroscopic time. Erd\"os and Yau proved the corresponding global in
macroscopic time result for the continuum model in ${\Bbb R}^d$, $d=2,3$,
and Gaussian randomness, \cite{erdyau}, which was extended by Erd\"os to the case of a
Schr\"odinger electron interacting with a phonon heat bath, \cite{erd}. The author
derived the corresponding result for the lattice ${\Bbb Z}^3$ and non-Gaussian
randomness, \cite{ch}, and proved that the mode of convergence can be
extended to $r$-th mean, for any $r\in{\Bbb R}_+$ (the previous works proved
convergence in expectation), \cite{ch1}.
Eng and Erd\"os proved the corresponding result for the kinetic macroscopic
and low density limit, \cite{engerd}.
Very recently, Erd\"os, Salmhofer and Yau established the breakthrough
result that beyond kinetic scaling, the macroscopic dynamics is governed
by a diffusion equation, \cite{erdsalmyau}.
\section{Definition of the model and statement of the main results}
\label{intro-sect-1}
We consider the discrete random Schr\"odinger operator
\begin{eqnarray}
H_\omega = \Delta + \lambda V_\omega \;
\label{Homega-def}
\end{eqnarray}
on $\ell^2({\Bbb Z}^2)$, with a radially decaying potential function
\begin{eqnarray}
V_{\omega}(x) =v_{\sigma}(x) \omega_x \;,
\end{eqnarray}
where
$\{\omega_x\}_{x\in{\Bbb Z}^2}$ are independent, identically distributed Gaussian random variables
normalized by ${\Bbb E}[\omega_x]=0$, ${\Bbb E}[\omega_x^2]=1$, for all $x\in{\Bbb Z}^2$.
Expectations of higher powers of $\omega_x$ satisfy Wick's theorem,
see \cite{erdyau}, and our discussion below.
We shall use the convention
\begin{eqnarray}
{\mathcal F}(f)(k)\;\equiv\;\hat f(k)&=&\sum_{x\in{\Bbb Z}^2}
e^{-2\pi i kx} f({ x})
\nonumber\\
{\mathcal F}^{-1}(g)(x)\;\equiv\;\check g(x)
&=&\int_{\Bbb T^2} dk\,
e^{2 \pi i k x} g(k)
\end{eqnarray}
for the Fourier transform and its inverse,
where $\Bbb T:=[-\frac12,\frac12]$.
We introduce a partition of unity $\sum_{j=0}^\infty P_j=1$ on ${\Bbb Z}^2$,
where $P_j\sim \chi(2^j<|x|\leq2^{j+1})$, $j\in{\Bbb N}_0$,
is an approximate characteristic functions for a dyadic shell of scale $2^j$.
We require that $|{\mathcal F}(P_j P_{j'})|$, for $|j-j'|\leq1$,
are bump functions on $\Bbb T^2$ at the dual scale $2^{-j}$ satisfying
$\|{\mathcal F}(P_j P_{j'})\|_{L^1(\Bbb T^2)}\sim 1$.
We shall assume that $v_{\sigma}$ is such that
for any $j,j'\in{\Bbb N}_0$ with $|j-j'|\leq 1$, the Fourier transform of $P_j P_{j'} v_{\sigma}^2$ satisfies
\begin{eqnarray}
|{\mathcal F}(P_j P_{j'} v_{\sigma}^2)| \leq C
2^{-2{\sigma} j}|{\mathcal F}(P_j P_{j'} )|
\sim C 2^{-2{\sigma} j}|{\mathcal F}(P_j^2)| \;,
\label{Fouv-dyad-est-1}
\end{eqnarray}
for a constant $C$ independent of $j,j'$.
Since
\begin{eqnarray}
\|P_jv_{\sigma}\|_{\ell^\infty({\Bbb Z}^2)}=
\|P_j^2 v_{\sigma}^2\|_{\ell^\infty({\Bbb Z}^2)}^{1/2}
\leq\|{\mathcal F}(P_j^2 v_\sigma^2)\|_{L^1(\Bbb T^2)}^{1/2}
\sim 2^{-{\sigma} j}\;,
\end{eqnarray}
this in particular implies that
\begin{eqnarray}
|x|^{{\sigma}}|v_{\sigma}(x)|\leq C \;,
\end{eqnarray}
for $0<{\sigma}\leq\frac12$.
The centered nearest neighbor lattice Laplacian $\Delta$
defines the Fourier multiplier
\begin{eqnarray}
{\mathcal F}(\Delta f) ({ k}) = {e_\Delta}({ k}) \hat f({ k}) \;,
\end{eqnarray}
where
\begin{eqnarray}
{e_\Delta}({ k}) = 2\cos(2\pi k_1)+2\cos(2\pi k_2)
\label{kinendef}
\end{eqnarray}
is the quantum mechanical kinetic energy of the electron.
For almost every realization of $V_\omega$, $H_\omega$ is a selfadjoint operator
on $\ell^2({\Bbb Z}^2)$.
We shall use the same argument for the determination of the localization
length of eigenfunctions of $H_\omega$ as in \cite{ch}.
Let $L> e^{\lambda^{-2}}$, and
\begin{eqnarray}
\Lambda_L:=[-L,L]^2\cap{\Bbb Z}^2 \;.
\end{eqnarray}
For $\ell\ll L$ and $x\in\Lambda_L$, let
\begin{eqnarray}
R_{x,\delta, \ell}\sim \chi\big(\,\big\{y\in{\Bbb Z}^2\big|\,
\frac{\delta\ell}{2}<|x_i-y_i|<\frac{\ell}{2}\,,\,
i=1,2\big\}\,\big)
\end{eqnarray}
denote an approximate characteristic function supported on a
cubical shell centered at $x$, of outer and inner side lengths $\ell$ and
$\delta\ell$, respectively. We shall adopt
the choice for $R_{x,\delta,\ell}$ from \cite{ch},
which is a product of differences of Fej\'er kernels with
\begin{eqnarray}
\|R_{x,\delta,\ell}\|_{\ell^\infty(\Lambda_L)}=1 \;.
\end{eqnarray}
It is not necessary here to specify $R_{x,\delta,\ell}$ in more detail, as
its explicit form only enters a result that can be
straightforwardly adapted from \cite{ch} (Eq. (~\ref{free-evol-est-1})).
Given a fixed realization of the random potential for which
$H_\omega$ is selfadjoint on $\ell^2({\Bbb Z}^2)$,
let ${H_\omega^{(\Lambda_L)}}$ denote the restriction of $H_\omega$ to $\Lambda_L$.
Moreover, let $\{\psi_\alpha^{(L)}\}_{\alpha\in{\mathfrak A}_L}$ denote an orthonormal ${H_\omega^{(\Lambda_L)}}$-eigenbasis in
$\ell^2(\Lambda_L)$
\begin{eqnarray}
({H_\omega^{(\Lambda_L)}}\psi_\alpha^{(L)})(x)&=&e_\alpha^{(L)}\psi_\alpha^{(L)}(x)
\;\;\;
(x\in\Lambda_L) \;,
\end{eqnarray}
satisfying Dirichlet boundary conditions
\begin{eqnarray}
\psi_\alpha^{(L)}(x)=0
\;\;\;
(x\in\partial\Lambda_L
:=\Lambda_{L+1}\setminus\Lambda_L)\;.
\label{eigenL}
\end{eqnarray}
The number of eigenfuntions is given by
\begin{eqnarray}
|{\mathfrak A}_L|=|\Lambda_L| \;.
\end{eqnarray}
Let, for $\tau>0$ arbitrary but fixed, and independent of $\lambda$ and ${\sigma}$,
\begin{eqnarray}
I_\tau:= (-4+\tau,-\tau)\cup(\tau,4-\tau) \;.
\end{eqnarray}
Let
\begin{eqnarray}
{\alg}_L(I_\tau):=\{\alpha\in{\mathfrak A}_L\big|\,e_\alpha^{(L)}\in
I_\tau\}\;,
\end{eqnarray}
and similarly as in \cite{ch}, let
for $\varepsilon$ small
\begin{eqnarray}
{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)&:=&\big\{\,\alpha\in{\alg}_L(I_\tau)\big|\,
\nonumber\\
&&\hspace{1cm}
\sum_{x\in\Lambda_L} |\psi_\alpha^{(L)}(x)| \,
\big\| R_{x, \delta, \ell} \psi_\alpha^{(L)}
\big\|_{\ell^2(\Lambda_L)} < \varepsilon\,\big\} \;.
\end{eqnarray}
As pointed out in \cite{ch}, the key observation is that
$\{\psi_\alpha^{(L)}\}_{\alpha\in{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}$
contains the class of localized eigenstates
with energies in $I_\tau$ that are concentrated in
balls of radius $O(\frac{ \delta \ell }{ \log \ell })$,
with $\delta$ independent of $\ell$.
Our main result is the following theorem.
\begin{theorem}
\label{thm-main-1}
For $\delta>0$ sufficiently small, $0<\lambda\ll\delta$,
any fixed $\tau$ with $\lambda\ll\tau<\delta$,
and any arbitrary $\eta>0$,
\begin{eqnarray}
\liminf_{L\rightarrow\infty}{\Bbb E} \left[
\frac {|{\mathfrak A}_L\setminus{\mathfrak A}_L(\delta^{\frac45},\delta,\ell_{\sigma}(\lambda);I_\tau)|}
{|{\mathfrak A}_L|} \right]\ge 1 - \delta^{\frac{1}{5}} \;.
\label{thm-main-est-1}
\end{eqnarray}
The lower bound on the localization length $\ell_{\sigma}(\lambda)$
satisfies the following estimates:
\begin{itemize}
\item
In the subcritical case $0<{\sigma}<\frac12$,
there exist positive constants $\lambda_0({\sigma},\eta)\ll1$ and $C_{\sigma}$
for every fixed $0<{\sigma}<\frac12$ such that
\begin{eqnarray}
\ell_{\sigma}(\lambda)\geq C_{\sigma} \lambda^{-\frac{2-\eta}{1-2{\sigma}}}
\end{eqnarray}
for all $\lambda<\lambda_0({\sigma},\eta)$.
\item
In the critical case ${\sigma}=\frac12$, there exists a positive constant
$\lambda_0(\eta)\ll1$ such that
\begin{eqnarray}
\ell_{{\sigma}=\frac12}(\lambda)\geq 2^{\lambda^{-\frac14+\eta}}
\end{eqnarray}
for all $\lambda<\lambda_0(\eta)$.
\end{itemize}
\end{theorem}
\noindent{We} add the following remarks.
\begin{itemize}
\item
(~\ref{thm-main-est-1}) trivially implies
\begin{eqnarray}
{\Bbb P}\Big[\liminf_{L\rightarrow\infty}
\frac {|{\mathfrak A}_L\setminus{\mathfrak A}_L(\delta^{\frac45},\delta,\ell_{\sigma}(\lambda);I_\tau)|}
{|{\mathfrak A}_L|}>1-\delta^{\frac{1}{10}}\Big]>1-\delta^{\frac{1}{10}} \;.
\end{eqnarray}
\item
Spectral restriction to the interval $I_\tau$ suppresses infrared singularities, and
enables one to apply certain smoothing procedures to $\frac{1}{{e_\Delta}-z}$, \cite{bo1}.
\item
Only a slight modification of the bounds used in our analysis of the subcritical case
along the lines of \cite{ch} is necessary to yield
the lower bound $\lambda^{-2+\eta}$ for ${\sigma}=0$. Inclusion of a classification of graphs
argument as in \cite{erdyau, ch} would improve the lower bound to
$\lambda^{-2}|\log\lambda|^{-1}$.
We shall not further discuss these matters here, since the argument is the same as
the one presented in \cite{ch} for the 3-D problem.
\end{itemize}
\section{Proof of Theorem {~\ref{thm-main-1}}}
Our starting point is the following key lemma. It is an extension of
a joint result with L. Erd\"os and H.-T. Yau in \cite{ch}.
\begin{lemma}\label{ceylemma}
Let $\varepsilon,\delta>0$ be small and $\lambda\ll1$.
Assume that there exists $t^*(\delta,\ell)>0$, such that
\begin{eqnarray}
\label{mainest}
&&{\Bbb E} \Big[\frac{1}{|{\mathfrak A}_L|}
\sum_{x\in\Lambda_L}\big\| R_{x, \delta, \ell}\chi_{I_\tau}({H_\omega^{(\Lambda_L)}})
e^{-i t^*(\delta,\ell) {H_\omega^{(\Lambda_L)}} }
\delta_x\big\|_{\ell^2(\Lambda_L)} ^2\Big]
\nonumber\\
&&\hspace{3cm}\ge 1- \varepsilon - {\Bbb E}\Big[\frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}\Big]
-C\frac{\ell}{L} \;.
\end{eqnarray}
Then,
\begin{eqnarray}
\liminf_{L\rightarrow\infty}{\Bbb E}\left[
\frac {|{\mathfrak A}_L\setminus {\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)| } {|{\mathfrak A}_L|}\right]\ge 1 - 4
\varepsilon^{\frac12}\;.
\end{eqnarray}
\end{lemma}
\noindent{\em Proof.}$\;$
The proof follows closely a line of arguments presented in \cite{ch},
but comprises key modifications due to the restriction of
the energy range to $I_\tau$.
We expand $\delta_x$ in the eigenbasis $\{\psi_\alpha^{(L)}\}$,
\begin{eqnarray*}
\delta_x &=& \sum_\alpha {a_x^\alpha} \psi_\alpha^{(L)}
\; \;\\
a_x^\alpha &=& \overline{\big\langle \delta_{ x} \, , \,
\psi_\alpha^{(L)} \big\rangle }
= \overline{\psi_\alpha^{(L)}(x)} \;,
\end{eqnarray*}
so that in particular,
\begin{eqnarray}
\|\delta_x\|_{\ell^2(\Lambda_L)}^2=\sum_{\alpha\in{\mathfrak A}_L}|a_x^\alpha|^2=1\;.
\label{axalphl2norm}
\end{eqnarray}
Applying the Schwarz inequality,
\begin{eqnarray}
\Big\| R_{x, \delta, \ell}\chi_{I_\tau}({H_\omega^{(\Lambda_L)}}) e^{-i t {H_\omega^{(\Lambda_L)}} }
\delta_x\Big\|_{\ell^2(\Lambda_L)}^2
\leq (1+ \varepsilon^{-\frac12} )(A)+
(1+\varepsilon^{\frac12}) (B) \; ,
\label{CSest1}
\end{eqnarray}
where
\begin{eqnarray}
(A)&:=&\Big\|R_{x, \delta, \ell}e^{-i t {H_\omega^{(\Lambda_L)}} }
\sum_{\alpha \in {\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}{a_x^\alpha}
\psi_\alpha^{(L)} \Big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&\leq& \Big\|R_{x, \delta, \ell}
\sum_{\alpha \in {\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}
e^{-i t e_\alpha^{(L)} }{a_x^\alpha}
\psi_\alpha^{(L)}\Big\|_{\ell^2(\Lambda_L)}
\nonumber\\
&\leq& \sum_{\alpha\in {\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}
|\psi_\alpha^{(L)}(x)| \big\| R_{x, \delta, \ell}
\psi_\alpha^{(L)} \big\|_{\ell^2(\Lambda_L)} \;,
\end{eqnarray}
using the a priori bound
\begin{eqnarray}
(A)&\leq&
\Big\|
\sum_{\alpha \in {\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}e^{-i t e_\alpha^{(L)} }{a_x^\alpha}
\psi_\alpha^{(L)} \Big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&=&\sum_{\alpha\in{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)}|a_x^\alpha|^2 \;\leq \; 1
\;,
\end{eqnarray}
which follows from $\|R_{x, \delta, \ell}\|_\infty=1$,
orthonormality of
$\{\psi_\alpha^{(L)}\}_{\alpha\in{\mathfrak A}_L}$,
and (~\ref{axalphl2norm}).
Moreover,
\begin{eqnarray}
(B)&:=&\Big\| R_{x, \delta, \ell} e^{-i t {H_\omega^{(\Lambda_L)}} }
\sum_{\alpha \in \Detau\setminus\Dell}
{a_x^\alpha} \psi_\alpha^{(L)}
\Big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&\leq&
\Big\|\sum_{\alpha \in \Detau\setminus\Dell}e^{-i t e_\alpha^{(L)} }
{a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&=&
\sum_{\alpha \in \Detau\setminus\Dell}|a_x^\alpha|^2
\nonumber\\
&=&
\sum_{\alpha \in \Detau\setminus\Dell}
|\psi_\alpha^{(L)}(x)|^2 \; .
\end{eqnarray}
Summing over $x\in\Lambda_L$,
\begin{eqnarray}
\sum_{x\in\Lambda_L} \big\| R_{x, \delta, \ell}
e^{-i t {H_\omega^{(\Lambda_L)}} }
\delta_x\big\|_{\ell^2(\Lambda_L)}^2
&\leq& (1+\varepsilon^{\frac12})\, \big|{\alg}_L(I_\tau)\setminus{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)\big|
\label{DbarDsplitest}\\
&+&\varepsilon (1+ \varepsilon^{-\frac12} )\, |{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)| \; ,
\nonumber
\end{eqnarray}
using the definition of ${\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau)$.
Let $I_\tau^c:={\Bbb R}\setminus I_\tau$. We thus get
\begin{eqnarray}
\frac {|{\mathfrak A}_L\setminus{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau) | } {|{\mathfrak A}_L|}
&=&\frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}
+ \frac {|{\alg}_L(I_\tau)\setminus{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau) | } {|{\mathfrak A}_L|}
\nonumber\\
&\geq& \frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}
\nonumber\\
&+&
\frac{1-\varepsilon^{\frac12}}{|{\mathfrak A}_L|}\sum_{x\in\Lambda_L} \big\| R_{x, \delta, \ell}
\chi_{I_\tau}({H_\omega^{(\Lambda_L)}})
e^{-i t {H_\omega^{(\Lambda_L)}} } \delta_x\big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&-& (1+ \varepsilon^{-\frac12})\, \varepsilon -C\frac{\ell}{L} \;.
\label{fracAcAlowbd}
\end{eqnarray}
Taking expectations and using (~\ref{mainest}),
\begin{eqnarray}
{\Bbb E}\Big[\frac {|{\mathfrak A}_L\setminus{\alg}_{L}^{(\omega)}(\varepsilon,\delta,\ell;I_\tau) | } {|{\mathfrak A}_L|}\Big]
&\geq&1-\varepsilon^{\frac{1}{2}}{\Bbb E}\Big[\frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}\Big]-3\varepsilon^{\frac12}
-C\frac{\ell}{L}\;.
\end{eqnarray}
Since $\frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}\leq1$, this implies the claim.
\hspace*{\fill}\mbox{$\Box$}
Our strategy therefore is to find large values for $\ell$ and $t^*(\delta,\ell)$
such that (~\ref{mainest}) is satisfied.
The following lemma controls the free Schr\"odinger evolution.
\begin{lemma}\label{fundestlemma}
Let for $\lambda$ small and $0<\delta<1$
\begin{eqnarray}
t^*(\delta ,\lambda):=\delta^{\frac45}\ell \;.
\end{eqnarray}
Then, the free evolution satisfies
\begin{eqnarray}\label{fundest0}
&& \frac{1}{|{\mathfrak A}_L|}\sum_{x\in\Lambda_L}
\big\| R_{x,\delta, \ell_{\sigma}(\lambda)}\chi_{I_\tau}({H_\omega^{(\Lambda_L)}})
e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\Lambda_L^2)}^2
\nonumber\\
&&\hspace{3.5cm}\geq 1 - \delta^{\frac{3}{10}} - \frac{|{\alg}_L(I_\tau^c)|}{|{\mathfrak A}_L|}
-C\frac{\ell }{L}\;.
\end{eqnarray}
\end{lemma}
\noindent{\em Proof.}$\;$
We note that
\begin{eqnarray}
&&\sum_{x\in\Lambda_L}
\big\| R_{x,\delta, \ell_{\sigma}(\lambda)}\chi_{I_\tau}({H_\omega^{(\Lambda_L)}})
e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\Lambda_L^2)}^2
\nonumber\\
&&\hspace{2cm}\geq \;(I)-(II)
\end{eqnarray}
where
\begin{eqnarray}
(I)&:=&\sum_{x\in\Lambda_L}\|R_{x,\delta, \ell_{\sigma}(\lambda)}
e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
(II)&:=&\sum_{x\in\Lambda_L} \big\|\chi_{I_\tau^c}({H_\omega^{(\Lambda_L)}})
e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\Lambda_L)}^2 \;.
\end{eqnarray}
This follows from $\chi R^2\chi=\chi^2-\chi\overline{R^2}\chi=
1-\overline{\chi^2}-\chi\overline{R^2}\chi
\geq 1-\overline{R^2}-\overline{\chi^2}=R^2-\overline{\chi^2}$, where
$R\equiv R_{x,\delta, \ell_{\sigma}(\lambda)}$, $\chi\equiv \chi_{I_\tau}({H_\omega^{(\Lambda_L)}})$,
and $\bar{A}:=1-A$ (so that $\overline{\chi^2}=\chi_{I_\tau^c}^2({H_\omega^{(\Lambda_L)}})$).
Replacing $\|\,\cdot\,\|_{\ell^2(\Lambda_L)}$ by $\|\,\cdot\,\|_{\ell^2({\Bbb Z}^2)}$
in $(I)$ costs a boundary term of size $O(\ell L)$ or smaller. Since $|{\mathfrak A}_L|\sim L^2$,
\begin{eqnarray}
&&\frac{1}{|{\mathfrak A}_L|}\sum_{x\in\Lambda_L}\|R_{x,\delta, \ell }
e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2(\Lambda_L)}
\nonumber\\
&=&\frac{1}{|{\mathfrak A}_L|}\sum_{x\in\Lambda_L}\|R_{x,\delta, \ell }
e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2({\Bbb Z}^2)}
+O(\frac{\ell}{L}) \;.
\end{eqnarray}
We then find
\begin{eqnarray}
\|R_{x,\delta, \ell }
e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2({\Bbb Z}^2)}
\geq 1-\delta^{\frac{3}{10}} \;,
\label{free-evol-est-1}
\end{eqnarray}
from a related argument in \cite{ch}, adapted to the present case.
On the other hand,
\begin{eqnarray}
(II)&\leq&\sum_{x\in\Lambda_L}
\big\| \chi_{I_\tau^c}({H_\omega^{(\Lambda_L)}})
e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\Lambda_L)}^2
\nonumber\\
&=&{\rm Tr}\Big[e^{i t^*(\delta, \lambda) \Delta }\chi_{I_\tau^c}({H_\omega^{(\Lambda_L)}})e^{-i t^*(\delta, \lambda) \Delta }\Big]
\nonumber\\
&=&{\rm Tr}\Big[\chi_{I_\tau^c}({H_\omega^{(\Lambda_L)}}) \Big]
\nonumber\\
&=&|{\alg}_L(I_\tau^c)| \;.
\end{eqnarray}
Recalling that $|\Lambda_L|=|{\mathfrak A}_L|$, this completes the proof.
\hspace*{\fill}\mbox{$\Box$}
Our result is implied by the following key lemma. It controls the
interaction of the electron with the impurity potential over
a time $t^*$ comparable to the lower bound on the localization
length $\ell_{\sigma}(\lambda)$.
\begin{lemma}
\label{Lemma-main-0}
Let for $0<\delta<1$
\begin{eqnarray}
t^*_{\delta,{\sigma},\lambda}=\delta^{\frac45}\ell_{{\sigma}}(\lambda) \;.
\label{tstar-def-1}
\end{eqnarray}
Then, for any arbitrary, but fixed $\tau>0$,
\begin{eqnarray}\label{fundest10}
&&\limsup_{L\rightarrow\infty}
{\Bbb E} \Big[\frac{1}{|{\mathfrak A}_L|}\sum_{x\in\Lambda_L}\big\|\chi_{I_\tau}({H_\omega^{(\Lambda_L)}})\big(
e^{-i t^*_{\delta,{\sigma},\lambda} {H_\omega^{(\Lambda_L)}} }\delta_x-
e^{-i t^*_{\delta,{\sigma},\lambda} \Delta } \delta_x
\big) \big\|_{\ell^2(\Lambda_L)}^2\Big]
\nonumber\\
&&\hspace{3cm}\leq C\tau^{\frac12}+\lambda^{\eta}
\, \;.
\end{eqnarray}
The definition of $\ell_{\sigma}(\lambda)$ is given in
Theorem {~\ref{thm-main-1}}.
\end{lemma}
To establish Lemma {~\ref{Lemma-main-0}},
it suffices to prove the following estimate.
\begin{lemma}
\label{Lemma-main-1}
Under the assumptions of Lemma {~\ref{Lemma-main-0}},
\begin{eqnarray}
&&\sup_{\phi\in\ell^2({\Bbb Z}^2)\atop\|\phi\|_{\ell^2(\Lambda_L)}=1}
{\Bbb E}\big[\|\chi_{I_\tau}(H_\omega)\big(
e^{-i t^*_{\delta,{\sigma},\lambda} H_\omega }-
e^{-i t^*_{\delta,{\sigma},\lambda} \Delta }
\big)\phi\|_{\ell^2({\Bbb Z}^3)}^2\big]<C\tau^{\frac12}+\lambda^{\eta}\;.
\label{Lemma-main-est-1}
\end{eqnarray}
\end{lemma}
The rest of this paper is devoted to the proof of Lemma {~\ref{Lemma-main-1}}.
\section{Resolvent expansion }
Let henceforth $t\equiv t^*_{\delta,{\sigma},\lambda}$.
We write
\begin{eqnarray}
\phi_t=\chi_{I_\tau}(H_\omega)e^{-itH_\omega}\phi_0
\end{eqnarray}
with $\phi_0\in\ell^2({\Bbb Z}^2)$
in resolvent representation
\begin{eqnarray}
\phi_t&=&\frac{1}{2\pi i}e^{\varepsilon t}\int_{{\Bbb R}} d\alpha e^{-it \alpha}
\frac{\chi_{I_\tau}(H_\omega)}{H_\omega-\alpha-i\varepsilon}\phi_0
\end{eqnarray}
where we will use the choice
\begin{eqnarray}
\varepsilon=\frac1t
\end{eqnarray}
in all that follows.
Due to the spectral restriction of $H_\omega$ to the disjoint union of
intervals $I_\tau$, the $\alpha$-integration
contour can be deformed into
\begin{eqnarray}
\phi_t&=&\frac{1}{2\pi i}e^{\varepsilon t}\int_{C_-\cup C_+}
d\alpha e^{-it \alpha}
\frac{\chi_{I_\tau}(H_\omega)}{H_\omega-\alpha-i\varepsilon}\phi_0\;,
\end{eqnarray}
where the loops
\begin{eqnarray}
C_-&:=&[-4+\tau/2,-\tau/2]\cup(-4+\tau/2-2i\varepsilon[0,1])\cup
\nonumber\\
&&([-4+\tau/2 ,-\tau/2 ]-2i\varepsilon)\cup(-\tau/2-2i\varepsilon[0,1])
\nonumber\\
C_+&:=&[\tau/2,4-\tau/2]\cup( 4-\tau/2-2i\varepsilon[0,1])\cup
\nonumber\\
&&([\tau/2 ,4-\tau/2 ]-2i\varepsilon)\cup(\tau/2-2i\varepsilon[0,1])
\end{eqnarray}
are taken in the clockwise direction.
$C_-$ and $C_+$ each enclose one of the components of $I_\tau-i\varepsilon$.
Let $C^{(v)}:=\{C^{(v)}_j\}_{j=1}^4$ denote the four vertical,
and $C^{(h)}:=\{C^{(h)}_j\}_{j=1}^4$ the four horizontal segments in $C_-$ and $C_+$.
Each segment carries an orientation accounting for the direction in which
the contour integration is taken.
Then,
\begin{eqnarray}
|\frac{1}{2\pi i}e^{\varepsilon t}\int_{C^{(v)}_j} d\alpha e^{-it \alpha}
\frac{\chi_{I_\tau}(H_\omega)}{H_\omega-\alpha-i\varepsilon}\phi_0|
&<&\frac14 |C^{(v)}_j|\sup_{z\in S_j\atop z'\in I_\tau-\varepsilon}|z-z'|
\nonumber\\
&=&\varepsilon\tau^{-1}\;,
\end{eqnarray}
as ${\rm dist}(C^{(v)}_j,I_\tau-i\varepsilon)=\tau/2$,
and $|C^{(v)}_j|=2\varepsilon$.
Henceforth, we shall omit the subscript "$\omega$" in the
random potential $V_\omega\equiv V$.
Defining
\begin{eqnarray}
\phi^{(h)}_t&:=&\frac{1}{2\pi i}e^{\varepsilon t}\int_{C^{(h)}}
d\alpha e^{-it \alpha}
\frac{1}{H_\omega-\alpha-i\varepsilon}\phi_0\;,
\end{eqnarray}
we have
\begin{eqnarray}
\|\phi_t\|_{\ell^2({\Bbb Z}^2)}^2&\leq&2\Big(\frac\varepsilon\tau\Big)^2+
2\|\chi_{I_\tau}(H_\omega)\phi^{(h)}_t\|_{\ell^2({\Bbb Z}^2)}^2 \;.
\label{phi-ell2-ircut-bound-1}
\end{eqnarray}
Next, we expand $\phi_t^{(h)}$ into
\begin{eqnarray}
\phi_t^{(h)}=\sum_{n=0}^N \phi_{n,t}+R_{N,t} \;,
\end{eqnarray}
where the $n$-th term is given by
\begin{eqnarray}
\phi_{n,t}&:=&\frac{e^{\varepsilon t}}{2\pi i}\int_{C^{(h)}} d\alpha e^{-it\alpha}
\tilde\phi_{n,\varepsilon}(\alpha)\;,
\end{eqnarray}
with
\begin{eqnarray}
\tilde\phi_{n,\varepsilon}(\alpha)&:=&(-\lambda)^n
\frac{1}{\Delta-\alpha-i\varepsilon}
\Big(V\frac{1}{\Delta-\alpha-i\varepsilon}\Big)^n
\phi_0 \;.
\label{phiNt-def-1}
\end{eqnarray}
In frequency space,
\begin{eqnarray}
{\mathcal F}( \phi_{n,t} )(k_0)&=&\frac{1}{2\pi i} e^{\varepsilon t}\int_{C^{(h)}} d\alpha
e^{-it\alpha}{\mathcal F}(\tilde\phi_{N,\varepsilon}(\alpha))(k_0)
\end{eqnarray}
where
\begin{eqnarray}
{\mathcal F}(\tilde\phi_{N,\varepsilon}(\alpha))(k_0)&=&(-\lambda)^n\int_{(\Bbb T^3)^n}dk_1\cdots dk_n
\frac{1}{{e_\Delta}(k_0)-\alpha-i\varepsilon}
\nonumber\\
&&\times\,
\Big[\prod_{j=1}^n\frac{1}{{e_\Delta}(k_j)-\alpha-i\varepsilon} \hat V(k_{j}-k_{j-1})\Big]
\hat \phi_0(k_n) \;,
\label{hatphint-expans}
\end{eqnarray}
and $\Bbb T=[-\frac12,\frac12]$.
We will refer to the Fourier multiplier $\frac{1}{{e_\Delta}(k)-\alpha-i\varepsilon}$
as a {\em particle propagator}.
The remainder term is given by
\begin{eqnarray}
R_{N,t}=- \lambda e^{\varepsilon t}\frac{1}{2\pi i} \int_{C^{(h)}}
d\alpha e^{-it\alpha} \frac{1}{H_\omega-\alpha-i\varepsilon} V\tilde\phi_{N,\varepsilon}(\alpha) \;.
\label{RNt-def-1}
\end{eqnarray}
The depth of the expansion $N$ remains to be optimized.
We remark that due to the truncation of the integration contour,
$\phi_{n,t}$ and $R_{N,t}$ cannot be written as time integrals
of the form
\begin{eqnarray}
\phi_{n,t}&\leftrightarrow& (-i\lambda)^n\int_{{\Bbb R}_+^{n+1}}\delta(t-\sum_{j=0}^n s_j)
e^{-s_0\Delta}V e^{-s_1\Delta}\cdots \cdots V e^{-is_n \Delta}\phi_0
\nonumber\\
R_{N,t}&\leftrightarrow&-i\lambda\int_0^t ds e^{-i(t-s)H_\omega}V\phi_{N,s}
\end{eqnarray}
as in the Duhamel expansions used in \cite{ch,erd,erdyau,erdsalmyau}.
While for $\phi_{n,t}$, this is not essential in the present work
(because we admit a polynomial error $O(\lambda^\eta)$, $\eta>0$, in
our bounds), our methods require an expression of the above form for $R_{N,t}$
(because we will apply the time partitioning
trick used in \cite{erdyau} and \cite{ch}).
To this end, we claim that
\begin{eqnarray}
R_{N,t}&=&R_{N,t}^{(0)}+R_{N,t}^{(1)}
\label{RNt-def-2}
\end{eqnarray}
with
\begin{eqnarray}
R_{N,t}^{(0)}&:=&e^{-itH_\omega}\frac{-\lambda}{2\pi i}\int_{C^{(h)}}
d\alpha \frac{1}{H_\omega-\alpha-i\varepsilon}V\tilde\phi_{N,\varepsilon}(\alpha)
\\
R_{N,t}^{(1)}&:=&-i\lambda\int_0^tds e^{-i(t-s)H_\omega}V \phi_{N,s}\;.
\end{eqnarray}
To see this, we note that (~\ref{RNt-def-1}) implies
\begin{eqnarray}
\partial_t R_{N,t} = -iH_\omega R_{N,t} -i\lambda V \phi_{N,t} \;,
\end{eqnarray}
which is solved by the variation of constants formula (~\ref{RNt-def-2}).
We note that $\chi_{I_\tau}(H_\omega)R_{N,t}^{(0)}$ would vanish if $C^{(h)}$ were
replaced by a connected $\alpha$-integration contour $C_{conn}$ that encloses
$I_\tau-i\varepsilon$. This is because
$C_{conn}$ can be deformed into a contour arbitrarily far away from the spectrum of
$\chi_{I_\tau}(H_\omega)H_\omega-i\varepsilon$,
as there is no obstructing phase factor $e^{-it\alpha}$.
Furthermore, due to the truncation of the integration contour to $C^{(h)}$, it is
also necessary to control
\begin{eqnarray}
&&\|\chi_{I_\tau}(H_\omega)\big(\phi_{0,t}-
e^{-it\Delta}\phi_0\big)\|_{\ell^2({\Bbb Z}^2)}^2
\nonumber\\
&&\hspace{2cm}\leq \int_{\Bbb T^2}dp
\Big|\int_{C\setminus C^{(h)}}d\alpha e^{-it\alpha}
\frac{1}{{e_\Delta}(p)-\alpha-i\varepsilon}\Big|^2 \;,
\label{free-evol-error-1}
\end{eqnarray}
where
\begin{eqnarray}
\tilde C&:=&[-4-\varepsilon,4+\varepsilon]\cup(4+\varepsilon-2i\varepsilon[0,1])\cup
\nonumber\\
&&\hspace{2cm}([-4-\varepsilon,4+\varepsilon]-2i\varepsilon)\cup(-4-\varepsilon-2i\varepsilon[0,1])\;.
\end{eqnarray}
We write $\tilde C\setminus C^{(h)} =\tilde C_-\cup \tilde C_0\cup \tilde C_+$, where
$\tilde C_{\pm}:=\{z\in \tilde C\setminusC^{(h)}\big|
\pm\Re(z)>2\}$. $\tilde C_-$ and $\tilde C_+$ are connected arcs,
while $\tilde C_0$ consists of two disjoint,
parallel lines, all of length $O(\tau)$. We claim that
\begin{eqnarray}
&&\Big|\int_{\tilde C_-\cup \tilde C_0\cup \tilde C_+}d
\alpha e^{-it\alpha}
\frac{1}{{e_\Delta}(p)-\alpha-i\varepsilon}\Big|
\nonumber\\
&&\hspace{2cm}<
C\Big[\chi(|{e_\Delta}(p)+4|<2\tau)+\chi(|{e_\Delta}(p)+4|<2\tau)
\nonumber\\
&&\hspace{3cm} +\chi(|{e_\Delta}(p)|<4\tau) + \frac{\varepsilon}{\tau}\Big]\;.
\end{eqnarray}
For fixed $p$, the size of
\begin{eqnarray}
\int_{\tilde C_- }d\alpha e^{-it\alpha}
\frac{1}{{e_\Delta}(p)-\alpha-i\varepsilon}
\end{eqnarray}
can be estimated as follows.
If $|{e_\Delta}(p)-4|<2\tau$, we deform
$\tilde C_-$ into a loop that encloses ${e_\Delta}(p)-i\varepsilon$, and a disjoint arc
of length $O(\varepsilon)$ connecting the endpoints of $\tilde C_-$.
The resolvent at ${e_\Delta}(p)-i\varepsilon$, due to the loop, yields a factor $e^{-it({e_\Delta}(p)-i\varepsilon)}$.
The integral over the arc is bounded by its length $O(\varepsilon)$, multiplied with the bound
$\frac1\varepsilon$ on the resolvent. Both contributions are $O(1)$.
If $|{e_\Delta}(p)+4|>2\tau$, we deform $\tilde C_-$
into a line of length $2\varepsilon$ connecting its endpoints, which has a distance
$\geq\tau$ from ${e_\Delta}(p)$. The modulus of the resolvent is therefore
$\leq O(\frac1\tau)$, and integrating, we get an error bound of order $O(\frac\varepsilon\tau)$.
The cases $\tilde C_0$ and $\tilde C_+$ are similar.
Thus,
\begin{eqnarray}
(~\ref{free-evol-error-1})&<&C\Big[{\rm mes}\{|{e_\Delta}(p)+4|<2\tau\}+
{\rm mes}\{|{e_\Delta}(p)|<4\tau\}
\nonumber\\
&&\hspace{2cm}+{\rm mes}\{|{e_\Delta}(p)-4|<2\tau\}+ \frac\varepsilon\tau\Big]
\nonumber\\
&<& C\tau^{\frac12}\;,
\end{eqnarray}
as $\varepsilon$ will be chosen $\ll\tau$ in the end.
The Schwarz inequality thus yields
\begin{eqnarray}
&&{\Bbb E}\Big[\|\chi_{I_\tau}(H_\omega)\big(\phi_t^{(h)}-
e^{-it\Delta}\phi_0\big)\|_{\ell^2({\Bbb Z}^2)}^2\Big]
\nonumber\\
&&\hspace{3cm}\leq\;C\tau^{\frac12} +
2\, {\Bbb E}\Big[ \big\| \sum_{n=1}^N \phi_{n,t} \big\|_2^2 \Big]
+2 \,{\Bbb E}\Big[ \big\| \chi_{I_\tau}(H_\omega) R_{N,t} \big\|_2^2 \Big]
\nonumber\\
&&\hspace{3cm}=\;C\tau^{\frac12} +
2\sum_{n,n'=1}^N {\Bbb E}\Big[ \langle\phi_{n',t},\phi_{n,t}\rangle \Big]
+2\, {\Bbb E}\Big[ \big\| \chi_{I_\tau}(H_\omega) R_{N,t} \big\|_2^2 \Big] \; .
\label{exp-phi-ell2-Schwarz-1}
\end{eqnarray}
Clearly, if $n+n'\not\in2{\Bbb N}$, ${\Bbb E}[\langle\phi_{n',t},\phi_{n,t}\rangle]=0$.
We partition $V$ into dyadic shells,
\begin{eqnarray}
V=\sum_{j=0}^{J+1} V_j \;,
\end{eqnarray}
where
\begin{eqnarray}
V_j(x)&=&P_j(x) v_{\sigma}(x)\omega_x
\end{eqnarray}
for $0\leq j\leq J$. The cutoff functions $P_j$ are defined at the beginning of section
{~\ref{intro-sect-1}}. For $j>J$, we rename $P_j\rightarrow \tilde P_j$, and define
\begin{eqnarray}
P_{J+1}&:=&\sum_{j=J+1}^\infty \tilde P_j
\label{tildPi-def-1}
\end{eqnarray}
Hence, the functions $V_j$ are supported on dyadic annuli of
radii and thicknesses $\sim 2^j$ centered at the origin, $j=1,\dots,J$, while $V_{J+1}$
is the part of $V$ supported in regions with a distance larger than $2^{J+1}$
from the origin.
Let
\begin{eqnarray}
R_z:=\frac{1}{\Delta-z}\;.
\end{eqnarray}
Then, we have
\begin{eqnarray}
{\Bbb E}\left[ \langle\phi_{n',t},\phi_{n,t}\rangle \right] &=&
\sum_{j_1,\dots,j_{2\bar n}=1}^{J+1} \frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\int_{C^{(h)}\times \overline C^{(h)}} d\alpha d\beta e^{-it(\alpha-\beta)}
\nonumber\\
&&\hspace{0.5cm}{\Bbb E}\Big[\langle\phi_0\,,\, R_{\alpha+i\varepsilon} V_{j_1}
R_{\beta-i\varepsilon}V_{j_2}
R_{\beta-i\varepsilon}\cdots\cdots
\nonumber\\
&&\hspace{1.5cm}\cdots\cdots V_{j_n} R_{\beta-i\varepsilon}
R_{\alpha+i\varepsilon} V_{j_{n+2}} \cdots
\cdots V_{j_{2\bar n}}
R_{\alpha+i\varepsilon} \phi_0\rangle\Big]
\label{exp-phinn-res-1}
\end{eqnarray}
for $1\leq n,n' \leq N$, and $\bar n:=\frac{n+n'}{2}\in{\Bbb N}$.
$\overline C^{(h)}$ denotes the complex conjugate of $C^{(h)}$, and is taken in the
counterclockwise direction by the variable $\beta$.
For $1\leq n,n' \leq N$, and $\bar n:=\frac{n+n'}{2}\in{\Bbb N}$, let
\begin{eqnarray}
\underline{{ p}}&=&(p_0,\dots,p_n,p_{n+1},\dots,p_{2\bar n+1})
\end{eqnarray}
and
\begin{eqnarray}
(\alpha_j,\sigma_j) &=& \left\{\begin{array}{ll}(\alpha,1)&
0\leq j\leq n\\
(\beta,-1)&n<j\leq 2n+1 \;. \end{array}\right.
\end{eqnarray}
Then, in frequency space representation,
\begin{eqnarray}
(~\ref{exp-phinn-res-1})
&=& \sum_{j_1,\dots,j_{2\bar n}=1}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\int_{C^{(h)}\times \overline C^{(h)}} d\alpha d\beta e^{-it(\alpha-\beta)}
\nonumber\\
&&\hspace{0.5cm}
\int_{(\Bbb T^3)^{2\bar n+2}} d\underline{{ p}} \,\delta (p_n-p_{n+1})
\overline{{\mathcal F}(\phi_0)(p_0)}{\mathcal F}(\phi_0)(p_{2\bar n+1})
\nonumber\\
&&\hspace{2cm}
\prod_{l=0}^{2\bar n+1}\frac{1}{{e_\Delta}(p_l)-\alpha_l-\sigma_l\varepsilon}
\nonumber\\
&&\hspace{3cm}
{\Bbb E}\Big[\prod_{\stackrel{i=1}{i\neq n+1}}^{2\bar n+1}
{\mathcal F}( V_{j_i})(p_i-p_{i-1})\Big]
\label{exppotgenexpr}
\end{eqnarray}
(noting that $\overline{{\mathcal F}( V)(k)}={\mathcal F}( V)(-k)$).
\section{Graph expansion}
We systematize the evaluation of the expectation value of products of
random potentials by use of {\em (Feynman) graphs}, which we represent as follows.
We consider two parallel, horizontal solid lines, which we refer to as {\em particle lines},
joined at a distinguished vertex which accounts for the $L^2$-inner product
(henceforth referred to as the "$L^2$-vertex").
The particle line to the left of the $L^2$ vertex shall contain $n$, and the one
its right shall contain $n'$ vertices,
accounting for copies of the random potential $\hat V$ (henceforth referred to
as "$V$-vertices").
The $n+1$ edges on
the left of the $L^2$-vertex
correspond to the propagators in $\hat\psi_{n,t}$, while the
$n'+1$ edges on the right correspond to those in
$\overline{\hat\psi_{n',t}}$. We shall refer to those edges
as {\em propagator lines}.
The expectation produces a sum over the
products of $\bar n=\frac{n+n'}{2}\in{\Bbb N}$ contractions between
all possible pairs of random potentials.
We insert an edge referred to as a {\em contraction line} between every pair of
mutually contracted random potentials. We then identify the contraction type with the
corresponding graph.
We let $\Pi_{n,n'}$ denote the set of all graphs comprising $n+n'$
$V$-vertices, one $L^2$-vertex, two particle lines, $\bar n$ contraction
lines, and $2\bar n+2$ propagator lines as defined above.
An example is given in Figure 1.
\subsection{Dyadic Wick expansion}
We shall next discuss the expectation of products of dyadically resolved
random potentials in detail.
It is evident that
\begin{eqnarray}
{\Bbb E}[V_j(x) V_{j'}(x')]&=&\delta_{|j-j'|\leq1}P_{j}(x)P_{j'}(x)
v_{\sigma}^2(x)\delta_{x,x'}
\nonumber\\
&\leq&C 2^{-2{\sigma} j}\delta_{x,x'}\;,
\end{eqnarray}
and
\begin{eqnarray}
{\Bbb E}[V_{J+1}(x) V_{J+1}(x')]&\leq&C 2^{-2{\sigma} J} \delta_{x,x'}\;.
\end{eqnarray}
The expectation of products $\prod_i\omega_{x_i}$ satisfies Wick's
theorem, and the same is true for the expectation of
products $\prod_i V_{j_i}(x_i)$. This can be formulated as follows.
There are $\bar n$ pairing contraction lines joining pairs of
$\hat V_\omega$-vertices in $\pi$.
We enumerate the contraction lines in an arbitrary, but fixed order
by $\{1,\dots,\bar n\}$.
We write $i\sim_{m} i'$ to express that
the $i$-th and the $i'$-th $V$-vertex are connected by the $m$-th
contraction line.
Given
\begin{eqnarray}
\underline{j}&:=&(j_1,\dots,j_{2\bar n})
\nonumber\\
\underline{x}&:=&(x_0,\dots,x_{2\bar n+1})\;,
\end{eqnarray}
let
\begin{eqnarray}
\delta_\pi(\underline{j},\underline{x}) :=\prod_{m=1}^{\bar n}
\Big[\delta_{|j_{i}-j_{i'}|\leq1}\delta_{x_i,x_{i'}}\Big]\Big|_{i\sim_m i'}\;.
\label{deltapi-x-def-1}
\end{eqnarray}
Then, in position space,
\begin{eqnarray}
{\Bbb E}\Big[\prod_{i=1}^{2\bar n}V_{j_i}(x_i)\Big]=
\sum_{\pi\in\Pi_{n,n'}}\delta_\pi(\underline{j},\underline{x})
\prod_{i=1}^{2\bar n}v_{\sigma}(x_i)\;.
\end{eqnarray}
On the other hand, we arrive at the frequency space picture as follows.
Let
\begin{eqnarray}
\underline{{ p}}&:=&(p_0,\dots,p_n,p_{n+1},\dots,p_{2\bar n+1}) \;.
\end{eqnarray}
If $i\sim_m i'$, contraction of
${\mathcal F}(P_{j_i} V)(p_{i+1}-p_{i})$ with
${\mathcal F}(P_{J_{i'}} V)(p_{i'+1}-p_{i'})$ yields
\begin{eqnarray}
&&{\Bbb E}\Big[ {\mathcal F}(P_{j_i} V)(p_{i+1}-p_{i})
{\mathcal F}(P_{j_{i'}} V)(p_{ i'+1}-p_{i'})\Big]
\nonumber\\
&&\hspace{1.5cm}
=\delta_{|j_{i}-j_{i'}|\leq1}
{\mathcal F}(P_{j_i}P_{j_{i'}} v_{\sigma}^2)\delta(p_{i+1}-p_{i}+p_{i'+1}-p_{i'})\;.
\end{eqnarray}
We define
\begin{eqnarray}
&&\delta_\pi(\underline{j},\underline{{ p}};v_{\sigma}):=
\nonumber\\
&&\hspace{1cm}\prod_{m=1}^{\bar n}
\Big[\delta_{|j_{i}-j_{i'}|\leq1}{\mathcal F}(P_{j_i} P_{j_{i'}} v_{\sigma}^2)
\delta(p_{i+1}-p_{i}+p_{i'+1}-p_{i'})\Big]
\Big|_{i\sim_m i'}\;.
\label{deltapi-def-1}
\end{eqnarray}
Then,
\begin{eqnarray}
{\Bbb E}\Big[\prod_{i=1\atop i\neq n+1}^{2\bar n+1}{\mathcal F}(V_{j_i})(p_i-p_{i-1})\Big]=
\sum_{\pi\in\Pi_{n,n'}}\delta_\pi(\underline{j},\underline{{ p}};v_{\sigma}) \;.
\label{Exp-prod-V-1}
\end{eqnarray}
We emphasize that the products (~\ref{deltapi-x-def-1}) and (~\ref{deltapi-def-1}) vanish unless the
scales of the contracted dyadic potentials pairwise coincide (up to overlap errors).
That is, $|j_i-j_{i'}|\leq1$ (where $|j_i-j_{i'}|=1$ accounts for overlap errors)
for every pair $i\sim_m i'$.
Expanding the expectation of the product of random potentials,
\begin{eqnarray}
{\Bbb E}[\langle \phi_{n',t},\phi_{n,t} \rangle ]
&=&\sum_{\pi\in\Pi_{n,n'}}{\rm Amp}(\pi)
\end{eqnarray}
where
\begin{eqnarray}
{\rm Amp}(\pi)&=&\sum_{j_1,\dots,j_{2\bar n}=1}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\int_{C^{(h)}\times \overline C^{(h)}} d\alpha d\beta e^{-it(\alpha-\beta)}
\nonumber\\
&&\hspace{0.5cm}
\int_{(\Bbb T^3)^{2\bar n+2}} d\underline{{ p}} \,\delta (p_n-p_{n+1})
\delta_\pi(\underline{j};\underline{{ p}};v_{\sigma})
\nonumber\\
&&\hspace{3.5cm}
\overline{{\mathcal F}(\phi_0)(p_0)}{\mathcal F}(\phi_0)(p_{2\bar n+1})
\nonumber\\
&&\hspace{2cm}
\prod_{l=0}^{2\bar n+1} \frac{1}{{e_\Delta}(p_l)-\alpha_l-\sigma_l\varepsilon}
\; . \label{exp-sum-graphs-1}
\end{eqnarray}
Here, $\delta(p_n-p_{n+1})$ corresponds to the $L^2$-vertex.
\section{Bounds on pairing graphs}
We shall use an analogy of the frequency space $L^1-L^\infty$ estimates on the
resolvents adapted to a spanning tree of $\pi$ from \cite{erdyau,ch}.
\begin{lemma}
\label{R-Lnorm-bounds-lemma-1}
Assume that $\alpha\in C^{(h)}$. Then, for the assumptions (~\ref{Fouv-dyad-est-1}) on $P_j$,
\begin{eqnarray}
&&\Big\| \,\Big|\frac{1}{{e_\Delta}-\alpha-i\varepsilon}\Big|*
|{\mathcal F}( P_j P_{j'} v_{\sigma}^2 )|\,\Big\|_{L^\infty(\Bbb T^2)}
\nonumber\\
&&\hspace{3cm}\leq
\left\{\begin{array}{ll}C_\tau 2^{j(1-2{\sigma})} &{\rm if}\; j \leq J\\
C\sigma^{-1}2^{-2{\sigma} J} \varepsilon^{-1}&{\rm if}\;j,j'=J+1\;,
\end{array}
\right.
\label{res-Linfty-bound-1}
\end{eqnarray}
where the constant $C_\tau$ only depends on $\tau$. Furthermore,
\begin{eqnarray}
\Big\| \,\Big|\frac{1}{{e_\Delta}-\alpha-i\varepsilon}\Big|*|{\mathcal F}( P_j P_{j'} v_{\sigma}^2)| \, \Big\|_{L^1(\Bbb T^2)}
\leq C\log\frac1\varepsilon\;.
\label{res-L1-bound-1}
\end{eqnarray}
for $0\leq j,j'\leq J+1$.
\end{lemma}
\noindent{\em Proof.}$\;$
We recall that by (~\ref{Fouv-dyad-est-1}),
\begin{eqnarray}
|{\mathcal F}(P_j P_{j'} v_{\sigma}^2)(p)|&\leq&
C 2^{-2 {\sigma} j}|{\mathcal F}(P_j P_{j'} )(p)|
\sim C 2^{-2{\sigma} j}
|{\mathcal F}(P_j^2)(p)|
\label{Pj-ass-Rnorm-1}
\end{eqnarray}
for $|j-j'|\leq1$, and any $j$. It thus suffices to discuss the diagonal term $j=j'$.
For $\alpha\inC^{(h)}$, it is shown in \cite{bo1} that given our assumptions on $P_j$,
convolution with $|{\mathcal F} (P_j^2)|$ acts like a smoothing operator on $\frac{1}{{e_\Delta}-\alpha-i\varepsilon}$,
on the scale dual to $2^j$, to the effect that
\begin{eqnarray}
\Big|\frac{1}{{e_\Delta}-\alpha-i\varepsilon}\Big|*|{\mathcal F}( P_j^2)|
\leq \frac{C_\tau}{|{e_\Delta}-\alpha|+\varepsilon+2^{-j} }\;.
\end{eqnarray}
The $L^\infty$-bounds (~\ref{res-Linfty-bound-1}) for $0\leq j\leq J$ then follow immediately.
For $j=J+1$,
\begin{eqnarray}
\Big|\frac{1}{{e_\Delta}-\alpha-i\varepsilon}\Big|*|{\mathcal F}( P_{J+1}^2)|
&\leq&\Big\|\frac{1}{{e_\Delta}-\alpha-i\varepsilon}\Big\|_{L^\infty(\Bbb T^2)}
\sum_{i=J+1}^\infty\|{\mathcal F}(\tilde P_i^2 v_{\sigma}^2)\|_{L^1(\Bbb T^2)}
\nonumber\\
&\leq&C\varepsilon^{-1}\sum_{i=J+1}^\infty 2^{-2{\sigma} i}\|{\mathcal F}(\tilde P_i^2)\|_{L^1(\Bbb T^2)}
\nonumber\\
&\leq&C\varepsilon^{-1}{\sigma}^{-1}2^{-2{\sigma} J} \;,
\end{eqnarray}
as $\|{\mathcal F}(P_i^2)\|_{L^1(\Bbb T^2)}\sim1$ ($\tilde P_i$ is defined in (~\ref{tildPi-def-1})).
The $L^1$-bound (~\ref{res-L1-bound-1}) has been proven in \cite{ch}.
\hspace*{\fill}\mbox{$\Box$}
\begin{lemma}
\label{amppi-nn-bound-lemma-1}
For $1\leq n,n'\leq N$, $\tau>0$ and $\pi\in\Pi_{n,n'}$, there exists a finite constant
$C_\tau$ depending only on $\tau$ such that defining
\begin{eqnarray}
{A_{{\sigma},\tau,J,\lambda,\varepsilon}}:=C_\tau(K_{\sigma}(J) \lambda^2\log\frac1\varepsilon
+\varepsilon^{-1}{\sigma}^{-1} 2^{-2{\sigma} J}\lambda^2\log\frac1\varepsilon)
\label{ampi-def-1}
\end{eqnarray}
and
\begin{eqnarray}
K_{\sigma}(J):=\left\{
\begin{array}{cl}
J+1&{\rm if}\;{\sigma}=\frac12\\
\frac{ 2^{(1-2{\sigma}){J+1}} -1 }{ 2^{(1-2{\sigma})}-1 }&{\rm if}\;0<{\sigma}<\frac12\;,
\end{array}\right.
\end{eqnarray}
one gets
\begin{eqnarray}
|{\rm Amp}(\pi)|<(\log\frac1\varepsilon)^2({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{\bar n} \;.
\end{eqnarray}
\end{lemma}
\noindent{\em Proof.}$\;$
We choose a spanning tree
$T$ on $\pi$ that contains all contraction lines between the pairs of random potentials,
and $\bar n$ out of all particle lines. In addition, $T$ shall include those particle
lines labeled by the momenta $p_n,p_{2\bar n+1}$,
but not those labeled by $p_0,p_{n+1}$. We then call $T$ {\em admissible}.
Momenta (resolvents) supported on $T$ are referred to as tree momenta (resolvents),
and momenta (resolvents) supported on its complement $T^c$ are called loop momenta (resolvents).
We shall then group together every tree resolvent with one adjacent
contraction line carrying a factor ${\mathcal F}(P_{j_i}P_{j_{i'}}v_{\sigma}^2)$, $|j_i-j_{i'}|\leq1$, and estimate
the corresponding convolution integral of the form (~\ref{convol-est-1}) below.
All loop resolvents supported on $T^c$ are estimated in $L^1(\Bbb T^2)$.
We recall that
\begin{eqnarray}
{\rm Amp}(\pi)&=&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\int_{C^{(h)}\times \overline C^{(h)}} d\alpha d\beta e^{-it(\alpha-\beta)}
\nonumber\\
&&\hspace{0.5cm}
\int_{(\Bbb T^3)^{2\bar n+2}} d\underline{{ p}} \,\delta (p_n-p_{n+1})
\delta_{\pi}(\underline{j};\underline{{ p}};v_{\sigma})
\nonumber\\
&&\hspace{3.5cm}
\overline{{\mathcal F}(\phi_0)(p_0)}{\mathcal F}(\phi_0)(p_{2\bar n+1})
\nonumber\\
&&\hspace{2cm}
\prod_{l=0}^{2\bar n+1} \frac{1}{{e_\Delta}(p_l)-\alpha_l-i\sigma_l\varepsilon}
\; . \label{exp-sum-graphs-2}
\end{eqnarray}
for $\underline{j}=(j_1,\dots,j_{2\bar n})$.
We integrate out the variable $p_{n+1}$, and apply the coordinate
transformation $p_j\mapsto p_j+p_n$, for all $j=0,\dots,n-1,n+2,\dots,2\bar n+1$.
It is easy to see that thereby, $\delta_{\pi}(\underline{j};\underline{{ p}};v_{\sigma})$ becomes
independent of $p_n$ and $p_{n+1}$.
We obtain
\begin{eqnarray}
{\rm Amp}(\pi)&=&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\int_{C^{(h)}\times \overline C^{(h)}} d\alpha d\beta e^{-it(\alpha-\beta)}
\nonumber\\
&&\hspace{0.5cm}
\int_{(\Bbb T^3)^{2\bar n}} d\underline{{ p}}' \,
\delta_{\pi}'(\underline{j};\underline{{ p}}';v_{\sigma})
\nonumber\\
&&\hspace{0.5cm}
\int_{\Bbb T^2}dp_n\frac{1}{{e_\Delta}(p_n)-\alpha-i\varepsilon}\frac{1}{{e_\Delta}(p_n)-\beta+i\varepsilon}
\nonumber\\
&&\hspace{3.5cm}
\overline{{\mathcal F}(\phi_0)(p_0+p_n)}{\mathcal F}(\phi_0)(p_{2\bar n+1}+p_n)
\nonumber\\
&&\hspace{2cm}
\prod_{l=0\atop l\neq n,n+1}^{2\bar n+1} \frac{1}{{e_\Delta}(p_l+p_n)-\alpha_l-i\sigma_l\varepsilon}
\; , \label{exp-sum-graphs-3}
\end{eqnarray}
where
\begin{eqnarray}
\underline{{ p}}':=(p_0,\dots,p_{n-1},p_{n+2},\dots,p_{2\bar n+1})
\end{eqnarray}
and
\begin{eqnarray}
\delta_{\pi}'(\underline{j};\underline{{ p}}';v_{\sigma}):=
\delta_{\pi}(\underline{j};\underline{{ p}};v_{\sigma})\Big|_{p_{n+1}, p_n\rightarrow0 }\;.
\end{eqnarray}
Clearly,
\begin{eqnarray}
|{\rm Amp}(\pi)|&\leq&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\Big[\sup_{q,q'\in\Bbb T^2}\int_{C^{(h)}\times \overline C^{(h)}} |d\alpha|\,|d\beta|
\nonumber\\
&&\hspace{0.5cm}
\int_{\Bbb T^2}dp_n\frac{1}{|{e_\Delta}(p_n)-\alpha-i\varepsilon|}\frac{1}{|{e_\Delta}(p_n)-\beta+i\varepsilon|}
\nonumber\\
&&\hspace{3.5cm}
\Big|\overline{{\mathcal F}(\phi_0)(p_0+q)}{\mathcal F}(\phi_0)(p_{2\bar n+1}+q')\Big|\Big]
\nonumber\\
&&\hspace{0.5cm}
\sup_{\alpha\inC^{(h)}}\sup_{\beta\in\overline{C^{(h)}}}\sup_{p_n\in\Bbb T^2}
\Big[\int_{(\Bbb T^3)^{2\bar n}} d\underline{{ p}}' \,
\delta_{\pi}'(\underline{j};\underline{{ p}}';v_{\sigma})
\nonumber\\
&&\hspace{3.5cm}
\prod_{l=0\atop l\neq n,n+1}^{2\bar n+1}
\frac{1}{|{e_\Delta}(p_l+p_n)-\alpha_l-\sigma_l\varepsilon|} \Big]
\; . \label{exp-sum-graphs-4}
\end{eqnarray}
Thus, dividing the resolvents into tree and loop terms and defining
\begin{eqnarray}
\delta_\pi(\underline{j}):=
\prod_{m=1}^{\bar n} \delta_{|j_{i}-j_{i'}|\leq1}
\Big|_{i\sim_m i'}\;,
\label{deltapi-def-2}
\end{eqnarray}
(see also (~\ref{deltapi-def-1})), one gets
\begin{eqnarray}
|{\rm Amp}(\pi)|&\leq&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1}
\frac{e^{2\varepsilon t} \lambda^{2\bar n}}{(2\pi)^2}
\delta_{\pi}(\underline{j})
\nonumber\\
&&\hspace{0.5cm}
\Big[\sup_{q,q'\in\Bbb T^2}\int_{\Bbb T^2}dp_{n}|\phi_0(p_n+q)|\,|\phi_0(p_n+q')|\Big]
\nonumber\\
&&\hspace{1cm}\Big[\sup_{p_n\in\Bbb T^2}\int_{C^{(h)}}|d\alpha|\,\frac{1}{|{e_\Delta}(p_n)-\alpha-i\varepsilon|}
\nonumber\\
&&\hspace{2.5cm}\int_{\overline{C^{(h)}}}|d\beta|\,
\frac{1}{|{e_\Delta}(p_{n})-\beta+i\varepsilon|}\Big]
\nonumber\\
&&
\sup_{\alpha\inC^{(h)}}\sup_{\beta\in\overline{C^{(h)}}}\sup_{p_n\in\Bbb T^2}
\Big\{\;
\Big[\prod_{ T^c}\Big\|\frac{1}{{e_\Delta}-\alpha_i\pm i\varepsilon}\Big\|_{L^1(\Bbb T^2)}\Big]
\nonumber\\
&&\hspace{2cm}
\Big[\prod_{ T}
\Big\|\,\Big|\frac{1}{{e_\Delta}-\alpha_i\pm i\varepsilon}\Big|*
\big|{\mathcal F}(P_{j_i}P_{j_{i'}} v_{\sigma}^2)\big|_{i\sim i'}\,\Big\|_{L^\infty(\Bbb T^2)}\Big] \; \Big\} \;,
\end{eqnarray}
where $i\sim i'$ implies that the vertices indexed by $i$ and $i'$ are linked by a contraction line.
$\prod_T$ and $\prod_{T^c}$ denote the products over all resolvents supported on
$T$ and $T^c$, respectively.
Assuming (~\ref{Pj-ass-Rnorm-1}), we can bound the off-diagonal terms $|j_i-j_{i'}|=1$
by the diagonal terms $j_i=j_{j'}$,
and due to Lemma {~\ref{R-Lnorm-bounds-lemma-1}}, we have
\begin{eqnarray}
\sup_{q\in\Bbb T^2}
\int_{\Bbb T^2} dp\Big|\frac{1}{{e_\Delta}(p)-\alpha-i\varepsilon}\Big|\,\big|{\mathcal F}(P_j^2 v_{\sigma}^2)(p-q)\big|
\leq C_\tau 2^{(1-2{\sigma})j}
\label{convol-est-1}
\end{eqnarray}
if $0\leq j\leq J$, and
\begin{eqnarray}
\sup_{q\in\Bbb T^2}
\int_{\Bbb T^2} dp\Big|\frac{1}{{e_\Delta}(p)-\alpha-i\varepsilon}\Big|\,\big|{\mathcal F}(P_{J+1}^2 v_{\sigma}^2)(p-q)\big|
\leq \varepsilon^{-1}{\sigma}^{-1}2^{-2{\sigma} J}
\label{convol-est-2}
\end{eqnarray}
if $j=J+1$. Hence,
\begin{eqnarray}
|{\rm Amp}(\pi)|&\leq&(C\log\frac1\varepsilon)^2\|\phi_0\|^2_{L^2(\Bbb T^2)}
\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1}
\delta_{\pi}(\underline{j})
(C\log\frac1\varepsilon)^{|T^c|}
\nonumber\\
&&\hspace{1cm}\prod_{i=1}^{2\bar n}\Big(2^{(1-2{\sigma}) j_i }\chi(j\leq J) +
{\sigma}^{-1}\varepsilon^{-1}2^{-2{\sigma} J} \delta_{j_i,J+1}\Big)^{1/2} \;,
\label{exp-sum-graphs-5}
\end{eqnarray}
where we have used
\begin{eqnarray}
\sup_{p\in\Bbb T^2}\int_{C^{(h)}}|d\alpha|\,\frac{1}{|{e_\Delta}(p)-\alpha-i\varepsilon|}<C\log\frac1\varepsilon\;.
\end{eqnarray}
The power $\frac12$ on the last line in (~\ref{exp-sum-graphs-5})
arises because the product extends over all random potentials,
while $T$ accounts only for the contraction lines (each adjacing to two random potentials).
We note also that $\delta_\pi(\underline{j})$ forces elements of $\underline{j}$ to be pairwise equal,
up to overlap terms.
Therefore,
\begin{eqnarray}
|{\rm Amp}(\pi)|&\leq&(C\log\frac1\varepsilon)^{2+|T^c|}
\Big(\sum_{j=0}^J 2^{(1-2{\sigma})j}+\sigma^{-1}\varepsilon^{-1}2^{-2{\sigma} J}\Big)^{|T|} \;,
\end{eqnarray}
where $|T|$ and $|T^c|$ denote the numbers of resolvents supported on $T$ and $T^c$, respectively.
From
\begin{eqnarray}
\sum_{j=0}^J 2^{(1-2{\sigma})j}=\left\{
\begin{array}{ll}
J+1&{\rm if}\;\sigma=\frac12\\
\frac{ 2^{(1-2{\sigma})(J+1)} -1 }{ 2^{(1-2{\sigma})}-1 }&{\rm if}\;0<\sigma<\frac12
\end{array}\right.
\end{eqnarray}
and $|T|=|T^c|=\bar n$,
the assertion of the lemma follows.
\hspace*{\fill}\mbox{$\Box$}
\section{Estimating the remainder term}
The remainder term of the resolvent expansion is given by
\begin{eqnarray}
R_{N,t}=- \lambda e^{\varepsilon t}\frac{1}{2\pi i} \int_{C^{(h)}}
d\alpha e^{-it\alpha} \frac{1}{H_\omega-\alpha-i\varepsilon} V\tilde\phi_{N,\varepsilon}(\alpha) \;,
\label{RNt-def-1-2}
\end{eqnarray}
as we recall from (~\ref{RNt-def-1}).
The trivial bound
\begin{eqnarray}
{\Bbb E}[\|R_{N,t}\|_{\ell^2({\Bbb Z}^2)}^2]&\leq&
C\lambda^2\varepsilon^{-2}{\Bbb E}[\|V\phi_{N,t}\|_{\ell^2({\Bbb Z}^2)}^2]
\nonumber\\
&\leq&
N!\lambda^2\varepsilon^{-2} (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
\label{RNt-triv-bound-lemma-1}
\end{eqnarray}
is insufficient in the subcritical case $0<{\sigma}<\frac12$.
We shall instead apply the time partitioning trick used in \cite{erdyau} and \cite{ch}.
In the critical case ${\sigma}=\frac12$, the time partitioning trick is not effective, but
the trivial bound (~\ref{RNt-triv-bound-lemma-1}) suffices.
\subsection{The subcritical case $0<{\sigma}<\frac12$}
We have
\begin{eqnarray}
R_{N,t}=R_{N,t}^{(0)}+R_{N,t}^{(1)}
\end{eqnarray}
with
\begin{eqnarray}
R_{N,t}^{(0)}&:=&e^{-itH_\omega}\frac{-\lambda}{2\pi i}\int_{C^{(h)}}
d\alpha \frac{1}{H_\omega-\alpha-i\varepsilon}V\tilde\phi_{N,\varepsilon}(\alpha)
\label{RNt-def-2-0}
\\
R_{N,t}^{(1)}&:=&-i\lambda\int_0^tds e^{-i(t-s)H_\omega}V \phi_{N,s}\;,
\label{RNt-def-2-1}
\end{eqnarray}
as was shown in (~\ref{RNt-def-2}).
\begin{lemma}
\label{RNt-0-bound-lemma-1}
\begin{eqnarray}
{\Bbb E}[\|R_{N,t}^{(0)}\|_{\ell^2({\Bbb Z}^2)}^2]&\leq&
N!\frac{\lambda^2}{\tau^2} (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N \;,
\end{eqnarray}
where ${A_{{\sigma},\tau,J,\lambda,\varepsilon}}$ is defined in (~\ref{ampi-def-1}).
\end{lemma}
\noindent{\em Proof.}$\;$
We can deform the contour $C^{(h)}$ of the $\alpha$-integration in (~\ref{RNt-def-2-0})
into
\begin{eqnarray}
\tilde C^{(h)}&:=&(-4+\tau/2+i[0,1])\cup([-4+\tau/2,-\tau/2]+i)\cup(-\tau/2+i[0,1])\cup
\nonumber\\
&&(\tau/2+i[0,1])\cup([4-\tau/2,\tau/2]-i)\cup(4-\tau/2+i[0,1]) \;,
\end{eqnarray}
as there is no obstructing phase factor $e^{-it\alpha}$.
One then immediately sees that
\begin{eqnarray}
{\Bbb E}[\|\chi_{I_\tau}(H_\omega)R_{N,t}^{(0)}\|_{\ell^2({\Bbb Z}^2)}^2]\leq
\frac{c\lambda^2}{\tau^2}{\Bbb E}[\|V\phi_{N,t}\|_{\ell^2({\Bbb Z}^2)}^2] \;,
\end{eqnarray}
since almost surely,
\begin{eqnarray}
\Big\|\chi_{I_\tau}(H_\omega)
\frac{1}{H_\omega-\alpha-i\varepsilon}\Big\|_{op}<c\tau^{-1}\;,
\end{eqnarray}
for any $\alpha\in \tilde C^{(h)}$. We note that by the effect of the
infrared regularization, use of unitarity of $e^{it H}$
in estimating (~\ref{RNt-def-2-0}) is {\em not} penalized by
the usual factor $t^2=\varepsilon^{-2}$.
\hspace*{\fill}\mbox{$\Box$}
Using unitarity in bounding the corresponding quantity for $R_{N,t}^{(1)}$,
however, costs a factor $\varepsilon^{-2}$, and we shall use the time partitioning trick
of \cite{erdyau} to account for it.
\begin{lemma} For $1\ll\kappa\ll \varepsilon^{-1}$, and $0<{\sigma}<\frac12$,
\begin{eqnarray}
{\Bbb E}[\|R_{N,t}^{(1)}\|_{\ell^2({\Bbb Z}^2)}^2]&\leq&
(3\kappa N)^2 (\log\frac1\varepsilon)^2\sum_{n=N+1}^{4N-1}
n! ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{n}
\nonumber\\
&&+ (4N)!
\frac{1}{\varepsilon^2\kappa^{(1-2{\sigma})N}} (\log\frac1\varepsilon)^2
C^{4N}({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{4N}
\label{RNt-1-est-1}
\end{eqnarray}
\end{lemma}
\noindent{\em Proof.}$\;$
The asserted estimate is obtained from application of the time partitioning
trick introduced in \cite{erdyau}. The details for the lattice model
are presented in \cite{ch}, and we shall here only sketch the strategy.
We choose $\kappa\in{\Bbb N}$ with
$1\ll\kappa\ll\varepsilon^{-1}$, and partition $[0,t]$ into $\kappa$ subintervals
\begin{eqnarray}
[0,t]=[0,\theta_1]\cup_{j=1}^{\kappa-1}(\theta_j,\theta_{j+1}]
\end{eqnarray}
with $\theta_j=\frac{jt}{\kappa}$, $j=1,\dots,\kappa$.
Thereby,
\begin{eqnarray}
R_{N,t}^{(1)}=-i\lambda\sum_{j=0}^{\kappa-1}e^{-i(t-\theta_{j+1})H_\omega}
\int_{\theta_j}^{\theta_{j+1}} ds \, e^{-is H_\omega}V \phi_{N,s} \;.
\label{RemNt-def-1}
\end{eqnarray}
Let
\begin{eqnarray}
\phi_{n,N,\theta}(s)&=&(-i\lambda)^{n-N}
\int_{{\Bbb R}_+^{n-N+1}} ds_{0}\cdots ds_{n-N}
\delta(\sum_{j=0}^{n-N}s_j-(s-\theta))
\nonumber\\
&&\times\,
e^{-is_0 \Delta}V\cdots V e^{-is_{n-N}\Delta}
V \phi_{N,\theta}\;.
\end{eqnarray}
That is, the first $N$ out of $n$ collisions happen in the time interval $[0,\theta]$, while
the remaining $n-N$ collisions occur in the time interval $(\theta,s]$.
Expanding $e^{-isH_\omega}$ in (~\ref{RemNt-def-1}) into a Duhamel series with $3N$ terms and
remainder, we find
\begin{eqnarray}
R_{N,t}^{(1)}=\tilde R_{N,t}^{(<4N)}+\tilde R_{N,t}^{(4N)}\;,
\end{eqnarray}
where
\begin{eqnarray}
\tilde R_{N,t}^{(<4N)}&=&\sum_{n=N+1}^{4N-1}\tilde\phi_{n,N,t} \;,
\\
\tilde\phi_{n,N,t}&:=&-i\lambda
\sum_{j=1}^{\kappa}
e^{-i(t-\theta_j)H_\omega}V\phi_{n,N,\theta_{j-1}}(\theta_{j})
\end{eqnarray}
and
\begin{eqnarray}
\tilde R_{N,t}^{(4N)}=-i\lambda \sum_{j=1}^{\kappa}e^{-i(t-\theta_j)H_\omega}
\int_{\theta_{j-1}}^{\theta_j}ds \;
e^{-i(\theta_j-s)H_\omega}
V \phi_{4N,N,\theta_{j-1}}(s) \;.
\end{eqnarray}
By the Schwarz inequality,
\begin{eqnarray}
\|\tilde R_{N,t}^{(<4N)}\|_{\ell^2({\Bbb Z}^2)} \leq (3N\kappa) \sup_{N<n<4N,1\leq j\leq\kappa}
\|\lambda V
\phi_{n,N,\theta_{j-1}}(\theta_{j})\|_{\ell^2({\Bbb Z}^2)}
\label{RNt-4N-est-1}
\end{eqnarray}
and
\begin{eqnarray}
\|\tilde R_{N,t}^{(4N)}\|_{\ell^2({\Bbb Z}^2)} \leq t \sup_{1\leq j\leq\kappa}
\sup_{s\in[\theta_{j-1},\theta_j]}
\|\lambda V
\phi_{4N,N,\theta_{j-1}}(s)\|_{\ell^2({\Bbb Z}^2)} \;.
\label{RNt-4N-est-2}
\end{eqnarray}
The functions $\phi_{n,N,\theta_{j-1}}(\theta_{j})$ and
$\phi_{4N,N,\theta_{j-1}}(s)$ have the following properties.
The expected value of $|(~\ref{RNt-4N-est-1})|^2$ is bounded by the first term after the
inequality sign in (~\ref{RNt-1-est-1}). This is a straightforward consequence of
Lemma {~\ref{amppi-nn-bound-lemma-1}}. For the detailed argument, see \cite{ch, erdyau}.
It remains to estimate (~\ref{RNt-4N-est-2}).
With $\theta'-\theta=\frac t\kappa$, we find
\begin{eqnarray}
(\hat\phi_{n,N,\theta}(\theta'))(k_0)
&=&\frac{i(-\lambda)^{n-N} e^{\frac{\varepsilon t}{\kappa}}}{2\pi}
\int_{I}d\alpha e^{-\frac{i\alpha t}{\kappa}}
\int_{(\Bbb T^2)^{n-N+1}} dk_{1}\cdots dk_{n-N}
\nonumber\\
&&\times\,
\frac{1}{{e_\Delta}(k_0)-\alpha-i\kappa\varepsilon}\hat V(k_1-k_0)\cdots
\nonumber\\
&&\hspace{1.5cm}\cdots\,
\frac{1}{{e_\Delta}(k_{n-N})-\alpha-i\kappa\varepsilon}
\hat V(k_{n-N+1}-k_{n-N})
\nonumber\\
&&\times\, \hat\phi_{N,\theta}(k_{n-N+1}) \;,
\end{eqnarray}
where we recall that
\begin{eqnarray}
\hat\phi_{N,\theta}(k_{n-N+1})&=&
\frac{i(-\lambda)^N e^{\varepsilon\theta}}{2\pi}\int_{C^{(h)}}d\alpha e^{-i\theta \alpha}
\int_{(\Bbb T^2)^N}\prod_{j=n-N+1}^{n+1}dk_j
\nonumber\\
&&\times\,\frac{1}{{e_\Delta}(k_{n-N+1})-\alpha-\frac i\theta}
\hat V(k_{n-N+2}-k_{n-N+1})\cdots
\nonumber\\
&&\hspace{1cm}\cdots\, \hat V(k_{n+1}-k_{n})
\frac{1}{{e_\Delta}(k_{n+1})-\alpha- \frac i\theta}
\hat\phi_{0}(k_{n+1}) \;.
\end{eqnarray}
The key observation here is that there are $n-N+1$ propagators with
imaginary parts $\pm i\kappa\varepsilon$ in the denominator, where $\kappa\varepsilon\gg\varepsilon$
(and $N+1$ propagators
whose denominators have an imaginary part $-\frac i\theta$, where $\frac1\theta$
and $\varepsilon$ can have a comparable size). For those $n-N+1$ propagators,
we have a bound
\begin{eqnarray}
\frac{1}{|{e_\Delta}-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_j^2 v_{\sigma}^2)|\leq
C 2^{-2{\sigma} j}\frac{1}{|{e_\Delta}(p)-\alpha|+\kappa\varepsilon+2^{-j}} \;.
\end{eqnarray}
We now separate the dyadic scales of the random potential into
\begin{eqnarray}
0\leq j\leq J'+1 \; \; , \; \; 2^{J'}\sim \frac{1}{\kappa}2^J \;.
\end{eqnarray}
Using
\begin{eqnarray}
\Big\|\,\frac{1}{|{e_\Delta}-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_j^2 v_{\sigma}^2)|
\,\Big\|_{L^\infty(\Bbb T^2)} \leq 2^{(1-2{\sigma})j}
\end{eqnarray}
for $j\leq J'$, we have
\begin{eqnarray}
\sum_{j=0}^{J'}\Big\|\,\frac{1}{|{e_\Delta}-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_j^2 v_{\sigma}^2)|
\,\Big\|_{L^\infty(\Bbb T^2)}&\leq& \frac{2^{(1-2{\sigma})(J'+1)}-1}{2^{(1-2{\sigma})}-1}
\nonumber\\
&\sim& \frac{1}{\kappa^{1-2{\sigma}}}\frac{2^{(1-2{\sigma})(J+1)}-1}{2^{(1-2{\sigma})}-1}
\;.
\end{eqnarray}
Furthermore,
\begin{eqnarray}
\sum_{j=J'+1}^{J+1}\Big\|\,\frac{1}{|{e_\Delta}-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_{j}^2 v_{\sigma}^2)|
\,\Big\|_{L^\infty(\Bbb T^2)} &\leq& \frac{1}{\kappa \varepsilon}
{\sigma}^{-1}2^{-2{\sigma} J'}
\nonumber\\
&\sim&\frac{1}{\kappa^{1-2{\sigma}}}
{\sigma}^{-1}\varepsilon^{-1}2^{-2{\sigma} J}
\end{eqnarray}
for $j=J'+1$.
Therefore, the estimates for resolvents with $\pm i\kappa\varepsilon$ in the denominators are
by a factor $\frac{1}{\kappa^{(1-2{\sigma})}}$ smaller than those for resolvents with $\pm i\varepsilon$
derived above.
\begin{eqnarray}
&&\sum_{j=0}^{J'+1}\Big\|\,\frac{1}{|{e_\Delta}(p)-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_j^2 v_{\sigma}^2)|
\,\Big\|_{L^\infty(\Bbb T^2)}
\nonumber\\
&&\hspace{3cm}\leq \frac{1}{\kappa^{1-2{\sigma}}}
\Big(K_{\sigma}(J)+\sigma^{-1}2^{(1-2{\sigma})J}\Big)\;.
\label{treeres-kappa-est-1}
\end{eqnarray}
As before, we systematize the evaluation of
\begin{eqnarray}
{\Bbb E}\Big[\|\lambda V
\phi_{4N,N,\theta_{j-1}}(s)\|_{\ell^2({\Bbb Z}^2)}^2\Big]
\end{eqnarray}
by invoking a graph expansion with $\pi\in\Pi_{4N,4N}$.
For every graph, we again introduce an admissible spanning
tree $T$, as in the proof of Lemma {~\ref{amppi-nn-bound-lemma-1}},
and use the estimate (~\ref{treeres-kappa-est-1})
for tree propagators with $\pm i\kappa\varepsilon$ in the denominators.
By the pigeonhole principle, there are at least $N$ of those for every $\pi$, and
any admissible spanning tree $T$ for $\pi$.
This gains a factor of at least
$\frac{1}{\kappa^{(1-2{\sigma})N}}$ in comparison to the bound in Lemma
{~\ref{amppi-nn-bound-lemma-1}}.
The $L^1(\Bbb T^2)$-bounds
on loop resolvents are estimated by $C\log\frac1\varepsilon$, as before.
Observing that the number of tree propagators is $\bar n$, and that there
are $\bar n+2$ propagators estimated in $L^1$, one concludes that the expected value of
$|(~\ref{RNt-4N-est-2})|^2$ is bounded by the second term after the inequality sign in
(~\ref{RNt-1-est-1}). A detailed exposition is given in
\cite{erdyau} and \cite{ch}.
\hspace*{\fill}\mbox{$\Box$}
\subsection{The critical case ${\sigma}=\frac12$}
The time partitioning only provides a logarithmic
improvement in $\kappa$,
\begin{eqnarray}
\sum_{j=0}^{J'}\Big\|\,\frac{1}{|{e_\Delta}-\alpha-i\kappa\varepsilon|}*|{\mathcal F}(P_j^2 v_{\sigma}^2)|
\,\Big\|_{L^\infty(\Bbb T^2)}&\leq& J'+1
\;\sim\; \frac{1}{\log\kappa}J
\end{eqnarray}
which is too small to produce a significant effect.
However, the trivial estimate (~\ref{RNt-triv-bound-lemma-1}) is sufficient
for our analysis,
because the large factor $2^J$ enters ${A_{{\sigma},\tau,J,\lambda,\varepsilon}}$ only logarithmically.
\section{Conclusion of the proof of Lemma {~\ref{Lemma-main-1}}}
To conclude the proof of Lemma {~\ref{Lemma-main-1}}, we make the following choices for
$\varepsilon,J,N,\kappa$ as functions of ${\sigma}$, $\lambda$ and $\eta$ (depending implicitly on $\tau$).
\subsection{The subcritical case $0<{\sigma}<\frac12$}
Recalling (~\ref{phi-ell2-ircut-bound-1}), (~\ref{exp-phi-ell2-Schwarz-1}), and
summarizing the estimates formulated in Lemmata {~\ref{amppi-nn-bound-lemma-1}} and
{~\ref{RNt-0-bound-lemma-1}},
our analysis infers that
\begin{eqnarray}
l.h.s.\;of\;(~\ref{Lemma-main-est-1})
&<&C\tau^{\frac12}+2\Big(\frac{\varepsilon}{\tau}\Big)^2+
\sum_{ n=1}^N n! (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{ n}
\nonumber\\
&&+N!\frac{\lambda^2}{\tau^2}(\log\frac1\varepsilon)^2
({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{N}
\nonumber\\
&&+\lambda^2 (3\kappa N)^2(\log\frac1\varepsilon)^2\sum_{n=N+1}^{4N-1}
n! ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{n}
\nonumber\\
&&+(4N)!\frac{\lambda^2}{\varepsilon^2\kappa^{(1-2{\sigma})N}}
(\log\frac1\varepsilon)^2
({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{4N} \;,
\end{eqnarray}
where we recall from (~\ref{ampi-def-1}) that
\begin{eqnarray}
{A_{{\sigma},\tau,J,\lambda,\varepsilon}}=C_\tau(K_{\sigma}(J) \lambda^2\log\frac1\varepsilon
+\varepsilon^{-1}{\sigma}^{-1}2^{-2{\sigma} J}\lambda^2\log\frac1\varepsilon) \;.
\end{eqnarray}
We have
\begin{eqnarray}
K_{{\sigma}}(J)=\frac{ 2^{(1-2{\sigma})(J+1)} -1}{2^{1-2{\sigma}}-1}\;.
\end{eqnarray}
Let $\eta>0$ be arbitrary but fixed. Setting
\begin{eqnarray}
\varepsilon&=&2^{-J}
\label{eps-def-lambda-1}\\
JK_{{\sigma}}(J)&=&\lambda^{-2+2\eta}
\label{eps-def-lambda-2}
\end{eqnarray}
we find
\begin{eqnarray}
K_{{\sigma}}(J)\lambda^2\log\frac1\varepsilon&=&J K_{\sigma}(J)\lambda^2\;\leq\;\lambda^{2\eta}
\nonumber\\
\varepsilon^{-1}{\sigma}^{-1}2^{-2{\sigma} J}\lambda^2\log\frac1\varepsilon&=&
{\sigma}^{-1}2^{(1-2{\sigma})J}\lambda^2\log\frac1\varepsilon
\nonumber\\
&=&{\sigma}^{-1}J K_{\sigma}(J) \;,
\end{eqnarray}
so that
\begin{eqnarray}
{A_{{\sigma},\tau,J,\lambda,\varepsilon}}&<&\lambda^{ 1.9\eta } \;,
\end{eqnarray}
for $\lambda$ sufficiently small (depending on ${\sigma}$).
Choosing
\begin{eqnarray}
N&=&\frac{\eta\log\frac1\lambda}{10\log\log\frac1\lambda}
\;,
\end{eqnarray}
one gets (noting that $\varepsilon>\lambda^2$)
\begin{eqnarray}
\sum_{ n=1}^N n! (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{ n}
&<&C(\log\frac1\lambda)^2\sum_{ n=1}^N (N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{ n}
\nonumber\\
&<&C(\log\frac1\lambda)^2\sum_{ n=1}^N \lambda^{1.5 \eta n}
\;<\;\lambda^{1.1\eta}
\end{eqnarray}
and
\begin{eqnarray}
N!\frac{\lambda^2}{\tau^2}(\log\frac1\varepsilon)^2
({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{N}&<&C(\log\frac1\lambda)^2
(N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
\;<\;\lambda
\end{eqnarray}
for $\tau\gg\lambda$.
Choosing
\begin{eqnarray}
\kappa&=&(\log\frac1\lambda)^{\frac{30}{\eta(1-2{\sigma})}} \;,
\end{eqnarray}
one gets
\begin{eqnarray}
\lambda^2 (3\kappa N)^2(\log\frac1\varepsilon)^2\sum_{n=N+1}^{4N-1}
n! ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{n}&<&C\lambda^2 (\log\frac1\lambda)^{\frac{100}{(1-2{\sigma})\eta}}
(4N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
\nonumber\\
&<&\lambda^{2\eta} \;.
\end{eqnarray}
Furthermore, since
\begin{eqnarray}
\kappa^{(1-2{\sigma})N}\;>\;\lambda^{-3}
\;,
\end{eqnarray}
one finds
\begin{eqnarray}
(4N)!\frac{\lambda^2}{\varepsilon^2\kappa^{(1-2{\sigma})N}}
(\log\frac1\varepsilon)^2
({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{4N}&<&(4N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{4N}
\;<\;\lambda^{2\eta}\;.
\end{eqnarray}
Thus, for $\lambda$ sufficiently small (depending on ${\sigma}$ and $\eta$),
\begin{eqnarray}
l.h.s.\;of\;(~\ref{Lemma-main-est-1})
<C\tau^{\frac12}+\lambda^{\eta}\;.
\end{eqnarray}
Moreover, (~\ref{tstar-def-1}), (~\ref{eps-def-lambda-1}) and
(~\ref{eps-def-lambda-2}) combined imply
that for every fixed $0<{\sigma}<\frac12$, there exists a positive constant $C_{\sigma}$
such that
\begin{eqnarray}
\ell_{\sigma}(\lambda)\geq C_{\sigma}\lambda^{-\frac{2-\eta}{1-2{\sigma}}} \;.
\end{eqnarray}
This proves the assertion of Lemma {~\ref{Lemma-main-1}} for $0<{\sigma}<\frac12$.
\subsection{The critical case ${\sigma}=\frac12$}
Using (~\ref{phi-ell2-ircut-bound-1}),
(~\ref{exp-phi-ell2-Schwarz-1}), Lemma
{~\ref{amppi-nn-bound-lemma-1}} and
({~\ref{RNt-triv-bound-lemma-1}}),
\begin{eqnarray}
l.h.s.\;of\;(~\ref{Lemma-main-est-1})
&<&C\tau^{\frac12}+2\Big(\frac{\varepsilon}{\tau}\Big)^2+
\sum_{ n=1}^N n! (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{ n}
\nonumber\\
&&+N!\frac{\lambda^2}{\tau^2}(\log\frac1\varepsilon)^2
({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{N}
\nonumber\\
&&+N! \lambda^2 \varepsilon^{-2} (\log\frac1\varepsilon)^2 ({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^{N} \;.
\end{eqnarray}
We have
\begin{eqnarray}
K_{\frac12}(J)=J+1\;.
\end{eqnarray}
Let $\eta>0$ be arbitrary (small) but fixed. Setting
\begin{eqnarray}
J&=&N\;=\;\lambda^{-\frac14+\eta}
\nonumber\\
\varepsilon&=&2^{-\lambda^{-\frac14+\eta}} \;=\;2^{-N}\;=\;2^{-J} \;,
\label{eps-def-lambda-3}
\end{eqnarray}
we get, for sufficiently small $\lambda>0$,
\begin{eqnarray}
{A_{{\sigma},\tau,J,\lambda,\varepsilon}}
&=&C_\tau\Big(J\lambda^2\log\frac1\varepsilon+2\varepsilon^{-1}2^{-J}\log\frac1\varepsilon\Big)
\nonumber\\
&<&2C_\tau N^2\lambda^2
\end{eqnarray}
and
\begin{eqnarray}
N^2{A_{{\sigma},\tau,J,\lambda,\varepsilon}}&<&\lambda^{3\eta}\;.
\end{eqnarray}
Then,
\begin{eqnarray}
\sum_{n=1}^N n!(\log\frac1\varepsilon)^2({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^n&<&N^2{A_{{\sigma},\tau,J,\lambda,\varepsilon}}+\sum_{n=2}^N N^2(N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^n
\nonumber\\
&<&\lambda^{2\eta}
\end{eqnarray}
and
\begin{eqnarray}
N!\frac{\lambda^2}{\tau^2}(\log\frac1\varepsilon)^2({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
&<&\frac{\lambda^{2}}{\tau^2}N^2(N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
\;<\;\lambda \;.
\end{eqnarray}
Furthermore,
\begin{eqnarray}
N!\varepsilon^{-2}\lambda^2(\log\frac1\varepsilon)^2({A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N&<&
\lambda^2 N^2 (4N{A_{{\sigma},\tau,J,\lambda,\varepsilon}})^N
\nonumber\\
&<&\lambda(4\lambda^{2\eta})^{\lambda^{-\frac14+\eta}}
\;<\;\lambda \;.
\end{eqnarray}
In conclusion,
\begin{eqnarray}
l.h.s.\;of\;(~\ref{Lemma-main-est-1})
<C\tau^{\frac12}+\lambda^{\eta}\;.
\end{eqnarray}
From (~\ref{tstar-def-1}) and (~\ref{eps-def-lambda-3}), we infer that
\begin{eqnarray}
\ell_{\sigma}(\lambda)\geq 2^{-\lambda^{-\frac14+\eta}} \;.
\end{eqnarray}
This concludes our proof of Lemma {~\ref{Lemma-main-1}} for ${\sigma}=\frac12$.
\subsection*{Acknowledgements}
I am deeply grateful to H.-T. Yau and L. Erd\"os for their support and
generosity. I have benefitted immensely from numerous discussions with H.-T. Yau
about topics closely related to those studied here
while being at the Courant Institute, NYU, as a Courant Instructor.
I also wish to thank M. Aizenman, S. Denissov, V. Jacsic, and S. Warzel for discussions.
This work was supported by NSF grant DMS-0524909.
|
{
"timestamp": "2005-10-26T19:55:58",
"yymm": "0503",
"arxiv_id": "math-ph/0503064",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503064"
}
|
\section*{Introduction}
An operator $A:\mathbb R^d\rightarrow\mathbb R^d$ is called additively homogeneous if it satisfies $A(x+a\textbf{1})=A(x)+a\textbf{1}$ for all $x\in\mathbb R^d$ and $a\in\mathbb R$, where $\textbf{1}$ is the vector $(1,\cdots,1)'$ in $\mathbb R^d$. It is called isotone if $x\le y$ implies $A(x)\le A(y)$, where the order is the product order on $\mathbb R^d$. It is called topical if it is isotone and homogeneous. The set of topical operators on $\mathbb R^d$ will be denoted by $Top_d$.
We recall that the action of matrices with entries in ${\mathbb R}_{\max}=\mathbb R\cup\{-\infty\}$ on ${\mathbb R}_{\max}^d$ is defined by $(Ax)_i=\max_j(A_{ij}+x_j)$. When matrix $A$ has no line of $-\infty$, the restriction of this action to $\mathbb R^d$ defines a topical operator, also denoted by $A$. Such operators are called $(\max,+)$ operators and composition of operators corresponds to the product of matrices in the $(\max,+)$ semi-ring.\\
Let $\left(A(n)\right)_{n\in\mathbb N}$ be a sequence of random topical operators on $\mathbb R^d$. Let $x(n,x_0)$ be defined by
\begin{equation}\label{defx}
\left\{\begin{array}{lcl}
x(0,x_0)&=&x_0\\
x(n,x_0)&=&A(n)x(n-1,x_0).
\end{array} \right.
\end{equation}
This class of system can modelize a wide range of situations. A review of applications can be found in the last section of~\cite{BM96}. When the $x(n,.)$ are daters, the isotonicity assumption expresses the causality principle, whereas the additive monotonicity expresses the possibility to change the origin of time. (See J.~Gunawardena and M.~Keane~\cite{GunawardenaKeane}, where topical applications have been introduced). Among other examples the $(\max,+)$ case has been applied to modelize queuing networks (J. Mairesse~\cite{Mairesse}, B.~Heidergott~\cite{CaractMpQueuNet}), train networks (B.~Heidergott and R.~De~Vries~\cite{HeidergottDeVriesPubTransNet}, H.~Braker~\cite{braker}) or Job-Shop (G. Cohen and al.~\cite{cohen85a}). It also computes the daters of some task resources models (S.~Gaubert and J.~Mairesse~\cite{gaumair95}) and timed Petri Nets including Events graphs (F.~Baccelli~\cite{Baccelli}) and 1-bounded Petri Nets (S.~Gaubert and J.~Mairesse~\cite{GaubertMairesseIEEE}). The role of the max operation is synchronizing different events. For devlopements on the max-plus modelizing power, see F.~Baccelli and al.~\cite{BCOQ} or B.~Heidergott, G.~J.~Olsder, and J.~van~der~Woude~\cite{MpAtWork}.
We are interested in the asymptotic behavior of $x(n,.)$. It follows from theorem~\ref{thVincent} that $\frac{1}{n}\max_ix_i(n,X_0)$ converges to a limit $\gamma$.
In many cases, if the system is closed, then every coordinate $x_i(n,X_0)$ also converges to $\gamma$. The value $\gamma$, which is often called cycle time, is the inverse of the throughput (resp. output) of the modelized network (resp. production system), therefore there has been many attempt to estimate it. (J.E.~Cohen~\cite{Cohen}, B.~Gaujal and A.~Jean-Marie~\cite{ComputIssuesSRS}, J.~Resing and al.~\cite{RVH}) Even when the $A(n)$'s are i.i.d. and take only finitely many values, approximating $\gamma$ is NP-hard (V.~Blondel and al.~\cite{LyapExpNP}). D.~Hong and its coauthors have obtained (\cite{BacHong1},\cite{BacHong2} ,\cite{GaubertHong} ) analyticity of $\gamma$ as a function of the law of $A(1)$. In this paper, we prove another type of stability, under the same assumptions.
We show that under suitable additional conditions, $x(n,.)$ satisfies a central limit theorem, a local limit theorem, a renewal theorem and a large deviations principle. When the $A(n)$ are $(\max,+)$ operators we give more explicit results. Those results justify the approximation of $\gamma$ by $\frac{1}{n}x_i(n,X_0)$ and lead the path to confidence intervals.\\
Products of random matrices in the usual sense have been intensively investigated. Let us cite H. Furstenberg~\cite{Furst63}, Y. Guivarc'h and A. Raugi~\cite{GR85} or I. Ya. Gol{$'$}dshe\u{\i}d and G. A. Margulis~\cite{GM2}. The interested reader can find a presentation of this theory in the book by Ph.~Bougerol and J.~Lacroix~\cite{BL}.
We investigate analogous problems to those studied by \'E. Le Page~\cite{LePage}, but for matrices in the $(\max,+)$ semi-ring and more generally for iterated topical operators.
This article is divided into three parts.
First we present the model of iterated topical operators, including a short review of known limit theorems and a sketch of the proof of our results. Second we state our theorems and comment on them. Finally we prove them.
\section{Iterated topical operators}\label{IFS}
\subsection{Memory loss property}
Dealing with homogeneous operators it is natural to introduce the quotient space of $\mathbb R^d$ by the equivalence relation $\sim$ defined by $x\sim y$ if $x-y$ is proportional to $\textbf{1}$. This space will be called projective space and denoted by $\mathbb{PR}_{\max}^d$. Moreover $\overline{x}$ will be the equivalence class of $x$.
The application $\overline{x}\mapsto (x_i-x_j)_{i<j}$ embeds $\mathbb{PR}_{\max}^d$ onto a subspace of $\mathbb R^{\frac{d(d-1)}{2}}$ with dimension $d-1$. The infinity norm of $\mathbb R^{\frac{d(d-1)}{2}}$ therefore induces a distance on $\mathbb{PR}_{\max}^d$ which will be denoted by $\delta$. A direct computation shows that $\delta(\overline{x},\overline{y})=\max_i(x_i-y_i)+\max_i(y_i-x_i)$. By a slight abuse, we will also write $\delta(x,y)$ for $\delta(\overline{x},\overline{y})$. The projective norm of $x$ will be $|x|_\mathcal{P}=\delta(x,0)$.
Let us recall two well known facts about topical operators. First a topical operator is non-expanding with respect to the infinity norm. Second the operator it defines from $\mathbb{PR}_{\max}^d$ to itself is non-expanding for $\delta$.
The key property for our proofs is the following:
\begin{defn}[MLP]\
\begin{enumerate}
\item A topical operator $A$ is said to have rank~1, if it defines a constant operator on $\mathbb{PR}_{\max}^d$ : $\overline{Ax}$ does not depend on $x\in\mathbb R^d$.
\item The sequence $\left(A(n)\right)_{n\in\mathbb N}$ of $Top_d$-valued random variables is said to have the memory loss (MLP) property if there exists an $N$ such that $A(N)\cdots A(1)$ has rank~1 with positive probability.
\end{enumerate}
\end{defn}
This notion has been introduced by J. Mairesse~\cite{Mairesse}, the $A(n)$ being $(\max,+)$ operators. The denomination rank~1 is natural for $(\max,+)$ operators.
We proved in~\cite{GM} that this property is generic for i.i.d. $(\max,+)$ operators: it is fulfilled when the support of the law of $A(1)$ is not included the union of finitely many affine hyperplanes.
Although this result could suggest the opposite, the MLP depends on the law of $A(1)$, and not only on its support :
if $\left(U(n)\right)_{n\in\mathbb N}$ is an i.i.d. sequence with the support of $U(1)$ equal to $[0,1]$, and $A(n)$ are the $(\max,+)$ operators defined by the matrices
$$A(n)=\left(
\begin{array}{cc}
-U(n) & 0 \\
0 & -U(n)
\end{array}\right),$$
then $\left(A(n)\right)_{n\in\mathbb N}$ has the MLP property iff $\mathbb P(U(n)=0)>0$.
The weaker condition that there is an operator with rank~1 in the closed semigroup generated by the support of the law of $A(1)$ has been investigated by J. Mairesse for $(\max,+)$ operators. It ensures the weak convergence of $\overline{x}(n,.)$ but does not seem appropriate for our construction.
\subsection{Known results}
Before describing our analysis, we give a brief review of published limit theorems about $x(n,X^0)$.
There has been many papers about the law of large numbers for products of random $(\max,+)$ matrices since it was introduced by J.E~Cohen~\cite{Cohen}. Let us cite F.~Baccelli~\cite{Baccelli}, the last one by T.~Bousch and J.~Mairesse~\cite{BouschMairesseEng} and our PhD thesis~\cite{theseGM} (in French). The last article proves results for a larger class of topical operators, called uniformly topical.
J.M. Vincent has proved a law of large number for topical operators, that will be enough in our case~:
\begin{thm}[\cite{vincent}]\label{thVincent
Let $\left(A(n)\right)_{n\in\mathbb N}$ be a stationary ergodic sequence of topical operators and $X^0$ an $\mathbb R^d$-valued random variable. If $A(1).0$ and $X^0$ are integrable, then there exists $\overline{\gamma}$ and $\underline{\gamma}$ in $\mathbb R$ such that
\begin{eqnarray*}
\lim_n \frac{\max_ix_i(n,X^0)}{n}&=&\overline{\gamma}~\textrm{a.s.}\\
\lim_n \frac{\min_ix_i(n,X^0)}{n}&=&\underline{\gamma}~\textrm{a.s.}
\end{eqnarray*}
\end{thm}
F. Baccelli and J. Mairesse give a condition to ensure $\overline{\gamma}=\underline{\gamma}$, hence the convergence of $\frac{x(n,X^0)}{n}$:
\begin{thm}[\cite{BM96}]\label{LGN
Let $\left(A(n)\right)_{n\in\mathbb N}$ be a stationary ergodic sequence of topical operators and $X^0$ an $\mathbb R^d$-valued random variable such that $A(1).0$ and $X^0$ are integrable. If there exists an $N$, such that $A(N)\cdots A(1)$ has a bounded projective image with positive probability, then there exists $\gamma$ in $\mathbb R$ such that
$$ \lim_n \frac{x(n,X^0)}{n}=\gamma\textbf{1}~\textrm{a.s.}$$
\end{thm}
In this case $\gamma$ is called the Lyapunov exponent of the sequence. We notice that the MLP property implies a bounded projective image with positive probability.
The following result has been proved by J. Mairesse when the $A(n)$ are $(\max,+)$ operators, but can be extended to topical operators with the same proof. It will be the key point to ensure the spectral gap.
\begin{thm}[Mairesse \cite{Mairesse}]\label{strcoupling}
If the stationary and ergodic sequence $\left(A(n)\right)_{n\in\mathbb Z}$ of random variables with values in $Top_d$ has memory loss property, then there exists a random variable $Y$ with values in $\mathbb{PR}_{\max}^d$ such that $Y_n:=A(n)\cdots A(1)Y$ is stationary. Moreover
$$\lim_{n\rightarrow\infty}\mathbb P\left(\exists x_0, Y_n\neq \bar{x}(n,x_0)\right)=0.$$
In particular $\overline{x}(n,x_0)$ converges in total variation uniformly in $x_0$.
\end{thm}
The law of $Y$ is called the invariant probability measure.
To end this section we mention two limit theorems, which are close to ours, but obtained by different ways. We will compare those results to ours in section~\ref{commentaires}.
With a martingale method J. Resing and al.~\cite{RVH} have obtained a central limit theorem for $x(n,X^0)$, when the Markov chain $\overline{x}(n,.)$ is aperiodic and uniformly $\Phi$-recurrent. The theorem has been stated for $(\max,+)$ operators, but it should make no difference to use topical ones.
With a subadditivity method, F. Toomey~\cite{toomey} has proved a large deviation principle for $x(n,x_0)$ when the projective image of $A(N)\cdots A(1)$ is bounded.
\subsection{Principle of the analysis}\label{principes}
From now on, $\left(A(n)\right)_{n\in\mathbb N}$ is an i.i.d sequence of topical operators with the MLP property.
The first step of the proof is to split our Markov chain $x(n,.)$ into another Markov chain and a sum of cocycles over this chain, following what \'E. Le Page made for products of random matrices. For any topical function $\phi$ from $\mathbb R^d$ to $\mathbb R$, $\phi(Ax)-\phi(x)$ only depends on $A$ and $\overline{x}$. Therefore $\phi(x(n,.))-\phi(x(n-1,.))$ only depends on $A(n)$ and $\overline{x}(n-1,.)$. Since $\mathbb{PR}_{\max}^d$ can be seen as an hyperplane of $\mathbb R^d$, $x(n,.)$ can be replaced by $\left(\phi(x(n,.)),\overline{x}(n,.)\right)$. (cf. lemma~\ref{bilip})
According to theorem~\ref{strcoupling}, we know that $\overline{x}(n,.)$ converges. On the other hand, by theorem~\ref{LGN} $x(n,X^0)$, goes to infinity (if $\gamma\neq 0$) in the direction of $\textbf{1}$, so $\phi(x(n,.))\sim \gamma n$. We investigate the oscillations of $\phi(x(n,.))-\gamma n $. Interesting $\phi$'s are defined by $\phi(x)=x_i$, $\phi(x)=\max_ix_i$, $\phi(x)=\min_ix_i$.\\
The second step is to prove the spectral gap for the operator defining the Markov chain $\left(A(n),\overline{x}(n-1,.)\right)_{n\in\mathbb N}$ and apply the results of~\cite{HH1} et~\cite{HH2} that give limit theorems for $\phi(x(n,X^0))-\phi(X^0)-\gamma n $. The spectral gap follows from the convergence of $\overline{x}(n,.)$, just like by \'E. Le Page~\cite{LePage}.
We use two series of results.
The first series are taken from the book by H. Hennion and L. Herv\'e~\cite{HH1} that sums up the classical spectral gap method developed since Nagaev~\cite{Nagaev} in a general framework. To apply it we demand integrability conditions on $\sup_x|\phi(A(1)x)-\phi(x)|$ to have a Doeblin operator on the space of bounded functions.
The second series are taken from the article~\cite{HH2} that is a new refinement of the method in the more precise framework of iterated Lipschitz operators. Since our model enters this framework, we get the same results with integrability conditions on $A(1)\,0$ that ensures that the Markov operator satisfy a Doeblin-Fortet condition on functions spaces defined by weights. The comparison between the two series of results will be made in section~\ref{commentaires}.
\section{Statement of the limit theorems}\label{statements}
\subsection{General case}
From now on, we state the results that we will prove in section~\ref{proofs}.
For local limit theorem and for renewal theorem we need non arithmeticity conditions.
There are three kind of non arithmeticity, depending if the theorem follows from~\cite{HH1} or~\cite{HH2}. We will denote them respectively by (weak-) non arithmeticity and algebraic non arithmeticity.
When $d=1$ they fall down to the usual non arithmeticity condition for real i.i.d. variables.
Algebraic non arithmeticity will be defined before the statement of LLT, but other non arithmeticity conditions will be defined in section~\ref{proofs} once we have given the definitions of the operator associated to the Markov chain. Unlike algebraic non arithmeticity, they depend on the 2-uple $\left(\left(A(n)\right)_{n\in\mathbb N},\phi\right)$, which will be called "the system".
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property. The sequence $\left(x(n,.)\right)_{n\in\mathbb N}$ is defined by equation (\ref{defx}) and $\gamma$ is the Lyapunov exponent defined by theorem~\ref{LGN}.
Since the topology of the uniform convergence over compact subset on $Top_d$ has an enumerable basis of open sets, the support of measures on it is well defined. We denote by $S_A$ the support of the law of $A(1)$ and by $T_A$ the semi-group generated by $S_A$ in $Top_d$.
\begin{thm}[CLT]\label{TCL}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$. Let $\phi$ be topical from $\mathbb R^d$ to $\mathbb R$.
Assume one of the following conditions:
\begin{enumerate}[i)]
\item $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ has a second moment,
\item $A(1)\,0$ has a $4+\epsilon$-th moment and $X^0$ has a $2+\epsilon$-th moment.
\end{enumerate}
Then there exists $\sigma^2\ge0$ such that $\frac{x(n,X^0)-n\gamma\textbf{1}}{\sqrt{n}}$ converges weakly to a random vector whose coordinates are equal and have law $\mathcal{N}(0,\sigma^2)$.\\
In the first case, or if $A(1)\,0$ has a $6+\epsilon$-th moment and $X^0$ has a $3+\epsilon$-th moment, then
\begin{itemize}
\item $\sigma^2=\lim\frac{1}{n}\mathbb E\left(\phi\left(x(n,X^0)\right)-n\gamma\right)^2 $
\item $\sigma=0$ iff there is a $\theta\in Top_d$ with rank~1 such that for any $A\in S_A$ and any $\theta'\in T_A$ with rank~1, $\theta A\theta'=\theta \theta' +\gamma\textbf{1}$.
\end{itemize}
\end{thm}
\begin{rem}
According to lemma~\ref{invtheta}, if there is such a $\theta$, then every $\theta\in T_A$ with rank~1 has this property.
\end{rem}
\begin{thm}[CLT with rate]\label{TCLV}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$. Let $\phi$ be topical from $\mathbb R^d$ to $\mathbb R$.
Assume one of the following conditions:
\begin{enumerate}[i)]
\item $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ has an $l$-th moment with $l\ge 3$,
\item $A(1)\,0$ has an $l$-th moment, with $l>6$.
\end{enumerate}
If $\sigma^2>0$ in theorem~\ref{TCL}, then there exists $C\ge 0$ such that for every initial condition $X^0$ with an $l$-th moment, we have
\begin{eqnarray}\label{vitTCL}
\lefteqn{\hspace{-3.5cm}\sup_{u\in\mathbb R}\left|\mathbb P[\phi(x(n,X^0))-n\gamma-\phi(X_i^0)\le\sigma u\sqrt{n}] -\mathcal{N}(0,1)(]-\infty,u])\right|}\nonumber\\
&&\le\frac{C\left(1+\mathbb E\left[\left\|X^0\right\|_\infty^l\right]\right)}{\sqrt{n}},
\end{eqnarray}
\begin{eqnarray*}
\lefteqn{\hspace{-2cm}\sup_{u\in\mathbb R^d}\left|\mathbb P[x(n,X^0)-n\gamma\textbf{1}\le\sigma u\sqrt{n}]-\mathcal{N}(0,1)(]-\infty,\min_iu_i])\right|}\\
&&\le \frac{C\left(1+\mathbb E\left[\left\|X^0\right\|_\infty^l\right]+\mathbb E\left[\left\|A(1)0\right\|_\infty^l\right]\right)}{n^{{\frac{l}{2(l+1)}}}}.\end{eqnarray*}
\end{thm}
\begin{defn}
We say that the sequence $\left(A(n)\right)_{n\in\mathbb N}$ is algebraically arithmetic if there are $a,b\in\mathbb R$ and a $\theta\in Top_d$ with rank~1 such that for any $A\in S_A$ and any $\theta'\in T_A$ with rank~1,
\begin{equation}\label{eqANA}
(\theta A\theta'-\theta \theta')(\mathbb R^d) \subset (a+b\mathbb Z)\textbf{1}.
\end{equation}
Otherwise the sequence is algebraically non arithmetic.
\end{defn}
\begin{rem}
According to lemma~\ref{invtheta}, if there is such a $\theta$, then every $\theta\in T_A$ with rank~1 has this property.
Moreover, for any $\theta,\theta'\in Top_d$ with rank~1 and any $A\in Top_d$, the function $\theta A\theta'-\theta \theta'$ is constant with value in $\mathbb R\textbf{1}$.
\end{rem}
\begin{thm}[LLT]\label{TLL}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$. Let $\phi$ be topical from $\mathbb R^d$ to $\mathbb R$.
Assume one of the following conditions:
\begin{enumerate}[i)]
\item $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ has a second moment, $\sigma>0$, and the system is non arithmetic
\item $A(1)\,0$ has a $4+\epsilon$-th moment, $X^0\in \mathbb{L}^\infty$ and the sequence $\left(A(n)\right)_{n\in\mathbb N}$ is algebraically non arithmetic.
\end{enumerate}
Then $\sigma>0$ and there exists a $\sigma$-finite measure $\alpha$ on $\mathbb R^d$, so that for any continuous function $h$ with compact support, we have:
$$\lim_n\sup_{u\in\mathbb R}\left|\sigma\sqrt{2\pi n}\mathbb E\left[h\left(x(n,X^0)-n\gamma\textbf{1}-u\textbf{1}\right)\right]-\mathbb E\left[e^{-\frac{(u+\phi(X^0))^2}{2n\sigma^2}}\right]\alpha(h)\right|=0.$$
Moreover the image of $\alpha$ by the function $x\mapsto (\overline{x},\phi(x))$ is the product of the invariant probability measure on $\mathbb{PR}_{\max}^d$ by the Lebesgue measure.
\end{thm}
\begin{rem}
Like in the usual LLT, this theorem says that the probability for $x(n,X^0)$ to fall in a box decreases like $\frac{1}{\sqrt{n}}$.
To replace the continuous functions by indicator functions of the box, we need to know more about the invariant probability measure on $\mathbb{PR}_{\max}^d$. In particular, numerical simulations show that some hyperplanes may have a weight for this probability measure, so those hyperplanes could not intersect the boundary of the box.
\end{rem}
The algebraic non arithmeticity is optimal in the following sense:
\begin{prop}\label{optNA}
If the conclusion of theorem~\ref{TLL} is true, then $\left(A(n)\right)_{n\in\mathbb N}$ is algebraically non arithmetic.
\end{prop}
\begin{thm}[Renewal theorem]\label{renouv}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$.
Assume that there is a topical $\phi$ from $\mathbb R^d$ to $\mathbb R$ such that $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ has a second moment. We denote by $\alpha$ the same measure as in theorem~\ref{TLL}. If $\gamma>0$ and the system is weakly non arithmetic, then for any function $h$ continuous with compact support and any initial condition $X^0$, we have:
$$\lim_{a\rightarrow-\infty} \sum_{n\ge1}\mathbb E\left[h\left(x(n,X^0)-a\textbf{1}\right)\right]=0,$$
$$\lim_{a\rightarrow+\infty} \sum_{n\ge1}\mathbb E\left[h\left(x(n,X^0)-a\textbf{1}\right)\right]=\frac{\alpha(h)}{\gamma}.$$
\end{thm}
\begin{rem}
The vector $\textbf{1}$ gives the average direction in which $x(n,X^0)$ is going to infinity. Like in the usual renewal theorem, this theorem says that the average number of $x(n,X^0)$ falling in a box is asymptotically proportional to the length of this box, when the box is going to infinity in that direction.
Like in the LLT, to replace the continuous functions by indicator functions of the box, we need to know more about the invariant probability measure on $\mathbb{PR}_{\max}^d$.
\end{rem}
\begin{thm}[Large deviations]\label{PGD}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of topical operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$. Let $\phi$ be topical from $\mathbb R^d$ to $\mathbb R$.
Assume that $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ has an exponential moment, and that $\sigma^2>0$ in theorem~\ref{TCL}. Then, there exists a non negative strictly convex function $c$, defined on a neighborhood of $0$ and vanishing only at $0$ such that for any bounded initial condition $X^0$ and any $\epsilon>0$ small enough we have:
$$\lim_{n}\frac{1}{n}\ln\left(\mathbb P\left[\phi\left(x(n,X^0)\right)-n\gamma>n\epsilon\right]\right)=-c(\epsilon),$$
$$\lim_{n}\frac{1}{n}\ln\left(\mathbb P\left[\phi\left(x(n,X^0)\right)-n\gamma<-n\epsilon\right]\right)=-c(-\epsilon).$$
\end{thm}
\subsection{Max-plus case}
When the $A(n)$ are $(\max,+)$ operators, it is natural to chose $\phi(x)=\max_ix_i$. In this case we get $\min_j\max_iA_{ij}\le\phi(Ax)-\phi(x)\le\max_{ij}A_{ij}$, so integrability condition can be checked on the last two quantities.
\begin{thm}[CLT]\label{TCL1mp}
Let $\left(A(n)\right)_{n\in\mathbb N}$ be an i.i.d sequence of $(\max,+)$ operators with the MLP property and $X^0$ an $\mathbb R^d$-valued random variable independent from $\left(A(n)\right)_{n\in\mathbb N}$.
If $\max_{ij}A(1)_{ij}$ and $\min_{j}\max_{i}A(1)_{ij}$ have a second moment, then there exists $\sigma^2\ge0$ such that for every initial condition $X^0$,
\begin{enumerate}[(i)]
\item $\frac{x(n,X^0)-n\gamma\textbf{1}}{\sqrt{n}}$ converges weakly to a random variable whose coordinates are equals and have law $\mathcal{N}(0,\sigma^2)$,
\item $\sigma^2=\lim\frac{1}{n}\int \left(\max_{i,j}A(n)\cdots A(1)_{ij}\right)^2d\mathbb P$.
\end{enumerate}
\end{thm}
Theorems~\ref{TCLV}~to~\ref{PGD} are specialized in the same way: the conclusion is valid if $\phi(x)=\max_ix_i$ and the moment hypothesis on $\sup_x|\phi(A(1)x)-\phi(x)|$ is satisfied by $\max_{ij}A(1)_{ij}$ and $\min_{j}\max_{i}A(1)_{ij}$.
In this case, we also get another condition to avoid degeneracy in the CLT.
To state it, we recall a few definitions and results about $(\max,+)$ matrices:
\begin{defn}\label{defprod}\
For any $k,l,m\in\mathbb N$, the product of two matrices $A\in{\mathbb R}_{\max}^{k\times l}$ and $B\in{\mathbb R}_{\max}^{l\times m}$ is the matrix $A B\in{\mathbb R}_{\max}^{k\times m}$ defined by~:
$$\forall 1\le i\le k, \forall 1\le j\le m, (AB)_{ij}:=\max_{1\le p\le l}A_{ip}+ B_{pj}.$$
\end{defn}
If those matrices have no line of $-\infty$, then the $(\max,+)$ operator defined by $AB$ is the composition of those defined by $A$ and the one defined by $B$.
\begin{defn}
A circuit on a directed graph is a closed path on the graph.
Let $A$ be a square matrix of size $d$ with entries in ${\mathbb R}_{\max}$.
\begin{enumerate}[i)]
\item The graph of $A$ is the directed weighted graph whose nodes are the integers from $1$ to $d$ and whose arcs are the $(i,j)$ such that $A_{ij}>-\infty$. The weight on $(i,j)$ is $A_{ij}$. The graph will be denoted by $\mathcal{G}(A)$ and the set of its elementary circuits by $\mathcal{C}(A)$.
\item The average weight of a circuit $c=(i_1,\cdots,i_n,i_{n+1})$ (where $i_1=i_{n+1}$) is $aw(A,c):=\frac{1}{n}\sum_{j=1}^n A_{i_ji_{j+1}}.$
\item The $(\max,+)$-spectral radius\footnote{this quantity is the maximal $(\max,+)$-eigenvalue of $A$, that is $$\rho_{\max}(A)=\max\{\lambda\in{\mathbb R}_{\max}|\exists V\in{\mathbb R}_{\max}^d\backslash\{(-\infty)^d\}, AV=V+\lambda\textbf{1}\}.$$ See~\cite{theseGaubert}.} of $A$ is $\rho_{\max}(A):=\max_{c\in\mathcal{C}(A)}aw(A,c)$.
\end{enumerate}
\end{defn}
\begin{thm}\label{s>0}
Assume the hypothesis of theorem~\ref{TCL1mp}, with $\gamma=0$. Then the variance $\sigma^2$ in theorem~\ref{TCL1mp} is $0$ if and only if $\left\{\rho_{\max}(B)|B\in T_A\right\}=\{0\}$.
\end{thm}
Theorem 3.2 of~\cite{GM} gives a condition to ensure the memory loss property. This condition also ensures that there are two matrices in $S_A$ with two distinct spectral radius. This proves the following corollary:
\begin{cor}\label{thgenesupp}
Let the law of $A(1)$ be a probability measure on the set of $d\times d$ matrices with finite second moment whose support is not included in the union of finitely many affine hyperplanes of $\mathbb R^{d\times d}$. Then $x(n,.)$ satisfies the conclusions of theorem~\ref{TCL1mp} with $\sigma >0$.
\end{cor}
We also give a sufficient condition to ensure the algebraic non arithmeticity:
\begin{thm}\label{NA}
Assume the hypothesis of theorem~\ref{TLL} $ii)$ except the algebraic non arithmeticity and $A(n)$ are $(\max,+)$ operators. If $\left(A(n)\right)_{n\in\mathbb N}$ is algebraically arithmetic, then there are $a,b\in\mathbb R$ such that $$\{\rho_{\max}(B)|B\in S_A, \mathcal{G}(B) \textrm{strongly connected}\}\subset a+b\mathbb Z.$$
\end{thm}
Together with corollary~\ref{thgenesupp}, this proves that the hypothesis are generic in the following sense:
\begin{cor}\label{thgeneNA}
If the law of $A(1)$ is a probability measure on the set of $d\times d$ matrices with $4+\epsilon$-th moment whose support is not included in the union of enumerably many affine hyperplanes of $\mathbb R^{d\times d}$, then $x(n,.)$ satisfies the conclusions of theorem~\ref{TLL}.
\end{cor}
\subsection{Comments}\label{commentaires}
The following table sums up the limit theorems. In each situation we assume that the sequence $\left(A(n)\right)_{n\in\mathbb N}$ has the memory loss property.
\begin{center}\begin{tabular}{|c|c|c|c|}
\hline Theorems: &\multicolumn{2}{|c|}{Moments of}&Additional \\
&$A(1)\,0$&$\max_{ij}A(1)_{ij}$ and $\min_{j}\max_{i}A(1)_{ij}$& condition\\
\hline CLT & $4+\epsilon$ & $2$ &\\
\hline CLT with rate & $6+\epsilon$ & $3$ & \\
\hline LLT & $4+\epsilon$ & $2$ &NA\\
\hline Renewal& -- &$2$ &NA\\
\hline LDP & -- & exp & $X^0\in L^\infty$\\
\hline \multicolumn{4}{c}{NA= non arithmeticity}\\
\end{tabular}\\
\end{center}
Let us first notice that the results of the second column are optimal in the sense that their restriction to $d=1$ are exactly the usual theorems for sum of i.i.d. real variable (except for LDP).
The results of the second column are stated for $(\max,+)$ operators because the $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ is not bounded for general topical operators. For other subclasses of topical operators, one has to choose $\phi$ such that $\sup_x\left|\phi(A(1)x)-\phi(x)\right|$ is integrable. For instance $\phi(x)=\min_ix_i$ is natural for $(\min,+)$ operators. Actually, it should be possible to derive renewal theorem and large deviation principle with the method of~\cite{HH2} but this has not been written down.
The results of the first column require stronger integrability conditions but they are also better for two reasons: they are true for any topical operators and the algebraic non arithmeticity does not depend on $\phi$. It is expressed without introducing the Markovian operator $Q$ although the system is algebraically non arithmetic iff $Q$ has an eigenvector with eigenvalue with modulus~$1$. Moreover for $(\max,+)$ operators the algebraic non arithmeticity can be deduced from theorem~\ref{NA}. \\
An important case for $(\max,+)$ operators is when $A_{ij}\in\mathbb R^+\cup\{-\infty\}$ and $A_{ii}\ge 0$ because it modelizes situations where $x_i(n,.)$ is the date of the $n$-th event of type $i$ and the $A_{ij}$ are delays. In this case the integrability of $\max_{ij}A(1)_{ij}$ and $\min_{j}\max_{i}A(1)_{ij}$ is equivalent to the integrability of $A(1)\,0$.\\
We mentioned earlier that J. Resing and al.~\cite{RVH} obtained a central limit theorem. In a sense our result is weaker because the MLP property implies that $\overline{x}(n,.)$ is uniformly $\Phi$-recurrent and aperiodic. But our integrability conditions are much weaker and the MLP property is easier to check.
F. Toomey's large deviations principle only requires the uniform bound of the projective image that is a very strong integrability condition. It suggests that the MLP property should not be necessary. But his formulation of the LDP is not equivalent to ours and in the $(\max,+)$ case it needs the fixed structure property, that is $\mathbb P(A_{ij}(1)=-\infty)\in\{0,1\}$.
\section{Proofs of the limit theorems}\label{proofs}
\subsection{From iterated topical functions to Markov chains}
In this section, we show that the hypothesis of the theorems on our model stated in section~\ref{statements} imply the hypothesis of the general theorems of~\cite{HH1} et~\cite{HH2}.
To apply the results of~\cite{HH1},
\begin{itemize}
\item the space $E$ will be $Top_d\times\mathbb{PR}_{\max}^d$ with the Borel $\sigma$-algebra,
\item the transition probability $Q$ will be defined by $$Q\left((A,\bar{x}),D\right)=\mathbb P\left((A(1),\overline{Ax})\in D\right),$$
\item the Markov chain $X_n$ will be $\left(A(n),\bar{x}(n-1,.)\right)$,
\item the function $\xi$ will be defined by $\xi(A,\bar{x})=\phi(Ax)-\phi(x)$, where $\phi$ is a topical function from $\mathbb R^d$ to $\mathbb R$.
\item $\sigma(A)=\sup_x|\xi(A,x)|=\sup_x\left|\phi(Ax)-\phi(x)\right|<\infty$ a.s.\,.
\end{itemize}
With these definitions, $S_n=\sum_{l=1}^n\xi(X_l)$ is equal to $\phi(x(n,X^0))-\phi(X^0)$.
To apply the results of~\cite{HH1}, we still need to define the space $\mathcal B$.
\begin{defn}
Let $\mathcal{L}^\infty$ be the space of complex valued bounded continuous functions on $\mathbb{PR}_{\max}^d$.
Let $j$ be the function from $Top_d\times\mathbb{PR}_{\max}^d$ to $\mathbb{PR}_{\max}^d$ such that $j(A,\bar{x})=\overline{A x}$ and $I$ the function from $\mathbb R^{\mathbb{PR}_{\max}^d}$ to $\mathbb R^{(Top_d\times\mathbb{PR}_{\max}^d)}$ defined by
$$I(\phi)=\phi\circ j.$$
We call $\mathcal B^\infty$ the image of $\mathcal{L}^\infty$ by $I$.
\end{defn}
Since $(\mathcal{L}^\infty,\|.\|_\infty)$ is a Banach space, $I$ is an injection, and $\|\phi\circ I\|_\infty =\|\phi\|_\infty$, $(\mathcal B^\infty,\|.\|_\infty)$ is also a Banach space.
\begin{defn}
The Fourier kernels denoted by $Q_t$ or $Q(t)$ are defined for any $t\in \mathbb C$ by
$$Q_t(x,dy)=e^{it\xi(y)}Q(x,dy).$$
We say that the system is non arithmetic if $Q(.)$ is continuous from $\mathbb R$ to the space $\mathcal{L}_{\mathcal B^\infty}$ of continuous linear operators on ${\mathcal B^\infty}$ and, for any $t\in\mathbb R^*$, the spectral radius of~$Q(t)$ is strictly less than~$1$.
We say that the system is weakly non arithmetic if $Q(.)$ is continuous from $\mathbb R$ to $\mathcal{L}_{\mathcal B^\infty}$ and, for all $t\in\mathbb R^*$, $Id-Q(t)$, where $Id$ is the identity on ${\mathcal B^\infty}$, is invertible.
\end{defn}
\begin{prop}\label{Hm}
If $\sigma\left(A(1)\right)$ has an $m$-th moment and if $A(n)$ has the MLP property, then $(Q,\xi,\mathcal B^\infty)$ satisfies condition $\mathcal{H}(m)$ of~\cite{HH1}.
Moreover, the interval $I_0$ in condition $(H3)$ is the whole $\mathbb R$ and $s(Q,\mathcal B^\infty)=1$.
\end{prop}
To prove $(H2)$, we will use theorem~\ref{strcoupling}.
In the sequel, $\nu_0$ will be the law of $Y$ in that theorem, that is the (unique) invariant probability measure.
\begin{proof}[Proof of proposition~\ref{Hm}]
Condition $(H1)$ is trivial, because of the choice of~$\mathcal B^\infty$.\\
To check condition $(H2)$, we take $\nu:=\mu\otimes \nu_0$. It is $Q$-invariant by definition of $\nu_0$. This proves $(i)$. To prove $(ii)$ and $(iii)$, we investigate the iterates of $Q$. For any $\phi\in\mathcal{L}^\infty$, and $x\in\mathbb R^d$ we have:
\begin{eqnarray*}
\left| Q^n(\phi\circ j)(A,\bar{x})-\nu(\phi\circ j)\right|
&=& \left|Q^n(\phi)(\overline{A x})-\nu_0(\phi)\right|\\
&=&\left|\int \phi\left(\bar{x}(n,\overline{A x})\right) d\mathbb P-\int \phi(Y_n)d\mathbb P\right|\\
&\le &\|\phi\|_\infty 2\mathbb P\left(\exists x_0, Y_n\neq \bar{x}(n,x_0)\right),
\end{eqnarray*}
If we denote $\psi\mapsto \nu(\psi)$ by $N$, we obtain:
\begin{equation}
\|Q^n-N\|\le 2\mathbb P\left(\exists x, Y_n\neq \bar{x}(n,x)\right)\rightarrow 0.
\end{equation}
This proves that the spectral radius $r(Q_{|Ker N})$ is strictly less than~$1$ and that $\sup_n\|Q^n\|<\infty$. Since $Q_{|Im N}$ is the identity, $dim (Im N)=1$ and $\mathcal B^\infty=Ker N\oplus Im N$, $Q$ is quasi-compact, so $(ii)$ is checked. Moreover $s(Q,\mathcal B^\infty)=1$.
It also proves $Ker(1-Q)\subset Im N$, which implies $(iii)$.\\
To prove $(H3)$ we set $Q_t^{(k)}:=e^{it\xi(y)}(i\xi(y))^kQ(x,dy)$ and
\begin{equation}
\Delta_h^{(k)}:= Q^{(k)}_{t+h}-Q^{(k)}_t-hQ^{(k+1)}_t.
\end{equation}
To prove that $Q^{k+1}$ is the derivative of $Q^{k}$, it remains to bound $\|\frac{1}{h}\Delta_h^{(k)}\|$ by a quantity that tends to zero with $h$.
To this aim, we introduce the following function:
$$\left\{\begin{array}{lccl}f:&\mathbb R &\rightarrow &\mathbb C\\&t&\mapsto &e^{it}-1-it.\end{array}\right.$$
The calculus will be based on the following estimations on $f$: $|f(t)|\le 2t$, and $|f(t)|\le t^2$.
Now everything follows from
\begin{equation}\label{introf}
\Delta_h^{(k)}(\phi\circ j)(A,\bar{x})
= \int \phi\left(\overline{BAx}\right)e^{it\xi(B,\overline{Ax})}\left(i\xi(B,\overline{A x})\right)^k f\left(h\xi(B,\overline{Ax})\right) d\mu(B) .
\end{equation}
First it implies that
\begin{equation}\label{norminfinie}
\left\|\Delta_h^{(k)}(\phi\circ j)\right\|_\infty
\le \|\phi\|_\infty \int \sigma^k(B)\left\|f\left(h\xi(B,.)\right) \right\|_\infty d\mu(B).
\end{equation}
Since $|f(t)|\le t^2$,
$$\frac{1}{|h|}\sigma(B)^{k} \left\| f\left(h\xi(B,.)\right) \right\|_\infty \le h \sigma^{k+2}(B) \rightarrow 0 .$$
Since $|f(t)|\le 2t$,
$$\frac{1}{|h|}\sigma^{k}(B) \left\| f\left(h\xi(B,.)\right) \right\|_\infty \le 2 \sigma^{k+1}(B).$$
When $k<m$, $\sigma^{k+1}$ is integrable so the dominate convergence theorem and the last two equations show that
\begin{equation}\label{term1}
\int\sigma(B)^{k} \left\| f\left(h\xi(B,.)\right) \right\|_\infty d\mu(B)=o(h).
\end{equation}
Finally for any $k<m$, $\|\frac{1}{h}\Delta_h^{(k)}\|$ tends to zero, so $Q_t^{(k+1)}$ is the derivative of $Q_.^{(k)}$ in $t$.
To prove that $Q_.^{(m)}$ is continuous, we notice that
\begin{equation}
\left(Q_{t+h}^{(m)}-Q_t^{(m)}\right)(\phi\circ j)(A,\overline{x})= \int \phi\left(\overline{BAx}\right)e^{it\xi(B,\overline{Ax})}\left(i\xi(B,\overline{A x})\right)^m g\left(h\xi(B,\overline{Ax})\right) d\mu(B) .
\end{equation}
where $g(t)=e^{it}-1$. Then we apply the same method as before, replacing the estimates on $f$ by $|g(t)|\le t$ to prove the convergence, and by $|g(t)|\le 2$ to prove the domination.
This proves $(H3)$ and the additional assumption of proposition~\ref{Hm}.
\end{proof}
In their article~\cite{HH2} H. Hennion and L. Herv\'e have proved limit theorems for sequences $\xi(Y_n,Z_{n-1})$, where $(Y_n)_{n\in\mathbb N}$ is an i.i.d. sequence of Lipschitz operators on a metric space $\mathcal M$, and $Z_n$ is defined by $Z_{n+1}=Y_{n+1}Z_n$. As explained in section~\ref{principes}, we take $\mathcal M=\mathbb{PR}_{\max}^d$, $Y_n=A(n)$ and again $\xi(A,\overline{x})=\phi(Ax)-\phi(x)$. In this case $Z_n=\overline{x}(n,X^0)$ and $S_n=\phi\left(x(n,X^0)\right)-\phi(X^0)$. Moreover in our situation, the $Y_n$, which are the projective function defined by $A(n)$, are 1-Lipschitz. Following the same proof as~\cite{HH2} with this additional condition, we get the CLT (resp. CLT with rate, LLT) for $S_n$ under the hypothesis of theorem~\ref{TCL} (resp.~\ref{TCLV},\ref{TLL}) on $A(1)0$.
The integrability conditions are weaker than in~\cite{HH2}, because the Lipschitz coefficient is uniformly bounded. The only difference in the proof is the H\"older inequality of the 4th part of proposition~7.3 of~\cite{HH2}: the exponents in the inequality should be changed to $1$ and $\infty$.\\
Let us give the notations of~\cite{HH2} we need to state the results. $G$ is the semi-group of the operators on $\mathcal M$, 1-Lipschitz for distance $\delta$. For a fixed $x_0\in\mathcal M$, every $\eta\ge 1$ and every $n \in \mathbb N$, we set $\mathcal M_\eta=\mathbb E[\delta^\eta(Y_1x_0,x_0)]$ and $\mathcal C_n=\mathbb E[c(Y_n\cdots Y_1)]$, where $c(.)$ is the Lipschitz coefficient.
When there is an $N\in\mathbb N$ such that $\mathcal C_N<1$, there is a $\lambda_0 \in]0,1[$, such that \mbox{$\int_Gc(g)\left(1+\lambda_0 \delta(gx_0,x_0)\right)^{2\eta}d\mu^{*N}(g)<1$.} We chose one such $\lambda_0$ and set the following notations:
\begin{enumerate}[(i)]
\item $\mathcal B_\eta$ is the set of functions $f$ from $\mathcal M$ to $\mathbb C$ such that $m_\eta(f)<\infty$, with norm $\| f\|_\eta=|f|_\eta+m_\eta(f)$, where
\begin{eqnarray*}
|f|_\eta&=&\sup_x\frac{|f(x)|}{(1+\lambda_0 \delta(x,x_0))^{1+\eta}}, \\
m_\eta(f)&=&\sup_{x\neq y}\frac{|f(x)-f(y)|}{\delta(x,y)\left(1+\lambda_0\delta(x,x_0)\right)^\eta\left(1+\lambda_0 \delta(y,x_0)\right)^\eta}.
\end{eqnarray*}
\item We say that the system is $\eta$-non arithmetic if there is no $t\in\mathbb R\backslash\{0\}$, no $\rho\in\mathbb C$, and no $w\in\mathcal B_\eta$ with non-zero constant modulus on the support $S_{\nu_0}$ of the invariant probability measure $\nu_0$ such that $|\rho|=1$ and for all $n\in\mathbb N$, we have
\begin{equation}\label{eqNA}
e^{itS_n}w(Z_n)=\rho^nw(Z_0) \mathbb P-\textrm{ a.s.,}
\end{equation}
when $Z_0$ has law $\nu_0$.
\end{enumerate}
\begin{rem}[non arithmeticity]
In the first frame the non arithmeticity condition is about the spectral radius of $Q_t$. Here we work with the associated $P_t$ that acts on $\mathcal M$ instead of $G\times\mathcal M$ (cf.~\cite{HH2}). If $P_t$ is quasi-compact, then the spectral radius $r(P_t)$ is~$1$ iff $P_t$ has an eigenvalue $\rho$ with modulus~$1$.
It is shown in proposition~9.1'~of~\cite{HH2} that if $r(P_t)=1$, then $P_t$ is quasi-compact as an operator on $\mathcal B_\eta$ and that an eigenvector $w$ with eigenvalue $\rho$ satisfies equation~(\ref{eqNA}).
\end{rem}
\begin{prop}\label{HS}\
\begin{enumerate}
\item If $A(1)\,0$ has an $\eta$-th moment, with $\eta\in\mathbb R^+$, then $\mathcal M_\eta<\infty$. If the sequence has the MLP property, then there is $n_0\in\mathbb N$ such that $\mathcal{C}_{n_0}<1$. If $\overline{X^0}$ has an $\eta$-th moment $\eta\in\mathbb R^+$, then $f\mapsto\mathbb E[f(\overline{X^0})]$ is continuous on $B_\eta$.
\item Algebraic non arithmeticity implies $\eta$-non arithmeticity for any $\eta>0$.
\end{enumerate}
\end{prop}
The first part of the proposition is obvious. The second part relies on the next two lemma that will be proved after the proposition:
\begin{lem}\label{suppnu}
The support of the invariant measure $\nu_0$ is $$S_{\nu_0}:=\overline{\{\overline{\theta\textbf{1}}|\theta\in T_A, \theta\textrm{ with rank~1}\}}.$$
\end{lem}
\begin{lem}\label{invtheta}
If equation~(\ref{eqANA}) is satisfied by some $\theta$ with rank~1, any $A\in S_A$ and any $\theta'\in T_A$ with rank~1, it is satisfied by any $\theta \in T_A$ with rank~1.
\end{lem}
\begin{proof}[Proof of proposition~\ref{HS}]
Let us assume that the system is $\eta$-arithmetic. Then there are $w\in\mathcal B_\eta$ and $t,a\in\mathbb R$ such that for $\mu$-almost every $A$ and $\nu_0$ almost every $\overline{x}$, we have:
\begin{equation}\label{eqNA1}
e^{it\left(\phi(Ax)-\phi(x)\right)}w(\overline{Ax})=e^{ita}w(\overline{x}).
\end{equation}
Since all functions in this equation are continuous, it is true for $\overline{x}\in S_{\nu_0}$ and $A\in T_A$. Since $S_{\nu_0}$ is $T_A$ invariant, we iterate equation~(\ref{eqNA1}) and get
\begin{equation}\label{eqNAn}
e^{it\left(\phi(Tx)-\phi(x)\right)}w(\overline{Tx})=e^{itan_T}w(\overline{x}),
\end{equation}
where $T\in T_A$ and $n_T$ is the number of operators of $S_A$ to be composed to obtain $T$.
Because of the MLP property, there is a $\theta\in T_A$ with rank~1. For any $A\in S_A$, $\theta A\in T_A$, so we apply equation~(\ref{eqNAn}) for $T=\theta A$ and $T=\theta$ and divide the first equation by the second one. Since $n_{\theta A}=n_{\theta}+1$ and $\overline{\theta Ax}=\overline{\theta x}$ , we get
$$e^{it\left(\phi(\theta Ax)-\phi(\theta x)\right)}=e^{ita}.$$
Setting $b=\frac{2\pi}{t}$, it means that $\phi(\theta Ax)-\phi(\theta x)\in a+b\mathbb Z$. Since $\theta$ has rank one, $(\theta Ax-\theta x)\in\mathbb R\textbf{1}$, so $\theta Ax-\theta x \in (a+b\mathbb Z)\textbf{1}$, and the algebraic arithmeticity follows by lemma~\ref{suppnu}.
\end{proof}
\begin{proof}[Proof of lemma~\ref{suppnu}]
By theorem $\ref{strcoupling}$, there is sequence of random variables $Y_n$ with law $\nu_0$, such that $Y_n=A(n)\cdots A(1)Y$. Let $K$ be a compact subset of ${\mathbb R}_{\max}$ such that $Y\in K$ with positive probability.
For any $\theta\in T_A$ and any $\epsilon>0$, the set $V$ of topical functions $A$ such that $\delta(\overline{Ax},\overline{\theta x})\le\epsilon$ for all $\overline{x}\in K$ is a neighborhood of $\theta$. Therefore the probability for $A(n_\theta)\cdots A(1)$ to be in $V$ is positive and by independence of $Y$, we have:
$$\mathbb P\left[Y\in K ,A(n_\theta)\cdots A(1)\in V\right]>0.$$
Since $\overline{\theta\textbf{1}}=\overline{\theta Y}$, this means that with positive probability, $$\delta\left(Y_{n_\theta},\overline{\theta \textbf{1}}\right)=\delta\left(A(n_\theta)\cdots A(1)Y,\overline{\theta Y}\right)\le\epsilon,$$ so $\overline{\theta \textbf{1}}\in S_ {\nu_0}.$
This proves that $\overline{\{\overline{\theta\textbf{1}}|\theta\in T_A, \theta\textrm{ with rank~1}\}}\subset S_ {\nu_0} $.\\
In~\cite{Mairesse}, $\nu_0$ is obtained as the law of $Z=\lim_n\overline{A(1)\cdots A(n)\textbf{1}}$. Indeed, the MLP property and the Poincar\'e recurrence theorem ensure that there are almost surely $M$ and $N$ such that $A(N)\cdots A(N+M)$ has rank~1. Therefore, for $n\ge N+M$, $\overline{A(1)\cdots A(n)\textbf{1}}=\overline{A(1)\cdots A(N+M)\textbf{1}}=Z$ . But $A(1)\cdots A(N+M)\in T_A$ almost surely, so \mbox{$Z\in\{\overline{\theta\textbf{1}}|\theta\in T_A, \theta\textrm{ with rank~1}\}$} almost surely and $S_ {\nu_0}\subset \overline{\{\overline{\theta\textbf{1}}|\theta\in T_A, \theta\textrm{ with rank~1}\}}$.
\end{proof}
\begin{proof}[Proof of lemma~\ref{invtheta}]
We assume that equation (\ref{eqANA}) is satisfied by $\theta=\theta_1$, any $A\in S_A$ and any $\theta'\in T_A$ with rank~1.
Let $A_1,\cdots, A_n \in S_A$, such that $\theta_2=A_1\cdots A_n$ has rank~1. For any $i\le n$, $A_i\cdots A_n\theta'$ has rank~1, so $(\theta_1A_{i}\cdots A_n\theta'-\theta_1A_{i+1}\cdots A_n\theta')(\mathbb R^d)\subset (a+b\mathbb Z)\textbf{1}$.
Summing these inclusions for $i=1$ to $i=n$, we get $(\theta_1\theta_2\theta'-\theta_1\theta')(\mathbb R^d)\subset (na+b\mathbb Z)\textbf{1}$ and
\begin{equation}\label{eq1}
\left((\theta_1\theta_2A\theta'-\theta_1A\theta')-(\theta_1\theta_2\theta'-\theta_1\theta')\right)(\mathbb R^d)\subset b\mathbb Z\textbf{1}.
\end{equation}
Now we write $\theta_2\theta'$ as
$$\theta_2\theta'=\theta_1\theta' +(\theta_1\theta_2\theta'-\theta_1\theta')- (\theta_1\theta_2\theta'-\theta_2\theta').$$
The last part does not depend on $\theta'$, so replacing $\theta'$ by $A\theta'$ and subtracting the first version, we get:
$$\theta_2A\theta'-\theta_2\theta'=\theta_1A\theta'-\theta_1\theta' +\left((\theta_1\theta_2A\theta'-\theta_1A\theta')-(\theta_1\theta_2\theta'-\theta_1\theta')\right).$$
With equation~(\ref{eq1}), this proves equation~(\ref{eqANA}) for $\theta=\theta_2$.
\end{proof}
\subsection{From Markov chains to iterated topical functions}
Propositions~\ref{Hm} and~\ref{HS} prove that under the hypothesis of section~\ref{statements} the conclusions of the theorems of~\cite{HH1} and~\cite{HH2} are true. This gives results about the convergence of $\left(\phi\left(x(n,X^0)\right)-\phi\left(X^0\right)-n\nu(\xi),\overline{x}(n,X^0)\right)$.
When $\overline{X^0}$ has law $\nu_0$, the sequence $\left(A(n),\overline{x}(n,X^0)\right)_{n\in\mathbb N}$ is stationary, so it follows from Birkhoff theorem that $\gamma=\int\xi(A,\overline{x}) d\nu_0(\overline{x})d\mu(A)=\nu(\xi)$.
The following lemma will be useful to go back to $x(n,.)$.
\begin{lem}\label{bilip}
If $\phi$ is a topical function from $\mathbb R^d$ to $\mathbb R$, the function $\psi:x\mapsto (\phi(x),\overline{x})$ is a Lipschitz homeomorphism with Lipschitz inverse from $\mathbb R^d$ onto $\mathbb R\times\mathbb{PR}_{\max}^d$.
\end{lem}
\begin{proof}
Let $(t,\overline{x})$ be an element of $\mathbb R\times\mathbb{PR}_{\max}^d$. Then $\psi(y)=(t,\overline{x})$ if and only if there is an $a\in\mathbb R$ such that $y=x+a\textbf{1}$ and $\phi(x)+a=t$. So the equation has exactly one solution $y=x+(t-\phi(x))\textbf{1}$ and $\psi$ is invertible.
It is well known that topical functions are Lipschitz, and the projection is linear, so it is Lipschitz and so is $\psi$.
For any $x,y\in\mathbb R^d$, we have $x\le y+\max_i(x_i-y_i)\textbf{1}$, so $\phi(x)-\phi(y)\le \max_i(x_i-y_i)$. Therefore, for any $1\le i\le d$, we have
$$\phi(x)-\phi(y)-(x_i-y_i)\le \max_i(x_i-y_i)-\min_i(x_i-y_i)=\delta(\overline{x},\overline{y}).$$
Permuting $x$ an $y$, we see that:
\begin{equation}\label{difftop}
|\phi(x)-\phi(y)-(x_i-y_i)|\le \delta(\overline{x},\overline{y}).
\end{equation}
Therefore $|x_i-y_i|\le |\phi(x)-\phi(y)|+\delta(\overline{x},\overline{y})$ and $\psi^{-1}$ is Lipschitz.
\end{proof}
\begin{proof}[Proof of theorem~\ref{TCL}]
Without lost of generality, we assume that $\gamma=0$.
Theorem~A of~\cite{HH1} and proposition~\ref{Hm} or theorem~A of~\cite{HH2} and proposition~\ref{HS} prove that
$\frac{\phi(x(n,X^0))-\phi(X^0)}{\sqrt{n}}$ converges to $\mathcal{N}(0,\sigma^2)$, which means that
$\frac{\phi(x(n,X^0))-\phi(X^0)}{\sqrt{n}}~\textbf{1}$ converges to the limit specified in theorem~\ref{TCL}\,. We just estimate the difference between the converging sequence and the one we want to converge:
\begin{equation}\label{majdiff}
\Delta_n:=\left\|\frac{x(n,X^0)}{\sqrt{n}}-\frac{\phi\left(x(n,X^0)\right)-\phi(X^0)}{\sqrt{n}}~\textbf{1}\right\|_\infty\le \frac{\left|\phi(X^0)\right|}{\sqrt{n}}+\frac{|\overline{x}(n,X^0)|_\mathcal{P}}{\sqrt{n}}.
\end{equation}
Each term of the last sum is a weakly converging sequence divided by $\sqrt{n}$ so it converges to zero in probability. This proves that $\Delta_n$ converges to zero in probability, which ensures the convergence of $\frac{x(n,X^0)}{\sqrt{n}}$ to the Gaussian law.\\
The expression of $\sigma^2$ is the direct consequence of theorems~A of~\cite{HH1} or theorem~S of~\cite{HH2}.\\
If $\sigma=0$, then again by theorem~A of~\cite{HH1}~or~S of~\cite{HH2}, there is a continuous function $\xi$ on $\mathbb{PR}_{\max}^d$ such that
\begin{equation}\label{eq2}
\phi(Ax)-\phi(x)=\xi(\overline{x})-\xi(\overline{Ax})
\end{equation} for $\mu$-almost every $A$ and $\nu_0$-almost every $\overline{x}$.
Since all functions are continuous in this equation, (\ref{eq2}) is true for every $A\in S_A$ and $\overline{x}\in S_{\nu_0}$.
By induction we get it for $A\in T_A$ and if $\theta\in T_A$ has rank~1 and $\overline{x}\in S_{\nu_0}$, $\overline{\theta Ax}=\overline{\theta x}$, so $\phi(\theta Ax)=\phi(\theta x)$.
Since $\theta Ax-\theta x\in\mathbb R\textbf{1}$, this means that $\theta Ax=\theta x.$
By lemma~\ref{suppnu}, it proves that $\theta A\theta'=\theta \theta'$ for any $\theta,\theta'\in T_A$ with rank~1 and $A\in S_A$.\\
Conversely, let us assume there is $\theta$ with rank one such that for any $\theta'\in T_A$ with rank~1 and $A\in S_A$, we have:
\begin{equation}\label{eq3}
\theta A\theta'=\theta \theta'.
\end{equation}
By lemma~\ref{invtheta} applied with $a=b=0$, it is true for any $\theta,\theta'\in T_A$ with rank~1, and any $A\in S_A$ and by induction, equation~(\ref{eq3}) is still true for $A\in T_A$.
Therefore, for any $m\in\mathbb N$ and $n\ge m+1$ and any $\theta'\in T_A$ with rank~1, if $A(n)\cdots A(n-m+1)$ has rank~1 ,then $x(n,\theta'\textbf{1})=A(n)\cdots A(n-m+1)\theta'\textbf{1}$ and for any $N\in\mathbb N$
\begin{eqnarray}\label{eqxborne}
\lefteqn{\mathbb P\left( \|x(n,\theta'\textbf{1})\|_\infty\le N\right)}\nonumber\\
&\ge & \mathbb P\left( A(n)\cdots A(n-m+1)\textrm{has rank~1}, \|A(n)\cdots A(n-m+1)\theta'\textbf{1} \|_\infty\le N\right)\nonumber\\
&\ge & \mathbb P\left( A(m)\cdots A(1)\textrm{has rank~1}, \|A(m)\cdots A(1)\theta'\textbf{1} \|_\infty\le N\right).
\end{eqnarray}
We fix a $\theta'\in T_A$ with rank one. The MLP property says there is an $m$ such that $\mathbb P(A(m)\cdots A(1) \textrm{has rank~1})>0$. Therefore, there is an $N\in\mathbb N$ such that the right member of~(\ref{eqxborne}) is a positive number we denote by $\beta$.
Equation~(\ref{eqxborne}) now implies that for any $\epsilon>0$, if $n\ge \max(m,N^2\epsilon^{-2})$, then $\mathbb P( \|\frac{1}{\sqrt{n}}x(n,\theta'\textbf{1})\|_\infty\le \epsilon )\ge \beta$, so $\mathcal{N}(0,\sigma^2)[-\epsilon,\epsilon]\ge\beta$. When $\epsilon$ tends to zero, we get that $\mathcal{N}(0,\sigma^2)(\{0\})\ge\beta>0$, which is true only if $\sigma=0$.
\end{proof}
\begin{proof}[Proof of theorem~\ref{TCLV}]
Without loss of generality, we assume that $\gamma=0$.
Equation~(\ref{vitTCL}) follows from theorem~B of~\cite{HH1} and proposition~\ref{Hm} or from theorem~B of~\cite{HH2} and proposition~\ref{HS}
The only fact to check is that the initial condition defines a continuous linear form on $\mathcal B_\eta$, with norm at most $C\left(1+\mathbb E(\|X^0\|^l_\infty\right)$, that is for any $f\in\mathcal B_\eta$, we have:
$$|\mathbb E(f(X^0))|\le C\left(1+\mathbb E(\|X^0\|^l_\infty)\right)\|f\|_\eta.$$
It easily follows from the fact that $|f(x)|\le \|f\|_\eta (1+|x|_\mathcal{P})^{1+\eta}$ and $1+\eta \le l $.
Taking $y=0$ in~(\ref{difftop}), we get $|\phi(x)-x_i|\le|x|_\mathcal{P}$. Together with~(\ref{majdiff}) it proves that for any $u\in\mathbb R^d$ and any $\epsilon>0$
\begin{eqnarray}\label{majvit}
\lefteqn{ \mathbb P[x(n,X^0)\le\sigma u\sqrt{n}]}\nonumber\\
&\le & \mathbb P\left[\min_i x_i(n,X^0)\le\sigma \min_iu_i\sqrt{n}\right]\nonumber\\
&\le &\mathbb P\left[\phi(x(n,X^0))\le(\sigma\min_iu_i+2\epsilon)\sqrt{n}\right]
+ \mathbb P\left[ \frac{\left|\phi(X^0)\right|}{\sqrt{n}}\ge\epsilon\right]
+ \mathbb P\left[\frac{|\overline{x}(n,X^0)|_\mathcal{P}}{\sqrt{n}}\ge\epsilon\right]\nonumber\\
&\le&\mathcal{N}(0,1)(]-\infty,\min_iu_i+\frac{2\epsilon}{\sigma}])+\frac{C}{\sqrt{n}}
+\frac{\mathbb E\left(\left|\phi(X^0)\right|^l\right)}{(\epsilon\sqrt{n})^{l}}
+\frac{\mathbb E\left(|\overline{x}(n,X^0)|^l_\mathcal{P}\right)}{(\epsilon\sqrt{n})^{l}}\nonumber\\
&\le&\mathcal{N}(0,1)(]-\infty,\min_iu_i])+\frac{C}{\sqrt{n}}+\frac{2\epsilon}{\sigma}
+\frac{\mathbb E\left(\left|\phi(X^0)\right|^l\right)}{(\epsilon\sqrt{n})^{l}}
+\frac{\mathbb E\left(|\overline{x}(n,X^0)|^l_\mathcal{P}\right)}{(\epsilon\sqrt{n})^{l}}.
\end{eqnarray}
Conversely,
\begin{eqnarray}\label{minvit}
\lefteqn{ \mathbb P[x(n,X^0)\le\sigma u\sqrt{n}]}\nonumber\\
&\ge & \mathbb P\left[\phi(x(n,X^0))\le\sigma \min_iu_i\sqrt{n}\right]\nonumber\\
&\ge &\mathbb P\left[\phi(x(n,X^0))\le(\sigma\min_iu_i-2\epsilon)\sqrt{n}\right]
- \mathbb P\left[ \frac{\left|\phi(X^0) \right|}{\sqrt{n}}\ge\epsilon\right]
- \mathbb P\left[\frac{|\overline{x}(n,X^0)|_\mathcal{P}}{\sqrt{n}}\ge\epsilon\right]\nonumber\\
&\ge &\mathcal{N}(0,1)(]-\infty,\min_iu_i-\frac{2\epsilon}{\sigma}])-\frac{C}{\sqrt{n}}
-\frac{\mathbb E\left(\left|\phi(X^0) \right|^l\right)}{(\epsilon\sqrt{n})^{l}}
- \frac{\mathbb E\left(|\overline{x}(n,X^0)|^l_\mathcal{P}\right)}{(\epsilon\sqrt{n})^{l}}\nonumber\\
&\ge &\mathcal{N}(0,1)(]-\infty,\min_iu_i])-\frac{C}{\sqrt{n}}-\frac{2\epsilon}{\sigma}
-\frac{\mathbb E\left(\left| \phi(X^0) \right|^l\right)}{(\epsilon\sqrt{n})^{l}}
- \frac{\mathbb E\left(|\overline{x}(n,X^0)|^l_\mathcal{P}\right)}{(\epsilon\sqrt{n})^{l}}\nonumber\\.
\end{eqnarray}
Taking $\epsilon=n^{-{\frac{l}{2(l+1)}}}$ in (\ref{majvit}) and (\ref{minvit}) will conclude the proof of theorem~\ref{TCLV} if we can show that $\mathbb E\left(|\overline{x}(n,X^0)|^l_\mathcal{P}\right)$ is bounded uniformly in $n$ and $X^0$. Without loss of generality, we assume $X^0=0$.
For $n_0\in\mathbb N$, we take $a\ge\left(\mathbb P\left[A(n_0)\cdots A(1)\textrm{ has not rank~1 }\right]\right)^{1/n_0}$. But if $A(n)\cdots A(m)$ has not rank~1, then for any integer less than $\frac{n-m-n_0}{n_0}$, the operator $A(1+in_0)\cdots A((i+1)n_0)$ has not rank~1 either. From the independence of the $A(n)$, we deduce
$$\mathbb P\left(A(n)\cdots A(m+1)\textrm{ has not rank~1 }\right)\le a^{n-m-n_0}.$$
We estimate $\delta\left(A(n)\cdots A(m+1)0,A(n)\cdots A(n_0+1+m)0\right)$: it is $0$ when $A(n+m)\cdots A(n_0+1+m)$ has rank~1, and it is always less than $\delta\left(A(n_0+m)\cdots A(m+1)0,0\right)$, that is less than $\textrm{1\hspace{-3pt}I}_{\{A(n)\cdots A(n_0+m+1)\textrm{ has not rank~1}\}} \left|A(n_0+m)\cdots A(m+1)0\right|_\mathcal{P}$, where $\textrm{1\hspace{-3pt}I}$ denotes the indicator function. Therefore, we have for any $n\ge m+n_0$
\begin{eqnarray}
\lefteqn{\mathbb E\left[\delta^l\left(A(n)\cdots A(m)0,A(n)\cdots A(n_0+1+m)0\right)\right]}\nonumber\\
&\le& \mathbb E\left[\textrm{1\hspace{-3pt}I}_{\{A(n)\cdots A(n_0+m+1)\textrm{ has not rank~1 }\}} \left|A(n_0+m)\cdots A(m+1)0\right|^l_\mathcal{P}\right]\nonumber\\
&=&a^{n-m-2n_0}\mathbb E\left[\left|A(n_0)\cdots A(1)0\right|^l_\mathcal{P}\right].
\end{eqnarray}
Let $n=qn_0+r$ be the Euclidean division of $n$ by $n_0$. Then we have
\begin{eqnarray*}
\left|x(n,0)\right|_\mathcal{P}&=&\delta\left(A(n)\cdots A(1)0,0\right)\\
&\le&\sum_{i=1}^{q}\delta\left(A(n)\cdots A(in_0+1)0,A(n)\cdots A((i-1)n_0+1)0\right)\\&&+\delta\left(A(n)\cdots A(n-r+1)0,0\right).\\
\end{eqnarray*}
Therefore we have:
\begin{eqnarray}
\left(\mathbb E\left[\left|x(n,0)\right|^l_\mathcal{P}\right]\right)^{1/l}
&\le&\sum_{i=1}^{q}\left(a^{n-in_0-2n_0}\mathbb E\left[\left|A(n_0)\cdots A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l}\nonumber\\
&&+\left(\mathbb E\left[\left|A(r)\cdots A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l}.\label{eq10}
\end{eqnarray}
We apply this decomposition again (with $n=r$, $n_0=1$ and $a=1$), to check that
$$ \left(\mathbb E\left[\left|A(r)\cdots A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l}\le r \left(\mathbb E\left[\left|A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l}\le n_0\left(\mathbb E\left[\left|A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l} .$$
It follows from the MLP property, that there is $n_0\in \mathbb N$ such that $a<1$. Introducing the last equation in equation~(\ref{eq10}), we see that $$\left(\mathbb E\left[\left|x(n,0)\right|^l_\mathcal{P}\right]\right)^{1/l}\le\left(1+\frac{a^{-2n_0l}}{1-a^{n_0l}}\right)n_0\left(\mathbb E\left[\left|A(1)0\right|^l_\mathcal{P}\right]\right)^{1/l}.$$
\end{proof}
To go from the abstract LLT and renewal theorem to ours, we will use the following classical approximation lemma.
\begin{lem}\label{lemmsuppcomp}
Let $h$ be a continuous function with compact support from \mbox{$\mathbb R^a\times\mathbb R^b$.} Then there are two continuous functions $f_0$ and $g_0$ with compact support in $\mathbb R^a$ and $\mathbb R^b$ respectively, so that for any $\epsilon>0$, there are $f_i$ and $g_i$ continuous functions with compact support satisfying:
$$\forall x\in\mathbb R^a,y\in\mathbb R^b, |h(x,y)-\sum_if_i(x)g_i(y)|\le \epsilon f_0(x)g_0(y).$$
\end{lem}
In the sequel, we denote by $\mathcal{L}$ the Lebesgue measure.
\begin{proof}[Proof theorem~\ref{TLL}]
By theorem~\ref{TCL}, the algebraic non arithmeticity ensures that $\sigma>0$.
We apply proposition~\ref{Hm} and theorem~C of~\cite{HH1} or proposition~\ref{HS} and theorem~C of~\cite{HH2}. This proves that, if $g\in\mathcal{C}_c(\mathbb R)$ and if $f$ is a bounded Lipschitz function on $\mathbb{PR}_{\max}^d$, then for any $x^0\in\mathbb R^d$,
\begin{equation}
\lim_n\sup_{u\in\mathbb R}\left|\sigma\sqrt{2\pi n}\mathbb E\left(f\left(\bar{x}(n,x^0)\right)g\left(\phi\left(x(n,x^0)\right)-\phi\left(x^0\right)-u\right)\right)-e^{-\frac{u^2}{2n\sigma^2}}\nu_0(f)\mathcal{L}(g)\right|=0.
\end{equation}
Moreover these convergences are uniform in $x^0$, because $\delta_{\overline{x^0}}$ is bounded independently of $x_0$ as a linear form on $\mathcal B^\infty$ and is in a disk of $\mathcal B_{\eta}$ with center $0$ and radius $\|X^0\|_\infty$ if $|x^0|_\mathcal{P}\le\|X^0\|_\infty$.
The uniformity allows us to take any random initial condition $X^0$ and get
\begin{equation}\label{eqTLL}
\lim_n \sup_{u\in\mathbb R}\left|\sigma\sqrt{2\pi n}\mathbb E\left[f\left(\bar{x}(n,X^0)\right)g\left(\phi\left(x(n,X^0)\right)-u\right)\right]-\mathbb E\left[e^{-\frac{(u+\phi(X^0))^2}{2n\sigma^2}}\right]\nu_0(f)\mathcal{L}(g)\right|=0.
\end{equation}
But the density of bounded Lipschitz functions in $(\mathcal{C}_c(\mathbb{PR}_{\max}^d),\|.\|_\infty)$ allows us to take $f$ and $g$ continuous functions with compact support in equation~(\ref{eqTLL}). Now, it follows from lemma~\ref{lemmsuppcomp}, that for any $h$ continuous with compact support
\begin{equation}
\lim_n \sup_{u\in\mathbb R}\left|\sigma\sqrt{2\pi n}\mathbb E\left[h\left(\bar{x}(n,X^0),\phi\left(x(n,X^0)\right)-u\right)\right]-\mathbb E\left[e^{-\frac{(u+\phi(X^0))^2}{2n\sigma^2}}\right]\nu_0\otimes \mathcal{L}(h)\right|=0.
\end{equation}
According to lemma~\ref{bilip} the function $\Phi:x\mapsto (\phi,\bar{x})$ is Lipschitz with a Lipschitz inverse, therefore $h$ has compact support iff $h\circ\Phi$ does. Since $\overline{x+u\textbf{1}}=\overline{x}$, this concludes the proof.
\end{proof}
\begin{proof}[Proof of proposition~\ref{optNA}]
Assume the sequence of random variables is algebraically arithmetic and the conclusion of theorem~\ref{TLL} holds.
There are $a,b\in\mathbb R$ and $\theta$ with rank~1, such that every $A\in S_A$ and $\theta'\in T_A$ with rank~1 satisfy equation~(\ref{eqANA}). We set $t=\frac{2\pi}{b}$ if $b\neq 0$ and $t=1$ otherwise, and for any $x\in\mathbb R^d$, $w(\overline{x})=e^{it\left(\phi(\theta x)-\phi(x)\right)}$.
Equation~(\ref{eqANA}) implies that, for any $A\in S_A$, $y\in\mathbb R^d$, and any $\theta'\in T_A$ with rank~1:
$$e^{it\left(\phi(A \theta'y)-\phi(\theta'y)\right)}w(\overline{A\theta'y})=e^{ita}w(\overline{\theta'y}).$$
We chose $y$ such that $\phi(\theta'y)=0$.
By induction, we get
\begin{equation}\label{eqw}
e^{it\phi\left(x(n,\theta'y)\right)}w(\overline{x}(n,\theta'y))=e^{itna}w(\overline{\theta'y}).
\end{equation}
For any $f:\mathbb R\mapsto \mathbb R$ and $g:\mathbb{PR}_{\max}^d\mapsto \mathbb R$ continuous with compact supports, the conclusion of theorem~\ref{TLL} for $h$ defined by $h(x)=f(\phi(x))(gw)(\overline{x})$ is that:
$$
\sigma\sqrt{2\pi n}\mathbb E\left[f(\phi(x(n,\theta'y)))g(\overline{x}(n,\theta'y))w(\overline{x}(n,\theta'y))\right]\rightarrow \mathcal{L}(f)\nu_0(gw). $$
Together with equation~(\ref{eqw}), it means
\begin{equation}\label{eq6}
e^{itna}w(\overline{\theta'y})\sigma\sqrt{2\pi n}\mathbb E\left[e^{-it.}f(\phi(x(n,\theta'y)))g(\overline{x}(n,\theta'y))\right]\rightarrow \mathcal{L}(f)\nu_0(gw)
\end{equation}
But conclusion of theorem~\ref{TLL} for $h$ defined by $h(x)=(fe^{-it.})(\phi(x))g(\overline{x})$ is that:
\begin{equation}\label{eq7}
\sigma\sqrt{2\pi n}\mathbb E\left[e^{-it.}f(\phi(x(n,\theta'y)))g(\overline{x}(n,\theta'y))\right]\rightarrow \mathcal{L}(fe^{-it.})\nu_0(g)
\end{equation}
Equations (\ref{eq6}) and (\ref{eq7}) together imply that $ta\in 2\pi\mathbb Z$ and that
$$w(\overline{\theta'y})\mathcal{L}(fe^{it.})\nu_0(g)=\mathcal{L}(f)\nu_0(gw).$$
The right side of the equation does not depend on $\theta'$ so by lemma~\ref{suppnu} $w$ is constant on $S_{\nu_0}$, this proves $\nu_0(gw)=w(\overline{\theta'y})$, so $\mathcal{L}(fe^{-it.})=\mathcal{L}(f)$ that is $e^{it.}=1$ or $t=0$. This is a contradiction, which concludes the proof.
\end{proof}
\begin{proof}[Proof of theorem~\ref{renouv}]
Applying proposition~\ref{Hm} and theorem~D of~\cite{HH1}, we have that, if $g\in\mathcal{C}_c(\mathbb R)$ and if $f$ is a bounded Lipschitz function on $\mathbb{PR}_{\max}^d$, then for any $x^0\in\mathbb R^d$,
$$\lim_{a\rightarrow-\infty} \sum_{n\ge1}\mathbb E\left[f\left(\phi\left(x(n,x^0)\right)-\phi\left(x^0\right)-a \right)g\left(\bar{x}(n,x^0)\right)\right]=0,$$
$$ \lim_{a\rightarrow+\infty} \sum_{n\ge1}\mathbb E\left[f\left(\phi\left(x(n,x^0)\right)-\phi\left(x^0\right)-a \right)g\left(\bar{x}(n,x^0)\right)\right]=\frac{\nu_0(f)\mathcal{L}(g)}{\gamma}.$$
Moreover these convergences are uniform in $x^0$, because $\delta_{\overline{x^0}}$ is bounded as a linear form on $\mathcal B^\infty$.
The uniformity allows us to remove the $\phi\left(x^0\right)$ in the last equations and take any random initial condition.
The result follows by the same successive approximations as in the proof of the LLT.
\end{proof}
\begin{proof}[Proof of theorem~\ref{PGD}]
Without lost of generality, we can assume that $\gamma=0$.
The exponential moment of $\sigma(A)$ means that there is a $\theta>0$ such that \mbox{$\int e^{\theta \sigma(A)} d\mu(A)<\infty$.}
An easy bound of the norm of $\xi^k(y)Q(.,dy)$ inspired by the proof of proposition~\ref{Hm} ensures that $z\mapsto Q_z$ is analytic on the open ball with center $0$ and radius $\theta$.
To prove that it is continuous on the domain $\{|\mathcal{R}z|<\theta/2\}$, we apply the same method.
Now theorem~E of~\cite{HH1} gives $$\lim_{n}\frac{1}{n}\ln \mathbb P\left[\phi\left(x(n,X^0)\right)-\phi(X^0)>n\epsilon\right]=-c(\epsilon).$$
Let $0<\eta<\epsilon$. For any $n\ge \|\phi(X^0)/\eta\|_\infty$, we have
$$\mathbb P\left[\phi\left(x(n,X^0)\right)-\phi(X^0)>n\epsilon\right]
\ge \mathbb P\left[\phi\left(x(n,X^0)\right)>n(\epsilon+\eta)\right],$$
which implies that
$$\liminf_{n}\frac{1}{n}\ln \mathbb P\left[\phi\left(x(n,X^0)\right)-n\gamma>n\epsilon\right]\ge-c(\epsilon+\eta).$$
The same method gives
$$\limsup_{n}\frac{1}{n}\ln \mathbb P\left[\phi\left(x(n,X^0)\right)-n\gamma>n\epsilon\right]\le-c(\epsilon-\eta).$$
By continuity of $c$, the first equality is proved. The second one follows from the same method applied to $-\phi$ instead of $\phi$.
\end{proof}
\subsection{Max-plus case}
Before proving the statements, we recall a few needed definitions and results about powers of matrices in the $(\max,+)$ algebra.
\begin{defn}
\begin{enumerate}
\item The critical graph of $A$ is obtained from $\mathcal{G}(A)$ by keeping only nodes and arcs belonging to circuits with average weight $\rho_{\max}(A)$. It will be denoted by $\mathcal{G}^c(A)$.
\item The cyclicity of a graph is the greatest common divisor of the length of its circuits if it is strongly connected (that is if any node can be reached from any other). Otherwise it is the least common multiple of the cyclicities of its strongly connected components. The cyclicity of $A$ is that of $\mathcal{G}^c(A)$ and is denoted by $c(A)$.
\end{enumerate}
\end{defn}
\begin{rem}\label{powerint} Interpretation of powers with $\mathcal{G}(A)$.\\
If $(i_1,i_2\cdots,i_n)$ is a path on $\mathcal{G}(A)$, its weight is $\sum_{1\le j\le n-1}A_{i_ji_{j+1}}$, so that $\left(A^{ n}\right)_{ij}$ is the maximum of the weights of length $n$ paths from $i$ to $j$.
\end{rem}
\begin{thm}[\cite{cohen83}]\label{proppuiss}
Assume $\mathcal{G}(A)$ is strongly connected, \mbox{$\rho_{\max}(A)=0$}. Then the sequence $\left(A^{ n}\right)_{n\in\mathbb N}$ is ultimately periodic and the ultimate period is the cyclicity of $A$.
\end{thm}
\begin{proof}[Proof of theorem~\ref{s>0}]
Suppose that $\sigma=0$. By proposition~\ref{Hm} we may apply theorem~A of~\cite{HH1}. The third point of the theorem says that there exists a bounded Lipschitz function $f$ such that for $\nu$-almost every $(B,\bar{x})$:
\begin{equation}\label{cobord}
\max_i(Bx)_i-\max_ix_i=f(\overline{x})-f(\overline{Bx})
\end{equation}
Since all functions in that equation are continuous, every $B\in S_A$ and $\overline{x}\in Supp(\nu_0)$ satisfy equation~(\ref{cobord}). If $B\in S_A$ and $\overline{x} \in Supp(\nu_0)$, then $\overline{Bx}\in Supp(\nu_0)$, so by induction equation~(\ref{cobord}) is satisfied by any $B$ in $T_A$. Since for $B\in T_A$, $B^n\in T_A$, $\max_iB^nx_i$ is bounded. But there exists a $k$ such that $c(B)\rho_{\max}(B)=B^{c(B)}_{kk}$, so $\max_i\left(B^{nc(B)}x\right)_i\ge nc(B)\rho_{\max}(B) +x_k$ and $\rho_{\max}(B)\le 0$.\\
Since every path on $\mathcal{G}(B)$ can be split into a path with length at most $d$ and closed paths whose average length are at most $\rho_{\max}(B)$, we have:
$$\max_i(B^nx)_i\le (n-d)\rho_{\max}(B) +d \max_{B_{ij}>-\infty} |B_{ij}|+\max_ix_i,$$
therefore $\rho_{\max}(B)\ge 0$.
So $\sigma=0$ implies that $\forall B\in T_A,\rho_{\max}(B)=0$.\\
Conversely, if $\rho_{\max}(B)=0$ for every $B\in T_A$, then
\begin{eqnarray*}
\max_ix_i(n,0)&=&\max_{ij}\left(A(n)\cdots A(1)\right)_{ij}\\
&\ge& \rho_{\max}\left(A(n)\cdots A(1)\right)=0 \textrm{ a.s.\,}.
\end{eqnarray*}
Therefore $\mathcal{N}(0,\sigma^2)(\mathbb R_+)\ge1$, and $\sigma=0$.
\end{proof}
\begin{proof}[Proof of theorem~\ref{NA}]
We assume the system is algebraically arithmetic.
Then there are $a,b\in\mathbb R$ and $\theta\in T_A$ such that for any $A \in S_A$ and $\theta'\in T_A$ with rank~1, we have:
$(\theta A\theta'-\theta \theta')(\mathbb R^d) \subset (a+b\mathbb Z)\textbf{1}.$
Replacing $\theta'$ by $A^n\theta'$, we get $(\theta A^{n+1}\theta'-\theta A^{n} \theta')(\mathbb R^d) \subset (a+b\mathbb Z)\textbf{1}$ and by induction
\begin{equation}\label{eq4}
(\theta A^{n+k}\theta'-\theta A^{n} \theta')(\mathbb R^d) \subset (ka+b\mathbb Z)\textbf{1}
\end{equation}
From now on, we assume that $\mathcal{G}(A)$ is strongly connected.
The matrix $\tilde{A}$ defined by $\tilde{A}_{ij}=A_{ij}-\rho_{\max}(A)$ satisfy $\rho_{\max}\left(\tilde{A}\right)=0$ and has a strongly connected graph. Therefore, by theorem~\ref{proppuiss}, there is an $n$ such that for any indices~$i,j$, $\tilde{A}^{n+c(A)}_{ij}=\tilde{A}^n_{ij}$.
Since for any $n\in\mathbb N$, $A^n_{ij}=\tilde{A}^n_{ij}+n\rho_{\max}(A)$, it means that $A^{(n+1)c(A)}_{ij}=A^{nc(A)}_{ij}+c(A)\rho_{\max}(A)$, and $(\theta A^{n+c(A)}\theta')_{ij}-(\theta A^{n} \theta')_{ij}=c(A)\rho_{\max}(A)$.
Together with equation~(\ref{eq4}), it says that $\rho_{\max}(A)\in a+\frac{b}{c(A)}\mathbb Z\subset a+\frac{b}{d!}\mathbb Z$, which concludes the proof.
\end{proof}
\section{Acknowledgements}
The author gratefully thanks Jean Mairesse for useful talks and suggestions of improvements to this article.
\bibliographystyle{alpha}
|
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"timestamp": "2007-01-08T14:52:18",
"yymm": "0503",
"arxiv_id": "math/0503634",
"language": "en",
"url": "https://arxiv.org/abs/math/0503634"
}
|
\section{Introduction}
In \cite{Wer1}, we introduced a theory of {\it contractive Markov
systems (CMS)} which provides a unifying framework in so-called
'fractal' geometry. It extends the known theory of {\it iterated
function systems (IFS) with place dependent probabilities}, which
are contractive on average, \cite{BDEG}\cite{Elton} in a way that it
also covers {\it graph directed constructions} of 'fractal' sets
\cite{MW}. In particular, Markov chains associated with such systems
naturally extend finite Markov chains and inherit some of their
properties.
By a {\it Markov system} we mean a structure on a metric space which
generates a Markov process on it and is given by a family
\[\left(K_{i(e)},w_e,p_e\right)_{e\in E}\]
(see Fig. 1) where $E$ is the set of edges of a finite directed
(multi)graph $(V,E,i,t)$ ($V:=\{1,...,N\}$ is the set of vertices of
the directed (multi)graph (we do not exclude the case $N=1$),
$i:E\longrightarrow V$ is a map indicating the initial vertex of
each edge and $t:E\longrightarrow V$ is a map indicating the
terminal vertex of each edge), $K_1,K_2,...,K_N$ is a partition of a
metric space $(K,d)$ into non-empty Borel subsets, $(w_e)_{e\in E}$
is a family of Borel measurable self-maps on the metric space such
that $w_e\left(K_{i(e)}\right)\subset K_{t(e)}$ for all $e\in E$ and
$(p_e)_{e\in E}$ is a family of Borel measurable probability
functions on $K$ (i.e. $p_e(x)\geq 0$ for all $e\in E$ and
$\sum_{e\in E}p_e(x)=1$ for all $x\in K$) (associated with the maps)
such that each $p_e$ is zero on the complement of $K_{i(e)}$.
\begin{center}
\unitlength 1mm
\begin{picture}(70,70)\thicklines
\put(35,50){\circle{20}} \put(10,20){\framebox(15,15)}
\put(40,20){\line(2,3){10}} \put(40,20){\line(4,0){20}}
\put(50,35){\line(2,-3){10}} \put(5,15){$K_1$} \put(34,60){$K_2$}
\put(61,15){$K_3$} \put(31,50){\framebox(7.5,5)}
\put(33,45){\framebox(6.25,9.37)} \put(50,28){\circle{7.5}}
\put(45,21){\framebox(6,5)} \put(10,32.5){\line(6,1){15}}
\put(10,32.5){\line(3,-5){7.5}} \put(17.5,20){\line(1,2){7.5}}
\put(52,20){\line(2,3){4}} \put(13,44){$w_{e_1}$}
\put(35,38){$w_{e_2}$} \put(49,42){$w_{e_3}$}
\put(33,30.5){$w_{e_4}$} \put(30,15){$w_{e_5}$}
\put(65,37){$w_{e_6}$} \put(0,5){ Figure 1. A Markov system.}
\put(0,60){$N=3$} \thinlines \linethickness{0.1mm}
\bezier{300}(17,37)(20,46)(32,52)
\bezier{50}(32,52)(30.5,51.7)(30,49.5)
\bezier{50}(32,52)(30,51)(28.7,51.7)
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\end{picture}
\end{center}
A Markov system is called {\it irreducible} or {\it aperiodic} iff
its directed graph is irreducible or aperiodic respectively.
We call a Markov system $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$
{\it contractive} with an {\it average contracting rate} $0<a<1$ iff
it satisfies the following {\it condition of contractiveness on
average}
\begin{equation}\label{cc}
\sum\limits_{e\in E}p_e(x)d(w_ex,w_ey)\leq ad(x,y)\mbox{ for all
}x,y\in K_i,\ i=1,...,N.
\end{equation}
This condition was discovered by Richard Isaac in 1961 for the case
$N=1$ \cite{Is}.
A Markov system $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ determines
a Markov operator $U$ on the set of all bounded Borel measurable
functions $\mathcal{L}^0(K)$ by
\[Uf:=\sum\limits_{e\in E}p_ef\circ w_e\mbox{ for all
}f\in\mathcal{L}^0(K)\] and its adjoint operator $U^*$ on the set of all Borel probability
measures $P(K)$ by
\[U^*\nu(f):=\int U(f)d\nu\mbox{ for all }f\in\mathcal{L}^0(K)\mbox{ and }\nu\in P(K).\]
\begin{Remark}
Note that each map $w_e$ and each probability $p_e$ need to
be defined only on the corresponding vertex set $K_{i(e)}$. This
is sufficient for the condition (\ref{cc}) and the definition of
$U^*$. For the definition of $U$, we can consider each $w_e$ to
be extended on the whole space $K$ arbitrarily and each $p_e$ to
be extended on $K$ by zero.
Also, the situation applies where each vertex set $K_i$ has its
own metric $d_i$. In this case, one can set
\[d(x,y)=\left\{\begin{array}{cc}
d_i(x,y) & \mbox{ if }x,y\in K_i \\
\infty & \mbox{otherwise}
\end{array}\right.
\] and use the convention $0\times\infty=0$.
\end{Remark}
We say a probability measure $\mu$ is an {\it invariant probability measure} of
the Markov system iff it is a stationary initial distribution of
the associated Markov process, i.e.
\[U^*\mu=\mu.\]
A Borel probability measure $\mu$ is called {\it
attractive} measure of the CMS if
\[{U^*}^n\nu\stackrel{w^*}{\to}\mu\mbox{ for all }\nu\in P(K),\]
where $w^*$ means weak$^*$ convergence.
Note that an attractive probability measure is a unique invariant
probability measure of the CMS if $U$ maps continuous functions on
continuous functions. We will denote the space of all bounded
continuous functions by $C_B(K)$.
The main result in \cite{Wer1} concerning the uniqueness of the
invariant measure is the following (see Lemma 1 and Theorem 2 in
\cite{Wer1}) (see also \cite{Wer2} for the case of constant
probabilities $p_e|_{K_{i(e)}}$ and compact state space).
\begin{theo}\label{Th}
Suppose $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ is an irreducible CMS such that $K_1$,$K_2$,
...,$K_N$ partition $K$ into non-empty open subsets, each $p_e|_{K_{i(e)}}$ is
Dini-continuous and there exists $\delta>0$ such that $p_e|_{K_{i(e)}}\geq\delta$
for all $e\in E$. Then:\\
(i) The CMS has a unique invariant Borel probability measure
$\mu$, and $\mu(K_i)>0$ for all $i=1,...,N$.\\
(ii) If in addition the CMS is aperiodic, then
\[U^nf(x)\to\mu(f)\mbox{ for all }x\in K\mbox{ and }f\in C_B(K)
\] and the convergence is uniform on bounded subsets, i.e. $\mu$ is an
attractive probability measure of the CMS.
\end{theo}
A function $h:(X,d)\longrightarrow\mathbb{R}$ is called {\it
Dini-continuous} iff
for some $c >0$ \[\int_0^c\frac{\phi(t)}{t}dt<\infty\]
where $\phi$ is {\it the modulus of uniform continuity} of $f$, i.e.
\[\phi(t):=\sup\{|h(x)-h(y)|:d(x,y)\leq t,\ x,y\in X\}.\]
It is easily seen that the Dini-continuity is weaker than the
H\"{o}lder and stronger than the uniform continuity. There is a
well known characterization of the Dini-continuity.
\begin{lemma}\label{Dc}
Let $0<c<1$ and $b>0$.
A function $h$ is Dini-continuous iff
\[\sum_{n=0}^\infty\phi\left(bc^n\right)<\infty\] where $\phi$ is
the modulus of uniform continuity of $h$.
\end{lemma}
The proof is simple (e.g. see \cite{Wer1}).
Furthermore, with a Markov system
$\left(K_{i(e)},w_e,p_e\right)_{e\in E}$, which has an invariant
Borel probability measure $\mu$, also is associated a measure
preserving transformation $S:(\Sigma,\mathcal{B}(\Sigma),
M)\longrightarrow(\Sigma,\mathcal{B}(\Sigma), M)$, which we call a
{\it generalized Markov shift}, where
$\Sigma:=\{(...,\sigma_{-1},\sigma_0,\sigma_1,...):\sigma_i\in E\
\forall i\in\mathbb{Z}\}$ is the {\it code space} provided with the
product topology, $\mathcal{B}(\Sigma)$ denotes Borel
$\sigma$-algebra on $\Sigma$ and $M$ is a {\it generalized Markov
measure} on $\mathcal{B}(\Sigma)$ given by
\[M\left(_m[e_1,...,e_k]\right):=\int
p_{e_1}(x)p_{e_2}(w_{e_1}x)...p_{e_k}(w_{e_{k-1}}\circ...\circ
w_{e_1}x)d\mu(x)\] for every cylinder set
$_m[e_1,...,e_k]:=\{\sigma\in\Sigma:\
\sigma_m=e_1,...,\sigma_{m+k-1}=e_k\}$, $m\in\mathbb{Z}$, and $S$ is
the usual left shift map on $\Sigma$. It is easy to verify that $S$
preserves measure $M$, since $U^*\mu=\mu$ (see \cite{Wer3}).
For a CMS, this two pictures, Markovian and dynamical, are related
by a {\it coding map} $F:(\Sigma,\mathcal{B}(\Sigma),
M)\longrightarrow K$ which was constructed in \cite{Wer3}. It is
defined, if $K$ is a complete metric space and each $p_e|_{K_i(e)}$
is Dini-continuous and bounded away from zero, by
\[F(\sigma):=\lim\limits_{m\to-\infty}w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_m}x_{i(\sigma_m)}\mbox{ for
}M\mbox{-a.e. }\sigma\in\Sigma\] where $x_i\in K_i$ for each
$i=1,...,N$ (the coding map does not depend on the choice of $x_i$'s
modulo an $M$-zero set). We show in this paper that $F$ is also well
defined under a weaker continuity condition on the probability
functions $p_e|_{K_i(e)}$.
Let denote by $\mathcal{F}$ the sub-$\sigma$-algebra on $\Sigma$
generated by cylinder sets of the form $_m[e_m,...,e_0]$, $m\leq 0$.
Note that $F$ is $\mathcal{F}$-$\mathcal{B}(K)$-measurable.
\begin{Example}\label{gm}
Let $G:=(V,E,i,t)$ be a finite irreducible directed
(multi)graph. Let $\Sigma^-_G:=\{(...,\sigma_{-1},\sigma_0):\
\sigma_m\in E\mbox{ and }
t(\sigma_m)=i(\sigma_{m-1})\ \forall
m\in\mathbb{Z}\setminus\mathbb{N}\}$ (be {\it one-sided
subshift of finite type} associated with $G$) endowed with the
metric $d(\sigma,\sigma'):=2^k$ where $k$ is the smallest
integer with $\sigma_i=\sigma'_i$ for all $k<i\leq 0$. Let $g$
be a positive, Dini-continuous function on $\Sigma_G$ such that
\[\sum\limits_{y\in T^{-1}(\{x\})}g(y)=1\mbox{ for all }x\in\Sigma_G\]
where $T$ is the right shift map on $\Sigma^-_G$. Set
$K_i:=\left\{\sigma\in\Sigma^-_G:t(\sigma_0)=i\right\}$ for
every $i\in V$ and, for
every $e\in E$,
\[w_e(\sigma):=(...,\sigma_{-1},\sigma_{0},e),\ p_e(\sigma):=g(...,\sigma_{-1},\sigma_{0},e)
\mbox{ for all }\sigma\in K_{i(e)}.\]
Obviously, maps $(w_e)_{e\in E}$ are contractions. Therefore,
$\left(K_{i(e)}, w_e, p_e\right)_{e\in E}$
defines a CMS. An invariant probability measure of this CMS is
called a $g$-measure. This notion was introduced by Keane
\cite {Ke} and further developed by Ledrappier \cite{Le}, Walters \cite{W1}, Berbee \cite{Ber} and others.
\end{Example}
\begin{Example}
Let $\mathbb{R}^2$ be normed by $\|.\|_1$. Let
$K_1:=\{(x,y)\in\mathbb{R}^2:\ y\geq 1\}$ and $K_2:=\{(x,y)\in\mathbb{R}^2:\ y\leq
-1\}$. Consider the following maps on $\mathbb{R}^2$:
\begin{eqnarray*}
&&w_1\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\begin{array}{c}
-\frac{1}{2}x-1\\-\frac{3}{2}y+\frac{1}{2}\end{array}\right),\
w_2\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\begin{array}{c}
-\frac{3}{2}x+1\\\frac{1}{4}y+\frac{3}{4}\end{array}\right),\\
&&w_3\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\begin{array}{c}
-\frac{1}{2}|x|+1\\-\frac{3}{2}y-\frac{1}{2}\end{array}\right),\mbox{
and }
w_4\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\begin{array}{c}
\frac{3}{2}|x|-1\\-\frac{1}{4}y+\frac{3}{4}\end{array}\right)
\end{eqnarray*}
with probability functions
\begin{eqnarray*}
&& p_1\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\frac{1}{15}\sin^2\|(x,y)\|_1+\frac{53}{105}
\right)1_{K_1}(x,y),\\
&& p_2\left(\begin{array}{c}x\\ y\end{array}\right)
:=\left(\frac{1}{15}\cos^2\|(x,y)\|_1+\frac{3}{7}\right)
1_{K_1}(x,y),\\
&& p_3\left(\begin{array}{c}x\\ y\end{array}\right):=\left(\frac{1}{15}\sin^2\|(x,y)\|_1+\frac{53}{105}
\right)1_{K_2}(x,y)\mbox{ and }\\
&& p_4\left(\begin{array}{c}x\\ y\end{array}\right)
:=\left(\frac{1}{15}\cos^2\|(x,y)\|_1+\frac{3}{7}\right) 1_{K_2}(x,y) .
\end{eqnarray*}
A simple calculation shows that $(K_{i(e)},w_e,p_e)_{e\in\{1,2,3,4\}}$, where $i(1)=i(2)=1$ and $i(3)=i(4)=2$,
defines a CMS with an average
contracting rate $209/210$ on $K_1\cup K_2$. Note that none of the maps are contractive (by Theorem \ref{Th}, it
has a unique (attractive) invariant Borel probability measure).
\end{Example}
\section{Main Part}
Let $\mathcal{M}:=\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ be a
contractive Markov system with an average contracting rate $0<a<1$
and an invariant Borel probability measure $\mu$. We assume that:
$(K,d)$ is a metric space in which sets of finite diameter are
relatively compact; the family $K_1,...,K_N$ partitions $K$ into
non-empty open subsets; each probability function $p_e|_{K_{i(e)}}$
is uniformly continuous and bounded away from zero by $\delta>0$;
the set of edges $E$ is finite and the map $i:E\longrightarrow V$ is
surjective. Note that the assumption on the metric space implies
that it is locally compact separable and complete.
In contrast to the papers \cite{Wer1}, \cite{Wer3}, \cite{Wer5} and
\cite{Wer6}, here we will prove some similar results under a weaker
continuity condition on the probability functions $p_e|_{K_{i(e)}}$
than the Dini-continuity. In Subsection 2.1, we show that the coding
map is well defined under this condition. In Subsection 2.2, using
the coding map, we set up a thermodynamic formalism for CMSs, which
is beyond the scope of the existing theory. Finally, using the
developed thermodynamic formalism, we show in Subsection 2.3 that
the irreducible CMS $\mathcal{M}$ with the probabilities
$p_e|_{K_{i(e)}}$ satisfying the following continuity condition has
a unique invariant Borel probability measure, which also can be
obtained empirically.
\begin{Definition}
We say that a function $h:K\longrightarrow\mathbb{R}$ has a {\it square summable variation} iff
for any $c >0$
\[\int_0^c\frac{\phi^2(t)}{t}dt<\infty\]
where $\phi$ is {\it the modulus of uniform continuity} of $h$, i.e.
\[\phi(t):=\sup\{|h(x)-h(y)|:d(x,y)\leq t,\ x,y\in X\}\]
or equivalently, for any $b>0$ and $0<c<1$,
\[\sum_{n=0}^\infty\phi^2\left(bc^n\right)<\infty,\] which is obviously a weaker condition than
the Dini-continuity and stronger than the uniform continuity.
\end{Definition}
This condition was introduced by A.
Johansson and A. \"{O}berg in \cite{JO}, where they proved the
uniqueness of the invariant probability measure for some iterated
function systems with place-dependent probabilities satisfying this
condition on a compact metric space. Shortly after that, it was
announced by Berger, Hoffman and Sidoravicius \cite{BHS} that the
condition of the square summability of variation is tight, in the
sense that for any $\epsilon>0$ there exists a $g$-function with a
summable variation to the power $2+\epsilon$ such that the
corresponding CMS as in Example \ref{gm} has several invariant Borel
probability measures.
\begin{Example}
Let $\alpha,\delta>0$ such that $\alpha+\delta<1$ and $0<c<1/e$.
Set
\begin{equation*}\label{ef}
p(x)=\left\{\begin{array}{cc}
\frac{\alpha}{\log\frac{1}{x}}+\delta& \mbox{if }0<x\leq\frac{1}{e} \\
\delta& \mbox{if }x=0.
\end{array}\right.
\end{equation*}
Let $\phi$ be the modulus of uniform continuity of $p$. Then
\[\phi(c^n)=\frac{\alpha}{n\log\frac{1}{c}}\mbox{ for all }n\in\mathbb{N}.\]
Hence $p$ is not Dini-continuous, but obviously it has a square
summable variation. See \cite{JO} for a discussion of the relation
between the Johansson-\"{O}berg condition and the Berbee condition
\cite{Ber}.
\end{Example}
Let $\Sigma^+:=E^{\mathbb{N}}$ endowed with the product topology of
the discreet topologies. For $x\in K$, let $P_x$ denote the Borel
probability measure on $\Sigma^+$ given by
\[P_x\left( _1[e_1,...,e_k]\right):=p_{e_1}(x)p_{e_2}(w_{e_1}x)...p_{e_k}(w_{e_{k-1}}\circ...\circ w_{e_1}x)\]
for every cylinder set $_1[e_1,...,e_k]\subset\Sigma^+$. Obviously,
$P_x$ represents the Markov process with the Dirac initial
distribution concentrated at the point $x$.
All results for CMSs with probabilities with a square summable
variation which we intend to prove in this paper follow from the
next lemma. It generalizes Lemma 3 in \cite{Elton} and Lemma 2.3 in
\cite{Wer5}. For the proof of it we use the methodology of Johansson
and \"{O}berg \cite{JO}.
\begin{lemma}\label{acl}
Suppose $\mathcal{M}$ is a CMS such that each probability function
$p_e|_{K_{i(e)}}$ has a square summable variation. Then $P_y$ is
absolutely continuous with respect to $P_x$ for all $x,y\in K_i$ and
$i=1,...,N$.
\end{lemma}
{\it Proof.} Let $\mathcal{A}_n$ be the $\sigma$-algebra on
$\Sigma^+$ generated by the cylinders of the form $_1[e_1,...,e_n]$
for all $n\in \mathbb{N}$. For each $n\in\mathbb{N}$, let $X_n$ be a
function of $\Sigma^+$ given by
\[X_n(\sigma):=\sum\limits_{e_1,...,e_2}\frac{P_y( _1[e_1,...,e_n])}{P_x( _1[e_1,...,e_n])}1_{ _1[e_1,...,e_n]}(\sigma)
\mbox{ for all }\sigma\in\Sigma^+\] with the convention that
$0/0=0$. Obviously, by the definition of $P_x$, we can restrict the
summation in the definition of $X_n$ on paths $(e_1,...,e_n)^*$ of
the digraph.
Now, observe that, for all $m\leq n$ and $C_m\in\mathcal{A}_m$,
\begin{equation}\label{me}
\int\limits_{C_m} X_n\ dP_x=\sum\limits_{C_n\subset
C_m}P_y(C_n)=P_y(C_m)=\int\limits_{C_m}X_m\ dP_x.
\end{equation}
Hence, $(X_n,\mathcal{A}_n)_{n\in\mathbb{N}}$ is a $P_x$-martingale.
If $X_n$ is uniformly bounded, then there exists
$X\in\mathcal{L}^1(P_x)$ such that $X_n\to X$ $P_x$-a.e. and in
$L^1$ sense, and $E_{P_x}(X|\mathcal{A}_m)=X_m$ $P_x$-a.e. for all
$m$ (see \cite{B}). Then, by (\ref{me}),
\[\int\limits_{C_m} X\ dP_x=\int\limits_{C_m}X_m\ dP_x=P_y(C_m)\mbox{ for all }C_m\in\mathcal{A}_m.\]
It means that the Borel probability measures $XP_x$ and $P_y$ agree
on all cylinder subsets of $\Sigma^+$, and therefore, are equal.
This implies the claim. So, it remains to show that the martingale
$(X_n,\mathcal{A}_n)_{n\in\mathbb{N}}$ is uniformly integrable, i.e.
\[\sup\limits_{n}\int\limits_{X_n>K}X_n dP_x\to 0\mbox{ as }K\to\infty.\]
By (\ref{me}), it is equivalent to show that
\[\sup\limits_{n}P_y(X_n>K)\to 0\mbox{ as }K\to\infty.\]
The latter is equivalent to
\[\sup\limits_{n}P_y(\log X_n>K)\to 0\mbox{ as }K\to\infty.\]
Let us use the following abbreviation:
\[p_i^x(\sigma):=p_{\sigma_i}(w_{\sigma_{i-1}}\circ...\circ w_{\sigma_1}x)\]
for $i\geq 2$ and $p_1^x(\sigma):=p_{\sigma_1}(x)$ for all
$\sigma\in\Sigma^+$. Then, since $\log x\leq x-1$,
\begin{eqnarray*}
\log X_n&=& \sum_{(e_1,...,e_n)^*}\log\frac{p_1^y...p_n^y}{p^x_1...p^x_n}1_{ _1[e_1,...,e_n]}\\
&=& \sum_{(e_1,...,e_n)^*}\sum\limits_{i=1}^n\log\frac{p_i^y}{p_i^x}1_{ _1[e_1,...,e_n]}\\
&\leq& \sum_{(e_1,...,e_n)^*}\sum\limits_{i=1}^n\frac{p_i^y-p_i^x}{p_i^x}1_{ _1[e_1,...,e_n]}.
\end{eqnarray*}
Now, observe that
\[\frac{p_i^y-p_i^x}{p_i^x}=\frac{p_i^y-p_i^x}{p_i^y}+\frac{(p_i^y-p_i^x)^2}{p_i^xp_i^y}.\]
Therefore,
\begin{equation}\label{JOE}
\log X_n\leq Y_n+Z_n,
\end{equation}
where
\[Y_n:=\sum_{(e_1,...,e_n)^*}\sum\limits_{i=1}^n\frac{p_i^y-p_i^x}{p_i^y}1_{ _1[e_1,...,e_n]}\]
and
\[Z_n:=\frac{1}{\delta^2}\sum_{(e_1,...,e_n)^*}\sum\limits_{i=1}^n(p_i^y-p_i^x)^2 1_{ _1[e_1,...,e_n]}.\]
Furthermore, observe that
\[Y_{i+1}-Y_i=\sum_{(e_1,...,e_n)^*}\frac{p_{n+1}^y-p_{n+1}^x}{p_{n+1}^y}1_{ _1[e_1,...,e_n]}\mbox{ for all }i\geq 1.\]
and
\begin{eqnarray*}
&&\int\limits_{ _1[e_1,...,e_n]}(Y_{n+1}-Y_n)\ dP_y\\
&=&\sum\limits_{e_{n+1}}\frac{p_{e_{n+1}}(w_{e_n}\circ...\circ
w_{e_1}y)-p_{e_{n+1}}(w_{e_n}\circ...\circ w_{e_1}x)}
{p_{e_{n+1}}(w_{e_n}\circ...\circ w_{e_1}y)}\\
&& \;\;\;\;\times p_{e_1}(y)...p_{e_{n}}(w_{e_{n-1}}\circ...\circ w_{e_1}y)p_{e_{n+1}}(w_{e_n}\circ...\circ w_{e_1}y)\\
&=&\sum\limits_{e_{n+1}}(p_{e_{n+1}}(w_{e_n}\circ...\circ
w_{e_1}y)-p_{e_{n+1}}(w_{e_n}\circ...\circ w_{e_1}x))\\
&& \;\;\;\;\times p_{e_1}(y)...p_{e_{n}}(w_{e_{n-1}}\circ...\circ w_{e_1}y)\\
&=&0.
\end{eqnarray*}
Hence, $(Y_n,\mathcal{A}_n)_{n\in\mathbb{N}}$ is a $P_y$-martingale.
Therefore, $Y_n-Y_{n-1}$, $Y_{n-1}-Y_{n-2}$,..., $Y_2-Y_1$, $Y_1$
are orthogonal in $\mathcal{L}^2(P_y)$. By the Pythagoras equality,
this implies that
\begin{eqnarray*}
\int Y^2_n\ dP_y&=&\int\left(\sum\limits_{i=2}^n(Y_i-Y_{i-1})+Y_1\right)^2\ dP_y\\
&=& \sum\limits_{i=2}^n\int(Y_i-Y_{i-1})^2\ dP_y+\int{Y_1}^2\ dP_y\\
&\leq& \sum\limits_{i=1}^n\int\frac{(p_i^y-p^x_{i})^2}{\delta^2}\ dP_y\\
&\leq& \frac{1}{\delta^2}\sum\limits_{i=1}^n\int \left[p_{\sigma_i}(w_{\sigma_{i-1}}\circ...\circ w_{\sigma_1}y)-
p_{\sigma_i}(w_{\sigma_{i-1}}\circ...\circ w_{\sigma_1}x)\right]^2\ dP_y.
\end{eqnarray*}
Let \[A_i:=\left\{\sigma\in\Sigma^+:\ d(w_{\sigma_{i}}\circ...\circ
w_{\sigma_1}y,w_{\sigma_{i}}\circ...\circ w_{\sigma_1}x)>
a^{\frac{i}{2}}d(x,y)\right\}\]
for $i\in\mathbb{N}$. By the contractiveness on average condition,
\[\int d(w_{\sigma_{i}}\circ...\circ w_{\sigma_1}y,w_{\sigma_{i}}\circ...\circ w_{\sigma_1}x)\ dP_y\leq a^id(x,y)\mbox{ for all }
i.\] Hence, by the Markov inequality,
\[P_y\left(A_i\right)\leq a^{\frac{i}{2}}\mbox{ for all }i.\]
Therefore,
\begin{eqnarray*}
\int {Y_n}^2\ dP_y&\leq&\frac{1}{\delta^2}\sum\limits_{i=1}^n\left({P_y(A_i)}+\phi^2\left(a^{\frac{i}{2}}d(x,y)
\right)\right)\\
&\leq&\frac{1}{\delta^2}\left(\sum\limits_{i=1}^\infty a^{\frac{i}{2}}+\sum\limits_{i=1}^\infty
\phi^2\left(a^{\frac{i}{2}}d(x,y)\right)\right)
\end{eqnarray*}
for all $n\in\mathbb{N}$. Hence \[\sup\limits_n
P_y\left(Y_n>K\right)\to 0\mbox{ as }K\to\infty.\]
Analogously,
\[\sup\limits_n P_y\left(Z_n>K\right)\to 0\mbox{ as }K\to\infty.\]
Since, by (\ref{JOE}),
\[\left\{\log X_n>K\right\}\subset\left\{Y_n>\frac{K}{2}\right\}\cup\left\{Z_n>\frac{K}{2}\right\},\]
we conclude that
\[\sup\limits_n P_y\left(\log X_n>K\right)\to 0\mbox{ as }K\to\infty,\]
as desired.\hfill$\Box$
\subsection{Coding map}
We shall denote by $d'$ the metric on $\Sigma$ defined by
$d'(\sigma,\sigma'):=(1/2)^k$ where $k$ is the largest integer with
$\sigma_i=\sigma'_i$ for all $|i|<k$. Denote by $\mathcal{A}$ the
finite $\sigma$-algebra generated by the zero time partition
$\{_0[e]:e\in E\}$ of $\Sigma$, and define, for each integer $m\leq
1$,
\[\mathcal{A}_m:=\bigvee\limits_{i=m}^{+\infty} S^{-i}\mathcal{A},\]
which is the smallest $\sigma$-algebra containing all finite
$\sigma$-algebras $\bigvee_{i=m}^{n}
S^{-i}\mathcal{A}$, $n\geq m$. Let $x\in K$. For each integer $m\leq
1$, let $P_x^m$ be the probability measure on the $\sigma$-algebra
$\mathcal{A}_m$ given by
\[P^m_x( _{m}[e_{m},...,e_n])=p_{e_{m}}(x)p_{e_{m+1}}(w_{e_{m}}(x))...p_{e_n}(w_{e_{n-1}}\circ...\circ
w_{e_{m}}(x))\] for all cylinder sets $_{m}[e_{m},...,e_n]$,
$n\geq{m}$.
\begin{lemma}
Let $m\leq 1$ and $A\in\mathcal{A}_m$.
Then $x\longmapsto P_x^m(A)$ is a Borel measurable function on $K$.
\end{lemma}
The proof is simple (see e.g. \cite{Wer1}).
\begin{Definition}
Let $\nu\in P(K)$. We call a probability measure $\Phi_m(\nu)$ on
$(\Sigma,\mathcal{A}_m)$ given by
\[\Phi_m(\nu)(A):=\int P_x^m(A)d\nu(x),\
A\in\mathcal{A}_m,\] {\it the $m$-th lift of} $\nu$.
\end{Definition}
\begin{Definition}
Set
\[\mathcal{C}(B):=\left\{(A_m)_{m=0}^{-\infty}:A_m\in\mathcal{A}_m\
\forall m\mbox{ and }B\subset\bigcup\limits_{m=0}^{-\infty}A_m
\right\}\] for $B\subset\Sigma$. Let $\nu\in P(K)$. We call a set
function given by
\[\Phi(\nu)(B):=\inf\left\{\sum\limits_{m=0}^{-\infty}\Phi_m(\nu)(A_m):(A_m)_{m\leq 0}\in\mathcal{C}(B)\right\}, B\subset\Sigma,\]
{ \it the lift of} $\nu$.
\end{Definition}
\begin{lemma}\label{om}
Let $\nu,\lambda\in P(K)$. Then\\
$(i)$ $\Phi(\nu)$ is an outer measure on $\Sigma$.\\
$(ii)$ If $\Phi_m(\nu)\ll\Phi_m(\lambda)$ for all $m\leq 0$ , then for all $\epsilon>0$
there exists $\alpha>0$ such that
\[\Phi(\lambda)(B)<\alpha\ \Rightarrow\ \Phi(\nu)(B)<\epsilon\mbox{ for all
}B\subset\Sigma.\]
\end{lemma}
For the proof see \cite{Wer3}.
Note that $M=\Phi(\mu)$ (see \cite{Wer3}), where $M$ is the
generalized Markov measure associated with the CMS $\mathcal{M}$ and
its invariant measure $\mu$.
Fix $x_i\in K_i$ for each $i\in\{1,...,N\}$ and
set
\[P^m_{x_1...x_N}:=\Phi_m\left(\frac{1}{N}\sum\limits_{i=1}^N\delta_{x_i}\right)\mbox{ and }
P_{x_1...x_N}:=\Phi\left(\frac{1}{N}\sum\limits_{i=1}^N\delta_{x_i}\right)\]
for every $m\in\mathbb{Z}\setminus\mathbb{N}$, where $\delta_x$
denotes the Dirac probability measure concentrated at $x$, i.e.
\[P^m_{x_1...x_N}( _m[e_m,...,e_n])=\frac{1}{N}p_{e_m}(x_{i(e_m)})p_{e_{m+1}}(w_{e_m}x_{i(e_m)})...p_{e_n}(w_{e_{n-1}}
\circ...\circ w_{e_m}x_{i(e_m)})\] for every cylinder set
$_m[e_m,...,e_n]$.
\begin{theo}\label{cml}
Let $x_i,y_i \in K_i$ for each $1\leq i\leq N$. Then the following hold.\\
(i)
\[\lim\limits_{m\to-\infty}d\left(w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(y_{i(\sigma_{m})})\right)=0\]
$P_{x_1...x_N}$-a.e..\\
(ii)
\[F_{x_1...x_N}:=\lim_{m\to-\infty}w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(x_{i(\sigma_{m})})\mbox{ exists }
P_{x_1...x_N}\mbox{-a.e.},\] and by $(i)$
$F_{x_1...x_N}=F_{y_1...y_N}$ $P_{x_1...x_N}$-a.e..
(iii) There exists a sequence of closed subsets $Q_1\subset
Q_2\subset...\subset\Sigma$ with \\ $\sum\limits_{k=1}^\infty P_{x_1...x_N}(\Sigma\setminus Q_k)<\infty$
such that $F_{x_1...x_N}|_{Q_k}$ is locally H\"{o}lder-continuous
with the same H\"{o}lder-constants for all $k\in\mathbb{N}$, i.e.
there exist $\alpha, C>0$ such that for every $k$ there exists
$\delta_k>0$ such that
\[ \sigma,\sigma'\in Q_k\mbox{ with }d'(\sigma,\sigma')\leq\delta_k\
\Rightarrow\ d(F_{x_1...x_N}(\sigma),F_{x_1...x_N}(\sigma'))\leq
Cd'(\sigma,\sigma')^\alpha.\]
\end{theo}
For the proof see \cite{Wer3}.
\begin{lemma}\label{acl2}
Suppose $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ is a CMS such that
$P_x$ is absolutely continuous with respect to $P_y$ for all $x,y\in
K_i$, $i=1,...,N$. Let $\nu_1\in P(K)$ such that $\nu_1(K_i)>0$ for
all $i=1,...,N$. Then $\Phi(\nu_2)$ is absolutely continuous with
respect to $\Phi(\nu_1)$ for all $\nu_2\in P(K)$.
\end{lemma}
{\it Proof.} By Lemma \ref{om} $(ii)$, it is sufficient to show that
$\Phi_m(\nu_2)$ is absolutely continuous with respect to
$\Phi_m(\nu_1)$ for all $m\leq 0$. Let $A\in\mathcal{A}_m$ such that
$\Phi_m(\nu_1)(A)=0$. Then for all $i=1,...,N$ there exists $x_i\in
K_i$ such that $P^m_{x_i}(A)=0$. Hence, by the hypothesis,
$P^m_{x}(A)=0$ for all $x\in K$. Therefore,
\[\Phi_m(\nu_1)(A)=\int P^m_x(A)\ d\nu_2(x)=0\mbox{ for all }\nu_2\in P(K),\]
as desired.\hfill$\Box$
\begin{cor}\label{cm}
Suppose $\left(K_{i(e)},w_e,p_e\right)_{e\in E}$ is a CMS with an
invariant Borel probability measure $\mu$ such that $P_x$ is
absolutely continuous with respect to $P_y$ for all $x,y\in K_i$,
$i=1,...,N$.
Let $x_i,y_i \in K_i$ for each $1\leq i\leq N$. Then the following hold.\\
(i)
\[\lim\limits_{m\to-\infty}d\left(w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(x_{i(\sigma_{m})}),w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(y_{i(\sigma_{m})})\right)=0\
M\mbox{-a.e.},\] (ii)
\[F_{x_1...x_N}:=\lim_{m\to-\infty}w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_{m}}(x_{i(\sigma_{m})})\mbox{ exists }
M\mbox{-a.e.},\] and by $(i)$ $F_{x_1...x_N}=F_{y_1...y_N}$
$M$-a.e..
(iii) There exists a sequence of closed subsets $Q_1\subset
Q_2\subset...\subset\Sigma$ with \\ $\lim\limits_{k\to\infty} M(\Sigma\setminus Q_k)=0$
such that $F_{x_1...x_N}|_{Q_k}$ is locally H\"{o}lder-continuous
with the same H\"{o}lder-constants for all $k\in\mathbb{N}$, i.e.
there exist $\alpha, C>0$ such that for every $k$ there exists
$\delta_k>0$ such that
\[ \sigma,\sigma'\in Q_k\mbox{ with }d'(\sigma,\sigma')\leq\delta_k\
\Rightarrow\ d(F_{x_1...x_N}(\sigma),F_{x_1...x_N}(\sigma'))\leq
Cd'(\sigma,\sigma')^\alpha,\] where $M$ is the generalized Markov
measure associated with the CMS and $\mu$.
\end{cor}
{\it Proof.} Set
\[\nu_1:=\frac{1}{N}\sum\limits_{i=1}^N\delta_{x_i}.\] Since $M=\phi(\mu)$, the claim follows by
Theorem \ref{cml} and Lemma \ref{acl2}.\hfill$\Box$
\subsection{The generalized Markov measure as an equilibrium state}
Now, we are going to set up a thermodynamic formalism for the CMS
$\mathcal{M}$. We will construct an energy function $u$ with respect
to which the generalized Markov measure $M$ will turn out to be a
unique equilibrium state if the probabilities $p_e|_{K_{i(e)}}$ of
the CMS have a square summable variation. This extends the results
from \cite{Wer6}.
\begin{Definition}
Let $X$ be a metric space and $T$ a continuous transformation on it.
Denote by $P(X)$ the set of all
Borel probability measures on $X$ and by $P_T(X)$
the set of all $T$-invariant Borel probability measures on $X$.
We call a Borel measurable
function $h :X \longrightarrow [-\infty,0]$ an {\it energy function}. Suppose that $T$ has a finite
topological entropy, i.e. $\sup_{\Theta\in
P_T(X)}h_\Theta(T)<\infty$, where $h_{\Theta}(T)$ is the
Kolmogorov-Sinai entropy of $T$ with respect to measure $\Theta$. We
call
\[P(h)=\sup\limits_{\Theta\in
P_T(X)}\left(h_\Theta(T)+\Theta(h)\right)\] the {\it pressure} of
$h$. We call $\Lambda\in P_T(X)$ {\it an equilibrium state} for $h$
iff
\[h_{\Lambda}(T)+\Lambda(h)=P(h).\]
Let's denote the set of all equilibrium states for $h$ by $ES(h)$.
\end{Definition}
The construction of the function $u$ goes through a definition of an
appropriate shift invariant subset of $\Sigma$ on which the energy
function shall be finite. This subset is exactly that on which the
coding map is defined. It means that the existence of an equilibrium
state is closely related with the existence of the coding map.
\begin{Definition}
Let
\[\Sigma_G:=\{\sigma\in\Sigma:\ t(\sigma_j)=i(\sigma_{j+1})\ \forall
j\in\mathbb{Z}\},\]
\begin{equation*}
D:=\{\sigma\in\Sigma_G:\
\lim\limits_{m\to-\infty}w_{\sigma_0}\circ
w_{\sigma_{-1}}\circ...\circ w_{\sigma_m}x_{i(\sigma_m)}\mbox{
exists}\}
\end{equation*} and
\[Y:=\bigcap\limits_{i=-\infty}^\infty S^i(D).\]
Now, set
\begin{equation}\label{ef}
u(\sigma)=\left\{\begin{array}{cc}
\log p_{\sigma_1}\circ F(\sigma)& \mbox{if }\sigma\in Y \\
-\infty& \mbox{if }\sigma\in\Sigma\setminus Y.
\end{array}\right.
\end{equation}
\end{Definition}
\begin{lemma}\label{pY}
(i) $F(\sigma)$ is defined for all $\sigma\in Y$ and $Y$ is a shift invariant subset of $\Sigma_G$.\\
(ii) If $P_x$ is absolutely continuous with respect to $P_y$ for all $x,y\in K_i$, $i=1,...,N$, then $M(Y)=1$.\\
(iii) If $P_x$ is absolutely continuous with respect to $P_y$ for
all $x,y\in K_i$, $i=1,...,N$, and the CMS has an invariant
probability measure $\mu$ such that $\mu(K_{i(e)})>0$ for all $e\in
E$. Then $Y$ is dense in $\Sigma_G$.
\end{lemma}
{\it Proof.} (i) is clear, by the definitions of $F$ and $Y$.
Note that the condition of the contractiveness on average and the boundedness away from zero of the
functions $p_e|_{K_{i(e)}}$ imply that each map $w_e|_{K_{i(e)}}$ is continuous (Lipschitz).
Therefore, $S(D)\subset D$. By Corollary \ref{cm}, $M(D)=1$ if $P_x$ is absolutely continuous with respect to $P_y$
for all $x,y\in K_i$, $i=1,...,N$. This implies $(ii)$. If in
addition $\mu(K_{i(e)})>0$ for all $e\in E$, then $M(O)>0$ for
every open $O\subset\Sigma_G$. This implies $(iii)$. \hfill$\Box$
\begin{Remark}
(i) Note that $Y$ and $F$ depend on the choice of $x_i$'s. By
Corollary \ref{cm} (ii), $Y$ changes only modulo $M$-zero set by a
different choice of $x_i$'s if $P_x$ is absolutely continuous with
respect to $P_y$
for all $x,y\in K_i$, $i=1,...,N$. \\
(ii) If all maps $w_e|_{K_{i(e)}}$ are contractive, then
$Y=\Sigma_G$ and $F|_{\Sigma_G}$ is H\"{o}lder-continuous (easy to
check).
\end{Remark}
\begin{Remark}
As far as the author is aware, the rigorous mathematical theory of
equilibrium states is developed only for upper semicontinuous energy
functions $h$ (see e.g. \cite{Kel}). This condition insures that the
convex space $ES(h)$ is non-empty and compact in the weak$^*$
topology. However, as the next example shows, $u$ is not upper
semicontinuous in general. Therefore, even the existence of an
equilibrium state for $u$ is not guaranteed by the existing theory.
\end{Remark}
\begin{Example}\label{ex}
Let $(K,d)=(\mathbb{R},|.|)$. Consider two maps
\[w_0(x):=\frac{1}{2}x,\ w_1(x):=2x\mbox{ for all }x\in\mathbb{R}\]
with probability functions
\[p_0(x):=\frac{1}{6}\sin^2x+\frac{17}{24},\ p_1(x):=\frac{1}{6}\cos^2x+\frac{1}{8}\mbox{
for all }x\in\mathbb{R}.\]
Then a simple
calculation shows that $(\mathbb{R},w_e,p_e)_{e\in\{0,1\}}$
defines a CMS with an average contracting rate $45/48$. In this
case, $\Sigma_G=\{0,1\}^\mathbb{Z}$. If we take $x=0$ for the definition of $Y$,
then, obviously, $Y=\Sigma_G$. Now, let $x\neq 0$. Let $N_{0n}(\sigma)$ and $N_{1n}(\sigma)$
be the numbers of zeros and ones in $(\sigma_{-n},...,\sigma_0)$
respectively for every $\sigma\in\Sigma_G$. Then, obviously,
$\sigma\notin Y$ if
$\left(N_{1n}(\sigma)-N_{0n}(\sigma)\right)\to\infty$. Hence,
$Y\neq\Sigma_G$ and, by Lemma \ref{pY} (iii), $Y$ is a
dense shift invariant subset of $\Sigma_G$. Since $Y$ is not closed, $u$ is not upper
semicontinuous.
Also, it is not difficult to see that in a general case there is
no hope to find $x_i$ such that $u$ becomes upper
semicontinuous, e.g. change $w_1$ to $w_1(x)=2x+1$, then, for any
choice of $x$ for the definition of $Y$, $Y\neq\Sigma_G$.
\end{Example}
\begin{prop}\label{egm}
There exists an invariant Borel probability measure $\mu$ of the CMS $\mathcal{M}$ such that
\[\sum\limits_{i=1}^N\int\limits_{K_i}d(x,x_i)\ d\mu(x)<\infty\mbox{ for all }x_i\in K_i,\ i=1,...,N.\]
\end{prop}
{\it Proof.} Fix $x_i\in K_i$ for all $i=1,...,N$ and set
\[f(x):=\sum\limits_{i=1}^N1_{K_i}(x)d(x,x_i)\mbox{ for all }x\in K.\]
Let $C:=\max\limits_{e\in E}d(w_ex_{i(e)},x_{t(e)})$. Then the
contractiveness on average condition (\ref{cc}) implies that
\[U^kf(x_i)\leq\frac{C}{1-a}\mbox{ for all }k\in\mathbb{N}\mbox{ and }i=1,...,N\]
(see the proof of Theorem 1 in \cite{Wer1}).
Now, set \[U_n:=\frac{1}{n}\sum\limits_{k=1}^n U^k\mbox{ for all
}n\in\mathbb{N}.\] By Theorem 1 (i) from \cite{Wer1}, the sequence
$({U^*}^k\delta_{x_i})_{k\in\mathbb{N}}$ is tight. Hence, the
sequence $({U_n}^*\delta_{x_i})_{n\in\mathbb{N}}$ is tight also. So,
it has a subsequence, say
$({U_{n_m}}^*\delta_{x_i})_{m\in\mathbb{N}}$, which converges
weakly$^*$ to a Borel probability measure, say $\mu$. By the setup
of $\mathcal{M}$, the operator $U$ has the Feller property. This
implies that $U^*\mu=\mu$.
Let $R>0$ and $f_R:=\min\{f,R\}$. Then
\[{U_{n_m}}^*\delta_{x_i}(f_R)\to \mu(f_R)\mbox{ as }m\to\infty.\]
On the other hand
\[{U_{n_m}}^*\delta_{x_i}(f_R)\leq \frac{1}{n_m}\sum\limits_{k=1}^{n_m} U^kf(x_i)\leq\frac{C}{1-a}\]
for all $m\in\mathbb{N}$. Hence
\[\mu(f_R)\leq\frac{C}{1-a}\mbox{ for all }R>0.\]
Therefore, by the Monotone Convergence Theorem,
\[\sum\limits_{i=1}^N\int\limits_{K_i}d(x,x_i)\ d\mu(x)=\mu(f)\leq\frac{C}{1-a}.\]
\hfill$\Box$
\begin{prop}\label{ex}
Suppose $\mathcal{M}$ is a CMS such that $P_x$ is absolutely continuous with respect to
$P_y$ for all $x,y\in K_i$, $i=1,...,N$. Then $ES(u)$ is nonempty.
\end{prop}
{\it Proof.} By Proposition \ref{egm}, the CMS has an invariant
Borel probability measure $\mu$ such that
\[\sum\limits_{i=1}^N\int\limits_{K_i}d(x,x_i)\ d\mu(x)<\infty\mbox{ for all }x_i\in K_i,\ i=1,...,N.\]
Furthermore, by Lemma \ref{pY} $(ii)$,
\[M(Y)=1,\] where $M$ is the generalized Markov measure associated with $\mathcal{M}$ and $\mu$.
Hence, by Proposition 1 in \cite{Wer6}, $M$ is an equilibrium state
for $u$. \hfill$\Box$
\begin{lemma}\label{ace}
Let $\Lambda\in ES(u)$ and $\tilde\Lambda\in P_S(\Sigma)$ such that $\tilde\Lambda$ is absolutely continuous with
respect to $\Lambda$. Then $\tilde\Lambda\in ES(u)$.
\end{lemma}
{\it Proof.} Let $\psi$ be the Radon-Nikodym density of
$\tilde\Lambda$ with respect to $\Lambda$. By the shift invariance
of $\tilde\Lambda$ and $\Lambda$, $\psi\circ S=\psi$. This implies
that
\[E_\Lambda(\psi|\mathcal{F})=E_\Lambda(\psi|\mathcal{F})\circ S\ \Lambda\mbox{-a.e.}.\]
Let $A\in\mathcal{F}$ and $e\in E$. Then
\begin{eqnarray*}
\int\limits_{A}1_{_1[e]}\ d\tilde\Lambda&=&\int\limits_{A}1_{_1[e]}\psi\circ S\ d\Lambda
=\int\limits_{S(A)}1_{_0[e]}E_\Lambda(\psi|\mathcal{F})\ d\Lambda\\
&=&\int\limits_{A}1_{_1[e]}E_\Lambda(\psi|\mathcal{F})\ d\Lambda
=\int\limits_{A}E_\Lambda(1_{_1[e]}|\mathcal{F})E_\Lambda(\psi|\mathcal{F})\ d\Lambda\\
&=&\int\limits_{A}E_\Lambda(1_{_1[e]}|\mathcal{F})\psi\ d\Lambda
=\int\limits_{A}E_\Lambda(1_{_1[e]}|\mathcal{F})\ d\tilde\Lambda.
\end{eqnarray*}
Hence
\[E_{\tilde\Lambda}(1_{_1[e]}|\mathcal{F})=E_\Lambda(1_{_1[e]}|\mathcal{F})\ \tilde\Lambda\mbox{-a.e.}.\]
Since $\Lambda$ is an equilibrium state for $u$,
\[\sum\limits_{e\in E}1_{_1[e]}\log E_\Lambda(1_{_1[e]}|\mathcal{F})=u\ \Lambda\mbox{-a.e.},\]
by Lemma 5 in \cite{Wer6}. Therefore, by the absolute continuity of
$\tilde\Lambda$ with respect to $\Lambda$,
\[\sum\limits_{e\in E}1_{_1[e]}\log E_{\tilde\Lambda}(1_{_1[e]}|\mathcal{F})=u\ \tilde\Lambda\mbox{-a.e.}.\]
Therefore,
\begin{eqnarray*}
h_{\tilde\Lambda}(S)&=&-\sum\limits_{e\in E}\int
E_{\tilde\Lambda}(1_{_1[e]}|\mathcal{F})\log
E_{\tilde\Lambda}(1_{_1[e]}|\mathcal{F})\
d\tilde\Lambda\\
&=&-\int\sum\limits_{e\in E}1_{_1[e]}\log E_{\tilde\Lambda}(1_{_1[e]}|\mathcal{F})\ d\tilde\Lambda\\
&=&-\int u\ d\tilde\Lambda.
\end{eqnarray*}
Since $P(u)=0$ (see e.g. \cite{Wer6}), it follows that
$\tilde\Lambda\in ES(u)$.\hfill$\Box$
\begin{prop}\label{ees}
Suppose $\mathcal{M}$ is CMS such that $P_x$ is absolutely continuous with respect to
$P_y$ for all $x,y\in K_i$, $i=1,...,N$. Then there exists an ergodic equilibrium state $\Lambda$ for the energy
function $u$ such that $\Lambda$ is absolutely continuous with
respect to $M$ where $M$ is a generalized Markov measure
associated with the CMS and its invariant Borel probability
measure $\mu$ with the property that $\sum_{i=1}^N\int_{K_i}d(x,x_i)\
d\mu(x)<\infty$ for all $x_i\in K_i$, $i=1,...,N$.
\end{prop}
{\it Proof.} By Proposition \ref{egm}, there exists an invariant
Borel probability measure $\mu$ of the CMS such that
$\sum_{i=1}^N\int_{K_i}d(x,x_i)\
d\mu(x)<\infty$ for all $x_i\in K_i$, $i=1,...,N$. By Proposition 1
in \cite{Wer6}, $M$ is an equilibrium state for $u$.
Since $\Sigma$ is a compact metric space, it is well known that
$P_S(\Sigma)$ is a compact (in the weak$^*$ topology) convex space.
By the Krein-Milman Theorem, $P_S(\Sigma)$ is a closed convex hull
of its extreme points, which are exactly the ergodic measures (see
e.g. Theorem 6.10 in \cite{W}). Furthermore, the ergodic measures
can be characterized as the minimal elements of the set
$P_S(\Sigma)$ with respect to the partial order given by the
absolute continuity relation (see e.g. \cite{Kel}, Lemma 2.2.2) (to
be precise, the absolute continuity relation is a partial order on
the equivalence classes consisting of equivalent measures, but every
equivalence class containing an ergodic measures consists of a
single element). Therefore, there exists an ergodic
$\tilde\Lambda\in P_S(\Sigma)$ such that $\tilde\Lambda$ is
absolutely continuous with respect to $M$. Thus, by Lemma \ref{ace},
$\tilde\Lambda\in ES(u)$.\hfill$\Box$
\begin{theo}\label{gMme}
Suppose CMS $\mathcal{M}$ has a unique invariant Borel probability measure $\mu$ and $M(Y)=1$, where $M$ is the associated
generalized Markov measure.
Then the following hold.\\
$(i)$ $M$ is a unique equilibrium state for the energy function $u$,\\
$(ii)$ $P(u)=0$,\\
$(iii)$ $F(M)=\mu$,\\
$(iv)$ $h_M(S)=-\sum_{e\in E}\int_{K_{i(e)}}p_e\log p_e\ d\mu$.
\end{theo}
For a proof see \cite{Wer6}.
\subsection{Uniqueness and empiricalness of the invariant probability
measure}
Now, we are going to show that an irreducible CMS with probabilities
with square summable variation has a unique invariant probability
measure and it can be obtained empirically.
Before we move to the main theorem, we need to clear up some
technical details.
Let $\nu\in
P(K)$. Since $x\longmapsto P_x(Q)$ is Borel measurable for all
$Q\in\mathcal{B}(\Sigma^+)$ (see e.g. Lemma 1 in \cite{Wer3}), we
can define
\[\tilde \phi(\nu)(A\times Q):=\int\limits_{A}P_x(Q)d\nu(x)\]
for all $A\in\mathcal{B}(K)$ and
$Q\in\mathcal{B}\left(\Sigma^+\right)$. Then $\tilde \phi(\nu)$
extends uniquely to a Borel probability measure on $K\times\Sigma^+$
with
\[\tilde \phi(\nu)(\Omega)=\int
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\Omega\right\}\right)d\nu(x)\]
for all $\Omega\in\mathcal{B}\left( K\times\Sigma^+\right)$. Note
that the set of all $\Omega\subset K\times\Sigma^+$ for which the
integrand in the above is measurable forms a Dynkin system which
contains all rectangles. Therefore, it is measurable for all
$\Omega\in\mathcal{B} \left(K\times\Sigma^+\right)$.
Now, consider the following map
\begin{eqnarray*}
\xi:\Sigma&\longrightarrow& K\times\Sigma^+\\
\sigma&\longmapsto&(F(\sigma),(\sigma_1,\sigma_2,...)).
\end{eqnarray*}
\begin{lemma}\label{tl}
Suppose $\mathcal{M}$ is a CMS with an invariant Borel probability
measure $\mu$ such that $\sum_{i=1}^N\int_{K_i}d(x,x_i)\
d\mu(x)<\infty$ for all $x_i\in K_i$, $i=1,...,N$, and $M(Y)=1$, where $M$ is the generalized Markov measure
associated with $\mathcal{M}$ and $\mu$. Let $\Lambda\in P_S(\Sigma)$ be absolutely
continuous with respect to $M$. Then
\[\xi(\Lambda)=\tilde \phi(F(\Lambda)).\]
\end{lemma}
{\it Proof.} We only need to check that
\[\xi(\Lambda)\left(A\times _{1}[e_{1},...,e_n]^+\right)=\tilde \phi(F(\Lambda))\left(A\times _{1}[e_{1},...,e_n]^+\right)\]
for all cylinder sets $_{1}[e_{1},...,e_n]^+\subset\Sigma^+$ and
$A\in\mathcal{B}(K)$. For such sets
\begin{eqnarray*}
\xi(\Lambda)\left(A\times _{1}[e_{1},...,e_n]^+\right)&=&
\Lambda\left(F^{-1}(A)\cap _{1}[e_{1},...,e_n]\right)\\
&=&\int\limits_{F^{-1}(A)}1_{_{1}[e_{1},...,e_n]}d\Lambda
\end{eqnarray*}
where $_{1}[e_{1},...,e_n]\subset\Sigma$ is the pre-image of
$_{1}[e_{1},...,e_n]^+$ under the natural projection.
Recall that $F^{-1}(A)\in\mathcal{F}$. Therefore, by Lemma 6 in \cite{Wer6},
\begin{equation*}
E_M(1_{_1[e]}|\mathcal{F})=p_e\circ F\ M\mbox{-a.e.}.
\end{equation*}
Since $\Lambda$ is absolutely continuous with respect to $M$,
\begin{equation}\label{ce}
E_\Lambda(1_{_1[e]}|\mathcal{F})=p_e\circ F\ \Lambda\mbox{-a.e.},
\end{equation}
analogously as in the proof of Lemma \ref{ace}. This implies, by
the shift invariance of $\Lambda$ and the pull-out property of the
conditional expectation, that
\[E_\Lambda(1_{_1[e_{1},...,e_n]}|\mathcal{F})=P_{F(\sigma)}\left(
_{1}[e_{1},...,e_n]^+\right)\ \Lambda\mbox{-a.e.}.\] Therefore,
\begin{eqnarray*}
\int\limits_{F^{-1}(A)}1_{_{1}[e_{1},...,e_n]}d\Lambda&=&\int\limits_{F^{-1}(A)}P_{F(\sigma)}\left(
_{1}[e_{1},...,e_n]^+\right)d\Lambda(\sigma)\\
&=&\int\limits_{A}P_{x}\left(
_{1}[e_{1},...,e_n]^+\right)dF(\Lambda)(x),
\end{eqnarray*}
as desired.\hfill $\Box$
\begin{theo}\label{uim}
Suppose $\mathcal{M}$ is an irreducible CMS such that $P_x$ is absolutely continuous with respect to
$P_y$ for all $x,y\in K_i$, $i=1,...,N$. Then
the following hold.\\
(i) The CMS has a unique invariant Borel probability measure $\mu$.\\
(ii) Let $f_e:K\longrightarrow[-\infty,+\infty]$ be Borel measurable
such that $f_e|_{K_{i(e)}}$ is bounded and uniformly continuous for
all $e\in E$. Then, for every $x\in K$,
\[\frac{1}{n}\sum\limits_{k=o}^{n-1}f_{\sigma_{k+1}}\circ{w_{\sigma_k}\circ...\circ w_{\sigma_1}(x)}\to
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ d\mu\mbox{ for
$P_x$-a.e. }\sigma\in\Sigma^+.\]
\end{theo}
{\it Proof.} By Proposition \ref{ees},
$u$ has an ergodic equilibrium state $\Lambda$ which is absolutely continuous with respect to $M$ where $M$
is the generalized Markov measure associated with the CMS and its invariant Borel probability measure $\mu$ with the
property that $\sum_{i=1}^N\int_{K_i}d(x,x_i)\
d\mu(x)<\infty$ for all $x_i\in K_i$, $i=1,...,N$.
This implies that $\Lambda(Y)=1$. Set
\begin{equation}\label{ef}
v(\sigma)=\left\{\begin{array}{cc}
\log f_{\sigma_1}\circ F(\sigma)& \mbox{if }\sigma\in Y \\
-\infty& \mbox{if }\sigma\in\Sigma\setminus Y.
\end{array}\right.
\end{equation}
Then, by Birkhoff's Ergodic Theorem,
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}v\circ S^k\to\int v\ d\Lambda\ \Lambda\mbox{-a.e.}.\]
Since $F\circ S^k(\sigma)=w_{\sigma_k}\circ...\circ
w_{\sigma_1}(F(\sigma))$ for all $\sigma\in Y$, we can assume,
without loss of generality, that
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ w_{\sigma_k}\circ...\circ w_{\sigma_1}(F(\sigma))\to
\int v\ d\Lambda \mbox{ for all }\sigma\in Y.\] By (\ref{ce}),
\begin{eqnarray*}
\int v\ d\Lambda&=&\lim\limits_{n\to\infty}\sum\limits_{e\in
E}\int 1_{_1[e]}n\wedge f_e\circ F\
d\Lambda=\lim\limits_{n\to\infty}\sum\limits_{e\in E}\int p_e\circ
Fn\wedge f_e\circ F\
d\Lambda\\
&=&\lim\limits_{n\to\infty}\sum\limits_{e\in
E}\int\limits_{K_{i(e)}} p_en\wedge f_e\
dF(\Lambda)=\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_e f_e\
dF(\Lambda).
\end{eqnarray*}
Now,
applying the map $\xi$, we get
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}f_{\sigma_{k+1}}\circ w_{\sigma_k}\circ...\circ w_{\sigma_1}(x)\to
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ dF(\Lambda)
\mbox{ for all }(x,\sigma)\in \xi(Y).\] By Lemma \ref{tl},
$\tilde\phi(F(\Lambda))(\xi(Y))=\xi(\Lambda)(\xi(Y))=\Lambda(\xi^{-1}(\xi(Y)))\geq\Lambda(Y)=1$.
Since $\tilde\phi(F(\Lambda))$ is a probability measure,
\begin{eqnarray*}
1&=&\tilde\phi(F(\Lambda))(\xi(Y))=\int
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)dF(\Lambda)(x)\\
&=&\sum\limits_{i=1}^N \int\limits_{K_i}
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)dF(\Lambda)(x).
\end{eqnarray*}
Furthermore, for each $i=1,...,N$, there exists
$x_i\in K_i$ such that
\begin{eqnarray*}
&&\int\limits_{K_i}
P_x\left(\left\{\sigma\in\Sigma^+:(x,\sigma)\in\xi(Y)\right\}\right)dF(\Lambda)(x)\\
&=&P_{x_i}\left(\left\{\sigma\in\Sigma^+:(x_i,\sigma)\in\xi(Y)\right\}\right)F(\Lambda)(K_i).
\end{eqnarray*}
Set
\[Q_i:=\left\{\sigma\in\Sigma^+:(x_i,\sigma)\in\xi(Y)\right\}.\]
Then
\[\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f_{\sigma_{k+1}}\circ
w_{\sigma_k}\circ...\circ w_{\sigma_1} \left(x_i\right) =
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ dF(\Lambda)\mbox{
for all }\sigma\in Q_i,\] and
\[\sum\limits_{i=1}^NP_{x_i}\left(Q_i\right)F(\Lambda)\left(K_i\right)=1.\]
Note that, since $\Lambda$ is an equilibrium state for $u$,
$U^*F(\Lambda)=F(\Lambda)$ (e.g. Proposition 1 in \cite{Wer6}).
Thus, by the irreducibility of the CMS $F(\Lambda)(K_i)>0$ for all
$i=1,...,N$ (see Lemma 1 in \cite{Wer1}). This implies that
$P_{x_i}\left(Q_i\right)=1$ for all $i=1,...,N$, as $F(\Lambda)$ and
$P_{x_i}$ are probability measures. Now, fix $x\in K_i$ for some
$i\in\{1,...,N\}$. Since $P_x$ is absolutely continuous with respect
to $P_{x_i}$, $P_{x}\left(Q_i\right)=1$. Furthermore, the
contractiveness on average condition (\ref{cc}) implies that
\[\int d(w_{\sigma_k}\circ...\circ w_{\sigma_1}(x),w_{\sigma_k}\circ...\circ w_{\sigma_1}(x_i))\ dP_x\leq a^kd(x,x_i).\]
Therefore, it follows, by the Borel-Cantelli argument, that
\[d(w_{\sigma_k}\circ...\circ w_{\sigma_1}(x),w_{\sigma_k}\circ...\circ w_{\sigma_1}(x_i))\to 0\ P_x\mbox{-a.e.}.\]
As each $f_e|_{K_{i(e)}}$ is uniformly continuous, we conclude that
\[\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f_{\sigma_{k+1}}\circ
w_{\sigma_k}\circ...\circ w_{\sigma_1} (x) = \sum\limits_{e\in
E}\int\limits_{K_{i(e)}} p_e f_e dF(\Lambda)\ P_x\mbox{-a.e.}.\]
Thus, the claim in (ii) will follow if we show that $F(\Lambda)$ is the unique
invariant Borel probability measure of the CMS.
Let $g\in C_B(K)$ and set $f_e=g$ for all $e\in E$. Then the
integration with respect to $P_x$ and Lebesgue's dominated
convergence theorem imply that
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}U^kg(x)\to\int g dF(\Lambda)\mbox{ for
all }g\in C_B(K).\] Since $x$ was arbitrary, again an integration and Lebesgue's dominated convergence theorem imply that
\begin{eqnarray}\label{e1}
\frac{1}{n}\sum\limits_{k=0}^{n-1}{U^*}^k\nu\stackrel{w^*}{\to}
F(\Lambda)\mbox{ for all }\nu\in P(K).
\end{eqnarray}
Suppose $U^*\lambda=\lambda$ for some $\lambda\in P(K)$. Then
\[\frac{1}{n}\sum\limits_{k=0}^{n-1}{U^*}^k\lambda=\lambda\mbox{ for
all }n\in\mathbb{N}.\] Therefore, $F(\Lambda)=\lambda$.
Hence, $F(\Lambda)$ is a unique invariant probability measure of the CMS, i.e.
$F(\Lambda)=\mu$.\hfill$\Box$
The following corollary generalizes corresponding results from
\cite{BDEG}, \cite{Elton}, \cite{Wer1} and \cite{Wer5}.
\begin{cor}\label{uc}
Suppose $\mathcal{M}$ is an irreducible CMS such that each $p_e|_{K_{i(e)}}$
has a square summable variation. Then the following hold.\\
(i) The CMS has a unique invariant Borel probability measure $\mu$.\\
(ii) Let $f_e:K\longrightarrow[-\infty,+\infty]$ be Borel measurable
such that $f_e|_{K_{i(e)}}$ is bounded and uniformly continuous for
all $e\in E$. Then, for every $x\in K$,
\[\frac{1}{n}\sum\limits_{k=o}^{n-1}f_{\sigma_{k+1}}\circ{w_{\sigma_k}\circ...\circ w_{\sigma_1}(x)}\to
\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_ef_e\ d\mu\mbox{ for
$P_x$-a.e. }\sigma\in\Sigma^+.\]
\end{cor}
{\it Proof.} By Lemma \ref{acl}, the measures $P_x$ and $P_y$ are
absolutely continuous with respect to each other for all $x,y\in
K_i$, $i=1,...,N$. Therefore, the claim follows by Theorem
\ref{uim}. \hfill$\Box$
\begin{cor}\label{DCc}
Suppose $\mathcal{M}$ is an irreducible CMS such that each
$p_e|_{K_{i(e)}}$ has a square summable
variation.
Then the following hold.\\
$(i)$ The generalized Markov measure $M$ is a unique equilibrium state for the energy function $u$,\\
$(ii)$ $P(u)=0$,\\
$(iii)$ $F(M)=\mu$,\\
$(iv)$ $h_M(S)=-\sum_{e\in E}\int_{K_{i(e)}}p_e\log p_e\ d\mu$.
\end{cor}
{\it Proof.} Since the CMS has a unique invariant Borel probability
measure $\mu$ (Corollary \ref{uc}) and $M(Y)=1$ (Lemma \ref{pY}
$(ii)$), the claim follows by Theorem \ref{gMme} and Theorem
\ref{uim}. \hfill$\Box$
Finally, we would like to give an application of Theorem \ref{uim}
(iv) which allows an empirical calculation of Kolmogorov-Sinai
entropy $h_M(S)$ of the generalized Markov shift associated with a
given CMS without knowing anything about its invariant measure.
\begin{cor}\label{Ec}
Suppose $\mathcal{M}$ is an irreducible CMS such that $P_x$ is absolutely continuous with respect to
$P_y$ for all $x,y\in K_i$, $i=1,...,N$. Then for every $x\in K$
\[\lim\limits_{n\to\infty}\frac{1}{n}\log P_x( _1[\sigma_1,...,\sigma_n])=-h_M(S) \mbox{ for
}P_x\mbox{-a.e. } \sigma\in\Sigma^+.\]
\end{cor}
{\it Proof.} By Theorem \ref{uim} (i) and Theorem \ref{gMme} (iv),
\[h_M(S)=-\int\sum\limits_{e\in E} p_e\log
p_ed\mu.\]
Set $f_e:=\log p_e$ for all $e\in E$. Then, by Theorem \ref{uim}
(ii), for every $x\in K$,
\[\lim\limits_{n\to\infty}\frac{1}{n}\log [p_{\sigma_1}(x)p_{\sigma_2}\circ w_{\sigma_1}(x)...
p_{\sigma_n}\circ w_{\sigma_{n-1}}\circ...\circ
w_{\sigma_1}(x)]=-h_M(S) \] for $P_x$-a.e. $\sigma\in\Sigma^+$, as
desired. \hfill$\Box$
\begin{Remark}
The author would like to point out that a similar entropy formula as that proved in Theorem \ref{gMme} (iv)
plays a central role in the recent book of Wojciech Slomczynski \cite{S}.
\end{Remark}
\begin{Remark}
This paper contributes more to the mystery of an ergodic classification of the generalized Markov shifts associated
with aperiodic CMSs. So far, we know only that they are strongly
mixing if the restrictions of the probability functions on their
vertex sets are Dini-continuous \cite{Wer5} and they are weak
Bernoulli if in addition all maps are contractive (the latter is not
difficult to see by reducing it to the corresponding result of P.
Walters for the natural extension of $g$-measures \cite{W1} since
$g:=\exp u|_{\Sigma_G}$ is Dini-continuous in this case). H. Brebee
\cite{Ber} has shown the very weak Bernoulli property of the natural
extension of $g$-measures on a full shift under a continuity
condition on $g$ which is weaker than the Dini-continuity and not
comparable with the square summability of variation (see \cite{JO}).
From this paper, we only can see that an aperiodic generalized
Markov shift is ergodic if the restrictions of the probability
functions on their vertex sets have a square summable variation
(Corollary \ref{DCc} and Lemma \ref{ees}).
\end{Remark}
\subsection*{Acknowledgements}
I would like to thank Barry Ridge, Wang Yang and Zhenkun Wang for
their help in the production of this paper.
|
{
"timestamp": "2005-08-18T12:13:14",
"yymm": "0503",
"arxiv_id": "math/0503633",
"language": "en",
"url": "https://arxiv.org/abs/math/0503633"
}
|
\section{Introduction}
Whether or not quantum information processing and quantum computing
\cite{QCcomp} become practical technologies crucially depends on the
ability to implement high-fidelity quantum logic gates in a scalable way
\cite{diVinc}. Among alternative routes to this challenging goal, are of
particular interest the schemes operating with photons as qubits
\cite{photqc,linopt}, since photons are ideal carriers of quantum
information in terms of transfer rates, distances and scalability.
A current trend makes use of linear optical elements and photodetectors
for the implementation of key components of quantum communications and
information processing in a probabilistic way \cite{linopt}. The desirable
objective though is a {\em deterministic} realization of entangling
operations between individual photons, which require sufficiently strong
nonlinearities or long interaction times. These are achievable,
at the single-photon level, by tight spatial confinement of the photons,
in the very demanding regime of strong atom-field coupling in high-$Q$
cavities \cite{phphcav}.
A promising alternative is to enhance both the nonlinear susceptibility
and interaction time, by employing the ultra-slow light propagation in
resonant media subject to electromagnetically induced transparency (EIT)
\cite{EIT,ScZub,vred}. In a pioneering work, Schmidt and Imamo\u{g}lu have
suggested the possibility of enhanced, non-absorptive, cross-phase modulation
of two weak fields in the EIT regime \cite{imam}, provided their interaction
time is long enough. However, upon entering the EIT medium light pulses
become spatially compressed by the ratio of group velocity $v$ to
the vacuum speed of light $c$ \cite{harhau}, so that the interaction time
of two colliding pulses is a constant independent of $v$. In order to maximize
this time, copropagating pulses with nearly matched group velocities have
been proposed \cite{lukimam,petmal}. The essential drawback of such an
approach is the spatial inhomogeneity of the conditional phase shift, causing
spectral broadening of the interacting pulses, thereby preventing the
realization of a high-fidelity quantum phase gate. Alternative approaches
free of spectral broadening have been suggested \cite{IFGKDP,lukin-pbg,MMMF}.
In all of them, however, a rather tight transverse confinement through
waveguiding or focusing of the pulses, close to the diffraction limit of
$\lambda^2$, is needed in order to attain a phase shift of $\pi$, which is
technically challenging.
When the light pulses enter EIT media, photonic excitations are temporarily
transferred to atomic excitations through the formation of quasi-particles,
the so-called dark-state (or slow-light) polaritons, which are superpositions
of light and matter degrees of freedom \cite{fllk}. The spatial compression
of the pulses leads to an {\em amplification} of the matter components of
polaritons. In this Letter we propose a hitherto unexplored mechanism for the
collisional entanglement of two single-quantum polaritons mediated by
the long-range interaction of their matter (atomic) components and
demonstrate its effectiveness. In contrast to the previous schemes which
employ {\em local} interactions, namely either two photons interact with
the same atom \cite{lukimam,petmal,IFGKDP,lukin-pbg} or two atoms after
absorbing the photons undergo $s$-wave scattering \cite{MMMF}, here the
two polaritons interact via the long-range dipole-dipole interactions
between their atomic components in the highly excited Rydberg states.
In a static electric field, these internal Rydberg states, populated
only in the presence of polaritons, possess large permanent dipole
moments \cite{RydAtoms}, which can further enhance the effective
interaction time between the polaritons. We will show that under
experimentally realizable conditions, the conditional phase shift
accumulated during a collision of two single-quantum polaritons
is {\em spatially homogeneous} and can be sufficiently large for
the implementation of the quantum phase gate, even for moderate
focusing or transverse confinement of interacting pulses.
We note that quantum gates for individual Rydberg atoms, coupled by
dipole-dipole interaction, has been proposed in \cite{JCZRCL}.
\begin{figure}[t]
\includegraphics[width=8.5cm]{alspls.eps}
\caption{(a)~Level scheme of atoms interacting with weak (quantum)
fields $E_{1,2}$ on the transitions $\ket{g} \to \ket{e_{1,2}}$
and strong driving fields of Rabi frequencies $\Omega_{1,2}$ on
the transitions $\ket{e_{1,2}} \to \ket{d_{1,2}}$, respectively.
$V_{\rm dd}$ denotes the dipole-dipole interaction between pairs
of atoms in Rydberg states $\ket{d}$.
(b)~Upon entering the medium, each field having Gaussian transverse
intensity profile is converted into the corresponding polariton
$\Psi_{1,2}$ representing a coupled excitation of the field and
atomic coherence. These polaritons propagate in the opposite directions
with slow group velocities $v_{1,2}$ and interact via the dipole-dipole
interaction.}
\label{fig:als}
\end{figure}
We consider an ensemble of cold alkali atoms with level configuration
as in Fig.~\ref{fig:als}. All the atoms are initially prepared
in the ground state $\ket{g}$. Two weak (quantum) fields $E_{1,2}$
having orthogonal polarizations and propagating in the opposite
directions along the $z$ axis resonantly interact with
the atoms on the transitions $\ket{g} \to \ket{e_{1,2}}$, respectively.
The intermediate states $\ket{e_{1,2}}$ are resonantly coupled by two
strong (classical) driving fields with Rabi frequencies $\Omega_{1,2}$
to the highly excited Rydberg states $\ket{d_{1,2}}$. In a static electric
field $E_{\rm st} \mathbf{e}_z$, the Rydberg states $\ket{d}$ possess large
permanent dipole moments $\mathbf{p}= \frac{3}{2} n q e a_0 \mathbf{e}_z$,
where $n$ and $q \equiv n_1 - n_2$ are, respectively, the (effective)
principal and parabolic quantum numbers, $e$ is the electron charge, and
$a_0$ is the Bohr radius \cite{RydAtoms}. A pair of atoms $i$ and $j$ at
positions $\mathbf{r}_i$ and $\mathbf{r}_j$ excited to states $\ket{d}$ interact
with each other via the dipole-dipole potential
\[
V_{\rm dd} = \frac{\mathbf{p}_i \cdot \mathbf{p}_j
- 3 (\mathbf{p}_i \cdot \mathbf{e}_{ij}) (\mathbf{p}_j \cdot \mathbf{e}_{ij})}
{4 \pi \epsilon_0 |\mathbf{r}_i -\mathbf{r}_j|^3} ,
\]
where $\mathbf{e}_{ij}$ is a unit vector along the interatomic direction.
This dipole-dipole interaction results in an energy shift of the pair of
Rydberg atoms, while we assume that the state mixing within the same $n$
manifold is suppressed by the proper choice of parabolic $q$ and magnetic
$m$ quantum numbers \cite{RydAtoms,JCZRCL}. In the frame rotating with the
frequencies of the optical fields, the interaction Hamiltonian has the
following form
\begin{equation}
H = V_{\rm af} + V_{\rm dd} ,
\end{equation}
where the atom-field and dipole-dipole interaction terms are given,
respectively, by
\begin{subequations}
\label{VafVdd}
\begin{eqnarray}
V_{\rm af} &=& - \hbar \sum_j^N [g_1^j \hat{\cal E}_1 \hat{\sigma}_{e_1 g}^j
+ \Omega_1 \hat{\sigma}_{d_1 e_1}^j
\nonumber \\ & & \;\;\;\;\;\;\;\;\;\;
+g_2^j \hat{\cal E}_2 \hat{\sigma}_{e_2 g}^j + \Omega_2 \hat{\sigma}_{d_2 e_2}^j
+ {\rm H. c.}], \\
V_{\rm dd} &=& \hbar \sum_{i > j}^N
\hat{\sigma}_{d d}^i \Delta(\mathbf{r}_i -\mathbf{r}_j) \hat{\sigma}_{d d}^j .
\end{eqnarray}
\end{subequations}
Here $N = \rho V$ is the total number of atoms, $\rho$ being the (uniform)
atomic density and $V$ the volume;
$\hat{\sigma}_{\mu \nu}^j \equiv \ket{\mu}_{jj}\bra{\nu}$
is the transition operator of the $j$th atom; $\hat{\cal E}_l$ is the
slowly-varying operator, corresponding to the electric field $E_l$ ($l=1,2$),
which obeys the commutation relations
$[\hat{\cal E}_l(\mathbf{r}),\hat{\cal E}^{\dagger}_{l^{\prime}}(\mathbf{r}^{\prime})]
= V \delta_{l l^{\prime}} \delta(\mathbf{r} - \mathbf{r}^{\prime})$;
$g_l^j$ is the corresponding atom-field coupling constant on the transition
$\ket{g}_j \to \ket{e_l}_j$; and $\hbar \Delta(\mathbf{r}_i -\mathbf{r}_j) \equiv
\, _i \bra{d} _j \bra{d} V_{\rm dd} \ket{d}_i \ket{d}_j$
is the dipole-dipole energy shift for a pair of atoms $i$ and $j$,
given by
\[
\Delta(\mathbf{r}_i -\mathbf{r}_j) = C \, \frac{1 - 3 \cos^2 \vartheta}{|\mathbf{r}_i -\mathbf{r}_j|^3} ,
\]
where $\vartheta$ is the angle between vectors $\mathbf{e}_z$ and
$\mathbf{e}_{ij}$,
and $C = \wp_{d_l} \wp_{d_{l^{\prime}}}/(4 \pi \epsilon_0 \hbar)$
is a constant proportional to the product of atomic dipole moments
$\wp_{d_l} = \bra{d_l} \mathbf{p} \ket{d_l}$ assumed the same for both
states $\ket{d_{1,2}}$, $\wp_{d_{1,2}} = \wp_d$.
Let us introduce collective atomic operators
$\hat{\sigma}_{\mu \nu}(\mathbf{r}) = \frac{1}{N_r} \sum_{j=1}^{N_r} \hat{\sigma}_{\mu \nu}^j$
averaged over the volume element $d^3 r$ containing $N_r = \rho \, d^3 r \gg 1$
atoms around position $\mathbf{r}$. Then Eqs.~(\ref{VafVdd}) can be cast in the
continuous form
\begin{subequations}
\label{VVcont}
\begin{eqnarray}
V_{\rm af} &=& - \hbar \rho \int d^3 r \sum_{l=1,2}
[g_l \hat{\cal E}_l \hat{\sigma}_{e_l g}(\mathbf{r})
+ \Omega_l \hat{\sigma}_{e_l d_l}(\mathbf{r})]
+ {\rm H. c.} ,\;\;\;\;\; \\
V_{\rm dd} &=& \hbar \rho^2 \int \! \! \! \int d^3 r \, d^3 r^{\prime}
\hat{\sigma}_{d d}(\mathbf{r}) \Delta(\mathbf{r} -\mathbf{r}^{\prime}) \hat{\sigma}_{d d} (\mathbf{r}^{\prime}) .
\end{eqnarray}
\end{subequations}
Using Eqs.~(\ref{VVcont}), one can derive a set of Heisenberg-Langevin
equations for the atomic operators $\hat{\sigma}_{\mu \nu}$ \cite{ScZub}.
When the number of photons in the quantum fields $\hat{\cal E}_l$ is much smaller
than the number of atoms, these equations can be solved perturbatively
in the small parameters $g_l \hat{\cal E}_l/\Omega_l$ and in the adiabatic
approximation for all the fields \cite{fllk}, with the result
\begin{subequations}
\label{sigmas}
\begin{eqnarray}
\hat{\sigma}_{ge_l}(\mathbf{r}) &=& -\frac{i}{\Omega_l}
\left[ \frac{\partial}{\partial t} + i \hat{\alpha}(\mathbf{r}) \right] \hat{\sigma}_{gd_l}(\mathbf{r}) , \\
\hat{\alpha}(\mathbf{r}) &= & \rho \int d^3 r^{\prime} \Delta(\mathbf{r} -\mathbf{r}^{\prime})
[\hat{\sigma}_{d_1 d_1}(\mathbf{r}^{\prime}) + \hat{\sigma}_{d_2 d_2}(\mathbf{r}^{\prime})] , \quad \\
\hat{\sigma}_{gd_l}(\mathbf{r}) &=& - \frac{g_l \hat{\cal E}_l}{\Omega_l^*} , \;\;\;\;
\hat{\sigma}_{d_l d_l}(\mathbf{r}) = \hat{\sigma}_{d_l g}(\mathbf{r}) \hat{\sigma}_{gd_l}(\mathbf{r}) .
\end{eqnarray}
\end{subequations}
Let us assume that the transverse profile of both quantum fields is
described by a Gaussian $e^{-r_{\bot}^2/w^2}$ of width $w$, where
$r_{\bot} = |\mathbf{r}_{\bot}|$ is the distance from the field propagation
axis, while the Rabi frequencies of classical driving fields $\Omega_l$
are uniform over the entire volume $V$. We may then write
$g_l \hat{\cal E}_l = g_l(\mathbf{r}_{\bot}) \hat{\cal E}_l(z)$, where the traveling-wave electric
field operators $\hat{\cal E}_l(z) = \sum_k a_l^k e^{ikz}$ are expressed through
the superposition of bosonic operators $a_l^k$ for the longitudinal field
modes $k$, while the (transverse-position-dependent) coupling constants
are given by $g_l(\mathbf{r}_{\bot}) = \tilde{g}_l e^{-r_{\bot}^2/2 w^2}$, with
$\tilde{g}_l = (\wp_{ge_l}/\hbar) \sqrt{\hbar \omega/2 \epsilon_0 V}$, $\wp_{ge_l}$
being the dipole matrix element on the transition $\ket{g} \to \ket{e_l}$,
$V = \pi w^2 L$, and $L$ the medium length. Under this approximation, the
propagation equations for the slowly-varying quantum fields have the form
\begin{equation}
\left(\frac{\partial}{\partial t} \pm c\frac{\partial}{\partial z}\right)
\hat{\cal E}_l(z,t) = i \tilde{g}_l N \hat{\sigma}_{g e_l}(z), \label{Eprop}
\end{equation}
the sign ``$+$'' or ``$-$'' corresponding to $l = 1$ or $2$, respectively.
Following \cite{fllk}, we introduce new quantum fields $\hat{\Psi}_l$---dark
state polaritons---via the canonical transformations
\begin{equation}
\hat{\Psi}_l = \cos \theta_l \hat{\cal E}_l - \sin \theta_l \sqrt{N} \hat{\sigma}_{gd_l} ,
\label{polars}
\end{equation}
where the mixing angles $\theta_l$ are defined through
$\tan^2 \theta_l = \tilde{g}_l^2 N/|\Omega_l|^2$. These polaritons
correspond to coherent superpositions of electric field $\hat{\cal E}_l$
and atomic coherence $\hat{\sigma}_{gd_l}$ operators. Employing the
plane-wave decomposition of the polariton operators, one can
show that in the weak-field limit, they obey the bosonic commutation
relations $[\hat{\Psi}_{l}(z),\hat{\Psi}_{l^{\prime}}^{\dagger}(z^{\prime})]
\simeq L\delta_{ll^{\prime}} \delta(z-z^{\prime})$. Using Eqs.~(\ref{sigmas})
and (\ref{Eprop}), we obtain the following propagation equations for
the polariton operators,
\begin{equation}
\left(\frac{\partial}{\partial t} \pm v_l\frac{\partial}{\partial z}\right)
\hat{\Psi}_l(z,t) = - i \sin^2 \theta_l \hat{\alpha}(z,t)\hat{\Psi}_l(z,t) . \label{Psiprop}
\end{equation}
Here $v_l = c \cos^2 \theta_l$ is the group velocity, while operator
$\hat{\alpha}(z,t)$ is responsible for the self- and cross-phase modulation between
the polaritons. It is related to the polariton intensity (excitation number)
operators $\hat{\cal I}_l \equiv \hat{\Psi}_l^{\dagger} \hat{\Psi}_l$ via
\begin{equation}
\hat{\alpha}(z,t) = \frac{1}{L} \int_0^L \!\! d z^{\prime} \Delta(z - z^{\prime})
[\sin^2 \theta_1 \hat{\cal I}_1(z^{\prime}, t) + \sin^2 \theta_2 \hat{\cal I}_2(z^{\prime}, t)] ,
\end{equation}
where the 1D dipole-dipole interaction potential $\Delta(z - z^{\prime})$ is obtained
after the integration over the transverse profile of the quantum fields,
\begin{eqnarray}
\Delta(z - z^{\prime}) &=& \frac{1}{\pi w^2} \int_{0}^{2 \pi} \!\! d \varphi^{\prime}
\!\! \int_{0}^{\infty} \!\! d r^{\prime}_{\bot} r^{\prime}_{\bot}
e^{-r^{\prime 2}_{\bot}/w^2} \Delta(z \mathbf{e}_z - \mathbf{r}^{\prime}) \nonumber \\
&=& \frac{2 C}{w^3}
\left[ \frac{2 |z -z^{\prime}|}{w}
-\sqrt{\pi} \left(1+ 2 \frac{|z -z^{\prime}|^2}{w^2} \right)
\right. \nonumber \\ & & \;\;\;\; \left. \times
\exp \left( \frac{|z -z^{\prime}|^2}{w^2} \right)
{\rm erfc}\left( \frac{|z -z^{\prime}|}{w} \right) \right] , \label{1Dddpot}
\end{eqnarray}
and is shown in Fig.~\ref{fig:phshgr}(a).
It follows from Eq.~(\ref{Psiprop}) that the intensity operators
$\hat{\cal I}_l$ are constants of motion: $\hat{\cal I}_l(z,t) = \hat{\cal I}_l(z \mp v_l t,0)$,
the upper (lower) sign corresponding to $l=1$ ($l=2$). Then the formal
solution for the polariton operators can be written as
\begin{eqnarray}
\hat{\Psi}_l(z,t) & = &
\exp \left[- i \sin^2 \theta_l \int_0^t \!\! d t^{\prime}
\hat{\alpha}(z \mp v_l(t-t^{\prime}),t^{\prime}) \right]
\nonumber \\ & & \;\; \times
\hat{\Psi}_l(z \mp v_l t,0) . \label{Psisolv}
\end{eqnarray}
Equation (\ref{Psisolv}) is our central result. Let us outline the
approximations involved in the derivation of this solution. In order
to accommodate the pulses in the medium with negligible losses, their
duration $T$ should exceed the inverse of the EIT bandwidth
$\delta \omega = |\Omega_l|^2 (\gamma_{ge_l} \sqrt{\kappa_0 L})^{-1}$,
where $\gamma_{ge_l}$ is the transversal relaxation rate and
$\kappa_0 \simeq 3 \lambda^2 /(2 \pi)\rho$ is the resonant absorption coefficient
on the transition $\ket{g} \to \ket{e_l}$. This yields the condition
$(\kappa_0 L)^{-1/2} \ll T v_l/L < 1$ which requires a medium with large
optical depth $\kappa_0 L \gg 1$ \cite{fllk}. In addition, the dipole-dipole
energy shift should lie within the EIT bandwidth $\delta \omega$ for all
$|z-z^{\prime}| \leq L$, which implies that $|\Delta(0)|=2 \sqrt{\pi} C/w^3 < \delta \omega$.
Finally, the propagation/interaction time of the two pulses
$t_{\rm out} = L/v_l$ is limited by the relaxation rate of the
Rydberg states $\gamma_{d_l}$ via $t_{\rm out} \gamma_{d_l} \ll 1$.
\begin{figure}[t]
\includegraphics[width=7cm]{ddphsh.eps}
\caption{(a)~The 1D dipole-dipole potential $\Delta({\zeta})$ of
Eq.~(\ref{1Dddpot}) as a function of dimensionless distance
$\zeta = (z - z^{\prime})/w$, in units of $2 C/w^3$ Hz.
(b)~ The resulting phase-shift $\phi(\tau) \equiv \phi(vt,L-vt,t)$
of Eq.~(\ref{phiphsh}) as a function of dimensionless time
$\tau = vt/w$, in units of $2 C/(v w^2)$ rad.}
\label{fig:phshgr}
\end{figure}
From now on, we assume that $\theta_{1,2} = \theta$, i.e.,
$\tilde{g}_1^2 N/|\Omega_1|^2 = \tilde{g}_2^2 N/|\Omega_2|^2$,
which yields $v_{1,2} = v = c \cos^2 \theta$.
We are interested in the evolution of input state
\begin{equation}
\ket{\Phi_{\rm in}} = \ket{1_1} \otimes \ket{1_2} ,
\end{equation}
composed of two single-excitation polariton wavepackets
\[
\ket{1_l} = \frac{1}{L} \int \! dz f_l(z) \hat{\Psi}_l(z)^{\dagger} \ket{0},
\]
where $f_l(z)$ define the spatial envelopes of the corresponding wavepackets
$l=1,2$ which initially (at $t=0$) are localized around $z=0,L$, respectively.
For such an initial state, all the relevant information is contained in the
expectation values of the polariton intensities
$\expv{\hat{\cal I}_l(z,t)} = \bra{\Phi_{\rm in}} \hat{\cal I}_l(z,t) \ket{\Phi_{\rm in}}$
and the two-particle wavefunction \cite{ScZub,lukimam,petmal}
\begin{equation}
F_{12}(z_1,z_2,t) = \bra{0} \hat{\Psi}_1(z_1,t) \hat{\Psi}_2(z_2,t)\ket{\Phi_{\rm in}}
\label{tpwv}.
\end{equation}
With the above solution, for the polariton intensities we have
$\expv{\hat{\cal I}_{1,2}(z,t)}=\expv{\hat{\cal I}_{1,2}(z \mp vt,0)}=|f_{1,2}(z \mp vt)|^2$,
which describes the shape-preserving counter-propagation of the two polaritons
with group velocity $v$. Substituting the operator solution (\ref{Psisolv})
into (\ref{tpwv}), after some algebra, we obtain the following expression
for the two-particle wavefunction
\begin{eqnarray}
F_{12}(z_1,z_2,t) &=& f_1(z_1 - vt) f_2(z_2 + vt) \exp[i \phi(z_1,z_2,t)] ,
\qquad \\
\phi(z_1,z_2,t) &=& - \sin^4 \theta \int_0^t \!\! d t^{\prime}
\Delta(z_1 - z_2 - 2 v (t - t^{\prime}) ) , \label{phiphsh}
\end{eqnarray}
which indicates that the dipole-dipole interaction between the two
single-excitation polaritons results in the conditional phase-shift
$\phi(z_1,z_2,t)$. We consider a situation in which at time $t=0$,
the first pulse is localized at $z_1 =0$ and the second pulse is at
$z_2 = L$, while after the interaction, at time $t_{\rm out} = L/v$,
the coordinates of the two pulses are $z_1 = L$ and $z_2 = 0$, respectively
[Fig.~\ref{fig:phshgr}(b)]. Then the phase-shift accumulated during the
interaction is spatially uniform, and is given by
\begin{equation}
\phi(L,0,L/v) = - \frac{\sin^4\theta}{v} \int_0^L \!\! d z^{\prime}
\Delta(2 z^{\prime} -L ) = \frac{2 C \sin^4 \theta}{v w^2} .
\end{equation}
This remarkably simple result is obtained upon replacing the variable
$(2 z^{\prime} -L)/w \to \zeta^{\prime}$ and extending the integration limits
to $L/w \to \infty$. The main limitation on the phase shift is imposed
by the condition $|\Delta(0)| < \delta \omega$. In terms of experimentally relevant
parameters, the group velocity is $v \simeq 2 |\Omega|^2 /(\kappa_0 \gamma_{ge}) \ll c$
($\sin^2 \theta \simeq 1$), and we have $\phi <\frac{1}{2}w\sqrt{\kappa_0/\pi L}$.
To relate the foregoing discussion to a realistic experiment, let us
assume an ensemble of cold alkali atoms in the ground state $\ket{g}$
with density $\rho \sim 10^{14}$~cm$^{-3}$ confined in a trap of length
$L \sim 100 \;\mu$m. The resonant quantum fields with $\lambda \sim 0.5\;\mu$m
have the transverse width $w \sim 30\;\mu$m. In the presence of driving
fields with appropriate frequencies, the single-photon pulses lead to the
(two-photon) excitation of single atoms to the Rydberg states $\ket{d}$
with quantum numbers $n \simeq 25$ and $q=n-1$. The corresponding dipole
moments are $\wp_d \simeq 900 e a_0$, while
$\gamma_d \sim 2 \times 10^3$~s$^{-1}$ \cite{RydAtoms}. With
$\gamma_{ge} \sim 10^7$~s$^{-1}$ and $\Omega \sim 1.6 \times 10^7$~rad/s,
the group velocity is $v \simeq 4$~m/s, and the accumulated phase shift
is $\phi \simeq \pi$ with the fidelity $F = \exp(-\gamma_d L/v)\gtrsim 0.95$.
To summarize, we have studied a novel highly-efficient scheme for
cross-phase modulation and entanglement of two counterpropagating
single-photon wavepackets, employing their ultra-small group velocities
in atomic vapors, under the conditions of electromagnetically induced
transparency, and the strong long-range dipole-dipole interactions of
the accompanying Rydberg-state excitations in a ladder-type field-atom
coupling setup. We have solved, in the weak-field and adiabatic
approximations, the effective one-dimensional propagation equations
for the polariton operators and have shown that the dipole-dipole
interaction leads to a {\em homogeneous} conditional phase shift
that reach the value of $\pi$ even if the transverse cross section of
the pulses $w^2$ is much (three orders of magnitude) larger than the
diffraction limit $\lambda^2$. This is the obvious merit of the present
proposal, as compared to previous schemes based on local interactions
of photons or slow-light polaritons
\cite{imam,harhau,lukimam,petmal,IFGKDP,lukin-pbg,MMMF},
which require the photonic beam cross section to be comparable to the cross
section for atomic resonant absorption. Hence, our proposal paves the
way to the coveted deterministic entanglement of two single-photon pulses
and the realization of the universal photonic phase gate \cite{IFGKDP}.
\begin{acknowledgments}
This work was supported by the EC (QUACS RTN and ATESIT network),
ISF, and Minerva.
\end{acknowledgments}
|
{
"timestamp": "2005-03-07T16:59:02",
"yymm": "0503",
"arxiv_id": "quant-ph/0503071",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503071"
}
|
\section{Introduction}
\label{sec:intro}
We consider a family of generalized nonlinear Schr\"odinger and
Hartree equations with a focusing nonlinearity. These equations have
solitary wave solutions, and, in this paper, we study the effective
dynamics of such solitary waves. The equations have the form:
\begin{equation}\label{eq:NLS}
{\rm i} \partial_t \psi(x,t) = -\Delta \psi(x,t) + V(x)\psi(x,t) - f(\psi)(x,t),
\end{equation}
where $t\in \mathbb{R}$ is time, $x\in\mathbb{R}^d$ denotes a point in
physical space, $\psi:\mathbb{R}^d\times\mathbb{R}\mapsto \mathbb{C}$
is a (one-particle) wave function, $V$ is the external potential,
which is a real-valued, confining, and slowly varying function on
$\mathbb{R}^d$, and $f(\psi)$ describes a nonlinear self-interaction
with the properties that $f(\psi)$ is ``differentiable'' in $\psi$,
$f(0)=0$, and $f(\bar{\psi})=\overline{f(\psi)}$. Precise assumptions
on $V$ and $f$ are formulated in Section~\ref{sec:ass}.
The family of nonlinearities of interest to us
includes local nonlinearities, such as
\begin{equation}
f(\psi)=\lambda|\psi|^{s}\psi, \ 0<s<\frac{4}{d},\ \lambda>0,
\end{equation}
and Hartree nonlinearities
\begin{equation}
f(\psi)=\lambda(\Phi*|\psi|^{2})\psi, \ \lambda>0,
\end{equation}
where the (two-body) potential $\Phi$ is real-valued, of positive
type, continuous, spherically symmetric, and tends to 0 as
$|x|\rightarrow \infty$. Here $\Phi*g:=\int \Phi(x-y)g(y)\diff^d y$ denotes
convolution. Such equations are encountered in the theory of Bose
gases (BEC), in nonlinear optics, in the theory of water waves and in
other areas of physics.
It is well known that \Eref{eq:NLS} has solitary wave solutions when
$V\equiv 0$.
Let $\sol_\mu\in\set{L}^2$ be a
spherically symmetric, positive solution of the nonlinear
eigenvalue problem
\begin{equation}\label{eq:1sol}
-\Delta \eta + \mu\eta-f(\eta)=0.
\end{equation}
The function $\sol_\mu$ is called a ``solitary wave profile''.
Among the solitary wave solutions of \eqref{eq:NLS} are Galilei transformations of $\sol_\mu$,
\begin{equation}\label{eq:solW}
\psi_{\text{sol}}:=\mathcal{S}_{a(t)p(t)\gamma(t)}\eta_{\mu(t)},
\end{equation}
where $\mathcal{S}_{a \mom \gamma}$ is defined by
\begin{equation}\label{eq:TB0}
(\mathcal{S}_{a \mom \gamma}\psi)(x):=\lexp{{\rm i}p\cdot(x-a)+{\rm i} \gamma}\psi(x-a).
\end{equation}
Let $\sigma:=\{a,p,\gamma,\mu\}$, where $\mu$ is as in
\Eref{eq:1sol}. For $\psi_{\text{sol}}$ to be a solution to
\eqref{eq:NLS} with $V\equiv 0$ the modulation parameters, $\sigma$,
must satisfy the equations of motion
\begin{equation}\label{eq:mod}
a(t)=2p t+a,\ p(t)=p,\ \gamma(t)=\mu t +p^2 t+\gamma, \
\mu(t)=\mu
\end{equation}
with $\gamma\in \mathbb{S}^1$, $a,p\in\mathbb{R}^d$, $\mu\in
\mathbb{R}_+$. In other words,
$\sigma$ satisfy \eqref{eq:mod}, then
\begin{equation}
\psi_{\text{sol}}(x,t)=(\mathcal{S}_{a(t)p(t)\gamma(t)}\eta_{\mu(t)})(x)
\end{equation}
solves \Eref{eq:NLS} with $V\equiv 0$. Thus \eqref{eq:solW}, with
$a(t),p(t),\gamma(t),\mu(t)$ as above, describes a
$2d+2$-dimensional family solutions of \Eref{eq:NLS} with $V\equiv 0$.
Let the {\bf soliton manifold}, $\set{M}_{\mathrm{s}}$ be defined by
\begin{equation}
\set{M}_{\mathrm{s}} := \{\mathcal{S}_{a \mom \gamma}\eta_{\mu}
: \{a,p , \gamma,\mu \} \in
\mathbb{R}^d\times
\mathbb{R}^d\times \mathbb{S}^1 \times I \} \; ,
\end{equation}
where $I$ is a bounded interval in $\mathbb{R}_+$.
Solutions to \eqref{eq:1sol} behave roughly like
$\lexp{-\sqrt{\mu}|x|}$, as $|x|\rightarrow
\infty$. So $\sqrt{\mu}$ is a reciprocal length
scale that indicates the ``size'' of the solitary wave.
We consider the Cauchy problem for \Eref{eq:NLS}, with initial
condition $\psi_0$ in a weighted Sobolev space. For Hartree
nonlinearities, global wellposedness is known \cite{Enno}. For local
nonlinearities, the situation is more delicate; see
Condition~\ref{con:GWP} and Remark~\ref{rem:GWP} in
Section~\ref{sec:ass}. Let the initial condition $\psi_0$ be
``close'' to $\set{M}_{\mathrm{s}}$. Then, we will show, the corresponding solution
$\psi$ will remain ``close'' to $\set{M}_{\mathrm{s}}$, over a long time interval.
A certain ``symplectically orthogonal'' projection of $\psi$ onto $\set{M}_{\mathrm{s}}$ is then
well defined and traces out a unique curve on $\set{M}_{\mathrm{s}}$. We denote this
curve by $\eta_{\sigma(t)}$, see Figure~\ref{fig:a}.
\begin{figure}[htbp]
\psfrag{M}{$\set{M}_{\mathrm{s}}$}
\psfrag{P}{$\psi(\cdot,t)$}
\psfrag{s}{$\eta_{\sigma(t)}$}
\centering
\centerline{\includegraphics{traj2}}
\parbox{\linewidth}{
\caption{The trajectory $\psi(\cdot,t)$ over the soliton Manifold $\set{M}_{\mathrm{s}}$.}
\label{fig:a}}
\end{figure}
An essential part of this paper is to
determine the leading order behavior of
$\sigma(t) = \{a(t),p(t),\gamma(t),\mu(t)\}$ and to estimate error
terms. To this end, let $W$ be a smooth, positive,
polynomially bounded function, and define
\begin{equation}\label{eq:18}
V(x)=W(\epsilon_{\sind{V}} x)
\end{equation}
where $\epsilon_{\sind{V}}$ is a small parameter. Furthermore, let $\psi_0$ be an
initial condition ``$\epsilon_{\sind{0}}$--close'' to $\eta_{\sigma_0}\in\set{M}_{\mathrm{s}}$, for some
$\sigma_0$. Roughly speaking, this initial condition has
length scale $1/\sqrt{\mu}_0$. We
will consider external potentials, $V$, as in \eqref{eq:18}, for a scaling
parameter $\epsilon_{\sind{V}}$ satisfying
\begin{equation}
\epsilon_{\sind{V}} \ll \sqrt{\mu_0},
\end{equation}
{\it i.e.\/}, we assume that the external potential varies very little over the
length scale of $\psi_0$. For simplicity, we choose $\mu=1$ and $\epsilon_{\sind{V}}\ll 1$,
at the price of re-scaling the nonlinearity.
We decompose the solution $\psi$ of
\eqref{eq:NLS} into a part which is a solitary wave and a small part,
a ``perturbation'', $w$. That is, we write $\psi$ as
\begin{equation}
\psi = \mathcal{S}_{a \mom \gamma}(\sol_\mu+w).
\end{equation}
This does not define a unique decomposition, unless $2d+2$ additional
conditions are imposed. These conditions say that the perturbation $w$
is `symplectically orthogonal' to the soliton manifold $\set{M}_{\mathrm{s}}$.
The main idea used to control the perturbation $w$ is to derive
differential equations in time for the modulation parameters, $\sigma$,
which depend on the external potential. These equations appear
naturally when one projects solutions of \eqref{eq:NLS} onto the
soliton manifold. To control the motions of $\sigma$ and $w$, we make
use of conserved quantities: the energy
\begin{equation}
\Hn_V(\psi):=\frac{1}{2}\int (|\nabla \psi|^2 + V|\psi|^2)
\diff^d x - F(\psi),
\end{equation}
where $F'(\psi)=f(\psi)$ (this is a variational derivative),
the mass (or charge)
\begin{equation}
\mathcal{N}(\psi):=\frac{1}{2}\int |\psi|^2\diff^d x,
\end{equation}
and the ``almost conserved'' momentum
\begin{equation}
\mathcal{P}(\psi):=\frac{1}{4}\int (\bar{\psi} \nabla \psi
- \psi\nabla \bar{\psi})\diff^d x.
\end{equation}
To achieve control over the perturbation $w$, we introduce a `Lyapunov
functional'
\begin{equation}
\Lambda(\psi,t):=K_{\sigma}(\psi)-K_{\sigma}(\mathcal{S}_{a \mom \gamma}\sol_\mu),
\end{equation}
where
$\sigma=\sigma(t) = \{ a(t), p(t), \gamma(t), \mu(t) \}$, and where
\begin{equation}
\begin{split}
K_{\sigma}(\psi) &:= \Hn_V(\psi)+(p^2+\mu)\mathcal{N}(\psi)-2p\cdot \mathcal{P}(\psi) \\
&-\frac{1}{2}\int\big( V(a)+\nabla V(a)\cdot (x-a)\big)
|\psi|^2\diff^d x,
\end{split}
\end{equation}
{\it i.e.\/}, $K_{\sigma}$ is essentially a linear combination of the conserved
and almost conserved quantities. Using the linear transformation
$u:=\mathcal{S}_{a \mom \gamma}^{-1}\psi$, we change questions about the size of fluctuations
around $\mathcal{S}_{a \mom \gamma}\sol_\mu$ to ones about the size of fluctuations around the
solitary wave profile $\eta_{\mu(t)}$.
In this ``moving frame'', the $K_{\sigma}(\psi)$ terms in the Lyapunov functional
introduced above take the form
\begin{equation}
K_{\sigma}(\mathcal{S}_{a \mom \gamma} u)=\En_{\freq}(u)+\frac{1}{2}\int \mathcal{R}_{V}|u|^2\diff^d x,
\end{equation}
where
\begin{equation}
\mathcal{R}_{V}(x):= V(x+a)-V(a)-\nabla V(a)\cdot (x-a)
\end{equation}
and
\begin{equation}\label{eq:Ew}
\En_{\freq}(u):=\Hn_{V=0}(u)+\mu \mathcal{N}(u).
\end{equation}
In the moving frame the Lyapunov functional depends on the parameters
$\mu$ and $a$, but not on $p$ and $\gamma$. Furthermore, $\sol_\mu$
is a critical point of $\En_{\freq}(\sol_\mu)$, {\it i.e.\/}, $\En_{\freq}'(\sol_\mu)=0$. The change
of frame discussed above simplifies the analysis leading to our main
result.
Simply stated, our main theorem shows that, for initial
conditions $\psi_0$ $\epsilon_{\sind{0}}$-close to $\set{M}_{\mathrm{s}}$, the perturbation $w$ is
of order $\epsilon:= \epsilon_{\sind{V}}+\epsilon_{\sind{0}}$, for all times smaller than $C\epsilon^{-1}$.
Furthermore, the center of mass of the solitary wave, $a$, and the
center of mass momentum $p$ satisfy the following equations
\begin{eqnarray}
\dot a = 2p + \mathcal{O}(\epsilon^2), && \dot p = - \nabla V(a)+\mathcal{O}(\epsilon^2).
\end{eqnarray}
The remaining modulation parameters $\mu$ and
$\gamma$ satisfy
\begin{eqnarray}
\dot \mu =\mathcal{O}(\epsilon^2), && \dot\gamma = \mu-V(a)+p^2+\mathcal{O}(\epsilon^2).
\end{eqnarray}
A precise statement is found in the next section.
This is the first result of its type covering
{\it confining} external potentials.
Indeed, we can exploit the confining nature of the potential
to obtain a {\it stronger} result than that of \cite{FGJS-I}
(and that stated above) for a certain class of initial conditions
which we now describe. Consider the classical
Hamiltonian function:
\begin{equation}
h(a,p):=\big(p^2+V(a)\big)/2.
\end{equation}
Given an initial
condition $\psi_0$ $\epsilon_{\sind{0}}$--close to $\eta_{\sigma_0}\in \set{M}_{\mathrm{s}}$, where
$\sigma_0=\{a_0,p_0,\gamma_0,\mu_0\}$, we require the initial
position $a_0$ and momentum $p_0$ to satisfy
\begin{equation}
h(a_0,p_0) - \min_{a} h(a,0) \leq \epsilon_{\sind{h}},
\end{equation}
with $\epsilon_{\sind{V}}\leq C\epsilon_{\sind{h}}\leq 1$, for some constant $C$. For this class of
initial conditions, our main result shows that the perturbation $w$
remains $\mathcal{O}(\epsilon)$ for longer times:
\begin{equation}
\label{eq:longer}
t<\frac{C}{\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}}+\epsilon^2}.
\end{equation}
This improvement is non-trivial. For example, it means that
we can control the perturbation of a solitary wave which undergoes
many oscillations near the bottom of a potential well.
\noindent{\bf Remark:}
We can also extend our analysis to a class of slowly time-dependent
external potentials without much additional work.
We introduce a scale parameter,
$\tau$, in time: $V(x,t):=W(\epsilon_{\sind{V}} x,\tau t)$. To
determine the size of $\tau$ heuristically we consider
\begin{equation}
\frac{d}{dt}h(a,p,t) = p\big(\dot p +\nabla V(a,t)\big)
+ \frac{1}{2}(\dot a-2p)\cdot \nabla V(a,t) + \partial_t V(a,t).
\end{equation}
We want the last two terms to have the same size. The second
but last term is of size $\epsilon^2\epsilon_{\sind{V}}$, since $\dot a$ satisfy the
classical equations of motion to order $\epsilon^2$. The last term is of
size $\tau$. Thus if $\tau$ is chosen to be $\tau=\mathcal{O}(\epsilon_{\sind{V}}^3)$
all our estimates will survive.
The following example suggests that accelerating solitary wave solutions of
\Eref{eq:NLS} in a confining external potential can, in fact, survive for
arbitrarily long times. Choose
$V(x):= x \cdot A x + d \cdot x + c \geq 0$ and $A>0$ (positive matrix).
Then \eqref{eq:NLS} has the following solution:
\begin{equation}\label{eq:124}
\psi(x,t)=\lexp{{\rm i} p(t)\cdot(x-a(t))+{\rm i} \gamma(t))}
\tilde{\eta}_\mu(x-a(t))
\end{equation}
with
\begin{equation}\label{eq:125}
\dot p = -\nabla V(a), \ \dot a=2p,\ \dot \gamma=p^2+\mu-V(a),
\end{equation}
where $\tilde{\eta}_\mu$ solves the equation
\begin{equation}
-\Delta \eta + \mu \eta - f(\eta) + (x \cdot A x) \eta = 0.
\end{equation}
Thus, given a solution of the equations of motion \eqref{eq:125}, a
family of solitary wave solutions is given by \eqref{eq:124}, for arbitrary
times $t$. For details see Appendix \ref{sec:fam}.
The first results of the above type, for bounded, time-independent
potentials were proved in \cite{Frohlich+Tsai+Yau2000,
Frohlich+Tsai+Yau2002} for the Hartree equation under a spectral
assumption. This result was later extended to a general class of
nonlinearities in \cite{FGJS-I}.
Neither of these works deals with a confining external potential.
In particular, their results do not extend to the longer
time interval~(\ref{eq:longer}) described above.
For local pure-power nonlinearities
and a small parameter $\epsilon_{\sind{V}}$, it has been shown in
\cite{Bronski+Jerrard2000} that if an initial condition is
of the form $\mathcal{S}_{a_0 \mom_0 \gamma_0}\eta_{\mu_0}$, then the solution $\psi(x,t)$ of
Eq.\/~\eqref{eq:NLS} satisfies
\begin{equation}
\epsilon_{\sind{V}}^{-d}|\psi(\frac{x}{\epsilon_{\sind{V}}},\frac{t}{\epsilon_{\sind{V}}})|^2
\rightarrow \nrm{\sol_\mu}^2 \delta_{a(t)}
\end{equation}
in the $\C{1*}$ topology (dual to $\C{1}$), provided $a(t)$ satisfies
the equation $\frac{1}{2}\ddot{a}=\nabla W(a)$, where $V(x)=W(\epsilon_{\sind{V}}
x)$. This result was strengthened in \cite{Keraani2002} for a bounded
external potential and in \cite{Carles2003} for a potential given by a
quadratic polynomial in $x$.
There have been many recent works on asymptotic properties for generalized
nonlinear Schr\"odinger equations. Asymptotic stability, scattering
and asymptotic completeness of solitary waves for bounded external
potential tending to 0 at $\infty$ has been shown under various
assumptions. See for example, \cite{Soffer+Weinstein1988,
Soffer+Weinstein1990,Soffer+Weinstein1992, BP92,
Buslaev+Perelman1995,Cuccagna2001,Cuccagna2002,
Buslaev+Sulem2002,
Tsai+Yau2002,Tsai+Yau2002b,Tsai+Yau2002c,RSS,Soffer+Weinstein2004,
Gustafson+Nakanishi+Tsai2004,Gang+Sigal2004,Perelman2004}.
Though these are all-time results, where ours is long (but finite)-time,
our approach has some advantages: we can handle confining potentials
(for which the above-described results are meaningless); we require a much
less stringent (and verifiable) spectral condition; we track the
finite-dimensional soliton dynamics (Newton equations); and our
methods are comparatively elementary.
Our paper is organized as follows. In Section~\ref{sec:ass}, we state
our hypotheses and the main result. In Section~\ref{sec:2}, we recall
the Hamiltonian nature of \Eref{eq:NLS} and describe symmetries of
\eqref{eq:NLS} for $V\equiv 0$. We give a precise definition of the
soliton manifold $\set{M}_{\mathrm{s}}$ and its tangent space. In Section~\ref{sec:3},
we introduce a convenient parametrization of functions in a small
neighborhood of $\set{M}_{\mathrm{s}}$ in phase space, and we derive equations for the
modulation parameters $\sigma=\{a,p,\gamma,\mu\}$ and the
perturbation $w$ around a solitary wave $\eta_{\sigma}=\mathcal{S}_{a \mom \gamma}\sol_\mu$. In
this parametrization, the perturbation $w$ is symplectically
orthogonal to the tangent space $\set{T}_{\sol_\sigma}\set{M}_{\mathrm{s}}$ to $\set{M}_{\mathrm{s}}$ at
$\sol_\sigma$. In Section \ref{sec:rel}, we similarly decompose the initial
condition $\psi_0$ deriving in this way the initial conditions,
$\sigma_0$ and $w_0$, for $\sigma$ and $w$, and estimating $w_0$. In
Section \ref{sec:5}, we derive bounds on the solitary wave position,
$a$, and the momentum, $p$, by using the fact that the Hamiltonian,
$h(a,p)$ is almost conserved in time. In Section \ref{sec:4}, we
construct the Lyapunov functional, $\Lambda(\psi,t)$, and compute its time
derivative. This computation is used in Section~\ref{sec:6} in order
to obtain an upper bound on $\Lambda(\psi,t)$. This bound, together with the
more difficult lower bound derived in Section~\ref{sec:7}, is used in
Section~\ref{sec:end} in order to estimate the perturbation $w$ and
complete the proof of our main result, Theorem~\ref{thm:main}. Some
basic inequalities are collected in
Appendices~\ref{app:RVbd}--\ref{sec:ene}. In Appendix~\ref{sec:fam},
we construct a family of time-dependent solutions with parameters
exactly satisfying the classical equations of motion.
\section{Notation, assumptions and main result}
\label{sec:ass}
Let $\Lp{s}$ denote the usual Lebesgue space of functions,
$\C{s}$ the space of functions with $s$ continuous
derivatives, and $\Sob{s}$ the Sobolev space of order $s$.
Abbreviate $\langle x\rangle^2:=1+|x|^2$.
\paragraph{Assumptions on the external potential.}
Let $W(x)$ be a $\C{3}$ function, and let $\min_x W(x)=0$.
Let $\beta\in\mathbb{Z}^d$ with $\beta_j\geq 0$
$\forall j=1,\ldots,d$ be a multi-index. Given a number $r\geq 1$
let $W$ be such that
\begin{eqnarray}\label{eq:Wup}
&|\partial_x^\beta W(x)|\leq C_{\max{V}}
\langle x\rangle^{r-|\beta|} \ \text{for} \ |\beta|\leq 3,
& \\
\label{eq:Wlow}
&\mathop{\mathrm{Hess}} W(x) \geq \rho_1 \langle x\rangle^{r-2},&
\end{eqnarray}
and
\begin{equation}\label{eq:Wfar}
W(x)\geq c_V|x|^r,\ \text{for}\ |x|\geq c_L
\end{equation}
for some positive constants $C_{\max{V}}$, $\rho_1$, $c_V$, $c_L$.
The number $r$ is called the growth rate of the external potential.
Here $\mathop{\mathrm{Hess}} W$ is the Hessian of $W$ with respect to spatial
variables. Define $V(x):=W(\epsilon_{\sind{V}} x)$. Then, for $r\geq 1$,
\begin{eqnarray}\label{eq:Vup}
& |\partial_x^\beta V(x)|\leq C_V \epsilon_{\sind{V}}^{|\beta|} \langle \eps x \rangle^{r-|\beta|},\
\text{for} \ |\beta|\leq 3, &
\\ \label{eq:Vlow}
& \mathop{\mathrm{Hess}} V(x) \geq \rho_1 \epsilon_{\sind{V}}^2 \langle \eps x \rangle^{r-2}, &
\end{eqnarray}
and
\begin{equation}\label{eq:Vfar}
V(x)\geq c_V(\epsilon_{\sind{V}}|x|)^r,\ \text{for}\ \epsilon_{\sind{V}}|x|\geq c_L.
\end{equation}
\paragraph{Assumptions on the initial condition $\psi_0$.}
The energy space, $\Espace$, for a given growth rate $r$ of the
external potential, is defined as
\begin{equation}\label{eq:Espace}
\Espace:=\{\psi\in \set{H}_1:\langle x\rangle^{r/2}\psi \in \Lp{2}\}.
\end{equation}
Let $\Espace'$ denote the dual space of $\Espace$.
The energy norm is defined as
\begin{equation}
\Enrm{\psi}^2:=\nrmHo{\psi}^2+\nrm{\langle \eps x \rangle^{r/2}\psi}^2
\end{equation}
We require $\psi_0\in \Espace$.
\medskip
In what follows, we identify complex functions
with real two-component functions
via
\[
\mathbb{C} \ni \psi(x) = \psi_1(x) + {\rm i} \psi_2(x) \;
\longleftrightarrow \; \vec{\psi}(x) = (\psi_1(x), \psi_2(x))
\in \mathbb{R}^2.
\]
Consider a real function $F(\vec{\psi})$ on a space
of real two-component functions, and let
$F'(\vec{\psi})$ denote its $L^2$-gradient.
We identify this gradient with a complex function
denoted by $F'(\psi)$. Then
\[
F'(\bar{\psi}) = \overline{F'(\psi)} \;
\longleftrightarrow \;
F(\sigma \vec{\psi}) = F(\vec{\psi}),
\]
where $\sigma :=\mathop{\mathrm{diag}}(1,-1)$, since the latter property
is equivalent to
$F'(\vec{\psi}) = \sigma F'(\sigma \vec{\psi})$.
\paragraph{Assumptions on the nonlinearity $f$.}
\begin{enumerate}
\item\label{con:GWP} (GWP \cite{Cazenave1996,Yajima+Zhang2001,Yajima+Zhang2004,Enno}) Equation~\eqref{eq:NLS} is globally
well-posed in the space $\set{C}(\mathbb{R},\Espace) \cap
\C{1}(\mathbb{R},\Espace')$. See Remark~\ref{rem:GWP} below.
\item\label{con}
The nonlinearity $f$ maps from $\set{H}_1$ to $\Sob{-1}$, with $f(0)=0$.
$f(\psi)=F'(\psi)$ is the $\set{L}^2$-gradient
of a $C^3$ functional $F : H_1 \to \mathbb{R}$
defined on the space of real-valued,
two-component functions, satisfying
the following conditions:
\begin{enumerate}
\item (Bounds)
\label{con:A}
\begin{equation}\label{eq:Taylor}
\sup_{\nrmFree{u}_{\set{H}_1}\leq M}
\nrmFree{F''(u)}_{\set{B(\set{H}_1,\Sob{-1})}}<\infty, \
\sup_{\nrmFree{u}_{\set{H}_1}\leq M}\ \nrmFree{F'''(u)}_{\set{H}_1 \mapsto
\set{B}(\set{H}_1,\Sob{-1})}<\infty,
\end{equation}
where $\set{B}(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$.
\item(Symmetries \cite{FGJS-I})
\label{con:sym}
$F(\mathcal{T}\psi)=F(\psi)$ where $\mathcal{T}$ is either translation
$\psi(x)\mapsto \psi(x+a)$ $\forall a\in\mathbb{R}^d$, or spatial
rotation $\psi(x)\mapsto \psi(R^{-1}x)$, $\forall R\in \set{SO}(d)$,
or boosts $\mathcal{T}_{\mom}^{\textrm{b}}: u(x)\mapsto \lexp{{\rm i} p\cdot x}u(x)$, $\forall
p\in\mathbb{R}^d$,
or gauge transformations
$\psi\mapsto \lexp{{\rm i} \gamma}\psi$, $\forall \gamma\in
\mathbb{S}^1$, or complex conjugation $\psi \mapsto \bar{\psi}$.
\end{enumerate}
\item\label{con:F} (Solitary waves) \label{con:Sol} There exists a
bounded open interval $\tilde{I}$ on the positive real axis such
that for all $\mu\in \tilde{I}$:
\begin{enumerate}
\item (Ground state
\cite{Berestycki+Lions+Peletier1981,Berestycki+LionsI1983,Berestycki+LionsII1983,McLeod1993})
The equation
\begin{equation}
\label{eq:gs}
-\Delta \psi + \mu \psi - f(\psi)=0.
\end{equation}
has a spherically symmetric, positive $\set{L}^2\cap \C{2}$ solution,
$\eta = \eta_\mu$.
\item\label{con:stab} (Stability: see {\it e.g.\/},
\cite{Grillakis+Shatah+Strauss1990}) This solution, $\eta$, satisfies
\begin{equation}
\partial_\mu \int \eta^2_\mu \diff^d x>0.
\end{equation}
\item\label{con:Null} (Null space condition: see {\it e.g.\/}, \cite{FGJS-I})
Let $\mathcal{L}_\sol$ be the linear operator
\begin{equation}
\mathcal{L}_\sol:=\begin{pmatrix} L_1 & 0 \\ 0 & L_2 \end{pmatrix}
\end{equation}
where $L_1:=-\Delta+\mu -f^{(1)}(\eta)$, and $L_2:=-\Delta +
\mu - f^{(2)}(\eta)$, with
$f^{(1)}:=\Big(\partial_{\mathop{\set{Re}}{\psi}}\big(\mathop{\set{Re}}(f)\big)\Big)(\eta)$,
and $f^{(2)}:=\Big(\partial_{\mathop{\set{Im}}{\psi}}\big(\mathop{\set{Im}}(f)\big)\Big)(\eta)$.
We require that
\begin{equation}
\Null{\mathcal{L}_\sol}=\mathop{\mathrm{span}}\{\begin{pmatrix}0 \\ \eta \end{pmatrix},
\begin{pmatrix} \partial_{x_j}\eta\\ 0\end{pmatrix},\ j=1,\ldots, d\}.
\end{equation}
\end{enumerate}
\end{enumerate}
Conditions~\ref{con}--\ref{con:F} on the nonlinearity are discussed
in \cite{FGJS-I}, where further references can be found. Examples of
nonlinearities that satisfy the above requirements are local
nonlinearities
\begin{equation}\label{eq:n1}
f(\psi)=\beta |\psi|^{s_1}\psi + \lambda |\psi|^{s_2}\psi, \ 0<s_1<s_2<\frac{4}{d},\ \beta\in \mathbb{R},\ \lambda>0,
\end{equation}
and Hartree nonlinearities
\begin{equation}\label{eq:n2}
f(\psi)=\lambda(\Phi*|\psi|^2)\psi, \ \lambda>0,
\end{equation}
where $\Phi$ is of positive type, continuous and spherically symmetric and
tends to 0, as $|x|\rightarrow \infty$.
Of course, $\lambda$ can be scaled out by rescaling
$\psi$. For precise conditions on $\Phi$ we refer to
\cite{Cazenave1996,Enno}.
\begin{remark}\label{rem:GWP}
For Hartree nonlinearities global well-posedness is known for
potentials $0\leq V\in \Lp{1}_{loc}$ \cite{Enno}. For local
nonlinearities, the situation is more delicate. Global
well-posedness and energy conservation is known for potentials with
growth-rate $r\leq 2$~\cite{Cazenave1996}. For $r>2$ and local
nonlinearities, local well-posedness has been shown in
the energy space~\cite{Yajima+Zhang2001,Yajima+Zhang2004}.
For local nonlinearities, a proof of the energy conservation
needed for global well-posedness, and the
application of this theory to our results, is missing.
\end{remark}
For $V\equiv 0$, \Eref{eq:NLS} is the usual generalized nonlinear
Schr\"odinger (or Hartree) equation. For self-focusing nonlinearities
as in examples \eqref{eq:n1} and \eqref{eq:n2}, it has stable solitary
wave solutions of the form
\begin{equation}\label{eq:solp}
\eta_{\sigma(t)}(x):=\lexp{{\rm i} p(t)\cdot (x-a(t))+{\rm i} \gamma(t)}\eta_{\mu(t)}(x-a(t)),
\end{equation}
where $\sigma(t):=\{a(t),p(t),\gamma(t),\mu(t)\}$, and
\begin{equation}
a(t)=2pt+a,\ \gamma(t)=\mu t + p^2 t+ \gamma,\
p(t)=p,\ \mu(t)=\mu,
\end{equation}
with $\gamma\in\mathbb{S}^1$, $a,p\in \mathbb{R}^d$ and $\mu\in
\mathbb{R}^+$, and where $\sol_\mu$ is the spherically symmetric,
positive solution of the nonlinear eigenvalue problem
\begin{equation}\label{eq:sol}
-\Delta \eta + \mu \eta - f(\eta)=0.
\end{equation}
Recall from \eqref{eq:TB0} that the linear map $\mathcal{S}_{a \mom \gamma}$ is defined as
\begin{equation}\label{eq:TB}
(\mathcal{S}_{a \mom \gamma} g)(x):= \lexp{{\rm i} p\cdot (x-a)+{\rm i} \gamma}g(x-a).
\end{equation}
In analyzing solitary wave solutions to \eqref{eq:NLS} we encounter
two length scales: the size $\propto \mu^{-1/2}$ of the support of
the function $\sol_\mu$, which is determined by our choice of initial
condition $\psi_0$, and a length scale determined by the potential, $V$,
measured by the small parameter $\epsilon_{\sind{V}}$. We consider the regime,
\begin{equation}
\frac{\epsilon_{\sind{V}}}{\sqrt{\mu}}\ll 1.
\end{equation}
We claim in the introduction that if $\psi_0$ is close to $\sol_\sigma$, for
some $\sigma$ then we retain control for times $\propto\epsilon^{-1}$.
Restricting the initial condition to a smaller class of $\sol_\sigma$, with
small initial energy, we retain control for longer times. In our
main theorem, which proves this claim, we wish to treat both
cases uniformly. To this end, let $\epsilon_{\sind{h}}$ and $K$ be positive numbers
such that $\epsilon_{\sind{h}}\in K[\epsilon_{\sind{V}},\min_{\mu\in I}\sqrt{\mu}]$
and assume
\begin{equation}
h(a_0,p_0):=\frac{1}{2}\big(p_0^2+V(a)\big)\leq \epsilon_{\sind{h}}
\end{equation}
(recall $\min_a V(a)=0$). The lower bound for $\epsilon_{\sind{h}}$
corresponds to our restricted class of initial data,
the upper bound to the larger class of data.
In particular, $\epsilon_{\sind{h}}\geq K \epsilon_{\sind{V}}$.
We are now ready to state our main result.
Fix an open proper sub-interval $I \subset \tilde{I}$.
\begin{theorem}\label{thm:main}
Let $f$ and $V$ satisfy the conditions listed above.
There exists $T>0$ such that for
$\epsilon:=\epsilon_{\sind{V}}+\epsilon_{\sind{0}}$ sufficiently small,
and $\epsilon_{\sind{h}}\geq K\epsilon_{\sind{V}}$,
if the initial condition $\psi_0$ satisfies
\begin{equation}\label{eq:ebnd}
\nrmHo{\psi_0-\mathcal{S}_{a_0 \mom_0 \gamma_0}\eta_{\mu_0}} +
\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\mathcal{S}_{a_0 \mom_0 \gamma_0}\eta_{\mu_0})}\leq \epsilon_{\sind{0}}
\end{equation}
for some $\sigma_0:=\{a_0,p_0,\gamma_0,\mu_0\}\in
\mathbb{R}^d\times \mathbb{R}^d\times\mathbb{S}^1\times I$ such that
\begin{equation}\label{eq:hbound}
h(a_0,p_0)\leq \epsilon_{\sind{h}},
\end{equation}
then for times $0\leq t\leq T(\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}}+\epsilon^2)^{-1}$,
the solution to \Eref{eq:NLS}
with this initial condition is of the form
\begin{equation}
\psi(x,t)=\mathcal{S}_{a(t) \mom(t) \gamma(t)}\big(\eta_{\mu(t)}(x)+w(x,t)\big),
\end{equation}
where
$\nrmHo{w}+\Bnrm{\langle \eps x \rangle^{r/2}w}\leq C\epsilon$. The modulation
parameters $a,p,\gamma$ and $\mu$ satisfy the differential
equations
\begin{align}\label{eq:thp}
\dot p& = -( \nabla V)(a) + \mathcal{O}(\epsilon^2), \\
\dot a& = 2p + \mathcal{O}(\epsilon^2), \\
\dot \gamma& = \mu -V(a)+p^2+\mathcal{O}(\epsilon^2), \\
\dot \mu &= \mathcal{O}(\epsilon^2).\label{eq:thf}
\end{align}
\end{theorem}
\begin{remark}[Remark about notation]
Fr\'echet derivatives are always understood to be defined on real
spaces. They are denoted by primes. $C$ and $c$ denote various
constants that often change between consecutive lines and
which do not depend on $\epsilon_{\sind{V}}$, $\epsilon_{\sind{0}}$ or $\epsilon$.
\end{remark}
\section{Soliton manifold}
\label{sec:2}
In this section we recall the Hamiltonian nature of \Eref{eq:NLS} and
some of its symmetries. We also define the soliton manifold and its
tangent space.
An important part in our approach is played by the variational
character of \eqref{eq:NLS}. More precisely, the nonlinear
Schr\"odinger equation \eqref{eq:NLS} is a Hamiltonian
system with Hamiltonian
\begin{equation}\label{eq:HV}
\Hn_V(\psi) := \frac{1}{2}\int (|\nabla \psi|^2 + V|\psi|^2)\diff^d x - F(\psi).
\end{equation}
The Hamiltonian $\Hn_V$ is conserved {\it i.e.\/},
\begin{equation}
\Hn_V(\psi)=\Hn_V(\psi_0).
\end{equation}
A proof of this can be found, for local nonlinearities and $r\leq 2$,
in {\it e.g.\/}, Cazenave~\cite{Cazenave1996}, and for Hartree nonlinearities in
\cite{Enno}. An important role is played by the mass
\begin{equation}\label{eq:N}
\mathcal{N}(\psi):= \int |\psi|^2 \diff^d x,
\end{equation}
which also is conserved,
\begin{equation}
\mathcal{N}(\psi(t))=\mathcal{N}(\psi_0).
\end{equation}
We often identify complex spaces, such as the Sobolev space
$\set{H}_1(\mathbb{R}^d,\mathbb{C})$, with real spaces; {\it e.g.\/},
$\set{H}_1(\mathbb{R}^d,\mathbb{R}^2)$,
using the identification
$\psi=\psi_1+{\rm i}\psi_2 \leftrightarrow (\psi_1,\psi_2)=:\vec{\psi}$.
With this identification, the complex structure ${\rm i}^{-1}$ corresponds to
the operator
\begin{equation}
J:=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}.
\end{equation}
The real $\set{L}^2$-inner product in the real notation is
\begin{equation}
\dotp{\vec u}{\vec w} := \int (u_1w_1 + u_2w_2) \diff^d x,
\end{equation}
where $\vec{u}:=(u_1,u_2)$.
In the complex notation it becomes
\begin{equation}
\dotp{u}{w} := \mathop{\set{Re}} \int u\bar{w} \diff^d x.
\end{equation}
We henceforth abuse notation and drop the arrows. The symplectic form is
\begin{equation}
\omega(u,w):=\mathop{\set{Im}}\int u\bar w \diff^d x.
\end{equation}
We note that $\omega(u,w)=\dotp{u}{J^{-1}v}$ in the real notation.
Equation~\eqref{eq:NLS} with $V\equiv 0$ is invariant
under spatial translations, $\mathcal{T}_a^{\textrm{tr}}$, gauge transformations, $\mathcal{T}^{\textrm{g}}_{\gamma}$, and
boost transformations, $\mathcal{T}_{\mom}^{\textrm{boost}}$, where
\begin{equation}
\mathcal{T}_a^{\textrm{tr}} :\psi(x,t) \mapsto \psi(x-a,t) \; , \
\mathcal{T}^{\textrm{g}}_{\gamma} : \psi(x,t)\mapsto \lexp{{\rm i}\gamma}\psi(x,t)
\label{eq:T1} \; ,
\end{equation}
\begin{equation}
\mathcal{T}_{\mom}^{\textrm{boost}}: \psi(x,t)\mapsto
\lexp{{\rm i}(p\cdot x - p^2 t) }
\psi(x-2 p t,t)\; .
\label{eq:T2}
\end{equation}
The transformations~\eqref{eq:T1}--\eqref{eq:T2}
map solutions of eq.~\eqref{eq:NLS} with $V\equiv 0$
into solutions of \eqref{eq:NLS} with $V\equiv 0$.
Let $\mathcal{T}_{\mom}^{\textrm{b}}:\psi(x)\mapsto \lexp{{\rm i}p\cdot x}\psi(x)$ be the $t=0$
slice of the boost transform. The combined symmetry transformations
$\mathcal{S}_{a \mom \gamma}$ introduced in \eqref{eq:TB} can be expressed as
\begin{align}\label{eq:Sym}
\mathcal{S}_{a \mom \gamma}\eta = \mathcal{T}_a^{\textrm{tr}}\mathcal{T}_{\mom}^{\textrm{b}}\mathcal{T}^{\textrm{g}}_{\gamma} \sol_\mu(x)
=\lexp{{\rm i} (p \cdot (x-a)+\gamma)}\sol_\mu(x-a).
\end{align}
We define the soliton manifold as
\begin{equation}
\set{M}_{\mathrm{s}} := \{\mathcal{S}_{a \mom \gamma}\eta_{\mu}
: \{a,p , \gamma,\mu \} \in
\mathbb{R}^d\times
\mathbb{R}^d\times \mathbb{S}^1 \times I \} \; .
\end{equation}
The tangent space to this manifold at the solitary wave
profile $\sol_\mu\in \set{M}_{\mathrm{s}}$ is given by
\begin{equation}
\set{T}_{\sol_\mu}\set{M}_{\mathrm{s}} = \mathop{\mathrm{span}}(\zvec_{\mathrm{t}},\zvec_{\mathrm{g}},\zvec_{\mathrm{b}},\zvec_{\mathrm{s}}) \; ,
\end{equation}
where
\begin{align}
\za :=&\left.\nabla_a \mathcal{T}_a^{\textrm{tr}} \sol_\mu \right|_{a=0} = \begin{pmatrix} -\nabla \sol_\mu\\ 0 \end{pmatrix}\; ,
\label{eq:t1} &&
\zg := \left. \frac{\partial}{\partial \gamma}
\mathcal{T}^{\textrm{g}}_{\gamma} \sol_\mu\right|_{\gamma=0} = \begin{pmatrix} 0 \\ \sol_\mu\end{pmatrix} \; ,\\
\zt := &\left. \nabla_p \mathcal{T}_{\mom}^{\textrm{boost}} \sol_\mu \right|_{p=0,t=0} =
\begin{pmatrix} 0 \\ x\sol_\mu \end{pmatrix}\;, && \label{eq:t4}
\zn := \begin{pmatrix} \partial_{\mu} \sol_\mu \\ 0 \end{pmatrix}\; .
\end{align}
Above, we have explicitly written the basis of tangent vectors in the real
space.
Recall that the equation~\eqref{eq:gs} can be written as
$\En_{\freq}'(\eta_\mu) = 0$ where
\[
\En_{\freq}(\psi) = \mathcal{H}_{V \equiv 0}(\psi) + \frac{\mu}{2} \mathcal{N}(\psi).
\]
Then the tangent vectors listed above are generalized zero modes of
the operator $\mathcal{L}_\mu := \En_{\freq}''(\eta_\mu)$. That is,
$(J\mathcal{L}_\mu)^2 z = 0$
for each tangent vector $z$ above.
To see this fact for $\zg$, for example,
recall that $\Ew'(\psi)$ is gauge-invariant.
Hence $\Ew'(\mathcal{T}^{\textrm{g}}_{\gamma} \sol_\mu)=0$. Taking the derivative with respect to the parameter
$\gamma$ at $\gamma=0$ gives $\mathcal{L}_\sol \zg=0$.
The other relations are derived analogously (see \cite{Weinstein1985}).
\section{Symplectically orthogonal decomposition}
\label{sec:3}
In this section we make a change of coordinates for the Hamiltonian
system $\psi\mapsto (\sigma,w)$, where $\sigma:=(a,p,\gamma,\mu)$.
We also give the equations in this new set of coordinates.
Let
\begin{equation}\label{eq:m}
m(\mu):=\frac{1}{2}\int \eta_\mu^2(x)\diff^d x.
\end{equation}
Let
\begin{equation}\label{eq:CI}
C_I:=\max_{\substack{z\in \{x\sol_\mu,\sol_\mu,\nabla\sol_\mu,\partial_\mu \sol_\mu\}\\ \mu\in \tilde{I}}}
(\nrmHo{z},\nrm{\langle \eps x \rangle^{r/2}z},\nrm{\mathcal{K} z}).
\end{equation}
When it will not cause confusion,
for $\sigma = \{ a, p, \gamma, \mu \}$ we will abbreviate
\[
\eta_{\sigma} := \mathcal{S}_{a \mom \gamma} \eta_\mu.
\]
Now define the neighborhood of $\set{M}_{\mathrm{s}}$:
\begin{equation}\label{eq:Udelta}
U_\delta := \{\psi\in L^2 :\inf_{\sigma\in \Sigma}
\nrm{\psi-\eta_{\sigma}}\leq \delta\},
\end{equation}
where $\Sigma := \{a,p,\gamma,\mu: a\in \mathbb{R}^d,
p\in \mathbb{R}^d,\gamma\in \mathbb{S}^1, \mu \in I\}$.
Our goal is to decompose a given function $\psi\in U_\delta$
into a solitary wave and a perturbation:
\begin{equation}\label{eq:splitt}
\psi = \mathcal{S}_{a \mom \gamma}(\sol_\mu + w).
\end{equation}
We do this according to the following theorem.
Let $\tilde{\Sigma} := \{a,p,\gamma,\mu: a\in \mathbb{R}^d,
p\in \mathbb{R}^d,\gamma\in \mathbb{S}^1, \mu \in \tilde{I}\}$.
\begin{theorem}\label{thm:splitt}
There exists $\delta > 0$ and a unique
map $\varsigma\in \C{1}(U_\delta,\tilde{\Sigma})$ such that
(i)
\begin{equation}
\dotp{\psi-\eta_{\varsigma(\psi)}}{J^{-1} z}=0, \;\;
\forall z\in \set{T}_{\eta_{\varsigma(\psi)}}\set{M}_{\mathrm{s}}, \;\;
\forall \psi \in U_\delta
\end{equation}
and (ii) if, in addition,
$\delta \ll (2C_I)^{-1}\min(m(\mu),m'(\mu))$
then there exists a constant
$c_I$ independent of $\delta$ such that
\begin{equation}\label{eq:Omega}
\sup_{\psi\in U_\delta} \nrm{\varsigma'(\psi)}\leq c_I.
\end{equation}
\end{theorem}
\begin{proof}
Part (i): Let the map $G:L^2 \times \tilde{\Sigma} \mapsto \mathbb{R}^{2d+2}$
be defined by
\begin{equation}
G_j(\psi,\varsigma):=\dotp{\psi-\eta_{\varsigma}}{J^{-1}z_{\varsigma,j}},
\ \forall j=1,\ldots 2d+2.
\end{equation}
Part (i) is proved by applying the implicit function theorem
to the equation $G(\psi,\varsigma)=0$, around a
point $(\eta_{\sigma},\sigma)$. For details we refer to Proposition~5.1 in
\cite{FGJS-I}.
Part (ii): Abbreviate:
\begin{equation}
\Omega_{jk}:=\dotp{\partial_{\varsigma_j}\eta_{\varsigma}}
{J^{-1}z_{\varsigma,k}},
\end{equation}
where $z_{\varsigma,k}$ is the $k$:th element of
$\mathcal{S}_{a \mom \gamma}\{\za,\zg,\zt,\zn\}$. By explicitly inserting the tangent
vectors, we find that $\nrm{\Omega} \geq \inf_{\mu\in
I}(m(\mu),m'(\mu))$. Thus, $\Omega$ is invertible by Condition
\ref{con:stab} in Section~\ref{sec:ass}.
From a variation of $\psi$ in $G(\psi,\varsigma(\psi))=0$ we find
\begin{equation}
\varsigma'_k(\psi)=\sum_{j=1}^{2d+2}
(J^{-1}z_{\varsigma})_j(\tilde{\Omega}^{-1})_{jk}.
\end{equation}
where
\begin{equation}
\tilde{\Omega}_{jk}:=\Omega_{jk} +
\dotp{\psi-\eta_{\varsigma(\psi)}}{J^{-1}\partial_{\varsigma_j}z_{\varsigma,k}}
\end{equation}
Using the upper bound of $\delta$, and the definition of $C_I$ above, we find
\begin{equation}
\sup_{\psi\in U_\delta} \nrm{\varsigma'(\psi)}\leq
\frac{2C_I}{\inf_{\mu\in I}(m(\mu),m'(\mu))}=:c_{I}.
\end{equation}
\end{proof}
We now assume $\psi(t) \in U_\delta\cap\Espace$,
and set $\sigma(t):=\varsigma(\psi(t))$ as defined by
Theorem~\ref{thm:splitt}.
Write
\begin{equation}\label{eq:udef}
u:=\mathcal{S}_{a \mom \gamma}^{-1}\psi = \eta_\mu + w
\end{equation}
so that $w$ satisfies
\begin{equation}
\dotp{w}{J^{-1} z}=0, \;\;
\forall z\in \set{T}_{\eta_{\mu}}\set{M}_{\mathrm{s}}.
\end{equation}
Here $u$ is the solution in a moving frame.
Denote the anti-self-adjoint infinitesimal generators of symmetries as
\begin{equation}
\mathcal{K}_{j} = \partial_{x_j}, \ \
\mathcal{K}_{d+j} = {\rm i} x_j, \ \ \mathcal{K}_{2d+1}={\rm i} ,\ \
\mathcal{K}_{2d+2}=\partial_\mu,\ \ j=1,...,d
\label{eq:gen}
\end{equation}
and define corresponding coefficients
\begin{equation}
\alpha_{j} = \dot{a}_j - 2p_j, \ \
\alpha_{d+j} = -\dot{p}_j - \partial_{x_j} V(a), \ \
j=1,...,d, \label{eq:mu}
\end{equation}
\begin{equation}
\alpha_{2d+1} =
\mu-p^2+\dot{a}\cdot p
-V(a)-\dot{\gamma},
\ \ \alpha_{2d+2} = -\dot{\mu}. \label{eq:nu}
\end{equation}
Denote
\begin{equation}
\pars\cdot\sgen := \sum_{j=1}^{2d+1} \alpha_j \mathcal{K}_j, \ \ \text{and}\ \
\spar\cdot\gen := \pars\cdot\sgen + \alpha_{2d+2}\partial_\mu.
\end{equation}
Substituting $\psi=\mathcal{S}_{a \mom \gamma} u$ into \eqref{eq:NLS} we obtain
\begin{equation}\label{eq:dut}
{\rm i} \dot u = \Ew'(u) + \mathcal{R}_{V} u +{\rm i} \pars\cdot\sgen u,
\end{equation}
where
\begin{equation}\label{eq:VR}
\mathcal{R}_{V}(x) = V(x+a) - V(a) - \nabla V(a) \cdot x.
\end{equation}
To obtain the equations for $(\sigma,w)$ we project Eqn.~\eqref{eq:dut}
onto $\set{T}_{\sol}\Mf$ and $(J\set{T}_{\sol}\Mf)^{\bot}$ and use \eqref{eq:udef}. We illustrate this
method of deriving the equations for $\sigma$, for the
projection of \eqref{eq:dut} along ${\rm i} \eta$:
\begin{equation}\label{eq:proj1}
\dotp{\eta}{\dot\mu\partial_\mu\eta+\dot w}=\dotp{{\rm i} \eta}{\mathcal{L}_\sol w+
\NII{w}+\mathcal{R}_{V} (\eta+w)+{\rm i} \pars\cdot\sgen (\eta+w)}.
\end{equation}
where we have used $u=\eta+w$ and $\Ew'(u)=\mathcal{L}_\sol w+\NII{w}$
where $\mathcal{L}_\sol := \En_{\freq}''(\eta)$ is given explicitly as
\begin{equation}\label{eq:LL}
\mathcal{L}_\sol w = -\Delta w +\mu w - f'(\eta)w.
\end{equation}
In particular, for local nonlinearities of the form $g(|\psi|^2)\psi$, we
have in the complex notation, since $\eta(x)\in \mathbb{R}$,
\begin{equation}
\mathcal{L}_\sol w := -\Delta w +\mu w - g(\eta^2)w - 2\eta g'(\eta^2)\mathop{\set{Re}} w.
\end{equation}
Here
\begin{equation}\label{eq:NII}
\NII{w} := - f(\eta + w ) + f(\eta) + f'(\eta)w.
\end{equation}
We find the equation for $\dot\mu$ once we note that
$\partial_t\dotp{\eta}{w}=0$, $\mathcal{L}_\sol{\rm i}\eta=0$, $\dotp{{\rm i}
\eta}{\mathcal{R}_{V}\eta}=0$, $\dotp{\eta}{\underline{\gen}\eta}=0$ and
$\adjoint{\underline{\gen}}=-\underline{\gen}$. Inserting this into \eqref{eq:proj1} gives
\begin{equation}
\dot\mu m'(\mu) =
\dotp{{\rm i} \eta}{\NII{w}+\mathcal{R}_{V} w}-\alpha\cdot\dotp{\mathcal{K}\eta}{w}.
\end{equation}
The projection along the other directions works the same way:
we use the fact that these directions are the generalized
zero modes of $\mathcal{L}_\sol$, and furthermore that they are orthogonal to $Jw$. The
calculations are worked out in detail in \cite{FGJS-I} (See
Eqns.~(6.20)--(6.22) in \cite{FGJS-I}.) We give the result:
\begin{align}
\dot{\gamma}
& = \mu-p^2+\dot{a}\cdot p- V(a)-
(m'(\mu))^{-1}\left(\dotp{\partial_\mu \eta}{\NII{w}+ \mathcal{R}_{V} w}
\right. \label{eq:gam} \\ & \left. \quad
- \alpha\cdot\dotp{\mathcal{K} \partial_\mu \eta}{{\rm i} w}
+ \dotp{\partial_\mu \eta}{\mathcal{R}_{V} \eta}
\right),\nonumber
\\ \nonumber \\
\dot{\mu}&=\big(m'(\mu)\big)^{-1} \left(
\dotp{{\rm i}\eta}{\NII{w}+\mathcal{R}_{V} w} - \alpha\cdot\dotp{\mathcal{K}
\eta}{w}\right), \label{eq:dotmu}
\end{align}
\begin{align}
\dot{a}_k&=2p_k+ \big(m(\mu)^{-1}\big)\left(\dotp{{\rm i} x_k \eta}{\NII{w}+ \mathcal{R}_{V}
w}-\alpha\cdot\dotp{\mathcal{K} x_k \eta}{w}
\right) , \label{eq:tr}
\\ \nonumber \\
\dot{p}_k & = -\partial_{a_k} V(a) + (m(\mu))^{-1}\big(-\frac{1}{2}\dotp{(\partial_{x_k}\mathcal{R}_{V})\eta}{\eta}+
\dotp{\partial_k\eta}{\NII{w}+\mathcal{R}_{V} w}\nonumber \\ & \quad -
\alpha\cdot\dotp{\mathcal{K} \partial_k \eta}{{\rm i} w} \big), \label{eq:bo}
\end{align}
and
\begin{equation}\label{eq:dw}
{\rm i} \dot{w} = \mathcal{L}_\sol w + N(w) + \mathcal{R}_{V}(\eta+w) + {\rm i} \pars\cdot\sgen(\eta+w) - {\rm i}
\dot{\mu} \partial_\mu \eta.
\end{equation}
Note that the first two terms on the right-hand side of
Eqn.~\eqref{eq:bo} can be written as $-\partial_{a_k}V_{\mathrm{eff}}(a,\mu)$, where
\begin{equation}\label{eq:Veff}
V_{\mathrm{eff}}(a,\mu):= \nrm{\sol_\mu}^{-2}
\int V(a+x)|\sol_\mu(x)|^2 \diff^d x.
\end{equation}
Hence,
\begin{equation}
\dot p_k = -\nabla_a V_{\mathrm{eff}}(a,\mu) + (m(\mu)^{-1}
\dotp{\partial_{x_k}\sol_\mu}{\NII{w}} + \mathcal{O}(\nrm{w}(\epsilon_{\sind{V}}^2+|\alpha|)),
\end{equation}
where $|\alpha|^2=\sum |\alpha_j|^2$.
Thus we have obtained the dynamical equations for
$(\sigma,w)$.
\begin{remark}
The transformation
\begin{equation}
\sigma:=(a,p,\gamma,\mu)\mapsto\hat{\sigma}:= (a,P,\gamma,m)
\end{equation} with $P:=\frac{1}{2}p\nrm{\sol_\mu}^2$ and
$m:=\frac{1}{2}\nrm{\sol_\mu}^2$ gives a canonical symplectic structure
and Darboux coordinates on $\set{M}_{\mathrm{s}}$, {\it i.e.\/}, for $w=0$
\begin{align}
\dot P &= -\partial_a \Hn_V(\mathcal{S}_{a \mom \gamma}\sol_\mu), &&
\dot a = \partial_P \Hn_V(\mathcal{S}_{a \mom \gamma}\sol_\mu), \\
\dot m & = \partial_\gamma \Hn_V(\mathcal{S}_{a \mom \gamma}\sol_\mu), &&
\dot \gamma = -\partial_{m} \Hn_V(\mathcal{S}_{a \mom \gamma}\sol_\mu).
\end{align}
Here $\nabla_{\hat{\sigma}} \Hn_V(\mathcal{S}_{a \mom \gamma}\sol_\mu) = (m\nabla_a
V_{\mathrm{eff}},2P/m,0,-P^2/m^2+V(a)-\mu)$.
\end{remark}
\section{Initial conditions $\tilde{\Par}_0$, $w_0$.}\label{sec:rel}
In this section we
use Theorem~\ref{thm:splitt} in order to decompose the initial condition $\psi_0$ as (see Figure~\ref{fig:1})
\begin{equation}
\psi_0 = \mathcal{S}_{\tilde{a}_0,\tilde{\mom}_0,\tilde{\gamma}_0}(\eta_{\tilde{\freq}_0}+w_0)
\end{equation}
so that $w_0 \bot J^{-1}\set{T}_{\eta_{\tilde{\freq}_0}}\set{M}_{\mathrm{s}}$. This decomposition
provides the initial conditions $\tilde{\Par}_0$ and $w_0$, for the parameters,
$\sigma$, and fluctuation, $w$ (determined for later times by
Theorem~\ref{thm:splitt}). The main work here goes into estimating
$w_0$.
\begin{figure}[htbp]
\psfrag{a}{$\psi_0$}
\psfrag{b}{$\eta_{\sigma_0}$}
\psfrag{c}{$\eta_{\varsigma(\psi_0)}=\sol_{\ParZ}$}
\psfrag{H}{$\set{H}_1$}
\psfrag{M}{$\set{M}_{\mathrm{s}}$}
\centering
\centerline{\includegraphics{orth}}
\parbox{\linewidth}{
\caption{Orthogonal decomposition versus skew-orthogonal decomposition.}\label{fig:1}}
\end{figure}
Let $\varsigma:U_{\delta}\mapsto \tilde{\Sigma}$
be the map established in
Theorem~\ref{thm:splitt}. Then $\tilde{\Par}_0=\{\tilde{a}_0,\tilde{\mom}_0,\tilde{\gamma}_0,\tilde{\freq}_0\}$
and $w_0$ are given as
$\tilde{\Par}_0:=\varsigma(\psi_0)$ and
\begin{equation}\label{eq:wZ}
w_0:=\mathcal{S}_{\tilde{a}_0\tilde{\mom}_0\tilde{\gamma}_0}^{-1}
(\psi_0-\sol_{\ParZ}), \ w_0 \bot J\set{T}_{\eta_{\tilde{\freq}_0}}\set{M}_{\mathrm{s}}.
\end{equation}
Recall the definitions of $\mathcal{K}$ \eqref{eq:gen}, and $C_I$
\eqref{eq:CI}. Theorem~\ref{thm:splitt} states $\sup_{\psi\in
U_\delta} \nrm{\varsigma'(\psi)} \leq c_{I}$.
Bounds for $w_0$ and $\tilde{\Par}_0$ are stated in the following
proposition
\begin{proposition}\label{prop:wo}
Let $w_0$ be defined as above. Let
$\sigma_0:=\{a_0,p_0,\gamma_0,\mu_0\}$ and let $\psi_0$ satisfy
$\|\psi_0-\eta_{\sigma_0}\|_{L^2} \leq \delta$ (where $\delta$
is from Theorem~\ref{thm:splitt}), and let
$\psi_0\in\Espace$. Then there exists positive constants $C_1$,
$C_2$, such that
\begin{align}\label{eq:N2}
|\tilde{\Par}_0-\sigma_0|&\leq c_{I}\nrm{\psi_0-\eta_{\sigma_0}},\\
\nrmHo{w_0}& \leq C_1(1+p_0^4+\nrm{\psi_0-\eta_{\sigma_0}}^4)\nrmHo{\psi_0-\eta_{\sigma_0}} \label{eq:stat1}
\end{align}
and
\begin{multline}
\nrm{\langle \eps x \rangle^{r/2}w_0}\leq 3^{r/2}\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\eta_{\sigma_0})}
\\+ C_2(1+|p_0|^2+\epsilon_{\sind{V}}^r|a_0|^r + \nrm{\psi_0-\eta_{\sigma_0}}^2+\epsilon_{\sind{V}}^r\nrm{\psi_0-\eta_{\sigma_0}}^r)\nrm{\psi_0-\eta_{\sigma_0}}. \label{eq:stat2}
\end{multline}
where $C_1$ and $C_2$ depend only on $C_I$, $c_I$ and $r$, where $C_I$
is defined in \eqref{eq:CI} and $c_I$ in Theorem~\ref{thm:splitt}.
\end{proposition}
\begin{proof}
First we consider inequality \eqref{eq:N2}. Abbreviate
$\tilde{\Par}_0:=\varsigma(\psi_0)$ and analogously for the components
$a,p,\gamma,\mu$ of $\varsigma$. Let
$|\varsigma|^2:=\sum_{j=1}^{2d+2}|\varsigma_j|^2$. From
Theorem~\ref{thm:splitt} we know that $\varsigma(\psi)$ is a
$\C{1}$-map. Thus, for $j\in 1,...,2d+2$ and some $\theta_1\in[0,1]$
\begin{equation}
(\tilde{\Par}_0-\sigma_0)_j=\dotp{\varsigma_j'(\theta_1 \psi_0+(1-\theta_1)\eta_{\sigma_0})}{(\psi_0-\eta_{\sigma_0})}.
\end{equation}
Since $\sup_{\psi\in U_{\delta}}\nrmFree{\varsigma'(\psi)}\leq c_{I}$
the inequality \eqref{eq:N2} follows.
Consider inequality \eqref{eq:stat1} and rewrite $w(\cdot,0)=:w_0$
from \eqref{eq:wZ} as
\begin{equation}\label{eq:woo}
w_0=\mathcal{S}_{\tilde{a}_0\tilde{\mom}_0\tilde{\gamma}_0}^{-1}
(\psi_0-\eta_{\sigma_0})
+\mathcal{S}_{\tilde{a}_0\tilde{\mom}_0\tilde{\gamma}_0}^{-1}(\eta_{\sigma_0}-\sol_{\ParZ}).
\end{equation}
To estimate this, we first estimate the linear operator $\mathcal{S}_{a \mom \gamma}^{-1}$:
\begin{equation}\label{eq:HoTB}
\nrmHo{\mathcal{S}_{a \mom \gamma}^{-1}\psi} \leq 2(1+|p|^2)^{1/2}\nrmHo{\psi}.
\end{equation}
The first term in \eqref{eq:woo} is in the appropriate form, for the second
term we recall that $\eta$ is a $\C{1}$-map. Thus for some
$\theta_2\in[0,1]$
\begin{equation}\label{eq:N1}
\sol_{\ParZ}-\eta_{\sigma_0}=\sum_{j=1}^{2d+2}\left.
(\tilde{\Par}_0-\sigma_0)_j
\partial_{\sigma_j}\eta_{\sigma}\right|_{\sigma=\theta_1\tilde{\Par}_0+(1-\theta_2)\sigma_0}.
\end{equation}
To calculate the norm of this expression, note that
\begin{equation}\label{eq:N3}
\partial_\sigma \sol_\sigma = \mathcal{S}_{a \mom \gamma} z_{\mu,p}, \ \text{where}\
z_{\mu,p}:=\{{\rm i} p\sol_\mu+\nabla \sol_\mu,{\rm i} x\sol_\mu,{\rm i} \sol_\mu,\partial_\mu \sol_\mu\}
\end{equation}
and $\nrmHo{z_{\mu,p}} \leq \sqrt{5}C_I(1+|p|^2)^{1/2}$. Let
$n(\sigma,\sigma_0):=(\sigma-\sigma_0)\theta_2+\sigma_0$, and define
$g^2:=1+|\tilde{\mom}_0-p_0|^2+p_0^2$. The $\set{H}_1$-norm of \eqref{eq:N1}, using
\eqref{eq:HoTB} and \eqref{eq:N3} is
\begin{equation}
\begin{split}
\label{eq:dHone}
\nrmHo{\sol_{\ParZ}-\eta_{\sigma_0}}
&\leq |\tilde{\Par}_0-\sigma_0| \big.\nrmHo{\partial_{\sigma}\eta_{\sigma}}
\big|_{\sigma=n(\tilde{\Par}_0,\sigma_0)} \\
&\leq \left. 2\sqrt{5}C_I(1+|p|^2)\right|_{p=n(\tilde{\mom}_0,p_0)}|\tilde{\Par}_0-\sigma_0|
\leq 9C_Ig^2|\tilde{\Par}_0-\sigma_0|.
\end{split}
\end{equation}
We now calculate the $\set{H}_1$ norm of $w_0$ (see \eqref{eq:woo})
using \eqref{eq:N2}, \eqref{eq:HoTB} with momentum
$p=\tilde{\mom}_0-p_0+p_0$ and \eqref{eq:dHone}. We find
\begin{equation}
\begin{split}
\nrmHo{w_0} &\leq 2g(\nrmHo{\psi_0-\eta_{\sigma_0}}+
\nrmHo{\sol_{\ParZ}-\eta_{\sigma_0}})\\
&\leq 2g\big(1+9C_Ic_{I}g^2\big)
\nrmHo{\psi_0-\eta_{\sigma_0}}.
\end{split}
\end{equation}
The coefficient above is less then $cg^4 +C$, and
$g^4\leq 3(1+c_{I}^4\nrm{\psi_0-\eta_{\sigma_0}}^4+|p_0|^4)$. Inserting
and simplifying gives the inequality \eqref{eq:stat1}.
The quantity appearing in the third and last inequality
\eqref{eq:stat2}, can be rewritten as
\begin{equation}\label{eq:N4}
\langle \eps x \rangle^{r/2}w_0 = \langle \eps x \rangle^{r/2}\mathcal{S}_{\tilde{a}_0\tilde{\mom}_0\tilde{\gamma}_0}^{-1}\big(
(\psi_0-\eta_{\sigma_0})+(\eta_{\sigma_0}-\sol_{\ParZ})\big).
\end{equation}
We begin our calculation of the norm of \eqref{eq:N4} by considering
the linear operator $\langle \eps x \rangle^{r/2}\mathcal{S}_{a \mom \gamma}$. We have
\begin{equation}\label{eq:N5}
\langle \eps x \rangle^{r/2}\mathcal{S}_{a \mom \gamma}\psi =\mathcal{S}_{a \mom \gamma}\kax{\epsilon_{\sind{V}}(x-a)}^{r/2}\psi
\end{equation}
and $\nrm{\mathcal{S}_{a \mom \gamma}\psi}=\nrm{\psi}$. From Lemma~\ref{lem:maxmin} we obtain
\begin{equation}\label{eq:xS}
\begin{split}
\nrm{\langle \eps x \rangle^{r/2}\mathcal{S}_{a \mom \gamma}\psi} &\leq \nrm{\kax{\epsilon_{\sind{V}}(x-(a-a_0)-a_0)}^{r/2}\psi}\\
&\leq 3^{\max(r/2,r-1)}\big(\nrm{\langle \eps x \rangle^{r/2}\psi}+g_2\nrm{\psi}\big),
\end{split}
\end{equation}
where $g_2:=(\epsilon_{\sind{V}}|a-a_0|)^{r/2}+(\epsilon_{\sind{V}}|a_0|)^{r/2})$.
Using this we find the $\set{L}^2$-norm of \eqref{eq:N4} to be
\begin{multline}\label{eq:q}
\nrm{\langle \eps x \rangle^{r/2}w_0}\leq
C\big(\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\eta_{\sigma_0})} + g_2\nrm{\psi_0-\eta_{\sigma_0}}
\\ + \nrm{\langle \eps x \rangle^{r/2}(\sol_{\ParZ}-\eta_{\sigma_0})}
+g_2\nrm{\sol_{\ParZ}-\eta_{\sigma_0}}\big).
\end{multline}
The first and second term of the above expression is in an appropriate
form. We bound the third term by using \eqref{eq:N1}, \eqref{eq:N3} and \eqref{eq:N5} to get
\begin{equation}\label{eq:xsol}
\begin{split}
\nrm{\langle \eps x \rangle^{r/2}(\sol_{\ParZ}-\eta_{\sigma_0})}
&\leq |\tilde{\Par}_0-\sigma_0|\left. \nrm{\kax{\epsilon_{\sind{V}}(x-a)}^{r/2}z_{p,\mu}}\right|_{\sigma=n(\tilde{\Par}_0,\sigma_0)} \\
&\leq 3^{\max(r/2,r-1)}\sqrt{5}C_Ig(1+g_2)
|\tilde{\Par}_0-\sigma_0|.
\end{split}
\end{equation}
The last term of \eqref{eq:q} is straight forward to bound:
\begin{equation}
\begin{split}
\nrm{\sol_{\ParZ}-\eta_{\sigma_0}} &\leq |\tilde{\Par}_0-\sigma_0|\left.\nrm{\partial_{\sigma}\eta_{\sigma}}\right|_{\sigma=n(\tilde{\Par}_0,\sigma_0)} \\
&\leq |\tilde{\Par}_0-\sigma_0|\big.\nrm{z_{p,\mu}}\big|_{\substack{p=n(\tilde{\mom}_0,p_0)
\mu=n(\tilde{\freq}_0,\mu_0)}}
\leq \sqrt{5}C_Ig|\tilde{\Par}_0-\sigma_0|.
\label{eq:solDiff}
\end{split}
\end{equation}
Inserting \eqref{eq:xsol} and \eqref{eq:solDiff} into \eqref{eq:q} gives
\begin{multline}
\nrm{\langle \eps x \rangle^{r/2}w_0}\leq C\Big(\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\eta_{\sigma_0})}
\\ +\big(g_2+g(1+2g_2)\big)
\nrm{\psi_0-\eta_{\sigma_0}}\Big),
\end{multline}
where $C$ depend only on $C_I$, $c_I$ and $r$.
We simplify this, by repeatedly using Cauchy's inequality and \eqref{eq:N2} on
the expression
in front of the $\nrm{\psi_0-\eta_{\sigma_0}}$-term, to obtain
\begin{multline}
\nrm{\langle \eps x \rangle^{r/2}w_0}\leq C\Big(\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\eta_{\sigma_0})}
+ \big(1+\nrm{\psi_0-\eta_{\sigma_0}}^2\\ +\epsilon_{\sind{V}}^r\nrm{\psi_0-\eta_{\sigma_0}}^r + |p_2|^2+(\epsilon_{\sind{V}}|a_0|)^r\big)
\nrm{\psi_0-\eta_{\sigma_0}}\Big).
\end{multline}
This gives the third inequality of the proposition.
\end{proof}
Recall the initial energy bound \eqref{eq:ebnd}
\begin{equation}
\nrmHo{\psi_0-\eta_{\sigma_0}}+\nrm{\langle \eps x \rangle^{r/2}(\psi_0-\eta_{\sigma_0})}\leq \epsilon_{\sind{0}},
\end{equation}
and the bound on the initial kinetic and potential energy for
the solitary wave \eqref{eq:hbound}
\begin{equation}
\frac{1}{2}(p_0^2+V(a_0))\leq \epsilon_{\sind{h}}.
\end{equation}
We have the corollary
\begin{corollary}\label{cor:wo}
Let \eqref{eq:ebnd}, \eqref{eq:hbound} and
\eqref{eq:Vup}--\eqref{eq:Vfar} hold
with $\epsilon_{\sind{0}} < \delta$. Then
\begin{equation}
|\tilde{\Par}_0-\sigma_0|\leq c_{I}\epsilon_{\sind{0}}, \
\nrmHo{w_0}\leq C_{1}\epsilon_{\sind{0}},
\end{equation}
\begin{align}
\nrm{\langle \eps x \rangle^{r/2}w_0}&\leq C_{2}\epsilon_{\sind{0}}
\end{align}
and
\begin{equation}\label{eq:hesta}
h(\tilde{a}_0,\tilde{\mom}_0)\leq C_3(\epsilon_{\sind{h}}+ \epsilon_{\sind{0}}^2+\epsilon_{\sind{V}}\epsilon_{\sind{0}}),
\end{equation}
where $C_1$, $C_2$ and $C_3$ depend only on $c_L$, $c_V$
(\Eref{eq:Vfar}), $C_E := \max (\epsilon_{\sind{V}},\epsilon_{\sind{0}},\epsilon_{\sind{h}})$ and the constants
in Proposition~\ref{prop:wo}.
\end{corollary}
\begin{proof}
Starting from Proposition~\ref{prop:wo} the first three inequalities
follow directly through the energy bounds \eqref{eq:ebnd},
\eqref{eq:hbound} together with the observation that either
$\epsilon_{\sind{V}}|a_0|\leq c_L$ or $c_V(\epsilon_{\sind{V}} |a_0|)^r\leq V(a_0)\leq 2\epsilon_{\sind{h}}$.
We also use that $\epsilon_{\sind{h}}$, $\epsilon_{\sind{0}}$ and
$\epsilon_{\sind{V}}$ are all bounded by a constant $C_E$.
The last inequality follows from the fact that
$h(a,p):=(p^2+V(a))/2$
is a $\C{1}$ function. For some $\theta\in[0,1]$
\begin{equation}
\begin{split}
h(a,p)-h(a_0,p_0) &= ((p-p_0)\theta+p_0)\cdot(p-p_0) \\
&+ \frac{1}{2} (a-a_0)\cdot \nabla V((a-a_0)\theta+a_0).
\end{split}
\end{equation}
Thus, using \eqref{eq:Vup}, and $\kax{x+y}^{r-1}\leq
3^{\max(0,(r-3)/2)}\big(1+2^{(r-1)/2}(|x|^{r-1}+|y|^{r-1})\big)$ gives
\begin{multline}
|h(a,p)-h(a_0,p_0)|\leq C\Big(|p-p_0|^2+|p_0|^2 +
\\
\epsilon_{\sind{V}}^2|a-a_0|\big(1+|\epsilon_{\sind{V}}(a-a_0)|^{r-1}+|\epsilon_{\sind{V}} a_0|^{r-1}\big)\Big).
\end{multline}
With $p=\tilde{\mom}_0$ and $a=\tilde{a}_0$ above, and $|\tilde{\Par}_0-\sigma_0|\leq c_{I}\epsilon_{\sind{0}}$,
$h(a_0,p_0)\leq \epsilon_{\sind{h}}$, \eqref{eq:hbound} and \eqref{eq:Vfar} we have
have shown \eqref{eq:hesta}.
\end{proof}
\section{Bounds on soliton position and momentum}\label{sec:5}
In this section we use the bounded initial soliton energy,
Corollary~\ref{cor:wo}, to find upper bounds on
position and momentum of the solitary wave. We express the norms first in
terms of $h(\tilde{a}_0,\tilde{\mom}_0)$ and the small parameters. In
Corollary~\ref{cor:apest} we state the final result,
where the bounds are just
constants times the small parameters $\epsilon_{\sind{0}}$, $\epsilon_{\sind{h}}$ and $\epsilon_{\sind{V}}$.
Recall (see \eqref{eq:Vup} and \eqref{eq:Vfar})
that the potential $V$ is non-negative and satisfies
the following upper and lower bounds:
\begin{equation}\label{eq:Vup_2}
|\partial_x^\beta V|\leq C_V\epsilon_{\sind{V}}\kax{\epsilon_{\sind{V}} a}^{r-1},
\ \text{for}\ |\beta|=1,
\end{equation}
and, if $\epsilon_{\sind{V}}|a|\geq c_L$ then
\begin{equation}\label{eq:Vfar_2}
V(a)\geq c_V(\epsilon_{\sind{V}}|a|)^r.
\end{equation}
To obtain the desired estimates on $a$ and $p$ we will use
the fact that the soliton energy,
\begin{equation}
h(a,p):=\frac{1}{2}\big(p^2+V(a)\big),
\end{equation}
is essentially conserved.
We abbreviate $\alpha:=\{\alpha^{\mathrm{tr}},\alpha^{\mathrm{b}},\alpha_{2d+1},\alpha_{2d+2}\}$.
The size of $\alpha$ is measured by $|\alpha|^2:=\sum_j |\alpha_j|^2$
and $\Anrm{\alpha}:=\sup_{s\leq t}|\alpha(s)|$.
We have the following:
\begin{proposition}\label{prop:apest2}
Let $V$ satisfy conditions \eqref{eq:Vup_2} and
\eqref{eq:Vfar_2}. Let $h_0:=h(\tilde{a}_0,\tilde{\mom}_0)$, and set
\begin{equation}
\label{eq:T11}
\tilde{T}_1:=\frac{C_{T_1}}{(\epsilon_{\sind{V}}^2+\Anrm{\alpha})(1+\epsilon_{\sind{V}}+h_0)},
\ \ C_{\tilde{T}_1}:=\frac{c_V}{2^{\max(2,r-1)/2}C_Vd},
\end{equation}
where the constants $C_V$ and $c_V$ are related to the growth rate of the
potential (see \eqref{eq:Vup} and \eqref{eq:Vfar}).
Then for times $t\leq \tilde{T}_1$:
\begin{equation}\label{eq:apest2}
|p|\leq C_{\tilde{p}}(\sqrt{h_0}+\Anrm{\alpha}t+\epsilon_{\sind{V}}) \ \text{and}\
\epsilon_{\sind{V}}|a|\leq C_{a},
\end{equation}
where $C_{a}$ and $C_{\tilde{p}}$ depend only on $c_L$, $c_V$,
$C_{\tilde{T}_1}$, $r$, $d$, $C_3$ and
$C_E = \max(\epsilon_{\sind{V}},\epsilon_{\sind{0}},\epsilon_{\sind{h}})$.
$C_3$ is the constant in Corollary~\ref{cor:wo} and
\end{proposition}
\begin{proof}
First we estimate $p$ in terms of $a$, using the almost conservation
of $h(a,p)$
\begin{equation}
\frac{d}{dt} h(a,p)= \frac{1}{2}\left(2p\cdot \left(\dot{p}+\nabla V(a)\right)+\nabla V(a)\cdot (\dot a-2p)\right).
\end{equation}
Now recall the definitions $\alpha^{\mathrm{b}}:=-\dot p-\nabla V(a)$ and
$\alpha^{\mathrm{tr}}:=\dot a-2p$ together with the upper bound \eqref{eq:Vup_2} of
the potential $|\nabla V|\leq d^{1/2}C_V \epsilon_{\sind{V}} \kax{\epsilon_{\sind{V}} a}^{r-1}$
to obtain
\begin{equation}
|\mathrm{d}_t h(a,p)|\leq |\alpha||p| +\frac{1}{2}C_V d^{1/2}\epsilon_{\sind{V}} |\alpha| \kax{\epsilon_{\sind{V}} a}^{r-1}.
\end{equation}
Integration in time and simplification gives
\begin{equation}\label{eq:hbd}
h(a(t),p(t))\leq h_0 + t(\Anrm{\alpha})\left(\Anrm{p}+2^{-1}d^{1/2}C_V\epsilon_{\sind{V}} \kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-1}\right).
\end{equation}
Recall that $h=2^{-1}(p^2+V(a))$ and that $V\geq 0$, thus
$|p|^2 \leq 2h$.
Solving the resulting quadratic inequality for $\Anrm{p}>0$ we find
that
\begin{equation}\label{eq:pbd}
\Anrm{p}\leq \sqrt{2 h_0} + 3t\Anrm{\alpha}+2^{-1}d^{1/2}C_V\epsilon_{\sind{V}}\kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-1}.
\end{equation}
The Eqn.~\eqref{eq:hbd} also implies
\begin{equation}\label{eq:Vint}
\sup_{s\leq t}V(a(s))\leq 2h_0 + 2t\Anrm{\alpha}
\left(\Anrm{p}+2^{-1}d^{1/2}C_V\epsilon_{\sind{V}}\kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-1}\right).
\end{equation}
As can be seen in \eqref{eq:pbd} we need to consider the possibility
of large $\epsilon_{\sind{V}}|a|$. Let $\epsilon_{\sind{V}}|a|\geq c_L$, with $c_L$ as in
\eqref{eq:Vfar_2} then $V(a)\geq c_V(\epsilon_{\sind{V}}|a|)^r$. Inserting this lower
bound and \eqref{eq:pbd} into \eqref{eq:Vint} we obtain
\begin{equation}
c_V(\epsilon_{\sind{V}}\Anrm{a})^{r}\leq 2h_0 + 2t\Anrm{\alpha}\left(
\sqrt{2h_0} + 3t\Anrm{\alpha}+d^{1/2}C_V\epsilon_{\sind{V}}\kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-1}
\right).
\end{equation}
Lemma~\ref{lem:maxmin} shows $\kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-1}\leq
2^{\max(0,r-3)/2}(1+(\epsilon_{\sind{V}}\Anrm{a})^{r-1})$ for $r\geq 1$.
If the maximal time satisfies
the inequality $t\leq \tilde{T}_1$ (see~(\ref{eq:T11})),
then the above inequality implies
\begin{equation}\label{eq:tCa}
\epsilon_{\sind{V}}\Anrm{a} \leq (\frac{2}{c_V}(C_4 + 2C_{\tilde{T}_1}+6C_{\tilde{T}_1}^2+
\frac{1}{2}c_V)^{1/r}=:\tilde{C}_a,
\end{equation}
where we have used that $h_0$ is bounded by the constant $C_E$. Thus,
either $\epsilon_{\sind{V}}|a|\leq c_L$ holds or, for the given time interval,
\eqref{eq:tCa} holds. In both cases $\epsilon_{\sind{V}}|a|\leq C_a$, where
the constant only depends on $C_4=C_3C_E$,
$C_{\tilde{T}_1}$, $c_V$, $c_L$ and $r$.
We insert this upper bound on $\epsilon_{\sind{V}}|a|$ into \eqref{eq:pbd} and
for times $t\leq \tilde{T}_1$ we find
\begin{equation}
\Anrm{p}\leq C_{\tilde{p}}(\sqrt{h_0}+\Anrm{\alpha}t+\epsilon_{\sind{V}}),
\end{equation}
where $C_{\tilde{p}}:=3+d^{1/2}C_VC_{\tilde{a}}^{r-1}$.
\end{proof}
Using the Corollary~\ref{cor:wo} we express the above proposition in
terms of $\epsilon_{\sind{h}}$ rather than $h_0$. Recall the requirement on $\delta$ from
Theorem~\ref{thm:splitt}
\begin{corollary}\label{cor:apest}
Let $V$ satisfy \eqref{eq:Vup}--\eqref{eq:Vfar} and let $\psi_0\in
U_{\delta}\cap \Espace$. Furthermore, let $\psi_0$ satisfy the
$\epsilon_{\sind{0}}$-energy bound \eqref{eq:ebnd} for $\eta_{\sigma_0}$ with
$\sigma_0=\{a_0,p_0,\gamma_0,\mu_0\}$, and let $h(a_0,p_0)\leq
\epsilon_{\sind{h}}$ ({\it i.e.\/}, \eqref{eq:hbound}). Let
\begin{equation}
\label{eq:T22}
T_1:=\frac{C_{T_1}}{(\epsilon_{\sind{V}}^2+\Anrm{\alpha})(1+\epsilon_{\sind{V}}+\epsilon_{\sind{h}}+\epsilon_{\sind{V}})},
\ \
T_2:=\frac{\sqrt{\epsilon_{\sind{h}}}}{\Anrm{\alpha}+\epsilon_{\sind{V}}^2},
\end{equation}
where
\begin{equation}
C_{T_1}:=\frac{C_{\tilde{T}_1}}{(1+C_3)(1+C_E^2)}.
\end{equation}
Then for times $t\leq \min(T_1,T_2)$:
\begin{equation}\label{eq2:apest}
|p|\leq C_p(\sqrt{\epsilon_{\sind{h}}}+\epsilon_{\sind{0}}+\epsilon_{\sind{V}}) \ \text{and}\
\epsilon_{\sind{V}}|a|\leq C_a,
\end{equation}
where $C_p$ depends on
$C_E = \max(\epsilon_{\sind{V}},\epsilon_{\sind{0}},\epsilon_{\sind{h}})$,
$C_V$, $d$, $r$ and $C_a$. $C_3$ is defined in Corollary~\ref{cor:wo}
and $C_a$ in Proposition~\ref{prop:apest2}. The constant $C_V$ is
defined in \eqref{eq:Vup}.
\end{corollary}
\begin{proof}
Under the assumptions of the corollary we have that
Corollary~\ref{cor:wo} holds and hence
\begin{equation}
h(\tilde{a}_0,\tilde{\mom}_0)\leq C_3(\epsilon_{\sind{h}}+\epsilon_{\sind{0}}^2+\epsilon_{\sind{V}}\epsilon_{\sind{0}}).
\end{equation}
We now modify the constants and estimates of Proposition~\ref{prop:apest2}
to take the upper bound of $h_0$ into account.
The new, maximal time derived from $\tilde{T}_1$ becomes
$T_1 \leq \tilde{T}_1$.
For times shorter than this time, $t\leq T_1$, the bound on $\epsilon_{\sind{V}}|a|$
remains the same. Using this estimate for $\epsilon_{\sind{V}}|a|$, we simplify the
$|p|$ estimate. Note first that $\sqrt{h_0}+\epsilon_{\sind{V}} \leq
(\sqrt{\epsilon_{\sind{h}}}+\epsilon_{\sind{0}}+\epsilon_{\sind{V}})(1+2\sqrt{C_3})$, inserted into
\eqref{eq:apest2} gives
\begin{equation}
|p|\leq \frac{1}{2}C_{p}(\sqrt{\epsilon_{\sind{h}}}+\epsilon_{\sind{V}}+\epsilon_{\sind{0}}+|\alpha|t),
\end{equation}
where $C_p$ depends on $C_3$, $C_E$, $C_a$ and $d$ and $r$.
With the choice of time interval $T_2$ such that $t\leq T_2$, where
$T_2$ is given in~(\ref{eq:T22}),
we obtain $|p|\leq C_p(\sqrt{\epsilon_{\sind{h}}}+\epsilon_{\sind{0}}+\epsilon_{\sind{V}})$.
\end{proof}
\section{Lyapunov functional}
\label{sec:4}
In this section we define the Lyapunov functional and calculate its time
derivative in the moving frame. Recall the definition of $\En_{\freq}(\psi)$ in
\eqref{eq:Ew} together with decomposition \eqref{eq:splitt}:
$\psi= \mathcal{S}_{a \mom \gamma}(\sol_\mu+w)$, with $w\bot J \set{T}_{\sol}\Mf$. Define the
Lyapunov functional, $\Lambda$, as
\begin{equation}\label{eq:L}
\Lambda := \En_{\freq}(\sol_\mu+w) + \frac{1}{2}\dotp{\mathcal{R}_{V} (\sol_\mu+w)}{\sol_\mu+w}
- \En_{\freq}(\sol_\mu) - \frac{1}{2}\dotp{\mathcal{R}_{V}\sol_\mu}{\sol_\mu}.
\end{equation}
Here we show that the Lyapunov functional $\Lambda$ is an almost
conserved quantity. We begin by computing its time derivative. Let
$\alpha^{\mathrm{b}}:=-\dotp-\nabla V(a)$ and $\alpha^{\mathrm{tr}}:=\dot a-2p$ (boost and
translation coefficients). We have the following proposition
\begin{proposition}\label{prop:dL}
Given a solution $\psi\in \Espace\cap U_\delta$ to \eqref{eq:NLS}, define
$\sol_\mu$ and $w$ as above. Then
\begin{equation}
\frac{d}{dt} \Lambda = p \cdot
\dotp{\nabla_a \mathcal{R}_{V} w}{w} - \alpha^{\mathrm{tr}}\cdot \mathrm{D}^2V(a)\cdot
\dotp{xw}{w} + R ,
\end{equation}
where
\begin{equation}
\begin{split}
R &: =\alpha^{\mathrm{b}}\cdot\dotp{{\rm i} w}{\nabla w} + 2p \cdot \dotp{\nabla_a
\mathcal{R}_{V}\sol_\mu}{w}-\frac{1}{2}\alpha^{\mathrm{tr}} \cdot \dotp{\nabla_a\mathcal{R}_{V}\sol_\mu}{\sol_\mu} \\
&+ \frac{\dot\mu}{2}\nrm{w}^2 -
\dot\mu\dotp{\mathcal{R}_{V}\sol_\mu}{\partial_\mu\sol_\mu}.
\end{split}
\end{equation}
\end{proposition}
Before proceeding to the proof, we recall the definition of the moving
frame solution $u$ defined by
\begin{equation}\label{eq:u}
u(x,t):= \lexp{-{\rm i} p \cdot x - {\rm i} \gamma}\psi(x+a,t).
\end{equation}
Here $a$, $p$ and $\gamma$ depend on time, in a way
determined by the splitting of Section~\ref{sec:3},
and the function $\psi$ is a solution of the nonlinear
Schr\"odinger equation \eqref{eq:NLS}. In the moving frame the
Lyapunov functional $\Lambda$ takes the form
\begin{equation}\label{eq:Lt}
\Lambda = \En_{\freq}(u) + \frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}
- \En_{\freq}(\sol_\mu) - \frac{1}{2}\dotp{\mathcal{R}_{V}\sol_\mu}{\sol_\mu}.
\end{equation}
We begin with some auxiliary lemmas.
\begin{lemma}\label{lem:Ehr}
Let $\psi\in \Espace$ be a solution to \eqref{eq:NLS}. Then
\begin{equation}\label{eq:Ehr}
\partial_t \dotp{\psi}{-{\rm i} \nabla \psi} = - \dotp{(\nabla V)\psi}{\psi}
\ \text{and}\ \
\partial_t\dotp{x \psi }{\psi}= 2\dotp{\psi}{-{\rm i} \nabla \psi}.
\end{equation}
\end{lemma}
\begin{proof}
The first part of this lemma was proved in \cite{FGJS-I}.
To prove the second part we use the equation
\begin{equation}
\partial_t(x_k|\psi|^2) = {\rm i} \nabla \cdot (x_k
\bar{\psi}\nabla \psi - x_k \psi\nabla \bar{\psi}) -
{\rm i}(\bar{\psi}\partial_k \psi - \psi \partial_k \bar{\psi}),
\end{equation}
understood in a weak sense, which follows from the nonlinear
Schr\"odinger equation \eqref{eq:NLS}. Formally, integrating this
equation and using that the divergence term vanishes gives the second
equation in \eqref{eq:Ehr}. To do this rigorously, let $\chi$ be a
$\C{1}$ function such that $|\nabla \chi(x)|\leq C$ and
\begin{equation}
\chi(x):=\left\{\begin{array}{ll} 1 & |x|\leq 1, \\ 0 & |x|>2,
\end{array}\right.
\end{equation}
and let $\chi_R(x):=\chi(\frac{x}{R})$.
Abbreviate $j_k:=(x_k \bar{\psi}\nabla \psi - x_k \psi\nabla
\bar{\psi})$ and let $R>1$. We multiply the divergence term by
$\chi_R$. Integration by parts gives
\begin{equation}
\left|\int (\nabla\cdot j_k) \chi_R\diff^d x\right|=\left|\int j_k\cdot \nabla
\chi_R(x) \diff^d x\right| \leq \frac{C}{R} \int |j_k| \diff^d x.
\end{equation}
We note that $j_k\in \Lp{1}$ for all $k$, and is independent of $R$, thus as
$R\rightarrow \infty$, this term vanishes. The remaining terms give in
the limit $R\rightarrow \infty$ the second equation in \eqref{eq:Ehr}.
\end{proof}
\begin{lemma}\label{lem:dEu}
Let $\psi\in\Espace$ be a solution to \eqref{eq:NLS}, and let $u$ be defined
as above. Then
\begin{equation}
\begin{split}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) &=
p \cdot \dotp{\nabla_a \mathcal{R}_{V} u}{u}
- \frac{1}{2} \alpha^{\mathrm{tr}} \cdot \mathrm{D}^2 V(a) \cdot \dotp{x u}{u} \\
&+ \frac{1}{2}\dot\mu \nrm{u}^2 + \alpha^{\mathrm{b}}\cdot \dotp{{\rm i} u}{\nabla u},
\end{split}
\end{equation}
where $\alpha^{\mathrm{tr}} := \dot a - 2 p$ and $\alpha^{\mathrm{b}} =-\dotp -\nabla V(a)$.
\end{lemma}
\begin{proof}
The functional $\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}$, is related to the
Hamiltonian functional by
\begin{equation}\label{eq:Ku}
\begin{split}
\En_{\freq}(u) + \frac{1}{2}\dotp{\mathcal{R}_{V} u}{u} &=
\Hn_V(\psi) + \frac{1}{2}(p^2+\mu)\nrm{\psi}^2 -
p \cdot \dotp{{\rm i} \psi}{\nabla \psi} \\ &-
\frac{1}{2}\int (V(a)+\nabla V(a)\cdot (x-a))|\psi|^2 \diff^d x ,
\end{split}
\end{equation}
which is obtained by substituting \eqref{eq:u} into
$\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}$.
Using the facts that the mass $\nrm{\psi}^2$ and
Hamiltonian $\Hn_V(\psi)$ are time independent,
together with the Ehrenfest relations, Lemma~\ref{lem:Ehr}, we
obtain
\begin{equation}\nonumber
\begin{split}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) &=
(\frac{\dot{\mu}}{2} + p\cdot\dotp)\nrm{\psi}^2 - \dot
p\cdot\dotp{{\rm i} \psi}{\nabla \psi} + p\cdot \dotp{(\nabla
V)\psi}{\psi} \\ &- \frac{\dot a}{2}\cdot \mathrm{D}^2V(a)\cdot \int (x-a)|\psi|^2 \diff^d x -
\nabla V(a)\cdot \dotp{{\rm i} \psi}{\nabla \psi}.
\end{split}
\end{equation}
Collecting $p\cdot\dotp$ and $p\cdot \nabla V$ together, and
combining $\dotp$ and $\nabla V(a)$ gives
\begin{multline}\label{eq:dt1}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) =
\frac{\dot\mu}{2}\nrm{\psi}^2 +
p\cdot\dotp{(\dotp+ \nabla V)\psi}{\psi} \\
- (\dotp + \nabla V(a))\cdot\dotp{{\rm i}\psi}{\nabla
\psi} - \frac{1}{2}\dot a\cdot
\mathrm{D}^2 V(a)\cdot \int (x-a) |\psi|^2
\diff^d x.
\end{multline}
From the definition of $u$, \eqref{eq:u}, the following relations hold
\begin{eqnarray}\label{eq:urel}
&\nrm{\psi} = \nrm{u},\qquad \dotp{{\rm i} \psi}{\nabla \psi} = p\nrm{u}^2+\dotp{{\rm i}
u}{\nabla u},& \\
&\dotp{(\nabla V)\psi}{\psi} = \dotp{(\nabla V_a) u}{u}, \qquad
\dotp{(x-a)\psi}{\psi} = \dotp{xu}{u}.& \label{eq:ures}
\end{eqnarray}
Substitution of \eqref{eq:urel}--\eqref{eq:ures} into \eqref{eq:dt1}
gives, after cancellation of the $p\cdot \dotp$ terms,
\begin{multline}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) =
\frac{\dot\mu}{2}\nrm{u}^2 + p\cdot\dotp{(\nabla V_a-\nabla
V(a))u}{u} \\ - (\dotp + \nabla V(a))\cdot \dotp{{\rm i} u}{\nabla u} -
\frac{1}{2}\dot a\cdot \mathrm{D}^2 V(a)\cdot \int x |u|^2 \diff^d x.
\end{multline}
The last remaining step is to rewrite the second last term as $\dot a
-2p+2p$ and combine its $p$ term with the difference of the
potentials, recalling the definition of $\mathcal{R}_{V}$, to obtain
\begin{multline}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) =
\frac{\dot\mu}{2}\nrm{u}^2 + p\cdot\dotp{(\nabla_a \mathcal{R}_{V})u}{u}
\\ -
(\dotp + \nabla V(a))\cdot \dotp{{\rm i} u}{\nabla u} +
\frac{1}{2}(2p-\dot a)\cdot \mathrm{D}^2 V(a)\cdot \int x |u|^2
\diff^d x.
\end{multline}
Identification of the boost coefficient $\alpha^{\mathrm{b}}:=-\dot p - \nabla
V(a)$ and the translation coefficient $\alpha^{\mathrm{tr}}:=\dot a-2p$ gives
the lemma.
\end{proof}
The time derivative of the second part of
the Lyapunov functional \eqref{eq:Lt} is computed in the next lemma.
\begin{lemma}\label{lem:dEsol}
Let $\sol_\mu$ be the solution of \eqref{eq:sol}, and let $\mu$ depend on $t$.
Then
\begin{multline}
\frac{d}{dt} \big(\En_{\freq}(\sol_\mu) + \frac{1}{2}\dotp{\mathcal{R}_{V}\sol_\mu}{\sol_\mu}\big) = \\
\frac{\dot\mu}{2}\nrm{\sol_\mu}^2 +
(p + \frac{1}{2}\alpha^{\mathrm{tr}})\cdot \dotp{\nabla_a \mathcal{R}_{V}\sol_\mu}{\sol_\mu} +
\dot\mu \dotp{\mathcal{R}_{V} \sol_\mu}{\partial_\mu \sol_\mu},
\end{multline}
where $\alpha^{\mathrm{tr}}:=\dot a -2p$.
\end{lemma}
\begin{proof}
The result follows directly, upon recalling that $\Ew'(\sol_\mu)=0$ and
$\frac{1}{2}\alpha^{\mathrm{tr}} + p=\frac{\dot a}{2}$.
\end{proof}
To proceed to the proof of Proposition~\ref{prop:dL}, we restate our
condition for unique decomposition of the solution to the
nonlinear Schr\"odinger equation, $\psi\in U_\delta \cap \Espace$, in
terms of $u$:
\begin{equation}\label{eq:usplitt}
u = \sol_\mu+w\quad\text{and}\quad w\bot J \set{T}_{\sol}\Mf.
\end{equation}
Given Lemma~\ref{lem:dEu} and Lemma~\ref{lem:dEsol}, Proposition~\ref{prop:dL}
follows directly.
\begin{proof}[Proof of Proposition~\ref{prop:dL}]
Lemma~\ref{lem:dEu} states
\begin{equation}
\begin{split}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) &=
p \cdot \dotp{\nabla_a \mathcal{R}_{V} u}{u}
- \frac{1}{2} \alpha^{\mathrm{tr}} \cdot \mathrm{D}^2 V(a) \cdot \dotp{x u}{u} \\
&+ \frac{1}{2}\dot\mu \nrm{u}^2 + \alpha^{\mathrm{b}}\cdot \dotp{{\rm i} u}{\nabla u}.
\end{split}
\end{equation}
Insert $u=\sol_\mu+w$ above, and use $w\bot \{\sol_\mu$, ${\rm i}
\nabla \sol_\mu$, $x\sol_\mu\}$. Recall that $\sol_\mu$ is a real valued symmetric
function, hence $\dotp{x\sol_\mu}{\sol_\mu}=0$ as well as $\dotp{{\rm i}
\sol_\mu}{\nabla \sol_\mu}=0$. We obtain
\begin{multline}
\frac{d}{dt} \big(\En_{\freq}(u)+\frac{1}{2}\dotp{\mathcal{R}_{V} u}{u}\big) = \\
p \cdot \big(\dotp{\nabla_a
\mathcal{R}_{V} w}{w} + 2\dotp{\nabla_a \mathcal{R}_{V} \sol_\mu}{w} + \dotp{\nabla_a \mathcal{R}_{V}
\sol_\mu}{\sol_\mu}\big)\\ - \frac{1}{2} \alpha^{\mathrm{tr}} \cdot \mathrm{D}^2 V(a)
\cdot \dotp{x w}{w} + \frac{1}{2}\dot\mu (\nrm{w}^2+\nrm{\sol_\mu}^2) +
\alpha^{\mathrm{b}}\cdot \dotp{{\rm i} w}{\nabla w}
\end{multline}
Subtracting the result of Lemma~\ref{lem:dEsol} we find
\begin{multline}
\frac{d}{dt} \Lambda =
p \cdot \dotp{(\nabla_a \mathcal{R}_{V})w}{w}
- \frac{1}{2}\alpha^{\mathrm{tr}}\cdot \mathrm{D}^2 V(a)\cdot \dotp{xw}{w} \\
+ \alpha^{\mathrm{b}} \cdot \dotp{{\rm i} w}{\nabla w}
+ 2p \cdot \dotp{\nabla_a \mathcal{R}_{V} \sol_\mu}{w} - \frac{1}{2}\alpha^{\mathrm{tr}} \cdot \dotp{\nabla_a \mathcal{R}_{V} \sol_\mu}{\sol_\mu} + \frac{\dot\mu}{2}\nrm{w}^2 \\
- \dot\mu \dotp{\mathcal{R}_{V} \sol_\mu}{\partial_\mu\eta} .
\end{multline}
Note that the terms on the second and third line are at least fourth
order in the small parameters. The last two lines is the definition
of $R$ in the proposition.
\end{proof}
\section{Upper bound on $\Lambda$}
\label{sec:6}
This section we estimate $\Lambda$ from above using
Corollary~\ref{cor:apest} in Proposition~\ref{prop:dL}. Taylor
expansion of $\En_{\freq}\big(\eta(t)+w(x,t)\big)$ around $\eta$ at
$t=0$, gives
\begin{equation}\label{eq:Eb}
|\mathcal{E}_{\mu(t)}(u(x,t)) - \En_{\freq}(\eta_{\mu(t)}(x))|_{t=0}\leq C \nrmHo{w_0}^2.
\end{equation}
The remaining terms in the
Lyapunov functional are estimated using the inequality $\mathop{\mathrm{Hess}} V(x)\leq
C\epsilon_{\sind{V}}^2|x|^2\langle \eps x \rangle^{r-2}$ together with Taylor's formula and
Lemma~\ref{lem:VRup2}. Furthermore, we use from Corollary~\ref{cor:apest}. that $|\epsilon_{\sind{V}} \tilde{a}_0|\leq C$. We obtain for a $\theta\in[0,1]$
\begin{multline}\label{eq:Rb}
\left|\dotp{\mathcal{R}_{V} u}{u)}-\dotp{\mathcal{R}_{V}\eta}{\eta}\right|_{t=0} =
\left|\dotp{\mathcal{R}_{V} w}{w)}+2\dotp{\mathcal{R}_{V}\eta}{w}\right|_{t=0} \\=
\epsilon_{\sind{V}}^2|\dotp{x\cdot \mathop{\mathrm{Hess}} V(x\theta+\tilde{a}_0)\cdot x}{2\eta_{\mu_0}
\mathop{\set{Re}}(w_0)} +|\dotp{\mathcal{R}_{V} w_0}{w_0}| \\
\leq C(\epsilon_{\sind{V}}^2\nrm{w_0} + \nrm{w_0}^2+\Bnrm{\epsilon_{\sind{V}} x \langle \eps x \rangle^{(r-2)/2}w_0}^2).
\end{multline}
We now use Corollary~\ref{cor:wo} and Lemma~\ref{lem:eqv} in \eqref{eq:Rb}
and \eqref{eq:Eb} to obtain
\begin{equation}\label{eq:L2}
\left|\dotp{\mathcal{R}_{V} u}{u)}-\dotp{\mathcal{R}_{V}\eta}{\eta}\right|_{t=0}
\leq C(\epsilon_{\sind{V}}^2 \epsilon_{\sind{0}} + \epsilon_{\sind{0}}^2)
\end{equation}
and
\begin{equation}
|\mathcal{E}_{\mu(t)}(u(x,t)) -
\En_{\freq}(\eta_{\mu(t)}(x))|_{t=0}\leq C\epsilon_{\sind{0}}^2.
\end{equation}
Thus, finally
\begin{equation}\label{eq:L0}
|\Lambda|_{t=0}\leq C (\epsilon_{\sind{0}}^2 + \epsilon_{\sind{V}}^2\epsilon_{\sind{0}}).
\end{equation}
\begin{proposition}\label{prop:better}
Let $\psi\in U_{\delta}\cap \Espace$, and let $\Lambda$, $w$ and
$\alpha$ be defined as above, and $\delta$ as defined in
Theorem~\ref{thm:splitt}. Then
\begin{multline}
|\frac{d}{dt}\Lambda|\leq C\Big((\epsilon_{\sind{V}}+\epsilon_{\sind{0}}
+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}\nrm{\epsilon_{\sind{V}} x\langle \eps x \rangle^{(r-2)/2}w}^2+|\alpha|\epsilon_{\sind{V}}\Bnrm{(\epsilon_{\sind{V}}|x|)^{1/2}w}^2
\\ + \big((\epsilon_{\sind{V}}+\epsilon_{\sind{0}}+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}^2+ |\alpha|\big)(\nrmHo{w}^2+
\epsilon_{\sind{V}}^2)\Big),
\end{multline}
for times $0\leq t\leq \min(T_1,T_2)$, where $T_1$ and $T_2$ are defined in
Corollary~\ref{cor:apest}.
\end{proposition}
\begin{proof}
Proposition~\ref{prop:dL} implies
\begin{multline}\label{R}
|\frac{d}{dt}\Lambda| \leq C \big( |p| |\dotp{\nabla_a \mathcal{R}_{V} w}{w}| +
|\alpha^{\mathrm{tr}}||\mathop{\mathrm{Hess}} V(a)| |\dotp{xw}{w}| \\ + |\alpha^{\mathrm{b}}|\nrm{w}\nrm{\nabla w} +
|p|\epsilon_{\sind{V}}^3 \nrm{w} + |\alpha^{\mathrm{tr}}|\epsilon_{\sind{V}}^3 + |\dot\mu|\nrm{w}^2 +
|\dot\mu|\epsilon_{\sind{V}}^2\big).
\end{multline}
An alternative form of Eqn.~\eqref{R} is
\begin{multline}\label{eq:alt}
|\frac{d}{dt} \Lambda| \leq C \big( |p| |\dotp{\nabla_a \mathcal{R}_{V} w}{w}|
+ |\alpha||\mathop{\mathrm{Hess}} V(a)| |\dotp{xw}{w}| \\ +
(|p|\epsilon_{\sind{V}}^2+ |\alpha|)(\nrmHo{w}^2+ \epsilon_{\sind{V}}^2)\big),
\end{multline}
where we have used $\epsilon_{\sind{V}}<C$ and $|\alpha_j|\leq |\alpha|$, $\forall j$.
Using Corollary~\ref{cor:RVup} we estimate the $\mathcal{R}_{V}$ terms to obtain
\begin{multline}
|\frac{d}{dt} \Lambda| \leq C \big(
|p|\epsilon_{\sind{V}}
\nrm{\epsilon_{\sind{V}}|x|\langle \eps x \rangle^{(r-2)/2}w}^2 +
|\alpha|\epsilon_{\sind{V}}\kax{\epsilon_{\sind{V}}\Anrm{a}}^{r-2}|\dotp{\epsilon_{\sind{V}} xw}{w}| \\ +
(|p|\epsilon_{\sind{V}}^2+ |\alpha|)(\nrmHo{w}^2+ \epsilon_{\sind{V}}^2)\big).
\end{multline}
The proposition now follows upon using $\epsilon_{\sind{V}}|a|\leq C_a$ and $|p|\leq
C(\epsilon_{\sind{V}}+\epsilon_{\sind{0}}+\sqrt{\epsilon_{\sind{h}}})$ for $t\leq \min(T_1,T_2)$ from
Corollary~\ref{cor:apest} and the inequality:
\begin{equation}
\dotp{\epsilon_{\sind{V}} xw}{w}\leq \nrm{(\epsilon_{\sind{V}}|x|)^{1/2}w}^2.
\end{equation}
\end{proof}
Equation~\eqref{eq:L0} and Proposition~\ref{prop:better} yield an
upper bound on $\Lambda$:
\begin{equation}\label{eq:Lup}
|\Lambda| \leq C\epsilon_{\sind{0}}^2 + C\epsilon_{\sind{V}}^2\epsilon_{\sind{0}} +
t\sup_{s\leq t} |\frac{d}{dt} \Lambda|.
\end{equation}
\section{Lower bound on $\Lambda$}
\label{sec:7}
In this section we estimate the Lyapunov-functional $\Lambda$ from
below. Recall the definition \eqref{eq:L} of $\Lambda$:
\begin{equation}
\Lambda:=\En_{\freq}(\eta+w)-\En_{\freq}(\eta) +
\frac{1}{2}\dotp{\mathcal{R}_{V}(\eta+w)}{\eta+w} -
\frac{1}{2}\dotp{\mathcal{R}_{V}\eta}{\eta}.
\end{equation}
We have the following result.
\begin{proposition}\label{prop:Llow}
Let $\Lambda$ and $w$ be defined as above.
Then for a positive constant $C$,
\begin{equation}\label{eq:Lcoer}
\Lambda\geq \frac{1}{2}\rho_2 \nrmHo{w}^2 + C_0\rho_1\nrm{\epsilon_{\sind{V}}|x|
\langle \eps x \rangle^{(r-2)/2}w}^2 - C\nrmHo{w}^3-C\epsilon_{\sind{V}}^2\nrm{w}.
\end{equation}
where $r$ and $\rho_1>0$ are defined in \eqref{eq:Vup}, $C_0$ is the
positive constant defined in Lemma~\ref{lem:RVbd} and $\rho_2>0$ is a
positive number. The constant $C_0$ depends on the constant $C_a$ defined in
Corollary~\ref{cor:apest} bounding the size of $\epsilon_{\sind{V}}|a|$.
\end{proposition}
\begin{proof}
By Taylor expansion we have
\begin{equation}
\En_{\freq}(\eta+w)-\En_{\freq}(\eta) = \frac{1}{2}\dotp{\mathcal{L}_\sol w}{w} + \RIII{w},
\end{equation}
where $\mathcal{L}_\sol:=(\mathop{\mathrm{Hess}} \En_{\freq})(\eta)$ and
by Condition~\ref{con:A}, $|\RIII{w}|\leq C\nrmHo{w}^3$. The
coercivity of $\mathcal{L}_\sol$ for $w\bot J\set{T}_{\sol}\Mf$ is proved in Proposition D.1
of \cite{FGJS-I} under
Conditions~\ref{con:GWP}--\ref{con:F} on the
nonlinearity (in Section~\ref{sec:ass}). Thus
\begin{equation}\label{eq:bl1}
\dotp{\mathcal{L}_\sol w}{w}\geq \rho_2 \nrmHo{w}^2 \ \text{for}\ w\bot J\set{T}_{\sol}\Mf.
\end{equation}
The remaining terms of $\Lambda$ can be rewritten as
\begin{equation}
\dotp{\mathcal{R}_{V}(\eta+w)}{\eta+w}-\dotp{\mathcal{R}_{V}\eta}{\eta} =
\dotp{\mathcal{R}_{V} w}{w}+2\dotp{\mathcal{R}_{V}\eta}{w}.
\end{equation}
In Lemma~\ref{lem:RVbd} we show that
\begin{equation}\label{eq:bl2}
\mathcal{R}_{V} \geq C_0\rho_1(\epsilon_{\sind{V}}|x|)^2\langle \eps x \rangle^{r-2} \ \text{for}\ r\geq 1.
\end{equation}
Using Lemma~\ref{lem:RVbd}, \eqref{eq:bl1}, \eqref{eq:bl2} and the fact that
$\dotp{\mathcal{R}_{V}\eta}{w}\leq C\epsilon_{\sind{V}}^2\nrm{w}$ we obtain the lower bound on $\Lambda$.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:end}
The upper bound \eqref{eq:Lup} together with the bound from below in
Proposition~\ref{prop:Llow} yield the inequality
\begin{multline}\label{eq:ml}
\frac{1}{2}\rho_2 \nrmHo{w}^2 + C_0\rho_1
\Bnrm{\epsilon_{\sind{V}} x\langle \eps x \rangle^{(r-2)/2}w}^2 -
C\nrmHo{w}^3-C\epsilon_{\sind{V}}^2\nrm{w}\leq C\epsilon_{\sind{0}}^2 + C\epsilon_{\sind{V}}^2\epsilon_{\sind{0}} \\ + tC \sup_{s\leq
t}\Big(
(\epsilon+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}\nrm{\epsilon_{\sind{V}} x\langle \eps x \rangle^{(r-2)/2}w}^2+
|\alpha| \epsilon_{\sind{V}}\nrm{(\epsilon_{\sind{V}} |x|)^{1/2}w}^2 \\+ ((\epsilon+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}^2+
|\alpha|)(\nrmHo{w}^2+ \epsilon_{\sind{V}}^2)\Big),
\end{multline}
for $0\leq t\leq \min(T_1,T_2)$, where $T_1$ and $T_2$ are defined in
Corollary~\ref{cor:apest} and $\epsilon:=\epsilon_{\sind{V}}+\epsilon_{\sind{0}}$.
The right-hand side is independent
of the operator $t\mapsto s$, $\sup_{s\leq t}$ in the given time interval,
we can therefore apply this to
both sides of \eqref{eq:ml}. To simplify, let
\begin{equation}
\rho:=\min(\frac{\rho_2}{8},\frac{C_0\rho_1}{3}).
\end{equation}
We absorb higher order terms into lower order ones. Furthermore, we assume
\begin{equation}
t \leq \min(T_1,T_2,T_3), \ \text{where}\
T_3:=\frac{\rho}{C(\Anrm{\alpha}+\epsilon_{\sind{V}}(\epsilon+\sqrt{\epsilon_{\sind{h}}}))
(1+\epsilon_{\sind{V}})},
\end{equation}
in agreement with Corollary~\ref{cor:apest}. Both $\rho$ and $C$ above depend on $I$, clarifying the need for $\epsilon\ll C(I)$. Note that
\begin{equation}
T_3C(\epsilon+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}\leq \rho, \ T_3C\Anrm{\alpha}\epsilon_{\sind{V}}\leq \rho, \
\text{and}\ T_3C((\epsilon+\sqrt{\epsilon_{\sind{h}}})\epsilon_{\sind{V}}^2+\Anrm{\alpha}\leq 2\rho.
\end{equation}
We obtain
\begin{multline}\label{eq:fint}
\rho \sup_{s\leq t} \left(4\nrmHo{w}^2 + 3\Bnrm{\epsilon_{\sind{V}}
x\langle \eps x \rangle^{(r-2)/2}w}^2 \right)\\ \leq C \big( \sup_{s\leq t} (
\nrmHo{w}^3+\epsilon_{\sind{V}}^2\nrm{w})+ \epsilon_{\sind{0}}^2 + \epsilon_{\sind{V}}^2\epsilon_{\sind{0}}\big) \\ +
\rho\sup_{s\leq t} \Big( \Bnrm{\epsilon_{\sind{V}} x\langle \eps x \rangle^{(r-2)/2}w}^2+
\Bnrm{|\epsilon_{\sind{V}} x|^{1/2}w}^2+2\epsilon_{\sind{V}}^2 + 2\nrmHo{w}^2\Big).
\end{multline}
Note that $g(y):=|y|-y^2\kax{y}^{-1}\leq 2^{-1}$, $y\in\mathbb{R}$.
Indeed $g(-y)=g(y)$ and $g$ is continuously differentiable on
$(0,\infty)$, $g(y)\geq 0$ since $|y|\geq y^2\kax{y}^{-1}$ with
$g(0)=g(\infty)=0$. The function $g(y)$ has one critical point on
$(0,\infty)$ at $y=(2^{-1}(\sqrt{5}-1))^{-1/2}$ with value $\max
g=(3-\sqrt{5})(2(\sqrt{5}-1))^{-1/2}\leq 2^{-1}$. This proves the claim.
We now use this intermediate function $g(x)$ to estimate the term
above with $|x|^{1/2}$. We have
\begin{equation}\label{eq:gex}
\epsilon_{\sind{V}} |x|-(\epsilon_{\sind{V}} |x|)^2\langle \eps x \rangle^{r-2}\leq g(\epsilon_{\sind{V}}|x|)\leq \frac{1}{2}.
\end{equation}
We also have the inequalities
\begin{equation}
C\nrmHo{w}^3\leq
\rho^{-1}C^2\nrmHo{w}^4+4^{-1}\rho\nrmHo{w}^2, \ \
C\epsilon_{\sind{V}}^2\nrmHo{w}\leq C^2\rho^{-1}\epsilon_{\sind{V}}^4+4^{-1}\rho\nrmHo{w}^2.
\end{equation}
Thus we have $3\rho\nrmHo{w}^2$ on the right-hand side and
$2\rho$ of terms containing $\langle \eps x \rangle$. Moving those to the
left-hand side of \eqref{eq:fint} using the above inequalities
and simplifying we obtain
\begin{equation}\label{eq:fint2}
\sup_{s\leq t} \left(\nrmHo{w}^2 + \Bnrm{\epsilon_{\sind{V}}
x\langle \eps x \rangle^{(r-2)/2}w}^2 \right) \leq C'\epsilon^2 +
C^2\rho^{-2}(\sup_{s\leq t}\nrmHo{w}^4).
\end{equation}
Abbreviate
$\kappa:=C'\epsilon^2$.
Let
\begin{equation}
X:=\sup_{s\leq t}\left(\nrmHo{w}^2 + \Bnrm{\epsilon_{\sind{V}}|x|\langle \eps x \rangle{\epsilon_{\sind{V}}
x}^{(r-2)/2}w}^2\right).
\end{equation}
Equation~\eqref{eq:fint2} implies
\begin{equation}
X \leq C^2\rho^{-2}X^2+ \kappa.
\end{equation}
Solving this inequality, we find
\begin{equation}\label{eq:eta}
X\leq 2
\kappa,\ \text{provided}\ \kappa\leq \frac{\rho^2}{4C^2}.
\end{equation}
The definition of $X$ and $\kappa$ implies
\begin{equation}\label{eq:1012}
\nrmHo{w}\leq c'\epsilon, \ \ \text{and}\
\Bnrm{\epsilon_{\sind{V}} x\langle \eps x \rangle{\epsilon_{\sind{V}} x}^{(r-2)/2}w}\leq c' \epsilon.
\end{equation}
Lemma~\ref{lem:eqv} allow us to rewrite \eqref{eq:1012} as
$\Enrm{w}\leq c'\epsilon$. Inserting \eqref{eq:1012} into the expressions
for our modulation parameters, the estimate of the $\alpha_j$-terms in
\eqref{eq:gam}--\eqref{eq:bo} gives us $|\alpha|\leq c\epsilon^2$ and
time interval $t\leq T'$, where
\begin{equation}
T' := c\min(\epsilon^{-2},\frac{\sqrt{\epsilon_{\sind{h}}}}{\epsilon^2},
\frac{1}{\epsilon^2+\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}}})
\end{equation}
Using $\epsilon_{\sind{h}}\geq K\epsilon_{\sind{V}}$ (that is, $\epsilon_{\sind{h}}$ is not
an order of magnitude smaller then $\epsilon_{\sind{V}}$), we can shorten
the time-interval to have an upper limit of
\begin{equation}
T'':= C (\epsilon^2+\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}})^{-1}.
\end{equation}
We now choose $\epsilon$ such that \eqref{eq:eta} holds and $c'\epsilon \leq
\frac{1}{2}\delta$, where $\delta$ is defined in
Theorem~\ref{thm:splitt}. Then there
is a maximum $T_0$ such that the solution $\psi$ of \eqref{eq:NLS} is
in $U_{\delta}$ for $t\leq T_0$. Thus the decomposition
\eqref{eq:splitt} is valid and the above upper bounds for $\nrmHo{w}$
and $\alpha$ are valid for $t\leq
\min(T_0,C(\epsilon^2+\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}})^{-1})$. Thus there exists a
constant $C_T$ such that $0<C_T\leq C$, such that for $t\leq
C_T(\epsilon^2+\epsilon_{\sind{V}}\sqrt{\epsilon_{\sind{h}}})^{-1}$ the theorem holds. This concludes
the proof of Theorem~\ref{thm:main}.\hfill\qed
|
{
"timestamp": "2005-03-07T01:01:06",
"yymm": "0503",
"arxiv_id": "math-ph/0503009",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503009"
}
|
\section{Introduction}
In our previous papers [1-6] published recently, a
time-independent novel perturbation theory has been developed in
the bound state domain, which is non-perturbative, self-consistent
and systematically improvable, and used to treat successfully
significant problems in different fields of physics. Gaining
confidence from these applications, we aim through the present
work to show that similar techniques can also be used in the
continuum.
In the next section we summarize the main ideas of our approach.
The extension of the model for scattering states and the
relationship to some other perturbation approaches are discussed
in Section 3. The paper ends with a brief summary and concluding
remarks.
\section{The Model}
Let us start with a brief introduction of the formalism to remind
the compact form of the method, which would provide an easy access
of the scheme to understanding of the treatments in the continuum.
For the consideration of spherically symmetric potentials, the
corresponding Schr\"{o}dinger equation in the bound state domain
for the radial wave function has the form $(\hbar=2m=1)$
\begin{equation}
\frac{\Psi''_{n}(r)}{\Psi_{n}(r)}=[V(r)-E_{n}],
~~~V(r)=\left[V_{0}(r)+\frac{\ell(\ell+1)}{r^{2}}\right]+\Delta{V(r)},~~~~n=0,1,2,...,
\end{equation}
where $V_{0}$ is an exactly solvable unperturbed potential
together with the angular momentum barrier while $\Delta V$ is a
perturbing potential. Expressing the wave function $\Psi_{n}$ as a
product
\begin{equation}
\Psi_{n}(r)=\chi_{n}(r)\phi_{n}(r),
\end{equation}
in which $\chi_{n}$ is the known normalized eigenfunction of the
unperturbed Schr\"{o}dinger equation whereas $\phi_{n}(r)$ is a
moderating function corresponding to the perturbing potential.
Substituting (2) into (1) yields
\begin{equation}
\left(\frac{\chi''_{n}}{\chi_{n}}+\frac{\phi''_{n}}{\phi_{n}}+2\frac{\chi'_{n}}{\chi_{n}}\frac{\phi'_{n}}{\phi_{n}}\right)=V-E_{n}.
\end{equation}
Instead of setting the functions $\chi_{n}$ and $\phi_{n}(r)$, we
will set their logarithmic derivatives
\begin{equation}
W_{n}=-\frac{\chi'_{n}}{\chi_{n}},
~~~\Delta{W_{n}}=-\frac{\phi'_{n}}{\phi_{n}}
\end{equation}
which leads to
\begin{equation}
\frac{\chi''_{n}}{\chi_{n}}=W_{n}^{2}-W'_{n}=\left[V_{0}(r)+\frac{\ell(\ell+1)}{r^{2}}\right]-\varepsilon_{n},
\end{equation}
where $\varepsilon_{n}$ is the eigenvalue of the exactly solvable
unperturbed potential, and
\begin{equation}
\left(\frac{\phi''_{n}}{\phi_{n}}+2\frac{\chi'_{n}}{\chi_{n}}\frac{\phi'_{n}}{\phi_{n}}\right)=
\Delta {W^{2}_{n}}-\Delta {W'_{n}}+2W_{n}\Delta {W_{n}}=\Delta
{V(r)}-\Delta\varepsilon_{n}
\end{equation}
in which $\Delta\varepsilon_{n}$ is the energy value for the
perturbed potential leading to
$E_{n}=\varepsilon_{n}+\Delta\varepsilon_{n}$. If the whole
potential, involving the perturbing piece $\Delta{V}$, can be
analytically solvable, then Eq.(1) through (5) and (6) reduces to
\begin{equation}
(W_{n}+\Delta {W_{n}})^{2}-(W_{n}+\Delta {W_{n}})'=V-E_{n},
\end{equation}
which is known as the usual supersymmetric quantum mechanical
treatment \cite{cooper} in the literature.
However, if the whole potential has no analytical solution as the
case considered in this Letter, which means Eq.(6) cannot be
exactly solvable for $\Delta{W}$, then one can expand the
functions in terms of the perturbation parameter $\lambda$,
\begin{equation}
\Delta{V(r;\lambda)}=\sum^{\infty}_{N=1}\lambda^{N}\Delta{V_{N}(r)},
~~\Delta{W_{n}(r;\lambda)}=\sum^{\infty}_{N=1}\lambda^{N}\Delta{W_{nN}(r)},\nonumber
~~\Delta\varepsilon_{n}(\lambda)=\sum^{\infty}_{N=1}\lambda^{N}\Delta\varepsilon_{nN}
\end{equation}
where $N$ denotes the perturbation order. Substitution of the
above expansion into Eq.(6) and equating terms with the same power
of $\lambda$ on both sides yields up to for instance
$O(\lambda^{3})$
\begin{equation}
2W_{n}\Delta {W_{n1}}-\Delta {W'_{n1}}=\Delta
{V_{1}}-\Delta\varepsilon_{n1},
\end{equation}
\begin{equation}
\Delta{W^{2}_{n1}}+2W_{n}\Delta{W_{n2}}-\Delta{W'_{n2}}=\Delta{V_{2}}-\Delta\varepsilon_{n2}
\end{equation}
\begin{equation}
2(W_{n}\Delta{W_{n3}}+\Delta{W_{n1}}\Delta{W_{n2}})-\Delta{W'_{n3}}=\Delta{V_{3}}-\Delta{\varepsilon_{n3}}
\end{equation}
Eq.(6)and its expansion through Eqs.(9-11) give a flexibility for
the easy calculations of the perturbative corrections to energy
and wave functions for the $\textit{nth}$ state of interest
through an appropriately chosen perturbed superpotential. It has
been shown [1-6] that this feature of the present model leads to a
simple framework in obtaining the corrections to all states
without using complicated mathematical procedures.
\section{Application to the scattering domain}
It is well known that there are many scattering problems in which
the interaction between the projectile and the target decomposes
naturally into two parts $(V=V_{0}+\Delta{V})$. This division is
especially useful if the scattering wave function under the action
one part can be obtained exactly $(V_{0})$, while the effect of
the other $(\Delta{V})$ can be treated in some approximation as in
the present formalism.
For simplicity, we here confine ourselves to $s-$wave scattering
from a potential which is assumed that vanishes beyond a finite
radius $R$. The associated total wavefunction behaves at large
distances
\begin{equation}
\Psi(r)=\frac{1}{k}\sin(kr+\delta) ,~~~r \geq R ,
\end{equation}
where $\delta$ is the $s-$wave phase shift.
Our present treatment of scattering has concerned itself primarily
with determining how the solutions of the free Schr\"{o}dinger
equation are affected by the presence of the interaction. Within
the framework of the present formalism we suppose that the
solutions of Eq.(5) are known, or are easily found, to give the
corresponding phase shift $\delta_{0}$. Considering the expansion
$\delta=\delta_{0}+\lambda\delta_{1}+\lambda^{2}\delta_{2}+...$,
as in Eq.(8), we aim here to derive explicitly solvable and easily
accessible expressions for the phase shift contributions at
successive perturbation orders.
\subsection{First-order phase shift correction}
Keeping in mind Eq.(12) and considering the discussion in Section
2, at the first perturbation order one has
\begin{equation}
(W+\lambda\Delta{W_{1}})=-k\cot(kr+\delta_{0}+\lambda\delta_{1}),~~~W_{n}=-\frac{\chi'}{\chi}=-k\cot(kr+\delta_{0}),
\end{equation}
from where the superpotential relating to the perturbing
interaction
\begin{equation}
\Delta{W_{1}(r)}=\frac{k\delta_{1}}{\sin^{2}(kr+\delta_{0})}~,
\end{equation}
is obtained assuming that
$\sin\lambda\delta_{1}\cong\lambda\delta_{1}$ and
$\cos\lambda\delta_{1}\cong1 $.
In the second step, one needs to employ Eq. (9) to arrive at
another expression for $\Delta{W_{1}}$. Rearranging the terms,
$\Delta{W'_{1}}-2W\Delta{W_{1}}=(\Delta\varepsilon_{1}-\Delta{V_{1})}$
and multiply both sides by the integrating factor
$\exp(-2\int^{r}_{0}W(z)dz)$, which is the square of the
unperturbed wave function $\chi^{2}(r)$ through Eq.(4), one obtain
\begin{equation}
\frac{d}{dr}\left[\chi^{2}(r)\Delta{W_{1}(r)}\right]=\chi^{2}(r)(\Delta\varepsilon_{1}-\Delta{V_{1}}).
\end{equation}
The integration, and the remove of $\Delta{\varepsilon_{1}}$ term
due to the consideration of elastic scattering process here,
yields
\begin{equation}
\Delta{W_{1}(r)}=-\frac{1}{\chi^{2}(r)}\int^{r}_{0}\chi^{2}(z)
\Delta{V_{1}(z)}dz.
\end{equation}
As $\chi=\frac{1}{k}\sin(kr+\delta_{0})$ in the asymptotic region,
comparison of Eqs.(14) and (16) reproduces the first-order change
in the phase shift
\begin{equation}
\delta_{1}=-k\int^{\infty}_{0}\chi^{2}(r) \Delta{V_{1}(r)}dr.
\end{equation}
If necessary, the corresponding change in the wavefunction can
easily be obtained by the substitution of Eq.(16) into (4),
$\phi_{1}=\exp(-\int\Delta{W_{1}})$. For the reliability of the
present expression obtained, Eq (17), one may compare it with that
reproduced by other methods. For example, in the limiting case
where the unperturbed potential vanishes, the unperturbed $s-$wave
function is reduced to a plane wave $\chi(r)=\sin(kr)/k$, and the
first-order change in the phase shift becomes
\begin{equation}
\delta_{1}=-\frac{1}{k}\int^{\infty}_{0}\sin^{2}(kr)
\Delta{V_{1}(r)}dr
\end{equation}
which is just the first Born approximation for the phase shift
\cite{thaler}. In addition, the well known expression for $s-$wave
scattering amplitude by the two-potential formula in scattering
theory \cite{thaler},
\begin{equation}
f_{1}=-e^{2i\delta_{0}}\int^{\infty}_{0}\chi^{2}(r)
\Delta{V_{1}(r)}dr
\end{equation}
where the phase factor in front of the integration arises because
of the standing wave boundary conditions, justifies once more our
result since $f_{1}=-e^{2i\delta_{0}}\delta_{1}/k$ and, equating
this to the above equation leads immediately to Eq.(17).
The present result has a widespread applicability, which may also
be used in the treatment of scattering length problems. At
low-energy limit, the phase shift is related to the scattering
length $\delta_{k\rightarrow{0}}\rightarrow{-ka}$ where
${a}={a_{0}}+\lambda{a_{1}}+\lambda^{2}{a_{2}}+...$ may be
expanded in a perturbation series similar to the phase shift.
Outside the range of the potential, the unperturbed wave function
behaves as $\chi\rightarrow(r-a_{0})$. Thus, the first correction
to the scattering length is
\begin{equation}
a_{1}=\lim_{r\rightarrow\infty}\left[\int^{r}_{0}(z-a_{0})^{2}\Delta{V_{1}(z)}dz\right]
\end{equation}
which can be calculated for a given $\Delta{V_{1}}$. The
scattering length has an important physical significance. In the
low-energy limit only the $s-$wave makes a nonzero contribution to
the cross section, so that the angular distribution of the
scattering is spherically symmetric and the total cross section is
$4\pi(a_{0}+\lambda{a_{1}}+...)^{2}$. This is also exactly the
result obtained in most textbooks for the low-energy scattering of
a hard sphere of radius Thus the scattering length is the
effective radius of the target at zero energy.
As a last example, consider the case of the angular momentum
barrier as the unperturbed potential $V_{0}=\ell(\ell+1)/r^{2}$
that produces $\left[rj_{\ell}(kr)\right]$ with a phase shift
$\delta_{0}=-\ell\pi/2$. For a trivial perturbation let us choose
$\Delta{V_{1}}=\lambda/r^{2}$, due to which the angular momentum
is slightly perturbed
$\overline{\ell}\approx\ell+\lambda/(2\ell+1)+O(\lambda^{2})$.
Therefore the phase shift correction at first-order is
$\delta_{1}=-\pi/2(2\ell+1)$. Again, this exact result confirms
the reliability of Eq.(17).
\subsection{Second-order phase shift correction}
To solve Eq.(10) for $\Delta{W_{2}}$ we mimic the preceding
calculation. The integration factor is the same. In fact,
examining Eqs.(9) and (10), the only difference is that the
quantity $\Delta{V_{1}}-\Delta\varepsilon_{1}$ is replaced by
$\Delta{V_{2}}-\Delta{W^{2}_{1}}-\Delta\varepsilon_{2}$. As
$\Delta\varepsilon_{2}$ term is zero due to the process of
interest, $\Delta{W_{2}}$ is thus
\begin{equation}
\Delta{W_{2}(r)}=-\frac{1}{\chi^{2}(r)}\int^{r}_{0}\chi^{2}(z)
\left[\Delta{W^{2}_{1}(z)}-\Delta{V_{2}(z)}\right]dz.
\end{equation}
Bearing in mind that $\chi=\frac{1}{k}\sin(kr+\delta_{0})$ for the
region $r\geq{R}$, the second-order expansion in the
superpotential similar to Eq.(13) provides another expression for
$\Delta{W_{2}}$ which is
\begin{equation}
\Delta{W_{2}(r)}=k\delta_{1}^{2}\frac{\cot(kr+\delta_{0})}{\sin^{2}(kr+\delta_{0})}+\frac{k\delta_{2}}{\sin^{2}(kr+\delta_{0})}
\end{equation}
Comparison of Eqs.(21) and (22), together with the substitution
of (14) in (21), leads to an auxiliary function for the second
order phase shift correction,
\begin{equation}
\delta_{2}(r)=-\frac{1}{k}\int^{r}_{0}\Delta{V_{2}(z)}\sin^{2}(kz+\delta_{0})
dz
+k\delta_{1}^{2}\int^{r}_{0}\frac{dz}{\sin^{2}(kz+\delta_{0})}-\delta_{1}^{2}\cot(kr+\delta_{0}),
\end{equation}
where a singularity appears in the second integral at $z=0$. This
problem can be circumvented by replacing the lower limit of the
integral with $R$. Assuming $\Delta{V}=\Delta{V_{1}}$ as in
realistic problems of nuclear physics, which means that
$\Delta{V_{2}}=0$, the $r-$dependent phase shift correction in the
second-order is given in the form of
\begin{equation}
\delta_{2}(r)=\delta_{1}^{2}\cot(kR+\delta_{0})-2\delta_{1}^{2}\cot(kr+\delta_{0}).
\end{equation}
As an alternative treatment, which leads to a concrete comparison,
one can go back to Eq.(21) and split $\chi^{2}\Delta{W_{1}^{2}}$
term in two parts as $(\chi^{2}\Delta{W_{1}})(\Delta{W_{1}})$
allowing to invoke Eq.(16). In this case the comparison of the
result with the expansion in (22) gives
\begin{equation}
\delta_{2}=-k\int^{\infty}_{0}\chi^{2}(r)\Delta{V_{1}(r)}dr
\int^{r}_{R}\frac{dz}{\chi^{2}(z)}
\left[\int^{R}_{z}\chi^{2}(y)\Delta{V_{1}(y)}dy-\frac{\delta_{1}}{k}\right]+\delta_{1}^{2}\cot(kR+\delta_{0})
\end{equation}
which is in agreement with the work in \cite{milward}. In
addition, the use of (17) in (24) transforms it into Eq. (25).
Furthermore, the reader is reminded that the second Born
approximation for the phase shift can be most easily derived using
the variable phase equation approach \cite{calegero},
\begin{equation}
\delta_{2}=2k^{2}\int^{\infty}_{0}\chi^{2}(r)\Delta{V_{1}(r)}\cot(kr)dr\int^{r}_{0}\chi^{2}(y)\Delta{V_{1}(y)}dy
\end{equation}
which, in the light of Eq. (15), is the same result as we find
from Eq (25), by putting $\delta_{0}=0$ . Higher order terms can
also be evaluated in the same manner.
\section{Concluding Remarks}
The recently introduced time-independent perturbation theory has
been successfully extended from the bound state region to the
scattering domain. For the clarification, the work has been
carried out with the consideration of $s-$wave scattering only.
However, generalization of the formalism to higher partial waves
in the scattering domain does not cause any problem. The inclusion
of the centrifugal barrier contribution in the effective potential
for instance leads to the replacement of the $s-$wave phase shift
with $\delta_{\ell}-\ell\pi/2$ due to the related wave function
$\chi(r)=\sin(kr+\delta_{\ell}-\ell\pi/2)/k$ in the asymptotic
region, supposing both the unperturbed and perturbed potentials
vanish at a large $r>R_{1}$ which means that in the region
$R_{1}<r\leq{R}$ there is then only the centrifugal barrier
contribution. This inclusion requires simply to repeat the present
calculations for the replacement in the phase shift.
It should be stress that, anything that can be achieved from the
present formalism must also be obtainable from the works [9,10] in
the literature. For instance, considering the bound state region,
Bender's formalism \cite{bender} can be simplified by introducing
the auxiliary function $F_{N}(r)$ such that the whole wave
function $\Psi_{N}(r)=\chi(r)F_{N}(r)$ where denotes the
perturbation order. The first-order correction can then be written
as
$\frac{d}{dr}\left[\chi^{2}\frac{dF}{dr}\right]=(\Delta{V_{1}}-\Delta\varepsilon_{1})\chi^{2}$
which corresponds exactly to the present treatment by Eq. (15)
when we identify $\Delta{W_{1}}=dF/dr$. The higher order
calculations can be linked to ours in the similar manner. Whereas,
the works of Milward and Wilkin \cite{milward} may be related to
the present formalism in both domain, the bound and scattering
region by making a relation between their probability density
distributions/derivatives and our $\Delta{W}$ functions, such as
$\Delta{W_{0}}=-P_{0}'/2P_{0}$ at the zeroth order,
$\Delta{W_{1}}=(-P_{1}/2P_{0})'$ at the first order and
$\Delta{W_{2}}=(-P_{2}/2P_{0})'$ at the second order etc.
Nevertheless, the present technique provides a clean and explicit
route for the calculations without tedious and cumbersome
integrals.
The energy variation of the scattering wave function and phase
shift can also be studied by perturbing in the energy. We wish to
stress that all these effects depend purely upon the perturbation
and the unperturbed wave function; explicit knowledge of the
unperturbed potential is not necessary. This exposition will be
deferred to a later publication.
|
{
"timestamp": "2005-03-21T13:21:05",
"yymm": "0503",
"arxiv_id": "nucl-th/0503055",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503055"
}
|
\section{Introduction} \label{Sec:Introduction}
Homotopy continuation methods provide reliable and efficient numerical
algorithms to compute accurate approximations to all isolated solutions
of polynomial systems, see e.g.~\cite{Li03} for a recent survey.
As proposed in~\cite{SW}, we can approximate a positive dimensional
solution set of a polynomial system by isolated solutions, which
are obtained as intersection points of the set with a generic
linear space of complementary dimension.
New homotopy algorithms have been developed in a series of
papers~\cite{SV,SVW1,SVW4,SVW9,SVW10} to give numerical
representations of positive dimensional solution sets of polynomial
systems. These homotopies are the main numerical algorithms in a
young field we call {\em numerical algebraic geometry}.
See~\cite{SW2} for a detailed treatment of this subject.
This paper provides an algorithm to compute numerical approximations
to positive dimensional solution sets of polynomial systems
by introducing the equations one at a time.
The advantage of working in this manner is that the special
properties of individual equations are revealed early in the
process, thus reducing the computational cost of later stages.
Consequently, although the new algorithm has more stages of
computation than earlier approaches, the amount of work in each stage
can be considerably less, producing a net savings in computing time.
This paper is organized in three parts. First we explain our method
to represent and to compute a numerical irreducible decomposition of
the solution set of a polynomial system. In the third section,
new diagonal homotopy algorithms will be applied to solve systems
subsystem by subsystem or equation by equation. Computational
experiments are given in the fourth section.
\section{A Numerical Irreducible Decomposition}
We start this section with a motivating illustrative example,
which shows the occurrence of several solution sets, of different
dimensions and degrees.
Secondly, we define the notion of witness sets, which we
developed to represent pure dimensional solution sets of polynomial
systems {\em numerically}. Witness sets are computed by
cascades of homotopies between embeddings of polynomial systems.
\subsection{An Illustrative Example}
Our running example (used also in~\cite{SVW1}) is
the following:
\begin{equation} \label{Eq:Illusex}
f(x,y,z)
= \left[
\begin{array}{r}
(y-x^2)(x^2+y^2+z^2-1)(x-0.5) \\
(z-x^3)(x^2+y^2+z^2-1)(y-0.5) \\
(y-x^2)(z-x^3)(x^2+y^2+z^2-1)(z-0.5) \\
\end{array}
\right].
\end{equation}
In this factored form we can easily identify the decomposition of the
solution set $Z=f^{-1}(\zero)$ into irreducible solution components,
as follows:
\begin{equation} \label{Eq:Illussol}
Z = Z_2 \cup Z_1 \cup Z_0
= \{Z_{21}\} \cup \{Z_{11} \cup Z_{12} \cup Z_{13} \cup Z_{14} \}
\cup \{Z_{01}\}
\end{equation}
where
\begin{center}
\begin{tabular}{l}
1. $Z_{21}$ is the sphere $x^2+y^2+z^2-1=0$, \\
2. $Z_{11}$ is the line $(x=0.5,z=0.5^3)$, \\
3. $Z_{12}$ is the line $(x=\sqrt{0.5},y=0.5)$, \\
4. $Z_{13}$ is the line $(x=-\sqrt{0.5},y=0.5)$, \\
5. $Z_{14}$ is the twisted cubic $(y-x^2=0,z-x^3=0)$, \\
6. $Z_{01}$ is the point $(x=0.5,y=0.5,z=0.5)$.
\end{tabular}
\end{center}
The sequence of homotopies in~\cite{SV} required to track 197 paths
to find a numerical representation of the solution set~$Z$.
With the new approach we will just have to trace 13 paths!
We show how this is done in Figure~\ref{Fig:Flowillusex}
in~\S \ref{Sec:Illusex} below, but we first describe a numerical
representation of~$Z$ in the next section.
\subsection{Witness Sets}
We define witness sets as follows. Let $f:{\mathbb C}^N\rightarrow{\mathbb C}^n$
define a system $f(\x) = \zero$ of $n$ polynomial equations $f =
\{f_1,f_2,\ldots,f_n \}$ in $N$ unknowns~$\x =
(x_1,x_2,\ldots,x_N)$. We denote the solution set of $f$ by
\begin{equation}
V(f) = \{ \ \x \in{\mathbb C}^N \ | \ f(\x)=\zero \ \}.
\end{equation}
This is a reduced\footnote{``Reduced'' means the set occurs with
multiplicity one, we ignore multiplicities $> 1$ in this paper.}
algebraic set. Suppose $X\subset V(f)\subset {\mathbb C}^N$ is a pure
dimensional\footnote{``Pure dimensional'' (or ``equidimensional'')
means all components of the set have the same dimension.} algebraic
set of dimension~$i$ and degree~$d$. Then, a witness set for $X$ is
a data structure consisting of the system $f$, a generic linear
space $L\subset{\mathbb C}^N$ of codimension~$i$, and the set of $d$ points
$X\cap L$.
If $X$ is not pure dimensional, then a witness set for $X$ breaks up
into a list of witness sets, one for each dimension. In our work, we
generally ignore multiplicities, so when a polynomial system has a
nonreduced solution component, we compute a witness set for the
reduction of the component. Just as $X$ has a unique decomposition
into irreducible components, a witness set for $X$ has a
decomposition into the corresponding irreducible witness sets,
represented by a partition of the witness set representation
for~$X$. We call this a \emph{numerical irreducible decomposition}
of~$X$.
The irreducible decomposition of the solution set~$Z$
in~(\ref{Eq:Illussol}) is represented by
\begin{equation}
[W_2 , W_1, W_0] =
[ [ W_{21} ], [ W_{11}, W_{12}, W_{13}, W_{14} ], [ W_{01} ] ],
\end{equation}
where the $W_i$ are witness sets for pure dimensional components,
of dimension~$i$, partitioned into witness sets $W_{ij}$'s corresponding
to the irreducible components of~$Z$.
In particular:
\begin{center}
\begin{tabular}{l}
1. $W_{21}$ contains two points on the sphere,
cut out by a random line, \\
2. $W_{11}$ contains one point on the line $(x=0.5,z=0.5^3)$,
cut out by a random plane, \\
3. $W_{12}$ contains one point on the line $(x=\sqrt{0.5},y=0.5)$,
cut out by a random plane, \\
4. $W_{13}$ contains one point on the line $(x=-\sqrt{0.5},y=0.5)$,
cut out by a random plane, \\
5. $W_{14}$ contains three points on the twisted cubic,
cut out by a random plane, \\
6. $W_{01}$ is still just the point $(x=0.5,y=0.5,z=0.5)$.
\end{tabular}
\end{center}
Applying the formal definition, the witness sets $W_{ij}$ consist of
witness points $\w = \{ i, f, L, \x \}$, for $\x \in Z_{ij} \cap L$,
where $L$ is a random linear subspace of codimension~$i$ (in this
case, of dimension $3-i$). Moreover, observe $\#W_{ij} =
\deg(Z_{ij}) = \#(Z_{ij} \cap L)$.
Witness sets are set-theoretically equivalent to {\em lifting
fibers} which occur in a {\em geometric resolution} of polynomial
system. This geometric resolution is a symbolic analogue to a
numerical irreducible decomposition. We refer to
\cite{GH93,GH01,GLS01,Lec03} for details about this symbolic
approach to solving polynomial system geometrically.
\subsection{Embeddings and Cascades of Homotopies}
A witness superset $\hatW_k$ for the pure $k$-dimensional part $X_k$
of~$X$ is a set in $X\cap L$, which contains $W_k:=X_k\cap L$ for a
generic linear space $L$ of codimension~$k$. The set of ``junk
points'' in $\hatW_k$ is the set $\hatW_k \setminus W_k$, which lies
in $ \left(\cup_{j>k}X_j\right)\cap L$.
The computation of a numerical irreducible decomposition for $X$
runs in three stages:
\begin{enumerate}
\item Computation of a \emph{witness superset} $\hatW$ consisting
of witness supersets $\hatW_k$ for each dimension $k = 1,2,\ldots,N$.
\item Removal of junk points from $\hatW$ to get a witness set $W$ for $X$.
\item Decomposition of $W$ into its irreducible components.
\newline In this stage, every witness set for a pure dimensional solution set
is partitioned into witness sets corresponding to the irreducible
components of the solution set.
\end{enumerate}
Up to this point, we have used the dimension of a component as the
subscript for its witness set, but in the algorithms that follow, it
will be more convenient to use codimension. The original algorithm
for constructing witness supersets was given in \cite{SW}. A more
efficient cascade algorithm for this was given in \cite{SV} by means
of an embedding theorem.
In \cite{SVW9}, we showed how to carry out the generalization of
\cite{SV} to solve a system of polynomials on a pure
$N$-dimensional algebraic set $Z\subset {\mathbb C}^m$. In the same
paper, we used this capability to address the situation where we
have two polynomial systems $f$ and $g$ on ${\mathbb C}^N$ and we wish to
describe the irreducible decompositions of $A\cap B$ where
$A\in{\mathbb C}^N$ is an irreducible component of $V(f)$ and $B\in{\mathbb C}^N$
is an irreducible component of $V(g)$. We call the resulting
algorithm a \emph{diagonal homotopy}, because it works by
decomposing the diagonal system $\bfu-\bfv=\zero$ on $Z=A\times B$, where
$(\bfu,\bfv)\in{\mathbb C}^{2N}$. In~\cite{SVW10}, we rewrote the homotopies
``intrinsically,'' which means that the linear slicing subspaces
are not described explicitly by linear equations vanishing on
them, but rather by linear parameterizations.
(Note that intrinsic forms were first used in a substantial way
to deal with numerical homotopies of parameterized linear spaces
in~\cite{HSS98}, see also~\cite{HV00}.)
This has always been allowed, even in \cite{SW},
but \cite{SVW10} showed how to do so consistently through the
cascade down dimensions of the diagonal homotopy, thereby
increasing efficiency by using fewer variables.
The subsequent steps of removing junk and decomposing the witness
sets into irreducible pieces have been studied in
\cite{SVW1,SVW2,SVW3,SVW4}. These methods presume the capability to
track witness points on a component as the linear slicing space is
varied continuously. This is straightforward for reduced solution
components, but the case of nonreduced components, treated in
\cite{SVW5}, is more difficult. An extended discussion of the basic
theory may be found in \cite{SW2}.
In this paper, we use multiple applications of the diagonal homotopy
to numerically compute the irreducible decomposition of $A\cap B$
for general algebraic sets $A$ and $B$, without the restriction that
they be irreducible. At first blush, this may seem an incremental
advance, basically consisting of organizing the requisite
bookkeeping without introducing any significantly new theoretical
constructs. However, this approach becomes particularly interesting
when it is applied ``equation by equation,'' that is, when we
compute the irreducible decomposition of $V(f)$ for a system
$f=\{f_1,f_2,\ldots,f_n\}$ by systematically computing $V(f_1)$, then
$A_1\cap V(f_2)$ for $A_1$ a component of $V(f_1)$, then $A_2\cap
V(f_3)$ for $A_2$ a component of $A_1\cap V(f_2)$, etc. In this way,
we incrementally build up the irreducible decomposition one equation
at a time, by intersecting the associated hypersurface with all the
solution components of the preceding equations. The main impact is
that the elimination of junk points and degenerate solutions at
early stages in the computation streamlines the subsequent stages.
Even though we use only the total degree of the equations---not
multihomogeneous degrees or Newton polytopes---the approach is
surprisingly effective for finding isolated solutions.
\section{Application of Diagonal Homotopies}
In this section, we define our new algorithms by means of two
flowcharts, one for solving subsystem-by-subsystem, and one that
specializes the first one to solving equation-by-equation. We then
briefly outline simplifications that apply in the case that only the
nonsingular solutions are wanted. First, though, we summarize the
notation used in the definition of the algorithms.
\subsection{Symbols used in the Algorithms}
A witness set $W$ for a pure $i$-dimensional component $X$ in $V(f)$
is of the form $W=\{i,f,L,\sX\}$, where $L$ is the linear subspace
that cuts out the $\deg X$ points $\sX=X \cap L$. In the following
algorithm, when we speak of a \emph{witness point} $\w\in W$, it
means that $\w=\{i,f,L,\x\}$ for some $\x\in\sX$. For such a $\w$
and for $g$ a polynomial (system) on ${\mathbb C}^N$, we use the shorthand
$g(\w)$ to mean $g(\x)$, for $\x \in \w$.
In analogy to $V(f)$, which acts on a polynomial system, we
introduce the operator $\sV(W)$, which means the solution component
represented by the witness set~$W$. We also use the same symbol
operating on a single witness point $\w=\{i,f,L,\x\}$, in which case
$\sV(\w)$ means the irreducible component of $V(f)$ on which point
$\x$ lies. This is consistent in that $\sV(W)$ is the union of
$\sV(\w)$ for all $\w\in W$.
Another notational convenience is the operator $\sW(A)$, which gives
a witness set for an algebraic set $A$. This is not unique, as it
depends on the choice of the linear subspaces that slice out the
witness points. However, any two witness sets $W_1,W_2\in\sW(A)$
are equivalent under a homotopy that smoothly moves from one set of
slicing subspaces to the other, avoiding a proper algebraic subset
of the associated Grassmannian spaces, where witness points diverge
or cross. That is, we have $\sV(\sW(A))=A$ and $\sW(\sV(W))\equiv W$,
where the equivalence in the second expression is under homotopy
continuation between linear subspaces.
The output of our algorithm is a collection of witness sets $W_i$,
$i=1,2,\ldots,N$, where $W_i$ is a witness set for the pure
\emph{codimension}~$i$ component of $V(f)$. (This breaks from our
usual convention of subscripting by dimension, but for this
algorithm, the codimension is more convenient.) Breaking $W_i$ into
irreducible pieces is a post-processing task, done by techniques
described in \cite{SVW1,SVW3,SVW4}, which will not be described
here.
The algorithm allows the specification of an algebraic set
$Q\in{\mathbb C}^N$ that we wish to ignore. That is, we drop from the
output any components that are contained in $Q$, yielding witness
sets for $V(f_1,f_2,\ldots,f_n)\in{\mathbb C}^N\setminus Q$. Set $Q$ can be specified
as a collection of polynomials defining it or as a witness point
set.
For convenience, we list again the operators used in our notation,
as follows:
\begin{description}
\item[$V(f)$] The solution set of $f(x)=0$.
\item[$\sW(A)$] A witness set for an algebraic set $A$,
multiplicities ignored, as always.
\item[$\sV(W)$] The solution component represented by witness
set $W$.
\item[$\sV(\w)$] The irreducible component of $V(f)$ on which
witness point $\w\in\sW(V(f))$ lies.
\end{description}
\begin{figure}
\centering
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\put(15,100){\makebox(0,0)[b]{Witness $W^A$ for $A=V(f^A)\setminus Q$}}
\put(0,92){\WitnessArrayBox{$W^A_j$}}
\put(75,100){\makebox(0,0)[b]{Witness $W^B$ for $B=V(f^B)\setminus Q$}}
\put(60,92){\WitnessArrayBox{$W^B_k$}}
\put(30,12){\WitnessArrayBox{$W^C_\ell$}}
\put(45,10){\makebox(0,0)[t]{Witness $W^C$ for $C=V(f^A,f^B)\setminus Q$}}
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\put(16,86){\makebox(0,0)[b]{$\w_A$}}
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\caption{Subsystem-by-subsystem generation of witness
sets for \hbox{$V(f^A,f^B)\setminus Q$.}}\label{Fig:SysBySys}
\end{figure}
\subsection{Solving Subsystem by Subsystem}
In this section, we describe how the diagonal homotopy can be
employed to generate a witness set $W=\sW(V(f^A,f^B)\setminus Q)$, given
witness sets $W^A$ for $A=V(f^A)\setminus Q$, and $W^B$ for $B=V(f^B)\setminus Q$.
Let us denote this operation as $W={\bf SysBySys}(A,B;Q)$.
Moreover, suppose
${\bf Witness}(f;Q)$ computes a witness set $\sW(V(f)\setminus Q)$ by any means
available, such as by working on the entire system $f$ as in our
previous works, \cite{SV,SW}, with junk points removed but not
necessarily decomposing the sets into irreducibles. With these two
operations in hand, one can approach the solution of any large
system of polynomials in stages. For example, suppose
$f=\{f^A,f^B,f^C\}$ is a system of polynomials composed of three
subsystems, $f^A$, $f^B$, and $f^C$, each of which is a collection
of one or more polynomials. The computation of a witness set
$W=\sW(V(f)\setminus Q)$ can be accomplished as
\begin{align*}
W^A = {\bf Witness}(f^A;Q),&\quad W^B={\bf Witness}(f^B;Q)\\
W^C = {\bf Witness}(f^C;Q),&\quad W^{AB} = {\bf SysBySys}(W^A,W^B;Q),\\
W = {\bf SysBySys}&(W^{AB},W^C;Q).
\end{align*}
This generalizes in an obvious way to any number of subsystems.
Although we could compute $W={\bf Witness}(f;Q)$ by directly working on
the whole system $f$ in one stage, there can be advantages to
breaking the computation into smaller stages.
The diagonal homotopy as presented in \cite{SVW9} applies to
computing $A\cap B$ only when $A$ and $B$ are each irreducible. To
implement {\bf SysBySys}, we need to handle sets that have more than one
irreducible piece. In simplest terms, the removal of the requirement
of irreducibility merely entails looping through all pairings of the
irreducible pieces of $A$ and $B$, followed by filtering to remove
from the output any set that is contained inside another set in the
output, or if two sets are equal, to eliminate the duplication. In
addition to this, however, we would like to be able to proceed
without first decomposing $A$ and $B$ into irreducibles. With a bit
of attention to the details, this can be arranged.
Figure~\ref{Fig:SysBySys} gives a flowchart for algorithm {\bf SysBySys}.
For this to be valid as shown, we require that the linear subspaces
for slicing out witness sets are chosen once and for all and used in
all the runs of {\bf Witness}\ and {\bf SysBySys}. In other words, the
slicing subspaces for $W^A$ and $W^B$ at the top of the algorithm
must be the same as each other and as the output~$W^C$. This ensures
that witness sets from one stage can, under certain circumstances,
pass directly through to the next stage. Otherwise, a continuation
step would need to be inserted to move from one slicing subspace to
another.
The setup of a diagonal homotopy to intersect two irreducibles
$A\in{\mathbb C}^N$ and $B\in{\mathbb C}^N$ involves the selection of certain random
elements. We refer to \cite{SVW9,SVW10} for the full details.
All we need to know at present is that in choosing these random
elements the only dependence on $A$ and $B$ is their dimensions,
$\dim A$ and $\dim B$. If we were to intersect another pair of
irreducibles, say $A'\in{\mathbb C}^N$ and $B'\in{\mathbb C}^N$, having the same
dimensions as the first pair, i.e., $\dim A'=\dim A$ and $\dim
B'=\dim B$, then we may use the same random elements for both. In
fact, the random choices will be generic for any finite number of
intersection pairs. Furthermore, if $A$ and $A'$ are irreducible
components of the solution set of the same system of polynomials,
$f^A$, and $B$ and $B'$ are similarly associated to system $f^B$,
then we may use exactly the same diagonal homotopy to compute $A\cap
B$ and $A'\cap B'$. The only difference is that in the former case,
the start points of the homotopy are pairs of points
$(\alpha,\beta)\in \sW(A)\times\sW(B)\subset{\mathbb C}^{2N}$, while in the
latter, the start points come from $\sW(A')\times\sW(B')$.
To explain this more explicitly, consider that the diagonal homotopy
for intersecting $A$ with $B$ works by decomposing $\bfu-\bfv$
on $A\times B$.
To set up the homotopy, we form the randomized system
\begin{equation}
\sF(\bfu,\bfv) = \left[
\begin{array}{l}
R_A f^A(\bfu)\\
R_B f^B(\bfv)
\end{array}
\right],
\end{equation}
where $R_A$ is a random matrix of size $(N-\dim A)\times \#(f^A)$
and $R_B$ is random of size $(N-\dim B)\times \#(f^B)$. [By
$\#(f^A)$ we mean the number of polynomials in system $f^A$ and
similarly for $\#(f^B)$.] The key property is that $A\times B$ is
an irreducible component of $V(\sF(\bfu,\bfv))$ for all $(R_A,R_B)$ in a
nonzero Zariski open subset of ${\mathbb C}^{(N-\dim A)\times
\#(f^A)}\times{\mathbb C}^{(N-\dim B)\times \#(f^B)}$, say $R_{AB}$. But
this property holds for $A'\times B'$ as well, on a possibly
different Zariski open subset, say $R_{A'B'}$. But $R_{AB}\cap
R_{A'B'}$ is still a nonzero Zariski open subset, that is, almost
any choice of $(R_A,R_B)$ is satisfactory for computing both $A\cap
B$ and $A'\cap B'$, and by the same logic, for any finite number of
such intersecting pairs.
The upshot of this is that if we wish to intersect a
pure dimensional set $A=\{A_1,A_2\}\subset V(f^A)$ with a
pure dimensional set $B=\{B_1,B_2\}\subset V(f^B)$, where $A_1$,
$A_2$, $B_1$, and $B_2$ are all irreducible, we may form one
diagonal homotopy to compute all four intersections $A_i\cap B_j$,
$i,j\in\{1,2\}$, feeding in start point pairs from all four
pairings. In short, the algorithm is completely indifferent as to
whether $A$ and $B$ are irreducible or not. Of course, it can
happen that the same irreducible component of $A\cap B$ can arise
from more than one pairing $A_i\cap B_j$, so we will need to take
steps to eliminate such duplications.
We are now ready to examine the details of the flowchart in
Figure~\ref{Fig:SysBySys} for computing $W^C=\sW(V(f^A,f^B)\setminus Q)$ from
$W^A=\sW(V(f^A)\setminus Q)$ and $W^B=\sW(V(f^B)\setminus Q)$. It is assumed that the
linear slicing subspaces are the same for $W^A$, $W^B$, and $W^C$.
The following items (a)--(g) refer to labels in that chart.
\begin{enumerate}
\item[(a)] Witness point $\w_A$ is a generic point of the component of
$V(f^A)$ on which it lies, $\sV(\w_A)$.
Consequently, $f^B(\w_A)=0$ implies, with
probability one, that $\sV(\w_A)$ is contained in some component
of $V(f^B)$. Moreover, we already know that $\w_A$ is not in any
higher dimensional set of $A$, and therefore it cannot be in any
higher dimensional set of $C$. Accordingly, any point $\w_A$ that passes
test~(a) is an isolated point in
witness superset $\hatW^C$. The containment of $\sV(\w_A)$ in $B$ means
that the dimension of the set is unchanged by intersection, so
if $\w_A$ is drawn from $W^A_j$, its correct destination is $W^C_j$.
On the other hand, if
$f^B(\w_A)\ne0$, then $\w_A$ proceeds to the diagonal homotopy as
part of the computation of $\sV(\w_A)\cap B$.
\item[(b)] This is the symmetric operation to (a).
\item[(c)] Witness points for components not completely contained in
the opposing system are fed to the diagonal homotopy in order to
find the intersection of those components. For each combination
$(a,b)$, where $a=\dim\sV(\w_A)$ and $b=\dim\sV(\w_B)$, there is a
diagonal homotopy whose random constants are chosen once and for
all at the start of the computation.
\item[(d)] This test, which appears in three places, makes sure
that multiple copies of a witness point do not make it into $W^C$.
Such duplications can arise when $A$ and $B$ have components in
common, when different pairs of irreducible components from $A$
and $B$ share a common intersection component, or when some
component is nonreduced.
\item[(e)] Since a witness point $\hatw$ is sliced out generically from
the irreducible component, $\sV(\hatw)$, on which it lies, if
$\hatw \in Q$, then $\sV(\hatw)\subset Q$. We have specified
at the start that we wish to ignore such sets, so we throw them
out here.
\item[(f)] In this test, ``singular'' means that the Jacobian
matrix of partial derivatives for the sliced system that cuts
out the witness point is rank deficient. We test this by a
singular value decomposition of the matrix. If the point is
nonsingular, it must be isolated and so it is clearly a witness
point. On the other hand, if it is singular, it might be either
a singular isolated point or it might be a junk point that lies
on a higher dimensional solution set, so it must be subjected to
further testing.
\item[(g)] Our current test for whether a singular test point is
isolated or not is to check it against all the higher dimensional
sets. If it is not in any of these, then it must be an isolated
point, and we put it in the appropriate output bin.
\end{enumerate}
In the current state of the art, the test in box~(g) is done using
homotopy membership tests. This consists of following the paths of
the witness points of the higher dimensional set as its linear
slicing subspace is moved continuously to a generically disposed one
passing through the test point. The test point is in the higher
dimensional set if, and only if, at the end of this continuation one
of these paths terminates at the test point, see~\cite{SVW2}.
In the future, it may
be possible that a reliable local test, based just on the local
behavior of the polynomial system, can be devised that determines if
a point is isolated or not. This might substantially reduce the
computation required for the test. As it stands, one must test the
point against all higher dimensional solution components, and so
points reaching box~(g) may have to wait there in limbo until all
higher dimensional components have been found.
The test~(e) for membership in $Q$ would entail a homotopy
membership test if $Q$ is given by a witness set. If $Q$ is given
as $V(f^Q)$ for some polynomial system $f^Q$, then the test is
merely ``$f^Q(\hatw)=0?$'' We have cast the whole algorithm on
${\mathbb C}^N$, but it would be equivalent to cast it on complex projective
space $\pn N$ and use $Q$ as the hyperplane at infinity.
As a cautionary remark, note that the algorithm depends on $A$ and
$B$ being complete solution sets of the given polynomial subsystems,
excepting the same set $Q$. It is not valid when $A$ or $B$ is a
partial list of components. In particular, suppose $A$ and $B$ are
distinct irreducible components of the same system, i.e., $f^A=f^B$.
The diagonal homotopy applies to finding $A\cap B$, but if we feed
these into the current algorithm, we will not get the desired
result. This is because of tests (a) and~(b), which would pass the
witness points around the diagonal homotopy block and directly into
the output. The algorithm is designed to compute $V(f)$, which in
this case includes $A\cup B$.
\begin{figure}
\centering
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\caption{Stage~$k$ of equation-by-equation generation of witness
sets for $V(f_1,\ldots,f_n)\in{\mathbb C}^N\setminus Q$}\label{Fig:EqByEq}
\end{figure}
\subsection{Solving Equation by Equation}
The equation-by-equation approach to solving a polynomial system is
a limiting case of the subsystem-by-subsystem approach, wherein one
subsystem is just a single polynomial equation. Accordingly, we
begin by computing a witness set $X^i$ for the solution set
$V(f_i)$, $i=1,2,\ldots,n$ of each individual polynomial. If any
polynomial is identically zero, we drop it and decrement~$n$. If any
polynomial is constant, we terminate immediately, returning a null
result. Otherwise, we find $X^i=(V(f_i)\cap L)\setminus Q$, where $L$ is a
1-dimensional generic affine linear subspace. A linear
parameterization of $L$ involves just one variable, so $X^i$ can be
found with any method for solving a polynomial in one variable,
discarding any points that fall in~$Q$.
Next, we randomly choose the affine linear subspaces that will cut
out the witness sets for any lower dimensional components that
appear in succeeding intersections.
The algorithm proceeds by setting $W^1=X^1$ and then computing
$W^{k+1}$ $=$ ${\bf SysBySys}$ $(W^k,X^{k+1};Q)$ for $k=1,2,\ldots,n-1$. The
output of stage~$k$ is a collection of witness sets $W^{k+1}_i$ for
$i$ in the range from 1 to $\min(N,k+1)$. (Recall, we are using the
codimension for the subscript.) Of course, some of these may be
empty, in fact, in the case of a total intersection, only the lowest
dimensional one, $W^{k+1}_{k+1}$, is nontrivial.
In applying the subsystem-by-subsystem method to this special case,
we can streamline the flowchart a bit, due to the fact that
$V(f_{k+1})$ is a hypersurface. The difference comes in the
shortcuts that allow some witness points to avoid the diagonal
homotopy.
The first difference is at the output of test~(a), which now sends
$\w$ directly to the final output without any testing for duplicates.
This is valid because we assume that on input $\sV(\w)$ is not
contained within any higher dimensional component of $\sV(W^k)$, and
in the intersection with hypersurface $V(f_{k+1})$ that is the only
place a duplication could have come from.
On the opposing side, test~(b) is now stronger than before. The
witness point $\x$ only has to satisfy one polynomial among
$f_1,f_2,\ldots,f_k$ in order to receive special treatment. This is
because we already threw out any polynomials that are identically
zero, so if $f_j(\x)=0$ it implies that $\sV(\x)$ is a factor of
$V(f_j)$. But the intersection of that factor with all the other
$V(f_i)$, $i\ne j$, is already in $V(f_1,f_2,\ldots,f_k)$, so nothing
new can come out of intersecting $\sV(\x)$ with $V(f_1,f_2,\ldots,f_k)$.
Accordingly, we may discard $\x$ immediately.
Another small difference from the more general algorithm is that the
test for junk at box~(g) never has to wait for higher dimensional
computations to complete. When carrying out the algorithm, we draw
witness points from $W^k$ in order proceeding from left to right so
that computations are performed by decreasing dimension. Moreover,
we should run all the witness points in $W^k_j$ through test~(a)
before proceeding to feed any of them to the diagonal homotopy. This
ensures that all higher dimensional sets are in place before we
begin computations on $W^k_{j+1}$. This is not a matter of much
importance, but it can simplify coding of the algorithm.
In the test at box~(d), we discard duplications of components,
including points that appear with multiplicity due to the presence
of nonreduced components. However, for the purpose of subsequently
breaking the witness set into irreducible components, it can be
useful to record the number of times each root appears. By the
abstract embedding theorem of~\cite{SVW9}, points on the same
irreducible component must appear the same number times, even though
we cannot conclude from this anything about the actual multiplicity
of the point as a solution of the system $\{f_1,f_2,\ldots,f_n\}$.
Having the points partially partitioned into subsets known to
represent distinct components will speed up the decomposition phase.
A final minor point of efficiency is that if $n>N$, we may arrive at
stage $k\ge N$ with some zero dimensional components, $W_N^k$. These
do not proceed to the diagonal homotopy: if such a point fails
test~(b), it is not a solution to system $\{f_1,f_2,\ldots,f_{k+1}\}=0$,
and it is discarded.
\subsection{Seeking only Nonsingular Solutions}
In the special case that $n\le N$, we may seek only the
multiplicity-one components of codimension $n$. (For $n=N$, this
means we seek only the nonsingular solutions of the system.) In this
case, we discard points that pass test~(a), since they give
higher dimensional components. Furthermore, we keep only the points
that test~(e) finds to be nonsingular and discard the singular ones.
This can greatly reduce the computation for some systems.
In this way, we may use the diagonal homotopy to compute nonsingular
roots equation-by-equation. This performs differently than more
traditional approaches based on continuation, which solve the entire
system all at once. In order to eliminate solution paths leading to
infinity, these traditional approaches use multihomogeneous
formulations or toric varieties to compactify ${\mathbb C}^N$. But this
does not capture other kinds of structure that give rise to positive
dimensional components. The equation-by-equation approach has the
potential to expose some of these components early on, while the
number of intrinsic variables is still small, and achieves
efficiency by discarding them at an early stage. However, it does
have the disadvantage of proceeding in multiple stages. For
example, in the case that all solutions are finite and nonsingular,
there is nothing to discard, and the equation-by-equation approach
will be less efficient than a one-shot approach. However, many
polynomial system of practical interest have special structures, so
the equation-by-equation approach may be commendable. It is too
early to tell yet, as our experience applying this new algorithm on
practical problems is very limited. Experiences with some simple
examples are reported in the next section.
\section{Computational Experiments}\label{Sec:Examples}
The diagonal homotopies are implemented in the software
package PHCpack~\cite{V99}. See~\cite{SVW7} for a description
of a recent upgrade of this package to deal with positive
dimensional solution components.
\subsection{An illustrative example}\label{Sec:Illusex}
The illustrative example (see Eq.~\ref{Eq:Illusex} for the system)
illustrates the gains made by our new solver. While our previous
sequence of homotopies needed 197 paths to find all candidate
witness points, the new approach shown in
Figure~\ref{Fig:Flowillusex} tracks just 13 paths. Many of the paths
take shortcuts around the diagonal homotopies, and five paths that
diverge to infinity in the first diagonal homotopy need no further
consideration. It happens that none of the witness points generated
by the diagonal homotopies is singular, so there is no need for
membership testing.
On a 2.4Ghz Linux workstation, our previous approach~\cite{SV}
requires a total of 43.3 cpu seconds (39.9 cpu seconds for solving
the top dimensional embedding and 3.4 cpu seconds to run the cascade
of homotopies to find all candidate witness points). Our new
approach takes slightly less than a second of cpu time. So for this
example our new solver is 40 times faster.
\begin{figure}
\begin{center}
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\caption{Flowchart for the illustrative example.}
\label{Fig:Flowillusex}
\end{center}
\end{figure}
\subsection{Adjacent Minors of a General 2-by-9 Matrix}
In an application from algebraic statistics~\cite{DES98}
(see also~\cite{HS00} for methods dedicated for these type of ideals)
one considers all adjacent minors of a general matrix.
For instance, consider this general 2-by-9 matrix:
\begin{displaymath}
\left[
\begin{array}{ccccccccc}
x_{11} & x_{12} & x_{13} & x_{14} &
x_{15} & x_{16} & x_{17} & x_{18} & x_{19} \\
x_{21} & x_{22} & x_{23} & x_{24} &
x_{25} & x_{26} & x_{27} & x_{28} & x_{29} \\
\end{array}
\right]
\end{displaymath}
Two minors are adjacent if they share one neighboring column.
Taking all adjacent minors from this general 2-by-9 matrix gives
8 quadrics in 18 unknowns. This defines a 10-dimensional surface,
of degree 256.
We include this example to illustrate that the flow of timings is
typical as in Table~\ref{tabminors}. Although we execute many
homotopies, most of the work occurs in the last stage, because both
the number of paths and the number of variables increases at each
stage. We are using the intrinsic method of \cite{SVW10} to reduce
the number of variables. With the older extrinsic method of
\cite{SVW9}, the total cpu time increases five-fold from 104s
to~502s.
\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c|rcr|c|r|} \hline
stage & \multicolumn{3}{c|}{\#paths}
& time/path & time \\ \hline
1 & 4 & = & 2 $\times$ 2 & 0.03s & 0.11s \\
2 & 8 & = & 4 $\times$ 2 & 0.05s & 0.41s \\
3 & 16 & = & 8 $\times$ 2 & 0.10s & 1.61s \\
4 & 32 & = & 16 $\times$ 2 & 0.12s & 3.75s \\
5 & 64 & = & 32 $\times$ 2 & 0.19s & 12.41s \\
6 & 128 & = & 64 $\times$ 2 & 0.27s & 34.89s \\
7 & 256 & = & 128 $\times$ 2 & 0.41s & 104.22s \\ \hline
\multicolumn{5}{|r}{total user cpu time\quad} & 157.56s \\ \hline
\end{tabular}
\caption{Timings on Apple PowerBook G4 1GHz for the $2\times9$
adjacent minors, a system of 8 quadrics in 18 unknowns.}
\label{tabminors}
\end{center}
\end{table}
\subsection{A General 6-by-6 Eigenvalue Problem}
Consider $f(\x,\lambda) = \lambda \x - A \x = \zero$, where $A \in
{\mathbb C}^{6 \times 6}$, $A$ is a random matrix. These 6 equations in 7
unknowns define a curve of degree~7, far less than what may be
expected from the application of B\'ezout's theorem: $2^6=64$.
Regarded as a polynomial system on ${\mathbb C}^7$, the solution set
consists of seven lines, six of which are eigenvalue-eigenvector
pairs while the seventh is the trivial line $\x = \zero$.
Clearly, as a matter of practical computation, one would employ an
off-the-shelf eigenvalue routine to solve this problem efficiently.
Even with continuation, we could cast the problem on
$\pn1\times\pn6$ and solve it with a seven-path two-homogeneous
formulation. However, for the sake of illustration, let us consider
how the equation-by-equation approach performs, keeping in mind that
the only information we use about the structure of the system is the
degree of each equation. That is, we treat it just like any other
system of 6 quadratics in 7 variables and let the
equation-by-equation procedure numerically discover its special
structure.
In a direct approach of solving the system in one total-degree
homotopy, adding one generic linear equation to slice out an
isolated point on each solution line, we would have 64 paths of
which 57 diverge. This does not even consider the work that would be
needed if we wanted to rigorously check for higher dimensional
solution sets.
Table~\ref{tabeigen} shows the evolution of the number of solution
paths tracked in each stage of the equation-by-equation approach.
The size of each initial witness set is $\#(X^i)=2$, so each new
stage tracks two paths for every convergent path in the previous
stage. If the quadratics were general, this would build up
exponentially to 64 paths to track in the final stage, but the
special structure of the eigenvalue equations causes there to be
only $i+2$ solutions at the end of stage~$i$. Accordingly, there are
only 12 paths to track in the final, most expensive stage, and only
40 paths tracked altogether. The seven convergent paths in the final
stage give one witness point on each of the seven solution lines.
\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c|rrrrr|c|} \hline
stage in solver & ~1 & ~2 & ~3 & ~4 & ~5 & total \\ \hline
\#paths tracked & ~4 & ~6 & ~8 & 10 & 12 & 40 \\ \hline
\#divergent paths & ~1 & ~2 & ~3 & ~4 & ~5 & 15 \\
\#convergent paths & ~3 & ~4 & ~5 & ~6 & ~\raise1pt\hbox to0pt{\hskip-3pt$\bigcirc$\hss}{7} & 25 \\ \hline
\end{tabular}
\caption{Number of convergent and divergent paths on a general
6-by-6 eigenvalue problem.}
\label{tabeigen}
\end{center}
\end{table}
\section{Conclusions}
The recent invention of the diagonal homotopy allows one to compute
intersections between algebraic sets represented numerically by
witness sets. This opens up many new possibilities for ways to
manipulate algebraic sets numerically. In particular, one may solve
a system of polynomial equations by first solving subsets of the
equations and then intersecting the results. We have presented a
subsystem-by-subsystem algorithm based on this idea, which when
carried to extreme gives an equation-by-equation algorithm. The
approach can generate witness sets for all the solution components
of a system, or it can be specialized to only seek the nonsingular
solutions at the lowest dimension. Applying this latter form to a
system of $N$ equations in $N$ variables, we come full circle in the
sense that we are using methods developed to deal with
higher dimensional solution sets as a means of finding just the
isolated solutions.
Experiments with a few simple systems indicates that the method can
be very effective. Using only the total degrees of the equations,
the method numerically discovers some of their inherent structure in
the early stages of the computation. These early stages are
relatively cheap and they can sometimes eliminate much of the
computation that would otherwise be incurred in the final stages.
In future work, we plan to exercise the approach on more challenging
problems, especially ones where the equations have
interrelationships that are not easily revealed just by examining
the monomials that appear. Multihomogenous homotopies and
polyhedral homotopies are only able to take advantage of that sort
of structure, while the equation-by-equation approach can reveal
structure encoded in the coefficients of the polynomials. One avenue
of further research could be to seek a formulation that uses
multihomogeneous homotopies or polyhedral homotopies in an
equation-by-equation style to get the best of both worlds.
|
{
"timestamp": "2005-03-29T18:46:35",
"yymm": "0503",
"arxiv_id": "math/0503688",
"language": "en",
"url": "https://arxiv.org/abs/math/0503688"
}
|
\section{Introduction}
Quantum computation and quantum information are subjects
of much continuing interest and study. An initial impetus
for this work was the realization that as computers became
smaller, quantum effects would become more important.
Additional interest arose with the discovery of
problems \cite{Shor,Grover} that could be solved more
efficiently on a quantum computer than on a classical
machine. Also quantum information, and possibly quantum
computation, \cite{Lloyd,Zizzi} is of recent
interest in addressing problems related to cosmology and
quantum gravity.
In all of this work qubits (or qudits for d-dimesional
systems) play a basic role. As quantum binary systems
the states $|0\rangle,|1\rangle$ of a qubit represent the binary
choices in quantum information theory. They also
represent the numbers $0$ and $1$ as numerical inputs
to quantum computers. For $n$ qubits, corresponding
product states, such as $|\underline{s}\rangle=
\otimes_{j=1}^{n}|s(j)\rangle$ where $s(j)=0 \mbox { or
}1,$ represent a specific $n$ qubit information state.
Since they also represent numbers, \begin{equation}\label{qubitnum}
|\underline{s}\rangle\rightarrow \sum_{j=1}^{n}\underline{s}(j)2^{j-1},
\end{equation} they and their linear superpositions are inputs to
quantum computers.
It is clear that states of qubits are very important to quantum
information theory. However, qubits and their states are not
essential to the representation of numbers in quantum mechanics.
This is based on the observation that in a state, such as
$|10010\rangle$, the $0s$ do not contribute to the
numerical value of the state. Instead they function more
like place holders. What is important is the
distribution of the $1s$ along a discrete lattice. This
is shown by Eq. \ref{qubitnum} where the value of the
number is determined by the distribution of $1s$ at the
values of $j$ for which $\underline{s}(j)=1$. The value, $0,$ of
$\underline{s}$ at other locations contributes nothing.
This suggests a different type of representation of
numbers that does not use qubits. It is based instead on
the distributions of $1s$ on an integer lattice. For
example the rational number $1001.01$ would be
represented here as $1_{3}1_{0}1_{-2}.$ In quantum
mechanics these states correspond to position eigenstates
of a system on a discrete lattice or path where the positions
are labelled by integers. Here the state $|j\rangle$ corresponds
to the number $2^{j}$ and $n$ product states, such as $|j_{1},\cdots
j_{n}\rangle,$ correspond to $\sum_{k=1}^{n}2^{j_{k}}.$
Here the representation of rational numbers
corresponds to those represented by finite strings
of binary digits or qubits and not as pairs of such
strings. This representation is much easier to
use and corresponds to that used in computers. It also
is dense in the set of all rational
numbers. For quantum states this means that any
representation of all nonnegative rational numbers as quantum states
would be approximated arbitrarily closely by a finite qubit
$|\underline{s}\rangle$ states or states of the form
$\otimes_{j\epsilon 1_{s}}|j\rangle$ where $1_{s} =
\{j|\underline{s}(j)=1\}.$ In what follows this type
of state will be referred to as a rational number state.
This representation is sufficient to describe
nonnegative rational numbers in quantum
mechanics. There are several ways to extend the
treatment to include negative and imaginary
rational numbers. These range from one type
of system with two internal binary degrees of freedom
to four different types of systems. Here an intermediate
approach is taken in which two types of systems which
have an internal binary degree of freedom are
considered. The two internal degrees of freedom
correspond to positive and negative and the two types of
systems correspond to real and imaginary. An example of
such a number in the representation considered here is
$r_{+,5}\: r_{-,3}\: i_{-,-2}\: i_{+,4}$
The goal of this paper is to use these ideas to give a
quantum mechanical representation of complex rational
numbers. Since states with varying numbers of $r+,r-,i+,i-$
systems will be encountered, a Fock space representation
is used. Both bosons and fermions will be considered.
The emphasis of this work is to describe a set of
quantum states that can be shown to represent complex
rational numbers. This requires definitions of the
basic arithmetical operations used in the axiomatic
definitions of rational numbers and showing that the
states have the desired properties.
An additional emphasis is that the state descriptions
and properties must be relatively independent of the
complex rational numbers that are part of
the complex number field $C$ on which the
Fock space is based. This means that the description
will not be based on a map from quantum states to $C$
that is used to define arithmetic properties of the
states. Instead the states and their properties will be
described independent of any such map.
It will be seen that the representation used here is more compact with
simpler representations of the basic arithmetic operations
than those based on qubit states with "binal" points, e.g.,
of the form $|\pm 10010.011\rangle$ \cite{BenRNQM,BenRNQMALG}.
It also extends the representation to complex numbers
which was not done in the earlier work.
Another (slight) advantage is that linear superpositions
of states containing just one system are not entangled
in the representation used here. This is not the case
for the qubit representation with $0s$ present. An
example is the Bell state $(1/\sqrt{2})(|10\rangle\pm
|01\rangle).$ Here this state is $(1/\sqrt{2})(|1\rangle\pm|0\rangle),$
which is not entangled. In this state $0$ and $1$ are the locations
of the $1s$ in the qubit state. This advantage is lost
when one considers states with more than one system present.
Another advantage of the representation
described here is that it may suggest new physical
models for quantum computation that are not qubit
based. Whether this is the case or not must await
future work.
The use of Fock spaces to describe quantum computation
and quantum information is not new. It has been used to
describe fermionic \cite{Kitaev,Ozhigov} and parafermionic
\cite{Lidar} quantum computation,
and quantum logic \cite{Gudder,DChiara}. The novelty of the
approach taken here is based on a description of complex
rational string numbers that is not based on logical or
physical qubits. In this sense if differs from
\cite{Kassman}. It also emphasizes basic arithmetic
operations instead of quantum logic gates. Also both
standard and nonstandard representations of numbers are
described. These follow naturally from the occupation
number description of quantum states.
Details of the description of the complex rational states
are given in the next three sections. The a-c operators are described
in Section \ref{CRSS}. The next section gives
properties of these and other operators and their use to
describe complex rational
states. Section \ref{BAORSS} describes the
arithmetic operations of addition, multiplication, and
division to any finite accuracy. The last section
summarizes some advantages of the approach used here.
Also a possible physical model of standard and
nonstandard numbers as pools of four types of Bose
Einstein condensates along an integer lattice is briefly
discussed.
\section{Complex Rational States}\label{CRSS}
The representation of complex rational states used here
is based on the notion of creating and annihilating
two types of systems, at various
locations. One type is used for real rational states
and the other for imaginary states. For bosons the degrees of
freedom associated with each type consist of a binary
internal degree, denoted by $+,-$, and a location $j$ on an
integer labelled lattice.
The creation operators for bosons are
$a^{\dag}_{+,j},a^{\dag}_{-,j}, b^{\dag}_{+,j},b^{\dag}_{-,j}$. The $a$ operators
create and annihilate bosons in states corresponding respectively
to positive and negative real rational states. The $b$
operators play the same role for imaginary states.
The state $|0\rangle$ is the vacuum state.
In this representation, the states $a^{\dag}_{+,j}|0\rangle,
a^{\dag}_{-,j}|0\rangle$ show an a (real) boson in states
$+,-$ at site $j$. The states $b^{\dag}_{+,j}|0\rangle,b^{\dag}_{-,j}
|0\rangle$ show a b (imaginary) boson in states $+,-$
at site $j$. In the order presented these states correspond to
the numbers $2^{j},-2^{j},i2^{j},$ and $-i2^{j}.$
For fermions the creation and annihilation operators
have an additional variable $h =0,1,2,\cdots.$ This
extra variable is needed to make fermions with the
same sign and $j$ value distinguishable. Thus boson
states of the form $a^{\dag}_{+,j}a^{\dag}_{+,j}|0\rangle$
become $a^{\dag}_{+,h,j}a^{\dag}_{+,h^{\prime},j}|0\rangle$ where $h\neq
h^{\prime}.$ Note that, as far as number properties are
concerned, $h$ is a dummy variable in that
$a^{\dag}_{+,h,j}|0\rangle$ and $a^{\dag}_{+,h^{\prime},j}|0\rangle$
both represent the same number. However in any physical
model it would represent some physical property.
One can also form linear superpositions of these
states. Simple boson examples and their equivalences in the
usual qubit based binary notation are,
\begin{equation}\label{a-ccorr}
\begin{array}{l}(1/\sqrt{2})(a^{\dag}_{+,j}\pm a^{\dag}_{-,j})|0\rangle
=(1/\sqrt{2})(|1\underline{0}^{j}\rangle\pm|-1\underline{0}^{j}\rangle)
\\ (1/\sqrt{2})(a^{\dag}_{+,j}\pm a^{\dag}_{-,k})|0\rangle =(1/\sqrt{2})(|1\underline{0}^{j}
\rangle\pm|-1\underline{0}^{k}\rangle) \\ (1/\sqrt{2})(a^{\dag}_{+,k} \pmb^{\dag}_{+,j}) =
(1/\sqrt{2})(|1\underline{0}^{k}
\rangle\pm|i1\underline{0}^{k}\rangle) \\ (1/\sqrt{2})(1\pm b^{\dag}_{-,j})|0\rangle
=(1/\sqrt{2})(|0\rangle\pm|-i1\underline{0}^{j}\rangle)
\\ (1/\sqrt{2})(1\pm a^{\dag}_{+,j})|0\rangle =(1/\sqrt{2})(|0\rangle\pm
|1\underline{0}^{j}\rangle).\end{array}\end{equation}
Here $\underline{0}^{j}$ represents a string of $j$ $0s.$
These states show one advantage of the representation used here
in that those on the left are valid for any value of $j$ or
$k$. The usual binary representations on the right are
valid only for $j,k\geq 0.$ Note also that $-$ and $i$
inside the qubit states denote the type and sign of the
number. They are not phase factors multiplying the
states.
\section{ Occupation Number States}\label{RNSSPACO}
The first step in representing states as products of
creation operators acting on $|0\rangle$ is to give the commutation
relations. Let $c^{\dag}_{j},\hat{c}^{\dag}_{j},c_{j},$
and $\hat{c}_{j}$ be variable a-c operators where
$c^{\dag}$ and $\hat{c}^{\dag}$ can take any one of
the four values $a^{\dag}_{+},a^{\dag}_{-},b^{\dag}_{+},b^{\dag}_{-}.$ Using
these the boson commutation relations can be given as \begin{equation}
\label{ccomm} [c_{j} ,c^{\dag}_{k} ]=\delta_{j,k}
\hspace{1cm} [c^{\dag}_{j} ,\hat{c}^{\dag}_{k}] = [c_{j}
,\hat{c}_{k}] =0.\end{equation} The first
equation stands for four equations as $c^{\dag}$ has any one
of four values. Each of the next two equations stands for
$16$ equations as $c^{\dag}$ and $\hat{c}^{\dag}$ each have
any one of four values.
For fermions the anticommutation relations are given
by\begin{equation}\label{canticomm} \{c_{g,j} ,c^{\dag}_{h,k}
\}=\delta_{j,k}\delta_{g,h} \hspace{0.5cm}
\{ c^{\dag}_{g,j} ,\hat{c}^{\dag}_{h,k} \} = \{c_{g,j}
,\hat{c}_{h,k}\} =0 \end{equation} where
$\{c,d\}=cd+dc.$ There are two sets of these relations,
one for $c^{\dag}$ and $\hat{c}^{\dag}$ each having the values
$a^{\dag}_{+},a^{\dag}_{-}$ and the other for $c^{\dag}$ and
$\hat{c}^{\dag}$ with the values $b^{\dag}_{+}$ or $b^{\dag}_{-}.$ Note that
because the $a$ and $b$ systems are two different types of
fermions, \emph{commutation} relations hold between their
operators, as in $[a^{\dag}_{+,g,j},b^{\dag}_{+,h,k}] =0,$ etc..
A complete basis set of states can be defined in terms of
occupation numbers of the various boson or fermion states. A general
basis can be defined as follows: Let $n_{r},m_{r},n_{i},m_{i}$
be any four functions that map the set of all integers to the
nonnegative integers. Each function has the value $0$
except possibly on finite sets of integers. Let
$s,s^{\prime},t,t^{\prime}$ be the four finite sets of integers
which are the nonzero domains, respectively, of the four
functions. Thus $n_{r,j}\neq 0[=0]$ if $j\epsilon s[j
\mbox{ not in }s],$ $m_{r,j}\neq 0[=0]$ if $j\epsilon s^{\prime}
[j \mbox{ not in }s^{\prime}],$ etc..
Let $\bigcup s,t$ be the set of all integers in one or more of the four sets.
Then a general boson occupation number state has the form
\begin{equation}\label{occno}
|n_{r},m_{r},n_{i},m_{i}\rangle = \prod_{j\epsilon \cup
s,t}|n_{r,j},m_{r,j}n_{i,j}m_{i,j}\rangle\end{equation}
where $|n_{r,j},m_{r,j}n_{i,j}m_{i,j}\rangle$ the
occupation number state for site $j$ is given by
\begin{equation}\label{occnost}\begin{array}{l}
|n_{r,j},m_{r,j}n_{i,j}m_{i,j}\rangle=\frac{1}{N(n,m,r,i,j)} \\
\hspace{1cm}\times (a^{\dag}_{+,j})^{n_{r,j}}
(a^{\dag}_{-,j})^{m_{r,j}}(b^{\dag}_{+,j})^{n_{i,j}}(b^{\dag}_{-,j})^{m_{i,j}}|0\rangle.
\end{array}\end{equation} The normalization factor
$N(n,m,r,i,j)=(n_{r,j}!m_{r,j}!n_{i,j}!m_{i,j}!)^{1/2}.$
Note that the product $\prod_{j\epsilon \cup
s,t}$ denotes a product of creation operators, and not
a product of states.
The interpretation of these states is that they are the
boson equivalent of \emph{nonstandard} representations of
complex rational numbers as distinct from \emph{standard}
representations. (This use of standard and nonstandard is
completely different from standard and nonstandard numbers
described in mathematical logic \cite{Chang}.) Such nonstandard
states occur often
in arithmetic operations and will be encountered later on.
They correspond to columns of binary numbers where
each number in the
column is any one of the four types, positive real,
negative real, positive imaginary, and negative
imaginary. In a boson representation individual systems,
are not distinguishable. The only measurable properties
are the number of systems of each type $+1,-1,+i,-i$ in
the single digit column at each site $j.$
An example would be a computation in which one
computes the value of the integral $\int_{a}^{b}f(x)dx$
of a complex valued function $f$ by computing in parallel,
or by a quantum computation, values of $f(x_{h})$ for
$h=1,2,\cdots,m$ and then combining the $m$ results to
get the final answer. The table, or matrix, of $m$ results
before combination is represented here by a state
$|n_{r},m_{r},n_{i},m_{i}\rangle$ where
$n_{r,j},m_{r,j}n_{i,j},m_{i,j}$ give the number of
$+1's$, $-1's,$ $+i's$, and $-i's$ in the column at site $j.$ This is
a nonstandard representation because it is numerically
equal to the final result which is a standard
representation consisting of one real and one imaginary rational
string number, often represented as a pair, $u,iv$.
The equivalent fermionic representation for the
state $|n_{r},m_{r},n_{i},m_{i}\rangle$ is based
on a fixed ordering of the a-c operators.
In this case the product $(a^{\dag}_{+,j})^{n_{r,j}}$ becomes
$a^{\dag}_{+1,j}\cdotsa^{\dag}_{+h,j}\cdotsa^{\dag}_{+n_{r,j},j}$
with similar replacements for $(a^{\dag}_{-,j})^{m_{r,j}},
(b^{\dag}_{+,j})^{n_{i,j}},(b^{\dag}_{-,j})^{m_{i,j}}.$
Each component state
$|n_{r,j},m_{r,j},n_{i,j},m_{i,j}\rangle$
in Eq. \ref{occnost} is given by
\begin{equation}\label{occferm}\begin{array}{c}
|n_{r,j},m_{r,j},n_{i,j},m_{i,j}\rangle=
a^{\dag}_{+n_{r,j},j}\cdots a^{\dag}_{+1,j}a^{\dag}_{-m_{r,j},j}\cdots \\
a^{\dag}_{-1,j} b^{\dag}_{+n_{i,j},j}\cdotsb^{\dag}_{+1,j}
b^{\dag}_{-m_{i,j},j}\cdotsb^{\dag}_{-1,j}|0\rangle\end{array}\end{equation}
The final state is given by an ordered product over the $j$ value,
\begin{equation}\label{occnoferm}|n_{r},m_{r},n_{i},m_{i}\rangle =
\prod_{j\epsilon \cup s,t}J|n_{r,j},m_{r,j}n_{i,j}m_{i,j}
\rangle.\end{equation} Here $J$ denotes a $j$ ordered
product where factors with larger values of $j$ are to
the right of factors with smaller $j$ values. The choice
of ordering, such as that used here in which
the ordering of the $j$ values is the opposite of that
for the $h$ values which increase to the left as in Eq.
\ref{occferm}, is arbitrary. However, it must remain fixed
throughout.
An example of a nonstandard representation is illustrated in
Figure \ref{fig1} for both bosons and fermions. The
integer values of $j$ are shown on the abcissa. The
ordinate shows the boson occupation numbers for each type of
system. Fermions are represented as two types
of systems each with two internal states $(+,-)$ on a two
dimensional lattice with $j$ any integer and $h$ any nonnegative
integer. The ordinate shows the range of $h$
values from $0$ to $n_{r,j},m_{r,j},n_{i,j},m_{i,j}$ for
each of the four types.
\begin{figure}[t]\begin{center}\vspace{1cm}
\resizebox{100pt}{100pt}{\includegraphics[230pt,120pt]
[530pt,420pt]{RCRNQMfig1.eps}}\end{center}
\caption{Example of a nonstandard complex rational
state for bosons and fermions for $4$ occupied
$j$ values. At each site $j$
the vertical bar shows the occupation numbers (bosons)
or extent of $h$ values (fermions) with different colors
and line slopes showing each of the four types of systems.
For instance the bar at site $j-1$ shows $13\; r+,$
$11\; r -,$ $4\; i+,$ and $2\; i -$ systems and the bar at
site $j$ shows $5\; r+$ and $8\; i+$ bosons. For fermions
the ordinate labels the ranges of the $h$ values for each
type.}\label{fig1} \end{figure}
The above shows the importance of nonstandard
representations, especially in cases where a large amount
of data or numbers is generated which must be combined
into a single numerically equivalent complex rational
number. This requires definition of standard
complex rational number states and of properties
to be satisfied by any conversion process.
For bosons a standard complex rational
state has the form of Eq. \ref{occnost} where
one of the functions $n_{r},m_{r}$ and one of $n_{i},m_{i}$
has the constant value $1$ on their nonzero domains. The other two
functions are $0$. The four possibilities are \begin
{equation}\label{stdbos}\begin{array}{c}
|\underline{1}_{s},0,\underline{1}_{t},0\rangle =(a^{\dag}_{+})^{s}
(b^{\dag}_{+})^{t}|0\rangle \\|\underline{1}_{s},0,0,
\underline{1}_{t^{\prime}}\rangle =(a^{\dag}_{+})^{s}
(b^{\dag}_{-})^{t^{\prime}}|0\rangle\\|0,\underline{1}_{s^{\prime}},
\underline{1}_{t},0\rangle = (a^{\dag}_{-})^{s^{\prime}}(b^{\dag}_{+})^{t}|0\rangle
\\|0,\underline{1}_{s^{\prime}},0,\underline{1}_{t^{\prime}}\rangle =
(a^{\dag}_{-})^{s^{\prime}}(b^{\dag}_{-})^{t^{\prime}}|0\rangle.
\end{array}\end{equation} Here $(a^{\dag}_{+})^{s}=\prod_{j\epsilon
s}a^{\dag}_{+,j}$ and $\underline{1}_{s}$ denotes the constant
$1$ function on $s$, etc. Pure real or imaginary standard
rational states are included if $t,\;t^{\prime}$ or
$s,\;s^{\prime}$ are empty. If $s,\;s^{\prime},\; t,\; t^{\prime}$ are
all empty one has the vacuum state $|0\rangle.$ Note that
Eq. \ref{stdbos} also is valid for fermions with the
replacements\begin{equation}\label{stdfer}\begin{array}{l}
(a^{\dag}_{\a} )^{s}\rightarrow a^{\dag}_{\a,1,j_{1}}a^{\dag}_{\a,1,j_{2}}
\cdotsa^{\dag}_{\a,1,j_{|s|}} \\ (b^{\dag}_{\b})^{t}\rightarrow
b^{\dag}_{\b,1,k_{1}}b^{\dag}_{\b,1,k_{2}}\cdotsb^{\dag}_{\b,1,k_{|t|}}.\end{array}
\end{equation} Here $\a=+,-$, $\b=+,-$, and
$s=\{j_{1},j_{2},\cdots,j_{|s|}\},\;
t=\{k_{1},k_{2},\cdots,k_{|t|}\}$. Also
$j_{1}<j_{2}<\cdots<j_{|s|},\;k_{1}<k_{2}<\cdots<k_{|t|},$
and $|s|,|t|$ denote the number of integers in $s,t.$
Standard states are quite important. All theoretical
predictions as computational outputs, and numerical
experimental results are represented by standard real
rational states. Nonstandard representations occur
during the computation process and in any situation where
a large amount of numbers is to be combined. Also qubit states
correspond to standard representations only.
This shows that it is important to describe the numerical
relations between nonstandard representations and standard
representations and to define numerical equality between states.
To this end let \begin{equation}\label{Nequ}
|n_{r},m_{r},n_{i},m_{i}\rangle
=_{N}|n^{\prime}_{r},m^{\prime}_{r},n^{\prime}_{i},m^{\prime}_{i}\rangle\end{equation}
be the statement of $N$ equality between the two
indicated states. This statement is satisfied if two
basic equivalences are satisfied.
For bosons the two $N$ equivalences are
\begin{equation}\label{abdjabj} a^{\dag}_{+,j}a^{\dag}_{-,j}=_{N}\tilde{1};\;\;\;\;
b^{\dag}_{+,j}b^{\dag}_{-,j}=_{N}\tilde{1}\end{equation}
and \begin{equation}\label{abjabj}\begin{array}{l} a^{\dag}_{\a,j}a^{\dag}_{\a,j}=_{N}
a^{\dag}_{\a,j+1}\;\;\;\; a_{\a,j}a_{\a,j}=_{N}a_{\a,j+1} \\ b^{\dag}_{\b,j}b^{\dag}_{\b,j}=_{N}
b^{\dag}_{\b,j+1}\;\;\;\; b_{\b,j}b_{\b,j}=_{N}b_{\b,j+1}.\end{array}\end{equation}
The first pair of equations says that any state that
has one or more $+$ and $-$ systems of either the $r$ or $i$ type at a site $j$ is
numerically equivalent to the state with one less $+$ and
$-$ system at the site $j$ of either type. This is the
expression here of $2^{j}-2^{j}=i2^{j}-i2^{j}=0.$
The second set of two pairs, Eq. \ref{abjabj}, says that any
state with two systems of the same type and in the same
internal state at site $j$, is numerically equivalent to a
state without these systems but
with one system of the same type and internal state at
site $j+1.$ This corresponds to $2^{j}+2^{j}=2^{j+1}$ or
$i2^{j}+i2^{j}=i2^{j+1}.$
From these relations one sees that any process whose
iteration preserves $N$ equality according to Eqs.
\ref{abdjabj} and \ref{abjabj} can be used to
determine if Eq. \ref{Nequ} is valid for two different
states. For example if \begin{equation}\label{jj1}
|n_{r},m_{r},n_{i},m_{i}\rangle
=a_{+,j}a_{-,j}|n^{\prime}_{r},m^{\prime}_{r},
n^{\prime}_{i},m^{\prime}_{i}\rangle\end{equation} or \begin{equation}\label{jjj+1}
|n_{r},m_{r},n_{i},m_{i}\rangle
=b^{\dag}_{+,j+1}b_{+,j}b_{+,j}|n^{\prime}_{r},m^{\prime}_{r},
n^{\prime}_{i},m^{\prime}_{i}\rangle,\end{equation} then Eq.
\ref{Nequ} is satisfied.
For fermions the corresponding $N$ equivalences are
\begin{equation}\label{fabdjabj}
a^{\dag}_{+,j,h}a^{\dag}_{-,j,h^{\prime}}=_{N}\tilde{1};\;\;\;\;
b^{\dag}_{+,j,h}b^{\dag}_{-,j,h^{\prime}}=_{N}\tilde{1}\end{equation}
and \begin{equation}\label{fabjabj}\begin{array}{l}
a^{\dag}_{\a,h,j}a^{\dag}_{\a,h^{\prime},j}=_{N}
a^{\dag}_{\a,h^{\prime\p},j+1}\;\;\;\; a_{\a,h,j}a_{\a,h^{\prime},j}
=_{N}a_{\a,h,^{\prime\p},j+1} \\ b^{\dag}_{\b,h,j}b^{\dag}_{\b,h^{\prime},j}=_{N}
b^{\dag}_{\b,h^{\prime\p},j+1}\;\;\;\; b_{\b,h,j}b_{\b,h^{\prime},j}
=_{N}b_{\b,h^{\prime\p},j+1}.\end{array}\end{equation} In Eq. \ref{fabjabj}
$h\neq h^{\prime}.$ Otherwise the values of $h,h^{\prime},h^{\prime\p}\geq
1$ are arbitrary except that removal of fermions is
restricted to occupied $h$ values and addition is
restricted to unoccupied values. To avoid poking holes in
the $h$ columns at each site $j,$ Fig. \ref{fig1},
it is useful to restrict system removal to the maximum
occupied h value and system addition to the nearest
unoccupied $h$ site. Numerically it does not matter where, in
the $h$ direction, the fermions are added or removed.
These equations have a meaning similar that that for the
corresponding boson equations. Eq. \ref{fabdjabj} says
that any state given by Eqs. \ref{occferm} and
\ref{occnoferm} is $N$ equal to a state with one
$a^{\dag}_{+}$ and one $a^{\dag}_{-}$ fermion removed from site $j$
i.e. $n_{r,j}\rightarrow n_{r,j}-1$ and
$m_{r,j}\rightarrow m_{r,j}-1.$ A similar situation holds
for removal of one $b^{\dag}_{+}$ and one $b^{\dag}_{-}$ fermion
from site $j.$ Eq. \ref{fabjabj} says that a state with two
$a^{\dag}_{+}$ or two $a^{\dag}_{-}$ fermions removed from site $j$
is $N$ equal to a state with one $a^{\dag}_{+}$ or $a^{\dag}_{-}$
fermion added to site $j+1.$ A similar situation holds
for the $b^{\dag}_{+}$ or $b^{\dag}_{-}$ fermions.
Corresponding to Eq. \ref{jj1} one has
the following: Let $|n_{r},m_{r},n_{i},m_{i}\rangle$ and
$|n^{\prime}_{r},m^{\prime}_{r},n^{\prime}_{i},m^{\prime}_{i}\rangle$ be such that
\begin{equation}\label{fjj1}
|n_{r},m_{r},n_{i},m_{i}\rangle =R_{a,+,j}R_{a,-,j}|n^{\prime}_{r}m^{\prime}_{r}
n^{\prime}_{i}m^{\prime}_{i}\rangle\end{equation} is satisfied
where \begin{equation}\label{fRRjj1}\begin{array}{l}R_{a,+,j}=
\sum_{h}a_{+,h+1,j}a^{\dag}_{+,h+1,j}a_{+,h,j}\\
R_{a,-,j}=\sum_{h}a_{-,h+1,j}a^{\dag}_{-,h+1,j}a_{-,h,j},\end{array}
\end{equation}then Eq. \ref{Nequ} is satisfied. A
similar statement holds if $b$ replaces $a$ in the above.
The presence of the factors $a_{+,h+1,j}a^{\dag}_{+,h+1,j}$
and $a_{-,h+1,j}a^{\dag}_{-,h+1,j}$ in Eq. \ref{fRRjj1} is to
ensure that the $a^{\dag}_{+}$ and $a^{\dag}_{-}$ operators with
the maximum $h$ values are deleted.
Corresponding to Eq. \ref{jjj+1} one has that if $|n_{r},m_{r},n_{i},m_{i}\rangle$ and
$|n^{\prime}_{r},m^{\prime}_{r},n^{\prime}_{i},m^{\prime}_{i}\rangle$
satisfy \begin{equation}\label{fjjj+1}
|n_{r},m_{r},n_{i},m_{i}\rangle
=_{\pm}S_{a,+,j}|n^{\prime}_{r},m^{\prime}_{r},
n^{\prime}_{i},m^{\prime}_{i}\rangle\end{equation} where
\begin{equation}\label{fSjjj+1}\begin{array}{l}
S_{a,+,j}=\sum_{h^{\prime}}a_{+,h^{\prime}+1,j}
a^{\dag}_{+,h^{\prime}+1,j}a^{\dag}_{+,h^{\prime},j} \\ \hspace{0.3cm}\times
\sum_{h}a_{+,h+1,j}a^{\dag}_{+,h+1,j}a_{+,h,j}a_{+,h,j}
,\end{array}\end{equation}
then Eq. \ref{Nequ} is satisfied. There are three other
equations one each for $S_{a,-,j},S_{b,+,j}$ and
$S_{b,-,j}.$
The expression $=_{\pm}$ denotes equality up to a
possible sign change. This can occur because the $S$
operators are products of an odd number of a-c operators.
If one wants to implement these state reduction steps
dynamically with operators that preserve fermion (or
boson) number, then a pool of additional fermions (or bosons) must be
available to serve as a source or sink of systems. This
is not included here because the emphasis is on defining
complex rational states and their arithmetic
properties.
It is worth noting that Eqs. \ref{jj1}, \ref{jjj+1},
\ref{fjj1}, and \ref{fjjj+1} can be regarded as axiomatic
definitions of $=_{N}$ with no reference to their
numerical meaning in terms of powers of $2$. The use of
numbers in their description is included as an aid to the
reader. It plays no role in their definition.Later
on a map from the complex states to $C$ will be
defined that shows that these properties of $=_{N}$ are
consistent with the map.
Reduction of a nonstandard representation to a standard
one proceeds by iteration of steps based on the above
equivalences. At some point the process stops when one
ends up with a state with at most one system of the $a$ or
$b$ type at each site $j$. This is the case for both
bosons and fermions. The possible
options for each $j$ can be expressed as
\begin{equation}\label{stdconv}\begin{array}{l}
|n_{r,j},m_{r,j},n_{i,j},m_{i,j}\rangle
=\left\{ \begin{array}{l}|1,0,0,1\rangle \\
|1,0,1,0\rangle\\|0,1,0,1\rangle\\|0,1,1,0\rangle\end{array}
\mbox { or }\left\{\begin{array}{l}|0,0,0,1\rangle\\
|0,0,1,0\rangle\\|1,0,0,0\rangle\\|0,1,0,0\rangle\end{array}\right.\right.
\\ \mbox{} \\ \hspace{3cm} \mbox{ or }|0,0,0,0\rangle.
\end{array}\end{equation} An example of such a
state for several $j$ is $|1_{+,3}i_{+,3}1_{-,2}i_{-,4}
1_{-,-6}\rangle.$ This state corresponds to the number
$2^{3}-2^{2}-2^{-6}+i(2^{3}-2^{4}).$
Conversion of a state in this form into a standard state requires
first determining the signs of the $a$ and $b$ systems occupying the
sites with the largest $j$ values. This determines the
signs separately for the real and imaginary components of
the standard representation.
In the example given above the real component is $+$ as
$3>2,-6$ and the imaginary component is $-$ as $4>3.$
Conversion of all a-c operators into the same kind, as
shown in Eq. \ref{stdbos}, is based on four relations
obtained by iteration of Eq. \ref{abjabj} and use of
Eq. \ref{abdjabj}. For $k<j$ and for bosons
they are \begin{equation}\label{abjk}
\begin{array}{l}a^{\dag}_{+,j}a^{\dag}_{-,k} =_{N}a^{\dag}_{+,j-1}\cdots
a^{\dag}_{+,k} \\ a^{\dag}_{-,j}a^{\dag}_{+,k}
=_{N}a^{\dag}_{-,j-1}\cdots
a^{\dag}_{-,k} \\ b^{\dag}_{+,j}b^{\dag}_{-,k}
=_{N}b^{\dag}_{+,j-1}\cdots
b^{\dag}_{+,k} \\ b^{\dag}_{-,j}b^{\dag}_{+,k}
=_{N}b^{\dag}_{-,j-1}\cdots
b^{\dag}_{-,k}.\end{array}\end{equation} These equations
are used to convert all $a$ and all $b$ operators to the
same type ($+$ or $-$) as the one at the largest occupied
$j$ value. Applied to the example $|1_{+,3}i_{+,3}1_{-,2}
i_{-,4}1_{-,-6}\rangle,$ gives
$|1_{+,2}1_{+,1}1_{+,0}1_{+,-1}1_{+,-3}\cdots
1_{+,-6}i_{-,3}\rangle.$ for the standard representation.
The same four equations hold for fermions provided $h$
subscripts are included. The values of $h$ are arbitrary
as they do not affect $=_{N}.$ However, physically,
application to a state of the form of Eq. \ref{stdconv}
requires that $h=1$ everywhere, as in $a^{\dag}_{+,1,j}a^{\dag}_{-,1,k}
=_{N}a^{\dag}_{+,1,j-1}\cdots a^{\dag}_{+,1,k}$ for
example.
\subsection{Some Useful Operators}\label{SUO}
Three unitary operators that allow changing between the types of
systems and moving the string states are useful. For bosons they are
defined by\begin{equation}\label{WQT} \begin{array}{c}\tilde{W}
a^{\dag}_{+,j}=a^{\dag}_{-,j}\tilde{W},\;\;\tilde{W}b^{\dag}_{+,j}=b^{\dag}_{-,j}\tilde{W}
\\ \tilde{Q}a^{\dag}_{+,j}=b^{\dag}_{+,j}\tilde{Q},\;\;\tilde{Q}a^{\dag}_{-,j}=
b^{\dag}_{-,j}\tilde{Q} \\ \tilde{T}c^{\dag}_{j}=c^{\dag}_{j+1}\tilde{T}
\\ \tilde{W}|0\rangle = |0\rangle,\;\;\tilde{Q}|0\rangle =
|0\rangle,\;\; \tilde{T}|0\rangle =|0\rangle. \end{array}\end{equation}
$\tilde{W}$ interchanges $+$ and $-$ states in $r$ and $i$ systems,
and $\tilde{Q}$ converts $r$ systems to $i$ systems and
conversely. $\tilde{T}$ is a translation operator that
shifts $a^{\dag}_{+},a^{\dag}_{-},b^{\dag}_{+},b^{\dag}_{-}$ operator
products one step along the line of $j$ values.
For fermions the equations become \begin{equation}\label{fWQT}
\begin{array}{c}\tilde{W} a^{\dag}_{+,h,j}=a^{\dag}_{-,h,j}\tilde{W},
\;\;\tilde{W}b^{\dag}_{+,h,j}=b^{\dag}_{-,h,j}\tilde{W}
\\ \tilde{Q}a^{\dag}_{+,h,j}=
b^{\dag}_{+,h,j}\tilde{Q},\;\;\tilde{Q}a^{\dag}_{-,h,j}=b^{\dag}_{-,h,j}\tilde{Q}
\\ \tilde{T}c^{\dag}_{h,j}=c^{\dag}_{h,j+1}\tilde{T}\\ \tilde{W}|0\rangle =
|0\rangle,\;\;\tilde{Q}|0\rangle =|0\rangle,\;\;\tilde{T}|0\rangle
=|0\rangle.\end{array}\end{equation} Note that
$\tilde{W},\tilde{Q},$ and $\tilde{T}$ commute with one
another for both bosons and fermions.
It is useful to define an operator $\tilde{N}$ that assigns to
each complex rational state a corresponding
complex rational number in $C$. For fermions $\tilde{N}$
can defined explicitly using a-c operators. One has
\begin{equation}\label{defN}\begin{array}{l}
\tilde{N}=\sum_{h,j}2^{j}[a^{\dag}_{+,h,j}a_{+,h,j}-
a^{\dag}_{-,h,j}a_{-,h,j} \\ \hspace{1cm}+i(b^{\dag}_{+,h,j}
b_{+,h,j}-b^{\dag}_{-,h,j}b_{-,h,j})].\end{array}\end{equation}
From this definition one can obtain the
following properties:\begin{equation}\label{Ndef}\begin{array}{c}
\tilde{N}\tilde{W}+\tilde{W}\tilde{N}=0 \\
\mbox{$[\tilde{N},a^{\dag}_{\a,h,j}]$}
=\a 2^{j}a^{\dag}_{\a,h,j} ;\;\;\;\;\;
[\tilde{N},b^{\dag}_{\b,h,j}]
=i\b 2^{j}b^{\dag}_{\b,h,j}\\ \tilde{N}|0\rangle
=0.\end{array}\end{equation} Here $\a = +,-$ and $\b =+,-.$
These equations also apply to bosons if the $h$ variable is
deleted.
The function of the operator $\tilde{N}$ is to provide a
link of complex rational states to the complex
numbers in $C.$ For each of these states, the $\tilde{N}$ eigenvalue
is the complex number equivalent, in $C,$ of the complex rational
number that $\tilde{N}$ associates to these states.
The eigenvalues of $\tilde{N}$ acting on states that are
products of $a^{\dag}$ and $b^{\dag}$ operators can be obtained from Eqs.
\ref{defN} or \ref{Ndef}. As an
example, for the state $a^{\dag}_{+,k_{1}}a^{\dag}_{-,k_{2}}b^{\dag}_{+,k_{3}}b^{\dag}_{-,k_{4}}
|0\rangle,$
\begin{equation}\label{Nex}\begin{array}{l}
\tilde{N}a^{\dag}_{+,k_{1}}a^{\dag}_{-,k_{2}}b^{\dag}_{+,k_{3}}b^{\dag}_{-,k_{4}}|0\rangle
= \\ \hspace{0.5cm}(2^{k_{1}}-2^{k_{2}}+i2^{k_{3}}-i2^{k_{4}}) \\
\hspace{1cm}\timesa^{\dag}_{+,k_{1}}a^{\dag}_{-,k_{2}}b^{\dag}_{+,k_{3}}
b^{\dag}_{-,k_{4}}|0\rangle.\end{array}\end{equation}
For standard representations in general one has
\begin{equation}\label{NNcs}\tilde{N}(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}
|0\rangle =N[(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}](a^{\dag}_{\alpha})^{s}
(b^{\dag}_{\beta})^{t}|0\rangle\end{equation} where \begin{equation}\label{Ncs}
\tilde{N}[(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}]=\left\{\begin{array}
{ll}2^{s}+i2^{t} & \mbox{ if } \alpha=+,\beta=+ \\ -2^{s}+i2^{t} &
\mbox{ if } \alpha=-,\beta=+ \\ 2^{s}-i2^{t} & \mbox{ if } \alpha=+,\beta=-
\\ -2^{s}-i2^{t} & \mbox{ if } \alpha=-,\beta=-.
\end{array}\right.\end{equation} Here $2^{s}=\sum_{j\epsilon
s}2^{j}$ and $2^{t}=\sum_{k\epsilon t}2^{j}$.
These results also hold for fermion states. For standard
states Eq. \ref{stdfer} gives an explicit representation
for $(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}(a^{\dag}_{\alpha})^{s}
(b^{\dag}_{\beta})^{t}|0\rangle.$
The operator $\tilde{N}$ has the satisfying property that any
two states that are $N$ equal have the same $\tilde{N}$
eigenvalue. If the state $|n_{r},m_{r},n_{i},m_{i}\rangle=_{N}
(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t} |0\rangle$ then \begin{equation}
\tilde{N}|n_{r},m_{r},n_{i},m_{i}\rangle =_{N}
\tilde{N} (a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}|0\rangle.\end{equation}
Here $\alpha =+,-$ and $\beta=+,-.$ This follows from Eqs.
\ref{abdjabj},\ref{abjabj}, and \ref{defN}.
These results show that the eigenspaces of $\tilde{N}$
are invariant for any process of reducing a
nonstandard state to a standard state using Eqs.
\ref{abdjabj}-\ref{fabjabj}. Any state
$|n_{r}m_{r}n_{i}m_{i}\rangle$ with $n_{r,j}\geq 1$ and
$m_{r,j}\geq 1$ for some $j$ has the same $\tilde{N}$ eigenvalue as the
state with both $n_{r,j}$ and $m_{r,j}$ replaced by
$n_{r,j}-1$ and $m_{r,j}-1.$ Also if $n_{r,j}\geq 2$ then
replacing $n_{r,j}$ by $n_{r,j}-2$ and $n_{r,j+1}$ by
$n_{r,j+1}+1$ does not change the $\tilde{N}$ eigenvalue.
Similar relations hold for $m_{r,j},n_{i,j},m_{i,j}.$
These results show that each eigenspace of $\tilde{N}$
is infinite dimensional. It is spanned by an infinite
number of nonstandard complex rational states
and exactly one standard state.
One may think that, because of the association of one
standard state to each eigenspace, one could
limit consideration to standard states only.
As is seen below, this is not the case as basic arithmetic operations
generate nonstandard states even when implemented on standard states.
\section{Basic Arithmetic Operations on Rational
States}\label{BAORSS}
\subsection{Addition and Subtraction}\label{AS}
In quantum mechanics $n-ary$ operations for $n\geq 2$
are usually represented as operators acting on $n$ fold
tensor product states of $n$ systems in $n$ different states.
For many operations $n=2$ or $n=3$ if unitarity is to be
preserved for the operations. Here, examples include arithmetic
addition and multiplication where $n=3.$ The operator
acts on two systems in different states and gives the result of the
operation as the state of a third system.
The question arises of how to represent this setup in $\mathcal H^{Ra}$
using products of boson or fermion a-c operators acting on the vacuum.
One way is to introduce additional distinguishable
particles. For example for fermions, besides the
operators $a^{\dag}_{\a,h,j},b^{\dag}_{\b,h,j}$ one has the
creation operators
$\hat{a^{\dag}}_{\a,h,j},\hat{b^{\dag}}_{\b,h,j}$ and
$\hat{\hat{a^{\dag}}}_{\a,h,j},\hat{\hat{b^{\dag}}}_{\b,h,j}$ and
corresponding annihilation operators to
represent the added fermions. In this case the operators
for the different types of fermions all commute with one
another just as the $a^{\dag}$ and $b^{\dag}$ operators do.
Another approach is to continue with the two types of
distinguishable $a$ and $b$
systems but add additional degrees of freedom to the
system states. An example would be to consider three
different regions of space parameterized by an
additional variable $z.$ In this case arithmetic
operations would be carried out on systems in $z=1$
and $z=2$ states and the result given as states of
systems in $z=3$ states. For fermions the relevant
creation operators would be $a^{\dag}_{\a,h,j,z},b^{\dag}_{\b,h,j,z}$
with the same type of commutation relations
as before (the $a^{\dag} s$ and $b^{\dag} s$ anticommute among themselves and
the $a^{\dag} s$ commute with the $b^{\dag} s$).
The above approaches also hold for distinguishable and
indistinguishable bosons except that all the a-c
operators commute. In this case the $h$ variable
is not needed
In what follows the usual product state representation
will be used because it is more familiar and is less cumbersome.
It is left up to the reader to convert
the states to an a-c operator representation based on
the above or any other choice of distinguishable and
indistinguishable systems.
Addition and multiplication operators that are unitary
can be defined for complex rational states. For
addition one has
\begin{equation}\label{defadd}\begin{array}{l}
\tilde{+}|n_{r},m_{r},n_{i},m_{i}\rangle|n^{\prime}_{r},m^{\prime}_{r},
n^{\prime}_{i},m^{\prime}_{i}\rangle|0\rangle = \\
|n_{r},m_{r},n_{i},m_{i}
\rangle|n^{\prime}_{r},m^{\prime}_{r},n^{\prime}_{i},m^{\prime}_{i}\rangle
|n+n^{\prime},m+m^{\prime}\rangle.\end{array}\end{equation} Here
$|n+n^{\prime},m+m^{\prime}\rangle$ denotes
$|n_{r}+n^{\prime}_{r},m_{r}+m^{\prime}_{r},n_{i}+n^{\prime}_{i},
m_{i}+m^{\prime}_{i}\rangle$ where the functions
$n_{r}+n^{\prime}_{r}$ correspond to addition of $n_{r}$ and
$n^{\prime}_{r}:$ \begin{equation}\label{compaddn}
(n_{r}+n^{\prime}_{r})_{j}=
\left\{\begin{array}{l} n_{r,j} \mbox{ if $j$ is in $s$
and not in $s^{\prime}$} \\ n^{\prime}_{r,j} \mbox{ if $j$ is not
in $s$ and is in $s^{\prime}$} \\ n_{r,j}+n^{\prime}_{r,j} \mbox
{if $j$ is in $s$ and in $s^{\prime}$} \\ 0 \mbox{ otherwise.}
\end{array}\right. \end{equation} where $s$ and
$s^{\prime}$ are the domains of $n_{r}$ and $n^{\prime}_{r}.$
Similar expressions hold for $(m_{r}+m^{\prime}_{r})_{j}
,(n_{i}+n^{\prime}_{i})_{j},(m_{i}+m^{\prime}_{i})_{j}.$
For bosons or fermions $|n_{r}+n^{\prime}_{r},m_{r}+m^{\prime}_{r},n_{i}+n^{\prime}_{i},
m_{i}+m^{\prime}_{i}\rangle$ is given by Eqs. \ref{occno} and
\ref{occnost} or \ref{occferm} and \ref{occnoferm} with
$n_{r,j}$ replaced by $n_{r,j}+n^{\prime}_{r,j},$ etc.. Also
the $j$ product is over all $j$ in the union of the $8$ sets
$s_{r},s_{i}, s^{\prime}_{r},\cdots$ which are the nonzero
domains of the respective $n$ and $m$ functions.
For fermions there may be a sign change in the above in case the
total number of systems in the states $|n_{r},m_{r},n_{i},
m_{i}\rangle$ and $|n^{\prime}_{r},m^{\prime}_{r},
n^{\prime}_{i},m^{\prime}_{i}\rangle$ is odd. This occurs because
in this case an odd number of additional systems is
created by the addition operation. As noted before, if
fermion number is to be preserved by dynamical operations,
then an additional supply needs to be available to serve
as a source or sink of fermions.
For standard representations the compact notation
$|\alpha s, \beta t\rangle = (a^{\dag}_{\alpha})^{s}
(b^{\dag}_{\beta})^{t}|0\rangle$ is useful where
$\alpha =+,-$ and $\beta =+,-.$ One has from Eq.
\ref{defadd}\begin{equation}\label{defplus}\begin{array}{l}
\tilde{+}|\alpha s,\beta t\rangle
|\alpha^{\prime}s^{\prime},\beta^{\prime}
t^{\prime}\rangle|0\rangle= \\ \hspace{1cm}|\alpha s,\beta t\rangle
|\alpha^{\prime}s^{\prime},\beta^{\prime} t^{\prime}\rangle|\alpha s,
\beta t+\alpha^{\prime} s^{\prime},\beta^{\prime}t^{\prime}\rangle\end{array}\end{equation}
where \begin{equation}\label{alpbetab}\begin{array}{l}|\alpha s,
\beta t+\alpha^{\prime} s^{\prime},\beta^{\prime}t^{\prime}\rangle \\ \hspace{0.5cm}=(a^{\dag}_{\alpha})^{s}
(b^{\dag}_{\beta})^{t}(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}
(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}|0\rangle =\\ \hspace{1cm}=(a^{\dag}_{\alpha})^{s}
(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}(b^{\dag}_{\beta})^{t}(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}
|0\rangle \\ \hspace{1.5cm}=|\alpha s+\alpha^{\prime} s^{\prime},\beta
t+\beta^{\prime}t^{\prime}\rangle .\end{array}\end{equation}
This result, which uses the commutativity of the $a$ and $b$
a-c operators, shows the separate addition of the $a$
and $b$ components of the states.
It is evident from this that the result of addition need
not be a standard representation even for standard input
states. A nonstandard result occurs if $s$ and $s^{\prime},$
or $t$ and $t^{\prime}$ contain one or more elements in common,
or $\a\neq\ap$ or $\b\neq\beta^{\p}.$ For
these cases the methods described would be used to
reduce the final result to a standard representation.
Here the reduction is fairly simple as there is at
most one application of Eqs. \ref{abdjabj} and \ref{abjabj}
for each $j$ value. More reduction steps are needed for
the results of iterated additions.
From now on the above notation will be used for both
fermions and bosons with the understanding that for
fermions the real component $(a^{\dag}_{\a})^{s}(a^{\dag}_{\ap})^{\sp}$
is given by Eqs. \ref{occnoferm} and \ref{stdfer} with
Eq. \ref{compaddn} applying if $\a =\ap.$ Recall that
for standard representations the functions $n_{r},n_{i},m_{r},m_{i}$
all have the constant value $1$ over their nonzero
domains. Also for fermions the equality sign in Eq.
\ref{defplus} is replaced by $=_{\pm}$ or equality up to
the sign. If the number of fermions in $|\alpha s,\beta t
+\alpha^{\p} s^{\p},\beta^{\p} t^{\p}\rangle$ is odd the sign is minus. Otherwise it
is even. The sign is always $+$ if the dynamical steps
of addition conserve the fermion number by use of a
sink or source of fermions. Also,
for fermions, the right hand operator products
$(a^{\dag}_{\a})^{s}(a^{\dag}_{\ap})^{\sp}(b^{\dag}_{\b})^{t}(b^{\dag}_{\beta^{\p}})^{t^{\p}}$
must be expressed in the standard order with $j$
increasing to the right and the appropriate values
of $n_{r,j}=2$ or $m_{r,j}=2$ in case $\a=\ap$ and $s$ and
$sp$ have elements in common. A similar situation holds
for the $b^{\dag}$ operator products.
Extension of $\tilde{+}$ to act on states that are
linear superpositions of rational string states generates
entanglement. The discussion will be limited to standard
states, but it also applies to linear superpositions over
all states, both standard and nonstandard.
Let $\psi=\sum_{\alpha,
s,\beta,t}d_{\alpha,s,\beta,t}|\alpha s,\beta t\rangle$
and $\psi^{\prime}=\sum_{\alpha^{\prime},
s^{\prime},\beta^{\prime},t^{\prime}}d^{\prime}_{\alpha^{\prime},s^{\prime},
\beta^{\prime},t^{\prime}}|\alpha^{\prime} s^{\prime},\beta^{\prime}
t^{\prime}\rangle.$ Then
\begin{equation}\label{plusentngl}\begin{array}{l}
\tilde{+}\psi\,\psi^{\prime}|0\rangle =\sum_{\alpha
,s,\beta,t}\sum_{\alpha^{\prime},s^{\prime},\beta^{\prime},t^{\prime}}
d_{\alpha, s,\beta,t}d^{\prime}_{\alpha^{\prime},
s^{\prime},\beta^{\prime},t^{\prime}} \\ \hspace{1cm}\times|\alpha s,\beta t\rangle
|\alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}\rangle|\alpha s,\beta t +\alpha^{\prime}
s^{\prime},\beta^{\prime}t^{\prime}\rangle\end{array} \end{equation} which is entangled.
To describe repeated arithmetic operations
it is useful to have a state that describes directly the
addition of $\psi$ to $\psi^{\prime}$. Since the overall
state shown in Eq. \ref{plusentngl} is entangled,
the desired state would be expected to be a mixed or
density operator state. This is indeed the case as can
be seen by taking the trace over the first two
components of $\tilde{+}\psi,\psi^{\prime}|0\rangle:$
\begin{equation}\label{rhoaddcmplx}\begin{array}{l}
\rho_{\psi+\psi^{\prime}}=Tr_{1,2}\tilde{+}
|\psi\rangle|\psi^{\prime}\rangle|0\rangle\langle 0|\langle\psi^{\prime}
|\langle\psi|\tilde{+}^{\dag}= \sum_{\alpha,\beta, s,t}
\\ \times\sum_{\alpha^{\prime},\beta^{\prime},s^{\prime},t^{\prime}}
|d_{\alpha, s,\beta,t}|^{2}|d^{\prime}_{\alpha^{\prime},s^{\prime},\beta^{\prime},t^{\prime}}|^{2}
\rho_{\alpha s,\beta t+ \alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}}.\end{array}
\end{equation} Here $\rho_{\alpha s,\beta t+
\alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}}$ is the pure state
density operator $|\alpha s,\beta
t+ \alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}\rangle\langle\alpha s,\beta
t+ \alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}|.$
The expectation value of $\tilde{N}$ on this state gives
the expected result:
\begin{equation}\label{Nplus} Tr(\tilde{N}\rho_{\psi+\psi^{\prime}})
=\langle\psi|\tilde{N} |\psi\rangle+\langle\psi^{\prime}
|\tilde{N}|\psi^{\prime}\rangle.\end{equation}
For subtraction use is made of the fact that
$|\alpha^{\prime}s,\beta^{\prime}t\rangle$ is the additive
inverse of $|\alpha s,\beta t\rangle$ if $\alpha^{\prime}\neq\alpha$
and $\beta^{\prime}\neq\beta.$ Then \begin{equation}\label{addinv}
|\alpha s,\beta t+\alpha^{\prime}s,\beta^{\prime}t\rangle =_{N}|0\rangle
\end{equation} where Eq. \ref{abdjabj} is used to give $(a^{\dag}_{\alpha})^{s}
(a^{\dag}_{\alpha^{\prime}})^{s}=_{N}1=_{N}(b^{\dag}_{\beta})^{t}
(b^{\dag}_{\beta^{\prime}})^{t}.$
A unitary subtraction operator, $\tilde{-},$
is defined by \begin{equation}\label{subtr}\begin{array}{l}
\tilde{-}|\alpha s,\beta t\rangle |\alpha^{\prime}s^{\prime},\beta^{\prime}
t^{\prime}\rangle|0\rangle= \\ \hspace{0.5cm}|\alpha s,\beta t\rangle
|\alpha^{\prime}s^{\prime},\beta^{\prime} t^{\prime}\rangle|\alpha
s,\beta t-\alpha^{\prime} s^{\prime},\beta^{\prime} t^{\prime}\rangle\end{array}\end{equation}
where $|\alpha s,\beta t-\alpha^{\prime} s^{\prime},\beta^{\prime} t^{\prime}\rangle
= |\alpha s,\beta t+\alpha^{\prime\p} s^{\prime},\beta^{\prime\p}
t^{\prime}\rangle$ and $\alpha^{\prime\p}\neq\alpha^{\prime}$ and
$\beta^{\prime\p}\neq \beta^{\prime}.$ Other properties of $\tilde{-},$
including extension to nonstandard states and linear state
superposition, are similar to those for addition.
One sees that the definition of $\tilde{+}$,
Eqs. \ref{defadd}-\ref{defplus}, satisfies the requisite
properties of addition. it is commutative
\begin{equation}\begin{array}{l}(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}
(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}|0\rangle=_{N}
\\ \hspace{1cm}(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}
(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}|0\rangle\end{array}\end{equation}
and associative\begin{equation}\begin{array}{l}
(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}
\{(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}
(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}(a^{\dag}_{\alpha^{\prime\p}})^{s^{\prime\p}}
(b^{\dag}_{\beta^{\prime\p}})^{t^{\prime\p}}\}|0\rangle \\ =_{N}
\{ (a^{\dag}_{\alpha})^{s} (b^{\dag}_{\beta})^{t}(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}
(b^{\dag}_{\beta^{\prime}})^{t^{\prime}}\}(a^{\dag}_{\alpha^{\prime\p}})^{s^{\prime\p}}
(b^{\dag}_{\beta^{\prime\p}})^{t^{\prime\p}}|0\rangle.\end{array}\end{equation}
Also $|0\rangle$ is the additive identity.
This is expressed here by noting that $(a^{\dag}_{\a})^{s}=(b^{\dag}_{\b})^{t}=1$
if $s$ or $t$ are empty. Note that these properties are expressed in terms
of $N$ equality, not state equality as these properties
may not hold for state equality. For example,
for fermions, the minus sign introduced by operator commutation
has no effect on the numerical value. but it can have a
nontrivial consequence for linear superposition states.
However, even in this case it does not affect the
numerical properties of states such as $\rho_{\psi+\psi^{\prime}}.$
For bosons there is no problem because the a-c operators commute.
Also the properties of $N$ equality are useful to show
that associativity, etc., also hold for addition of nonstandard states.
\subsection{Multiplication}\label{M}
The description of multiplication is more complex
because it is an iteration of addition, and
complex rational states are involved. The operator
$\tilde{\times}$ is defined by
\begin{equation}\label{timescmplx}\begin{array}{l}
\tilde{\times}|\alpha s, \beta t\rangle|\alpha^{\prime}
s^{\prime},\beta^{\prime} t^{\prime}\rangle |0\rangle = \\ \hspace{0.5cm}|\alpha
s,\beta t\rangle|\alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}\rangle
|\alpha s, \beta t \times \alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime}\rangle.\end{array} \end{equation}
The definition of the state $|\alpha s,\beta t\times \alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime}\rangle$ is most easily expressed
as follows. Let $c_{j}$ stand for any one of
$a^{\dag}_{+,j},a^{\dag}_{-,j},b^{\dag}_{+,j},b^{\dag}_{-,j}$ and $c^{s}$ for any one of
$(a^{\dag}_{+})^{s},(a^{\dag}_{-})^{s},(b^{\dag}_{+})^{s},
(b^{\dag}_{-})^{s}.$ This use of a variable without a dagger
to represent any one of the four creation operators is
done in the following to avoid symbol clutter.
The definition of multiplication is divided into two
steps: converting the product $c^{s}\times \hat{c}^{t}$
into a product of $c_{0}$ times some operator product and
then defining $c_{0}\times --$ to take account of
complex numbers. Note that the $\tilde{N}$ eigenvalues of
$c_{0}|0\rangle$ range over the numbers $1,-1,i,-i.$
The first step uses Eq. \ref{WQT} to define $c_{j}\times
\hat{c}^{s}$ by
\begin{equation}\label{cjmult}\begin{array}{l}
c_{j}\times \hat{c}^{s}=c_{0}\times
\tilde{T}^{j}\hat{c}^{s}(\tilde{T}^{\dag})^{j}= \\ \hspace{0.5cm}(c_{0}\times
\hat{c}_{k_{1}+j})\cdots (c_{0}\times
\hat{c}_{k_{n}+j}).\end{array}\end{equation}
Here $s=\{k_{1},k_{2},\cdots,k_{n}\}$ where
$k_{1}<k_{2}<\cdots<k_{n}$ for fermions. This equation
shows that multiplication by a power of $2$
is equivalent to a $j$ translation by that power.
Extension of this to multiplication by a product of the
$c$ operators gives \begin{equation}\label{cscspmult}
\begin{array}{l}c^{s}\times\hat{c}^{s^{\prime}} = (c_{k_{1}}\times
\hat{c}^{s^{\prime}})(c_{k_{2}}\times \hat{c}^{s^{\prime}})\cdots (c_{k_{n}}\times
\hat{c}^{s^{\prime}}) \\ =(c_{0}\times\tilde{T}^{k_{1}}\hat{c}^{s^{\prime}}
(\tilde{T}^{\dag})^{k_{1}})\cdots (c_{0}\times\tilde{T}^{k_{n}}\hat{c}^{s^{\prime}}
(\tilde{T}^{\dag})^{k_{n}}).\end{array} \end{equation}Here
$s=k_{1},k_{2},\cdots ,k_{n}.$
For the second step, all four cases of multiplication by
$c_{0}$ can be expressed as: \begin{equation}\label{ab0mult}
\begin{array}{l}b^{\dag}_{-,0}\times c^{s} =\left\{\begin{array}{ll}\tilde{Q}
\tilde{W}c^{s} \tilde{W}^{\dag}\tilde{Q}^{\dag} & \mbox{
if }c=a^{\dag}_{+}, a^{\dag}_{-} \\ \tilde{Q}c^{s}\tilde{Q}^{\dag} & \mbox{ if
}c=b^{\dag}_{+},b^{\dag}_{-}.\end{array}\right. \\
b^{\dag}_{+,0}\times c^{s} =\left\{\begin{array}{ll}\tilde{Q}
c^{s}\tilde{Q}^{\dag} & \mbox{ if }c=a^{\dag}_{+}, a^{\dag}_{-} \\\tilde{Q}
\tilde{W}c^{s} \tilde{W}^{\dag}\tilde{Q}^{\dag} & \mbox{ if }
c=b^{\dag}_{-},b^{\dag}_{+}.\end{array}\right.\end{array} \end{equation} For all $c$
\begin{equation}\label{a0mult}\begin{array}{l} a^{\dag}_{-,0}\times c^{s}=
\tilde{W}c^{s}\tilde{W}^{\dag} \\
a^{\dag}_{+,0}\times c^{s}=c^{s} \\ c_{0}\times\tilde{1}=
\tilde{1}\times c_{0}=\tilde{1}. \end{array}\end{equation}
This gives
\begin{equation}\label{asxbt}\begin{array}{l}
|\alpha s, \beta t \times \alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime}\rangle =\\ |(\alpha s\times
\alpha^{\prime}s^{\prime}+\beta t\times
\beta^{\prime}t^{\prime}),(\alpha s\times \beta^{\prime}t^{\prime}+\beta
t\times\alpha^{\prime}s^{\prime}) \rangle.\end{array}\end{equation}
where $(\alpha s\times\alpha^{\prime}s^{\prime}+\beta t\times
\beta^{\prime}t^{\prime})$ and $ (\alpha s\times \beta^{\prime}t^{\prime}+\beta
t\times\alpha^{\prime}s^{\prime})$ denote the real and
imaginary components of the product state. Note that if $s$
is empty, then $c^{s}= \tilde{1}.$ From the above and Eq. \ref{ab0mult}
one has $\tilde{1}\times (c)^{s_{\prime}}|0\rangle =
(c)^{s_{\prime}}\times\tilde{1}|0\rangle = |0\rangle.$
This corresponds to a proof for the number
representation constructed here that multiplication
of any number by $0$ gives $0$.
Extension of $\tilde{\times}$ to cover nonstandard states
is straight forward. To see this one notes that any
nonstandard state can be written in the form
$c_{1}^{s_{1}}c_{2}^{s_{2}}\cdots c_{n}^{s_{n}}|0\rangle$
where for each $\ell=1,2,\cdots,n$ $c^{s_{\ell}}_{\ell}$ is any
one of $(a^{\dag}_{+})^{s_{\ell}},(a^{\dag}_{-})^{s_{\ell}},
(b^{\dag}_{+})^{s_{\ell}},(b^{\dag}_{-})^{s_{\ell}}$ and $s_{\ell}$ is a finite
set of integers. The product of this state with another
nonstandard state $\hat{c}_{1}^{t_{1}}\hat{c}_{2}^{t_{2}}\cdots
\hat{c}_{m}^{t_{m}}|0\rangle$ is the state
$$\prod_{j=1}^{n}\prod_{k=1}^{m}c_{j}^{s_{j}}\times
\hat{c}_{k}^{t_{k}}|0\rangle.$$ Each component $c_{j}^{s_{j}}\times
\hat{c}_{k}^{t_{k}}$ is evaluated according to the description
in Eqs. \ref{cscspmult} \emph{et seq}. The large number of
multiplications needed here suggests that it may be more
efficient to convert each nonstandard state to a standard
state and then carry out the multiplication.
Extension of multiplication to linear superpositions of
complex rational string states is straightforward.
Following Eq. \ref{rhoaddcmplx} the result of multiplying
$\psi$ and $\psi^{\prime}$ is the density operator
$\rho_{\psi\times\psi^{\prime}}$ where \begin{equation}
\label{rhomultcmplx}\begin{array}{l}
\rho_{\psi\times\psi^{\prime}}=Tr_{1,2}\tilde{\times}
|\psi\rangle|\psi^{\prime}\rangle|0\rangle\langle 0|\langle\psi^{\prime}
|\langle\psi|\tilde{\times}^{\dag}= \sum_{\alpha,\beta,
s,t} \\ \times\sum_{\alpha^{\prime},\beta^{\prime},s^{\prime},t^{\prime}}
|d_{\alpha, s,\beta,t}|^{2}|d^{\prime}_{\alpha^{\prime},s^{\prime},\beta^{\prime},t^{\prime}}|^{2}
\tilde{P}_{\alpha s\beta t\times \alpha^{\prime}s^{\prime}\beta^{\prime}t^{\prime}}.\end{array}
\end{equation} This is the same as Eq. \ref{rhoaddcmplx} for
addition except that the projection operator is for the product state
$|\alpha s\beta t\times
\alpha^{\prime}s^{\prime}\beta^{\prime}t^{\prime}\rangle$.
From the definition of $\tilde{N}$ one obtains
\begin{equation}\label{Nmult}
Tr\tilde{N}\rho_{\psi\times\psi^{\prime}}=\langle\psi|\tilde{N}
|\psi\rangle\langle\psi^{\prime}|\tilde{N}|\psi^{\prime}\rangle.
\end{equation} Here $N(\alpha s\beta t\times \alpha^{\prime}
s^{\prime}\beta^{\prime}t^{\prime})= N(\alpha s\beta t)
N(\alpha^{\prime}s^{\prime}\beta^{\prime}t^{\prime})$ has been used.
The above results also show that multiplication is commutative in that
\begin{equation}\label{multcomm}\begin{array}{l}
(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}\times
(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}b^{\dag}_{\beta^{\prime}})^{t^{\prime}}|0\rangle
=_{N}\\ \hspace{1cm}(a^{\dag}_{\alpha^{\prime}})^{s^{\prime}}b^{\dag}_{\beta^{\prime}})^{t^{\prime}}\times
(a^{\dag}_{\alpha})^{s}(b^{\dag}_{\beta})^{t}|0\rangle.\end{array}\end{equation}
Distributivity of multiplication over addition for complex
rational states follows
from Eqs. \ref{cjmult} and \ref{cscspmult}. To see this
let $s^{\prime}=s_{1}\bigcup s_{2}$ be a partition of
$s^{\prime}$ into two sets where all integers in $s_{1}$ are
larger than those in $s_{2}.$ Then the equations show
that \begin{equation}\begin{array}{c}c^{s}\times \hat{c}^{s^{\prime}}=
c^{s}\times (\hat{c}^{s_{1}}\hat{c}^{s_{2}})=c^{s}\times(\hat{c}^{s_{1}}+
\hat{c}^{s_{2}}) \\ =_{N}(c^{s}\times\hat{c}^{s_{1}})(c^{s}\times
\hat{c}^{s_{2}})=(c^{s}\times\hat{c}^{s_{1}})+c^{s}\times \hat{c}^{s_{2}}).
\end{array}\end{equation} Note again that $N$ equality is
used, not state equality.
\subsection{Division}\label{D}
As is well known the complex rational string states and linear
superpositions of these states are
not closed under division. However they just escape
being closed in that division can be approximated to any
desired accuracy. One defines an $\ell$ accurate division operator
$\tilde{\div}_{\ell}$ by
\begin{equation}\label{defdiv}\begin{array}{l}
\tilde{\div}_{\ell}|\alpha s, \beta t\rangle|\alpha^{\prime}
s^{\prime},\beta^{\prime} t^{\prime}\rangle |0\rangle = \\ \hspace{0.5cm}|\alpha
s,\beta t\rangle|\alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime}\rangle
|\alpha s, \beta t /( \alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime})_{\ell}\rangle\end{array} \end{equation} where \begin{equation}
|\alpha s, \beta t /( \alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime})_{\ell}\rangle =|\alpha s, \beta t\times
(\alpha^{\prime}s^{\prime},\beta^{\prime}t^{\prime})^{-1}_{\ell}\rangle\end{equation}
and \begin{equation}|(\alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime})^{-1}_{\ell}\rangle=|(\alpha^{\prime}s^{\prime},
\beta^{\prime\p}t^{\prime}\times(q^{-1})_{\ell}\rangle.\end{equation} Here
$\beta^{\prime\p}\neq \beta^{\prime}$ and
$q=((\alpha^{\prime}s^{\prime})^{2}+\beta^{\prime}t^{\prime}\times
\beta^{\prime\p}t^{\prime})^{1/2}_{\ell}.$ This is the complex
rational state expression of
$(u+iv)^{-1}=(u-iv)/(u^{2}+v^{2})^{1/2}.$
Determination of the real rational state
$|q^{-1}_{\ell}\rangle$ involves two computations to
accuracy $\ell,$ a square root and an inverse. Since
"accuracy $\ell$" is common to both, The discussion here
will be limited to the inverse as the square root is
calculated in a similar way but with a different
algorithm.
The above shows it is sufficient to consider
states of the form $(a^{\dag}_{+})^{s}|0\rangle$ or $(a^{\dag}_{-})^{s}|0\rangle$
in detail as extension to imaginary and complex rational
states uses these results. The main goal is to show that
any string state $(a^{\dag}_{+})^{s}|0\rangle$ or $(a^{\dag}_{-})^{s}|0\rangle$ has
an inverse string state to an accuracy of at least
$a^{\dag}_{+,-\ell}|0\rangle$ for any $\ell.$ Accuracy is defined by means
of an ordering relation $<_{N}$ on the rational string number states.
A few details are given in the next section. The inverse can be used with
the definition of multiplication to show
that\begin{equation}\label{cdivcp}(c^{s}/\hat{c}^{s^{\prime}})_{\ell}
|0\rangle=(c^{s}\times(\hat{c}^{s^{\prime}})^{-1}_{\ell})|0\rangle
\end{equation} where $c,\hat{c}$ are each either
$a^{\dag}_{+}$ or $a^{\dag}_{-}.$ This would be applied to
$[(\alpha^{\prime}s^{\prime},\beta^{\prime\p}t^{\prime}))/q]_{\ell}$
to evaluate $|(\alpha s,\beta t)/(\alpha^{\prime}s^{\prime},
\beta^{\prime}t^{\prime})\rangle.$
Let $a^{\dag}_{+,[-1,-\ell]}=a^{\dag}_{+,-1}a^{\dag}_{+,-2}\cdots
a^{\dag}_{+,-\ell}.$ One has to show that for each operator
product $(a^{\dag}_{\alpha})^{s}$ and each $\ell$ there exists a
product $(a^{\dag}_{\alpha})^{t},$ where \begin{equation}\label{definv}
(a^{\dag}_{\alpha})^{s}\times(a^{\dag}_{\alpha})^{t}|0\rangle =_{N}
a^{\dag}_{+,[-1,-\ell]}a^{\dag}_{+,<-\ell}|0\rangle.\end{equation}
Here $a^{\dag}_{+,<-\ell}$ is
an arbitrary product of $a^{\dag}_{+}$ operators at locations $<-\ell.$
The arithmetic difference between the states $a^{\dag}_{+,[-1,-\ell]}
a^{\dag}_{+,<-\ell}|0\rangle$
and $a^{\dag}_{+,0}|0\rangle$ is less than $a^{\dag}_{+,-\ell}|0\rangle.$
For each $(a^{\dag}_{\alpha})^{s}$ and each $\ell$, the inverse
product $(a^{\dag}_{\alpha})^{t}$ can be constructed inductively.
Details are given in the appendix. Extension of the
definition to cover division to accuracy $\ell$ by a
nonstandard state may be possible in principle but an
inductive construction of the inverse of a nonstandard
state seems prohibitive. In this case it is much more
efficient to convert the nonstandard state to a standard
one and then construct the inverse following the methods
in the appendix. The result obtained will be $N$ equal
to the direct inverse of the nonstandard state.
As was done for addition and multiplication, a unitary
operator for division to accuracy $\ell$ on linear superposition
states can be defined. The result, $\rho_{\psi/\psi^{\prime}},$
given by Eq. \ref{rhomultcmplx} with
$\tilde{P}_{(\alpha s\beta t)/
(\alpha^{\prime}s^{\prime}\beta^{\prime}t^{\prime})_{\ell}}$ replacing
$\tilde{P}_{\alpha s\beta t\times
\alpha^{\prime}s^{\prime}\beta^{\prime}t^{\prime}},$ is obtained by
tracing over the fist two states.
It is to be noted that arithmetic operations on complex
rational states satisfy the necessary properties, such as commutativity,
distributivity, existence of an $\ell$ inverse, etc. However
linear superposition states do not satisfy all these properties.
No triple $\psi,\psi^{\prime},\psi^{\prime\p},$ satisfies the distributive law
\begin{equation}\label{supdist}\psi\times_{N}\psi^{\prime}+_{N}\psi
\times_{N}\psi^{\prime\p}=_{N}\psi\times_{N}(\psi^{\prime}+_{N}\psi^{\prime\p}).
\end{equation} Also linear superposition states do
not have $\ell$ inverses. Given $\psi$ there is no state
$\psi^{\prime}_{\ell}$ that satisfies\begin{equation}\label{supinv}
\psi\times\psi^{\prime}=a^{\dag}_{+,[-1,-\ell]}a^{\dag}_{+,<-\ell}|0\rangle.
\end{equation}
\section{Discussion}
In this paper a binary quantum mechanical representation of complex
rational numbers was presented that did not use
qubits. It is based on the observation that the
numerical value of a qubit state such as
$|10010.01\rangle$ depends on the distribution of $1s$
only with the $0s$ functioning merely as place holders.
The representation described here extends the literature
representations \cite{BenRNQM,BenRNQMALG,Kitaev} to include
boson and fermion representations of complex rational
numbers. The representation is compact and seems well
suited to represent complex
rational numbers. Since both standard and nonstandard representations
are included, arithmetic combinations of different types of numbers
are relatively easy to represent. This is not the case for
qubit product states, which are limited to standard
representations. For example the qubit
representation of the nonstandard state
$a^{\dag}_{-,-1}b^{\dag}_{-,6}b^{\dag}_{+,2}a^{\dag}_{+,3}|0\rangle$
is the pair of states, $|111.1\rangle,|-i111100.0\rangle.$
These qubit states correspond to the standard representations,
$a^{\dag}_{+,2}a^{\dag}_{+,1}a^{\dag}_{+,0}a^{\dag}_{+,-1}|0\rangle$ and
$b^{\dag}_{-,5}b^{\dag}_{-,4}b^{\dag}_{-,3}b^{\dag}_{-,2}|0\rangle$ of
$a^{\dag}_{-,-1}a^{\dag}_{+,3}|0\rangle$ and $b^{\dag}_{-,6}b^{\dag}_{+,2}|0\rangle$.
This flexibility makes the arithmetic operations
relatively easy to express in that the various steps can
be shown in a compact form. For instance addition of
several complex rational states consists of converting
a product of creation operator products, to a standard
form. This conversion process is equivalent to the steps
one goes through in carrying out the addition of several
product qubit states where each product state can be any
one of the four types of numbers.
Another advantage for the number representation shown here
is that it may expand the search horizon for
implementable physical models of quantum computers. An
example of such a model using two types of bosons
that have two different internal states, $+,-$, consists of a
string of Bose Einstein condensate (BEC) pools along an
integer $j$ lattice. Each pool can contain up to four
different BECs where the pool at site $j$ contains $n_{+,j}$
and $n_{-,j}$ bosons of type $r$ and $m_{+,j}$ and $m_{-,j}$
bosons of type $i.$
Such a string of BEC pools is a possible physical model of a
nonstandard complex rational state. For example, one might
imagine starting out a quantum computation of $\int_{a}^{b}f(x)dx$
with all pools empty, coherently computing many values of
$f(x_{j})$ for $j=1,\cdots,M,$ and putting the results into the
pools by adding bosons of the appropriate type and state at
specified $j$ locations. The resulting string of BEC
pools is a nonstandard representation of the value of
the integral. It is converted to a standard
representation by removing bosons according to rules
based on Eqs. \ref{abdjabj} and \ref{abjabj}. This
corresponds to carrying out the sum indicated by the integral.
It would be very useful for this conversion if bosons could be
found that interact physically according to one or more of these rules.
Then part of the conversion process could happen
automatically.
It should be emphasized that the operator $\tilde{N}$
was introduced early in the development as an aid to
understanding. It is not essential in that the whole
development here can be carried out with no reference to
$\tilde{N}.$ The advantage of this is that complex
rational states and the arithmetic operations
can be defined independently of and without
reference to corresponding properties on $C$.
In this case Eqs. \ref{abdjabj} and
\ref{abjabj}, or Eqs. \ref{fabdjabj} and \ref{fabjabj},
become definitions of $=_{N}.$ Also the
definitions of operators for basic arithmetic operations,
Eqs. \ref{defplus}, \ref{subtr}, \ref{timescmplx}, and
\ref{defdiv} do not depend on $\tilde{N}.$ As an
operator on the states in $\mathcal H^{Ra},$
$\tilde{N}$ corresponds to a map from states $\psi$
where the expectation value $\langle
\psi|\tilde{N}|\psi\rangle$ is the number in $C$
associated to $\psi.$ Eqs. \ref{Nplus} and \ref{Nmult},
give the satisfactory result that
$\tilde{N}$ is a morphism from states in $\mathcal H^{Ra}$ to $C$
in that it preserves the basic arithmetic operations.
If $\tilde{N}$ is not used, one needs to define an ordering $<_{N}$
that satisfies ordering axioms for rational numbers separately on the
real and imaginary parts. The ordering is defined on the standard
positive rational states and extended to the standard negative
states by reflection. Extension to nonstandard states uses $=_{N}$
as in \begin{equation}\begin{array}{l} \mbox{If $|n_{+},n_{-},m_{+},m_{-}
\rangle =_{N}|\alpha s,\beta t\rangle,$}\\ \hspace{0.25cm}\mbox{
$|n^{\prime}_{+},n^{\prime}_{-},m^{\prime}_{+},m^{\prime}_{-}\rangle
=_{N}|\alpha^{\prime} s^{\prime},\beta^{\prime} t^{\prime}\rangle,$} \\ \hspace{0.5cm}
\mbox{ and $|\alpha s,\beta t\rangle<_{N}|\alpha^{\prime}
s^{\prime},\beta^{\prime} t^{\prime}\rangle,$ } \\ \hspace{1cm}
\mbox{ then $|n_{+},n_{-},m_{+},m_{-} \rangle <_{N}|n^{\prime}_{+},
n^{\prime}_{-},m^{\prime}_{+},m^{\prime}_{-}\rangle.$}\end{array}\end{equation}
The description of division used $<_{N}$
implicitly in referring to division to accuracy
$a^{\dag}_{-\ell}|0\rangle$ instead of accuracy $2^{-\ell}.$
The description given here is not limited
to binary representations. For a $k-ary$ representation
one replaces Eq. \ref{abjabj} by \begin{equation}\label{kary}
(c^{\dag}_{j})^{k}=_{N} c^{\dag}_{j+1};\;\;\;\; (c_{j})^{k}=_{N}c_{j+1}
\end{equation} and changes Eq. \ref{abjk} to reflect this
difference. Appropriate changes would be needed in any
results depending on these equations.
\section*{Acknowledgements}
This work was supported by the U.S. Department of Energy,
Office of Nuclear Physics, under Contract No. W-31-109-ENG-38.
|
{
"timestamp": "2005-06-20T21:21:04",
"yymm": "0503",
"arxiv_id": "quant-ph/0503154",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503154"
}
|
\section{Introduction}
\indent \indent In complex dynamics there exists an extensive
study of polynomials as dynamical systems acting on ${\mathbb C}$. The
orbit of a point $z_0\in{\mathbb C}$ under a polynomial $f \in {\mathbb C}[z]$ is
the sequence $z_0,z_1,z_2,\ldots$ defined by
$$z_n =f^n(z_0).$$
\hspace{0pt} Subsets of ${\mathbb C}$ which are of particular interest are the
\textbf{filled Julia set}, which is the set of points with bounded
orbit; the \textbf{Julia set}, which corresponds to the boundary
of the filled Julia set; and the \textbf{Fatou set}, the
complement of the Julia set.
\hspace{0pt} A important result of Sullivan \cite{Su} says that for all
polynomials $f \in {\mathbb C}[z]$ there is no wandering component of the
Fatou set, i.e. every connected component of the Fatou set is
pre-periodic under the action of $f$ (This result holds also for
rational functions but our emphasis will be on polynomials).
\hspace{0pt} Recently the study of iterations of rational functions over
${\mathbb C}$ has been extended to the study of rational functions with
coefficient in the field ${\mathbb C}_p$ (\cite{BE}, \cite{BE3},
\cite{RL2}, \cite {RL3}). This field is the smallest complete
algebraically closed extension of ${\mathbb Q}$ with respect to the
$p$-adic valuation. The construction of ${\mathbb C}_p$ is analogous to
that of the complex numbers starting with rational numbers and the
usual absolute value, some interesting differences arise between
${\mathbb C}$ and ${\mathbb C}_p$.
\hspace{0pt} The field ${\mathbb C}_p$, endowed with the $p$-adic valuation, is an
\textbf{ ultrametric space}, i.e. for all $x,y \in {\mathbb C}_p$
$$|x+y|\leq \max \{|x|,|y|\}.$$
\hspace{0pt} From the above inequality, known as the \textbf{strong triangle
inequality}, it follows that ${\mathbb C}_p$ is totally disconnected, so
the connected component notion used in complex dynamics must be
replaced by the concept of \textbf{infraconnected component} (see
\cite{ES}).
\hspace{0pt} The motivation of this work arises from a result of Benedetto
\cite{BE}, who studied the family of polynomials in ${\mathbb C}_p[z]$
defined by
$$P_{\lambda}(z)= \frac{\lambda}{p} \ z^p
+\left(1-\frac{\lambda}{p}\right)\ z^{p+1},$$ where $\lambda \in
\Lambda =\{\lambda \in {\mathbb C}_p: |\lambda-1|_p<1\}$, obtaining the
following result:
\bigskip
\noindent {\bf Theorem (Benedetto).} {\em There is a dense set of
parameters $\lambda \in \Lambda$ such that the polynomial
$P_\lambda $ has a wandering disc contained in the filled Julia
set which is not attracted to an attracting cycle.}
\bigskip
\hspace{0pt} From the above theorem we conclude that there exist polynomials
in ${\mathbb C}_p[z]$ with wandering infraconnected components of the Fatou
set, in contrast with the result of Sullivan for complex rational
functions.
\hspace{0pt} In this work, we will study perturbations of the polynomials
$P_\lambda$, of the form $$Q_\lambda = P_\lambda + Q,$$ where $Q$
is a polynomial with $\|Q\|\leq
\left(\frac{1}{p}\right)^{\frac{1}{p-1}}$ and we will obtain the
following result (for the definition of $\|Q\|$ see Section 2.1).
\bigskip
\noindent {\bf Theorem.} {\em There is a dense set of parameters
$\lambda \in \Lambda$ such that $Q_\lambda $ has a wandering disc
contained in the filled Julia set which is not attracted to an
attracting cycle.}
\bigskip
\hspace{0pt} Let $Pol_d$ be the space of monic centered polynomials of
degree $d\geq 2$ and coefficients in ${\mathbb C}_p$. The parameter space
$Pol_d$ is naturally identified with ${\mathbb C}_p^{d-1}$. If we call
$E_d$ the set of polynomials in $Pol_d$ that have a wandering
disc, from the above theorem, we obtain directly the following
consequence.
\bigskip
\noindent {\bf Corollary.} {\em For all $\lambda \in \Lambda$, the
polynomial $ P_\lambda $ belong to the interior of
$\overline{E}_{p+1}$.}
\bigskip
\hspace{0pt} In addition, we will prove that the above theorem is also true
for a wider class of perturbations of the polynomials $P_{\lambda}$. In
fact, if we consider $R_B$ the set of rational functions without
poles in the fixed ball $B= \{z: |z|\leq r\}\ \ (r>1)$ and the
subset $R_B^{E}$ of functions in $R_B$ with a wandering disc, we
obtain the following consequence.
\bigskip
\noindent {\bf Corollary.} {\em For all $\lambda \in \Lambda$, the
polynomial $ P_\lambda $ belong to the interior of
$\overline{R_{B}^E}$.}
\medskip
\hspace{0pt} In sections 2.1 and 2.2 we recall some basic concepts and facts
from ultrametric analysis and dynamics. In Section 2.3 we will
present in detail some results and techniques used in \cite{RL}
since they are essential for our study of the perturbation
$Q_{\lambda}$. Our study is not done directly on $Q_{\lambda}$, but it is
more convenient to work with an affinely conjugated map
$Q_{\lambda}^*$. In Section 3.1 we study the behavior of
$Q_{\lambda}^*$ over the filled Julia set. Finally, in Section 3.2
we mimic \cite{BE} to study $Q_\lambda^*$ as a function of the
parameter $\lambda$ and prove our main result.
\newpage
\section{Preliminaries}
\indent \indent In this section we recall definitions and results
that are used throughout this work.
\hspace{0pt} The field ${\mathbb C}_p $ endowed with the $p$-adic valuation denoted
by $|\cdot|$ is an ultrametric space, i.e. for $z_0,z_1 \in {\mathbb C}_p$
we have that $|z_0-z_1|\leq\max \{|z_0|,|z_1|\}$. From this
inequality and the completeness of ${\mathbb C}_p$ arise interesting
topological and geometrical results, some of them are:
\begin{itemize}
\item[(i)] The {\bf value group} of the valuation is the set
$|{\mathbb C}_p^*|:=\{|z|:z\in {\mathbb C}_p^{*} \}=\{p^r:r\in{\mathbb Q} \}$.
\item[(ii)]{\bf The isosceles triangle principle}: If
$|z_1|\neq|z_2|$, then $|z_1+z_2|=\max \{ |z_1|,|z_2|\}$.
\item[(iii)] For $z_0 \in {\mathbb C}_p$, $r\in |{\mathbb C}_p^{*}|$, the {\bf open
ball} with radius $r$ and center $z_0$ is the set
$$B_r(z_0)=\{ z \in \mathbb{C}_p : |z-z_0|< r\}$$ and the {\bf closed ball}
with center $z_0$ and radius $r$ is the set
$$\overline{B_{r}}(z_0)=\{ z \in \mathbb{C}_p: |z-z_0|\leq r \}.$$ These are open and closed sets in the topology of
$\mathbb{C}_p$. By definition, we have that the diameter
of $B$ denoted $\operatorname{diam}(B)$ belong to $|{\mathbb C}_p^*|$, where $B$ is an open or closed ball.
\hspace{0pt} We denote by $\mathcal{O}_{{\mathbb C}_p}$ the closed ball $
\{z:|z|\leq1\}.$
\item[(iv)] Every point of a ball is a center, that is, if $z_1
\in B_r(z_0)$ (resp. $z_1 \in \overline{B_r}(z_0)$), then
$B_r(z_1) =B_r(z_0)$ (resp.
$\overline{B_r}(z_1)=\overline{B_r}(z_0)$).
\item[(v)] If two balls have not empty intersection then one is
contained in the other one.
\end{itemize}
\hspace{0pt} The properties below are about convergence in ultrametric
spaces, some of them are different than in archimedean analysis.
Let $a_1, a_2,\dots,a_n,\dots $ be a sequence in ${\mathbb C}_p$. Then
\begin{itemize}
\item[(vi)] If $\underset{n\rightarrow \infty}{\lim} a_n=a$ and
$a\neq 0$, then there exists a $n_0$ such that $|a_n|=|a|$ for all
$n>n_0$.
\item[(vii)]The series $\underset{n=1}{\overset{\infty}{\sum}}a_n$
converges if and only if $\underset{n\rightarrow \infty}{\lim} a_n
=0$.
\item[(viii)] The power series
$\underset{n=1}{\overset{\infty}{\sum}}a_nx^n$ has convergence
radius $r:= \left(\underset{n \rightarrow \infty}{\limsup}
\sqrt[n]{|a_n|}\right)^{-1}$.
\item[(ix)] If $r$ is the convergence radius of
$\underset{n=1}{\overset{\infty}{\sum}}a_nx^n$, then the map
$$x\longmapsto
\underset{n=1}{\overset{\infty}{\sum}}a_nx^n$$
\hspace{0pt} is differentiable in $B_r(0)$ and its derivative is
$$x\longmapsto
\underset{n=1}{\overset{\infty}{\sum}}na_nx^{n-1}.$$
\end{itemize}
\subsection{Ultrametric analysis.}
\indent \indent Let $B$ be a ball with radius $r$ (open or
closed), we denote by $\mathcal{H}(B)$ the ring of power series
which converge in $B$. The space $\mathcal{H}(B)$ endowed with the
norm
$$\| f\|_{B}=\underset{i\geq0}{\sup}|a_i|r^{i}$$ is a complete ultrametric valued ring.
\hspace{0pt} As in the complex case, a rational function can be written as a
power series around every point $z_0$ which is not a pole. But
observe that if we consider the function $f:{\mathbb C}_p \longrightarrow
{\mathbb C}_p$ given by
$$f(z)=\left\{ \begin{array}{ll}
1, & |z|<1\\
0, & |z|\geq 1 \\
\end{array} \right.$$ we have that also can be written as power series around every point of ${\mathbb C}_p$.
Hence, it is clear that the idea of holomorphic functions in
${\mathbb C}_p$ is different from the one in ${\mathbb C}$, which it is defined by a
local property. Indeed, a function defined in a subset $X$ of
${\mathbb C}_p$ is holomorphic if it is the uniform limit of rational
functions without poles in $X$ (see \cite{TJ}). In this work, we
only consider holomorphic functions defined on a ball $B$ and, in
this case, the definition coincides with the complex one: a
function $f$ is \textbf{holomorphic} in ${B}$ if and only if $f$
can be written as a convergent power series in $B$. Thus,
$\mathcal{H}(B)$ is the space of holomorphic functions in the disc
$B$.
\bigskip
\hspace{0pt} Now, we will show an analogous to the Newton's method, in order
to guarantee the existence of roots in a holomorphic function.
\begin{lema}{\bf(Hensel)} {\hspace*{-3pt}\bf .} Let $f\in \mathcal{O}_{{\mathbb C}_{p}}[[z]]$.
If there exists $z_{0}\in \mathcal{O}_{{\mathbb C}_p}$ with $|f(z_{0})|<
|f^{'}(z_0)|^2$, then there is an unique root $w$ of $f$ such
that $|w-z_0|\leq \displaystyle
\frac{|f(z_0)|}{|f^{'}(z_0)|}$.\label{Hensel}
\end{lema}
\textbf{Proof:}
\hspace{0pt} We define recursively the sequence,
$$z_1= z_0 -\frac{f(z_0)}{f^{'}(z_0)},
\ z_{n+1}= z_n-\frac{f(z_n)}{f^{'}(z_n)}.$$ We will show
inductively that:
\begin{itemize}
\item[(i)] $|f(z_n)|\leq C^{2^{n}}|f^{'}(z_0)|^{2}$, where $C=
\displaystyle\frac{\left |f(z_0)\right |}{\left |f^{'}(z_0)\right |^{2}}<1.$
\item[(ii)]$|f^{'}(z_n)|=|f^{'}(z_0)|.$
\item[(iii)]$|z_n-z_0|=|z_1-z_0|.$
\end{itemize}
\hspace{0pt} Now
$$f(z_1)=f(z_0+z_1-z_0)=f(z_0)+f'(z_0)(z_1-z_0)+d (z_1-z_0)^{2}=
d(z_1-z_0)^2$$ for some $d=d(z_0,z_1) \in
\mathcal{O}_{{\mathbb C}_p}$. Therefore
\begin{equation*}
|f(z_1)| = |d(z_1-z_0)^2| \leq |z_1-z_0|^2 = C^2|f'(z_0)|^2.
\end{equation*}
and
$$f'(z_1)= f'(z_0)+e (z_1-z_0)$$ for some $e=e(z_0,z_1)\in \mathcal{O}_{{\mathbb C}_p}$. Hence $$|f'(z_1)-f'(z_0)|\leq |z_1-z_0|
\leq\left | \displaystyle \frac{f(z_0)}{f'(z_0)}\right |<|f'(z_0)|.$$
From the previous inequality, and the isosceles triangle principle
applied to $|f'(z_1)-f'(z_0)+f'(z_0)|$ we have that
$$|f'(z_1)|=|f'(z_0)|.$$
\hspace{0pt} The inductive steps for (i) and (ii) are analogous to the
previous one, (iii) is direct consequence of (i) and (ii).
\hspace{0pt} Since $|f'(z_n)|=|f'(z_0)|$, we have that $|z_{n+1}-z_n|=\displaystyle
\frac{|f(z_n)|}{|f'(z_n)|}
\leq C^{2^n}|f'(z_0)|$, so $\{ z_n\}_{n\in \mathbb{N}}$ is a Cauchy sequence and if we
denote its limit by $w$ we get, from (i) and (iii), that $w$ is a
root of
$f$ and $|w-z_0|\leq \displaystyle \frac{|f(z_0)|}{|f^{'}(z_0)|}$. \hfill $\Box$
\bigskip
\hspace{0pt} We now enumerate some interesting properties of holomorphic
functions (see \cite{BE2}).
\begin{teo}{\hspace*{-6pt}\bf .}
Let $f(z)=\overset{\infty}{\underset{i=0}{\sum}}a_iz^{i}$ with
$a_i \in \mathbb{C}_p$ and $r \in |\mathbb{C}_p^{*}|$ such that
$\underset{i\rightarrow\infty}{\lim}|a_i|r^{i}=0$. Then $f$ has a
root $\alpha \in \mathbb{C}_p$ with $|\alpha|=r$ if only if there
exist $n, m \in {\mathbb Z} $ with $n<m$ and such that
\begin{equation}
|a_n|r^n=|a_m|r^m=\underset {i\geq 0}{\sup}\{|a_i|r^{i}
\}\label{ec m y n}.
\end{equation}
\hspace{0pt} Moreover if $n,m$ are the smallest and the greatest integers,
respectively, that make \emph{(\ref{ec m y n})} true, then $f$ has
exactly $m-n$ roots with absolute value $r$, counting
multiplicity.\label{teorema de soluciones contar}
\end{teo}
\begin{coro}{\hspace*{-6pt}\bf .}
Let $B$ be a closed ball, $f\in \mathcal{H}(B)$, and $D$ an open
ball (resp. closed) contained in $B$. Then $f(D)$ is an open ball
(resp. closed).\label{imagen de bola}
\end{coro}
\begin{coro}{\hspace*{-6pt}\bf .}
Let $f\in \mathcal{H}(B)$, $w_0 \in \mathbb{C}_p$, $\delta \in
|\mathbb{C}_p^{*}|$ such that $\overline{B_{\delta}}(w_0)\subset B
$. If $|f(w)-f(w_0)|= \alpha$ for all $w$ in $\{ w:|w-w_0|=\delta
\}$, then
$$f(\{ w:|w-w_0|=\delta\})=\{w:|w-f(w_0)|=\alpha \}.$$\label{esfera}
\end{coro}
\begin{coro}{\hspace*{-6pt}\bf .} Let $B$ be a closed ball and $f\in \mathcal{H}(B)$.
Then there exists $d\in{\mathbb N}$ such that the series $f- w$ has exactly
$d$ roots in $B$, counting multiplicity, for all $w \in f(B)$.
\end{coro}
\hspace{0pt} For $f\in \mathcal{H}(B)$, we define the \textbf{degree of the
map} $f$ as the number $d$ from the last corollary.
\subsection{Polynomial dynamics over ${\mathbb C}_p.$}
\indent\indent This section contains some important definitions
and dynamical properties that will be needed later.
\bigskip
\hspace{0pt} Let $P \in {\mathbb C}_p[z]$. For $z \in {\mathbb C}_p$, we define the
\textbf{orbit} of $z$, denoted by $\mathcal{O}(z)$, as the
sequence $\{P^n(z)\}_{n\in {\mathbb N}}$. If $P(z)=z$, we say that $z$ is a
\textbf{fixed point} of $P$; if for $z$ there exists $n\in {\mathbb N}$
such that $P^n(z)=z$, we will say $z$ is a \textbf{periodic
point}.
\hspace{0pt} Let $P'$ be the formal derivative of $P$, $z_0$ a fixed point
of $P$ and $\theta =|P'(z_0)|$. Then:
\begin{itemize}
\item[(i)] If $\theta < 1$, we say that $z_0$ is an
\textbf{attracting fixed point}. \item[(ii)] If $\theta > 1$, we
say that $z_0$ is a \textbf{repelling fixed point}. \item[(iii)]
If $\theta =1$, we say that $z_0$ is an \textbf{indifferent fixed
point}.
\end{itemize}
\bigskip
\hspace{0pt} Other object of study is the \textbf{filled Julia set} denoted
by $K(P)$, that correspond to the set of points of ${\mathbb C}_p$ with
bounded orbit. Some properties of the filled Julia set are:
\begin{enumerate}
\item[(i')] $K(P) \neq \emptyset$. \item[(ii')] $K(P)$ is closed
and bounded. \item[(iii')]$P^{-1}(K(P))= K(P)$, i.e. $K(P)$ is
completely invariant.
\end{enumerate}
\bigskip
\hspace{0pt} Another important set is the \textbf{Julia set}, $J(P)$, which
is the boundary of the filled Julia set. The Julia set can be also
defined as follows
$$\{z \in {\mathbb C}_p: \text{ for every neighbourhood } U \text{ of } z,\,
\underset{n \in {\mathbb N}}{\bigcup}P^n (U)= {\mathbb C}_p\}.$$
\hspace{0pt} Finally, we define the \textbf{Fatou set} as the complement of
the Julia set. We denote it by $F(P)$.
\bigskip
\hspace{0pt} We are not going to study just polynomials, so we have to
introduce the concept of polynomial like maps. If $U$ and $V$ are
open balls in ${\mathbb C}_p$ such that $U\subsetneq V$ and
$f:U\longrightarrow V$ is a holomorphic function of degree $d$
with $d\geq 1$, we say that $(f, U)$ is a \textbf{polynomial like
map} of degree $d$.
\hspace{0pt} All the preceding concepts can be also defined, in a similar
way, for polynomial like maps. That is, the filled Julia set of
$(f,U)$ is the set
$$K(f,U)=\{z\in U: f^n(z)\in U \text{ for all } n\in {\mathbb N}\}.$$
\hspace{0pt} The Julia set is $$J(f,U)=\partial K(f,U),$$ and the Fatou set
is
$$F(f,U)= U\setminus J(f,U).$$
\hspace{0pt} These new definitions will allow us to study the dynamical
behavior of some holomorphic functions restricted to balls.
\hspace{0pt} Let $D$ be a subset of ${\mathbb C}_p, \,a \in D$ and $I_a:{\mathbb C}_p
\longrightarrow {\mathbb R}$ the map defined by $I_a(x)=|x-a|$. We say that
$D$ is \textbf{ infraconnected} if and only if for all $a \in
{\mathbb C}_p$ the set $\overline{I_a(D)}$ is an interval (see \cite{ES}).
In particular, for $(f,U)$, we are interested in understanding the
behavior of the filled Julia set. If we consider $B$, the smallest
ball that contains $K(f,U)$, then $f^{-n}(B)$ is a collection of
disjoint closed balls, named balls of level $n$. Then for $w\in
K(f,U)$ there is an unique sequence $\{B_n\}_{n\in{\mathbb N}}$ of nested
closed balls, where $B_n$ is a ball of level $n$, such that $w\in
\underset{n\in {\mathbb N}}{\cap} B_n$. The set $C(w):=\underset{n}{\cap}
B_n$ is the infraconnected component of $K(f,U)$ that contains $w$
(see \cite{TJ}).
\hspace{0pt} Now let $(f,U)$ be a polynomial like map. We say that
$E\subseteq U$ is a \textbf{wandering set} if $f^n(E)\bigcap
f^m(E)\neq \emptyset$ only when $n=m$.
\hspace{0pt} Furthermore, if $(f,U)$ and $(g,U)$ are polynomial like maps,
if there is a homomorphism $h:U\longrightarrow U$ such that
$g=h^{-1}f h$, we say that $f$ and $g$ are \textbf{topologically
conjugated}. In these case, if $E$ is a wandering set of $f$, then
$h(E)$ is a wandering set of $g$. Therefore, the existence of
wandering set is invariant under conjugacy. This fact will turn
out to be very important to obtain our results.
\newpage
\subsection{The family of polynomials $P_\lambda$.}
\indent\indent For $\lambda \in \Lambda = \{ \lambda \in
\mathbb{C}_p : |\lambda -1|<1\}$ let
$$P_{\lambda}(z)= \frac{\lambda}{p}z^p
+\left( 1-\frac{\lambda}{p} \right) z^{p+1}.$$
\begin{teo}[Benedetto]{\hspace*{-6pt}\bf .} There is a dense set of parameters $\lambda \in \Lambda$, such that
the polynomial $P_{\lambda}$ has a wandering disc contained in
$K(P_{\lambda})$, which is not attracted to an attracting cycle.
\end{teo}
\hspace{0pt} Now we will sketch the proof of this theorem (see \cite{RL}),
paying attention to the techniques which will be important later.
\hspace{0pt} First we notice that $\overline{B_{\rho}}(0)$, with $\rho = \displaystyle
p ^{\frac{-1}{p-1}}$, is invariant under the action of $P_{\lambda}$ and
that $z=1$ is a repelling fixed point. Now, we considere $B_1(0)$
and $B_1(1)$, which are neighbourhoods of the fixed ball
$\overline{B_\rho(0)}$ and the repelling fixed point $z=1$
respectively. From the strong triangle inequality and Corollary
\ref{imagen de bola}, we see that the set $K(P_{\lambda})$ is contained in
$B_1(0) \sqcup B_1(1)$. This allows us to define the itinerary of
a point $x \in K(P_{\lambda} )$ as the sequence
$$\theta _1 \theta_2 \ldots \theta _n \ldots $$ with
$\theta_i \in \{0,1\}$ and $ P_{\lambda}^{i}(x) \in B_1(\theta_i)$ for $i
\in {\mathbb N}$. Furthermore, we obtain that all the points of a ball
contained in the filled Julia set have the same itinerary. If this
itinerary is not pre-periodic, then $D$ is a wandering disc. In
order to find such disc is necessary to study the behavior of the
$P_{\lambda}$ in the filled Julia set. The lemmas below describe such
behavior.
\hspace{0pt} We define $S>0$ by $pS^{p-1}=\rho$ and the sequence $\{\rho
_n\}_{n \in {\mathbb N}}$ by
$$\rho_{0} =1,\ \ p\rho_{n}^p = \rho_{n-1}.$$ \vspace*{-60pt}
\begin{lema}{\hspace*{-6pt}\bf .}
$\begin{array}{rl} \\*[55pt] 1)& \text{ Let } m\geq 1, z_0 \text{
and } z_1 \text{ such that }|z_{0}|=| z_{1}|=\rho _{m}.\text{ If }
| z_{0}-z_{1}| \leq S, \text{ then: }\\*[8 pt] & \hspace*{2
cm}|P_{\lambda}(z_{0})-P_{\lambda}(z_{1})| \leq \rho _{m-1}
|z_{0}-z_{1}|.\\*[8 pt] 2) & \text{ If } z_{0},z_{1} \in
B_{1}(1), \text{ then: } \\*[8pt] &\hspace*{2
cm}|P_{\lambda}(z_{0}) -P_{\lambda}(z_{1})| =p\ |z_{0}-z_{1} |.
\end{array}$\label{comp de P}
\end{lema}
\textbf{Proof:}
\begin{itemize}
\item[{\em1)}]We observe that
$P_{\lambda}(z_{0})-P_{\lambda}(z_{1}) =\displaystyle \frac{\lambda}{p}\left
(p\, \varepsilon \,z_0^{p-1}+\cdots
+p\,\varepsilon^{p-1}z_0+\varepsilon^p\right
)+\left(1-\frac{\lambda}{p}\right )((p+1)\,\varepsilon\,
z_0^p+\cdots +\varepsilon ^{p+1}),$
with $\varepsilon =z_1-z_0$, since $|\varepsilon |= |z_0-z_1|\leq
S< \rho_m$ and
$$|P_{\lambda}(z_0)-P_{\lambda}(z_1)|\leq |\varepsilon |\max
\{z_0^{p-1}, p\,|\varepsilon |^{p-1},\rho_{m-1}\}= |\varepsilon |\
\rho_{m-1},$$
we have that $$|P_{\lambda}(z_0)-P_{\lambda}(z_1)|\leq \rho_{m-1}\
|z_0-z_1|.$$
\item[{\em2)}] The proof is straightforward from the previous one and will be omitted.\hfill
$\Box$
\end{itemize}
\hspace{0pt} With the previous lemma it is possible to find a necessary and
sufficient condition for the existence of wandering discs in
$K(P_{\lambda})$, this condition is:
\begin{lema}{\hspace*{-6pt}\bf .}
Let $\{ m_{i}\} _{i\geq 0},\{ M_{i}\} _{i\geq 0}$ be two sequences
of positive integers such that, for all $i\geq0$ we have that
$\rho_{m_{i}-1} \cdot \ldots \cdot \rho_{1}\cdot p^{M_{i}}\leq1.$
Suppose that for $\lambda_{0} \in \Lambda $ there exists $x\in
K(P_{\lambda_{0}})$ with itinerary
$$\underset{m_0}{\underbrace{0\ldots0}}\,\underset{M_0}{\underbrace{1\ldots1}}\,
\underset{m_1}{\underbrace{0\ldots0}}
\,\underset{M_1}{\underbrace{1\ldots1}}\ldots\underset{m_i}{\underbrace{0\ldots0}}\,\underset{M_i}{\underbrace{1\ldots1}}\ldots,$$
then the ball $U=\{z:|z-x|\leq S \}$ is contained in
$K(P_{\lambda_{0}})$.\label{ex para P}
\end{lema}
\hspace{0pt} Therefore, to prove Theorem 2.6 it suffices to find $x \in
K(P_{\lambda})$ and sequences $\{M_i\}_{i \in {\mathbb N}},\{m_i\}_{i \in {\mathbb N}}$ with
$\lim M_i = \infty$ such that the hypothesis of the previous lemma
are satisfied. In order to do this we study the function $P_{\lambda}(z)$
as a function of $\lambda$.
\hspace{0pt} Now, we will see two lemmas that will allow us to find such
sequences $\{M_i\}_{i\in {\mathbb N}},\{ m_i\}_{i\in {\mathbb N}}$ with $\lim M_i=
\infty$ implying the existence of wandering discs in $K(P_{\lambda})$.
\begin{lema}{\hspace*{-6pt}\bf .}
Let $m\geq1$, and $z_0,z_1 \in B_{p^{-m}}(1)$. If $\lambda_0 ,
\lambda_1 \in \Lambda$ satisfy $|z_0 -z_1|=|\lambda_0
-\lambda_1|$, then \label{sn1}
$$|P_{\lambda_0}^{m}(z_0)-P_{\lambda_1}^{m}(z_1)|=p^{m}|\lambda_0 -\lambda_1|.$$
\end{lema}
\textbf{Proof:} \ We proceed by induction. From
\begin{eqnarray}
&&|P_{\lambda_0}(z_0)-P_{\lambda_0}(z_1)|=p\,|z_0-z_1|=p\,|\lambda_0
-\lambda_1|, \label{eq1}\\*[.3 cm]
&&|P_{\lambda_0}(z_1)-P_{\lambda_1}(z_1)|=p\,|\lambda_0
-\lambda_1||z_1-1|< p\,|\lambda_0-\lambda_1|, \label{eq2}
\end{eqnarray}
we have that $|P_{\lambda_0}(z_0)-P_{\lambda_1}(z_1)|= p\,
|\lambda_0 -\lambda_1|$, therefore the lemma is true for $m=1$.
\hspace{0pt} Now, for the inductive step, we suppose that $z_0, z_1 \in \{
z: |z-1|\leq p^{-m}\}$. By hypothesis we have that
$$|P^{m-1}_{\lambda_0}(z_0)-P^{m-1}_{\lambda_1}(z_1)|=
p^{m-1}|\lambda_0- \lambda_1|.$$
$$|P^{m-1}_{\lambda_1}(z_1)-1|<p^{m-1}.$$
\hspace{0pt} Therefore,
\begin{eqnarray}
&&|P_{\lambda_0}(P^{m-1}_{\lambda_0}(z_0))-P_{\lambda_0}(P^{m-1}_{\lambda_1}(z_1))|=p^{m
}|\lambda_0-\lambda_1|,\label{eq3}\\*[.3cm]
&&|P_{\lambda_0}(P^{m-1}_{\lambda_1}(z_1))-P_{\lambda_1}(P^{m-1}_{\lambda_1}(z_1))|=
p\,|\lambda_0-\lambda_1||P_{\lambda_1}^{m-1}(z_1)-1|<p^{m}|\lambda_0
-\lambda_1|.\label{eq4}
\end{eqnarray}
\hspace{0pt} From (\ref{eq3}) and (\ref{eq4}) we obtain
$$|P_{\lambda_0}^{m}(z_0)-P_{\lambda_1}^{m}(z_1)|=p^{m}|\lambda_0
-\lambda_1|.$$
\hfill $\Box$
\begin{lema}{\hspace*{-6pt}\bf .}
Let $m\geq 1 $ and $z_0, z_1$ with $|z_0|=|z_1|=\rho _m$ and such
that $|z_0-z_1|\leq S.$ If $\lambda_0 ,\lambda_1\in \Lambda$ are
such that
$$\rho_{m-1}\cdot \ldots\cdot \rho_1 \cdot |z_0-z_1|< |\lambda_0-
\lambda _1|\leq S,$$ then
$$|P_{\lambda_0}^{m}(z_0)-P_{\lambda_1}^{m}(z_1)|=|\lambda_0
-\lambda_1|.$$ \label{lema en P inductivo}
\end{lema}
\hspace{0pt} \textbf{Proof:} \, First we will show inductively that, for $1\leq i\leq m$,
\begin{eqnarray}
&&|P_{\lambda_0}^{i}(z_0)-P_{\lambda_1}^{i}(z_1)|\leq \max \{
\rho_{m-1}\cdot\ldots\cdot \rho_{m-i}|z_0-z_1|,
\rho_{m-i}|\lambda_0-\lambda_1|\}.\label{eq6}
\end{eqnarray}
Observe that
\begin{eqnarray}
&&|P_{\lambda_0}(z_0)-P_{\lambda_0}(z_1)|\leq
\rho_{m-1}|z_0-z_1|,\label{eq7}
\\&&|P_{\lambda_0}(z_1)-P_{\lambda_1}(z_1)|=
\rho_{m-1}|\lambda_0-\lambda_1|.\label{eq8}
\end{eqnarray}
Using the ultrametric inequality, (\ref{eq7}) and (\ref{eq8}), we
get (\ref{eq6}) for $i=1$.
If we assume (\ref{eq6}) as the inductive hypothesis, we have
\begin{eqnarray}
&&|P_{\lambda_0}(P^{i}_{\lambda_0}(z_0))-P_{\lambda_0}(P^{i}_{\lambda_1}(z_1))|\leq
\rho_{m-i-1}|P^{i}_{\lambda_0}(z_0)-P^{i}_{\lambda_1}(z_1)|,\label{eq9}\\
&&|P_{\lambda_0}(P^{i}_{\lambda_1}(z_1))-P_{\lambda_1}(P^{i}_{\lambda_1}(z_1))|=\rho_{m-
i-1}|\lambda_0-\lambda_1|.\label{eq10}
\end{eqnarray}
\hspace{0pt} From (\ref{eq9}) and (\ref{eq10}) we obtain (\ref{eq6}) for
$i+1$. Notice that for the inductive step from $m-1$ to $m$, the
hypothesis of the lemma gives us that
$$|P_{\lambda_0}^{m}(z_0)-P_{\lambda_1}^{m}(z_1)|=|\lambda_0
-\lambda_1|.$$ \hfill $\Box$
\hspace{0pt} Let $\lambda \in \Lambda$ and $M_0 \in {\mathbb N}$ with $p^{-M_0} \leq
S$, we choose $m_0 \in {\mathbb N}$ such that $$\rho_{m_0-1} \cdot \ldots
\cdot \rho_1 p^{M_0} \leq 1.$$
\hspace{0pt} Now, if we choose $x \in K(P_\lambda)$ with itinerary
$$\underset{m_0}{\underbrace{0\ldots 0}}\,1\,1\,1\,1\,1 \ldots,$$
we obtain Lemma \ref{sn1} hypothesis with $z_0 =
P_{\lambda_0}^{m_0}(x), z_1=P_{\lambda_1}^{m_0}(x)$ and $M=M_1$,
for all $\lambda_0, \lambda_1 \in \{z: |\lambda- z|\leq
p^{-M_0}\}$, and we have
$$|P_{\lambda_0}^{m_0+M_0}(x)-P_{\lambda_1}^{m_0+M_0}(x)|=
p^{M_0}|\lambda_0 - \lambda_1|.$$ Hence, there exists $w_0 \in
\Lambda$ with $P_{w_0}^{m_0+M_0}(x)=0$ such that the itinerary of
$x$ for $P_w$ is
$$\underset{m_0}{\underbrace{0\ldots 0}}\,\underset{M_0}{\underbrace{1\ldots 1}}\,0 \ldots$$
for all $w \in \{z: |z-w_0|< p^{-M_0}\}$.
\hspace{0pt} As before, we choose $M_1$ such that $P^{M_1-M_0}\leq S$, and
$m_1$ such that
$$\rho_{m_1-1} \cdot \ldots
\cdot \rho_1 p^{M_1} \leq 1$$ obtaining that there exists
$\lambda' \in \Lambda$ with $|\lambda'- w_0|= \rho_{m_1}p^{-M_0}$
such that $P_{\lambda'}(x)=1$.
\hspace{0pt} Therefore, the itinerary of $x$ for $P_{\lambda'}$ is
$$\underset{m_0}{\underbrace{0\ldots
0}}\,\underset{M_0}{\underbrace{1\ldots
1}}\,\underset{m_1}{\underbrace{0\ldots 0}}\,1\,1\,1 \ldots$$
\hspace{0pt} Lemma \ref{lema en P inductivo} allows us to make this process
inductively, obtaining the Lemma \ref{ex para P} hypothesis.
\section{Results.}
\indent\indent In this section, we establish some properties of
the perturbations of the polynomials $P_\lambda$. Throughout,
$$\rho_0 =1,\ \ p\ \rho_n^p =\rho_{n-1}.$$
\hspace{0pt} Recall that $ p\rho^p=\rho$ and that $pS^{p-1}=\rho$. For the
rest of this work we fix $\widehat{r}\in |{\mathbb C}_p^*|$, with
$\widehat{r}>1$ and $B=\{z\in{\mathbb C}_p:
|z|\leq \widehat{r}\}$. The perturbations are:
$$Q_\lambda^*(z)=P_\lambda(z)+Q(z),$$
where $Q\in \mathcal{H}(B)$ with $\|Q\|< \rho$. For this family we will obtain the following result:
\begin{teo}{\hspace*{-6pt}\bf .}
There is a dense set of parameter $\lambda \in \Lambda$ such that
the function $Q_{\lambda}^*$ has a wandering disc contained in the
filled Julia set, which is not attracted to an attracting
cycle.\label{teo pa estrella}
\end{teo}
\hspace{0pt} To prove this theorem, we will study a topological conjugation
of $Q_\lambda^*$.
\hspace{0pt} Notice that $$p(Q_\lambda^*(z)-z) \in \mathcal{O}_{{\mathbb C}_p}[[z]],$$
in addition $$\left|p\ Q_\lambda^*(1)-p \,\right|=\frac{1}{p}\
|Q(1)|<\frac{\rho}{p}$$ and
$$|p\ (Q^*_\lambda)'(1)-1|=\frac{1}{p}\ |P'_\lambda(1)+Q'(1)-1|=1.$$
From Hensel's Lemma, there is an unique root of $p\ (Q_\lambda^*(z)-z)$ in $B_{r_0}(1)$, where $r_0= \frac{|Q(1)|}{p}$.
We denote this root by $z_\lambda$ and observe that $z_\lambda $ is a fixed point of $Q_\lambda^*$.
\hspace{0pt} Now, we define the function
\begin{equation*}
\begin{array}{rccl}
h:& \Lambda& \longrightarrow& \displaystyle \left\{z:|z-1|\leq \frac{|Q(1)|}{p}\right\} \\
& \lambda & \longmapsto & \ \ z_\lambda
\end{array}
\end{equation*}
obtaining that
\begin{prop}{\hspace*{-6pt}\bf .}
The function $h$ is holomorphic in $\Lambda$.\label{res1}
\end{prop}
\hspace{0pt} \textbf{Proof:}
\smallskip
\hspace{0pt} Let $\{ h_{n} \}_{n\geq 0}$ be the sequence of functions
defined recursively as follows:
\begin{align*} &h_{0}(\lambda)=1\\
&h_{n}(\lambda)=
h_{n-1}(\lambda)-\frac{Q^{*}_{\lambda}(h_{n-1}(\lambda))}{(Q^{*}_{\lambda})'(h_{n-1}(\lambda))}\end{align*}
\hspace{0pt} Then for all $n\in {\mathbb N},\ h_{n}$ is a rational function without
poles in $\Lambda$.
\hspace{0pt} As in the proof of Lemma \ref{Hensel}, we have
\begin{equation*}
\begin{array}{rcccl}
| h_{n}(\lambda)-h(\lambda) |& = &\left| \displaystyle\underset{i\geq
n}{\sum}\displaystyle\frac{Q^{*}_{\lambda}(h_{i}(\lambda))}{(Q^{*}_{\lambda})'(h_{i}(\lambda))}\right|
&\leq& \displaystyle\underset{i\geq n}{\max}
\left\{\left|\frac{Q^{*}_{\lambda}(h_{i}(\lambda))}{(Q^{*}_{\lambda})'(h_{i}(\lambda))}\right|
\right\}\\\\& =& \displaystyle\underset{i\geq
n}{\max}{|Q^{*}_{\lambda}(h_{i}(\lambda) )|}&<& \rho ^{2^n}.
\end{array}
\end{equation*}
\hspace{0pt} Hence $h_n$ converges to $h$ uniformly in $\Lambda$. Therefore,
$h\in \mathcal{H}(\Lambda).$ \hfill $\Box$
\bigskip
\bigskip
\hspace{0pt} We may now introduce the affine map
$$A_\lambda(z)=z +h(\lambda)-1$$ we will work with the map
$$Q_\lambda(z)=A_\lambda^{-1}
(Q_\lambda^*(A_\lambda(z)))=P_\lambda(A_\lambda(z))+Q(A_\lambda(z))+1-h(\lambda).
$$ which is affinely conjugated to $Q_\lambda ^*$.
\bigskip
\hspace{0pt} Notice that
\begin{equation*}
\begin{array}{rcl}
Q_\lambda(1)&=&P_\lambda(A_\lambda(1))+Q(A_\lambda(1))+1-h(\lambda)\\\\
&=&P_\lambda(h(\lambda))+Q(h(\lambda))+1-h(\lambda)\\\\
&=&1.
\end{array}
\end{equation*}
Moreover,
\begin{equation*}
\begin{array}{rcl}
|Q_\lambda'(1)|&=&|P'_\lambda(A_\lambda(1))+ Q'(A_\lambda(1))|\\\\
&=&|P'_\lambda(h(\lambda))+ Q'(h(\lambda))|\\\\
&=&p.
\end{array}
\end{equation*}
\hspace{0pt} Thus, just as to the polynomials $P_\lambda$, $z=1$ is a
repelling fixed point of $Q_\lambda$ for all $\lambda \in
\Lambda$.
\bigskip
\hspace{0pt} For the family $Q_\lambda$ we will obtain the following
theorem.
\bigskip
\begin{teo}{\hspace*{-6pt}\bf .}There is a dense set of parameter $\lambda \in \Lambda$ such that
the function $Q_{\lambda}$ has a wandering disc contained in the
filled Julia set, which is not attracted to an attracting
cycle.\label{teo que hago}
\end{teo}
\bigskip
\bigskip
\hspace{0pt} {\bf Proof of Theorem \ref{teo pa estrella}.}
\hspace{0pt} Recall that $Q_\lambda
(z)=A^{-1}_{\lambda}(Q^*_\lambda(A_\lambda(z)))$, i.e.
$Q_\lambda^*(z)= A_\lambda(Q_\lambda(A^{-1}_\lambda(z))).$
\hspace{0pt} If $D$ is a ball, then $A_{\lambda} ^{-1}(D)$ and
$A_\lambda(D)$ are balls, and, obtaining directly that
$K(Q_\lambda,B)=A_\lambda^{-1}(K(Q_\lambda^*,A_\lambda(B)))$, it
is sufficient to show that if $D$ is a wandering disk for
$Q_\lambda$, then $A^{-1}(D)$ is a wandering disk for
$Q_\lambda^*.$
\hspace{0pt} Suppose that $D$ is a wandering ball for $Q_\lambda$, i.e.
$Q_\lambda^m(D)\cap Q^n_\lambda(D) = \emptyset$ when $n\neq m$. It
follows that for $n\neq m$ we have that $A_\lambda^{-1}((Q_\lambda
*)^n(A_\lambda (D))) \cap A_\lambda^{-1}((Q_\lambda *)^m(A_\lambda
(D))) = \emptyset$, then $(Q_\lambda *)^n(A_\lambda
(D))\cap(Q_\lambda *)^m(A_\lambda (D))= \emptyset$. Therefore
$A_\lambda(D) $ is a wandering disk for $Q^*_\lambda$.\hfill$\Box$
\subsection{Properties of $\mathbf{Q_{\lambda}(}z\mathbf{)}$.}
\indent\indent The next proposition states a property of the
function $h$ that will be used several times.
\begin{prop}{\hspace*{-6pt}\bf .} If $\la{0},\la{1}
\in \Lambda$, then $|h(\la{0})-h(\la{1})|\leq \rho
|\la{0}-\la{1}|$.\label{res2}
\end{prop}
\hspace{0pt} \textbf{Proof:}
\hspace{0pt} Let $\lambda _{0}, \lambda _{1}\in \Lambda$. Since
$$Q_{\lambda_0}^*(h(\lambda_0))-Q_{\lambda_1}^*(h(\lambda_1))=h(\lambda_0)-h(\lambda_1)$$
we have that
$$|Q_{\lambda_0}^*(h(\lambda_0))-Q_{\lambda_1}^*(h(\lambda_1))|\leq
\frac{|Q(1)|}{p}$$ and from
$$|Q(h(\la{0})) -Q(h(\la{1}))|<\displaystyle \frac{\rho}{p} \
|h(\la{0})-h(\la{1})|,$$ it follows that
$$| P_{\lambda _{0}}(h(\la{0}))-P_{\la{1}}(h(\la{1}))|< \frac{\rho}{p} \
|h(\la{0})-h(\la{1})|.$$
In addition, from
$$| P_{\lambda _{0}}(h(\la{0}))-P_{\lambda _{0}}(h(\la{1}))|=p\ | h(\la{0})-h(\la{1}) |>
\frac{\rho}{p} \ |h(\la{0})-h(\la{1})|$$
$$| P_{\lambda
_{0}}(h(\la{1}))-P_{\lambda _{1}}(h(\la{1})) |=p\ | \lambda _{0}-
\lambda _{1} |\,
|h(\la{1}) -1| $$ and by isosceles triangle principle, necessarily we have that $$p\ | \lambda
_{0}- \lambda _{1} | \,| h(\lambda_{1}) -1| =p\ |
h(\la{0})-h(\la{1}) |.$$
Finally from $|h(\lambda_1)-1|\leq \frac{\rho}{p}$, we have that
$| h(\la{0})-h(\la{1})|\leq \rho \,|\la{0}-\la{1}|.$\hfill $\Box$
\bigskip
\newpage
\begin{lema}{\hspace*{-6pt}\bf .} Let $z\in \mathbb{C}_p.$
\begin{itemize}
\item[ i)] If $\rho < |z |<1$, then $|Q_{\lambda}(z) |=p\ | z|^{p}
> |z|.$
\item[ii)] If $|z|\leq \rho$, then $|Q_{\lambda}(z)|\leq \rho.$
\item[iii)] If $|z-1|<1$, then $|Q_{\lambda}(z)-1|=p\ | z-1|.$
\item[iv)] If $1<|z|<\widehat{r}$, then $|Q_{\lambda}(z)|=p\
|z|^{p+1}$.
\end{itemize}\label{res3}
\end{lema}
\hspace{0pt} \textbf{Proof:}
\hspace{0pt} From Proposition \ref{res2}, for every $\lambda \in \Lambda$ we
have that $|h(\lambda)-1|\leq \frac{\rho}{p}< \rho$.
\begin{itemize}
\item[{\em i)}] Since $\rho<|z |<1$, we have that
$|A_{\lambda}(z)| =|z|$. In addition, $|A_{\lambda}(z)|^{p+1}<
|A_{\lambda}(z)|^{p}$. Hence $|P_{\lambda}(A_{\lambda}(z))|=p\
|A_{\lambda}(z) |^{p}>p\rho^{p}=\rho$. Furthermore,
$|Q(A_{\lambda}(z))|< \rho$ and $|1-h(\lambda)|<\rho$, therefore
$$ |Q_{\lambda}(z) |=p\ | z |^{p}.$$
\item[{\em ii)}] Observe that $|A_{\lambda}(z)|\leq \rho$ since
$|z |\leq \rho $. It follows $$| P_{\lambda}(A_{\lambda}(z))|=p\
|A_{\lambda} (z)|^{p}\leq p\rho^{p}=\rho.$$ In addition, from
$|Q(A_{\lambda}(z))|< \rho$, $|1-h(\lambda)|<\rho$ and the strong
triangle inequality, we have that
$$|Q_{\lambda}(z)|\leq \rho.$$
\item[{\em iii)}]$$\begin{array}{rl}
|Q_{\lambda}(z)-1|&=|Q_{\lambda}(z)-Q_{\lambda}(1)|\\
&=|P_{\lambda}(A_{\lambda}(z))+ Q(A_{\lambda}(z))-
P_{\lambda}(A_{\lambda}(1))-Q(A_{\lambda}(1))|.\end{array}$$ Since
$|A_{\lambda}(z)-A_{\lambda}(1)|= |z-1|$, we have that
$|P_{\lambda}(A_{\lambda}(z))-
P_{\lambda}(A_{\lambda}(1))|=p\,|z-1|.$
Moreover,
$|Q_{\lambda}(A_{\lambda}(z))-Q_{\lambda}(A_{\lambda}(1))| \leq
\rho \,|z-1|$. Again, from the strong triangle inequality, we have
that
$$|Q_{\lambda}(z)-1|=p\ |z-1|.$$
\item[{\em iv)}] Since $|z|>1$, it follows that
$|P_{\lambda}(A_\lambda (z))|=p\ |z|^{p+1}$. Furthermore,
$|Q(A_\lambda (z))|< \rho$ and $|h(\lambda)-1|< \rho$, therefore
$$|Q_{\lambda}(z)|=p\ |z|^{p+1}.$$\hfill $\Box$
\end{itemize}
\hspace{0pt} Recall that $B$ is the closed ball defined by $\{z \in {\mathbb C}_p
:|z| \leq \widehat{r}\}$, where $\widehat{r}$ is an element of
$|{\mathbb C}_p^*|$ chosen in the beginning of this section.
\begin{prop}{\hspace*{-6pt}\bf .}
For each $\lambda \in \Lambda$, $(Q_{\lambda}, B)$ is an polynomial like map of degree $p+1$.\label{res4}
\end{prop}
\hspace{0pt} \textbf{Proof:}
\hspace{0pt} Let $\lambda \in \Lambda$. From the previous lemma we deduce
that $Q_{\lambda}(B)=\{z: |z|<p \ \widehat{r} ^{p+1}\}$, we will
prove that $(Q_{\lambda}, B)$ is of degree $p+1$.
\hspace{0pt} Since $|P_\lambda(A_\lambda(z))-P_\lambda(z)|=p\
|h(\lambda)-1|<\rho$ we conclude that
$$Q_\lambda(z)-P_\lambda(z)=
P_\lambda(A_\lambda(z))-P_\lambda(z)+Q(A_\lambda(z))-h(\lambda)+1,$$
using that $\|Q_\lambda -P_\lambda \|< \rho$, the power series of
$Q_\lambda$ is
$$Q_{\lambda}(z)= a_0+a_1z+\ldots+\left (a_p+\frac{\lambda}{p}\right )z^p
+\left(a_{p+1}+1-\frac{\lambda}{p}\right )z^{p+1}+a_{p+2}z^{p+2}\ldots$$
where $\underset{i\geq 0}{\sup}\{ |a_i|z^{i}\}<\rho$. From
Theorem \ref{teorema de soluciones contar} it is possible to count
the solutions of $Q_{\lambda}(z)-w_0=0$.
\hspace{0pt} If $p\leq|w_0|<pr^{p+1}$ and $f(z_0)=w_0$, then
$|z_0|=\left(\displaystyle \frac{|w_0|}{p}\right )^{p+1}$, by Lemma
\ref{res3}{(\em iv)}. Therefore $w_0$ has $p+1$ pre-images in $B$.
Hence $(Q_{\lambda},B)$ is a polynomial like map of degree
$p+1$.\hfill $\Box$
\begin{prop}{\hspace*{-6pt}\bf .}
$K(Q_{\lambda},B)\subset B_{1}(0)\sqcup B_{1}(1).$\label{res5}
\end{prop}
\hspace{0pt} \textbf{Proof:}
\hspace{0pt} Suppose that $z\notin B_1(0) \sqcup B_1(1)$.
\hspace{0pt} If $|z|>1$ then $|Q_{\lambda}(z)|= p|z|^{p+1}$. It follows that
there exists $n\in \mathbb{N}$, such that $Q_{\lambda}^{n}(z)
\notin B.$
\hspace{0pt} If $|z|=1$ and $|z-1|=1$, then
$|Q_{\lambda}(z)|=|P_{\lambda}(A_{\lambda}(z))+Q(A_{\lambda}(z))|=p$.
Hence $z\notin K(Q_{\lambda},B).$
\hspace{0pt} Therefore $K(Q_{\lambda},B)\subset B_{1}(0)\sqcup
B_{1}(1).$\hfill $\Box$
\bigskip
\hspace{0pt} This result allow us to define the itinerary of a point in
$K(Q_{\lambda},B)$. To simplify notation let $B_{0}=B_1(0)$ and
$B_1= B_1(1)$.
\hspace{0pt} For any $z \in K(Q_{\lambda},B)$, the itinerary of $z$ for
$Q_{\lambda}$ is defined by
$$\theta _{0}\theta _{1}\ldots \theta _{n}\ldots \in \{ 0,1\}^{{\mathbb N} \cup \{0\}} \text{\
where }Q_{\lambda}^{n}(z) \in B_{\theta _{n}} \text{\ for\ all \
}n \geq 0.$$
\begin{lema}{\hspace*{-6pt}\bf .}
Let $\lambda \in \Lambda $ and $D$ a ball contained in
$K(Q_{\lambda},B)$. Then:\label{res6}
\begin{itemize}
\item[1)]All points in $D$ have the same itinerary for
$Q_{\lambda}. $ \item[2)] If the common itinerary of points in
$D$ is not pre-periodic, under the one side shift, then $D$ is a
wandering disc which is not attracted to an attracting periodic
point.
\end{itemize}
\end{lema}
\newpage \noindent \textbf{Proof:}
\begin{itemize}
\item[{\em1)}] We proceed by contradiction. Assume that there
exist $z_0\text{\ and\ }z_1\in D$ with different itineraries. Then
there exists $n_0 \in \mathbb{N}$ such that
$Q_{\lambda}^{n_{0}}(z_0)\in B_{0}$ and $
Q_{\lambda}^{n_{0}}(z_1)\in B_{1}$. Since $Q_{\lambda}^{n_{0}}(D)$
is a ball which has non-trivial intersection with $B_{0}$ and
$B_{1}$ we have that $\{z:| z | \leq 1 \} \subset
Q_{\lambda}^{n_{0}}(D)\subset K(Q_{\lambda},B)$, obtaining a
contradiction with Proposition \ref{res5}.
\item[{\em2)}] Now, we suppose that $D$ is not a wandering disc, that is, there exist $n> m \geq 0$ such that $Q_{\lambda}^{n}(D) \cap Q_{\lambda}^{m}(D) \neq
\emptyset$. Hence $Q_{\lambda}^{m}(z_0)\in Q_{\lambda}^{n}(D)$ for some $z_0\in D $. Therefore $Q_{\lambda}^{k}(z_0)\in
Q_{\lambda}^{k+n-m}(D)$ for all $k$ in $ \mathbb{N}$.
\hspace{0pt} Since every point in $D$ has the same itinerary
we conclude that the itinerary of the points in $D$ is pre-periodic with eventual period $n-m$.
\hspace{0pt} We must show that $D$ is not attracted to a periodic orbit. We suppose that there is an attracting periodic point $z_0$ and $s>0$ such that $B_s(z_0)$ is contained in the attracting
basin of $z_0$, and $z_1\in D$ such that $z_1 \in
B_s(z_0)$, from the first part of the proposition we have that
every points of $B_s(z_0)$ have a common itinerary, and it is periodic.\hfill $\Box$
\end{itemize}
\hspace{0pt} From the previous lemma we conclude that in order to prove
Theorem \ref{teo que hago} it is sufficient to find a wandering
disc in the filled Julia set of $(Q_\lambda,B)$ whose itinerary is
not pre-periodic, for a dense subset in $\Lambda$. Therefore, we
need to study the behavior of the points in $K(Q_\lambda,B)$ such
that its orbit visits both $B_0$ and $B_1$.
\hspace{0pt} From Lemma \ref{res3} we know that the open ball $B_\rho(0)$ is
fixed under the action of $Q_\lambda$ and we have that a point
$x\in B$ has itinerary
$$\underset{n}{\underbrace{0\ldots0}}\,1\ldots$$ if and only if
$|x| =\rho_n$. Recall $\rho_0=1, \ p\rho^p_n=\rho_{n-1}$. This
crucial fact holds already for the family $P_\lambda$ \cite{RL2}.
\hspace{0pt} The following lemma describe the local behavior of $Q_\lambda$
in the set $\{z:|z|=\rho_n\}$ and in $B_1$. \vspace*{-70pt}
\begin{lema}{\hspace*{-6pt}\bf .} $ \begin{array}{cl} \\*[59pt]
{\it 1}. & \text{Let } m\geq 1,|z_{0}|=| z_{1}|=\rho _{m}.\text{
If } | z_{0}-z_{1}| \leq S\text{, then}\\*[10pt] & \hspace*{3cm}
|Q_{\lambda}(z_{0})-Q_{\lambda}(z_{1})| \leq \rho _{m-1}
|z_{0}-z_{1}|.\\*[10pt] {\it 2}. & \text{ If } z_{0},z_{1} \in
B_{1}(1) \text{, then:}\\*[10pt] & \hspace*{3.4 cm
}|Q_{\lambda}(z_{0}) -Q_{\lambda}(z_{1})| =p\ |z_{0}-z_{1} |.
\end{array}$
\label{diferg}
\end{lema}
\hspace{0pt} \textbf{Proof:}
\begin{itemize}
\item[{\em1)}]We observe that $|A_{\lambda}( z_{i})|=|z_{i}
|=\rho_{m}$, hence
\\
$|Q(A_{\lambda}(z_{0}))- Q(A_{\lambda}(z_{1})) |< \rho \ |
A_{\lambda}(z_{0})-A_{\lambda}(z_{1})|=\rho \ |z_0-z_1|$.
Letting $\epsilon = A_{\lambda}(z_1)-A_{\lambda}(z_0)$ we have
\begin{equation*}
\begin{array}{rcl}
P_{\lambda}(A_{\lambda}(z_{0}))-P_{\lambda}(A_{\lambda}(z_{1}))
=&\displaystyle \frac{\lambda}{p}\left (p\epsilon
A_{\lambda}(z_0)^{p-1}+\ldots
+p\epsilon^{p-1}A_{\lambda}(z_0)+\epsilon^p\right )\\\\
&+\left(1-\frac{\lambda}{p}\right )((p+1)\epsilon
A_{\lambda}(z_0)^p+\ldots +\epsilon ^{p+1}).
\end{array}
\end{equation*}
Moreover,
$|P_{\lambda}(A_{\lambda}(z_0))-P_{\lambda}(A_{\lambda}(z_1))|\leq
|\epsilon|\max \{A_{\lambda}(z_0)^{p-1},
p\,|\epsilon|^{p-1},\rho_{m-1}\}= |\epsilon|\ \rho_{m-1},$ since
$|\epsilon|= |z_0-z_1|\leq S< \rho_m=|A_{\lambda}(z_0)|.$
Therefore $$|Q_{\lambda}(z_0)-Q_{\lambda}(z_1)|\leq \rho_{m-1}\
|z_0-z_1|.$$
\item[{\em2)}] We note that if $y \in B_{1}$, then
$A_{\lambda}(y) \in B_{1}$. Now
\begin{equation*}
\begin{array}{rcl}
P_{\lambda}(A_{\lambda}(z_{0}))-P_{\lambda}(A_{\lambda}(z_{1}))
=&\displaystyle \frac{\lambda}{p}\left (p\,\epsilon
A_{\lambda}(z_0)^{p-1}+\ldots
+p\, \epsilon^{p-1}A_{\lambda}(z_0)+\epsilon^p\right )\\\\
&+\left(1-\frac{\lambda}{p}\right )((p+1)\,\epsilon \,
A_{\lambda}(z_0)^p+\ldots +\epsilon ^{p+1}),
\end{array}
\end{equation*}
where $\epsilon = z_0 -z_1 $, thus
$|P_{\lambda}(A_{\lambda}(z_{0}))-P_{\lambda}(A_{\lambda}(z_{1})|=p\
| z_{0}-z_{1}|$ since $|\epsilon|<1$. Furthermore we have that
$$|Q(A_{\lambda}(z_{0}))-Q(A_{\lambda}(z_{1})|\leq \rho \
|z_{0}-z_{1} | <p\ |z_{0}-z_{1} |.$$ Therefore
$$|Q_{\lambda}(z_{0})-Q_{\lambda}(z_{1})|= p\ |
z_{0}-z_{1}|.$$\hfill $\Box$
\end{itemize}
\bigskip
\hspace{0pt} The following lemma gives a sufficient condition for the
existence of a wandering disc in $K(Q_\lambda,B)$.
\begin{lema}{\hspace*{-6pt}\bf .}
Let $\{ m_{i}\} _{i\geq 0},\{ M_{i}\} _{i\geq 0}$ be sequences of
positive integers such that if $i\geq0$
$$\rho_{m_{i}-1}\cdot ...\cdot \rho_{1}\cdot p^{M_{i}}\leq1.$$
\hspace{0pt} If for $\lambda_{0} \in \Lambda $ there exists $z_0\in
K(Q_{\lambda_{0}})$ with itinerary
$$\underset{m_0}{\underbrace{0\ldots0}}\underset{M_0}{\underbrace{1\ldots1}}\underset{m_
1}{\underbrace{0\ldots0}}
\underset{M_1}{\underbrace{1\ldots1}}\ldots\underset{m_i}{\underbrace{0\ldots0}}\underset{M_i}{\underbrace{1\ldots1}}\ldots,$$
then the closed ball $D=\{|z-z_0|\leq S \}$ is contained in
$K(Q_{\lambda_{0}})$.\label{res8}
\hspace{0pt} If we add the hypothesis $ \lim M_i= \infty$, then by Lemma
\ref{res6} we have that $D$ is a wandering disc contained in
$K(Q_{\lambda},B)$ which is not attracted to an attracting cycle.
\end{lema}
\hspace{0pt} \textbf{Proof:}
\hspace{0pt} We now define the sequence $\{ N_{i}\}_{i\geq 0}$
recursively:
$$N_{0}=0$$
$$N_{i}=N_{i-1}+m_{i-1}+M_{i-1}.$$
\hspace{0pt} We will prove inductively that $\operatorname{diam} (Q_{\lambda
_{0}}^{N_{i}}(D))\leq S$. For $i=0$ the claim is true because the
definition of $D$. \hspace{0pt} Suppose that $\operatorname{diam} (Q_{\lambda
_{0}}^{N_{i}}(D))\leq S$, since $Q_{\lambda
_{0}}^{N_{i}+j}(z_0)\in B_{0}$ for all $ j$ in ${\mathbb N}$ \linebreak
with $0\leq j\leq m_i $ and $Q_{\lambda _{0}}^{N_i+m_{i}}(z_0) \in
B_{1}$, it follows that $|Q_{\lambda _{0}}^{N_{i}}(z_0)| =\rho
_{m_{i}}$. Therefore $|Q_{\lambda _{0}}^{N_{i}}(y)| =\rho
_{m_{i}}$ and $Q_{\lambda _{0}}^{N_i+m_{i}}(y) \in B_{1} $ for all
$ y \in D.$
\hspace{0pt} From the first statement of the previous lemma we have that
$$\operatorname{diam}(Q_{\lambda _{0}}^{N_{i}+m_{i}}(D))\leq \rho _{m_{i-1}}\cdot ...\cdot \rho _{1}
\operatorname{diam}(Q_{\lambda _{0}}^{N_{i}}(D))\leq p^{-M_{i}}S.$$ \hspace{0pt} Now,
using the second statement of the same lemma, we obtain
$$\operatorname{diam}(Q_{\lambda _{0}}^{N_{i+1}}(D))\leq p^{M_{i}}\operatorname{diam}(Q_{\lambda
_{0}}^{N_{i}+m_{i}}(D))\leq S.$$ \hspace{0pt} Therefore $D\subset
K(Q_{\lambda _{0}},B)$. \hfill $\Box$
\newpage
\subsection{Parameter selection.}
\indent\indent In this section we will prove results that describe
the behavior of the iterates $Q_{\lambda}^{n}(z) $, not just as a
function of $z$ but also as a function of $\lambda$.
\begin{lema}{\hspace*{-6pt}\bf .}
Let $\lambda_0,\lambda_1 \in \Lambda$. \label{res9}
\begin{itemize}
\item[1)] If $z\in B_0$ then
$|P_{\lambda_0}(z)-P_{\lambda_1}(z)|=p\
|z|^p|\lambda_0-\lambda_1|.$ \item[2)] If $z\in B_1$ then
$|P_{\lambda_0}(z)-P_{\lambda_1}(z)|=p\ |\lambda_0-\lambda_1|\
|z-1|.$
\end{itemize}
\end{lema}
\hspace{0pt} \textbf{Proof:}
\begin{itemize}
\item[{\em1)}]
\begin{align*}
&|P_{\lambda_0}(z)-P_{\lambda_1}(z)| =\left |z^p\left (\displaystyle
\frac{\lambda_0-\lambda_1}{p}\right )-z^{p+1}\left
(\displaystyle\frac{\lambda_0-\lambda_1}{p}\right )\right | =p\
|\lambda_0-\lambda_1|\ |z-1|.
\end{align*}
\item[{\em 2)}]
\begin{align*}
&|P_{\lambda_0}(z)-P_{\lambda_1}(z)| =\left|z^p\left (\displaystyle
\frac{\lambda_0-\lambda_1}{p}\right )-z^{p+1}\left (\displaystyle
\frac{\lambda_0-\lambda_1}{p}\right )\right | =p\ |x|^p\
|\lambda_0-\lambda_1|.
\end{align*}
\hfill $\Box$
\end{itemize}
\begin{lema}{\hspace*{-6pt}\bf .}
Let $M\in \mathbb{N}$ and $x_{0},x_{1}\in \{ x:|x-1 | \leq
p^{-M}\}$. If the parameters $\lambda _{0}, \lambda _{1}\in
\Lambda$\label{res10} are such that $| \lambda _{0}- \lambda _{1}|
= | x_{0}-x_{1}|$, then
$$| Q_{\lambda_{0}}^{M}(x_{0})-Q_{\lambda_{1}}^{M}(x_{1})| =p^{M} |\lambda _{0}- \lambda
_{1} |.$$
\end{lema}
\hspace{0pt} \textbf{Proof:} \setcounter{equation}{0} \ First we prove the lemma for
$M=1$.
\hspace{0pt} By the second part of Lemma \ref{diferg} we have
\begin{equation}| Q_{\lambda _{0}}(x_{0})-Q_{\lambda
_{0}}(x_{1}) |= p\ | x_{0}-x_{1}|=p \ |
\lambda_{0}-\lambda_{1}|.\label{c1}
\end{equation}
\hspace{0pt} Furthermore by Lemma \ref{comp de P} and Lemma \ref{res2} we
obtain
\begin{equation}| P_{\lambda _{0}}(A_{\lambda_0}(x_{1}))-P_{\lambda
_{0}}(A_{\lambda_1}(x_{1}))| =p\ | h(\lambda _{0})-h(\lambda
_{1})| < p \ |\lambda _{0}-\lambda_{1}|.\label{c2}
\end{equation}
\hspace{0pt} By Lemma \ref{res9}, we conclude that
\begin{equation}
| P_{\lambda _{0}}(A_{\lambda_1}(x_{1}))-P_{\lambda
_{1}}(A_{\lambda_1}(x_{1}))|=p\ |\lambda _{0}-\lambda_{1}|
\,\,|A_{\lambda_1}(x_{1})-1 |< p \ |\lambda
_{0}-\lambda_{1}|,\label{c3}
\end{equation}
\hspace{0pt} and from equations (\ref{c2}) and (\ref{c3}) \begin{equation} |
P_{\lambda _{0}}(A_{\lambda _{0}}(x_{1}))-P_{\lambda
_{1}}(A_{\lambda _{1}}(x_{1}))|<p\ |\lambda _{0}-\lambda _{1}|.
\label{c4}
\end{equation}
\hspace{0pt} Moreover,
\begin{equation}
|Q(A_{\lambda_0}(x_{1}))-
Q(A_{\lambda_1}(x_{1}))|< \rho \ | h(\lambda _{0})-h(\lambda _{1})
| <p\ |\lambda _{0}-\lambda _{1}|. \label{c5}\end{equation}
\hspace{0pt} From (\ref{c4}) and (\ref{c5}) we have that
\begin{equation}
| Q_{\lambda _{0}}(x_{1})-Q_{\lambda _{1}}(x_{1}) |< p \ |
\lambda_{0}-\lambda_{1}|. \label{c6}
\end{equation}
\hspace{0pt} Finally, from (\ref{c1}) and (\ref{c6}) we obtain that $|
Q_{\lambda_{0}}(x_{0})-Q_{\lambda _{1}}(x_{1}) |= p \ |
\lambda_{0}- \lambda_{1}|. $
\hspace{0pt} Now let us prove that the proposition is true for $M+1$. By the
inductive hypothesis and the third statement of Lemma \ref{res3}
we have that
$Q_{\lambda_{0}}^{M}(x_{0}),Q_{\lambda_{1}}^{M}(x_{1})$ belong to
$B_{1}$ and using Lemma \ref{diferg} with
$z_0=Q_{\lambda_0}^M(x_0) $ and $z_1=Q_{\lambda_1}^M(x_1)$ we
obtain that
\begin{equation}
|
Q_{\lambda_{1}}(Q_{\lambda_{0}}^{M}(x_{0}))-Q_{\lambda_{1}}(Q_{\lambda_{1}}^{M}(x_{1}))|=
p^{M+1}| \lambda _{0}- \lambda _{1}|.\label{c7}
\end{equation}
\hspace{0pt} Furthermore
\begin{equation}
|P_{\lambda_{0}}(A_{\lambda_0}(Q_{\lambda_{0}}^{M}(x_{0})))-P_{\lambda_{0}}(A_{\lambda_1
}(Q_{\lambda_{0}}^{M}(x_{0})))|=p\
\label{c8}|h(\lambda_{0})-h(\lambda_{1})| <p \
|\lambda_{0}-\lambda_{1} |,
\end{equation}
just as before, from the previous lemma
\begin{equation}
|P_{\lambda_{0}}(A_{\lambda_1}(Q_{\lambda_{0}}^{M}(x_{0})))-P_{\lambda_{1}}(A_{\lambda_1
}(Q_{\lambda_{0}}^{M}(x_{0})))|<p\
\label{c9}|\lambda_{0}-\lambda_{1}|
\end{equation}
\hspace{0pt} and
\begin{equation}
| Q(A_{\lambda_0}\label{c10}(Q_{\lambda_{0}}^{M}(x_{0})))-
Q(A_{\lambda_1}(Q_{\lambda_{0}}^{M}(x_{0}))) |< \rho \
|h(\lambda_{0})-h(\lambda_{1})|<p \ |\lambda_{0}-\lambda_{1}|.
\end{equation}
The strong triangle principle applied to (\ref{c8}), (\ref{c9})
and (\ref{c10}) gives us
\begin{equation}
|Q_{\lambda_0}(Q_{\lambda_0}^M(x_0))-Q_{\lambda_1}(Q_{\lambda_0}^M(x_0))|<p\
|\lambda_0-\lambda_1|, \label{c11}
\end{equation}
and from (\ref{c7}) and (\ref{c11}) we conclude that
$$| Q_{\lambda_{0}}^{M+1}(x_{0})-Q_{\lambda_{1}}^{M+1}(x_{1})|
=p^{M+1} |\lambda _{0}- \lambda _{1} |.$$ \hfill $\Box$
\begin{lema}{\hspace*{-6pt}\bf .}
Let $m \in \mathbb{N}$ and let $x_{0},x_{1}$ be such that
$|x_0|=|x_1|=\rho _{m}$ and $| x_{0}-x_{1}|\leq S$. If $\lambda
_{0}, \lambda _{1}\in \Lambda$ are such that $$\rho _{m-1} \cdot
\ldots \cdot \rho_1| x_{0}-x_{1}| < | \lambda _{0}- \lambda _{1}|
\leq S,$$ then
$$| Q_{\lambda_{0}}^{m}(x_{0})-Q_{\lambda_{1}}^{m}(x_{1})|= | \lambda _{0}- \lambda
_{1}|.$$\label{res11}
\end{lema}
\hspace{0pt} \textbf{Proof:}
We start by inductively proving that if $1\leq i\leq m $, then
$$| Q_{\lambda_{0}}^{i}(x_{0})- Q_{\lambda_{1}}^{i}(x_{1}) | \leq \max \{ \rho_{m-i}|
\lambda_{0}-\lambda_{1}|, \rho_{m-1} \cdot \ldots \cdot \rho _{m-i}| x_{0}-x_{1}| \}.$$
From the first part of Lemma \ref{diferg} we have
\begin{equation}| Q_{\lambda_{0}}(x_{0})-Q_{\lambda_{0}}(x_{1})|
\leq \rho_{m-1}| x_{0}-x_{1}|. \label{c12}
\end{equation}
Since $|A_{\lambda_0}( x_{1})|=|A_{\lambda_1}( x_{1})|= \rho
_{m}$, and $\rho _{m-1} \cdot \ldots \cdot \rho_{1}|
h(\lambda_{0})-h(\lambda_{1})| <| \lambda_{0}-\lambda_{1}| \leq
S$, we obtain
\begin{equation}
|P_{\lambda_0}(A_{\lambda_0}(x_1))-P_{\lambda_1}(A_{\lambda_1}(x_1))|
\leq \max \{\rho_{m-1}|h(\lambda_0)-h(\lambda_1)|,
\rho_{m-1}|\lambda_0-\lambda_1|\}, \label{c13}
\end{equation}
by the equation (\ref{eq6}) of Lemma \ref{lema en P inductivo}.
\hspace{0pt} Moreover
\begin{equation}
|Q(A_{\lambda_0}(x_{1}))-Q(A_{\lambda_1}(x_{1}))|<\rho \,|
h(\lambda_{0})-h(\lambda_{1})|<\rho\,|\lambda_0-\lambda_1|.
\label{c14}
\end{equation}
From inequalities (\ref{c13}) and (\ref{c14}), together with
Proposition \ref{res2} we have
\begin{equation}
\label{c16}| Q_{\lambda_{0}}(x_{1})-Q_{\lambda_{1}}(x_{1})|\leq
\rho_{m-1}|\lambda_0 -\lambda _1|.
\end{equation}
Now inequalities (\ref{c12}) and (\ref{c16}) give us
$$| Q_{\lambda_{0}}(x_{0})-Q_{\lambda_{1}}(x_{1})|\leq \max \{\rho_{m-1}\, |x_0-x_1|,
\rho_{m-1}\, |\lambda_0-\lambda_1|\}.$$
\hspace{0pt} Therefore which one is true for $i=1$.
\hspace{0pt} Now suppose $| Q_{\lambda_{0}}^{i}(x_{0})-
Q_{\lambda_{1}}^{i}(x_{1}) | \leq \max \{ \rho_{m-i}|
\lambda_{0}-\lambda_{1}|, \rho_{m-1} \cdot \ldots \cdot \rho
_{m-i}| x_{0}-x_{1}| \}$.
\hspace{0pt} Notice that
$|Q_{\lambda_{0}}^{i}(x_{0})|=|Q_{\lambda_{1}}^{i}(x_{1})|=\rho
_{m-i}$, therefore, using Lemma \ref{diferg} with
$z_0=Q_{\lambda_{0}}^{i}(x_{0})$ and
$z_1=Q_{\lambda_{1}}^{i}(x_{1})$, we obtain that
\begin{equation}
|
Q_{\lambda _{1}}(Q_{\lambda_{0}}^{i}(x_{0}))-Q_{\lambda
_{1}}(Q_{\lambda_{1}}^{i}(x_{1}))|\leq \rho _{m-(i+1)} |
Q_{\lambda_{0}}^{i}(x_{0})- Q_{\lambda_{1}}^{i}(x_{1})
|.\label{c17}
\end{equation}
\hspace{0pt} From Lemma \ref{res9} with
$x=A_{\lambda_0}(Q_{\lambda_{0}}^{i}(x_{0}))$ we have that
\begin{equation} |
P_{\lambda_{0}}(A_{\lambda_0}(Q_{\lambda_{0}}^{i}(x_{0})))-P_{\lambda_{1}}(A_{\lambda_0
(Q_{\lambda_{0}}^{i}(x_{0})) )|=p\,|
\lambda_{0}-\lambda_{1}|\,\rho_{m-i}^{p}= \rho_{m-(i+1)}|
\lambda_{0}-\lambda_{1}|.\label{c18}
\end{equation}
Using the first part of Lemma \ref{comp de P} we obtain that
\begin{equation*}
|P_{\lambda_{1}}(A_{\lambda_0}(Q_{\lambda_{0}}^{i}(x_{0})))-P_{\lambda_{1}}(A_{\lambda_1}
(Q_{\lambda_{1}}^{i}(x_{1})) )|\\ \leq \rho _{m-(i+1)}|
Q_{\lambda _{0}}^{i}(x_{0})-Q_{\lambda
_{1}}^{i}(x_{1})+h(\lambda_{0})-h(\lambda_{1})|
\end{equation*}
\begin{equation*}
|Q(A_{\lambda_0}(Q_{\lambda_0}^{i}(x_0)))-Q(A_{\lambda_1}(Q_{\lambda_0}^{i}(x_0)))|<\rho
\,|z_{\lambda_0}-z_{\lambda_1}|<\rho\,|\lambda_0-\lambda_1|.
\end{equation*}
\hspace{0pt} From the inequalities above and Proposition \ref{res2} we have
that
$$| Q_{\lambda_{0}}^{i+1}(x_{0})- Q_{\lambda_{1}}^{i+1}(x_{1})
| \leq \max \{ \rho_{m-(i+1)}| \lambda_{0}-\lambda_{1}|,\,
\rho_{m-1} \cdot \ldots \cdot \rho _{m-(i+1)}| x_{0}-x_{1}| \}.$$
\hspace{0pt} Notice that in the inductive step for $i=m-1$, we have that
$$|Q_{\lambda_{0}}^{m-1}(x_{0})- Q_{\lambda_{1}}^{m-1}(x_{1}) | \leq
\max \{ \rho_{1}| \lambda_{0}-\lambda_{1}|, \rho_{m-1} \cdot
\ldots \cdot \rho _{1}| x_{0}-x_{1}| \}.$$ From the above
inequality we obtain
$$|Q_{\lambda_{1}}(Q_{\lambda_{0}}^{m-1}(x_{0}))-Q_{\lambda
_{1}}(Q_{\lambda_{1}}^{m-1}(x_{1}))|\leq \rho _{1} |
Q_{\lambda_{0}}^{m-1}(x_{0})- Q_{\lambda_{1}}^{m-1}(x_{1}) |< |
\lambda_{0}-\lambda_{1}|$$ and from lemmas \ref{res9} and
\ref{comp de P} we have the following inequalities
\begin{equation*}
\begin{array}{rcl}
|
P_{\lambda_{1}}(A_{\lambda_0}(Q_{\lambda_{0}}^{m-1}(x_{0})))-P_{\lambda_{1}}(A_{\lambda_
1}(Q_{\lambda_{1}}^{m-1}(x_{1})) )|&\leq &| Q_{\lambda
_{0}}^{m-1}(x_{0})-Q_{\lambda
_{1}}^{m-1}(x_{1})+h(\lambda_{0})-h(\lambda_{1})|\\\\
&<& | \lambda_{0}-\lambda_ {1}|.
\end{array}
\end{equation*}
\begin{equation*}|
P_{\lambda_{0}}(A_{\lambda_0}(Q_{\lambda_{0}}^{m-1}(x_{0})))-P_{\lambda_{1}}(A_{\lambda_
0}(Q_{\lambda_{0}}^{m-1}(x_{0})) )|=p\,|
\lambda_{0}-\lambda_{1}|\,\rho_{1}^{p}= | \lambda_{0}-\lambda_{1}|
\end{equation*}
Moreover
\begin{equation*}
| Q(A_{\lambda_0}(Q_{\lambda_{0}}^{m-1}(x_{0})))-
Q(A_{\lambda_1}(Q_{\lambda_{0}}^{m-1}(x_{0})))|< \rho |
h(\lambda_{0})-h(\lambda_{1})|<| \lambda _{0}-\lambda_{1}|
\end{equation*}
and using the four previous inequalities and Proposition
\ref{res9} we have
$$| Q_{\lambda_{0}}^{m}(x_{0})-Q_{\lambda_{1}}^{m}(x_{1})|= |
\lambda _{0}- \lambda _{1}|. $$ \hfill$\Box$
\bigskip
\begin{prop}{\hspace*{-6pt}\bf .}
Let $\lambda \in \Lambda$ and consider $x\in K(Q_{\lambda},B)$
with itinerary $$\theta _{0}\,
\theta_{1}\ldots\theta_{n-1}\,1\,1\,1\ldots,$$ for $Q_\lambda$,
i.e. $Q^{n}_{\lambda}(x)=1$, for some $n\geq 1$.
\hspace{0pt} Suppose that there exists $\epsilon \in (0,1)$ such that for
all $\lambda_{0},\lambda_{1}$ in $\{\omega:| \omega -\lambda|\leq
\epsilon \}$, is true that
$|Q^{n}_{\lambda_{0}}(x)-Q^{n}_{\lambda_{1}}(x)|=|
\lambda_{0}-\lambda_{1}|.$
Let $M,m \in \mathbb{N}$ be such that $p^{-M}\leq \epsilon $ and
$$p^{M} \cdot \rho _{m-1}\cdot\ldots\cdot \rho_{1}< 1.$$
Then there exists $\lambda' \in \Lambda$ with $|\lambda -\lambda'|
\leq p^{-M}$ such that $x$ has itinerary
$$\theta _{0}\, \theta_{1}\ldots\theta_{n-1}\,\underset{M}{\underbrace{1\ldots
1}}\,\underset{m}{\underbrace{0\ldots 0}}\,1\,1\ldots\text{ for }
Q_{\lambda '}$$ and such that for all pairs of elements
$\lambda_{0},\lambda_{1} $ in $\{\omega:| \omega -\lambda' |\leq
Sp^{-M} \}$, we have that
$$ | Q_{\lambda_{0}}^{n+m+M}(x)-Q_{\lambda_{1}}^{n+m+M}(x)|=|
\lambda_{0}-\lambda_{1}|.$$\label{paso inductivo}
\end{prop}
\textbf{Proof:} \ Let $\phi: \Lambda\longrightarrow {\mathbb C}_p$ be the function
defined by $\phi(w)= Q_{w}^{n+M}(x)$. By Proposition \ref{res1} we
have that $\phi$ is holomorphic in $\Lambda$. Furthermore, by
hypothesis, if $\lambda_{0},\lambda_{1}\in B_{p^{-M}}(\lambda)$,
then $|Q_{\lambda_{0}}(x)-Q_{\lambda_{1}}(x)|=| \lambda_{0}-
\lambda_{1}|\leq p^{-M }.$
\hspace{0pt} Applying Lemma \ref{res10} to $z_0=Q_{\lambda_{0}}^n(x)$ and
$z_1=Q_{\lambda_{1}}^n(x)$, we have that
\begin{equation}| \phi(\lambda_{0}) -\phi(\lambda_{1}) |= p^{M}
|\lambda_{0}- \lambda_{1}|.\label{c20}\end{equation}
Then, by Corollary \ref{esfera}, we have that $\phi (\{ w:|
w-\lambda|= p^{-M}\})= \{ w:| w-1|= 1\}$. Therefore, there exists
$w_{0}\in \Lambda$ such that $\phi(w_{0})=0$.
\hspace{0pt} If $w\in \{z: |z-w_0|< p^{-M}\}$, the itinerary of $x$ for
$Q_w$ is
$$\theta _0 \, \theta_{1}\ldots\theta_{n-1}\,\underset{M}{\underbrace{1\ldots
1}}\,0\ldots $$ this is direct consequence of (\ref{c20}).
\hspace{0pt} Now let us consider the function $\psi:\Lambda\longrightarrow
{\mathbb C}_p$ defined by $\psi(w)=Q_{w}^{n+M+m}(x)$.
\hspace{0pt} Since $Q_\lambda$ leaves $B_\rho(0)$ fixed, we obtain that
$|\psi(w_0)|<\rho$. Now, by (\ref{c20}) we have that
$|\phi(w)|=\rho_m$ if $w$ is such that $|w-w_0|=\rho_m p^{-M}$,
hence $|\psi(w)|=1$. Using again Corollary \ref{esfera}, we
observe that
$$\psi(\{ w:| w-w_{0}|= p^{-M}\rho_{m}\})= \{ w:| w|= 1\}.$$
\hspace{0pt} Therefore, there exists $\lambda'$ such that
$\psi(\lambda')=1$.
\hspace{0pt} Thus, the itinerary of $x$ for $Q_{\lambda'}$ is
$$\theta _0 \, \theta_{1}\ldots\theta_{n-1}\,\underset{M}{\underbrace{1\ldots
1}}\underset{m}{\underbrace{0\ldots 0}}\,1\,1\ldots$$
\hspace{0pt} Notice that $|\phi(\lambda')|=\rho_m$. If $\lambda_0, \lambda_1
$ belong to $\{w:|w-\lambda'|\leq S\,p^{-M}\}$, then the points
$z_0=\phi(\lambda_0)$ y $z_1=\phi(\lambda_1)$ are such that
$|z_0|=|z_1|=\rho_m$ and $|z_0-z_1|=p^M |\lambda_0-\lambda_1|\leq
S$, by (\ref{c20}). Moreover, by hypothesis, we have
$$\rho_{m-1}\cdots \rho_1 |z_0-z_1|=\rho_{m-1}\cdots\rho_1 p^M
|\lambda_0-\lambda_1|< |\lambda_0-\lambda_1|.$$
\hspace{0pt} Now, applying Lemma \ref{res11} we obtain that
$$|Q_{\lambda_0}^{n+M+m}(x)-Q_{\lambda_1}^{n+M+m}(x)|=|\lambda_0-\lambda_1|.$$
\hfill$\Box$
\noindent {\bf Proof of Theorem \ref{teo que hago}.}
\hspace{0pt} We define the sequence $\{M_i\}_{i \in {\mathbb N}}$ recursively. Choose
$M_0 \in {\mathbb N}$ such that $p^{-M_0} \leq S$, and suppose that $M_i$
is already defined. Now choose $M_{i+1} \in {\mathbb N}$ satisfying
$p^{M_{i+1}-M_i}\leq S$.
\hspace{0pt} Furthermore, we define $\{m_i\}_{i\in {\mathbb N}}$ such that for each
$i \in {\mathbb N}$,
$$\rho _{m_i -1}\cdot\ldots\cdot\rho_1 \cdot p^{M_i}\leq 1.$$
\hspace{0pt} For an arbitrary $\lambda_0 \in \Lambda$ there exists $x \in
B_0$ such that $Q_{\lambda_0}^{m_0}(x)=1$, i.e. its itinerary for
$Q_{\lambda_0}$ is
$$\underset{m_0}{\underbrace{0\ldots0}}\,1\,1\,1\ldots$$
\hspace{0pt} By Lemma \ref{res11}, for $\lambda \in \Lambda$ with $|\lambda-
\lambda_0|\leq S$, we have
$$|Q_{\lambda}^{m_0}(x)-Q_{\lambda_0}^{m_0}(x)|=|\lambda-\lambda_0|.$$
Since $\rho_{m_{i+1}-1}\cdot\ldots\cdot\rho_1\cdot p^{M_{i+1}}\leq
1$, we have that
$$p^{M_i}\cdot\rho_{m_{i+1}-1}\cdot\ldots\cdot\rho_1\leq S <1.$$
\hspace{0pt} Therefore for $\lambda=\lambda_0,n=m_0, m=m_1$ and $\epsilon
=p^{-M_0}$ the hypothesis of Proposition \ref{paso inductivo}
hold. Hence we may consider $\lambda_1$ with
$|\lambda_0-\lambda_1|\leq p^{-M_0}$ and such that the itinerary
of $x$ for $Q_{\lambda_1}$ is
$$\underset{m_0}{\underbrace{0\ldots 0}}\, \underset{M_0}{\underbrace{1\ldots 1}}\,
\underset{m_1}{\underbrace{0\ldots 0}}\, 1\, 1\ldots.$$
\hspace{0pt} In view of the second part of Proposition \ref{paso inductivo},
for all pairs of elements $\omega_0, \omega_1$ in
$\{\omega:|\omega - \lambda_1|\leq Sp^{-M_0} \}$ we have that
$|Q_{\omega_0}^{m_0+M_0+m_1}(x)-Q_{\omega_1}^{m_0+M_0+m_1}(x)|=|\omega_0-\omega_1|$,
then we can use this proposition recursively. For the i-th step we
consider $n=n_0+M_0+\ldots+m_{i-1}+M_{i-1}+m_{i},\, \lambda =
\lambda_{i-1}$ y $\epsilon_i =Sp^{-M_{i-1}}$, obtaining
$\lambda_i \in \Lambda$ with $|\lambda_i-\lambda_{i-1}|\leq
Sp^{-M_i}$ and such that the itinerary of $x$ for $Q_{\lambda_i}$
is
$$\underset{m_0}{\underbrace{0\ldots0}}\,\underset{M_0}{\underbrace{1\ldots1}}\ldots
\underset{M_{i-1}}{\underbrace{1\ldots1}}\,
\underset{m_i}{\underbrace{0\ldots0}}\,1\,1\,1\ldots$$
\hspace{0pt} By definition $\underset{i\rightarrow \infty}{\lim}M_i=
\infty$, and since $|\lambda_{i+1}-\lambda_i|\leq Sp^{-M_i}$,
$\{\lambda_i\}_{i\in {\mathbb N}}$ is a Cauchy sequence. If we call its limit $\lambda$ we have that $|\lambda-\lambda_0|\leq
p^{-M_0}$ and the itinerary of $x$ for $Q_{\lambda}$ is
$$\underset{m_0}{\underbrace{0\ldots0}}\,\underset{M_0}{\underbrace{1\ldots1}}\ldots
\underset{M_{i-1}}{\underbrace{1\ldots1}}\,
\underset{m_i}{\underbrace{0\ldots0}}\,\underset{M_{i}}{\underbrace{1\ldots1}}\ldots.$$
Moreover the sequences $\{M_i\}_{i\in{\mathbb N}},\{m_i\}_{i\in{\mathbb N}}$ satisfy
the hypothesis of Lemma \ref{res8}, therefore $Q_\lambda$ has a
wandering disc contained in $K(Q_{\lambda},B)$, which is not
attracted to an attracting cycle.
\hspace{0pt} Finally, recall that $\lambda_0\in \Lambda$ and $M_0 \in {\mathbb N}$
were chosen arbitrarily and since $|\lambda-\lambda_0|\leq
p^{-M_0}$ we have that for a dense set of parameters $\lambda \in
\Lambda$ the function $Q_\lambda$ has a wandering disc in
$K(Q_\lambda, B)$ which is not attracted to an attracting
cycle.\hfill$\Box$
\newpage
|
{
"timestamp": "2005-03-30T23:37:03",
"yymm": "0503",
"arxiv_id": "math/0503720",
"language": "en",
"url": "https://arxiv.org/abs/math/0503720"
}
|
\section{Introduction}
\label{Sect:1}
It has long been recognised that the ultimate accuracy of optical measurements
is set by the quantum nature of light. Indeed the desire to approach these
quantum limits was a strong motivation for the study of nonclassical and
particularly squeezed states of light \cite{LoudonKnight}. The use of coherent
laser sources typically provides a limiting resolution that is inversely
proportional to the square root of the mean number of photons used in the
measurement ($N^{-1/2}$). This can be improved upon by the use of squeezed
states which enhances the resolution by the square root of the degree of
squeezing ($N^{-1/2}e^{-r}$). The full quantum limit is reached by complete
control of the photon number and gives a quantum limited resolution that it
inversely proportional to the photon number ($N^{-1}$) \cite{Giovannetti}.
One of the earliest proposals for the application of squeezed light was to
improve the sensitivity of optical interferometry \cite{Caves}, which was
demonstrated very soon after the first successful squeezing experiments
\cite{Xiao, Grangier}. This was followed by a demonstration of enhanced
sensitivity in a spectroscopic measurement \cite{Polzik}. More recently, it
has been suggested that squeezed light can be used to enhance the resolution
of measurements of small displacements in optical images, or beam
displacements \cite{Claude}. An experimental demonstration, based on
squeezed light prepared in a novel `flipped' mode, followed soon afterwards
\cite{treps2002}.
The quantum limit for detection of phase shifts can be approached using
a balanced interferometer with equal intensity inputs \cite{Holland}. It
has also been suggested that the same degree of resolution could be achieved
by means of special beam-splitters that send all of the light though one arm
of the interferometer so that a two-mode `Schr\"odinger-cat' state is prepared
\cite{Jacobson, Nobu}. The same $N^{-1}$-limited resolution can be obtained
for beam displacements by use of a pair of specially shaped modes, each
having precisely the same number of photons \cite{Barnett2003}.
In this paper we examine the factors limiting our ability to measure the rotation
of a beam about the optical axis. We will find that, as with interferometric and
beam-displacement measurements, the resolution depends on the number
of photons used and can be improved by the use of suitable nonclassical
states of light. The resolution also depends, however, on the orbital angular
momentum of the light used to make the observation \cite{allen92, OAMbook}.
We will find that it is the product of the orbital angular momentum per photon,
$\hbar\ell$, and the total photon number, $N$, that determines the limiting resolution.
Hence it is the total number of quanta of orbital angular momentum, $N\ell$, that
sets the minimum detectable rotation.
After some general considerations, Sect.~\ref{Sect:2}, we present two
different schemes to measure small rotations, Sect.~\ref{Sect:3} and
Sect.~\ref{Sect:4}. A comparison of the resolution achievable by different
measurements concludes the paper, Sect.~\ref{Sect:5}.
\section{General considerations}
\label{Sect:2}
Let us consider a light beam propagating through an {\em image rotator}, that
is a device that rotates an input image about the optical axis.
It is not necessary to specify the form of the rotator, but elementary examples
include a rotating Dove prism \cite{hecht}, or a pair of stationary
Dove prisms with a fixed relative orientation. The latter arrangement has recently
been used to detect optical angular momentum at the single-photon
level \cite{Leach}. A further example of a beam rotator is a light beam passing
{\em off-axis} through a rotating glass disc, which induces a tangential
displacement, or rotation, of the beam \cite{Jones}
\footnote{It has recently been suggested that the dual phenomenon, i.e. light
carrying orbital angular momentum exerting a torque on a transparent medium,
should also exist \cite{miles.rodney.steve}.}.
In this work we consider a beam with an image, or transverse spatial profile,
$u_I(x,y)$ propagating in the $z$ direction through an image rotator. The
beam after passing through the rotator has a transverse profile
\begin{eqnarray}\label{eq:rot}
u_O(x,y)=u_I(x\cos\delta\phi +y \sin\delta\phi,y\cos\delta\phi -x
\sin\delta\phi),
\end{eqnarray}
where $\delta \phi $ is the azimuthal rotation angle and we fix the $z$ axis
as the rotation axis. In Sect.~\ref{Sect:3} and \ref{Sect:4} we will consider
two different beams $u_I$.
It is natural to describe the beam $u_I$ as superposition of Laguerre-Gaussian
modes as these are eigenmodes of the $z-$component of angular momentum, which is
the generator of the rotation. This means that the only effect of a rotator on
these modes is to add a constant phase shift.
Laguerre-Gaussian modes, which at the beam waist have the form
\cite{siegman}:
\begin{equation}\label{eq:normLGmodes}
u_{p\ell}(r,\phi)=\frac{1}{w_0}\sqrt{\frac{p!}{\pi
(|\ell|+p)!}}\exp\left[-\frac{r^2}{2w_0^2} \right]
\left(\frac{r}{w_0}\right)^{|\ell |}L_p^{|\ell |}\left(\frac{r^2}{w_0^2}\right)
e^{i\ell \phi},
\end{equation}
are labelled
by an angular index, $\ell$, associated with the angular momentum carried by the
beam \cite{allen92}, and by a radial index, $p$, giving $p+1$ bright rings in the
intensity profile (Fig.~\ref{fig:2}). Modes with $p=0$ have a single intense ring
with radius \cite{Allen2000}
\begin{eqnarray}\label{eq:r.max}
\bar{r}=w_0\sqrt{|\ell|}.
\end{eqnarray}
Modes with non-vanishing $p$ have a less compact spatial distribution in the
transverse plane (see Fig.~\ref{fig:2}c-d).
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{GLM.p1.eps}
\end{center}
\caption{\label{fig:2}
Intensity $\left(\rho^{|l|}e^{-\frac{\rho^2}{2}}L^{|l|}_p(\rho^2)\right)^2$, with
radial coordinate normalized with the beam waist $\rho=r/w_0$. The dashed
circle, with radius $8$ represents the transverse extension of a rotator. Beams
with $p=0$ have the maximum intensity at $\rho=\sqrt{|l|}$. a) Intensity for
$l=49,~p=0$, showing a bright circle with radius $7$. b) For the mode
$l=64,~p=0$ the maximum intensity is at the boundary of the device. For
increasing value of $p$ we observe a spreading in the intensity, as shown in
c) $l=49,~p=1$ and d) $l=49,~p=2$.}
\end{figure}
Our study of rotation measurements starts with the realization that the optics used
will, inevitably, have a maximum distance from the optical axis beyond which light
will be lost by the experiment. For simplicity, we suppose that this limit is set
by the radius $R$ of the rotator. This, in turn, sets a maximum value for the
angular momentum that can be carried by a mode propagating through it
\cite{zambrini2004}. The Laguerre-Gaussian modes with non-zero $p$ extend to a
larger radius than those with the same value of $\ell$ but $p=0$ (see
Fig.~\ref{fig:2}). This means that the largest allowed angular momentum will occur
for a $p=0$ mode. For a mode with a bright ring of radius (\ref{eq:r.max}) at the
edges of the device ($\bar{r}=R$), the beam would be strongly diffracted.
The radial intensity distribution of the Laguerre-Gaussian modes,
for large values of $|\ell|$, has the form
\begin{eqnarray}\label{eq:LG.radially.Gauss}
|u_{0\ell} (\bar{r}+d)|^2\simeq |u_{0\ell} (\bar{r})|^2
e^{-d^2/w_0^2},
\end{eqnarray}
so that the intensity tends to be radially distributed like a Gaussian
centred in $\bar{r}$ and with a waist $w_0$.
Hence we can set the limit for a transmitted Laguerre-Gaussian mode for
\begin{eqnarray}
\bar{r}+w_0=R.
\end{eqnarray}
From Eq.~(\ref{eq:r.max}) we obtain the maximum angular momentum index
transmitted by a device with maximum effective radius $R$ as:
\begin{eqnarray}\label{eq:lm}
\ell_{M}=\left(\frac{R}{w_0}-1\right)^2.
\end{eqnarray}
We can use this result to suggest a probable limit for the smallest detectable
rotation $\delta\phi$. Consider the uncertainty relation for rotation angle and
angular momentum \cite{B+P}
\begin{eqnarray}\label{eq:uncert}
\Delta\phi\Delta L\geq\frac{\hbar}{2}|1-2\pi P(\pi)|,
\end{eqnarray}
where the values of $\phi$ are in the range $[-\pi,\pi]$.
The form of this uncertainty relation has recently been confirmed
experimentally \cite{sonja}, and states minimizing the uncertainty product
(\ref{eq:uncert}) have been derived \cite{pegg}.
For small angular uncertainties we have
\begin{eqnarray}\label{eq:uncert2}
\Delta\phi\geq\frac{\hbar}{2\Delta L},
\end{eqnarray}
which gives a bound on the minimum possible $\Delta\phi$:
\begin{eqnarray}\label{eq:uncert3}
\Delta\phi\geq\frac{1}{2 \ell_M}.
\end{eqnarray}
For the analogous problem of the optical phase \cite{P+B}
the minimum achievable uncertainty is inversely proportional to the mean (or
maximum) photon number ($N$) \cite{summy}. The minimum resolvable phase shift also
seems to be inversely proportional to $N$ \cite{Holland,Barnett2003}.
This suggests that the minimum resolvable rotation given a single photon will be
\begin{eqnarray}\label{eq:prec}
\delta\phi\propto\ell_M^{-1}.
\end{eqnarray}
We expect that the optimal use of $N$ photons will give a limit
\begin{eqnarray}\label{eq:prec2}
\delta\phi\propto(N\ell_M)^{-1}.
\end{eqnarray}
The analogy between the uncertainty, $\Delta\phi$, and the resolution,
$\delta\phi$, leads us to refer to (\ref{eq:prec2}) as the `Heisenberg' limit.
\section{Displacement scheme}
\label{Sect:3}
A natural way to measure small angles imparted by an image rotator is through
the displacement of a beam shining the rotator far from the axis, as in Jones
experiment \cite{Jones}. In this scheme the azimuthal displacement gives the
measure of the rotation angle, as shown in Fig.~\ref{fig:1}. Clearly the
resolution is increased by working at the edges of the device, that is at the
maximum distance from the device axis, and with a small size of the light
spot. In the following we consider a beam with a Gaussian transverse profile,
centred in $x=r_0,y=0$
\begin{eqnarray}\label{eq:Gauss.in}
u_I(x,y)=\frac{1}{\pi^{1/2}w_0}\exp\left[-\frac{(x-r_0)^2+y^2}{2w_0^2} \right],
\end{eqnarray}
with a small beam waist $w_0$ and large $r_0$, `near' to the edge of the
device. Clearly there are limits for the achievable experimental precision due
simply to the finite size of the optical elements used. Given a device with a
radial size $R$, than the off-axis Gaussian (\ref{eq:Gauss.in}) will be
transmitted if $r_0+w_0\sim R$.
The rotated output beam obtained by
Eqs.~(\ref{eq:rot}) and (\ref{eq:Gauss.in}) is
\begin{eqnarray}\label{eq:Gauss.out}
u_O(x,y)=\frac{1}{\pi^{1/2}w_0}
\exp\left[-\frac{(x-r_0\cos\delta\phi)^2+(y-r_0\sin\delta\phi)^2}{2w_0^2} \right].
\end{eqnarray}
The effect of the rotation is to displace the output beam by $\Delta
x=\left[r_0^2(\cos\delta\phi-1)^2+r_0^2\sin^2\delta\phi\right]^{1/2} $.
For small $\delta\phi$ we find
\begin{eqnarray}\label{eq:displ}
\delta\phi=\frac{\Delta x}{r_0},
\end{eqnarray}
so that the resolution achieved measuring small angles in this
scheme depends on the lateral beam position $r_0$ and on the
precise measurement of the displacement $\Delta x$ between the input and the
rotated light spots.
\begin{figure}
\begin{center}
\end{center}
\caption{\label{fig:1} Scheme based on displacement measurement (picture NA).}
\end{figure}
Small displacements $\Delta x$ are measured with high resolution by shining a
split detector and taking the difference of the light intensities on the two
halves \cite{Claude}. For a perfectly aligned beam the signal detected is
zero, while any small misalignment gives an imbalance in the intensities.
Given a Gaussian mode in a coherent state with mean photon number equal to
$N$, the minimum displacement measurable is
\begin{eqnarray}\label{eq:1coh}
\Delta x=\frac{\sqrt{\pi}w_0}{2}\frac{1}{ \sqrt{N}}.
\end{eqnarray}
The standard quantum limit (\ref{eq:1coh}) can be beaten by engineering the spatial
mode impinging on the detector and its statistics. In particular the input beam is
prepared by superposing an even Gaussian mode (\ref{eq:Gauss.in}) with an odd $
flipped$ mode $u_I^{odd}(x,y)=u_I(x,y) {\rm sign}(y)$. We note that a flipped
mode is not stable under propagation as it has a discontinuity in $y=0$ that would
be smoothed by diffraction. Nevertheless, it was experimentally possible to beat the
shot noise limit in displacement measurements by shaping this kind of beam
\cite{treps2002}.
In general we have \cite{Barnett2003}
\begin{eqnarray}\label{eq:1numb}
\Delta x=\frac{\sqrt{\pi}w_0}{2}f(N)
\end{eqnarray}
with $f(N)$ depending on the state in which the modes $u_I^{odd}$ and $u_I$ are
prepared. If the Gaussian mode is in a coherent state with average intensity
$N$ and the flipped mode is in vacuum then $f(N)=N^{-1/2}$, as in
Eq.~(\ref{eq:1coh}). This is the limit resolution obtained with classical
states, i.e. the standard quantum limit. Better resolution can be achieved if
the flipped mode is prepared in a strongly squeezed state, leading to
$f(N)=N^{-3/4}$. The best resolution is obtained with highly non-classical
states, for instance by preparing the two modes in number states $|N/2\rangle$. In this
case $f\sim N^{-1}$ and the displacement $\Delta x \sim N^{-1}$ is the
`Heisenberg limit' mentioned in the previous Section.
From these results for displacement measurements we obtain the
maximum angle resolution of the scheme in Fig.~\ref{fig:1}:
\begin{eqnarray}
\label{eq:dphi} \delta\phi=\frac{ \sqrt{\pi}w_0}{2r_0}f(N).
\end{eqnarray}
Clearly, $\delta\phi$ depends both on the spatial characteristics of the mode
($w_0$ and lateral displacement $r_0$) and also on the state
of light (through $f(N)$). A
decomposition of (\ref{eq:Gauss.in}) in angular momentum eigenmodes allows us to
write $\delta\phi$ in terms of the angular momentum index $\ell$. In
particular for a Gaussian spot centred far from the axis $z$ ($r_0\gg w_0$)
there is a large dispersion in the angular momentum spectrum.
We can see this either by writing $u_I(x,y)$ in terms of its angular Fourier
components \cite{vasnetsov}
\begin{eqnarray}
u_I(x,y)=\frac{1}{\pi^{1/2}w_0}\exp\left(-\frac{x^2+y^2+r_0^2}{2w_0^2}\right)
\sum_{\ell=-\infty}^{\ell=+\infty}I_{|\ell|}
\left(\frac{r_0\sqrt{x^2+y^2} }{w_0^2}\right)e^{i\ell\phi}
\end{eqnarray}
or by explicitly constructing its decomposition in
terms of the Laguerre-Gaussian modes
(see Fig.~\ref{fig:histo-gauss}b). The latter procedure is carried out in the
Appendix.
Due to the dispersion in the angular momentum spectrum,
it is important to consider the constraint, imposed by the extension $R$
of the rotator, found in Sect.~\ref{Sect:2}. From Eq.~(\ref{eq:lm}) and setting
$r_0+w_0= R$ we find the maximum resolution in the displacement scheme
\begin{eqnarray}\label{eq:dphi2}
\delta\phi=\frac{\sqrt{\pi}}{2}\frac{1}{\sqrt{\ell_{M}}}f(N).
\end{eqnarray}
In Eq.~(\ref{eq:dphi2}) we immediately identify a `geometrical' factor
depending on the angular momentum index and the statistical factor $f$. In
analogy with the standard quantum limit, obtained by using Gaussian coherent
states in interferometry, we consider the dependence $\sim{1}/{\sqrt{\ell}}$
in Eq.~(\ref{eq:dphi2}) as the standard $optical$ limit for rotation
measurements, as it is obtained with Gaussian spatial distributions. A spatial
Gaussian mode prepared in a Gaussian coherent state then gives a combined
`standard quantum limit' in which the minimum resolvable rotation, $\propto
(N\ell_M)^{-1/2}$, is the inverse of the root square of the number of quanta of
angular momentum. For $r_0 \gg w_0$, the Gaussian mode becomes a good
approximation to an angle-angular momentum minimum uncertainty product state
\cite{pegg} with $<\ell>=0$, $\Delta \ell=r_0/\sqrt{2}w_0=\sqrt{\ell_M/2}$ and
$\Delta \phi =1/\sqrt{2\ell_M}$. In Fig.~\ref{fig:histo-gauss} the $P(\ell)$
are plotted for $r_0 =3 w_0$ and $r_0 =10 w_0$ and are compared with Gaussians
having the same variance. The approach to a Gaussian form is an indication of
reaching the minimum uncertainty product limit \cite{pegg}.
\begin{figure}
\begin{center}
\includegraphics[width=5.5cm]{Pl.3w0.eps}
\includegraphics[width=5.5cm]{Pl.10w0.eps}
\end{center}
\caption{\label{fig:histo-gauss} The histograms show the
probabilities $P(\ell)$ given in Eq.~(\ref{eq:Pofell}). The symbols (a) and
smooth line (b) are Gaussians with width
given by the variance $\Delta \ell = r_0/\sqrt{2}w_0$.
a) $r_0 =3 w_0$. b) $r_0 =10 w_0$.}
\end{figure}
\section{Interferometric scheme}
\label{Sect:4}
If the incoming beam is an angular momentum eigenstate then the only effect of the
rotator is to add a constant phase shift.
Interferometers form the basis of phase shift measurements \cite{loudon} and
so it is natural to consider the interferometer shown in Fig.~\ref{fig:3} to measure
rotations. The rotator is placed along one of the paths inside the
interferometer. Here the shift is in the azimuthal spatial profile of the field
and this contrasts with well-known interferometers \cite{interf} designed to
measure shift in the longitudinal phase of the light beam.
\begin{figure}
\begin{center}
\end{center}
\caption{\label{fig:3} Interferometric phase measurement using
angular momentum eigenstates.
The single mode annihilation operators are
$\hat a=\int d\vec x v_I(\vec x)\hat a(\vec x)$,
$\hat b=\int d\vec x v_I(\vec x)\hat b(\vec x)$,
where $\hat a(\vec x)$ and $\hat b(\vec x)$ are continuum
annihilation operators \cite{PRA97}. (picture NA)}
\end{figure}
Given any mode of the form
\begin{eqnarray}\label{eq:AMeigen.in}
v_I(x,y)=v(r)\exp(i\ell\phi)
\end{eqnarray}
entering in the rotator, the beam at the output will be
\begin{eqnarray}\label{eq:AMeigen.out}
v_O(x,y)=v_I(x,y)\exp(i\ell\delta\phi).
\end{eqnarray}
We note that the interferometer considered here has recently been used to detect
the angular momentum of single photons \cite{Leach}. In the context of
rotation resolution, we are interested in the smallest angles
$\delta\phi$ that can be measured with this device.
The rotation through an angle $\delta\phi$ on the beam (\ref{eq:AMeigen.in})
introduces only a homogeneous phase shift $\ell\delta\phi$ on the whole beam,
and so it follows that the description of the interferometer in
Fig.~\ref{fig:3} -- illuminated by angular momentum eigenmodes -- is
completely equivalent to standard interferometers \cite{interf} measuring
longitudinal phase shifts. We note that to have interference the input modes $
a$ and $ b$ need to have the same angular momentum index ($\ell$).
The difference in the intensities of the two beams emerging from the
interferometer depends both on the phase shift, here $\ell\delta\phi$, and on
the quantum state of the incoming beams. In particular, when the noise level
has the size of the signal we are at the limit of the smallest detectable
phase shift
\begin{eqnarray}\label{eq:dphi4}
\delta\phi=\frac{1}{\ell}f(N),
\end{eqnarray}
with $f(N)= N^{-1/2}, N^{-3/4}, N^{-1}$ depending on the input states of the
modes $ a$ and $b$. We have seen in Sect.~\ref{Sect:2} how the
transverse size of the device sets the limit of the maximum value of $\ell$
of the beam that can be transmitted. By using the maximum allowed angular
momentum we reach the limiting angle resolution $\propto{1}/{\ell_{M}}$.
It is particularly interesting to consider
the case in which the beams entering in the
interferometer are prepared in the states
$|N/2\rangle|N/2\rangle$ \cite{Holland,Barnett2003}. The angle resolution is then
\begin{eqnarray} \label{eq:limit}
\delta\phi =2.24 \frac{1}{\ell_M N},
\end{eqnarray}
which is the `Heisenberg limit' anticipated in Section \ref{Sect:2}.
\section{Conclusions}
\label{Sect:5}
The resolution attainable in an optical measurement of rotations, $\delta\phi$,
depends on two factors, the number of photons and the orbital angular momentum
content of the beam. For a displaced Gaussian spot we find,
for a single photon, that
$\delta\phi\propto\ell_{M}^{-1/2}$ where $\ell_M$ is the largest angular
momentum index supportable by the image rotator. If the measurement is
performed by using a coherent state with mean photon number $N$ than we find
that $\delta\phi\propto(N\ell_M)^{-1/2} $, i.e., that it is inversely
proportional to the square root of the number of quanta of angular momentum.
Use of nonclassical states of light can enhance the sensitivity by changing the
functional dependence on $N$. In particular, use of correlated number states
can produce a resolution that is proportional to $N^{-1}$. We can also increase
the sensitivity by changing the functional dependence on $\ell_M$. Using
eigenmodes of orbital angular momentum leads to a resolution proportional to
$\ell_M^{-1}$, with the ultimate `Heisenberg' limit being
$\propto(N\ell_M)^{-1} $.
We have demonstrated a clear analogy between orbital angular momentum
in rotation measurements and photon number in interferometry.
There are, however, very important practical differences.
Creating states of well defined orbital angular momentum is relatively
straightforward, while making photon number states is very difficult.
Secondly, enhancement of resolution based on controlling the photon number
requires extremely high efficiencies of photon detection as any losses rapidly
degrade the signal by changing the expected photon number.
Using eigenmodes of orbital angular momentum, however, is relatively robust as no
matter how many photon are lost, each of the remaining photons still carries
$\ell\hbar$ units of angular momentum.
\section{Acknowledgements}
This work was supported by the Engineering and Physical Sciences Research
Council (GR/S03898/01).
|
{
"timestamp": "2005-03-30T02:50:23",
"yymm": "0503",
"arxiv_id": "quant-ph/0503224",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503224"
}
|
\section{Introduction}
Low-energy ${\bar K}N$ and ${\bar K}A$
interactions have gained substantial interest during the last
two decades. It is known from the time-honored Martin analysis~\cite{Martin}
that the isoscalar $s$-wave $K^-N$ scattering length is large and repulsive,
Re$a_0{=}{-}1.7$~fm, while
the isovector length is moderately attractive, Re$a_1{=}0.37$~fm.
It is clear that such a strong repulsion in the
${\bar K}N$ isoscalar channel leads
also to a repulsion in the low-energy $K^-p$ system, since
Re$a_{K^- p}{=}0.5\mathrm{Re}(a_0{+}a_1)$=$-$0.74~fm.
It should be noted that Conboy's analysis~\cite{Conboy} of
low energy ${\bar K}N$ data gives a solution with Re$a_0$=$-$1.03~fm
and Re$a_1$=0.94~fm, that also results in repulsion in the
$K^-p$ channel, but with substantially smaller strength,
Re$a_{K^- p}$=$-$0.05 fm. Data from KEK
show that the energy shift of the 1$s$
level of kaonic hydrogen is repulsive~\cite{Ito}.
Very recent results for kaonic hydrogen from the DEAR experiment
\cite{Guaraldo1} also indicate a repulsive energy shift.
However, the consistency of the bound state with the scattering
data can be questioned, as first pointed out in Ref.~\cite{MRR}.
Nevertheless, it is possible that the actual $K^-p$ interaction is
attractive if the isoscalar $\Lambda(1405)$ resonance is a bound state
of ${\bar K}N$ system~\cite{Dalitz,Weise1}. A fundamental reason
for such a scenario is provided by the leading order term in the chiral
expansion for the $K^-N$ amplitude which is attractive.
New developments in the analysis of the ${\bar K}N$ interaction based on
chiral Lagrangians can be found in Refs.\cite{Weise,Oset,Oller,LuKo}.
These results provide further support for the description
of the $\Lambda$(1405) as a meson-baryon bound state. More recently,
it has even been argued that there are indeed two poles in the complex
plane in the vicinity of the nominal $\Lambda$(1405) pole~\cite{Jido}.
For recent evidence to support this scenario, see e.g.~\cite{OsetJ}.
A different view seems to be taken in Ref.\cite{BNW}.
Such a non-trivial dynamics of the ${\bar K}N$ interaction leads to
very interesting in-medium phenomena in interactions of anti-kaons
with finite nuclei as well as with dense nuclear matter,
including neutron stars, see e.g.
Refs.~\cite{Sibirtsev1,Lutz,Sibirtsev2,Ramos,Heiselberg,Cieply}.
Recently, exotic few-body nuclear systems involving the $\bar K$-meson as
a constituent were studied by Akaishi and Yamazaki~\cite{Akaishi}.
They proposed a phenomenological ${\bar K}N$ potential model, which
reproduces the $K^-p$ and $K^-n$ scattering lengths from the Martin
analysis~\cite{Martin}, the kaonic hydrogen atom data from
KEK~\cite{Ito,Iwasaki} and the mass and width of the
$\Lambda$(1405) resonance. The ${\bar K}N$ interaction in this model
is characterized by a strong $I{=}0$ attraction, which allows the
few-body systems to form dense nuclear objects. As a result, the nuclear
ground states of a $K^-$ in $(pp)$, $^3$He, $^4$He and $^8$Be were
predicted to be discrete states with binding energies of 48, 108, 86
and 113 MeV and widths of 61, 20, 34 and 38 MeV, respectively.
More recent work on this subject can be found e.g.
in Refs.~\cite{Dote1,Dote2}.
Furthermore, very recently a strange tribaryon $S^0(3115)$ was detected
in the interaction of stopped $K^-$-mesons with
$^4$He~\cite{Suzuki}. Its width was found to be less
than 21 MeV. In principle, this state may be interpreted
as a candidate of a deeply bound state $({\bar K}NNN)^{Z{=}0}$ with
$I{=}1,I_3{=}{-}1$. However, the observed tribaryon $S^0(3115)$ is
about 100 MeV
lighter than the predicted mass. Moreover, in the experiment
an isospin~1 state was detected at a position where no peak was predicted.
Further searches for bound kaonic nuclear states as well as new
data on the interactions of $\bar K$-mesons with lightest nuclei are thus
of great importance.
Up to now the $s$-wave $K^-\alpha$ scattering
length, which we denote as $A(K^- \alpha)$, has not been measured and
relevant theoretical calculations have not yet
been done. In this paper we present a first calculation
of $A(K^-\alpha)$ within the framework of the multiple scattering
approach (MSA).
We investigate the pole position of the $K^-\alpha$ scattering amplitude
within the zero range approximation (ZRA) in order to find out whether
the formation of a bound state in $\bar K \alpha$ system is possible.
Furthermore, we discuss the possibility to measure the ${\bar K}\alpha$
scattering length through the ${\bar K}\alpha$ final state
interaction (FSI). Recently it was proposed to measure the reaction
$dd{\to}\alpha{K^+ K^-}$ near the threshold at
COSY-J\"ulich~\cite{Buescher02}.
We apply our approach to calculate the $K^- \alpha$
FSI effect in this reaction and demonstrate that the
$K^-\alpha$ invariant mass distribution is sensitive enough to
the $K^-\alpha$ FSI and may be used for a
determination of the $s$-wave $K^-\alpha$ scattering length.
Our paper is organized as follows: In Sect.~2 we calculate
the $K^-\alpha $ scattering length within the MSA and determine the
pole position of the amplitude in the zero range approximation.
In Sect.~3 an analysis of the FSI
in the reaction $dd{\to}\alpha{K^+ K^-}$ is considered.
Our conclusions are given in Sect.~4.
\section{The {\boldmath $K^-\alpha$} scattering length}
\subsection{Multiple scattering formalism}
To calculate the $s$-wave $K^-\alpha$ scattering length as well as
the FSI enhancement factor, we use the Foldy--Brueckner adiabatic approach
based on the multiple scattering (MS) formalism~\cite{Goldberger}.
Note that this method has already been used for the calculation of the
enhancement factor in the reactions
$pd {\to} ^3$He$\eta$ \cite{Faldt2}, $pn {\to} d\eta$ \cite{Grishina1} and
$pp {\to} d\bar{K^0}K^+$ \cite{Grishina_a0_04}.
In the Foldy--Brueckner adiabatic approach, the continuum
$K^-\alpha$ wave function, which is defined at fixed coordinates
of the four nucleons in $^4$He, can be written as the sum of the incident
plane wave of the kaon and waves emerging from the four fixed
scattering centers. Keeping only the $s$-wave contribution, we can
express the total wave function $\Psi_k$ through the $j$-channel wave
functions ${\psi}_j({\mathbf r}_j)$ in the following way
\begin{eqnarray}
\Psi_k({\mathbf r}_{K^{-}}{;}{\mathbf r}_1,{\mathbf r}_2,
{\mathbf r}_3,{\mathbf r}_4){=}
{\mathrm e}^{i{\mathbf k}{\cdot}
\mathbf {r}_{K^-}}\!\!
{+} \!\!\sum_{j=1}^{4} \!t_{K^-N_j} \frac{{\mathbf e}^{i k R_j}}{R_j}
\ {\psi}_j({\mathbf r}_j), \label{fixcenter4}
\end{eqnarray}
where $R_j{=}\left|\mathbf{r}_{K^-}{-}\mathbf{r}_j\right|$ and the
t-matrix, $t_{K^-N_j}$, is related to the elastic scattering amplitude
$f_{K^- N}$ via~\cite{Grishina1,Grishina_a0_04}
\begin{equation}
t_{K^- N}(k_{K^- N}) =
(1+\frac{m_{K^-}}{m})\, f_{K^- N}(k_{K^- N}),
\end{equation}
with $m \,(m_{K^-})$ the nucleon (charged kaon) mass,
and $k_{\bar{K}N}$ is the modulus of the relative $\bar{K}N$ momentum.
Note that we use the unitarized scattering length approximation
for the latter, {\it i.e.}
\begin{equation}
f^{I}_{\bar{K} N}(k_{\bar{K} N})=
\left[(a^{I}_{\bar{K} N})^{-1} - ik_{\bar{K} N}\right]^{-1},
\end{equation}
where $I$ is the isospin of the $\bar{K}N$ system. For each
scattering center $j$ an effective wave ${\psi}_j(\mathbf {r}_j)$
is defined as the sum of the incident plane wave and the waves
scattered from the three other centers
\begin{eqnarray}
{\psi}_j({\mathbf r}_j)= {\mathrm e}^{i{\mathbf k} \cdot
{\mathbf r}_j}+ \sum_{l \neq j} t_{K^- N_l} \frac{{\mathbf e}^{i k
R_{jl}}}{R_{jl}} \ {\psi}_l({\mathbf r}_l)\ ,
\label{fixcenterwave}
\end{eqnarray}
where $R_{jl}{=}\left |\mathbf{r}_l{-}\mathbf{r}_j \right|$.
Therefore, the channel wave functions ${\psi}_j({\mathbf r}_j)$ can
be found by solving the system of the four linear
equations~(\ref{fixcenterwave}).
To obtain the FSI factor we calculate the total wave
function $\Psi_k$ given by Eq.~(\ref{fixcenter4}) at
$\mathbf{r}_{K^-}{=}\sum_{j=1}^4\mathbf{r}_j{=}0$ and average
it over the coordinates of the nucleons ${\mathbf{r}}_j$ in $^4$He.
Thus the FSI enhancement factor is~\cite{Goldberger}
\begin{eqnarray}
\lambda^{\mathrm{MS}}(k_{K^- \alpha}) {=}
\left| \left\langle \Psi_{q_{K^-}^{\mathrm{lab}}}
(\mathbf{r}_{K^{-}}\!{=}\!\!\sum_{j{=}1}^4\!
\mathbf{r}_j{=}0;\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,
\mathbf{r}_4)\right\rangle \right|^2\!\!.
\label{enhancement}
\end{eqnarray}
For the nuclear density function we use the factorized form
\begin{eqnarray}
&&\left|\mathrm
{\Phi}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,\mathbf{r}_4)
\right|^2= \prod_{j=1}^4 \rho_{j}({\mathbf {r}}_j) \ , \label{he4}
\end{eqnarray}
where the single nucleon density is taken in Gaussian form as
\begin{eqnarray}
\rho (\mathbf{r})= \frac{1}{(\pi \, R^2)^{3/2}} \
\mathrm{e}^{-r^2/R^2} \ ,
\label{radiushe4}
\end{eqnarray}
with $R^2/4{=}0.62$~fm$^{2}$. Note that the independent
particle model formulated by Eqs.~(\ref{he4}-\ref{radiushe4})
provides a rather good description of the $^4$He electromagnetic
form factor up to momentum transfer
$\mathbf{q}^2{=}8$~fm${}^{-2}$~\cite{Boitsov}.
The integration in Eq.~(\ref{enhancement}) over the
nucleon coordinates~$\mathbf{r}_j$ was performed
using the Monte-Carlo method. This approach provides us with a
possibility to include all configurations of the nucleons
in ${}^4 \mathrm{He}$. Within this method we can also take into
account in Eq.~(\ref{fixcenter4}) the dependence of the $t_{K^- N_j}$
amplitude on the type of nucleonic scatterer, {\it i.e.} proton or neutron.
Note that the simple version of the multiple scattering approach
used in Ref.~\cite{Wycech} can be applied only to the case of
identical scatterers.
The s-wave $K^-\alpha$ scattering length can be derived from the
asymptotic expansion of Eq.~(\ref{fixcenter4}) at
$r_{K^-}{\to}\infty$ and it is
\begin{eqnarray}
A(K^- \alpha)=\frac{m_{\alpha}}{m_{\alpha}+m_{K^{-}}}
\left.\left\langle\sum_{j=1}^{4} t_{K^- N} \
{\psi}_j({\mathbf r}_j) \label{akhe4}\right\rangle
\right|_{\sum_{j=1}^4
\mathbf{r}_j =0}~,
\end{eqnarray}
with $m_\alpha$ the $\alpha$-particle mass.
Here the procedure of averaging over the coordinates of the nucleons
is similar to Eq.~(\ref{enhancement}).
\subsection{S-wave scattering length and the pole position
of the amplitude in the zero range approximation}
The basic uncertainties of the MSA calculations are given by the
next-to-leading order model corrections such as recoil corrections,
contributions from inelastic double and triple scattering terms,
{\it etc.} and due to the uncertainties of the elementary $I$=0 and
$I$=1 ${\bar K}N$ scattering lengths.
The calculations of the $K^-\alpha$ scattering length were done for
five sets of parameters for the ${\bar K}N$ lengths shown in
the Table~\ref{Tab1}.
Here we used the results from a $K$-matrix fit (Set 1) and a
separable fit (Set 2)~\cite{Barret}. We also study the constant
scattering length fit
(CSL) given by Dalitz and Deloff~\cite{Dalitz}, which we
denoted as Set 3 and the CSL fit from Conboy~\cite{Conboy} (Set 4).
The recent predictions for ${\bar K}N$ scattering lengths based on the chiral
unitary approach of Ref.\cite{Oller} are denoted as Set~5.
\renewcommand{\arraystretch}{1.2}
\begin{table*}[t]
\begin{center}
\begin{tabular}{|l|c|l|l|l|}
\hline Set & Reference & $a_0(\bar{K}N) [{\mbox{fm}}]$&
$a_1(\bar{K}N)[{\mbox{fm}}]$& $A(K^- \alpha) [{\mbox{fm}}]$
\\ \hline
1 & \cite{Barret} & $-1.59+i0.76$ & $0.26 + i .57$ & $-1.80+ i0.90$ \\ \hline
2 & \cite{Barret} & $-1.61+i0.75$ & $0.32 + i0.70$ & $-1.87 + i 0.95$ \\ \hline
3 & \cite{Dalitz} & $-1.57+i0.78$ & $0.32 + i0.75$ & $-1.90 + i 0.98$ \\ \hline
4 & \cite{Conboy} & $-1.03+i0.95$ & $0.94 + i0.72$ & $-2.24+ i 1.58$ \\ \hline
5 & \cite{Oller} & $-1.31+i1.24$ & $0.26 + i0.66$ & $-1.98+ i 1.08$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{Tab1} The $K^- \alpha$ scattering length for various
sets of the elementary $\bar K N $ scattering lengths
$a({\bar{K}N})$ ($I=0,1$).}
\end{table*}
The results of our calculations are listed in the last
column of Table~\ref{Tab1}. These results are very similar for
the Sets 1--3 giving the real and imaginary parts of the
scattering length $A(K^-\alpha)$ within the range of $-1.8 \div -1.9$~fm and
$0.9 \div 0.98$~fm, respectively. The results for Set 4 are
quite different: Re$A(K^-\alpha)$ =-2.24~fm and
Im$A(K^- \alpha)$=1.58 fm. Furthermore, our
calculations with Set~5 are close to the results obtained with
Sets 1--3.
Unitarizing the constant scattering length, we can
reconstruct the ${\bar K}\alpha$ scattering amplitude within
the zero range approximation (ZRA) as
\begin{equation}
f_{\bar{K} \alpha}(k)=
\left[A(\bar{K}\alpha)^{-1} - ik\right]^{-1},
\label{f_KHepole}
\end{equation}
where $k{=}k_{\bar{K}\alpha} $
is the relative momentum of the $K^-\alpha$ system.
The denominator of the amplitude of Eq.(\ref{f_KHepole}) has a zero at
the complex energy
\begin{equation}
E^*=E_R - \frac{1}{2}i \Gamma_R=\frac{k^2}{2\mu} ,
\label{pole}
\end{equation}
where $E_R$ and $\Gamma_R$ are the binding energy and width
of the possible $K^-\alpha$ resonance, respectively. Here
$\mu$ is reduced mass of the system with $\alpha$ mass taken
as 3.728~GeV.
For Set 1 and Set 4 we find a pole at the complex energies of
$E^*{=}(-6.7{-}i 18/2)$ MeV
and $E^*{=}(-2.0{-}i 11.3/2)$ MeV, respectively. The result
for Set 5 is $E^*{=}(-4.8{-}i 14.9/2)$ MeV.
Note that assuming a strongly attractive phenomenological
$\bar K N$ potential, Akaishi and Yamazaki~\cite{Akaishi}
predicted a deeply bound ${\bar K}\alpha$ state at
$E^*{=}(-86{-}i 34/2)$ MeV, which is far from our solutions.
With a very similar elementary $\bar K N$ scattering
length given by Set~1 and used in both calculations, we predict
a loosely bound state. It is not clear if medium effects and higher
order corrections might be so strong in order to change so drastically
the ${\bar K}\alpha$ scattering length predicted by
our calculations within the multiple scattering approach.
In any case it is very important
to measure the $s$-wave ${\bar K}\alpha$ scattering length
experimentally and to clarify the situation concerning the
possible existence of a (deeply) bound ${\bar K}\alpha$ state.
Note that in the limit of small absorption,
{\it i.e.} when the imaginary part of $A(\bar K \alpha)$ approaches
zero, the real part of the scattering length should be much larger
for the case of a loosely bound state as compared to the case of
a deeply bound state. Such a situation is supported by the
calculations within ZRA (even in
the presence of absorption) where in
the case of a deeply bound state we found that
$A_{\bar K \alpha}$=$-$0.07{+}$i$0.72 fm.
We expect that the ZRA can be applied for the description of the
amplitude which is generated by the short range potential
used in Ref.\cite{Akaishi}.
\section{ The reaction {\boldmath $dd{\to}\alpha{K^- K^+}$} near
threshold and the {\boldmath $K^-\alpha$} final-state interaction}
It is well known~\cite{Buescher02,Grishina2001} that the reaction
\begin{equation}
dd \to \alpha K^- K^+
\label{ddHeKK}
\end{equation}
provides an opportunity to study
$I{=}0$ mesonic resonances in the $ K^-K^+$ sector.
At the same time near the reaction threshold it might be sensitive to
the to $K^-\alpha $ final state interaction. Here we study
whether it is possible to evaluate the $s$-wave $K^- \alpha$
scattering length from the $K^-\alpha$ final-state interaction.
Similar evaluation of the $d{\bar K^0}$ FSI and relevant scattering length
was done in our previous study~\cite{Sibirtsev04} of the
$pp{\to}d{\bar K^0}K^+$ reaction. As has been stressed in
Ref.~\cite{Oset9} this reaction should be very sensitive to the
${\bar K^0}d$ FSI. Through our analysis we extracted a
new limit for the $K^-d$ scattering length from the
$\bar{K^0} d$ invariant mass spectrum from the
$pp{\to}d{\bar K^0}K^+$ reaction measured recently
at COSY-J\"ulich~\cite{Kleber}.
It is clear that the FSI effect is essential at low invariant
masses of the interacting particles,
where the relative $s$-wave contribution is expected to be dominant.
One can also safely assume that the range
of the FSI is much larger as compared to the range of the basic hard
interaction related to the production of the $\bar K K$-meson pair.
This means that the basic production amplitude and the FSI term can be
factorized~\cite{Goldberger,Wycech,Sibirtsev96,Sibirtsev3,Hanhart} and the
FSI can be taken into account by multiplying the production
operator by the FSI enhancement factor defined by Eq.(\ref{enhancement}).
\begin{figure}[tb]
\vspace*{-5mm}
\centerline{\psfig{file=kon1.ps,width=8.5cm,height=9.5cm}}
\vspace*{-3mm}
\caption{The $K^- \alpha$ FSI enhancement
factor $\lambda^{\mathrm{MS}}(k)$, Eq.(\ref{enhancement}),
as a function of the relative momentum $k$ of the $K^- \alpha$ system.
The solid lines in the lower and upper part of the figure
show our calculations with Set~1 and Set~4 for the $\bar K N$ scattering
lengths, respectively. The dashed lines
illustrate the Watson--Migdal enhancement factor
normalized to $\lambda^{\mathrm{MS}}(k)$ at $k=0$.}
\label{fig:fsi_k}
\end{figure}
Fig.\ref{fig:fsi_k} shows the dependence of the
$K^-\alpha$ FSI enhancement factor $\lambda^{\mathrm{MS}}(k)$
given by Eq.~(\ref{enhancement})
on the relative momentum of the $K^-\alpha$ system,
$k$. The solid lines in the upper (lower) part
of Fig.\ref{fig:fsi_k} show the results obtained with Set 1 (Set 4)
for the $\bar K N$ scattering length. The calculations with Set 1
result in $\lambda^{\mathrm{MS}}(k){\simeq}$0.55 at $k{=}0$ and
FSI factor smoothly decreases with $k$. The calculations with Set 4
give $\lambda^{\mathrm{MS}}(k){>}$1 at $k{=}0$ and show a much stronger
$k$-dependence.
\begin{figure}[tb]
\vspace*{-4mm}
\centerline{\psfig{file=kon2.ps,width=8.5cm,height=7.5cm}}
\vspace*{-3mm}
\caption{The $K^-\alpha$ FSI factor averaged over the
three body phase space of the reaction $dd{\to}\alpha K^+ K^-$
as a function of excess energy. The solid and dashed lines
show the calculations with parameters of Set~1 and 4, respectively.}
\label{fig2}
\end{figure}
Following the Watson--Migdal approximation~\cite{Watson,Migdal}
the $k$-dependence of the enhancement factor is generally
described in terms of the on-shell scattering amplitude as
\begin{equation}
\lambda_{WM}=\frac{C}{|1-iqA_{\bar K \alpha}|^2},
\end{equation}
where $C$ is normalization constant.
Now, the dashed lines in Fig.~\ref{fig:fsi_k} illustrate the
Watson--Migdal enhancement factor
normalized to $\lambda^{\mathrm{MS}}(k)$ at $k{=}0$. The upper and
lower parts of Fig.~\ref{fig:fsi_k} are calculated using
the scattering lengths $A_{\bar K \alpha}$ obtained with
parameters of Set 1 (Set 4), respectively, and listed in
Table~\ref{Tab1}. It is clear that the momentum dependence of
$\lambda^{\mathrm{WM}}(k)$ and $\lambda^{\mathrm{MS}}(k)$
is different at different $k$. However, the absolute difference between
$\lambda^{\mathrm{WM}}(k)$ and $\lambda^{\mathrm{MS}}(k)$ at
$k{\leq}100$ MeV/c is relatively small.
Obviously, the energy dependence of the total cross section for the
$dd{\to}\alpha{K^+K^-}$ reaction is also distorted by the the $K^-\alpha$
FSI. In Fig.~\ref{fig2} we show the enhancement factor
$\lambda^{\mathrm{MS}}(k)$ averaged over the 3-body phase space
as a function of the excess energy $\epsilon$ for
the $dd{\to}\alpha{K^+K^-}$ reaction. The results for the Sets~2,~3 and 5
are practically the same as for Set~1. It is interesting to note that
there is essentially enhancement of the cross section at small $\epsilon$
for the Set~4, while for the Set~1 we obtain suppression.
The experiment would provide only a convolution of the
production amplitude and FSI factor. Since the production amplitude
is model dependent it is difficult to extract the absolute
value of the FSI factor from the data. However, the dependence
of the FSI on the relative momentum $k$ is very well
defined because the dependence of the basic hard interaction
on $k$ can be neglected at small $k$. According to
Ref.\cite{Buescher02} the total cross section of the reaction
$dd{\to}\alpha{K^+K^-}$ might be about 0.4$\ldots$1~nb
at $\epsilon{=}40\ldots~50$ MeV.
\begin{figure}[t]
\vspace*{-2mm}
\centerline{\epsfig{file=kon3a.ps,width=8.5cm,height=6.9cm}}
\vspace*{-9mm}
\centerline{\psfig{file=kon3b.ps,width=8.5cm,height=6.9cm}}
\vspace*{-3mm}
\caption{The invariant $K^-\alpha$ mass spectra produced in the
$dd{\to}\alpha{K^+ K^-}$ reaction at excess energies
30 and 50 MeV. The solid lines describe the pure phase space
distribution, while the dashed and dotted lines show our
calculations with $K^-\alpha$ FSI given by parameters of
Set 1 and 4, respectively.}
\label{fig3}
\end{figure}
Finally, we calculated the ~$K^-\alpha$
invariant mass spectra at excess energies $\epsilon$=30 and $50$~MeV
which are shown in Fig.~\ref{fig3}. The solid lines
show the calculations for the pure
phase space, {\it i.e.} for the constant production
amplitude and neglecting FSI.
The dashed and dotted lines in Fig.~\ref{fig3} show the results
obtained with the $K^-\alpha$ FSI calculated with the parameters of
the Set 1 and 4, respectively. All lines at each figures are
normalized to the same value, given by the reaction cross section
at a certain excess energy. At $\epsilon$=50 MeV the invariant mass spectra are
normalized to the $dd{\to}\alpha{K^+ K^-}$ cross section of 1~nb.
It is clear that the FSI significantly changes the $K^-\alpha$ mass
spectra. The most pronounced effect is observed
at low invariant masses available in the first 10~MeV bin.
To draw quantitative conclusions, one can compare the ratio of the
cross sections at the lowest $K^-\alpha$ invariant masses, within the
first 10~MeV bin, calculated with and without FSI.
We found that this ratio $R{=}1.26{\ldots}1.34$ at $\epsilon{=}30$ MeV,
$1.49{\ldots}\\ 1.56$ at $\epsilon{=}50$ MeV and $1.84{\ldots}2.18$ at
$\epsilon{=}100$ MeV.
Here the limits of the ratio at each excess energy
are given by the calculations with the ${\bar K}N$ scattering
length from the Set~1 and Set~4. With these estimates it is clear that
reasonable determination of the $K^-\alpha$ scattering length requires
sufficient statistical accuracy at $K^-\alpha$ invariant masses
below 4.23~GeV, at least 100 events. Such a high
precision experiment apparently can be done at COSY.
\section{Conclusions}
The findings of this study can be summarized as follows:
\begin{itemize}
\item
We have investigated the $s$-wave $K^-\alpha$ scattering length
and the $K^-\alpha $ FSI enhancement factor within the Foldy--Brueckner
adiabatic approach based on the multiple scattering formalism.
We have studied uncertainties of the calculations due to the elementary $K^-N$
scattering length available presently. The resulting $s$-wave $K^-\alpha$
scattering lengths for the various input parameters are collected in
Tab.~\ref{Tab1}.
\item
Through the determination of the pole position of the
$K^-\alpha$ scattering amplitude within ZRA, we found a loosely
bound state with binding energy
$E_R{=}-2{\ldots}-7$MeV and width $\Gamma_R{=}11{\ldots}18 $~MeV.
Our result differs from the prediction of Akaishi and
Yamazaki \cite{Akaishi} obtained under the assumption of a
strongly attractive phenomenological $\bar K N$ potential.
\item
We have analyzed the $K^-\alpha$ FSI in the reaction
$dd{\to}\alpha K^+ K^-$ and discussed the possibility to evaluate the
$K^-\alpha$ scattering length from the $K^-\alpha$ invariant mass spectra.
We have demonstrated that the measurement of the
$K^-\alpha$ mass distribution near the reaction threshold may
provide a new tool for the determination of the $s$-wave
$K^-\alpha$ scattering length.
\item
Furthermore, we have investigated the momentum dependence of the enhancement
factor $\lambda^{\mathrm{MS}}(k)$ calculated within MSA and compared it
with the one obtained utilizing the Watson--Migdal formalism. It was found that
the absolute difference between both calculations is relatively small
at momenta $q{\leq}100$ MeV/c.
\end{itemize}
It is important to stress that for kaonic helium atoms, energy shifts
can be measured for the $2p$ state and widths for the $2p$ and $3d$ states.
The $np{\to}1s$ transitions for $^4$He cannot be observed since the
absorption from the $p$ states is almost complete~\cite{Batty}.
Therefore the possibility to determine the $s$-wave $\bar K \alpha $
scattering length from experiments with kaonic atoms is questionable.
With this respect a measurement at COSY provides an unique opportunity
to determine $s$-wave $K^-\alpha$ scattering length.
\subsection*{Acknowledgements}
We appreciate discussions with
C.~Hanhart, M.~Hartmann, R.~Lemmer and P.~Winter.
This work was partially supported by Deutsche Forschungsgemeinschaft
through funds provided to the SFB/TR 16 ``Subnuclear Structure of Matter''
and by the DFG grant 436 RUS 113/787.
This research is part of the EU Integrated Infrastructure Initiative Hadron
Physics Project under contract number RII3-CT-2004-506078.
V.G. acknowledges support by the COSY FFE grant No. 41520739 and A.S.
acknowledges support by the COSY FFE grant No. 41445400 (COSY-067).
|
{
"timestamp": "2005-03-30T08:42:06",
"yymm": "0503",
"arxiv_id": "nucl-th/0503076",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503076"
}
|
\section{Quantum catastrophe models}
We begin by constructing the quantum catastrophe models and first consider
those derived from catastrophes occurring in a single variable,
such as the cusp.
We take as our model a system of two interacting bosonic modes. Let
($x_1$, $p_{x_1}$) and ($x_2$, $p_{x_2}$) be the (abstract) position
and momentum coordinates representing these modes.
We assume an interaction between these modes such that the interacting
system is separable in a description in
terms of two {\it collective} bosonic excitations,
the coordinates of which we denote $(y_1,p_{y_1})$ and $(y_2,p_{y_2})$.
We construct the Hamiltonian of one of these collective modes $y_1$
so that it undergoes the catastrophe.
The question that we shall address is then: given the structure
of the system in terms of the collective modes $\mathbf{y}$,
what is the entanglement between the original bare modes $\mathbf{x}$?
We write the Hamiltonian of the collective mode
in which the catastrophe occurs as
\begin{eqnarray}
H_1 = \frac{1}{2m} p_{y_1}^2 + m \omega^2 U_\mathrm{cat} (y_1)
\end{eqnarray}
with $m$ and $\omega$ the characteristic mass and
frequency of the mode.
The potential $U_\mathrm{cat} (y_1)$ is taken from elementary
catastrophe theory, and can be written as a power series
$U_\mathrm{cat} (y) = \sum_{n=1}^{\infty} A_n y_1^n$. We rescale the
coordinate $y_1 \rightarrow y_1 \sqrt{\hbar/m\omega} $, and measure the
energy in units $\hbar \omega$, such that
\begin{eqnarray}
H_1 &=&-\frac{1}{2} \frac{d^2}{dy_1^2}
+ \sum_{n=1}^{\infty} \frac{A_n}{\mu^{n/2-1}} y_1^n
\nonumber \\
&=& -\frac{1}{2} \frac{d^2}{dy_1^2} + V_\mathrm{cat}(y_1),
\label{H1}
\end{eqnarray}
which defines the rescaled catastrophe potential
$V_\mathrm{cat}(y_1)$.
Here, $\mu \equiv m \omega / \hbar$ is our explicit ``macroscopy''
parameter, which
is meant in the sense that the limit $\mu \to \infty$ can be thought
of either as the limit in which the system size (and hence mass $m$) becomes
macroscopic, or as the semi-classical limit $\hbar \to 0$.
The limit $\mu\to\infty$ is analogous to the thermodynamic
limit in the QPT models, and therein lies the correspondence
between these quantum catastrophes and the QPT work cited in the
introduction.
The behaviour of the mode described by the $H_1$
is largely governed by the fixed points of the classical catastrophe
potential $V_\mathrm{cat}(y_1)$, and this is especially true in the
limit $\mu \to \infty$. By construction the fixed points of
$V_\mathrm{cat}(y_1)$, which
we denote $\tilde{y}$, are of the order $\tilde{y} \sim \sqrt{\mu}$, and
are thus ``macroscopic''. Expanding $V_\mathrm{cat}(y_1)$ in Eq. (\ref{H1})
about $\tilde{y}$ and taking the limit $\mu \to \infty$ we obtain
\begin{eqnarray}
\tilde{H} &=&-\frac{1}{2} \frac{d^2}{dy_1^2}
+ \frac{1}{2} \left.\frac{d^2V}{dy_1^2}\right|_{y_1=\tilde{y}} y_1^2
+ V(\tilde{y}).
\end{eqnarray}
This effective Hamiltonian describes small $O(1)$ fluctuations
about fixed point $\tilde{y}$.
The second derivative determines the excitation spectrum around the fixed
point, and $ V(\tilde{y}) \sim O(\mu)$ is the energy of the bottom of the
harmonic potential
well in which the system is localised. In general, the potential will
have more than one fixed point and an independent effective Hamiltonian
may be derived for each.
The way in which contributions from different fixed points
combine to give the overall ground state of the quantum system will be
treated for individual catastrophes.
The second collective mode $y_2$ is assumed to be simple harmonic,
and thus the full Hamiltonian of the catastrophe model is
\begin{eqnarray}
H_\mathrm{cat} (\mathbf{y}) &=&
-\frac{1}{2} \frac{d^2}{dy_1^2}
-\frac{1}{2} \frac{d^2}{dy_2^2}
+ V_\mathrm{cat}(y_1) + \frac{1}{2} y_2^2,
\label{Hcat}
\end{eqnarray}
We relate the coordinates of the two
collective modes $\mathbf{y}$ to those of the bare modes
$\mathbf{x}$ via the rotation
\begin{eqnarray}
y_1 = c x_1 + s x_2,\quad y_2 = -s x_1 +c x_2,
\end{eqnarray}
where $c=\cos(\theta/2)$
and $s= \sin(\theta/2)$, and $\theta$ reflects the degree of mixing. In
terms of the ${\bf x}$-representation, $H_\mathrm{cat}(\mathbf{x})$
is not separable,
and this
rotation generates an interaction between the two bare modes
${\bf x}$. We quantise the collective coordinates $y_i$ and
the bare coordinates $x_i$ according to
\begin{eqnarray}
y_i = 2^{-1/2}(b_i^\dag + b_i),\quad x_i = 2^{-1/2}(a_i^\dag + a_i),
\end{eqnarray}
with momenta defined canonically.
In this second quantised notation, the two representations are
related through a two-mode SU(2) squeezing transformation described by
the unitary operator
$W = \exp(-\frac{\theta}{2} a_1^\dag a_2 + \frac{\theta}{2} a_1 a_2^\dag)$.
To make the connexion with a familiar model: the above scheme is very similar to
the Dicke model in the thermodynamic limit. Here, the two bare modes are
the photon field and the collective atomic coordinate, and these
are related to the collective excitations (polaritons) by just such a
squeezing \cite{cetb03,note}.
In this paper, we consider two one-dimensional
catastrophes --- the cuspoids $A_{+3}$ and $A_{+5}$, commonly referred to as
the cusp and the butterfly.
We shall also consider a catastrophe that occurs in two dimensions,
$V_\mathrm{cat}(y_1,y_2)$ and is non-separable.
In this case, we calculate the entanglement between the modes $y_1$
and $y_2$ with the catastrophe itself providing the interaction
between the modes. In selecting which
catastrophes to study, we require that the spectra of the catastrophe
be bounded from below for all values of the control parameters at finite
$\mu$.
\section{Entanglement about fixed points:
$\mu \to \infty$ limit \label{secinf}}
For the one-dimensional catastrophes, the two-mode
Hamiltonian that determines the excitations
about $\tilde{y}_1$ in the
$\mu\to \infty$ limit is
\begin{eqnarray}
H = -\frac{1}{2}\frac{d^2}{dy_1^2} - \frac{1}{2}\frac{d^2}{dy_2^2}
+\frac{1}{2}\epsilon_1^2 y_1^2 + \frac{1}{2} y_2^2
+V(\tilde{y}_1)
\label{eHam_wfn}
\end{eqnarray}
with $\epsilon_1^2 = \left.d^2 V/dy_1^2\right|_{y_1=\tilde{y}}$.
The ground state wave function of the system is thus the Gaussian
\begin{eqnarray}
\Psi({\bf y}) = (\pi^2/\epsilon_1 )^{-1/4}
\exp \rb{-\frac{\epsilon_1}{2}y_1^2 - \frac{1 }{2}y_2^2},
\end{eqnarray}
which in the ${\bf x}$-representation reads
\begin{eqnarray}
\Psi({\bf x}) = \rb{\frac{\pi^2}{\epsilon_1}}^{1/4}
\exp
\left\{
-\frac{\epsilon_1}{2} (c x_1 + s x_2)^2
-\frac{1 }{2} (s x_1 -c x_2)^2
\right\}.
\end{eqnarray}
To find the entanglement of this wave function, we require the reduced
density matrix (RDM) of one of the bare modes, $x_1$, say. This is obtained
through $\rho(x_1,x_1') = \int dx_2 \Psi(x_1, x_2) \Psi^*(x_1',x_2)$ as
\begin{eqnarray}
\rho(x_1,x_1') = \frac{\pi}
{\sqrt{\epsilon_1 (s^2 \epsilon_1 +c^2 )}}
\exp\left\{- \alpha (x_1^2 + {x_1'}^2) + \beta x_1 x_2\right\},
\end{eqnarray}
where $\alpha$ and $\beta$ are coefficients, only the ratio of which
is important for the entanglement:
\begin{eqnarray}
\frac{2\alpha}{\beta} =
\frac{(\epsilon_1 +1 )^2 + 2 \epsilon_1
\left[ \cot^2(\theta/2)+\tan^2(\theta/2) \right]}
{(\epsilon_1 - 1)^2}.
\label{ab}
\end{eqnarray}
We shall quantify the entanglement in our two mode system with the von
Neumann entropy $S$. The entropy of the density matrix $\rho(x_1,x_1')$
is evaluated by comparison with the density matrix of a
harmonic oscillator at finite temperature. Details of this approach have
been given elsewhere \cite{NLCETB04}, and we just give the result here:
\begin{eqnarray}
S = \frac{1}{\log 2}
\left\{
\frac{\Omega}{2T} \coth \rb{\frac{\Omega}{2T}}
-\ln \left[2 \sinh \rb{\frac{\Omega}{2T}} \right]
\right\},
\end{eqnarray}
where the ratio of frequency to temperature of the fictitious oscillator
is given by $\Omega/T = \mathrm{arccosh} (2 \alpha / \beta)$.
For the one-dimensional catastrophes, the entanglement is maximised
when the squeezing angle is $\theta=\pi/2$.
For this choice, Eq. (\ref{ab}) simplifies to
\begin{eqnarray}
\frac{2\alpha}{\beta} =
\frac{\epsilon_1^2 + 6 \epsilon_1 + 1}
{(\epsilon_1 - 1)^2}.
\label{ab2}
\end{eqnarray}
This procedure is easily adapted to calculate the entanglement in
the two-dimensional catastrophe.
We now consider our three example catastrophes in turn.
\section{Cusp}
The cusp catastrophe, $A_{+3}$ is the most familiar and,
from the point of view of
applications, the most important catastrophe.
With coefficients chosen for convenience, the scaled cusp
potential is
\begin{eqnarray}
V_{+3}(y_1) = \frac{1}{4\mu} y_1^4 + \frac{A}{2} y_1^2.
\end{eqnarray}
We shall only consider a harmonic perturbation here, and reserve until
later a discussion of the effects of linear perturbations.
We also shall set $\theta=\pi/2$ here to give maximum
mixing between the modes. This leaves us with a single control
parameter $A$.
The full two-mode Hamiltonian in
terms of the creation and annihilation operators of the
$\mathbf{x}$-modes is
\begin{eqnarray}
H_{+3} ({\bf a}) &=&
\frac{ A+3}
{4}
(a_1^\dag a_1 + a_2^\dag a_2 + 1)
\nonumber\\
&&+
\frac{A-1}
{8}
({a_1^\dag}^2 + a_1^2 + {a_2^\dag}^2 + a_2^2)
\nonumber\\
&&+
\frac{A-1}
{4}
(a_1^\dag a_2 + a_1 a_2^\dag+ a_1^\dag a_2^\dag + a_1 a_2 )
\nonumber\\
&&+
\frac{1}{64 \mu} (a_1^\dag + a_1 + a_2^\dag + a_2)^4.
\label{xcusp}
\end{eqnarray}
It may at first appear unusual that the
coefficient of $a_i^\dag a_i$ should depend
on the parameter $A$.
However, it can be shown that, by individually squeezing the collective
modes before applying the two-mode SU(2) transformation,
this dependence on $A$ can be removed. If both modes are
squeezed identically, the entanglement properties of the system are
left invariant, since this squeezing then represents a global rescaling
of the phase space. For simplicity though, we retain the form of
Eq. (\ref{xcusp}).
\begin{figure}[t]
\begin{center}
\psfrag{S}{$S$}
\psfrag{A}{$A$}
\psfrag{Al0}{$A<0$}
\psfrag{Ag0}{$A>0$}
\psfrag{Ss}{$S^*$}
\psfrag{As}{$A^*$}
\psfrag{m}{$\mu$}
\psfrag{m=10}{$\mu=10$}
\psfrag{m=20}{$\mu=20$}
\psfrag{m=40}{$\mu=40$}
\psfrag{m=70}{$\mu=70$}
\psfrag{tdl}{$\mu\to \infty$}
\includegraphics[width=1\linewidth,clip=true]{./cusp1.eps}
\caption{
Entanglement properties of the cusp catastrophe.
The von Neumann entropy $S$ in the macroscopic limit
$\mu\to \infty$ (thick line)
shows a divergence at the critical value $A=0$ where the potential
changes from a double- to a single-well structure (inset sketches).
Numerical results for finite $\mu$ show a peak near this point.
Inset {\bf (a)} shows the scaling with $\mu$ of the parameter value $A^*$
at which the entanglement maximum occurs, and {\bf (b)}
shows the value of the entropy $S^*$ at this point.
\label{figENT1}
}
\end{center}
\end{figure}
We now consider the fixed points.
For $A > 0$, only one stable fixed point exists and this lies at the
origin. Taking $\mu \to \infty$, we see that the
excitation energy about this fixed point is $\epsilon_1 = \sqrt{A}$.
For $A<0$, the origin becomes unstable, and two new stable fixed
points appear at $y_1 =\pm \sqrt{\mu |A|}$.
In the $\mu\to \infty$ limit, these two fixed points are
degenerate and have the same excitation energy $\epsilon_1 = 2\sqrt{|A|}$.
The shape of the potential is sketched as insets in Fig. \ref{figENT1}
and shows clearly the change of the potential from double to single
well structure.
Note that the form $V_{+3}$, Eq. (11), is also used in Landau theory
of phase transition in statistical mechanics, or in quantum field
theory ($\phi^4$-model).
Describing the vanishing of the excitation energy as
$\epsilon_1 \sim A^{z\nu}$, and the divergence of the ``correlation length''
as $\xi \equiv \epsilon^{-1/2} \sim A^\nu$, we find exponents
$\nu = 1/4$ and $z=2$.
We now consider the entanglement.
For $A>0$, the entropy follows directly from the approach outlined in
section \ref{secinf}. For $A<0$, the situation is complicated slightly by
the existence of two fixed points.
With the limit $\mu\to\infty$ taken in correspondence
with the thermodynamic limit, the ground state
of the system would be an equal mixture of density matrices
localised at the two
fixed points.
We prefer here to use the limit $\mu\to \infty$
to calculate an approximate wave function for finite but large
$\mu$. This is obtained by taking a coherent superposition of
the two localised wave functions and allows direct comparison
with the numerical results for finite $\mu$. Since the two lobes
are orthogonal, the reduced density matrix of the total system is equal
to the sum of the reduced density matrices for the two lobes:
$\rho_1 = 1/2 ( \rho_+ + \rho_-)$.
This is the same result as is obtained if one
takes the ground-state to be the incoherent mixture;
so the difference between these two approaches is unimportant. However,
this will be seen not to be the case when we consider the
two-dimensional catastrophe.
From the general theory of entropy \cite{wehrl} we know that for
$\rho = \sum_i \lambda_i \rho_i$ with $\lambda_i$ probabilities, the total
entropy $S(\rho)$ is bounded by
\begin{eqnarray}
\sum_i \lambda_i S(\rho_i)
\le S(\rho)
\le \sum_i \lambda_i S(\rho_i) - \lambda_i \lg \lambda_i
\label{Sbound}.
\end{eqnarray}
In the current situation, since $\rho_+$ is orthogonal to $\rho_-$,
the upper bound becomes an equality. Furthermore, since
$S(\rho_+) = S(\rho_-)$,
we have $S(\rho_1) = S_{\rm mix} + S(\rho_+)$ with $ S_{\rm mix}=1$.
The mixing entropy represents the contribution from the 'global', i.e.
macroscopic, structure of the wave function, whereas local structure
enters through the individual $S(\rho_+)$ terms.
If the parity symmetry $V_{+3}(y_1)=V_{+3}(-y_1)$ is broken
by an additional linear term $\propto y_1$ in the potential,
the degeneracy of the two fixed points would be lifted
and the contribution from the mixing entropy $S_{\rm mix}=1$ would disappear.
The single-well entropy $S(\rho_+)$ is calculated as in section
\ref{secinf}, and we plot the total entropy $S(\rho)$ in
Fig. \ref{figENT1}.
The similarity between the behaviour of this simple cusp model and the
QPT models is apparent. At the critical point, the entropy diverges as
\begin{eqnarray}
S \sim \nu \lg A = \lg \xi,
\end{eqnarray}
i.e., with the correlation length $\xi$,
and we thus see ``critical entanglement'' \cite{ON02}.
Numerically obtained results for finite $\mu$ are shown
alongside the $\mu\to\infty$ result.
The value of A for which the peak in the
entanglement occurs at finite $\mu$, $A^*$,
scales with $\mu$ to a very good
approximation as $A^*= c \mu^{0.75}$ with a numerically
determined constant of $c = 4.1$. This relation is plotted in
Fig. \ref{figENT1}a. We mention that the exponent of $0.75 \approx 3/4$
has been observed numerically for the entropy in the Dicke model
\cite{NLCETB03}. We also investigated the value of the entropy $S^*$
at its peak (Fig. \ref{figENT1}b) but found no convincing
scaling relation for finite $\mu$.
\section{Butterfly}
The second one-dimensional catastrophe that we study is
the butterfly, $A_{+5}$, which gives rise to the potential
\begin{eqnarray}
V_{+5}(y_1) = \frac{A_2}{2} y_1^2 + \frac{A_4}{4 \mu} y_1^4
+ \frac{1}{6 \mu^2} y_1^6.
\end{eqnarray}
The parameter space is two-dimensional ($A_2,A_4$), and rather than give a
full account of this space, we simply look at two representative values
of $A_4$
{\it Case (i):} $A_4=0$. For $A_2>0$, $y_1=0$ is the only fixed point
and this has
excitation energy $\epsilon_1 = \sqrt{A_2}$.
For $A_2<0$, $\tilde{y} = \pm \sqrt{\mu} |A_2|^{1/4}$ are the two
stable fixed points, both with $\epsilon_1 = \sqrt{2|A_2|}$.
Apart from numerical coefficients,
the behaviour here is the same as that of the
cusp. This result generalises to all $A_{+k}$ catastrophes:
for $V_{+k}$ with $A_i=0, \forall i>2$ the excitation energy is
$\sqrt{A_2}$ for $A_2>0$, and $ \sqrt{(k-3) |A_2|}$ for $A<0$, with
behaviour like that of the cusp.
{\it Case (ii):} $A_4 = - 4 /\sqrt{3}$. Here we see new behaviour absent
in the cusp. The $A_2$ parameter range is divided up
into three regions by the fixed points,
\begin{eqnarray}
A_2<0; && \tilde{y}=\pm \left[\frac{\mu}{\sqrt{3}}
\rb{2+\sqrt{4 - 3 A_2}}
\right]^{1/2} \equiv \tilde{y}_\pm
\nonumber\\
0<A_2<4/3; && \tilde{y}=0
\nonumber\\
&& \tilde{y}= \tilde{y}_\pm
\nonumber\\
A_2 > 4/3; && \tilde{y}=0.
\end{eqnarray}
Thus, increasing $A_2$ from below zero upwards, the potential moves
through a sequence of first a double, then triple, then single well
structures, as shown by the insets in Fig. \ref{figENT2}.
The stability or otherwise of the fixed points is only part of the
story in determining the $\mu \to \infty$ ground state of the system.
For $A_2>4/3$ and
$A_2<0$, the situation is straightforward and the ground state
is obtained exactly as for the two phases in the cusp.
\begin{figure}[t]
\begin{center}
\psfrag{S}{$S$}
\psfrag{A2}{$A_2$}
\psfrag{A2=0}{$A_2=0$}
\psfrag{A2=1}{$A_2=1$}
\psfrag{A2=2}{$A_2=2$}
\psfrag{Ss}{$S^*$}
\psfrag{As}{$A_2^*$}
\psfrag{m}{$\mu$}
\psfrag{m=5}{$\mu=5$}
\psfrag{m=7}{$\mu=7$}
\psfrag{m=10}{$\mu=10$}
\psfrag{m=20}{$\mu=20$}
\psfrag{tdl}{$\mu\to \infty$}
\includegraphics[width=1\linewidth,clip=true]{./V6.eps}
\caption{
The von Neumann entropy of the Butterfly catastrophe with
$A_4=-4/\sqrt{3}$ as the
potential undergoes a double-triple-single well transition, both for
$\mu \to \infty$ and finite $\mu$.
The profile of the entanglement is very different to that of the cusp
as the transition here is induced by a level crossing in the spectrum.
Inset shows scaling of $A_2^*$ as a function of $\mu$.
\label{figENT2}
}
\end{center}
\end{figure}
In the central region $0<A_2<4/3$, however, we have three fixed points,
and their weight in determining the ground state depends
on the energy $V(\tilde{y})$ of the bottom of the well at $\tilde{y}$.
In the
$\mu \to \infty$ limit, the system will be completely localised
in whichever of the fixed points has the lowest base energy, or, if the
energies are degenerate, we take an equal superposition
to describe the large-$\mu$ wave function.
For $A_2>1$,
$y=0$ is the fixed point with lowest energy, and for $A_2<1$ the
two fixed points at finite displacements $y=\tilde{y}_\pm$ have the
lowest energy and are degenerate. Only at $A=1$ are all
three points degenerate and we
have a three-lobed wave function.
This structure is induced by a level crossing in
the $\mu\to \infty$ spectrum, with the energy of the double
well crossing the energy of the single well at $A=1$.
For finite $\mu$, the level-crossing is actually
avoided, due to the overlap of all three wells.
This situation therefore bears some similarity to that
described in Ref. \cite{vid04}, where a discontinuous entanglement was
observed at a level crossing associated with a first-order QPT.
Away from the level crossing, the entanglement is calculated just
as for the cusp. In the region of $A_2=1$, we need to exercise a
little care, because the entanglement is discontinuous at $A_2=1$.
Exactly at this point, the
excitation energies of the three wells do not disappear, but
rather take the finite values $\epsilon_1 = (1,2,2)$.
The entanglement in the central well (with $\epsilon_1=1$) is zero,
$S_0=0$, since the wave function is circularly symmetric about the origin
($\epsilon_2=1$ as well) and can thus be written as a product state with
respect to all co-ordinate systems.
The entanglement for each of the displaced wells
is $S_\pm \approx 0.197$. Thus, by combining the appropriate
density matrices,
we find that for $A_2$ slightly less than unity, the double-well state has
$S=1.197$. For $A_2$ just slightly bigger than unity we have $S=0$,
due to the product state in the single well. Directly at $A_2=1$ we have
the three-lobed wave function, and
$S = 2/3 S_+ + 1/2 S_- + \lg 3 \approx 1.716$.
These results plus the corresponding finite $\mu$ data are shown in
Fig. \ref{figENT2}. The approach of the finite $\mu$ results to
the $\mu\to\infty$ limit is nicely seen, and in particular
to the limiting value of $S\approx 1.716$ at $A_2=1$.
We stress that the entanglement maximum occurs not at the value of $A_2$
at which the fixed point becomes unstable, but rather at the level
crossing. Moving through the points $A_2=0$ and $A_2=4/3$, where
fixed point stability does change, nothing special happens to the
entropy (or any other ground-state property), since these
fixed points do not contribute to the determination of the ground state
at these values of $A_2$.
By examining the finite $\mu$ data (Fig. \ref{figENT2}b), we determine that
the value of $A_2$ at which the entanglement peak occurs scales
as $A^*-1\sim c_0 \mu^{-c_1}$ with numerical
parameters $(c_0,c_1)$ determined to be $(-3.55,1.90)$ to within
a few percent.
\section{Two-dimensional Catastrophe}
\begin{figure}[t]
\begin{center}
\psfrag{S}{$S$}
\psfrag{g}{$\gamma$}
\psfrag{gl1}{$\gamma<1$}
\psfrag{gg1}{$\gamma>1$}
\psfrag{gs}{$\gamma^*$}
\psfrag{m}{$\mu$}
\psfrag{m=10}{$\mu=10$}
\psfrag{m=20}{$\mu=20$}
\psfrag{m=30}{$\mu=30$}
\psfrag{m=40}{$\mu=40$}
\psfrag{tdl}{$\mu\to \infty$}
\includegraphics[width=1\linewidth,clip=true]
{./combfig2D.eps}
\caption{
The von Neumann entropy of the two-dimensional molar
catastrophe with
$A=-1$ as a function of $\gamma$.
Plots of the potential for $\gamma<1$ and $\gamma>1$ are shown at
the top of the figure. The origin of the potential is unstable
and there are four stable potential wells satellite to this.
Lower right inset shows scaling of $\gamma^*$ as a function of $\mu$.
\label{figENT3}
}
\end{center}
\end{figure}
The most familiar two-dimensional catastrophes are the umbillics
with the germs $y_1^2 y_2 \pm y_2^3$.
However, these are unsuitable for our purpose as their
spectra are not bounded from below and this, in fact,
is true of all the two-dimensional,
elementary catastrophes of Thom \cite{thom}.
Therefore, we consider the non-simple catastrophe
\begin{eqnarray}
V_{\mathrm{m}} = \frac{1}{2} A(y_1^2 + y_2^2)
+ \frac{1}{4\mu}(y_1^4 +2 \gamma y_1^2 y_2^2 +y_2^4),
\end{eqnarray}
where we have only included harmonic perturbations as before.
This catastrophe
is described as non-simple because the germ (that part proportional to
$\mu^{-1}$ in the above) depends
irreducibly on a modulus, $\gamma$, whereas
simple germs have no free parameters.
The fixed point structure of $V_{\mathrm{m}}$ divides the behaviour into three
regimes in the $\mu\to \infty$ limit. For $A>0$, we obtain a
single fixed point at the origin, and since the ground-state
of the system is a product state of two Gaussians with the same width,
there is no entanglement.
For $A<0$, the origin is unstable; for $\gamma \ne 1$,
the system possesses four fixed points,
as is readily observed from the molar-shaped potentials plotted as
insets of Fig. \ref{figENT3}. For all $\gamma>1$,
the four stable fixed points
lie on the lines $y_1=0$ and $y_2=0$, whereas for $\gamma<1$ they lie
on the diagonals $y_1=\pm y_2$. In the following,
we set $A_2=-1$ throughout, as the entanglement
properties are the same for all $A_2<0$. We calculate
the entanglement between modes $y_1$ and $y_2$ induced by the
interaction in the catastrophe itself, and do not apply the two-mode
squeezing.
We first study $\gamma >1$ as this is
the simpler of the two cases. The stable
fixed points are given by
\begin{eqnarray}
(y_1, y_2) = (\pm \sqrt{\mu},0)
;\quad
(y_1, y_2) = (0,\pm \sqrt{\mu}).
\end{eqnarray}
At each fixed point, $y_1$ and $y_2$ are the excitation
coordinates with excitation energies
\begin{eqnarray}
\epsilon_+^2 = 2;\quad \epsilon_-^2 = \gamma-1.
\end{eqnarray}
Excitations in the direction of the displacement $\pm \sqrt{\mu}$
are described $\epsilon_+$.
The individual wave functions localised around any of these fixed points
are unentangled, since they are just products of Gaussians is the $y_1$
and $y_2$ directions. However, combining these four functions into the
four-lobed wave function that describes the large $\mu$ limit,
the total system is entangled.
This is solely due to the mixing entropy of its four lobed structure.
We can not calculate the entanglement of this structure in the way we did
for the one-dimensional catastrophes, because the four reduced density
matrices of each lobe are not orthogonal. This means that
the upper bound in Eq. (\ref{Sbound}) remains as an upper bound,
and is not equality.
Nevertheless, we can proceed as follows.
Writing $\ket{\tilde{y}_1,\tilde{y}_2}$ for the wave function
of the system localised at $(\tilde{y}_1,\tilde{y}_2)$,
the four-lobed large-$\mu$ wave function can be written as
\begin{eqnarray}
\ket{\Psi} &= &
\frac{1}{2}
\left\{
\ket{\tilde{y}, 0}
+\ket{-\tilde{y},0}
+\ket{0,\tilde{y}}
+\ket{0,-\tilde{y}}
\right\}
\end{eqnarray}
with $\tilde{y}=\sqrt{\mu}$.
Given that the individual lobes contribute nothing to the
entanglement by themselves, we ignore their individual structure
in this description.
In the limit $\mu\to \infty$, the three single-mode states
$\ket{0},\ket{\pm\tilde{y}}$ are all orthogonal, and thus the RDM of
one of the modes
$\rho_1 = \mathrm{Tr}_2\ket{\Psi}\bra{\Psi}$
is
\begin{eqnarray}
\rho_1 = \frac{1}{4}
\left\{
\rb{\frac{}{}\ket{\tilde{y}}+\ket{-\tilde{y}}}
\rb{\frac{}{}\bra{\tilde{y}}+\bra{-\tilde{y}}}
+ 2 \ket{0}\bra{0}
\right\}.
\end{eqnarray}
Furthermore, the orthogonality of these states means that this
density matrix can be simply treated as a three-by-three matrix
and the
entropy is simply $S=1$, independent of $\gamma$ for $\gamma>1$.
It is interesting to note that had we taken as the ground-state
density matrix the incoherent mixture of the four
contributions,
\begin{eqnarray}
\rho &=& \frac{1}{4}
\left\{
\op{\tilde{y},0}{\tilde{y},0} + \op{-\tilde{y},0}{-\tilde{y},0}
\right.
\nonumber\\
&&~~~~~~~~~~~~
\left.
+\op{0,\tilde{y}}{0,\tilde{y}} + \op{0,-\tilde{y}}{0,-\tilde{y}}
\right\},
\end{eqnarray}
leading to the RDM
\begin{eqnarray}
\rho_1 =\frac{1}{4}
\left\{
\op{\tilde{y}}{\tilde{y}} + \op{-\tilde{y}}{-\tilde{y}} + 2\op{0}{0}
\right\}
\end{eqnarray}
and a value of the von Neumann entropy of $S=3/2$, which
is clearly at variance with the numerical results.
We now consider the region $\gamma <1$, and for simplicity we also assume
$\gamma>0$. The four fixed points are
\begin{eqnarray}
(y_1,y_2) = \rb{\pm\sqrt{\frac{\mu}{1+\gamma}},
\pm\sqrt{\frac{\mu}{1+\gamma}}}
\end{eqnarray}
where the two $\pm$ signs are independent. Each fixed point has
the excitation energies
\begin{eqnarray}
\epsilon_+^2 = 2
;\quad
\epsilon_-^2 = 2\frac{1-\gamma}{1+\gamma}.
\end{eqnarray}
The eigenmodes of the system are not $y_1$ and $y_2$, but rather lie
along, and perpendicular to, the diagonals of the $y_1$-$y_2$ plane.
Each individual fixed-point wave function is thus entangled with
respect to modes $y_1$ and $y_2$.
This entanglement can be calculated as in section \ref{secinf}, but here
with two excitation energies and the rotation between the
eigenmodes and the ${\bf y}$ coordinates. The entanglement determining
parameter
$2\alpha/\beta$ is evaluated to be
\begin{eqnarray}
\frac{2\alpha}{\beta} = \frac{4 - 3 \gamma^2 + 4 \sqrt{1-\gamma^2}}
{\gamma^2},
\end{eqnarray}
from which the single-lobe entanglement follows directly.
The contribution of the four-lobed structure of the large-$\mu$
superposition can be assessed as follows.
From a macroscopic point of view, we can ignore the structure of
the individual lobes, and write the wave function as
\begin{eqnarray}
\ket{\Psi} &=& \frac{1}{2}
\left\{
\ket{\tilde{y},\tilde{y}} + \ket{\tilde{y}-,\tilde{y}}
+ \ket{-\tilde{y},\tilde{y}} + \ket{-\tilde{y},-\tilde{y}}
\right\}
\nonumber\\
&=& \rb{\frac{}{} \ket{\tilde{y}} + \ket{-\tilde{y}}}\otimes
\rb{\frac{}{} \ket{\tilde{y}} + \ket{-\tilde{y}}}.
\end{eqnarray}
The second forms clearly shows this wave function to be a product
state from the macroscopic viewpoint. Thus the mixing entropy of
forming the four-lobed structure is zero, and the entropy of the
system is just the single lobe entropy above.
In Fig. \ref{figENT3} we plot these results alongside
the numerical data for finite $\mu$.
The scaling of $\gamma^*$ with $\mu$ is observed to be
$\gamma^*-1 = c_0 \mu^{-c_1}$ with coefficients fitted as
$(c_0,c_1) = (4.93\times10 ^{4},4.09)$.
\section{Conclusions}
We have constructed and studied a family of quantum catastrophe
models, and investigated their ground-state entanglement properties.
The cusp catastrophe, with its bifurcating fixed point, demonstrates
behaviour that is remarkable similar to the QPT models, such as the Dicke
model --- underlining the importance of bifurcations of classical
fixed points in this context. It should be noted that whilst this
bifurcation occurs for all values of $\mu$, a peak in the entanglement
is only observed when $\mu$ is sufficiently large ($\mu >10$ here). This
illustrates that the bifurcation is not, in itself, a sufficient condition
for the occurrence of the entanglement maximum,
but that the system must also be capable of sufficient delocalisation.
The butterfly catastrophe displays very different behaviour to the cusp ---
namely a discontinuous entropy induced by a level crossing
in the macroscopic limit.
The cusp and the two-dimensional catastrophe demonstrate that a
mixing term in the entropy can contribute to the total entanglement
in cases where a wave function is split up into localisation areas
that are separated within (abstract) position space. In
particular the two-dimensional catastrophe suggests a
distinction between `global' and `local' (within the lobes)
entanglement, and one could speculate that in more complex
situations, with wave functions split up further and
further, a hierarchy of entanglement entropies might emerge.
Our results also have a bearing on the issue of quantum chaos
and entanglement in such systems, as the model here is capable of
emulating the behaviour of more sophisticated nonlinear Hamiltonians,
despite being separable --- and thus integrable.
It is clear that there is
no unequivocal relation between delocalization and the onset
of quantum chaos on one hand and the peaking of entanglement on the other.
This work was supported by the Dutch Science Foundation NWO/FOM
and the UK EPSRC Network `Transport, Dissipation, and Control
in Quantum Devices'.
|
{
"timestamp": "2005-03-17T18:39:50",
"yymm": "0503",
"arxiv_id": "quant-ph/0503160",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503160"
}
|
\section{Introduction}
\label{1}
A comprehensive understanding of Hamiltonian dynamics is a long outstanding
problem in nonlinear and statistical physics, which has important applications
in various other areas of physics. Typical Hamiltonian systems are
nonhyperbolic as they exhibit mixed phase space with coexisting regular and
chaotic regions. Over the past years, a number of ground-breaking works
\cite{chirikov1,asymp,meiss1,greene1,pikovsky1,christiansen1,lau1,zasl,uptodate} have
increasingly elucidated the asymptotic behavior of such systems and it is now
well understood that, because of the stickiness due to Kolmogorov-Arnold-Moser
(KAM) tori, the chaotic dynamics of typical Hamiltonian systems is fundamentally
different from that of hyperbolic, fully chaotic systems. Here ``asymptotic''
means in the limit of large time scales and small length scales. But in
realistic situations, the time and length scales are limited. In
the case of hyperbolic systems, this is not a constraint because the
(statistical) self-similarity of the underlying invariant sets guarantees the
fast convergence of the dynamical invariants (entropies, Lyapunov exponents,
fractal dimensions, escape rates, etc) and the asymptotic dynamics turns out to
be a very good approximation of the dynamics at finite scales. In nonhyperbolic
systems, however, the self-similarity is usually lost because the invariant sets
are not statistically invariant under magnifications. As a result, the
finite-scale behavior of a Hamiltonian system may be fundamentally different
from the asymptotic behavior considered previously, which is in turn hard to
come by either numerically \cite{uptodate,fluid} or experimentally \cite{nh_h}.
The aim of this paper is to study the dynamics of Hamiltonian systems at
finite, physically relevant scales. To the best of our knowledge, this problem
has not been considered before. Herewith we focus on Hamiltonian chaotic
scattering, which is one of the most prevalent manifestations of chaos in open
systems, with examples ranging from fluid dynamics \cite{fluid,nh_h} to
solid-state physics \cite{sol_stat} to general relativity \cite{relat}. We show
that the finite-scale dynamics of a Hamiltonian system is characterized by {\it
effective} dynamical invariants (e.g., effective fractal dimension), which: (1)
may be significantly different from the corresponding invariants of the
asymptotic dynamics; (2) depend on the resolution but can be regarded
as constants over many decades in a given region of the phase space; and (3) may
change drastically from one region to another of the {\it same} dynamically
connected (ergodic) component. These features are associated with the slow and
nonuniform convergence of the invariant measure due to the breakdown of
self-similarity in nonhyperbolic systems. To illustrate the mechanism behind
the properties of the effective invariants, we introduce a simple deterministic
model which we build on the observation that a Hamiltonian system can be
represented as a chain of hyperbolic systems.
The paper is organized as follows. We start, in Sec. \ref{s2}, with the
analysis of the invariant measure and the outline of the transport structures
underlying its convergence.
Our chain model is introduced and analyzed in Sec. \ref{3}. The
effective fractal dimension is defined in Sec. \ref{4} and its properties are
verified for a specific system in Sec. \ref{5}. Conclusions are presented
in the last section.
\section{Invariant Measure}
\label{s2}
For concreteness, consider a two-dimensional area preserving map with a major
KAM island surrounded by a chaotic region. One such map captures all the main
properties of a wide class of Hamiltonian systems with mixed phase space. When
the system is open (scattering), almost all particles initialized in the chaotic
region eventually escape to infinity. We first study this case with a diffusive
model for the transversal motion close to the main KAM island, obtaining an
analytical expression for the probability density $\rho(x,t)$
of particles remaining in the scattering region at time $t$ and distance $x$
from the island [see APPENDIX]. We find that, in the case of chaotic scattering, a singularity
develops and the invariant measure, given by
$\lim_{t\rightarrow\infty}\rho(x,t)$, accumulates on the outermost KAM torus of
the KAM island [APPENDIX]. Physically, this corresponds to the tendency
of nonescaping particles to concentrate around the regular regions.
Dynamically, the stickiness due to KAM tori underlies two major features of
Hamiltonian chaotic scattering, namely the algebraic decay of the survival
probability of particles in the scattering region
\cite{asymp,meiss1,greene1,pikovsky1,christiansen1} and the integer
dimension of the chaotic saddle \cite{lau1}, and distinguishes this phenomenon
from the hyperbolic chaotic scattering characterized by exponential decay and
noninteger fractal dimension. However, the convergence of the measure is rather
slow and highly nonuniform, as shown in Fig.~\ref{fig1} for typical parameters,
which is in sharp contrast with the fast, uniform convergence observed in
hyperbolic systems. Our main results are ultimately related to this slow and
nonuniform convergence of the invariant measure.
Previous works on transport in Hamiltonian systems have used stochastic models,
where invariant structures around KAM islands are smoothened out and the
dynamics is given entirely in terms of a diffusion equation
\cite{chirikov1,greene1} or a set of transition probabilities (Markov chains or
trees) \cite{meiss1,other_chains}. The stochastic approach is suitable to
describe transport properties (as above), but cannot be used to predict the
behavior of dynamical invariants such as Lyapunov exponents and fractal
dimensions. Here we adopt a deterministic approach where we use the Cantori
surrounding the KAM islands to split the nonhyperbolic dynamics of the
Hamiltonian system into a chain of hyperbolic dynamical systems.
Cantori are invariant structures that determine the transversal transport close
to the KAM islands \cite{asymp,meiss1}. There is a hierarchy of infinitely many
Cantori around each island. Let $C_1$ denote the area of the
scattering region outside the outermost Cantorus, $C_2$ denote the annular area
in between the first and second Cantorus, and so on. As $j$ is increased, $C_j$
becomes thinner and approaches the corresponding island. For simplicity, we
consider that there is a single island \cite{hier} and that, in each iteration,
a particle in $C_j$ may either move to the outer level $C_{j-1}$ or the inner
level $C_{j+1}$ or stay in the same level \cite{meiss1}. Let $\Delta_j^{-}$ and
$\Delta_j^{+}$ denote the transition probabilities from level $j$ to $j-1$ and
$j+1$, respectively. A particle in $C_1$ may also leave the scattering region,
and in this case we consider that the particle has escaped. The escaping region
is denoted by $C_0$. The chaotic saddle is expected to have points in $C_j$ for
all $j\geq 1$. It is natural to assume that the transition probabilities
$\Delta_j^{-}$ and $\Delta_j^{+}$ are constant in time. This means that each
individual level can be regarded as a hyperbolic scattering system, with its
characteristic exponential decay and noninteger chaotic saddle dimension.
Therefore, a nonhyperbolic scattering is in many respects similar to a sequence
of hyperbolic scatterings.
\begin{figure}[pt]
\begin{center}
\epsfig{figure=fig1.eps,width=6.0cm}
\caption{Snapshots of the probability density $\rho$ as a
function of $x$, for $\rho(x,0)=\delta(x-x_0)$,
$x_0=1$, $x_1=2$, $\alpha=3$, and the outermost torus of
the KAM island at $x=0$ [APPENDIX].
The time $t$ is indicated in the figure.}
\label{fig1}
\end{center}
\end{figure}
\section{Chain Model}
\label{3}
We now introduce a simple deterministic model that incorporates
the above elements and reproduces essential features of the Hamiltonian dynamics.
Our model is depicted in Fig.~\ref{fig2} and consists of a semi-infinite
chain of 1-dimensional ``$/\backslash/$-shaped'' maps,
defined as follows:
\begin{equation}
M_j(x) = \left \{
\begin{array}{lll}
&\xi_j x, & 0\leq x<1/\xi_j\\
- &\xi_j (x - \Delta_j^{-}) +2, & 1/\xi_j< x - \Delta_j^{-} < 2/\xi_j\\
&\xi_j (x-1)+1, & -1/\xi_j < x -1 \leq 0 ,
\end{array} \right. \nonumber
\end{equation}
where $\xi_j> 3$ and $0< \Delta_j^{-} < 1-3/\xi_j\;$
($j=1,2,\ldots$). If $x$ falls in the interval $ 1/\xi_j\leq x \leq
1/\xi_j + \Delta_j^{-}$, where $M_j$ is not defined, the ``particle''
is considered to have crossed a Cantorus to the ``outer level''
$j-1$. This interval is mapped uniformly to $[0,1]$, and the iteration
proceeds through $M_{j-1}$. Symbolically, this is indicated by
$j\rightarrow j-1$. Similarly, if $x$ falls into $ 1-1/\xi_j -
\Delta_j^{+}\leq x\leq 1-1/\xi_j$, where $\Delta_j^{+} = 1-3/\xi_j -
\Delta_j^{-}$, the particle goes to the ``inner level'', and
$j\rightarrow j+1$.
Particles
that reach $ 1/\xi_1\leq x
\leq 1/\xi_1 + \Delta_1^{-}$ are considered to have escaped.
The domain of $M_j$ is denoted by $I_j$ and is analogous to $C_j$ in a Hamiltonian system,
where $\Delta_j^{-}$ and $\Delta_j^{+}$ represent the transition probabilities.
The transition rate ratios
$\mu = \Delta_j^{+}/\Delta_j^{-}$ and $\nu = \Delta_{j+1}/\Delta_j$ are taken in
the interval $0 <\mu < \nu <1$ and are set to be independent of $j$, where
$\Delta_{j}= \Delta_j^{+} + \Delta_j^{-}$.
The parameter $\mu$ is a measure of the fraction of particles
in a level $j$ that will move to the inner level $j+1$ when leaving level $j$, while $\nu$
is a measure of how much longer it takes for the particles in the inner level to escape.
The nondependence on $j$ corresponds to
the approximate scaling of the Cantori suggested by the renormalization theory \cite{meiss1}.
Despite the hyperbolicity of each map, the entire chain behaves as a nonhyperbolic system.
For a uniform initial distribution in $I_1$,
it is not difficult to show \cite{details} that
the number of particles remaining in the chain after a long time $t$
decays algebraically as $Q(t)\sim t^{-\ln \mu /\ln \nu}$,
and that the initial conditions of never escaping particles form a zero Lebesgue
measure fractal set
with box-counting dimension 1.
However, the finite-scale behavior may deviate considerably from these
asymptotics, as shown in Fig.~\ref{fig3}.
\begin{figure}[bt]
\begin{center}
\epsfig{figure=fig2.eps,width=5.0cm}
\caption{Semi-infinite chain of hyperbolic maps $M_j$, $j=1,2,\ldots$}
\label{fig2}
\end{center}
\end{figure}
In Fig.~\ref{fig3}(a) we show the survival probability $Q$ as a function of
time. For small $\mu$ and $\nu$, the curve is composed of a discrete sequence
of exponentials with scaling exponents $\ln (1-\Delta_j^-)$,
which decrease (in absolute value) as we go forward in the sequence.
The length of each exponential segment is of the
order of $\mu$ in the decay of $Q$ and $-\ln\nu$ in the variation of $\ln t$.
This striking behavior is related to the time evolution of the density of
particles inside the chain. This is shown in Fig.~\ref{fig3}(b), where we plot
the average position $\langle j\rangle$ of an ensemble of particles initialized
in $I_1$ (i.e., $j=1$). The transitions between successive exponentials in the
decay of $Q$ [Fig.~\ref{fig3}(a)] match the transitions from a level $j$ to the
next in the average position of the remaining particles [Fig.~\ref{fig3}(b)].
In a Hamiltonian system, the increase of $\langle j\rangle$ in time is related
to the development of the singular invariant measure anticipated in our
diffusion analysis [see Fig.~\ref{fig1}]. The piecewise exponential behavior
of $Q$ is smoothened out for large $\mu$ and $\nu>\mu$ [Figs.~\ref{fig3}(a) and
\ref{fig3}(b)].
In Fig.~\ref{fig3}(c) we show the fractal dimension of the set
of initial conditions of never escaping particles as computed from the
uncertainty algorithm \cite{uncert}, which consists in measuring the scaling
of the fraction $f(\varepsilon)$ of {\it $\varepsilon$-uncertain} points (initial points
whose escaping time is different from the escaping time
of points taken $\varepsilon$ apart).
The scaling is statistically well defined
over decades and the exponent $\alpha=\Delta \ln f(\varepsilon)/\Delta \ln
\varepsilon$ can be computed accurately.
However, the resulting dimension $1-\alpha$ is not only significantly smaller
than 1 but also depends critically
on the region $L$ of the phase space where it is computed. The convergence of
the dimension is indeed so slow that it can only be noticed when observed over
very many decades of resolution, as shown in Fig.~\ref{fig3}(d) where data of
Fig.~\ref{fig3}(c) is plotted over 35 decades! Initially smaller, the dimension
measured for $L=I_1$ approaches the dimension measured for $L=I_2$ as the scale
$\varepsilon$ is reduced beyond $10^{-15}$ (i.e. the corresponding curves in
Fig.~\ref{fig3}(d) become parallel). As shown in Fig.~\ref{fig3}(d), this
behavior is related to a transition in the average innermost level $\langle
j_{max} \rangle$ reached by the particles launched from $\varepsilon$-uncertain
points. As $\varepsilon$ is further reduced, new transitions are expected. The
dimension measured in between transitions is mainly determined by the dimension
$D=\ln 3/\ln \xi_k$, $k=\langle j_{max} \rangle$, of the corresponding element
of the chain. For given $j$ and $\varepsilon$,
the measured dimension is larger when
$L$ is taken in a denser part of the invariant set,
such as in the subinterval of $I_1$ first mapped into $I_2$ [Fig.~\ref{fig3}(c); diamonds],
because $\langle j_{max} \rangle$ is larger in these regions.
In some regions, however, the measured dimension is quite different
from the asymptotic value even at scales as small as $\varepsilon =10^{-30}$.
This slow convergence of the dimension is due to the slow increase of $\langle j_{max} \rangle$,
which in a Hamiltonian system is related to the slow convergence of the invariant
measure [Fig.~\ref{fig1}].
The convergence is even slower for smaller $\mu$ and larger $\nu$.
Incidentally, the experimental measurements of the fractal dimension are usually
based on scalings over less than two decades \cite{avnir1}. Therefore, at
realistic scales the dynamics is clearly not governed by the asymptotic
dynamical invariants.
\begin{figure}[pt]
\begin{center}
\epsfig{figure=fig3.eps,width=8.0cm}
\caption{Chain model for $\xi_1=4.1$.
(a) Survival probability $Q$ and (b) average position $\langle j\rangle$ as a function of time
for $\mu=0.01$ and $\nu=0.02$ (full line), $\mu=0.01$ and $\nu=0.1$ (dashed, bottom),
and $\mu=0.08$ and $\nu=0.1$ (dashed, top).
(c) Fraction $f(\varepsilon)$ of uncertain points as a
function of the scale $\varepsilon$ for points taken from $L=I_1$ (circles), $L=I_2$ (squares),
and the subinterval $L$ of $I_1$ first mapped into $I_2$ (diamonds),
where $\mu=0.01$ and $\nu=0.1$.
Circles in (c) are shifted vertically upward for clarity.
(d) The same as in (c) for $\varepsilon \geq 10^{-35}$ and $L=I_1$ (circles), $L=I_2$ (squares),
and $L=I_3$ (triangles). Dashed line (right-side axis): average maximum $j$ of
orbits started from $\varepsilon$-uncertain points, for $L=I_1$.
}
\label{fig3}
\end{center}
\end{figure}
\section{Effective Dynamical Invariants}
\label{4}
Our results on the chain model motivate us to introduce the concept of
effective dynamical invariants. As a specific example, we consider the
{\it effective} fractal dimension, which, for the intersection of a fractal set $S$
with a $n$-dimensional region $L$, we define as
\begin{equation}
\left. D_{eff}(L;\varepsilon)= n-\frac{d \ln f(\varepsilon')}{d\ln \varepsilon'}\right|_{\varepsilon'=\varepsilon},
\label{2}
\end{equation}
where $f(\varepsilon')= N(\varepsilon')/N_0(\varepsilon')$, and
$N(\varepsilon')$ and $N_0(\varepsilon')$ are the number of cubes of edge length
$\varepsilon'$ needed to cover $S\cap L$ and $L$, respectively \cite{prev_work}.
We take $L$ to be a generic segment of line [i.e., $n=1$ in Eq. (\ref{2})] intersected
by $S$ on a fractal set.
In the limit $\varepsilon\rightarrow 0$, we recover the usual box-counting
dimension $D=1-\lim_{\varepsilon\rightarrow 0}\Delta \ln f(\varepsilon)/\Delta
\ln \varepsilon$ of the fractal set $S\cap L$, which is known to be 1 for all
our choices of $L$. However, for any practical purpose, the parameter
$\varepsilon$ is limited and cannot be made arbitrarily small (e.g., it cannot
be smaller than the size of the particles, the resolution of the experiment, and
the length scales neglected in modeling the system). At scale $\varepsilon$ the
system behaves as if the fractal dimension were $D_{eff}(L;\varepsilon)$
(therefore ``effective'' dimension). In particular, the final state sensitivity
of particles launched from $L$, with the initial conditions known within
accuracy $\varepsilon^*$, is determined by $D_{eff}(L;\varepsilon^*)$ rather
than $D$: as $\varepsilon$ is variated around $\varepsilon^*$, the fraction of
particles whose final state is uncertain scales as
$\varepsilon^{1-D_{eff}(L;\varepsilon^*)}$, which is different from the
prediction $\varepsilon^{1-D}$. This is important in this context because, as
shown in Fig.~\ref{fig3} (where the effective dimension is given by $1-\alpha$),
the value of
$D_{eff}(L;\varepsilon)$
may be significantly different from the
asymptotic value $D=1$ even for unrealistically small $\varepsilon$ and may also
depend on the region of the phase space. Similar considerations apply to many
other invariants as well.
We now return to the Hamiltonian case. Consider a scattering process in which
particles are launched from a line $L$ transversal to the stable manifold $W_s$
of the chaotic saddle. Based on the construction suggested by the chain model,
it is not difficult to see that $W_s\cap L$ exhibits a hierarchical structure
which is not self-similar and is composed of infinitely many nested Cantor sets,
each of which is associated with the dynamics inside one of the regions $C_j$.
As a consequence, the effective dimension $D_{eff}(L;\varepsilon)$ in
Hamiltonian systems is expected to behave similarly to the effective dimension
in the chain model [Figs.~\ref{fig3}(c) and \ref{fig3}(d)]. In particular,
$D_{eff}(L;\varepsilon)$ is expected to display a strong dependence on $L$ and a
weak dependence on $\varepsilon$.
\section{Numerical Verification}
\label{5}
We test our predictions on the area preserving H\'enon map: $f(x,y)=
(\lambda -y -x^2,x)$, where $\lambda$ is the bifurcation parameter. In this
system, typical points outside KAM islands are eventually mapped to infinity.
Because of the symmetry $f^{-1}=g\circ f\circ g$, where $g(x,y)=(y,x)$,
the stable and unstable manifolds of the chaotic saddle are obtained from each other
by exchanging $x$ and $y$.
For $\lambda=0.05$, the system displays a period-one and a period-four major island, as shown in
Fig.~\ref{fig4}(a). In the same figure we also show the complex invariant structure
around the islands, the stable manifold of the chaotic saddle, and three different
choices for the line of starting points: a large interval away from the islands
($L_a$), a small subinterval of this interval where the stable manifold
appears to be denser ($L_b$), and an interval closer to the islands ($L_c$).
The corresponding effective dimensions are computed
for a wide interval of $\varepsilon$. The results are shown in Fig.~\ref{fig4}(b):
$D_{eff}(L_a;\varepsilon)= 0.84$,
$D_{eff}(L_b;\varepsilon)= 0.90$, and
$D_{eff}(L_c;\varepsilon)= 0.97$ for $10^{-8}<\varepsilon< 10^{-5}$.
These results agree with
our predictions that the effective fractal dimension has the following
properties: $D_{eff}$ may be significantly different from the asymptotic
value $1$ of the fractal dimension; $D_{eff}$ depends on the
resolution $\varepsilon$ but is nearly constant over decades; $D_{eff}$ depends
on the region of the phase space under consideration and, in particular,
is larger in regions closer to the islands and in regions where the stable
manifold is denser.
Similar results are expected for any typical Hamiltonian system with mixed phase space.
\begin{figure}[pt]
\begin{center}
\epsfig{figure=fig4a.eps,width=5.0cm}
\epsfig{figure=fig4b.eps,width=5.0cm}
\caption{(a) KAM islands (blank), stable manifold (gray),
and the lines of initial conditions ($L_b$ is a subinterval of $L_a$).
(b) Effective dimension for $L=L_a$ (circles), $L=L_b$ (squares),
and $L=L_c$ (triangles). The data in (b) are shifted vertically for clarity.}
\label{fig4}
\end{center}
\end{figure}
\section{Conclusions}
We have shown that the finite-scale dynamics of Hamiltonian systems,
relevant for realistic situations, is governed by effective dynamical
invariants. The effective invariants are not only different from the asymptotic
invariants but also from the usual hyperbolic invariants because they strongly
depend on the region of the phase space. Our results are generic and expected
to meet many practical applications. In particular, our results are expected to
be relevant for fluid flows, where the advection dynamics of tracer particles is
often Hamiltonian \cite{fluid}. In this context, a slow nonuniform convergence
of effective invariants is expected not only for time-periodic flows, capable of
holding KAM tori, but also for a wide class of time-irregular incompressible
flows with nonslip obstacles or aperiodically moving vortices.
\acknowledgements
This work was supported by MPIPKS, FAPESP, and CNPq.
A. E. M. thanks Rainer Klages for illuminating discussions.
|
{
"timestamp": "2005-03-29T02:56:47",
"yymm": "0503",
"arxiv_id": "nlin/0503060",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503060"
}
|
\section{}
\section{Introduction}
As a century old theory, quantum mechanics has provided the most
effective description of the physical world. Recently, new
discoveries were found for its applications to information and
computation science \cite{Nielsen}, \textit{e.g.}, the efficient
prime factorization of larger numbers \cite{Shor} and the perfectly
secure quantum cryptography \cite{Bb84}. These, and related
developments, have highlighted a general theme that quantum
mechanics often makes impossible tasks in the classical world
possible. Conversely, some possible operations in the classical
world become impossible in the quantum world \cite{Pati2}. For
example, an unknown quantum state cannot be perfectly cloned
\cite{Wootters,Dieks}, while copies of an unknown quantum state
cannot be deleted except for being swapped into the subspace of an
ancilla \cite{Pati}.
The principle of linear superposition of states is an important
feature of quantum mechanics. A significant consequence is that an
unknown quantum state cannot be perfectly cloned, which has been
known for quite some time \cite{Wootters,Dieks}. This impossibility
can also be understood from the causality requirement that no signal
can be transmitted faster than the speed of light, even with the aid
of nonlocal quantum resource such as entanglement. With the rapid
development of quantum information science in recent years, we have
come to realize the essential role of this simple, yet profound,
limitation in quantum information processing, especially in quantum
cryptography \cite{Bb84}. Intuitively, the no-cloning theorem
implies there exists an essential difference between one copy and an
ensemble of such copies of an unknown quantum state. One cannot
obtain any information from only one copy of the quantum state
without any prior knowledge of the state. Extensive research has
focused on the no-cloning theorem related topics in quantum
information science \cite{Yuen,Barnum,Duan}. Recently, Pati
discovered another important theorem of impossibilities for an
unknown quantum state based on the principle of linear
superposition: no linear transformations on two copies of an unknown
quantum state can delete a copy except for being swapped into an
ancilla state \cite{Pati}.
In this letter, we show that yet another theorem of impossibilities
exists: quantum information of an unknown qubit cannot be split
into two complementing qubits, \textit{i.e.}, the information in one
qubit is an inseparable entity. Our paper is organized as follows:
in Sec. II we present our no-splitting problem in terms of a common
scenario from quantum secret sharing. We show that if our discussion
is restricted to only product pure final states, then the
no-splitting statement is apparently valid. Following, in Sec. III,
we consider the nontrivial case of the no-splitting problem,
\textit{i.e.}, for pure entangled final states. We then present a
no-splitting theorem for a two-qubit case and argue that the
no-splitting theorem also should be true in more general cases.
Finally, we discuss several effects and applications of our
no-splitting problem and point out possible future directions.
We note that Pati and Sanders have independently developed a similar
idea -- the no-partial erasure of quantum information -- in a recent
paper \cite{Pati3}. They claim that our non-splitting theorem
becomes a straightforward corollary of their no-partial eraser
theorem. This, however, is not the case. As demonstrated in their
example of Eq.(8), if the final state is allowed to be a mixed state
(for example due to entanglement with an ancilla), their no-partial
eraser becomes invalid. On the contrary, the final pure state can
contain entanglement between of the two (complementary) qubits for
our theorem, thus our result must supersedes their no-partial
erasure theorem. In fact, as we show in Sec. II, the no-partial
erasure theorem is valid for product states, but not for the more
general case of entangled states in Sec. III. We emphasize that the
possible existence of entanglement between the two qubits is what
makes our theorem on non-splitting of quantum information more
important.
\section{The No-splitting problem}
We start by presenting our non-splitting idea in terms of a common
scenario from quantum secret sharing: we assume that Alice and Bob
want to store and share a secret, say, an unknown spatial direction
of a qubit on the Bloch sphere, specified by its Euler angle
$(\theta,\phi)$. If this secret is initially held by Alice, she can
simply send the unknown value of $\theta$ or $\phi$ to Bob in the
classical world, and this would accomplish one simple scheme of the
secret sharing as they now each possess the complementary part of
the secret $\theta$ or $\phi$. However, this scheme as well as all
other classically allowed more sophisticated schemes is impossible
in the quantum world.
With the pseudo-spin representation on the Bloch sphere, the
unknown qubit initially held by Alice can be denoted as
\begin{equation}
|v(\theta, \phi)\rangle_A=\cos\frac {\theta} {2} |0\rangle_A +
\sin \frac {\theta} {2} e^{i \phi}|1\rangle_A.\label{unkstat}
\end{equation}
In terms of this state, the no-cloning theorem says that there
exists NO unitary transformation $\mathcal{U}$ such that
\begin{equation}
\mathcal{U}|v(\theta, \phi)\rangle_A|w\rangle_B =|v(\theta,
\phi)\rangle_A|v(\theta, \phi)\rangle_B,
\end{equation}
where $|w\rangle_B$ denotes an arbitrary given state of the
ancilla qubit $B$. The no-deleting theorem of Pati states that
there exists NO unitary transformation $\mathcal{U}$ either to
achieve the following
\begin{equation}
\mathcal{U}|v(\theta, \phi)\rangle_A|v(\theta,
\phi)\rangle_B|w\rangle_C =|v(\theta,
\phi)\rangle_A|x\rangle_B|y\rangle_C,
\end{equation}
where for clarity we have assumed two copies of the unknown state.
And, $|x\rangle_B$ and $|y\rangle_C$ are any known states.
A restricted form of the no-splitting theorem, \textit{the two real
parameters $(\theta,\phi)$ contains in one qubit can not be split
into two complementary qubits in a product state}, can be
mathematically stated as follows. There does not exist any unitary
transformation $\mathcal{U}$ such that
\begin{equation}
|\Psi\rangle_{AB}:=\mathcal{U}|v(\theta,\phi)\rangle_A|w\rangle_B=|x(\theta)\rangle_A
|y(\phi)\rangle_B.\label{distribu}
\end{equation}
When we use the linearity of $\mathcal{U}$ (from quantum mechanics),
the plausible forms for states on the right hand side of Eq.
(\ref{distribu}) are
\begin{eqnarray}
|x(\theta)\rangle_A&=&\cos \frac {\theta} {2} |x_1\rangle_A+ \sin
\frac {\theta} {2} |x_2\rangle_A,\\
|y(\phi)\rangle_B&=&|y_1\rangle_B+e^{i\phi}|y_2\rangle_B,
\end{eqnarray}
with un-normalized states $|x_1\rangle_A$, $|x_2\rangle_A$,
$|y_1\rangle_B$, and $|y_2\rangle_B$, all independent of $\theta$
and $\phi$. It is an easy exercise to conclude this kind of linear
transformation cannot exist in quantum mechanics by comparing the
LHS with the RHS of Eq. (\ref{distribu}).
The above version of no-splitting theorem for product pure final
states is valid also for more general cases with higher dimensions
and more parameters. This restricted version can indeed be derived
from the no-partial erasure theorem (Theorem 4) in Ref. [11], but
the converse is not true (Corollary 5 in Ref. [11]). We will show in
the following section that the no-partial erasure theorem is invalid
for the more general case of entangled pure final states. In
contrast, our no-splitting theorem remains valid for both cases.
\section{No-splitting theorem}
The above restricted version of the theorem is limited to
separable pure states in the RHS of Eq. (\ref{distribu}). More
generally, $|\Psi\rangle_{AB}$ can take the form of an entangled
pure state. For example, when the unitary transformation
$\mathcal{U}$ corresponds to a control-NOT gate with qubit $A$ as
the control qubit and $|w\rangle_B=|0\rangle_B$, we obtain
\begin{eqnarray}
|\Psi\rangle_{AB}&=&\frac {1} {2} \left(\cos \frac {\theta} {2}
|0\rangle_A+ \sin \frac {\theta} {2}
|1\rangle_A\right)\left(|0\rangle_B+e^{i\phi}|1\rangle_B\right) \nonumber\\
&+& \frac {1} {2} \left(\cos \frac {\theta} {2} |0\rangle_A- \sin
\frac {\theta} {2}
|1\rangle_A\right)\left(|0\rangle_B-e^{i\phi}|1\rangle_B\right),\nonumber \\
\label{examp}
\end{eqnarray}
which consists of coherent superpositions where each contains a
split state of $\theta$ and $\phi$. Does this example point to a
failure of our non-splitting idea when $|\Psi\rangle_{AB}$ is an
entangled state? No. In fact, in this case we only need to examine
the reduced density matrix of qubit $A$ and $B$, respectively. For
the state (\ref{examp}), the reduced density matrix for qubit $A$
({or} $B$) is
\begin{equation}
\rho_{A(B)}=\cos^2{\frac {\theta}
{2}}|0\rangle_{A(B)}\mbox{}_{A(B)}\!\langle0|+\sin^2{\frac
{\theta} {2}}|1\rangle_{A(B)}\mbox{}_{A(B)}\!\langle 1|,
\end{equation}
both independent of $\phi$. Thus, the above example does not provide
a counterexample to our non-splitting idea.
It is also straightforward to show that the no-partial erasure
theorem of Pati and Sanders \cite{Pati3} is no longer valid in this
case, since for the state (\ref{examp}), simply discarding one qubit
will result in a mixed state with parameter $\theta$. This
observation is trivial because a simple measurement in the
computational basis will erase the information of $\phi$. On the
other hand, as shown by the above observation, our no-splitting
theorem remains valid. We formulated our no-splitting idea into the
following theorem, which constitutes the central result of this
letter.
\begin{theorem}
There exists no two-qubit unitary transformation $\mathcal{U}$
capable of splitting an unknown qubit. In mathematical terms, the
transformed state is
\begin{equation}
|\Psi\rangle_{AB}:=\mathcal{U}|v(\theta,\phi)\rangle_A|w\rangle_B,
\label{transstat}
\end{equation}
where $|v(\theta,\phi)\rangle_A$ is defined in Eq. (\ref{unkstat}),
and $|w\rangle_B$ is an arbitrarily given pure state of qubit $B$.
This theorem then states that
\begin{eqnarray}
\textrm{
tr}_B\left(|\Psi\rangle_{AB}\mbox{}_{AB}\!\langle\Psi|\right)&=&\rho_A(\theta)\label{requir1}
\end{eqnarray}
and \begin{eqnarray}
\textrm{
tr}_A\left(|\Psi\rangle_{AB}\mbox{}_{AB}\!\langle\Psi|\right)&=&\rho_B(\phi)\label{requir2}
\end{eqnarray}
cannot be satisfied simultaneously.
\end{theorem}
We now prove this general result.\\
\textbf{Proof}: Inserting Eq. (\ref{unkstat}) into Eq.
(\ref{transstat}), we obtain
\begin{eqnarray}
|\Psi\rangle_{AB}=\cos \frac {\theta } {2}
\mathcal{U}|0\rangle_A|w\rangle_B+\sin \frac {\theta} {2}
e^{i\phi} \mathcal{U}|1\rangle_A|w\rangle_B.
\end{eqnarray}
Applying the Schmidt decomposition of a two-qubit pure state, we
immediately find
\begin{equation}
\mathcal{U}|1\rangle_A|w\rangle_B=r_0|\tilde{0}\tilde{0}\rangle_{AB}
+r_1|\tilde{1}\tilde{1}\rangle_{AB},
\end{equation}
where $|\tilde{0}\rangle_{A(B)}$ and $|\tilde{1}\rangle_{A(B)}$ are
the corresponding orthogonal basis states of the Schmidt
decomposition for qubits $A$ and $(B)$, and $r_0$ and $r_1$ are
real parameters which satisfy the normalization condition
\begin{equation}
r_0^2+r_1^2=1.
\end{equation}
Because the state $\mathcal{U}|0\rangle_A|w\rangle_B$ is
orthogonal to state $\mathcal{U}|1\rangle_A|w\rangle_B$, we deduce
that
\begin{equation}
\mathcal{U}|0\rangle_A|w\rangle_B=\alpha r_1
|\tilde{0}\tilde{0}\rangle_{AB} -\alpha r_0
|\tilde{1}\tilde{1}\rangle_{AB}
+c|\tilde{0}\tilde{1}\rangle_{AB}+d|\tilde{1}\tilde{0}\rangle_{AB},
\end{equation}
where $\alpha$, $c$, and $d$ are generally complex. They satisfy
the normalization condition
\begin{eqnarray}
|\alpha|^2+|c|^2+|d|^2=1.
\end{eqnarray}
The conditions of Eqs. (\ref{requir1}) and (\ref{requir2}) are
summarized in the following equivalent set of equations:
\begin{eqnarray}
d^* r_0 &=&0,\\
c r_1&=&0,\\
c^* r_0&=&0,\\
d r_1&=&0,\\
\alpha r_0 r_1&=&0,\\
|\alpha|^2 r_1^2 +|d|^2-r_0^2&=&0,\\
c^* \alpha r_1 -d \alpha^* r_0&=&0.
\end{eqnarray}
Suppose $r_0\neq 0$, then $c=d=\alpha r_1=0$, but $r_0^2=|\alpha|^2
r_1^2 +|d|^2=0$; therefore, $r_0=0$, which is contradictory. Now
assume $r_0=0$, which leads to $r_1\neq 0$ and $c=d=0$, then
$|\alpha|^2=( {r_0^2-|d|^2})/ {r_1^2}=0$, thus
$|\alpha|^2+|c|^2+|d|^2=0$. Again this is contradictory. Thus, there
is no self-consistent solution to Eqs. (\ref{requir1}) and
(\ref{requir2}), \textit{i.e.}, we have completed the proof of our
theorem.
When $|\Psi\rangle_{AB}$ is a product pure state, Eqs.
(\ref{requir1}) and (\ref{requir2}) reduces to Eq. (\ref{distribu}).
Theorem $1$ further indicates that the information of the amplitude
($\theta$) and the phase ($\phi$) cannot be split into two qubits by
any two-qubit unitary transformation, even for more general
(entangled) pure final two qubit states.
We speculate that the no-splitting theorem is valid for more general
cases of higher dimensional Hilbert spaces with more parameters.
This is based on the observation that the number of constraining
equations grows faster than the number of parameters; hence, in
general no solution could be expected just as we show above for the
case of two qubits.
\section{Applications and future directions}
It has been debated that some tasks of quantum information
processing can only be implemented in real Hilbert space or
restricted to equatorial states (states with the same amplitude on
all the computational basis but different phases). However, the
tasks never would work in the complete complex Hilbert space, for
example, Pati's remote state preparation protocol \cite{pati} and
its higher dimensional generalizations \cite{zz}, the $(2,2)$
quantum secret sharing protocol with pure states \cite{cleve}, and
Yao's self-testing quantum apparatus \cite {yao}. Our theorem,
therefore, provides a stronger evidence that all such tasks can
never be implemented in the whole complex Hilbert space, even
including the potential effort of transferring complex states into
real or equatorial ones. Furthermore, Grover's algorithm
\cite{grover} only calls for rotations of real angles, and Shor's
algorithm \cite{shor} requires discrete Fourier transform which only
needs transformation between equatorial states. Our theorem thus
implies that in some cases, the restricted quantum information and
computation schemes in real or equatorial space may have the same
power \cite{Ber}, or even more power, than schemes in the whole
complex Hilbert space.
Interestingly, despite such strong restrictions from the restricted
version of our no-splitting theorem or the no-partial erasure
theorem \cite{Pati3} that there exists even no probabilistic
approach for splitting or partially erasing an unknown state, the
converse procedure, \textit{i.e.}, to combine two states
\begin{equation}
\cos{\frac{\theta}{2}}|0\rangle+\sin{\frac{\theta}{2}}|1\rangle,
\frac{1}{\sqrt{2}}(|0\rangle+e^{i\varphi}|1\rangle)
\end{equation}
into one can be easily accomplished. As a simple example, we give
the following protocol starting from
\begin{equation}
\left(\cos{\frac{\theta}{2}}|0\rangle+\sin{\frac{\theta}{2}}|1\rangle\right)\otimes
\frac{1}{\sqrt{2}}(|0\rangle+e^{i\varphi}|1\rangle),
\end{equation}
executing a parity detection measurement ($ZZ$), followed by an XOR
gate, then discarding the ancillary qubit, we will reach either
\begin{equation}
\cos{\frac{\theta}{2}}|0\rangle+\sin{\frac{\theta}{2}}e^{i\varphi}|1\rangle,
\end{equation}
or
\begin{equation}
\cos{\frac{\theta}{2}}e^{i\varphi}|0\rangle+\sin{\frac{\theta}{2}}|1\rangle,
\end{equation}
both with the probability of $1/2$. We believe this interesting
observation will shed light on future investigations of the
``quantum nature" of quantum information.
In summary, we have shown that the unknown information of one copy
of a qubit cannot be split into two complementary qubits, whether
the final pure state of the two qubits is separable or entangled.
Our result demonstrates the inseparable property for quantum
information in terms of an unknown single qubit and is schematically
illustrated in Fig. \ref{fig1}. Together with the no-cloning
theorem, the no-splitting theorem shows that one qubit is an entity
that corresponds to the basic unit in quantum computation and
quantum information.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.25in]{nsfig1.eps}
\end{center}
\caption{Schematic illustration of our result for no-splitting is
in the second row, as compared to the no-cloning and its inverse
no-deleting theorems in the first row. The unknown initial qubit
is represented by the ying-yang circle together with the known
ancilla qubit represented by the empty circle on the left.}
\label{fig1}
\end{figure}
We thank Mr. P. Zhang, Ms. J. S. Tang, Prof. C. P. Sun, and Prof.
Z. Xu for useful discussions. This work was supported by the US
National Science Foundation and by the National Science Foundation
of China.
|
{
"timestamp": "2006-07-03T20:48:46",
"yymm": "0503",
"arxiv_id": "quant-ph/0503168",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503168"
}
|
\section{Introduction\label{introsec}}
A collection of journal papers is a database of papers that
comprehensively samples the journal literature of a scientific
specialty. As such, the social and epistemological processes of the
specialty are manifested in the complex network of linkages among
entities within the collection of papers. These manifestations are
studied by bibliometricians and subject matter experts to assess the
state of research in a specialty, and such studies are used to
advise managers and policy makers in both government and industry to
facilitate research management.
It is important to develop both complex network models and network
analysis tools that can be applied to collections of papers. Such
tools must be used for the problem of predicting how the underlying
processes of a research specialty are manifested in a collection of
papers, and more importantly, to perform the inverse problem of
modeling research specialty processes from their manifestations in
collections of papers. Examples of useful information about research
specialties to be extracted from collections of papers include: 1)
identifying social structures such as research teams, groups of
experts, and leaders of 'schools of thought', 2) identifying
knowledge structure, such as research subtopics, base knowledge, and
exemplars, and 3) identifying temporal trends and events such as
discoveries, emergence of new specialties and research teams,
knowledge accretion, and creation and obsolescence of concepts and
exemplars.
This paper introduces a structural model of coupled networks in
collections of journal papers and proposes a construction method for
bipartite and unipartite weighted networks from such collections.
The methods presented here constitute an important step in the
effort to apply the developing science of complex networks theory
to collections of papers and eventually to the study of scientific
specialties as complex social networks and knowledge networks.
As complex networks, collections of papers have three distinguishing
characteristics: 1) they are formed from coupled networks of many
different types of entities, e.g., papers, references, authors, 2)
both unipartite and bipartite networks in collections of papers are
best expressed as weighted networks, where strength of linkage
between pairs of entities is expressed as a positive real link
weight, and 3) collections of papers are best represented as
collections of bipartite networks.
To date, the phenomenon of coupled networks has received little
attention in the physics literature. Zheng and Ergun \cite{zheng03}
model the simultaneous growth of two loosely coupled sections of a
unipartite network and show conditions for power-law link
distributions in the crosslinks between network sections. Borner,
\emph{et al}, model the simultaneous growth of citation networks and
author collaboration networks by modeling behavior of authors
\cite{borner04}.
In contrast to the paucity of research on coupled networks, recently
a great deal of study has been focused on weighted networks. Yook,
\emph{et al} \cite{yook01}, originally investigated growing weighted
networks using preferential attachment rules and random attachment
rules. Newman \cite{newman04analysis} showed that weighted networks
could be expressed as multigraphs, and explained how this treatment
allows generalization of many analysis techniques of unweighted
networks to weighted networks. Barrat, \emph{et al}
\cite{barrat04c}, studied a large weighted author collaboration
network, and the weighted world airline network, and showed that
these networks have differences in correlations of node degrees to
strength and clustering. Other studies focus on the statistical
properties of weighted networks \cite{barrat04, bianconi04,
almaas04, jezewski04}, transport models of weighted networks
\cite{bagler04, goh05, goh04}, or growth models of weighted networks
\cite{barthelemy05, barrat04b, dorogovtsev04, antal05, fu04}. Fan,
\emph{et al} \cite{fan04},and Li, \emph{et al} \cite{fan04a},
gathered a collection of papers on the specialty of econophysics,
and studied a weighted unipartite collaboration network of authors
from that collection.
On the topic of bipartite networks, recently several papers have
reported on structural models and growth models. Ergun
\cite{Ergun02} models the human sexual contact network as a
bipartite graph, with growth having preferential attachment rules
similar to a Yule process. Ramasco, \emph{et al}, present a
bipartite Yule model for paper to author networks \cite{ramasco04}.
Guillaueme and Latapy \cite{guillaume04a} also present a bipartite
Yule model and propose a method of deriving a bipartite expression
of any unipartite network. Morris \cite{morris05a} proposes the use
of general bipartite Yule processes for entity-type pairs in
collections of journal papers, and gives examples for paper to
reference networks and paper to author networks. Morris
\cite{morris04a} also gives a detailed analysis of a bipartite Yule
model for paper to reference networks that models heavily cited
exemplar references in emerging specialties. Goldstein, \emph{et
al}, \cite{goldstein04group} and Morris, \emph{et al},
\cite{morris04b} propose bipartite Yule models for paper to author
networks that model the success-breeds-success phenomenon for teams
of authors.
As shown in Figure \ref{coupled}, a collection of journal papers
constitutes a series of coupled bipartite networks. As diagrammed in
the figure, a collection of papers contains 6 direct bipartite
networks: 1) papers to paper authors, 2) papers to references, 3)
papers to paper journals, 4) papers to terms, 5) references to
reference authors, and 6) references to reference journals.
Additionally, there are 15 indirect bipartite networks in
collections of papers as defined by the diagram. Examples of
interesting indirect networks are paper authors to reference
authors, and paper journals to reference journals networks, which
can be used for author co-citation analysis \cite{white81} and
journal co-citation analysis \cite{mccain91} respectively.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{bipart1.eps}}%
\caption{Diagram showing a collection of papers as a series of
coupled bipartite networks.\label{coupled}}
\end{figure}
This paper introduces a formal matrix-based treatment of coupled
bipartite structures in collections of papers. This treatment is
used to calculate the weights of indirect bipartite networks and is
extended to calculation of weights of unipartite co-occurrence
networks in the collection. For example, the proposed method can be
used to calculate the weights of a bipartite paper author to
reference network, or, it can be used to find weights of the
unipartite co-occurrrence network of authors that link to common
papers (a co-authorship network).
The proposed matrix-based technique is similar to multi-port
analysis using ABCD parameters in electrical networks
\cite{chirlian69}. The method is also very similar to methods used
in multi-layer neural networks \cite{hagan96}.
In conjunction with simple bipartite Yule growth models
\cite{morris05a}, the proposed weight calculation method produces
simple models of weighted network growth, growing as it does from
unweighted direct links that occur as papers are added to the
collection.
\section{Collections of journal papers}
\subsection{Research specialties}
A \textit{research specialty} is a self-organized social
organization whose members tend to study a common research topic,
attend the same conferences, publish in the same journals, cite each
other's work, and belong to the same social networks that are known
as \textit{invisible colleges} \cite{crane72}. Thomas Kuhn, the
pioneer of the study of research processes, considered specialties
to be quite small, "100 members, sometimes considerably less"
\cite{kuhn70}.
The processes that drive research specialties are twofold: 1) social
processes of research teams, communication networks, and
collaboration, and 2) epistemological processes of the discovery,
emergence, accretion, and obsolescence of knowledge. As described by
Kuhn, the distinguishing feature of a specialty is its
\textit{paradigm}, which is the researchers' "way of thinking" about
their problem: models, analytical techniques, validation standards
and so forth. Progress in a specialty is characterized by long and
stable periods of \textit{puzzle-solving} within the specialty's
paradigm, punctuated by discoveries that accompany the overthrow
and/or creation of new paradigms \cite{kuhn70}. This characteristic
of specialties is similar to \textit{punctuated equilibria}
phenomena \cite{eldredge72} that characterize self-organizing
systems \cite{bak96}.
Specialties create their own \emph{literature}, i.e., a body of
journal papers and books that broadly focus on the specialty's
research topic. We define a \textit{collection of papers} as a list
of journal papers that constitutes a comprehensive sample of a
specialty's journal literature. As a working definition, define a
collection of papers as a database of records, one record per paper,
that contains information about the individual papers in such a
list.
Although the range of size of such collections is large, the size of
such collections is much smaller than the immense databases of
papers that are often studied in the physics literature. Morris
\cite{morris04a}, using back-of-envelope style approximations,
suggests that collections of papers should range from as few as 100
papers to as many as 5000 papers. Huge heterogeneous datasets, such
as the SPIRES database \cite{redner98}, 20 years of PNAS papers
\cite{borner04}, 100 years of Physical Review journals
\cite{redner04}, or all the chemistry publications of the
Netherlands \cite{vanraan01}, are not collections of papers as
defined here, because they all sample more than one specialty's
literature. Despite this conceptual constraint, the weight
calculation method proposed here can still be applied to such huge
collections.
\subsection{Definition of collections of journal papers} For
discussion in this paper, a collection of journal papers is a
database where each record corresponds to a journal paper. For each
paper, its associated authors, cited references, journal, index
terms and publication year are listed. Furthermore, for each
reference, a reference author, reference journal, and reference year
are listed. As defined here, collections of papers are constructed
to comprehensively sample the literature of a scientific specialty.
For our purposes, collections of papers are typically downloaded
from the Science Citation Index using Thompson/ISI's Web of Science
product \footnote{http://www.isinet.com}. Queries and seed
references are used to gather topic specific collections that cover
a specialty. The records for these papers are typically collected
into text files using a tagged file format and downloaded for
analysis. For the purpose of demonstrating the concepts proposed in
this paper, a fictitious collection of four papers is given in the
Appendix that covers the fictional specialty of
\textit{improbability generation}. (Apologies to humor author
Douglas Adams.) This example collection is provided to allow readers
to understand the extraction of entities and links from the source
data of the collection. For illustrative purposes the entities in
this example are more densely linked than would normally be found in
such a small collection of papers.
A collection of papers can be considered as a network of
\textit{bibliographic entities} of various \textit{entity-types}
\cite{morris04crossmaps}. Bibliographic entities may correspond to
\textit{physical entities} in the real world, and more than one
bibliographic entity may correspond to the same physical entity. For
example, a paper and a reference in a collection of papers may both
correspond to the same physical paper in the real world.
It is common in studies of networks in journal literature to match
references to papers to build a model of "papers citing papers",
usually referred to as a \textit{citation network} \cite{albert02}.
There are both methodological and theoretical reasons to avoid this
type of treatment: 1) on one hand, a collection of papers typically
has 20 times more references than papers, making such citation
network models grossly incomplete because unmatched papers and
references (including references corresponding to books), have
unknown incoming and outgoing links, 2) the second problem is that
references, especially highly cited references, can be considered as
\textit{concept symbols} \cite{morris04a, small78}, and therefore
should be considered as separate entity-types from papers, which
merely represent undifferentiated research reports. Figuratively, it
is inappropriate to use an "apples-citing-apples" model when the
actual network is "apples-citing-oranges." Further discussion of
citation networks is outside the scope of this paper.
For our proposed structural model of collections of journal papers
presented in this paper, we will limit our discussion to a model
comprised of 7 entity-types: 1) papers, 2) paper authors, 3) paper
journals, 4) index terms, 5) references, 6) reference authors and 7)
reference journals. Index terms are terms supplied by authors or
abstract services to associate with papers for search and
classification purposes. Paper authors are the authors of papers,
while reference authors are the authors associated with references.
Paper journals are the journals that papers are published in, while
reference journals are the journals associated with references.
References corresponding to books, films, web pages, and eprint
archive articles have no associated reference journal.
Using the 7 entity-types given in our structural model, Figure
\ref{coupled} illustrates that a collection of journal papers
constitutes a series of coupled bipartite networks. As noted in
Section \ref{introsec}, there are 6 direct bipartite networks and 15
indirect bipartite networks in this structural model. These indirect
bipartite networks are best analyzed as weighted networks and those
weights can be calculated from the paths of direct links that
connect entities in the two partitions of interest.
Note the fictitious collection of papers in the Appendix. The source
file for this collection, which consists of 4 papers, is listed in
ISI tagged file format. See footnote \footnote{A set of MATLAB
routines that can extract several types of bipartite networks from
ISI tagged files is available from the authors. Please contact one
of the authors for further information}. The extracted entities for
this collection consists of 4 papers, 3 paper authors, 4 paper
journals, 7 index terms, 10 references, 6 reference authors, and 7
reference journals. These entities and their corresponding index
numbers are listed in the Appendix.
\section{Bipartite networks in collections of journal papers}
\subsection{Dyad definitions} In a dyad, the two entities can be: 1) \textit{like entities}, that is,
entities of the same entity-type, or 2) \textit{unlike entities},
that is, entities of different entity-types. \textit{Direct links}
are defined as direct associations. A paper has direct links to its
authors (paper authors), its associated index terms, the references
the paper cites, and the journal the paper was published in. A
reference is directly linked to the papers that cite it, the author
associated with the reference (reference author), and the journal
that is associated with the reference (reference journal).
\textit{Indirect links} are links between two unlike entities that
occur over a path of two or more direct links. For example, a paper
author is indirectly linked to a reference author if he or she
authors a paper that cites a reference that is associated with that
reference author.
The first entity of interest in a dyad is the \textit{primary
entity} while the other entity is the \textit{secondary entity}.
Designation of primary entity-type and secondary entity-type in
direct and indirect bipartite networks is arbitrary and is assumed
to be based on the interest of the investigator. For
\emph{co-occurrence networks}, the primary and secondary
entity-types are explicitly defined, as will be explained in Section
\ref{cooccursec}. \textit{Co-occurrence links} are between like
primary entities and occur when both entities link to the same
secondary entity. For example, two papers have a co-occurrence link
when they both cite a common reference, or, in another example, two
paper authors have a co-occurrence link if they coauthor a paper. In
co-occurrence links the like entities of the dyad are primary
entities, while the unlike entities to which they co-link are the
secondary entities.
\subsection{Dyad identifier notation} Table \ref{deftab} lists the
conventions used here to denote entity-type variables within a
collection of papers. The variables $x_1$, $x_2$, and so forth will
be used to denote unspecified entity-types. \textit{Dyad notation}
is used to specify dyad types in the collection of papers. The
symbols of primary and secondary entity-types associated with dyads
are separated by a comma and placed between square brackets, e.g.,
$[x_1,x_2]$, where $x_1$ denotes the primary entity-type, and $x_2$
denotes the secondary entity-type. This notation will be referred to
as the \textit{dyad identifier}, and will be used as a suffix to
variables to specify the entity-types of interest. However, the dyad
identifier will be dropped to reduce clutter in the notation when
the primary and secondary entity-types are obvious from context.
Some examples of the use of dyad identifiers:
\begin{itemize}
\item $\mathbf{O}[p,r]$ denotes an occurrence matrix listing the links of
papers, the primary entity-type, to references, the secondary
entity-type.
\item$\mathbf{C}[ap,p]$ denotes the co-occurrence matrix listing the
co-authorship counts of pairs of paper authors, the primary
entity-type, in papers, the secondary entity-type.
\end{itemize}
\begin{table}
\caption{Variable conventions used for entities in collections of
papers.\label{deftab}}
\begin{tabular}{|p{.2\textwidth}l|p{.2\textwidth}l|} \hline
$p$: paper & $r$: reference\\
$ap$: paper author & $ar$: reference author\\
$jp$: paper journal & $jr$: reference journal\\
$yp$: paper year & $yr$: reference year\\
$t$: term & \\
$x_i$: unspecified entity & \\ \hline
\multicolumn{2}{|p{.45\textwidth}|}{Prefix '$n$' to any entity
variable to denote the number of entities in the
collection of that entity-type, e.g., $np$ denotes the number of papers in the collection} \\
\hline
\end{tabular}
\end{table}
\subsection{Bipartite networks} Bipartite
networks are comprised of two distinct partitions of nodes, where
all links in the network are from entities in the first partition to
entities in the second partition. For our purposes, the first
partition exclusively holds entities of some entity-type, while the
other partition exclusively holds entities of some other
entity-type. As an example, Figure \ref{f3} shows a diagram of a
bipartite network of a partition of papers linked to a partition of
references. Note that links only occur between papers and
references and that there are no links between pairs of papers or
pairs of references.
\begin{figure}
\resizebox{0.25\textwidth}{!}{%
\includegraphics{figure3.eps}}%
\caption{A collection of papers and references as a bipartite
network. References are linked to papers in which they are
cited.\label{f3}}
\end{figure}
Assume the diagrammatic convention as shown in Figure \ref{f4}, that
entities of $x_1$, the primary entity-type, are the entities in the
group on the left and the entities of $x_2$, the secondary
entity-type, are the entities in the group to the right. There are
$nx_1$ primary entities and $nx_2$ secondary entities. The strength
of the link between $x_1$ entity $i$ and $x_2$ entity $j$ is the
link weight, $o_{ij}[x_1,x_2]$.
\subsection{Occurrence matrices} Mathematically, the links in a
bipartite network are described by a rectangular adjacency matrix,
which we'll define as an \textit{occurrence matrix}. This is an
$nx_1$ by $nx_2$ matrix that lists all the link weights between the
entities of the two partitions:
\begin{equation}
\mathbf{O}[x_1,x_2]=
\left[
\begin{array}{cccc}
o_{11} & o_{12} & \dots & o_{1nx_2} \\
o_{21} & \ddots & & \vdots\\
\vdots & & \ddots & \vdots \\
o_{nx_11} & \dots & \dots & o_{nx_1nx_2} \\
\end{array}
\right]
\end{equation}
Figure \ref{f4} shows how the links in a bipartite network
correspond to elements in its occurrence matrix. There is a
bipartite network for every possible pair of entity-types in the
collection of papers. Occurrence matrices for entity-type pairs with
direct relations are derived directly from the tables in the
collection's database. For the example collection of papers
discussed in this paper, the occurrence matrices for the 6 direct
bipartite networks in the collection are given in the Appendix.
Occurrence matrices for entity-type pairs with indirect links are
calculated by cascading bipartite networks of direct links, as will
be shown later.
\begin{figure}
\resizebox{0.25\textwidth}{!}{%
\includegraphics{figure4.eps}}%
\caption{Diagram of a general bipartite network and conventions for
labeling link weights in the occurrence matrix of the network.
\label{f4}}
\end{figure}
Note the following property of occurrence matrices:
\begin{equation}
\mathbf{O}[x_1,x_2]=\mathbf{O}[x_2,x_1]^T \label{eq26}
\end{equation}
Using dyad identifier notation, exchanging the variables is
equivalent to transposing the occurrence matrix.
\subsection{Coupled and cascaded bipartite
networks\label{coupledsec}} \textit{Coupled bipartite networks} are
pairs of bipartite networks that share a common partition. Figure
\ref{coupled1} shows an author to paper network coupled to a paper
to reference network through common papers using the example
collection of papers in the Appendix. \textit{Cascaded bipartite
networks} are comprised of a series of two or more coupled bipartite
networks. Figure \ref{cascade} shows an example of such a cascade,
where a reference author to reference network is coupled to a
reference to paper network that is in turn coupled to a paper to
paper author network. We define the extreme left and right
partitions as the \textit{outer partitions} and all other partitions
as the \textit{inner partitions}.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{couple_ap_p_r.eps}}%
\caption{An example of coupled bipartite networks. A paper
author-paper network is coupled to a paper-reference network through
common papers. This example is taken from the example collection in
the Appendix.\label{coupled1}}
\end{figure}
Assume that we are interested in describing the links between two
different types of entities as a weighted bipartite network. We
first find a cascade of networks where the two entity-types of
interest are the outer partitions. Then it is necessary to apply
some algorithm that meaningfully reduces the indirect links between
pairs of opposite outer entities as weights in a bipartite network
joining those outer entities. Intuitively, we want pairs of outer
entities that have many indirect links through the inner partitions
to have more weight than those pairs of outer entities with few or
no connecting links.
For example, suppose that we wish to find a weighted bipartite
network between reference authors and paper authors for the purpose
of conducting author co-citation analysis \cite{white81}. We can
find a cascade of bipartite networks as shown in Figure
\ref{cascade}, where reference authors are linked to their
references, the references are linked to the papers that cite them,
and those papers are linked to the paper authors that authored them.
The weights of a bipartite network of reference authors to paper
authors are found by finding the indirect links between each
reference author and paper author through references and papers, and
applying an algorithm that produces a weight from those identified
indirect links. The more indirect links between a reference author
and a paper author, the more weight should be assigned to the link
between them in the resulting bipartite network.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{cascade.eps}}%
\caption{An example of a cascade of bipartite networks. A reference
author to reference network is coupled to a reference to paper
network that is, in turn, coupled to a paper to paper author
network.\label{cascade}}
\end{figure}
\section{Algorithm for construction of weighted bipartite networks\label{constructsec}}
\subsection{Reducing a cascade of bipartite networks to a single
weighted bipartite network}
Given a cascade of bipartite networks with occurrence matrices
$\mathbf{O}[x_1,x_2]$, $\mathbf{O}[x_2,x_3],\dots,
\mathbf{O}[x_{n-1},x_n]$, this cascade can be reduced to a single
bipartite network with occurrence matrix $\mathbf{O}[x_1,x_n]$
listing the link weights between the $x_1$ entities and the $x_n$
entities in the network. The proposed weight algorithm is iterative
and works by sequentially reducing two adjacent networks to a single
network, then reducing that weighted network and its adjacent
network. This process continues until only a single bipartite
network remains.
The algorithm is based on using a generalized form of matrix
arithmetic. Given a pair of opposite outer entities, the algorithm
finds all unique paths from the left outer entity to the right outer
entity, and assigns a weight to each of those paths. The weights of
these parallel paths are then combined to calculate the weight of
the link between the two entities.
\subsection{Reducing adjacent coupled bipartite networks to a single weighted bipartite network}
Consider a pair of coupled bipartite networks, with entity-types
$x_1$, $x_2$, and $x_3$, as shown in Figure \ref{f5}. Occurrence
matrices $\mathbf{O}[x_1,x_2]$ and $\mathbf{O}[x_2,x_3]$ enumerate
the links in the two bipartite networks in this figure. Each link in
the figure is labeled with its corresponding occurrence matrix
element. There are $nx_1$, $nx_2$, and $nx_3$ entities of the
entity-types $x_1$, $x_2$, and $x_3$ respectively. A pair of links
that connects an $x_1$ entity to an $x_3$ entity is defined as a
\emph{path}. Figure \ref{f6}, part (a) shows a path from $x_1$
entity $i$ to $x_3$ entity $j$, connected through $x_2$ entity $k$
by links $o_{ik}[x_1,x_2]$ and $o_{kj}[x_2,x_3]$. There are $nx_2$
possible paths from $x_1$ entity $i$ to $x_3$ entity $j$ as shown in
Figure \ref{f6} part (b).
\begin{figure}
\resizebox{0.4\textwidth}{!}{%
\includegraphics{figure5.eps}}%
\caption{Diagram of adjacent bipartite networks and conventions for
naming entities and links.\label{f5}}
\end{figure}
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure6.eps}}%
\caption{ a) Example path between $x_1$ entity $i$ and $x_3$ entity
$j$ through $x_2$ entity $k$. b) Shows $nx_2$ possible paths between
$x_1$ entity $i$ and $x_3$ entity $j$ through $x_2$
entities.\label{f6}}
\end{figure}
The \textit{path weight} associated with a path is calculated from
the weights of the path's two links using a \textit{path weight
function}:
\begin{equation}
p_{ij}(k)=f_2(o_{ik}[x_1,x_2], o_{kj}[x_2,x_3])\label{eq1},
\end{equation}
where $f_2$ is the path weight function, to be defined later. The
resulting link weight from $x_1$ entity $i$ to $x_3$ entity $j$ is
calculated from the path weights of all possible paths between those
two entities using a \textit{path combining function}:
\begin{equation}
o_{ij}[x_1,x_3]=f_1\Big(p_{ij}(1), p_{ij}(2), \dots
p_{ij}(nx_2)\Big), \label{eq2}
\end{equation}
where $f_1$ is the path combining function, to be defined later.
Substituting Equation (\ref{eq1}) into Equation (\ref{eq2}) gives
the \textit{link weight function} which defines the rules for
calculating link weights of cascaded bipartite networks:
\begin{multline}
o_{ij}[x_1,x_3]=\\
f_1 \Big( f_2(o_{i1},o_{1j}), f_2(o_{i2},o_{2j}) ,\dots,
f_2(o_{i\,nx_2},o_{nx_2\, j})\Big). \label{eq3}
\end{multline}
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure7.eps}}%
\caption{Diagram illustrating vector operation of the link weight
function.\label{f7}}
\end{figure}
The link weight function of Equation \ref{eq3} is a matrix function
that is used to compute all the $nx_1$ times $nx_3$ possible weights
of the occurrence matrix $\mathbf{O}[x_1,x_3]$ according to the
rules for weight computation given by $f_1$ and $f_2$. Consider
Figure \ref{f7} which illustrates how the link weight function uses
row $i$ of $\mathbf{O}[x_1,x_2]$ and column $j$ of
$\mathbf{O}[x_2,x_3]$ to produce element $o_{ij}$ of matrix
$\mathbf{O}[x_1,x_3]$. As shown, the function $f_2$ is applied to
matching elements of the row vector and column vector to produce
$nx_2$ scalar results. The function $f_1$ operates on all these
$nx_2$ results to produce the final scalar result $o_{ij}[x_1,x_3]$.
The concepts of 1) bipartite networks of entities, 2) cascaded
bipartite networks, and 3) link weight functions, provide a
systematic means of finding multiple indirect links between outer
entities in cascades of bipartite networks, and combining those
multiple links as a weight in a bipartite network between the outer
entities. The choice of path weight function and path combining
function is generally driven by the application. In the case of
cascades of unweighted bipartite networks, matrix multiplication
makes a good link weight function because it yields weights that are
equal to occurrence counts. For example, for a paper to reference
network coupled to a reference to reference author network, matrix
multiplication as a link weight function will produce weights,
$o_{ij}[p,ar]$, that are the the number of times paper $i$ cites
reference author $j$.
In other situations, however, other link weight functions are more
appropriate. For example, when reducing cascades of weighted
bipartite networks, it is necessary to consider how to compute path
weights from the two links in a path. Suppose we have a weighted
bipartite network of \emph{linguistic terms} to papers in a
collection of papers. The weights, $o_{ij}[t,p]$, in this network
are the number of times term $i$ appears in the body of paper $j$.
Now assume this matrix is coupled to a paper to reference author
network, and that there is a path from term $i$ to reference author
$j$ that corresponds to 10 occurrences of term $i$ in paper $k$,
which cites reference author $j$ 2 times. If we use multiplication
as the path weight function, then this yields $10 \times 2 = 20 $
for the path weight. This has no meaning as an occurrence count
between term $i$ and reference author $j$. In this case we may want
to simply use a link weight equal to the number of times reference
author $j$ is cited by paper $k$, or use a link weight equal to the
minimum of the number of times paper $k$ cites reference author $j$
and the number of times term $i$ occurs in paper $k$. We can also
express the two links in the path as electrical conductances and
calculate the path weight as the resulting conductance of those two
conductances in series.
The next three subsections will describe three link weight
functions: 1) matrix multiplication, appropriate for cascades of
unweighted networks, 2) the overlap function, appropriate for
cascades of weighted occurrence networks, and 3) the inverse
Minkowski function, used to compute paths weights as similar to
conductances in series.
\subsection{Link weight function using matrix multiplication}
For applications where at least one of the matrix arguments is
binary, matrix multiplication is often used as the link weight
function because it directly yields weights that are simple
occurrence and co-occurrence counts in the resulting reduced
bipartite matrix.
If the path weight function $f_2$ is defined as a product:
\begin{equation}
f_2\biggl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\biggr)=o_{ik}[x_1,x_2]\cdot
o_{kj}[x_2,x_3] \label{eq5}
\end{equation}
and the path combining function $f_1$ is a summation:
\begin{multline}
f_1\Bigl(f_2\bigl(o_{i1}[x_1,x_2],o_{1j}[x_2,x_3]\bigr),\dots, \\
f_2\bigl(o_{i\,nx_2}[x_1,x_2],o_{nx_2\,j}[x_2,x_3]\bigr)\Bigl) \\
= \sum_{k=1}^{nx_2}f_2\bigl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\bigr).\label{eq6}
\end{multline}
Then the link weight function is simply standard matrix
multiplication:
\begin{equation}
o_{ij}[x_1,x_3]=\sum_{k=1}^{nx_2} o_{ik}[x_1,x_2]\cdot
o_{kj}[x_2,x_3]. \label{eq7}
\end{equation}
As an example, assume that $x_1$, $x_2$, and $x_3$ are paper
authors, papers and references respectively, taken from the example
collection of papers in the Appendix. The binary matrix
$\mathbf{O}[ap,p]$, the transpose of $\mathbf{O}[p,ap]$, Equation
(\ref{opap}), lists the links of the individual paper authors to
each paper, while the binary matrix $\mathbf{O}[p,r]$, Equation
(\ref{opr}), lists the links of individual papers with each
reference. Using matrix multiplication:
\begin{equation}
\mathbf{O}[ap,r]=\mathbf{O}[ap,p]\cdot \mathbf{O}[p,r]. \label{eq10}
\end{equation}
This yields:
\begin{eqnarray}
\mathbf{O}[ap,r] &=& \left[
\begin{array}{cccc}
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
\end{array}
\right]
\left[\begin{array}{cccccccccc}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{array}\right] \nonumber \\
\nonumber \\
&=& \left[
\begin{array}{cccccccccc}
2 & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
3 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 \\
\end{array}
\right]. \label{eq11}
\end{eqnarray}
This is a matrix, $\mathbf{O}[ap,r]$, in which weight,
$o_{ij}[ap,r]$, is the number of times that paper author $i$ cites
reference $j$.
Suppose we wish to find the paper author to reference author
occurrence matrix of the example collection of papers in the
Appendix. Consulting Figure \ref{coupled}, the direct links from
paper authors to reference authors go from paper author to paper to
reference to reference author. Calculation of the occurrence matrix,
$\mathbf{O}[ap,ar]$, from paper author to reference author is
performed by the matrix multiplication:
\begin{equation}
\resizebox{0.35\textwidth}{!}{%
\includegraphics{eq30.eps}}%
.\end{equation}
Using the example paper collection in the Appendix, first find the
paper author to reference matrix by multiplying the paper author to
paper matrix and the paper to reference matrix. This was done in
Equation (\ref{eq11}). Then multiply the paper author to reference
matrix with the reference to reference author matrix:
\begin{eqnarray}
\mathbf{O}[ap,ar]=\mathbf{O}[ap,r]\cdot\mathbf{O}[r,ar]&=& \nonumber \\
\nonumber \\
\left[
\begin{array}{cccccccccc}
2 & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
3 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 \\
\end{array}\right]
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}\right] &=& \nonumber \\
\nonumber \\
= \left[\begin{array}{cccccc}
2 & 3 & 2 & 1 & 0 & 0 \\
1 & 1 & 2 & 1 & 0 & 0 \\
3 & 3 & 4 & 4 & 1 & 1 \\
\end{array}\right]\label{eq32}
.\end{eqnarray}
The result in Equation (\ref{eq32}) gives the desired occurrence
matrix of paper authors to reference authors for the example. In
this matrix, the weight $o_{ij}[ap,ar]$ is the number of times that
paper author $i$ cites reference author $j$.
\subsection{Link weight function using the overlap
function\label{overlapsec}} The overlap function is useful for
calculating weights of links when reducing cascades of weighted
bipartite networks. This is appropriate for calculating bipartite
networks involving linguistic terms, and is also useful for
calculating weights in co-occurrence networks of reference authors
and reference journals.
Think of the two links in a path as conduits, each with a maximum
capacity. The maximum capacity of these two conduits in series is
equal to that of the conduit with the smallest capacity. Considering
this series capacity as the path weight, the path weight function
becomes the minimum of the weights of the two links on the path:
\begin{equation}
f_2=min\Bigl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\Bigr)\label{eq12}
.\end{equation}
Using a path combining function that sums the path
weights:
\begin{equation}
f_1=\sum_{k=1}^{nx_2} f_2\Bigl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\Bigr)
\label{eq13} ,\end{equation} yields the overlap function
\cite{salton89} as the link weight function:
\begin{equation}
f_1=\sum_{k=1}^{nx_2} min\Bigl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\Bigr)
\label{eq14} .\end{equation}
This can be defined as a matrix
operation "OVL":
\begin{equation}
\mathbf{O}[x_1,x_3]=OVL\Bigl(
\mathbf{O}[x_1,x_2],\mathbf{O}[x_2,x_3] \Bigr) \label{eq15}
.\end{equation}
Discussion of the application and characteristics of
this function can be found in \cite{jones87}.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure9.eps}}%
\caption{Example of cascaded bipartite networks with non-binary link
weights. Terms to paper network cascaded with paper to reference
author network.\label{f9}}
\end{figure}
As an example, assume that $x_1$, $x_2$, and $x_3$ are linguistic
terms, papers and reference authors respectively, as shown in Figure
\ref{f9}. The matrix $\mathbf{O}[t,p]$ lists the occurrence counts
of the individual terms with each paper:
\begin{equation}
\mathbf{O}[t,p]=
\left[
\begin{array}{cccc}
3 & 5 \\
2 & 6 \\
1 & 9 \\
\end{array}
\right]\label{eq16} ,
\end{equation} and the matrix
$\mathbf{O}[p,ar]$ lists the associations of individual papers with
each reference author:
\begin{equation}
\mathbf{O}[p,ar]=
\left[
\begin{array}{cccc}
2 & 3 & 0 \\
0 & 4 & 1 \\
\end{array}
\right]\label{eq17} .\end{equation} Using the overlap function to
calculate the link weights of $\mathbf{O}[t,ar]$:
\begin{eqnarray}
\mathbf{O}[t,ar]&=&OVL\Bigl(\mathbf{O}[t,p],\mathbf{O}[p,ar]\Bigr)\nonumber \\
\nonumber\\
\mathbf{O}[t,ar]&=&OVL\left(
\left[
\begin{array}{cc}
3 & 5 \\
2 & 6 \\
1 & 9 \\
\end{array}\right],
\left[
\begin{array}{ccc}
2 & 3 & 0 \\
0 & 4 & 1 \\
\end{array}
\right] \right)
=\left[\begin{array}{ccc}
2 & 7 & 1 \\
2 & 6 & 1 \\
1 & 5 & 1 \\
\end{array}\right].
\label{eq19}\nonumber \\
\end{eqnarray}
\subsection{Link weight function using the inverse Minkowski
function\label{minkowskisec}}
The \textit{inverse Minkowski function}, an adaptation of the
well-known Minkowski distance metric \cite{cios98}, can be used when
it is desired to model path weights as if the link weights were
electrical conductances in series. In this case use the inverse
Minkowski metric as the path weight function:
\begin{equation}
f_2=\left[{ \Bigl( {o_{ik}[x_1,x_2]} \Bigr) }^{-p} +
{\Bigl(o_{kj}[x_2,x_3]\Bigr)}^{-p} \right]^{-\frac{1}{p}}
\label{eq20} ,\end{equation} where $p$ ranges from zero to positive
infinity. Note that, in contrast to the Minkowski metric as
normally expressed, the exponents in the inverse Minkowski metric
are negative. This function will always generate a path weight that
is less than or equal to the smallest link weight in the path,
modeling a situation where indirect links tend to be weaker than
direct links. Using a path combining function that sums the path
weights:
\begin{equation}
f_1=\sum_{k=1}^{nx_2} f_2\Bigl(o_{ik}[x_1,x_2],o_{kj}[x_2,x_3]\Bigr)
\label{eq21}
\end{equation}
yields the final inverse Minkowski link weight function:
\begin{equation}
o_{ij}[x_1,x_3]=\sum_{k=1}^{nx_2}{ \left[{ \Bigl( {o_{ik}[x_1,x_2]}
\Bigr) }^{-p} + {\Bigl(o_{kj}[x_2,x_3]\Bigr)}^{-p}
\right]^{-\frac{1}{p}} }.\label{eq22}
\end{equation}
This can be defined as a matrix operation "$INVMINK$":
\begin{equation}
\mathbf{O}[x_1,x_3]=INVMINK
\Bigl(\mathbf{O}[x_1,x_2],\mathbf{O}[x_2,x_3]\Bigr) \label{eq23}
.\end{equation}
When this function is used with $p=\infty$, Equation (\ref{eq20})
produces the minimum of its arguments and so reverts to Equation
(\ref{eq12}), making the inverse Minkowski link weight function
revert to the overlap link weight function. When p = 1, then the
path weight function, Equation (\ref{eq20}), becomes:
\begin{equation}
f_2=\left[{ \frac{1}{o_{ik}[x_1,x_2]} + \frac{1}{o_{kj}[x_2,x_3]} }
\right]^{-1} \label{eq24} .\end{equation} This makes the path weight
function produce a value that is twice the harmonic average of the
link weights of the path. This is equivalent to calculating the
path weight by modeling the link weights as electrical conductances
in series.
The inverse Minkowski path weight function always produces a path
weight that is less than the smallest weight on the path. This is
appropriate in situations where indirect paths should have less
weight than direct paths, and mathematically expresses a sensed
diffusion, or weakening, of the strength of linkage when linkage is
indirect.
\subsection{Weights in unipartite co-occurrence
networks\label{cooccursec}}
\emph{Co-occurrence networks} are weighted unipartite networks of
like entities where the links between pairs of entities is the count
of the number of common secondary entities that the two primary
entities both link to. For example, in a \textit{bibliographic
coupling network}, the nodes are papers, and the link weights are
the number of common references cited by each pair of papers. A
\textit{co-occurrence matrix} is the adjacency matrix of a
co-occurrence network. For binary occurrence matrices the
co-occurrence matrix can be found by post multiplying the occurrence
matrix by its transpose. Using Equation (\ref{eq26}):
\begin{equation}
\mathbf{C}[x_1,x_2]=\mathbf{O}[x_1,x_2]\cdot\mathbf{O}[x_2,x_1]\label{eq44}
,\end{equation} where $\mathbf{C}[x_1,x_2]$ is the co-occurrence
matrix listing the number of common associations of pairs of $x_1$
entities with $x_2$ entities. For example, to calculate the
co-occurrence of papers by their links to references using the paper
to reference matrix from the example collection in the Appendix, use
Equation (\ref{opr}):
\begin{eqnarray}
\mathbf{C}[p,r]=\mathbf{O}[p,r]\cdot\mathbf{O}[r,p]&=& \nonumber \\
\nonumber \\
\left[\begin{array}{cccccccccc}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{array}\right]
\left[\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
\end{array}\right] &=& \nonumber \\
\nonumber \\
= \left[\begin{array}{cccc}
3 & 2 & 2 & 2 \\
2 & 5 & 3 & 2 \\
2 & 3 & 5 & 2 \\
2 & 2 & 2 & 6 \\
\end{array}\right]
.\label{eq45}
\end{eqnarray}
The diagonal of the co-occurrence matrix $c_{ii}[x_1, x_2]$ lists
the number of links that each $x_1$ has with entities of the $x_2$
entity-type. For example, in the bibliographic coupling matrix,
$\mathbf{C}[p,r]$, calculated in Equation (\ref{eq45}), the diagonal
lists the number of references each papers cites.
Computation of co-occurrences can be viewed, similar to the
discussion of Section \ref{coupledsec}, as the calculation of link
weights in a cascade of two bipartite networks. Given a bipartite
network of two unlike entity-types, mirror the network across the
secondary entity-type partition to obtain a cascade of two networks.
For example, the paper to reference network shown in Figure \ref{f3}
has been mirrored on the references to produce the
paper-reference-paper cascade of two bipartite networks shown in
Figure \ref{f13} (a). Calculating the weights of this cascade using
matrix multiplication will produce the co-occurrence counts of
papers' links to references, bibliographic coupling strength
\cite{kessler63}, as was done in Equation (\ref{eq45}).
\begin{figure}
\resizebox{0.5\textwidth}{!}{%
\includegraphics{figure13.eps}}%
\caption{Mirror of paper to reference bipartite network to calculate
weights in a unipartite co-occurrence network as a cascade of two
bipartite networks. (a) Mirror across references to calculate
bibliographic coupling. (b) Mirror across papers to calculate
co-citation.\label{f13}}
\end{figure}
The same network of Figure \ref{f3} can be mirrored on the papers to
produce the reference-paper-reference cascade of bipartite networks
shown in Figure \ref{f13}(b). Calculating the link weights in this
network using matrix multiplication yields the co-occurrence counts
of references links to papers, co-citation strength \cite{small73}.
Note that each occurrence matrix has two co-occurrence matrices
associated with it. Figure \ref{f14} illustrates this for a sample
paper to reference occurrence matrix, $\mathbf{O}[p,r]$. To the
right of $\mathbf{O}[p,r]$ is the square symmetric bibliographic
coupling matrix $\mathbf{C}[p,r]$, whose size is number of papers in
$\mathbf{O}[p,r]$. Similarly, below $\mathbf{O}[p,r]$ is the square
symmetric co-citation matrix, $\mathbf{C}[r,p]$ whose size is the
number of references in $\mathbf{O}[p,r]$.
\begin{figure*}
\resizebox{0.9\textwidth}{!}{%
\includegraphics{figure14.eps}}%
\caption{Diagram showing that each occurrence matrix is associated
with a pair of co-occurrence matrices. Upper left matrix is paper
to reference occurrence matrix $\mathbf{O}[p,r]$, below is reference
co-occurrence matrix relative to papers (co-citation matrix),
$\mathbf{C}[r,p]$. Upper right matrix is paper co-occurrence matrix
relative to references (bibliographic coupling matrix),
$\mathbf{C}[p,r]$.\label{f14}}
\end{figure*}
Linguistic terms to paper networks, reference author to paper
networks and reference journal to paper networks are weighted
networks. Because of this, it is not desirable to calculate their
co-occurrence matrices using matrix multiplication because the
resulting link weights cannot be interpreted. Noting that
calculation of co-occurrence matrices is analogous to computing link
weights for a pair of cascaded bipartite networks, as was
demonstrated in Figure \ref{f13} and the discussion above, other
link weight functions can be used to find their co-occurrence
matrices. This can be done, for example, using the overlap function
of Section \ref{overlapsec}.
As an example, assume the paper to linguistic term matrix:
\begin{equation}
\mathbf{O}[p,t]=
\left[\begin{array}{cccccc}
8 & 9 & 5 & 3 & 1 & 0 \\
5 & 4 & 9 & 2 & 0 & 1 \\
0 & 0 & 2 & 6 & 5 & 4 \\
1 & 1 & 0 & 5 & 2 & 5 \\
\end{array}\right]
\label{46} .\end{equation}
Using the overlap function, the
co-occurrence matrix of papers linked to terms is:
\begin{eqnarray}
\mathbf{C}[p,t]&=&OVL\Big(\mathbf{O}[p,t],\mathbf{O}[t,p]\Big) \nonumber \\
\nonumber \\
&=&OVL\left(
\left[\begin{array}{cccccc}
8 & 9 & 5 & 3 & 1 & 0 \\
5 & 4 & 9 & 2 & 0 & 1 \\
0 & 0 & 2 & 6 & 5 & 4 \\
1 & 1 & 0 & 5 & 2 & 5 \\
\end{array}\right],
\left[\begin{array}{cccc}
8 & 5 & 0 & 1 \\
9 & 4 & 0 & 1 \\
5 & 9 & 2 & 0 \\
3 & 2 & 6 & 5 \\
1 & 0 & 5 & 2 \\
0 & 1 & 4 & 5 \\
\end{array}\right]\right)\nonumber \\
\nonumber \\
&=& \left[\begin{array}{cccc}
26 & 16 & 6 & 6 \\
16 & 21 & 5 & 5 \\
6 & 5 & 17 & 11 \\
6 & 5 & 11 & 14 \\
\end{array}\right]
\label{eq47}
\end{eqnarray}
\section{Recursive matrix growth}
The recursive growth equations presented in this section are a
natural outgrowth of the proposed matrix-based mathematical
treatment of collections of journal papers. They are useful for the
purpose of providing insight into the character of occurrence
distributions in the collections, as will be explained.
The basic record in a collection of journal papers is the paper. The
collection grows paper by paper in the temporal order of the
publication dates of the papers. When a new paper is added, it is
associated with the existing entities in the collection and
additionally, new entities, e.g., new paper authors or new
references, and new terms that enter into the collection.
This section will present a recursive model of the growth of both
occurrence and co-occurrence matrices as papers are added to the
collection. The recursive model of matrix growth is found by
examination of matrix partitions in occurrence and co-occurrence
matrices as papers are added to the collection.
It is easiest to consider the growth of an example occurrence
matrix. For convenience, the paper-reference matrix will be studied.
The results can be easily extended to other occurrence matrices, for
example the paper to paper author matrix \cite{morris04b}. In the
matrix the rows correspond to papers and are ordered in the sequence
of publication of the papers to which they correspond. The columns
correspond to references and are ordered in the sequence in which
their corresponding references first appear. As shown in Figure
\ref{f23}, the matrix contains a descending stair step sequence of
ones from its upper left corner diagonally to its lower right
corner. This sequence of ones corresponds to the initial appearance
of references as papers are added to the collection. Below this
diagonal sequence of ones is a roughly lower triangular region
sparsely populated with ones that correspond to citations to
existing references as each paper is added. Above the diagonal
sequence of ones is a roughly upper triangular area of zeros.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure23.eps}}%
\caption{Diagram of the structure of a paper to reference
matrix.\label{f23}}
\end{figure}
Considering the collection of journal papers dynamically, the
collection grows from an initial paper by sequential addition of
papers in the order in which they were published. In this sense the
paper-reference matrix $\bm{\Omega}$ grows dynamically one paper at
a time. Assume $i$ to be the number of papers, while $nr_i$ is the
number of references that have appeared in all papers up to and
including paper $i$. Assume $\bm{\Omega}_i$, whose size is $i$ by
$nr_i$, as the paper-reference matrix after the addition of paper
$i$, then consider the addition of paper $i+1$. A new row vector,
$i+1$, is added to $\bm{\Omega}_i$. This vector is partitioned into
a 1 by $i$ vector $\bm{\delta}_i$ listing the paper's citations to
existing references, and $\mathbf{1}$, a 1 by $nr_{i+1}-nr_i$ vector
of ones occurring in new columns added for the new references that
have appeared in paper $i+1$. Figure \ref{f23} shows a pictorial
representation of this addition. In the new columns, $\mathbf{0}$,
an $i$ by $nr_{i+1}-nr_i$ zero matrix appears. The recursive matrix
equation for growth of the paper-reference equation is:
\begin{equation}
\mathbf{\Omega}_{i+1}= \left[
\begin{array}{cc}
\mathbf{\Omega}_i & \mathbf{0} \\
\bm{\delta}_i & \mathbf{1} \\
\end{array}\right] \label{eq64}
.\end{equation}
Figure \ref{f24} shows a map of a typical
paper-reference matrix, where each dot shows the location of a one
in the matrix.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure24.eps}}%
\caption{Example of a typical paper to reference matrix.\label{f24}}
\end{figure}
As papers are added to the collection, note that individual papers
collect no links after their initial appearance, while references
cumulate links (citations from newly appearing papers) as papers are
added. Entity-types that cumulate links in collections of papers
usually have a power-law frequency distribution relative to papers.
Three such power-law distributions are well-known: 1) papers per
paper author distribution (Lotka's law) \cite{white89}, 2) papers
per paper journal distribution (Bradford's law) \cite{white89}, and
papers per reference distribution (reference power law)
\cite{naranan71}. Papers, which don't cumulate links, tend to have
exponential tailed distributions relative to other entity-types. Two
examples are authors per paper distribution (1-shifted Poisson)
\cite{morris04b}, and references per paper distribution (lognormal)
\cite{morris04a}.
The bibliographic coupling matrix, which will be designated
$\bm{\beta}$, is a symmetric matrix that lists the bibliographic
coupling counts of all pairs of papers within the data collection.
The diagonal of $\bm{\beta}$ contains the counts of the number of
references cited in each paper. The bibliographic coupling matrix
can be obtained by multiplying the paper-reference matrix by its
transpose:
\begin{equation}
\bm{\beta} = \bm{\Omega}\cdot\bm{\Omega}^T \label{eq65}
.\end{equation}
The recursive growth equations for the bibliographic
coupling matrix can be derived by substituting (\ref{eq64}) into
(\ref{eq65}):
\begin{eqnarray}
\bm{\beta}_{i+1}&=& \bm{\Omega}_{i+1}\cdot\bm{\Omega}_{i+1}^T = \nonumber \\
\nonumber \\
&=& \left[ \begin{array}{cc}
\bm{\Omega}_i\cdot\bm{\Omega}_i^T &
\bm{\Omega}_i\cdot\ \bm{\delta}_i^T \\
\bm{\delta}_i\cdot\bm{\Omega}_i^T &
\bm{\delta}_i\cdot\bm{\delta}_i^T + \bm{1}\cdot\bm{1}^T
\end{array}\right] \nonumber \\
\nonumber \\
&=&\left[\begin{array}{cc}
\bm{\beta}_i & \bm{\Omega}_i\cdot\bm{\delta}_i^T \\
\bm{\delta}_i\cdot\bm{\Omega}_i^T & m_{i+1} \\
\end{array}\right]\label{eq66}
,\end{eqnarray} where $m_{i+1}$ is the number of references cited by
paper $i+1$. Figure \ref{f25} shows a pictorial representation of a
typical bibliographic coupling matrix with the partitions in
Equation (\ref{eq66}) identified. It is easy to see from Equation
(\ref{eq66}) and Figure \ref{f25} that bibliographic coupling counts
between pairs of papers are static, and do not change as more papers
are added to the collection.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure25.eps}}%
\caption{Diagram of a bibliographic coupling matrix.\label{f25}}
\end{figure}
The co-citation matrix, designated as $\bm{\Gamma}$, is a symmetric
$nr$ by $nr$ matrix that lists the co-citation counts of all pairs
of references within the data collection. The diagonal of
$\bm{\Gamma}$ contains the counts of the number of papers that cite
each reference. The co-citation matrix can be obtained by
multiplying the transpose of the paper-reference matrix by itself:
\begin{equation}
\bm{\Gamma} = \bm{\Omega}^T\cdot\bm{\Omega} \label{eq67}
.\end{equation}
The recursive growth equations for the co-citation
matrix can be derived by substituting Equation (\ref{eq64}) into
Equation (\ref{eq67}):
\begin{eqnarray}
\bm{\Gamma}_{i+1}&=& \bm{\Omega}_{i+1}^T\cdot\bm{\Omega}_{i+1} \nonumber \\
\nonumber \\
&=& \left[ \begin{array}{cc}
\bm{\Omega}_i^T\cdot\bm{\Omega}_i +
\bm{\delta}_i^T\cdot\bm{\delta}_i &
\bm{\delta}_i^T\cdot\ \bm{1} \\
\bm{1}^T\cdot\bm{\delta}_i^T & \bm{1}^T\cdot\bm{1}
\end{array}\right] \nonumber \\
\nonumber \\
&=&\left[\begin{array}{cc}
\bm{\Gamma}_i + \bm{\delta}_i^T\cdot\bm{\delta}_i & \bm{\delta}_i^T\cdot\ \bm{1} \\
\bm{1}^T\cdot\bm{\delta}_i & \bm{1}^T\cdot\bm{1} \\
\end{array}\right]\label{eq68}
.\end{eqnarray}
Figure \ref{f26} shows a pictorial representation of a typical
co-citation matrix with the partitions in Equation (\ref{eq68})
identified. It is easy to see that the co-citation count between two
references is not static, but can be increased with the addition of
each new paper to the collection.
\begin{figure}
\resizebox{0.45\textwidth}{!}{%
\includegraphics{figure26.eps}}%
\caption{Diagram of a co-citation matrix.\label{f26}}
\end{figure}
\section{Example}
An illustrative example of the techniques outlined here uses a
collection of 902 papers on the topic of complex network theory.
This collection was gathered in 2003 by finding all papers that cite
key references in the specialty. A detailed analysis of the paper
to reference network for this collection was presented by Morris
\cite{morris04a}, while analysis of the paper author to paper
network for this collection was presented by Goldstein, \emph{et
al}, \cite{goldstein04group} and Morris, \emph{et al},
\cite{morris04b}.
Figure \ref{fap_ar} shows a weighted occurrence matrix,
$\mathbf{O}[ap,ar]$, for the paper author to reference author
network from this collection. In this diagram, the paper authors are
rows, reference authors are columns, and the size of the circle at
position $(i,j)$ in the diagram is proportional to the link weight
from paper author $i$ to reference author $j$. In this case the link
weight is equal to the number of times that paper author $i$ cited
reference author $j$.
In order to visualize the structure of links in the network, the
rows and columns of the matrix have been arranged using a seriation
algorithm \cite{morris04optimal} and clustering dendrograms have
been added on the left and top of the figure
\cite{morris04crossmaps}. The figure is meant to show collaboration
groups of paper authors and their links to reference authors as
symbols of 'schools of thought' \cite{white97authors}. The
visualization technique of Figure \ref{fap_ar} is explained in
Morris and Yen \cite{morris04crossmaps}.
\begin{figure*}
\resizebox{1.0\textwidth}{!}{%
\includegraphics{figureap_ar1.eps}}%
\caption{Visualization of the occurrence matrix of a weighted paper
author to reference author network from a collection of papers from
the specialty of complex networks theory.\label{fap_ar}}
\end{figure*}
Only paper authors that authored 6 or more papers were visualized.
For clustering paper authors, the co-occurrence matrix of
co-authorship counts, $\mathbf{C}[ap,p]$, was calculated using
matrix multiplication:
$\mathbf{C}[ap,p]=\mathbf{O}[ap,p]\cdot\mathbf{O}[p,ap]$. These
co-authorship counts were converted to distances and a hierarchical
clustering routine was applied to produce the dendrogram on the left
of the figure. Groups of paper authors clustered this way can be
regarded as 'research teams.'
Only reference authors that were cited 50 or more times were
visualized. For clustering reference authors, the co-occurrence
matrix of co-citation counts, $\mathbf{C}[ar,p]$, was calculated
using the overlap function:
$\mathbf{C}[ar,p]=OVL(\mathbf{O}[ar,p],\mathbf{O}[p,ar])$. These
co-citation counts were converted to distances and a hierarchical
clustering routine was applied to produce the dendrogram at the top
of the figure. Groups of reference authors clustered this way can be
regarded as representing 'schools of thought.'
The paper author to reference author matrix, $\mathbf{O}[ap,ar]$,
was calculated using matrix multiplication
$\mathbf{O}[ap,ar]=\mathbf{O}[ap,p]\cdot\mathbf{O}[p,r]\cdot\mathbf{O}[r,ar]$.
The matrix clearly shows that dominant reference authors in the
specialty, who are cited by authors to represent key ideas in the
specialty, are heavily linked across all paper authors. Note that
there is evidence of correlation of groups of paper authors to
groups of reference authors. For example, paper authors Choi, Hong,
Kim and Holme are all heavily connected to reference authors Newman
and Watts, while paper authors Pastor-Satorras, Vespignani, Vazquez,
and Moreno are all heavily connected to reference authors
Pastor-Satorras and Albert.
This example illustrates the usefulness of the matrix-based
mathematical treatment of cascades of bipartite networks in
collection of journal papers. In the example, we have shown this
treatment can be used for construction of weighted unipartite
co-occurrence networks for clustering purposes: 1) paper authors
linked by co-authorship, and 2) reference authors linked by common
papers. Additionally, the method was used to calculate a weighted
bipartite network of paper authors to reference authors.
\section{Conclusion}
We have introduced several valuable methods that can be used to
apply complex networks theory to collections of journal papers:
\begin{itemize}
\item \textbf{The structural model of coupled bipartite networks for
collections of papers.} This is a novel model that allows analysis
of any bipartite network in the collection in a general,
standardized, manner. Further, it allows building a \emph{multiple
entity-type} growth model of this system of networks, a technique
not generally studied by complex networks researchers.
\item \textbf{The matrix-based method of calculating weighted bipartite
networks.} Using the general concept of link weight functions, we
have shown that this matrix-based technique can be applied to
cascades of unweighted bipartite networks using matrix
multiplicaiton. Additionally, the technique can be applied to
cascades of weighted bipartite networks using the overlap function
or the inverse Minkowski function.
\item \textbf{The calculation of weighted unipartite co-occurrence networks.}
Considering co-occurrence networks as coupled bipartite networks
made by mirroring around a bipartite partition, calculation of
weighted co-occurrence networks uses the same matrix-based
calculation method as weighted bipartite networks.
\item \textbf{The construction of simple models of weighted matrix
growth.} This structural model of coupled bipartite networks, when
considered with unweighted bipartite growth models, such as the
bipartite Yule model, yields a simple model of growth of weighted
bipartite networks and weighted unipartite co-occurrence networks.
Morris \cite{morris05a} has shown that simple bipartite Yule
processes effectively simulate the statistics of bipartite and
weighted unipartite networks in collections of papers.
\end{itemize}
The structural model and matrix-based techniques introduced here
provide a unified framework of all entities in networks of papers,
e.g., paper to author networks that are manifestations of social
collaboration processes, or paper to reference networks that are
manifestations of epistemological processes such as knowledge
accretion and exemplar knowledge in a specialty. Such networks are
often studied as decoupled processes despite their almost certain
interdependence. For example, note that the paper author to
reference author network example of Figure \ref{fap_ar} shows
correlations between groups of paper authors and groups of reference
authors. A realistic model of processes in a research specialty
should be able to predict that such correlations will occur, but the
model must also predict the characteristics of the paper author to
paper network (such as Lotka's law), and simultaneously predict the
characteristics of the paper to reference network (such as the
reference power law.) All of these bipartite networks are
interdependent and those interdependencies cannot be modeled using
simple unipartite or bipartite growth models. The structural model
introduced here is a step toward modeling the complex
interdependencies in a research specialty.
Furthermore, and importantly, these techniques can be applied to
other report-based structures that can be expressed as collections
of entities. For example, a collection of intelligence reports
about terrorist events can, after application of an entity
extraction program, be expressed as a collection of entities:
reports, place names, terrorist group leader names, terrorist group
names, government officials' names, and incident types. These
entities are linked in a coupled bipartite structure, similar to
Figure \ref{coupled} and analysis of those linkages could produce
useful information about networks of terrorists. So the structural
model introduced here may allow the study of other self-organizing
social organizations as well, through their manifestations in
collections of reports.
\section{Acknowledgements} We would like to thank Michel Goldstein,
now of Amazon.com, for many discussions and ideas that contributed
to this work over the last year.
|
{
"timestamp": "2005-03-08T18:48:08",
"yymm": "0503",
"arxiv_id": "physics/0503061",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503061"
}
|
\section*{Introduction.}
Let $(M^n,g)$ be a Riemannian manifold and $(TM^n,g_s)$ its
tangent bundle equipped with the Sasaki metric \cite{Sk}. Let
$\xi$ be a given smooth vector field on $M^n$. Then $\xi$
naturally defines a mapping $\xi:M^n\to TM^n$ such that the
submanifold $\xi(M^n)\subset TM^n$ is transverse to the fibers.
This fact allows to ascribe to the vector field $\xi$ some
geometrical characteristics from the geometry of submanifolds. We
say that the vector field $\xi$ is \textit{minimal, totally
umbilic} or \textit{totally geodesic} if $\xi(M^n)$ possesses the
same property. In a similar way we can say about the
\textit{sectional, Ricci} or \textit{scalar curvature} of a vector
field. For the case of a \textit{unit} vector field this approach
has been proposed by H.Gluck and W.Ziller \cite{G-Z}. They proved
that the Hopf vector field $h$ on three-sphere $S^3$ is one with
globally minimal volume, i.e. $h(S^3)$ is a globally minimal
submanifold in the unit tangent bundle $T_1S^3$. Corresponding
local consideration leads to the notion of the \textit{ mean
curvature of a unit vector field} and a number of examples of
locally minimal unit vector fields were found based on a preprint
version of \cite{GM} (see \cite{BX-V1,BX-V2,GD-V1} and
references). In a different way, the second author found examples
of unit vector fields of \textit{constant mean curvature}
\cite{Ym1} and completely described the \textit{totally geodesic }
unit vector fields on 2-dimensional manifolds of constant
curvature \cite{Ym2}. The energy of a mapping $\xi:M^n\to T_1M^n$
can also be ascribed to the vector field $\xi$ and we can say
about the \textit{energy } of a unit vector field (see \cite{Wd,
GMLF, Wk-Bt} and references).
In contrast to unit vector fields, there are few results (both of
local or global aspects) on the geometry of general vector fields
treated as submanifolds in the \textit{tangent bundle}. It is
known \cite{Liu} that if $\xi$ is the zero vector field, then
$\xi(M^n)$ is totally geodesic in $TM^n$. Walczak~P. \cite{Wk}
treated the case when $\xi$ is a non-zero vector field on $M^n$
and proved that if $\xi$ is a parallel vector field on $M^n$, then
$\xi(M^n)$ is totally geodesic in $TM^n$. Moreover, if $\xi$ is of
constant length, then $\xi(M^n)$ is totally geodesic in $TM^n$ if
and only if $\xi$ is a parallel vector field on $M^n$. The latter
condition is rather burdensome. The basic manifold $M^n$ should be
a metrical product $M^{n-k}\times E^k\ (k\geq1)$, where $E^k$ is a
Euclidean (flat) factor.
Remark that $\xi(M^n)$ has maximal dimension among submanifolds in
the tangent bundle, transverse to the fibers. In this paper, we
study submanifolds $N^l$ of $TM^n$ with $l\le n$ which are
transverse to the fibers. We show in section 2 that any transverse
submanifold $N^l$ of $TM^n$ can be realized locally as the image
of a submanifold $F^l$ of $M^n$ under some vector field $\xi$
defined along $F^l$. We also investigate some cases when the image
can be globally realized. Mainly, we are interested in
submanifolds among this class which are totally geodesic. In this
way, we get a chain of inclusions:
$$\xi(F^l)\subset\xi(M^n)\subset TM^n. $$ In comparison with the
case when $\xi$ is defined over the whole $M^n$ or, at least, over
a domain $D^n\subset M^n$ as in \cite{Wk}, the picture becomes
different, because $\xi(F^l)$ can be totally geodesic in $TM^n$
while $\xi(M^n)$ is not. Our considerations include also the case
when the vector field is defined only on $F^l$, so that $\xi$
defines a ``direct" embedding $\xi:F^l\to TM^n$.
For $l=1$ we get nothing else but a vector field along a curve in
$M^n$ which generates a geodesic in $TM^n$. Sasaki S. \cite{Sk}
described geodesic lines in $TM^n$ in terms of vector fields along
curves in $M^n$ and found the differential equations on the curve
and the corresponding vector field. Moreover, in the case when
$M^n$ is of constant curvature, Sato K. \cite{St} explicitly
described the curves and the vector fields.
Evidently, our approach takes an intermediate position between the
above mentioned considerations for $l=1$ and $l=n$.
Necessary and sufficient conditions on $\xi(F^l)$ to be totally
geodesic, that we make explicit in section~3 (Proposition
\ref{Pr5}), have a clearer geometrical meaning if we suppose that
$\xi$ is of constant length along $F^l$ (Theorem \ref{Th1}) or is
a normal vector field along $F^l$ (Theorem \ref{Th2}). Indeed, an
application of Theorem \ref{Th2} to the specific case of foliated
Riemannian manifolds allows us to clarify the geometrical
structure of $\xi(M^n)$ (Corollary \ref{Foli}).
The case of a base space $M^n$ of constant curvature is discussed
in detail in section~4. An application to the case of a Riemannian
manifold of constant curvature enlightens us as to the non
rigidity of the totally geodesic property of $\xi(F^l)$, $l < n$,
contrary to the case $l=n$.
Finally, an application of our results to Lie groups endowed with
bi-invariant metrics gives a clear geometrical picture of our
problem.
\begin{remark} Throughout the paper
\begin{itemize}\itemsep=0ex
\item[-] $M^n$ is a given Riemannian manifold with metric $\bar g$,
$F^l$ is a submanifold of $M^n$ with the induced metric $g$,
$TM^n$ is the tangent bundle of $M^n$ equipped with the Sasaki
metric $g_s$;
\item[-] $\bar\nabla , \, \nabla , \, \tilde \nabla $ are the
Levi-Civita connections with respect to $\bar g,\, g,\, g_s$
respectively;
\item[-] the indices range is fixed as $a,b,c=1\dots n; \ \
i,j,k=1\dots l$;
\item[-] all the vector fields are supposed sufficiently smooth,
say of class $C^\infty$.
\end{itemize}
\end{remark}
\section{Local geometry of $\xi (F^l)$.}
\subsection{Tangent bundle of $\xi(F^l)$.}
Let $(M^n,\bar g)$ be an $n$-dimensional Riemannian manifold with metric $\bar g$.
Denote by $\bar g (\cdot\,,\cdot)$ the scalar product with respect to $\bar g$. The
{\it Sasaki metric} $g_s$ on $TM^n$ is defined by the following scalar product: if
$\tilde X,\tilde Y$ are tangent vector fields on $TM^n$, then
\begin{equation}
\label{Eqn1}
g_s(\tilde X,\tilde Y)=
{\bar g}(\pi_* \tilde X, \pi_* \tilde Y)+{\bar g}(K \tilde X,K \tilde Y)
\end{equation}
where $\pi_*:TTM^n \to TM^n $ is the differential of the projection $\pi:TM^n \to M^n
$ and $K: TTM^n \to TM^n$ is the {\it connection map} \cite{Dmb}. The local
representations for $\pi_*$ and $K$ are the following ones. Let $(x^1,\dots ,x^n)$ be
a local coordinate system on $M^n$. Denote by $\partial/\partial x^a $ the natural
tangent coordinate frame. Then, at each point $x\in M^n$, any tangent vector $\xi$ can
be decomposed as $\xi=\xi^a \frac{\partial}{\partial x^a}(x)$. The set of parameters
$\{x^1,\dots ,x^n;\,\xi^1,\dots,\xi^n\}$ forms the natural induced coordinate system
in $TM^n$, i.e. for a point $z=(x,\xi )\in TM^n$, with $x\in M^n, \ \ \xi \in T_xM^n$,
we have $x=(x^1,\dots ,x^n), \, \xi =\xi ^a\frac{\partial}{\partial x^a}(x)$. The
natural frame in $T_{z}TM^n$ is formed by $\left\{ \frac{\partial}{\partial x^a}(z),
\frac{\partial}{\partial \xi ^a}(z)\right\}$ and for any $\tilde X\in T_{z}TM^n$ we
have the decomposition $\tilde X=\tilde X^a\frac{\partial}{\partial x^a}(z)+\tilde
X^{n+a}\frac{\partial}{\partial \xi ^a}(z)$. Now locally, the \textit{horizontal}
and \textit{vertical} projections of $\tilde X$ are given by
\begin{equation} \label{Eqn2}
\begin{array}{l}
\pi_* \tilde X= \tilde X^a\frac{\partial}{\partial x^a}(\pi(z)), \\[1ex]
K \tilde X= (\tilde X^{n+a}+\bar\Gamma^a_{bc}(\pi(z))\,\xi^b \tilde X^c)\,
\frac{\partial}{\partial x^a}(\pi(z)), \\[1ex]
\end{array}
\end{equation}
where $\bar\Gamma^a_{bc}$ are the Christoffel symbols of the metric $\bar g$. The
inverse operations are called \textit{lifts }. If $\bar X=\bar X^a\,\partial/\partial
x^a$ is a vector field on $M^n$ then the vector fields on $TM$ given by
$$
\begin{array}{l}
\bar X^h=\bar X^a \partial/\partial x^a-\bar\Gamma^a_{bc}\,\xi^b\bar X^c\,\partial/\partial
\xi^a ,\\[1ex]
\bar X^v=\bar X^a\partial/\partial \xi^a
\end{array}
$$
are called the \textit{horizontal} and \textit{vertical} lifts of $X$ respectively.
Remark that for any vector field $\bar X$ on $M^n$ it holds
\begin{equation}\label{Pr}
\begin{array}{ll}
\pi_* {\bar X}^h=\bar X,& K {\bar X}^h=0, \\[1ex]
\pi_* {\bar X}^v=0, & K {\bar X}^v=\bar X.
\end{array}
\end{equation}
Let $F^l$ be an $l$-dimensional submanifold in $M^n$ with a local representation
given by
$$
x^a= x^a(u^1,\dots ,u^l).
$$
Let $\xi$ be a vector field on $M^n$ defined in some neighborhood of (or only on) the
submanifold $F^l$. Then the restriction of $\xi$ to the submanifold $F^l$, called
\textit{a vector field on $M^n$ along $F^l$}, generates a submanifold $\xi(F^l)\subset
TM^n$ with a local representation of the form
\begin{equation}\label{Sub}
\xi(F^l):
\left\{
\begin{array}{ll}
x^a=& x^a(u^1,\dots ,u^l), \\
\xi ^a=&\xi ^a(x^1(u^1,\dots ,u^l),\dots ,x^n(u^1,\dots ,u^l)).
\end{array}
\right.
\end{equation}
In what follows we will refer to the submanifold (\ref{Sub}) as to one
\textit{generated by a vector field on $M^n$ along $F^l$.}
The following Proposition describes the tangent space of $\xi(F^l)$.
\begin{proposition}\label{Pr1}
A vector field $\tilde X$ on $TM^n$ is tangent to $\xi(F^l)$ along $\xi(F^l)$
if and only if its horizontal-vertical decomposition is of the form
$$
\tilde X = X^h+(\bar \nabla _X\, \xi)^v,
$$
where $X$ is a tangent vector field on $F^l$, $\bar \nabla _X\, \xi$ is the covariant
derivative of $\xi$ in the direction of $X$ with respect to the Levi-Civita connection
of $M^n$ and the lifts are considered as those on $TM^n$.
\end{proposition}
\begin{proof}
Let us denote by $\tilde e_i$ the vectors of the coordinate frame
of $\xi (F^l)$. Then, evidently, $$\textstyle \tilde e_i=\left\{
\frac{\partial x^1}{\partial u^i}, \dots , \frac{\partial
x^n}{\partial u^i}; \ \ \frac{\partial \xi ^1}{\partial u^i},
\dots , \frac{\partial \xi ^n}{\partial u^i} \right\}. $$ Applying
(\ref{Eqn2}), we have $$
\begin{array}{rl}
\pi_*\tilde e_i= &\frac{\partial x^a}{\partial
u^i}\frac{\partial}{\partial x^a}= \frac{\partial}{\partial
u^i},\\[2ex] K\tilde e_i =& (\frac{\partial \xi^a}{\partial u^i} +
\bar \Gamma ^a_{bc} \, \xi^b \, \frac {\partial x^c}{\partial
u^i})\frac{\partial}{\partial x^a}= (\frac{\partial
\xi^a}{\partial x^c}\frac{\partial x^c}{\partial u^i} + \bar
\Gamma ^a_{bc} \, \xi^b \, \frac {\partial x^c}{\partial
u^i})\frac{\partial}{\partial x^a}\\[1ex] =&\frac {\partial
x^c}{\partial u^i}(\frac{\partial \xi^a}{\partial x^c} + \bar
\Gamma ^a_{bc} \, \xi^b \,) \frac{\partial}{\partial x^a}=
\bar\nabla_i\xi,
\end{array}
$$ where $\bar \Gamma ^a_{bc}$ are the Christoffel symbols of the
metric $\bar g$ taken along $F^l$ and $\bar \nabla _i$ means the
covariant derivative of a vector field on $M^n$ with respect to
the Levi-Civita connection of $\bar g$ along the $i$-th
coordinate curve of the submanifold $F^l \subset M^n$. Summing up,
we have
\begin{equation}\label{Eqn3}
\tilde e_i= \left(\frac{\partial}{\partial u^i}\right)^h +(\bar
\nabla _i \xi)^v.
\end{equation}
Let $\tilde X$ be a vector field on $TM^n$ tangent to $\xi(F^l)$
along $\xi(F^l)$. Then the following decomposition holds
$
\tilde X \,= \tilde X^i \tilde e_i.
$
Set $ X=\tilde X^i\partial/\partial u^i$. The vector field $X$ is
tangent to $F^l$ and, taking into account (\ref{Eqn3}), the
decomposition of $\tilde X$ can be represented as
$
\tilde X=X^h+(\bar \nabla _X \, \xi)^v,
$
which completes the proof.
\end{proof}
\begin{corollary}\label{Cor1} Let $(F^l,g)$ be a submanifold of a
Riemannian manifold\linebreak $(M^n,\bar g)$ with the induced
metric. Let $\xi$ be a vector field on $M^n$ along $F^l$. Then the
metric on $\xi(F^l)$, induced by the Sasaki metric of $TM^n$, is
defined by the following scalar product $$ {g_s}(\tilde X,\tilde
Y)=g\,(X,Y)+{\bar g}\,(\bar \nabla_X\, \xi ,\bar \nabla_Y\, \xi),
$$ for all vector fields $\tilde X=X^h+(\bar \nabla _X \, \xi)^v$
and $\tilde Y=Y^h+(\bar \nabla _Y \, \xi)^v$ on $\xi(F^l)$, where
$X,Y$ are vector fields on $F^l$.
\end{corollary}
\subsection{ Normal bundle of $\xi (F^l)$.}
To describe the normal bundle of $\xi(F^l)$, we need one auxiliary
notion. Let $\xi$ be a given vector field on a submanifold $F^l
\subset~M^n$. Then $\bar \nabla $ enables us to define a
point-wise linear mapping $\bar \nabla \xi: T_xF^l \to T_xM^n$, $X
\to \bar \nabla _X \xi$, for all $x \in M^n$. Its dual mapping,
with respect to the corresponding scalar products induced by $g$
and $\bar g$, gives rise to the linear mapping $(\bar \nabla
\xi)^*: T_xM^n \to T_xF^l$ defined by the formula
\begin{equation}\label{Eqn4}
g\,((\bar \nabla \xi)^*W,X)={\bar g}\,(\bar \nabla_X \xi,W) \mbox{
for all $W \in T_xM^n$ and $X \in T_xF^l$}.
\end{equation}
We call the mapping $(\bar \nabla \xi)^*: T_xM^n \to T_xF^l$ the
{\it conjugate derivative mapping}, or simply {\it conjugate
derivative}. Remark, that if $W$ is a vector field on $M^n$, then
the application of $(\bar\nabla\xi)^*$ gives rise to a vector
field $(\bar\nabla\xi)^*W$ on $F^l$ by
$
[(\bar \nabla\xi)^*W]_x=(\bar\nabla\xi)^*W_x\in T_xF^l \mbox{ for
all $x\in F^l$}.
$
Now we can prove
\begin{proposition}\label{Pr4}
Let $\eta$ and Z be normal and tangent vector fields on $F^l$
respectively. Then the lifts $$ \eta^h, \ \eta^v-((\bar \nabla
\xi)^* \eta)^h,\ Z^v-((\bar \nabla \xi)^* Z)^h $$ to the points of
$\xi(F^l)$ span the normal bundle of $\xi (F^l)$ in $TM^n$.
\end{proposition}
\begin{proof}
Let $\tilde X=X^h+(\bar \nabla_X \xi)^v$ be a vector field on $\xi
(F^l).$ Let $\eta $ and $Z$ be vector fields on $F^l$ which are
normal and tangent to $F^l$ respectively. Taking into account
(\ref{Eqn1}), (\ref{Pr}) and (\ref{Eqn4}), we have $$
\begin{array}{l}
{g_s}(\tilde X, \eta^h)={\bar g}\,(X, \eta)=0 \\[2ex]
\begin{array}{rl}
{g_s}(\tilde X, \eta^v - [(\bar \nabla \xi)^* \eta]^h)= &-{\bar
g}\,(X,(\bar \nabla \xi)^*\eta) + {\bar g}\,(\bar \nabla_X \xi,
\eta) \\[1ex]
=&-{\bar g}\,(\bar \nabla_X \xi,\eta) + {\bar
g}\,(\bar \nabla_X \xi, \eta)=0
\end{array}\\[3ex]
\begin{array}{rl}
{g_s}(\tilde X, Z^v-[(\bar \nabla \xi)^*Z]^h)=&-{\bar g}\,(X,(\bar
\nabla \xi)^* Z) +{\bar g}\,(\bar \nabla_X \xi, Z) \\[1ex]
=&-{\bar g}\,(\bar \nabla_X \xi,Z)+{\bar g}\,(\bar \nabla_X
\xi,Z)=0
\end{array}
\end{array}
$$
Let $\eta _1, \dots , \eta_p$ ($p=1,\dots, n-l$) be a normal
frame of $F^l$ while $f_1, \dots ,f_l$ span $T_xF^l$ at each
point $x\in F^l$. Consider the vector fields $$
N_\alpha=\eta^h_\alpha , \ \ P_\alpha=\eta^v_\alpha -((\bar \nabla
\xi)^* \eta_\alpha)^h, \ \ F_i=f_i^v-((\bar \nabla \xi)^* e_i)^h,
$$ where $\alpha=1,\dots, n-l; \ i=1,\dots,n$. Let us show that
these are linearly independent. Indeed, suppose that $$
\lambda^\alpha N_\alpha + \mu^\alpha P_\alpha + \nu^iF_i=
\{\lambda^\alpha \eta_\alpha - \mu^\alpha(\bar \nabla
\xi)^*\eta_\alpha - \nu^i(\bar \nabla \xi)^*e_i\}^h+\{\mu^\alpha
\eta_\alpha + \nu^i f_i\}^v=0. $$ Because of the fact that the
horizontal and vertical components are linearly independent, we
see that $\mu^\alpha \eta_\alpha+\nu^if_i=0$ which is possible iff
$\mu^\alpha=0, \nu^i=0.$ Then, from the horizontal part of the
decomposition above we see that $\lambda^\alpha=0.$ So, $N_\alpha,
~P_\alpha$ and $F_i$ are linearly independent, which completes the
proof.
\end{proof}
\begin{remark}
In the case when $\xi $ is a normal vector field, the images
$(\bar\nabla \xi )^* \eta $ and $(\bar\nabla \xi )^*Z$ have a
simple and natural meaning, namely $$
\begin{array}{l}
(\bar\nabla \xi )^* \eta = g^{ik}{\bar g}\,(\nabla _k^\perp \xi,
\eta)\frac{\partial}{\partial u^i},
\ \
(\bar\nabla \xi )^* Z = -A_\xi Z,
\end{array}
$$ where $ \nabla ^\perp $ is the normal bundle connection of
$F^l$ and $A_\xi$ is the shape operator of $F^l$ with respect to
the normal vector field $\xi$. In fact, $(\bar\nabla \xi )^* \eta$
is the vector field on $F^l$ dual to the 1-form ${\bar g}\,(\nabla
_k^\perp \xi, \eta)\,du^k$.
\end{remark}
\section{Characterization of submanifolds of $TM^n$ transverse to fibers.}
It is clear that all totally geodesic vector fields along
submanifolds of $M^n$ generate submanifolds in $TM^n$ which are
transverse to the fibers of $TM^n$. We study in this section the
converse question. We start with the local case.
\begin{proposition}\label{Transv}
Let $N^l$ be an embedded submanifold in the tangent bundle of a
Riemannian manifold $M^n$, which is transverse to the fiber at a
point $z\in N^l$, then there is a submanifold $F^l$ of $M^n$
containing $x=\pi(z)$, a neighborhood $U$ of $x$ in $M^n$, a
neighborhood $V$ of $z$ in $TM^n$ and a vector field $\xi$ on
$M^n$ along $F^l \cap U$ such that $N^l \cap V=\xi(F^l \cap U)$.
\end{proposition}
\begin{proof}
Since $T_z N^l$ is transverse to the vertical subspace $V_z TM^n$ of
$TTM^n$ at $z$, $\pi_* \upharpoonright T_z N^l :T_z N^l \to T_x
M^n$ is injective, and so there is an open neighborhood $W$ of $z$
in $TM^n$ such that $\pi_* \upharpoonright T_{z'} N^l:T_{z'} N^l
\to T_{\pi(z')} M^n$ is injective for all $z'\in W \cap N^l$.
Hence $\pi \upharpoonright {W\cap N^l}:W\cap N^l \to M^n$ is an
immersion, and thus there exist a cubic centered coordinate system
$(U,\varphi)$ about $x=\pi(z)$ and a neighborhood $V$ of $z$ in
$W$ such that $\pi \upharpoonright {V\cap N^l}$ is 1:1 and $\pi(V
\cap N^l)$ is a part of a slice $F^l$ of $(U,\varphi)$ (\cite{Wr},
p. 28). The slice $F^l$ is a submanifold of $M^n$ and we have $\pi
\upharpoonright {V\cap N^l}: V\cap N^l \to U\cap F^l$ is an
imbedding onto, and so there is a $C^\infty$-mapping $\xi:F^l \cap
U \to N^l \cap V$ such that $\pi \circ \xi=Id_{F^l \cap U}$. In
other words, $\xi$ is a vector field on $M^n$ along $F^l \cap U$
such that $N^l \cap V=\xi(F^l \cap U)$.
\end{proof}
The global version of the last result requires further conditions.
\begin{theorem} \label{Trans-max}
Let $N^n$be a connected compact $n$-dimensional submanifold of the
tangent bundle of a connected simply connected Riemannian manifold
$M^n$, which is everywhere transverse to the fibers of $TM^n$.
Then $M^n$ is also compact, and there is a vector field $\xi$ on
$M^n$ such that $\xi(M^n)=N^n$.
\end{theorem}
\begin{proof}
The fact that $N^n$ is everywhere transverse to the fibers of
$TM^n$ implies that $\pi\upharpoonright {N^n}:N^n \to M^n$ is an
immersion. Since $M^n$ and $N^n$ are connected of the same
dimension and $N^n$ is compact, then $M^n$ is compact and
$\pi\upharpoonright {N^n}$ is a covering projection (cf.
\cite{KoNz}, Vol.~1, p.178). Now, $M^n$ is simply connected and so
$\pi \upharpoonright {N^n}$ is a diffeomorphism. Let $\xi:M^n \to
N^n$ be the inverse of $\pi \upharpoonright {N^n}$. Then $\xi$ is
a vector field on $M^n$ and $\xi(M^n)=N^n$.
\end{proof}
In a similar way, we can show the following:
\begin{theorem}
Let $N^l$ be a connected compact submanifold of the tangent bundle
of a connected simply connected manifold $M^n$, which is
transverse to the fibers it meets and projects onto a simply
connected submanifold $F^l$ of $M^n$. Then $F^l$ is compact and
there is a vector field $\xi$ on $M^n$ along $F^l$ such that
$\xi(F^l)=N^l$.
\end{theorem}
In the particular case of horizontal totally geodesic submanifolds
of $TM^n$, i.e. whose tangent space at any point is horizontal, we
can state the following:
\begin{theorem}\label{Hor-Sub}
Let $N^l$ be a connected complete totally geodesic horizontal
submanifold of the tangent bundle of a connected Riemannian
manifold $M^n$ which projects into a simply connected Riemannian
submanifold $F^l$ of $M^n$. Then $F^l$ is also complete and
totally geodesic in $M^n$ and there is a parallel vector field
$\xi$ on $M^n$ along $F^l$ such that $\xi(F^l)=N^l$.
\end{theorem}
\begin{proof}
By hypothesis, for all $z\in N^l$, $T_z N^l$ is a horizontal
subspace of $T_z TM^n$ with respect to the Levi-Civita connection
of $\bar g$. Hence $\pi \upharpoonright {N^l}:N^l \to F^l$ is an
isometric submersion of $N^l$ into $F^l$, with $N^l$ and $F^l$
connected and of the same dimension. Since $N^l$ is complete, also
$F^l$ is complete and $N^l$ is a covering space of $F^l$ (cf.
\cite{KoNz}, Vol.1, p.176). The fact that $F^l$ is simply
connected implies that $\pi \upharpoonright {N^l}:N^l \to F^l$ is
an isometry, and there is an isometry $\xi:F^l \to N^l$ such that
$\pi \upharpoonright {N^l}\circ \xi=Id_{F^l}$, i.e. $\xi$ is a
vector field on $M^n$ along $F^l$.
Now, $F^l$ is totally geodesic. Indeed, let $X$ and $Y$ be vector
fields on $F^l$, and denote by the same letters some of their
extensions to $M^n$. If we denote by $X^h$ and $Y^h$ their
horizontal lifts to $TM^n$, then $X^h \upharpoonright {N^l}$ and
$Y^h \upharpoonright {N^l}$ are vector fields on $TM^n$ along
$N^l$. For all $z\in N^l$, $T_z N^l$ being horizontal, $\pi_*
\upharpoonright {T_z N^l}:T_z N^l \to T_x M^n$ is bijective. Since
$\pi_*(X^h(z))=X(\pi (z))$ and $\pi_*(Y^h(z))=Y(\pi (z))$, we have
that $X^h(z)$ and $Y^h(z)$ are tangent to $N^l$. Thus $(\tilde
\nabla_{X^h} Y^h)\upharpoonright {N^l}$ is tangent to $N^l$ and
hence horizontal. Consequently $(\tilde \nabla_{X^h} Y^h)
\upharpoonright {N^l}=(\bar \nabla_{X} Y)^h \upharpoonright {N^l}$
and is tangent to $N^l$. Hence $\bar \nabla_X Y=\pi_*\circ(\bar
\nabla_X Y)^h$ is tangent to $F^l$ and so $F^l$ is totally
geodesic. It remains to prove that $\xi$ is parallel along $F^l$.
In fact, for all $x\in F^l$ and $X\in T_x F^l$, the vector $X^h +
(\bar \nabla_X \xi)^v$ is tangent to $\xi(F^l)=N^l$ at $\xi(x)$
and is mapped onto $X$. Since $T_{\xi(x)}N^l$ is a horizontal
space, $\bar \nabla_X \xi=0$. Therefore, $\xi$ is parallel along
$F^l$.
\end{proof}
\begin{corollary}
Let $N^n$ be a connected complete totally geodesic horizontal
$n$-dimensional submanifold of the tangent bundle of a connected
simply connected Riemannian manifold $M^n$. Then $M^n$ is also
complete and there is a parallel vector field $\xi$ on $M^n$ such
that $\xi(M^n)=N^n$.
\end{corollary}
\section{ The conditions on $\xi (F^l)$ to be totally geodesic.}
Evidently, geometrical properties of the submanifold $\xi(F^l)$
depend on the submanifold $F^l$ and the vector field $\xi$. If one
does not pose any restrictions on them, the geometry of $\xi(F^l)$
becomes rather intricate. Nevertheless, it is possible to
formulate the conditions on $\xi(F^l)$ to be totally geodesic in
more or less geometrical terms.
To do this, we introduce the notion of a $\xi$-connection on the
Riemannian manifold $M^n$.
\begin{definition}
Let $M^n$ be a Riemannian manifold with Riemannian connection
$\bar\nabla$ and curvature tensor $\bar R$. Let $\xi$ be a fixed
smooth vector field on $M^n$. Denote by $\mathfrak{X}(M^n)$ the
set of all smooth vector fields on $M^n$. The mapping
$\stackrel{*}{\nabla}:\mathfrak{X}(M^n)\times \mathfrak{X}(M^n)\to
\mathfrak{X}(M^n)$ defined by
\begin{equation}\label{Conn}
\stackrel{*}{\nabla}_{\bar X}{\bar Y}=\bar\nabla_{\bar X}\bar Y+\frac12\Big[ \bar R(\xi,\bar\nabla_{\bar
X}\xi)\bar Y+ \bar R(\xi,\bar\nabla_{\bar Y}\xi)\bar X\Big]
\end{equation}
is a torsion-free affine connection on $M^n$. It is called the $\xi$-connection.
\end{definition}
Remark that if $\xi$ is a parallel vector field or the manifold $M^n$ is flat, then
the $\xi$-connection is the same as the Levi-Civita connection of $M^n$.
It is easy to check that (\ref{Conn}) indeed defines a torsion-free affine connection.
Now we can state the main technical tool for the further considerations.
\begin{proposition}\label{Pr5}
Let $F^l$ be a submanifold in a Riemannian manifold $M^n.$ Let $\xi$ be a vector field
on $M^n$ along $F^l$. Then $\xi(F^l)$ is totally geodesic in $TM^n$ if and only if
\begin{itemize}
\item[(a)] $F^l$ is totally geodesic with respect to
the $\xi$-connection (\ref{Conn});
\item[(b)] for any vector fields $X,Y$ on $F^l$
$$
\bar \nabla_X \bar \nabla_Y \xi =
\bar \nabla_{\stackrel{*}{\nabla}_X Y} \xi +\frac{1}{2} \bar R(X,Y)\xi.
$$
\end{itemize}
\end{proposition}
\begin{proof}
By definition, the submanifold $\xi(F^l)$ is totally geodesic in $TM^n$ if and only
if $g_s\,(~\tilde \nabla_{\tilde X}\tilde Y,\tilde N)~=~0$ for any vector fields
$\tilde X,\tilde Y$ tangent to $\xi(F^l)$ along $\xi(F^l)$ and $\tilde N$ normal to
$\xi (F^l)$. To calculate $\tilde \nabla_{\tilde X}\tilde Y$, we use the Kowalski
formulas \cite{Kow}.
{\it For any vector fields $\bar X,\bar Y$ on $M^n$, the covariant derivatives of various
combinations of lifts to the point $(x,\xi) \in TM^n$ can be found as follows}
\begin{equation}\label{Kow}
\begin{array}{ll}
\tilde \nabla_{\bar X^h}\bar Y^h = (\bar \nabla_{\bar X} \bar Y)^h-
\frac{1}{2}(\bar R (\bar X,\bar Y) \xi)^v, \
&\tilde \nabla_{\bar X^v}\bar Y^h = \frac{1}{2} (\bar R (\xi ,\bar X) \bar Y)^h,\\[2ex]
\tilde \nabla_{\bar X^h}\bar Y^v = (\bar \nabla_{\bar X} \bar Y)^v+
\frac{1}{2}(\bar R (\xi ,\bar Y) \bar X)^h, \
& \tilde \nabla_{\bar X^v}\bar Y^v = 0.
\end{array}
\end{equation}
{\it where $\bar \nabla$ and $\bar R$ are the Levi-Civita connection and the curvature
tensor of $M^n$ respectively}.
Let $\tilde X=X^h+(\bar \nabla_X \xi)^v$ and $\tilde Y=(Y)^h+(\bar \nabla_Y \xi)^v$ be
vector fields tangent to $\xi(F^l).$ Then, applying (\ref{Kow}), we easily
find
$$
\tilde \nabla_{\tilde X} \tilde Y = (\bar \nabla_X Y+\frac{1}{2} \bar R(\xi ,\bar
\nabla _X \xi)Y+\frac{1}{2}\bar R(\xi,\bar\nabla_Y\xi)X)^h+ (\bar \nabla_X \bar
\nabla_Y\, \xi - \frac{1}{2}\bar R(X,Y)\xi)^v
$$
or
$$
\tilde \nabla_{\tilde X} \tilde Y = (\stackrel{*}{\nabla}_X Y)^h+ (\bar \nabla_X \bar
\nabla_Y\, \xi - \frac{1}{2}\bar R(X,Y)\xi)^v.
$$
Using Proposition \ref{Pr4}, we see that the totally geodesic property of $\xi(F^l)$
is equivalent to
\begin{equation}\label{Cond}
\left\{
\begin{array}{rl}
{\bar g}\,(\stackrel{*}{\nabla}_X Y ,\eta)&=0,\\[2ex]
{\bar g}\,(\stackrel{*}{\nabla}_X Y,(\nabla \xi)^* \eta)&={\bar g}\,(\bar \nabla_X \bar \nabla_Y
\xi-\frac{1}{2}\bar R(X,Y)\xi ,\eta),\\[2ex]
{\bar g}\,(\stackrel{*}{\nabla}_X Y,(\nabla \xi)^*Z)&={\bar g}\,(\bar \nabla_X \bar \nabla_Y
\xi-\frac{1}{2}\bar R(X,Y)\xi ,Z),
\end{array}
\right.
\end{equation}
for any vector fields $X,Y,Z$ tangent to $F^l$ and any vector field $\eta$ orthogonal
to $F^l$.
From $(\ref{Cond})_1$ we see that $F^l$ must be autoparallel with respect to
$\stackrel{*}{\nabla}$ and hence totally geodesic \cite{KoNz}. Thus, $\stackrel{*}{\nabla}_XY$ is tangent to
$F^l$ and it is possible to apply (\ref{Eqn4}). Therefore, we can rewrite the
equations $(\ref{Cond})_2$ and $(\ref{Cond})_3$ as
$$
\left\{
\begin{array}{l}
{\bar g}\,(\bar \nabla_{\stackrel{*}{\nabla}_X Y} \xi - \bar \nabla_X \bar \nabla_Y \xi+
\frac{1}{2} \bar R(X,Y) \xi,\eta) =0, \\[1ex]
{\bar g}\,(\bar \nabla_{\stackrel{*}{\nabla}_X Y} \xi -\bar \nabla_X \bar \nabla_Y \xi +
\frac{1}{2}\bar R(X,Y) \xi,Z) =0
\end{array}
\right.
$$
for any vector fields $\eta $ normal and $Z$ tangent to $F^l$ along $F^l$. Thus, we conclude
$$
\bar \nabla_X \bar \nabla_Y \xi =\bar \nabla_{\stackrel{*}{\nabla}_X Y} \xi
+ \frac{1}{2} \bar R(X,Y) \xi,
$$
which completes the proof.
\end{proof}
For the cases when $l=1$ and $l=n$, we get the known conditions for the totally
geodesic property of $\xi(F^l)$.
\begin{corollary}\label{l=1}
If $l=1$ and $\xi(F^l)$ is a curve $\Gamma$ in $TM^n$ then
this curve is a geodesic if and only if
$$
\left\{
\begin{array}{l}
x''+\bar R(\xi,\xi')x'=0, \\[1ex]
\xi''=0,
\end{array}
\right.
$$
where $(')$ means the covariant derivative with respect to the natural parameter of
$\Gamma$ and $x(\sigma)=(\pi\circ\Gamma)(\sigma)$ \emph{(cf. \cite{Sk})};
\end{corollary}
\begin{proof}
Indeed, in this case $\tilde X=\tilde Y=\Gamma'=(x')^h+(\xi')^v$, $\bar X=\bar Y=x'$
and $\stackrel{*}{\nabla}_{\bar X}{\bar Y}=x''+\bar R(\xi,\xi')x'$. Thus, $x(\sigma)$ is geodesic
with respect to the $\xi$-connection iff $x''+\bar R(\xi,\xi')x'=0$ and the rest of
the proof is evident.
\end{proof}
\begin{corollary}\label{l=n}
If $l=n$ and $F^l=M^n$, then $\xi(M^n)$ is totally geodesic in $TM^n$ if and only if
for any vector fields $\bar X,\bar Y$ on $M^n$ \emph{(cf. \cite{Wk})}
$$
\bar \nabla_{\bar X} \bar \nabla_{\bar Y} \xi =
\bar \nabla_{\stackrel{*}{\nabla}_{\bar X}\bar Y} \xi +\frac{1}{2} \bar R(\bar X,\bar Y)\xi.
$$
\end{corollary}
\begin{proof}
In this case, only $(b)$ of Proposition \ref{Pr5} should be checked, which completes
the proof.
\end{proof}
The result of Corollary \ref{l=n} can be expressed in more
geometrical terms. To do this, introduce a symmetric bilinear
mapping $h_\xi: \mathfrak{X}(M^n)\times \mathfrak{X}(M^n)\to
\mathfrak{X}(M^n)$ by
\begin{equation}\label{h}
h_\xi(\bar X,\bar Y)=\frac12 \Big[\bar R(\xi,\nabla_{\bar X}\xi)\bar Y+
\bar R(\xi,\nabla_{\bar Y}\xi)\bar X\Big],
\end{equation}
for all $\bar X$, $\bar Y \in \mathfrak{X}(M^n)$. Then the
definition of the $\xi$-connection takes as similar form as the
Gauss decomposition
\begin{equation}\label{Conn1}
\stackrel{*}{\nabla}_{\bar X}{\bar Y}=\bar\nabla_{\bar X}{\bar Y}+h_\xi(\bar X,\bar Y).
\end{equation}
Define a \textit{``shape operator"} $A_\xi$ for the field $\xi$ by
\begin{equation}\label{Shp}
A_\xi\bar Y=-\bar \nabla_{\bar Y}\xi,\;\textup{for all}\; \bar Y
\in \mathfrak{X}(M^n).
\end{equation}
Then the covariant derivative of the $(1,1)$-tensor field $A_\xi$ is given by
$$
(\bar\nabla_{\bar X}A_\xi)\bar Y=-\bar\nabla_{\bar X}\bar \nabla_{\bar
Y}\xi+\bar\nabla_{\bar\nabla_{\bar X}\bar Y}\xi.
$$
Hence we see that the Codazzi-type equation
$
\bar R(\bar X,\bar Y)\xi=(\bar\nabla_{\bar Y}A_\xi)\bar X-(\bar\nabla_{\bar X}A_\xi)\bar Y
$
holds. In these notations $$ \bar\nabla_{\stackrel{*}{\nabla}_{\bar X}\bar Y}
\xi +\frac{1}{2} \bar R(\bar X,\bar Y)\xi-\bar \nabla_{\bar X}
\bar \nabla_{\bar Y} \xi= \bar\nabla_{h_\xi(\bar X,\bar Y)}\xi+
\frac{1}{2}\Big[(\bar\nabla_{\bar X}A_\xi)\bar Y+(\bar\nabla_{\bar
Y}A_\xi)\bar X\Big]. $$ If we introduce a symmetric bilinear
mapping $ \Omega _\xi: \mathfrak{X}(M^n)\times
\mathfrak{X}(M^n)\to \mathfrak{X}(M^n)$ defined by $$ \Omega _\xi
(\bar X,\bar Y)=\bar\nabla_{h_\xi(\bar X,\bar Y)}\xi+
\frac{1}{2}\Big[(\bar\nabla_{\bar X}A_\xi)\bar Y+(\bar\nabla_{\bar
Y}A_\xi)\bar X\Big], $$ then Corollary \ref{l=n} can be
reformulated as
\begin{corollary}
If $\xi$ is a smooth vector field on a Riemannian manifold $M^n$ then
$\xi(M^n)$ is totally geodesic in $TM^n$ if and only if for any vector fields
$\bar X,\bar Y$ on $M^n$
\begin{equation}\label{Omega}
\Omega _\xi (\bar X,\bar Y)=\bar\nabla_{h_\xi(\bar X,\bar Y)}\xi+
\frac{1}{2}\Big[(\bar\nabla_{\bar X}A_\xi)\bar Y+(\bar\nabla_{\bar Y}A_\xi)\bar
X\Big]\equiv 0,
\end{equation}
where $h_\xi$ and $A_\xi$ are defined by (\ref{h}) and (\ref{Shp}) respectively.
\end{corollary}
\begin{remark} The statement of Proposition \ref{Pr5} can also be reformulated in
these terms, namely, {\it let $F^l$ be a submanifold in a Riemannian manifold $M^n$
and $\xi $ be a vector field on $M^n$ along $F^l$. Then $\xi (F^l)$ is totally
geodesic in $TM^n$ if and only if $F^l$ is totally geodesic with respect to the
$\xi$-connection (\ref{Conn}) and $\Omega_\xi$ vanishes on the tangent bundle of
$F^l$}
\end{remark}
Now, combining Theorem \ref{Trans-max} with Proposition \ref{Pr5},
we obtain
\begin{corollary}
On a connected simply connected compact $n-$dimensional Riemannian
manifold, vector fields satisfying $(b)$ of Proposition \ref{Pr5}
generate the only connected compact totally geodesic
$n$-dimensional submanifolds of the tangent bundle which are
transverse to fibers.
\end{corollary}
As has been shown in \cite{Ym}, for the case of the unit tangent bundle, the Hopf
vector fields on odd dimensional spheres generate totally geodesic submanifolds in
$T_1S^n$. For the tangent bundle the situation is different.
\begin{theorem}
A non-zero Killing vector field on a space of non-zero constant curvature $(M^n,c)$ never
generates a totally geodesic submanifold in $TM^n$. Moreover, a manifold with
positive sectional curvature does not admit a non-zero Killing vector field with
totally geodesic property.
\end{theorem}
\begin{proof}
Let $\xi$ be a Killing vector field on a space $M^n$ of constant curvature $c$. Then
$A_\xi $ is a skew-symmetric linear operator, i.e.
\begin{equation}\label{Killing}
\bar g(A_\xi \bar X,\bar Y)+\bar
g(\bar X,A_\xi \bar Y)=0,
\end{equation}
and moreover,
\begin{equation}\label{KillProp}
(\bar\nabla_{\bar X}A_\xi)\bar Y=\bar R(\xi,\bar X)\bar Y
\end{equation}
for all vector fields $\bar X,\bar Y$ on $M^n$ (cf. \cite{KoNz}). Since $M^n$ is of
non-zero constant curvature, the equation (\ref{Omega}) can be simplified in the
following way.
$$
\begin{array}{rl}
(\bar\nabla_{\bar X}A_\xi)\bar Y+(\bar\nabla_{\bar Y}A_\xi)\bar X=&\bar R(\xi,\bar X)\bar Y+\bar R(\xi,\bar Y)\bar
X=\\[1ex]
&c\,\Big[2\bar g(\bar X,\bar Y)\,\xi-\bar g(\xi,\bar X)\bar Y-\bar g(\xi,\bar Y)\bar
X\Big]
\end{array}
$$
$$
\begin{array}{l}
\bar R(\xi,\bar\nabla_{\bar X}\xi)\bar Y+\bar R(\xi,\bar\nabla_{\bar Y}\xi)\bar X=
c\,\Big[\bar g(\bar\nabla_{\bar X}\xi,\bar Y)+\bar g(\bar X,\bar\nabla_{\bar Y}\xi)\bar X)\Big]\xi-\\[1ex]
c\,\Big[(\bar g(\xi,\bar X)\bar\nabla_{\bar Y}\xi+\bar g(\xi,\bar Y)\bar\nabla_{\bar
X}\xi)\Big]= c\,\Big[\bar g(\xi,\bar X)A_\xi\bar Y+\bar g(\xi,\bar Y)A_\xi\bar X\Big].
\end{array}
$$
So, $\xi$ is totally geodesic if
$$
\bar g(\xi,\bar X)\bar Y+\bar g(\xi,\bar Y)\bar X
-\bar\nabla_{\bar g(\xi,\bar X)A_\xi\bar Y+\bar g(\xi,\bar Y)A_\xi\bar X}\xi=
2\bar g(\bar X,\bar Y)\,\xi,
$$
or
$$
\bar g(\xi,\bar X)\Big[\bar Y+A_\xi(A_\xi\bar Y)\Big]+\bar g(\xi,\bar Y)\Big[\bar X
+A_\xi(A_\xi\bar X)\Big]=2\bar g(\bar X,\bar Y)\,\xi,
$$
for all vector fields $\bar X,\bar Y$ on $M^n$. Choosing $\bar X,\bar Y$ such that
$\bar X_x\ne 0$ and $\bar X_x=\bar Y_x \perp \xi_x,$ we get $2|\bar X_x|^2\xi_x=0$.
Therefore, $\xi=0$ for all $x\in M^n.$
Let $\xi$ be a non-zero Killing vector field on a manifold with \textit{positive}
(non-constant) sectional curvature. From (\ref{Killing}) it follows that
$A_\xi\xi\perp\xi$. If $A_\xi\xi=0$, then, after setting $Y=\xi$ in (\ref{Killing}),
we conclude that $\xi$ has a constant length and therefore can be totally geodesic if
it is a parallel vector field \cite{Wk}. In this case, $M^n=M^{n-1}\times E^1$ and we
come to a contradiction. Suppose that $A_\xi\xi\ne0$. Then $\xi\wedge A_\xi\xi$ is a
non-zero bivector field. Setting $\bar Y=\bar X$ in (\ref{Omega}) and using
(\ref{KillProp}), we have
$$
A_\xi\Big[\bar R(\xi,A_\xi \bar X)\bar X\Big]+\bar R(\xi, \bar X)\bar X=0.
$$
Taking a scalar product in both sides with $\xi$ and applying (\ref{Killing}), we get
$$
-\bar g(\bar R(\xi,A_\xi \bar X)\bar X,A_\xi\xi)+K_{\xi\wedge \bar X}|\xi\wedge \bar
X|^2=0.
$$
Finally, setting $\bar X=A_\xi\xi$, we have $K_{\xi\wedge \bar X}=0$ and come to a
contradiction.
\end{proof}
The next Theorem is analogous to the one proved by Walczak P. \cite{Wk}, but does not
have similar rigid consequences for the structure of $M^n$.
\begin{theorem}\label{Th1}
Let $\xi$ be a vector field of constant length along a submanifold $F^l \subset M^n$.
Then $\xi(F^l)$ is a totally geodesic submanifold in $TM^n$ if and only if $F^l$ is
totally geodesic in $M^n$ and $\xi$ is a parallel vector field on $M^n$ along $F^l$.
\end{theorem}
\begin{proof}
The condition $|\,\xi\, |=const$ implies ${\bar g}\,(\bar \nabla_X \xi, \xi)=0$ for
any vector field $X$ tangent to $F^l$ . As $\xi(F^l)$ is supposed to be totally
geodesic, it follows from the second condition of Proposition \ref{Pr5} that ${\bar
g}\,(\bar \nabla_X \bar \nabla_Y \xi, \xi)=0$. Hence ${\bar g}\,(\bar \nabla_X \xi,
\bar \nabla_Y \xi)=0$ for any $X,Y \in T_xF^l$, $x \in F^l$. Supposing $X=Y$, we see
that $\bar \nabla_X \xi =0$, i.e. $\xi$ is parallel along $F^l$ in the ambient space
and the second condition of Proposition \ref{Pr5} is fulfilled. Moreover, the
condition $\bar \nabla_X \xi =0$ means that the $\xi$-connection (\ref{Conn})
coincides with the Levi-Civita connection of $M^n$, so that by Proposition \ref{Pr5}
$F^l$ is totally geodesic in $M^n$.
On the other hand, if $F^l$ is totally geodesic in $M^n$ and $\bar \nabla_X \xi=0$ for
any tangent vector field $X$ on $F^l$, then both conditions from Proposition \ref{Pr5}
are satisfied evidently.
\end{proof}
Giving more restrictions on the vector field, we can a more geometrical result.
\begin{theorem}\label{Th2}
Let $\xi $ be a normal vector field on a submanifold $~F^l \subset~M^n,$ which is
parallel in the normal bundle. Then $\xi (F^l)$ is totally geodesic in $TM^n$ if and
only if $F^l$ is totally geodesic in $M^n.$
\end{theorem}
\begin{proof}
If $\xi$ is a normal vector field to $F^l$ and parallel in the normal bundle, then
$\bar \nabla_X \xi=-A_{\xi} X$ for each vector field $X$ on $F^l$, where $A_{\xi}$ is
the shape operator of $F^l$ with respect to $\xi,$ and hence ${\bar g}\,(\bar
\nabla_X \xi, \xi)=0.$ This means that $|\xi |$=const along $F^l$.
Let $\xi(F^l)$ be totally geodesic in $TM^n$. Then from (b) of Proposition \ref{Pr5}
we see that $\bar g \,(\bar\nabla_X\bar\nabla_Y\xi,\xi)=0$, which implies $|\bar
\nabla_X\xi|=0$ for each $X$ tangent to $F^l$. In this case, along $F^l$ the
$\xi$-connection (\ref{Conn}) coincides with the Levi-Civita connection of $M^n$ and
(a) of Proposition \ref{Pr5} implies the totally geodesic property of $F^l$.
Conversely, if $\xi$ is a normal vector field which is parallel in the normal bundle
of $F^l$ and $F^l$ is totally geodesic, then $\bar \nabla_X\xi=0$ for any vector field
$X$ tangent to $F^l$. Evidently, both conditions of Proposition \ref{Pr5} are
fulfilled.
\end{proof}
The application of Theorem \ref{Th2} to the specific case of a foliated Riemannian
manifold allows to clarify the geometrical structure of $\xi(M^n)$. The manifold $M^n$
is said to be \textit{$\nu$-foliated} if it admits a family $\mathcal{F}$ of connected
$\nu$-dimensional submanifolds $\{\mathcal{F}_\alpha; \alpha\in A\}$ called
\textit{leaves} such that (i) $M^n=\bigcup\limits_{\alpha\in A}\mathcal{F}_\alpha$;
(ii) $\mathcal{F}_\alpha\cap\mathcal{F}_\beta=\emptyset$ for $\alpha\ne\beta$; (iii)
there exists a coordinate covering $\mathcal{U}$ of $M^n$ such that in each local
chart $U\in\mathcal{U}$ the leaves can be expressed locally as level submanifolds,
i.e. $u^{\nu+1}=c_{\nu+1},\dots,u^n=c_n$.
The family $\mathcal{F}$ is called a \textit{$\nu$-foliation} and
\textit{hyperfoliation} for $\nu=n-1$. The hyperfoliation is said to be
\textit{transversally orientable }if $M^n$ admits a vector field $\xi$ transversal to
the leaves. Moreover, with respect to the Riemannian metric on $M^n$, this vector
field can be chosen as a field of unit normals for each leaf.
A submanifold $F^{k+\nu}\subset M^n$ is called \textit{$\nu$-ruled} if $F^{k+\nu}$
admits a $\nu$-foliation $\big\{\mathcal{F}_\alpha;\, \alpha\in A\big\}$ such that
each leaf $\mathcal{F}_\alpha$ is totally geodesic in $M^n$. The leaves
$\mathcal{F}_\alpha$ are called \textit{elements} or \textit{generators} \cite{Rov}.
\begin{corollary}\label{Foli}
Let $M^n$ be a Riemannian manifold admitting a totally geodesic transversally
orientable hyperfoliation $\mathcal{F}$. Let $\xi$ be a field of normals of the
foliation having constant length. Then $\xi(M^n)$ is an $(n-1)$-ruled submanifold in
$TM^n$ with the elements $\xi(\mathcal{F}_\alpha)$.
\end{corollary}
\begin{proof}
Indeed, let $\mathcal{F}_\alpha$ be a leaf of the hyperfoliation and $\xi$ be a vector
field of constant length on $M^n$ which is a field of normals along each leaf.
Applying Theorem \ref{Th2}, we get that $\xi(\mathcal{F}_\alpha)$ is totally geodesic
in $TM^n$ for each $\alpha$. Since $\xi:M^n\to \xi(M^n)$ is a homeomorphism,
$\xi(\mathcal{F}_\alpha)\cap\xi(\mathcal{F}_\beta)=\emptyset$ for $\alpha\ne\beta$ and
$\xi(M^n)=\bigcup\limits_{\alpha\in A}\xi(\mathcal{F}_\alpha)$. Finally, if
$\mathcal{F}_\alpha$ is given by $u^{n}=c_n$ within a local chart $U$ then from
(\ref{Sub}) we see that $\xi(\mathcal{F}_\alpha)$ is given by the same equalities
within the local chart $\xi(U)$. So, $\xi(\mathcal{F})=\big\{\xi(\mathcal{F}_\alpha);
\alpha\in A\big\}$ form a hyperfoliation on $\xi(M^n)$ with totally geodesic leaves in
$TM^n$.
\end{proof}
\section{The case of a base space of constant curvature.}
If the ambient space is of constant curvature $c\ne0$ and $\xi$ is a normal vector
field on a submanifold $F^l \subset M^n$, then the necessary and sufficient condition
on $\xi$ to generate a totally geodesic submanifold in $TM^n$ takes a rather simple
form.
\begin{theorem}\label{Th3}
Let $F^l$ be a submanifold of a space $M^n(c)$ of constant curvature $c\ne 0$. Let
$\xi$ be a normal vector field on $F^l.$ Then $\xi (F^l)$ is totally geodesic in
$TM^n$ if and only if $F^l$ is totally geodesic in $M^n(c)$ and $\xi$ is parallel in
the normal bundle.
\end{theorem}
\begin{proof}
The curvature tensor of $M^n (c)$ is of the form
\begin{equation}\label{R}
\bar R(\bar X,\bar Y)\bar Z=
c\ ( {\bar g}\,(\bar Y,\bar Z)\bar X-{\bar g}\,(\bar X,\bar Z)\bar Y\,).
\end{equation}
If $\xi $ is a normal vector field on $F^l$ then
$
\bar \nabla_X \xi=-A_{\xi} X+\nabla^{\perp}_X \xi.
$
As $A_{\xi} X$ is tangent and $\nabla^{\perp}_X \xi$
is normal to $F^l$, from (\ref{R}) we find
$$
\bar R(\xi, \bar \nabla_X \xi) Y= -c\,g\,(A_{\xi} X,Y)\,\xi
$$
for any vector fields $X,Y$ on $F^l.$ Thus, the conditions from Proposition \ref{Pr5}
mean that
\begin{equation}\label{Eqn5}
\left\{
\begin{array}{l}
\bar \nabla_X Y-c\,g\,(A_{\xi} X,Y) \xi
\mbox{ \ \ is tangent to \ } F^l, \\[2ex]
\bar \nabla_{\bar \nabla_X Y-c\,g\,(A_{\xi} X,Y) \xi} \xi = \bar \nabla_X
\bar \nabla_Y \xi.
\end{array}
\right.
\end{equation}
Multiplying $(\ref{Eqn5})_1$ by $\xi$ and by normal vector field $\eta$ orthogonal to
$\xi$, we have
$$
\left\{
\begin{array}{r}
g\,(A_{\xi} X,Y)(1-c \, |\xi |^2)=0,\\[1ex]
g\,(A_{\eta} X,Y)=0.
\end{array}
\right.
$$
If $\xi$ is of constant length $|\xi |^2=\frac{1}{c} \ \ (c>0)$
then by Theorem \ref{Th1}, $F^l$ is totally geodesic in $M^n,$ otherwise $F^l$
is totally geodesic immediately.
So, $F^l$ is totally geodesic and therefore $\bar \nabla_X \xi= \nabla^{\perp}_X \xi$,
$\bar\nabla_XY=\nabla_XY$. The condition $(\ref{Eqn5})_2$ now takes the form
\begin{equation}\label{Eqn6}
\nabla^{\perp}_{\nabla_X Y} \xi = \nabla^{\perp}_X
\nabla^{\perp}_Y \xi.
\end{equation}
Set $Y=\nabla_V Z$, where $V$ and $Z$ are arbitrary vector fields tangent to $F^l$.
Then from (\ref{Eqn6}), we get
$$
\nabla^{\perp}_{\nabla_X \nabla_V Z} \xi = \nabla^{\perp}_X \nabla^
{\perp}_{\nabla_V Z} \xi.
$$
Applying (\ref{Eqn6}) to $\nabla^{\perp}_{\nabla_V Z} \xi$ in the right-hand side of
the above equation, we see that $\nabla^{\perp}_{\nabla_V Z} \xi =
\nabla^{\perp}_V\nabla^{\perp}_Z \xi$ and therefore,
\begin{equation}\label{Eqn7}
\nabla^{\perp}_{\nabla_X \nabla_V Z} \xi = \nabla^{\perp}_X \nabla^{\perp}_V
\nabla^{\perp}_Z \xi.
\end{equation}
Interchanging the roles of $X$ and $V$, we get
\begin{equation}\label{Eqn8}
\nabla^{\perp}_{\nabla_V \nabla_X Z} \xi = \nabla^{\perp}_V \nabla^{\perp}_X
\nabla^{\perp}_Z \xi.
\end{equation}
Finally, applying again (\ref{Eqn6}) to the bracket $[X,V]$ and $Z$, we get
\begin{equation}\label{Eqn9}
\nabla^{\perp}_{\nabla_{[X,V]}Z} \xi = \nabla^{\perp}_{[X,V]}
\nabla^{\perp}_Z \xi.
\end{equation}
Combining (\ref{Eqn7}),(\ref{Eqn8}) and (\ref{Eqn9}), we obtain
$$
\nabla^{\perp}_{R(X,V)Z} \xi = R^\perp(X,V) \nabla^{\perp}_Z \xi
$$
where $R$ is the curvature tensor of $F^l$ and $R^\perp$ is the normal curvature
tensor. Since $F^l$ is totally geodesic and $M^n(c)$ is of constant curvature,
$R^\perp(X,Y)\eta \equiv 0$ for any normal vector field $\eta$ and, moreover,
$$
R(X,Y)Z = c\,(g\,(Y,Z)X - g\,(X,Z)Y).
$$
So, we have
$$
c\, \nabla^{\perp}_{g\,(Y,Z)X-g\,(X,Z)Y} \xi = 0.
$$
Setting $X$ orthogonal to $Y$ and $Y=Z$ we get
$
\nabla^{\perp}_X \xi = 0
$
for any vector field $X$ on $F^l$, which completes the necessary part of the proof.
The sufficient part is trivial.
\end{proof}
The application of Theorem \ref{Th3} to the case of a space of
constant curvature shows the difference between our considerations
and Walczak's \cite{Wk}. Let $S^n$ be the unit sphere and
$S^{n-1}$ be the unit totally geodesic great sphere in $S^n$.
Denote by $D^n$ an open equatorial zone around $S^{n-1}$ where the
unit geodesic vector field orthogonal to $S^{n-1}$ is regularly
defined. Then $D^n$ is a Riemannian manifold of constant positive
curvature and $S^{n-1}$ is a totally geodesic submanifold in
$D^n$.
\textit{Let $\xi$ be a unit (or of constant length) geodesic
vector field on $D^n\subset S^n$ which is normal to the totally
geodesic great sphere $S^{n-1}$. Then $\xi(D^n)$ is not totally
geodesic in $TD^n$ while the restriction of $\xi$ to $S^{n-1}$
generates the totally geodesic submanifold $\xi(S^{n-1})$ in
$TD^n$.}
Indeed, $\xi$ is of constant length and by Walczak's result,
$\xi(D^n)$ can be totally geodesic in $TD^n$ only if $\xi$ is a
parallel vector field on $D^n$ \cite{Wk}, which is impossible due
to positive curvature of $D^n$. On the other hand, $\xi$ is
parallel in the normal bundle of $S^{n-1}\subset D^n$ and we can
apply Theorem \ref{Th3} to see that $\xi(S^{n-1})$ is totally
geodesic in $TD^n$.
As concerns flat Riemannian manifolds, Walczak has shown that
every totally geodesic vector field on a flat Riemannian manifold
is harmonic (cf. \cite{Wk}) and that, consequently, on a compact
flat Riemannian manifold, a vector field is totally geodesic if
and only if it is parallel. We shall give a similar result for
vector fields along submanifolds.
\begin{theorem}\label{Flat}
Let $F^l$ be a compact oriented submanifold in a flat Riemann\-ian
manifold $M^n$. Let $\xi$ be a vector field on $F^l$. Then
$\xi(F^l)$ is totally geodesic in $TM^n$ if and only if $F^l$ is
totally geodesic in $M^n$ and $\xi$ is parallel along $F^l$.
\end{theorem}
\begin{proof}
Since $M^n$ is flat, the $\xi$-connection is the same as the
Levi-Civita connection on $M^n$. So, by Proposition \ref{Pr5},
$\xi(F^l)$ is totally geodesic if and only if $F^l$ is totally
geodesic and
\begin{equation}\label{A19}
\bar \nabla_{X}\bar \nabla_{Y}\xi=\bar \nabla_{\bar \nabla_{X}Y} \xi
\end{equation}
for all vector fields $X$ and $Y$ on $F^l$.
Suppose now that $\xi(F^l)$ is totally geodesic. Then $F^l$ is
totally geodesic and is thus flat. Hence locally we can choose
vector fields $X_1$, $X_2$,...,$X_l$ tangent to $F^l$ such that
$\bar \nabla_{X_i}X_j=\nabla_{X_i}X_j=0$, and $\bar
g(X_i,X_j)=g(X_i,X_j)=\delta_{ij}$, for all $i,j=1,...,l$. Putting
$X=Y=X_i$ in the identity (\ref{A19}), we obtain $\bar
\nabla_{X_i}\bar \nabla_{X_i}\xi=0$. Hence, $\sum_{i=1}^l \bar
g(\bar \nabla_{X_i}\bar \nabla_{X_i}\xi,\xi)=0$, i.e.
\begin{equation}\label{A20}
\sum_{i=1}^l X_i.\bar g( \bar \nabla_{X_i}\xi,\xi)=\sum_{i=1}^l |
\bar\nabla_{X_i}\xi |^2.
\end{equation}
If we consider the function $f$ defined by $f(x)=\frac {1}{2} \bar
{g}_x(\xi,\xi)$, for all $x\in F^l$, then we can define a global
vector field $X_f$ on $F^l$ by the local formula $X_f =g( \bar
\nabla_{X_i}\xi,\xi)X_i$. Formula (\ref{A20}) can thus be written
locally as div$X_f$=$\sum_{i=1}^l | \bar \nabla_{X_i}\xi |^2$.
Integrating both sides of the last equality and applying Green's
theorem, we obtain $\sum_{i=1}^l \int_{F^l}| \bar \nabla_{X_i}\xi
|^2 dv=0$, and hence $\bar\nabla_{X_i}\xi=0$, for all $i=1,...,l$.
Therefore $\xi$ is parallel along $F^l$.
The sufficient part of the theorem is trivial.
\end{proof}
\begin{remarks}
1.\ If in Theorem \ref{Flat} the field $\xi$ is a normal vector field along $F^l$,
then $\bar \nabla_{X}\xi$ is also normal for each vector field $X$ on $F^l$. Indeed,
for the $X_i$'s constructed in the proof of the theorem, we have $\bar g( \bar
\nabla_{X_i}\xi,X_j)=X_i.\bar g(\xi,X_j)=0$, and so $ \bar \nabla_{X_i}\xi$ is normal
to $F^l$. Hence the identity (\ref{A19}) can be written as
\begin{equation}\label{A21}
\bar \nabla_{X}^{\perp}\bar \nabla_{Y}^{\perp}\xi=\bar
\nabla_{\nabla_X Y}\xi.
\end{equation}
Also, $\xi$ is parallel if and only if it is parallel in the normal bundle. Hence
$\xi(F^l)$ is totally geodesic if and only if $F^l$ is totally geodesic and $\xi$ is
parallel in the normal bundle.
2.\ The condition of compactness is necessary. Indeed, if we consider $\mathbb{R}^n$
with its canonical coordinates $(x_1,x_2,...,x_n)$ and its canonical Euclidean metric,
and the hypersurface $\mathbb{R} ^{n-1}$ which is identified with the subspace given
by: $x_n=0$, then $\mathbb{R} ^{n-1}$ is an oriented totally geodesic submanifold of
$\mathbb{R} ^n$. We have $\bar \nabla_{\partial/\partial x_i}
\partial/\partial x_j=0$ for all $i,j=1,...,n$. We consider the
vector field $\xi$ on $\mathbb{R} ^n$ along $\mathbb{R} ^{n-1}$
defined by $\xi(x)=x_1 \partial/\partial x_n (x)$, where $x_1$ is
the first component of $x$. Now, to show that $\xi(\mathbb{R}
^{n-1})$ is totally geodesic in $T \mathbb{R} ^n$, it suffices to
check that (\ref{A19}) is verified. In fact,$\bar
\nabla_{\partial/\partial x_i} \bar \nabla_{\partial/\partial
x_j}\xi=\bar \nabla_{\partial/\partial x_i} \delta_{1j}
\partial/\partial x_n =0$. But $\bar \nabla_{\partial/\partial
x_1}\xi=\partial/\partial x_n$, and so $\xi$ is not parallel.
\end{remarks}
\section{The case of Lie groups with bi-invariant metrics}
Let us consider a connected Lie group $G^n$ equipped with a
bi-invariant metric $\bar g$, i.e. invariant by both left and
right translations. We shall generalize the results of Walczak~P.
\cite{Wk} on totally geodesic left invariant vector fields on
$G^n$ to left invariant vector fields along Lie subgroups.
Let $H^l$ be a Lie subgroup of $G^n$. The metric $g$ induced from
$\bar g$ on $H^l$ is a bi-invariant metric. If we denote by $\bar
\nabla$ and $\nabla$ the Levi-Civita connections on $G^n$ and
$H^l$ respectively, then we have $\bar \nabla_X
Y=\frac{1}{2}[X,Y]$, for all $X$,$Y$ of $\mathfrak{g}$, the Lie
algebra of $G^n$, and $\nabla_X Y=\frac{1}{2}[X,Y]$, for all
$X$,$Y$ of $\mathfrak{h}$, the Lie algebra of $H^l$.
\begin{lemma}\label{Subalg}
A connected complete submanifold $F^l$ of $G^n$ containing the
identity element $e$ of $G^n$, such that $T_e F^l$ is a subalgebra
of $\mathfrak{g}$, is totally geodesic if and only if $F^l$ is a
Lie subgroup $H^l$ of $G^n$.
\end{lemma}
\begin{proof}
If we denote by $\exp$ the exponential mapping $\exp:\mathfrak{g}
\to G^n$ of the Lie group $G^n$, and by $\exp_x:T_x G^n \to G^n$
the exponential map at a point $x$ of $G^n$ with respect to the
Levi-Civita connection of the metric $g$, then for all $x\in G^n$,
$\exp_x=\exp \circ (L_{x^{-1}})_*$, where $L_x$ is the left
translation of $G^n$ by $x$. Indeed, we show firstly that
$\exp_e=\exp$. Let $X\in \mathfrak{g}\equiv T_eG^n$ and
$\gamma(t)=\exp tX$. It suffices to check that $\gamma$ is a
geodesic. We have $\dot \gamma (t)=(L_{\gamma(t)})_*(\dot
\gamma(0))=(L_{\gamma(t)})_*(X)$, and thus $\bar \nabla_{\dot
\gamma (t)}\dot \gamma (t)=\bar
\nabla_{X(\gamma(t))}X(\gamma(t))$, where $X$ denotes also the
left invariant vector field on $G^n$ corresponding to $X$. Hence
$\bar \nabla_{\dot \gamma (t)}\dot \gamma
(t)=\frac{1}{2}[X,X](\gamma(t))=0$, and so $\exp_e=\exp$. Now, our
assertion follows from the fact that left translations are
isometries.
We consider a Lie subgroup $H^l$ of $G^n$ and $\mathfrak{h} =T_e
H^l$ its Lie algebra. If $X\in\mathfrak{h}$, then $\exp_e tX=\exp
tX \in H^l$, for all $t$ in a neighborhood of $0$, i.e. $H^l$
contains the geodesic starting from $e$ and with initial condition
$X$, and by the left translations, $H^l$ contains all geodesics
starting from points of $H^l$ with initial vectors tangent to
$H^l$ at these points. Thus $H^l$ is totally geodesic.
Conversely, suppose that $F^l$ is a connected complete
submanifold of $G^n$ such that $e\in F^l$ and $T_e
F^l=:\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. Let
$H^l$ be the connected subgroup of $G^n$ with Lie algebra
$\mathfrak{h}$. $H^l$ is then a connected totally geodesic
submanifold of $G^n$ with $T_e H^l=T_e F^l$. Therefore $H^l=F^l$.
\end{proof}
\begin{proposition}\label{L-Invar}
A left invariant vector field on $G^n$ along a submanifold $F^l$
generates a totally geodesic submanifold of $TG^n$ if and only if
it is parallel along $F^l$ and $F^l$ is totally geodesic.
\end{proposition}
\begin{proof}
A left invariant vector field on $G^n$ is necessarily of constant
length, and we apply Theorem \ref{Th1}.
\end{proof}
\begin{corollary}
A left invariant vector field $\xi$ on $G^n$ along a Lie subgroup
$H^l$ is totally geodesic if and only if it is an element of the
centralizer of $\mathfrak{h}$ in $\mathfrak{g}$.
\end{corollary}
\begin{proof}
By Lemma \ref{Subalg}, $H^l$ is a totally geodesic submanifold in
$G^n$. Thus, by virtue of Proposition \ref{L-Invar}, $\xi$ is
totally geodesic if and only if $\xi$ is parallel along $H^l$.
Suppose that $\xi$ is totally geodesic. Then $\bar \nabla_X \xi
=0$, for all $X \in \mathfrak{h}$; i.e. $\xi$ is in the
centralizer of $\mathfrak{h}$ in $\mathfrak{g}$.
Conversely, if $\xi$ is in the centralizer of $\mathfrak{h}$ in
$\mathfrak{g}$, then $\bar \nabla_X \xi =0$, for all $X \in
\mathfrak{h}$. Let $x \in H^l$ and $z \in T_x H^l$. It suffices to
prove that $\bar \nabla_z \xi =0$. But $X:=(L_{x^{-1}})_*(z) \in
T_e H^l \equiv \mathfrak{h}$, and consequently $\bar \nabla_z \xi=
(\bar \nabla _X \xi)(x) =0$.
\end{proof}
\begin{corollary}
(a)\ There are no non-zero left invariant totally geodesic vector fields on a
semi-simple Lie subgroup of a Lie group with a bi-invariant Riemannian metric.
(b)\ Every left invariant vector field along a subgroup of an abeli\-an Lie group with
a bi-invariant Riemannian metric generates a totally geodesic submanifold of the
tangent bundle.
\end{corollary}
\begin{theorem}\label{CompCon}
Let $N^l$ be a connected complete totally geodesic embedded
submanifold of the tangent bundle of a connected Lie group $G^n$
equipped with a bi-invariant Riemannian metric such that
$H^l=\pi(N^l)$ is a Lie subgroup of $G^n$. Suppose that $N^l$ is
horizontal at a point $z$ of $T_e G^n$.
(a)\ If $z \in T_e H^l$, then $N^l$ is the image of $H^l$ by a left invariant vector
field on $H^l$ which belongs to the center of $\mathfrak{h}$. In particular, if $H^l$
is semi-simple, then $H^l$ is the only connected totally geodesic embedded submanifold
of $TG^n$ which is tangent to $H^l$ at $e$ and orthogonal to the fiber at a point of
$T_eG^n$.
(b)\ If $H^l$ is simple, then $N^l$ is the image of $H^l$ by a left invariant vector
field on $G^n$ along $H^l$ which belongs to the centralizer of $\mathfrak{h}$ in
$\mathfrak{g}$.
\end{theorem}
\begin{proof}
Using Proposition \ref{Transv}, there is a neighborhood $U$ of $e$
in $G^n$, a neighborhood $V$ of $z$ in $TG^n$ and a vector field
$Y$ on $M^n$ along $H^l \cap U$ such that $N^l \cap V=Y(H^l \cap
U), Y(e)=z$. We have $T_z N^l=T_z (N^l \cap V)=T_z Y(H^l \cap U)$.
Then each vector of $T_z N^l$ can be written as $X^h + (\bar
\nabla_X Y)^v$, for some $X\in \mathfrak{h}$. But $T_z N^l$ is a
subset of the horizontal subspace of $TTG^n$ at $z$, so at $e$ we
have $\bar \nabla_X Y=0$ for all $X\in \mathfrak{h}$. On the other
hand, since $N^l \cap V=Y(H^l \cap U)$ is totally geodesic, the
second assertion of Proposition \ref{Pr5} reduces at $e$ to the
identity $$ \bar \nabla_{X_1}\bar \nabla_{X_2}Y=\frac{1}{2}\bar
R(X_1,X_2)Y , \mbox{ for all vector fields } X_1,X_2 \mbox{ on }
\, H^l. $$ Then for all $W\in \mathfrak{g}=T_e G^n$, we have $$
\bar g(\bar \nabla_{X_1(e)}\bar \nabla_{X_2}Y,W)= \frac{1}{2} \bar
g(\bar R(X_1(e),X_2(e))Y(e),W). $$ If we extend $W$ to a vector
field $X_3$ along $H^l$, which is orthogonal to $\bar
\nabla_{X_2}Y$ in a neighborhood of $e$ in $H^l$, then we can
write $$ \bar g(\bar \nabla_{X_1(e)}\bar \nabla_{X_2}Y,W)=-\bar
g(\bar \nabla_{X_2(e)}Y,\bar \nabla_{X_1(e)}X_3)=0, $$ and
consequently, $ \bar g(\bar R(X_1(e),X_2(e))Y(e),W)=0, $ for all
$X_1(e),X_2(e)\in \mathfrak{h}=T_e H^l$ and $W\in \mathfrak{g}=T_e
G^n$. Therefore we have $$R(\cdot,\cdot)Y(e)=0, \mbox { when
applied to vectors in $T_e H^l$}. $$
Let us denote by $\xi$ the left invariant vector field on $G^n$
along $H^l$ such that $Y(e)=\xi(e)$. Then $\bar
R(\cdot,\cdot)\xi(e)=0$ when applied to vectors in $T_e H^l$, and
hence
\begin{equation}\label{center}
\bar R(\cdot,\cdot)\,\xi=0, \mbox{ when applied to elements of
$\mathfrak{h}$.}
\end{equation}
Consider now two cases.
(a)\ If $\xi(e)=z \in T_e H^l$, then $\xi \in \mathfrak{h}$, and we have, by virtue of
(\ref{center}), $\bar R(X,\xi)\xi=0$, for all $X \in \mathfrak{h}$. Thus
$|\,[\xi,X]\,|\,^2=4\bar g(\bar R(\xi,X)X,\xi)=0$ for all $X\in \mathfrak{h}$. It
follows that $\xi$ belongs to the center of $\mathfrak{h}$.
(b)\ If $H^l$ is simple, then $[\mathfrak{h},\mathfrak{h}]= \mathfrak{h}$. But $\bar
\nabla_{[X_1,X_2]} \xi= \frac12 [[X_1, X_2], \xi]= -2R(X_1,X_2)\xi=0$, for all $X_1$,
$X_2 \in \mathfrak{h}$, by virtue of (\ref{center}). Since
$[\mathfrak{h},\mathfrak{h}]= \mathfrak{h}$, we deduce easily that $\bar \nabla _X
\xi=0$, for all $X \in \mathfrak{h}$, or equivalently $[X,\xi]=0$, for all $X \in
\mathfrak{h}$. It follows that $\xi$ belongs to the centralizer of $\mathfrak{h}$ in
$\mathfrak{g}$.
In both cases, $\xi$ belongs to the centralizer of $\mathfrak{h}$ in $\mathfrak{g}$.
Hence, by Lemma \ref{Subalg}, $H^l$ is totally geodesic in $G^n$, and Proposition
\ref{L-Invar} implies then that $\xi(H^l)$ is a complete totally geodesic submanifold
of $TG^n$. Therefore $\xi(H^l)=N^l$, because $ \xi_* (T_e H^l)=T_z N^l$ and $N^l$ and
$H^l$ are connected.
\end{proof}
\begin{corollary}
Let $N^l$ be a connected complete horizontal totally geodesic
submanifold of the tangent bundle of a connected Lie group $G^n$
equipped with a bi-invariant Riemannian metric such that
$H^l=\pi(N^l)$ is a simply connected submanifold of $G^n$
containing the identity element. Suppose that $\mathfrak{h}:=
\pi_*(T_z N^l)$ is a Lie subalgebra of $\mathfrak{g}$ for a point
$z$ of $T_e G^n\cap N^l$. If $Z \in T_e H^l$ (resp. $\mathfrak{h}$
is simple), then $H^l$ is a Lie subgroup of $G^n$ and $N^l$ is the
image of $H^l$ by a left invariant vector field on $H^l$ (resp. on
$G^n$ along $H^l$) which belongs to the center of $\mathfrak{h}$
(resp. centralizer of $\mathfrak{h}$ in $\mathfrak{g}$).
\end{corollary}
\begin{proof}
By Theorem \ref{Hor-Sub}, $H^l$ is complete and totally geodesic.
It follows from Lemma \ref{Subalg} that $H^l$ is a Lie subgroup of
$G^n$. Now, our corollary follows from Theorem \ref{CompCon}.
\end{proof}
|
{
"timestamp": "2005-03-24T21:52:54",
"yymm": "0503",
"arxiv_id": "math/0503561",
"language": "en",
"url": "https://arxiv.org/abs/math/0503561"
}
|
\section{INTRODUCTION}
Due to the fundamental importance of the waves and instabilities
in plasma and hydrodynamics investigations, computational
researchers have devoted great efforts in developing appropriate
tools. One of the main challenges after developing numerically
stable algorithms in fluid models has been generation of the waves
in the linear, nonlinear as well as unstable modes; i.e. waves
which preserve analytic dispersion relations\footnote{The
waves'propagation characteristics are encoded in the dispersion
relations\cite{whitham}}\cite{Tam}. Furthermore extending the case
of hydrodynamics to that of MHD and or plasma physics one deals
with waves with considerably more complicated propagation
characteristics than the hydrodynamics cases treated by those
authors; i.e. dispersion, polarization, oblique propagations, etc.
The main problems in generating a wave spectrum from small
amplitude disturbances in fluid equations are: (1) the highly
nonlinear nature of those equations; (2) the lack of an initial
thermal velocity distribution. The first problem could cause any
small amplitude configuration space disturbance to grow to very
large amplitudes in relatively short times and result in wave
breaking and non-propagation. Also when there does exist a thermal
distribution, there are always a distribution of thermalized
particles in phase with most waves; they can therefore excite the
allowed modes to at least half their thermal level. Therefore in a
case without thermal equilibrium, a disturbance of arbitrary
wavelength cannot strictly speaking apportion its energy to other
allowed modes. For example in purely electrostatic cases, we know
from equilibrium statistical mechanics that when there exist a
thermal distribution each mode $E_l(k)$ can acquire an energy
\cite{dawson}:
\begin{equation}
\frac{<\mid E_l(k)\mid^2>}{8\pi}\propto kT.
\end{equation}
To investigate MHD wave spectra therefore magnetohydrodynamic
particle codes have served as powerful tools\cite{lebof},
\cite{tajima}, \cite{brunel}, and \cite{kazemi}. For other plasma
waves PIC \cite{birdsal} and \cite{hockney} or hybrid codes \cite
{kazemi}, \cite{winske}, and \cite{hono} have served as the main
wave investigation tools; i.e., basically codes which start from
thermal equilibrium. In these codes the random particle
distribution acts like a disturbance in velocity space and
configuration space remains unaltered at the beginning of each
simulation.
In our case we initiate each simulation by a perturbation in
configuration space. Despite the initial shape of the
perturbation, we observe other allowed modes to develop similar to
PIC simulations. We believe that the mesh discretization and the
finite differencing contribute in the following ways: (i) round of
errors alter the initial perturbation shape and can drive other
wavelength; (ii) as the nonlinear effects grow amplitudes and
shorten wavelengths to the numerical dissipation and dispersion
scale lengths, these effects can act to dampen and initiate the
propagation of the different modes and prevent indefinite
nonlinear growth. These effects can therefore explain the observed
wave spectra. With this then we can use fluid instead of PIC codes
as a convenient alternative to investigate many waves.
The organization of the paper is as follows: in section II the
model is treated analytically; in section III the numerical scheme
(algorithm, stability and conservation laws) are presented; in
section IV the various tests of the model are presented (test of
the dispersion relation, two stream instability, screening effect
and nonlinear harmonic generation). At the end a brief summary and
conclusion with future direction are presented.
\section{ANALYTICAL\ TREATMENT}
We focus on the investigation of the high frequency (hf)
longitudinal waves; i. e. a frequency domain where ions can be
safely assumed to form an immobile background ($n_{0}$ represents
their uniform density). The appropriate equations are then
Poisson's and the electron fluid equations:
\begin{equation}
\frac{\partial n}{\partial t}+\frac{\partial }{\partial x}(nv)=0, \label{1}
\end{equation}
\begin{equation}
\frac{\partial v}{\partial t}+v\frac{\partial }{\partial x}v=\frac{e}{m}%
\frac{\partial }{\partial x}\varphi -\frac{1}{nm}\frac{\partial P}{\partial x%
}, \label{2}
\end{equation}
\begin{equation}
\frac{\partial ^{2}\varphi }{\partial x^{2}}=4\pi e(n-n_{0}). \label{3}
\end{equation}
Here $\varphi $ is the self-consistent electric potential, and
$n$, $v$, $P$ and $m$ represent the electron density, velocity,
pressure and rest mass respectively. Without any loss of
generality this problem is treated in one dimension. These basic
equations are supplemented by an ''equation of state'' according
to the particular thermodynamic properties of the fluid of
interest. Here, isothermal equation of state is used:
\begin{equation}
P=nT, \label{4}
\end{equation}
where $T$ is the electron temperature and is assumed to be
constant and Boltzmann's constant, $k$, is assumed to be unity.
The minimum requirement of any computational model lies in its
ability to preserve conservation laws; for that fluid equations
are cast in flux conservative form. Equation (\ref{2}) in
conservative form upon using Eq. (\ref{4}) in Eq. (\ref{2})
becomes:
\begin{equation}
\frac{\partial v}{\partial t}+\frac{\partial }{\partial x}\left( \frac{1}{2}%
v^{2}-\frac{e}{m}\varphi +\frac{T}{m}\ln n\right) =0. \label{5}
\end{equation}
Note that the logarithmic term is caused by the electron pressure.
Therefore the three equations that form the basis of our model
are:
\begin{equation}
\frac{\partial n}{\partial t}+\frac{\partial }{\partial x}(nv)=0, \label{6}
\end{equation}
\begin{equation}
\frac{\partial v}{\partial t}+\frac{\partial }{\partial x}\left( \frac{1}{2}%
v^{2}-\frac{e}{m}\varphi +\frac{T}{m}\ln n\right) =0, \label{7}
\end{equation}
\begin{equation}
\frac{\partial ^{2}\varphi }{\partial x^{2}}=4\pi e(n-n_{0}). \label{8}
\end{equation}
We will next derive a dispersion relation for wave propagation using Eqs. (%
\ref{6}), (\ref{7}), and (\ref{8}). To do this, linearizing Eqs.
(\ref{6}), (\ref{7}), and (\ref {8}) about a spatially uniform
equilibrium ($n=n_{0}+\delta n$, $v=\delta v$ and $\varphi =\delta
\varphi $), we obtain the following set:
\begin{equation}
\frac{\partial \delta n}{\partial t}+n_{0}\frac{\partial }{\partial x}\delta
v=0, \label{9}
\end{equation}
\begin{equation}
\frac{\partial \delta v}{\partial t}+\frac{\partial }{\partial x}\left( -%
\frac{e}{m}\delta \varphi +\frac{1}{n_{0}}\delta n\right) =0, \label{10}
\end{equation}
\begin{equation}
\frac{\partial ^{2}\delta \varphi }{\partial x^{2}}=4\pi e\delta n.
\label{11}
\end{equation}
Assuming simple plane wave solutions, Eqs. (\ref{9}), (\ref{10}),
and (\ref{11}) reduce to the following set of equations:
\begin{equation}
-i\omega \delta n+ikn_{0}\delta v=0, \label{12}
\end{equation}
\begin{equation}
-i\omega \delta v+ik(-\frac{e}{m}\delta \varphi +\frac{1}{n_{0}}\delta n)=0,
\label{13}
\end{equation}
\begin{equation}
-k^{2}\delta \varphi =4\pi e\delta n. \label{14}
\end{equation}
Eqs. (\ref{12}), (\ref{13}), and (\ref{14}) yield nontrivial
solution if the following is obeyed:
\begin{equation}
\omega ^{2}=\omega _{p}^{2}+k^{2}v_{T}^{2}, \label{15}
\end{equation}
where
\begin{equation}
\omega _{p}^{2}=\frac{4\pi e^{2}n_{0}}{m}\text{ and }v_{T}^{2}=\frac{T}{m}
\label{16}
\end{equation}
are the electron plasma frequency and the thermal velocity,
respectively.
Studies of Langmuir waves (hf electron waves) are of particular importance.
Aside from the applications to real experimental situations which will
become evident in the application section, they serve as excellent probes
for testing the validity of the fluid code that we have developed.
\section{NUMERICAL\ ALGORITHM}
Our model is simply an intuitive construct based on well-known
fluid dynamics and Poisson's equations, geared toward plasma
physics applications, where many different wave phenomena in
dispersive media are of interest. Its physical ''conceptual
basis'' can be regarded as a model that treats non-stationary
electron wave motion for hf domain where $\omega \gg kv_{T}$ in
linear and nonlinear regions. Besides, it can predict electron
wave spectrum more accurately than ''particle in cell simulation''
as here we expect less numerical noise.
\subsection{Normalization}
In these calculations we use the following normalizations:
\begin{equation}
\omega _{p}t\rightarrow t,\quad \frac{x}{r_{D}}\rightarrow x,\quad \frac{v}{%
v_{T}}\rightarrow v,\quad \frac{n}{n_{0}}\rightarrow n,\quad \frac{e\varphi
}{T}\rightarrow \varphi ,\quad \label{17}
\end{equation}
where
\begin{equation}
\text{ }r_{D}^{2}=\frac{T}{4\pi e^{2}n_{0}} \label{18}
\end{equation}
is the electron Debye length. Using these definitions, Eqs. (\ref{1}), (\ref{3}), (\ref{5}%
), and (\ref{15}) can now be rewritten as follows:
\begin{equation}
\frac{\partial n}{\partial t}+\frac{\partial }{\partial x}(nv)=0, \label{19}
\end{equation}
\begin{equation}
\frac{\partial v}{\partial t}+\frac{\partial }{\partial x}\left( \frac{1}{2}%
v^{2}-\varphi +\ln n\right) =0, \label{20}
\end{equation}
\begin{equation}
\frac{\partial ^{2}\varphi }{\partial x^{2}}=n-1, \label{21}
\end{equation}
\begin{equation}
\omega ^{2}=1+k^{2}. \label{22}
\end{equation}
It is already mentioned, logarithmic term in Eq. (\ref{20}) is caused by the
electron pressure.. Thus the code has the flexibility of being easily
converted to the case when electron pressure is negligible.
\subsection{The Numerical Scheme}
Next we shall describe the numerical scheme. The steps of the
scheme are summarized in Table I. A Lax-Wendroff method is used to
push $n$ and $v$, while a poisson solver at the end of each step
updates the electric potential.
The grid spacing and time step are denoted by $\Delta $ and $\Delta t$
respectively. The fluid velocity and density are known at integer time step $%
l$. To complete the initial conditions, $\varphi $ is computed at
the same time step ($l$) by the help of a Poisson solver that is
based on tridiagonal matrix method. Then $n$ and $v$ are pushed
from $l$ to $l+1/2$ as the auxiliary step of the Lax-Wendroff
scheme using Eqs. (\ref{19}) and (\ref{20}) (please refer to item
3 of the Table I). Then again $\varphi $ is computed in the
auxiliary step ($l+1/2$) using the value of $n$ in the mentioned
step. Having known $n$, $v$, and $\varphi $ at the time step
$l+1/2$, we push $n$ and $v$ all the way to time step $l+1$ as the
main step of the Lax-Wendroff scheme in Eqs. (\ref{19}) and
(\ref{20}) (items 5 and 6 in Table I). The electric potential
$\varphi $ is then computed at the time step $l+1$ using $n^{l+1}$.
\begin{center}
\begin{tabular}{|l|}
\hline
\begin{tabular}{l}
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
TABLE\ I \\
\quad \quad \quad Numerical Algorithm of the Fluid Model for Plasma Waves
\end{tabular}
\\ \hline
\begin{tabular}{l}
Initially we have: $n_{m}^{l}$, $v_{m}^{l}$ \\
\quad 1. Compute electric potential, $\varphi _{m}^{l}$, using Poisson
solver. \\
\quad 2. Compute fluxes in continuity and momentum equation in main step: \\
\quad $\quad (f_{n})_{m}^{l}=n_{m}^{l}v_{m}^{l},$ \\
\quad \quad $(f_{v})_{m}^{l}=\frac{1}{2}(v_{m}^{l})^{2}-\varphi _{m}^{l}+\ln
n_{m}^{l}.$ \\
\quad 3. Push velocity and density half a time step: \\
\quad \quad $n_{m+1/2}^{l+1/2}=\frac{1}{2}(n_{m+1}^{l}+n_{m}^{l})-\frac{%
\Delta t}{2\Delta }\left[ (f_{n})_{m+1}^{l}-(f_{n})_{m}^{l}\right] ,$ \\
\quad $\quad v_{m+1/2}^{l+1/2}=\frac{1}{2}(v_{m+1}^{l}+v_{m}^{l})-\frac{%
\Delta t}{2\Delta }\left[ (f_{v})_{m+1}^{l}-(f_{v})_{m}^{l}\right] .$ \\
\quad 4. Compute electric potential in half step, $\varphi
_{m+1/2}^{l+1/2}$, using $n_{m+1/2}^{l+1/2}$.
\\
\quad 5. Compute fluxes in continuity and momentum equations in half step:
\\
$\quad \quad (f_{n})_{m+1/2}^{l+1/2}=n_{m+1/2}^{l+1/2}v_{m+1/2}^{l+1/2},$ \\
\quad \quad $(f_{v})_{m+1/2}^{l+1/2}=\frac{1}{2}(v_{m+1/2}^{l+1/2})^{2}-%
\varphi _{m+1/2}^{l+1/2}+\ln n_{m+1/2}^{l+1/2}.$ \\
\quad 6.Push the velocity and density another half a time step: \\
$\quad \quad n_{m}^{l+1}=n_{m}^{l}-\frac{\Delta t}{\Delta }\left[
(f_{n})_{m+1/2}^{l+1/2}-(f_{n})_{m-1/2}^{l+1/2}\right] ,$ \\
\quad $\quad v_{m}^{l+1}=v_{m}^{l}-\frac{\Delta t}{\Delta }\left[
(f_{v})_{m+1/2}^{l+1/2}-(f_{v})_{m-1/2}^{l+1/2}\right] $.\\
\quad 7.Compute electric potential in the main step, $\varphi
_{m}^{l+1}$, using $n_{m}^{l+1}$.\\
\end{tabular}
\\ \hline
\end{tabular}
\end{center}
\subsection{Conservation Laws}
Equations (19), (20) are in conservative form, and we demand that
the corresponding difference equations to be equally conservative.
More specifically, we expect finite difference scheme to conserve
the mass density ($\int_{-\infty}^{+\infty} n dx$), momentum and
the energy of the system, irrespective of the errors incurred by
the finite difference lattice.
To investigate the conservation laws, in what follows, a method
compatible with both the auxiliary and the main steps will be
presented \cite{Potter}. That is, Eqs. (\ref{19}) and (\ref{20})
are integrated over each space-time cell ($m$) of area $\Delta
t\Delta_m $ ($\Delta t=t^{l+1}-t^{l}$) as follows:
\begin{equation}
\int_{t^{l}}^{t^{l+1}}dt\int_{\Delta _{m}}dx\frac{\partial n}{\partial t}%
=-\int_{t^{l}}^{t^{l+1}}dt\int_{\Delta _{m}}dx\frac{\partial }{\partial x}%
(nv), \label{23}
\end{equation}
\begin{equation}
\int_{t^{l}}^{t^{l+1}}dt\int_{\Delta _{m}}dx\frac{\partial v}{\partial t}%
=-\int_{t^{l}}^{t^{l+1}}dt\int_{\Delta _{m}}dx\frac{\partial }{\partial x}%
\left( \frac{1}{2}v^{2}-\varphi +\ln n\right) . \label{24}
\end{equation}
Here $\int_{\Delta _{m\text{ }}}$denotes integral over the cell labelled by $%
m $. Carrying out trivial integration over $dt$ and $dx$ on the
left and right sides respectively Eqs. (\ref{23}) and (\ref{24})
become:
\begin{equation}
\int_{\Delta _{m}}n^{l+1}dx-\int_{\Delta
_{m}}n^{l}dx=-\int_{t^{l}}^{t^{l+1}}dt\sum_{\alpha }(nv)_{m}, \label{25}
\end{equation}
\begin{equation}
\int_{\Delta _{m}}v^{l+1}dx-\int_{\Delta
_{m}}v^{l}dx=-\int_{t^{l}}^{t^{l+1}}dt\sum_{\alpha }\left( \frac{1}{2}%
v^{2}-\varphi +\ln n\right) _{m}, \label{26}
\end{equation}
where $\alpha $ stands for the boundaries of every cell (the right
and the left). Using
\begin{equation}
\int_{\Delta _{m}}\left(
\begin{array}{l}
n^{l} \\
v^{l}
\end{array}
\right) dx =\Delta \left(
\begin{array}{l}
n_{m}^{l} \\
v_{m}^{l}
\end{array}
\right). \label{27}
\end{equation}
the following equations are thus obtained:
\begin{equation}
n_{m}^{l+1}=n_{m}^{l}-\int_{t^{l}}^{t^{l+1}}dt\frac{1}{\Delta }\sum_{\alpha
}(nv)_{m} \label{28}
\end{equation}
\begin{equation}
v_{m}^{l+1}=v_{m}^{l}-\int_{t^{l}}^{t^{l+1}}dt\frac{1}{\Delta
}\sum_{\alpha }\left( \frac{1}{2}v^{2}-\varphi +\ln n\right) _{m}.
\label{29}
\end{equation}
Summing over cells ($m$) in the system results in:
\begin{equation}
\sum_{m=1}^{M}\left( n_{m}^{l+1}-n_{m}^{l}\right)
=-\sum_{m=1}^{M}\int_{t^{l}}^{t^{l+1}}dt\frac{1}{\Delta }\sum_{\alpha
}(nv)_{m}, \label{30}
\end{equation}
\begin{equation}
\sum_{m=1}^{M}\left( v_{m}^{l+1}-v_{m}^{l}\right)
=\sum_{m=1}^{M}\int_{t^{l}}^{t^{l+1}}dt\frac{1}{\Delta }\sum_{\alpha }\left(
\frac{1}{2}v^{2}-\varphi +\ln n\right) _{m}. \label{31}
\end{equation}
Since finite differences were used in computing all the
derivatives, then if one sums over all the grid cells in the
system, each such quantities will appear twice with opposite signs
corresponding to the cell boundaries that are being shared between
the neighboring cells, and they will thus add up to zero. There
can, however, be contributions from the walls of the computation
box. For the periodic boundary condition the walls contributions
gives zero; for other cases appropriate boundary conditions are
implemented to insure good conservation using guard cells.
\subsection{Numerical Stability Analysis}
In order to obtain the Courant-Fredricks-Lewy (CFL) condition for
the model, the difference equations (obtained from the
differential equations for the problem by discretizing them) must
be considered. We follow the method of Potter \cite{Potter};
\textit{i.e.} obtain the integration time pusher operator from the
difference equations assuming a spatially uniform system and solve
them in Fourier space and obtain a non-local result. We shall do
the stability analysis with the pressure term.
Recall that the differential equations (\ref{19}), (\ref{20}) and
(\ref{21}) formed the basis of the model. These equations upon
linearization, give:
\begin{equation}
\frac{\partial \delta n}{\partial t}+\frac{\partial }{\partial x}\delta v=0,
\label{32}
\end{equation}
\begin{equation}
\frac{\partial \delta v}{\partial t}+\frac{\partial }{\partial x}(-\delta
\varphi +\delta n)=0, \label{33}
\end{equation}
\begin{equation}
\frac{\partial ^{2}\delta \varphi }{\partial x^{2}}=\delta n. \label{34}
\end{equation}
Next using Eqs. (\ref{32}), (\ref{33}), and (\ref{34}), after
combining the
auxiliary and the main steps of the Lax-Wendroff scheme and assuming $n$, $v$%
, and $\varphi $ to have the form ($l$ refers to the time step and $m$
inside the parenthesis to the grid location along $x$)
\begin{equation}
(n^{l}\text{, }v^{l}\text{, }\varphi ^{l})=(\hat{n}^{l}\text{, }\hat{v}^{l}%
\text{, }\hat{\varphi}^{l})e^{i(km\Delta )}, \label{35}
\end{equation}
we obtain the following integration matrix ($\sigma =k\Delta /2$):
\begin{equation}
\left(
\begin{array}{l}
n \\
v \\
\varphi
\end{array}
\right) ^{l+1}=\left(
\begin{array}{ccc}
1-\frac{2\Delta t^{2}}{\Delta ^{2}}\sin ^{2}\sigma & \frac{-2i\Delta t}{%
\Delta }\sin \sigma \cos \sigma & \frac{2\Delta t^{2}}{\Delta ^{2}}\sin
^{2}\sigma \\
\frac{-2i\Delta t}{\Delta }\sin \sigma \cos \sigma -\frac{i\Delta \Delta t}{2%
}\cot \sigma & 1-\frac{\Delta t^{2}}{2}-\frac{2\Delta t^{2}}{\Delta ^{2}}%
\sin ^{2}\sigma & 0 \\
\frac{\Delta t^{2}}{2}-\frac{\Delta ^{2}}{4\sin ^{2}\sigma } & \frac{i\Delta
\Delta t}{2}\cot \sigma & -\frac{\Delta t^{2}}{2}
\end{array}
\right) \left(
\begin{array}{l}
n \\
v \\
\varphi
\end{array}
\right) ^{l}
\end{equation}
Thus, according to Von Neumann stability condition the following
inequality should be held:\footnote{$()^{l+1}=g()^l$ where
$g=e^{-i \omega \Delta t}$; Von Neumann stability condition holds
for $\omega$ real.}
\begin{equation}
\left| g_{\mu }\right| \leq 1, \label{37}
\end{equation}
where $g_{\mu }$ are the eigenvalues of the integration matrix and
subscript refer to different eigenvalues (here $\mu =1,2,3$). The
value of $g_{\mu }$ is then determined by setting the following
determinant equal to zero; i.e.,
\begin{equation}
\left|
\begin{array}{ccc}
1-\frac{2\Delta t^{2}}{\Delta ^{2}}\sin ^{2}\sigma -g & \frac{-2i\Delta t}{%
\Delta }\sin \sigma \cos \sigma & \frac{2\Delta t^{2}}{\Delta ^{2}}\sin
^{2}\sigma \\
\frac{-2i\Delta t}{\Delta }\sin \sigma \cos \sigma -\frac{i\Delta \Delta t}{2%
}\cot \sigma & 1-\frac{\Delta t^{2}}{2}-\frac{2\Delta t^{2}}{\Delta ^{2}}%
\sin ^{2}\sigma -g & 0 \\
\frac{\Delta t^{2}}{2}-\frac{\Delta ^{2}}{4\sin ^{2}\sigma } & \frac{i\Delta
\Delta t}{2}\cot \sigma & -\frac{\Delta t^{2}}{2}-g
\end{array}
\right| =0 \label{38}
\end{equation}
The corresponding solutions for $g$ are simply:
\[
g_{1}=0,
\]
\begin{equation}
g_{2,3=}1-\frac{1}{2}\Delta t^{2}-\frac{2\Delta t^{2}}{\Delta ^{2}}\sin
^{2}\sigma \pm i\sqrt{\Delta t^{2}\cos ^{2}\sigma \left( 1+\frac{4}{\Delta
^{2}}\sin ^{2}\sigma \right) }. \label{39}
\end{equation}
$g_{1}$ fulfills the inequality (\ref{37}). For the two other eigenvalues,
we have:
\begin{equation}
\left| g_{2}\right| =\left| g_{3}\right| =\left[ 1-\Delta t^{2}\left( 1+%
\frac{4}{\Delta ^{2}}\sin ^{2}\sigma \right) +\Delta t^{4}\left( \frac{1}{4}%
+\cos ^{4}\sigma \right) \left( 1+\frac{4}{\Delta ^{2}}\sin ^{2}\sigma
\right) ^{2}\right] ^{1/2} \label{40}
\end{equation}
Equation (\ref{37}) is then obeyed if the following inequality is
held:
\begin{equation}
\Delta t^{2}\left( \frac{1}{4}+\cos ^{4}\sigma \right) \left( 1+\frac{4}{%
\Delta ^{2}}\sin ^{2}\sigma \right) \leq 1. \label{41}
\end{equation}
Since $\Delta t$ and $\Delta $ are small values ($0<\Delta t\ll 1$ and $%
0<\Delta \ll 1$) the inequality (\ref{41}) will be satisfied if$:$
\begin{equation}
\frac{\Delta t}{\Delta }\leq \frac{2}{\sqrt{4+\Delta ^{2}}}. \label{42}
\end{equation}
Inequality (\ref{42}) is exact up to the scheme accuracy, however,
taking into account the smallness of $\Delta t$ and $\Delta $
the following stability condition results:
\[
\frac{\Delta t}{\Delta }\leq 1.
\]
\section{TESTING\ THE\ CODE}
As mentioned, we have constructed the one-dimensional version of
the code and have tested it by looking at small and large
amplitude (nonlinear) effects in an initially uniform plasma. In
what follows, a review of the results will be given.
\subsection{Dispersion relation}
The most basic requirement of a computational model aside from
conservation laws is its ability to predict the linear theory;
e.g. the waves dispersion relation. The degree to which the
analytic dispersion relation is obeyed acts as a gauge of the
computational model and serves to determine its limitations.
From Eq. 40, the dispersion relation of the corresponding difference
equation is:
$$
e^{-\omega _{I}\Delta t}\sin (\omega _{R}\Delta t)=\sqrt{(\Delta t)^{2}\cos
^{2}\sigma (1+\frac{4}{\Delta ^{2}}\sin ^{2}\sigma )}.
$$
where $\omega=\omega_R + \omega_I$. Comparison of this with the
analytic dispersion relation shows that by changing
$k\longrightarrow k\sin (k\Delta )/(k\Delta )$ in the analytic
case
one roughly recovers the above result for $\Delta t\ \omega _{R}\ll 1$ , $%
k\Delta \ll 1$ . The fact that $\omega _{I}$ does not have any $k$
dependence, implies no part of the $k$ space to be more
susceptible to
numerical instability than others\footnote
Many PIC algorithms show $\omega _{I}\propto k^{2}$; i.e. intense
short wavelength noise or instability.}. The difference dispersion
relation above also indicates that for $\sin (k\Delta )/(k\Delta
)\longrightarrow 1$ the numerical dispersion to disappear; i.e.
for modes with wavelengths long compared with the grid spacing it
should be negligible.
For the initial perturbations, small fluctuations in the density from a
uniform background were implemented. Table 2. shows three different initial
perturbations used in the simulations; i.e. :
\begin{center}
\begin{tabular}{|l|}
\hline
$ n(x) = 1 + 0.01 \sin (k_0 x)$ \cr
\hline
$ n(x) = 1 + 0.01(-x+x^3)e^{-x^2} $ \cr
\hline
$ n(x)= 1 + 0.01\left\{
\begin{tabular}{rr}
$-1+x$ & $-1\le x < 0$ \cr
$1-x$ & $0\le x \le 1$ \cr
$ 0 $ & Else where
\end{tabular}
\right.$
\cr
\hline
\end{tabular}
\end{center}
The reason for these choices is that the first perturbation
maintain harmonics with wave numbers very close to $k_{0}$ while
the latter two maintain harmonics more uniformly distributed in
the $k$ space. The most important reason for such choices was to
determine the impact of the initial perturbations on the final
wave spectra; strictly speaking the latter two are expected to
give rise to more uniform spectra. The initial velocity profiles
corresponding to these three profiles are drawn in Figs. 1(a), (b)
and (c). These velocity profiles indicate broader and more uniform
distribution of bulk flow velocities in the latter two; i.e. the
volume of phase space available to wave propagations are
considerably larger.
\begin{figure}
\epsfxsize=9truecm
\centerline{\epsfbox{vx.eps}}
\caption{Velocity profile for a) $n(x) = 1 + 0.01(-x+x^3)e^{-x^2}$,
b)$n(x) = 1 + 0.01 \sin (k_0 x)$ and c) Saw-tooth function}
\end{figure}
Given these two facts though, the plots of the power
spectra\footnote{ The power spectrum is determined in two steps:
First, the spatial FFT is used in a quantity (e.g. E(x,t)) and
stored E($k_i$,t), next for each $k_i$ temporal FFT is performed
on E($k_i$,t)} of the modes versus $\omega$ (their frequency)
indicate very close agreement in all the cases; i.e. regardless of
the initially excited modes and phase velocities, most the allowed
$k$-space tends to get excited. This supports our earlier claim
that the discretization procedure and the numerical dispersion and
dissipation have in effect broadened and stabilized the initial
spectrum.
Finally the plots of the dispersion relation for a system size of
1024$\Delta$ with $\Delta=0.01$ are shown in Fig 2 and 3. The
close agreement between the analytic theory (solid lines) and the
model (circles) for wave numbers $k$ as large as 6 indicate
resolution of the modes with wave lengths of the order of grid
spacing with negligible numerical dispersion. Comparison of these
with the corresponding PIC simulations for a system 256$\Delta$
length (Fig. 4) clearly indicate resolution of much shorter
wavelengths here and considerably less numerical dispersion. This
is understandable since in the PIC models the finite particle size
effects introduce additional numerical dispersion which cause
smaller allowed $k$'s.
\begin{figure}
\epsfxsize=9truecm
\centerline{\epsfbox{presure.eps}}
\caption{Dispersion relation for Langmuir wave.}
\end{figure}
\begin{figure}
\epsfxsize=6truecm
\centerline{\epsfbox{dopler.eps}}
\caption{Dispersion relation for Langmuir wave with Doppler
effect}
\end{figure}
\begin{figure}
\epsfxsize=6truecm
\centerline{\epsfbox{sim_data.eps}}
\caption{Dispersion relation for Langmuir wave for a typical PIC
simulation}
\end{figure}
One last remark about the cases corresponding to Figs. 2 and 3 is
that the latter involves the case in which the bulk plasma had an
initial flow velocity. Fig. 3 not only shows that the doppler
shifted waves also obey their respected dispersion relation, it
also shows how any "resulting" plasma flow could impact those
waves. That is if any nonzero average flow should arise from the
initial perturbations (i.e. if the scheme does not preserve
momentum conservation ) the dispersion relation would be impacted
as in Fig. 3. A glance at Fig. 2 though points that there could
not have been any doppler shift and therefore no net plasma flow
must have resulted from the initial perturbations. Calculations
also showed that $\langle v_f\rangle =0$ initially remained so to
round off errors throughout the simulation. So these plots also
probe the momentum to be conserved in the model.
\subsection{Wave Launching on the Boundary}
In the next example a wave is launched from the boundary and its
behavior is followed. Theoretically, recall that in an
unmagnetized plasma and in the linear regime the plasma shields
any incoming AC density perturbation whose frequency is less than
plasma frequency ($\omega_p$). This effect is shown in Fig. 5(b)
and Fig. 6. In this example the frequency of the applied density
perturbation is half of the plasma frequency. The wave is launched
at $x=-25 \lambda_D$. The amplitude of the density perturbation
has the following range: nonlinear (0.2,1.8) Fig. 5(a) and linear
(0.99,1.01) Fig. 5(b)\footnote{In these particular shots the wave
trough fall at launch points.}. The penetration depth is from
$x=(-25,-20)$ in the linear and $x=(-25,-15)$ in the nonlinear
case. Furthermore as Fig. 6(a) indicates, upon penetration, after
one wave period following the first crest ($x=-18$), the second
crest steepens with its wavelength decreasing to grid cell
scale.\footnote{The oscillations are numerical in nature. The
model should be modified to include FCT filter\cite{Boris} to
eliminate these spurious oscillations.} In the linear regime
though [Fig. 6(b)] no steepening can be seen.
\begin{figure}
\vspace{2cm}
\epsfxsize=10truecm
\centerline{\epsfbox{depth1.eps}}
\caption{Non-linear and linear penetration of electric field
(both plots are sketched at t=10).
a) Nonlinear case b) linear case }
\end{figure}
In the other case, with the same initial condition (respect to
linear case), we launched a wave whose frequency was larger than
the plasma frequency( $\omega
> \omega_p$). This time the density perturbation propagated into the
plasma with its wavelength and amplitude unchanged as it
penetrated the plasma. Its behavior also conformed with the
analytic dispersion relation. The results are shown in Fig. 7.
\begin{figure}
\vspace{3cm}
\epsfxsize=12truecm
\centerline{\epsfbox{screen1.eps}}
\caption{Density versus the position when the external frequency is half of
the plasma frequency. To give a time evolution feeling, they are plotted for
five different normalized time. }
\end{figure}
\begin{figure}
\vspace{3cm}
\epsfxsize=12truecm
\centerline{\epsfbox{propag1.eps}}
\caption{Density versus the position when the external frequency is two times of
the plasma frequency. To give a time evolution feeling, they are plotted for
five different normalized time.}
\end{figure}
\subsection{Two Stream Instability}
As a more severe test of the code, we treated the two stream
instability. Although the instability arises under a wide range of
beam conditions, we shall consider only the simple case of two
countrastreaming uniform beams of electrons with the same number
density $n_0$. The first beam travels in the x direction with
drift velocity $v_d$ and the second beam in the opposite direction
with same drift velocity, i.e. the countrastreaming beams have the
same speed. The dispersion relation is as follows:
\begin{equation}
\frac{\omega_p^2}{(kv_d-\omega)^2}+\frac{\omega_p^2}{(kv_d+\omega)^2}=1
\end{equation}
where $\omega_p^2=4\pi e^2 n_0/m$ is the same plasma frequency for
both beams. One can then obtain the following expression for
$\omega^2$:
\begin{equation}
\omega^2=\omega_p^2+k^2v_d^2\pm
\omega_p{(\omega_p^2+4k^2v_d^2)}^{1/2}.
\end{equation}
This relationship between $\omega^2$ and $k^2$ is shown
graphically in Fig. 8. It is clear that, there exists a critical
wave number $k_c$ which separates the stable and unstable modes.
In fact , for $k^2<k_c^2$ two values of $\omega$ are complex, one
of which represents a growing wave; i.e. an instability. Moreover,
there exists a wave number $k_m$ that corresponds to the most
unstable mode.
\begin{figure}
\epsfxsize=6truecm
\centerline{\epsfbox{dipers.eps}}
\caption{Representation of relationship between $\omega^2$ and $k^2$.}
\end{figure}
These effects are examined by the fluid code. In this case the
code was generalized to a two countrastreaming fluid model. As the
two countrastreaming beams emerging from the opposite ends meet
half way into the simulation box, a growing wavelike disturbance
develops. Figs. 9 and 10 show the evolution of this disturbance
for the cases with and without the pressure terms respectively. In
both cases the disturbance grows locally while in the latter it
also begins to propagate in both directions; i.e. a result of the
dispersion due to the pressure term.
\begin{figure}
\epsfxsize=10truecm
\centerline{\epsfbox{instable.eps}}
\caption{Electric field versus the position in absence of pressure.
Time is normalized by $\omega_p$.}
\end{figure}
\begin{figure}
\epsfxsize=12truecm
\centerline{\epsfbox{instablepres.eps}}
\caption{Electric field versus the position in presence of pressure.
Time is normalized by $\omega_p$.}
\end{figure}
Furthermore, the instability of each mode was investigated using
the mode energy discussed in the previous section: i.e.
\begin{equation}
P(k,t)={|E(k,t)|}^2
\end{equation}
The time derivative of this function with respect to $k$ is shown
in Fig. 11. As expected, there exists a critical wave number
bellow which unstable modes can grow. Furthermore we observed the
the most unstable mode corresponding to $k=k_m$ as the maximum in
the Fig. 11. Also the dynamic evolution of the beam-beam
interaction was observed as a movie and both the disturbance
growth and upstream propagations (when pressure term was included)
were observed.
\begin{figure}
\epsfxsize=8truecm
\centerline{\epsfbox{power.eps}}
\caption{$dp(k,t)/dt$ versus $k$. Cutoff and
maximum wave numbers ($k_c$,$k_m$ ) are comparable with theory.}
\end{figure}
\section{conclusion}
The result of this paper demonstrates that fluid model can be used
to investigate any waves predicted by their basic set of equation.
This can include waves of kinetic nature with and without
dispersion with resolution far greater than the corresponding PIC
codes. It was demonstrated that appropriate initial perturbations
coupled with difference algorithms of sufficient but not excessive
numerical dispersion and dissipation can give rise to wave spectra
spanning all the allowed k-space. Many areas of plasma and or
space research can greatly benefit from these techniques.
|
{
"timestamp": "2005-03-05T14:09:47",
"yymm": "0503",
"arxiv_id": "physics/0503043",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503043"
}
|
\section{Introduction}
The characterization and elimination of decoherence and other
noise sources has emerged as one of the major challenges
confronting the coherent experimental control of increasingly
large multi-body quantum systems. Decoherence arising from
undesired interactions with background (or environment) systems
and imprecision in the classical control fields lead to severe
limits on the observation of mesoscopic and macroscopic quantum
phenomena, such as interference effects, and, in particular, the
realization of quantum communication and computation algorithms.
Measurement of the strength and other detailed properties of the
noise mechanisms affecting a physical implementation is a critical
part of optimizing, improving, and benchmarking the physical
device and experimental protocol \cite{Nicolas,Yaakov}. Moreover,
in the case of quantum devices capable of universal control,
knowledge of specific characteristics of the noise enables the
selection and optimization of passive and active error-prevention
strategies \cite{Knill,KLZ,VL,AB,Kempe,CN}.
The exact method for characterizing the noise affecting an
implementation is quantum process tomography (QPT) \cite{CN}. Let
$D$ denote the dimensionality of the Hilbert space (constituted,
e.g., from $n_q = \log_2(D)$ qubits). For QPT, the desired
transformation (usually a unitary operator) must be applied to
each member of a complete set of $D^2$ input states (spanning the
state space), followed by tomographic measurement of the output
state. This allows for a complete reconstruction of the
superoperator (completely positive linear map) representing the
imperfect implementation of the target transformation. From this
superoperator the cumulative noise superoperator can be extracted
from conventional analysis of the matrix. The QPT approach to
noise estimation suffers from several practical deficiencies.
First, often the intrinsic properties of the noise operators are
of interest, but the noise superoperator determined from QPT will
depend on the symmetries between the noise mechanisms and the
choice of target transformation. Second, the number of experiments
that must be carried out grows exponentially in the number of
qubits $D^4 = 2^{4n_q}$. Third, conventional numerical analysis of
the tomographic data requires the manipulation of matrices of
exponentially increasing dimension ($D^2 \times D^2$). For these
last two reasons QPT becomes infeasible for processes involving
more than about a dozen qubits, far fewer than the one thousand or
so qubits required for the fault-tolerant implementation of
quantum algorithms that outperform conventional computation. Hence
the infeasibility of complete noise estimation via tomography
prompts the question of whether there exist efficient methods by
which specific features of the noise may be determined.
We show below that the overall noise strength and the associated
accuracy of an implementation may be estimated by a scalable
experimental method. Specifically we show that the average gate
fidelity (\ref{avegatefid}), and some more generalized fidelities
described below, can be estimated directly with an accuracy
$O(1/\sqrt{DN})$ where $N$ is the number of independent
experiments. This method provides a solution to the important
problem of efficiently measuring which member of a set of
experimental configurations and algorithmic techniques produces
the most accurate implementation of an arbitrary target
transformation. By varying over different experimental methods and
noise-reduction algorithms and then directly measuring the
variation in the associated fidelity this method enables
estimation of more detailed characteristics of the noise.
\section{Efficient Estimation of the Average Gate Fidelity}
A convenient starting point for our analysis is the average gate
fidelity
\begin{equation} \label{avegatefid}
\overline{F_g}(\Lambda) \equiv \mathbb{E}_\psi
\left(F_{\mathrm{g}}(U, \Lambda, \psi) \right) \equiv \int d\psi
\; \< \psi |U^{-1} ( \Lambda( U |\psi \>\< \psi | U^{-1}) ) U |
\psi \>
\end{equation}
where
\begin{equation} \label{krausform}
\Lambda(\rho) = \sum_k A_k \rho A_k^\dagger
\end{equation}
is a completely positive (CP) map characterizing the noise.
The gate fidelity $F_g$ is the inner-product of the state obtained
from the actual implementation with the state that would be
ideally obtained under the target unitary. The measure $d\psi$
denotes the natural, unitarily invariant (Fubini-Study) measure on
the set of pure states and hence the average gate fidelity
provides an indicator that is independent of the choice of initial
state. If the implementation is perfect then $\overline{F_g}=1$
and under increasing noise $\overline{F_g}$ decreases. Due to the
invariance of the Fubini-Study measure the average fidelity
depends only on the noise operator and can been expressed in the
form \cite{H3,Bowdery,Nielsen}
\begin{equation}
\label{knownaverage}
\overline{F_{\mathrm{g}}}(\Lambda) = \frac{\sum_k |\mathrm{Tr}(A_k)|^2 + D}{D^2
+D}.
\end{equation}
Hence the average fidelity can be determined if the noise operator
is known. The noise operator can be determined experimentally by
measuring the CP map $\Lambda(U \cdot U^{-1})$ tomographically and
then factoring out the inverse of the target map $U^{-1} \cdot U$.
This procedure has been carried out recently for 3 qubits in
recent a implementation of the quantum Fourier transform using
liquid-state NMR techniques \cite{Yaakov}. As noted above, this
method requires $\mathcal{O}(D^4)$ experiments and the
conventional manipulation of matrices of dimension $D^2 \times
D^2$. Recently Nielsen has proposed a method \cite{Nielsen} for
the direct measurement of $ \overline{F_\mathrm{g}}$ that requires
$D^4$ experiments but analysis of matrices of dimension only $D
\times D$ (rather than $D^2 \times D^2$).
We now describe how the average gate fidelity (\ref{avegatefid})
can be estimated accurately from a simple experimental protocol.
Our method requires the physical implementation of the ``motion
reversal" transformation $U^{-1} U | \psi \> \< \psi | U^{-1} U$
on an arbitrary state $| \psi \> \< \psi |$.
Under this transformation, the CP map $\Lambda$ in the gate
fidelity (\ref{avegatefid}) can be interpreted as the decoherence
and experimental control errors arising under the imperfect
implementation of the motion reversal experiment, i.e., $\Lambda =
\Lambda_{U^{-1} U }$, rather than as the noise associated with
only the forward transformation $U$, i.e, $\Lambda =
\Lambda_{U}$.
The key idea is to choose the target transformation $U$ randomly
from the Haar measure \cite{PZK98}. This will earn us the
advantage of the concentration of measure in large Hilbert spaces,
as described further below, and leads to a universal form of the
gate fidelity depending only on the intrinsic strength of the
cumulative noise. This universal form will allow us to evaluate
the average fidelity for more generalized motion reversal
protocols.
Our starting point is the gate fidelity uniformly averaged over
all unitaries,
\begin{equation}
\mathbb{E}_U(F_{\mathrm{g}})
= \int_{U(D)} dU \; \mathrm{Tr}[ \rho U^{-1} \Lambda( U \rho U^{-1}) U ],
\end{equation}
where in the above $dU$ denotes the unitarily-invariant Haar
measure on $U(D)$ and $\rho = |\psi \> \< \psi |$.
In order to evaluate this integral we use the superoperator
representation of the map (\ref{krausform}),
\begin{equation} \label{superop}
\hat{\Lambda} = \sum_k A_k \otimes A_k^*,
\end{equation}
and similarly $\hat{U} = U \otimes U^*$, where $^*$ denotes
complex conjugation.
The Haar averaged gate fidelity takes the form
\begin{eqnarray}
\mathbb{E}_U(F_{\mathrm{g}}) & = & \mathrm{Tr} \left( \rho
\left[ \int dU \ \hat{U} \hat{\Lambda} \hat{U}^{-1} \right] \rho
\right)\\
& = & \mathrm{Tr} \left( \rho
\hat{\Lambda}^{\mathrm{ave}} \rho \right) = F_{\mathrm{g}}(\hat{\Lambda}^{\mathrm{ave}}).
\end{eqnarray}
where $\hat{\Lambda}^{\mathrm{ave}} \equiv \int dU \ \hat{U}
\hat{\Lambda} \hat{U}^{-1}$. As shown in the Appendix, the
Haar-averaged superoperator $\hat{\Lambda}^{\mathrm{ave}}$ is
$U(D)$-invariant and thus can be expressed as a depolarizing
channel
\begin{equation}
\hat{\Lambda}^{\mathrm{ave}}\rho =
p \rho + (1-p)\frac{ \mathbbm{1}}{D},
\end{equation}
(assuming $\mathrm{Tr}(\rho) = 1$) characterized by the single
``strength'' parameter
\begin{equation}
p = \frac{\sum_k |\mathrm{Tr}(A_k)|^2 - 1}{D^2 -1},
\end{equation}
where $ p \in [0,1]$ and we have made use of the fact that
$\mathrm{Tr}(\hat{\Lambda}^{\mathrm{ave}}) =
\mathrm{Tr}(\hat{\Lambda}) = \sum_k |\mathrm{Tr}(A_k)|^2$.
Direct substitution leads to
\begin{equation}
\mathbb{E}_U(F_{\mathrm{g}})
= F_{\mathrm{g}}(\hat{\Lambda}^{\mathrm{ave}}) = p
+\frac{(1-p)}{D}.
\end{equation}
Hence the gate fidelity for the Haar-averaged operator resulting
from a motion reversal experiment depends only on the single
parameter $\mathrm{Tr}(\hat{\Lambda})$ which represents the
intrinsic strength of the cumulative noise. We remark that this
result holds for general (possibly non-unital) noise. Furthermore,
suppressing the arguments of $F$ we note that the unitary
invariance of the natural measure on pure states implies the
equivalence
\begin{equation}
\overline{F_g}(\Lambda) = \mathbb{E}_\psi( F_{\mathrm{g}}) =
\mathbb{E}_U(F_{\mathrm{g}}),
\end{equation}
and hence we recover Eq.~\ref{knownaverage}.
We now describe why and how the intrinsic noise strength
(characterized by $p$ or $\mathrm{Tr}(\hat{\Lambda})$) can be
estimated via an efficient experimental protocol. By implementing
a single target transformation $U$ that is randomly drawn from the
Haar measure, we gain the advantage of the concentration of
measure in large Hilbert spaces: the motion reversal (gate)
fidelity for the single random $U$ is exponentially close to the
Haar-averaged motion reversal (gate) fidelity. From the unitary
invariance of the Fubini-Study measure we know that
\begin{equation}
\mathbb{E}_\psi(F_{\mathrm{g}}^2) =
\mathbb{E}_U(F_{\mathrm{g}}^2).
\end{equation}
As will be shown in Ref.~\cite{BKE}, the typical fluctuation for a
random initial state $| \psi \>$, given a fixed $U$ and $\Lambda$,
decreases exponentially with the number of qubits,
\begin{equation}
(\Delta_\psi F_g)^2 \equiv \overline{F_{\mathrm{g}}^2} -
\overline{F_{\mathrm{g}}}^2 \leq O(1/D).
\end{equation}
Therefore it follows that,
\begin{equation}
\label{fluct} (\Delta F)_U^2 \equiv \mathbb{E}_U(F_{\mathrm{g}}^2)
- \mathbb{E}_U(F_{\mathrm{g}})^2 \leq O(1/D).
\end{equation}
Hence the fidelity under motion reversal of a single random $U$
and arbitrary (non-random) initial state is exponentially close to
the Haar-averaged fidelity
\begin{equation}
F_{\mathrm{g}}(U,\Lambda,\psi) =
F_{\mathrm{g}}(\Lambda^{\mathrm{ave}}) + O(1/\sqrt{D}) = p
+\frac{(1-p)}{D} + O(1/\sqrt{D}).
\end{equation}
The protocol is now clear: after the motion reversal sequence has
been applied experimentally, the single parameter $p$
characterizing the average gate fidelity appears as the residual
population of the initial state. Due to the invariance of the Haar
measure we may choose the initial state to be the computational
basis state $(|0\>\<0|)^{\otimes n_q}$. Hence the gate fidelity
can be determined directly from a standard readout (projective
measurement) of the final state in the computational basis. When
the noise strength is actually non-negligible (e.g., the noise
strength does not decrease as a polynomial function of $1/D$) an
accurate estimate of $p$ is possible with only a few experimental
trials. If in each of $N$ repetitions of the motion-reversal
experiment an independent random unitary is applied, then the
observed average will approach the Haar-average as
$\mathcal{O}(1/\sqrt{DN})$.
\section{Generalized Fidelities in a Discrete-Time Scenario}
More generally we imagine the ability to implement a set of
independent random unitary operators $\{ U_j\}$ and their
inverses. The entire sequence is subject to some unknown noise,
consisting of the decoherence processes and control errors
affecting the implementation. Such generalized motion reversal
sequences are relevant not only for noise-estimation, but also
have important applications in studies of fidelity decay
\cite{Emerson02} and decoherence rates \cite{ALPZ04} for quantum
chaos and many-body complex systems.
We first consider the fidelity loss arising under an iterated
motion reversal sequence of the form
\begin{equation}
\rho(n) = \hat{U}_n^{-1} \hat{\Lambda} \hat{U}_n \dots \
\hat{U}_2^{-1} \hat{\Lambda} \hat{U}_2 \ \hat{U}_1^{-1}
\hat{\Lambda} \hat{U}_1 \rho(0),
\end{equation}
where here $\hat{\Lambda}_j = \hat{\Lambda}_{U_j^{-1} U_j}$
denotes the cumulative noise from the motion reversal of $U_j$ and
we now allow arbitrary (possibly mixed) initial states $\rho(0)$.
The fidelity of this iterated transformation is,
\begin{equation}
F_n(\psi,\{U_j\}) = \mathrm{Tr}\left( \rho(0) \hat{U}_n^{-1}
\hat{\Lambda}_n \hat{U}_n \dots \hat{U}_1^{-1} \hat{\Lambda}_1
\hat{U}_1 \rho(0) \right).
\end{equation}
Averaging over the Haar measure for each $U_j$ takes the form,
\begin{eqnarray}
\overline{F_n} \equiv \mathbbm{E}_{\{U_j\}}(F_n(\psi,\{U_j\}))
& \equiv & \int_{U(D)^{\otimes n}} \left( \Pi_{j=1}^n dU_j \right)
F_n(\psi,\{U_j\}) \\
& = & \mathrm{Tr}\left( \rho(0) \left[ \Pi_{j=1}^n
\hat{\Lambda}_j^{\mathrm{ave}} \right] \rho(0) \right) ,
\end{eqnarray}
where $dU_j$ denotes the Haar measure and we have defined the Haar
averaged noise operator,
\begin{equation}
\hat{\Lambda}_j^{\mathrm{ave}} \equiv
\mathbbm{E}_{U_j}(\hat{\Lambda}_j) \equiv \int_{U(D)} dU
\hat{U}^{-1} \hat{\Lambda}_j \hat{U}.
\end{equation}
As noted above and shown in the Appendix,
$\hat{\Lambda}^{\mathrm{ave}} \equiv \int dU \hat{U} \hat{\Lambda}
\hat{U}^{-1}$ is a depolarizing channel
\begin{equation}\label{lambdaave}
\hat{\Lambda}_j^{\mathrm{ave}}\rho =
p_j \rho + (1-p_j) \frac{\mathbbm{1}}{D},
\end{equation}
with strength parameter
\begin{equation}
p_j = \frac{\mathrm{Tr}(\hat{\Lambda}_j) - 1}{D^2 -1}.
\end{equation}
Because each $U_j$ is random, we can further simplify this result
by assuming that the cumulative noise for each $U_j$ has the same
strength $p_j = p$, in which case we obtain for arbitrary noise a
universal exponential decay of the averaged fidelity
\begin{equation}
\overline{F_n} = p^n \mathrm{Tr}[\rho(0)^2] + \frac{(1-p^n)}{D}.
\end{equation}
depending only on the noise strength. In the limit of large $n$,
we see that $\overline{F_n} \rightarrow D^{-1}$, as may be
expected from the average fidelity between random states
\cite{ZS05}. Most importantly, due to the concentration of measure
($\ref{fluct}$), for large $D$ the fidelity loss under iterated
motion reversal of a single sequence of random unitary operators
will be exponentially close to the Haar-average, and hence the
noise strength can be estimated with only a few experimental runs.
Another important generalized fidelity is the one obtained under
the imperfect `Loschmidt echo' sequence \cite{Pastawski,Emerson02}
\begin{equation}
\rho(n) = \hat{U}_1^{-1} \dots \hat{U}_n^{-1} \hat{\Lambda}_n
\hat{U}_n \dots \hat{\Lambda}_1 \hat{U}_1 \rho(0),
\end{equation}
where the superoperator $\hat{\Lambda}_j$ represents the
cumulative noise during the implementation of each $U_j$. The
fidelity between the initial state and final state in the
Loschmidt echo experiment takes the form,
\begin{equation}
F_n^{\mathrm{echo}}(\psi,\{\Lambda_j\},\{U_j\}) =
\mathrm{Tr}\left( \rho(0) \hat{U}_1^{-1} \hat{U}_2^{-1} \dots
\hat{U}_n^{-1} \hat{\Lambda}_n \hat{U}_n \dots \hat{\Lambda}_1
\hat{U}_1 \rho(0) \right).
\end{equation}
Moving to the interaction picture we define
\begin{equation}
\hat{\Lambda}_j(j) = \hat{U}_1^{-1} \dots \hat{U}_j^{-1}
\hat{\Lambda}_j \hat{U}_j \dots \hat{U}_1,
\end{equation}
so that,
\begin{equation}
F_n^{\mathrm{echo}}(\psi,\{\Lambda_j\},\{U_j\}) =
\mathrm{Tr}\left( \rho(0) \hat{\Lambda}_n(n)
\hat{\Lambda}_{n-1}(n-1) \dots \hat{\Lambda}_1(1) \rho(0) \right).
\end{equation}
From the invariance of the Haar measure the average fidelity
simplifies to
\begin{equation}
\overline{F_n^{\mathrm{echo}}} = \mathrm{Tr}\left( \rho(0)
\hat{\Lambda}_n^{\mathrm{ave}} \hat{\Lambda}_{n-1}^{\mathrm{ave}}
\dots \hat{\Lambda}_1^{\mathrm{ave}} \rho(0) \right)
\end{equation}
with $\hat{\Lambda}_j^{\mathrm{ave}}$ given by
Eq.~\ref{lambdaave}. As before, we can simplify this result by
assuming that the cumulative noise for each step has the same
strength ($p_j=p$), in which case we obtain for arbitrary noise a
universal exponential form for the decay of fidelity
\begin{equation}
\label{expdecay} \overline{F_n^{\mathrm{echo}}}(p) = p^n
+\frac{(1-p^n)}{D}.
\end{equation}
A generalized version of this Loschmidt echo that is more relevant
to noise estimation is one for which noise appears in both the
forward and backward sequence of the motion reversal. The
associated fidelity is,
\begin{equation}
F_n^{\mathrm{gen}}(\psi,\Lambda,\{U_j\}) = \mathrm{Tr}\left(
\rho(0) \hat{\Lambda} \hat{U}_1^{-1} \hat{\Lambda} \hat{U}_2^{-1}
\dots \hat{\Lambda} \hat{U}_n^{-1} \hat{\Lambda} \hat{U}_n \dots
\hat{\Lambda} \hat{U}_1 \rho(0) \right).
\end{equation}
While we have not directly evaluated the average of this fidelity
analytically in the general case, for the special case of unitary
noise we have analytic and numerical evidence supporting the
relation
\begin{equation}
F^{\mathrm{gen}}_n \simeq F^{\mathrm{echo}}_{2n}
\end{equation}
for large $n$, which we conjecture should hold under general
noise.
\section{Generalized Fidelities for Continuous-Time Weak Noise}
We describe our system by the Markovian Master Equation
\cite{GKS,Al}
\begin{equation}
\frac{d}{dt}\rho = -i[H_C(t),\rho] + \epsilon{\hat L}(\rho)
\label{MME}
\end{equation}
where $H_C(t)$ governs a controlled reversible part of the
dynamics and the generator
\begin{equation}
{\hat L} \, \rho \equiv L(\rho)= -i[H,\rho]+\frac{1}{2} \sum
_{\alpha}\bigl( [V_{\alpha} ,\rho V_{\alpha}^{\dagger}]
+[V_{\alpha}\rho , V_{\alpha}^{\dagger}]\bigr) \label{GKLS}
\end{equation}
with the condition $\mathrm{Tr}H =\mathrm{Tr}V_{\alpha}=0$ (which
fixes the decomposition of ${\hat L}$ into Hamiltonian and
dissipative parts \cite{Al}) describes all sources of
imperfections and noise. Here $0 <\epsilon \ll 1$ is a small
parameter characterizing noise strength.
The time dependent fidelity of the initial state $\phi$
is given by
\begin{equation}
F_{\phi}(t) = \<\phi| {\bf T}\exp\Bigl\{ \epsilon\int_0^t {\hat
L}(s)ds\Bigr\}(|\phi\>\<\phi|)|\phi\> \label{fid}
\end{equation}
where ${\bf T}$ denotes the chronological order, and
\begin{equation}
{\hat L}(s)= {\hat U}^{\dagger}(s,0){\hat L}{\hat U}(s,0), \quad
U(t,s) = {\bf T}\exp\Bigl\{ -i\int_s^t H_C(u) du\Bigr\} .
\label{gen}
\end{equation}
Using the notation
\begin{equation}
{\hat \Gamma}(t) = {\bf T}\exp\Bigl\{ \epsilon\int_0^t {\hat
L}(s)ds \Bigr\}\label{prop}
\end{equation}
we can write down the following "cumulant expansion" of the
dynamics with respect to the small parameter $\epsilon$
\begin{equation}
{\hat \Gamma}(t)= \exp\Bigl\{ \epsilon {\hat K}_1(t) + \epsilon^2
{\hat K}_2(t) + \cdots\Bigr\}\ . \label{cum}
\end{equation}
Using the Wilcox formula for the matrix-valued functions
\begin{equation}
\frac {d}{dx}\exp A(x)= \Bigl( \int_0^1 \exp(\lambda A(x))
\frac{d}{dx} A(x)\exp(-\lambda A(x))d\lambda\Bigr) \exp A(x)
\label{Wil}
\end{equation}
one obtains
\begin{equation}
{\hat K}_1(t) = \int_0^t {\hat L}(s) ds, \quad {\hat K}_2(t) =
\frac{1}{2}\int_0^t ds\int_0^s du [{\hat L}(s),{\hat L}(u)] \ .
\label{cum1}
\end{equation}
We assume now the following {\sl ergodic hypothesis}: a) the
ergodic mean exists and is equal to the Haar average
\begin{equation}
\lim_{T\to\infty} \frac{1}{T}\int_0^T {\hat L}(t)dt = {\hat
L}^{\mathrm{ave}} = \int_{U(D)} dU \, \, {\hat U}{\hat L}{\hat
U}^{\dagger}\ , \label{erg}
\end{equation}
b) the fluctuations $\delta{\hat L}(t)\equiv {\hat L}(t)-{\hat
L}^{\mathrm{ave}}$ around ergodic mean are {\sl normal}, i.e. for
long $t$
\begin{equation}
\|\int_s^{s+t}\delta {\hat L}(u)du \|\sim t^{1/2} . \label{norm}
\end{equation}
These conditions are satisfied , for instance if the
time-dependent dynamics $t\mapsto U(t)$ can be modelled by a
random walk on the group $U(D)$ or by a trajectory on $U(D)$ given
by a certain deterministic dynamics with strong enough ergodic
properties. The norm of ${\hat K}_2(t)$ can be estimated using
(\ref{norm})
\begin{equation}
\|{\hat K}_2(t)\| = \frac{1}{2}\|\int_0^t ds\int_0^s du
\Bigl([\delta{\hat L}(s),\delta{\hat L}(u)] + [\delta{\hat
L}(s),{\hat L}^{\mathrm{ave}}]+[{\hat
L}^{\mathrm{ave}},\delta{\hat L}(u)]\Bigr)\|\sim t^{3/2} .
\label{K2}
\end{equation}
Therefore
for small enough $\epsilon$ and long enough times $t$ such that
$\epsilon t$ is fixed the first term dominates and we can write
\begin{equation}
{\hat \Gamma}(t)\simeq \exp\bigl( \epsilon \int_0^t {\hat
L}(s)ds\bigr) \ . \label{cum2}
\end{equation}
Then replacing ${\hat\Gamma}(t)$ by $\exp (\epsilon {\hat
L}_{av}t)$ and using the explicit expression
(\ref{dgen},\ref{dgen1}) we obtain the universal exponential decay
of the fidelity
\begin{equation}
F_{\phi}(t) \simeq e^{-\gamma t} +\frac{1}{D} \bigl(1- e^{-\gamma
t}\bigr), \quad \gamma = \frac {D}{2(D^2 -1)}\sum_{\alpha}{\rm
tr}(|V_{\alpha}|^2) \ .
\label{fidfin}
\end{equation}
\section{Discussion}
We have described how generalized Haar-averaged fidelities may be
directly estimated with only a few experimental measurements. By
implementing a motion reversal sequence with a Haar-random unitary
transformation, the observed fidelity decay provides a direct
experimental estimate of the intrinsic strength of the noise.
Moreover, because the target transformation is a Haar-random
unitary, the cumulative noise measured by this method will not be
biased by any special symmetries of the target transformation.
The only inefficiency of our protocol is the requirement of
experimentally implementing a Haar-random unitary: the
decomposition into elementary one and two qubit gates requires an
exponentially long gate sequence \cite{PZK98}. However, the
randomization provided by Haar-random unitary operators may be
unnecessarily strong and this leads to the open question of
whether efficient sets of random unitaries, e.g. the random
circuits studied in Refs.~\cite{Emerson03,ELL}, can provide an
adequate degree of randomization for the above protocols.
Indeed the experimental results of Ref.~\cite{Yaakov} suggest that
even a structured transformation such as the quantum Fourier
transform is sufficiently complex to approximately average the
cumulative noise to an effective depolarizing channel, and from
studies of quantum chaos it is known that efficient chaotic
quantum maps are faithful to the universal Haar-averaged fidelity
decay under imperfect motion-reversal \cite{Emerson02}. While more
conclusive evidence is needed to answer this question, it appears
likely that the inefficiency associated with implementing
Haar-random unitary unitary operators may be overcome.
An additional question is whether the implementation of random
unitary operators (e.g., Haar-random unitary operators or even
efficient random circuits) leads to an even stronger form of
averaging. We have throughout our analysis made the usual
assumption that the noise superoperator $\Lambda$ is independent
of the specific target transformation but depends only on the
duration of the experiment.
However it is known that the actual noise in general depends
sensitively on the choice of target transformation $U$. Moreover,
the cumulative noise operator generally also depends on the
particular sequence of elementary one and two qubit gates applied
to generate $U$. For example, the implementation of the quantum
Fourier transform \cite{Yaakov,Nicolas} will generate very
different cumulative noise than the trivial implementation of the
identity operator $U = {\mathbbm 1}$ for the same time $\tau$.
However, it appears likely that the cumulative noise operators,
and in particular their intrinsic noise strength, under a specific
but random gate sequence should become concentrated about an
average value depending only on the length of the sequence. If
this is the case, then the usual assumption that the noise is
independent of the actual gate sequence becomes statistically well
motivated, and the measured fidelity under motion reversal can
provide a benchmark of an intrinsic noise strength that is fully
independent of the target unitary.
{\bf Note added in proof:} additional evidence for the conjectured
relation (31) can be found in Ref.~\cite{Bettelli}.
\section{APPENDIX: Haar Averaged Superoperators}
We consider a linear superoperator ${\hat\Lambda}$ acting on the
space ${\bf M}_D$ of $D\times D$ complex matrices treated as a
Hilbert space with a scalar product $(X,Y) =
\mathrm{Tr}(X^{\dagger} Y)$. The superoperator ${\hat\Lambda}$
has a $D^2 \times D^2$ dimensional matrix representation and
$\mathrm{Tr} {\hat\Lambda}$ denotes the usual sum over the
diagonal elements of the matrix. For clarity of notation we will
sometimes express the linear operation ${\hat\Lambda} \rho$ in the
form $\Lambda(\rho)$. By $\{|k\>\}$ we denote an orthonormal basis
in ${\bf C}^D$ while $\{E_{kl} = |k\>\<l|\}$ is a corresponding
basis in ${\bf M}_D$. The group $U(D)$ of unitary $D\times D$
matrices has its natural unitary representation on ${\bf M}_D$
defined by
\begin{equation}
U(D) \ni U\mapsto {\hat U}\ ,\quad {\hat U}X = U X U^{\dagger}\ .
\label{rep}
\end{equation}
This representation is reducible and implies the decomposition of
${\bf M}_D$ into two irreducible invariant subspaces
\begin{equation}
{\bf M}_D = {\bf M}_D^c\oplus {\bf M}_D^0\ ,\quad {\bf M}_D^0 = \{
X\in {\bf M}_D ; \mathrm{Tr}X =0\}\ ,\quad {\bf M}_D^c = \{ X = c
\, \mathbbm{1} \}, \label{rep1}
\end{equation}
where $c$ is an arbitrary complex number.
Any superoperator ${\hat\Lambda}$ possesses exactly two linear
$U(D)$ invariants, i.e. the linear functionals on superoperator
space which are invariant with respect to all transformation of
the form ${\hat\Lambda}\mapsto {\hat U}{\hat\Lambda}{\hat
U}^{\dagger}$ :
\begin{equation}
\mathrm{Tr}[\Lambda(\mathbbm{1})] = \sum_{k=1}^D \<
k|\Lambda(\mathbbm{1})| k\> \label{inv1}
\end{equation}
and
\begin{equation}
\mathrm{Tr}({\hat\Lambda}) \equiv \sum_{k,l=1}^D
(E_{kl},\Lambda(E_{kl})) = \sum_{k,l=1}^D \<k| \, \Lambda(E_{kl})
\,| l\> \label{inv2}
\end{equation}
\noindent {\bf Example} Take $\Lambda(X) = A X B $, then
$\mathrm{Tr} [\Lambda(\mathbbm{1})] = \mathrm{Tr}(AB)$
and $\mathrm{Tr}({\hat\Lambda})=\mathrm{Tr}(A) \mathrm{Tr}(B)$.
A $U(D)$-invariant operator satisfies
${\hat\Lambda}^{\mathrm{inv}}= {\hat
U}{\hat\Lambda}^{\mathrm{inv}}{\hat U}^{\dagger}$ for any $U\in
U(D)$. The following lemma completely characterizes
$U(D)$-invariant trace-preserving superoperators
\noindent
{\bf Lemma 1} Let ${\hat\Lambda}^{\mathrm{inv}}$ be a
$U(D)$-invariant trace-preserving operator. Then
\begin{equation}
{\hat\Lambda}^{\mathrm{inv}}\, X \equiv \Lambda^{\mathrm{inv}}(X)
= p \, X + (1-p)\, \mathrm{Tr}(X) \frac{\mathbbm{1}}{D} \ ,
\label{invform}
\end{equation}
where
\begin{equation}
p = \frac{\mathrm {Tr}({\hat\Lambda}^{\mathrm{inv}})- 1 }{D^2 -1}.
\label{invform1}
\end{equation}
\noindent
{\bf Proof} Schur's lemma implies the form
(\ref{invform}) for $U(D)$-invariant trace-preserving operators.
From the normalization
$\mathrm{Tr}[\Lambda^{\mathrm{inv}}(\mathbbm{1})] = D$ for the
trace, the detailed expression (\ref{invform1}) can be explicitly
calculated by comparing $U(D)$-invariants for both sides of
eq.(\ref{invform}). $\Box$
The Haar-averaged superoperator corresponding to the noise under
the imperfect motion-reversal protocol, averaged over all possible
unitary operators, is a $U(D)$-invariant superoperator
\begin{equation}
{\hat\Lambda}^{\mathrm{ave}} = \int_{U(D)} dU \ {\hat
U}{\hat\Lambda}{\hat U}^{\dagger}\ . \label{av}
\end{equation}
where $dU$ is the normalized Haar measure on $U(D)$. Using Lemma 1
we can easily compute the averaged form of the dynamical map for
both the Schr\"odinger operator
\begin{equation}
\Lambda(\rho) = \sum _{\alpha} A_{\alpha}\rho
A_{\alpha}^{\dagger}\ ,\quad
\sum_{\alpha}A_{\alpha}^{\dagger}A_{\alpha}= \mathbbm{1}
\label{dynmap}
\end{equation}
and for the semigroup generator
\begin{equation}
{\hat L} \, \rho \equiv L(\rho) = -i[H,\rho]+\frac{1}{2} \sum
_{\alpha}\bigl( [V_{\alpha} ,\rho V_{\alpha}^{\dagger}]
+[V_{\alpha}\rho , V_{\alpha}^{\dagger}]\bigr) \label{GKLS1}
\end{equation}
with the condition $\mathrm{Tr}H =\mathrm{Tr}V_{\alpha}=0$ which
fixes the decomposition of ${\hat L}$ into Hamiltonian and
dissipative parts. From the fact that $\mathrm
{Tr}({\hat\Lambda}^{\mathrm{ave}}) = \mathrm {Tr}({\hat\Lambda})$
we obtain
\begin{equation}
\Lambda^{\mathrm{ave}} (\rho) = p \, \rho + (1-p)
\mathrm{Tr}(\rho)\frac{\mathbbm{1}}{D} \label{dmap}
\end{equation}
where
\begin{equation}
p = \frac{\mathrm {Tr}({\hat\Lambda}) -1}{D^2-1}
= \frac{\sum_{\alpha}|\mathrm{Tr}(A_{\alpha})|^2 -1}{D^2-1}
\label{dmap1}.
\end{equation}
Similarly for the generator we obtain
\begin{equation}
{\hat L}^{\mathrm{ave}}\, \rho \equiv L^{\mathrm{ave}}(\rho)=
-{\gamma} \left(\rho -
\mathrm{Tr}(\rho)\frac{\mathbbm{1}}{D}\right) \label{dgen}
\end{equation}
where
\begin{equation}
\gamma = \frac {D}{2(D^2
-1)}\sum_{\alpha}\mathrm{Tr}(|V_{\alpha}|^2) \ . \label{dgen1}
\end{equation}
\section{Acknowledgements}
We would like to thank David Cory for the many discussions that
stimulated this work. R.A. would like to acknowledge the
hospitality of the Perimeter Institute for Theoretical Physics
where part of this work was completed.
We acknowledge financial support from the National Sciences and
Engineering Research Council of Canada, the Polish Ministry of
Science and Information Technology - grant PBZ-MIN-008/P03/2003,
and the EC grant RESQ IST-2001-37559.
|
{
"timestamp": "2005-12-16T02:56:07",
"yymm": "0503",
"arxiv_id": "quant-ph/0503243",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503243"
}
|
\section{Introduction}
\setcounter{equation}{0}
Let $\lambda_1 ,\lambda_2 ,\lambda_2, \lambda_4 >0$ be given.
We consider the following system for $(u, \eta )$ in $\Bbb R^2$.
\begin{eqnarray}
\label{11}
\Delta u&=&-\lambda_1 e^{\eta} -\lambda_2 e^{u}+4\pi \sum_{j=1} ^{N} \delta (z-z_{j} ),\\
\label{12}
\Delta \eta &=& -\lambda_3 e^{\eta} -\lambda_4 e^{u}
\end{eqnarray}
equipped with the boundary condition
\begin{equation}
\label{13}
\int_{\Bbb R^2} e^{u}dx +\int_{\Bbb R^2} e^{\eta} dx < \infty ,
\end{equation}
where we denoted $z=x_1+ix_2 \in \Bbb C=\Bbb R^2$.
The system
(\ref{11})-(\ref{12}) is the reduced form of the Bogomol'nyi type of
equations
modeling the cosmic strings with matter field given by the
massive $W-$boson of the electroweak theory, if we choose the coefficients as,
\begin{equation}
\label{13a}
\lambda_1=2m_W^2 , \quad \lambda_2=4e^2, \lambda_3 =\frac{16\pi G
m_W^4}{e^2},\quad \lambda_4=32\pi G m_W ^2,
\end{equation}
where $m_W$ is the mass of the $W-$boson, $e$ is the charge of
the electron, and $G$ is the gravitational constant.
The points $\{ z_1, \cdots,
z_N\}$ corresponds to the location on the $(x_1,x_2)-$plane
of parallel (along the $x_3-$axis) strings. See \cite{yan,amb}
for the derivation of this system from the corresponding
Einstein-Weinberg-Salam theory as well as interesting physical backgrounds of the model.
There are many previous mathematical studies on the planar
electroweak theory recently(\cite{spr1,spr2,bar2,
cha2}). In particular in \cite{cha3} the authors considered full
electroweak field as the matter field coupled with the
gravitation. In the model from which our system is derive the
matter field coupled to gravity is the massive $W-$boson.
In \cite{yan} the construction of radially symmetric solutions(in
the case $z_1=\cdots=z_N$) of (\ref{11})-(\ref{13}) is discussed by
further reduction
the system into a single equation, and solving the ordinary
differential equation.
When the locations of strings are different to each other, however, we
cannot assume the radial symmetry of the solutions,
and no existence theory is available. In particular, the
author of \cite{yan} left the construction of solution in this case as an open
problem. One of our main purpose in
this paper is to solve this problem. Actually, we solve the
existence problem for more general coefficient cases as in
(\ref{11})-(\ref{12}). The following is our main theorem.
\begin{theorem}
Let $N \in \Bbb N \cup \{0\}$, and $ \mathcal{Z}=\{ z_{j}\}_{j=1} ^{N} $
be given in $\Bbb R^2$
allowing multiplicities. Suppose the coefficients, $\lambda_1,
\lambda_2,\lambda_3,\lambda_4 $ satisfy one of the conditions;
either
\begin{equation}\label{condition}
\lambda_1\lambda_4 -\lambda_2\lambda_3=0, \end{equation}
or
\begin{equation}\label{conditionA}
\lambda_1\lambda_4 -\lambda_2\lambda_3\neq 0 \quad \mbox{and}\quad \frac{ \lambda_2}{2\lambda_4} < N+1.
\end{equation}
Then, there exists a
constant $\varepsilon_1
>0$
such that
for any $\varepsilon \in (0, \varepsilon_1 )$ and any $c_0 >$ there exists a family of solutions to
(\ref{11})-(\ref{13}),
$(u,\eta )$. Moreover, the solutions we constructed
have the following representations:
\begin{eqnarray}
\label{14}
u (z)&=&\ln \rho^I _{\varepsilon, a^* _{ \varepsilon} } (z)+ \varepsilon ^{2}
w_1 (\varepsilon |z|) +\varepsilon ^{2} v^* _{1,\varepsilon} (\varepsilon z), \\
\label{15}
\eta (z)&=&\ln \rho^{II} _{\varepsilon, a^* _{\varepsilon} } (z)+ \varepsilon ^{2}
w_2 (\varepsilon |z|) +\varepsilon ^{2} v^* _{2, \varepsilon} (\varepsilon z),
\end{eqnarray}
where the functions $\rho^{I}_{\varepsilon , a} (z), \rho^{II}_{\varepsilon , a}
(z)$ are defined by
\begin{equation}
\label{16}
\rho^{I}_{\varepsilon ,
a}(z) = \frac{8\varepsilon^{2{N+2}} \vert f (z) \vert^2}{\lambda_2\left(
1+\varepsilon^{2N +2} \vert F (z) +
\frac{a}{\varepsilon^{N+1}}\vert^2\right)^2} ,
\end{equation}
and
\begin{equation}
\label{17}
\rho^{II}_{\varepsilon ,
a}(z) = \frac{ c_0 \varepsilon^4}{\left( 1+\varepsilon^{2N +2} \vert F
(z) + \frac{a}{\varepsilon^{N+1}}\vert^2\right)^{\frac{2\lambda_4}{\lambda_2}}}
\end{equation}
with
\begin{equation}
\label{18}
f(z) = (N +1)
\prod\limits_{j=1}^{N}(z-z_{j}), \quad F (z) = \int_0^z f (\xi )
d\xi
\end{equation}
for $k=1,2$, $\varepsilon > 0 $ and $a=a_1+i a_2\in \Bbb C$.
The smooth radial functions, $w_1, w_2$ in (\ref{14}) and (\ref{15}) respectively
satisfy the asymptotic formula,
\begin{equation}
\label{19}
w_1 (|z|)=-C_1 \ln |z| + O(1), \qquad w_2 (|z|)=-C_2\ln |z| +O(1)
\end{equation}
as $|z|\to \infty$, where
\begin{eqnarray}
\label{19a}
C_1&=&\frac{c_0\lambda_1\lambda_2\lambda_4}{2(N+1)(\lambda_2+\lambda_4)(\lambda_2 +2\lambda_4 )},\\
\label{19b}
C_2&=
&\frac{C_1\lambda_4}{\lambda_2}-\frac{(\lambda_1\lambda_4 -\lambda_2\lambda_3)c_0}{2(N+1)\lambda_2 }
B\left(\frac{1}{N+1}, \frac{2\lambda_4}{\lambda_2} -\frac{1}{N+1}\right)
\end{eqnarray}
with the beta function(Euler's integral of the first kind)
defined by
$$
B(x,y)=\int_0 ^1 t^{x-1} (1-t)^{y-1} dt. \quad \forall x,y >0
$$
(see \cite{gra}.)
The function $v^* _{1,\varepsilon}, v^* _{2,\varepsilon} $ in (\ref{14}) and
(\ref{15}) respectively
satisfy
\begin{equation}
\label{110}
\sup_{z\in \Bbb R^2 } \frac{ |v^* _{1, \varepsilon} (\varepsilon z)|+|v^* _{2, \varepsilon} (\varepsilon z)|}{ \ln (e+|z|
)}
\leq o(1) \qquad \mbox{as $\varepsilon \to 0$}.
\end{equation}
\end{theorem}
{ \textsf{Remark} 1.1.} In
the physical model of the cosmic strings of $W-$boson we note that
the coefficients in (\ref{13a}) satisfy (\ref{condition}), and the
term containing Euler's integral
vanishes in (\ref{19b}) to yield
$$C_2=\frac{C_1\lambda_4}{\lambda_2}=\frac{c_0\lambda_1\lambda_4 ^2}{2(N+1)(\lambda_2+\lambda_4)(\lambda_2 +2\lambda_4
)} >0$$
as well as $C_1 >0$. Thus, we have extra(additional) contributions from the second terms
of to the
decays of $u$ and $\eta $ in (\ref{14}) and (\ref{15})
respectively.\\
\ \\
{ \textsf{Remark} 1.2.} In our cosmic strings of $W-$boson we do not need smallness condition
of the constant $G$ for the existence of condition, contrary to the other matter models of
cosmic strings(see \cite{yan1,yan2, cha2}.)
\section{Proof of Theorem 1.1}
\setcounter{equation}{0}
We note that for any $\varepsilon > 0 $ and $ a \in \Bbb C $, $\ln
\rho^{I}_{\varepsilon , a}(z)$, is a solution of the Liouville
equation(\cite{lio}).
\begin{equation}
\label{21}
\Delta \ln \rho^{I}_{\varepsilon , a}(z)=-\lambda_2\rho^{I}_{\varepsilon
, a }(z)+4\pi \sum_{j=1} ^{N} \delta
(z-z_{1,j} ).
\end{equation}
We consider the following equation for $\rho^{II}_{a, \varepsilon} (z) $
\begin{equation}
\label{21a}
\Delta \ln \rho^{II}_{a, \varepsilon} (z) =-\lambda_4 \rho^{I}_{ a, \varepsilon} (z).
\end{equation}
From (\ref{21}) we have
\begin{equation}
\label{21b}
\Delta \left[ \ln \rho^{I}_{ a, \varepsilon} (z) -\sum_{j=1}^N \ln |z-z_j|^2 \right]=-\lambda_2 \rho^{I}_{ a, \varepsilon} (z)
.
\end{equation}
Combining (\ref{21a}) with (\ref{21b}), we obtain
$$
\Delta \left\{ \lambda_4 \left[ \ln\rho^{I}_{ a, \varepsilon} (z) -\sum_{j=1}^N \ln |z-z_j|^2\right] -\lambda_2 \ln
\rho^{II}_{a, \varepsilon} (z) \right\}=0,
$$
from which we derive
$$
\ln \rho^{II}_{a, \varepsilon} (z) =\frac{\lambda_4}{\lambda_2} \left[ \ln \rho^{I}_{ a, \varepsilon} (z) -\sum_{j=1}^N \ln |z-z_j|^2 \right]
+h(z),
$$
where $h(z)$ is a harmonic function.
Choosing $h(z)$ as the constant,
$$ h(z)\equiv \frac{\lambda_4}{\lambda_2} \ln \left(\varepsilon ^{\frac{4\lambda_2}{\lambda_4} -2N-2}
\lambda_2^{\frac{\lambda_2}{\lambda_4}}[8(N+1)^2]^{-1} c_0 ^{\frac{\lambda_2}{\lambda_4}}\right),$$
we get the form of $\rho^{II}_{a, \varepsilon} (z)$ given in (\ref{17}).
We set
$$
g^{I}_{\varepsilon , a}(z) =\frac{1}{\varepsilon ^2}
\rho^{I}_{\varepsilon , a }\left(\frac{z}{\varepsilon}\right), \quad
g^{II}_{\varepsilon , a}(z) =\frac{1}{\varepsilon ^4}
\rho^{II}_{\varepsilon , a}\left(\frac{z}{\varepsilon}\right),
$$
and define $\rho_1 (r)$ and $\rho_2 (r)$ by
$$
\rho_1(r)=\frac{8(N +1)^2r^{2N}}{ \lambda_2 (1+r^{2N +2} )^2}
=\lim_{\varepsilon \to 0} g^{I}_{\varepsilon , 0}(z) ,
$$
and
$$
\rho_2(r)=\frac{c_0}{ (1+r^{2N +2} )^{\frac{2\lambda_4}{\lambda_2}}}
=\lim_{\varepsilon \to 0} g^{II}_{\varepsilon , 0}(z)
$$
respectively.
We transform $(u, \eta )\mapsto (v_1, v_2)$ by the
formula
\begin{equation}
\label{22}
u (z)=\ln \rho^{I}_{\varepsilon ,
a}(z) +\varepsilon^{2} w_1 (\varepsilon |z|) +\varepsilon ^{2} v_1 (\varepsilon z),
\end{equation}
\begin{equation}
\label{23}
\eta (z) =\ln \rho^{II}_{\varepsilon ,
b}(z) +\varepsilon^{2} w_2 (\varepsilon |z|) +\varepsilon ^{2} v_2 (\varepsilon z),
\end{equation}
where $w_1$ and $w_2$ are the radial functions to be determined
below.
Then, using (\ref{21}), the system can be written as the functional equation,
$P(v_1,v_2, a, \varepsilon )=(0,0)$, where
\begin{equation}
\label{24}
P_1 (v_1, v_2, a,\varepsilon )=
\Delta v_1 + \lambda_1 g^{II}_{a,\varepsilon }(z) e^{\varepsilon^2 (w_2 +v_2 )} +
\lambda_2\frac{ g^{I}_{\varepsilon , a}(z)}{\varepsilon^2} (e^{\varepsilon^{2} (w_1+v_1)} -1)
+\Delta w_1,
\end{equation}
and
\begin{equation}
\label{25}
P_2 (v_1, v_2, a, \varepsilon )=
\Delta v_2 +\lambda_3 g^{II}_{\varepsilon , a}(z)
e^{\varepsilon^{2}(w_2+v_2)} +\lambda_4\frac{ g^{I}_{\varepsilon , a}(z)}{\varepsilon^2} (e^{\varepsilon^{2} (w_1+v_1)} -1)
+\Delta w_2.
\end{equation}
Now we introduce the functions spaces introduced in \cite{cha1}.
For $\a >0$ the Banach spaces $X_\alpha$
and $Y_\alpha$ are defined as
\[ X_\alpha =\{ u \in L_{loc}^2 ({\mathbb R}^2) \mid
\int_{{\mathbb R}^2} (1+|x|^{2+\alpha})|u(x)|^2 dx <\infty \} \]
equipped with the norm $\| u \|^2_{X_\alpha} =
\int_{{\mathbb R}^2} (1+|x|^{2+\alpha})|u(x)|^2 dx$, and
\[ Y_\alpha =\{ u\in W_{loc}^{2,2}({\mathbb R}^2) \mid \| \Delta u \|_{X_\alpha}^2
+\Big\| \frac{u(x)}{1+|x|^{1+\frac{\alpha}{2}}}\Big\|_{L^2({\mathbb R}^2)}^2 < \infty \} \]
equipped with the norm $\| u \|_{Y_\alpha}^2= \| \Delta u
\|_{X_\alpha}^2
+ \big\| \frac{u(x)}{1+|x|^{1+\frac{\alpha}{2}}}
\big\|_{L^2({\mathbb R}^2)}^2$.
We recall the following propositions proved in \cite{cha1}.
\begin{pro}
Let $Y_\alpha$ be the function space introduced above. Then we
have the followings.
\begin{enumerate}
\item[(i)] If $v\in Y_\alpha$ is a harmonic function, then $v \equiv
constant.$
\item[(ii)] There exists a constant $C>0$ such that for all $v \in Y_\alpha$
\[ |v(x)| \le C\| v \|_{Y_\alpha} \ln (e +|x|),
\qquad \forall x\in {\mathbb R}^2 . \]
\end{enumerate}
\end{pro}
\begin{pro}
Let $\a \in (0, \frac12 )$, and let us set
\begin{equation}
\label{26}
L=\Delta + \rho :{Y_\alpha} \to {X_\alpha} .
\end{equation}
where
$$ \rho (z)=\rho (|z|)=\frac{8(N+1)^2 |z|^{2N}}{(1+|z|^{2N+2} )^2}. $$
We have
\begin{equation}
\label{27}
Ker L=\mbox{Span} \left\{ \varphi_{+}, \varphi_{-} ,
\varphi_{0}
\right\},
\end{equation}
where we denoted
\begin{equation}
\label{28}
\varphi_+ (r,\theta)= \frac{ r^{N+1} \cos
(N+1)\theta}{1+r^{2N+2}},\quad \varphi_- (r,\theta )=
\frac{r^{N+1} \sin (N+1)\theta}{1+r^{2N+2}},
\end{equation}
and
\begin{equation}
\label{29}
\varphi_{0}=\frac{1-r^{2N+2}}{1+r^{2N+2}}.
\end{equation}
Moreover, we have
\begin{equation}
\label{210}
Im L =\{ f\in X_\alpha | \int_{\Bbb R^2} f\varphi_{\pm}
=0\}.
\end{equation}
\end{pro}
\ \\
Hereafter, we fix $\alpha=\frac14$, and set $X_{\frac14} =X$ and
$Y_{\frac14}=Y$.\\
Using Proposition 2.1 (ii), one can check easily that for $\varepsilon >0$
$P$ is a well defined continuous mapping from
$B_{\varepsilon_0}$ into
$X ^2$, where we set $B_{\varepsilon_0}=\{ \|v_1\|^2_{Y }+\|v_2\|^2_{Y
}+|a|^2 <\varepsilon_0\}$, for sufficiently small $\varepsilon_0$.
In order to extend continuously $P$ to $\varepsilon=0$ the radial functions $w_1 (r), w_2 (r)$
should satisfy
\begin{eqnarray}
\label{211}
&&\Delta w_1 +\lambda_2 \rho_1 w_1 +\lambda_1 \rho_2 =0\\
\label{212}
&&\Delta w_2 +\lambda_4 \rho_1 w_1 +\lambda_3 \rho_2 =0
\end{eqnarray}
For the existence and asymptotic properties of $w_1$ and $w_2$ we
have the following lemma, which is a part of Theorem 1.1.
\begin{lemma}
There exist radial solutions $w_1 (|z|), w_2 (|z|)$
of (\ref{211})-(\ref{212}) belonging to $Y $, which
satisfy the asymptotic formula in (\ref{19}),(\ref{19a}),(\ref{19b}).
\end{lemma}
\noindent{\bf Proof:}
Let us set $f(r)=\rho_1 (r)$. Then, it is found in \cite{bar1, cha1}
that the ordinary differential equation(with respect to $r$),
$\Delta w_1 +C_1 \rho_1 w_1 =f(r)$ has a solution $w_1 (r)\in Y$
given by
\begin{equation}
\label{213}
w_1(r) = \varphi_0 (r) \left\{\int_0 ^r \frac{\phi_{f} (s)
-\phi_{ f}(1)}{(1-s)^2} ds + \frac{\phi_{ f }(1) r}{1-r} \right\}
\end{equation}
with
$$
\,\, \phi_{ f} (r) := \left(\frac{
1+r^{2N+2}}{1-r^{2N+2}}\right)^2 \frac{(1-r)^2}{r} \int_0^r
\varphi_{0}(t) t{ f}(t) dt,
$$
where $\phi_{ f} (1)$
and $w_1(1)$ are defined as limits of $\phi_{ f} (r)$ and $w_1(r)$
as $r\to 1$. From the formula (\ref{213}) we find that
$$
w_1 (r)=
\varphi_0 (r) \int_2^r \left(\frac{ 1+s^{2N+2}}{1-s^{2N+2
}}\right)^2 \frac{I(s)}{s} ds +\mbox{(bounded function of $r$)}
$$
as $r\to
\infty$, where
$$ I(s)= \lambda_1 \int_0^s
\varphi_0(t) t\rho_2 (t) dt.
$$
Since $\varphi_0 (r) \rightarrow -1$ as $r\to \infty$, the first
part of (\ref{19}) follows if we show
$$
I =I(\infty )=\lambda_1
\int_0^\infty \varphi_0 (r) r\rho_2 (r) dr
=C_1.
$$
Changing variable $r^{2N +2}=t$, we evaluate
\begin{eqnarray}
\label{214}
I&=& \lambda_1 \int_0 ^\infty \varphi_0
(r)\rho_2(r)
rdr\nonumber\\
&=&c_0 \lambda_1 \int_0 ^\infty
\left[ \frac{r^{2N}}{(1+r^{2N_2 +2})^{3+\frac{2\lambda_4}{\lambda_2}}}
-\frac{r^{4N+2}}{(1+r^{2N_2 +2})^{3+\frac{2\lambda_4}{\lambda_2}}}\right]r
dr\nonumber\\
&=&\frac{c_0\lambda_1 }{2(N+1)}
\left[\int_0 ^\infty \frac{1}{ (1+t)^{3+\frac{2\lambda_4}{\lambda_2}} }dt-\int_0 ^\infty \frac{t}{(1+t)^{3+\frac{2\lambda_4}{\lambda_2}} }dt\right]\nonumber \\
&=&\frac{c_0\lambda_1 }{2(N+1)}\left[ \frac{1}{2+\frac{2\lambda_4}{\lambda_2}} - \frac{1}{\left(2+\frac{2\lambda_4}{\lambda_2} \right)\left(1+\frac{2\lambda_4}{\lambda_2}
\right)}\right]\nonumber \\
&=&
\frac{c_0\lambda_1\lambda_2\lambda_4}{2(N+1)(\lambda_2+\lambda_4)(\lambda_2 +2\lambda_4 )} =C_1.
\end{eqnarray}
In order to obtain $C_2$ we find from (\ref{211}) and (\ref{212})
that
$$
\Delta (\lambda_4 w_1 -\lambda_2 w_2 ) =(-\lambda_1\lambda_4 +\lambda_2 \lambda_3 )\rho_2 ,
$$
from which we have
\begin{eqnarray}
\label{214a}
w_2 (z)&=&\frac{\lambda_4}{\lambda_2} w_1 (z) +\frac{\lambda_1\lambda_4 -\lambda_2\lambda_3}{2\pi \lambda_2}\int_{\Bbb R^2} \ln (|z-y|)
\rho_2 (|y|)dy \nonumber \\
&=& -\frac{\lambda_4 C_1}{\lambda_2} \ln|z| +\frac{\lambda_1\lambda_4 -\lambda_2\lambda_3}{2\pi \lambda_2}\left[\int_{\Bbb R^2} \rho_2
(|y|)dy\right]
\ln|z| +O(1)\nonumber \\
\end{eqnarray}
as $|z|\to \infty$. In the case $\lambda_1\lambda_4 -\lambda_2\lambda_3=0$, we have $C_2
=\frac{\lambda_4 C_1}{\lambda_2}$.
In the case $\frac{2\lambda_4}{\lambda_2} >\frac{1}{N+1}$, we compute the integral as follows.
\begin{eqnarray}
\label{214b}
\int_{\Bbb R^2}\rho_2 (|y|)dy
&=&2\pi c_0 \int_0 ^\infty \frac{r}{(1+r^{2N +2})^{\frac{2\lambda_4}{\lambda_2}}}dr\nonumber \\
&=& \frac{\pi c_0}{N+1}\int_0 ^\infty \frac{t^{-\frac{N}{N+1}}}{(1+t)^{\frac{2\lambda_4}{\lambda_2}}}
dt \quad (r^{2N+2}=t )\nonumber \\
&=& \frac{\pi c_0}{N+1}B\left(\frac{1}{N+1}, \frac{2\lambda_4}{\lambda_2} -\frac{1}{N+1}
\right),
\end{eqnarray}
where we used the formula(See pp.
322\cite{gra}) for the beta function
$$
\int_0 ^\infty \frac{x^{\mu -1}}{(1+x)^{\nu}} dx =B (\mu , \nu- \mu ), \qquad
\mbox{where $\nu > \mu$}.
$$
Substituting (\ref{214b}) into (\ref{214a}), we have
$w_2 (z)=-C_2 \ln |z| +O(1)$ as $|z|\to \infty$, where $C_2$ is
given by (\ref{19b}).
This completes the proof of Lemma 2.1 $\square$\\
\ \\
Now we compute the linearized operator of $P$.
By direct computation we have
\[
\lim_{\varepsilon \to 0} \left.\frac{\partial g^{I}_{a,\varepsilon}(z) }{\partial
a_1}\right|_{a= 0}
=-4 \rho_1 \varphi_+ , \quad
\lim_{\varepsilon \to 0} \left. \frac{\partial g^{I}_{a,\varepsilon}(z)} {\partial
a_2}\right|_{a =0}
=-4 \rho_1 \varphi_- ,
\]
\[
\lim_{\varepsilon \to 0} \left. \frac{\partial g^{II}_{a,\varepsilon }(z)}{\partial
a_1}\right|_{a =0}
=-4\rho_2 \varphi_+ ,\quad
\lim_{\varepsilon \to 0} \left.\frac{\partial g^{II}_{a,\varepsilon }(z)}{\partial
a_2}\right|_{a=0}
=-4\rho_2 \varphi_- .
\]
Let us set $P'_{u,\eta, a }(0,0,0,0)=\mathcal{A}$. Then, using the
above preliminary computations, we obtain
$$
\mathcal{A}_1[\nu_1,\nu_2, \alpha ]=
\Delta \nu_1 +\lambda_2 \rho_1 \nu_1-4 (\lambda_2 w_1\rho_1 +\lambda_1 \rho_2 )(\varphi_+ \a_1
+\varphi_- \a_2),
$$
and
$$
\mathcal{A}_2[\nu_1,\nu_2, \alpha ]=
\Delta \nu_2 +\lambda_4\rho_1 \nu_1 -4 (\lambda_4 w_1\rho_1 +\lambda_3 \rho_2 )(\varphi_+ \a_1
+\varphi_- \a_2).
$$
We establish the following lemma for the operator $\mathcal{A}$.
\begin{lemma}
The operator $\mathcal{A}:Y^2 \times \Bbb C \times \Bbb R_+ $
defined above is onto. Moreover, kernel of $\mathcal{A}$ is given
by
$$
Ker \mathcal{A}= Span\{ (0,1);
(\varphi_\pm , \frac{\lambda_4}{\lambda_2} \varphi_\pm ),
(\varphi_0 , \frac{\lambda_4}{\lambda_2} \varphi_0 )\}
\times \{(0,0)\}.
$$
Thus, if we decompose $Y ^2\times \Bbb C=
U
\oplus Ker \mathcal{A}$, where we set $U=(Ker \mathcal{A})^\bot$, then
$\mathcal{A}$ is an isomorphism from $U$ onto $X ^2$.
\end{lemma}
In order to prove the above lemma we need to establish the following.
\ \\
\begin{pro}
\begin{equation} \label{214c}
I_\pm := \int_{\Bbb R^2} ( \lambda_2 w_1 \rho_1 +\lambda_1\rho_2 )\varphi_\pm dx \neq 0.
\end{equation}
\end{pro}
{\bf Proof:}
In order to transform the integrals we use the formula
$$ L \left[
\frac{1}{16(1+r^{2N+2})^2}\right] = \frac{(N+1)^2
r^{4N+2}}{(1+r^{2N+2})^4}, \qquad \forall N\in \Bbb Z_+$$ which
can be verified by an elementary computation. Using this, we
have the following
\begin{eqnarray*}
\lefteqn{\int_{{\Bbb R}^2} (\lambda_2 w_1 \rho_1 +\lambda_1\rho_2 )
\varphi_\pm ^2 dx = \int_0^{2\pi} \int_0 ^\infty (\lambda_2 w_1 \rho_1
+\lambda_1\rho_2 ) \frac{r^{2N+2}}{ (1+r^{2N+2})^2}\left\{
\begin{array}{c}
\cos^2 (N+1)\theta \\
\sin^2 (N+1)\theta \end{array}
\right\}
rdrd\theta }\hspace{.0in}\\
&&=\pi\int_0^\infty \left[ \frac{8(N+1)^2
r^{2N}}{(1+r^{2N+2})^2} w_1 +\lambda_1\rho_2 \right]
\frac{r^{2N+2}}{(1+r^{2N+2})^2} rdr\\
&&= \pi \int_0^\infty \left[ \frac{1}{2} L \left\{
\frac{1}{(1+r^{2N+2})^2}\right\} w_1 +\frac{\lambda_1\rho_2
r^{2N+2}}{(1+r^{2N+2})^2} \right]rdr\\
&&= \pi\int_0^\infty \left[\frac{1}{2} L w_1 \cdot
\frac{1}{(1+r^{2N+2})^2} +\frac{\lambda_1\rho_2
r^{2N+2}}{(1+r^{2N+2})^2} \right] rdr\\
&&= \pi\lambda_1 c_0
\int_0^\infty\left[-\frac{\rho_2 }{2(1+r^{2N+2})^2}+\frac{\rho_2
r^{2N+2}}{(1+r^{2N+2})^2}\right] rdr\\
&&=\frac{\pi \lambda_1 c_0}{2}\int_0 ^\infty \frac{r^{2N+2}
-1}{(1+r^{2N+2})^{2+\frac{2\lambda_4}{\lambda_2}}} rdr =\frac{\pi \lambda_1 c_0}{4}\int_0 ^\infty \frac{
t^{N +1}-1}{(1+t^{N+1} )^{2+\frac{2\lambda_4}{\lambda_2}}}dt \quad
(\mbox{$r^2=t$})\\
&&=\frac{\pi \lambda_1 c_0}{4}\left[\int_0 ^1 \frac{ t^{N
+1}-1}{(1+t^{N+1} )^{2+\frac{2\lambda_4}{\lambda_2}}}dt +\int_1 ^\infty \frac{ t^{N
+1}-1}{(1+t^{N+1}
)^{2+\frac{2\lambda_4}{\lambda_2}}}dt\right]\\
&&\quad (\mbox{Changing variable $t\to 1/t$ in the second
integral,}) \\
&&=\frac{\pi \lambda_1 c_0}{4}\left[\int_0 ^1 \frac{ t^{N
+1}-1}{(1+t^{N+1} )^{2+\frac{2\lambda_4}{\lambda_2}}}dt +\int_0 ^1
\frac{(1-t^{N+1})t^{\frac{2\lambda_4}{\lambda_2}} }{(1+t^{N+1}
)^{2+\frac{2\lambda_4}{\lambda_2}}}dt\right]\\
&&=\frac{\pi \lambda_1 c_0}{4}\int_0 ^\infty \frac{(t^{N+1} -1)(1-t^{\frac{2\lambda_4}{\lambda_2}}
)}{(1+t^{N+1} )^{2+\frac{2\lambda_4}{\lambda_2}}}dt <0 .
\end{eqnarray*}
This completes the proof of the proposition.$\square$\\
\ \\
We are now ready to prove Lemma 2.2.\\
\noindent{\bf Proof of Lemma 2.2:} Given $(f_1 ,f_2 )\in X ^2$, we
want first to show that there exists
$(\nu_1, \nu_2 )\in Y ^2 $, $\a_1,\a_2\in {\Bbb R}$ such that
$$
\mathcal{A}(\nu_1, \nu_2, \a_1,\a_2 )=(f_1, f_2 ),
$$
which can be rewritten as
\begin{equation}
\label{214d}
\Delta \nu_1 +\lambda_2 \rho_1 \nu_1 -4 (\lambda_2 w_1\rho_1 +\lambda_1 \rho_2 )(\varphi_+ \a_1
+\varphi_- \a_2) =f_1,
\end{equation}
and
\begin{equation}
\label{214e}
\Delta \nu_2 +\lambda_4\rho_1 \nu_1 -4 (\lambda_4 w_1\rho_1 +\lambda_3 \rho_2 )(\varphi_+
\a_1
+\varphi_- \a_2) =f_2 .
\end{equation}
Let us set
\begin{equation}
\a_1 =
\frac{1}{4I_+} \int_{\Bbb R^2} f_1 \varphi_+ dx, \qquad \a_2 =
\frac{1}{4I_-} \int_{\Bbb R^2} f_2 \varphi_- dx,
\end{equation}
where $I_\pm \neq 0$ is defined in (\ref{214c}). We introduce $\tilde{f}$
by
\begin{equation}
\tilde{f_1}= f_1-\a_1 \varphi_+ -\a_2 \varphi_- .
\end{equation}
Using the fact
\begin{eqnarray}
\int _0 ^{2\pi} \varphi_+\varphi_- d\theta
=0,
\end{eqnarray}
we find easily
\begin{equation}
\int_{\Bbb R^2} \tilde{f_1}\varphi_\pm dx =0.
\end{equation}
Hence, by (\ref{210}) there exists $\nu_1 \in Y$ such that
$\Delta \nu_1 +\lambda_2 \rho_1 \nu_1=\tilde{f_1}$. Thus we have found
$(\nu_1 , \a_1, \a_2)\in Y \times \Bbb R^2$ satisfying
(\ref{214d}). Given such $(\nu_1 , \a_1, \a_2)$, the function
\begin{eqnarray}
\nu_2 (z)=\frac{1}{2\pi } \int_{\Bbb R^2} \ln (|z-y|) g (y) dy
+c_1,
\end{eqnarray}
where
$$
g=f_2-\lambda_4\rho_1 \nu_1 +4 (\lambda_4 w_1\rho_1 +\lambda_3 \rho_2 )(\varphi_+
\a_1
+\varphi_- \a_2),
$$
and $c_1$ is any constant,
satisfies (\ref{214e}), and belongs to $Y$.
We have just finished the proof that $\mathcal{A}: Y ^2 \times \Bbb
R^2 \to X ^2$ is onto.\\
We now show that the restricted operator(denoted by the same
symbol),
$\mathcal{A}: (Ker L\oplus
Span\{1\})^\bot \times \Bbb R^2 \to X ^2$ is one
to one.
Given $ (\nu_1,\nu_2, \a_1, \a_2)\in (Ker L\oplus
Span\{1\})^\bot \times \Bbb R^2 $,
let us consider the equation, $\mathcal{A}(\nu_1,\nu_2, \a_1 ,\a_2 )=(0, 0 )$, which
corresponds to
\begin{equation}
\label{214f}
\Delta \nu_1 +\lambda_2 \rho_1 \nu_1 -4 (\lambda_2 w_1\rho_1 +\lambda_1 \rho_2 )(\varphi_+ \a_1
+\varphi_- \a_2) =0,
\end{equation}
and
\begin{equation}
\label{214g}
\Delta \nu_2 +\lambda_4\rho_1 \nu_1 -4 (\lambda_4 w_1\rho_1 +\lambda_3 \rho_2 )(\varphi_+
\a_1
+\varphi_- \a_2) =0 .
\end{equation}
Taking $L^2 (\Bbb R^2 )$ inner product of (\ref{214f}) with $\varphi_\pm$,
and using (\ref{214c}), we find
$\a_1=\a_2 =0$. Thus, (\ref{214f}) implies
$\nu_1 \in Ker L$. This, combined with the hypothesis
$\nu_1 \in (Ker L)^\bot$ leads to $\nu_1 =0$. Now, (\ref{214g}) is reduced to
$\Delta \nu_2=0$. Since $\nu_2\in Y$, Proposition 2.1 implies
$\nu_2=$ constant. Since $\nu_2\in (Span\{1\})^\bot $ by hypothesis, we
have $\nu_2=0$.
This completes the proof of the lemma.
$\square$\\
\ \\
We are now ready to prove our main theorem.\\
\noindent{\bf Proof of
Theorem 1.1:} Let us set
\[ U =
(\mathrm{Ker} L \oplus
\mathrm{Span}\{1\} )^\bot
\times \Bbb \Bbb R^2 .
\]
Then, Lemma 2.2 shows that
$P'_{(v_1 , v_2 , \a )} (0,0,0,0) : U \to X ^2$ is an
isomorphism. Then, the standard implicit function theorem(See e.g.
\cite{zei}), applied to the functional $P : U \times
(-\varepsilon_0, \varepsilon_0) \to X ^2$, implies that there
exists a constant $\varepsilon_1\in (0,\varepsilon_0)$ and a
continuous function $\varepsilon \mapsto \psi ^*_\varepsilon :=
(v_{1,\varepsilon}^*, v_{2,\varepsilon}^*, a_\varepsilon^* )$ from
$(0, \varepsilon_1)$ into a neighborhood of 0 in $U$ such that
\[ P(v_{1,\varepsilon}^*, v_{2,\varepsilon}^*, a_\varepsilon^* )
=(0,0), \quad\mbox{for all } \varepsilon\in (0, \varepsilon_1). \]
This completes the proof of Theorem 1.1.
The
representation of solutions $u_1,u_2$, and the explicit form of
$\rho^I _{\varepsilon, a^ * _{ \varepsilon} } (z),$
$ \rho^{II} _{\varepsilon, a ^ * _{\varepsilon} } (z),$ ,
together with the asymptotic behaviors of $w_1, w_2$
described in Lemma 2.1, and the fact that
$v_{1,\varepsilon}^*, v_{2,\varepsilon}^* \in
Y$, combined with Proposition 2.1, implies that the solutions
satisfy the boundary condition in (\ref{13}).
Now,
from Proposition 2.1 we obtain that for each $j=1,2$,
\begin{equation}
\label{225}
|v^*
_{j, \varepsilon} ( x)|\leq C \Vert v^* _{j,\varepsilon} \Vert_{Y} (\ln^+ | x|
+1) \leq C \Vert \psi_\varepsilon \Vert_{U} (\ln^+ | x| +1).
\end{equation}
This implies then
$$
|v^* _{j, \varepsilon} ( \varepsilon x)|\leq C \Vert \psi _{\varepsilon}
\Vert_{U}(\ln^+ | \varepsilon x| +1) \leq C \Vert \psi_\varepsilon
\Vert_{U}(\ln^+ | x| +1).cxxc
$$
From the continuity of the function $\varepsilon\mapsto
\psi_{\varepsilon}$ from $(0, \varepsilon_0 )$ into $U$ and the fact $\psi^*
_0 =0$ we have
\begin{equation}
\label{226}
\|\psi_\varepsilon \|_{U} \to 0\qquad
\mbox{ as $\varepsilon \to 0$} .
\end{equation}
The proof of (\ref{110}) follows from (\ref{225})
combined with (\ref{226}).
This
completes the proof of Theorem 1.1$\square$\\
$$\mbox{\bf Acknowledgements} $$
This work was supported by Korea Research Foundation Grant
KRF-2002-015-CS0003.
|
{
"timestamp": "2005-03-23T15:47:36",
"yymm": "0503",
"arxiv_id": "math/0503493",
"language": "en",
"url": "https://arxiv.org/abs/math/0503493"
}
|
\section*{Introduction}
Let $(M,g)$ be an $n+1$ -- dimensional Riemannian manifold with
metric $g$. A vector field $\xi$ on it is called {\it holonomic}
if $\xi$ is a field of normals of some family of regular
hypersurfaces in $M$ and {\it non-holonomic} otherwise. The
foundation of the classical geometry of unit vector fields was
proposed by A.Voss at the end of the nineteenth century. The
theory includes the {\it Gaussian} and {\it the mean curvature} of
a vector field and their generalizations (see \cite{Am} for
details). Here we will consider a unit vector field from another
point of view. Namely, let $T_1M$ be the unit tangent sphere
bundle of $M$ endowed with the Sasaki metric \cite{S}. If $\xi$ is
a unit vector field on $M$, then one may consider $\xi$ as a
mapping $\xi : M \to T_1M $ so that the image $\xi (M) $ is a
submanifold in $T_1M$ with the metric induced from $T_1M$.
H.Gluck and W.Ziller \cite{G-Z} called $\xi$ {\it a minimal vector
field} if $ \xi(M)$ is of minimal volume with respect to induced
metric. They considered the unit vector field on $S^3$ tangent to
the fibers of a Hopf fibration $S^3
\stackrel{S^1}{\longrightarrow} S^2$ and proved that these (Hopf)
vector fields are unique ones with global minimal volume. Note
that this result is not true for greater dimensions where Hopf
vector fields are still critical points for the volume functional
but do not provide the global minimum among all unit vector fields
\cite{Jon,Ped}. The local aspect of the problem was considered
first in \cite{GM-LF}. The authors have found the necessary and
sufficient condition for a unit vector field to generate locally a
minimal submanifold in the tangent sphere bundle. In fact, that
condition implies that {\it the mean curvature } of the
submanifold $\xi(M)$ is zero. Using that criterion, a number of
examples of local minimal vector unit fields have been found (~see
lab2 \cite{ BX-V1, BX-V2, GD-V1, GD-V2, TS-V1,TS-V2}).
In this paper, we give an {\it explicit formula} for the mean curvature of
$\xi(M)$ using some special but natural normal frame for $\xi(M)$ and give an
example of a unit vector field of {\it constant mean curvature}
on a Lobachevsky space. We shall state the main result after some preliminaries.
Let $\nabla$ denote the Levi-Civita connection on $M$. Then $\nabla_ X \xi$ is
always orthogonal to $\xi$ and hence,
$(\nabla\xi)(X)=\nabla_X\xi :T_pM \to \xi^\perp_p$ is a linear operator at each
$p\in M$. We define the adjoint operator $ (\nabla\xi)^*(X) :\xi^\perp_p \to
T_pM$ by
$$
\left< (\nabla\xi)^*X,Y\right>_g = \left< X,\nabla_Y\xi \right>_g
$$
Then there is an orthonormal frame $e_0, e_1, \dots , e_n $ in $T_pM$ and an
orthonormal frame $f_1, \dots , f_n $ in $\xi_p^\perp$ such that
$$
(\nabla\xi)(e_0)=0 , \quad (\nabla\xi)(e_\alpha )=\lambda_\alpha f_\alpha,
\quad (\nabla\xi)^*(f_\alpha)=\lambda_\alpha e_\alpha ,
\qquad \alpha=1, \dots , n ,
$$
where $\lambda_1\ge \lambda_{2}\ge \dots \ge \lambda_n\ge 0$ are the singular
values of $\nabla\xi$. As we will see, the vectors
$$
\tilde n_{\sigma |} =\frac{1}{\sqrt{1+\lambda_\sigma^2}}\big(-\lambda_\sigma
e_\sigma^h +f_\sigma^v \ \big),\mbox{\hspace{3em}} \sigma=1,\dots , n ,
$$
where $H$ and $V$ are the horizontal and vertical lifts
respectively, form an orthonormal frame in the normal bundle of $\xi(M)$.
Furthermore, we introduce the notation
$$
r(X,Y)\xi=\nabla_X\nabla_Y\xi-\nabla_{\nabla_XY}\xi.
$$
Then $R(X,Y)\xi=r(X,Y)\xi-r(Y,X)\xi$ , where $R$ is the Riemannian curvature
tensor. Now we are able to state our main result.
\vspace{1ex}
{\bf Theorem \ref{Th1}} {\it Let $H_{\sigma |}$ be the components of
the mean curvature vector of $\xi(M)$ with respect to the
orthonormal frame $\tilde n_\sigma$. Then
$$
\begin{array}{c}
(n+1)H_{\sigma |}=\\[1ex]\displaystyle
\frac{1}{\sqrt{1+\lambda_\sigma^2}}
\left\{ \big<r(e_0,e_0)\xi,f_\sigma\big> + \sum_{\alpha =1}^n
\frac{\big<r(e_\alpha,e_\alpha)\xi,f_\sigma\big>+
\lambda_\sigma\lambda_\alpha \big<R(e_\sigma,
e_\alpha)\xi,f_\alpha)\big>}{1+\lambda_\alpha^2} \right\}.
\end{array}
$$ }. \vspace{1ex}
The following very simple example gives a unit vector field of
{\it constant mean curvature}.
\vspace{1ex}
{\bf Proposition \ref{Ex}} {\it Let $M$ be the Lobachevsky 2-plane
with the metric
$$
ds^2=du^2+e^{2u}dv^2.
$$
Let $X_1=\{1,0\}$ and $X_2=\{0,e^{-u}\}$. Then $\xi=\cos \omega X_1+\sin\omega X_2$,
where $\omega=au+b $, generates a hypersurface $\xi(M)\subset T_1M$ of constant
mean curvature
$$
H=\frac{a}{2\sqrt{2+a^2}}.
$$
}
\vspace{1ex}
{\bf Index convention.} Throughout the paper we take $i, j, k, \ldots =0, \dots, n$
and $\alpha, \beta, \ldots = 1, \dots, n .$
\section{ Basic concepts from the geometry of the unit tangent sphere bundle.}
Let $(u^0,\dots ,u^n)$ be a local coordinate system on $M$ and
let $\partial /\partial u^i $ be the vectors of a natural frame on
$M^n.$ The points of the tangent bundle $TM$ are the pairs
$\tilde Q=(Q,\xi)$, where $Q\in M$ and $\xi\in T_QM$. Each point
$\tilde Q\in TM$ is uniquely determined by the set of parameters
$(u^0,\dots ,u^n;\xi^0,\dots ,\xi^n)$, where $(u^0, \dots ,u^n)$
fix the point $Q$ and $\{\xi^0, \dots, \xi^n\}$ are the
coordinates of $\xi$ with respect to the frame $\{ \partial
/\partial u^0, \dots ,\partial /\partial u^n \}$. The local
coordinates $(u^0,\dots ,u^n;\xi^0, \dots ,\xi^n)$ are called
{\it natural induced coordinates } in the tangent bundle. Each
smooth tangent vector field $\xi=\xi(u^0, \dots ,u^n)$ generates a
smooth submanifold $\xi(M)\subset TM$ having a parametric
representation of the form
\begin{equation}
\label{lab1}
\left\{
\begin{array}{lcl}
u^i & = & u^i,\\
\xi^i & = & \xi ^i(u^0, \dots, u^n).
\end{array}
\right.
\end{equation}
Setting $|\xi |=1$, we get a submanifold in the unit tangent sphere bundle
$\xi(M^n)\subset T_1M^n.$
A natural Riemannian metric on the tangent bundle has been defined by S.Sasaki
\cite{S}. We describe it in terms of the {\it connection map}.
The tangent space $T_{\tilde Q}TM$ can be split into {\it vertical} and {\it
horizontal} parts:
$$
T_{\tilde Q}TM^n=H_{\tilde Q}TM^n \oplus V_{\tilde Q}TM^n.
$$
The vertical part $V_{\tilde Q}TM$ is tangent to the fiber, while the
horizontal part is transversal to it. For $\tilde X \in T_{\tilde Q}TM^n$ we
have
\begin{equation}
\label{lab2}
\tilde X=\tilde X^i \partial /\partial u^i + \tilde X^{n+i} \partial /\partial
\xi^i
\end{equation}
with respect to the natural frame $\{ \partial /\partial u^i, \partial /\partial
\xi^i \}$ on $TM$.
Let $\pi:TM \to M$ be the projection map. It is easy to check that the
differential $\pi_*:T_{\tilde Q}TM \to T_QM $ of the mapping $\pi$ acts on
$\tilde X$ as follows:
\begin{equation}
\label{lab3}
\pi_*\tilde X=\tilde X^i \partial /\partial u^i,
\end{equation}
and is a linear isomorphism between $V_{\tilde Q}TM$ and $T_QM$.
The {\it connection map} $K: T_{\tilde Q}TM \to T_QM$ acts on $\tilde X$ by
\begin{equation}
\label{lab4}
K\tilde X=(\tilde X^{n+i}+\Gamma_{jk}^{i}\xi^j\tilde X^k) \partial /\partial u^i
\end{equation}
and it is a linear isomorphism between $H_{\tilde Q}TM$ and $T_QM$.
Moreover, it is easy to see that $V_{\tilde Q}TM=\ker \pi_*$, $H_{\tilde Q}TM=\ker K$.
The images $\pi_*\tilde X$ and $K\tilde X$ are called {\it horizontal} and
{\it vertical } projections of $\tilde X$, respectively.
The {\it Sasaki metric} on $TM$ is defined by the following scalar product:
if $\tilde X,\tilde Y \in T_{\tilde Q}TM$, then
\begin{equation}
\label{lab5}
\big<\big< \tilde X,\tilde Y \big>\big>_S=
\big<\pi_* \tilde X, \pi_* \tilde Y\big>_g+\big<K \tilde X,K \tilde Y\big>_g
\end{equation}
where $\big<,\big>_g$ is the scalar product with respect to the metric $g $
on the initial manifold (the base space of tangent bundle).
Horizontal and vertical subspaces are mutually orthogonal with respect to Sasaki
metric.
The inverse operations of projections (\ref{lab3}) and (\ref{lab4})
are called {\it lifts}. Namely, if $X \in T_QM^n$, then
$$
X^H=X^i \partial /\partial u^i -\Gamma_{jk}^i\xi^j X^k \partial /\partial \xi^i
$$
is in $H_{\tilde Q}TM$ and is called the {\it horizontal lift } of X, and
$$
X^V=X^i \partial /\partial \xi^i
$$
is in $V_{\tilde Q}TM$) and is called the {\it vertical lift } of $ X$.
Among all lifts of various vectors from $T_QM$ into $T_{(Q,\xi)}TM$, one can
naturally distinguish two of them, namely $\xi^H$ and $\xi^V$. The vector field
$\xi^H$ is the {\it geodesic flow} vector field, while $\xi^V$ (being
normalized) is a {\it unit normal} vector field of $T_1M \subset TM$.
In the geometry of the {\it unit tangent sphere bundle} it appears to be
convenient to introduce the notion of {\it tangential lift} \cite{BX-V3}:
\begin{equation}\label{lab5_1}
X^t=X^V-\big<X,\xi\big>\xi^V.
\end{equation}
In other words, the tangential lift is the projection of the vertical lift onto the
tangent space of $T_1M$.
We denote by $\tilde\nabla$ the Levi-Civita connection of the Sasaki metric on
$T_1M$. In terms of horizontal and tangential lifts we then have \cite{BX-V3}:
\begin{equation}
\label{lab6}
\begin{array}{ll}
\tilde\nabla_{X^H}Y^H = (\nabla_XY)^H - \frac{1}{2}(R(X,Y)\xi)^t,
&\tilde\nabla_{X^t}Y^H = \frac{1}{2}(R(\xi,X)Y)^H, \\
\tilde\nabla_{X^H}Y^t = (\nabla_XY)^t \ + \frac{1}{2}(R(\xi_1,Y)X)^H, &
\tilde\nabla_{X^t}Y^t = -\big<Y,\xi\big>X^t.
\end{array}
\end{equation}
\begin{remark} \rm It is evident that if $Z \perp \xi $, the vertical and tangential
lifts of $Z$ coincide, particulary $ (\nabla_X\xi)^t=(\nabla_X\xi)^V$ for any
$X$. We will use this fact throughout the paper without special comments.
\end{remark}
\section{The mean curvature formula for a unit vector field}
\subsection{The structure of tangent and normal bundles of $\xi(M)$}
Let $\xi$ be the unit tangent vector field on $M$. We denote by $T\xi(M)$ the
tangent bundle of $\xi(M)\subset T_1M$. The structure of $T\xi(M)$ can be
described as follows:
\begin{lemma}
\label{L1} \it The vector $\tilde X \in T_{(Q,\xi)}T_1M$ is tangent to $\xi(M)$
at $(Q,\xi)$ if and only if
\begin{equation}
\label{lab7}
\tilde X = X^H + (\nabla_X\xi)^V
\end{equation}
where $X \in T_QM$.
\end{lemma}
\begin{proof} Using the local representation (\ref{lab1}) of $ \xi (M)$, we
consider the coordinate frame of $T_{(Q,\xi)}\xi (M)$:
$$
\tilde e_i = \left\{0,\dots , 1, 0, \dots , 0; \frac{\partial \xi^0}{\partial
u^i},
\dots , \frac{\partial \xi^n}{\partial u^i}\right\}.
$$
Let $\tilde X \in T_{(Q,\xi)}TM$ be tangent to $\xi(M)$.
Then
$$
\tilde X = \tilde X^i \tilde e_i.
$$
Applying (\ref{lab3}) and (\ref{lab4}), we obtain
$$
\begin{array}{lcl}
\pi_*\tilde e_i & = & \partial /\partial u^i, \\
K\tilde e_i & = & \nabla_i\xi.
\end{array}
$$
From this we get
$$
\begin{array}{lcl}
\pi_* \tilde X & = & \tilde X^i \partial /\partial u^i, \\
K \tilde X & = & \nabla_{\pi_* \tilde X} \xi.
\end{array}
$$
Setting $X = \pi_* \tilde X$ and taking into account the remark, we get
(\ref{lab7}).
\end{proof}
To describe the structure of the normal bundle of $\xi(M)$, we use the {\it
adjoint covariant derivative operator}. As $\xi $ is a fixed unit vector field,
$\nabla_X\xi$ can be considered as a pointwise linear operator
$(\nabla\xi):T_QM \to \xi^{\perp}$, where $\xi^{\perp}$ is the orthogonal
complement of $\xi$ in $T_QM$, acting as
$$
(\nabla \xi)(X) = \nabla _X \xi.
$$
The matrix of this operator is formed by the covariant derivatives $\nabla_i
\xi^k$.
The {\it adjoint covariant derivative} linear operator $(\nabla\xi)^*:
\xi^{\perp} \to T_QM$ can be defined in a standard way:
\begin{equation}
\label{lab8}
\big<(\nabla \xi)^*X,Y\big> = \big<X,(\nabla\xi)(Y)\big>
\end{equation}
for each $X \in \xi^{\perp}$. The matrix of $(\nabla \xi)^*$ has the form
$$
\left[ (\nabla \xi)^* \right]_j^i = g^{im} \nabla _m \xi^kg_{kj}.
$$
As $\nabla$ is the Riemannian connection for $g$, we obtain for $(\nabla\xi)^*$
the formally transposed matrix
$$
\left[(\nabla \xi)^* \right]_k^i = \nabla^i \xi_k.
$$
Now the structure of $\xi(M)$ can be described as follows:
\begin{lemma}
\label{L2}
The vector $\tilde N \in T_{(Q,\xi)}T_1M$ is normal to
$\xi(M)$ if and only if $$ \tilde N = - \left[(\nabla
\xi)^*N\right]^H + N^V $$ where $N \in T_QM$ and $N\perp \xi$.
\end{lemma}
The proof follows easily from (\ref{lab5}), (\ref{lab7}) and (\ref{lab8})
\subsection{Second fundamental form of $\xi(M)$ in $T_1M$}
We denote by $\tilde \Omega_{\tilde N}$ the second fundamental form of $\xi(M)$
in $T_1M^n$ with respect to the normal vector field $\tilde N$ defined in Lemma
\ref{L2}.
Then the following statement holds.
\begin{lemma} \label{L3}
For $\tilde X, \tilde Y $ being tangent to $\xi(M)$ we have
$$
\tilde \Omega_{\tilde N}(\tilde X, \tilde Y) = \frac{1}{2}
\big< r(X,Y) \xi + r(Y,X) \xi -\nabla_{R(\xi, \nabla_X
\xi)Y+R(\xi, \nabla_Y \xi)X} \xi ,N \big>,
$$
where $r(X,Y)\xi =\nabla_X \nabla_Y \xi - \nabla_{\nabla_X Y}\xi$
\end{lemma}
\begin{proof}
By definition we have
$$
\tilde \Omega_{\tilde N}(\tilde X,\tilde
Y) = \big<\big< \tilde \nabla_{\tilde X}\tilde Y,\tilde
N\big>\big>
$$
where $\tilde X,\tilde Y \in T_{(Q,\xi)} \xi(M)$.
Using Lemma \ref{L1}, we put $\tilde X = X^H + (\nabla_X\xi)^V; \
\tilde Y = Y^H +(\nabla_Y\xi)^V$. Then applying (\ref{lab6}) and
(\ref{lab5_1}), we
have
$$
\begin{array}{l}
\tilde \nabla_{\tilde X} \tilde Y = \tilde
\nabla_{X^H+(\nabla_X\xi)^t} (Y^H +(\nabla_Y\xi)^t) = \\[1ex]
\left[\nabla_X Y + \frac{1}{2}R(\xi,\nabla_X \xi)Y +
\frac{1}{2}R(\xi,\nabla_Y \xi)X\right]^H+
\left[\nabla_X \nabla_Y \xi - \frac{1}{2}R(X,Y) \xi\right]^t= \\[1ex]
\left[\nabla_X Y + \frac{1}{2}R(\xi,\nabla_X \xi)Y +
\frac{1}{2}R(\xi,\nabla_Y \xi)X\right]^H + \left[\nabla_X \nabla_Y \xi -
\frac{1}{2}R(X,Y) \xi\right]^V - \\[1ex]
\big<\nabla_X \nabla_Y \xi,\xi\big>\xi^V.
\end{array}
$$
Let $N$ be orthogonal to $\xi$. Then $\tilde N = - \left[
(\nabla \xi)^*N \right]^H + N^V $ is normal to $\xi(M)$.
Therefore
\begin{eqnarray}
\tilde \Omega_{\tilde N}(\tilde X, \tilde Y) = -\big<\nabla_XY +
\frac{1}{2}R(\xi,\nabla_X\xi)Y + \frac{1}{2}R(\xi,\nabla_Y\xi)X,
(\nabla \xi)^*N\big> +\nonumber\\[1ex]
\big<\nabla_X \nabla_Y \xi - \frac{1}{2}R(X,Y)\xi,N\big>=\nonumber\\[1ex]
\big<\nabla_X \nabla_Y \xi - \frac{1}{2}R(X,Y)\xi
-\nabla_{\nabla_XY+\frac{1}{2}R(\xi,\nabla_X \xi)Y+
\frac{1}{2}R(\xi,\nabla_Y \xi)X} \xi, N\big>.\hspace{1em}\label{lab9}
\end{eqnarray}
To simplify the expression (\ref{lab9}), we introduce the
following tensor $r$:
\begin{equation}\label{lab10}
r(X,Y)\xi = \nabla_X \nabla_Y \xi - \nabla_{\nabla_X Y}\xi.
\end{equation}
Then for the Riemannian tensor, we get
$$
R(X,Y)\xi = r(X,Y)\xi - r(Y,X)\xi
$$
and (\ref{lab9}) can be rewritten as
\begin{equation}\label{lab11}
\tilde \Omega_{\tilde N}(\tilde X, \tilde Y) = \frac{1}{2} \big<
r(X,Y) \xi + r(Y,X) \xi -\nabla_{R(\xi, \nabla_X \xi)Y+R(\xi,
\nabla_Y \xi)X} \xi ,N \big>.
\end{equation}
\end{proof}
Next, we determine the components of $\tilde \Omega$ with respect to some special
frame.
As $(\nabla \xi): T_QM \to \xi^\perp$ and $(\nabla \xi)^*:
\xi^\perp \to T_QM$ are mutually adjoint, then in
$T_QM$ and $\xi^\perp$, respectively, there exist orthonormal frames
$\{e_0, e_1,\dots ,e_n\}$ and $\{f_1, \dots ,f_n\}$ such that
$$
\left\{
\begin{array}{lrl}
(\nabla \xi)e_0 & = & 0, \\
(\nabla \xi)e_\alpha & = & \lambda_\alpha f_\alpha, \\
(\nabla \xi)^*f_\alpha & = & \lambda_\alpha e_\alpha,
\end{array}
\right.
$$
where $\lambda_n \ge \lambda_{n-1} \dots \ge \lambda_1 \ge 0$ is a set of
singular values (functions) of the linear operator $\nabla \xi$.
Then
\begin{equation}
\label{lab9'}
\left\{
\begin{array}{l}
\tilde e_0 = e_0^H, \\ \tilde e_\alpha =
e_\alpha^H+(\nabla_{e_\alpha}\xi)^V = e_\alpha^H + \lambda_\alpha
f_\alpha^V
\end{array} \right.
\end{equation}
form an orthogonal frame of the tangent space of
$T_{(Q,\xi)}\xi(M)$ while
\begin{equation}\label{n12}
\tilde n_{\sigma} = \frac{1}{\sqrt{1+\lambda_
\sigma^2}}\left(\lambda_\sigma e_\sigma^H - f_\sigma^V \right)
\end{equation}
form the orthonormal frame in $\xi(M)^\perp$.
\begin{lemma}\label{L4}
The components of second fundamental form of
$\xi(M)\subset T_1M$ with respect to the frames (\ref{lab9'}) and
(\ref{n12}) are given by
$$
\begin{array}{rcl}
\tilde \Omega_{\sigma | 00} &=&
\frac{1}{\sqrt{1+\lambda_\sigma^2}} \big\{ \big< r(e_0,e_0)
\xi,f_\sigma \big> \big\}, \\[1ex]
\tilde \Omega_{\sigma | \alpha
0} &=& \frac{1}{2}\frac{1}{\sqrt{1+\lambda_\sigma^2}} \frac{1}{\sqrt{1+\lambda_\alpha^2}}
\big\{ \big<
r(e_\alpha,e_0) \xi + r(e_0,e_\alpha) \xi,f_\sigma \big> +
\lambda_\sigma \lambda_\alpha \big< R(e_\sigma,e_0) \xi, f_\alpha
\big> \big\}, \\[1ex]
\tilde \Omega_{\sigma | \alpha \beta}
&=& \frac{1}{2}\frac{1}{ \sqrt{1+ \lambda_\sigma^2}} \frac{1}{\sqrt{1+\lambda_\alpha^2}}
\frac{1}{\sqrt{1+\lambda_\beta^2}}
\big\{ \big<
r(e_\alpha, e_\beta) \xi+ r(e_\beta, e_\alpha) \xi, f_\sigma \big>\\
&&+ \lambda_\alpha \lambda_\sigma \big< R(e_\sigma, e_\beta)
\xi, f_\alpha \big> + \lambda_\beta \lambda_\sigma \big<
R(e_\sigma, e_\alpha) \xi, f_\beta \big> \big\},
\end{array}
$$
where $\sigma,\alpha,\beta=1,\dots,n$
\end{lemma}
\begin{proof}
Indeed, with respect to (\ref{lab9'}) and (\ref{n12}) the
components of $\tilde\Omega $ are $$ \tilde \Omega_{\sigma |
ik} = \tilde \Omega_{\tilde n_\sigma}(\tilde e_i, \tilde e_k). $$
Using (\ref{lab11}), we have
$$ \tilde \Omega_{\sigma | ik}
=\frac{1}{2}\frac{1}{\sqrt{1+\lambda_\sigma^2}} \big<r(e_i,e_k)
\xi+r(e_k,e_i)\xi -\nabla_{R(\xi, \nabla_{e_i} \xi)e_k+R(\xi,
\nabla_{e_k} \xi)e_i} \xi ,f_{\sigma} \big>. $$
Setting $i=k=0$ and applying (\ref{lab9'}), we get
$$
\tilde \Omega_{\sigma | 00}= \frac{1}{\sqrt{1+\lambda_\sigma^2}}\big\{
\big< r(e_0,e_0)\xi,f_\sigma \big> \big\}.
$$
Setting $i= \alpha,\, k = 0$ and applying (\ref{lab9'}) again, we obtain
$$
\begin{array}{rl}
\tilde \Omega_{\sigma | \alpha 0} = & \frac{1}{2}
\frac{1}{\sqrt{1+ \lambda_\sigma^2}} \big\{ \big< r(e_\alpha, e_0)
\xi, f_\sigma \big> + \big< r(e_0, e_\alpha) \xi, f_\sigma \big> -
\big< \nabla_{R(\xi,( \nabla \xi)e_\alpha) e_0} \xi, f_\sigma
\big> \big \} =\\[1ex]
& \frac{1}{2} \frac{1}{\sqrt{1+\lambda_\sigma^2}} \big\{ \big<
r(e_\alpha,e_0) \xi, f_\sigma \big> + \big< r(e_0,e_\alpha)
\xi,f_\sigma \big> + \lambda_\sigma \lambda_\alpha \big<
R(e_\sigma,e_0) \xi, f_\alpha \big> \big\}.
\end{array}
$$
Finally, setting $i=\alpha ,\, k=\beta $ applying again (\ref{lab9'}), we obtain
$$
\begin{array}{lrl}
\tilde \Omega_{\sigma | \alpha \beta} =
&\frac{1}{2}\frac{1}{\sqrt{1+\lambda_\sigma^2}} &\left\{\big<
r(e_\alpha, e_\beta) \xi + r(e_ \beta, e_\alpha) \xi - \right.\\
& &\left.\quad\quad\quad\quad\quad \nabla_{R(\xi,( \nabla \xi)(e_\alpha))e_\beta +R(\xi,
(\nabla \xi)(e_ \beta))e_\alpha} \xi, f_\sigma \big> \right\}=
\\[1ex] & \frac12\frac{1}{\sqrt{1+\lambda_\sigma^2}} &\left\{\big<
r(e_\alpha, e_\beta) \xi + r(e_\beta, e_\alpha) \xi,f_\sigma \big>
- \right.\\
&&\left.\quad\quad\big< \lambda_\alpha R(\xi,
f_\alpha)e_\beta + \lambda_\beta R(\xi, f_\beta) e_\alpha,(\nabla
\xi)^*(f_\sigma)\big> \right\}= \\[1ex]
&\frac12\frac{1}{\sqrt{1+\lambda_\sigma^2}} &\left\{\big<
r(e_\alpha,e_\beta) \xi + r(e_\beta, e_\alpha) \xi, f_\sigma \big>
- \right.\\ &&\left.\quad\lambda_\alpha\lambda_\sigma \big<R(\xi,
f_\alpha)e_\beta, e_\sigma \big> - \lambda_\beta\lambda_\sigma
\big<R(\xi, f_\beta) e_\alpha, e_\sigma \big> \right\} = \\[1ex]
&\frac{1}{2} \frac{1}{\sqrt{1+\lambda_\sigma^2}} &\left\{ \big<
r(e_\alpha,e_\beta) \xi, f_\sigma \big> +\big< r(e_\beta,
e_\alpha) \xi, f_\sigma \big> +\right.\\ &&\left.\quad
\lambda_\alpha \lambda_\sigma \big< R(e_\sigma,e_\beta) \xi, f_
\alpha \big> + \lambda_\beta \lambda_\sigma \big< R(e_\sigma,
e_\alpha) \xi, f_\beta \big> \right\}.
\end{array}
$$
So, the lemma is proved.
\end{proof}
\subsection{The mean curvature formula}
Now we are able to prove the main result.
\begin{theorem}\label{Th1}
The components of the mean curvature vector of
$\xi(M)\subset T_1M$ with respect to the frames (\ref{lab9'}) and
(\ref{n12}) are given by
\begin{equation}\label{H}
\begin{array}{cc}
(n+1)H_{\sigma |}=\\[1ex]\displaystyle
\frac{1}{\sqrt{1+\lambda_\sigma^2}}\left\{
\big<r(e_0,e_0)\xi,f_\sigma\big> + \sum\limits_{\alpha =1}^n
\frac{\big<r(e_\alpha,e_\alpha)\xi,f_\sigma\big>+
\lambda_\sigma\lambda_\alpha \big<R(e_\sigma,e_\alpha)\xi,
f_\alpha)\big>}{1+\lambda_\alpha^2} \right\}.
\end{array}
\end{equation}
\end{theorem}
\begin{proof} With respect to the frames (\ref{lab9'}) and (\ref{n12}) the
matrix of the first fundamental form $\tilde G$ of $\xi(M)$ is
\begin{equation}
\label{n10} \tilde G = \left(
\begin{array}{cccc}
1 & 0 & \ldots & 0 \\ 0
& 1+\lambda_1^2 & \ldots &0 \\ \vdots & \vdots
& \ddots & \vdots \\ 0 & 0 & \ldots
& 1+\lambda_{n}^2 \\
\end{array}
\right).
\end{equation}
For the inverse matrix we have
\begin{equation}
\label{n11} \tilde G^{-1} = \left(
\begin{array}{cccc}
1 & 0 & \ldots & 0 \\ 0
&\frac{1}{1+\lambda_1^2} & \ldots & 0 \\ \vdots &
\vdots &\ddots & \vdots \\ 0 & 0
&\ldots & \frac{1}{1+\lambda_n^2}
\\
\end{array}
\right).
\end{equation}
So we have $$
\begin{array}{lcl}
\tilde \Omega_{\sigma | 00} & = &
\frac{1}{\sqrt{1+\lambda_\sigma^2}} \big< r(e_0,e_0) \xi,f_\sigma \big>
,\\[1ex] \tilde \Omega_{\sigma | \alpha \alpha} & = &
\frac{1}{\sqrt{1+\lambda_\sigma^2}} \big[ \big< r(e_\alpha,e_\alpha) \xi,
f_\sigma \big> + \lambda_ \sigma \lambda_ \alpha \big<
R(e_\sigma,e_\alpha) \xi, f_\alpha \big> \big].
\end{array}
$$
Taking (\ref{n11})into account, we have:
$$ H_\sigma |
=\frac{1}{(n+1)}\tilde G^{ii} \tilde \Omega_{\sigma | ii} = $$
$$ \frac{1}{(n+1)\sqrt{1+\lambda_\sigma^2}} \left\{ \big<
r(e_0,e_0)\xi,f_\sigma \big>+ \sum_{\alpha=1}^{n} {\frac{ \big<
r(e_\alpha,e_\alpha) \xi,f_\sigma +\lambda_\sigma \lambda_\alpha
R(e_\sigma,e_\alpha) \xi,f_\alpha \big>} {1+\lambda_\alpha^2}}
\right\}. $$
So we get the result.
\end{proof}
\subsubsection{Simplified formula for the mean curvature of a unit
vector field.}
It is possible to simplify the formula (\ref{H}). To do this, we introduce
the following notations:
$$
E_{i| j k}=\big<\nabla_{\displaystyle e_i}e_j,e_k\big>, \quad
F_{i| j k}=\big<\nabla_{\displaystyle e_i}f_j,f_k\big>,
$$
where $f_0$ is supposed to be zero. Evidently, $E_{i| j k}=-E_{i| kj}$
and $F_{i| j k}=-F_{i| kj}$.
Then it is simple to check that
$$
\big<r(e_i,e_j)\xi,f_k\big>=e_i(\lambda_j)\delta_{jk}+
\lambda_jF_{i| jk}-\lambda_kE_{i| jk}.
$$
Therefore,
$$
\begin{array}{l}
\big<r(e_j,e_j)\xi,f_i\big>=e_j(\lambda_j)\delta_{ij}
+\lambda_jF_{j| ji}-\lambda_i E_{j| ji}, \\[1ex]
\big<r(e_i,e_j)\xi,f_j\big>=e_i(\lambda_j), \\[1ex]
\big<r(e_i,e_j)\xi,f_i\big>=e_i(\lambda_j)\delta_{ij}+\lambda_j F_{i| ji}-
\lambda_i E_{i| ji}
\end{array}
$$
From this it follows that
$$
\begin{array}{l}
\big<R(e_i,e_j)\xi,f_j\big>=\big<r(e_i,e_j)\xi,f_j\big>-\big<r(e_j,e_i)\xi,f_j\big>=\\[1ex]
e_i(\lambda_j)-e_i(\lambda_j)\delta_{ij}-\lambda_j F_{i| ji}+
\lambda_i E_{i| ji} =\\[1ex]
e_i(\lambda_j) - e_j(\lambda_j)\delta_{ij}-\lambda_jF_{j| ji}+\lambda_i E_{j| ji}+
(\lambda_i+\lambda_j)(E_{j| ij}-F_{j| ij})= \\[1ex]
e_i(\lambda_j)-\big<r(e_j,e_j)\xi,f_i\big>-
(\lambda_i+\lambda_j)(E_{j| ji}-F_{j| ji}).
\end{array}
$$
So, we see that
$$
\big<r(e_j,e_j)\xi,f_i\big>=e_i(\lambda_j)-
(\lambda_i+\lambda_j)(E_{j| ji}-F_{j| ji})
-\big<R(e_i,e_j)\xi,f_j\big>.
$$
Finally, introducing the matrix $G_{i| j}$ with the components
$$
G_{i| j}=E_{i| ij}-F_{i| ij},
$$
we can rewrite the mean curvature formula as follows
\begin{equation}\label{SH}
\begin{array}{c}
\displaystyle (n+1)H_{\sigma |}=\\[2ex]
\displaystyle \frac{1}{\sqrt{1+\lambda_\sigma^2}}
\sum_{i=0}^n\frac{e_\sigma(\lambda_i)-(\lambda_i+\lambda_\sigma)G_{i| \sigma}
+(\lambda_i\lambda_\sigma-1)\big<R(e_\sigma,e_i)\xi,f_i\big>}
{1+\lambda_i^2},
\end{array}
\end{equation}
where $\lambda_0=0$ and $f_0=0$ is supposed.
\section{Some special cases and examples}
\subsection{Normal vector field of a Riemannian foliation}
We consider an important special case of a unit {\it geodesic } vector field
$\xi$ such that the orthogonal distribution $\xi^\perp$ is integrable.
In other words, suppose that a given Riemannian manifold admits a Riemannian
transversally orientable hyperfoliation. Then the following holds.
\begin{theorem} Let $M^{n+1}$ admit a Riemannian transversally orientable
hyperfoliation. Let $\xi$ be a unit normal vector field of the foliation. Then
the components of the mean curvature vector of $\xi(M)$ are
$$
H_{\sigma |}=\frac{1}{(n+1)\sqrt{1+k_\sigma^2}}
\sum_{\alpha=1}^{n}\left\{\frac{-e_\sigma(k_\alpha)+
(1-k_\alpha k_\sigma)\big<R(\xi,e_\alpha)e_\alpha,e_\sigma\big>}{1+k_\alpha^2}
\right\}
$$
where $e_\alpha$ determine the principal directions and $k_\alpha $ are the principal
curvatures of the fibers.
\end{theorem}
\begin{remark} \rm The analogous problem was treated in \cite{BX-V1}, where the
authors considered the {\it minimality} condition for the vector
field. The corresponding conditions in \cite{BX-V1} differ from the mean curvature
components by a factor. We refer to \cite{BX-V4} for applications of this conditions.
\end{remark}
\begin{proof} For the given situation, the singular frame is simple. As $\xi$ is geodesic
vector field, we have $e_0=\xi $, while the others are principal vectors of the second
fundamental form of the fibers. If we denote the corresponding shape operator by $A_\xi$,
then
$$
\nabla_{e_\alpha}\xi=-A_\xi e_\alpha=-k_\alpha e_\alpha
$$
So, neglecting the condition on the $\lambda_\alpha $ to be {\it positive} (in fact, we never used
this condition
in proof of the formula (\ref{SH})), we may put $f_\alpha=e_\alpha $ and $\lambda_\alpha=-
k_\alpha$.
Therefore, in (\ref{SH}) we obtain $G_{i| j}=0$ and the result follows immediately.
\end{proof}
\subsection{Strongly normal vector field.}
A unit vector field $\xi$ is called {\it normal} if
$R(X,Y)\xi=\alpha \xi $ and {\it strongly normal} if
$r(X,Y)\xi=\alpha \xi$ for all $X,Y \in \xi^\perp $. Our result
(\ref{H}) allows to prove easily \cite{GD-V1}:
\vspace{1ex}
{ \it Every unit strongly normal geodesic vector
field is minimal}
\vspace{1ex}
Indeed, since $\xi$ is geodesic, $\nabla_\xi\xi=0$ and therefore
$e_0=\xi$. Hence, $r(e_0,e_0)\xi=0$ and $e_1, \dots , e_{n} \in
\xi^\perp , \quad f_1, \dots , f_{n} \in \xi^\perp$. Evidently, a
strongly normal vector field is always normal. So, each term in
(\ref{H}) vanishes.
\subsection{Geodesic vector fields on 2-dimensional manifolds}
For $dim M=2$ the mean curvature of $\xi(M) \subset T_1M$ equals
$$
H= \frac{1}{2 \sqrt{1+\lambda^2}} \left\{ \big< r(e_0,e_0) \xi +
\frac{r(e_1,e_1) \xi}{1+\lambda^2},f_1 \big> \right\}
$$
or
\begin{equation}\label{H_2}
H=\frac{1}{2\sqrt{1+\lambda^2}}\left\{-\big<\nabla_{e_0}e_0,e_1\big>
\lambda+\frac{e_1(\lambda)}{1+\lambda^2}\right\}.
\end{equation}
The above formula allows to prove the following statement.
\vspace{1ex}
{\it A unit geodesic vector field on a 2-dimensional manifold is
minimal if and only if it is strongly normal} (see \cite{GD-V1}).
\vspace{1ex}
Indeed, in this case we can set $e_0=\xi$, $f_1=\pm e_1$. So,
up to a sign,
$$
H=\frac{1}{2(1+\lambda^2)^{3/2}}\big<r(e_1,e_1)\xi,e_1\big>
$$
and the statement follows immediately.
In \cite{GD-V1}, the authors give an example of a geodesic but not
strongly normal vector field and hence not minimal. Here we can
easily find the mean curvature of that field. Namely, consider the
2-dimensional manifold of non-positive curvature with metric
$$
ds^2=du^2+e^{2uv}dv^2.
$$
Set $\xi=\{1,0\}$. Then, up to a sign, the
singular frame is
$$
e_0=\xi \mbox{ and } e_1=\{0,e^{-uv}\}=f_1.
$$
It is easy to see that $$ \nabla _{e_1}\xi=ve_1. $$ Hence
$\lambda=v$ and $e_1(\lambda)=e^{-uv}$.
So, the mean curvature of $\xi(M)$ is given by
$$
H=\frac{e^{-uv}}{2(1+v^2)^{3/2}}.
$$
\vspace{1ex}
\subsection{Examples of non-geodesic minimal vector fields on some 2-dimensional
Riemannian manifolds}
Next, we consider a Riemannian 2-manifold $M$ with the metric
$$
ds^2=du^2+e^{2g(u)}dv^2.
$$
As it was shown in \cite{GD-V1} for
the general situation, the vector field $\partial/\partial u$ is
minimal. Here we shall consider the vector field which makes a
constant angle with $\partial/\partial u$ along each $u$ - geodesic.
\begin{proposition}Up to a sign,
the mean curvature of the vector field $\xi$ on a 2-dimensional
Riemannian manifold with metric $ ds^2=du^2+e^{2g(u)}dv^2$ which
is parallel along each $u$ - geodesic, is
$$
H=\frac{e^{-2g}\omega_{vv}}{2\Big(1+(e^{-g}\omega_v+g')^2\Big)^{3/2}},
$$
where $\omega(v)$ is the angle function of $\xi$ with respect
to the direction of $u$ - geodesics.
\end{proposition}
\begin{proof}
Consider the mutually orthogonal unit vector fields $ X_1= \{ 1,0
\} $ and $ X_2= \{ 0,e^{-g} \} $. A direct calculation gives $$
\begin{array}{lclclcl}
\nabla_{X_1}X_1 & = & 0, & & \nabla_{X_1}X_2 & = & 0 ,\\
\nabla_{X_2}X_1 & = & g' X_2, & & \nabla_{X_2}X_2 & = & -g'X_1.
\end{array}
$$ Let $\omega (u,v)$ be the angle function defining the vector
field $\xi$ by $$ \xi=\cos \omega X_1 + \sin \omega X_2 $$ Let
$\eta $ be a unit vector field orthogonal to $\xi$: $$ \eta=-\sin
\omega X_1 + \cos \omega X_2. $$ Then $$ \nabla_{X_1}\xi =
X_1(\omega) \eta , \mbox{ }
\nabla_{X_2}\xi=-(X_2(\omega)+g')\eta. $$
Now, suppose $\xi$ to be parallel along a $u$ - geodesic, that is,
set $X_1(\omega)=0$. Then the singular frame is : $e_0=X_1$ and $
e_1=X_2$. The singular function is $\lambda=-(X_2(\omega)+g')$ and
we see that, up to a sign, $f_1$ coincides with $\eta$. So
$$
H=\frac{e_1(\lambda )}{2(1+\lambda^2)^{3/2}}.
$$
For $e_1(\lambda)$ we obtain
$$
e_1(\lambda)=X_2(-X_2(\omega)+g')=-X_2(X_2(\omega))+X_2(g')=-e^{-2g}\omega_{vv}
$$
since $g$ does not depend on $v$. Therefore
$$
H=\frac{e^{-2g}\omega_{vv}}{2\Big(1+(e^{-g}\omega_v+g')^2\Big)^{3/2}},
$$
what was claimed.
\end{proof}
From the above formula we conclude:
\vspace{1ex}
{\it On a 2-dimensional manifold with metric $ds^2=du^2+e^{2g(u)}dv^2$ the unit
vector field $\xi$which is parallel along $u$ -- geodesics, is minimal if its angle
increment along $v$ -- curves is not higher then the linear one. }
\vspace{1ex}
Particularly, if $\omega=const$, then $\xi$ is minimal.
\subsection{The mean curvature of a general unit vector field on
2-dimensional manifolds}
In the case of $dim M=2,$ the mean curvature of a unit vector field can
be expressed in terms of the geodesic curvature of integral curves
of the given field and their orthogonal trajectories.
\begin{proposition}
Let $\xi$ and $\eta$ be unit mutually orthogonal vector fields
on a 2-dimensional Riemannian manifold. Denote by $k$ and $\kappa$
the geodesic curvatures of the integral curves of the field $\xi$
and $\eta$, respectively. The mean curvature $H$ of the vector
field $\xi$ is given, up to a sign, by
$$
H=\frac12\left[\xi\left(\frac{k}{\sqrt{1+k^2+\kappa^2}}\right)-
\eta\left(\frac{\kappa}{\sqrt{1+k^2+\kappa^2}}\right)
\right].
$$
\end{proposition}
\begin{remark} \rm The analogous expression can be found in
\cite{GM-LF} as a condition of minimality of the unit vector field
on 2-dimensional manifolds.
\end{remark}
\begin{proof}
From (\ref{H_2}) one can see that after the replacement $\xi\to -\xi$
the mean curvature $H$ just changes its sign. Therefore, we may
choose the direction of $\xi$ in such a way that it will be the
field of principal normals of the $\eta $ -- curves. The same arguments
allow us to consider $\eta$ as the field of principal normals
of the $\xi$ -- curves. Denote by $\omega$ an angle between $\xi$ and the
field $e_0$ of the singular frame. Then
$$ e_0=\cos\omega\xi+ \sin\omega\eta. $$
As $\nabla_{e_0}\xi=0$, we have
$$\cos\omega\nabla_\xi\xi+\sin\omega\nabla\eta\xi=0.$$
The Frenet formulas give
$$\nabla_\xi\xi=k\eta, \quad \nabla_\eta\xi=-\kappa\eta.$$
Therefore, we obtain
\begin{equation}
\label{par}
k\cos\omega-\kappa\sin\omega=0.
\end{equation}
Denote by $e_1$ and $f_1$ the other vectors of the singular
frame. It is easy to check that the change of directions of these
vectors induces a sign change of $H$. Therefore, we can always
set $f_1=\eta$ and $e_1=\pm\sin\omega\xi \mp \cos\omega\eta$
to satisfy the equation $\nabla_{e_1}\xi=\lambda f_1$ with
$\lambda\geq 0$. Taking all of this into account, set
$$
\begin{array}{c}
e_0=\cos\omega\xi+\sin\omega\eta, \\
e_1=\sin\omega\xi-\cos\omega\eta.
\end{array}
$$
Then we have
$$
\begin{array}{l}
\nabla_{e_0}\xi=\cos\omega\nabla_\xi\xi
+\sin\omega\nabla_\eta\xi=0, \\[1ex]
\nabla_{e_1}\xi=\sin\omega\nabla_\xi\xi-\cos\omega\nabla_\eta\xi=\lambda\eta.
\end{array}
$$
From these equations we derive
$$
\begin{array}{l}
\nabla_\xi\xi=\lambda\sin\omega \,\eta,\\[1ex]
\nabla_\eta\xi=-\lambda\cos\omega\,\eta.
\end{array}
$$
Comparing this with the Frenet formulas, we conclude that
$k=\lambda\sin\omega,\ \kappa=\lambda\cos\omega$. Therefore,
\begin{equation}
\label{om}
\lambda^2=k^2+\kappa^2,\quad \sin\omega=\frac{k}{\lambda},
\quad \cos\omega=\frac{\kappa}{\lambda}
\end{equation}
To use the formula (\ref{H_2}), we should find $e_1(\lambda)$ and
$\big<\nabla_{e_0}e_0,e_1\big>$. Now, keeping in mind (\ref{par}), we have
$$ e_1(\lambda)
=\frac{k}{\lambda}\xi(\lambda)-\frac{\kappa}{\lambda}\eta(\lambda)
$$
and
$$
\begin{array}{rl}\displaystyle
\nabla_{e_0}e_0=&\cos{\omega}\nabla_\xi(\cos{\omega}\,\xi+\sin{\omega}\,\eta)
+\sin{\omega}\nabla_\eta(\cos{\omega}\,\xi
+\sin{\omega}\,\eta)=\\\displaystyle
&-(\xi(\omega)\cos\omega+\eta(\omega)\sin\omega)e_1-
(k\cos\omega-\kappa\sin\omega)e_1=\\\displaystyle
&-\big(\xi(\sin\omega)-\eta(\cos\omega)\big)e_1.
\end{array}
$$
Therefore, using (\ref{om}), we get
$$
-\big<\nabla_{e_0}e_0,e_1\big>=\xi\left(\frac{k}{\lambda}\right)-
\eta\left(\frac{\kappa}{\lambda}\right).
$$
Substituting these expressions into (\ref{H_2}), we
obtain
$$
\begin{array}{l}
\displaystyle H=\!\frac12\frac{1}{\sqrt{1+\lambda^2}}
\left[\left(\xi\Big(\frac{k}{\lambda}\Big)-
\eta\Big(\frac{\kappa}{\lambda}\Big)\right)\lambda+
\frac{1}{1+\lambda^2}\left(\frac{k}{\lambda}\xi(\lambda)-
\frac{\kappa}{\lambda}\eta(\lambda)\right)\right]=\\[2ex]\displaystyle
\qquad\frac{1}{2}\frac{1}{(1+\lambda^2)^{3/2}}
\left[\big((1+\lambda^2)\,\xi(k)-k\lambda\,\xi(\lambda)\big)-
\big((1+\lambda^2)\,\eta(\kappa)-\kappa\lambda\,\eta(\lambda)\big)\right]=\\[3ex]\displaystyle
\qquad\frac12\left[\xi\left(\frac{k}{\sqrt{1+\lambda^2}}\right)-
\eta\left(\frac{\kappa}{\sqrt{1+\lambda^2}}\right)\right].
\end{array}
$$
Taking into account (\ref{om}), we get what was claimed.
\end{proof}
\vspace{1ex}
{\bf Corollary. }{\it If $\xi$ is a geodesic vector field then
$$
H=-\frac12\frac{\partial}{\partial\sigma}\left(\frac{\kappa}{\sqrt{1+\kappa^2}}\right)
$$
where $\sigma$ is the arc-length parameter of the orthogonal trajectories of the field $\xi$
and $\kappa$ is their geodesic curvature.}
\vspace{1ex}
A unit geodesic vector field is said to be {\it radial} if it is a tangent vector field of
geodesics starting at a fixed point. Now we can confirm the following statement \cite{BX-V1}.
\begin{proposition}
If each radial vector field on a 2-dimensional Riemannian
manifold $M$ is minimal, then $M$ has constant curvature.
\end{proposition}
\begin{proof}
Indeed, if such a vector field is minimal, then its orthogonal
trajectories are Gauss circles of constant geodesic curvature,
which means that those circles are Darboux ones. Therefore,
$M$ is of constant Gaussian curvature ( see \cite{Bl}).
\end{proof}
\subsection{Some examples of vector fields of constant mean curvature.}
\subsubsection{The example on the Lobachevsky 2-space.}
Consider the Lobachevsky plane $L^2$ with the metric
$$
ds^2=du^2+e^{2u}dv^2.
$$
The coordinate lines of $L^2$ are $u$ -geodesics and their orthogonal trajectories.
\begin{proposition}
\label{Ex}
The unit vector field on $L^2$ whose angle function with respect to
$u$ - geodesics is $\omega=au+b \ (a,b=const)$ has constant mean
curvature
$$
H=\frac{a}{2\sqrt{2+a^2}}.
$$
\end{proposition}
\begin{proof}
Indeed, consider the field $\xi=\cos\omega X_1+\sin\omega X_2$ where
$\omega=au+b$ and $X_1=\{1,0\}, \ X_2=\{0,e^{-u}\}$. Then
$$
\begin{array}{ll}
\nabla_{X_1}X_1=0, & \nabla_{X_1}X_2=0, \\[1ex]
\nabla_{X_2}X_1=X_2, & \nabla_{X_2}X_2=-X_1.
\end{array}
$$
Now we define the singular frame for $\xi$. To do this, we
introduce the vector field $\eta=-\sin\omega X_1+\cos\omega X_2$.
Then
$$
\begin{array}{l}
\nabla_{X_1}\xi=\frac{\partial\omega}{\partial u}\eta=a\eta ,\\[1ex]
\nabla_{X_2}\xi=\eta.
\end{array}
$$
Therefore, setting
$$
e_0=\frac{1}{\sqrt{1+a^2}} ( X_1-aX_2), \ \ \
e_1=\frac{1}{\sqrt{1+a^2}} (a X_1+X_2),
$$
we have
$$
\nabla_{e_0}\xi=0, \ \ \ \nabla_{e_1}\xi=\sqrt{1+a^2}\eta.
$$
Hence, $f_1=\eta$ and $ \lambda=\sqrt{1+a^2}=const$. So,
$e_1(\lambda)=0$. Moreover,
$$
\nabla_{e_0}e_0=-\frac{a}{\sqrt{1+a^2}}\,e_1.
$$
Substituting this into (\ref{H_2}), we have
$$
H=\frac{a}{2\sqrt{2+a^2}}.
$$
So, the statement is proved.
\end{proof}
\subsubsection{The generalized examples on the Lobachevsky $(n+1)$- space.}
Consider the $(n+1)$ - dimensional Lobachevsky space endowed with
horospherical coordinates $( u, v^1, \dots, v^n)$. Then
$$
ds^2=du^2+e^{2u} [(dv^1)^2+ \dots + (dv^n)^2 ].
$$
Consider the unit vector fields
\begin{equation}\label{base}
X_0=\{1,0,\dots, 0\}, X_1=\{0,e^{-u},\dots,0\},\dots, X_n=\{0,0,\dots,e^{-u}\}.
\end{equation}
It is easy to check that
$$
\begin{array}{ll}
\nabla_{X_{\scriptstyle 0}}X_0=0, &\nabla_{X_{\scriptstyle 0}}X_{\alpha}=0,\\
\nabla_{X_{\scriptstyle \alpha}}X_0=X_{\alpha} &
\nabla_{X_{\scriptstyle\alpha}}X_\alpha=-X_0.
\end{array}
$$
Define the unit vector field $\xi$ as follows:
\begin{equation}\label{vf1}
\xi=\cos\theta X_0+\sin\theta\cos u X_1+\sin\theta\sin uX_2,
\end{equation}
where $\theta\in [0, \pi/2]$ is constant.
\begin{proposition}\label{Lob_n}
The unit vector field which is given by (\ref{vf1}) with respect to the
frame (\ref{base}) on Lobachevsky $(n+1)$ - space with the metric
$$
ds^2=du^2+e^{2u} [(dv^1)^2+ \dots + (dv^n)^2 ],
$$
is a field of constant mean curvature. Namely, we have
$$
\begin{array}{l}
H_{1|}=\displaystyle\frac{n-2}{n+1}\frac{\sqrt{2}\sin\theta\cos\theta}{1+\cos^2{\theta}},\\[2ex]
H_{2|}=\displaystyle\frac{n\sqrt{2}\sin\theta}{2(n+1)}, \\[2ex]
H_{\sigma|}=0 \quad \sigma\geq 3.
\end{array}
$$
\end{proposition}
\begin{proof}
With respect to the frame $\{X_0,X_1,\dots,X_n\}$, the matrix
$(\nabla\xi)$ has the form
$$
\left[
\begin{array}{cccccc}
0 & -\sin\theta\cos u & -\sin\theta \sin u &0&\dots&0\\
-\sin\theta\sin u &\cos\theta & 0 &0&\dots&0\\
\sin\theta\cos u & 0 & \cos\theta &0&\dots&0\\
0 & 0 & 0 &\cos\theta & \dots&0 \\
\vdots &\vdots & \vdots &0&\ddots&0\\
0 &0 & 0 &0&\dots &\cos\theta
\end{array}
\right].
$$
It is easy to find that the matrix $(\nabla\xi)^t(\nabla\xi)$ has the following
expression
$$
\left[
\begin{array}{cc}
A&0\\
0&B
\end{array}
\right],
$$
where $A$ is the $3\times 3$ matrix
$$
\left[
\begin{array}{ccc}
\sin^2\theta & -\sin\theta\cos\theta\sin u & \sin\theta\cos\theta\cos u\\
-\sin\theta\cos\theta\sin u &\cos^2\theta +\sin^2\theta\cos^2u
&\sin^2\theta\sin u\cos u \\
\sin\theta\cos\theta\cos u & sin^2\theta\sin u\cos u & \cos^2\theta
+\sin^2\theta\sin^2(u)\\
\end{array}
\right]
$$
and $B$ is the diagonal $(n-2)\times(n-2)$ matrix of the form
$$
\left[
\begin{array}{ccc}
\cos^2\theta &\dots&0\\
\vdots &\ddots&\vdots \\
0 & \dots &\cos^2\theta
\end{array}
\right].
$$
The eigenvalues of the matrix $(\nabla\xi)^t(\nabla\xi)$ are
$$
\lambda_0^2=0, \lambda_1^2=\lambda_2^2=1,
\lambda_3=\dots=\lambda_n^2=cos^2\theta.
$$
Now it is easy to find the vectors of the singular frame. We get
$$
\begin{array}{l}
\begin{array}{ccl}
e_0&=&\cos\theta X_0+\sin\theta\sin uX_1-\sin\theta\cos uX_2, \\
e_1&=&\cos uX_1+\sin uX_2, \\
e_2&=&\sin\theta X_0-\cos\theta \sin uX_1+\cos\theta \cos uX_2, \\
\end{array}\\
\ \, e_3= X_3,\dots , e_n= X_n
\end{array}
$$
and
$$
\begin{array}{l}
\begin{array}{ccl}
f_1&=&-\sin\theta X_0+\cos\theta \cos u X_1+\cos\theta\sin uX_2,\\
f_2&=&-\sin u X_1+\cos uX_2, \\
\end{array}\\
\ \, f_3=e_3, \dots , f_n =e_n.
\end{array}
$$
So, we have
$$
\begin{array}{lcl}
\nabla_{\displaystyle e_0}\xi=0, & \nabla_{\displaystyle e_1}\xi=f_1,
& \nabla_{\displaystyle e_2}\xi= f_2,\\[1ex]
\nabla_{\displaystyle e_3}\xi=\cos\theta f_3, & \dots
& \nabla_{\displaystyle e_n}\xi=\cos\theta f_n
\end{array}
$$
Straightforward computation gives the following components for the
matrix $G_{i| j}$:
$$
\left[
\begin{array}{cccccc}
0 & \sin\theta\cos\theta & -\sin\theta & 0 & \dots & 0 \\
-\cos\theta & 0 & -\sin\theta & 0 & \dots & 0 \\
-\cos\theta & -\sin\theta\cos\theta & 0 & 0 &\dots & 0 \\
-\cos\theta & -\sin\theta & -\sin\theta &0 & \dots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-\cos\theta & -\sin\theta & -\sin\theta &0 & \dots & 0
\end{array}
\right].
$$
As all $ \lambda_i$ are constants, we have
$$
\begin{array}{cl}
H_{1|}=&\displaystyle\frac{1}{(n+1)\sqrt{1+\lambda_1^2}}
\sum\limits_{i=0}^{n}\frac{-(\lambda_1+\lambda_i)G_{i| 1} +
(\lambda_i-\lambda_1)\big<R(e_1,e_i)\xi,f_i\big>}{1+\lambda_i^2}=\\[3ex]
&\displaystyle \frac{1}{(n+1)\sqrt{2}}\left[
\sum\limits_{i=0}^2 (-G_{i| 1})+\sum\limits_{i=3}^n
\frac{-(1+\lambda_i)G_{i| 1}+(\lambda_i-1)\big<\xi,e_1\big>}
{1+\cos^2\theta}\right] = \\[3ex]
&\displaystyle\frac{1}{(n+1)\sqrt{2}}\left[0+(n-2)\frac{(1+\cos\theta \sin\theta+(\cos\theta -
1)\sin\theta} {1+\cos^2\theta}\right]= \\[3ex]
&\displaystyle\frac{n-2}{n+1}\frac{\sqrt{2}\sin\theta\cos\theta }{1+\cos^2\theta }.
\end{array}
$$
Analogously, we get
$$
\begin{array}{cl}
H_{2|}=&\displaystyle\frac{1}{(n+1)\sqrt{1+\lambda_2^2}}
\sum\limits_{i=0}^{n}\frac{-(\lambda_2+\lambda_i)G_{i| 2} +
(\lambda_i-\lambda_2)\big<R(e_2,e_i)\xi,f_i\big>}{1+\lambda_2^2}=\\[3ex]
&\displaystyle\frac{1}{(n+1)\sqrt{2}}\left[
\sum\limits_{i=0}^2 (-G_{i| 2})+\sum\limits_{i=3}^n
\frac{-(1+\lambda_i)G_{i| 2}+(\lambda_i-1)\big<\xi,e_2\big>}
{1+\cos^2\theta}\right] = \\[3ex]
&\displaystyle\frac{\sqrt{2}}{2(n+1)}\left[2\sin\theta+(n-2)
\frac{(1+\cos\theta )\sin\theta+(\cos\theta -1)\sin\theta\cos\theta }
{1+\cos^2\theta}\right]=\\[3ex]
&\displaystyle\frac{\sqrt{2}}{2(n+1)}\left[2\sin\theta+(n-2)
\frac{\sin\theta+\sin\theta\cos^2\theta}
{1+\cos^2\theta}\right]=
\frac{n\sqrt{2}\sin\theta}{2(n+1)}.
\end{array}
$$
and $H_{\sigma|}=0$ for all $\sigma\geq 3$.
\end{proof}
A similar but more complicated computation shows that there exist a
family of vector fields of constant mean curvature on the Lobachevsky space.
Namely, let $\xi$ be a vector field given by
\begin{equation} \label{vf2}
\xi=\cos\theta X_0+\sin\theta\cos{au} X_1+\sin\theta\sin{au} X_2,
\end{equation}
where $a$ and $\theta$ are constants and the frame $X_0, X_1, \dots , X_n$
is chosen as above. Then the following statement is true.
\begin{proposition}\label{Lob_n2}
The unit vector field which is given by (\ref{vf2}) with respect to the
frame (\ref{base}) on the Lobachevsky $(n+1)$ - space with the metric
$$
ds^2=du^2+e^{2u} [(dv^1)^2+ \dots + (dv^n)^2 ],
$$
is a field of constant mean curvature. Namely, we have
$$
\begin{array}{l}
H_{1|}=\displaystyle\frac{\sqrt{2}\sin\theta\cos\theta }{n+1}
\left(\frac{1-a^2}{1+\cos^2\theta+a^2\sin^2\theta}+\frac{n-
2}{1+\cos^2\theta}\right),\\[2ex]
H_{2|}=\displaystyle\frac{an\sin\theta}{(n+1)\sqrt{1+\cos^2\theta+a^2\sin^2\theta}},\\[2ex]
H_{\sigma|}=0 \quad \sigma\geq 3.
\end{array}
$$
\end{proposition}
The proof is based on the fact that the singular values of $(\nabla\xi)$
are the following constants:
$$
\lambda_1=1, \lambda_2=\sqrt{\cos^2\theta+a^2\sin^2\theta}, \lambda_3=
\ldots =\lambda_n=\cos\theta .
$$
{\large Acknowledgement.} The author expresses his thanks to
P.Nagy who invited him to take part in a fruitful workshop
(Debrecen, 2000) on the geometry of tangent sphere bundle. The talks with L.Vanhecke and
E.Boeckx gave the starting impulse to the article. The author
also thanks A.Borisenko who was the first who asked on
examples of vector fields of constant mean curvature.
|
{
"timestamp": "2005-03-24T22:24:41",
"yymm": "0503",
"arxiv_id": "math/0503567",
"language": "en",
"url": "https://arxiv.org/abs/math/0503567"
}
|
\section{Introduction}
Very much physics is sometimes contained in simple and basic results
of optics and electromagnetics. In this paper I shall focus on the
character of electromagnetic waves reflected from a planar surface. As
is well known, many everyday light phenomena that we can observe with
plain eyes \cite{Minnaert} can be justified and explained with basic
wave theory which is is being taught to freshmen in physics and
engineering schools. As examples in optics we could mention the glare
on road surfaces on a sunny day which can be reduced by use of
Polaroid sun glasses, or the way the images reflected from a water
surface differ from those that are direcly observed.
The polarization state of light changes in refrection and refraction
processes. Since our eyes are not capable of sensing polarization, and
natural light very often is rather unpolarized, the subtleties of the
outdoor images, as they appear to us, may only be present in very
indirect ways. But one especially interesting phenomenon in this
respect is the possibility of light to become fully polarized in
reflection. This happens when light impinges on a surface in a certain
direction, from the Brewster angle. In the following, let us
concentrate on the dependence of Brewster angle on the fundamental
material parameters. In particular, the emphasis shall be on the way
how the Brewster angle can be visualized in a geometrical way which
contains pedagogical and physical insight.
In the following, the materials to be analyzed are assumed isotropic
and lossless. However, in one respect the analysis is more general
than that encountered in basic textbooks in optics which often
restrict the treatment to non-magnetic media: here also magnetic
permeability is taken as a material parameter that can vary. Presently
in many engineering applications, composite materials research, and
nanotechnology, great interest is in the magnetic properties of
matter, which gives motivation to allow magnetic contrasts in the
studies of canonical problems.
Hence, if both electric and magnetic responses are present, the
material from which the wave reflects is characterized by two
parameters, the relative permittivity and permeability $\epsilon$ and
$\mu$. These are assumed in the present paper to be real and positive
\footnote{It is perhaps important to emphasize here the explicit assumption
of positiveness of the material parameters. In recent years very much
research has been and is still being focused on materials with
negative permittivity and permeability values, so-called
negative-phase-velocity media, left-handed media, or metamaterials
\cite{Veselago,Pendry}. Large research programs have been launched in
the U.S.\ and in Europe which target on design and exploitation of
metamaterials; see, for example, {\tt
http://www.darpa.mil/dso/thrust/matdev/metamat.htm} and {\tt
http://www.metamorphose-eu.org}}. But to ease the analysis, instead of
using these parameters, it appears more convenient to apply the
refractive index $n$ and relative impedance $\eta$ of the material:
\begin{equation}
n = \sqrt{\epsilon\mu}, \quad \eta = \sqrt{\mu/\epsilon}
\end{equation}
Obviously the inverse relations are $\epsilon=n/\eta$ and $\mu=n\eta$.
The following sections give the reflection coefficients from such a
material and a way to visualize them.
\section{Reflection coefficients}
The geometry of the problem to be analyzed is very simple and shown in
Figure~\ref{fig:iim1}. An incident electromagnetic wave is impinging
from free space and faces a planar interface. On the other side of the
boundary, there is a homogeneous half space of dielectric--magnetic
medium with refractive index and impedance parameters $n$ and
$\eta$. After the collision with the boundary, part of the energy is
refracted and penetrates into the medium, and the remaining part
reflects away form the interface.
\begin{figure}[h]
\psfragscanon
\psfrag{t1}[][]{{$\theta_1$}}
\psfrag{t2}[][]{{$\theta_2$}}
\psfrag{nh}[][]{{$n,\quad \eta$}}
\centerline{\includegraphics[width=8cm]{iim1.eps}}
\caption{Plane wave hitting a boundary between free space and a
dielectric--magnetic material with refractive index $n$ and
impedance $\eta$.}
\label{fig:iim1}
\end{figure}
In general, the wave changes its polarization state in
reflection. Only for two eigenpolarizations of the incident wave do
the reflected and refracted waves remain with the same polarization as
the incoming wave. These two are parallel (P) and perpendicular (S)
polarizations, meaning that the linearly polarized electric field
vector is in the plane of incidence (P) or perpendicular to it
(S). The plane of incidence is spanned by the incident wave direction
and the normal of the interface (the plane of paper in
Figure~\ref{fig:iim1}).
The reflection coefficients for the two polarizations can be written
in many equivalent forms \cite{Jackson,Born_Wolf}; the following
electric field Fresnel coefficients are quite symmetric:
\begin{eqnarray}
R_{\rm P} & = & \frac{\eta \cos\theta_2 - \cos\theta_1}{\eta \cos\theta_2 +
\cos\theta_1} \label{1} \\
R_{\rm S} & = & \frac{\eta \cos\theta_1 - \cos\theta_2}{\eta
\cos\theta_1 + \cos\theta_2} \label{2}
\end{eqnarray}
In using these formulas, the value for the refraction angle
$\theta_2$ is needed. It is determined by the Snell's law
\begin{equation}\label{3}
\sin\theta_1 = n \sin\theta_2
\end{equation}
These expressions give the reflected electric field vector for unit
incident field. The magnitudes of the reflection coefficients are
always between zero and unity. Note, however, that the reflection
coefficients can attain complex values even in the case of real values
for $n$ and $\eta$; this happens for total internal reflection with the
associated Goos--H\"anchen phenomenon.
Of course, very interesting is the case when the reflection
vanishes. It is easy to solve from (\ref{1})--(\ref{3}) the incidence
angle for which the reflection coefficient is zero. This is called the
Brewster angle, and it is for the parallel polarization
\begin{equation}\label{BrP}
\theta_{\rm Br,P} = \arcsin \left( n \sqrt{\frac{1-\eta^2}{n^2-\eta^2}} \right)
\end{equation}
For the perpendicular polarization the Brewster angle can be written as
\begin{equation}\label{BrS}
\theta_{\rm Br,S} = \arcsin \left( n \sqrt{\frac{\eta^2-1}{n^2\eta^2-1}} \right)
\end{equation}
Note that only for one polarization there exists a Brewster angle; the
requirements are (see Figure~\ref{fig:plane1})
\begin{itemize}
\item Parallel polarization: $n>1$ and $\eta<1$, or $n<1$ and $\eta>1$
\item Perpendicular polarization: $n>1$ and $\eta>1$, or $n<1$ and $\eta<1$
\end{itemize}
\begin{figure}[h!]
\psfragscanon
\psfrag{n}[][]{{$n$}}
\psfrag{h}[][]{{$\eta$}}
\psfrag{1}[][]{{$1$}}
\psfrag{P}[][]{{{\tt P}}}
\psfrag{S}[][]{{\tt S}}
\centerline{\includegraphics[width=8cm]{plane1.eps}}
\caption{Regions of the ($n$-$\eta$)-plane where the Brewster angle
can be observed for parallel and perpendicular polarizations.}
\label{fig:plane1}
\end{figure}
Note that the expression (\ref{BrP}) is a generalization from the
familiar Brewster-angle relation $\tan\theta_{\rm Br,P}=n$ which is valid
for non-magnetic media $(\eta=1/n)$, and naturally only exists for the
parallel polarization. When magnetic response is allowed, the relation
for the polarizing angle has one more degree of freedom. It can be
written, of course, also in forms other than (\ref{BrP})--(\ref{BrS}),
see, for example \cite{Futterman}.
An interesting observation is that the Brewster angle can attain any
values between zero and 90$^\circ$, as can be seen from
Figure~\ref{fig:mmm30} in case of parallel polarization. Note that
for ordinary dielectric materials where $n=1/\eta$ the Brewster angle
$\theta_{\rm Br}=\arctan(n)$ is larger than 45$^\circ$. For the parallel
polarization, the impedance as function of the refractive index and
the Brewster angle is
\begin{equation}
\eta = \frac{n \cos\theta_{\rm Br}}{\sqrt{n^2 - \sin^2\theta_{\rm Br}}}
\end{equation}
\begin{figure}[h]
\centerline{\includegraphics[width=9cm]{mmm30.eps}}
\caption{Equi-Brewster-angle curves in the ($n$-$\eta$) -plane for
parallel polarization. Four curves are shown. The thick curve
$\eta=1/n$ divides the plane into a upper ``paramagnetic part'' where
$\mu>1$, and the lower ``diamagnetic part'' where $\mu<1$.}
\label{fig:mmm30}
\end{figure}
The simple law for the non-magnetic Brewster angle $\tan\theta_1=n$,
combined with the Snell's law $\sin\theta_1=n\sin\theta_2$ yields
$\cos\theta_1=\sin\theta_2$. This means that the incidence and refracted
angles are complementary angles $(\theta_1+\theta_2=90^\circ)$. Therefore
(see Figure~\ref{fig:iim1}) the direction of the reflected wave is
orthogonal to the refracted wave. In such a geometric constellation
the dipoles induced in the medium by the refracted ray, which have a
radiation null along their axis direction, do not cause reradiation
into the direction of the reflected ray. Hence physical intuition
agrees with the result of Brewster angle formula \cite{Sastry,DeSmet},
although the interpretation has been also criticized
\cite{Nitzan,Merzbacher}.
But let us return to the more general case of the properties of the
wave that reflects from a dielectric--magnetic interface.
\section{Geometric interpretation}
The square roots of differences of squares in the relations
(\ref{BrP}) and (\ref{BrS}) for the two Brewster angles remind of the
Pythagorean theorem. And indeed, after some time of trigonometric play
with these relations, beautiful geometric interpretations can be
discovered from right triangles that are built from the three basic
measures $n$, $\eta$, and $n\eta$. Further, an arrangement of these
triangles in three dimensions reveals structures with which the
Brewster angles can be grasped in a very visual sense.
This geometric construction is illustrated in Figure~\ref{fig:tetra}
for the relations expressing the Brewster angle for parallel
polarization. From the magnitudes of $n$ and $\eta$, a tetrahedron is
uniquely determined. The faces of this geometrical object are four
right triangles. The Brewster angle can be read from the bottom of the
tetdahedron.
Figure~\ref{fig:tetra2} shows the same for the perpendicular
polarization.
\begin{figure*}[h!]
\psfragscanon
\psfrag{t}[][]{{$\theta_{\rm Br,P}$}}
\psfrag{1}[][]{{$n$}}
\psfrag{2}[][]{{$\eta$}}
\psfrag{3}[][]{{$n\eta$}}
\psfrag{4}[][]{{$n\sqrt{1-\eta^2}$}}
\psfrag{5}[][]{{$\eta\sqrt{n^2-1}$}}
\psfrag{6}[][]{{$\sqrt{n^2-\eta^2}$}}
\psfrag{X}[][]{{$\begin{array}{ll}{\bf Parallel\ polarization}\\
(n>1,\;\; \eta<1) \end{array}$}}
\psfrag{X2}[][]{{$\begin{array}{ll}{\bf Parallel\ polarization}\\
(n<1,\;\; \eta>1) \end{array}$}}
\centerline{\includegraphics[width=13cm]{tetraBr.eps}}
\vspace{10mm}
\psfrag{1}[][]{{$\eta$}}
\psfrag{2}[][]{{$n$}}
\psfrag{3}[][]{{$\eta n$}}
\psfrag{4}[][]{{$\eta\sqrt{1-n^2}$}}
\psfrag{5}[][]{{$n\sqrt{\eta^2-1}$}}
\psfrag{6}[][]{{$\sqrt{\eta^2-n^2}$}}
\centerline{\includegraphics[width=13cm]{tetraBr2.eps}}
\caption{A geometrical view of the Brewster angle determined by the
primary material constants $n$ and $\eta$. Parallel polarization,
$n>1,\eta<1$ (upper figure); $n<1,\eta>1$ (lower figure). Note the four
right-triangular faces of the tetrahedra.}
\label{fig:tetra}
\end{figure*}
\begin{figure*}[h]
\psfragscanon
\psfrag{t}[][]{{$\theta_{\rm Br,S}$}}
\psfrag{1}[][]{{$n\eta$}}
\psfrag{2}[][]{{$1$}}
\psfrag{3}[][]{{$n$}}
\psfrag{4}[][]{{$n\sqrt{\eta^2-1}$}}
\psfrag{5}[][]{{$\sqrt{n^2-1}$}}
\psfrag{6}[][]{{$\sqrt{n^2\eta^2-1}$}}
\psfrag{X}[][]{{$\begin{array}{ll}{\bf Perpendicular\ polarization}\\
(n>1,\;\; \eta>1) \end{array}$}}
\psfrag{X2}[][]{{$\begin{array}{ll}{\bf Perpendicular\ polarization}\\
(n<1,\;\; \eta<1) \end{array}$}}
\centerline{\includegraphics[width=13cm]{tetraBr.eps}}
\vspace{10mm}
\psfrag{1}[][]{{$1$}}
\psfrag{2}[][]{{$n\eta$}}
\psfrag{3}[][]{{$n$}}
\psfrag{4}[][]{{$\sqrt{1-n^2}$}}
\psfrag{5}[][]{{$n\sqrt{1-\eta^2}$}}
\psfrag{6}[][]{{$\sqrt{1-\eta^2n^2}$}}
\centerline{\includegraphics[width=13cm]{tetraBr2.eps}}
\caption{The same as in Figure~\ref{fig:tetra}, for the perpendicular
polarization. Upper figure: $n>1,\eta>1$; lower figure: $n<1,\eta<1$.}
\label{fig:tetra2}
\end{figure*}
\section{Conclusion}
Sir David Brewster performed his studies on the character of reflected
light during the second decade of the 19th century. Therefore the
concept of polarizing angle is nearly as old as the understanding of
the transverse nature of light. The fascinating manner how the
material properties affect the appearance of the Brewster angle is
very interesting still today, both from experimental application point
of view and also pedagogically when we are learning physics, optics,
and electromagnetism. Hopefully the present article can give a helpful
contribution to a modern understanding of the Brewster angle.
|
{
"timestamp": "2005-03-29T12:50:47",
"yymm": "0503",
"arxiv_id": "physics/0503216",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503216"
}
|
\section{Introduction}
Given a graph $G$ with $m$ edges, the Max-Cut problem is to determine
(the size of) the maximum cut in
$G$. For complete graphs, the largest cut has size $m/2+o(m)$.
On the other hand, it is well known that a cut of
size at least $m/2$ in a graph $G$ can be found using the natural greedy
algorithm. Improving this, Edwards~\cite{Edwards73, Edwards75} showed that every graph
with $m$ edges has a cut of size
$$m/2+\sqrt{\frac{m}{8}+\frac{1}{64}}-\frac{1}{8},
$$
which is
best possible. The Max-Cut problem is equivalent to finding a bipartition
$V_1,V_2$ of the vertex set of $G$ which minimizes $e_G(V_1)+e_G(V_2)$, where
$e_G(V_i)$ denotes the number of edges in the subgraph of $G$ induced by~$V_i$.
The related problem when one is looking for a partition into $k$ classes $V_1,\dots,V_k$
which minimizes all $e_G(V_i)$ simultaneously, i.e. which minimizes
$\max\{e_G(V_1),\dots,e_G(V_k)\}$, was studied by Bollob\'as and Scott~\cite{BS93,BS99,BS_JGT}
as well as Porter~\cite{Porter92, Porter94, Porter99}, see also~\cite{BS02}
for a survey.
Here, we suppose that we are given several graphs on the same vertex set
and we want to find a bipartition which maximizes the sizes of the cuts
for all these graphs simultaneously. This problem was posed by Bollob\'as
and Scott~\cite{BS_JGT}. More precisely, they asked the following
question: What is the largest integer $f(m)$ such that whenever $G_1$ and
$G_2$ are two graphs with $m$ edges on the same vertex set $V$, there exists
a bipartition of $V$ in which for both $i=1,2$ at least $f(m)$ edges of
$G_i$ go across (i.e.~their endvertices lie in different partition classes).
They suggested that perhaps even $f(m)=(1-o(1))m/2$, i.e.~that we can
almost do as well as in the case where we
only have a single graph.
Theorem~\ref{main} shows that this is indeed the case.
Given a graph $G$ and disjoint subsets $A,B$ of its vertex set, let
$e_G(A,B)$ denote the number of edges between $A$ and~$B$.
\begin{thm} \label{main}
Consider graphs $G_1,\dots,G_\ell$ on the same vertex set $V$ and suppose that
$G_i$ has $m_i$ edges. Then there is a bipartition of $V$ into two classes
$A$ and $B$ so that for all $i=1,\dots,\ell$ we have
$$e_{G_i}(A,B) \ge \frac{m_i}{2}-\sqrt{\ell m_i/2}.$$
\end{thm}
Rautenbach and Szigeti~\cite{RS} observed that even for $\ell=2$ we cannot
guarantee that $e_{G_i}(A,B) \ge m_i/2$ for all~$i$. Indeed, let $G_1$
and $G_2$ be two edge-disjoint cycles of length~5 on the same vertex set.
(So $G_1\cup G_2=K_5$.) They also proved that $f(m)\ge m/2-\Delta^3$
if $\Delta(G_i)\le \Delta$ for $i=1,2$. (This answers the problem of
Bollob\'as and Scott if $(\Delta(G_i))^3=o(m)$ for $i=1,2$.)
The following result for partitions of graphs into more than
two parts shows that simultaneously for all graphs we can ensure that the number
of crossing edges is almost as large as one would expect in a random partition
(and almost the value one can ensure if one partitions only a single graph).
\begin{thm} \label{kpartite}
Let $k\ge 2$.
Consider graphs $G_1,\dots,G_\ell$ on the same vertex set $V$ and suppose that
$G_i$ has $m_i$ edges. Then there is a partition of $V$ into $k$ classes $V_1,\dots,V_k$
so that for all $i=1,\dots,\ell$ the number of edges spanned by the
$k$-partite subgraph of $G_i$ induced by $V_1,\dots,V_k$ is at least
$$\frac{(k-1)m_i}{k}-\sqrt{2\ell m_i}.$$
\end{thm}
In fact, if $\Delta(G_i)=o(m_i)$ for each $i$, then we can strengthen the conclusion:
The next theorem shows that there is a partition of $V$ into $k$ classes where each
of the $\binom{k}{2}$ bipartite graphs spanned by two of the partition classes contains almost
$2m_i/k^2$ edges for all $i=1,\dots,\ell$ simultaneously. Again, this is
about the number of edges which one would expect in a random partition.
\begin{thm} \label{kpartite2}
Let $k\ge 2$ and $0<{\varepsilon}\le 1/(9\ell^2 k^4)$.
Consider graphs $G_1,\dots,G_\ell$ on the same vertex set $V$. Suppose that
$G_i$ has $m_i$ edges and that $\Delta(G_i) \le {\varepsilon} m_i$ for all $i=1,\dots,\ell$.
Then there is a partition of $V$ into $k$ classes $V_1,\dots,V_k$
so that for all $i=1,\dots,\ell$ and for all $s,t$ with $1\le s < t\le k$
we have
$$e_{G_i}(V_s,V_t)\ge \frac{2m_i}{k^2}-{\varepsilon}^{1/4} m_i$$
and $$e_{G_i}(V_s)\ge \frac{m_i}{k^2}-{\varepsilon}^{1/4}m_i.$$
\end{thm}
Note that even for $\ell=1$ the condition that $\Delta(G_i)\le {\varepsilon} m_i$
cannot be omitted completely.
For example, the result is obviously false if $G$ is a star.
On the other hand, a result of Bollob\'as and
Scott~\cite[Thm.~3.2]{BS_JGT} implies that in the case when the maximum degree of
each $G_i$ is bounded by a constant~$\Delta$, the bound on $e_{G_i}(V_s,V_t)$ in
Theorem~\ref{kpartite2} can be improved to $2m_i/k^2-C$ where $C=C(\ell,\Delta)$
(and similarly for $e_{G_i}(V_s)$). Note that this implies that if $G$ has bounded maximum
degree, then one can achieve a bounded error term in Theorems~\ref{main} and~\ref{kpartite} as
well.
The proofs of Theorems~\ref{main}--\ref{kpartite2} can be derandomized to
yield polynomial time algorithms which
find the desired partitions (see Section~\ref{sec:alg}).
\section{An open problem}
Consider an $r$-uniform hypergraph $\mathcal{H}$ with $m$ hyperedges.
It is easy to see that there is a partition $V_1,\dots,V_r$ of the vertex
set of $\mathcal{H}$ such that at least $r!m/r^r$ hyperedges of $\mathcal{H}$ meet every~$V_i$
(in other words, each $r$-uniform hypergraph contains an $r$-partite subhypergraph with
at least $r!m/r^r$ hyperedges). To verify this, consider the expected number of hyperedges
which meet every~$V_i$ in a random partition of the vertices.
We believe that one does not loose much if one considers several
hypergraphs simultaneously:
\begin{conj}\label{hypergraphconj}
Suppose that $\mathcal{H}_1,\dots,\mathcal{H}_\ell$ are $r$-uniform hypergraphs on the same
vertex set~$V$ such that $\mathcal{H}_i$ has $m_i$ hyperedges. Then there exists a partition
of $V$ into $r$ classes $V_1,\dots,V_r$ such that for all $i=1,\dots,\ell$ at
least $r!m_i/r^r-o(m_i)$ hyperedges of $\mathcal{H}_i$ meet each of the classes $V_1,\dots,V_r$.
\end{conj}
Given an $r$-uniform hypergraph $\mathcal{H}$ and distinct vertices $x,y\in \mathcal{H}$, denote
by $N_{\mathcal{H}}(x,y)$ the number of hyperedges which contain both $x$ and $y$.
Let $\Delta_2(\mathcal{H})$ denote the maximum of $|N_{\mathcal{H}}(x,y)|$ over all pairs
$x\neq y$.
One can adapt our proof of Theorem~\ref{kpartite2} to show that
Conjecture~\ref{hypergraphconj} holds in the case when $\Delta_2(\mathcal{H}_i)=o(m_i)$ for each~$i$.
We omit the details.
\section{Proofs}
The proofs all proceed by considering a random partition and analyzing this using
the second moment method.
\begin{lemma}\label{randompart}
Let $c \in \mathbb{R}$ with $c > 1/2$. Suppose that $G$ is a graph with $m$ edges
whose vertex set is $V$.
Consider a random bipartition of $V$ into two classes $A$ and $B$ which is obtained
by including each $v \in V$ into $A$ with probability $1/2$ independently
of all other vertices in $V$. Then with probability at least $1-1/(2c)$ we have
$$e_{G}(A,B) \ge \frac{m}{2}-\sqrt{cm/2}.$$
\end{lemma}
If we apply the above result with $c=\ell$ (say) to the graphs in Theorem~\ref{main},
the failure probability for each of them is less than $1/(2\ell)$. Summing up
all these failure probabilities immediately implies Theorem~\ref{main}.
{\removelastskip\penalty55\medskip\noindent{\bf Proof of Lemma~\ref{randompart}. }}
For every edge $e$ of the graph $G$, define an indicator variable
$X_e$ as follows: if one endvertex of $e$ is in $A$ and the other one is in $B$, then
let $X_e:=1$, otherwise let $X_e:=0$. Clearly, $\mathbb{P}[X_e=1]=1/2$.
Also, for $e,e' \in E(G)$ with $e \neq e'$, we have
$$
\mathbb{E} [X_e \cdot X_{e'}]= \mathbb{P}[X_e=1,\, X_{e'}=1]= \frac{1}{2} \mathbb{P}[X_e=1\mid X_{e'}=1]=\frac{1}{4}.
$$
Note that the final equality holds regardless of whether $e$ and $e'$ have an endvertex in
common or not.
Now let $X:=\sum_{e \in E(G)} X_e$.
Thus $X$ counts the number of edges between $A$ and $B$ and $\mathbb{E} X=m/2$.
Let $\sum_{e,e' \in E(G) \atop e \neq e'}$ denote the sum over all
ordered pairs $e,e'$ of distinct edges in~$G$. Then,
using the fact that $\mathbb{E}[X_e^2]=\mathbb{E}[X_e]$, we have
\begin{align*}
\mathbb{E}[X^2] & = \sum_{e \in E(G)} \mathbb{E}[X_e]+\sum_{e,e' \in E(G) \atop e \neq e'} \mathbb{E}[X_e \cdot X_{e'}]\\
& = \mathbb{E}[X]+ \sum_{e,e' \in E(G) \atop e \neq e'} \frac{1}{4} = \frac{m}{2}+ \frac{m(m-1)}{4}
= \frac{m(m+1)}{4}.
\end{align*}
This in turn implies that the variance of $X$ satisfies
${\rm Var} X=\mathbb{E}[X^2]-(\mathbb{E} X)^2=m/4$.
The result now follows from a straightforward application of Chebyshev's inequality:
$$
\mathbb{P}[X \le m/2-\sqrt{c m/2}] \le \mathbb{P}[ |X -\mathbb{E} X| \ge \sqrt{c m/2}]
\le \frac{{2\rm Var}X}{c m}=\frac{1}{2c}.
$$
\noproof\bigskip
\removelastskip\penalty55\medskip\noindent{\bf Proof of Theorem~\ref{kpartite}.}
As in Lemma~\ref{randompart}, we first consider a single graph $G$
with $m$ edges and vertex set~$V$. Consider a random partition of $V$ into
$k$ disjoint sets $V_j$ which is obtained by including each
$v\in V$ into $V_j$ with probability $1/k$ independently
of all other vertices. Let $X_e:=0$ if the edge $e$ has both its
endpoints in some $V_j$ and let $X_e:=1$ otherwise. So $\mathbb{P} [X_e=1]=(k-1)/k$.
Also, it is easy to check that $\mathbb{E} [X_e \cdot X_{e'}]=(k-1)^2/k^2$.
Again, this holds regardless of whether $e$ and $e'$ have an endvertex in
common or not. Let $X$ denote the number of edges whose endvertices lie in different
vertex classes. Thus $\mathbb{E} X=\frac{k-1}{k}m$ and
\begin{align*}
\mathbb{E}[X^2] & = \sum_{e \in E(G)} \mathbb{E}[X_e]+
\sum_{e,e' \in E(G) \atop e \neq e'} \mathbb{E}[X_e \cdot X_{e'}]\\
& =\frac{k-1}{k}m+ m(m-1) \frac{(k-1)^2}{k^2}\le m+(\mathbb{E}[X])^2.
\end{align*}
Therefore ${\rm Var} X\le m$ and so Chebyshev's inequality implies that
$$
\mathbb{P}[X \le \frac{k-1}{k}m-\sqrt{2\ell m}] \le \mathbb{P}[ |X -\mathbb{E} X| \ge \sqrt{2\ell m}]
\le \frac{{\rm Var}X}{2\ell m}\le \frac{1}{2\ell}.
$$
Theorem~\ref{kpartite} now follows by summing up this bound on the failure
probability for each of the graphs $G_i$.
\noproof\bigskip
\removelastskip\penalty55\medskip\noindent{\bf Proof of Theorem~\ref{kpartite2}. }
Let ${\varepsilon}$ be as in the statement of the theorem.
As in the previous proof, we first consider a single graph $G$, this time with $m$ edges
and maximum degree $\Delta\le {\varepsilon} m$.
Consider a random partition of $V:=V(G)$ into $k$ disjoint sets $V_j$ which is obtained
by including each vertex $v\in V$ into $V_j$ with probability $1/k$
independently of all other vertices. Fix some $s$ and $t$ with $1 \le s <t \le k$.
This time let $X_e:=1$ if one endvertex of $e$ is contained
in $X_s$ and the other in $X_t$. Put $X_e:=0$ otherwise. So $\mathbb{P}[X_e=1]=2/k^2=:\alpha$.
Now the value of $\mathbb{E} [X_e \cdot X_{e'}]$ depends on whether $e$ and $e'$ have an
endvertex in common or not:
If they do have an endvertex in common, we will use the trivial bound
$\mathbb{E} [X_e \cdot X_{e'}] \le 1 < 1+\alpha^2$. Note that the number of ordered
pairs $e, e'$ of distinct edges for which this can happen is trivially at most $2\Delta m$.
If $e$ and $e'$ have no vertex in common, then it is easy to see that
$$
\mathbb{E} [X_e \cdot X_{e'}]=\mathbb{P}[X_e=1]\mathbb{P}[X_{e'}=1]=\alpha^2.
$$
Let $X:=\sum_{e \in E(G)} X_e$. Thus $\mathbb{E}[X]=2m/k^2=\alpha m$.
Moreover
\begin{align*}
\mathbb{E}[X^2] & = \sum_{e \in E(G)} \mathbb{E}[X_e]+\sum_{e,e' \in E(G) \atop e \neq e'} \mathbb{E}[X_e \cdot X_{e'}]\\
& < \mathbb{E}[X]+ 2\Delta m+ \sum_{e,e' \in E(G) \atop e \neq e'} \alpha^2 \\
& \le \alpha m+2\Delta m+ \alpha^2 m^2 \le 3\Delta m+(\mathbb{E}[X])^2.
\end{align*}
Thus ${\rm Var}X \le 3\Delta m \le 3 {\varepsilon} m^2$. So we can conclude that
$$
\mathbb{P}[X \le \alpha m-{\varepsilon}^{1/4} m] \le \mathbb{P}[ |X -\mathbb{E} X| \ge {\varepsilon}^{1/4} m]
\le \frac{{\rm Var}X}{\sqrt{{\varepsilon}} m^2} \le 3\sqrt{{\varepsilon}}\le \frac{1}{\ell k^2}.
$$
In exactly the same way one can show that $\mathbb{P}[e_G(V_s)\le m/k^2-{\varepsilon}^{1/4}m]\le 1/(\ell k^2)$.
(This time $\alpha:=1/k^2$.)
Now sum up these failure probabilities for all the $\binom{k}{2}$ pairs $s,t$ and all
the $k$ values of $s$ to see that
the probability that a random partition does not have the required properties for $G$ is
at most $3/(4\ell)$. Again, Theorem~\ref{kpartite2} follows from summing up
this probability for all $G_i$.
\noproof\bigskip
We remark that at the expense of increasing the error terms the
partition classes in Theorems~\ref{main}--\ref{kpartite2} can be chosen to have
almost equal sizes. Indeed, Chernoff's inequality implies that in a
random partition of the vertex set as considered in the proofs with high
probability the vertex classes have almost equal sizes.
\section{Algorithmic aspects}\label{sec:alg}
Papadimitriou and Yannakakis~\cite{PY91} showed that the Max-Cut problem is APX-complete.
On the other hand,
as mentioned in the introduction, the obvious greedy algorithm always guarantees a
cut whose size is at least $m/2$.
Moreover, the proofs described in the previous section can be derandomized
to yield polynomial algorithms which construct partitions
satisfying the bounds in Theorems~\ref{main}--\ref{kpartite2}.
As the derandomization argument is similar for all three results, we only
only describe it for Theorem~\ref{main}.
More background information on derandomization can be found for instance
in the books~\cite{ASp,MRbook}
and in Fundia~\cite{Fundia} (in particular, the framework
described in the latter applies to our situation). For simplicity, we consider
Theorem~\ref{main} only for~$\ell=2$, i.e.~in the case of two graphs.
So let $G_1$ and $G_2$ be two graphs whose vertex set is $V$
with $e(G_i)=m_i$. Consider a
random partition of $V$ into sets $A$ and $B$
as described in the proof of Theorem~\ref{main} (cf.~Lemma~\ref{randompart}).
For $i=1,2$ define random variables $X_i:=e_{G_i}(A,B)$ and put
$\mu_i:=m_i/2=\mathbb{E}[X_i]$. Set
$$Z_i:=\frac{\mu_i^2-2\mu_i X_i +X_i^2}{m_i}
$$ for $i=1,2$ and
$Z:=Z_1+Z_2$. The proof of Theorem~\ref{main}
shows that for each $i$
$$\mathbb{P} [ X_i < \mu_i -\sqrt{m_i}]\le \frac{{\rm Var}X_i}{m_i}<1/2.
$$
But $\mathbb{E}[Z_i]={\rm Var}X_i/m_i$ and so
$\mathbb{E}[Z]=\mathbb{E} [Z_1]+\mathbb{E}[Z_2]<1$.
Let $v_1,\dots,v_n$ be an enumeration of the vertices in~$V$.
Let $A_i$ denote the event that the vertex $v_i$
is contained in $A$. Then
$$
1>\mathbb{E}[Z]=(\mathbb{E}[Z \mid A_1]+\mathbb{E}[Z \mid A_1^c] )/2 \ge \min
\{\mathbb{E}[Z \mid A_1],\mathbb{E}[Z \mid A_1^c] \}.
$$
Thus at least one of $\mathbb{E}[Z\mid A_1]$, $\mathbb{E}[Z \mid A_1^c]$ has
to be less than~1. Let $C_1\in\{A_1,A_1^c\}$ be such that
$\mathbb{E}[Z\mid C_1]<1$. Note that both $\mathbb{E}[Z \mid A_1]$ and $\mathbb{E}[Z \mid A_1^c]$
can be computed in polynomial time and so also $C_1$ can be determined in
polynomial time. Now
$$
1>\mathbb{E}[Z\mid C_1]=(\mathbb{E}[Z \mid C_1\cap A_2]+\mathbb{E}[Z \mid C_1\cap A_2^c] )/2.
$$
So similarly as before there exists $C_2\in\{A_2,A_2^c\}$ such that
$\mathbb{E}[Z\mid C_1\cap C_2]<1$ and $C_2$ can be determined in polynomial time.
We continue in this fashion until we have obtained events $C_k\in\{A_k,A_k^c\}$
for all $k=1,\dots,n$ such that
$$
\mathbb{E}[Z\mid C_1\cap \dots \cap C_n]<1.
$$
The proof of Chebyshev's inequality shows that
for each $i=1,2$ and for any event $U$ which has positive probability, we have
\begin{equation*}
\mathbb{P} [ X_i < \mu_i -\sqrt{m_i} \mid U] \le
\frac{\mu_i^2-2\mu_i\mathbb{E} [X_i \mid U] +\mathbb{E}[X_i^2 \mid U]}{m_i}=\mathbb{E}[Z_i\mid U]
\end{equation*}
(the above also follows from Corollary~4 in~\cite{Fundia}).
Taking $U:=C_1\cap \dots \cap C_n$ this implies that
\begin{align}\label{eqfinal}
\sum_{i=1,2}\mathbb{P} [ X_i< \mu_i-\sqrt{m_i} \mid U]\le \sum_{i=1,2}\mathbb{E}[Z_i\mid U]
=\mathbb{E}[Z\mid U]<1.
\end{align}
But $U:=C_1\cap \dots \cap C_n$ means that for each vertex $v_k\in V$ we
have decided whether $v_k\in A$ or $v_k\in B$. So the left hand side of~(\ref{eqfinal})
is either $0$ or~$1$, i.e.~it has to be~0. This means that the unique partition corresponding
to $C_1\cap \dots \cap C_n$ is as desired in Theorem~\ref{main}. Since each
$C_k$ can be determined in polynomial time this gives us a polynomial algorithm.
\section*{Acknowledgement}
We are grateful to Dieter Rautenbach for telling us about the problem.
|
{
"timestamp": "2005-03-21T15:25:40",
"yymm": "0503",
"arxiv_id": "math/0503403",
"language": "en",
"url": "https://arxiv.org/abs/math/0503403"
}
|
\section{Introduction}
One of the key features of a physical system for quantum information
processing (QIP) is quantum entanglement. The problem of entanglement
of multipartite systems is far from being completely understood, and
it has numerous interesting aspects.
One of the possible approaches to multipartite entanglement is to
search for quantum states with prescribed bipartite entanglement
properties~\cite{KoashiBI00,PleschB03,PleschB02}. This is a nontrivial
task as there exist limitations on the bipartite entanglement in
multipartite systems, which were quantified by Coffmann, Kundu and
Wootters~\cite{CoffmanKW00}. In a pioneering work, O'Connor and
Wootters~\cite{OConnorW01} have considered a system of quantum bits,
and have searched for an entangled state of these with maximal
bipartite entanglement. This state appears to be the ground state of
the antiferromagnetic Ising model, the spins representing the qubits.
This illustrates the relation between states of maximal bipartite
entanglement and the spin couplings known from statistical physics. We
will refer to this approach as the question of \emph{direct bipartite
entanglement}, as the relevant quantity is the bipartite
entanglement present in the system as it is.
Another approach to the problem of multipartite entanglement is
related to cluster~\cite{BriegelR01} and graph~\cite{HeinEB04} states.
These are genuine multipartite entangled states, which can be
projected onto a maximally entangled state of any chosen two spins by
a von Neumann measurement on the others. Such states arise dynamically
in a system of spins with pairwise Ising couplings. They constitute
the fundamental entangled resource for one-way quantum
computers~\cite{RaussendorfB01,RaussendorfBB03}. It is an interesting
property of the Ising dynamics in this case, that it transforms a
whole basis of product states into a basis which consists of cluster
or graph states. In this way a basis transformation from a product
state basis to a special -- in a sense maximally -- entangled basis is
realized.
These states are the starting points for the second approach, the
bipartite entanglement in multipartite systems available via assistive
measurements on all but two subsystems. The two key concepts in its
quantitative description are entanglement of
assistance~\cite{DiVincensoFMSTU} (or concurrence of
assistance~\cite{LaustsenVV03}, quantifying the entanglement available
via assistive measurements, and localizable
entanglement~\cite{VerstraetePC04b,quantph0411123}. The computational
feasibility of concurrence of assistance for a pair of qubits makes
the quantitative study of a part of this question feasible.
One of our aims is to relate the above two approaches. We will show
that the optimizations of direct and measurement assisted bipartite
entanglement are indeed related. Our other task is to study these
generic features in actual spin systems, as such systems do appear
quite naturally in this context.
Coupled spin systems have attracted a vast amount of research interest
in the quantum information community recently. The couplings studied
in statistical physics allow for performing certain tasks in QIP such
as e.g. quantum state
transfer~\cite{Bose03,ChristandlDEL04,OsborneN04}, realization of
quantum gates~\cite{SchuchS03,YungLB04}, and quantum
cloning~\cite{ChiaraFMMM04}. As the systems of coupled spins are
appropriate models for solid state systems, and also for quantum
states in optical lattices in certain cases~\cite{Garcia-RipollC03},
they bear actual practical relevance.
In the second part of this paper we focus on dynamical generation of
entanglement. We consider a system initially in a pure product state,
and investigate the entanglement of the states of the system
throughout the evolution. The ``prototype'' of such entanglement
generation is that of cluster and graph states. The various aspects of
the dynamical behavior of entanglement in spin systems has been
considered by several authors
recently~\cite{AmicoOPRP04,Subrahmanyam04a,PlastinaAOF04,quantph0409039,quantph0409048,VidalPA04}.
In addition to interpolating between the two approaches to bipartite
entanglement in multipartite systems, we consider the possibility of
controlling the process through the initial state of the system. We
address the following question. Is it possible to dynamically generate
states with optimal direct bipartite entanglement? We find a positive
answer, and also that the same couplings are capable of producing
states with high bipartite entanglement available via measurements, if
a different initial state is chosen. Our main tool of describing
measurement assisted bipartite entanglement will be concurrence of
assistance. We will examine the possibility of controlling the
behavior of this entanglement generation by the initial state of the
system. This is analogous to the control of quantum operations in
programmable quantum
circuits~\cite{quantph0102037,prl79_321,pra65_022301,pra66_042302}.
Finally we show that a suitably chosen magnetic field can enable
couplings different from Ising to create whole entangled bases
resembling those of cluster states regarding concurrence of
assistance. (Note that the generation of cluster states with non-Ising
couplings was considered very recently in Ref.~\cite{quantph0410145})
In addition, the application of magnetic field in the case of Ising
couplings can temporally enhance the presence of high pairwise
concurrence of assistance.
As we are mainly interested in illustrating generic features and
certain examples of entanglement behavior, a part of our results
concerning actual spin systems is simply computed by numerical
diagonalization of the appropriate Hamiltonians, even though we
present some analytical considerations where we find them appropriate.
Thus some of our considerations are limited to an order of 10 spins,
even though according to the numerical experience, they seem to be
scalable. This number coincides with that of the quantum bits expected
to be available in quantum computers in the near future. As the
realization of the discussed couplings is not necessarily restricted
to spins, our results may become directly applicable in such systems.
We consider two topologies of the pairwise interactions: a \emph{ring}
where each spin interacts with its two neighbors, and also the
\emph{star} topology where the interaction is mediated by a central
spin interacting with all the others. This was found interesting from
the point of view of entanglement distribution~\cite{HuttonB04} and
also from other aspects of its dynamics~\cite{BreuerBP04} recently.
The paper is organized as follows: in the introductory
Section~\ref{sect:entangmeas} we briefly review the entanglement
measures we use in the following. Section~\ref{sect:graphstates} is
devoted to the review of the dynamical generation of cluster and graph
states in spin systems, which is the background of the second part of
the paper. In Section~\ref{sect:upb} we present two interesting
properties of concurrence of assistance, which relates the two above
mentioned approaches to bipartite entanglement in multipartite systems,
and will be useful in the following. In Section~\ref{sec:control}, the
controlled generation of specific entangled states is addressed.
Section~\ref{sect:bases} is devoted to the enhanced generation of
certain entangled bases with the help of magnetic field.
Section~\ref{sect:concl} summarizes our results.
\section{Entanglement measures}
\label{sect:entangmeas}
In this Section we give an overview in a nutshell of the entanglement
measures and related quantities that will be used throughout this
paper.
\paragraph{One-tangle.}
For a bipartite system $A\bar{A}$ (A being a qubit, $\bar{A}$ being
the rest of the system) in the pure state $\Ket{\Psi}_{A\bar{A}}$, the
one-tangle~\cite{HillW97} of either of the subsystems
\begin{equation}
T\left(\Ket{\Psi}_{A{\bar{A}}}\right)=
4\det(\varrho_{A})
\label{eq:entanglement}
\end{equation}
(where $\varrho_{A}=\mathop{\mbox{tr}}\nolimits_{\bar{A}}\Ket{\Psi}_{A\bar{A}}\Bra{\Psi}$), is a
measure of entanglement. It quantifies the entanglement between the
qubit $A$ and the rest of the system, including all multipartite
entanglement between qubit A and the sets all the subsystems in
$\bar{A}$.
Although there is an extension of one-tangle to mixed states, it is
not computationally feasible except for the case of 2 qubits, in which
case one-tangle is equal to the square of concurrence. This justifies
the following interpretation: the square root of one-tangle is the
concurrence of such a two-qubit system in a pure state, for which the
density matrix of one of the qubits is equal to that of qubit A. This
means, it would be the concurrence itself if the subsystem $\bar{A}$
were also a qubit.
\paragraph{Concurrence.}
Having a bipartite system in a mixed state, a way of defining their
entanglement is to consider the average entanglement of all the pure
state decompositions of the state. This quantity is termed as the
\emph{entanglement of formation}:
\begin{equation}
E(\varrho)=\min\sum_{i}p_{i}E(\Ket{\Psi_{i}}),\quad\text{so
that}\,\,\sum_{i}p_{i}\Ket{\Psi_{i}}\Bra{\Psi_{i}}=\varrho.
\label{eq:entform}
\end{equation}
This is a kind of generalization of the entanglement defined in
Eq.~\eqref{eq:entanglement}. Its additivity is one of the most
interesting open questions of QIT.
The definition of entanglement of formation supports the following
interpretation: imagine that the bipartite system as a whole is a
subsystem of a large system. Entanglement of formation measures the
bipartite entanglement available on average if everything but the
bipartite subsystem is simply dropped.
If the system in argument consists of two qubits, there is a closed
form for entanglement of formation found by
Wootters~\cite{Wootters98}. This consideration includes another
entanglement measure.
Given the two-qubit density matrix $\varrho$, one calculates the
matrix \begin{equation}
\tilde{\varrho}=(\sigma^{(y)}\otimes\sigma^{(y)})\varrho^{*}(\sigma^{(y)}\otimes\sigma^{(y)}),
\label{eq:wootterstilde}
\end{equation}
where $*$ stands for complex conjugation in the product-state basis.
$\tilde{\rho}$ describes a very unphysical state for an entangled
state, while it is a density matrix for product states.
In the next step one calculates the eigenvalues $\lambda_{i}$
($i=1\ldots4$) of the Hermitian matrix
\begin{equation}
\label{eq:rhomatrix}
\hat{R}=\sqrt{\sqrt{\varrho}\tilde{\varrho}\sqrt{\varrho}},
\end{equation}
which are in fact square roots of the eigenvalues of the non-Hermitian
matrix
\begin{equation}
\hat{R}_{2}=\varrho\tilde{\varrho}.
\label{eq:R2}
\end{equation}
Concurrence is then defined as
\begin{equation}
C(\varrho)=\max(0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}),
\label{eq:concurrence}
\end{equation}
where the eigenvalues are put into a decreasing order.
Entanglement of formation is a monotonously increasing function of
concurrence:
\begin{eqnarray}
E(\varrho)=h\left(\frac{1+\sqrt{1-C(\varrho)^{2}}}{2}\right),\nonumber \\
h(x):=-x\log_{2}(x)-(1-x)\log_{2}(1-x).
\end{eqnarray}
Thus concurrence can be used as an entanglement measure on its own
right.
In multipartite systems the one-tangle and concurrence are linked by
the Coffmann-Kundu-Wootters inequalities
\begin{equation}
\label{eq:CKW}
T_k \geq \sum\limits_{l\neq k} C_{kl}^2
\end{equation}
which have be proven initially for three qubits in a pure state and
certain classes of multi-qubit states. For a long time they were
conjectured to be true in general. This conjecture was very recently
proven~\cite{quantph0502176}. These inequalities set limitations to
the bipartite entanglement that can be present in a multipartite
system.
\paragraph{Concurrence of assistance.}
Consider again a bipartite system described by the density operator
$\varrho$. One can follow a route complementary to that in case of
entanglement of formation and ask what is the \emph{maximum} average
entanglement available amongst the pure state realizations, termed as
the \emph{entanglement of assistance}~\cite{Wootters98}:
\begin{eqnarray}
E_{\text{assist}}(\varrho)=\max\sum_{i}p_{i}E(\Ket{\Psi_{i}}),
\nonumber \\
\text{so that}\,\,\sum_{i}p_{i}\Ket{\Psi_{i}}\Bra{\Psi_{i}}=\varrho,&
\label{eq:entass}
\end{eqnarray}
c.f. Eq.~\eqref{eq:entform}.
Interpreting again the bipartite system as a subsystem of a larger
system, one can consider that the whole system is in a pure state,
that is, we have a purification of $\varrho$ at hand. In this case
entanglement of assistance describes the maximum entanglement
available on average in the bipartite system, when a collaborating
third party, instead of omitting the rest of the system as in the case
of entanglement of formation, makes optimal von Neumann measurements
on it. Although entanglement of assistance is not an entanglement
measure according to some definitions, it is a very informative
quantity regarding entanglement.
Having a system of two qubits, one can also use concurrence instead of
entanglement in Eq.~\eqref{eq:entass}, yielding the definition of
\emph{concurrence of assistance}:
\begin{eqnarray}
C_{\text{assist}}(\varrho)=\max\sum_{i}p_{i}C(\Ket{\Psi_{i}}\Bra{\Psi_{i}}),
\nonumber \\
\text{so that}\,\,\sum_{i}p_{i}\Ket{\Psi_{i}}\Bra{\Psi_{i}}=\varrho.&
\label{eq:concass}
\end{eqnarray}
The advantage of this quantity is, that it can be easily calculated
for two qubits. As it is shown in~\cite{LaustsenVV03}, it is simply
\begin{equation}
C_{\text{assist}}(\varrho)=
\mathop{\mbox{tr}}\nolimits\sqrt{\sqrt{\varrho}\tilde{\varrho}\sqrt{\varrho}}=
\sum_{i=1}^{4}\lambda_{i},
\label{eq:cassist}
\end{equation}
c.f. Eq.~\eqref{eq:concurrence}. Note that this quantity is
essentially a fidelity between the physical density matrix $\varrho$
and the matrix $\tilde{\varrho}$, which is physical for separable
states only.
Thanks to the formula in Eq.~\eqref{eq:cassist}, concurrence of
assistance is not only an informative quantity, but it is as feasible
as concurrence itself in the case of qubit pairs.
\section{Graph states revisited}
\label{sect:graphstates}
In this Section we briefly review the properties of the Ising dynamics
for spin-1/2 particles without magnetic field, which are known from
Refs.~\cite{BriegelR01,HeinEB04}. We will talk about spins in this
context, and the $\hat \sigma^{(z)}$ eigenstates will represent the computational
basis: $\ket{0}=\ket{\uparrow}$, $\ket{1}=\ket{\downarrow}$. Consider a
set of spins, with pairwise interactions between them:
\begin{equation}
\label{eq:IsingnoB}
\hat H = -\sum\limits_{\langle k,l \rangle}
\hat \sigma^{(x)}_k \otimes \hat \sigma^{(x)}_l
\end{equation}
where the summation ${\langle k,l \rangle}$ goes over those spins
which interact with each other. (Hence the name graph states for the
states to be considered here: the geometry can be envisaged as a
graph, where the vertices are the spins, and the edges represent
pairwise Ising interactions.) As the summands in
Eq.~\eqref{eq:IsingnoB} commute, the time evolution can be written as
a product of two-spin unitaries
\begin{equation}
\label{eq:U}
\hat U(\tau) =e^{-i\hat H \tau}=
\prod\limits_{\langle k,l \rangle}
\hat U_{k,l}(\tau),
\end{equation}
where
\begin{equation}
\label{eq:Ukltau}
\hat U_{k,l}(\tau)=e^{i \hat \sigma^{(x)}_k \otimes \hat \sigma^{(x)}_l \tau}.
\end{equation}
Here $\tau$ stands for the scaled time measured in arbitrary units.
First we study the time instant $\tau=\frac{\pi}{4}$: one may directly
verify that
\begin{equation}
\label{eq:Ukl}
\hat U_{k,l}=
\frac{1}{\sqrt{2}}
\left(
\hat 1 +i \hat \sigma^{(x)} _k \otimes \hat \sigma^{(x)} _l
\right).
\end{equation}
The evolution operators without a time argument will denote those for
$\tau=\frac{\pi}{4}$ in what follows. These describe conditional
phase gates in a suitably chosen basis. Let us assume that the system
is initially in a state $\ket{e_m}$ of the computational basis, a
common eigenvector of all the $\hat \sigma^{(z)}$-s:
\begin{equation}
\label{eq:szeig}
\hat \sigma^{(z)}_n \ket{e_m} = e_{n,m} \ket{e_m}, \qquad e_{n,m}=\pm 1.
\end{equation}
The state $\hat U \ket{e_m}$ will be an eigenvector of the following
complete set of commuting observables:
\begin{equation}
\label{eq:K}
\hat K_n=\hat U \hat \sigma^{(z)} _n \hat U^\dag,
\end{equation}
with the same eigenvalues as the $e_{n,m}$-s in Eq.~\eqref{eq:szeig}.
The operators $\hat K_n$ in Eqs.~\eqref{eq:K} depend on the geometry of
the graph. They can be evaluated simply by utilizing the following
relations:
\begin{eqnarray}
\label{eq:Ucomm}
\hat U_{k,l} \hat \sigma^{(x)} _k \hat U_{k,l}^\dag &=& \hat \sigma^{(x)} _k \nonumber \\
\hat U_{k,l} \hat \sigma^{(y)} _k \hat U_{k,l}^\dag &=& -\hat \sigma^{(z)} _k \otimes \hat \sigma^{(x)} _l
\nonumber \\
\hat U_{k,l} \hat \sigma^{(z)} _k \hat U_{k,l}^\dag &=& \hat \sigma^{(y)} _k \otimes \hat \sigma^{(x)} _l
\nonumber \\
\hat U_{k,l} \hat \sigma^{(x,y,z)} _m \hat U_{k,l}^\dag &=&
\sigma_m^{(x,y,z)}
\quad (m\neq k,l).
\end{eqnarray}
which can be verified directly by substituting Eq.~\eqref{eq:Ukl} into
Eq.~\eqref{eq:K}. The joint eigenstates of these operators are termed
as \emph{graph states}~\cite{HeinEB04}. It can be shown that many of
the so arising states corresponding to different graphs are local
unitary equivalent.
As an example, consider a ring of $N$ spins with pairwise Ising
interaction. In this case
\begin{equation}
\label{eq:Kring}
\hat K^{\text{(ring)}}_l=
-\hat \sigma^{(x)}_{l-1}\otimes \hat \sigma^{(z)}_{l} \otimes \hat \sigma^{(x)}_{l+1},
\end{equation}
where the arithmetics in the indices is understood in the modulo N
sense. The common eigenstates of these commuting variables are termed
as \emph{cluster states}, and they were introduced in
Ref.~\cite{BriegelR01}, although in a different basis. They are
suitable as an entangled resource for one-way quantum
computers~\cite{RaussendorfB01}.
Note that $\hat U(\pi)=-\hat 1$ in general. Specially for a ring
topology, $\hat U(\pi/2)=-\hat 1$ holds too. This means that the
evolution is periodic: at such time instants the initial state appears
again, which is a computational basis state. Thus the Ising dynamics
without magnetic field produces oscillations between the computational
basis state and a graph (or in some of the cases, cluster) state. The
achieved graph state is selected by the initial basis state.
To obtain a more complete picture on the whole process of the
entanglement oscillations, we plot the temporal behavior of the
entanglement quantities in Fig.~\ref{fig:entangosc} for the ring
topology.
\begin{figure}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig1.eps}
\caption{(Color online.)
Overlap with the initial state and entanglement measures
for the first two qubits, during the entanglement oscillations for
five spins in a ring, generated by the Ising Hamiltonian without
magnetic field in~\eqref{eq:IsingnoB}. In the initial state all
spins are up, thus in state $\ket{0}$ if we consider qubits. The
plotted quantities are dimensionless.}
\label{fig:entangosc}
\end{figure}
In the figure we observe that the concurrence of assistance of the
qubit pair is almost equal to the square root of one-tangle of one of
the constituent spins. We will show later in this paper that the
square root of one tangle is an upper bound for concurrence of
assistance. Thus for the states in argument, the entanglement of a
subsystem with the rest of the system can be indeed ``focused'' to a
pair of qubits via suitably chosen measurement on the rest of the
system. This is obvious for the cluster states, but it appears to hold
for the most of the time evolution.
The dynamical entanglement behavior of the systems in argument can be
controlled by the appropriate choice of the initial state. Consider
for instance the following polarized initial state:
\begin{equation}
\label{eq:instate_Ising}
\Ket{\Psi_{\text{A}}(t=0)} =
\mathop{\otimes}\limits_{k=1}^N
\left(
\cos \left(\frac{\theta}{2}\right) \ket{0}_k +
\sin \left(\frac{\theta}{2}\right) \ket{1}_k
\right).
\end{equation}
The ``A''index reflects that \emph{all} the spins are rotated from the
$z$ direction in the same way. This state can be prepared by a
simultaneous one-qubit rotation, which is available even in optical
lattice systems. If $\theta=l\pi$ ($l$ being integer), we obtain the
graph state periodically, while for $\theta=l\pi/2$ the state is
stationary, thus no entanglement will be generated. Between these
values, the entanglement measured by one-tangle or concurrence of
assistance is a monotonous and continuous function of $\theta$ for all
values of time. Thus by varying this parameter of the initial state,
one can control the amount of the generated entanglement.
From the above discussion we find that Ising dynamics without magnetic
field has the following properties from the point of view of
entanglement generation:
\begin{enumerate}
\item The generated bipartite entanglement is always small.
\item In the case of the cluster states one can project the state with
certainty to a maximally entangled pair of two spins by a
measurement on the others. Moreover, required measurement is a local
one.
\item \emph{All} the states of the computational basis are
periodically transferred into states which have properties 1-2.
\item One can control the amount of the dynamically generated
entanglement by a parameter of the initial state, which can be
altered by the same local rotation applied on all the spins.
\end{enumerate}
During our investigations we will check which of these properties may
arise under different couplings, initial states and topologies.
\section{Two properties of concurrence of assistance}
\label{sect:upb}
In this Section we present two properties of concurrence of
assistance for multi-qubit systems.
Our first proposition formulates an upper bound of concurrence of
assistance.
\begin{theorem}
\label{thm:upb}
For an arbitrary state of two qubits $A$ and $B$, square root of the
one-tangle of either qubits serves as an upper bound for concurrence
of assistance, i.e.:
\begin{equation}
\label{eq:lemmst}
\sqrt{T_A}\geq C^{\text{assist}}_{AB}.
\end{equation}
\end{theorem}
Proof: Consider the ensemble realization of the state $\varrho_{AB}$
of the qubits A,B
\begin{equation}
\label{eq:bndpr1}
\varrho_{AB}=\sum_k p_k \ket{\xi_k} \bra{\xi_k}
\end{equation}
which provides the maximum in
Eq.~\eqref{eq:concass}, and use the notation
\begin{equation}
\label{eq:rhok}
\varrho_k=\mathop{\mbox{tr}}\nolimits_B \ket{\xi_k} \bra{\xi_k},
\end{equation}
thus
\begin{equation}
\label{eq:rhoa}
\varrho_{A}=\mathop{\mbox{tr}}\nolimits_B \varrho_{AB}=\sum_k p_k\varrho_k,
\end{equation}
due to the linearity of the partial trace. Substituting
Eq.~\eqref{eq:rhoa} into the definition in Eq.~\eqref{eq:entanglement}
we obtain
\begin{equation}
\label{eq:sqt}
\sqrt{T_A}=2\sqrt{\det\left( \sum_k p_k \varrho_k\right)},
\end{equation}
while according to the definition in Eq.~\eqref{eq:concass},
\begin{equation}
\label{eq:cassp}
C^{\text{assist}}_{AB}=2\sum_k \sqrt{\det(p_k\rho_k)},
\end{equation}
where we have exploited the fact that for pure states
\begin{equation}
C( \ket{\xi_k})=2\sqrt{\det \varrho_k}.
\end{equation}
Substituting Eqs.~\eqref{eq:sqt} and ~\eqref{eq:cassp} into the
statement of the Proposition in inequality~\eqref{eq:lemmst}, what we
have to show is that
\begin{equation}
\sum_k \sqrt{\det(p_k\varrho_k)} \leq
\sqrt{\det\left( \sum_k p_k \varrho_k\right)}.
\end{equation}
This is a consequence of the recursive application of the
inequality~\eqref{eq:mainineq}, which is proven in
Appendix~\ref{app:ineqproof}. \hfill QED.
Intuitively, in the spirit of the considerations concerning lower
bound of localizable entanglement in Ref.~\cite{quantph0411123}, we can
claim that a local measurement on the ancillary systems of a
purification of $\varrho_{AB}$ cannot create additional entanglement
between the spin $A$ and the rest of the system $\bar{A}$, as such a
measurement is an operation on the complementary system. Thus, by
choosing the optimal measurement we can, at best, concentrate all of
the originally available entanglement ($\sqrt{T_{A}}$) into the
entanglement between the qubits $A$ and $B$.
The appearance of the one-tangle in the context of concurrence of
assistance suggests that there might be some relation with the CKW
inequalities, and this is the case indeed. Nevertheless, it is simple
to prove the following:
\begin{theorem}
\label{thm:ckw}
For a system of three qubits $A$,$B$,$C$ in a pure state,
\begin{equation}
C_{AB}=C^{\text{assist}}_{AB}\ {\mathrm{and}}\
C_{AC}=C^{\text{assist}}_{AC}
\end{equation}
implies that the Coffmann-Kundu-Wootters inequalities in
Eq.~\eqref{eq:CKW} are saturated, thus
\begin{equation}
C_{AB}^2+C_{AC}^2=T_A
\end{equation}
holds
\end{theorem}
This immediately follows from the same derivation as in
Ref.~\cite{CoffmanKW00} by exploiting the fact that the matrices $R_2$
of Eq.~\eqref{eq:R2} for subsystems $AB$ and $AC$ have rank one due to
the conditions of the proposition. (C.f. Eqs.~\eqref{eq:concurrence}
and~\eqref{eq:cassist}).
Proposition~\ref{thm:ckw} relates the direct and measurement assisted
approach to bipartite entanglement in multipartite systems. The
question remains open, of course, whether it is true for more
parties, too.
As already pointed out in Section~\ref{sect:graphstates}, for the
graph states themselves $\sqrt{T_A} = C^{\text{assist}}_{AB}=1$, and
besides $\sqrt{T_A} \approx C^{\text{assist}}_{AB}$ holds throughout
the whole time evolution generated by Ising couplings. According to
Proposition~\ref{thm:upb} it is correct to call such states as those with
maximal concurrence of assistance. Meanwhile $C_{AB}\ll
C^{\text{assist}}_{AB}$, which suggests that CKW inequalities are far
from being saturated, which is indeed the case. The generated
entanglement is essentially multipartite, but it can be converted to
bipartite via a measurement. On the other hand, if CKW inequalities
are saturated, then we can expect concurrence of assistance being
below the square-root of one-tangle. Besides, the question naturally
arises, whether it is possible to dynamically create entanglement
oscillations in spin systems which saturate CKW inequalities instead.
\section{Controlled generation of concurrence and concurrence of assistance}
\label{sec:control}
Now we turn our attention to spin-1/2 systems as those naturally
realize multi-qubit systems. We assign the $\hat \sigma^{(z)}$ eigenstates as the
computational basis states as $\ket{0}=\ket{\uparrow}$,
$\ket{1}=\ket{\downarrow}$. We will use the qubit notation for
simplicity.
We have seen in Section~\ref{sect:graphstates} that certain states
with maximal concurrence of assistance can be generated in dynamical
oscillations, and the control over the available entanglement is
realized by the altering of the initial state. This control requires
a simultaneous operation on all the spins, and as for bipartite
entanglement, it effects the entanglement available via assistive
measurements only, as concurrence itself takes low values throughout
the evolution. First we consider whether it is possible to control the
concurrence itself too, and if it is possible to control the evolution
by varying a single spin only.
Consider first a system of $N+1$ spins with XY couplings:
\begin{equation}
\hat H_{XY}=-\sum\limits_{<i,j>} \hat \sigma^{(x)}_i \hat \sigma^{(x)}_j + \hat \sigma^{(y)}_i \hat \sigma^{(y)}_j,
\label{eq:XYnoB}
\end{equation}
in a star topology: spin $0$ is the middle one, while spins $1$ to $N$
are the outer ones, each coupled to the central one. Even though the
summands of the Hamiltonian do not commute, the eigenvalues and
eigenvectors can be calculated. One would expect that the state of the
middle spin can control the entanglement behavior, as the interaction
of the outer spins is mediated by this one. Indeed, if one considers
the initial state where only the middle spin is rotated, the others
point upwards, i.e. they are in the state $\ket{0}$:
\begin{widetext}
\begin{equation}
\label{eq:inXY}
\Ket{\Psi_{\text{M}}(t=0)}=
\left(\cos \left(\frac{\theta}{2}\right) \ket{0}_0 +
\sin \left(\frac{\theta}{2}\right) \ket{1}_0 \right)
\otimes
\mathop{\otimes}\limits_{k=1}^N \ket{0}_k,
\end{equation}
the time evolution, as shown in
Appendix~\ref{app:andyn}, reads
\begin{eqnarray}
\label{eq:XYtime}
\Ket{\Psi_{\text{M}}(t)}=
&\cos \left(\frac{\theta}{2}\right)&
\left(
\ket{0}_0 \otimes \mathop{\otimes}\limits_{k=1}^N \ket{0}_k
\right)
\nonumber \\
+
&\sin \left(\frac{\theta}{2}\right)&
\left(
\cos(2\sqrt{N}t)
\ket{1}_0 \otimes \mathop{\otimes}\limits_{k=1}^N \ket{0}_k
-i\sin(2\sqrt{N}t)
\ket{0}_0 \otimes
\frac{1}{\sqrt{N}}\sum\limits_{l=1}^N
\ket{0,\ldots 0,1_l,0\ldots}
\right).
\end{eqnarray}
\end{widetext}
The rotation of the central spin indeed controls the entanglement
behavior of the system: for $\theta=0$ no entanglement is created,
while for $\theta=\pi$ the maximal entanglement oscillation will
appear. The state is a superposition of a product and an entangled
state depending on $\theta$, thus this parameter controls the
available entanglement continuously.
These entanglement oscillations are different than those in case of
Ising couplings. As shown in Appendix~\ref{app:rankone}, concurrence
is equal to concurrence of assistance in the case of any superposition
of the computational basis states with all spins up and one down. This
means that in the states arising throughout this evolution
measurements do not facilitate ``focusing'' entanglement onto two
spins. Besides, it has been proven in Ref.~\cite{CoffmanKW00} that
these states saturate CKW inequalities in Eq.~\eqref{eq:CKW}, thus the
bipartite entanglement present in the states is maximal. This scheme
provides a dynamical way of preparing multipartite states with maximal
bipartite entanglement, which is controlled by the initial state of
one spin. In addition, it illustrates that Proposition~\ref{thm:ckw}
works for more than two subsystems, which is shown exactly in this
specific case. Note that at certain times the central spin gets
disentangled from the outer ring, which is meanwhile in a state with
highest pairwise concurrence possible. Such a maximally entangled
state is reached for the whole system, too, at different times, see
also in Fig.~\ref{fig:XYfig}/a).
In Fig.~\ref{fig:XYfig} we present the behavior of concurrence and
square root of one tangles for a ring topology, and for an outer spin
in a state different from the others, as an illustration. Here we
consider the initial state producing the maximal entanglement, that
is, one spin is considered to point downwards, while all the others
point upwards. An analytical solution similar to that in
Appendix~\ref{app:andyn} would be feasible too, but more energy
eigenstates have nonzero weights in the initial state. Of course the
functions are not equal for all the spins in such case, but their
behavior is similar to the star topology. According to
Appendix~\ref{app:rankone}, concurrence is equal to concurrence of
assistance, and of course CKW inequalities are saturated.
\begin{figure*}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig2.eps}
\caption{(Color online.)
Concurrence and one-tangle for spins coupled by XY
interactions in the absence of magnetic field. In Figs. a)-d)
6+1 spins are ordered into a star topology, while in e)-f) a
ring of 6 spins is considered. In the initial state all spins
are up, except for one, which is down. In a)-b) the central spin
while in c)-d) an outer spin is flipped to point upwards.
Figures on the left display concurrences of qubit pairs, those
on the right display square roots of one-tangles as a function
of time. Legend: c: the central central spin, f: an outer spin
which is flipped initially, o$_k$: an outer spin which is the
$k$-th neighbor of the initially flipped one. Time is measured
in arbitrary units, the other quantities are dimensionless. The
figure is obtained from exact numerical diagonalization and
direct calculations.}
\label{fig:XYfig}
\end{figure*}
From the above discussion one might conclude that the XY couplings
``prefer'' to generate pure bipartite entanglement. This is however
not the case. In order to examine this issue, we have plotted the
behavior of entanglement quantities for an XY-coupled star
configuration with the initial state in
Eq.~\eqref{eq:instate_Ising}, that is, the polarized state arising
as a product of all the spins in the same state which is a
superposition of $\ket{0}$ and $\ket{1}$. It appears that in this
case concurrence between two outer spins is heavily suppressed, but
concurrence of assistance takes rather high values for certain
initial states. Moreover, concurrence of assistance is very close to
the square-root of one-tangle, just as in the case of the Ising
couplings. Thus XY couplings can, if the initial state is suitably
chosen, produce states with a high amount of bipartite entanglement
available via assistive measurements. Notice however, that the
square-root of one-tangle is higher than concurrence of assistance,
thus there is also some multipartite entanglement present in the
system which cannot be accessed by assistive measurements.
\begin{figure*}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig3.eps}
\caption{(Color online.)
Comparison of rotating all spins or the central spin in the
initial state of a 6+1 spin star with XY couplings. Fig. a)
displays the temporal behavior of concurrence if the central spin
is rotated, i.e. the initial state in Eq.~\eqref{eq:inXY} is used,
while the other three figures display the evolution of
concurrence, concurrence of assistance and square-root of
one-tangle with an initial state in Eq.~\eqref{eq:instate_Ising},
that is, all spins in the same superposition of $\ket{0}$ and
$\ket{1}$. All the bipartite quantities correspond to two outer
spins, square-root of one-tangle is that of one of these. $\theta$
stands for the dimensionless parameter of the input state.}
\label{fig:xyallcontrol}
\end{figure*}
Consider now Ising interactions, and ask whether it is sufficient to
rotate just one spin in order to control the amount of available
entanglement, e.g. disable entanglement oscillations. For the
rotation of an outer spin in the star configuration or the ring
topology we have found that entanglement cannot be completely
suppressed. However, if we rotate the central spin in a star topology,
it is possible to control entanglement behavior. This is illustrated
in Fig.~\ref{fig:Isingcontrol}. Similarly to the case of initial state
of~\eqref{eq:instate_Ising}, concurrence of assistance is almost equal
to the square root of one-tangle, while concurrence itself is close to
zero.
\begin{figure*}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig4.eps}
\caption{(Color online.)
Control of entanglement generation in a system of 6+1
Ising-coupled spins in a star configuration. The central spin is
rotated, i.e. initial state is that in Eq.~\eqref{eq:inXY}, the
others are in the state $\ket{0}$. Figures a) and c) display
temporal behavior of concurrence as a function of parameter
$\theta$ of the initial state, for a) two outer spins and
b) an outer and a central spin. Figure b) shows the difference
between square root of one tangle and concurrence of assistance
for two outer spins. Figure d) shows concurrence for the central
and an outer spin. This quantity is zero for the outer spins.}
\label{fig:Isingcontrol}
\end{figure*}
It is important to note that the possible high value of concurrence of
assistance appears to have nothing to do with the bipartite nature of
the couplings. In order to see this, consider a ring of spins with the
``weird'' threepartite couplings
\begin{equation}
\label{eq:weird}
\hat H_{\text{weird}}= -\sum_k \hat \sigma^{(x)}_{k-1} \hat \sigma^{(y)}_{k} \hat \sigma^{(x)}_{k+1}.
\end{equation}
The temporal behavior of concurrence of assistance and square-root of
one-tangle for neighbors is shown in
Fig.~\ref{fig:weird}. Concurrence of assistance apparently reaches its
upper limit showing that threepartite interaction can also generate
maximal focusable bipartite entanglement.
\begin{figure}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig5.eps}
\caption{(Color online.)
Time evolution of concurrence of assistance and one-tangle for the
``weird'' Hamiltonian in Eq.~\eqref{eq:weird}, for 6 spins. In the
initial product state all spins point upwards.}
\label{fig:weird}
\end{figure}
In this Section we have shown that it is possible to generate
entanglement oscillations not only between product and graph (or
cluster) states, but also between product states, and states with
maximal possible bipartite entanglement, and control this entanglement
behavior by the initial state.
\section{Entangled bases in the presence of a magnetic field}
\label{sect:bases}
In Section~\ref{sect:graphstates} we have seen that in the absence of
magnetic field the Ising couplings induce such dynamics that
\emph{all} the states of the computational basis evolve into graph
states periodically. In the Heisenberg picture we may interpret this
so that the product of the $\hat \sigma^{(z)}$ operators evolves to such a joint
observable, which has an eigenbasis formed fully by graph states. One
of the key features of such states is that they can be projected onto
a maximally entangled state of any pair of selected spins by a von
Neumann measurement on the rest of the spins. We show here that this
property is preserved, moreover enhanced if the magnetic field is
present.
First we consider the Ising Hamiltonian with a magnetic field pointing
towards a direction characterized by the angle $\phi$:
\begin{equation}
\label{eq:Ising}
\hat H _\text{Ising}= -\sum\limits_{\langle k,l \rangle}
\hat \sigma^{(x)}_k \otimes \hat \sigma^{(x)}_l -
B\sum_k e^{i\frac{\phi}{2}\hat \sigma^{(x)}_k} \hat \sigma^{(z)}_k e^{-i\frac{\phi}{2}\hat \sigma^{(x)}_k}.
\end{equation}
Thus we have two free parameters characterizing the magnetic field,
its magnitude $B$ and direction $\phi$. Note that the rotation of the
magnetic field is equivalent to a rotation of the initial state in
this case.
In particular, we are interested in the temporal behavior of the
concurrence of assistance $C_{\text{assist}}$ for certain pairs of
spins. Therefore we calculate the time evolution of all the states
$\ket{e_i}$ of the computational basis:
\begin{equation}
\label{eq:isingtrstates}
\Ket{e_i'(B,t)}=
\exp\left(-i\hat H _{\text{Ising}}t\right)\Ket{e_i},
\quad i=1\ldots 2^N,
\end{equation}
Then we can evaluate the average
\begin{equation}
\label{eq:ensavg}
{\overline{C_{\text{assist}}}}(B,t)= \frac{1}{2^N}
\sum_i C_{\text{assist}}\left( \Ket{e_i'(B,t)} \right),
\end{equation}
and also the standard deviation
\begin{equation}
\label{eq:ensdev}
\sigma_{C_{\text{assist}}}(B,t)= \sqrt{\overline{C_{\text{assist}}^2}-\overline{C_{\text{assist}}}^2}
\end{equation}
of concurrence of assistance over the computational basis states as
initial states. The deviation is informative regarding the deviation
of the quantity from the average for the different initial states.
A typical result of the calculation is plotted in
Fig.~\ref{fig:Isingbasis}
\begin{figure*}[htbp]
\centering
\includegraphics{Koniorczyk_spins_fig6.eps}
\caption{(Color online.)
Average (a,c) and standard deviation (b,d)
of concurrence of assistance for a pair of outer spins of a star
topology, taken over all the possible computational basis states
as initial states. Ising Hamiltonian with a magnetic field as in
Eq.~\eqref{eq:Ising}, 4+1 spins in a ring topology. In Figs. a)
and b), $\phi=0$, $B$ dependence is plotted in Figs. c) and d),
$B=1$, $\phi$-dependence is plotted. Similar figures are obtained
for different choice of the spin pair, and ring topologies too.}
\label{fig:Isingbasis}
\end{figure*}
For $B=0$ the expected entanglement oscillations are present. If the
magnetic field is nonzero, the system does not tend to return to the
initial product states. Magnetic field resolves many of the the high
degeneracies of the Ising Hamiltonian, and the eigenvalues become
incommensurable. Therefore, even though the evolution of the system
will be almost periodic according to the quantum recurrence
theorem~\cite{BocchieriL57}, the reasonable approximate recurrences
occur after an extremely long time.
For $B\neq 0$, the ensemble average of concurrence of assistance
appears to be rather strictly close to one for quite long time
intervals, while its standard deviation is low. The deviation can be
further suppressed by the suitable choice of magnetic field. This
behavior of concurrence of assistance is very similar to that in
Fig.~\ref{fig:Isingbasis} also for different chosen pair of qubits,
for qubit pairs of a ring topology, and also for different
computationally feasible number of qubits. From this we can conclude
that the elements of the computational basis are transformed into
states which can be projected into nearly maximally entangled states
of chosen two spins via a von Neumann measurement on the rest of the
spins. Otherwise speaking, Ising couplings do take the products of
$\hat \sigma^{(z)}$ matrices to such complete set of commuting operators, whose
eigenstates have the above mentioned property. The temporal duration
of the presence of this property is significantly enhanced by the
magnetic field.
The so arising entanglement is essentially multipartite: the
appearance of the magnetic field does not enhance concurrence of the
qubit pairs as it can be verified by performing the same calculation
with concurrence. Note that the characteristic behavior of the
entanglement as reflected by the Meyer-Wallach measure for the kicked
Ising model, also in the case of the presence of a magnetic field
pointing towards an arbitrary direction was also reported in
\cite{quantph0409039}.
Another relevant question might be whether the required measurements
are local, i.e. how much localizable entanglement is present. To
illustrate this issue in our numerical framework, we have evaluated a
lower bound for localizable entanglement by the mere consideration of
a measurement on the computational basis. According to our experience,
the behavior of the so available bipartite entanglement resembles
that of concurrence of assistance, but takes lower values. However,
quite remarkable bipartite entanglement is still available, which is
in most of the cases still higher than the limit that CKW inequalities
would allow for, without measurements.
Next we investigate the properties of the $XY$-model from the same
point of view: into Eq.~\eqref{eq:isingtrstates} we substitute the
Hamiltonian
\begin{eqnarray}
\label{eq:XY}
\hat H_{\text{XY}} = -\sum\limits_{\langle k,l \rangle}
\left( \hat \sigma^{(x)}_k \otimes \hat \sigma^{(x)}_l +\hat \sigma^{(y)}_k \otimes \hat \sigma^{(y)}_l\right) \nonumber \\
- \sum_k e^{i\frac{\phi}{2}\hat \sigma^{(x)}_k} \hat \sigma^{(z)}_k e^{-i\frac{\phi}{2}\hat \sigma^{(x)}_k}.
\end{eqnarray}
A homogeneous magnetic field parallel to the $z$ does not have any
effect on the entanglement behavior of the system, as
\begin{equation}
\label{eq:commut}
\left[ \sum_l \hat \sigma^{(z)};\sum\limits_{\langle k,l \rangle}
\left( \hat \sigma^{(x)}_k \otimes \hat \sigma^{(x)}_l +\hat \sigma^{(y)}_k \otimes \hat \sigma^{(y)}_l\right)\right]=0
\end{equation}
thus the local rotations generated by $\sum_l \hat \sigma^{(z)}$ can be taken into
account after calculating the effect of the couplings. Therefore we
pick $B=1$, and investigate the dependence of concurrence and
concurrence of assistance on the direction $\phi$ of the field.
The quantities evaluated are again those in Eqs.~\eqref{eq:ensavg}
and~\eqref{eq:ensdev}, both for concurrence and concurrence of
assistance. A typical result is displayed in Fig.~\ref{fig:XYbasis}.
\begin{figure*}
\centering
\includegraphics{Koniorczyk_spins_fig7.eps}
\caption{(Color online.)
Time evolution of averages (a,c) and deviations (b,d)
of concurrence (a,b) and concurrence of assistance (c,d) for two
outer spins of a star configuration of 4+1 spins coupled by the XY
Hamiltonian with magnetic field in \eqref{eq:XY}. Parameter $\phi$
describes the direction of the magnetic field. Similar behavior
was observed for ring topologies and different choice of the qubit
pair too.}
\label{fig:XYbasis}
\end{figure*}
It appears that for $\phi=0$ we obtain oscillations in the average
concurrence, too, while concurrence of assistance is not significantly
higher than concurrence itself. The appropriate choice of the
direction of the magnetic field can suppress concurrence,
significantly enhance concurrence of assistance and decrease its
deviation. Thus even though the couplings are not Ising type, at least
the feature of the Ising couplings that it produces bases with high
concurrence of assistance can be retained.
\section{Conclusions}
\label{sect:concl}
In this paper we have related the problems of maximizing pairwise
concurrence and pairwise concurrence of assistance in a system of
multiple qubits. We have shown that the square root of one tangle of
a qubit is an upper bound for the concurrence of assistance of a qubit
pair containing the particular qubit. We have also shown that for a
certain set of states for which the CKW inequality is known to be
saturated, the concurrence is equal to the concurrence of assistance.
This means that the bipartite subsystem under consideration is not
correlated with the rest of the system via intrinsic multipartite
entanglement.
We have also studied the entanglement behavior of spin-1/2 systems
modeling qubits, from this perspective. We have shown that in a star
configuration of an XY coupled spins entanglement oscillations between
product states and states with maximal bipartite entanglement
according to CKW inequalities can be dynamically generated. The
oscillations can be controlled by rotating the spin which mediates the
interaction, and at some points it gets disentangled from the rest of
the outer ring, which is maximally entangled in the CKW sense. This
maximal entanglement is reached for the whole system, too. We have
shown numerically that the star topology facilitates the similar
control of entanglement oscillations between product and graph states.
The rotation of all the qubits of the initial state on the other hand
leads to different behavior of concurrence of assistance, as the
enhancement of bipartite entanglement to the measurement appears. We
have found similar behavior for different topologies numerically.
According to our numerical results magnetic field can lead to the
temporal enhancement of concurrence of assistance in the entanglement
oscillations starting from the states of the computational basis, in
the case of spins coupled by Ising interactions, arranged into ring or
star topologies. Thereby a special entangled basis can be accessed.
We have found similar behavior for the case of XY couplings: magnetic
field applied along properly chosen direction suppresses concurrence
and enhances concurrence of assistance.
According to the presented results, pairwise couplings between spins
and qubits can be used effectively for different tasks of distributing
bipartite entanglement between multiple parties. It is also possible
to control the dynamical behavior of entanglement by local quantum
operations such as rotation of control qubits. Besides, magnetic
field can be utilized to temporally enhance certain entanglement
features, or to chose between qualitatively different kinds of
entanglement behavior. It would be also interesting to investigate
whether the entangled bases available in the described means are
useful for quantum information processing tasks.
\begin{acknowledgments}
This work was supported by the European Union projects QGATES and
CONQUEST, and by the Slovak Academy of Sciences via the project CE-PI.
M.~K. acknowledges the support of National Scientific Research Fund of
Hungary (OTKA) under contracts Nos. T043287 and T034484. The authors
thank G\'eza T\'oth for useful discussions.
\end{acknowledgments}
|
{
"timestamp": "2005-03-15T10:31:08",
"yymm": "0503",
"arxiv_id": "quant-ph/0503133",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503133"
}
|
\section{Introduction}
In this paper we prove a slight variant of the hypergraph removal lemma established recently and independently by Gowers \cite{gowers}
and Nagle, R\"odl, Schacht and Skokan \cite{nrs}, \cite{rodl}, \cite{rodl2}. To motivate this lemma, let us first recall the more well-known triangle removal lemma from graph theory of
Ruzsa and Szemer\'edi \cite{rsz}. It will be convenient to work in the setting of tripartite graphs, though we will comment about the generalization
to general graphs shortly. We adopt the following $o()$ and $O()$ notation: If $x,y_1,\ldots,y_n$ are parameters,
we use $o_{x \to 0; y_1,\ldots,y_n}(X)$ to denote any quantity bounded in magnitude by $X c(x,y_1,\ldots,y_n)$, where $c()$ is a function which goes to zero as $x \to 0$ for each fixed choice of $y_1,\ldots,y_n$. Similarly, we use $O_{y_1,\ldots,y_n}(X)$ to denote any quantity bounded by
$X C(y_1,\ldots,y_n)$, for some function $C()$ of $y_1,\ldots,y_n$. If $A$ is a finite set, we use $|A|$ to denote the cardinality of $A$.
\begin{theorem}[Triangle removal lemma, tripartite graph version]\label{triangle-removal}\cite{rsz} Let $V_1, V_2, V_3$ be finite non-empty sets of vertices, and let $G = (V_1,V_2,V_3,E_{12}, E_{23}, E_{31})$ be a tri-partite graph on these sets of vertices, thus $E_{ij} \subseteq V_i \times V_j$ for $ij = 12,23,31$. Suppose that the number of triangles in this graph does not exceed $\delta |V_1| |V_2| |V_3|$ for some $0 < \delta < 1$.
Then there exists a graph $G' = G'(V_1,V_2,V_3,E'_{12}, E'_{23}, E'_{31})$ which contains no triangles whatsoever, and
such that $|E_{ij} \backslash E'_{ij}|= o_{\delta \to 0}(|V_i \times V_j|)$ for $ij=12,23,31$.
\end{theorem}
One can view $G'$ as a ``triangle-free approximation'' to $G$.
Note that we do not assume that $G'$ is a subgraph of $G$, but one can easily obtain this conclusion by replacing $E'_{ij}$ with $E'_{ij} \cap E_{ij}$ if desired (i.e. one replaces $G'$ by $G' \cap G$). As we shall see, however, it will be convenient to allow the
possibility that $G'$ is not a subgraph of $G$.
\begin{remark}
The above theorem is phrased for tri-partite graphs, but it quickly implies an analogous version for non-partite graphs $G = (V,E)$, by
taking three copies $V_1 = V_2 = V_3 = V$ of the vertex set $V$, and constructing the
bipartite graph $\tilde G = (V_1,V_2,V_3,E_{12},E_{23},E_{31})$, where $E_{ij}$ consists of those pairs $( x, y )$ which
are the endpoints of an edge in $E$. We omit the details.
\end{remark}
It was observed in \cite{rsz} that Theorem \ref{triangle-removal} implies Roth's famous theorem \cite{roth} that subsets of integers of positive density
contain infinitely many progressions of length three. In \cite{soly-roth} it was also observed that Theorem \ref{triangle-removal} also implies
that subsets of ${\hbox{\bf Z}}^2$ with positive density contain infinitely many right-angled triangles (a result
first obtained in \cite{AS}). It was observed earlier (for instance in \cite{rodl-icm} or \cite{frankl02})
that an extension of the triangle removal lemma to hypergraphs would similarly imply
Szemer\'edi's famous theorem \cite{szemeredi} on progressions of arbitrary length; by modifying the observation in \cite{soly-roth}, it
would also imply a multidimensional extension of that theorem due to Furstenberg and Katznelson \cite{fk}. We shall return to this issue
in the sequel \cite{tao-multiprime} to this paper, and discuss the above hypergraph removal lemma in detail later in this introduction.
Theorem \ref{triangle-removal} was proven using the \emph{Szemer\'edi regularity lemma} (see e.g. \cite{szemeredi-reg}, \cite{komlos} for a survey of this lemma and its applications), which roughly speaking allows one to approximate an arbitrary large and complex graph to arbitrary accuracy by a much simpler object; see also \cite{van}, \cite{shkredov} for further refinements of Theorem \ref{triangle-removal}. This proof in fact yields a little bit
more information on the triangle-free approximation $G'$ to $G$, namely that $G'$ can be chosen to be ``bounded complexity''. More precisely:
\begin{theorem}[Strong triangle removal lemma, tripartite graph version]\label{triangle-removal-2}\cite{rsz} Let $V_1, V_2, V_3$ be finite non-empty sets of vertices, and let $G = (V_1,V_2,V_3,E_{12}, E_{23}, E_{31})$ be a tri-partite graph on these sets of vertices. Suppose that $G$ contains
at most $\delta |V_1| |V_2| |V_3|$ triangles.
Then there exists a graph $G' = G'(V_1,V_2,V_3,E'_{12}, E'_{23}, E'_{31})$ which contains no triangles whatsoever, and
such that $|E_{ij} \backslash E'_{ij}|= o_{\delta \to 0}(|V_i \times V_j|)$ for $ij=12,23,31$. Furthermore, there exists a quantity
$M = O_\delta(1)$, and partitions $V_i = V_{i,1} \cup \ldots V_{i,M}$ for each $i=1,2,3$ into sets $V_{i,a}$ (some of which may be empty)
such that for each $ij=12,23,31$, $E'_{ij}$ is the union of sets of the form $V_{i,a} \times V_{j,b}$.
\end{theorem}
Note that the graph $G'$ constructed in Theorem \ref{triangle-removal-2} will typically not be a subgraph of $G$. One could make the sets
$V_{i,1},\ldots,V_{i,M}$ to be the same size (with at most one exception for each $i$) without much difficulty but we will not endeavour to do so here.
There is also a version of this lemma for non-tripartite graphs which is well known (and essentially equivalent to the tripartite version)
but we will not reproduce it here.
It turns out that Theorem \ref{triangle-removal} and Theorem \ref{triangle-removal-2} can be rephrased in a more ``probabilistic''
manner. One reason for doing this is because in our arguments we will need two basic concepts
from probability theory, which are \emph{conditional expectation} and \emph{complexity} respectively. It
seems that with the aid of these concepts, the proofs become somewhat cleaner to give\footnote{For a more traditional combinatorial approach to these problems, see \cite{rs}.}.
To explain these concepts we need some notation. For reasons which will become clearer later, we shall use a rather general notation which
incorporates the above Theorems as a special case.
\begin{definition}[Hypergraphs] If $J$ is a finite set and $d \geq 0$, we define ${J \choose d} := \{ e \subseteq J: |e| = d \}$ to be the set of all
subsets of $J$ of cardinality $d$. A \emph{$d$-uniform hypergraph} on $J$ is then defined to be any subset $H_d \subseteq {J \choose d}$ of
${J \choose d}$. For instance, an undirected graph $G = (V,E)$ without loops can be viewed as a $2$-uniform hypergraph on $V$.
\end{definition}
\begin{example}\label{triangle-ex} If $J := \{1,2,3\}$, then the triangle $H_2 := {J \choose 2} =
\{\{1,2\}, \{2,3\}, \{3,1\}\}$ is a 2-uniform hypergraph on $J$.
\end{example}
\begin{definition}[Hypergraph systems] A \emph{hypergraph system} is a quadruplet $V = (J, (V_j)_{j \in J}, d, H_d)$, where $J$ is a finite set,
$(V_j)_{j \in J}$ is a collection of finite non-empty sets indexed by $J$, $d \geq 1$ is positive integer, and $H_d \subseteq {J \choose d}$ is a
$d$-uniform hypergraph. For any $e \subseteq J$, we set $V_e := \prod_{j \in e} V_j$, and let $\pi_e: V_J \to V_e$ be the canonical projection map.
\end{definition}
\begin{remark}
Very roughly speaking, a hypergraph system corresponds to the notion of a \emph{measure-preserving system}\footnote{A measure preserving system is a probability space $(X, {\mathcal B}, \mu)$ together with a shift $T: X \to X$ that preserves the measure $\mu$. The ergodic approach to Szemer\'edi's theorem, as introduced by Furstenberg\cite{furst}, recasts the problem of finding arithmetic progressions as that of understanding averages such as $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \mu(A \cap T^n A \cap \ldots \cap T^{(k-1)n} A)$. This can in turn be viewed as the problem of understanding shift operators such as $(T, T^2,\ldots,T^{k-1})$ on a product space $X \times \ldots \times X$. This has some intriguing parallels with the combinatorial approach, in which the problem of obtaining arithmetic progressions in a set $V$ is reduced to that of analyzing Cayley-type graphs or hypergraphs, which can be viewed as subsets of $V \times \ldots \times V$. We do not know of any formal connection between these two approaches, nevertheless there do appear to be some interesting similarities.}
in ergodic theory, though with the notable difference that no analogue of the shift operator exists in a hypergraph system. Indeed the $V_j$ are simply finite sets, and need not have any additive structure whatsoever.
\end{remark}
\begin{definition}[Conditional expectation] Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system.
If $f: V_J \to {\hbox{\bf R}}$ is a function, we define the expectation ${\hbox{\bf E}}(f) = {\hbox{\bf E}}(f(x) | x \in V_J)$ by the formula
$$ {\hbox{\bf E}}(f) = {\hbox{\bf E}}(f(x) | x \in V_J) := \frac{1}{|V_J|} \sum_{x \in V_J} f(x).$$
Similarly, if ${\mathcal{B}}$ is a $\sigma$-algebra\footnote{Of course, since $V_J$ is finite, we do not need to distinguish finite unions and countable unions, and could simply call ${\mathcal{B}}$ an ``algebra'', or even a ``partition''; the latter notation is in fact used in most treatments of the regularity lemma. However we prefer the notation of $\sigma$-algebra as being highly suggestive, evoking ideas and insights from probability theory, measure theory, and information theory.}
on $V_J$, i.e. a collection of sets in $V_J$ which contains $\emptyset$ and $V_J$, and
is closed under unions, intersections, and complementation, we define the \emph{conditional expectation} ${\hbox{\bf E}}(f|{\mathcal{B}}): V_J \to {\hbox{\bf R}}$ by the formula
$$ {\hbox{\bf E}}(f|{\mathcal{B}})(x) := \frac{1}{|{\mathcal{B}}(x)|} \sum_{y \in {\mathcal{B}}(x)} f(y),$$
where ${\mathcal{B}}(x)$ is the smallest element of ${\mathcal{B}}$ which contains $x$. For each $e \subseteq J$, let ${\mathcal{A}}_e$ be the $\sigma$-algebra on $V_J$
defined by ${\mathcal{A}}_e := \{ \pi_e^{-1}(E): E \subseteq V_e \}$. In other words, ${\mathcal{A}}_e$ consists of those subsets of $V_J$,
membership of which is determined solely by the co-ordinates of $V_J$ indexed by $e$.
\end{definition}
One can interpret the usage of these averages as imposing the uniform probability distribution on each $V_e$, which basically amounts
to introducing a set $(x_j)_{j \in J}$ of independent random variables, with each $x_j$ ranging uniformly in $V_j$.
If ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are two $\sigma$-algebras on $V_J$, we use ${\mathcal{B}}_1 \vee {\mathcal{B}}_2$ to denote the smallest $\sigma$-algebra that contains both ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$; this corresponds to the familiar concept of the \emph{common refinement} of two partitions. We can more generally define $\bigvee_{i \in I} {\mathcal{B}}_i$ for any collection $({\mathcal{B}}_i)_{i \in I}$ of $\sigma$-algebras.
\begin{example} For any finite non-empty sets $V_1,V_2,V_3$, the quadruplet $V = (J, (V_j)_{j \in J}, 2, H_2)$ is a hypergraph system,
where $J := \{1,2,3\}$ and $H_2 := {J \choose 2}$ are as in Example \ref{triangle-ex}.
The $\sigma$-algebra ${\mathcal{A}}_{\{1,2\}}$ is the algebra of all subsets of $V_1 \times V_2 \times V_3$
which do not depend on the third variable, and thus take the form $E \times V_3$ for some $E \subseteq V_1 \times V_2$. Similarly for ${\mathcal{A}}_{\{2,3\}}$
and ${\mathcal{A}}_{\{3,1\}}$.
\end{example}
\begin{definition}[Complexity] Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system.
If ${\mathcal{B}}$ is a $\sigma$-algebra in $V_J$, we define the \emph{complexity} ${\hbox{\roman complex}}({\mathcal{B}})$ of ${\mathcal{B}}$ to be the least number of sets in $V_J$
needed to generate ${\mathcal{B}}$ as a $\sigma$-algebra; this can be viewed as a simplified version of the Shannon entropy ${\hbox{\bf H}}({\mathcal{B}})$, which we will not use here. We observe the obvious inequalities
\begin{equation}\label{complex-jump}
{\hbox{\roman complex}}({\mathcal{B}}_1 \vee {\mathcal{B}}_2) \leq {\hbox{\roman complex}}({\mathcal{B}}_1) + {\hbox{\roman complex}}({\mathcal{B}}_2) \hbox{ for arbitrary } {\mathcal{B}}_1, {\mathcal{B}}_2
\end{equation}
and
\begin{equation}\label{b-card}
|{\mathcal{B}}| \leq 2^{2^{{\hbox{\roman \scriptsize complex}}({\mathcal{B}})}}.
\end{equation}
\end{definition}
\begin{remark} If one views ${\mathcal{B}}$ as a partition, the complexity is essentially the logarithm of the number of cells in the partition.
From an information-theoretic perspective, the complexity measures how many bits of information are needed to know which atom of ${\mathcal{B}}$ a given point in $V_J$ lies in.
\end{remark}
If $E$ is a subset of $V_J$, we let $1_E: V_J \to {\hbox{\bf R}}$ be the indicator function, thus $1_E(x) := 1$ when $x\in E$ and $1_E(x) := 0$ otherwise.
In particular, ${\hbox{\bf E}}(1_E) = |E|/|V_J|$ can be viewed as the ``density'' or ``probability'' of $E$ in $V_J$.
With all this notation, Theorem \ref{triangle-removal-2} becomes
\begin{theorem}[Strong triangle removal lemma, $\sigma$-algebra version]\label{triangle-main} Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be
a hypergraph system with $J = \{1,2,3\}$, $d = 2$, and $H_d = {J \choose d} = \{ \{1,2\}, \{2,3\}, \{3,1\} \}$.
For each $e \in H_d$, let $E_e$
be a set in ${\mathcal{A}}_e$ such that
$$ {\hbox{\bf E}}( \prod_{e \in H_d} 1_{E_e} ) \leq \delta$$
for some $0 < \delta < 1$. Then there exist sets $E'_e \in {\mathcal{A}}_e$ for $e \in H_d$
such that
$$ \bigcap_{e \in H_d} E'_e = \emptyset$$
and
$${\hbox{\bf E}}( 1_{E_e \backslash E'_e} ) = o_{\delta \to 0}(1) \hbox{ for all } e \in H_d.$$
Furthermore, for each $i \in J$ there exists sub-algebras ${\mathcal{B}}_i \subseteq {\mathcal{A}}_{\{i\}}$
such that
$$ {\hbox{\roman complex}}({\mathcal{B}}_i) = O_\delta(1) \hbox{ for } i \in J$$
and
$$ E'_e \in \bigvee_{i \in e} {\mathcal{B}}_i \hbox{ for } e \in H_d.$$
\end{theorem}
It is easy to see that Theorem \ref{triangle-removal-2} and Theorem \ref{triangle-main} are equivalent. The notation here may appear
quite cumbersome, but the advantages of these notations will hopefully become more apparent when we prove a generalization of this result shortly.
The case of $d=2$, and $J$ and $H_d$ arbitrary, was treated in \cite{efr}. It was then conjectured in that paper that a result of the above
type should also hold for higher $d$. The generalization of Theorem \ref{triangle-removal} to the higher $d$ case
was accomplished only recently and independently by Gowers \cite{gowers-hyper} and Nagle, R\"odl, Schacht, Skokan \cite{nrs}, \cite{rodl}, \cite{rodl2},
using the language of hypergraphs. It turns out that Theorem \ref{triangle-removal-2} or Theorem \ref{triangle-main} can similarly be generalized, and with the notation already developed, the extension is very easy to state:
\begin{theorem}[Hypergraph removal lemma]\label{main-2}\cite{gowers-hyper}, \cite{nrs}, \cite{rodl}, \cite{rodl2}
Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system. For each $e \in H$, let $E_e$ be a set in ${\mathcal{A}}_e$ such that
\begin{equation}\label{E-dens}
{\hbox{\bf E}}( \prod_{e \in H_d} 1_{E_e} ) \leq \delta
\end{equation}
for some $0 < \delta < 1$. Then for each $e \in H_d$ there exists a set $E'_e \in {\mathcal{A}}_e$ such that
\begin{equation}\label{E-cap}
\bigcap_{e \in H_d} E'_e = \emptyset
\end{equation}
and
\begin{equation}\label{E-error}
{\hbox{\bf E}}( 1_{E_e \backslash E'_e} ) = o_{\delta \to 0; J}(1) \hbox{ for all } e \in H_d.
\end{equation}
Furthermore, there exist sub-algebras ${\mathcal{B}}_{e'} \subseteq {\mathcal{A}}_{e'}$ whenever $e' \subset J$ and $|e'| < d$ obeying the complexity estimate
\begin{equation}\label{E-complex}
{\hbox{\roman complex}}({\mathcal{B}}_{e'}) = O_{J, \delta}(1) \hbox{ whenever } e' \subseteq J \hbox{ and } |e'| < d
\end{equation}
(so in particular $|{\mathcal{B}}_{e'}| = O_{J, \delta}(1)$, thanks to \eqref{b-card})
and
\begin{equation}\label{E-meas}
E'_e \in \bigvee_{e' \subsetneq e} {\mathcal{B}}_{e'} \hbox{ for all } e \in H_d.
\end{equation}
\end{theorem}
Clearly Theorem \ref{triangle-main} is a special case of Theorem \ref{main-2}.
We have attributed this theorem to Gowers \cite{gowers-hyper} and Nagle-R\"odl-Schacht-Skokan \cite{nrs}, \cite{rodl}, \cite{rodl2} because it
follows from their methods, although a theorem of this type is not stated explicitly in those papers. One can formulate variants of this
removal lemma in the case when $H_d$ is not $d$-uniform but we will not do so here. A related result has recently been obtained
in \cite{rs}, using techniques similar in spirit to those here (though with substantially different notation).
The main purpose of this paper is
to explicitly prove Theorem \ref{main-2} in a completely self-contained manner.
In a subsequent paper \cite{tao-multiprime}, we will then transfer this theorem (as in \cite{gt-primes}) to
obtain a relative version of Theorem \ref{main-2}, restricted to a suitably pseudorandom subset of $\prod_j V_j$. This will
then be used (again following \cite{gt-primes}) to deduce
the existence of infinitely many constellations of a prescribed shape in
the Gaussian primes and similar sets.
As a corollary of Theorem \ref{main-2}, we obtain the hypergraph removal lemma in a formulation closer to that of Gowers or Nagle-R\"odl-Schacht-Skokan:
\begin{corollary}[Hypergraph removal lemma, partite hypergraph version]\label{hypergraph-removal}\cite{gowers-hyper}, \cite{nrs},\cite{rodl}, \cite{rodl2} Let $(V_j)_{j \in J}$ be a collection of finite non-empty sets. Let $0 \leq d \leq |J|$, and let $H_d \subseteq {J \choose d}$
be a $d$-uniform hypergraph on $J$.
For each $e \in H_d$, let $E_e$ be a subset of $\prod_{j \in e} V_j$. Suppose that
$$ |\{ (x_j)_{j \in J} \in \prod_{j \in J} V_j: (x_j)_{j\in e} \in E_e \hbox{ for all } e \in H_d \}|
\leq \delta \prod_{j \in J} |V_j|$$
for some $0 < \delta \leq 1$; in other words, the $J$-partite hypergraph $G = ((V_j)_{j \in J},(E_e)_{e \in H_d})$ contains at most
$\delta \prod_{j \in J} |V_j|$ copies of $H_d$. Then for each $e \in H_d$ there exists $E'_e \subset \prod_{j \in e} V_j$ such that
$$ \{ (x_j)_{j \in J} \in \prod_{j \in J} V_j: (x_j)_{j\in e} \in E'_e \hbox{ for all } e \in H_d \} = \emptyset$$
(i.e. the $J$-partite hypergraph $G' = G'((V_j)_{j \in J},(E'_e)_{e \in H_d})$ contains no copies of $H_d$ whatsoever), and
such that
$|E_e \backslash E'_e| = o_{\delta \to 0; |J|}(\prod_{j \in e} |V_j| )$ for all $e \in H_d$.
\end{corollary}
The deduction of Corollary \ref{hypergraph-removal} from Theorem \ref{main-2} is analogous to the deduction of Theorem \ref{triangle-removal}
from Theorem \ref{triangle-main} and is omitted. It seems quite likely that we can obtain similar analogues for non-partite hypergraphs,
just as was the case with the non-partite version of Theorem \ref{triangle-removal}; see \cite{gowers-hyper}, \cite{nrs}, \cite{rodl}, \cite{rodl2} for
some examples of this, though for applications to Szemer\'edi-type theorems it is the partite version which is of importance.
It should be unsurprising that
Theorem \ref{triangle-removal} is then the special case of Corollary \ref{hypergraph-removal} applied to the (hyper)graph
in Example \ref{triangle-ex}.
The case $|J|=4$ and $H_3 = {J \choose 3}$ was
treated in \cite{frankl02}. Just as Theorem \ref{triangle-removal} implies Roth's theorem, Corollary \ref{hypergraph-removal} implies Szemer\'edi's theorem \cite{szemeredi} on arithmetic progressions, as well as the multidimensional generalization of that theorem due to Furstenberg and Katznelson \cite{fk}; see \cite{soly-2}, \cite{frankl02}, \cite{gowers-hyper}, \cite{rodl2} for further discussion\footnote{It was also recently observed that this hypergraph removal result also implies another theorem of Furstenberg and Katznelson \cite{fk2} on affine subspaces of dense subsets of high-dimensional finite field vector spaces; see \cite{rstt}.}. Thus this paper provides
a moderately short and self-contained proof of these theorems, although we emphasize that this goal was already achieved in the prior work
of \cite{gowers-hyper}, \cite{nrs}, \cite{rodl}, \cite{rodl2}.
The remainder of this paper is devoted to proving Theorem \ref{main-2}. As one might expect from
the previous proofs of these types of results, our proof shall proceed by proving a ``hypergraph regularity lemma'' and a ``hypergraph counting
lemma''. The arguments are broadly along similar lines to those of Gowers or Nagle, R\"odl, Schacht, and Skokan, although it seems that using the notation of
$\sigma$-algebras and probability theory allows for slightly cleaner arguments.
The author thanks Fan Chung Graham, Vojt\v{e}ch R\"odl, Mathias Schacht, and Jozsef Solymosi
for helpful comments and references. He is particularly indebted to
Mathias Schacht for supplying the recent preprint \cite{rs}, and to the anonymous referees for a careful reading of the paper and
many cogent suggestions and corrections. The author is supported by a grant from the Packard foundation.
\section{Pseudorandomness and the regularity lemma}
Henceforth the hypergraph system $V = (J, (V_j)_{j \in J}, d, H_d)$ will be fixed. In this section we shall state and prove
a $\sigma$-algebra version of the hypergraph regularity lemma (Lemma \ref{full-regularity}).
This lemma establishes a dichotomy between pseudorandomness (or $\varepsilon$-regularity, or small discrepancy) on one hand, and bounded complexity\footnote{This is very similar to the dichotomy between weak mixing and compactness in ergodic theory, which is of great utility in proving statements such as Szemer\'edi's theorem; it seems of interest to explore these connections further.} on the other; the regularity lemma then asserts, very roughly speaking,
that any given set or $\sigma$-algebra (or family of $\sigma$-algebras) can be split into a component with bounded complexity, and a
component which is pseudorandom (has small discrepancy).
In order to state the regularity lemma we need to formalize the notion of pseudorandomness (or more precisely, of discrepancy). We shall
also need a notion of the \emph{energy} of a $\sigma$-algebra in order to keep track of the inductions that go into the
proof of the regularity lemma, and also in the final statement of our regularity lemma.
We shall not state the final regularity lemma we need (Lemma \ref{full-regularity}) immediately. To begin with, we set out
our notation for discrepancy and energy. Initially we shall be focusing primarily on a single edge $e \subseteq J$, as opposed to an entire
hypergraph $H_d$, though this hypergraph shall emerge later in this section.
\begin{definition}[$e$-discrepancy]
For any $e \subseteq J$, we define the \emph{skeleton} $\partial e$ of $e$ to be the set $\{ f \subsetneq e: |f| = |e|-1\}$.
If $e \subseteq J$, $E_e \subseteq V_J$, and ${\mathcal{B}}$ is a $\sigma$-algebra on $V_J$,
we define the \emph{$e$-discrepancy} $\Delta_e(E_e|{\mathcal{B}})$ of the set $E_e$ with respect to the $\sigma$-algebra ${\mathcal{B}}$ to be
the quantity\footnote{This quantity is related to the Gowers uniformity norms used for instance in \cite{gowers}, \cite{gowers-hyper}, \cite{gt-primes}, but we will not explicitly introduce those norms here. This quantity is also related to the notion of a pseudorandom hypergraph, studied for instance in \cite{krs-hyper}.}
\begin{equation}\label{Psie}
\Delta_e(E_e|{\mathcal{B}}) := \sup_{E_f \in {\mathcal{A}}_f \forall f \in \partial e}
| {\hbox{\bf E}}\left( (1_{E_e} - {\hbox{\bf E}}(1_{E_e}|{\mathcal{B}})\right) \prod_{f \in \partial e} 1_{E_f} )|
\end{equation}
where the supremum is over all collections of sets $(E_f)_{f \in \partial e}$, where each $E_f$ lies in the $\sigma$-algebra ${\mathcal{A}}_f$.
Note that since $V_J$ is finite, so is $\Delta_e(E_e|{\mathcal{B}})$.
\end{definition}
Roughly speaking, the $e$-discrepancy $\Delta_e(E_e|{\mathcal{B}})$ measures the amount of ``structure'' in $E_e$ which is not already captured by the $\sigma$-algebra
${\mathcal{B}}$. By ``structure'', we mean sets which can be easily described by sets from the lower order $\sigma$-algebras ${\mathcal{A}}_f$, as opposed to a
generic set in ${\mathcal{A}}_e$ which in general is likely to have no good decomposition (or approximate decomposition) into sets from the ${\mathcal{A}}_f$.
Thus if $\Delta_e(E_e|{\mathcal{B}})$ is small, we expect $E_e$ to behave randomly (i.e. in an unstructured way) on most atoms of ${\mathcal{B}}$.
The $\Delta_e(E_e|{\mathcal{B}})$
generalize the concept of $\varepsilon$-regularity, as the following
example shows:
\begin{example}\label{gve} Let $G = (V_1, V_2, E_{12})$ be a bipartite graph between two finite non-empty sets $V_1, V_2$; we can thus view
$E_{12}$ as a set in ${\mathcal{A}}_{\{1,2\}}$,
where $V$ is the hypergraph system $V = (J, (V_j)_{j \in J}, d, H_d)$ with $J = \{1,2\}$, $d=2$, and $H_d = {J \choose d} =
\{ \{1,2\} \}$. Suppose that
$E_{12}$ has density ${\hbox{\bf E}}(1_{E_{12}}) = \sigma$ (i.e. $\sigma = |E_{12}|/|V_1| |V_2|$), and that
$$ \Delta_{\{1,2\}}(E_{12}|{\mathcal{A}}_\emptyset) \leq \varepsilon$$
for some $\varepsilon > 0$. Then by definition we have
$$ |{\hbox{\bf E}}( (1_{E_{12}} - \sigma) 1_{E_1} 1_{E_2} )| \leq \varepsilon \hbox{ whenever } E_1 \in {\mathcal{A}}_{\{1\}}, E_2 \in {\mathcal{A}}_{\{2\}}.$$
In the original setting of the bipartite graph $G$, this is equivalent to asserting that
$$ \bigl| |E_{12} \cap (E_1 \times E_2)| - \sigma |E_1| |E_2| \bigr| \leq \varepsilon |V_1| |V_2|$$
for all $E_1 \subseteq V_1$ and $E_2 \subseteq V_2$. The reader may recognize this as a pseudorandomness condition or
$\varepsilon$-regularity condition on the graph $G$. If we replace ${\mathcal{A}}_\emptyset$ by a finer $\sigma$-algebra such as ${\mathcal{B}}_1 \vee {\mathcal{B}}_2$ for some
${\mathcal{B}}_1 \subseteq {\mathcal{A}}_{\{1\}}$ and ${\mathcal{B}}_2 \subseteq {\mathcal{A}}_{\{2\}}$, where the complexity of ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ is small compared to $1/\varepsilon$,
then a condition such as $\Delta_{\{1,2\}}(E_{12}|{\mathcal{B}}_1 \vee {\mathcal{B}}_2) \leq \varepsilon$ states, roughly speaking, that the graph $G$ is $\varepsilon$-regular on ``most''
of the atoms $A_1 \times A_2$ in the partition associated to ${\mathcal{B}}_1 \vee {\mathcal{B}}_2$.
\end{example}
If ${\mathcal{B}}$ is a $\sigma$-algebra on $V_J$ and $E$ is a set in $V_J$ (not necessarily in ${\mathcal{B}}$),
we define the \emph{$E$-energy} of ${\mathcal{B}}$ to be the quantity
$$ {\mathcal{E}}_E({\mathcal{B}}) := {\hbox{\bf E}}( |{\hbox{\bf E}}(1_E|{\mathcal{B}})|^2 ).$$
Clearly, the $E$-energy ${\mathcal{E}}_E({\mathcal{B}})$ ranges between 0 and $1$; intuitively, ${\mathcal{E}}_E({\mathcal{B}})$ is a measure of how much information
about $E$ is captured by ${\mathcal{B}}$, and is thus in many ways complementary to the $e$-discrepancy $\Delta_e(E|{\mathcal{B}})$. From Pythagoras' theorem we can verify the identity
\begin{equation}\label{pythagoras}
{\mathcal{E}}_E({\mathcal{B}}') = {\mathcal{E}}_E({\mathcal{B}}) + {\hbox{\bf E}}( |{\hbox{\bf E}}(1_E|{\mathcal{B}}') - {\hbox{\bf E}}(1_E|{\mathcal{B}})|^2 ) \hbox{ whenever } {\mathcal{B}} \subseteq {\mathcal{B}}',
\end{equation}
thus finer $\sigma$-algebras have larger $E$-energy.
\begin{remark} In the setting of Example \ref{gve} with ${\mathcal{B}} = {\mathcal{B}}_1 \vee {\mathcal{B}}_2$ for some ${\mathcal{B}}_1 \subseteq {\mathcal{A}}_{\{1\}}$
and ${\mathcal{B}}_2 \subseteq {\mathcal{A}}_{\{2\}}$, the energy is a familiar quantity in the theory of the regularity lemma, and is usually referred
to as the \emph{index} of the partition; see \cite{szemeredi-reg}.
\end{remark}
Let us informally say that a set $E_e \in {\mathcal{A}}_e$ is \emph{$e$-pseudorandom with respect to ${\mathcal{B}}$} if the $e$-discrepancy $\Delta_e(E_e|{\mathcal{B}})$ is small.
A fundamental fact (which was already exploited in \cite{szemeredi}, \cite{szemeredi-reg})
is that if $E$ is \emph{not} $e$-pseudorandom with respect to ${\mathcal{B}}$, then we can find a refinement of ${\mathcal{B}}$ with
higher energy and not much larger complexity:
\begin{lemma}[Large discrepancy implies energy increment]\label{increment}
Let $e \subseteq J$, let $E_e \in {\mathcal{A}}_e$ be a set, and for each $f \in \partial e$ let
${\mathcal{B}}_f \subseteq {\mathcal{A}}_f$ be a $\sigma$-algebra such that
$$ \Delta_e(E_e|\bigvee_{f \in \partial e} {\mathcal{B}}_f) \geq \varepsilon$$
for some $\varepsilon > 0$. Then there exists a $\sigma$-algebra ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for all $f \in \partial e$ such that
\begin{equation}\label{complexity-double}
{\hbox{\roman complex}}( {\mathcal{B}}'_f ) \leq {\hbox{\roman complex}}( {\mathcal{B}}_f ) + 1
\end{equation}
and
\begin{equation}\label{energy-increment}
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) \geq {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) + \varepsilon^2.
\end{equation}
\end{lemma}
\begin{proof} By \eqref{Psie} (and the finiteness of $V_J$) we can find sets $E_f \in {\mathcal{A}}_f$ for all $f \in \partial e$ such that
$$ |{\hbox{\bf E}}\left( \bigl(1_{E_e} - {\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}_f)\bigr) \prod_{f \in \partial e} 1_{E_f} \right)| \geq \varepsilon.$$
For each $f \in \partial e$, let ${\mathcal{B}}'_f$ be the $\sigma$-algebra
$$ {\mathcal{B}}'_f := {\mathcal{B}}_f \vee {\mathcal{B}}(E_f)$$
then we have ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$, and obtain \eqref{complexity-double} from \eqref{complex-jump}.
Since $\prod_{f \in \partial e} 1_{E_f}$ is measurable with respect to $\bigvee_{f \in \partial e} {\mathcal{B}}'_f$,
and $1_{E_e} - {\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}'_f)$ has zero conditional expectation with respect to $\bigvee_{f \in \partial e} {\mathcal{B}}'_f$ we see that
$$ {\hbox{\bf E}}\left( \bigl(1_{E_e} - {\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}'_f)\bigr) \prod_{f \in \partial e} 1_{E_f} \right) = 0$$
and hence
$$ |{\hbox{\bf E}}\left( \bigl({\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}'_f) - {\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}_f)\bigr)
\prod_{f \in \partial e} 1_{E_f} \right)| \geq \varepsilon.$$
By the boundedness of $\prod_{f \in \partial e} 1_{E_f}$ and the Cauchy-Schwarz inequality we conclude
$$ {\hbox{\bf E}}\left( \bigl|{\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}'_f) - {\hbox{\bf E}}(1_{E_e}|\bigvee_{f \in \partial e} {\mathcal{B}}_f)\bigr|^2 \right) \geq \varepsilon^2,$$
and \eqref{energy-increment} then follows from \eqref{pythagoras}.
\end{proof}
By iterating Lemma \ref{increment}, one expects to be able to show that any given set $E_e \in {\mathcal{A}}_e$ must be $e$-pseudorandom with respect to a $\sigma$-algebra ${\mathcal{B}}$ of bounded complexity, since otherwise we could create a tower of $\sigma$-algebras whose energy increments indefinitely.
Such statements can be viewed as $\sigma$-algebra analogues of the Szemer\'edi regularity lemma. There are several such lemmas available; the
final lemma which we need is a bit lengthy to state, so we begin by stating some simpler regularity lemmas which we will then iterate to obtain
the stronger lemmas which we need. We first obtain a preliminary iteration of Lemma \ref{increment}, in which the
single set $E_e \in A_e$ is replaced by an ensemble of sets, or more precisely an
ensemble $({\mathcal{B}}_e)_{e \in H}$ of $\sigma$-algebras with bounded complexity.
If $H_d$ is a $d$-uniform hypergraph, we define $\partial H_d$ to be the $(d-1)$-uniform hypergraph
$\partial H_d := \bigcup_{e \in H_d} \partial e$.
\begin{lemma}[Dichotomy between randomness and structure]\label{dichotomy} Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system.
For each $e \in H_d$, let ${\mathcal{B}}_e \subseteq {\mathcal{A}}_e$ be a $\sigma$-algebra with the complexity bounds
$$ {\hbox{\roman complex}}({\mathcal{B}}_e) \leq m \hbox{ for all } e \in H_d$$
for some $m > 0$, and for each $f \in \partial H_d$, let ${\mathcal{B}}_f \subseteq {\mathcal{A}}_f$ be a $\sigma$-algebra with the complexity bounds
$$ {\hbox{\roman complex}}({\mathcal{B}}_f) \leq M \hbox{ for all } f \in \partial H_d$$
for some $M > 0$. Let $\varepsilon, \delta > 0$. Then one of the following statements must hold.
\begin{itemize}
\item (Randomness) There exists $\sigma$-algebras ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for all $f \in \partial H_d$ such that
\begin{equation}\label{ebe-1}
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) < {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) + \varepsilon^2 \hbox{ for all } e \in H_d \hbox{ and } E_e \in {\mathcal{B}}_e
\end{equation}
and
\begin{equation}\label{ebe-2}
\Delta_e(E_e|\bigvee_{f \in \partial e} {\mathcal{B}}'_f) \leq \delta \hbox{ for all } e \in H_d \hbox{ and } E_e \in {\mathcal{B}}_e.
\end{equation}
\item (Structure) There exist $\sigma$-algebras ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for all $f \in \partial H_d$ such that
\begin{equation}\label{ebe-3}
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) \geq {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) + \varepsilon^2 \hbox{ for some } e \in H_d \hbox{ and } E_e \in {\mathcal{B}}_e
\end{equation}
and
\begin{equation}\label{ebe-4}
{\hbox{\roman complex}}({\mathcal{B}}'_f) \leq M + O_{|J|, m, \varepsilon, \delta}(1) \hbox{ for all } f \in \partial H_d.
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof} We run the following algorithm:
\begin{itemize}
\item Step 0. Initialize ${\mathcal{B}}'_f := {\mathcal{B}}_f$ for all $f \in \partial H_d$. Note that \eqref{ebe-1} and \eqref{ebe-4} currently hold.
\item Step 1. If \eqref{ebe-2} holds, then we halt the algorithm (we are in the ``randomness'' half of the dichotomy).
Otherwise, there exists an $e \in H$ and $E_e \in {\mathcal{B}}_e$ such that
$$ \Delta_e(E_e|\bigvee_{f \in \partial e} {\mathcal{B}}'_f) > \delta.$$
We can then invoke Lemma \ref{increment} to locate
refinements ${\mathcal{B}}'_f \subseteq {\mathcal{B}}''_f \subseteq {\mathcal{A}}_f$ for all $f \in \partial H_d$ (note that ${\mathcal{B}}''_f$ will just equal ${\mathcal{B}}'_f$ if
$f \not \subset e$) such that
$$ {\hbox{\roman complex}}({\mathcal{B}}''_f) \leq {\hbox{\roman complex}}({\mathcal{B}}'_f) + 1 \hbox{ for all } f \in \partial H_d$$
and
$$ {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}''_f) \geq {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) + \delta^2.$$
\item Step 2. We replace ${\mathcal{B}}'_f$ with ${\mathcal{B}}''_f$ for all $f \in \partial H_d$. If \eqref{ebe-1} fails (i.e. \eqref{ebe-3} holds), then we halt the algorithm (we are in the ``structure'' half of the dichotomy). Otherwise,
we return to Step 1.
\end{itemize}
Observe that every time we return from Step 2 to Step 1, the quantity
$$ \sum_{e \in H_d} \sum_{E_e \in {\mathcal{B}}_e} {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f)$$
increases by at least $\delta^2$. On the other hand, if this quantity ever increases by more than $|H_d| 2^{2^m} \varepsilon^2 = O_{|J|, m, \varepsilon}(1)$, then
by \eqref{b-card} and the pigeonhole principle \eqref{ebe-1} will necessarily fail. Since we only
return to Step 1 when \eqref{ebe-1} holds, we see that the algorithm can only iterate at most $O_{|J|, m, \varepsilon, \delta}(1)$ times.
Thus when we terminate we must have \eqref{ebe-4}. The claim then folows.
\end{proof}
We now iterate Lemma \ref{dichotomy} to obtain the following preliminary regularity lemma.
Define a \emph{growth function} to be an increasing function $F: {\hbox{\bf R}}^+ \to {\hbox{\bf R}}^+$ such that $F(x) \geq 1+x$ for all $x$.
\begin{lemma}[Preliminary regularity lemma]\label{partial-regularity} Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system.
For each $e \in H_d$ let ${\mathcal{B}}_e \subseteq {\mathcal{A}}_e$ be a $\sigma$-algebra, and suppose that we have the bound
$$ {\hbox{\roman complex}}({\mathcal{B}}_e) \leq m \hbox{ for all } e \in H_d$$
for some $m > 0$. Let $\varepsilon > 0$, and let $F$ be a growth function (possibly depending on $\varepsilon$).
Then there exists $M > 0$, and for each $f \in \partial H_d$ there exists a pair of $\sigma$-algebras
${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ such that we have the estimates
\begin{align}
F(m) \leq M &\leq O_{|J|, \varepsilon, m, F}(1) \label{M-bound} \\
{\hbox{\roman complex}}( {\mathcal{B}}_f ) &\leq M \hbox{ for all } f \in \partial H_d \label{coarse-complex} \\
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) - {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) &\leq \varepsilon^2
\hbox{ for all } e \in H_d, E_e \in {\mathcal{B}}_e\label{coarse-fine} \\
\Delta_e( E_e | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) &\leq \frac{1}{F(M)} \hbox{ for all } e \in H_d, E_e \in {\mathcal{B}}_e \label{fine-accurate}
\end{align}
\end{lemma}
\begin{remark}
Lemma \ref{partial-regularity} provides a coarse low-order approximation $({\mathcal{B}}_f)_{f \in \partial H_d}$ and
a fine low-order approximation $({\mathcal{B}}'_f)_{f \in \partial H_d}$
to the high-order $\sigma$-algebras $({\mathcal{B}}_e)_{e \in H_d}$. The coarse approximation has bounded complexity,
the fine approximation is close to the coarse approximation in an $L^2$ sense, and the high order $\sigma$-algebras are pseudorandom
with respect to the fine approximation. The key point here is that the discrepancy control on the fine approximation given by \eqref{fine-accurate} is superior to the complexity control on the coarse approximation given by \eqref{coarse-complex} by an \emph{arbitrary} growth function $F$. If one were to try to use a single approximation instead of a pair of coarse and fine approximations, it appears impossible to obtain such a crucial gain.
\end{remark}
\begin{proof} We perform the following iteration.
\begin{itemize}
\item Step 0. Initialize ${\mathcal{B}}_f = \{ \emptyset, V_J\}$ to be the trivial $\sigma$-algebra for all $f \in \partial H_d$, thus ${\mathcal{B}}_f$ has complexity 0
initially.
\item Step 1. Set $M := \max(F(m), \sup_{f \in \partial H_d} {\hbox{\roman complex}}({\mathcal{B}}'_f))$, and $\delta := 1/F(M)$.
We apply Lemma \ref{dichotomy}, and end up in either the randomness or structure half of the dichotomy. In either
case we generate $\sigma$-algebras ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for each $f \in \partial H_d$.
\item Step 2. If we are in the randomness half of the dichotomy, we terminate the algorithm. Otherwise, if we are in the structure half of
the dichotomy, we replace ${\mathcal{B}}_f$ with ${\mathcal{B}}'_f$ for each $f \in \partial H_d$, and return to Step 1.
\end{itemize}
Observe that every time we return from Step 2 to Step 1, the quantity
$$ \sum_{e \in H_d} \sum_{E_e \in {\mathcal{B}}_e} {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f)$$
increases by at least $\varepsilon^2$. On the other hand, this quantity is non-negative and does not exceed $|H_d| 2^{2^m} = O_{|J|,m}(1)$, thanks to \eqref{b-card}.
Thus this algorithm terminates after $O_{|J|, m, \varepsilon}(1)$ steps. By \eqref{ebe-4}, we see that at each of these steps, the quantity $M$
increases to be at most $M + O_{J, m, \varepsilon, F(M)}(1)$, while initially $M$ is equal to $F(m)$. Thus at the end of the algorithm we have
\eqref{M-bound} as desired. The remaining claims \eqref{coarse-complex}, \eqref{coarse-fine}, \eqref{fine-accurate} follow from
construction (and \eqref{ebe-1}, \eqref{ebe-2}).
\end{proof}
\begin{remark} Lemma \ref{partial-regularity} already implies the Szemer\'edi regularity lemma in its usual form (and with
the usual tower-exponential bounds); see \cite{tao:regularity} for further discussion.
The above lemma is also similar in spirit to the modern regularity lemmas that appear for instance in \cite{rs} (except for an issue of obtaining regularity at all orders less than $d$, which we shall address in Lemma \ref{full-regularity} below).
In such lemmas, the objective is not to obtain a partition for which the original graph or hypergraph is regular, but instead to obtain a partition for which a \emph{modified} graph or hypergraph is \emph{very} regular, where the modification consists of adding or subtracting a small number of edges. The analogue of such a modification in our context is the decomposition
$$ 1_{E_e} = F_{\operatorname{regular}} + F_{\operatorname{small}}$$
where
$$ F_{\operatorname{regular}} := {\hbox{\bf E}}( 1_{E_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f ) + (1_{E_e} - {\hbox{\bf E}}(1_{E_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f))$$
and
$$ F_{\operatorname{small}} := {\hbox{\bf E}}(1_{E_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f) - {\hbox{\bf E}}(1_{E_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f).$$
The function $F_{\operatorname{small}}$ is small thanks to \eqref{coarse-fine} and \eqref{pythagoras}. Now consider $F_{\operatorname{regular}}$. On a typical
atom of $\bigvee_{f \in\partial e} {\mathcal{B}}_f$, the first term is constant, and the second term is going to be very pseudorandom (have small correlation
with sets of the form $\bigcap_{f \in \partial e} E_f$ for $E_f \in {\mathcal{A}}_f$) thanks to \eqref{fine-accurate} and \eqref{Psie}.
\end{remark}
Lemma \ref{partial-regularity}
regularizes the $\sigma$-algebras ${\mathcal{B}}_e$ on the $d$-uniform hypergraph $H_d$ in terms of $\sigma$-algebras ${\mathcal{B}}_f$, ${\mathcal{B}}'_f$ on
the $(d-1)$-uniform hypergraph $\partial H_d$. However it does not regularize the $\sigma$-algebras on $\partial H_d$. This
can be accomplished by one final iteration, which gives our final regularity lemma (which is essentially the same lemma\footnote{In contrast, the earlier regularity lemmas of
Chung \cite{chung} and Frankl-Rodl \cite{frankl} are closer to Lemma \ref{partial-regularity}, with $\partial H_d$ generalized
to $\partial^l H_d$ for any fixed $l$. The case $l=d-1$ in particular is essentially a routine generalization of the ordinary regularity lemma and appears to have been folklore for quite some time.} as that in \cite{gowers-hyper}, \cite{rodl}, or \cite{rs}).
\begin{lemma}[Full regularity lemma]\label{full-regularity} Let $V = (J, (V_j)_{j \in J}, d, H_d)$ be a hypergraph system, and define
the $j$-uniform hypergraphs $H_j$ for all $0 \leq j < d$ recursively backwards from $j=d$ by the formula $H_j := \partial H_{j+1}$.
(In particular, if $H_d$ is non-empty then we have $H_0 = \{\emptyset\}$.)
For all $e \in H_d$ let ${\mathcal{B}}_e \subseteq {\mathcal{A}}_e$ be a $\sigma$-algebra, and suppose that we have the bound
$$ {\hbox{\roman complex}}({\mathcal{B}}_e) \leq M_d \hbox{ for all } e \in H_d$$
for some $M_d > 0$. Let $F$ be a growth function. Then there exists numbers
\begin{equation}\label{growth-cond}
M_d \leq F(M_d) \leq M_{d-1} \leq F(M_{d-1}) \leq \ldots \leq M_0 \leq F(M_0) \leq O_{|J|, M_d, F}(1)
\end{equation}
and for each $0 \leq j < d$ and $f \in H_j$ there exist $\sigma$-algebras
${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$, such that we have the estimates
\begin{align}
{\hbox{\roman complex}}( {\mathcal{B}}_f ) &\leq M_j \hbox{ for all } 0 \leq j < d, f \in H_j \label{coarse-complex-2} \\
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_f) - {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) &\leq \frac{1}{F(M_j)^2}
\hbox{ for all } 1 \leq j \leq d, e \in H_j, E_e \in {\mathcal{B}}_e\label{coarse-fine-2} \\
\Delta_e( E_e | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) &\leq \frac{1}{F(M_0)} \hbox{ for all } 1 \leq j \leq d, e \in H_j, E_e \in {\mathcal{B}}_e.
\label{fine-accurate-2}
\end{align}
\end{lemma}
\begin{remark} At every order $0 \leq j \leq d$, Lemma \ref{full-regularity} gives coarse and fine approximations $({\mathcal{B}}_f)_{f \in H_{j-1}}$, $({\mathcal{B}}'_f)_{f \in H_{j-1}}$
at the $(j-1)$-uniform level to the $\sigma$-algebras $({\mathcal{B}}'_e)_{e \in H_j}$ at the $j$-uniform level. As one goes down in order, the
$\sigma$-algebras rapidly become more complex\footnote{At the zeroth order $j=0$, all $\sigma$-algebras have complexity zero, but this is a degenerate exception to the above general rule.} (though lower order, of course). However, the bounds in \eqref{coarse-fine-2} and
\eqref{fine-accurate-2} will keep apace with this growth in complexity (see \cite{rs} for some related discussion concerning the desirability
of having the constants grow along such a hierarchy). Indeed the bound \eqref{fine-accurate-2} is extremely strong,
as $F(M_0)$ dominates all the other quantities which appear in the above lemma; it is effectively as if the fine approximation was perfectly accurate
(so that $1_{E_e}$ is approximable by ${\hbox{\bf E}}(1_{E_e} |\bigvee_{f \in \partial e} {\mathcal{B}}'_f )$ with only negligible error). The main remaining
difficulty when using this lemma is to exploit the estimate \eqref{coarse-fine-2} measuring the gap between the coarse and fine approximations;
one has to take some care here because
the error bound $1/F(M_j)^2$ here safely exceeds the complexity\footnote{We will only need to bound the complexity of the coarse algebras ${\mathcal{B}}_e$. Some (very weak) bounds on the complexity of the fine algebras ${\mathcal{B}}'_e$ are available but they seem to be useless for applications and so we have not stated them explicitly here.} of the higher-order objects $({\mathcal{B}}_e)_{e \in H_j}$, but not that of
the lower-order objects $({\mathcal{B}}_e)_{e \in H_{j-1}}$.
\end{remark}
\begin{proof} We induct on $d$ (keeping $J$ fixed); the implicit constants in \eqref{growth-cond} will change when one does this, but the induction
will only run for at most $|J|$ steps and so this will not cause a difficulty.
When $d=0$ the claim is trivial (and the claim \eqref{coarse-complex-2} has an enormous amount of room available!)
so assume that
$d \geq 1$ and the claim has already been proven for all smaller $d$.
We will need a growth function $F^{\operatorname{fast}}$ to be chosen later; as the name suggests, this function will grow substantially faster than $F$,
in particular we assume $F^{\operatorname{fast}}(n) \geq F(n)$ for all $n$.
Applying Lemma \ref{partial-regularity} with $m$ equal to $M_d$, with $\varepsilon$ equal to $1/F(M_d)$, and
the growth function $F^{\operatorname{fast}}$, we can create $\sigma$-algebras ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for all
$f \in H_{d-1}$ and a quantity $M_{d-1}$ such that
\begin{align}
F(M_d) \leq F^{\operatorname{fast}}(M_d) \leq M_{d-1} &\leq O_{|J|, \varepsilon, M_d, F^{\operatorname{fast}}}(1)
= O_{|J|, M_d, F, F^{\operatorname{fast}}}(1) \label{M-bound-0} \\
{\hbox{\roman complex}}( {\mathcal{B}}_f ) &\leq M_{d-1} \hbox{ for all } f \in H_{d-1} \nonumber \\
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_e) - {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) &\leq \frac{1}{F(M_d)^2}
\hbox{ for all } e \in H_d, E_e \in {\mathcal{B}}_e\nonumber \\
\Delta_e( E_e | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) &\leq \frac{1}{F^{\operatorname{fast}}(M_{d-1})} \hbox{ for all } e \in H_d, E_e \in {\mathcal{B}}_e. \label{fine-accurate-0}
\end{align}
Now we apply the induction hypothesis with $d$ replaced by $d-1$, and $H_d$ replaced by $H_{d-1}$. This generates numbers
\begin{equation}\label{mmm}
M_{d-1} \leq F(M_{d-1}) \leq \ldots \leq M_0 \leq F(M_0) \leq O_{|J|, M_{d-1}, F}(1)
\end{equation}
and for each $0 \leq j < d-1$ and $f \in H_j$ there exist $\sigma$-algebras
${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$, such that we have the estimates
\begin{align*}
{\hbox{\roman complex}}( {\mathcal{B}}_f ) &\leq M_j \hbox{ for all } 0 \leq j < d-1, f \in H_j \\
{\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}'_e) - {\mathcal{E}}_{E_e}(\bigvee_{f \in \partial e} {\mathcal{B}}_f) &\leq \frac{1}{F(M_j)^2}
\hbox{ for all } 1 \leq j \leq d-1, e \in H_j, E_e \in {\mathcal{B}}_e\\
\Delta_e( E_e | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) &\leq \frac{1}{F(M_0)} \hbox{ for all } 1 \leq j \leq d-1, e \in H_j, E_e \in {\mathcal{B}}_e.
\end{align*}
Comparing this with the conclusion of Lemma \ref{full-regularity}, we see that we can obtain all the claims we need except for
\eqref{fine-accurate-2} when $j=d$, as well as the final bound in \eqref{growth-cond}. To obtain \eqref{fine-accurate-2}, we see from
\eqref{fine-accurate-0} that it would suffice to ensure that
$$ F^{\operatorname{fast}}(M_{d-1}) \geq F(M_0).$$
But since $F(M_0) = O_{|J|,M_{d-1}, F}(1)$, this can be achieved simply by choosing the growth function $F^{\operatorname{fast}}$ to be sufficiently large and
rapidly increasing depending on $F$ and $|J|$. By \eqref{mmm}, \eqref{M-bound-0}, we then have
$$ F(M_0) = O_{|J|,M_{d-1}, F}(1) = O_{|J|, M_d, F, F^{\operatorname{fast}}}(1) = O_{|J|, M_d, F}(1)$$
and the claim \eqref{growth-cond} follows.
\end{proof}
\begin{remark} The dependence of constants here is quite terrible. Typically $F$ will be an exponential function. In the graph case
$d=2$ one can take $M_0$ to be a tower of exponentials, whose height is bounded by some polynomial of $F(M_2)$; a
modification of the arguments in \cite{gowers-sz}
shows that this tower bound is essentially best possible. However, for $d=3$, both $M_0$ and $M_1$ will be an \emph{iterated} tower of exponentials of iterated height equal to a polynomial in $F(M_3)$, basically because of the need for $F^{\operatorname{fast}}$ to exceed the bounds one obtains from the $d=2$ case. The situation of course gets even worse for larger values of $d$, though
for any fixed $d$ the bounds are still primitive recursive. As stated earlier, the complexity bounds for the fine approximations
${\mathcal{B}}'_f$ will be even worse than this, perhaps by yet another layer of iteration.
Nevertheless, this regularity lemma is still sufficient for applications in which one is willing to have qualititative control only on the error
terms (e.g. $o(1)$ type bounds) rather than quantitative control. (As we shall see in \cite{tao-multiprime}, obtaining infinitely many constellations in the Gaussian primes will be one such application.) In view of recent results on effective bounds on Szemer\'edi-type theorems (see e.g. \cite{gowers}, \cite{shkredov}) it seems quite possible that these very rapid bounds, while perhaps necessary in order to have a regularity lemma, are not needed for the hypergraph removal lemma.
\end{remark}
\section{Statement of counting lemma}
As is customary in these arguments, the regularity lemma must be complemented with a counting lemma in order for it to be applicable
to proving results such as Theorem \ref{main-2}. In the $\sigma$-algebra language, the setup is as follows. Suppose
we start with $\sigma$-algebras $({\mathcal{B}}_e)_{e \in H_d}$ as in the hypotheses of Lemma \ref{full-regularity}.
Then, among other things, this lemma yields further $\sigma$-algebras $({\mathcal{B}}_e)_{e \in H_j}$ for $0 \leq j < d$, each of which has some complexity bound.
Combining all of these $\sigma$-algebras together, one obtains a somewhat large (but still bounded complexity) $\sigma$-algebra
$ \bigvee_{e \in H} {\mathcal{B}}_e$, where $H := \bigcup_{0 \leq j \leq d} H_j$. In particular, if $E_e$ are sets in ${\mathcal{B}}_e$ for all
$e \in H_d$, then $\bigcap_{e \in H_d} E_e$ is the union of atoms in $\bigvee_{e \in H} {\mathcal{B}}_e$. Here, of course, an atom of a $\sigma$-algebra ${\mathcal{B}}$
is a non-empty set in ${\mathcal{B}}$ of minimal size; since the ambient space $V_J$ is finite, every point is contained in exactly one atom of ${\mathcal{B}}$.
Roughly speaking, the counting lemma we give below (Lemma \ref{count-lemma})
gives a formula for computing the probability of atoms in $\bigvee_{e \in H} {\mathcal{B}}_e$, or at least those atoms which are ``good''.
It can be informally described as follows. For each $e \in H$, let $A_e$ be an atom of ${\mathcal{B}}_e$, thus
$\bigcap_{e \in H} A_e$ will be an atom of $\bigvee_{e \in H} {\mathcal{B}}$ (if it is non-empty). The counting lemma then says that under most circumstances we have the approximate formula\footnote{The reader may wish to interpret ${\hbox{\bf E}}(1_A)$ as being the ``probability'' of the ``event'' $A$, thus
for instance ${\hbox{\bf E}}( \prod_{e \in H} 1_{A_e})$ is the probability of the joint event $\bigcap_{e \in H} A_e$. Similarly, many of the arguments in the
sequel also have a strongly probabilistic flavour.}
\begin{equation}\label{counting}
{\hbox{\bf E}}( \prod_{e \in H} 1_{A_e} ) \approx \prod_{e \in H} {\hbox{\bf E}}( 1_{A_e} | \bigcap_{f \in \partial e} A_f )
\end{equation}
where we use ${\hbox{\bf E}}(f|A)$ to denote the conditional expectation
$$ {\hbox{\bf E}}(f|A) := \frac{1}{|A|} \sum_{x \in A} f(x).$$
This can be viewed as an assertion that higher order atoms $A_e$ are approximately independent of each other, conditioning on lower order
atoms $A_f$, although a precise formulation of this heuristic is somewhat difficult to quantify. In particular, if we remove those ``bad''
atoms $\bigcap_{e \in H} A_e$
for which ${\hbox{\bf E}}( 1_{A_e} | \bigcap_{f \in \partial e} A_f )$ is small for at least one $e \in H$, then all the remaining non-empty atoms will have fairly large size. Thus if the set $\bigcap_{e \in H} E_e$ has very small size, then after removing all the bad atoms we expect
this set to in fact be empty. This is the strategy behind proving Theorem \ref{main-2}.
We now formalize the above discussion. We begin by describing the good atoms. Informally speaking, the good atoms are going to be those which are fairly large (at all orders) and also fairly regular (at all orders). This is consistent with previous
experience with counting lemmas (say in the graph case), in which one must first throw away all cells of the partition which are
too small (or have too few edges), as well as all pairs of cells for which the graph is irregular, before one can obtain a useful estimate for (say) the number of triangles in a graph.
\begin{definition}[Good atoms]\label{good-def} Let the notation, assumptions, and conclusions be as in Lemma \ref{full-regularity}, and let $H := \bigcup_{0 \leq j \leq d} H_j$. Let $\bigcap_{e \in H} A_e$ be a (possibly empty) atom of $\bigvee_{e \in H} {\mathcal{B}}_e$, where for each $e \in H$, $A_e$ is an atom of ${\mathcal{B}}_e$. We say that this atom is \emph{good} if
for all $0 \leq j \leq d$ and $e \in H_j$ we have the largeness estimates
\begin{equation}\label{e-large}
{\hbox{\bf E}}( 1_{A_e} \prod_{f \in \partial e} 1_{A_f} ) \geq \frac{1}{\log F(M_j)} {\hbox{\bf E}}(\prod_{f \in \partial e} 1_{A_f})
\end{equation}
as well as the regularity estimates
\begin{equation}\label{e-regularity}
{\hbox{\bf E}}\left( \bigl|{\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f )\bigr|^2 \prod_{f \subsetneq e} 1_{A_f} \right)
\leq \frac{1}{F(M_j)} {\hbox{\bf E}}( \prod_{f \subsetneq e} 1_{A_f} ).
\end{equation}
\end{definition}
\begin{remark}
While the definition of a good atom allows for $\bigcap_{e\in H} A_e$ to be empty, the counting lemma we prove below will show that in fact good atoms are always non-empty (assuming $F$ is sufficiently rapid). The reader should not take the logarithmic factor in \eqref{e-large} too seriously; the point is that $\log F(M_j)$ is smaller than any power of $F(M_j)$ but still much larger
than any given function of $M_j$.
\end{remark}
One can easily verify that most atoms are good in the following sense. For any $0 \leq j \leq d$, $e \in H_j$, and any atom
$A_e$ of ${\mathcal{B}}_e$, let $B_{e,A_e}$ be the union of all the sets $\bigcap_{f \subsetneq e} A_f$ for which \eqref{e-large} or \eqref{e-regularity} fails. We remark for future reference that the set $B_{e,A_e}$ lies in $\bigvee_{f \subsetneq e} {\mathcal{B}}_f$.
Note also that if the atom $\bigcap_{e \in H} A_e$ is not good, then there exists $e \in H$ such that
$\bigcap_{e' \in H} A_{e'} \subseteq A_e \cap B_{e,A_e}$.
\begin{lemma}[Most atoms are good]\label{good-lots} Let the notation, assumptions, and conclusions be as in Lemma \ref{full-regularity}
and Definition \ref{good-def}.
For any $0 \leq j \leq d$, $e \in H_j$, and any atom $A_e$ of ${\mathcal{B}}_e$,
we have ${\hbox{\bf E}}(1_{A_e} 1_{B_{e,A_e}}) = O(1 / \log F(M_j))$.
\end{lemma}
\begin{proof} Consider the contribution to ${\hbox{\bf E}}( 1_{A_e} 1_{B_{e,A_e}} )$ from the case where \eqref{e-large} fails. This contribution is bounded by\footnote{Note that \eqref{e-large} depends only on those $A_f$ for which $f \in \partial e$, as opposed to the larger class of events $A_f$ for which $f \subsetneq e$.}
$$ \sum_{(A_f)_{f \in \partial e} \hbox{\scriptsize atoms in } ({\mathcal{B}}_f)_{\partial e}: \hbox{\scriptsize \eqref{e-large} fails}}
{\hbox{\bf E}}( 1_{A_e} \prod_{f \in \partial e} 1_{A_f} )$$
which by failure of \eqref{e-large} is bounded by
$$ \leq \sum_{(A_f)_{f \in \partial e} \hbox{\scriptsize atoms in } ({\mathcal{B}}_f)_{\partial e}} \frac{1}{\log F(M_j)} {\hbox{\bf E}}( \prod_{f \in \partial e} 1_{A_f} )
= \frac{1}{\log F(M_j)}.$$
Next, consider the contribution to ${\hbox{\bf E}}( 1_{A_e} 1_{B_{e,A_e}})$ arising from the case when \eqref{e-regularity} fails. The total contribution
of this case is
$$ \sum_{(A_f)_{f \subsetneq e}: \hbox{\scriptsize \eqref{e-regularity} fails}} {\hbox{\bf E}}( \prod_{f \subsetneq e} 1_{A_{f}} )$$
which by failure of \eqref{e-regularity} is at most
$$ F(M_j) \sum_{(A_{f})_{f \subsetneq e}} {\hbox{\bf E}}\left( \bigl|{\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f )\bigr|^2 \prod_{f \subsetneq e} 1_{A_{f}} \right)$$
which in turn is at most
$$ F(M_j) {\hbox{\bf E}}\left( |{\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f )|^2 \right).$$
But by \eqref{pythagoras}, \eqref{coarse-fine-2} we have
$$ {\hbox{\bf E}}\left( |{\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f )|^2 \right) \leq \frac{1}{F(M_j)^2}.$$
Combining all of these estimates, the claim follows.
\end{proof}
We can now state the counting lemma; closely related results appear in the work of Gowers \cite{gowers}, Nagle, R\"odl, and Schacht \cite{nrs},
and R\"odl and Schacht \cite{rs}.
\begin{lemma}[Counting lemma]\label{count-lemma} Let the notation, assumptions, and conclusions be as in Lemma \ref{full-regularity} and
Definition \ref{good-def}, and let $H := \bigcup_{0 \leq j \leq d} H_j$. Let $\bigcap_{e \in H} A_e$ be a good atom of $\bigvee_{e \in H} {\mathcal{B}}_e$. Then, if the growth function $F$ is sufficiently rapid depending on $|J|$, we have that $\bigcap_{e \in H} A_e$ is non-empty, and more precisely
$${\hbox{\bf E}}( \prod_{e \in H} 1_{A_e} ) = (1 + o_{M_d \to \infty; |J|}(1)) \prod_{e \in H} {\hbox{\bf E}}( 1_{A_e} | \bigcap_{f \in \partial e} A_f )
+ O_{|J|, M_0}\left(\frac{1}{F(M_0)}\right)
$$
(compare with \eqref{counting}).
\end{lemma}
This lemma is a little lengthy (though straightforward) to prove, and we defer it to the next section. Let us assume it for now, and conclude
the proof of Theorem \ref{main-2}.
\begin{proof}[of Theorem \ref{main-2} assuming Lemma \ref{count-lemma}] Let $V = (J, (V_j)_{j \in J}, d, H_d)$,
$(E_e)_{e \in H_d}$, $\delta$ be as in Theorem
\ref{main-2}. We define $H_j$ recursively for $0 \leq j < d$ by setting $H_j := \partial H_{j+1}$, and then set $H := \bigcup_{0 \leq j \leq d} H_j$.
For any $e \in H_d$ we set ${\mathcal{B}}_e := {\mathcal{B}}(E_e)$,
thus each ${\mathcal{B}}_e$ has complexity at most 1. Let $M_d \geq 1$ be a quantity to be chosen later, and let $F$ be a growth function depending on $|J|$
(but not on $\delta$) to be chosen later. We apply the regularity lemma, Lemma \ref{full-regularity}, to obtain quantities \eqref{growth-cond}
and $\sigma$-algebras ${\mathcal{B}}_f \subseteq {\mathcal{B}}'_f \subseteq {\mathcal{A}}_f$ for all $f \in H$.
Suppose that $\bigcap_{e \in H} A_e$ is a (possibly empty) atom of $\bigvee_{e \in H} {\mathcal{B}}_e$ such that $A_e = E_e$ for $e \in H_d$. If this atom is good, then by the counting Lemma (Lemma \ref{count-lemma}) and Definition \ref{good-def}
we have
$$
{\hbox{\bf E}}( 1_{\bigcap_{e \in H} A_e} ) = (1 + o_{M_d \to \infty; |J|}(1)) \prod_{0 \leq j \leq d} \prod_{e \in H_j} \frac{1}{F(M_j)^{1/10}}
+ O_{|J|, M_0}\left(\frac{1}{F(M_0)}\right),
$$
if $F$ is sufficiently rapid depending on $|J|$. Using \eqref{growth-cond}, we thus see
that (if $M_d$ is sufficiently large depending on $J$)
$$ {\hbox{\bf E}}( 1_{\bigcap_{e \in H} A_e} ) \geq c(|J|, M_d, F)$$
for some $c(|J|, M_d, F) > 0$. On the other hand, $\bigcap_{e \in H} A_e$ is contained in $\bigcap_{e \in H_d} E_e$, which has density
at most $\delta$ by the hypothesis \eqref{E-dens}. Thus if $\delta$ is sufficiently small depending on $|J|$, $M_d$,
$F$, we see that no atom $\bigcap_{e \in H} A_e$ with $A_e = E_e$ for $e \in H_d$ can possibly be good.
Now let $B_{e,A_e}$ be as in Lemma \ref{good-lots}. Let us define
$$ E'_e := V_J \backslash \bigl( B_{e, E_e} \cup \bigcup_{f \subsetneq e} \bigcup_{A_f} A_f \cap B_{f, A_f} \bigr)$$
for all $e \in H_d$, where for brevity we adopt the convention that $A_f$ is always understood to range over the atoms of ${\mathcal{B}}_f$. Then we observe that $E'_e \in \bigvee_{f \subsetneq e} {\mathcal{B}}_f$. The claims \eqref{E-complex}, \eqref{E-meas}
then follow from \eqref{coarse-complex-2}.
Also, from Lemma \ref{good-lots}, \eqref{coarse-complex-2} we see that for any $e \in H_d$,
\begin{align*}
{\hbox{\bf E}}( 1_{E_e \backslash E'_e} ) &\leq {\hbox{\bf E}}( 1_{E_e} 1_{B_{e,E_e}} ) +
\sum_{f \subsetneq e} \sum_{A_f} {\hbox{\bf E}}( 1_{A_f} 1_{B_{f, A_f}} ) \\
&\leq O(F(M_d)^{-1/10}) + \sum_{0 \leq j < d} \sum_{f \in H_j} \sum_{A_f} O( 1 / \log F(M_j) ) \\
&\leq O(F(M_d)^{-1/10}) + \sum_{0 \leq j < d} \sum_{f \in H_j} O_{M_j}( 1 / \log F(M_j) ) \\
&\leq \sup_{0 \leq j \leq d} O_{M_j, |J|}(1 / \log F(M_j)).
\end{align*}
If one chooses $F$ sufficiently rapidly growing (depending only on $|J|$), we conclude from \eqref{growth-cond} that we have
$$ {\hbox{\bf E}}(1_{E_e \backslash E'_e}) = o_{M_d \to 0; |J|}(1).$$
By choosing $M_d$ sufficiently large depending on $|J|$, and then letting $\delta$ be sufficiently small depending on $M_d$ and $|J|$, we
conclude \eqref{E-error}.
The final thing to verify is \eqref{E-cap}. To see this, first observe that this set lies in $\bigvee_{f \in H \backslash H_d} {\mathcal{B}}_f$ and thus is the union of atoms of the form $\bigcap_{f \in H \backslash H_d} A_f$. Suppose for contradiction that $\bigcap_{e \in H_d} E'_e$
contains a non-empty atom of the form $\bigcap_{f \in H \backslash H_d} A_f$. Set $A_e := E_e$ for $e \in H_d$. By the preceding discussion
we know that $\bigcap_{e \in H} A_e$ cannot be good, thus there exists an $f' \in H$ such
that $\bigcap_{g \subsetneq f'} A_g$ lies in $B_{f',A_{f'}}$.
From construction of $H$, there exists $e \in H_d$ which contains $f'$.
But then by definition of $E'_e$, $\bigcap_{f \in H \backslash H_d} A_f$ cannot lie in $E'_e$, contradiction. Thus
$\bigcap_{e \in H_d} E'_e$ is empty, which is \eqref{E-cap}, and Theorem \ref{main-2} follows.
\end{proof}
It remains to prove the counting lemma. This will be accomplished in the next section.
\section{Proof of counting lemma}
We now prove Lemma \ref{count-lemma}. Fix a good collection $(A_e)_{e \in H}$ of atoms. We introduce the numbers $p_e \in {\hbox{\bf R}}$,
the functions $b_e, c_e: V_J \to {\hbox{\bf R}}$, and the sets $A_{<e} \subseteq V_J$ for all $e \in H$ by the formulae
\begin{align*}
p_e &:= {\hbox{\bf E}}( 1_{A_e} | \bigcap_{f \in \partial e} A_f ) \\
b_e &:= {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}_f ) \\
c_e &:= 1_{A_e} - {\hbox{\bf E}}( 1_{A_e} | \bigvee_{f \in \partial e} {\mathcal{B}}'_f ) \\
A_{<e} &:= \bigcap_{f \subsetneq e} A_f.
\end{align*}
Note that we have not yet shown that $\bigcap_{f \in \partial e} A_f$ is non-empty; for now, let us just assign an arbitrary value
to $p_e$ (e.g. $p_e = 1$) when $\bigcap_{f \in \partial e} A_f$ is empty. We thus have the decomposition
\begin{equation}\label{e-decomp}
1_{A_e} = p_e + b_e + c_e
\end{equation}
on the set $\bigcap_{f \in \partial_e} A_f$. One should think of the constant $p_e$ as the main term, and the other two
terms as error terms. The $c_e$ error term will be very easy to handle, whereas the $b_e$ error term will cause
somewhat more difficulty. Since $(A_e)_{e \in H}$ is good, we have the estimates
\begin{equation}\label{pe-big}
p_e \geq 1 / \log F(M_j) \hbox{ for all } 0 \leq j \leq d \hbox{ and } e \in H_j
\end{equation}
and
\begin{equation}\label{ge-small}
{\hbox{\bf E}}( |b_e|^2 1_{A_{<e}} ) \leq F(M_j)^{-1} {\hbox{\bf E}}( 1_{A_{<e}} ) \hbox{ for all } 0 \leq j \leq d \hbox{ and } e \in H_j.
\end{equation}
From \eqref{fine-accurate-2} and \eqref{Psie}, we also have
\begin{equation}\label{he-small}
|{\hbox{\bf E}}( c_e \prod_{f \in \partial e} 1_{E_f} )| \leq \frac{1}{F(M_0)} \hbox{ whenever } E_f \in {\mathcal{A}}_f \hbox{ for } f \in \partial e.
\end{equation}
Our objective is to use the above estimates \eqref{e-decomp}, \eqref{pe-big}, \eqref{ge-small}, \eqref{he-small} to conclude that
\begin{equation}\label{ae}
{\hbox{\bf E}}( \prod_{e \in H} 1_{A_e} ) = (1 + o_{M_d \to \infty; |J|}(1)) \prod_{e \in H} p_e + O_{|J|, M_0}(\frac{1}{F(M_0)}).
\end{equation}
This will be achieved by several applications of the Cauchy-Schwarz and triangle inequalities. However, there is a certain amount of notational
burden in order to keep track of the expressions in the succesive applications of these inequalities. It will be convenient to
return to the original sets $(V_j)_{j \in J}$. We can identify $A_e \in {\mathcal{B}}_e$ as a subset $\overline{A_e}$ of $V_e = \prod_{j \in e} V_j$, and similarly we can view
the ${\mathcal{A}}_e$-measurable $b_e$ and $c_e$ as functions $\overline{b_e}$ and $\overline{c_e}$
on $V_e$. One can then write \eqref{ae} in the form
\begin{equation}\label{vj-form}
\begin{split}
\frac{1}{\prod_{j \in J} |V_j|} &\sum_{(v_j)_{j \in J} \in \prod_{j \in J} V_j}\ \prod_{e \in H} 1_{\overline{A_e}}\bigl( (v_j)_{j \in e} \bigr)
\\
&=
\bigl(1 + o_{M_d \to \infty; |J|}(1)\bigr) \prod_{e \in H} p_e + O_{|J|, M_0}\left(\frac{1}{F(M_0)}\right).
\end{split}
\end{equation}
For inductive purposes we will need to generalize\footnote{The basic problem is that we need the Cauchy-Schwarz inequality to eliminate each of the $\overline{b_e}$ factors in turn (using \eqref{ge-small}), but each time we apply this inequality we essentially double the number of free variables that one has to sum or average over. In particular, one ends up sampling more than one point from
each vertex class $V_j$, which forces us to leave the probabilistic framework that has been so convenient for us in preceding sections and return to a combinatorial framework. One could stay in the probabilistic framework using the machinery of tensor products (and conditional tensor products) of probability spaces, but this would introduce even more excessive notation into an already notation-heavy argument and would probably not be helpful to the reader.}
this formula.
\begin{definition}[Hypergraph bundle] A \emph{hypergraph bundle} over $H$ is a hypergraph $G \subseteq 2^K$ on a finite set $K$, together
with a map $\pi: K \to J$ (which we call the \emph{projection map} of the bundle), which is a hypergraph homomorphism (i.e. for each edge $g \in G$, the function $\pi$ is injective on $g$ and $\pi(g) \in H$). For any $g \subseteq K$, we write $V_g$ for the product set $V_g := \prod_{k \in g} V_{\pi(k)}$. We say that the bundle is \emph{closed under set inclusion} if whenever $g \in G$ and $g' \subset g$, we have $g' \in G$.
\end{definition}
\begin{remark} From a probabilistic viewpoint, the probability space $V_J$ corresponds to sampling one vertex independently from each of the vertex classes $V_j$ of $V_J$, whereas the more general spaces $V_g$ correspond to the possibility of sampling more than one vertex independently from each of the vertex classes.
\end{remark}
The generalization of the formula \eqref{vj-form} is then
\begin{lemma}[Generalized counting lemma]\label{gencount} Let $G \subseteq 2^K$ be a hypergraph bundle over $H$ which is closed under set inclusion,
with projection map $\pi: K \to J$. Let
$d' := \sup_{g \in G} |g|$ be the order of $G$. Then, if $F$ is sufficiently rapidly growing depending on $d'$, $|J|$ and $|K|$, we have
\begin{equation}\label{vk-count}
\begin{split}
&\frac{1}{|V_K|} \sum_{(v_k)_{k \in K} \in V_K}\ \prod_{g \in G} 1_{\overline{A_{\pi(g)}}}( (v_k)_{k \in g} ) \\
&=
\bigl(1 + o_{M_d \to \infty; d', |J|, |K|}(1)\bigr) \prod_{g \in G} p_{\pi(g)} + O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right).
\end{split}
\end{equation}
\end{lemma}
Observe that \eqref{vj-form} is the special case of this lemma with $G = H$ (and $K = J$, and $\pi$ being the identity map); note from construction of $H$ that $H$ is automatically closed under set inclusion.
\begin{proof} We shall use a double induction. Firstly, we shall induct on the order $d'$ of the bundle $G$. When $d' = 0$ the claim is
vacuously true (the left-hand side and the main term of the right-hand side is equal to 1), so we may assume $d' \geq 1$ and the claim has already been proven for $d'-1$ and for all choices of hypergraph bundle $G \subseteq 2^K$ which are closed under set inclusion.
Next, we fix $K$ and induct on the quantity $r := |\{ g \in G: |g| = d' \}|$, which is a positive integer between $1$ and $2^{|K|}$. We thus assume
that the claim has already been proven for all smaller values of $r$ (note that for $r=0$ this follows from the previous induction hypothesis).
The constants may change as we progress in this induction, but since the number of steps in the induction cannot exceed $2^{|K|}$, this will not be a concern.
Let $g_0 \in G$ be such that $|g_0| = d'$. We use \eqref{e-decomp} to split
\begin{align*}&\prod_{g \in G} 1_{\overline{A_{\pi(g)}}}( (v_k)_{k \in g} ) =\\
&\left[\prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}( (v_k)_{k \in g} )\right]
\left( p_{\pi(g_0)} + \overline{b_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr) + \overline{c_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr) \right)
\end{align*}
and consider the contribution of the three terms separately.
We first consider the contribution of the $p_{\pi(g)}$ term, which is the main term. Applying the second induction hypothesis to $G \backslash \{g_0\}$ we see from
\eqref{vk-count} that
\begin{align*}
&\frac{1}{|V_K|} \sum_{(v_k)_{k \in K} \in V_K} \prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr)
\\
&=
\bigl(1 + o_{M_d \to \infty; d', |J|, |K|}(1)\bigr) \prod_{g \in G \backslash \{g_0\}} p_{\pi(g)} + O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right).
\end{align*}
Multiplying this by the quantity $p_{\pi(g_0)}$, which is between 0 and 1, we see that the contribution of this term to \eqref{vk-count} is
\begin{equation}\label{contrib-1}
(1 + o_{M_d \to \infty; d', |J|, |K|}(1)) \prod_{g \in G} p_{\pi(g)} + O_{d', |J|, |K|, M_0}(\frac{1}{F(M_0)}).
\end{equation}
Next we consider the $\overline{c_{\pi(g_0)}}$ term. We split $V_K = V_{g_0} \times V_{K \backslash g_0}$. Let us temporarily freeze the values of $v_k$
for $k \in K \backslash g_0$, and consider the expression
$$
\frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}}
\left[ \prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]
\overline{c_{\pi(g_0)}}\bigl((v_k)_{k \in g_0} \bigr).$$
Observe that for each $g \in G \backslash \{g_0\}$, we have $g \neq g_0$ and $|g| \leq d' = |g_0|$. Thus $g \cap g_0$ is a proper subset of $g_0$, and
thus there exists an element of $\partial g_0$ which contains $g \cap g_0$. Thus one can rewrite the product $\prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr)$ in the form
$$ \prod_{f \in \partial g_0} 1_{E_f}\bigl( (v_k)_{k \in \pi(f)} \bigr)$$
for some sets $E_f \subseteq V_f$ whose exact form is not important here (we allow the $E_f$ to depend on the frozen $v_k$). Applying \eqref{he-small}, we conclude that
$$
\left|\frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}}
\left[ \prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]
\overline{c_{\pi(g_0)}}\bigl((v_k)_{k \in g_0} \bigr)\right| \leq 1/F(M_0).$$
Averaging this over all choices of the frozen variables $k \in K \backslash g_0$, we conclude that the contribution of this term to \eqref{vk-count}
is at most
\begin{equation}\label{fm0}
1/F(M_0).
\end{equation}
Finally we consider the contribution of the $\overline{b_{\pi(g_0)}}$ term, which is the most difficult from a notational viewpoint to handle,
mainly because of the need to invoke the Cauchy-Schwarz inequality. We expand this contribution as
$$
\frac{1}{|V_K|} \sum_{(v_k)_{k \in K} \in V_K}
\left[ \prod_{g \in G \backslash \{g_0\}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]
\overline{b_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr).$$
We take absolute values and discard\footnote{This discarding step is important as it lowers the total order of the expression being computed, which compensates for a certain doubling of the hypergraph bundle which shall occur shortly when we apply Cauchy-Schwarz. We can get away with this step because the smallness of $b_{\pi(g_0)}$, as given by \eqref{ge-small}, safely dominates any loss we absorb by discarding these high-order factors.} the bounded factors $1_{\overline{A_{\pi(g)}}}( (v_k)_{k \in g} )$ with $|g| = d'$, to estimate this expression by
$$
O\left( \frac{1}{|V_K|} \sum_{(v_k)_{k \in K} \in V_K} \left[ \prod_{g \in G_{\subsetneq g_0} \cup G'} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]
\bigl|\overline{b_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr)\bigr| \right)$$
where $G_{\subsetneq g_0} := \{ g: g \subsetneq g_0 \}$ and $G' := \{g \in G \backslash G_{\subsetneq g_0}: |g| \leq d'-1 \}$. We factorize this
as
\begin{equation}\label{precauchy}
\begin{split}
O\biggl( \frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}}&
\left[ \prod_{g \in G_{\subsetneq g_0}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right] \left|\overline{b_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr)\right|\\
&
\left[ \frac{1}{|V_{K \backslash g_0}|} \sum_{(v_k)_{k \in K \backslash g_0} \in V_{K \backslash g_0}}
\prod_{g \in G'} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right] \biggr).
\end{split}
\end{equation}
On the other hand, from \eqref{ge-small} we have
\begin{align*}
\frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}} &
\left[\prod_{g \in G_{\subsetneq g_0}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr)\right]\\
& \bigl|\overline{b_{\pi(g_0)}}\bigl((v_k)_{k \in g_0}\bigr)\bigr|^2
\leq \frac{1}{F(M_{d'})} {\hbox{\bf E}}(1_{\overline{A_{<\pi(g_0)}}}),
\end{align*}
and hence by Cauchy-Schwarz we can estimate \eqref{precauchy} by
\begin{equation}\label{post-cauchy}
\begin{split}
O\Biggl( F(M_{d'})^{-1/2} &{\hbox{\bf E}}(1_{\overline{A_{<\pi(g_0)}}})^{1/2}
\biggl(\frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}}
\left[\prod_{g \in G_{\subsetneq g_0}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr)\right]\\
&\left[ \frac{1}{|V_{K \backslash g_0}|} \sum_{(v_k)_{k \in K \backslash g_0} \in V_{K \backslash g_0}}
\prod_{g \in G'} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} ) \bigr) \right]^2 \biggr)^{1/2} \Biggr.
\end{split}
\end{equation}
From the first induction hypothesis we have
$$ {\hbox{\bf E}}(1_{\overline{A_{<\pi(g_0)}}}) = \bigl(1 + o_{M_d \to \infty; d', |J|}(1)\bigr) \prod_{g \in G_{\subsetneq g_0}} p_{\pi(g)} + O_{d', |J|, M_0}\left(\frac{1}{F(M_0)}\right)$$
and thus
\begin{equation}\label{pag}
{\hbox{\bf E}}(1_{\overline{A_{<\pi(g_0)}}}) = O_{M_d, d', |J|}(\prod_{g \in G_{\subsetneq g_0}} p_{\pi(g)}) + O_{d', |J|, M_0}\left(\frac{1}{F(M_0)}\right).
\end{equation}
Now we estimate the expression in parentheses in \eqref{post-cauchy}. As we shall see, this expression can be rewritten in a form
which can be handled by the induction hypothesis, but with the hypergraph bundle $G$ replaced by a hypergraph of approximately
twice the size (roughly speaking, we throw away all edges of top order $d'$, and double all the remaining edges that are not
contained in $G_{\subsetneq g_0}$). It is this doubling which forces us to work with a generalized counting lemma\footnote{There is a possible alternate approach which avoids the Cauchy-Schwarz inequality, and hence the need to work with hypergraph bundles. One can attempt to use the lower-order induction hypothesis to show some uniform distribution properties concerning the intersections of the lower-order atoms with each other, in order that the contribution of the $b_{g_0}$ error be shown to be negligible. A model example of such a statement, in the graph setting, would be the assertion that in an $\varepsilon$-regular graph $H$, the number of copies of a fixed small graph $G$ in $H$, with one edge specified to be $(x,y)$, is usually close to a fixed quantity independent of $x$ and $y$, except for a small number of exceptional pairs $(x,y)$. We will not pursue such an alternate approach here.}
rather than the original counting lemma.
Let $\tilde K = K \oplus_{g_0} K$ be the set $K \times \{0,1\}$, with the
elements $(k,0)$ and $(k,1)$ identified for all $k \in g_0$. There is an obvious projection $\phi: \tilde K \mapsto K$, and hence a map $\pi \circ \phi: \tilde K \to H$. On $\tilde K$ we also place a hypergraph bundle $\tilde G$, defined as the set $\{ g \times \{i\}: g \in G_{\subsetneq g_0} \cup G', i \in 1,2\}$; note that $g \times \{0\}$ and $g \times \{1\}$ will be identified when $g \in G_{\subsetneq g_0}$. From the definitions we observe that
\begin{align*}
&\frac{1}{|V_{g_0}|} \sum_{(v_k)_{k \in g_0} \in V_{g_0}}
\left[\prod_{g \in G_{\subsetneq g_0}} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]
\left[ \frac{1}{|V_{K \backslash g_0}|} \sum_{(v_k)_{k \in K \backslash g_0} \in V_{K \backslash g_0}}
\prod_{g \in G'} 1_{\overline{A_{\pi(g)}}}\bigl( (v_k)_{k \in g} \bigr) \right]^2 \\
&= \frac{1}{|V_{\tilde K}|} \sum_{(v_{\tilde k})_{\tilde k \in \tilde K} \in V_{\tilde K}}
\prod_{\tilde g \in \tilde G} 1_{\overline{A_{\pi \circ \phi(\tilde g)}}} \bigl( (v_{\tilde k})_{\tilde k \in \tilde g} \bigr).
\end{align*}
Applying the first induction hypothesis, we can write this expression as
\begin{equation}\label{moo}
(1 + o_{M_d \to \infty; d', |J|, |K|}(1)) \prod_{\tilde g \in \tilde G} p_{\pi \circ \phi(\tilde g)} + O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right).
\end{equation}
By the definition of $\tilde G$, we can write
$$ \prod_{\tilde g \in \tilde G} p_{\pi \circ \phi(\tilde g)} = \prod_{g \in G_{\subsetneq g_0}} p_{\pi(g)} \times
[\prod_{g \in G'} p_{\pi(g)}]^2$$
and thus by \eqref{pe-big} and \eqref{growth-cond} we can rewrite \eqref{moo} as
$$ O_{M_d, d', |J|, |K|}\left(\prod_{g \in G_{\subsetneq g_0}} p_{\pi(g)}
\left[\prod_{g \in G'} p_{\pi(g)}\right]^2
\right) + O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right).$$
Inserting this and \eqref{pag} back into \eqref{post-cauchy}, we can estimate \eqref{post-cauchy} by
$$ O_{M_d, d', |J|, |K|}\left( F(M_{d'})^{-1/2} \prod_{g \in G_{\subsetneq g_0}} p_{\pi(g)} \prod_{g \in G'} p_{\pi(g)} \right)
+ O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right) .$$
Re-inserting those elements $g$ of $G$ for which $|g| = d'$ using \eqref{pe-big}, we can estimate this by
$$ O_{M_d, d', |J|, |K|}( F(M_{d'})^{-1/4} \prod_{g \in G} p_{\pi(g)})
+ O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right)$$
(for instance). By choosing $F$ sufficiently rapid depending on $d'$, $|J|$, $|K|$, we can write this as
$$ o_{M_d \to \infty; d', |J|, |K|}(\prod_{g \in G} p_{\pi(g)}) + O_{d', |J|, |K|, M_0}\left(\frac{1}{F(M_0)}\right).$$
Combining this with the bounds \eqref{contrib-1}, \eqref{fm0} we obtain \eqref{vk-count}, which closes the induction.
This completes the proof of Lemma \ref{gencount}, and hence Lemma \ref{count-lemma}.
\end{proof}
|
{
"timestamp": "2005-11-16T18:07:49",
"yymm": "0503",
"arxiv_id": "math/0503572",
"language": "en",
"url": "https://arxiv.org/abs/math/0503572"
}
|
\section{\label{sec:level1}First-level heading:\protect\\ The line
Implementing atom-optical devices often requires a strong confinement for all except one
degree of freedom~\cite[and Refs therein]{intro}. Examples of physical situations where a
strong confinement is needed are guided matter-wave interferometers~\cite{intro}, one
dimensional optical lattices~\cite{morsch2002a}, cold gases in very elongated traps for
studies of superfluidity~\cite{cataliotti2003a}, the Tonks-Girardeau
gas~\cite{paredes2004a} or phase fluctuations of quasi-condensates~\cite{dettmer2001a}. A
proper description of the dynamics of such reduced quasi-1D systems should account for the
nature of the discrete transverse states. Therefore one needs to deduce the effective 1D
interaction between the remaining longitudinal degrees of freedom from the real 3D
free-space interaction potential $V(r)$. Collisions under confinement different from the
1D case are treated in~\cite{petrov2000b,diffg}. Apart from ultracold {\em atom-atom
collisions}, scattering in confined geometries also occurs in various physical situations
such as scattering of guided atomic matter waves or of guided electromagnetic and
accoustic waves~\cite{olsson1981} from {\em obstacles inside a guide}, e.g., (heavy)
impurity atoms or material defects, respectively. The latter is of importance for the
propagation of radiation or sound within transmission lines or resonators.
As for atom-atom collisions, resonant quasi-1D scattering in the transverse ground state
of the guide (single mode regime) was first considered for bosons in an harmonic guide
employing for the interaction potential $V(r)$ a delta-like zero-range
approximation~\cite{olshanii1998a}. Numerical simulations~\cite{bergeman2003} confirmed
for certain finite range potentials $V(r)$ the existence of the so-called confinement
induced resonance (CIR) originally predicted in~\cite{olshanii1998a}. A further
investigation of the CIR is provided in~\cite{granger2004a} dealing for the first time
with a general finite-range $V(r)$ for both bosons and fermions under harmonic
confinement. Effects of the non-parabolicity of the confinement are considered
in~\cite{peano2004a}, with focus on the center of mass dynamics and employing a zero-range
approximation for the interaction $V(r)\,$.
The present work extends the above approaches and gives an alternative and complementary
description of scattering under confinement, treating both the cases of collisions and of
scattering by fixed obstacles. We develop a general formalism based on the Green's
functions that allows us to express the scattering properties in confined geometries in
terms of the phase-shifts $\delta_l$ of free-space scattering. The coupling between these
phase-shifts is explicitly taken into account. A general initial scattering state can be
treated properly, describing in particular the ``multi-channel'' regime, in the sense that
the total energy allows several transversal excited states to be effectively occupied.
In the case of collisions where $V(r)$ is the atom-atom interaction potential, the center
of mass motion is known to separate from the relative one only for a {\em parabolic
confinement}. Our approach then provides a deeper understanding of this collision
process. On the other hand, for an {\em arbitrary confinement}, scattering processes that
can naturally be described by the formalism include, e.g., the quantum scattering of
individual cold atoms, or other equivalent systems, by a central field $V(r)$ fixed in the
center of the guide at $\bm{r}=0$. As for atom-atom scattering, the relative coordinates
$\bm{r}$ are not exactly separable from the center of mass coordinates $\bm{R}\,$ if the
confinement is no longer parabolic. Nevertheless, in such a situation of coupled center of
mass and relative motion, the formalism provides in the ultracold regime a distinct
starting-point to account for this coupling.
Under the above restrictions concerning atom-atom collisions, our investigation confirms
that the CIR~\cite{olshanii1998a} is a general consequence of the dominant terms of the
scattering amplitudes. The main requirements are a large positive $s$-wave scattering
length $a$, $a\sim l_\perp$ [$l_\perp$ is the length scale of the confining
potential $U(\rho)\,$, such that $U\approx 0$ for $\rho\ll l_\perp$, and equals the
cylinder radius for a square-well type confinement], a short-ranged scattering potential
$V(r)$, $R_V\ll l_\perp$ [$R_V$ is the range of $V(r)\,$, such that $V\approx 0$ for $r\gg
R_V$], small longitudinal momenta and small phase-shifts $\delta_l$, as described below.
The resonance is accompanied by a $l=0$ bound-state of $V(r)$ strongly distorted by the
confinement $U(\rho)$ and pushed towards the continuum. This modified bound
state~\cite{bergeman2003} is shown to be a herald of the CIR. In the context of scattering
of individual guided atoms by a central field, these conclusions hold irrespective of
restrictions due to anharmonicities and imply the unambiguous strong effects of
confinement on the scattering process.
{\em Phase-Shifts}.\, The Schr\"odinger equation for the scattering wave function
$\Psi(\bm{r})$, with $\bm{r}=(\bm{\rho},z)$, reads
\begin{equation}
\label{diff}
\left[\nabla^2 - u(\rho) + k^2\right]\Psi(\bm{r}) = v(r)\Psi(\bm{r}),
\end{equation}
where $u(\rho)\equiv 2\mu U(\rho)/\hbar^2$, $v(r)\equiv 2\mu V(r)/\hbar^2$, and
$E=\hbar^2k^2/2\mu > 0$ is the total energy. In the case of atomic collisions, $\bm{r}$
is the relative coordinate and the relation between $U(\rho)$ and the confining potential
$U_c(\rho_i)$ of the $i$-th particle in the laboratory reference frame is given by
$U(\rho)=2\,U_c(\rho/2)\,$. Note that this relation is no longer exact for non-parabolic
$U_c$ (but provides a first uncoupled description of the relative motion, by quenching the
center of mass at the origin $\bm{R}=0$). The cylindrical boundary condition is met by
expanding the solution in the transverse eigenstates $\varphi_n(\rho)$, with energies
$\epsilon_n\equiv \hbar^2q_n^2/2\mu$ and normalized to $\int
dxdy\,\varphi_n(\rho)^\ast\varphi_m(\rho)=\delta_{nm}$. As a result, one obtains the
integral equation
\begin{equation}
\label{int}
\Psi(\bm{r}) = \Psi_i(\bm{r})-\int d^3\bm{r}^\prime
G_c(\bm{r},\bm{r}^\prime) v(r^\prime)\Psi(\bm{r}^\prime).
\end{equation}
For a given $k$ low enough such that $k\sim 1/l_\perp$, let $n_E$ be the integer obeying
$k^2=q_{n_E}^2+k_{n_E}^2\leq q_{1+n_E}^2$. The following study includes the situations of
ground state scattering ($n_E=0$) as well as scattering in the {\em transversally excited}
modes ($n_E\geq 1$). In both cases, transverse states with $n>n_E$ can only be {\em
virtually} occupied, since $k^2<q_n^2\,$. The general initial state is
$\Psi_i(\bm{r})=\sum_{n=0}^{n_E}b_ne^{ik_nz}\varphi_n(\rho)$ for some constants $b_n$,
with $k^2=q_n^2+k_n^2$. In Eq.(\ref{int}),
\begin{equation}
\label{gc}
G_c(\bm{r},\bm{r}^\prime) =
\sum_{n=0}^\infty\varphi_n(\rho)\varphi_n(\rho^\prime)^\ast G_n(z-z^\prime)
\end{equation}
is an axially symmetric Green's function and $G_n(z)=-e^{ik_n|z|}/2ik_n$ (for $n\leq n_E$) and
$G_n(z)=e^{-p_n|z|}/2p_n$ (with $k^2=q_n^2-p_n^2$, for $n\geq 1+n_E$) are 1D Green's
functions. The excited states with quantum numbers larger than $n_E$ decrease
exponentially with increasing distance from the scattering region. In the asymptotic
limit $|z|\rightarrow\infty$, one has for $n\leq n_E$
\begin{subequations}
\label{c-boundary}
\begin{eqnarray}
\label{asymptotic}
\hspace{-1.5em}
\Psi(\bm{r}) &\approx&
\sum_{n=0}^{n_E}\left[b_ne^{ik_nz} + f_n^\pm\,e^{ik_n|z|}\right]\varphi_n(\rho),
z\rightarrow\pm\infty\, , \\
\label{f1d}
\hspace{-1em}
f_n^\pm &\equiv & \frac{1}{2ik_n}\int d\bm{r^\prime}\left[e^{\pm ik_nz^\prime}
\varphi_n(\rho^\prime)\right]^\ast v(r^\prime)\Psi(\bm{r}^\prime),
\end{eqnarray}
\end{subequations}
where $f_n^\pm$ is the $n$-th channel {\em effective 1D scattering amplitude} for forward
$z>0$ and backward $z<0$ scattering.
Consider next $G_c(\bm{r},\bm{r}')$ for $r'< r \ll l_\perp$. In this region
$U(\rho)\approx 0$ and one should be able to approximate $G_c$ by the free 3D Green's
functions $G_{1,2}(\bm{r},\bm{r}^\prime)\equiv e^{\pm
ik|\bm{r}-\bm{r}^\prime|}/4\pi|\bm{r}-\bm{r}^\prime|$. Thus, we write
\begin{subequations}
\begin{eqnarray}
\label{green-3D}
G_c(\bm{r},\bm{r}^\prime) & = & \frac{1}{2\pi}\int d\phi^\prime
\left(\gamma_+ \frac{e^{ik|\bm{r}-\bm{r}'|}}{4\pi|\bm{r}-\bm{r}'|}
+ \gamma_- \frac{e^{-ik|\bm{r}-\bm{r}'|}}{4\pi|\bm{r}-\bm{r}'|}\right)
\nonumber\\
& & \hspace{0.5em}
+ \hspace{0.5em} \Delta_c(\bm{r},\bm{r}^\prime) \\
\label{green}
&=& ik\sum_l j_l(kr')
\left[ \gamma_+ h_l^{(1)}(kr) - \gamma_- h_l^{(2)}(kr) \right]
\nonumber\\
& & \hspace{5em}
\times\frac{2l+1}{4\pi}P_l(\cos{\theta})P_l(\cos{\theta'})
\nonumber\\
& & \hspace{1em}
+ \hspace{0.5em} \Delta_c(\bm{r},\bm{r}^\prime)\, ,
\hspace{3.5em} r'< r \ll l_\perp.
\end{eqnarray}
\end{subequations}
In Eq.(\ref{green}), we have used the well known expansion of $G_{1,2}$ in spherical
coordinates~\cite[Prob.7.5]{morse1953}. The value of $\gamma_{\pm}$ and $\Delta_c$ can be
explicitly obtained if $U(\rho)$ is approximated by a {\em square-well} type
confinement for $r\ll l_\perp$. Indeed, the eigenstates are then close to Bessel
functions, $\varphi_n(\rho)\approx N_nJ_0(q_n\rho)/\pi^{1/2}l_\perp$, normalized on a disc
of radius $l_\perp$, $N_n=1/|J_1(r_{n+1})|$, $r_{n+1}$ being the $(n+1)$-th root of
$J_0$. Separating from $G_c$ the terms $n\leq n_E$, the series for $n>n_E$ can be
approximated by an integral over $q$ emerging from the continuum limit $q_n\rightarrow q$
and valid when $r',r\ll l_\perp$. Note that $q$ starts at $q_{1+n_E}>k$. One then compares
real and imaginary parts of $G_c$ in Eq(\ref{gc}) with a suitable expansion of $G_{1,2}$
in {\em cylindrical} coordinates~\cite[Prob.7.9]{morse1953} in Eq(\ref{green-3D}). This
comparison leads to
\begin{subequations}
\label{parameters}
\begin{eqnarray}
\label{gamma}
& & \hspace{-1em}
\gamma_{\pm} = 1/2 \pm\gamma/2\, ,
\hspace{2em}\gamma\equiv \sum_{n=0}^{n_E}2N_n^2/kk_nl_\perp^2\,,\\
\label{deltac}
& & \hspace{-2.4em} \Delta_c(\bm{r},\bm{r}^\prime)\equiv
- \frac{1}{4\pi}\int_0^{p_c}dp\, e^{-p|z-z'|} J_0(q\rho)J_0(q\rho')\, ,
\end{eqnarray}
\end{subequations}
with $q=\sqrt{k^2+p^2}$ and $q_{1+n_E}\equiv\sqrt{k^2+p_c^2}\,$. The homogeneous
(Helmholtz) term $\Delta_c$ corrects the Green's function $\gamma_+G_1+\gamma_-G_2$, with
$\gamma_++\gamma_-=1$, in order to account for the discreteness due to the
confinement. Within the flatness condition, the above approach is valid for arbitrary
$U(\rho)$. It yields an intrinsic connection between the confined and the free space
scattering approaches (see \cite{olshanii1998a} for parabolic confinement).
In order to obtain the scattering phases $\delta_l$ that are associated with the spherical
symmetry, we expand the incident state in spherical coordinates employing
$e^{ik_nz}\varphi_n(\rho) = \sum_l i^l(2l+1)\alpha_{nl}j_l(kr)P_l(\cos{\theta})$, with
$\alpha_{nl}=N_nP_l(k_n/k)/\pi^{1/2}l_\perp$~\cite{morse1953}. Analogously in $\Delta_c$,
the equivalent expansion is given by $e^{-pz}J_0(q\rho)=\sum_l
i^l(2l+1)P_l(ip/k)j_l(kr)P_l(\cos{\theta})$ stemming from an analytic continuation into
the complex $\theta$-plane ($\theta\rightarrow\pi/2-i\theta$). Inserting these
expressions and Eq.(\ref{green}) into Eq.(\ref{int}) and using Eq.(\ref{gamma}) yields,
for $R_V\ll r\ll l_\perp$,
\begin{eqnarray}
\label{spherical}
\Psi(\bm{r}) &\approx& \sum_l
i^l(2l+1) \left[\, \alpha_l + \gamma_l(z) - i\gamma kT_l \,\right]j_l(kr)P_l(\cos{\theta})
\nonumber\\
& & \hspace{1em} + \sum_l i^l(2l+1)
\left[\, kT_l \,\right] n_l(kr)P_l(\cos{\theta})\, ,
\end{eqnarray}
with $\alpha_l=\sum_{n=0}^{n_E}b_n\alpha_{nl}$. Here $4\pi T_l\equiv i^{-l}\int d^3\bm{r}'
[j_l(kr')P_l(\cos{\theta'})] v(r')\Psi(\bm{r}')$ and $4\pi\gamma_l(z)\equiv\int_0^{p_c}dp
\int_{(z)} d^3\bm{r}'P_l(\pm ip/k)e^{\pm pz'}J_0(q\rho')v(r')\Psi(\bm{r}')$. The
integration over $\bm{r}'$ for $\gamma_l(z)$ is performed in a finite volume $\Omega$
covering the range of $v(r')$. If $z$ is outside $\Omega$, the positive sign refers to a
positive $z$ and vice-versa. Inside $\Omega$, both signs are needed according to whether
$z\gtrless z'$. Except for this $z$-dependence of $\gamma_l(z)$ in Eq.(\ref{spherical}),
we have now succeeded in representing the total scattering wave function in spherical
coordinates.
Noteworthy at this point is the fact that $\gamma_l(z)$ accounts for {\em couplings}
between different angular momenta. Indeed, by using $e^{\pm pz'}J_0(q\rho') =\sum_{l'}
i^{l'}(2l'+1)P_{l'}(\mp ip/k)j_{l'}(kr')P_{l'}(\cos{\theta}')$ and the property
$P_{l'}(\mp u)=(-)^{l'}P_{l'}(\pm u)$ in the definition of $\gamma_l(z)$, one gets a
constant $\gamma_l(z)$ if, for each $l$, only $l'$-waves are kept such that
$l+l'=\mathrm{even}$. The latter condition is also necessary to obtain non-zero matrix
elements $\langle l\left|U(\rho)\right|l'\rangle$ due to the parity symmetry
$\bm{r}\rightarrow-\bm{r}$. Therefore, a constant $\gamma_l(z)\approx\gamma_l$ arises
\begin{equation}
\label{gammal}
\gamma_l = \sum_{l'[l]} (2l'+1)P_{ll'}T_{l'}\, , \hspace{1.5em} l=0,1,2,\dots\, ,
\end{equation}
where $P_{ll'} \equiv k\int_0^{p_c/k}du\, P_l(iu)P_{l'}(iu)$ and $l'[l]$ denotes the sum
over even (odd) $l'$ for even (odd) $l$. Eq.(\ref{gammal}) is equivalent to the condition
that the ``perturbation'' $U(\rho)$ to the free space scattering does not couple even and
odd angular momenta.
It is now possible to introduce the phase-shifts $\delta_l$. The solution
Eq.(\ref{spherical}) can be written as ($R_V\ll r\ll l_\perp$)
\begin{subequations}
\label{sphe-boundary}
\begin{eqnarray}
\label{spherical-delta}
& & \hspace{-2.5em} \Psi(\bm{r}) \approx \sum_l
c_l'\left[\cos{\delta_l}\,j_l(kr) - \sin{\delta_l}\,n_l(kr)\right]P_l(\cos{\theta}), \\
\label{constants}
& & \hspace{-1.5em}
c_l' \equiv \frac{(2l+1)(\alpha_l+\gamma_l)\,i^l}{\cos{\delta_l} - i\gamma \sin{\delta_l}},
\hspace{1em}
T_l \equiv \frac{\alpha_l+\gamma_l}{i\gamma k - k\cot{\delta_l}}\, ,
\end{eqnarray}
\end{subequations}
where the last two relations {\em define} formally $c_l'$ and $\delta_l$. That this
$\delta_l$ is the actual phase-shift can be seen as follows. On one hand,
Eq.(\ref{spherical-delta}) is the (intermediate) asymptotics $R_V\ll r\ll l_\perp$ of the
solution $\Psi(\bm{r})=\sum_lc_l'R_l(r)P_l(\cos{\theta})$ in the region of $V(r)$. On the
other hand, the free-space scattering solution in this region, i.e., not taking into
account the boundary, is just a {\em different superposition}
$\Psi_{3D}(\bm{r})=\sum_lc_lR_lP_l$ with the {\em same} radial part $R_l$. In other words,
the effect of the confinement $U(\rho)$ is to change the superposition coefficients from
$c_l$ to $c_l'$ while keeping the scattering phases of the free-scattering problem. Then
the second relation in Eq.(\ref{constants}) together with Eq.(\ref{gammal}) gives a {\em
matrix equation for $T_l$} in terms of $\delta_l$, i.e., for $l=0,1,2,\dots$
\begin{subequations}
\label{main}
\begin{equation}
\label{tmatrix}
\left(i\gamma k - k\cot{\delta_l}\right)T_l = \alpha_l
+ \sum_{l'[l]} (2l'+1)P_{ll'}T_{l'}\, .
\end{equation}
Finally, the effective amplitude $f_n^\pm$ is given by expanding $e^{\pm
ik_nz'}\varphi_n(\rho')$ in the integrand of Eq.(\ref{f1d}), thus
\begin{equation}
\label{f1d-spherical}
f_n^\pm = f_{ng} \pm f_{nu} \equiv
\left( \sum_{l\,\mathrm{even}} \pm \sum_{l\,\mathrm{odd}} \right)
\frac{(2l+1)4\pi \alpha_{nl}}{2ik_n}T_l\, .
\end{equation}
The relationship between the amplitudes in Eq.(\ref{f1d-spherical}) and the matrix
elements $T_l$ of Eq.(\ref{tmatrix}) constitutes the main result of our formalism.
{\em Current Conservation}. Inserting Eqs.(\ref{tmatrix},\ref{f1d-spherical}) into
Eq.(\ref{c-boundary}), the probability conservation should follow. From the total current
along the $z$-axis, the conservation condition is
\begin{equation}
\label{conservation}
\hspace{-0.15em}
0=\sum_{n=0}^{n_E}(|f_{ng}|^2 + \mathrm{Re}\{b_n^\ast f_{ng}\}
+ |f_{nu}|^2 + \mathrm{Re}\{b_n^\ast f_{nu}\})k_n.
\end{equation}
\end{subequations}
In the remainder of this paper, we analyse the scattering process given by the leading
terms of Eqs.(\ref{main}). We consider first the case of the single mode regime in more
detail, followed by the case of transverse excitations and angular momenta couplings.
{\em Single Mode Resonances}. When only the ground state ($n_E=0$, $b_n=\delta_{0n}$,
$k^2=q_0^2+k_0^2$) represents an open channel, the symmetric and antisymmetric sectors of
Eq.(\ref{asymptotic}), $\Psi(\bm{r})=[\psi_g(z)+\psi_u(z)]\,\varphi_0(\rho)$, are given
respectively by (for $z\gtrless 0$)
\begin{subequations}
\label{sectors}
\begin{eqnarray}
\label{sectors-g}
\hspace{-2em}
\psi_g(z) &=& (1+f_{0g})\cos{(k_0z)}+if_{0g}\sin{(k_0|z|)},\\
\hspace{-2em}
\psi_u(z) &=& i(1+f_{0u})\sin{(k_0z)}\pm f_{0u}\cos{(k_0z)}.
\end{eqnarray}
\end{subequations}
In the context of collisions between identical particles, it is clearly seen that, at
resonance $f_{0g}=-1$, the bosonic sector $\psi_g$ is mapped into a non-interacting
$f_{0u}=0$ pair of (spin-polarized) fermions, the well known fermionization of
impenetrable bosons. Now, the inverse is also seen to occur for $\psi_u$ at the fermionic
resonance, $f_{0u}=-1$, first obtained in~\cite{granger2004a}. A further insight is gained
by setting
\begin{equation}
\label{sectors-normalized}
f_{0g,u} = - \left[ 1 + i\cot{\delta_{g,u}}\right]^{-1}.
\end{equation}
The conservation condition Eq.(\ref{conservation}) is then fulfilled for real 1D
phase-shifts $\delta_{g,u}$ and one can rewrite
$\psi_g=e^{i\delta_g}\cos{(k_0|z|+\delta_g)}$ and
$\psi_u=ie^{i\delta_u}\sin{(k_0z\pm\delta_u)}$. Thus at resonance $|\delta_{g,u}|=\pi/2$
and the above discussed boson-fermion and fermion-boson mappings exist also under
longitudinal confinement, e.g., by imposing $\psi_{g,u}(z=l_\parallel)=0$, as numerically
verified in Ref.~\cite{granger2004a}.
{\em CIR and bound-states}. The resonance $f_{0g}=-1$ can be calculated from a general
potential $V(r)$ by solving Eq.(\ref{tmatrix}) for even $l$. Since $kR_V\sim
R_V/l_\perp\ll 1$, the phase-shifts $\tan{\delta_l}=\tan{\delta_l(k)}\sim k^{2l+1}\sim
1/l_\perp^{2l+1}$ are generally small~\cite{mott1965} for large $l_\perp$. From
Eq.(\ref{tmatrix}), it follows that $l=0$ is the leading contribution and $f_{0g}$ has the
form compatible with Eq.(\ref{sectors-normalized})
\begin{subequations}
\label{leading}
\begin{equation}
\label{swave}
f_0^\pm \approx f_{0g} \approx - \frac{1}
{1+i\left[-\,\frac{d_\perp^2}{2a}\left(1-aP_{00}\right)\right]k_0 }\,,
\end{equation}
where $d_\perp\equiv l_\perp/N_0\,$,
$P_{00}=p_c$, and $a$ is the 3D $s$-wave scattering length, $k\cot{\delta_0}\approx
-1/a$. This corresponds to solving for $z$ under an effective 1D pseudopotential
$V_{1D}(z)=g_{1D}\delta(z)$, with the coupling strength
\begin{equation}
\label{gswave}
g_{1D} =
\frac{\hbar^2}{\mu} \frac{2a}{d_\perp^2}\left(1-\frac{C'a}{d_\perp}\right)^{-1},
\hspace{1em} C'\equiv d_\perp p_c\, .
\end{equation}
\end{subequations}
As in previous works in the single mode regime (for atom-atom collisions in parabolic
confinement)~\cite{olshanii1998a,bergeman2003,granger2004a}, the resonance
$|g_{1D}|\rightarrow\infty$ at $d_\perp\approx C'a$ requires low longitudinal momenta
$k_0\ll k\sim 1/l_\perp\,$, such that
$p_c\stackrel{k_0\rightarrow 0}{\longrightarrow}\sqrt{q_1^2-q_0^2}$
is not negligible, and large positive scattering length $0<a\sim l_\perp\,$ (meaning that
a weak bound-state of $V(r)$ approaches the threshold~\cite{mott1965}).
For scattering by a central field, not only $V(r)$ but also $U(\rho)$ can be quite
general.
Viewing CIR as a low energy resonant scattering, one could say that bound-states close to
threshold are neither probed at ``high'' energies $k_0\sim 1/l_\perp$
($k\rightarrow q_1$, $p_c\rightarrow 0$), nor do they exist for small scattering lengths
($a\ll d_\perp$). However, by calculating the bound-state with energy $E_B'$, this
interpretation for the physical mechanism behind CIR is not accurate: $f_{0g}\approx -1$
occurs before $E_B'$ approaches zero (threshold without confinement), whereas
$E_B'\rightarrow \epsilon_0$ (threshold under confinement) occurs only if $l_\perp$ is
decreased much further below its CIR value. This is explicitly verified e.g. when
$U(\rho)$ is a square-well box of radius $l_\perp\,$: using a cosine approximation to
$J_0$ for its roots, $q_0\approx 3\pi/4l_\perp$ and $q_1\approx 7\pi/4l_\perp$, whence
$C'=d_\perp\sqrt{q_1^2-q_0^2}=\sqrt{20/3}=2.58$ (see~\cite{bergeman2003} for parabolic
$U(\rho)$ and zero-range atom-atom interaction).
In fact, the outer $l=0$ bound-state of $V(r)$ in the absence of the confinement has the
energy $E_B\equiv -\,\hbar^2\kappa_B^2/2\mu$ that is related to $a$ via $\kappa_B\approx
1/|a|$, when $a\gg R_V$~\cite{mott1965}. Under lateral confinement, its tail
$e^{-\kappa_Br}$ is changed to be zero at the edge $r=\rho=l_\perp$. By the uncertainty
principle, this slight squeeze lifts $E_B<0$ by an amount $\epsilon_0$, which can be
sufficient for this state to pass the limit $E=0$ as $l_\perp$ decreases further.
This new confined bound-state $E_B'$ satisfies Eq.(\ref{diff}) with $k^2$ replaced by
$2\mu E_B'/\hbar^2$, i.e., $k_0\equiv\pm i\sqrt{q_0^2-2\mu E_B'/\hbar^2}$. Since the
diverging term $e^{ik_0z}$ should be absent from Eq.(\ref{asymptotic}) and $e^{ik_0|z|}$
should decay, $1/f_0^\pm$ must vanish at $\mathrm{Im}\{k_0\}>0$. From Eq.(\ref{swave}),
for $a<0$, the virtual bound-state with energy $E_B$ turns into a real one with energy
$E_B'$, which starts at zero for $a/d_\perp=0$ and goes to a positive fraction of
$\epsilon_0$ as $a/d_\perp\rightarrow -\infty$. This bound-state exists only under
confinement and its experimental measurement is reported in~\cite{moritz2005a}. For $a>0$,
one obtains $E_B'\rightarrow E_B$ for $d_\perp\rightarrow\infty$, as expected. For
$a\rightarrow +\infty$ (or $d_\perp\rightarrow 0$), $E_B'$ tends to a positive fraction of
$\epsilon_0$. It turns out that the CIR condition (at $a/d_\perp=1/C'=\sqrt{3/20}\approx
0.39$) occurs before $E_B'$ reaches zero (at $a/d_\perp\approx 0.82$). On the other hand,
the CIR almost coincides with the condition $E_B'+(\epsilon_1-\epsilon_0)=\epsilon_0$ (at
$a/d_\perp\approx 0.35$). In Ref.~\cite{bergeman2003}, this last coincidence is exact,
since $E_B'+(\epsilon_1-\epsilon_0)$ can be associated with a bound-state of the excited
channels $n\geq 1$ due to a special property of the harmonic oscillator. However, despite
this coincidence, a general mechanism behind CIR needs further study, since
$E_B'+(\epsilon_1-\epsilon_0)$ has no clear meaning yet beyond parabolic guides and
zero-range pseudopotentials.
{\em Excited Channels}. At energies $k^2=q_{n_E}^2+k_{n_E}^2> q_0^2$, the case is more
complex. Keeping only the $l=0$ wave as before, the $n$-th scattering amplitude
$f_n^\pm$ is
\begin{equation}
\label{high-energy}
f_n^\pm\approx f_{ng} \approx - \frac{\sum_m b_mN_m/N_n}
{1 + \sigma_n + i\left[-\frac{d_\perp^2}{2}(-k\cot{\delta_0} - P_{00})\right]k_n},
\end{equation}
where $0\leq m,n\leq n_E$, $P_{00}=(q_{1+n_E}^2-k^2)^{1/2}$ and
in $\sigma_n\equiv\sum_{m\neq n}N_m^2k_n/N_n^2k_m$, $m=n$ is excluded. For the {\em
single} incoming excited channel $n_E$, i.e., $b_n=\delta_{n,n_E}$, the amplitude
$f_{n_Eg}$ does have the form Eq.(\ref{sectors-normalized}) at small $k_{n_E}$. Thus, CIR
at {\em threshold energies} $k\rightarrow q_{n_E}$ can occur when
$-\tan{\delta_0}/k=d_\perp/C'$ as first indicated in Ref.~\cite{granger2004a} for
parabolic confinement. In a more realistic situation of finite temperatures $T$, however,
for a given energy each $b_n$ has the same weight (depending on $E/T$ and with random
phases). Since $f_{ng}=-b_n$ cannot be met for all $n$ simultaneously, one expects no
sharp resonance, with the transmission and reflection probabilities being distributed
among all channels according to~Eq.(\ref{conservation}).
{\em $l$-couplings}. In the single mode regime, Eq.(\ref{tmatrix}) is also an equation for
$t_l\equiv T_l/k_0$ without the singularity $\gamma\sim k_0^{-1}$. If then $\sum_{l'[l]}
(2l'+1)P_{ll'}t_{l'}$ on the r.h.s converges, one can neglect it compared to
$\alpha_l$ for $k_0\rightarrow 0$, and $t_l\approx \alpha_l/[i\gamma
k_0k-(2l+1)k_0P_{ll}-k_0k\cot{\delta_l}]$ is well behaved. Thus, angular momentum
{\em couplings} should be negligible for $k_0\rightarrow 0$ and the series
Eq.(\ref{f1d-spherical}) of individual momenta $l$ is dominated by $l=0$ since
$\delta_l\sim k^{2l+1}\sim 1/l_\perp^{2l+1}$ are small, justifying Eq.(\ref{swave}).
This does not apply straightforwardly to the excited channel case, whose approximation is
based only on the smallness of $\delta_l\,$.
{\em Discussion}. Consider now the case $U(\rho)=\mu\omega_\perp^2\rho^2/2$ of harmonic
confinement, $\mu$ being the reduced mass. In Eq.(\ref{gswave}), the oscillator length
$a_\perp\equiv(\hbar/\mu\,\omega_\perp)^{1/2}$ should replace $d_\perp\equiv l_\perp/N_0$
instead of $l_\perp\,$. This is due to tunneling, since $|\varphi_n(\rho)|^2\sim
e^{-\rho^2/a_\perp^2}$ is small at $\rho\approx l_\perp$ (as in the square-well case) only
if $l_\perp>a_\perp$. Then
$\epsilon_1-\epsilon_0\equiv\hbar^2(q_1^2-q_0^2)/2\mu=2\hbar\omega_\perp$ and
$C'=d_\perp\sqrt{q_1^2-q_0^2}=2\,$. The difference to $C=1.4603\dots$ of
Ref.~\cite{olshanii1998a} originates from the continuum limit in
Eq.(\ref{green-3D}) and Eq.(\ref{deltac}). Indeed, from Eq.(9) of
Ref.~\cite{olshanii1998a}, the continuum approximation for $C$ is
$C\equiv\mathrm{lim}_{s\rightarrow\infty}(\int_0^sds'/\surd{s'}-\sum_{s'=1}^s 1/\surd{s'})
\approx\int_0^1 ds'/\surd{s'}=2\,$. In addition, this comparison reveals the nature of the
``irregular'' part $1/z$ of $\Psi(\bm{r})$ for the pseudopotential approximation (see
Eq.(8) of Ref.~\cite{olshanii1998a} or the equivalent $s$-wave expansion in Eq.(9) of
Ref.~\cite{petrov2000b}). This is the singular part of the free-space Green's function
$\gamma_+G_1+\gamma_-G_2$, with $\gamma_++\gamma_-=1$, and originates from the sum of the
excited transverse levels. As a result, one expects certain details of the guide to be
unimportant, except for the low lying levels which account for the terms $\gamma$ and
$\Delta_c$ and the bound-state $E_B'$.
We have provided a general treatment of quantum scattering in confined geometries. For
scattering by obstacles inside the guide, the treatment should be applicable to a variety
of central force fields $V(r)$ and confining potentials $U(\rho)$. For ultracold atomic
collisions, non-parabolic guides can be considered with restrictions due to the center of
mass. The 1D scattering amplitude is given in terms of the free-space phase shifts
$\delta_l$ and their couplings among each other. This covers the case of higher energies
and a transversal multi-channel incident state. In the single mode regime, we have shown
that the CIR is closely related to the behaviour of a confined bound state.
The Brazilian Agency CNPq, the German A. v. Humboldt Foundation and the DFG
Schwerpunktprogramm: ``Wechselwirkung in Ultrakalten Atom- und Molek\"ulgasen'' are
acknowledged for financial support.
|
{
"timestamp": "2005-06-24T15:05:21",
"yymm": "0503",
"arxiv_id": "quant-ph/0503196",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503196"
}
|
\section{Introduction}
\label{sec:intro}
The multivector fields on a smooth manifold $M$ can be seen as
multidifferential operators on the algebra $\mathcal{C}^\infty(M)$ of smooth
functions on $M$. This assignment is a particular case of the following
general construction: given a graded associative and commutative algebra
$A$, one defines the Hochschild--Kostant--Rosenberg map
$$\mathrm{HKR}\colon\mathcal{V}^\bullet (A)\to
\mathsf{Hoch}^\bullet(A)$$ from the space of multivector fields $\mathcal{V}^\bullet(A):=
S^{\bullet}(\mathsf{Der}(A)[-1])[1]$ to the Hochschild complex
$\mathsf{Hoch}^\bullet(A)$, as the map which regards a multiderivation of $A$
as a multilinear operator. Actually the image of HKR is contained
in the subcomplex $\mathcal{D}^\bullet(A)\subset \mathsf{Hoch}^\bullet(A)$ of
multidifferential operators.
If one considers $\mathcal{V}^\bullet(A)$ as a complex
with trivial differential, then the HKR map is a morphism of complexes, and
the classical Hochschild--Kostant--Rosenberg Theorem \cite{HKR} states that
when $A$ is a smooth algebra, e.g., a polynomial algebra, the HKR map
induces isomorphisms in cohomology $\mathcal{V}^\bullet(A)\simeq
{\mathsf H}^\bullet(\mathcal{D}^\bullet(A)) \simeq \mathsf{HHoch}^\bullet(A)$.
In this paper we are primarily concerned with the case in which $A$ is
the algebra of smooth functions on a graded manifold $N$. In this case
it is known that HKR still induces an isomorphism
$\mathcal{V}^\bullet(N)\simeq {\mathsf H}^\bullet
(\mathcal{D}^\bullet(N))$, where we used the short-hand notations $\mathcal{V}^\bullet(N)$
for $\mathcal{V}^\bullet(\mathcal{C}^\infty(N))$ and $\mathcal{D}^\bullet(N)$ for
$\mathcal{D}^\bullet(\mathcal{C}^\infty(N))$; for a proof, see \cite{V} in case $N$ is
an ordinary manifold and \cite{CF} for the general case.
Many interesting algebraic structures can be defined on the objects
introduced above. It is well known that $\mathcal{V}^\bullet(A)$ and
$\mathsf{HHoch}^\bullet(A)$ are Gerstenhaber algebras \cite{Gerst}, that
${\mathsf H}^\bullet (\mathcal{D}^\bullet(A))$ is a sub-Gerstenhaber algebra
of $\mathsf{HHoch}^\bullet(A)$, and that HKR preserves these structures.
Moreover, when $A$ is a finite dimensional algebra endowed with a
non-degenerate symmetric inner product compatible with the
multiplication of $A$, then
$\mathcal{V}^\bullet(A)$, ${\mathsf H}^\bullet(\mathcal{D}^\bullet(A))$ and
$\mathsf{HHoch}^\bullet(A)$ become Batalin--Vilkovisky (BV) algebras \cite{Tr}.
The purpose of this paper is to extend this construction to the
case in which $A$ is the algebra of smooth functions on a graded manifold $N$.
In this case the algebra is not finite dimensional but we can remedy when
$N$ has a Berezinian volume. We prove in fact the following
\begin{Thma}
Let $N$ be a graded manifold endowed with a fixed Berezinian volume $v$
and whose body is a closed smooth manifold. Then $\mathcal{V}^\bullet
(N)$ and ${\mathsf H}^\bullet(\mathcal{D}^\bullet(N))$
can be endowed with BV algebra structures compatible with their classical
Gerstenhaber structures. Moreover $\mathrm{HKR}$ is a map of BV algebras.
\end{Thma}
The BV algebra structure on multidifferential operators is inspired by
\cite{Tr}, whereas the BV structure on $\mathcal{V}^\bullet(N)$ is the
standard one on the space of multivector fields of a graded manifold
$N$. Both structures depend on the choice of a Berezinian volume on
$N$ \cite{KSM}. The HKR map lifts to an $L_\infty$ map \cite{K,CF}
and, at least in the non graded case, to a $G_\infty$ map \cite{Ta}
between complexes.
One may conjecture that it also lifts to a $BV_\infty$ map \cite{TT}.
This would be the analogue, for a graded manifold, of Kontsevich's
cyclic formality conjecture \cite{S}.\\
In the second part of the paper, we generalize our results to
differential graded manifolds $(N,Q)$. From an
algebraic point of view, this corresponds to considering differential
graded commutative associative algebras $(A,\dd)$. In this case, the
Hochschild complex is actually a bicomplex
with differentials $\delta_0$ and $\delta_1$,
and the Hochschild cohomology will be the cohomology of the total
complex. The Hochschild bicomplex and its cohomology will be denoted by
$\mathsf{Hoch}_{\mathsf{DG}}^\bullet(A)$ and $\mathsf{HHoch}_{\mathsf{DG}}^\bullet(A)$
to distinguish them from the Hochschild complex and cohomology of
$A$ seen as a graded algebra. The differential $\dd$ gives rise
to the differential $\{\dd,\cdot\}$ on the space
${\mathcal{V}}^\bullet(A)$ of multivector fields; the HKR map
$({\mathcal{V}}^\bullet(A),\{\dd,\cdot\},0)\to (\mathsf{Hoch}_{\mathsf{DG}}^\bullet(A),\delta_0,\delta_1)$
(see Lemma~\ref{lem:dr})
is a map of bicomplexes. We show by an example that the induced map
in cohomology is not an isomorphism in general. In particular we
consider the differential graded manifold $N=T[1]M$, where
$M$ is a smooth manifold, with $\dd$ given by the de~Rham differential,
so that
$\mathcal{C}^\infty(T[1]M)$ is the de~Rham algebra $\Omega^\bullet(M)$ of
$M$, and we prove the following
\begin{Thmb}
If $M$ is a simply connected closed oriented smooth manifold of
positive dimension, then the HKR map ${\mathsf H}^\bullet(\mathcal{V}^\bullet
(\Omega^\bullet(M)),\{\dd,\cdot\})\to\mathsf{HHoch}_{\mathsf{DG}}^\bullet
(\Omega^\bullet(M))$ is not an isomorphism.
\end{Thmb}
The key ingredient of the proof is the isomorphism \cite{Chen} between the
(shifted) homology $H_\bullet(\mathcal{L} M)[\dim M]$ of the free loop space
$\mathcal{L} M$ of $M$ and the Hochschild cohomology of the differential graded
algebra $\Omega^\bullet(M)$ .
We remark that when only ordinary smooth manifolds are considered, it is
not known whether the space of multivector fields is quasi-isomorphic to
the Hochschild cohomology. Up to our knowledge, only a partial result in
this direction is known
\cite{N}, namely, when $M$ is a smooth manifold,
$\mathcal{V}^\bullet(M)$ is quasi-isomorphic to
the topological Hochschild complex $\mathsf{HHoch}_{\mathsf
{top}}(\mathcal{C}^\infty(M))$ consisting of continuous multilinear
homomorphisms (with respect to the Fr\'echet topology).
If we further assume that $(N,Q)$ is an SQ-manifold, i.e., that the
vector field
$Q$ is divergence-free, then a BV structure is induced on the cohomology
${\mathsf H}^\bullet(\mathcal{V}^\bullet (A),\{\dd, \cdot\})$ and on the
Hochschild cohomology
$\mathsf{HHoch}_{\mathsf{DG}}^\bullet (A)$, and the HKR map is a morphism of BV
algebras (although, as remarked above, not an isomorphism in general).
An example is the de Rham algebra $(\Omega^\bullet(M),\dd)$ of a closed
manifold $M$. In this case, the BV structure on
$\mathsf{HHoch}_{\mathsf{DG}}^\bullet (\Omega^\bullet(M))$ corresponds to the one
found in \cite{CS} on the homology of the free loop space \cite{CJ,M}, whereas
the BV structure on ${\mathsf H}^\bullet
(\mathcal{V}^\bullet (\Omega^\bullet(M),\{\dd,\cdot\}))$ is the trivial one.
The plan of the paper is as follows. We begin by
constructing the BV structure on the space of multivector fields in
Section~\ref{sec:mvf}. Next we recall some facts on
Hochschild cohomology in Section~\ref{sec:hoch}.
Then we discuss BV structures on the space of multidifferential
operators in Section~\ref{sec:mdo},
and in Section~\ref{sec:hkr} we define the HKR map,
describe its main properties, and prove Theorem~\ref{thm:hkr}. Finally
in Sections~\ref{sec:dgm} and \ref{sec:sq} we present a generalization of
these results to the case of differential graded manifolds and prove
Theorem~\ref{thm:derham}.
\\
\begin{Ack}
We thank Thomas Tradler and the Referee for useful comments on a first draft
of the paper. R.~L.\ thanks the Universit\"at Z\"urich--Irchel and D.~F.\
thanks the IH\'ES for their hospitality.
\end{Ack}
\section{BV structure on multivector fields}
\label{sec:mvf}
Let $A$ be a graded commutative and associative algebra and let $\mathsf{Der}(A)
=
\oplus_{j\in{\mathbb Z}} \mathsf{Der}^j(A)$ be the graded Lie algebra of derivations of $A$,
namely $\mathsf{Der}^j(A)$ consists of linear maps $\phi\colon A\to A$ of degree
$j$ such that $\phi(ab) = \phi(a)b+ (-1)^{j\, |a|} a\phi(b)$ and the bracket
is $\{\phi,\psi\}=\phi\circ\psi -(-1)^{|\phi||\psi|}\psi\circ\phi$.
The space of multiderivations
$\mathcal{V}^\bullet(A):=S^\bullet(\mathsf{Der}(A)[-1])[1]$
can be endowed with a Gerstenhaber structure, with the wedge product and
the bracket which is the extension of the graded commutator
$\{\cdot,\cdot\}$ on $\mathsf{Der}(A)$ to $\mathcal{V}^\bullet(A)$ by the Leibnitz
rule. Since $A$ is graded, the space $\mathcal{V}^\bullet(A)$ has a natural
double grading given by
\[
\mathcal{V}^{i,j}(A)= \{\phi\in
S^i(\mathsf{Der}(A)[1])[-1]\,|\,\deg(\phi)=j\}.
\]
We want to construct an operator $\Delta$ on
$\mathcal{V}^\bullet(A)$ which makes this Gerstenhaber algebra into a BV
algebra. We will use as an auxiliary tool the complex ${\mathcal
I}^\bullet(A)$ of integral forms of $A$, closely following
\cite{D}; a different approach to the BV
algebra structures on $\mathcal{V}^\bullet(A)$ can be found in
\cite{KSM}. Denote by $\Omega^1(A)$ the space of $1$-forms of $A$, namely,
the space ${\rm Hom}(\mathcal{V}^1(A),A)$, and assume that the Berezinian ${\rm
Ber}(\Omega^1(A))$ is free and generated by one element $v$.
To a \emph{divergence operator} $\div$, viz.\ an
even linear map
$\div\colon \mathsf{Der}(A)\to A$ satisfying
\[
\div(fX)=f\div(X)+(-1)^{|f||X|}X(f),
\]
we associate a linear operator
$L\colon \mathcal{V}^1(A)
\otimes_A {\rm Ber}(\Omega^1(A)) \to {\rm Ber}(\Omega^1(A))$ by the rule
\[
L(X\otimes v)=\div(X)\,v.
\]
Observe that
for every $f\in A$ and every
$X\in\mathsf{Der}(A)$, we have $L_X(fv)=X(f)\,v + (-1)^{|f||X|}f\,L_X(v)$ where we
are using the notation $L_X(v):=L(X\otimes v)$.
We now introduce the space ${\mathcal I}^\bullet(A)$ of integral forms
\cite{D} as the
$A$-module generated by the elements of ${\rm
Ber}(\Omega^1(A))$ and by the operations $\iota_X$ with
$X\in\mathcal{V}^1(A)$, acting on the left and subject to the rules
$[\iota_X,\iota_Y]=0$
and $\iota_{fX}=f\iota_X$.
The action of $L_X$ is extended to ${\mathcal I}^\bullet(A)$ by the rule
$[L_X,\iota_Y]=\iota_{\{X,Y\}}$
One can define an exterior derivative $\dd$ on
${\mathcal I}^\bullet(A)$ by imposing
$\dd v=0$ and forcing Cartan's identity $\dd \iota_X+\iota_X\dd=L_X$.
Indeed, a consequence of Cartan's formula is that
$\dd(\iota_{X_1}\cdots\iota_{X_k}v)=L_{X_1}(\iota_{X_2}\cdots\iota_{X_k}
v)-\iota_{X_1}\dd(\iota_{X_2}\cdots\iota_{X_k}
v)$, and the action of $\dd$ on elements of ${\mathcal I}^\bullet(A)$ can
be computed inductively. The exterior derivative $\dd$ defined by this
procedure is a differential precisely when
$[L_X,L_Y]=L_{\{X,Y\}}$. This is equivalent to the
vanishing of the curvature of $\div$; namely,
\[
\div(\{X,Y\}) - X(\div(Y)) + (-1)^{|X|\,|Y|}
Y(\div(X))=0.
\]
Once the generator $v$ of ${\rm
Ber}(\Omega^1(A))$ is fixed, iterated ``contractions'' $\iota_X$ induce an
isomorphism
\[
\mathcal{V}^\bullet(A)\xrightarrow{\sim}{\mathcal I}^{\bullet}(A)
\]
and the differential $\dd$ induces on the space of multivector fields an
operator $\Delta$ of degree $-1$ such that $\Delta^2=0$.
An easy computation shows that $\Delta(X)=\div(X)$ for any
$X\in\mathsf{Der}(A)$,
and that $\Delta$ satisfies the seven term relation
\begin{multline}
\label{eq:seven}
\Delta(a\wedge b\wedge c) + \Delta(a)\wedge b\wedge c + (-1)^{|a|} a\wedge
\Delta(b)\wedge c + (-1)^{|a|+|b|} a\wedge b\wedge \Delta(c) =\\
= \Delta(a\wedge b)\wedge c + (-1)^{|a|}a\wedge\Delta(b\wedge c) +
(-1)^{(|a|+1)|b|}b\wedge \Delta(a\wedge c)
\end{multline}
and the compatibility with the bracket
\begin{equation}
\label{eq:defbv}
\{a,b\} := (-1)^{|a|}\left(\Delta(a\wedge b) - \Delta(a)\wedge b -
(-1)^{|a|}a\wedge\Delta(b)\right).
\end{equation}
Therefore we have proved
\begin{Lem}
If the Berezinian ${\rm Ber}(\Omega^1(A))$ is a free
$A$-module of rank one
and $\div$ is a curvature-free divergence operator,
then the operator $\Delta$ defined as above endows $\mathcal{V}^\bullet(A)$
with a BV structure compatible with the usual Gerstenhaber structure.
\end{Lem}
\par
The main example of this construction is when $A=\mathcal{C}^\infty(N)$, $N$
being a graded manifold endowed with a Berezinian volume $v$.
In this case the operators
$L_X$ and $\iota_X$ are just the classical Lie derivatives and contraction
operators, and the complex ${\mathcal I}^\bullet(N)$ is the
complex of integral forms of the graded manifold.
Since the Berezinian is a line
bundle and $v$ is a nowhere zero section,
there exists an operator $\div$ defined uniquely by the equation $L_Y(v) =
\div(Y)\,v$, which is indeed a divergence operator whose curvature
vanishes. Observe that in the case when $N$ is an oriented smooth
manifold, this
amounts to choosing an ordinary volume form $v$. In the case when $N=T[1]M$,
with $M$ an oriented smooth manifold, there is a canonical Berezinian
volume $v$ characterized by
\[
\int_N \alpha\,v = \int_M \alpha,
\qquad\forall\alpha\in C^\infty(N) = \Omega^\bullet(M).
\]
\begin{Rem}\label{rem:Fourier}
The geometry of $T[1]M$ is closely related to the geometry of the formal
neighborhood of $M$ inside its cotangent bundle $T^*M$. Namely, the
Liouville volume form on $T^*M$ induces a curvature-free divergence
operator $\Delta$ on $\mathcal{V}^\bullet(T^*M)$, which makes it a
BV algebra. The algebra $A=\Gamma(S^\bullet TM)$ of smooth
functions on
$T^*M$ which are polynomial along the fibers is a BV subalgebra of
$\mathcal{V}(T^*M)$; it can be considered as the algebra of multivector
fields on $T^*M$ which are ``infinitesimal in the cotangent direction''.
As a consequence of the ``Fourier transform''
\cite{CF,R}, the Gerstenhaber algebras $\mathcal{V}^\bullet(T[1]M)$ and
$\mathcal{V}^\bullet(A)$ are isomorphic. But it can be easily verified that
they are also isomorphic as BV algebras.
\end{Rem}
\begin{Rem}
For a smooth manifold $M$, integral forms are
just ordinary differential forms and $\mathcal{I}^\bullet(M)$ is naturally
identified with $\Omega^\bullet(M)$.
On the other hand, for a graded manifold $N$ which is non trivial in odd
degrees, the complex $\mathcal{I}^\bullet(N)$ of integral forms is not isomorphic to
the de~Rham complex of $N$ (see \cite{D} for details).
\end{Rem}
\section{BV structure on Hochschild cohomology}
\label{sec:hoch}
The aim of this Section is to recall some standard facts about Hochschild
cohomology and fix notations for the rest of the paper. We address the reader
to \cite{L} and \cite{Tr} for a comprehensive treatment.
\subsection{Hochschild cohomology}
Let $A=\oplus_{j\in{\mathbb Z}}A_j$ be a graded
algebra over ${\mathbb R}$, with a graded commutative associative product $\mu$
and a unit
${\bf 1}$.
We also suppose that $A$ is endowed with a non degenerate symmetric inner
product compatible with the algebra multiplication, namely such that $\langle
a,b\rangle = (-1)^{|a|\,|b|} \langle b,a\rangle$ and
$\langle\mu(a\otimes b), c\rangle = \langle a, \mu(b\otimes c)
\rangle$. Finally, a graded bimodule $B$ over the algebra $A$ is given.
Let us set $T(A):=\bigoplus_{k\ge 0} A^{\otimes k}$
and $T^B(A) := {\mathbb R}\oplus \bigoplus_{k,l\ge 0} A^{\otimes k}
\otimes B \otimes A^{\otimes l}$.
It is well known that $T(A)$ is a coalgebra and $T^B(A)$ a bi-comodule over
$T(A)$ with the coproducts
\begin{align*}
T(A)& \to T(A)\otimes T(A)\\
(a_1,\ldots, a_n)&\mapsto \sum_{i=0}^{n}\left(a_1,\ldots,a_i\right)
\otimes \left(a_{i+1},\ldots,a_n\right)
\end{align*}
and
\begin{align*}
T^B(A)&\to (T(A)\otimes T^B(A))\oplus (T^B(A)\otimes T(A)) \\
(a_1,\ldots,a_k,b,a_{k+1},\ldots,a_n)&\mapsto \sum_{i=0}^k(a_1,\ldots,a_i)
\otimes (a_{i+1},\ldots,b,\ldots,a_n)+\\ &\phantom{mn}
+ \sum_{i=k}^n(a_1,\ldots,b,\ldots,a_i) \otimes (a_{i+1},\ldots,a_n).
\end{align*}
Hence we can define the space $\mathsf{Coder}(T(A),T^B(A))$,
of coderivations from $T(A)$ to $T^B(A)$, with
respects to the above coproducts.
The Hochschild cochain complex of $A$ with values in $B$ is defined as
\[
\mathsf{Hoch}^\bullet(A,B):= \mathsf{Coder}(T(A[1]), T^{B[1]}(A[1]))[-1]
\]
where by $A[1]$ we mean the graded algebra obtained by shifting the degrees of
$A$ by 1; namely, $A[1] = \oplus_{j\in{\mathbb Z}}(A[1])_j$ with
$(A[1])_j := A_{j+1}$.
As usual one can make the identification $$\mathsf{Hoch}^\bullet(A,B) =
\prod_{n}\mathsf{Hom}(A[1]^{\otimes
n},B[1])[-1]=
\prod_{n}\mathsf{Hom}(A^{\otimes n},B)[-n].$$
Let us denote by $\widetilde{\mu^B}$ and $\widetilde{\mu}$ the lifts of
the bimodule structure $\mu^B \colon A\otimes B
\otimes A\to B$ and of the multiplication $\mu\colon A\otimes A\to A$
to coderivations of $T(A[1])$
with values in $T^{B[1]}(A[1])$.
Then, on the Hochschild cochain complex we can define a
degree 1 differential
$\delta^B \colon \mathsf{Hoch}^\bullet(A,B) \to \mathsf{Hoch}^\bullet(A,B)$, by setting
$\delta^B(f):=\widetilde{\mu^B}\circ f -(-1)^{|f|} f\circ
\widetilde{\mu}$. It is easy to check that $(\delta^B)^2=0$; the
cohomology of the Hochschild complex with respect to the differential
$\delta^B$ is called Hochschild cohomology of $A$ with values in $B$ and
it is denoted by $\mathsf{HHoch}^\bullet(A,B)$. When $B=A$ with the canonical
bimodule structure we write $\mathsf{HHoch}^\bullet(A)$ for
$\mathsf{HHoch}^\bullet(A,A)$; moreover $\delta^A$ is simply denoted by
$\delta$.
\begin{Rem} Since $A$ and $B$ are graded objects, the Hochschild complex
$\mathsf{Hoch}(A,B)$ is a bigraded object: in the identification
$\mathsf{Hoch}^\bullet(A,B) =
\prod_{n}\mathsf{Hom}(A[1]^{\otimes
n},B[1])[-1]$, the horizontal
degree is provided by the number of $A$-factors, and
the vertical degree by the degree of the maps:
\[
\mathsf{Hoch}^{i,j}(A,B) = \{f\in\mathsf{Hom}(A[1]^{\otimes
i},B[1])[-1]|\, \deg(f)=j\}.
\]
The differential $\delta^B$ is a horizontal differential, since it
increases the number of factors by one, leaving the degree of the
maps unchanged. So one can think of the Hochschild complex as a
bicomplex, with horizontal differential $\delta_1^B(f):=\widetilde{\mu^B}\circ f -(-1)^{|f|}
f\circ \widetilde{\mu}$ and trivial vertical differential
$\delta_0^B:=0$, and to consider $\delta^B$ as the total differential
$\delta^B=\delta_0^B+\delta_1^B$. We will come back to this point
of view when we will be discussing the Hochschild cohomology of
differential graded algebras in Section \ref{sec:sq}.
\end{Rem}
\subsection{Operations on the Hochschild cochain complex}
On the Hochschild co\-chain complex
$\mathsf{Hoch}^\bullet(A)$ one can define various operations.
First, there is a composition $f\circ g$ whose
graded antisymmetrization $\{f,g\}:=f\circ g - (-1)^{|f|\, |g|} g\circ f$
gives rise to a graded odd Lie bracket of degree $+1$,
also known as the Gerstenhaber
bracket. Notice that the associativity of the product $\mu$ of $A$
is equivalent to
$\{\widetilde\mu,\widetilde\mu\} = 0$, which immediately implies that
the Hochschild differential $\delta(f) =
\{\widetilde\mu, f\}$ indeed squares to zero. Similar
relations holds for $\widetilde{\mu^B}$ and $\delta^B$.
Next, using the identification of $\mathsf{Hoch}^\bullet(A)$ with
$\prod_{n\ge 0} \mathsf{Hom}(A^{\otimes n}, A)[-n]$ we define a
product between $\phi\in \mathsf{Hom}(A^{\otimes k}, A)[-k]$ and
$\psi\in \mathsf{Hom}(A^{\otimes l}, A)[-l]$ as
$$(\phi\cup \psi) (a_1\otimes \cdots \otimes a_{k+l}):=
(-1)^\epsilon
\mu(\phi(a_1\otimes \cdots \otimes a_k) \otimes \psi(a_{k+1}\otimes \cdots
\otimes a_{k+l})),$$
where $\epsilon=l(|a_1|+\cdots+|a_k|+k)$.
This associative product is non-commuta\-tive but it gives rise to a
graded commutative product in cohomology. The cup product and the
Gerstenhaber bracket satisfy in cohomology the graded Leibnitz rule
\[
\{a,b\cup c\} = \{a,b\}\cup c + (-1)^{(|a|+1)|b|}b\cup\{a,c\}.
\]
Therefore $(\mathsf{HHoch}^\bullet(A),\cup,\{\cdot,\cdot\})$ is
a Gerstenhaber algebra \cite{Gerst}.
In addition, on the complex $\mathsf{Hoch}^\bullet(A,A^*)$ one has an
operator $\beta$ given by the dual to Connes' $B$-operator
\cite{Co}. More explicitly, one defines $\beta
\colon \mathsf{Hoch}^\bullet(A,A^*) \to \mathsf{Hoch}^{\bullet-1}(A,A^*)$ as
\[
(\beta(f)(a_1 ,\ldots,a_n ))(a_{n+1}) :=
\sum_{i=1}^{n+1}
(-1)^\epsilon
(f(a_i ,\ldots,a_{n+1},a_1,\ldots,a_{i-1}))({\bf 1})
\]
where ${\bf 1}$ is the unit of $A$ and $\epsilon = |f|+ |a_1|+\cdots +
|a_{n+1}| + (|a_i |+\cdots + |a_n|)(|a_1|+\cdots +|a_{i-1}|)$.
The inner product on $A$ gives rise to an injection $P\colon A\to A^*$ which
is an $A$-bimodule map, and, by composing the Hochschild
cochains with the injection $P$, one obtains an injective map $\wp\colon
\mathsf{Hoch}^\bullet(A) \to \mathsf{Hoch}^\bullet(A,A^*)$. If moreover $\wp$ is a
quasi-isomorphism, i.e., induces an isomorphism $H(\wp)$ in cohomology,
then we can define an operator $\Delta_\beta$ of degree $-1$
on $\mathsf{HHoch}^\bullet(A)$ by setting
$\Delta_\beta = H(\wp)^{-1}\circ\beta\circ H(\wp)$.
As shown in \cite{Tr} (see also \cite{Men}), the operator $\Delta_\beta$
squares to zero in cohomology and is compatible with
the Gerstenhaber structure on $\mathsf{HHoch}^\bullet(A)$ in the sense that
(cf.\ equation~\eqref{eq:seven})
\begin{multline*}
\Delta_\beta(a\cup b\cup c) + \Delta_\beta(a)\cup b\cup c + (-1)^{|a|} a\cup
\Delta_\beta(b)\cup c + (-1)^{|a|+|b|} a\cup b\cup \Delta_\beta(c) =\\
= \Delta_\beta(a\cup b)\cup c + (-1)^{|a|}a\cup\Delta_\beta(b\cup c) +
(-1)^{(|a|+1)|b|}b\cup \Delta_\beta(a\cup c)
\end{multline*}
and (cf.\ equation~\eqref{eq:defbv})
\begin{equation*}
\{a,b\} = (-1)^{|a|}\left(\Delta_\beta(a\cup b) - \Delta_\beta(a)\cup b -
(-1)^{|a|}a\cup\Delta_\beta(b)\right).
\end{equation*}
In other words $(\mathsf{HHoch}^\bullet(A),\cup,\{\cdot,\cdot\},
\Delta_\beta)$ is a BV algebra. Summing up, we have
\begin{Prop}
\label{prop:bv}
If the map $\wp\colon \mathsf{Hoch}^\bullet(A) \to
\mathsf{Hoch}^\bullet(A,A^*)$ induced by the inner product of $A$ is a
quasi-isomorphism, then
$\mathsf{HHoch}^\bullet(A)$ is endowed with a BV algebra structure, compatible with
its Gerstenhaber structure.
\end{Prop}
A trivial example is when $A$ is finite dimensional, and hence $\wp$
is an isomorphism. A more interesting case is the algebra of functions
on a graded manifold $N$ endowed with a Berezinian volume $v$.
In this case the pairing is defined by
\begin{equation}\label{e:pairing}
\langle f_1,f_2\rangle=\int_N f_1 f_2\, v.
\end{equation}
In general, when $N$ is a graded manifold,
$\mathsf{Hoch}^\bullet(\mathcal{C}^\infty(N))$ is not necessarily quasi-isomorphic to
$\mathsf{Hoch}^\bullet(\mathcal{C}^\infty(N), \mathcal{C}^\infty(N)^*)$, and hence we do not
know whether we can define a BV structure on $\mathsf{Hoch}^\bullet(\mathcal{C}^\infty(N))$.
However we will see in Section~\ref{sec:mdo} that a version of
Proposition~\ref{prop:bv} can be applied to a certain subcomplex of the
Hochschild complex, namely to the subcomplex of multidifferential
operators.
\section{BV structure on multidifferential operators}
\label{sec:mdo}
The Hochschild complex of $A$ has a sub-Gerstenhaber algebra
$\mathcal{D}^\bullet(A)$ consisting of multidifferential operators, namely sums
of cochains of the form $(a_1,\ldots,a_n)\mapsto \prod_{i=1}^n\phi_i(a_i)$
where $\phi_i$ are compositions of derivations. The bigrading on the
Hochschild complex induces a bigrading on the subalgebra of
multidifferential operators: \[
\mathcal{D}^{i,j}(A):=\mathcal{D}^\bullet(A)\cap
\mathsf{Hoch}^{i,j}(A).\]
We now want to discuss under
which conditions the cohomology of ${\mathcal D}^{\bullet}(A)$ admits a
natural BV structure. As above we are assuming that there
exists a non degenerate symmetric inner product on $A$ compatible with the
multiplication, and hence an injective
map $\wp\colon \mathsf{Hoch}^\bullet(A) \to \mathsf{Hoch}^\bullet(A,A^*)$. The point is to
determine when the Connes cyclic $\beta$-operator $\beta\colon
\mathsf{Hoch}^\bullet(A,A^*) \to \mathsf{Hoch}^{\bullet-1} (A,A^*)$ induces an operator
$\Delta_\beta\colon{\mathcal D}^{\bullet}(A)\to {\mathcal D}^{\bullet-1}(A)$
making the diagram
\[
\xymatrix{
{\mathcal D}^\bullet(A) \ar[r]^\wp
\ar@{-->}[d]_{\Delta_\beta} &
\mathsf{Hoch}^\bullet(A,A^*) \ar[d]_\beta\\
{\mathcal D}^{\bullet-1}(A) \ar[r]^\wp &
\mathsf{Hoch}^{\bullet-1}(A,A^*)\\
}
\]
commutative. To answer this question, we look at the problem from a more
general perspective; namely, let
$C^\bullet(A)$ be any sub-Gerstenhaber algebra of
$\mathsf{Hoch}^\bullet(A)$ whose $\wp$-image in $\mathsf{Hoch}^\bullet(A,A^*)$ is closed under
$\beta$. Since $\wp$ is injective, $\beta$ induces a well-defined
operator $\Delta_\beta$ on the complex $C^\bullet(A)$.
Following \cite{Tr} and \cite{Men}, the operator $\Delta_\beta$
squares to zero
in the cohomology of $C^\bullet(A)$, and endows
${\mathsf H}^\bullet(C^\bullet(A))$ with a BV algebra
structure compatible with its Gerstenhaber
structure.
\par
We now specialize to the case when $A=C^\infty(N)$, where $N$ is a
graded manifold endowed with a Berezinian volume $v$.
In order to prove that the
cohomology ${\mathsf H}^\bullet({\mathcal
D}^\bullet(N))$ of the algebra of multidifferential operator admits a
natural BV structure, we only need to prove that
$(\beta\circ\wp)({\mathcal D}^\bullet(N))\subseteq
\wp({\mathcal D}^\bullet(N))$
with $\wp$ induced by the pairing \eqref{e:pairing}.
We first need the following ``integration-by-parts'' Lemma.
\begin{Lem}\label{l:multi}
Let $D$ be a multidifferential operator. Then there exist a
multidifferential operator $\tilde{D}$ such that
\[
\langle D(f_1,\dots,f_n),{\bf 1}\rangle = \langle
\tilde{D}(f_1,\dots,f_{n-1}), f_n\rangle
\]
\end{Lem}
Then we observe that for every $D\in {\mathcal D}^n(N)$ and for every
$i=1,\dots,n$, the operator
\[
D_i(f_1,\dots,f_n):= D(f_i,\dots,f_n,f_1,\dots,f_{i-1}),
\qquad f_1,\dots,f_n\in A,
\]
is still in ${\mathcal D}^n(N)$.
Finally
\begin{multline*}
(\beta\circ\wp(D))
(f_1 ,\ldots,f_{n-1})(f_{n}) =
\sum_{i=1}^{n}
(-1)^\epsilon
\langle D(f_i ,\ldots,f_{n},f_1,\ldots,f_{i-1}),{\bf 1}\rangle=\\
=\sum_{i=1}^{n}
(-1)^\epsilon
\langle D_i(f_1 ,\ldots,f_{n}),{\bf 1}\rangle=
\sum_{i=1}^{n}
(-1)^\epsilon
\langle \tilde D_i(f_1 ,\ldots,f_{n-1}),{f_n}\rangle=\\
=\wp\left(\sum_{i=1}^{n}
(-1)^\epsilon\tilde D_i\right)(f_1,\dots,f_{n-1})(f_n).
\end{multline*}
\begin{proof}[Proof of Lemma \ref{l:multi}]
The proof is by induction on the order of the multidifferential
operator $D$. If $D$ is homogeneous of order zero,
\[
D(f_1,\dots,f_{n})=\lambda f_1\cdots f_{n}
\]
for some constant $\lambda$, so that
\[
\langle D(f_1,\dots,f_{n}), {\bf 1}\rangle =
\int_N \lambda f_1\cdots f_{n}\, v=
\langle \lambda f_1\cdots f_{n-1},f_n\rangle
\]
and we are done. Now assume the claim
proved for operators up to order $k$ and prove it for order $k+1$
operators by the following argument. A homogeneous component of an order
$k+1$ multidifferential operator can be written as
\[
D(f_1,\dots,f_{n})=D_0(f_1,\dots f_{i-1},X(f_i),f_{i+1},
\dots,f_{n})
\]
for a suitable multidifferential operator $D_0$ of order $k$, some index
$i$ and some vector field $X$.
We compute
\[
\langle D(f_1,\dots,f_n),{\bf 1} \rangle=
\langle D_0(f_1,\dots,X(f_i), \dots, f_{n}),{\bf 1}\rangle
\]
Here we have to distinguish two cases. If $i\neq n$, by the induction
hypothesis applied to $D_0$, we can write
\[
\langle D_0(f_1,\dots,X(f_i), \dots, f_{n}),{\bf 1}\rangle
=
\langle\tilde{D}_0(f_1,\dots,X(f_i), \dots, f_{n-1}), f_n\rangle
\]
and we are done. If $i=n$ then the induction hypothesis gives
\[
\langle D_0(f_1,\dots, f_{n-1},X(f_{n})), {\bf 1}\rangle
=
\langle\tilde{D}_0(f_1,\dots,f_{n-1}), X(f_n)\rangle.
\]
For any vector field $Y$, Cartan's formula gives
$L_Y(v) = \dd i_Y(v) + i_Y \dd(v)=\dd i_Y(v) $,
since $\dd(v)=0$ \cite{D}. Hence, by Stokes' Theorem we have that
\[
0=\int_N \dd i_Y (f \, v) = \int_N Y(f)\, v + (-1)^{|f|\, |Y|}
\int_N f\, L_Y(v).
\]
Recall for Section \ref{sec:mvf} that
there exists an operator $\div$ defined uniquely by the equation $L_Y(v) =
\div(Y)\,v.$ Therefore
\begin{equation}
\label{eq:sc}
\langle Y(f),{\bf 1}\rangle =
\int_N Y(f)\, v = - (-1)^{|f|\, |Y|} \int_N f\, \div(Y)\, v = -
\langle \div(Y),f\rangle.
\end{equation}
Going back to our problem with $D_0$, we apply the previous formula to the
vector field $Y=\tilde{D}_0(f_1,\dots,f_{n-1})X$ and obtain
\begin{align*}
\langle \tilde{D}_0(f_1,\dots,f_{n-1}),X(f_n)\rangle
&=\int_N \tilde{D}_0(f_1,\dots,f_{n-1})X(f_n)\, v\\
&=\langle \div(\tilde{D}_0(f_1,\dots,f_{n-1})X),f_n \rangle.
\end{align*}
The map $(f_1,\cdots,f_{n-1})
\mapsto \div(\tilde{D}_0(f_1,\dots,f_{n-1})X)$ is a multidifferential
operator, and the Lemma is proved by setting
$\tilde D (f_1,\dots,f_{n-1})=\div(\tilde{D}_0(f_1,\dots,f_{n-1})X)$.
\end{proof}
\section{The Hochschild--Kostant--Rosenberg map}
\label{sec:hkr}
The
Hochschild--Kostant--Rosenberg (HKR) map is defined as follows:
\begin{equation}
\label{eq:hkr-m}
\begin{array}{ccc}
\mathcal{V}^\bullet(A) & \longrightarrow & \mathsf{Hoch}^\bullet(A)\\
\phi_1\wedge\cdots\wedge\phi_n & \mapsto &
\displaystyle{\frac1{n!}\sum_{\sigma\in S_n}
\mathrm{sign}(\sigma)\ \phi_{\sigma(1)}\cup\cdots\cup\phi_{\sigma(n)}}.
\end{array}
\end{equation}
Note that the HKR map is actually a map of bigraded vector spaces:
$\mathcal{V}^{i,j}(A)\to \mathsf{Hoch}^{i,j}(A)$. We have already observed that both
$\mathcal{V}^\bullet(A)$ and
$\mathsf{HHoch}^\bullet(A)$ are Gerstenhaber algebras, and it is well known
that the HKR map in fact preserves these structures. More explicitly
\begin{Thm}
\label{thm:hkrg}
If $\mathcal{V}^\bullet(A)$ is endowed with the zero differential, then $\mathrm{HKR}$ is a
morphism of complexes. Moreover the induced map in cohomology is a
morphism of Gerstenhaber algebras.
\end{Thm}
\begin{proof}
This is a standard result: the fact that
$\mathrm{HKR}$ respects the product structures in cohomology follows directly
from the fact that the cup product is commutative in cohomology \cite{Gerst}.
An easy check shows that for $X,Y\in\mathsf{Der}(A)$ we have
\[
\{\mathrm{HKR}(X), \mathrm{HKR}(Y)\} - \mathrm{HKR}(\{X,Y\}) = 0
\]
and hence, by the compatibility between the bracket and the product,
$\mathrm{HKR}$ induces in cohomology a map of Gerstenhaber algebras.
\end{proof}
The classical Theorem of Hochschild, Kostant and Rosenberg \cite{HKR}
states that when $A$ is a smooth algebra (e.g. for the coordinate ring
of a smooth affine algebraic variety) then the HKR map is a
quasi-isomorphism, i.e., induces an isomorphism
$\mathcal{V}^\bullet(A)\xrightarrow{\sim}\mathsf{HHoch}^\bullet(A)$.
\par
One sees from equation (\ref{eq:hkr-m}) that the HKR map actually takes
its values in the subcomplex
$\mathcal{D}^\bullet(A)$ of multidifferential operators.
For a smooth algebra $A$, the inclusion
$\mathcal{D}^\bullet(A)\hookrightarrow \mathsf{Hoch}^\bullet(A)$ is a
quasi-isomorphism, so the classical Hochschild-Kostant-Rosenberg
theorem can then be stated as follows.
\begin{Thm}
\label{thm:hkr-iso}
If $A$ is a smooth algebra, then
$\mathrm{HKR}\colon\mathcal{V}^\bullet(A)\to
{\mathsf H}^\bullet(\mathcal{D}^\bullet(A))$ is
an isomorphism of Gerstenhaber algebras.
\end{Thm}
Our main result is a version of Theorem \ref{thm:hkr-iso} for graded manifolds,
namely, we prove
\begin{Thm}
\label{thm:hkr}
Let $N$ be a graded manifold endowed with a fixed Berezinian volume $v$
and whose body is a smooth closed manifold. Then $\mathcal{V}^\bullet
(N)$ and ${\mathsf H}^\bullet(\mathcal{D}^\bullet(N))$
can be endowed with BV algebra structures compatible with their classical
Gerstenhaber structures. Moreover
$\mathrm{HKR}\colon\mathcal{V}^\bullet(N)\to
{\mathsf H}^\bullet(\mathcal{D}^\bullet(N))$ is
an isomorphism of BV algebras.
\end{Thm}
\begin{proof}
We have seen in Sections~\ref{sec:mvf} and~\ref{sec:mdo} that, in case
$A=\mathcal{C}^\infty(N)$ is the
algebra of smooth functions of a graded manifold $N$ endowed with a Berezinian
volume form, then both $\mathcal{V}^\bullet(N)$ and ${\mathsf H}^\bullet(
\mathcal{D}^\bullet(N))$ are BV
algebras in a way compatible with their classical Gerstenhaber structures.
We know from Theorem~\ref{thm:hkrg} that $\mathrm{HKR}$ induces
in cohomology a morphism of Gerstenhaber algebras. Moreover we know from
\cite{CF} that $\mathrm{HKR}\colon \mathcal{V}^\bullet(N) \to \mathcal{D}^\bullet(N)$ is a
quasi-isomorphism. Therefore, by the compatibility between the BV Laplacian and
the Gerstenhaber bracket, we only need
to prove that for every vector field $X\in\mathcal{V}^1(N)$ on a graded manifold $N$,
we have
$$\mathrm{HKR}(\Delta(X)) = \Delta_\beta(\mathrm{HKR}(X)).$$
To see this, consider the diagram
\[
\xymatrix{
\mathcal{V}^1(N) \ar[r]^{\mathrm{HKR}} \ar[d]_{\Delta} &
\mathcal{D}^1(N) \ar[r]^{\wp\phantom{mmmmm}} \ar[d]_{\Delta_\beta}&
\mathsf{Hoch}^1(\mathcal{C}^\infty(N),\mathcal{C}^\infty(N)^*) \ar[d]_\beta\\
\mathcal{V}^0(N) \ar[r]^{\mathrm{HKR}} &
\mathcal{D}^0(N) \ar[r]^{\wp\phantom{mmmmm}} &
\mathsf{Hoch}^0(\mathcal{C}^\infty(N),\mathcal{C}^\infty(N)^*)
}
\]
Since the diagram on the right commutes and $\wp$ is injective,
commutativity of the diagram on the
left follows from the commutativity of the external diagram. This is indeed
the case since on the one side,
for $X\in\mathcal{V}^1(N)$ and $f\in\mathcal{C}^\infty(N)$, we have that
\begin{equation}
\label{eq1}
\left(\beta(\wp(\mathrm{HKR}(X)))\right)(f) = - \langle X(f), {\bf 1}\rangle,
\end{equation}
on the other side
\begin{equation}
\label{eq2}
\left(\wp(\mathrm{HKR}(\Delta(X)))\right)(f) = \langle \Delta(X), f\rangle.
\end{equation}
By Section~\ref{sec:mvf}, $\Delta(X)=\div(X)$,
and the right-hand sides of equations~\eqref{eq1} and \eqref{eq2} coincide
by means of equation~\ref{eq:sc}.
\end{proof}
\section{The HKR theorem for differential graded
manifolds}\label{sec:dgm}
We now consider the more general case of differential graded manifolds,
i.e., of graded manifolds $N$ endowed with a degree 1 integrable vector
field $Q$. Note that, since the degree of $Q$ is 1, the integrability
condition $\{Q,Q\}=0$ is equivalent to $Q^2=0$. The algebraic counterpart
of a differential graded manifold $(N,Q)$ is a differential graded
algebra $(A,\dd)$, where $\dd$ is a degree one differential on the
graded algebra $A$. A classical example is given by the
de Rham algebra
$(\Omega^\bullet(M),\dd)$ of a differential manifold $M$ with the de
Rham differential. The corresponding graded manifold is
$T[1]M$; the de Rham differential on differential forms corresponds to
a degree 1 integrable vector field on $T[1]M$. Note that ordinary graded
manifolds can be considered as differential graded manifolds with
the trivial vector field $Q=0$.
\par
The construction of the Hochschild
complex of a graded algebra $A$ with values in $B$ described in Section
\ref{sec:hoch} generalizes to the case of a differential graded
algebra $(A,\dd)$. In this case one actually gets a nontrivial vertical
differential by setting $\delta^B_0(f):=
\widetilde\dd\circ f -(-1)^{|f|} f\circ\widetilde\dd$, where
$\widetilde\dd$ denotes the lift of the differential $\dd\colon A\to
A$, to coderivations of $T(A[1])$ with values in $T^{B[1]}(A[1])$.
The horizontal differential $\delta_1^B$ is the same as in the case of
graded algebras described in Section \ref{sec:hoch}. One easily checks
that the total differential $\delta^B=\delta_0^B+\delta_1^B$
squares to zero. We show this in the particular case $B=A$, the
general case being similar. By definition,
$\delta_1=\{\widetilde\mu,\cdot\}$ and
$\delta_0=\{\widetilde\dd,\cdot\}$; the associativity of the
product $\mu$ is equivalent to
$\{\widetilde\mu,\widetilde\mu\} = 0$,
the fact that $\dd$ is a derivation for $\mu$ is equivalent to
$\{\widetilde\dd,\widetilde\mu\} = 0$,
and $\dd^2=0$ is equivalent to
$\{\widetilde \dd,\widetilde \dd\}=0$.
These three properties immediately imply that the
Hochschild differential $\delta(f) = \{\widetilde\mu+\widetilde\dd,
f\}$ indeed squares to zero.
The total complex will be denoted by $\mathsf{Hoch}_{\mathsf{DG}}(A,B)$; its
cohomology is called Hochschild cohomology of $A$ with values in
$B$ and it is denoted by $\mathsf{HHoch}_{\mathsf {DG} }^\bullet(A,B)$, where the
subscript
${\mathsf {DG} }$ means that we are working in the category of differential
graded algebras. Clearly, one recovers the Hochschild cohomology of a
graded algebra $A$ by considering it as a differential graded algebra
with trivial differential. When $B=A$ with the canonical bimodule
structure, we write $\mathsf{HHoch}_{\mathsf {DG} }^\bullet(A)$ for
$\mathsf{HHoch}_{\mathsf {DG} }^\bullet(A,A)$. As in the graded case, the differential
graded Hochschild complex $\mathsf{Hoch}_{\mathsf {DG} }(A)$ has a graded Lie algebra
structure, and both $\delta_0$ and $\delta_1$ are operators of adjoint
type for this Lie algebra structure.
\par
Since the vector field $Q$ squares to zero, it induces a differential
on the algebra of multivector fields of the differential graded
manifold $(N,Q)$. Algebraically, this amounts to saying that the
operator $\{\dd,\cdot\}$ acts as a differential on
$\mathcal{V}^\bullet(A)$. We can therefore look at
$\mathcal{V}^\bullet(A)$ as a bicomplex: the horizontal
differential is zero, and the vertical differential is
$\{\dd,\cdot\}$. We have a HKR map $\mathcal{V}^\bullet(A)\to \mathsf{Hoch}_{\mathsf {DG} }(A)$,
which is defined as in the case of differential algebras.
\begin{Lem}
\label{lem:dr}
The HKR map $(\mathcal{V}^\bullet(A),\{\dd,\cdot\},0)\to
(\mathsf{Hoch}_{\mathsf {DG} }(A),\delta_0,\delta_1)$ is a map of bicomplexes.
\end{Lem}
\begin{proof}
What we have said on the HKR map for graded algebras implies that
$\mathrm{HKR}\colon (\mathcal{V}^\bullet(A),0)\to (\mathsf{Hoch}_{\mathsf {DG}
}(A),\delta_1)$ is a map of complexes. So we are left with checking the
compatibility of $\{\dd,\cdot\}$ with the differential
$\delta_0$. This follows from the following more general fact: given a
vector field $X$ and a multivector field $Y$, then
$\mathrm{HKR}(\{X,Y\})=\{\mathrm{HKR}(X),\mathrm{HKR}(Y)\}$, as one can easily verify. Note that
for an arbitrary multivector field $X$, the above identity only holds
up to homotopy. Since
$\delta_0(\mathrm{HKR}(Y))=\{\mathrm{HKR}(\dd),\mathrm{HKR}(Y)\}$, this concludes the proof.
\end{proof}
Being compatible with the differentials, the HKR map induces a map
between the cohomologies of the total complexes
$
{\mathsf H}^\bullet(\mathcal{V}^\bullet(A),\{\dd,\cdot\})\to
\mathsf{HHoch}_{\mathsf {DG} }^\bullet(A),
$
which is a map of graded Lie algebras. In contrast with the case of
smooth algebras which are the subject of the classical HKR theorem, this
map is not an isomorphism in general, as the next theorem shows.
\begin{Thm}\label{thm:derham}
If $M$ is a simply connected closed oriented smooth manifold of
positive dimension, then the HKR map ${\mathsf H}^\bullet(\mathcal{V}^\bullet
(\Omega^\bullet(M)),\{\dd,\cdot\})\to\mathsf{HHoch}_{\mathsf {DG} }^\bullet
(\Omega^\bullet(M))$ is not an isomorphism.
\end{Thm}
We need the following Lemma, relating the $\{\dd,\cdot\}$-cohomology of
multivector fields on $T[1]M$ to the de Rham cohomology of $M$:
\begin{Lem}\label{lemma:mv-de-rham}
For any differential manifold $M$, there is an isomorphism
\[
{\mathsf H}^\bullet(\mathcal{V}^\bullet
(\Omega^\bullet(M),\{\dd,\cdot\}))\simeq {\mathsf H}^\bullet_{{\rm de
Rham}}(M).\]
\end{Lem}
\begin{proof}
Recall that $\mathcal{V}^\bullet
(\Omega^\bullet(M))$ is the algebra of multivector fields on the graded
manifold $T[1]M$. We fix local
coordinates $\{x^i,\theta^j\}$ on $T[1]M$, where $x^i$ are (even)
coordinates on $M$ and $\theta^j$ (odd) coordinates on the fibers.
Consider the globally well-defined
derivation $\iota_E$ which on the local generators of multivector fields acts
as
\begin{equation*}
\iota_E\left(x^i\right)=0\,;\quad
\iota_E\left(\theta^i\right)=0\,;\quad
\iota_E\left(\frac{\partial}{\partial x^i}\right)=\frac{\partial}{\partial
\theta^i}\,;\quad
\iota_E\left(\frac{\partial}{\partial \theta^i}\right)=0\,.
\end{equation*}
The derivation $\{\dd,\cdot\}$ acts as
\begin{equation*}
\left\{\dd, x^i\right\}=\theta^i\,;\quad
\left\{\dd, \theta^i\right\}=0\,;\quad
\left\{\dd, \frac{\partial}{\partial x^i}\right\}=0\,;\quad
\left\{\dd, \frac{\partial}{\partial \theta^i}\right\}=\frac{\partial}
{\partial x^i}\,.
\end{equation*}
It follows that $L_E=\{\dd,\cdot\}\circ\iota_E + \iota_E\circ \{\dd,\cdot\}$
is a derivation on $\mathcal{V}(T[1]M)$ which, when restricted
to the fields of degree $m$, is the multiplication by $m$; namely
\begin{equation*}
L_E\left(x^i\right)=0\,;\quad
L_E\left(\theta^i\right)=0\,;\quad
L_E\left(\frac{\partial}{\partial x^i}\right)=\frac{\partial}{\partial
x^i}\,;\quad L_E\left(\frac{\partial}{\partial
\theta^i}\right)=\frac{\partial}{\partial
\theta^i}\,.
\end{equation*}
Now, suppose that $\Psi$ is a $\{\dd,\cdot\}$-closed multivector field of
degree $m\ge 1$. Then it is also $\{\dd,\cdot\}$-exact:
\begin{align*}
\Psi&=\frac{1}{m}L_E(\Psi)=\frac{1}{m}\{\dd,\iota_E\Psi\} +
\frac{1}{m}\iota_E(\{\dd,\Psi\})\\
&=\{\dd,\frac{1}{m}\iota_E\Psi\}
\end{align*}
This shows that higher cohomology groups vanish, and we are left to prove that
${\mathsf H}^0(\mathcal{V}^\bullet(T[1]M),\{\dd,\cdot\})={\mathsf H}^\bullet_{{\rm de
Rham}}(M)$. To see this, just notice that the
$0$-vector fields on $T[1]M$ are the differential forms on $M$ and the action
of
$\{\dd,\cdot\}$ on $\mathcal{V}^0(T[1]M)$ is precisely the action of the de Rham
differential on
$\Omega^\bullet(M)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:derham}]
Let $\mathcal{L} M$ be the free loop space on $M$. On the one hand we have
Chen's isomorphism
\cite{Chen, GJP}
\[
{\mathsf H}_\bullet(\mathcal{L} M)[\dim M]
\simeq
\mathsf{HHoch}_{\mathsf {DG} }^\bullet(\Omega^\bullet(M)).
\]
On the other hand, we have the isomorphism
\[
{\mathsf H}^\bullet(\mathcal{V}^\bullet
(\Omega^\bullet(M),\{\dd,\cdot\}))\simeq {\mathsf H}^\bullet_{{\rm de
Rham}}(M)
\]
from Lemma \ref{lemma:mv-de-rham}.
Finally, $
{\mathsf H}_\bullet(\mathcal{L} M)[\dim M]
\not\simeq {\mathsf H}^\bullet_{{\rm de Rham}}(M)$
for any simply connected closed oriented smooth manifold of
positive dimension \cite{SV}.
\end{proof}
\begin{Rem}
Observe that
another way of proving
Lemma \ref{lemma:mv-de-rham}
goes through the Gerstenhaber
isomorphism described in Remark \ref{rem:Fourier}.
In fact, it is not difficult to see that the image of the multivector field
$\dd$ under this isomorphism is the restriction to $A=\Gamma(S^\bullet TM)$
of the canonical Poisson bivector field on the symplectic manifold $T^*M$.
Thus, ${\mathsf H}^\bullet(\mathcal{V}^\bullet(T[1]M),\{\dd,\cdot\})$ is
isomorphic to the Poisson cohomology of $T^*M$ (restricted to functions
polynomial along the fibers)
which in turn (by nondegeneracy of the Poisson structure) is isomorphic to
the de Rham cohomology of the total space
and hence of the base.
\end{Rem}
\section{BV structures in the differential graded case}\label{sec:sq}
By forgetting the differential, i.e., by looking at a differential
graded manifolds simply as a graded manifolds, we obtain a BV
structure on the space of their multivector fields, as in Section
\ref{sec:mvf}. In general, this BV structure does not induce a BV
structure on the
$\{Q,\cdot\}$-cohomology of multivector fields. Indeed, the BV generator
$\Delta$ is a derivation of the BV bracket, so it does not map
$\{Q,\cdot\}$-closed vector fields to $\{Q,\cdot\}$-closed vector fields.
Rather, if $X$ is a $\{Q,\cdot\}$-closed vector field, then
\[
\{Q,\Delta(X)\}=\{\Delta(Q),X\}.
\]
Yet, this implies that, if the vector field $Q$ is divergence-free,
i.e., if $\Delta(Q)=0$ then $\Delta$ induces a BV structure on the
$\{Q,\cdot\}$-cohomology, since
\[
\Delta\{Q,X\}=-\{Q,\Delta(X)\}
\]
and so $\{Q,\cdot\}$-exact multivector fields are mapped to
$\{Q,\cdot\}$-exact multivector
fields. Note that, since the divergence operator $\Delta$ we are
considering in this paper is defined as the variation of the
Berezinian volume form of $N$ along a vector field, the condition
$\Delta(Q)=0$ means that the volume form is $Q$-invariant.
A differential graded manifold $(N,Q)$ with a $Q$-invariant Berezinian
volume form is usually called an SQ-manifold \cite{schwarz,sch2}.
\begin{Rem}
In case $N$ is an odd symplectic manifold and the vector field $Q$ is
Hamiltonian, one speaks of PQ-manifolds \cite{AKSZ}. Note that,
if $Q=H_S$, i.e., if $S$ is the
function on $N$ whose Hamiltonian vector field is $Q$, then $\div(Q)=\Delta(S)$
and $\{Q,Q\}=H_{\{S,S\}}$ where on the right we have the odd Poisson bracket
associated to the odd symplectic structure on $N$. Therefore, under the mild
assumption that $S$ has at least one critical point, the two equations
$\{Q,Q\}=0$ and $\div(Q)=0$
imply the quantum master equation for
$S$, namely $\Delta(S)+\frac{\sqrt{-1}}{2\hbar}\{S,S\}=0.$
\end{Rem}
As far as concerns the BV structures on Hochschild cohomology, the
same construction we described in Section~\ref{sec:hoch} also works
in the differential graded case: if $(A,\dd)$ is the differential
graded algebra of functions on the SQ-manifold $(N,Q)$, then we have
a BV structure on $\mathsf{HHoch}_{\mathsf {DG} }^\bullet(A)$ under the
hypothesis that $\wp\colon \mathsf{Hoch}_{\mathsf {DG} }^\bullet(A) \to
\mathsf{Hoch}_{\mathsf {DG} }^\bullet(A,A^*)$ is a quasi-isomorphism.
Moreover, by the same argument used in Section
\ref{sec:hkr}, the HKR map ${\mathsf
H}^\bullet(\mathcal{V}^\bullet(A),\{\dd,\cdot\})
\to \mathsf{Hoch}^\bullet_{\mathsf {DG} }(A)$ is a BV map in this case.
\par
An example is given by the de Rham algebra $(\Omega^\bullet(M),\dd)$ of
a smooth closed manifold $M$. In the coordinates $\{x^i,\theta^j\}$ on
$T[1]M$, the de Rham differential on $\Omega^\bullet(M)$ is written
\[
\dd=\sum_{i=1}^{\dim M}\theta^i\frac{\partial}{\partial x^i},
\]
so that its divergence is
\[
\div(\dd)=\sum_{i=1}^{\dim M}\frac{\partial \theta^i}{\partial
x^i}=0.
\]
The pairing on $\Omega^\bullet(M)$ induced by the canonical Berezinian
volume form on $T[1]M$ is the usual Poincar\'e duality pairing:
\[
\langle\omega_1 ,\omega_2 \rangle=\int_M\omega_1\wedge\omega_2.
\]
The induced map $\wp\colon
\mathsf{Hoch}_{\mathsf {DG} }^\bullet(\Omega^\bullet(M) ) \to
\mathsf{Hoch}_{\mathsf {DG} }^\bullet(\Omega^\bullet(M), \Omega^\bullet(M)^*)$ is a
quasi-isomor\-phism \cite{M}, and so there exists
a BV algebra structure on $\mathsf{HHoch}_{\mathsf {DG} }^\bullet
(\Omega^\bullet(M))$. This BV algebra structure coincides,
via Chen's isomorphism
\[
\mathsf{HHoch}_{\mathsf {DG} }^\bullet(\Omega^\bullet(M))\simeq
{\mathsf H}_\bullet(\mathcal{L} M)[\dim M],
\]
with the Chas--Sullivan BV structure on the
homology of the free loop space of
$M$ \cite{CS, Chen, CJ, GJP, M, Tr}. Also the $\{\dd,\cdot\}$-cohomology
of
$\mathcal{V}^\bullet(\Omega^\bullet(M))$ has a nice geometrical
interpretation: we have shown in the proof of Lemma
\ref{lemma:mv-de-rham} that
\[
{\mathsf H}^{p}(\mathcal{V}^\bullet(\Omega^\bullet(M)),\{\dd,\cdot\}) =
\begin{cases}
0 & \text{ if }p\neq 0\\
{\mathsf H}^\bullet_{{\rm de
Rham}}(M) & \text{ if }p=0.
\end{cases}
\]
Note that, since the $\{\dd,\cdot\}$-cohomology
of
$\mathcal{V}^\bullet(\Omega^\bullet(M))$ is concentrated in degree zero, the
BV structure on ${\mathsf
H}^\bullet(\mathcal{V}^\bullet(\Omega^\bullet(M)),\{\dd,\cdot\})$ is the
trivial one. Finally, the BV map ${\mathsf
H}^\bullet(\mathcal{V}^\bullet(\Omega^\bullet(M)),\{\dd,\cdot\})\to
\mathsf{HHoch}_{\mathsf {DG} }^\bullet(\Omega^\bullet(M))$ is the natural map
\[
{\mathsf H}^\bullet_{{\rm de
Rham}}(M)\simeq {\mathsf H}_\bullet(M)[\dim M]\to{\mathsf H}_\bullet(\mathcal{L} M)[\dim M]
\]
induced by the natural embedding $M\hookrightarrow {\mathcal{L} }M$ which
identifies the points of $M$ with the constant loops in ${\mathcal{L}} M$.
\begin{Rem}
The constructions of Section \ref{sec:mdo} also work in the differential
graded case: a BV structure is defined on the total cohomology of any
sub-Gerstenhaber algebra $C_{\mathsf {DG} }^\bullet(A)$ of $\mathsf{Hoch}_{\mathsf {DG} }^\bullet(A)$, whose $\wp$-image in $\mathsf{Hoch}_{\mathsf {DG} }^\bullet(A,A^*)$ is
closed under $\beta$. This way we obtain a BV structure on the total
cohomology of multidifferential operators on an SQ-manifold. Moreover,
the HKR map ${\mathsf H}^\bullet(\mathcal{V}^\bullet(A),\{\dd,\cdot\})\to {\mathsf H}_{\mathsf {DG} }^\bullet (\mathcal{D}^\bullet(A))$ is a BV map.
\end{Rem}
|
{
"timestamp": "2006-01-12T13:17:51",
"yymm": "0503",
"arxiv_id": "math/0503380",
"language": "en",
"url": "https://arxiv.org/abs/math/0503380"
}
|
\section{Introduction}
The parallelogram law states that $\|x+y\|^2+\|x-y\|^2=2(\|x\|^2 +
\|y\|^2)$ holds for all vectors $x$ and $y$ in a Hilbert space.
This law implies that the so-called parallelogram inequality
$\|x+y\|^2\leq 2(\|x\|^2 + \|y\|^2)$ trivially holds. S. Saitoh
\cite{SAI} noted the inequality $\|x+y\|^2\leq
2(\|x\|^2+\|y\|^2)$ may be more suitable than the usual triangle
inequality. He used this inequality to the setting of a natural
sum Hilbert space for two arbitrary Hilbert spaces.
Obviously the classical triangle inequality in an arbitrary
normed space implies the above inequality. This motivates us to
introduce an apparently extension of the triangle inequality.
More precisely, we introduce the notion of a $q$-norm, by
replacing, in the definition of a norm, the triangle inequality
by $\| x+y\| ^{q}\leq 2^{q-1}\left( \| x\| ^{q}+\| y\|
^{q}\right)$, where $q \geq 1$. We establish that every q-norm is
a norm in the usual sense, and that the converse is true as well.
The reader is referred to \cite{J-L} for undefined terms and
notations.
\begin{definition} Let ${\mathcal X}$ be a real or complex linear space and $q \in [1, \infty)$. A mapping $%
\| \cdot \| :{\mathcal X}\rightarrow \left[ 0,\infty \right) $ is
called a $q$-norm on ${\mathcal X}$\ if it satisfies the following
conditions:
\begin{enumerate}
\item $\| x\| =0\Leftrightarrow x=0,$
\item $\| \lambda x\| =\| \lambda \|
\| x\| \ \ $for all $x\in {\mathcal X}$ and all scalar $\lambda ,$
\item $\| x+y\| ^{q}\leq 2^{q-1}\left( \|
x\| ^{q}+\| y\| ^{q}\right) \ $for all $x,y\in {\mathcal X}.$
\end{enumerate}
\end{definition}
We first prove a rather trivial result.
\begin{proposition}
Every norm in the usual sense is a $q$-norm.
\end{proposition}
\begin{proof} One can easily verify that the
function $f(t) = \frac{1 + t^q}{2} - (\frac{1 + t}{2})^q$ has a
nonnegative derivative and so it is monotonically increasing on
$[0, \infty)$. It follows that $(\frac{1 +
\frac{\|y\|}{\|x\|}}{2})^q \leq \frac{1 +
(\frac{\|y\|}{\|x\|})^q}{2}$ whenever $\|x\| \leq \|y\|$.
Therefore $\|\frac{x + y}{2}\|^q \leq (\frac{\|x\| + \|y\|}{2})^q
\leq \frac{\|x\|^q + \|y\|^q}{2}$ for all $x, y \in {\mathcal X}$.
It follows that $\|.\|$ is a $q$-norm.
\end{proof}
Now we state the following lemma which is interesting on its own
right.
\begin{lemma} Let ${\mathcal X}$ be a real or complex linear space. Let $%
\| \cdot \| :{\mathcal X}\rightarrow \left[ 0,\infty \right) $ be
a mapping satisfying (1) and (2) in the definition of a $q$-norm.
Then $\| \cdot \| $ is a norm if and only if the set $B=\left\{
x\mid \| x\| \leq 1\right\}$ is convex.
\end{lemma}
\begin{proof} If $\| \cdot \| $ is a norm, then $B$ is clearly a
convex set. Conversely, let $B$ be convex and $x,y\in {\mathcal
X}.$ We can assume that
$x \neq 0, y\neq 0$. Putting $x^{\prime }=\frac{x}{\| x\| }$ and $%
y^{\prime }=\frac{y}{\| y\| }$ we have $x^{\prime },y^{\prime
}\in B.$
Now $\lambda x^{\prime }+\left( 1-\lambda \right) y^{\prime }\in
B$ for all $0 \leq \lambda \leq 1.$ In particular, for $\lambda
=\frac{\|x\|}{\|x\| +\| y\| }$ we obtain%
\[
\| \frac{x}{\| x\| +\| y\| }+\frac{%
y}{\| x\| +\| y\| }\| =\| \lambda x^{\prime }+\left( 1-\lambda
\right) y^{\prime }\| \leq 1.
\]
So that $\| x+y\| \leq \| x\| +\| y\|.$
\end{proof}
We are just ready to prove our main result.
\begin{theorem}
Every $q$-norm is a norm in the usual sense.
\end{theorem}
\begin{proof} We shall show that $B = \left\{ x : \|
x\|
\leq 1\right\} $ is convex. Let $x,y\in B.$ Then we have%
\[
\| x+y\| ^{q}\leq 2^{q-1}\left( \| x\| ^{q}+\| y\| ^{q}\right)
\leq 2^{q-1}\left( 1+1\right) =2^{q}.
\]%
whence $\| \frac{x+y}{2}\| ^{2}\leq 1,$ so $\frac{1}{2}%
x+\left( 1-\frac{1}{2}\right) y\in B.$ Thus if $A=\left\{ \frac{k}{2^{n}}%
\mid n=1,2,\ldots ;k=0,1,\ldots ,n\right\}$, then for each
$\lambda \in A$ we have $\lambda x+\left( 1-\lambda \right) y\in
B.$
Let $0\leq \lambda \leq 1$ and $z=\lambda x+\left( 1-\lambda
\right) y.$ Since $A$\ is dense in $\left[ 0,1\right]$, there
exists a decreasing
sequence $\left\{ r_{n}\right\} $ in $A$ such that $\lim\limits_{n}r_{n}=%
\lambda .$ Put $\beta _{n}=\frac{1-r_{n}}{1-\lambda }.$ Obviously $0\leq
\beta _{n}\leq 1,$ $\lim\limits_{n}\beta _{n}=1$ and $\frac{r_{n}+\beta
_{n}-1}{r_{n}} \leq 1.$ Since $\frac{r_{n}+\beta _{n}-1}{r_{n}}x\in B$ and $%
r_{n}\in A$ we conclude that%
\[
\beta _{n}z = \lambda \beta _{n}x+\left( 1-\lambda \right) \beta _{n}y=r_{n}%
\frac{r_{n}+\beta _{n}-1}{r_{n}}x+\left( 1-r_{n}\right) y\in B.
\]%
Thus $\beta _{n} \| z\| =\| \beta _{n}z\| \leq 1$ for all $n.$
Tending $n$ to infinity we get $\| z\| \leq 1,$ i.e. $z\in B.$
\end{proof}
{\bf Acknowledgment.} We would like to sincerely thank Professor
Saburou Saitoh for his encouragement.
|
{
"timestamp": "2005-12-23T11:29:41",
"yymm": "0503",
"arxiv_id": "math/0503616",
"language": "en",
"url": "https://arxiv.org/abs/math/0503616"
}
|
\section{Introduction}
This paper summarizes some results of work originally initiated by
Peter Carr. It supposes to investigate various numerical and
analytical methods of option pricing using VG model in order to find
out which algorithm is most efficient.
Let us first give a brief overview of the VG model. The Variance Gamma
(VG for short) process was proposed by Madan and Seneta (see
\cite{MadanSeneta1990}) to describe stock price dynamics instead of the
Brownian motion in the original Black-Scholes model. Two new
parameters: $\theta$ skewness and $\nu$ kurtosis are introduced in
order to describe asymmetry and fat tails of real life distributions.
The VG process is defined by evaluating Brownian motion with drift at a
random time specified by gamma process. In other words, the VG model
with parameter vector $(\sigma, \nu, \theta)$ assumes that the forward
price satisfies the following equation
\begin{equation}} \def\eeq{\end{equation} \label{underVG} \ln F_{t} = \ln F_{0} + X_{t} + \omega t, \eeq
where \begin{equation}} \def\eeq{\end{equation} X_{t} = \theta \gamma_{t}(1, \nu) + \sigma W_{\gamma_{t}(1,
\nu)}, \eeq
\noindent and $\gamma_{t}(1, \nu)$ is a Gamma process playing the role of
time in this case with unit mean rate and density function given by
\begin{equation}} \def\eeq{\end{equation} \label{VGdensity} f_{\gamma_{t}(1, \nu)}(x) =
\frac{x^{\frac{t}{\nu}-1}e^{-\frac{x}{\nu}}}
{\nu^{\frac{t}{\nu}}\Gamma\left(\frac{t}{\nu}\right)}. \eeq
In the Eq.~(\ref{underVG}) $\omega$ is chosen to make $F_{t}$ a
martingale.
The probability density function for the VG process may be written as
\begin{equation}\label{pdfVG}
h_t(x) = \int_0^\infty \dfrac{dg}{\sqrt{2\pi g}}\exp \left[
- \dfrac{(x-\theta g)^2}{2 \sigma ^2g}\right]
\frac{g^{\frac{t}{\nu}-1}e^{-\frac{g}{\nu}}}
{\nu^{\frac{t}{\nu}}\Gamma\left(\sfrac{t}{\nu}\right)}
\end{equation}
\noindent or after integration over $g$
\begin{equation}\label{pdfVGfin}
h_t(x) = \dfrac{2\exp \left( \theta x /\sigma ^2 \right)}{\sqrt{2}\pi \sigma
\nu^{\frac{t}{\nu}} \Gamma\left(\sfrac{t}{\nu}\right)} \left( \dfrac{x^2}{\theta ^2 +
\sfrac{2\sigma ^2}{\nu}} \right)^{\frac{t}{2\nu} - \frac{1}{4}}
K_{\frac{t}{\nu} - \frac{1}{2}}\left( \dfrac{1}{\sigma ^2}
\sqrt{x^2\left( \theta ^2 + \frac{2\sigma ^2}{\nu}\right)} \right),
\end{equation}
\noindent where $K$ is the modified Bessel function of the second kind. The
characteristic function $ \phi_{\gamma_{t}(1, \nu)}(u)$ for the VG
process has remarkably simple form
\begin{equation}\label{char1}
\phi _t(u) \equiv \left< E^{iux} \right> \equiv \int_0^\infty
h_t(x)e^{iux} dx = \dfrac{1} {(1 - i\theta \nu u + \frac{1}{2}\sigma ^2\nu
u^2)^\frac{t}{\nu}}.
\end{equation}
Another derivation of this expression could be obtained when
conditioning on time change like in Romano-Touzi for stochastic
volatility models
\begin{eqnarray}} \def\beaz{\begin{eqnarray*} \label{charFunc}
\phi_{\gamma_{t}(1, \nu)}(u) &=&
{\mathbb E}[e^{iu\gamma_{t}(1, \nu)}] = \int_{0}^{\infty}e^{iux}f_{\gamma_{t}(1, \nu)}(x)dx
= \int_{0}^{\infty}e^{iux}\frac{x^{\frac{t}{\nu}-1}e^{-\frac{x}{\nu}}}{\nu^{\frac{t}{\nu}}
\Gamma\left(\frac{t}{\nu}\right)}dx
\nonumber} \def\leeq{\lefteqn\\ &=&
\int_{0}^{\infty}\frac{x^{\frac{t}{\nu}-1}e^{-\frac{x(1 - iu\nu)}{\nu}}}{\nu^{\frac{t}{\nu}}
\Gamma\left(\frac{t}{\nu}\right)}dx
\nonumber} \def\leeq{\lefteqn\\ &=&
(1 - iu\nu)^{-\frac{t}{\nu}}\int_{0}^{\infty}\frac{(x(1 - iu\nu))^{\frac{t}{\nu}-1}
e^{-\frac{x(1 - iu\nu)}{\nu}}}{\nu^{\frac{t}{\nu}}\Gamma\left(\frac{t}{\nu}\right)}
d(x(1 - iu\nu))
\nonumber} \def\leeq{\lefteqn\\ &=&
(1 - iu\nu)^{-\frac{t}{\nu}}\int_{0}^{\infty}\frac{y^{\frac{t}{\nu}-1}
e^{-\frac{y}{\nu}}}{\nu^{\frac{t}{\nu}}\Gamma\left(\frac{t}{\nu}\right)}dy
= (1 - iu\nu)^{-\frac{t}{\nu}}.
\end{eqnarray}} \def\eeaz{\end{eqnarray*}
\begin{eqnarray}} \def\beaz{\begin{eqnarray*} \label{interm1}
\phi_{X_{t}}(u) &=& {\mathbb E}[e^{iuX_{t}}] =
{\mathbb E}[{\mathbb E}[e^{iuX_{t}} \mid \gamma_{t}(1, \nu)]] =
{\mathbb E}[{\mathbb E}[e^{iu\left(\theta \gamma_{t}(1, \nu) + \sigma W_{\gamma_{t}(1, \nu)}\right)}
\mid \gamma_{t}(1, \nu)]]
\nonumber} \def\leeq{\lefteqn\\ &=&
{\mathbb E}[e^{iu\theta \gamma_{t}(1, \nu) - \frac{1}{2}u^{2}\sigma^{2}\gamma_{t}(1,
\nu)}] = {\mathbb E}[e^{i\left(u\theta + i\frac{1}{2}u^{2}\sigma^{2}\right)\gamma_{t}(1,
\nu)}] \nonumber} \def\leeq{\lefteqn\\&=& \phi_{\gamma_{t}(1, \nu)}(u\theta + i\frac{1}{2}u^{2}\sigma^{2})
= \left( 1 - i\theta\nu u + \frac{1}{2}\sigma^{2}\nu u^{2}\right)^{-\frac{t}{\nu}}.
\end{eqnarray}} \def\eeaz{\end{eqnarray*}
Now, to prevent arbitrage, we need $F_{t}$ be a martingale, and, since
$F_{t}$ is already an independent increment process, all we need is
\begin{equation}} \def\eeq{\end{equation} {\mathbb E}[F_{t}] = F_{0}, \eeq
\noindent or
\begin{equation}} \def\eeq{\end{equation} {\mathbb E}[F_{0}e^{X_{t} + \omega t}] = F_{0}\phi_{X_{t}}(-i)e^{\omega t}
= F_{0}. \eeq
This tells us that
\begin{equation}} \def\eeq{\end{equation} \label{omega} \omega = - \frac{\ln \phi_{X_{t}}(-i)}{t} =
-\frac{-\frac{t}{\nu} \ln\left( 1 - \theta\nu -
\frac{1}{2}\sigma^{2}\nu \right)}{t} =\frac{1}{\nu}\ln\left( 1 -
\theta\nu - \frac{1}{2}\sigma^{2}\nu \right). \eeq
Note that from the definition of $\omega$ above, in order to have a
risk neutral measure for VG model, its parameters must obey an
inequality:
\begin{equation}} \def\eeq{\end{equation} \label{constrain} \dfrac{1}{\nu} > \theta + \frac{\sigma^{2}}{2}.
\eeq
Note that risk neutral parameters $\theta , \nu, \sigma $ do not have
to be equal to their statistical counterparts.
Accordingly, the characteristic function of the $x_T \equiv \log
S_T$ VG process is
\begin{equation}} \def\eeq{\end{equation} \label{charS}
\phi(u) = \dfrac{S_0 e^{(r-q+\omega)T}}{ \left( 1 - i\theta\nu u + \frac{1}{2}\sigma^{2}\nu u^{2}\right)^{\frac{T}{\nu}}}.
\eeq
Statistical parameters of VG distribution may be calculated from the
historical data on stock prices. In particular we have to find the
values of the parameters $\theta ^*, \nu^*$ and $\sigma ^*$ such that
the folloiwng expression is maximized:
\begin{equation}\label{calibr}
\prod_{j=1}^{n} h_{\tau _j} (x_j),
\end{equation}
\noindent where $h_{\tau _j} (x_j)$ are given by Eq.\ref{pdfVGfin} and $x_j$
are observed returns per time $\tau _j$, i.e. $x_j =
\log(S_j/S_{j-1})$.
\section{Pricing European option}
The value of European option on a stock when the risk neutral
dynamics is given by Eq.~(\ref{underVG}) is
\begin{equation}\label{EurVG}
V = \exp(-rT) \int_{-\infty}^{\infty} h_T\left(x-(r-q+\omega )T\right)
W(e^x)dx,
\end{equation}
\noindent where $T$ is time until expiration, $q$ is continuous dividend and
$W(e^x)$ is payoff function that has the following form
\begin{equation}\label{payoff}
W(e^x) = (S_0e^x - K)^+ - \mbox{call}, \quad W(e^x) = (K - S_0e^x)^+ -
\mbox{put}.
\end{equation}
Direct calculation allows us to derive the put-call parity relation
identical to Black-Scholes case
\begin{equation}\label{parity}
C = S_0 e^{-qT} - Ke^{-rT} + P.
\end{equation}
There are several methods to price a European option under the VG
model. One method uses the closed form solution derived in
\cite{MadCarrChang}. Although the expression is analytic it requires
computation of modified Bessel functions, and hence may not be as fast
as we would like our pricing model to be. Therefore, FFT method has
been widely utilized to obtain a more efficient pricer. Few flavors of
the FFT method has been previously discussed with regard to the VG model.
First of all the FFT method of Carr and Madan \cite{CarrMadan:99a},
nowadays almost standard in math finance, was applied to the VG model
to price the European vanilla option since the characteristic function
of the log-return process has a very simple form given above. Further
we intend to show, that unfortunately this method blows up at some
values of the VG parameters.
Mike Konikov and Dilip Madan \cite{MadanKonikov:2004a} proposed another
interesting method based on the definition of the VG process as being a
time changed Brownian motion, where the time change is assumed
independent of the Brownian motion. This method was described in detail
in \cite{MadanKonikov:2004a} while has not been implemented yet.
Also Mike Konikov and I independently implemented a modification of the FFT
method - the Fractional Fourier Transform, which is described in detail
in \cite{Bailey_Swarztrauber_1991, Chourdakis2004}. This method usually allows
acceleration of the pricing function by factor 8-10, while for the VG model it
still demonstrates same problem as the original FFT.
Below we discuss why the Carr and Madan FFT approach fails for the VG model.
We propose another method, which originally has been developed in a general form by Lewis \cite{Lewis:2001},
that seems to be free of such problems.
\section{Carr-Madan's FFT approach and the VG model}
Let us start with a short description of the Carr-Madan FFT method. It was
worked out for models where the characteristic function of underlying price
process ($S_t$) is available. Therefore, the vanilla options can be priced very
efficiently using FFT as described in Carr and Madan ~\cite{CarrMadan:99a}. The
characteristic function of the price process is given by
\begin{equation}\label{cFunc}
\phi(u,t)={\mathbb E}(e^{iuX_t}),
\end{equation}
\noindent where $X_t=\log(S_t)$. Note that the above representation holds for all
models and is not just restricted to L\'evy models where the characteristic
functions have a time homogeneity constraint that $\phi(u,t)=e^{-t\psi(u)}$,
where $\psi(u)$ is the L\'evy characteristic exponent.
Once the characteristic function is available, then the vanilla call option can
be priced using Carr-Madan's FFT formula:
\begin{equation} \label{callFFT}
C(K,T)=\frac{e^{-\alpha\log(K)}}{\pi} \int_0^{\infty}\mathrm{Re}
\left[e^{-iv\log(K)}\omega(v)\right]dv,
\end{equation}
\noindent where
\begin{equation} \label{omega}
\omega(v)=\frac{e^{-rT}\phi(v-(\alpha+1)i,
T)}{\alpha^2+\alpha-v^2+i(2\alpha+1)v}
\end{equation}
The integral in the first equation can be computed using FFT, and as a result
we get call option prices for a variety of strikes. For complete details, see
Carr \& Madan paper \cite{CarrMadan:99a}.
The put option values can just be constructed from Put-Call
symmetry.
\begin{figure}[ht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{FRFT_90_002_001.eps}
\caption{European option values in VG model at $T=0.02 yrs, K = 90, \sigma = 0.01$
obtained with FRFT.}
\label{MikeFRFT1}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{Integr_90_002_001.eps}
\caption{European option values in VG model at $T=0.02 yrs, K = 90, \sigma = 0.01$
obtained with the adaptive integration.}
\label{MikeIntegr1}
\end{minipage}
\end{flushleft}
\end{figure}
Parameter $\alpha $ in Eq.~(\ref{callFFT}) must be positive. Usually
$\alpha
= 3$ works well for various models. It is important that the denominator in
Eq.~(\ref{omega}) has only imaginary roots while integration in
Eq.~(\ref{callFFT}) is provided along real $v$. Thus, the integrand of
Eq.~(\ref{callFFT}) is well-behaved.
But as it turned out, this is not the case for the VG model. To show
this let us consider the European call option values obtained by
Mike Konikov by computing FFT of the VG characteristic function
according to Eq.~(\ref{callFFT}).
In Fig.~\ref{MikeFRFT1} the results of that test obtained using the FRFT
algorithm are given for strike $K=90$, maturity $T = 0.02$ yrs and volatility
$\sigma = 0.01$. It is seen that at positive coefficients of skew $\Theta
\approx 2$ and coefficients of kurtosis $\nu \approx 0.5$ the option value has
a delta-function-wise pick that doesn't seem to be a real option value
behavior. In Fig.~\ref{MikeIntegr1} similar results are obtained using a
different method of evaluation of the integral in Eq.~(\ref{callFFT}) - an
adaptive integration. Eventually, in Fig.~\ref{MikeFFT1} same test was provided
using a standard FFT method. The results look quite different that allows a
guess that something is wrong with FRFT and the adaptive integration. One could
also note that this test plays with an option with a very short maturity.
Therefore, to let us make another test with a longer maturity. In
Fig.~\ref{MikeFFT2}-\ref{MikeIntegr2} the results of the test that uses
same integration procedures, but for the option with $K = 90, T=1, \sigma = 1$,
are presented. It is seen that for longer maturities FFT also blows up almost
at the same region of the model parameters. Moreover, it occurs not only at
positive value of the skew coefficient but at negative as well. Thus, the
problem lies not in the numerical method that was used to evaluate the integral
in the Eq.~(\ref{callFFT}), but in the integral itself.
\begin{figure}[ht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{FFT_90_002_001.eps}
\caption{European option values in VG model at $T=0.02 yrs, K = 90, \sigma = 0.01$
obtained with FFT.}
\label{MikeFFT1}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{FFT_90_100_100.eps}
\caption{European option values in VG model at $T=1.0 yrs, K = 90, \sigma = 1.0$
obtained with the FFT.}
\label{MikeFFT2}
\end{minipage}
\end{flushleft}
\end{figure}
\begin{figure}[ht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{FRFT_90_100_100.eps}
\caption{European option values in VG model at $T=1.0 yrs, K = 90, \sigma = 1.0$
obtained with the FRFT.}
\label{MikeFRFT2}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{Integr_90_100_100.eps}
\caption{European option values in VG model at $T=1.0 yrs, K = 90, \sigma = 1.0$
obtained with the adaptive integration.}
\label{MikeIntegr2}
\end{minipage}
\end{flushleft}
\end{figure}
Now having expression Eq.~(\ref{char1}) for the VG characteristic
function let us substitute it and Eq.~(\ref{omega}) into the
Eq.~(\ref{callFFT}) that gives
\begin{equation} \label{callFFT1}
C(K,T) \propto \dfrac{e^{-\alpha\log(K) - rT}}{\pi}\int_0^{\infty}
\Re\left\{\dfrac{e^{-iv\log(K)}}{\left[\alpha^2+\alpha-v^2+i(2\alpha+1)v
\right]\left( 1 - i\theta\nu u + \frac{1}{2}\sigma^{2}\nu
u^{2}\right)^{\frac{t}{\nu}} }\right\}dv,
\end{equation}
\noindent where $u \equiv v-(\alpha+1)i$. At small $T$ close to zero the second term
in the denominator of the Eq.~(\ref{callFFT1}) is close to 1. Therefore at
small $T$ the denominator has no real roots. To understand what happens at
larger maturities, let us put $T = 0.8, \nu = 0.1, \alpha =3, \sigma = 1$ and
see how the denominator behaves as a function of $v$ and $\Theta $. The results
of this test obtained with the help of Mathematica package are given in
Fig~\ref{Math1}.
It is seen that at $v=0$ at positive $\Theta $ the characteristic function has
a singularity. To investigate it in more detail, we assume $v=0$ and plot the
denominator as a function of $\sigma $ and $\Theta$ (see Fig.~\ref{Math2}). As
follows from this Figure in the interval $0 < \sigma < 2$ there exists a value
of $\Theta $ that makes the integrand in the Eq.~(\ref{callFFT1}) singular.
This means that singularity of the integrand can not be eliminated, and thus
the Carr-Madan FFT method can not be used together with the VG model for
pricing European vanilla options. Using FRFT or adaptive integration that both
are slight modifications of the FFT, also doesn't help.
Note that for the VG model the authors of \cite{CarrMadan:99a} derived
condition which keeps the characteristic function to be finite, that reads
\begin{equation}\label{cond}
\alpha < \sqrt{\dfrac{2}{\nu \sigma ^2} + \dfrac{\Theta ^2}{\sigma ^4}} -
\dfrac{\Theta }{\sigma ^2} - 1.
\end{equation}
\begin{figure}[ht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{math11.eps}
\caption{Denominator of the Eq.~(\ref{callFFT1}) at $T = 0.8, \nu = 0.1,
\alpha =3, \sigma = 1$ as a function of $v$ and $\Theta $.}
\label{Math1}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{math21.eps}
\caption{Denominator of the Eq.~(\ref{callFFT1}) at $T = 0.8, \nu = 0.1,
\alpha =3, v=0$ as a function of $\sigma$ and $\Theta $.} \label{Math2}
\end{minipage}
\end{flushleft}
\end{figure}
Also as can be seen, for $\Theta, \nu $ and $\sigma $ corresponding to the
above mentioned tests $\alpha $ becomes negative that doesn't allow using this
method to price the options in terms of strike.
In order to solve these problems one needs to find another way how to
regularize the integrand, i.e. eliminate doing it in the way as Carr and Madan
did it using a regularization factor $e^{-\alpha k}$.
\section{Lewis's regularization}
Another approach of how to apply FFT to the pricing of European options was
proposed by Alan Lewis \cite{Lewis:2001}. Lewis notes that a general integral
representation of the European call option value with a vanilla payoff is
\begin{equation}\label{genInt}
C_T(x_0, K) = e^{-rT} \int_{-\infty}^{\infty} \left( e^x - K\right)^+ q(x, x_0,
T)dx,
\end{equation}
\noindent where $x = \log S_T$ is a stock price that under a pricing measure evolves
as $S_T = S_0\exp[(r-q)T + X_T$, $r-q$ is the cost of carry, $T$ is the
expiration time for some option, $X_T$ is some Levy process satisfying
${\mathbb E}[exp(i u X_T)] =1$, and $q$ is the density of the log-return
distribution $x$.
The central point of the Lewis's work is to represent the
Eq.~(\ref{genInt}) as a convolution integral and then apply a Parseval
identity
\begin{equation}\label{parseval}
\int_{-\infty}^{\infty} f(x) g(x_0-x)dx = \dfrac{1}{2\pi}
\int_{-\infty}^{\infty} e^{-i u x_0}\hat{f}(u)\hat{g}(u)du,
\end{equation}
\noindent where the hat over function denotes its Fourier transform.
The idea behind this formula is that the Fourier transform of a transition
probability density for a Levy process to reach $X_t = x$ after the elapse of
time $t$ is a well-known characteristic function, which plays an important role
in mathematical finance. For Levy processes it is $\phi _t(u) =
{\mathbb E}[\exp(iuX_t)], u \in \Re$, and typically has an analytic extension (a
generalized Fourier transform) $u \rightarrow z \in {\mathbb C}$, regular in
some strip ${\cal S}_X$ parallel to the real z-axis.
Now suppose that the generalized Fourier transform of the payoff
function $\hat{w}(z) = \int_{-\infty}^{\infty} e^{izx}(e^x - K)^+dx$
and the characteristic function $\phi _t(z)$ both exist (we will
discuss this below). Then from a chain of equalities the call option
value can be expressed as follows
\begin{eqnarray} \label{chain}
C_T(x_0, K) &=& e^{-rT} {\mathbb E}\left[ \left( e^x - K\right)^+\right]
= \dfrac{e^{-rT}}{2\pi}{\mathbb E} \left[\int_{i\mu -\infty}^{i\mu +\infty}
e^{-izx_T} \hat{w}(z)dz \right]
\\
&=& \dfrac{e^{-rT}}{2\pi}{\mathbb E} \left[\int_{i\mu -\infty}^{i\mu +\infty} e^{-iz[x_0
+ (r-q + \omega)T]} e^{-izX_T} \hat{w}(z)dz \right]
\nonumber} \def\leeq{\lefteqn \\
&=& \dfrac{e^{-rT}}{2\pi}\int_{i\mu -\infty}^{i\mu +\infty}
e^{-iz[x_0 + (r-q+ \omega)T]} {\mathbb E}[e^{-izX_T}] \hat{w}(z)dz =
\dfrac{e^{-rT}}{2\pi}\int_{i\mu -\infty}^{i\mu +\infty}
e^{-izY} \phi_{X_T} (-z) \hat{w}(z)dz.
\nonumber} \def\leeq{\lefteqn
\end{eqnarray}
Here $Y = x_0 + (r-q+ \omega)T$, $\mu \equiv$ Im $z$. This is a formal
derivation which becomes a valid proof if all the integrals in
Eq.~(\ref{chain}) exist.
The Fourier transform of the vanilla payoff can be easily found by a direct
integration
\begin{equation}\label{FTpayoff}
\hat{w}(z) = \int_{-\infty}^{\infty} e^{izx}(e^x - K)^+dx = -
\dfrac{K^{iz+1}}{z^2 - iz}, \qquad \mathrm {Im} z > 1.
\end{equation}
Note that if z were real, this regular Fourier transform would not exist. As
shown in \cite{Lewis:2000}, payoff transforms $\hat{w}(z)$ for typical claims
exist and are regular in their own strips ${\cal S}_w$ in the complex z-plane,
just like characteristic functions.
Above we denoted the strip where the characteristic function $\phi (z)$ is
well-behaved as ${\cal S}_X$. Therefore, $\phi (-z)$ is defined at the
conjugate strip ${\cal S}^*_X$. Thus, the Eq.~(\ref{chain}) is defined at the
strip ${\cal S}_V = {\cal S}^*_X \bigcap {\cal S}_w$, where it has the form
\begin{equation}\label{callFFTfin}
C(S,K,T) = - \dfrac{Ke^{-rT}}{2\pi}\int_{i\mu -\infty}^{i\mu +\infty}
e^{-izk} \phi_{X_T} (-z) \dfrac{dz}{z^2-iz}, \quad \mu \in {\cal S}_V,
\end{equation}
\noindent and $k = \log(S/K) + (r-q+ \omega)T$.
The characteristic function of the VG process has been given by the
Eq.~(\ref{charS}) and is defined in the strip $\beta - \gamma <$ Im
$z < \beta + \gamma $, where
\begin{equation} \label{beta}
\beta = \dfrac{\Theta }{\sigma ^2}, \quad
\gamma = \sqrt{\dfrac{2}{\nu \sigma ^2} + \dfrac{\Theta ^2}{\sigma ^4} +
2(\mathrm {Re} z)^2}.
\end{equation}
This condition can be relaxed by assuming in the Eq.~(\ref{beta}) $\mathrm {Re}
z = 0$ \footnote{In other words, if it is valid at $\mathrm {Re} z = 0$, it
will be valid for any $\mathrm {Re} z$}. Accordingly, $\phi (-z)$ is defined in
the strip $\gamma - \beta >$ Im $z > - \beta - \gamma $.
Now let us choose Im $z$ in the form
\begin{equation}\label{ImzForm}
\mu \equiv \mathrm{Im} \ z = \sqrt{1 + \dfrac{2\Theta }{\sigma ^2} +
\dfrac{\Theta ^2}{\sigma ^4}} - \dfrac{\Theta }{\sigma ^2}.
\end{equation}
Taking into account the Eq.~(\ref{constrain}) which makes a constrain on the
available values of the VG parameters, it is easy to see that $\mu $ defined in
such a way obeys the inequality $\mu < \gamma - \beta $. On the other hand, as
also can be easily seen, $\mu \ge 1$ at any value of $\Theta $ and positive
volatilities $\sigma$, and the equality is reached when $\Theta =0$. It means,
that Im $z = \mu$ lies in the strip ${\cal S}^*_X$ as well as in the strip
${\cal S}_w$, i. e. $\mu \in {\cal S}_V$.
Now one more trick with contour integration. The integrand in
Eq.~(\ref{callFFTfin}) is regular throughout ${\cal S}^*_X$ except for simple
poles at $z = 0$ and $z = i$. The pole at $z = 0$ has a residue
$-Ke^{-rT}i/(2\pi)$, and the pole at $z = i$ has a residue $Se^{-qT}i/(2\pi)$
\footnote{This is because $\phi _T(-i) = e^{-\omega T}$}. The analysis of the
previous paragraph shows that the strip ${\cal S}^*_X$ is defined by the
condition $\gamma - \beta
> \mathrm{Im} z > - \beta - \gamma$, where $\gamma - \beta > 1$, and $- \beta -
\gamma < 0$. Therefore we can move the integration contour to $\mu_1 \in
(0,1)$. Then by the residue theorem, the call option value must also equal the
integral along Im $z = \mu_1$ minus $2\pi i$ times the residue at $z=i$. That
gives us a first alternative formula
\begin{equation} \label{altFFT}
C(S,K,T) =
Se^{-qT} - \dfrac{Ke^{-rT}}{2\pi}\int_{i\mu_1
-\infty}^{i\mu_1 +\infty} e^{-izk} \phi_{X_T} (-z) \dfrac{dz}{z^2-iz}
\end{equation}
For example, with $\mu _1 = 1/2$ which is symmetrically located between the two
poles, this last formula becomes
\begin{equation} \label{altFFT2}
C(S,K,T) =
Se^{-qT} - \dfrac{1}{\pi}\sqrt{SK}e^{-(r+q)T/2}
\int_{0}^{\infty} \mathrm{Re}\left[e^{-iu \kappa } \Phi \left(-u -\dfrac{i}{2}
\right) \right]\dfrac{du}{u^2+ \frac{1}{4}} \\
\end{equation}
\noindent where $\kappa = \ln (S/K) + (r-q)T, \quad \Phi(u) = e^{i u \omega T} \phi_{X_T}(u)$ and
it is taken into account that the integrand is an even function of
its real part. The last integral can be rewritten in the form
\begin{equation} \label{intF}
\int_{0}^{\infty} e^{-iu \ln \kappa } \phi_1(u) du, \qquad
\phi _1(u) = \dfrac{4}{4u^2+ 1}\Phi\left(-u - \dfrac{i}{2}\right) .
\end{equation}
This can be immediately recognized as a standard inverse Fourier
transform, and by derivation the integrand is regular everywhere.
Indeed, $\phi _{X_T}(-u-i/2)|_{u=0} = (1-\frac{\sigma ^2 \nu }{8} -
\frac{\nu \theta }{2})^{-t/\nu}$, therefore the denominator vanishes
if $\frac{2}{\nu} = \theta + \frac{\sigma ^2}{4}$. Now using the
Eq.~(\ref{constrain}) one finds that $\theta + \frac{\sigma ^2}{4} >
2h + \sigma ^2$ or $\frac{\sigma ^2}{4} < -\frac{\theta }{3}$. Thus,
$\theta $ must be negative to turn the denominator to zero. The last
equality could be also rewritten as $\theta +\frac{\sigma ^2}{4} <
\frac{2\theta }{3}$. Thus, the denominator vanishes if $\frac{1}{\nu } < \frac{2\theta }{3}$,
i.e. $\nu $ must be negative, but it is not! Therefore, the characteristic function in Eq.~(\ref{intF})
doesn't have singularity at $u=0$. Thus, a standard FFT or FRFT method can be applied to get the value
of the integral.
In Fig.~\ref{MyFFT1} -\ref{MyFFT2} the results of the European vanilla option
pricing with the VG model conducted by using this new FFT method are displayed.
Two test has been provided with parameters $T=1$ yr, $K=90, \sigma = 0.1$
(Fig.~\ref{MyFFT1}) and $T=1$ yr, $K=90, \sigma = 0.5$ (Fig.~\ref{MyFFT2}). It
is seen that the option value surface is regular in both cases. Zero values
indicates that region, where the VG constrain Eq.~(\ref{constrain}) is not
respected. The higher values of $\sigma $ and $\Theta$ are the lower values of
$\nu $ are required to obey this constraint. Therefore, at higher values of
$\nu $ the model is not defined that produces irregularity in the graph. This
effect is better observable in Fig.~\ref{MyFFT2_2} that is obtained by rotation
of the Fig.~\ref{MyFFT2}. The above means that the new FFT method can be used
with no essential problem. A generalization of this method for FRFT is also
straightforward.
In the region of the VG parameters values where an application of the
Carr-Madan FFT procedure doesn't cause the problem the results of that
method are almost identical to what the described above method gives.
An example of such a comparison is given in Fig.~\ref{diff} (my NewFFT
Matlab code vs Mike's FFT code). It is seen that the difference is of
the order of $10^{-7}$.
\begin{figure}[bht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{Lewis_90_100_010.eps}
\caption{European option values in VG model at $T=1.0 yr, K = 90, \sigma = 0.1$
obtained with the new FFT method.} \label{MyFFT1}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{Lewis_90_100_050.eps}
\caption{European option values in VG model at $T=1.0 yrs, K = 90, \sigma =
0.5$ obtained with the new FFT method.} \label{MyFFT2}
\end{minipage}
\end{flushleft}
\begin{center}
\includegraphics[totalheight=2in]{Lewis_90_100_050_1.eps}
\caption{European option values in VG model at $T=1.0 yr, K = 90, \sigma = 0.5$
obtained with the new FFT method (rotated graph).} \label{MyFFT2_2}
\end{center}
\end{figure}
\begin{figure}[bht]
\begin{center}
\includegraphics[totalheight=1.5in]{diff.eps}
\caption{The difference between the European call option values for the VG
model obtained with Carr-Madan FFT method and the new FFT method. Parameters of
the test are: $S=100, T=0.5 yr, \sigma = 0.2, \nu =0.1, \Theta =-0.33, r=q=0.$
at various strikes).} \label{diff}
\end{center}
\end{figure}
\section{Black-Scholes-wise method}
One more method of regularization of the Fourier kernel for the VG
model has been proposed by Sepp \cite{Sepp2003} and is also discussed in \cite{Iddo2004VG}, \cite{ContTankov2004}. The idea is
as follows.
Given characteristic function $\phi_{X_t} (z)$ of the model $M$ the
price of a European option can be expressed as
\begin{eqnarray}\label{HEuropPrice}
\Pi_1^M &=& \dfrac{1}{2} + \dfrac{\xi}{2\pi} \int_{-\infty}^{\infty}
\dfrac{e^{-iu \ln K} e^{iu [\ln S +(r-q+\omega)T]} \phi_{X_T}(u-i)}{i u \phi_{X_T}(-i)}du, \\
\Pi_2^M &=& \dfrac{1}{2} + \dfrac{\xi}{2\pi} \int_{-\infty}^{\infty}
\dfrac{e^{-iu \ln K}e^{iu [\ln S +(r-q+\omega)T]}\phi_{X_T}(u)}{i u}du, \nonumber} \def\leeq{\lefteqn \\
V^M &=& \xi \left[e^{-q T}S_0 \Pi_1^M - e^{-r T} K \Pi_2^M\right]
\nonumber} \def\leeq{\lefteqn,
\end{eqnarray}
\noindent where $\xi = 1(-1)$ for a call(put). Eq.~(\ref{HEuropPrice}) is
a generalization of the Black-Scholes option pricing formula. Note
that $\phi_{X_t} (0) = 1$ by definition, and $\phi_{X_t} (-i)$ is a
function of time to expiry $T$ and parameters of the model only.
{\bf Proof}: Assume that $\phi _T(-z)$ has a strip of regularity $0
\le \mu \le 1$. First we rewrite Eq.~(\ref{callFFTfin}) as
\begin{eqnarray} \label{BSFFT}
C(S,K,T) &=& - \dfrac{K e^{-rT}}{2\pi}\int_{i\mu -\infty}^{i\mu
+\infty} e^{-izk}\phi_{X_T} (-z) \dfrac{dz}{z^2-i z} \\
&=& - \dfrac{Ke^{-rT}}{2\pi}\left[\int_{i\mu -\infty}^{i\mu
+\infty} e^{-izk} \phi_{X_T} (-z) \dfrac{i dz}{z} - \int_{i\mu -\infty}^{i\mu
+\infty} e^{-izk} \phi_{X_T}(-z) \dfrac{i dz}{z-i} \right]\nonumber} \def\leeq{\lefteqn \\
&=& - \dfrac{Ke^{-rT}}{2\pi}({\cal R}(I_1) - {\cal R}(I_2)) \nonumber} \def\leeq{\lefteqn
\end{eqnarray}
\begin{figure}[ht]
\begin{center}
\includegraphics[totalheight=2 in]{contour.eps}
\caption{Integration contour for ${\cal R}(I_1)$}. \label{Contour}
\end{center}
\end{figure}
In order to evaluate $I_1$ we employ a contour integral over the
contour given by 6 parametric curves (see Fig.~(\ref{Contour}):
$\Gamma_1: z=u, u \in (q,R), q,R > 0; \Gamma_2 : z = R + ib, b \in
(0, v); \Gamma_3: z = u + iv, u \in (R,-R); \Gamma_4 : z = -R + ib,
b \in (v, 0); \Gamma_5 : z = u, u \in (-R,-q); \Gamma_6 : z =
qe^{i\theta }, \theta \in (\pi, 0)$. As the integrand is analytic on
this contour we can apply the Cauchy theorem. Also note that the
integral along curve $\Gamma_6$ is a half of the integral along the
whole circle around zero which in turn is equal to $2\pi i^2
Res(e^{-izk}\phi_t(-z)/z)$. As the integrals along vertical lines
vanish at $R \rightarrow \infty$ and at $q \rightarrow 0$ the
integral along the real axis tends to an integral from $-\infty$ to
$\infty$, eventually changing variable $u \rightarrow - u$ we
obtain
\begin{equation} \label{FirstInt}
{\cal R}(I_1) = \pi + \int_{-\infty}^{\infty}e^{- iu \ln K }e^{iu [\ln S +(r-q+\omega)T]}\dfrac{\phi_{X_T}(u)}{iu}du.
\end{equation}
To compute the ${\cal R}(I_2)$ we use a similar contour build around
the point $z=i$, i.e. $\Gamma _1 : z = u + i, u \in (q,R), q,R
> 0; \Gamma _2 : z = R+ib, b \in (1, 1+v); \Gamma _3 : z = u+i(1+v), u \in (R,-R);
\Gamma _4 : z = -R+ib, b \in (v, 1); \Gamma _5 : z = u + i, u \in
(-R,-q); \Gamma _6 : z = i + qe^{i\theta}, \theta \in (0, \pi)$.
Again taking limits $R \rightarrow \infty$ and $q \rightarrow 0$,
changing variable $ u \rightarrow u-i$, we obtain
\begin{equation} \label{SecondInt}
{\cal R}(I_2) = \dfrac{S}{K}e^{(r-q)T}\left( \pi + \int_{-\infty}^{\infty}e^{-iu \ln K}e^{iu [\ln S +(r-q+\omega)T]}
\dfrac{\phi_{X_T}(u-i)}{iu \phi_{X_T}(-i)}du \right).
\end{equation}
Substituting these integrals into the Eq.~(\ref{BSFFT}) we obtain
the Eq.~(\ref{HEuropPrice}) $\blacksquare$.
The difficulty in using FFT to evaluate the
Eqs.~(\ref{HEuropPrice}), as noted by Carr and Madan is the
divergence of the integrands at $u=0$. Specifically, let us develop
the characteristic function $\phi_{X_t} (z)$ with $z = u +iv$ as
Taylor series in $u$
\begin{equation}\label{Taylor}
\phi_{X_t} (z) = {\mathbb E}[e^{-v X_t}] + iu{\mathbb E}[x e^{-v X_t}] - \frac{1}{2}
u^2 {\mathbb E}[x^2e^{-v X_t}] + ...
\end{equation}
In Eq.~(\ref{altFFT}) we have to chose $z=u-i$ in the first
expression, and $z=u$ in the second one. As it is easy to check in
both cases that the leading term in the expansion under both
integrals is $1/(iu)$ which is just a source of the divergence.The
source of this divergence is a discontinuity of the payoff function
at $K=S_T$. Accordingly the Fourier transform of the payoff function
has large high-frequency terms. The Carr-Madan solution is in fact
to dampen the weight of the high frequencies by multiplying the
payoff by an exponential decay function. This will lower the
importance of the singularity, but at the cost of degradation of the
solution accuracy.
As the Eqs.~(\ref{HEuropPrice}) can be used whenever the
characteristic function of the given model is known, we can apply it
to the Black-Scholes model as well that gives us the Black-Scholes
option price $V^{BS}$ which is a well known analytic expression. Now
the idea is to rewrite representation of the option price in
the Eqs.~(\ref{HEuropPrice}) in the form
\begin{equation}\label{newRepres}
V^M = [V^M - V^{BS}] + V^{BS}.
\end{equation}
The term in braces can now be computed with FFT as
\begin{eqnarray}\label{NewEuropPrice}
\Pi_1^{M-BS} &=& \dfrac{\xi}{2\pi}
\int_{-\infty}^{\infty}\dfrac{e^{-iu \kappa} \left[ \phi_{X_t} (u-i) e^{i(u-i)\omega T} -
\phi_{BS} (u-i)e^{- \frac{\sigma ^2}{2}T}\right] }{i u} du, \\
\Pi_2^{M-BS} &=& \dfrac{\xi}{2\pi} \int_{-\infty}^{\infty}
\dfrac{e^{- iu \kappa } \left[\phi_{X_t} (u)e^{iu \omega T} - \phi_{BS} (u)\right]}{i u}du, \nonumber} \def\leeq{\lefteqn \\
V^{M} - V^{BS} &=& \xi \left[e^{-q T}S_0 \Pi_1^{M -BS} - e^{-r T} K
\Pi_2^{M - BS}\right], \nonumber} \def\leeq{\lefteqn
\end{eqnarray}
\noindent where $\kappa = \ln (K/S) - (r-q)T$, $\phi_{BS} (u) = \exp\left(-\frac{\sigma ^2 T}{2}u^2 \right)$
and $\phi _{X_T}(-i) = e^{-\omega T}$. This is possible
because we have removed the divergence in the integrals. In addition
the magnitude of $\phi_{X_t} (z) - \phi_{BS} (z)$ is smaller than
that of $\phi_{X_t} (z)$ that increases accuracy of the solution.
In more detail, first terms of the expansion of $\phi_{X_t} (u)e^{iu \omega T} - \phi_{BS} (u)$ and
$\phi_{X_t} (u-i) e^{i(u-i)\omega T} - \phi_{BS} (u-i)e^{- \frac{\sigma ^2}{2}T}$ in series at small $u$ are
\begin{eqnarray} \label{expan1}
D_1|_{u=0} &\equiv& \phi_{X_t} (u)e^{iu \omega T} - \phi_{BS} (u) = T ( \theta + \omega + \dfrac{\sigma^2}{2} )i u + O(u^2) \\
D_2|_{u=0} &\equiv&\phi_{X_t} (u-i) e^{i(u-i)\omega T} - \phi_{BS} (u-i)e^{- \frac{\sigma ^2}{2}T} =
- \left( \sigma ^2 + \frac{\theta +\sigma ^2}{-1 + \nu(\theta + \sigma ^2/2) } - \omega \right)iu + O(u^2) \nonumber} \def\leeq{\lefteqn
\end{eqnarray}
However, an usage of these expressions in the Eq.~(\ref{NewEuropPrice}) together with the FFT method produces
an error of the order of $O(u)$. That is why it is better to choose a small $u=\epsilon $, for instance $\epsilon =10^{-6}$,
then computing integrands in the Eq.~(\ref{NewEuropPrice}) exactly and substituting
$D_{1,2}|_{u=0} \approx D_{1,2}|_{u=\epsilon }$.
\begin{figure}[bht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{CM_BS_expan.eps}
\caption{European option values in VG model. Difference between the Carr-Madan solution and Black-Scholes-wise solution
with $D_{1,2}(u=0)$ at $T=1.0 yr, \sigma = 0.1, \theta =0.1, \nu = 0.1$}
\label{BSFFT1}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{CM_BS_epsilon.eps}
\caption{European option values in VG model. Difference between the Carr-Madan solution and Black-Scholes-wise solution
with $D_{1,2}(u=\epsilon)$ at $T=1.0 yr, \sigma = 0.1, \theta =0.1, \nu = 0.1$}
\label{BSFFT2}
\end{minipage}
\end{flushleft}
\end{figure}
Fig.~\ref{BSFFT1}, \ref{BSFFT2} show the results of our computation of the European option values
under the VG model. Difference between the Carr-Madan solution and Black-Scholes-wise solution
with $D_{1,2}(u=\epsilon)$ and $D_{1,2}(u=0)$ at $T=1.0 yr, \sigma = 0.1, \theta =0.1, \nu = 0.1$ are plotted for 200 strikes.
It is seen that for the first method the difference is of the order of 0.5\%.
\section{Convergency and performance}
Artur Sepp reported in \cite{Sepp2003} that the convergency of the
Black-Scholes-wise method is approximately 3 times faster than that
of the Lewis method. It could be understood because as we mentioned
above in the limit of small $u$ the difference between the VG
solution and the Black-Scholes formula which is under the Fourier
integral is of the second order in $u$ while in the Lewis method it
is of the zero order. In other words using the Black-Scholes-wise
formula allows us to remove a part of the FFT error instead substituting it
with the exact analytical solution of the Black-Scholes problem.
We also fulfilled investigation of how all three methods converge for the VG model.
The results are given in Fig.~\ref{convBS},\ref{convLewis},\ref{convCM}. We display $\log_{10}$
difference between the option price obtained with $N=8192$, and that with $N=4096, 1024,512,256$.
We don't see much difference in the convergency of the Lewis and Black-Scholes-wise method while
the Carr-Madan methods behaves better at low $N$. In Fig.~\ref{conv3} we also present the ratio
$(C_{N=8192} - C_{N=4096})/C_{N=8192}$ for all three methods. The Carr-Madan still converges better for
out of the money spot prices while convergency of two other methods is similar.
\begin{figure}[bht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{convergencyBS.eps}
\caption{Convergency of the Black-Scholes-wise method. Difference between the
option price obtained with $N=8192$, and that with $N=4096, 1024,512,256$}.
\label{convBS}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{convergencyLewis.eps}
\caption{Convergency of the Lewis method. Difference between the
option price obtained with $N=8192$, and that with $N=4096, 1024,512,256$}.
\label{convLewis}
\end{minipage}
\end{flushleft}
\end{figure}
\begin{figure}[bht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{convergencyCM.eps}
\caption{Convergency of the Carr-Madan method. Difference between the
option price obtained with $N=8192$, and that with $N=4096, 1024,512,256$}.
\label{convCM}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\includegraphics[totalheight=2in]{convergency3.eps}
\caption{Convergency of all three methods}.
\label{conv3}
\end{minipage}
\end{flushleft}
\end{figure}
Cont and Tankov also analyze the Lewis method. They emphasize the
fact that the integral in the Eq.~(\ref{altFFT}) is much easier to
approximate at infinity than that in the Carr-Madan method, because
the integrand decays exponentially (due to the presence of
characteristic function). However, the price to pay for this is
having to choose $\mu _1$. This choice is a delicate issue because
choosing big $\mu _1$ leads to slower decay rates at infinity and
bigger truncation errors and when $\mu _1$ is close to one, the
denominator diverges and the discretization error becomes large. For
models with exponentially decaying tails of Levy measure, $\mu _1$
cannot be chosen a priori and must be adjusted depending on the
model parameters.
Carr and Madan in \cite{CarrMadan:99a} compare performance of 3 methods for computing VG prices:
VGP which is the analytic formula in Madan, Carr, and Chang;
VGPS which computes delta and the risk-neutral probability of finishing
in-the-money by Fourier inversion of the distribution function, i.e. according to the
Eq.~(\ref{HEuropPrice}); VGFFTC which is a Carr-Madan method using FFT to invert the dampened call price;
VGFFTTV which uses FFT to invert the modified time value. The results are given in Tab.~(\ref{comparison}). The computation times for the first
two methods involve 160 strike levels. The first 4 rows of Tab.~(\ref{comparison}) display
4 combinations of parameter settings, while the last 4 rows show computation times in seconds.
\begin{table}[ht]
\begin{flushleft}
\begin{minipage}[ht]{0.4\textwidth}
\begin{tabular}{|l|r|r|r|r|}
\hline
& case 1 & case 2 & case 3 & case 4 \\
\hline
$\sigma$ & .12 & .25 & .12 & .25 \\
$\nu$ & .16 & 2.0 & .16 & 2.0 \\
$\theta$ & -.33 & -.10 & -.33 & -.10 \\
$T$ & 1 & 1 & .25 & .25 \\
\hline
VGP & 22.41 & 24.81 & 23.82 & 24.74 \\
VGPS & 288.50 & 191.06 & 181.62 & 197.97 \\
VGFFTC & 6.09 & 6.48 & 6.72 & 6.52 \\
VGFFTTV & 11.53 & 11.48 & 11.57 & 11.56 \\
\hline
\end{tabular}
\caption{CPU times for VG pricing. Represented from \cite{CarrMadan:99a}. }
\label{comparison}
\end{minipage}
\hspace{0.1\textwidth}
\begin{minipage}[ht]{0.4\textwidth}
\begin{tabular}{|l|r|r|r|r|}
\hline
& case 1 & case 2 & case 3 & case 4 \\
\hline
$\sigma$ & .12 & .25 & .12 & .25 \\
$\nu$ & .16 & 2.0 & .16 & 2.0 \\
$\theta$ & -.33 & -.10 & -.33 & -.10 \\
$T$ & 1 & 1 & .25 & .25 \\
\hline
Lewis & 0.031 & 0.031 & 0.031 & 0.031 \\
Carr-Madan & 0.047 & 0.047 & 0.032 & 0.032 \\
BS-wise & 0.078 & 0.078 & 0.062 & 0.062 \\
\hline
\end{tabular}
\caption{CPU times for VG pricing. Our calculations.}
\label{OurCalc}
\end{minipage}
\end{flushleft}
\end{table}
It is seen that the analytic formula is slow while the slowest (and least
accurate in case 4) method inverts for the delta and for the probability of
paying off.
However, this is not true if one uses a modified method given in the Eq.~(\ref{NewEuropPrice}).
Our calculations show that the performance of the Lewis method is same as the Carr-Madan method, and
the performance of the Black-Scholes-wise method is only twice worse (because we need 2 FFT to compute 2 integrals)
(see Tab.~\ref{OurCalc}).
\section{Conclusion}
We discussed various analytic and numerical methods that have been
used to get option prices within a framework of VG model. We showed
that a popular Carr-Madan's FFT method \cite{CarrMadan:99a} blows up
for certain values of the model parameters even for European vanilla option. Alternative methods -
one originally proposed by Lewis, and Black-Scholes-wise method were
considered that seem to work fine for any value of the VG
parameters. Convergency and accuracy of these methods is comparable with that of the Carr-Madan
method, thus making them suitable for being used to price options with the VG model.
\newpage
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
\newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
|
{
"timestamp": "2010-01-15T20:29:12",
"yymm": "0503",
"arxiv_id": "physics/0503137",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503137"
}
|
\section{Introduction}
A classical problem in geometry is to determine whether a Riemannian
manifold ${\mathcal V}$ can be isometrically immersed in another Riemaniann
manifold $\bar{\mathcal V}$. We will restrict ourselves to the case of codimension $1$
immersions, i.e., ${\mathcal V}$ has dimension $n$ and $\bar{\mathcal V}$ has
dimension $n+1$.
It is well known that the Gauss and Codazzi equations
are necessary
conditions relating the Riemann curvature tensor $\bar\mathrm{R}$ of $\bar{\mathcal V}$,
the Riemann curvature tensor $\mathrm{R}$ of ${\mathcal V}$ and the shape operator
$\mathrm{S}$ of ${\mathcal V}$. Denoting by $\nabla$ the Riemannian connection of
${\mathcal V}$, these equations are the following:
\begin{equation*}
\langle\mathrm{R}(X,Y)Z,W\rangle-\langle\bar\mathrm{R}(X,Y)Z,W\rangle
=\langle\mathrm{S} X,Z\rangle\langle\mathrm{S} Y,W\rangle
-\langle\mathrm{S} Y,Z\rangle\langle\mathrm{S} X,W\rangle
\end{equation*}
\begin{equation*}
\nabla_X\mathrm{S} Y-\nabla_Y\mathrm{S} X-\mathrm{S}[X,Y]=\bar\mathrm{R}(X,Y)N,
\end{equation*}
for all vector fields $X$, $Y$, $Z$ and $W$ on ${\mathcal V}$.
Moreover, in the case where $\bar{\mathcal V}$ is a space-form,
i.e., the sphere $\mathbb{S}^{n+1}$, the Euclidean space $\mathbb{R}^{n+1}$ or the
hyperbolic space $\mathbb{H}^{n+1}$, the Gauss and Codazzi equations are also a
sufficient condition for ${\mathcal V}$ to be locally isometrically immersed
in $\bar{\mathcal V}$ with $\mathrm{S}$ as shape operator. In this case the Gauss and
Codazzi equations involve only the metric and the shape operator of ${\mathcal V}$.
The author studied this problem when $\bar{\mathcal V}$ is a product manifold
$\mathbb{S}^n\times\mathbb{R}$ or $\mathbb{H}^n\times\mathbb{R}$ (\cite{codazzi}). Then the Gauss and
Codazzi equations involve the metric of ${\mathcal V}$, its shape operator $\mathrm{S}$,
the projection $T$ of the vertical vector field (i.e., the unit vector
field corresponding to the factor $\mathbb{R}$) on the tangent space of
${\mathcal V}$ and the normal component $\nu$ of the vertical vector field (i.e.,
its inner product with the unit normal of ${\mathcal V}$). The author proved that
the Gauss and Codazzi equations, together with two other compatibility
equations coming from the fact that the vertical vector field is parallel,
are a necessary and sufficient condition for ${\mathcal V}$ to be locally
isometrically immersed in $\bar{\mathcal V}$ with $\mathrm{S}$ as shape operator, $T$ as
tangent projection of the vertical vector field and $\nu$ as normal
component of the vertical vector field.
It is natural to try to generalize this result to other homogeneous manifolds.
We will investigate the case of surfaces in manifolds of dimension $3$, i.e.,
$n=2$. Indeed, the classification of simply connected
$3$-dimensional homogeneous manifolds is well known. Such a
manifold has an isometry group of dimension $3$, $4$ or $6$. When the dimension
of the isometry group is $6$, then we have a space form. When the dimension
of the isometry group is $3$, the manifold has the
geometry of the Lie group $\mathrm{Sol}_3$.
In this paper we will consider the homogeneous
manifolds whose isometry groups have dimension
$4$: such a manifold is a Riemannian fibration over a $2$-dimensional space
form, the fibers are geodesics and there exists a one-parameter family of
translations along the fibers, generated by a unit Killing field $\xi$
which will be called the vertical vector field.
These manifolds are classified, up to isometry, by the curvature $\kappa$ of
the base surface of the fibration
and the bundle curvature $\tau$, where $\kappa$ and $\tau$ can be any real
numbers satisfying $\kappa\neq 4\tau^2$. The bundle curvature is the number
$\tau$ such $\bar\nabla_X\xi=\tau X\times\xi$ for any vector field $X$
on $\bar{\mathcal V}$, where $\bar\nabla$ denotes the Riemannian connection of
$\bar{\mathcal V}$.
When the bundle curvature $\tau$ vanishes (and then $\kappa\neq 0$),
we get a product manifold
$\mathbb{M}^2(\kappa)\times\mathbb{R}$ where $\mathbb{M}^2(\kappa)$ is the simply connected
$2$-manifold of constant
curvature $\kappa$. Their isometry group has $4$ connected components.
The vertical vector $\xi$ is simply the vector
corresponding to the factor $\mathbb{R}$. This case was treated in
\cite{codazzi}.
When $\tau\neq 0$, the isometry group has $2$ connected components: an
isometry either preserves the orientations of both the fibers and the
base of the fibration, or reverses both orientations. These
manifolds are of three types: they have the isometry
group of the Berger spheres for $\kappa>0$, of the Heisenberg space
$\mathrm{Nil}_3$ for $\kappa=0$, and of $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$ for
$\kappa<0$. In this paper we will deal with these three types of manifold.
Like for $\mathbb{M}^2(\kappa)\times\mathbb{R}$, the Gauss and Codazzi equations
involve the metric of
${\mathcal V}$, its shape operator $\mathrm{S}$, the tangential projection $T$ of
$\xi$ and the normal component $\nu$ of $\xi$. Denoting by $K$ the
curvature of $\mathrm{d} s^2$, these equations become
$$K=\det\mathrm{S}+\tau^2+(\kappa-4\tau^2)\nu^2,$$
$$\nabla_X\mathrm{S} Y-\nabla_Y\mathrm{S} X-\mathrm{S}[X,Y]=
(\kappa-4\tau^2)\nu(\langle Y,T\rangle X-\langle X,T\rangle Y)$$
The first theorem is the following one.
\begin{thmintro}[theorem \ref{isometry}]
Let ${\mathcal V}$ be a simply connected oriented
Riemannian manifold of dimension $2$,
$\mathrm{d} s^2$ its metric (which we also denote by
$\langle\cdot,\cdot\rangle$), $\nabla$ its
Riemannian connection and $\mathrm{J}$ the rotation of angle $\frac\pi2$
on $\mathrm{T}{\mathcal V}$. Let $\mathrm{S}$ be a field of symmetric operators
$\mathrm{S}_y:\mathrm{T}_y{\mathcal V}\to\mathrm{T}_y{\mathcal V}$, $T$ a vector field on ${\mathcal V}$
and $\nu$ a smooth function on ${\mathcal V}$ such that
$||T||^2+\nu^2=1$.
Let $\mathbb{E}$ be a $3$-dimensional homogeneous manifold with a $4$-dimensional
isometry group and $\xi$ its vertical vector field.
Let $\kappa$ be its base curvature and $\tau$ its bundle curvature.
Then there exists an isometric immersion $f:{\mathcal V}\to\mathbb{E}$ such that
the shape operator with respect to the normal $N$ associated to $f$ is
$$\mathrm{d} f\circ\mathrm{S}\circ\mathrm{d} f^{-1}$$ and such that
$$\xi=\mathrm{d} f(T)+\nu N$$
if and only if $(\mathrm{d} s^2,\mathrm{S},T,\nu)$ satisfies the
Gauss and Codazzi equations for $\mathbb{E}$ and, for all vector fields
$X$ on ${\mathcal V}$, the following equations:
$$\nabla_XT=\nu(\mathrm{S} X-\tau\mathrm{J} X),\quad
\mathrm{d}\nu(X)+\langle\mathrm{S} X-\tau\mathrm{J} X,T\rangle=0.$$
In this case, the
immersion is unique up to a global isometry of $\mathbb{E}$ preserving
the orientations of both the fibers and the base of the fibration.
\end{thmintro}
The two additional conditions come from the fact that
$\bar\nabla_X\xi=\tau X\times\xi$ for all vector fields $X$.
We notice that this theorem seems specific to dimension $2$, since the
operator of rotation $\mathrm{J}$ is involved.
The method to prove this theorem is similar to that of \cite{codazzi}
and was inspired by that of Tenenblat
(\cite{tenenblat}): it is based on differential forms, moving frames
and integrable distributions.
However, things are technically much more complicated
here: in \cite{codazzi} the proof was simplified by the fact that
$\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ can be included in $\mathbb{R}^{n+2}$ and
in the Lorentz space $\mathbb{L}^{n+2}$ respectively. We will first present
the models used for the $3$-dimensional homogeneous manifolds, and then we will prove
the theorem.
Finally we will give two applications of the main theorem
to constant mean curvature (CMC)
surfaces in $3$-dimensional homogeneous manifolds
with $4$-dimensional isometry group.
The first application is the existence of an isometric correspondence
between certain CMC surfaces in homogeneous $3$-manifolds with the same
anisotropy coefficient $\kappa-4\tau^2$. This correspondence
generalizes the classical Lawson correspondence between certain
CMC surfaces in space-forms. This is the following theorem.
\begin{thmintro}[see theorem \ref{sisters}]
Let $\mathbb{E}_1$ and $\mathbb{E}_2$ be two $3$-dimensional homogeneous manifolds
with $4$-dimensional isometry groups, of base curvatures $\kappa_1$
and $\kappa_2$ and bundle curvatures $\tau_1$ and $\tau_2$
respectively, and such that
$$\kappa_1-4\tau_1^2=\kappa_2-4\tau_2^2.$$
Let $H_1$ and $H_2$ be two real numbers such that
$$\tau_1^2+H_1^2=\tau_2^2+H_2^2.$$
Then there exists an isometric
correspondence between simply connected
CMC $H_1$ surfaces in $\mathbb{E}_1$ and
simply connected
CMC $H_2$ surfaces in $\mathbb{E}_2$.
This correspondence is called the correspondence of the sister
surfaces.
\end{thmintro}
The second application is the existence of ``twin immersions'' of
non-minimal CMC immersions in homogeneous $3$-manifolds with non-vanishing
bundle curvature. This twin immersion might be useful to prove an
Alexan-drov-type theorem in these manifolds.
\begin{notation}
In this paper we will use the following index conventions: Latin
letters $i$, $j$, etc, denote integers between $1$ and $n$ (or
the integers $1$ and $2$), Greek
letters $\alpha$, $\beta$, etc, denote integers between $1$ and
$n+1$ (or between $1$ and $3$).
The set of vector fields on a Riemannian manifold ${\mathcal V}$ will be
denoted by $\mathfrak{X}({\mathcal V})$.
The Riemann curvature tensor $\mathrm{R}$ of a Riemannian manifold ${\mathcal V}$
of Riemannian connection $\nabla$ is
defined using the following convention:
$$\mathrm{R}(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z.$$
The shape operator of a hypersurface ${\mathcal V}$ of a Riemannian manifold
$\bar{\mathcal V}$ associated to its unit normal $N$ is
$$\mathrm{S} X=-\bar\nabla_XN$$ where $\bar\nabla$ is the Riemannian
connection of $\bar{\mathcal V}$.
\end{notation}
\section{$3$-dimensional homogeneous manifolds with $4$-dimensional isometry group}
In this section we will give the general setting for
simply connected homogeneous
$3$-manifolds with $4$-dimensional isometry group and we will describe the
models used. We will consider only those having non-vanishing bundle curvature
(since the product manifolds $\mathbb{M}^2(\kappa)\times\mathbb{R}$ were treated in
\cite{codazzi}).
The reader can refer to \cite{scott} for the geometry of
$3$-dimensional homogeneous manifolds.
\subsection{Canonical frame} \label{canonicalframe}
Let $\mathbb{E}$ be a simply connected
$3$-dimensional homogeneous manifold with a $4$-dimensional isometry group.
Such a manifold is a Riemannian fibration over a
simply connected $2$-manifold of constant
curvature $\kappa$. The fibers are geodesics. We will denote by
$\xi$ a unit vector field on $\mathbb{E}$ tangent to the fibers; it will be called
the vertical vector field. It is a Killing field (corresponding to
translations along the fibers).
We will denote by $\bar\nabla$ and $\bar\mathrm{R}$ the Riemannian connection
and the Riemannian curvature tensor of $\mathbb{E}$ respectively.
We assume that $\mathbb{E}$ is not a product manifold $\mathbb{M}^2(\kappa)\times\mathbb{R}$.
The manifold $\mathbb{E}$ locally has a direct orthonormal
frame $(E_1,E_2,E_3)$ with $$E_3=\xi$$
whose non-vanishing Christoffel symbols
$\bar\Gamma^\alpha_{\beta\gamma}
=\langle\nabla_{E_\alpha}E_\beta,E_\gamma\rangle$ are the following:
$$\bar\Gamma^3_{12}=\bar\Gamma^1_{23}=-\bar\Gamma^3_{21}
=-\bar\Gamma^2_{13}=\tau,$$
$$\bar\Gamma^1_{32}=-\bar\Gamma^2_{31}=\tau-\sigma,$$
for some real numbers $\sigma$ and $\tau\neq 0$
(this will be explicited in the sequel).
Then we have
$$[E_1,E_2]=2\tau E_3,\quad
[E_2,E_3]=\sigma E_1,\quad
[E_3,E_1]=\sigma E_2.$$
We will call $(E_1,E_2,E_3)$ the canonical frame of $\mathbb{E}$.
For all vector field $X$ we have $$\bar\nabla_XE_3=\tau X\times E_3$$
where $\times$ denotes the vector product in $\mathbb{E}$, i.e., for all
vector fields $X$, $Y$, $Z$,
$\langle X\times Y,Z\rangle=\det_{(E_1,E_2,E_3)}(X,Y,Z)$.
Setting $$\langle\bar\mathrm{R}(X\wedge Y),Z\wedge W\rangle
=\langle\bar\mathrm{R}(X,Y)Z,W\rangle,$$
the matrix of $\bar\mathrm{R}$ in the basis
$(E_2\wedge E_3,E_3\wedge E_1,E_1\wedge E_2)$ is
$$\bar\mathrm{R}=\diag(a,a,b)$$ with $$a=\tau^2,\quad
b=-3\tau^2+2\sigma\tau.$$
We now compute the curvature $\kappa$
of the base of the fibration. If $\bar M\to M$ is a Riemannian submersion,
then the sectional curvature of a $2$-plane $\Pi$ in $M$ generated by
an orthonormal pair $(X,Y)$ is
$$K(\Pi)=\bar K(\bar\Pi)+
\frac34\left|\left|[\bar X,\bar Y]^{\mathrm v}\right|\right|^2$$
where $\bar X$ and $\bar Y$ are horizontal lifts of $X$ and $Y$ in $\bar M$,
$\bar K(\bar\Pi)$ is the sectional curvature of a $2$-plane $\bar\Pi$
in $\bar M$ generated by $(\bar X,\bar Y)$, and where $Z^{\mathrm v}$
denotes the vertical part of a vector field $Z$ in $\bar M$
(see \cite{docarmo}, chapter 8).
In our case we get
$$\kappa=\langle\bar\mathrm{R}(E_1,E_2)E_1,E_2\rangle
+\frac34\left|\left|[E_1,E_2]^{\mathrm v}\right|\right|^2
=b+\frac34\left|\left|2\tau E_3^{\mathrm v}\right|\right|^2
=b+3\tau^2.$$
Thus we have $b=\kappa-3\tau^2$, and so
$$\sigma=\frac\kappa{2\tau}.$$
\begin{prop} \label{exprbarR}
For all vector fields $X,Y,Z,W$ on $\mathbb{E}$ we have
$$\langle\bar\mathrm{R}(X,Y)Z,W\rangle=
(\kappa-3\tau^2)\langle \mathrm{R}_0(X,Y)Z,W\rangle
+(\kappa-4\tau^2)\langle\mathrm{R}_1(\xi;X,Y)Z,W\rangle$$
with $$\mathrm{R}_0(X,Y)Z=\langle X,Z\rangle Y-\langle Y,Z\rangle X,$$
\begin{eqnarray*}
\mathrm{R}_1(V;X,Y)Z & = &
\langle Y,V\rangle\langle Z,V\rangle X
+\langle Y,Z\rangle\langle X,V\rangle V \\
& & -\langle X,Z\rangle\langle Y,V\rangle V
-\langle X,V\rangle\langle Z,V\rangle Y.
\end{eqnarray*}
\end{prop}
\begin{proof}
We set $X=\tilde X+x\xi$ with $\tilde X$ horizontal and $x=\langle
X,\xi\rangle$, etc. Using the multilinearity
of the Riemann curvature tensor, we get a sum of 16 terms; the terms
where $\xi$ appears three or four times, or twice at positions $1,2$
or $3,4$, vanish by antisymmetry. The terms where $\xi$ appears once
vanish because the matrix of $\bar\mathrm{R}$ in the basis
$(E_2\wedge E_3,E_3\wedge E_1,E_1\wedge E_2)$ is diagonal. Hence we have
\begin{eqnarray*}
\langle\bar\mathrm{R}(X,Y)Z,W\rangle & = &
\langle\bar\mathrm{R}(\tilde X,\tilde Y)\tilde Z,\tilde W\rangle \\
& & +yw\langle\bar\mathrm{R}(\tilde X,\xi)\tilde Z,\xi\rangle
+yz\langle\bar\mathrm{R}(\tilde X,\xi)\xi,\tilde W\rangle \\
& & +xw\langle\bar\mathrm{R}(\xi,\tilde Y)\tilde Z,\xi\rangle
+xz\langle\bar\mathrm{R}(\xi,\tilde Y)\xi,\tilde W\rangle \\
& = & (\kappa-3\tau^2)(
\langle\tilde X,\tilde Z\rangle\langle\tilde Y,\tilde W\rangle
-\langle\tilde X,\tilde W\rangle\langle\tilde Y,\tilde Z\rangle) \\
& & +\tau^2(yw\langle\tilde X,\tilde Z\rangle
-yz\langle\tilde X,\tilde W\rangle
-xw\langle\tilde Y,\tilde Z\rangle
+xz\langle\tilde Y,\tilde W\rangle) \\
& = & (\kappa-3\tau^2)(
\langle X,Z\rangle\langle Y,W\rangle
-\langle X,W\rangle\langle Y,Z\rangle) \\
& & -(\kappa-4\tau^2)
(\langle X,Z\rangle\langle Y,\xi\rangle\langle W,\xi\rangle
+\langle Y,W\rangle\langle X,\xi\rangle\langle Z,\xi\rangle \\
& & \quad-\langle X,W\rangle\langle Y,\xi\rangle\langle Z,\xi\rangle
-\langle Y,Z\rangle\langle X,\xi\rangle\langle W,\xi\rangle).
\end{eqnarray*}
\end{proof}
\subsection{The manifolds with the isometry group of the Berger spheres}
\label{bergerspheres}
They occur when $\tau\neq 0$ and $\kappa>0$; they are
fibrations over round $2$-spheres.
They are obtained by deforming the metric of a round sphere in a way
preserving the Hopf fibration but modifying the length of the fibers.
Their isometry group is included in that of the round sphere. The reader
can refer to \cite{petersen}.
The sphere $\mathbb{S}^3$ is the univeral covering of $\mathrm{SO}_3(\mathbb{R})$, which
can be identified with the unitary
tangent bundle to the $2$-sphere $\mathrm{U}\mathbb{S}^2$.
Indeed, the group $\mathrm{SO}_3(\mathbb{R})$ acts
transitively on $\mathrm{U}\mathbb{S}^2$, and the stabilizer of any point in $\mathrm{U}\mathbb{S}^2$
is trivial. The unitary tangent bundle $\mathrm{U}\mathbb{S}^2$ can be endowed
with the metric induced by
the standard metric on the tangent bundle $\mathrm{T}\mathbb{S}^2$. We will give an
expression of this metric.
Let $(x,y)\mapsto\varphi(x,y)$ be a conformal parametrization
of a domain $D$ in $\mathbb{S}^2$
and let $\lambda$ be the conformal factor, i.e., the metric of $D$ is
given by $\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)$. Then a parametrization of
$\mathrm{U} D$ is the following:
$$(x,y,\theta)\mapsto
\left(\varphi(x,y),\frac1\lambda(\cos\theta\partial_x
+\sin\theta\partial_y)\right).$$
Let $p=\varphi(x,y)\in D$, $v\in\mathrm{T}_pD$ and
$V\in\mathrm{T}_{(p,v)}(\mathrm{U} D)$. Let $\alpha(t)=(p(t),v(t))$ be
a curve such that $v(t)\in\mathrm{T}_{p(t)}\mathbb{H}^2$, $p(0)=p$, $v(0)=v$ and
$\alpha'(0)=V$. Then the norm of $V$ is given by
$$||V||_{(p,v)}^2=||\mathrm{d}\pi(V)||_p^2+
\left|\left|\frac{\mathrm{D}v}{\mathrm{d} t}(0)\right|\right|_p^2$$
where $\pi:\mathrm{U} D\to D$ is the canonical projection.
We set $\alpha(t)=(x(t),y(t),\theta(t))$. Then we have
$$v(t)=\frac1\lambda(\cos\theta(t)\partial_x+\sin\theta(t)\partial_y),$$
and thus
\begin{eqnarray*}
\frac{\mathrm{D}v}{\mathrm{d} t} & = &
-\frac{\dot\lambda}{\lambda^2}
(\cos\theta\partial_x+\sin\theta\partial_y)
+\frac{\dot\theta}\lambda(-\sin\theta\partial_x
+\cos\theta\partial_y) \\
& & +\frac1\lambda(\cos\theta(\dot x\nabla_{\partial_x}\partial_x
+\dot y\nabla_{\partial_y}\partial_x)
+\sin\theta(\dot x\nabla_{\partial_x}\partial_y
+\dot y\nabla_{\partial_y}\partial_y)),
\end{eqnarray*}
where the dot denotes the derivation with respect to $t$.
Since $\dot\lambda=\dot x\lambda_x+\dot y\lambda_y$,
$\nabla_{\partial_x}\partial_x=\frac{\lambda_x}\lambda\partial_x
-\frac{\lambda_y}\lambda\partial_y$,
$\nabla_{\partial_y}\partial_y=-\frac{\lambda_x}\lambda\partial_x
+\frac{\lambda_y}\lambda\partial_y$
and $\nabla_{\partial_x}\partial_y=
\nabla_{\partial_y}\partial_x=\frac{\lambda_y}\lambda\partial_x
+\frac{\lambda_x}\lambda\partial_y$, we get
$$\frac{\mathrm{D}v}{\mathrm{d} t}=\frac1{\lambda^2}
(\lambda\dot\theta+\dot y\lambda_x-\dot x\lambda_y)
(\cos\theta\partial_y-\sin\theta\partial_x).$$
Thus $$||V||^2_{(p,v)}=\lambda^2(\dot x^2+\dot y^2)
+\frac1{\lambda^2}(\lambda\dot\theta+\dot y\lambda_x-\dot x\lambda_y)^2.$$
Setting $z=\theta$ on the universal covering, we get the following
expression for the metric of $\widetilde{\mathrm{U} D}$:
$$\mathrm{d} s^2=\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)
+\left(-\frac{\lambda_y}\lambda\mathrm{d} x
+\frac{\lambda_x}\lambda\mathrm{d} y+\mathrm{d} z\right)^2.$$
We now choose $D=\mathbb{S}^2\setminus\{\infty\}$ with the metric of constant
curvature $4$ (i.e., the metric of the round sphere of radius $\frac12$)
given by the stereographic projection, i.e.,
$$\lambda=\frac 1{1+x^2+y^2}.$$ Then we get
$$\mathrm{d} s^2=\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)
+(2\lambda(y\mathrm{d} x-x\mathrm{d} y)+\mathrm{d} z)^2.$$
More generally, $\mathbb{R}^3$ endowed with the metric
$$\mathrm{d} s^2=
\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)
+\left(\tau\lambda(y\mathrm{d} x-x\mathrm{d} y)+\mathrm{d} z\right)^2$$ with
$$\lambda=\frac1{1+\frac\kappa4(x^2+y^2)}$$
is the universal cover of a homogeneous manifold $\mathbb{E}$ of bundle curvature $\tau$
and of base curvature $\kappa>0$ minus the fiber
corresponding to the point $\infty\in\mathbb{S}^2$.
The fibers are given by $\{x=x_0,y=y_0\}$ in these coordinates.
The canonical frame is $(E_1,E_2,E_3)$ with
\begin{equation} \label{canonicalbergerspheres}
\begin{array}{c}
E_1=\lambda^{-1}(\cos(\sigma z)\partial_x+\sin(\sigma z)\partial_y)
+\tau(x\sin(\sigma z)-y\cos(\sigma z))\partial_z, \\
E_2=\lambda^{-1}(-\sin(\sigma z)\partial_x+\cos(\sigma z)\partial_y)
+\tau(x\cos(\sigma z)+y\sin(\sigma z))\partial_z, \\
E_3=\partial_z
\end{array}
\end{equation}
with $$\sigma=\frac\kappa{2\tau},$$
which satisfy
$$[E_1,E_2]=2\tau E_3,\quad [E_2,E_3]=\frac\kappa{2\tau}E_1,
\quad [E_3,E_1]=\frac\kappa{2\tau}E_2.$$
This frame is defined on the open set $\mathbb{E}'$ which is $\mathbb{E}$ minus the fiber
corresponding to the point $\infty\in\mathbb{S}^2$.
The Berger spheres in the strict sense are the manifolds such that $\kappa=4$.
\subsection{The manifolds with the isometry group of the
Heisenberg space $\mathrm{Nil}_3$}
\label{heisenberg}
They occur when $\tau\neq 0$ and $\kappa=0$; they are fibrations
over the Euclidean plane.
The Heisenberg space is the Lie group
$$\mathrm{Nil}_3=\left\{\left(\begin{array}{ccc}
1 & a & c \\
0 & 1 & b \\
0 & 0 & 1
\end{array}\right);(a,b,c)\in\mathbb{R}^3\right\}$$
endowed with a left invariant metric.
It is useful to use exponential coordinates. In this model, the Heisenberg
space $\mathrm{Nil}_3$ is $\mathbb{R}^3$ endowed with the following metric:
$$\mathrm{d} s^2=\mathrm{d} x^2+\mathrm{d} y^2+
(\tau(y\mathrm{d} x-x\mathrm{d} y)+\mathrm{d} z)^2.$$
The fibers are given by $\{x=x_0,y=y_0\}$ in these coordinates.
The canonical frame is $(E_1,E_2,E_3)$ with
\begin{equation} \label{canonicalheisenberg}
E_1=\partial_x-\tau y\partial_z,\quad
E_2=\partial_y+\tau x\partial_z,\quad
E_3=\partial_z,
\end{equation}
which satisfy
$$[E_1,E_2]=2\tau E_3,\quad [E_2,E_3]=0,
\quad [E_3,E_1]=0.$$
The reader can refer to \cite{mercuri} (where $\tau=\frac12$).
\subsection{The manifolds with the isometry group of $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$}
\label{psl}
They occur when $\tau\neq 0$ and $\kappa<0$; they are fibrations over
hyperbolic planes.
The Lie group $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$ with its standard metric
can be identified with the universal covering of the unitary
tangent bundle to the hyperbolic plane $\mathrm{U}\mathbb{H}^2$ equipped with its
canonical metric. Indeed, the group $\mathrm{PSL}_2(\mathbb{R})$ acts
transitively on $\mathrm{U}\mathbb{H}^2$, and the stabilizer of any point
in $\mathrm{U}\mathbb{H}^2$ is trivial. The unitary tangent bundle $\mathrm{U}\mathbb{H}^2$
can be endowed with the metric induced by
the standard metric on the tangent bundle $\mathrm{T}\mathbb{H}^2$.
The reader can refer to \cite{scott}. We will give an
expression of this metric.
Let $(x,y)\mapsto\varphi(x,y)$ be a conformal parametrization of $\mathbb{H}^2$
and let $\lambda$ be the conformal factor, i.e., the metric of $\mathbb{H}^2$ is
given by $\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)$. Then, proceeding as in section
\ref{bergerspheres}, we obtain that a metric on $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$ is
$$\mathrm{d} s^2=\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)
+\left(-\frac{\lambda_y}\lambda\mathrm{d} x
+\frac{\lambda_x}\lambda\mathrm{d} y+\mathrm{d} z\right)^2.$$
This metric defines a homogeneous manifold with $\kappa=-1$ and
$\tau=-\frac12$.
More generally, we can take the Poincar\'e disk model for the hyperbolic
plane of constant curvature $\kappa<0$. The manifold
$\mathbb{D}^2\left(\frac2{\sqrt{-\kappa}}\right)\times\mathbb{R}$, where
$\mathbb{D}^2(\rho)=\{(x,y)\in\mathbb{R}^2;x^2+y^2<\rho^2\}$,
endowed with the metric
$$\mathrm{d} s^2=
\lambda^2(\mathrm{d} x^2+\mathrm{d} y^2)
+\left(\tau\lambda(y\mathrm{d} x-x\mathrm{d} y)+\mathrm{d} z\right)^2$$ with
$$\lambda=\frac1{1+\frac\kappa4(x^2+y^2)}$$
is a homogeneous manifold of bundle curvature $\tau$ and of base curvature
$\kappa<0$.
The fibers are given by $\{x=x_0,y=y_0\}$ in these coordinates.
The canonical frame
is $(E_1,E_2,E_3)$ with
\begin{equation} \label{canonicalpsl}
\begin{array}{c}
E_1=\lambda^{-1}(\cos(\sigma z)\partial_x+\sin(\sigma z)\partial_y)
+\tau(x\sin(\sigma z)-y\cos(\sigma z))\partial_z, \\
E_2=\lambda^{-1}(-\sin(\sigma z)\partial_x+\cos(\sigma z)\partial_y)
+\tau(x\cos(\sigma z)+y\sin(\sigma z))\partial_z, \\
E_3=\partial_z
\end{array}
\end{equation}
with $$\sigma=\frac\kappa{2\tau},$$
which satisfy
$$[E_1,E_2]=2\tau E_3,\quad [E_2,E_3]=\frac\kappa{2\tau}E_1,
\quad [E_3,E_1]=\frac\kappa{2\tau}E_2.$$
\section{Preliminaries}
\subsection{The compatibility equations for surfaces in
$3$-dimensional homogeneous manifolds} \label{compatibilityE}
We consider a $3$-dimensional homogeneous manifold $\mathbb{E}$ with an
isometry group of dimension $4$, of bundle curvature $\tau$ and of base
curvature $\kappa$. Let $\bar\mathrm{R}$ be the Riemann curvature tensor of $\mathbb{E}$.
Let ${\mathcal V}$ be an oriented surface in $\mathbb{E}$,
$\nabla$ the Riemannian connection of ${\mathcal V}$,
$\mathrm{J}$ the rotation of angle $\frac\pi2$ on $\mathrm{T}{\mathcal V}$,
$N$ the unit normal to ${\mathcal V}$
and $\mathrm{S}$ the shape operator of ${\mathcal V}$.
\begin{prop}
For $X,Y,Z,W\in\mathfrak{X}({\mathcal V})$ we have
$$\langle\bar\mathrm{R}(X,Y)Z,W\rangle=
(\kappa-3\tau^2)\langle \mathrm{R}_0(X,Y)Z,W\rangle
+(\kappa-4\tau^2)\langle\mathrm{R}_1(T;X,Y)Z,W\rangle,$$
$$\bar\mathrm{R}(X,Y)N=(\kappa-4\tau^2)\nu
(\langle Y,T\rangle X-\langle X,T\rangle Y),$$
where $$\nu=\langle N,\xi\rangle,$$
$T$ is the projection of $\xi$ on
$\mathrm{T}{\mathcal V}$, i.e., $$T=\xi-\nu N,$$
and $\mathrm{R}_0$ and $\mathrm{R}_1$ are as in proposition \ref{exprbarR}.
\end{prop}
\begin{proof}
This is a consequence of proposition \ref{exprbarR}, using the fact
that $X$, $Y$ and $Z$ are tangent to the
surface and $N$ is normal to the surface.
\end{proof}
\begin{cor}
The Gauss and Codazzi equations in $\mathbb{E}$ are
$$K=\det\mathrm{S}+\tau^2+(\kappa-4\tau^2)\nu^2,$$
$$\nabla_X\mathrm{S} Y-\nabla_Y\mathrm{S} X-\mathrm{S}[X,Y]=
(\kappa-4\tau^2)\nu(\langle Y,T\rangle X-\langle X,T\rangle Y),$$
where $K$ is the Gauss curvature of ${\mathcal V}$.
\end{cor}
\begin{prop}
For $X\in\mathfrak{X}({\mathcal V})$ we have
$$\nabla_XT=\nu(\mathrm{S} X-\tau\mathrm{J} X),\quad
\mathrm{d}\nu(X)+\langle\mathrm{S} X-\tau\mathrm{J} X,T\rangle=0.$$
\end{prop}
\begin{proof}
On the one hand we have
\begin{eqnarray*}
\bar\nabla_X\xi & = & \bar\nabla_X(T+\nu N) \\
& = & \bar\nabla_XT+\mathrm{d}\nu(X)N+\nu\bar\nabla_XN \\
& = & \nabla_XT+\langle\mathrm{S} X,T\rangle N+\mathrm{d}\nu(X)N-\nu\mathrm{S} X.
\end{eqnarray*}
On the other hand we have
\begin{eqnarray*}
\bar\nabla_X\xi & = & \tau X\times\xi \\
& = & \tau X\times(T+\nu N) \\
& = & \tau(\langle\mathrm{J} X,T\rangle N-\nu\mathrm{J} X).
\end{eqnarray*}
We conclude taking the tangential and normal parts in both expressions.
\end{proof}
\subsection{Moving frames} \label{movingframes}
In this section we introduce some material about the technique of
moving frames.
Let ${\mathcal V}$ be a Riemannian manifold of dimension $n$,
$\nabla$ its Levi-Civita connection, and
$\mathrm{R}$ the Riemannian curvature tensor.
Let $\mathrm{S}$ be a field of symmetric operators
$\mathrm{S}_y:\mathrm{T}_y{\mathcal V}\to\mathrm{T}_y{\mathcal V}$.
Let $(e_1,\dots,e_n)$ be a local orthonormal frame on ${\mathcal V}$ and
$(\omega^1,\dots,\omega^n)$ the dual basis of $(e_1,\dots,e_n)$,
i.e., $$\omega^i(e_k)=\delta^i_k.$$ We also set
$$\omega^{n+1}=0.$$
We define the forms $\omega^i_j$, $\omega^{n+1}_j$, $\omega^i_{n+1}$
and $\omega^{n+1}_{n+1}$ on ${\mathcal V}$ by
$$\omega^i_j(e_k)=\langle\nabla_{e_k}e_j,e_i\rangle,\quad
\omega^{n+1}_j(e_k)=\langle\mathrm{S} e_k,e_j\rangle,$$
$$\omega^j_{n+1}=-\omega^{n+1}_j,\quad
\omega^{n+1}_{n+1}=0.$$
Then we have
$$\nabla_{e_k}e_j=\sum_i\omega^i_j(e_k)e_i,\quad
\mathrm{S} e_k=\sum_j\omega^{n+1}_j(e_k)e_j.$$
Finally we set $R^i_{klj}=\langle\mathrm{R}(e_k,e_l)e_j,e_i\rangle$.
\begin{prop} \label{differentiation}
We have the following formulas:
\begin{equation} \label{diffomega1}
\mathrm{d}\omega^i+\sum_p\omega^i_p\wedge\omega^p=0,
\end{equation}
\begin{equation} \label{diffomega2}
\sum_p\omega^{n+1}_p\wedge\omega^p=0,
\end{equation}
\begin{equation} \label{diffomega3}
\mathrm{d}\omega^i_j+\sum_p\omega^i_p\wedge\omega^p_j=
-\frac{1}{2}\sum_k\sum_lR^i_{klj}\omega^k\wedge\omega^l,
\end{equation}
\begin{equation} \label{diffomega4}
\mathrm{d}\omega^{n+1}_j+\sum_p\omega^{n+1}_p\wedge\omega^p_j=
\frac{1}{2}\sum_k\sum_l\langle\nabla_{e_k}\mathrm{S} e_l
-\nabla_{e_l}\mathrm{S} e_k-\mathrm{S}[e_k,e_l],e_j\rangle\omega^k\wedge\omega^l.
\end{equation}
\end{prop}
For a proof of these classical formulas,
the reader can refer to \cite{codazzi}, proposition 2.4.
\subsection{Some facts about hypersurfaces}
\label{hypersurfaces}
In this section we consider an orientable hypersurface ${\mathcal V}$ of an
$(n+1)$-dimensionnal Riemannian manifold $\bar{\mathcal V}$.
Let $(e_1,\dots,e_n)$ be a local
orthonormal frame on ${\mathcal V}$, $e_{n+1}$ the normal to ${\mathcal V}$, and
$(E_1,\dots,E_{n+1})$ a local orthonormal frame on $\bar{\mathcal V}$. We denote
by $\nabla$ and $\bar\nabla$ the Riemannian connections on
${\mathcal V}$ and $\bar{\mathcal V}$ respectively, and by $\mathrm{S}$ the shape operator of
${\mathcal V}$ (with respect to the normal $e_{n+1}$). We define the forms
$\omega^\alpha$, $\omega^\alpha_\beta$ on ${\mathcal V}$ as in
section \ref{movingframes}. Then we have
$$\bar\nabla_{e_k}e_\beta=\sum_\gamma\omega^\gamma_\beta(e_k)e_\gamma.$$
Let $A\in\mathrm{SO}_{n+1}(\mathbb{R})$ be the matrix whose columns are the
coordinates of the $e_\beta$ in the frame $(E_\alpha)$, namely
$A^\alpha_\beta=\langle e_\beta,E_\alpha\rangle$. Let
$\Omega=(\omega^\alpha_\beta)\in{\mathcal M}_{n+1}(\mathbb{R})$.
\begin{lemma} \label{diffA}
The matrix $A$ satisfies the following equation:
$$A^{-1}\mathrm{d} A=\Omega+L(A)$$
with $$L(A)^\alpha_\beta=\sum_k
\left(\sum_{\gamma,\delta,\varepsilon}A^\varepsilon_\alpha
A^\gamma_kA^\delta_\beta
\bar\Gamma_{\gamma\varepsilon}^\delta\right)\omega^k,$$
where the $\bar\Gamma_{\gamma\varepsilon}^\delta$ are the Christoffel
symbols of the frame $(E_\alpha)$.
\end{lemma}
\begin{proof}
We have $$e_\beta=\sum_\alpha A^\alpha_\beta E_\alpha.$$
Then, on the one hand we have
\begin{eqnarray*}
\bar\nabla_{e_k}e_\beta
& = & \sum_\delta\mathrm{d} A^\delta_\beta(e_k)E_\delta
+\sum_\delta A^\delta_\beta\bar\nabla_{e_k}E_\delta \\
& = & \sum_\varepsilon\mathrm{d} A^\varepsilon_\beta(e_k)E_\delta
+\sum_\gamma\sum_\delta\sum_\varepsilon
A^\delta_\beta A^\gamma_k\bar\Gamma^\varepsilon_{\gamma\delta}
E_\varepsilon,
\end{eqnarray*}
and on the other hand we have
$$\bar\nabla_{e_k}e_\beta=
\sum_\gamma\sum_\varepsilon\omega^\gamma_\beta(e_k)
A^\varepsilon_\gamma E_\varepsilon.$$
Identifying the coefficients we get
\begin{eqnarray*}
\mathrm{d} A^\varepsilon_\beta(e_k) & = &
-\sum_\gamma\sum_\delta
A^\delta_\beta A^\gamma_k\bar\Gamma^\varepsilon_{\gamma\delta}
+\sum_\gamma\omega^\gamma_\beta(e_k)A^\varepsilon_\gamma \\
& = & \sum_\gamma\sum_\delta
A^\delta_\beta A^\gamma_k\bar\Gamma^\delta_{\gamma\varepsilon}
+\sum_\gamma\omega^\gamma_\beta(e_k)A^\varepsilon_\gamma
\end{eqnarray*}
since the frame $(E_\alpha)$ is orthonormal.
We conclude using the fact that $A^{-1}$ is the transpose of $A$.
\end{proof}
\section{Isometric immersions of surfaces into $3$-dimensional
homogeneous manifolds}
We consider a simply connected oriented
Riemannian manifold ${\mathcal V}$ of dimension
$2$. Let $\mathrm{d} s^2$ be the metric on ${\mathcal V}$ (we will also denote it by
$\langle\cdot,\cdot\rangle$), $\nabla$
the Riemannian connection of ${\mathcal V}$, $\mathrm{R}$ its Riemann curvature
tensor and $\mathrm{J}$ the rotation of angle $\frac\pi2$ on $\mathrm{T}{\mathcal V}$.
Let $\mathrm{S}$ be a field of symmetric operators
$\mathrm{S}_y:\mathrm{T}_y{\mathcal V}\to\mathrm{T}_y{\mathcal V}$, $T$ a vector field on ${\mathcal V}$
such that $||T||\leqslant 1$ and $\nu$ a smooth function on ${\mathcal V}$
such that $\nu^2\leqslant 1$.
The compatibility equations for surfaces in $3$-dimensional
homogeneous manifolds with $4$-dimensional isometry group
established in section \ref{compatibilityE} suggest to introduce
the following definition.
\begin{defn}
Let $\mathbb{E}$ be a $3$-dimensional homogeneous manifold with a $4$-dimensional
isometry group. Let $\kappa$ be its base curvature and $\tau$ its
bundle curvature.
We say that $(\mathrm{d} s^2,\mathrm{S},T,\nu)$ satisfies the
compatibility equations for $\mathbb{E}$
if $$||T||^2+\nu^2=1$$ and, for all $X,Y,Z\in\mathfrak{X}({\mathcal V})$,
\begin{equation} \label{gaussE}
K=\det\mathrm{S}+\tau^2+(\kappa-4\tau^2)\nu^2,
\end{equation}
\begin{equation} \label{codazziE}
\nabla_X\mathrm{S} Y-\nabla_Y\mathrm{S} X-\mathrm{S}[X,Y]=
(\kappa-4\tau^2)\nu(\langle Y,T\rangle X-\langle X,T\rangle Y),
\end{equation}
\begin{equation} \label{conditionT1}
\nabla_XT=\nu(\mathrm{S} X-\tau\mathrm{J} X),
\end{equation}
\begin{equation} \label{conditionT2}
\mathrm{d}\nu(X)+\langle\mathrm{S} X-\tau\mathrm{J} X,T\rangle=0.
\end{equation}
\end{defn}
\begin{rem}
We notice that \eqref{conditionT1} implies \eqref{conditionT2} except
when $\nu=0$ (by differentiating the identity $\langle T,T\rangle
+\nu^2=1$ with respect to $X$).
\end{rem}
\begin{thm} \label{isometry}
Let ${\mathcal V}$ be a simply connected oriented
Riemannian manifold of dimension $2$,
$\mathrm{d} s^2$ its metric and $\nabla$ its Riemannian connection.
Let $\mathrm{S}$ be a field of symmetric operators
$\mathrm{S}_y:\mathrm{T}_y{\mathcal V}\to\mathrm{T}_y{\mathcal V}$, $T$ a vector field on ${\mathcal V}$ and $\nu$
a smooth function on ${\mathcal V}$ such that $||T||^2+\nu^2=1$.
Let $\mathbb{E}$ be a $3$-dimensional homogeneous manifold with a $4$-dimensional
isometry group
and $\xi$ its vertical vector field.
Let $\kappa$ be its base curvature and $\tau$ its bundle curvature.
Then there exists an isometric immersion $f:{\mathcal V}\to\mathbb{E}$
such that
the shape operator with respect to the normal $N$ associated to $f$ is
$$\mathrm{d} f\circ\mathrm{S}\circ\mathrm{d} f^{-1}$$ and such that
$$\xi=\mathrm{d} f(T)+\nu N$$
if and only if $(\mathrm{d} s^2,\mathrm{S},T,\nu)$ satisfies the
compatibility equations for $\mathbb{E}$.
In this case, the
immersion is unique up to a global isometry of $\mathbb{E}$ preserving
the orientations of both the fibers and the base of the fibration.
\end{thm}
The fact that the compatibility equations are necessary was proved in
section \ref{compatibilityE}.
To prove that they are sufficient, we consider a local orthonormal frame
$(e_1,e_2)$ on ${\mathcal V}$ and the forms $\omega^i$, $\omega^3$,
$\omega^i_j$, $\omega^3_j$, $\omega^i_3$ and
$\omega^3_3$ as in section \ref{movingframes} (with $n=2$).
From now on we assume that $\tau\neq 0$ since the case $\tau=0$
was treated in \cite{codazzi}.
We denote by
$(E_1,E_2,E_3)$ the canonical frame of $\mathbb{E}$ (see section
\ref{canonicalframe}); in particular we
have $E_3=\xi$. We denote by $\mathbb{E}'$ the open set where
the canonical frame is defined (in particular we have $\mathbb{E}'=\mathbb{E}$ when
$\kappa=0$ or $\kappa<0$; see sections \ref{bergerspheres},
\ref{heisenberg} and \ref{psl}).
We set
$$T^k=\langle T,e_k\rangle,\quad T^3=\nu.$$
We define the one-form $\eta$ on ${\mathcal V}$ by
$$\eta(X)=\langle T,X\rangle.$$
In the frame $(e_1,e_2)$ we have $\eta=\sum_kT^k\omega^k$.
We define the following matrix of one-forms:
$$\Omega=(\omega^\alpha_\beta)\in{\mathcal M}_3(\mathbb{R}).$$
For $Z\in\mathrm{SO}_3(\mathbb{R})$, we set
$$L(Z)^\alpha_\beta=\sum_k
\left(\sum_{\gamma,\delta,\varepsilon}Z^\varepsilon_\alpha
Z^\gamma_kZ^\delta_\beta
\bar\Gamma_{\gamma\varepsilon}^\delta\right)\omega^k,$$
where the $\bar\Gamma_{\gamma\varepsilon}^\delta$ are the Christoffel
symbols of the frame $(E_\alpha)$ (see section \ref{hypersurfaces}).
This defines an antisymmetric matrix of $1$-forms.
We also set $\sigma=\frac\kappa{2\tau}$.
From now on we assume that the hypotheses of theorem \ref{isometry}
are satisfied. We first prove some technical lemmas that are
consequences of the compatibility equations.
\begin{lemma} \label{diffeta}
We have $$\mathrm{d}\eta=-2\tau\nu\omega^1\wedge\omega^2.$$
\end{lemma}
\begin{proof}
By \eqref{conditionT1} we have
$\mathrm{d}\eta(X,Y)=\langle\nabla_XT,Y\rangle-\langle\nabla_YT,X\rangle
=2\tau\nu\langle X,\mathrm{J} Y\rangle$. Thus $\mathrm{d}\eta(e_1,e_2)=-2\tau\nu$.
\end{proof}
\begin{lemma} \label{diffT}
We have
$$\mathrm{d} T^1=\sum_\gamma T^\gamma\omega^\gamma_1+\tau T^3\omega^2,$$
$$\mathrm{d} T^2=\sum_\gamma T^\gamma\omega^\gamma_2-\tau T^3\omega^1,$$
$$\mathrm{d} T^3=\sum_\gamma T^\gamma\omega^\gamma_3-\tau T^1\omega^2
+\tau T^2\omega^1.$$
\end{lemma}
\begin{proof}
The first two identities are a consequence of condition
\eqref{conditionT1} and the last one of condition \eqref{conditionT2}.
\end{proof}
\begin{lemma} \label{diffOmega}
We have
\begin{eqnarray*}
\mathrm{d}\Omega+\Omega\wedge\Omega & = & \left(\begin{array}{ccc}
0 & \tau^2 & 0 \\
-\tau^2 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\omega^1\wedge\omega^2 \\
& & +(\kappa-4\tau^2)T^3\left(\begin{array}{ccc}
0 & T^3 & -T^2 \\
-T^3 & 0 & T^1 \\
T^2 & -T^1 & 0
\end{array}\right)\omega^1\wedge\omega^2.
\end{eqnarray*}
\end{lemma}
\begin{proof}
We set $\Psi=\mathrm{d}\Omega+\Omega\wedge\Omega$ and
$R^i_{klj}=\langle\mathrm{R}(e_k,e_l)e_j,e_i\rangle$. By proposition
\ref{differentiation} we have
$$\Psi^i_j=-\frac12\sum_k\sum_lR^i_{klj}\omega^k\wedge\omega^l
+\omega^i_3\wedge\omega^3_j,$$
and by the Gauss equation \eqref{gaussE} we have
$R^i_{klj}=\bar R^i_{klj}+\omega^3_j\wedge \omega^3_i(e_k,e_l)$ with
$$\bar R^i_{klj}=
(\kappa-3\tau^2)(\delta^k_j\delta^l_i-\delta^l_j\delta^k_i)
+(\kappa-4\tau^2)(T^lT^j\delta^k_i+T^kT^i\delta^l_j-T^lT^i\delta^k_j
-T^kT^j\delta^l_i).$$
Thus we get
$$\Psi^i_j=(\kappa-3\tau^2)\omega^i\wedge\omega^j
+(\kappa-4\tau^2)(T^i\omega^j-T^j\omega^i)\wedge\eta.$$
In the same way, by proposition \ref{differentiation} we have
$$\Psi^3_j=\frac12\sum_k\sum_l
\langle\nabla_{e_k}\mathrm{S} e_l-\nabla_{e_l}\mathrm{S} e_k-\mathrm{S}[e_k,e_l],e_j\rangle
\omega^k\wedge\omega^l,$$
and by the Codazzi equation \eqref{codazziE} we have
$$\langle\nabla_{e_k}\mathrm{S} e_l-\nabla_{e_l}\mathrm{S} e_k
-\mathrm{S}[e_k,e_l],e_j\rangle=
(\kappa-4\tau^2)T^3(T^l\delta^k_j-T^k\delta^l_j).$$
Thus we get
$$\Psi^3_j=(\kappa-4\tau^2)T^3\omega^j\wedge\eta.$$
Hence we have
\begin{eqnarray*}
\Psi & = & (\kappa-3\tau^2)\left(\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\omega^1\wedge\omega^2 \\
& & +(\kappa-4\tau^2)\left(\begin{array}{ccc}
0 & -T^2 & -T^3 \\
T^2 & 0 & 0 \\
T^3 & 0 & 0
\end{array}\right)\omega^1\wedge\eta \\
& & +(\kappa-4\tau^2)\left(\begin{array}{ccc}
0 & T^1 & -0 \\
-T^1 & 0 & -T^3 \\
0 & T^3 & 0
\end{array}\right)\omega^2\wedge\eta.
\end{eqnarray*}
We conclude using that $\omega^1\wedge\eta=T^2\omega^1\wedge\omega^2$,
$\omega^2\wedge\eta=-T^1\omega^1\wedge\omega^2$ and
$(T^1)^2+(T^2)^2+(T^3)^2=1$.
\end{proof}
\begin{lemma} \label{exprL}
We have
\begin{eqnarray*}
L(Z) & = & (2\tau-\sigma)\left(
\begin{array}{ccc}
0 & -T^3 & T^2 \\
T^3 & 0 & -T^1 \\
-T^2 & T^1 & 0
\end{array}\right)\eta \\
& & +\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & \tau \\
0 & -\tau & 0
\end{array}\right)\omega^1+\left(
\begin{array}{ccc}
0 & 0 & -\tau \\
0 & 0 & 0 \\
\tau & 0 & 0
\end{array}\right)\omega^2.
\end{eqnarray*}
\end{lemma}
\begin{proof}
We compute that
\begin{eqnarray*}
L(Z)^\alpha_\beta & = &
\sum_k\left(\sum_\gamma\sum_\delta\sum_\varepsilon
Z^\varepsilon_\alpha Z^\gamma_k Z^\delta_\beta
\bar\Gamma^\delta_{\gamma\varepsilon}\right)\omega^k \\
& = & \sum_k(\tau(Z^2_\alpha Z^1_k Z^3_\beta
+Z^3_\alpha Z^2_k Z^1_\beta-Z^1_\alpha Z^2_k Z^3_\beta
-Z^3_\alpha Z^1_k Z^2_\beta) \\
& & \quad
+(\tau-\sigma)(Z^2_\alpha Z^3_k Z^1_\beta-Z^1_\alpha Z^3_k Z^2_\beta)
)\omega^k \\
& = & \sum_k(
\tau T^\beta(Z^1_k Z^2_\alpha-Z^1_\alpha Z^2_k)
+\tau T^\alpha(Z^1_\beta Z^2_k-Z^1_k Z^2_\beta) \\
& & \quad+(\tau-\sigma)T^k(Z^1_\beta Z^2_\alpha-Z^1_\alpha Z^2_\beta)
)\omega^k.
\end{eqnarray*}
Moreover the matrix $Z$ lies in $\mathrm{SO}_3(\mathbb{R})$, so it is equal to
its comatrix. Using this fact we compute that
$$L(Z)^1_2=-(2\tau-\sigma)T^3(T^1\omega^1+T^2\omega^2),$$
$$L(Z)^1_3=(2\tau-\sigma)T^1T^2\omega^1+(2\tau-\sigma)(T^2)^2\omega^2
-\tau\omega^2,$$
$$L(Z)^2_3=-(2\tau-\sigma)(T^1)^2\omega^1-(2\tau-\sigma)T^1T^2\omega^2
+\tau\omega^1,$$
which proves the lemma.
\end{proof}
\begin{lemma} \label{LwedgeL}
We have
\begin{eqnarray*}
L\wedge L & = & \tau(2\tau-\sigma)T^3\left(\begin{array}{ccc}
0 & -T^3 & T^2 \\
T^3 & 0 & -T^1 \\
-T^2 & T^1 & 0
\end{array}\right)\omega^1\wedge\omega^2 \\
& & +\tau(\tau-\sigma)\left(\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\omega^1\wedge\omega^2.
\end{eqnarray*}
\end{lemma}
\begin{proof}
We compute that
\begin{eqnarray*}
L\wedge L & = & \tau(2\tau-\sigma)\left(\begin{array}{ccc}
0 & T^1 & 0 \\
-T^1 & 0 & -T^3 \\
0 & T^3 & 0
\end{array}\right)\eta\wedge\omega^2 \\
& & +\tau(2\tau-\sigma)\left(\begin{array}{ccc}
0 & -T^2 & -T^3 \\
T^2 & 0 & 0 \\
T^3 & 0 & 0
\end{array}\right)\eta\wedge\omega^1 \\
& & +\tau^2\left(\begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\omega^1\wedge\omega^2.
\end{eqnarray*}
We conclude using that $(T^1)^2+(T^2)^2+(T^3)^2=1$.
\end{proof}
\begin{lemma} \label{LwedgeOmega}
We have
\begin{eqnarray*}
L\wedge\Omega+\Omega\wedge L & = &
(2\tau-\sigma)\eta\wedge\left(\begin{array}{ccc}
0 & -\mathrm{d} T^3 & \mathrm{d} T^2 \\
\mathrm{d} T^3 & 0 & -\mathrm{d} T^1 \\
-\mathrm{d} T^2 & \mathrm{d} T^1 & 0
\end{array}\right) \\
& & +\tau(2\tau-\sigma)T^3\left(\begin{array}{ccc}
0 & T^3 & -T^2 \\
-T^3 & 0 & T^1 \\
T^2 & -T^1 & 0
\end{array}\right)\omega^1\wedge\omega^2 \\
& & +\tau(2\tau-\sigma)\left(\begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\omega^1\wedge\omega^2 \\
& & +\tau\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -1 \\
0 & 1 & 0
\end{array}\right)\mathrm{d}\omega^1
+\tau\left(\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0
\end{array}\right)\mathrm{d}\omega^2.
\end{eqnarray*}
\end{lemma}
\begin{proof}
We compute that
\begin{eqnarray*}
L\wedge\Omega+\Omega\wedge L & = &
(2\tau-\sigma)\eta\wedge M \\
& & +\tau\omega^2\wedge\left(\begin{array}{ccc}
0 & -\omega^3_2 & 0 \\
-\omega^2_3 & 0 & \omega^2_1 \\
0 & \omega^1_2 & 0
\end{array}\right) \\
& & +\tau\omega^1\wedge\left(\begin{array}{ccc}
0 & \omega^1_3 & -\omega^1_2 \\
\omega^3_1 & 0 & 0 \\
-\omega^2_1 & 0 & 0
\end{array}\right)
\end{eqnarray*}
with $$M=\left(\begin{array}{ccc}
0 & T^2\omega^3_2-T^1\omega^1_3 & -T^3\omega^2_3+T^1\omega^1_2 \\
-T^1\omega^3_1+T^2\omega^2_3 & 0 & T^3\omega^1_3-T^2\omega^2_1 \\
T^1\omega^2_1-T^3\omega^3_2 & -T^2\omega^1_2+T^3\omega^3_1 & 0
\end{array}\right).$$
We conclude using lemma \ref{diffT}, formulas \eqref{diffomega1} and
\eqref{diffomega2}, and the fact that $(T^1)^2+(T^2)^2+(T^3)^2=1$.
\end{proof}
For $y\in{\mathcal V}$, let ${\mathcal Z}(y)$ be the set of matrices
$Z\in\mathrm{SO}_3(\mathbb{R})$ such
that the coefficients of the last line of $Z$ are the $T^\beta(y)$.
It is diffeomorphic to the circle $\mathbb{S}^1$.
We now prove the following proposition.
\begin{prop} \label{matrixA}
Assume that the compatibility equations for $\mathbb{E}$ are
satisfied. Let $y_0\in{\mathcal V}$ and $A_0\in{\mathcal Z}(y_0)$. Then there exist a
neighbourhood $U_1$ of
$y_0$ in ${\mathcal V}$ and a unique map $A:U_1\to\mathrm{SO}_3(\mathbb{R})$ such that
$$A^{-1}\mathrm{d} A=\Omega,$$
$$\forall y\in U_1,\quad A(y)\in{\mathcal Z}(y),$$
$$A(y_0)=A_0.$$
\end{prop}
\begin{proof}
Let $U$ be a coordinate neighbourhood in ${\mathcal V}$. The set
$${\mathcal F}=\{(y,Z)\in U\times\mathrm{SO}_3(\mathbb{R});Z\in{\mathcal Z}(y)\}$$ is a
manifold of dimension $3$, and
$$\mathrm{T}_{(y,Z)}{\mathcal F}=\{(u,\zeta)\in\mathrm{T}_yU\oplus\mathrm{T}_Z\mathrm{SO}_3(\mathbb{R});
\zeta^3_\beta=(\mathrm{d} T^\beta)_y(u)\}.$$
Let $Z$ denote the projection
$U\times\mathrm{SO}_3(\mathbb{R})
\to\mathrm{SO}_3(\mathbb{R})\subset{\mathcal M}_3(\mathbb{R})$.
We consider on ${\mathcal F}$ the
following matrix of $1$-forms:
$$\Theta=Z^{-1}\mathrm{d} Z-\Omega-L(Z)$$
where $L(Z)$ is defined in lemma \ref{diffA},
namely for $(y,Z)\in{\mathcal F}$ we have
$$\Theta_{(y,Z)}:\mathrm{T}_{(y,Z)}{\mathcal F}\to{\mathcal M}_3(\mathbb{R}),$$
$$\Theta_{(y,Z)}(u,\zeta)=Z^{-1}\zeta-\Omega_y(u)-L(Z)(u).$$
We claim that, for each $(y,Z)\in{\mathcal F}$, the space
$${\mathcal D}(y,Z)=\ker\Theta_{(y,Z)}$$ has dimension $2$.
We first notice that
the matrix $\Theta$ belongs to $\mathfrak{so}_3(\mathbb{R})$ since
$\Omega$, $L(Z)$ and $Z^{-1}\mathrm{d} Z$ do. Moreover we have
$$(Z\Theta)^3_\beta
=\mathrm{d} Z^3_\beta-\sum_\gamma Z^3_\gamma\omega^\gamma_\beta
-\sum_\gamma Z^3_\gamma L(Z)^\gamma_\beta
=\mathrm{d} T^\beta-\sum_\gamma T^\gamma\omega^\gamma_\beta
-\sum_\gamma T^\gamma L(Z)^\gamma_\beta.$$
Using lemmas \ref{diffT} and \ref{exprL} we compute that
$$(Z\Theta)^3_\beta=0.$$
Thus the values of
$\Theta_{(y,Z)}$ lie in the space
$${\mathcal H}=\{H\in\mathfrak{so}_3(\mathbb{R});(ZH)^3_\beta=0\},$$
which has dimension $1$ (indeed, the map
$F:\mathrm{SO}_3(\mathbb{R})\to\mathbb{S}^2,
Z\mapsto(Z^3_\beta)_\beta$ is a submersion, and we
have $H\in{\mathcal H}$ if and
only if $ZH\in\ker(\mathrm{d} F)_Z$).
Moreover, the space $\mathrm{T}_{(y,Z)}{\mathcal F}$ contains the subspace
$\{(0,ZH);H\in{\mathcal H}\}$, and the restriction of $\Theta_{(y,Z)}$ on this
subspace is the map $(0,ZH)\mapsto H$. Thus
$\Theta_{(y,Z)}$ is onto
${\mathcal H}$, and consequently the linear map $\Theta_{(y,Z)}$ has rank
$1$. This finishes proving the claim.
We now prove that the distribution ${\mathcal D}$ is involutive. We first compute
that
\begin{eqnarray*}
\mathrm{d}\Theta & = & -Z^{-1}\mathrm{d} Z\wedge Z^{-1}\mathrm{d} Z-\mathrm{d}\Omega-\mathrm{d} L \\
& = & -(\Theta+\Omega+L)\wedge(\Theta+\Omega+L)-\mathrm{d}\Omega-\mathrm{d} L \\
& = & -\Theta\wedge\Theta-\Theta\wedge\Omega
-\Omega\wedge\Theta-\Omega\wedge L-L\wedge\Omega \\
& & -\Omega\wedge\Omega-\mathrm{d}\Omega-L\wedge L-\mathrm{d} L.
\end{eqnarray*}
Using lemmas \ref{diffeta}, \ref{diffOmega}, \ref{LwedgeL},
\ref{LwedgeOmega} and the relation $\sigma=\frac\kappa{2\tau}$, we
obtain
$$\mathrm{d}\Theta=-\Theta\wedge\Theta-\Theta\wedge\Omega-\Omega\wedge\Theta.$$
From this formula we deduce that if $\xi_1,\xi_2\in{\mathcal D}$, then
$\mathrm{d}\Theta(\xi_1,\xi_2)=0$, and so
$\Theta([\xi_1,\xi_2])=\xi_1\cdot\Theta(\xi_2)
-\xi_2\cdot\Theta(\xi_1)-\mathrm{d}\Theta(\xi_1,\xi_2)=0$, i.e.,
$[\xi_1,\xi_2]\in{\mathcal D}$. Thus the distribution ${\mathcal D}$ is involutive, and
so, by the theorem of Frobenius, it is integrable.
Let ${\mathcal A}$ be the integral manifold through $(y_0,A_0)$.
If $\zeta\in\mathrm{T}_{A_0}\mathrm{SO}_3(\mathbb{R})$ is such that
$(0,\zeta)\in\mathrm{T}_{(y_0,A_0)}{\mathcal A}={\mathcal D}(y_0,A_0)$,
then we have $0=\Theta_{(y_0,A_0)}(0,\zeta)=A_0^{-1}\zeta$. This
proves that
$$\mathrm{T}_{(y_0,A_0)}{\mathcal A}\cap
\left(\{0\}\times\mathrm{T}_{A_0}\mathrm{SO}_3(\mathbb{R})\right)=\{0\}.$$
Thus the manifold
${\mathcal A}$ is locally the graph of a function
$A:U_1\to\mathrm{SO}_3(\mathbb{R})$ where $U_1$ is a neighbourhood of
$y_0$ in $U$. By construction, this map satisfies the properties of
proposition \ref{matrixA} and is unique.
\end{proof}
\begin{prop} \label{functionf}
Let $x_0\in\mathbb{E}$ (without loss of generality we can assume that
$x_0\in\mathbb{E}'$). There exist a neighbourhood $U_2$ of $y_0$ contained in
$U_1$ and a unique function $f:U_2\to\mathbb{E}'$ such that
$$\mathrm{d} f=(B\circ f)A\omega,$$
$$f(y_0)=x_0,$$
where $\omega$ is the column $(\omega^1,\omega^2,0)$ and, for $x\in\mathbb{E}'$,
$B(x)\in{\mathcal M}_3(\mathbb{R})$ is the matrix of the coordinates of the frame
$(E_\alpha(x))$ in the frame
$(\partial_{x^\alpha})$.
\end{prop}
\begin{proof}
We consider on $U_1\times\mathbb{E}'$ the following matrix of $1$-forms:
$$\Lambda=B^{-1}\mathrm{d} x-A\omega,$$
namely, for $q\in U_1$ and $x\in\mathbb{E}'$ we have
$$\Lambda_{(q,x)}:\mathrm{T}_qU_1\oplus\mathrm{T}_x\mathbb{E}\to{\mathcal M}_{3,1}(\mathbb{R}),$$
$$\Lambda_{(q,x)}(u,v)=B(x)^{-1}v-A(q)\omega_q(u).$$
We first notice that for all $(q,x)\in U_1\times\mathbb{E}'$ the linear
map $\Lambda_{(q,x)}$ is onto ${\mathcal M}_{3,1}(\mathbb{R})$. Consequently the space
$${\mathcal E}(q,x)=\ker\Lambda_{(q,x)}$$ has dimension $2$. We will prove that this
distribution ${\mathcal E}$ is integrable.
We have
$$\mathrm{d}\Lambda=-B^{-1}\mathrm{d} BB^{-1}\wedge\mathrm{d} x
-\mathrm{d} A\wedge\omega-A\wedge\mathrm{d}\omega.$$
By equations \eqref{diffomega1} and \eqref{diffomega2} we have
$\mathrm{d}\omega=-\Omega\wedge\omega$; and by proposition \ref{matrixA} we
have $\mathrm{d} A=A\Omega+AL(A)$. Thus we get
$$\mathrm{d}\Lambda=-B^{-1}\mathrm{d} B\wedge\Lambda-B^{-1}\mathrm{d} B\wedge A\omega
-AL(A)\wedge\omega.$$
Using lemma \ref{exprL} we compute that
$$L(A)\wedge\omega=-(2\tau-\sigma)T^3\left(\begin{array}{c}
T^1 \\
T^2 \\
T^3
\end{array}\right)\omega^1\wedge\omega^2
-\left(\begin{array}{c}
0 \\
0 \\
\sigma
\end{array}\right)\omega^1\wedge\omega^2,$$
and thus, using the fact that $A^3_\beta=T^\beta$ and
$A=\mathrm{com}A$, we get
$$AL(A)\wedge\omega=\left(\begin{array}{c}
-\sigma A^1_3 \\
-\sigma A^2_3 \\
-2\tau T^3
\end{array}\right)\omega^1\wedge\omega^2.$$
We will use the notation $(x,y,x)$ instead of $(x^1,x^2,x^3)$ for the
coordinates in $\mathbb{E}$ and we will use the local models described in
sections \ref{bergerspheres}, \ref{heisenberg} and \ref{psl}. Using
formulas \eqref{canonicalbergerspheres},
\eqref{canonicalheisenberg} and \eqref{canonicalpsl}, we get that the
matrix $B$ is
$$B=\left(\begin{array}{ccc}
\lambda^{-1}\cos(\sigma z) & -\lambda^{-1}\sin(\sigma z) & 0 \\
\lambda^{-1}\sin(\sigma z) & \lambda^{-1}\cos(\sigma z) & 0 \\
\tau(x\sin\sigma z-y\cos\sigma z) & \tau(x\cos\sigma z+y\sin\sigma z)
& 1
\end{array}\right),$$
with $$\lambda=\frac1{1+\frac\kappa4(x^2+y^2)}.$$
We will write $$A\omega=\left(\begin{array}{c}
\alpha^1 \\
\alpha^2 \\
\eta
\end{array}\right)$$ with
$$\alpha^j=A^j_1\omega^1+A^j_2\omega^2.$$
Then we have
$$\Lambda=B^{-1}\mathrm{d} X-A\omega=\left(\begin{array}{c}
\lambda(\cos(\sigma z)\mathrm{d} x+\sin(\sigma z)\mathrm{d} y)-\alpha^1 \\
\lambda(-\sin(\sigma z)\mathrm{d} x+\cos(\sigma z)\mathrm{d} y)-\alpha^2 \\
\tau\lambda(y\mathrm{d} x-x\mathrm{d} y)+\mathrm{d} z-\eta
\end{array}\right).$$
We also compute that
$$B^{-1}\mathrm{d} B=\left(\begin{array}{ccc}
\frac\kappa2\lambda(x\mathrm{d} x+y\mathrm{d} y) & -\sigma\mathrm{d} z & 0 \\
\sigma\mathrm{d} z & \frac\kappa2\lambda(x\mathrm{d} x+y\mathrm{d} y) & 0 \\
a & b & 0
\end{array}\right)$$
with $$a=\frac{\tau\kappa}2\lambda(y\cos(\sigma z)-x\sin(\sigma z))
(x\mathrm{d} x+y\mathrm{d} y)+\tau(\sin(\sigma z)\mathrm{d} x-\cos(\sigma z)\mathrm{d} y),$$
$$b=-\frac{\tau\kappa}2\lambda(x\cos(\sigma z)+y\sin(\sigma z))
(x\mathrm{d} x+y\mathrm{d} y)+\tau(\cos(\sigma z)\mathrm{d} x+\sin(\sigma z)\mathrm{d} y).$$
Thus we have
\begin{eqnarray*}
B^{-1}\mathrm{d} B\wedge A\omega+AL(A)\wedge\omega & = &
\left(\begin{array}{c}
\frac\kappa2\lambda(x\mathrm{d} x+y\mathrm{d} y)\wedge\alpha^1
-\sigma\mathrm{d} z\wedge\alpha^2 \\
\sigma\mathrm{d} z\wedge\alpha^1
+\frac\kappa2\lambda(x\mathrm{d} x+y\mathrm{d} y)\wedge\alpha^2 \\
a\wedge\alpha^1+b\wedge\alpha^2
\end{array}\right) \\
& & +\left(\begin{array}{c}
-\sigma A^1_3 \\
-\sigma A^2_3 \\
-2\tau T^3
\end{array}\right)\omega^1\wedge\omega^2.
\end{eqnarray*}
Using the above expression for $\Lambda$ we get
$$\lambda\mathrm{d} x=\cos(\sigma z)\Lambda^1-\sin(\sigma z)\Lambda^2
+\cos(\sigma z)\alpha^1-\sin(\sigma z)\alpha^2,$$
$$\lambda\mathrm{d} y=\sin(\sigma z)\Lambda^1+\cos(\sigma z)\Lambda^2
+\sin(\sigma z)\alpha^1+\sin(\sigma z)\alpha^2,$$
$$\mathrm{d} z=\Lambda^3+\eta-\tau\lambda(y\mathrm{d} x-x\mathrm{d} y).$$
The term in the first line of the matrix
$B^{-1}\mathrm{d} B\wedge A\omega+AL(A)$ is
\begin{eqnarray*}
\frac\kappa2(y\cos(\sigma z)-x\sin(\sigma z))\alpha^2\wedge\alpha^1
+\sigma\tau(y\cos(\sigma z)-x\sin(\sigma z))\alpha^1\wedge\alpha^2 \\
\quad-\sigma\eta\wedge\alpha^2-\sigma A^1_3\omega^1\wedge\omega^2
+\chi^1
\end{eqnarray*}
where $\chi^1$ is a linear combination of the $\Lambda^\alpha$ (the
coefficients being $1$-forms).
Since $\sigma=\frac{\kappa}{2\tau}$, the first two terms in this expression
cancel. Moreover we have $\eta\wedge\alpha^2
=(A^3_1A^2_2-A^3_2A^2_1)\omega^1\wedge\omega^2
=-A^1_3\omega^1\wedge\omega^2$, hence the term in the first line of
the matrix $B^{-1}\mathrm{d} B\wedge A\omega+AL(A)$ is $\chi^1$. In the same
way, the term in the second line of
the matrix $B^{-1}\mathrm{d} B\wedge A\omega+AL(A)$ is
a linear combination of the $\Lambda^\alpha$ which will be
denoted by $\chi^2$. Finally we compute that the term in the
third line of the matrix $B^{-1}\mathrm{d} B\wedge A\omega+AL(A)$ is
$$\left(\frac{2\tau}{\lambda}-\frac{\tau\kappa}2(x^2+y^2)\right)
\alpha^1\wedge\alpha^2-2\tau T^3\omega^1\wedge\omega^2+\chi^3$$
where $\chi^1$ is a linear combination of the $\Lambda^\alpha$.
Since $\lambda^{-1}=1+\frac\kappa4(x^2+y^2)$ and
$\alpha^1\wedge\alpha^2=(A^1_1A^2_2-A^1_2A^2_1)\omega^1\wedge\omega^2
=T^3\omega^1\wedge\omega^2$, this term is simply $\chi^3$.
We conclude that
$$B^{-1}\mathrm{d} B\wedge A\omega+AL(A)=\chi$$
where $\chi$ is a matrix of $2$-forms which are linear combinations
of the coefficients of $\Lambda$. Finally we have
$$\mathrm{d}\Lambda=-B^{-1}\mathrm{d} B\wedge\Lambda-\chi.$$
From this formula we deduce that if $\xi_1,\xi_2\in{\mathcal E}$, then
$\mathrm{d}\Lambda(\xi_1,\xi_2)=0$, and so
$[\xi_1,\xi_2]\in{\mathcal E}$. Thus the distribution ${\mathcal E}$ is involutive, and
so, by the theorem of Frobenius, it is integrable.
Let ${\mathcal A}$ be the integral manifold through $(y_0,x_0)$.
If $v\in\mathrm{T}_{x_0}\mathbb{E}$ is such that
$(0,v)\in\mathrm{T}_{(y_0,x_0)}{\mathcal A}={\mathcal D}(y_0,x_0)$,
then we have $0=\Lambda_{(y_0,x_0)}(0,v)=B(x_0)^{-1}v$. This
proves that
$$\mathrm{T}_{(y_0,x_0)}{\mathcal A}\cap
\left(\{0\}\times\mathrm{T}_{x_0}\mathbb{E}\right)=\{0\}.$$
Thus the manifold
${\mathcal A}$ is locally the graph of a function
$A:U_2\to\mathbb{E}'$ where $U_2$ is a neighbourhood of
$y_0$ in $U_1$. By construction, this map satisfies the properties of
proposition \ref{matrixA} and is unique.
\end{proof}
We now prove the theorem.
\begin{proof}[Proof of theorem \ref{isometry}]
Let $y_0\in{\mathcal V}$, $A_0\in{\mathcal Z}(y_0)$ and $x_0\in\mathbb{E}'$.
We consider on ${\mathcal V}$ a local orthonormal frame $(e_1,e_2)$ in
the neighbourhood of $y_0$ and we keep the same notations. Then by
propositions \ref{matrixA} and \ref{functionf} there exists a unique map
$A:U_2\to\mathrm{SO}^3(\mathbb{R})$ such that
$$A^{-1}\mathrm{d} A=\Omega+L(A),$$
$$\forall y\in U_1,\quad A(y)\in{\mathcal Z}(y),$$
$$A(y_0)=A_0,$$
and a unique map $f:U_2\to\mathbb{E}'$ such that
$$\mathrm{d} f=(B\circ f)A\omega,$$
$$f(y_0)=x_0,$$
where $U_2$ is a neighbourhood of $y_0$, which we can
assume simply connected. We will check that $f$ has the properties
required in the theorem on $U_2$.
We have $\mathrm{d} f^\alpha(e_k)=(B(f)A)^\alpha_k$, so in the frame
$(\partial_{x^\alpha})$ the vector
$\mathrm{d} f(e_k)$ is given by the column $k$ of the matrix $BA$, which is
invertible. Hence $\mathrm{d} f$ has rank $2$, and thus $f$ is an immersion.
Moreover, in the frame $(E_\alpha)$ the vector
$\mathrm{d} f(e_k)$ is given by the column $k$ of the matrix $A$, which is
orthogonal, and thus we have
$\langle\mathrm{d} f(e_p),\mathrm{d} f(e_q)\rangle
=\delta^p_q$, which means that $f$ is an isometry.
The columns of $A(y)$ form a direct orthonormal frame of
$\mathbb{E}$. The first and second columns form a direct orthonormal frame of
$\mathrm{T}_{f(y)}f({\mathcal V})$
Thus the third column gives, in the frame $(E_\alpha)$,
the unit normal $N(f(y))$ to
$f({\mathcal V})$ in $\mathbb{E}$ at the point $f(y)$.
We set $X_j=\mathrm{d} f(e_j)$. Then we have
\begin{eqnarray*}
\mathrm{d} A^\alpha_j(e_k) & = & \langle\bar\nabla_{X_k}X_j,E_\alpha\rangle
+\langle X_j,\bar\nabla_{X_k}E_\alpha\rangle \\
& = & \langle\bar\nabla_{X_k}X_j,E_\alpha\rangle
+\sum_\gamma\sum_\delta A^\gamma_kA^\delta_j
\bar\Gamma^\delta_{\gamma\alpha} \\
& = & \langle\bar\nabla_{X_k}X_j,E_\alpha\rangle
+(AL(A))^\alpha_j(e_k),
\end{eqnarray*}
so
\begin{eqnarray*}
\langle\bar\nabla_{X_k}X_j,N\rangle & = &
\sum_\alpha\langle\bar\nabla_{X_k}X_j,E_\alpha\rangle A^\alpha_3
=\sum_\alpha A^\alpha_3(\mathrm{d} A-AL(A))^\alpha_j(e_k) \\
& = & \sum_\alpha A^\alpha_3(A\Omega)^\alpha_j(e_k)
=\sum_\alpha\sum_\gamma A^\alpha_\gamma
A^\alpha_3\omega^\gamma_j(e_k) \\
& = & \omega^3_j(e_k)=\langle\mathrm{S} e_k,e_j\rangle.
\end{eqnarray*}
This means that the shape operator of $f({\mathcal V})$ in $\mathbb{E}$ is
$\mathrm{d} f\circ\mathrm{S}\circ\mathrm{d} f^{-1}$.
Finally, the coefficients of the vertical vector
$\xi=E_3$ in
the orthonormal frame $(X_1,X_2,N)$ are given by the last
line of $A$. Since $A(y)\in{\mathcal Z}(y)$ for all $y\in U_2$ we get
$$\xi=\sum_j T^jX_j+T^3N
=\mathrm{d} f(T)+\nu N.$$
We now prove that the local immersion is unique up to a global
isometry of $\mathbb{E}$ preserving $\xi$ (and also, consequently, the
orientation of the base of the fibration). Let $\tilde f:U_3\to\mathbb{E}$ be
another immersion satisfying the conclusion of the theorem, where
$U_3$ is a simply connected neighbourhood of $y_0$ included in $U_2$,
let $(\tilde X_\beta)$ be the associated frame (i.e., $\tilde
X_j=\mathrm{d}\tilde f(e_j)$ and $\tilde X_3$ is the normal of
$\tilde f({\mathcal V})$) and let $\tilde A$ the
matrix of the coordinates of the frame $(\tilde X_\beta)$ in the
frame $(E_\alpha)$. Up to an isometry of $\mathbb{E}$ (which is necessarily
direct), we can
assume that $f(y_0)=\tilde f(y_0)$ and that the frames
$(X_\beta(y_0))$ and $(\tilde X_\beta(y_0))$ coincide, i.e.,
$A(y_0)=\tilde A(y_0)$. We notice that this isometry necessarily fixes
$\xi$ since the $T^\alpha$ are the same for $x$
and $\tilde x$. The matrices $A$ and $\tilde A$ satisfy
$A^{-1}\mathrm{d} A=\Omega+L(A)$ and
$\tilde A^{-1}\mathrm{d}\tilde A=\Omega+L(\tilde A)$ (see
section \ref{hypersurfaces}),
$A(y),\tilde A(y)\in{\mathcal Z}(y)$ and $A(y_0)=\tilde A(y_0)$,
thus by the uniqueness of the solution of the equation in proposition
\ref{matrixA} we get $A(y)=\tilde A(y)$. We conclude similarly that
$f=\tilde f$ on $U_3$.
The proof that this local immersion $f$ can be extended to the whole ${\mathcal V}$
(since ${\mathcal V}$ is simply connected) is exactly the same as the proof
of the corresponding statement in theorem 3.3 in \cite{codazzi} (it is
a standard argument).
\end{proof}
\begin{rem} \label{changeofsigns}
If $(\mathrm{d} s^2,\mathrm{S},T,\nu)$ satisfies the compatibilty equations and
correspond to an immerion $f:\Sigma\to\mathbb{E}$, then
$(\mathrm{d} s^2,\mathrm{S},-T,-\nu)$ also satisfies the compatibilty equations and
corresponds to the immersion $\sigma\circ f$ where $\sigma$ is an
isometry of $\mathbb{E}$
reversing the orientations of both the fibers and the base of the
fibration.
\end{rem}
\section{Constant mean curvature surfaces in $3$-dimensional homogeneous manifolds}
In this section we will give an application of theorem \ref{isometry}
to constant mean curvature surfaces (CMC) in $3$-dimensional homogeneous manifolds with
$4$-dimensional isometry group. Abresch and Rosenberg proved that there
exists a holomorphic quadratic differential for CMC surfaces in
$\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, generalizing the Hopf differential for
CMC surfaces in $3$-dimensional space forms (\cite{abresch}). Since the
Hopf differential is a very useful tool for CMC surfaces, this
motivated many works on CMC surfaces in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$.
Recently, Abresch announced the existence of a holomorphic quadratic
differential for CMC surfaces in all $3$-dimensional homogeneous manifolds with
$4$-dimensional isometry group (\cite{abreschsurvey}).
This indicates that the theory of CMC
surfaces in these manifolds may be particularily interesting.
We will consider constant mean curvature immersions of
oriented surfaces. Consequently the
mean curvature will be defined
with a sign: it will be positive
if the mean curvature vector induces the same orientation as the initial
orientation, and it will be negative
if the mean curvature vector induces the opposite orientation.
We will denote by $\mathrm{I}$ and $\mathrm{J}$ the identity and the rotation of
angle $\frac\pi2$ on the tangent bundle of a surface.
\subsection{A generalized Lawson correspondence}
It is well known that there exists an isometric correspondence
between certain simply connected CMC surfaces in space-forms:
more precisely, every simply connected CMC $H_1$ surface in
$\mathbb{M}^3(K_1)$ is isometric to a simply connected CMC $H_2$ surface in
$\mathbb{M}^3(K_2)$ with $K_1-K_2=H_2^2-H_1^2$, and the shape operators of these
two surfaces differ by $(H_2-H_1)\mathrm{I}$.
Two such surfaces are called
cousin surfaces. This correspondence is often called the Lawson
correspondence. In particular, any simply connected minimal surface
in $\mathbb{S}^3$ is isometric to a CMC $1$ surface in $\mathbb{R}^3$, and any
minimal surface
in $\mathbb{R}^3$ is isometric to a CMC $1$ surface in $\mathbb{H}^3$.
The Lawson correspondence is a consequence of the Gauss and Codazzi
equations in the space-forms.
In this section we will use the compatibility equations for
homogeneous $3$-manifolds with $4$-dimensional isometry group
and theorem \ref{isometry} to prove the existence of an
isometric correspondence between certain simply connected
CMC surfaces in these $3$-manifolds. Hence this will be a generalisation
of the Lawson correspondence.
The technique will be to start with some data $(\mathrm{d} s^2,\mathrm{S},T,\nu)$
on a surface satisfying the compatibility equations for some homogeneous
$3$-manifold and to modify them in order to get data
satisfying the compatibility equations for another homogeneous
$3$-manifold. An important fact is that the space of symmetric traceless
operators is globally invariant by rotation.
The easiest change is to keep $\mathrm{d} s^2$ and $\nu$, and to rotate $T$
and the traceless part of $\mathrm{S}$ by some fixed angles; the Codazzi
equation then implies that we need to take the same angle for $T$ and
the traceless part of $\mathrm{S}$.
\begin{prop} \label{correspondence}
Let $\mathbb{E}_1$ and $\mathbb{E}_2$ be two $3$-dimensional homogeneous manifolds
with $4$-dimensional isometry groups, of base curvatures $\kappa_1$
and $\kappa_2$ and bundle curvatures $\tau_1$ and $\tau_2$
respectively. Assume that
$$\kappa_1-4\tau_1^2=\kappa_2-4\tau_2^2.$$
Let $H_1$ and $H_2$ be two real numbers such that
$$\tau_1^2+H_1^2=\tau_2^2+H_2^2.$$
Let ${\mathcal V}$ be a surface with a quadruple $(\mathrm{d} s^2,\mathrm{S}_1,T_1,\nu)$
satisfying the compatibility equations for $\mathbb{E}_1$ and such that
$$\tr\mathrm{S}_1=2H_1.$$
Let $$\theta\in\mathbb{R},$$
$$T_2=e^{\theta\mathrm{J}}T_1,$$
$$\mathrm{S}_2=e^{\theta\mathrm{J}}(\mathrm{S}_1-H_1\mathrm{I})+H_2\mathrm{I}.$$
In particular $\mathrm{S}_2$ is symmetric and satisfies $$\tr\mathrm{S}_2=2H_2.$$
If the real number $\theta$ satisfies
\begin{equation} \label{phase}
\tau_2+iH_2=e^{i\theta}(\tau_1+iH_1),
\end{equation}
then the quadruple $(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$
satisfies the compatibility equations for $\mathbb{E}_2$.
Conversely, if the function $\nu$ is not identically zero
and if the quadruple $(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$
satisfies the compatibility equations for $\mathbb{E}_2$, then
\eqref{phase} holds.
\end{prop}
\begin{proof}
The fact that $\mathrm{S}_2$ is symmetric comes from the fact that the space
of symmetric traceless operators is invariant by a rotation.
We have $$\det(\mathrm{S}_k-H_k\mathrm{I})=\det\mathrm{S}_k-H_k^2$$ for $k=1,2$,
and so $$\det\mathrm{S}_1=\det\mathrm{S}_2+H_1^2-H_2^2.$$
Let $K$ be the Gauss curvature of the metric $\mathrm{d} s^2$.
By the Gauss equation \eqref{gaussE} we have
\begin{eqnarray*}
K & = & \det\mathrm{S}_1+\tau_1^2+(\kappa_1-4\tau_1^2)\nu^2 \\
& = & \det\mathrm{S}_2+H_1^2-H_2^2+\tau_1^2+(\kappa_1-4\tau_1^2)\nu^2 \\
& = & \det\mathrm{S}_2+\tau_2^2+(\kappa_2-4\tau_2^2)\nu^2
\end{eqnarray*}
since $\kappa_1-4\tau_1^2=\kappa_2-4\tau_2^2$ and
$\tau_1^2+H_1^2=\tau_2^2+H_2^2$. Thus the quadruple
$(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$ satisfies the Gauss equation for
$\mathbb{E}_2$.
Since
$\mathrm{J}$ commutes with $\nabla_X$ for all vector fields $X$, we have
$$\nabla_X\mathrm{S}_2 Y-\nabla_Y\mathrm{S}_2 X-\mathrm{S}_2[X,Y]=
e^{\theta\mathrm{J}}(\nabla_X\mathrm{S}_1 Y-\nabla_Y\mathrm{S}_1 X-\mathrm{S}_1[X,Y]).$$
On the other hand, a computation done in the proof of proposition 4.1 in
\cite{codazzi} shows that
$$\langle Y,T_2\rangle X-\langle X,T_2\rangle Y=
e^{\theta\mathrm{J}}(\langle Y,T_1\rangle X-\langle X,T_1\rangle Y).$$
Hence the Codazzi equation
for $\mathbb{E}_2$ is satisfied by $(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$.
To prove that the quadruple $(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$
satisfies the compatibility equations \eqref{conditionT1}
and \eqref{conditionT2} for $\mathbb{E}_2$, it suffices to prove that
\begin{equation} \label{rotationshape}
\mathrm{S}_2-\tau_2\mathrm{J}=e^{\theta\mathrm{J}}(\mathrm{S}_1-\tau_1\mathrm{J}).
\end{equation}
Using the expression of $\mathrm{S}_2$, equation \eqref{rotationshape}
is equivalent to
\begin{equation} \label{rotationshape2}
H_2\mathrm{I}-\tau_2\mathrm{J}=e^{\theta\mathrm{J}}(H_1\mathrm{I}-\tau_1\mathrm{J}).
\end{equation}
We notice that this is a purely algebraic condition: the shape
operators are not involved anymore.
We consider a local direct orthonormal frame and we will
identify the operators with their matrix in this frame.
Then we have
$$\mathrm{J}=\left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right).$$ Then equation \eqref{rotationshape2}
is equivalent to
$$\left\{\begin{array}{ccc}
H_2 & = & H_1\cos\theta+\tau_1\sin\theta, \\
\tau_2 & = & \tau_1\cos\theta-H_1\sin\theta.
\end{array}\right.,$$
i.e., it is equivalent to equation \eqref{phase}. This proves
the first assertion of the theorem.
Conversely, if $(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$ satisfies the compatibility
equations for $\mathbb{E}_2$, then the compatibility equations
\eqref{conditionT1} for $(\mathrm{d} s^2,\mathrm{S}_1,T_1,\nu)$ and
$(\mathrm{d} s^2,\mathrm{S}_2,T_2,\nu)$ imply that
\eqref{rotationshape} holds at every point where $\nu\neq 0$.
If there exists a point where $\nu\neq 0$, this implies that
\eqref{phase} holds.
\end{proof}
\begin{thm} \label{sisters}
Let $\mathbb{E}_1$ and $\mathbb{E}_2$ be two $3$-dimensional homogeneous manifolds
with $4$-dimensional isometry groups, of base curvatures $\kappa_1$
and $\kappa_2$ and bundle curvatures $\tau_1$ and $\tau_2$
respectively, and such that
$$\kappa_1-4\tau_1^2=\kappa_2-4\tau_2^2.$$
Let $\xi_1$ and $\xi_2$ be the vertical vector fields of $\mathbb{E}_1$ and
$\mathbb{E}_2$ respectively.
Let $\Sigma$ be a simply connected
Riemann surface and let $x_1:\Sigma\to\mathbb{E}_1$ be
a conformal constant mean curvature $H_1$ immersion with
$H_1^2\geqslant\tau_2^2-\tau_1^2$.
Let $N_1$ be the induced normal (compatible with the orientation
of $\Sigma$). Let
$\mathrm{S}_1$ be the symmetric operator on $\Sigma$ induced by the shape
operator of $x_1(\Sigma)$ associated to the normal $N_1$. Let $T_1$
be the vector field on $\Sigma$ such that $\mathrm{d} x_1(T_1)$ is the
projection of $\xi_1$ onto $\mathrm{T}(x_1(\Sigma))$. Let
$\nu=\langle N_1,\xi_1\rangle$.
Let $H_2\in\mathbb{R}$ such that
$$\tau_1^2+H_1^2=\tau_2^2+H_2^2.$$
Let $\theta\in\mathbb{R}$ such that
$$\tau_2+iH_2=e^{i\theta}(\tau_1+iH_1).$$
Then there exists a conformal immersion
$x_2:\Sigma\to\mathbb{E}_2$ such that:
\begin{enumerate}
\item the metrics induced on $\Sigma$ by $x_1$ and $x_2$ are the same,
\item the symmetric operator on $\Sigma$ induced by the shape operator
of $x_2(\Sigma)$ is $e^{\theta\mathrm{J}}(\mathrm{S}_1-H_1\mathrm{I})+H_2\mathrm{I}$,
\item $\xi_2=\mathrm{d} x_2(e^{\theta\mathrm{J}}T_1)+\nu N_2$ where
$N_2$ is the unit normal to $x_2$.
\end{enumerate}
Moreover, this immersion $x_2$ is unique up to isometries of $\mathbb{E}_2$
preserving the orientations of both the fibers and the base of the
fibration, and it has constant mean curvature $H_2$.
The immersions $x_1$ and $x_2$ are called sister immersions. The
number $\theta$ is called the phase of $(x_1,x_2)$.
\end{thm}
This means that there exists an isometric
correspondence between CMC $H_1$ simply connected surfaces
in $\mathbb{E}_1$ and CMC $H_2$ simply connected surfaces in $\mathbb{E}_2$.
\begin{proof}
Let $\mathrm{d} s^2$ be the metric on $\Sigma$ induced by $x_1$. Then $(\mathrm{d}
s^2,\mathrm{S}_1,T_1,\nu)$ satisfies the compatibility equations for
$\mathbb{E}_1$. Thus, by proposition \ref{correspondence}, the quadruple
$(\mathrm{d} s^2,\mathrm{S}_2,e^{\theta\mathrm{J}}T_1,\nu)$
with $\mathrm{S}_2=e^{\theta\mathrm{J}}(\mathrm{S}_1-H_1\mathrm{I})+H_2\mathrm{I}$ also does. Thus by
theorem \ref{isometry} there exists an immersion $x_2$
satisfying properties 1, 2, and 3, and this immersion is unique up to
isometries of $\mathbb{E}_2$
preserving the orientations of both the fibers and the base of the
fibration.
Moreover, we have
$\tr\mathrm{S}_2=2H_2$, i.e., the immersion
$x_2$ has mean curvature $H_2$.
\end{proof}
\begin{figure}[htbp]
\begin{center}
\input{sisters.pstex_t}
\caption{The correspondence of the sister surfaces}
\label{figuresisters}
\end{center}
\end{figure}
Figure \ref{figuresisters} helps visualizing which classes of CMC surfaces
are related by the sister surface correspondence. We start from a CMC
surface in some homogeneous $3$-manifold.
Then we can go horizontally on
the graph. We can go to the left until reaching a manifold with $\tau=0$;
in this case the absolute mean curvature
$|H|$ increases. We can go to the right until reaching
$H=0$; in this case $|H|$ decreases.
A particularily interesting case is when $\mathbb{E}_1$ is the Heisenberg space
$\mathrm{Nil}_3$ with its standard metric ($\kappa_1=0$, $\tau_1=\frac12$) and
$\mathbb{E}_2=\mathbb{H}^2\times\mathbb{R}$ ($\kappa_2=-1$, $\tau_2=0$). Then CMC $H_1$
surfaces in $\mathrm{Nil}_3$ correspond isometrically
to CMC $H_2$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ with
$H_2^2=H_1^2+\frac14$. In particular we have the following corollary.
\begin{cor} \label{sistersheisenberg}
There exists an isometric correspondence with phase $\theta=\frac\pi2$
between simply connected
minimal surfaces in the Heisenberg space $\mathrm{Nil}_3$ and
simply connected
CMC $\frac12$ surfaces in $\mathbb{H}^2\times\mathbb{R}$.
\end{cor}
The fact that $\theta=\frac\pi2$ suggests that this correspondence
looks like the conjugate cousin correspondence between
minimal surfaces in $\mathbb{R}^3$ and CMC $1$ surfaces in $\mathbb{H}^3$
(\cite{bryant}, \cite{umehara}). This correpondence
has nice geometric properties, and is useful to construct
CMC $1$ surfaces in $\mathbb{H}^3$ with some prescribed
geometric properties starting from a solution of a Plateau
problem in $\mathbb{R}^3$ (see for example
\cite{karcher}, \cite{troisdroites}).
In particular, if a minimal surface $\Sigma_1$ in $\mathrm{Nil}_3$ contains an
ambient geodesic $\gamma$, then the normal curvature of $\gamma$
vanishes, and so
$$0=\langle\gamma',\mathrm{S}_1\gamma'\rangle
=\langle\gamma',-\mathrm{J}\mathrm{S}_2\gamma'+\frac12\mathrm{J}\gamma'\rangle
=-\langle\gamma',\mathrm{J}\mathrm{S}_2\gamma'\rangle.$$
This means that $\mathrm{S}\gamma'$ is colinear to $\gamma'$, i.e.,
$\gamma$ is a geodesic line of curvature in the
sister CMC $\frac12$ surface in $\mathbb{H}^2\times\mathbb{R}$.
We describe two examples of sister CMC $\frac12$ surfaces in $\mathbb{H}^2\times\mathbb{R}$
of minimal surfaces in $\mathrm{Nil}_3$. We will use the exponential coordinates
given in section \ref{heisenberg} (with $\tau=\frac12$).
We will denote between parentheses ( )
the coordinates of a vector in the coordinate frame
$(\partial_x,\partial_y,\partial_z)$, and between brackets [ ]
the coordinates of a vector in the canonical frame $(E_1,E_2,E_3)$;
with these notations one has
$$\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)=\left[\begin{array}{c}
a \\
b \\
\frac12(ya-xb)+c
\end{array}\right].$$
\begin{example}[vertical plane]
A vertical plane ${\mathcal P}$ in $\mathrm{Nil}_3$ is a flat minimal surface (but not totally
geodesic). A conformal parametrisation is
$$\varphi:(u,v)\mapsto\left(\begin{array}{c}
v \\
0 \\
u
\end{array}\right).$$
We have $$\varphi_u=E_3,\quad\varphi_v=E_1,\quad
N=E_2,$$
and so $$\nu=0,$$
$$\langle T,\partial_u\rangle=\langle\xi,\varphi_u\rangle
=1,$$
$$\langle T,\partial_v\rangle=\langle\xi,\varphi_v\rangle
=0,$$
i.e., $$T=\partial_u.$$
We also have
$$\bar\nabla_{\varphi_u}N=\frac12E_1=\frac12\varphi_u,\quad
\bar\nabla_{\varphi_v}N=\frac12E_3=\frac12\varphi_v,$$
so in the direct orthonormal frame $(\partial_u,\partial_v)$
we have $$\mathrm{S}=-\frac12\left(\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right).$$
We now show that the CMC $\frac12$ sister in $\mathbb{H}^2\times\mathbb{R}$ of ${\mathcal P}$
is the product ${\mathcal H}\times\mathbb{R}$ where
${\mathcal H}$ is a horocycle in $\mathbb{H}^2$. We will use the upper half-plane model
for $\mathbb{H}^2$. Then $\mathbb{H}^2\times\mathbb{R}=\{(x,y,z)\in\mathbb{R}^3;y>0\}$ and the
metric is $\mathrm{d} s^2=\frac1{y^2}(\mathrm{d} x^2+\mathrm{d} y^2)+\mathrm{d} z^2$. We consider
the direct orthonormal frame $(E_1,E_2,E_3)$ defined by
$E_1=y\partial_x$, $E_2=y\partial_y$, $E_3=\partial_z$; it satisfies
$\bar\nabla_{E_1}E_1=E_2$, $\bar\nabla_{E_1}E_2=-E_1$, and the
other derivatives vanish. For ${\mathcal H}$, we can choose the curve of
equation $y=1$ in $\mathbb{H}^2$. A conformal parametrization of ${\mathcal H}\times\mathbb{R}$
is
$$\tilde\varphi:(u,v)\mapsto\left(\begin{array}{c}
-u \\
1 \\
v
\end{array}\right).$$
We have $$\tilde\varphi_u=-E_1,\quad\tilde\varphi_v=E_3,\quad
N=E_2,$$
and so $$\tilde\nu=0,\quad\tilde T=\partial_v.$$
We also have
$$\bar\nabla_{\tilde\varphi_u}N=E_1=-\tilde\varphi_u,\quad
\bar\nabla_{\tilde\varphi_v}N=0,$$
so in the direct orthonormal frame $(\partial_u,\partial_v)$
we have $$\tilde\mathrm{S}=\left(\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}\right).$$
Hence, $\tilde\varphi$ induces on $\mathbb{R}^2$ the same metric as $\varphi$,
and we have $\tilde\nu=\nu$, $\tilde T=\mathrm{J} T$ and
$\tilde\mathrm{S}=\mathrm{J}\mathrm{S}+\frac12\mathrm{I}$, so
$\tilde\varphi$ is the sister immersion of $\varphi$.
The vertical lines in ${\mathcal P}$ are mapped to horizontal horocycles
in ${\mathcal H}\times\mathbb{R}$, and horizontal lines in ${\mathcal P}$ are mapped to
vertical lines in ${\mathcal H}\times\mathbb{R}$.
\end{example}
\begin{example}[surface of equation $z=0$]
The surface ${\mathcal A}$ of equation $z=0$ in the exponential coordinates is
a minimal surface in $\mathrm{Nil}_3$ which is invariant by rotation about the
$z$-axis (but it is not invariant by any translation; see
\cite{mercuri}). We consider
the following parametrisation:
$$\varphi:(u,v)\mapsto\left(\begin{array}{c}
u\cos v \\
u\sin v \\
0
\end{array}\right),$$
for $u>0$ (the origin in ${\mathcal A}$ is excluded).
We have $$\varphi_u=\left(\begin{array}{c}
\cos v \\
\sin v \\
0
\end{array}\right)=\left[\begin{array}{c}
\cos v \\
\sin v \\
0
\end{array}\right],$$
$$\varphi_v=\left(\begin{array}{c}
-u\sin v \\
u\cos v \\
0
\end{array}\right)=\left[\begin{array}{c}
-u\sin v \\
u\cos v \\
-\frac12u^2
\end{array}\right],$$
so $$\langle\varphi_u,\varphi_u\rangle=1,$$
$$\langle\varphi_v,\varphi_v\rangle=u^2\left(1+\frac{u^2}4\right),$$
$$\langle\varphi_u,\varphi_v\rangle=0.$$
The unit normal vector is
$N=\frac{\varphi_u\times\varphi_v}{||\varphi_u\times\varphi_v||}$;
we compute that $$\nu=\frac1{\sqrt{1+\frac{u^2}4}}.$$
A direct orthonormal frame $(e_1,e_2)$ is given by
$$e_1=\partial_u,\quad e_2=\frac1{u\sqrt{1+\frac{u^2}4}}\partial_v.$$
We compute that $$T=-\frac u{2\sqrt{1+\frac{u^2}4}}\partial_v.$$
We now show that the CMC $\frac12$ sister in $\mathbb{H}^2\times\mathbb{R}$ of ${\mathcal A}$
is the CMC $\frac12$ graph ${\mathcal B}$ of theorem D in \cite{nellicmc}. This
surface ${\mathcal B}$ is also invariant by rotation about a vertical axis.
If we take for $\mathbb{H}^2$ the Poincar\'e unit disk model, then ${\mathcal B}$ is
the graph of the function $(x,y)\mapsto\frac2{\sqrt{1-x^2-y^2}}$.
We will use the Lorentzian for $\mathbb{H}^2\times\mathbb{R}$, i.e.,
$$\mathbb{H}^2\times\mathbb{R}=\{(x^0,x^1,x^2,x^3)\in\mathbb{L}^3\times\mathbb{R};
-(x^0)^2+(x^1)^2+(x^2)^2=-1,x_0>0\}$$ with the restriction of the
quadratic form $-(\mathrm{d} x^0)^2+(\mathrm{d} x^1)^2
+(\mathrm{d} x^2)^2+(\mathrm{d} x^3)^2$. In this model, we consider
the map
$$\tilde\varphi:(u,v)\mapsto\left(\begin{array}{c}
1+\frac{u^2}2 \\
u\sqrt{1+\frac{u^2}4}\cos v \\
u\sqrt{1+\frac{u^2}4}\sin v \\
2\sqrt{1+\frac{u^2}4}
\end{array}\right),$$
for $u>0$. We can check that it is a parametrization of ${\mathcal B}$ minus
the origin (using that the correspondence between the Poincar\'e model
and the Lorentzian model is given by $x+iy=\frac{x^1+ix^2}{1+x^0}$,
$z=x^3$). We have
$$\tilde\varphi_u=\frac1{\sqrt{1+\frac{u^2}4}}
\left(\begin{array}{c}
u\sqrt{1+\frac{u^2}4} \\
1+\frac{u^2}2\cos v \\
1+\frac{u^2}2\sin v \\
\frac u2
\end{array}\right),\quad
\tilde\varphi_v=\left(\begin{array}{c}
0 \\
-u\sqrt{1+\frac{u^2}4}\sin v \\
u\sqrt{1+\frac{u^2}4}\cos v \\
0
\end{array}\right),$$
so $$\langle\tilde\varphi_u,\tilde\varphi_u\rangle=1,$$
$$\langle\tilde\varphi_v,\tilde\varphi_v\rangle
=u^2\left(1+\frac{u^2}4\right),$$
$$\langle\tilde\varphi_u,\tilde\varphi_v\rangle=0,$$
so $\tilde\varphi$ induces the same metric as $\varphi$.
We compute that $$\tilde T=\frac u{2\sqrt{1+\frac{u^2}4}}e_1
=\mathrm{J} T.$$
Thus we also have $\tilde\nu^2=\nu^2$. Moreover, $\tilde\varphi_u$
points outwards and $\tilde\varphi_v$
points in the counter-clockwise direction, so the normal $\tilde N$
points up, i.e., $\tilde\nu>0$. So we get $$\tilde\nu=\nu.$$
It remains to check that $\tilde\mathrm{S}=\mathrm{J}\mathrm{S}+\frac12\mathrm{I}$.
Since $\nu\neq 0$, the
compatibility equations \eqref{conditionT1} for $\varphi$ and
$\tilde\varphi$ imply that
$\tilde\mathrm{S}=\mathrm{J}(\mathrm{S}-\frac12\mathrm{J})=\mathrm{J}\mathrm{S}+\frac12\mathrm{I}$.
Hence $\tilde\varphi$ is the sister immersion of $\varphi$.
The straight lines in ${\mathcal A}$ passing through the origin are mapped
to the generatrices of ${\mathcal B}$, which are lines of curvatures lying in
vertical planes. Thus the symmetries of ${\mathcal B}$ with respect to these
vertical planes correspond to the symmetries of ${\mathcal A}$ with respect
to the straight lines passing through the origin.
\end{example}
\begin{example}[CMC rotational spheres]
The sister of the CMC $H_1$ rotational sphere in $\mathrm{Nil}_3$ is the
CMC $\sqrt{H_1^2+\frac14}$ rotational sphere in $\mathbb{H}^2\times\mathbb{R}$. Indeed,
the sister of this sphere is a possibly immersed CMC
sphere in $\mathbb{H}^2\times\mathbb{R}$, which is necessarily rotational by a
theorem of Abresch and Rosenberg (\cite{abresch}).
\end{example}
\begin{rem}
CMC $H$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ have very different properties
when $H\leqslant\frac12$ and when $H>\frac12$; for example compact
embedded CMC
$H$ surfaces exist only for $H>\frac12$. The reader can refer for
example to \cite{nellicmc}. An explanation is that CMC $H$ surfaces
in $\mathbb{H}^2\times\mathbb{R}$ arise from minimal surfaces in a Berger sphere when
$H>\frac12$, in $\mathrm{Nil}_3$ when $H=\frac12$, and in a space $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$ when
$H<\frac12$.
\end{rem}
\begin{rem}
When $\kappa-4\tau^2=0$, the sister relation is the composition
of the classical cousin relation between the round $3$-spheres and $\mathbb{R}^3$
and of the conjugation by a phase $\theta$ in the associate
family. The hyperbolic $3$-space
does not appear in this classification since it is not a fibration
over a $2$-manifold of constant curvature.
\end{rem}
\begin{rem}
A classical problem in the theory of minimal surfaces is the
question of the existence of minimal isometric deformations of
a given minimal surface.
The compatibility equations show that an associated family of a
given minimal surface (i.e., a one-parameter family of minimal isometric
deformation of this surface obtained by rotating the shape operator)
in a homogeneous $3$-manifold
$\mathbb{E}$ when $\tau\neq 0$ cannot be obtained in a simple way as in
$\mathbb{S}^3$, $\mathbb{R}^3$, $\mathbb{H}^3$, $\mathbb{S}^2\times\mathbb{R}$ or $\mathbb{H}^2\times\mathbb{R}$
(see \cite{codazzi}). Indeed, if the quadruple
$(\mathrm{d} s^2,\mathrm{S},T,\nu)$ satisfies
the compatibility equations for $\mathbb{E}$, then, in general, the quadruple
$(\mathrm{d} s^2,e^{\theta\mathrm{J}}\mathrm{S},e^{\theta\mathrm{J}}T,\nu)$ where
$\theta\in\mathbb{R}\setminus2\pi\mathbb{Z}$ does not. The question of the existence
of the associate family for minimal surfaces in $\mathbb{E}$ when $\tau\neq 0$
remains open.
\end{rem}
\subsection{Twin immersions}
In this section we will study the special case of sister immersions lying
in the same homogeneous $3$-manifold. They necessarily have
opposite mean curvatures.
\begin{thm} \label{twins}
Let $\mathbb{E}$ be a homogeneous $3$-manifold with a $4$-dimensional
isometry group, of base curvature $\kappa$ and bundle curvature $\tau$.
Let $\xi$ be its vertical vector field.
Let $\Sigma$ be a simply connected
Riemann surface and let $x:\Sigma\to\mathbb{E}$ be
a conformal constant mean curvature $H\neq 0$ immersion.
Let $N$ be the induced normal (compatible with the orientation
of $\Sigma$). Let
$\mathrm{S}$ be the symmetric operator on $\Sigma$ induced by the shape
operator of $x(\Sigma)$ associated to the normal $N$. Let $T$
be the vector field on $\Sigma$ such that $\mathrm{d} x(T)$ is the
projection of $\xi$ onto $\mathrm{T}(x(\Sigma))$. Let
$\nu=\langle N,\xi\rangle$. Let $$\theta=-2\arctan\frac H\tau.$$
Then there exists a unique conformal immersion
$\hat x:\Sigma\to\mathbb{E}$ such that:
\begin{enumerate}
\item the metrics induced on $\Sigma$ by $x$ and $\hat x$ are the same,
\item the symmetric operator on $\Sigma$ induced by the shape operator
of $\hat x(\Sigma)$ is $\tilde\mathrm{S}=e^{\theta\mathrm{J}}(\mathrm{S}-H\mathrm{I})-H\mathrm{I}
=e^{\theta\mathrm{J}}(\mathrm{S}-\tau\mathrm{J})+\tau\mathrm{J}$,
\item $\xi=\mathrm{d}\hat x(e^{\theta\mathrm{J}}T)+\nu\hat N$ where
$\hat N$ is the unit normal to $\hat x$.
\end{enumerate}
Moreover, this immersion $\hat x$ is unique up to isometries of $\mathbb{E}$
preserving the orientations of both the fibers and the base of the
fibration, and it has constant mean curvature $-H$.
It is called the twin immersion of the immersion $x$.
\end{thm}
\begin{proof}
This is a particular case of theorem \ref{sisters} with
$\mathbb{E}_1=\mathbb{E}_2=\mathbb{E}$, $\tau_1=-\tau_2=\tau$, $H_1=-H_2=H$.
It sufficies to check that the phase $\theta$
satisfies $\tau-iH=e^{i\theta}(\tau+iH)$.
The equivalence of the two expressions of $\tilde\mathrm{S}$ is a
consequence of \eqref{rotationshape2}.
\end{proof}
We notice that when $\tau\to 0$, then
$\theta\to\pi$, i.e., $\tilde T\to-T$, and also $\tilde\mathrm{S}\to-\mathrm{S}$.
This limit corresponds to
the image of the initial surface by a horizontal symmetry in
$\mathbb{M}^2(\kappa)\times\mathbb{R}$.
Moreover, we notice that the twin surface of a multigraph (over a part of
the base of the fibration) is also a multigraph
(since a surface is a multigraph if and only if $\nu$ does not vanish).
This suggests that the twin surface could be used to get an
Alexandrov reflection-type principle
in homogeneous manifolds with non-vanishing bundle curvature, since there is no
Alexandrov reflection principle (see \cite{alexandrov})
in these manifolds (the horizontal and
vertical ``symmetries'' are not isometries). Such an Alexandrov reflection
principle would be very useful for the theory of CMC surfaces
in homogeneous manifolds, in particular for proving that any closed
embedded CMC surface in the Heisenberg space or in $\widetilde{\mathrm{PSL}_2(\mathbb{R})}$
is a rotational sphere (this was proved
for CMC surfaces in $\mathbb{R}^3$, $\mathbb{H}^3$, a $3$-hemisphere, $\mathbb{H}^2\times\mathbb{R}$ and
a $2$-hemisphere cross $\mathbb{R}$ using the Alexandrov reflection principle).
We now give some examples of twin surfaces in the Heisenberg space
$\mathrm{Nil}_3$ with its standard metric (i.e., $\kappa=0$, $\tau=\frac12$). We
will use the exponential coordinates described in section
\ref{heisenberg}. Figueroa, Mercuri and Pedrosa classified CMC surfaces
in $\mathrm{Nil}_3$ invariant by a one-parameter family of translations or
rotations (see \cite{mercuri}; note that in their article the mean
curvature is defined as the trace of the shape operator, whereas in this
paper it is defined as the half of the trace). We will compute the
twin surfaces of these examples.
We will denote between parentheses ( )
the coordinates of a vector in the coordinate frame
$(\partial_x,\partial_y,\partial_z)$, and between brackets [ ]
the coordinates of a vector in the canonical frame $(E_1,E_2,E_3)$.
\begin{example}[translational tubes] \label{tube}
Let $H>0$. The map
$$\varphi:(u,v)\mapsto\left(\begin{array}{c}
u \\
\frac{\cos v}{2H} \\
u\frac{\cos v}{4H}+\frac1{4H}f(v)
\end{array}\right),$$ with
$$f(v)=\sqrt{1+\frac{\cos^2v}{4H^2}}\sin v
+\frac{1+4H^2}{2H}\arcsin\left(\frac{\sin v}{\sqrt{1+4H^2}}\right),$$
for $(u,v)\in\mathbb{R}^2$, is a CMC $H$ immersion defining
a surface which is invariant by
horizontal translations in the $x$-direction. This surface is an annulus,
and it is a bigraph over a part of the minimal surface of equation
$z=\frac{xy}2$; moreover it is ``symmetric'' with respect to this
minimal surface.
We have $$\varphi_u=\left(\begin{array}{c}
1 \\
0 \\
\frac{\cos v}{4H}
\end{array}\right)=\left[\begin{array}{c}
1 \\
0 \\
\frac{\cos v}{2H}
\end{array}\right],$$
$$\varphi_v=\left(\begin{array}{c}
0 \\
-\frac{\sin v}{2H} \\
-u\frac{\sin v}{4H}+\frac1{4H}f'(v)
\end{array}\right)=\left[\begin{array}{c}
0 \\
-\frac{\sin v}{2H} \\
\frac1{4H}f'(v)
\end{array}\right],$$
$$f'(v)=2\cos v\sqrt{1+\frac{\cos^2v}{4H^2}},$$
and so
$$\langle\varphi_u,\varphi_u\rangle=1+\frac{\cos^2v}{4H^2},$$
$$\langle\varphi_v,\varphi_v\rangle
=\frac1{4H^2}\left(1+\frac{\cos^4v}{4H^2}\right).$$
$$\langle\varphi_u,\varphi_v\rangle
=\frac{\cos^2v}{4H^2}\sqrt{1+\frac{\cos^2v}{4H^2}}.$$
The unit normal vector is given by
$N=\frac{\varphi_u\times\varphi_v}{||\varphi_u\times\varphi_v||}$;
we compute that $$\nu=-\frac{\sin v}
{\sqrt{1+\frac{\cos^4v}{4H^2}}}.$$
We have $$\langle T,\partial_u\rangle
=\langle\xi,\varphi_u\rangle=\frac{\cos v}{2H},$$
$$\langle T,\partial_v\rangle
=\langle\xi,\varphi_v\rangle
=\frac{\cos v}{2H}\sqrt{1+\frac{\cos^2v}{4H^2}},$$
We notice that $\nu(u_1,-v)=-\nu(u_2,v)$ for all $(u_1,u_2,v)$.
This indicates that the twin immersion could be an
orientation-reversing reparametrization of the surface.
For this reason we set
$$\tilde\varphi:(u,v)\mapsto\varphi(u+h(v),-v)
=\left(\begin{array}{c}
u+h(v) \\
\frac{\cos v}{2H} \\
(u+h(v))\frac{\cos v}{4H}-\frac1{4H}f(v)
\end{array}\right)$$ where $h$ is a function.
This is a CMC $-H$ immersion defining globally
the same surface as $\varphi$.
We compute that $$\tilde\varphi_u=\left[\begin{array}{c}
1 \\
0 \\
\frac{\cos v}{2H}
\end{array}\right],\quad
\tilde\varphi_v=\left[\begin{array}{c}
h'(v) \\
-\frac{\sin v}{2H} \\
h'(v)\frac{\cos v}{2H}-\frac1{4H}f'(v)
\end{array}\right],$$
and so
$$\langle\tilde\varphi_u,\tilde\varphi_u\rangle
=1+\frac{\cos^2v}{4H^2},$$
\begin{eqnarray*}
\langle\tilde\varphi_v,\tilde\varphi_v\rangle & = &
\left(1+\frac{\cos^2v}{4H^2}\right)h'(v)^2
-\frac{\cos^2v}{2H^2}h'(v)\sqrt{1+\frac{\cos^2v}{4H^2}} \\
& & +\frac1{4H^2}\left(1+\frac{\cos^4v}{4H^2}\right),
\end{eqnarray*}
$$\langle\tilde\varphi_u,\tilde\varphi_v\rangle
=\left(1+\frac{\cos^2v}{4H^2}\right)h'(v)
-\frac{\cos^2v}{4H^2}\sqrt{1+\frac{\cos^2v}{4H^2}}.$$
Thus $\tilde\varphi$ induces on $\mathbb{R}^2$ the same metric as $\varphi$ if
and only if
$$h'(v)=\frac{\cos^2v}{2H^2\sqrt{1+\frac{\cos^2v}{4H^2}}}.$$
We now assume that this condition is satisfied; we can also assume that
$h(0)=0$. The function $h$ is increasing. We have
$$\tilde\nu=\nu,$$
$$\langle\tilde T,\partial_u\rangle
=\langle\xi,\tilde\varphi_u\rangle=\frac{\cos v}{2H},$$
$$\langle\tilde T,\partial_v\rangle
=\langle\xi,\tilde\varphi_v\rangle
=\frac{\cos v}{2H\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(\frac{\cos^2v}{4H^2}-1\right).$$
The direct orthonormal frame $(e_1,e_2)$ obtained from the frame
$(\partial_u,\partial_v)$ by the Gram-Schmidt process satisfies
$$e_1=\frac{\partial_u}{||\partial_u||},$$
$$e_2=
\frac{-\langle\partial_u,\partial_v\rangle\partial_u
+||\partial_u||^2\partial_v}
{||\partial_u||
\sqrt{||\partial_u||^2||\partial_u||^2
-\langle\partial_u,\partial_v\rangle^2}}.$$
A computation gives
$$||\partial_u||^2||\partial_u||^2
-\langle\partial_u,\partial_v\rangle^2
=\frac1{4H^2}\left(1+\frac{\cos^2v}{4H^2}\right).$$
Thus we get
$$e_1=\frac1{\sqrt{1+\frac{\cos^2v}{4H^2}}}\partial_u,$$
$$e_2=-\frac{\cos^2v}{2H\sqrt{1+\frac{\cos^2v}{4H^2}}}\partial_u
+2H\partial_v.$$
So we have $$T=\frac{\cos v}{\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(\frac1{2H}e_1+e_2\right),$$
$$\tilde T=\frac{\cos v}{\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(\frac1{2H}e_1-e_2\right).$$
Let $\theta=-2\arctan(2H)$. Then we have
$$\cos\theta=\frac{1-4H^2}{1+4H^2},\quad
\cos\theta=-\frac{4H}{1+4H^2}.$$
Since $\mathrm{J} e_1=e_2$ and $\mathrm{J} e_2=-e_1$, we get
$$e^{\theta\mathrm{J}}T=\tilde T.$$
Finally, the compatibility equation \eqref{conditionT1} implies that
$$\tilde S=e^{\theta\mathrm{J}}(\mathrm{S}-\tau\mathrm{J})+\tau\mathrm{J}$$
at points where $\nu\neq 0$; and by continuity this identity
holds everywhere. This proves that $\tilde\varphi$ is the twin
immersion of $\varphi$.
Thus the translational tube is \emph{globally} invariant by the
twin relation, but it is \emph{not pointwise} invariant: the
correspondence is
$$\varphi(u,v)\mapsto\varphi(u+h(v),-v).$$
Geometrically, this correspondence maps a point of the tube to the
other point of the tube lying in the same fiber
and then translates it by $h(v)$ in the $x$-direction.
In particular, the closed curve $v\mapsto\varphi(u_0,v)$ is
mapped to the curve $v\mapsto\varphi(u_0+h(v),-v)$,
which is \emph{not} closed.
\end{example}
\begin{example}[rotational spheres] \label{sphere}
Let $H>0$. The map
$$\varphi:(u,v)\mapsto\left(\begin{array}{c}
\frac1H\cos u\cos v \\
\frac1H\sin u\cos v \\
\frac1{2H}f(v)
\end{array}\right),$$ with
$f$ as in example \ref{tube},
for $(u,v)\in\mathbb{R}\times(-\frac{\pi}2,\frac{\pi}2),$
is a CMC $-H$ immersion defining a rotational sphere minus the top and
bottom points (the normal of the immersion points outside whereas the
mean curvature vector points inside).
It is a bigraph over a part of the minimal surface of equation
$z=0$; moreover it is ``symmetric'' with respect to this
minimal surface.
We have $$\varphi_u=\frac1H\left[\begin{array}{c}
-\sin u\cos v \\
\cos u\cos v \\
-\frac1{2H}\cos^2v
\end{array}\right],\quad
\varphi_v=\frac1H\left[\begin{array}{c}
-\cos u\sin v \\
-\sin u\sin v \\
\frac1{2}f'(v)
\end{array}\right],$$
and so
$$\langle\varphi_u,\varphi_u\rangle
=\frac{\cos^2v}{H^2}\left(1+\frac{\cos^2v}{4H^2}\right),$$
$$\langle\varphi_v,\varphi_v\rangle
=\frac1{H^2}\left(1+\frac{\cos^2v}{4H^2}\right),$$
$$\langle\varphi_u,\varphi_v\rangle
=-\frac{\cos^3v}{2H^3}\sqrt{1+\frac{\cos^2v}{4H^2}}.$$
The unit normal vector is given by
$N=\frac{\varphi_u\times\varphi_v}{||\varphi_u\times\varphi_v||}$;
we compute that $$\nu=\frac{\sin v}
{\sqrt{1+\frac{\cos^4v}{4H^2}}}.$$
We have $$\langle T,\partial_u\rangle
=\langle\xi,\varphi_u\rangle=-\frac{\cos^2v}{2H^2},$$
$$\langle T,\partial_v\rangle
=\langle\xi,\varphi_v\rangle
=\frac{\cos v}{H}\sqrt{1+\frac{\cos^2v}{4H^2}}.$$
Let $$\tilde\varphi:(u,v)\mapsto\varphi(u+g(v),-v)
=\left(\begin{array}{c}
\frac1H\cos(u+g(v))\cos v \\
\frac1H\sin(u+g(v))\cos v \\
-\frac1{2H}f(v)
\end{array}\right)$$
where $g$ is a function. This is a CMC $H$ immersion defining globally
the same surface as $\varphi$.
We compute that $$\tilde\varphi_u=\frac1H\left[\begin{array}{c}
-\sin(u+g(v))\cos v \\
\cos(u+g(v))\cos v \\
-\frac1{2H}\cos^2v
\end{array}\right],$$
$$\tilde\varphi_v=\frac1H\left[\begin{array}{c}
-\cos(u+g(v))\sin v-g'(v)\sin(u+g(v))\cos v \\
-\sin(u+g(v))\sin v+g'(v)\cos(u+g(v))\cos v \\
-\frac12f'(v)-\frac1{2H}g'(v)\cos^2v
\end{array}\right],$$
and thus
$\tilde\varphi$ induces on $\mathbb{R}\times(\frac{\pi}2,\frac{\pi}2)$
the same metric as $\varphi$ if
and only if
$$g'(v)=-\frac{\cos v}{H\sqrt{1+\frac{\cos^2v}{4H^2}}}.$$
We now assume that this condition is satisfied; we can also assume that
$g(0)=0$. The function $g$ is odd and $2\pi$-periodic. We have
$$\tilde\nu=\nu,$$
$$\langle\tilde T,\partial_u\rangle
=\langle\xi,\tilde\varphi_u\rangle=-\frac{\cos^2v}{2H^2},$$
$$\langle\tilde T,\partial_v\rangle
=\langle\xi,\tilde\varphi_v\rangle
=\frac{\cos v}{H\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(\frac{\cos^2v}{4H^2}-1\right).$$
The direct orthonormal frame $(e_1,e_2)$ obtained from the frame
$(\partial_u,\partial_v)$ by the Gram-Schmidt process satisfies
$$e_1=\frac H{\cos v\sqrt{1+\frac{\cos^2v}{4H^2}}}\partial_u,$$
$$e_2=-\frac{\cos v}{2\sqrt{1+\frac{\cos^2v}{4H^2}}}\partial_u
+H\partial_v.$$
So we have $$T=\frac{\cos v}{\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(-\frac1{2H}e_1+e_2\right),$$
$$\tilde T=\frac{\cos v}{\sqrt{1+\frac{\cos^2v}{4H^2}}}
\left(-\frac1{2H}e_1-e_2\right).$$
Let $\theta=2\arctan(2H)$. We check as in example \ref{tube}
that $$e^{\theta\mathrm{J}}T=\tilde T,$$
$$\tilde S=e^{\theta\mathrm{J}}(\mathrm{S}-\tau\mathrm{J})+\tau\mathrm{J}.$$
This proves that $\tilde\varphi$ is the twin
immersion of $\varphi$.
Thus the rotational sphere is \emph{globally} invariant by the
twin relation, but it is \emph{not pointwise} invariant: the
correspondence is
$$\varphi(u,v)\mapsto\varphi(u+g(v),-v).$$
Geometrically, this correspondence maps a point of the sphere to the
other point of the sphere lying in the same fiber
and then rotates it by the angle $g(v)$ about the $z$-axis.
In particular, the circle $v\mapsto\varphi(u_0,v)$ lying in a
vertical plane is
mapped to the curve $v\mapsto\varphi(u_0+g(v),-v)$,
which is closed but not contained in a vertical plane.
\end{example}
\bibliographystyle{alpha}
|
{
"timestamp": "2005-03-23T18:45:51",
"yymm": "0503",
"arxiv_id": "math/0503500",
"language": "en",
"url": "https://arxiv.org/abs/math/0503500"
}
|
\section{The Hidden Subgroup Problem}
One of the principal quantum algorithmic paradigms is the use of the abelian
Fourier transform to discover a function's hidden periodicities. In the
examples relevant to quantum computing, an oracle function $f$ defined on an
abelian group $G$ has ``hidden periodicity'' if there is a ``hidden'' subgroup
$H$ of $G$ so that $f$ is precisely invariant under translation by $H$ or,
equivalently, $f$ is constant on the cosets of $H$ and takes distinct values on
distinct cosets. The \emph{hidden subgroup problem} is the problem of
determining the subgroup $H$ from such a function. Algorithms for these problems
typically adopt the approach detailed below, called \emph{Fourier sampling}
\cite{BernsteinV93}:
\begin{description}
\item[Step 1.] Prepare two registers, the first in a uniform superposition over
the elements of a group $G$ and the second with the value zero, yielding the
state
$$
\psi_1 = \frac{1}{\sqrt{|G|}} \sum_{g \in G} \ket{g} \otimes \ket{0} \enspace.
$$
\item[Step 2.] Calculate (or if it is an oracle, query) the function $f$ defined on
$G$ and XOR it with the second register. This entangles the two registers and
results in the state
$$
\psi_2 = \frac{1}{\sqrt{|G|}} \sum_{g \in G} \ket{g} \otimes \ket{f(g)} \enspace.
$$
\item[Step 3.] Measure the second register. This produces a uniform
superposition over one of $f$'s level sets, i.e., the set of group elements $g$
for which $f(g)$ takes the measured value $f_0$. As the level sets of $f$ are
the cosets of $H$, this puts the first register in a uniform distribution over
superpositions on one of those cosets, namely $cH$ where $f(c)=f_0$ for some
$f_0$. Moreover, it disentangles the two registers, resulting in the state
$\psi_3 \otimes |f_0\rangle$ where
$$
\psi_3 = \frac{1}{\sqrt{|H|}} \ket{cH} = \frac{1}{\sqrt{|H|}} \; \sum_{h \in H} \ket{ch} \enspace.
$$
Alternately, since the value $f_0$ we observe has no bearing on the algorithm,
we can use the formulation in which the
environment, rather than the user, measures $f$. In that case, tracing over $f$
yields a mixed state with density matrix
\[ \frac{1}{[G:H]} \sum_{f} \ket{\psi_3} \bra{\psi_3}
= \frac{1}{|G|} \sum_c \ket{cH} \bra{cH} \enspace ,
\]
i.e., a classical mixture consisting of one pure state
$\psi_3$ for each coset. Kuperberg refers to this as the {\em coherent}
hidden subgroup problem~\cite{Kuperberg03}.
\item[Step 4.] Carry out the quantum Fourier transform on $\psi_3$ and measure the
result.
\end{description}
For example, in Simon's algorithm \cite{Simon97},
the ``ambient'' group $G$ over which the Fourier transform is performed
is ${\mathbb Z}_2^n$, $f$ is an oracle with the promise that $f(x)=f(x+y)$ for some $y$,
and $H=\{0,y\}$ is a subgroup of order $2$. In Shor's factoring algorithm
\cite{Shor97} $G$ is the group ${\mathbb Z}_n^*$ where $n$ is the number we wish to
factor, $f(x) = r^x \bmod n$ for a random $r < n$, and $H$ is the subgroup of
${\mathbb Z}_n^*$ of index order$(r)$. (However, since $|{\mathbb Z}_n^*|$ is unknown, Shor's
algorithm actually performs the transform over ${\mathbb Z}_q$ where $q$ is polynomially
bounded by $n$; see \cite{Shor97} or \cite{HalesH99,HalesH00}.)
These are all abelian instances of the \emph{hidden subgroup problem} (HSP).
Interest in \emph{nonabelian} versions of the HSP evolved from the relation to
the elusive \textsc{Graph Automorphism} problem: it would be sufficient to solve
efficiently the HSP over the symmetric group $S_n$ in order to have an
efficient quantum algorithm for graph automorphism (see, e.g.,
Jozsa~\cite{Jozsa00} for a review). This was the impetus behind the development
of the first nonabelian quantum Fourier transform~\cite{beals} and is,
in part, the reason that the nonabelian HSP has remained such an active
area of research in quantum algorithms.
In general, we will say that the HSP for a family of groups $G$ has a
\emph{Fourier sampling} algorithm if a procedure similar to that outlined above
works. Specifically, the algorithm prepares a superposition of the form
$$
\frac{1}{\sqrt{|H|}} \sum_{h \in H} |ch\rangle,
$$
over a random coset $cH$ of the hidden subgroup $H$,
computes the (quantum) Fourier transform of this state, and measures the result.
After a polynomial number of such trials, a polynomial amount of classical
computation, and, perhaps, a polynomial number of classical queries to the
function $h$ to confirm the result, the algorithm produces a set of generators
for the subgroup $H$ with high probability.
When $G$ is abelian, measuring a state's Fourier transform has a clear meaning:
one observes the frequency $\chi$ with probability equal to the squared
magnitude of the transform at that frequency. In the case where $G$ is a
\emph{nonabelian} group, however, it is necessary to select bases for each
representation of $G$ to perform full measurement. (We explain this in more
detail below.) The subject of this article is the relationship between this
choice of basis and the information gleaned from the measurement: are some bases
more useful for computation than others?
Since we are typically interested in exponentially large groups, we will take
the size of our input to be $n = \log |G|$. Throughout, ``polynomial'' means
polynomial in $n$, and thus polylogarithmic in $|G|$.
\subsection{Nonabelian Hidden Subgroup Problems}
Although a number of interesting results have been obtained on the nonabelian HSP,
the groups for which efficient solutions are known remain woefully
few. On the positive side, Roetteler and Beth~\cite{RoettelerB98} give
an algorithm for the wreath product ${\mathbb Z}_2^k\; \wr\; {\mathbb Z}_2$. Ivanyos, Magniez, and
Santha~\cite{IvanyosMS01} extend this to the more general case of semidirect
products $K \ltimes {\mathbb Z}_2^k$ where $K$ is of polynomial size, and also give an
algorithm for groups whose commutator subgroup is of polynomial size. Friedl,
Ivanyos, Magniez, Santha and Sen solve a problem they call Hidden Translation,
and thus generalize this further to what they call ``smoothly solvable'' groups:
these are solvable groups whose derived series is of constant length and whose
abelian factors are each the direct product of an abelian group of bounded
exponent and one of polynomial size~\cite{FriedlIMSS02}. (See also
Section~\ref{sec:closure}.)
In another vein, Ettinger and H{\o}yer~\cite{EttingerH98} show that the HSP is
solvable for the dihedral groups in an \emph{information-theoretic} sense;
namely, a polynomial number of quantum queries to the function oracle gives
enough information to reconstruct the subgroup, but the best known
reconstruction algorithm takes exponential time. More generally, Ettinger,
H{\o}yer and Knill~\cite{EttingerHK04} show that for \emph{arbitrary} groups the
HSP can be solved information-theoretically with a finite number of quantum
queries. However, their algorithm calls for a quantum measurement for each
possible subgroup, and since there might be $|G|^{\Omega(\log |G|)}$ of these,
it requires an exponential number of quantum operations.
Our current understanding of the HSP, then, divides group families into three
classes.
\begin{description}
\label{classification}
\item[I.] \textbf{Fully Reconstructible.} Subgroups of a family of groups $\{
G_i \}$ are \emph{fully reconstructible} if the HSP can be solved with high
probability by a quantum circuit of size polynomial in $\log |G_i|$.
\item[II.] \textbf{Information-Theoretically Reconstructible.}
Subgroups of a family of groups $\{ G_i \}$ are
\emph{information-theoretically reconstructible} if the solution to
the HSP for $G_i$ is determined information-theoretically by the
fully measured result of a quantum circuit of size polynomial in
$\log |G_i|$.
\item[III.] \textbf{Quantum Information-Theoretically
Reconstructible.} Subgroups of a family of groups $\{ G_i \}$ are
\emph{quantum information-theoretically reconstructible} if the
solution to the HSP for $G_i$ is determined by the quantum state
resulting from a quantum circuit of polynomial size in $\log |G_i|$,
in the sense that there exists a positive operator-valued
measurement (POVM) that yields the subgroup $H$ with constant
probability, but where it may or may not be possible to carry out
this POVM with a quantum circuit of polynomial size.
\end{description}
In each case, the quantum circuit has oracle access to a function $f : G \to S$,
for some set $S$, with the property that $f$ is constant on each left coset of a
subgroup $H$, and distinct on distinct cosets.
In this language, then, subgroups of abelian groups are fully reconstructible,
while the result of \cite{EttingerHK04} shows that subgroups of arbitrary groups
are quantum information-theoretically reconstructible. The other work cited
above has labored to place specific families of nonabelian groups into the more
algorithmically meaningful classes I and II.
\subsection{Nonabelian Fourier transforms}
In this section we give a brief review of nonabelian Fourier analysis, but only
to the extent needed to set down notation. We refer the reader
to~\cite{Serre77} for a more complete exposition.
Fourier analysis over a finite abelian group $A$ expresses a
function $\phi: A \to {\mathbb C}$ as a linear combination of homomorphisms $\chi: A \to
{\mathbb C}$. If $A = {\mathbb Z}_p$, for example, these are the familiar basis functions
$\chi_t: z \mapsto \omega_p^{tz}$,
where $\omega_p$ denotes the $p$th root of unity ${\rm e}^{2 \pi i/p}$. Any function
$\phi : A \to {\mathbb C}$ can be uniquely expressed as a linear combination of these
$\chi_t$, and this change of basis is the Fourier transform.
When $G$ is a nonabelian group, however, this same procedure cannot work: in
particular, there are not enough homomorphisms of $G$ into ${\mathbb C}$ to
span the space of all ${\mathbb C}$-valued functions on $G$. To define a sufficient
basis, the representation theory of finite groups considers more general
functions, namely homomorphisms from $G$ into groups of unitary matrices.
A \emph{representation} of a finite group $G$ is a homomorphism $\rho: G \to
\textrm{U}(d)$, where $\textrm{U}(d)$ denotes the group of unitary $d \times d$ matrices (with
entries from ${\mathbb C}$); the dimension $d = d_\rho$ is referred to as the
\emph{dimension} of $\rho$. If $\rho: G \to \textrm{U}(d)$ is a representation, a
subspace $W$ of ${\mathbb C}^d$ is said to be \emph{invariant} if $\rho(g)(W) \subset W$
for all $g$. A representation is said to be \emph{irreducible} if the only
invariant subspaces are the trivial subspace ${\mathbb C}^d$ and $\{ \vec{0} \}$.
For a function $\phi: G \to {\mathbb C}$ and an irreducible representation $\rho$,
$\hat{\phi}(\rho)$ denotes \emph{the Fourier transform of $\phi$ at $\rho$} and
is defined by
\[
\hat{\phi}(\rho) = \sqrt{\frac{d_\rho}{|G|}} \,\sum_g \phi(g)\rho(g) \enspace.
\]
Note that $\phi$ takes values in ${\mathbb C}$ while $\rho$ is matrix-valued. It is a
fact that a finite group has a finite number of distinct irreducible
representations (up to isomorphism), and the \emph{Fourier transform} of a
function $\phi: G \to {\mathbb C}$ is the collection of matrices $\hat{\phi}(\rho)$,
taken over all distinct irreducible representations $\rho$.
Fixing a group $G$ and a subgroup $H$, we shall focus primarily on the functions
$\varphi_{c}: G \to {\mathbb C}$ of form
$$
\varphi_{c}(g) = \begin{cases} \frac{1}{\sqrt{|H|}} & \text{if}\;g \in cH,\\
0 & \text{otherwise,}
\end{cases}
$$
corresponding to the first register of the state $\psi_3$ resulting from Step 3
above, which is a uniform superposition over the coset $cH$. The Fourier
transform of such a function is then
\[
\widehat{\varphi_{c}}(\rho) = \sqrt{\frac{d_\rho}{|G||H|}} \,\rho(c) \cdot
\sum_{h \in H} \rho(h)\enspace.
\]
Note, as above, that $\widehat{\varphi_{c}}(\rho)$ is a $d_\rho \times d_\rho$
matrix.
For any subgroup $H$, the sum $\sum_h \rho(h)$ is precisely $|H|$
times a projection operator (see, e.g., \cite{HallgrenRT00}); we write
$$
\sum_h \rho(h) = |H| \,\pi_H(\rho) \enspace.
$$
With this notation, we can express $\widehat{\varphi_{c}}(\rho)$ as
$\sqrt{n_\rho}
\,\rho(c) \cdot \pi_H(\rho)$ where $n_\rho = d_\rho |H|/|G|$. For a $d \times d$
matrix $M$,
we let $\norm{M}$ denote the matrix norm given by
$$
\norm{M}^2 = \textbf{tr} \left( M^\dag{} M \right) = \sum_{ij} \abs{M_{ij}}^2,
$$
where $M^{\dag}$ denotes the conjugate transpose of $M$. Then the
probability that we observe the representation $\rho$ is
\begin{align*}
\norm{\widehat{\varphi_c}(\rho)}^2 &= \norm{\sqrt{n_\rho} \,\rho(c)
\,\pi_H(\rho)}^2\\
&= n_\rho \norm{\pi_H(\rho)}^2 \\
&= n_\rho \,\textbf{rk}\; \pi_H(\rho)\enspace,
\end{align*}
where $\textbf{rk}\; \pi_H(\rho)$ denotes the rank of the projection operator
$\pi_H(\rho)$.
See~\cite{HallgrenRT00} for more discussion.
\subsection{Weak vs.\ strong sampling and the choice of basis}
Hallgren, Russell, and Ta-Shma~\cite{HallgrenRT00} show that by measuring only
the \emph{names} of representations---the so-called \emph{weak standard method}
in the terminology of~\cite{GrigniSVV01}---it is possible to reconstruct normal
subgroups (and thus solve the HSP for \emph{Hamiltonian groups}, all of whose
subgroups are normal). More generally, this method reconstructs the
\emph{normal core} of a subgroup, i.e., the intersection of all its conjugates.
On the other hand, they show that this is insufficient to solve Graph
Automorphism, since even in an information-theoretic sense this method cannot
distinguish between the trivial subgroup of $S_n$ and subgroups of order 2 consisting
of the identity and an involution.
Therefore, in order to solve the HSP for nonabelian groups, we need to measure
not just the name of the representation we are in, but also the row and column.
In order for this measurement to be well-defined, we need to choose a basis for
$U(d_\rho)$ for each $\rho$. Grigni, Schulman, Vazirani and
Vazirani~\cite{GrigniSVV01} call this the \emph{strong standard method}. They
show that if we measure using a uniformly \emph{random} basis, then trivial and
non-trivial subgroups are still information-theoretically indistinguishable.
However, they leave open the question of whether the strong standard method with
a clever choice of basis, rather than a random one, allows us to solve the HSP
in nonabelian groups, yielding an algorithm for Graph Automorphism.
Indeed, in representation theory certain bases are ``preferred'', and have very
special computational properties, because they give the matrices $\rho(g)$ a
highly structured or sparse form. In particular, Moore, Rockmore and
Russell~\cite{MRR03} showed that so-called \emph{adapted bases} yield highly
efficient algorithms for the quantum Fourier transform.
\subsection{Contributions of this paper}
As stated above, \cite{HallgrenRT00} and~\cite{GrigniSVV01} leave an important
open question: namely, whether there are cases where the \emph{strong standard
method}, with the proper choice of basis, offers an advantage over a simple
abelian transform or the \emph{weak standard method}. We settle this question
in the affirmative. Our results deal primarily with the \emph{$q$-hedral}
groups, i.e., semidirect products of the form ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ where $q \mid
(p-1)$, and in particular the \emph{affine} groups $A_p \cong {\mathbb Z}_p^* \ltimes
{\mathbb Z}_p$.
We begin in Section~\ref{sec:full-reconstruction} by focusing on full
reconstructibility. We define the \emph{Hidden Conjugate Problem} (HCP) as
follows: given a group $G$, a non-normal subgroup $H$, and a function which is
promised to be constant on the cosets of some conjugate $bHb^{-1}$ of $H$ (and
distinct on distinct cosets), determine the subgroup $bHb^{-1}$ by finding an
element $c \in G$ so that $cHc^{-1} = bHb^{-1}$. We adopt the above
classification (fully, information-theoretically, quantum information-
theoretically) for this problem in the natural way. Then we show that given a
subgroup of sufficiently small (but still exponentially large) index, hidden
conjugates in $A_p$ are fully reconstructible (Theorem~\ref{thm:hcp}). This
almost immediately implies that, for prime $q = (p-1)/{\rm polylog}(p)$, subgroups of
the $q$-hedral groups ${\mathbb Z}_q \ltimes Z_p$ are fully reconstructible
(Theorem~\ref{thm:hsp}).
Section~\ref{sec:info} concerns itself with information-theoretic
reconstructibility. We generalize the results of Ettinger and H{\o}yer on the
dihedral group and show that hidden conjugates of any subgroup are information-
theoretically reconstructible in the affine groups, and more generally the $q$-
hedral groups for all $q$ (Theorem~\ref{thm:infohcp}). We then show that we can
identify the order, and thus the conjugacy class, of a hidden subgroup, and this
implies that all subgroups of the affine and $q$-hedral groups are information-
theoretically reconstructible (Theorem~\ref{thm:infohsp}).
The results of Sections~\ref{sec:full-reconstruction} and~\ref{sec:info} rely
crucially on measuring the high-dimensional representations of the affine and
$q$-hedral groups in a well-chosen basis, namely an \emph{adapted} basis that
respects the group's subgroup structure. We show in Section~\ref{sec:random}
that we lose information-theoretic reconstructibility if we measure using a
\emph{random} basis instead. Specifically, we need an exponential number of
measurements to distinguish conjugates of small subgroups of $A_p$. This
establishes for the first time that the strong standard method is indeed
stronger than measuring in a random basis: some bases provide much more
information about the hidden subgroup than others.
For some nonabelian groups, the HSP can be solved with a ``forgetful''
approach, where we erase the group's nonabelian structure and perform
an abelian Fourier transform instead. In Section~\ref{sec:abelian} we
show that this is not the case for the affine groups: specifically, if
we treat $A_p$ as a direct product rather than a semidirect one, its
conjugate subgroups become indistinguishable.
As an application, in Section~\ref{sec:shift} we consider \emph{hidden
shift} problems. In the setting we consider, one must reconstruct a
``hidden shift'' $s \in {\mathbb Z}_p$ from an oracle $f_s(x)=f(x-s)$, where $f$ is
any function that is constant on the (multiplicative) cosets of a
known multiplicative subgroup of ${\mathbb Z}_p^*$. These functions have been
studied in some depth for their pseudorandom properties, and several
instances have been suggested as cryptographically strong pseudorandom
generators. By associating $f_s$ with its isotropy subgroup, and using
our reconstruction algorithm to find that subgroup, we give an
efficient quantum algorithm for the hidden shift problem in the case
where $f(x)$ is a function of $x$'s multiplicative order mod $r$ for
some $r={\rm polylog}(p)$. This generalizes the work of van Dam, Hallgren,
and Ip~\cite{vanDamHI03}, who give an algorithm for hidden shift
problems in the case where $f$ is precisely a multiplicative character.
Finally, in Section~\ref{sec:closure} we show that the set of
groups for which the HSP can be solved in polynomial time has the
following closure property: if ${\mathcal H} = \{ H_n \}$ is a family of
groups for which we can efficiently solve the HSP and ${\mathcal K} =
\{ K_n \}$ is a family of groups for which $|K_n| = {\rm polylog} | H_n |$,
we can also efficiently solve the HSP for the family $\{ G_n \}$, where
each $G_n$ is any extension of $K_n$ by $H_n$.
This subsumes the results of~\cite{HallgrenRT00} on Hamiltonian
groups, and also those of~\cite{IvanyosMS01} on groups with commutator
subgroups of polynomial size.
\section{The affine and $q$-hedral groups}
Let $A_p$ be the \emph{affine group}, consisting of ordered pairs $(a,b) \in
{\mathbb Z}_p^* \times {\mathbb Z}_p$, where $p$ is prime, under the multiplication rule
$(a_1,b_1) \cdot (a_2, b_2) = (a_1a_2, b_1 + a_1b_2)$. $A_p$ can be viewed as
the set of affine functions $f_{(a,b)} : {\mathbb Z}_p \to {\mathbb Z}_p$ given by $f_{(a,b)} : x
\mapsto ax + b$ where multiplication in $A_p$ is given by function composition.
Structurally, $A_p$ is a semidirect product ${\mathbb Z}_p^* \ltimes {\mathbb Z}_p \cong {\mathbb Z}_{p-1}
\ltimes {\mathbb Z}_p$. Its subgroups are as follows:
\begin{itemize}
\item Let $N \cong {\mathbb Z}_p$ be the normal subgroup of size $p$ consisting of
elements of the form $(1,b)$.
\item Let $H \cong {\mathbb Z}_p^* \cong {\mathbb Z}_{p-1}$ be the non-normal subgroup of size $p-
1$ consisting of the elements of the form $(a,0)$. Its conjugates $H^b = (1,b)
\cdot H \cdot (1,-b)$ consist of elements of the form $(a,(1-a)b)$. In the
action on ${\mathbb Z}_p$, $H^b$ is the stabilizer of $b$.
\item More generally, if $a \in {\mathbb Z}_p^*$ has order $q$, let $N_q \cong {\mathbb Z}_q
\ltimes {\mathbb Z}_p$ be the normal subgroup consisting of all elements of the form
$(a^t,b)$, and let $H_a$ be the non-normal subgroup $H_a = \langle (a,0)
\rangle$ of size $q$. Then $H_a$ consists of the elements of the form $(a^t,0)$
and its conjugates $H_a^b=(1,b) \cdot H_a \cdot (1,-b)$ consist of the elements
of the form $(a^t,(1-a^t)b)$.
\end{itemize}
Construction of the representations of $A_p$ requires that we fix a generator
$\gamma$ of ${\mathbb Z}_p^*$. Define $\log: {\mathbb Z}_p^* \to {\mathbb Z}_{p-1}$ to be the isomorphism
$\log \gamma^t = t$. Let $\omega_p$ denote the $p$th root of unity ${\rm e}^{2 \pi
i/p}$. Then $A_p$ has $p-1$ one-dimensional representations $\sigma_s$, which
are simply the representations of ${\mathbb Z}_p^* \cong {\mathbb Z}_{p-1}$, given by
$\sigma_t((a,b)) = \omega_{p-1}^{t \log a}$. Moreover, it has one $(p-1)$-
dimensional representation $\rho$ given by
\begin{equation}
\label{eq:adaptedbasis}
\rho((a,b))_{j,k} = \left\{ \begin{array}{ll}
\omega_p^{bj} & k = aj \bmod p \\
0 & \mbox{otherwise}
\end{array} \right. ,
\; 1 \leq j,k < p
\enspace ,
\end{equation}
where the indices $i$ and $j$ are elements of ${\mathbb Z}_p^*$.
See~\cite[\S8.2]{Serre77} for a more detailed discussion.
Similarly, given prime $p$ and $q \mid p-1$, we consider the \emph{$q$-hedral
groups}, namely semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$. These embed in $A_p$
a natural way: namely, as the normal subgroups $N_q$ defined above. The
\emph{dihedral} groups are the special case where $q=2$.
The representations of ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ include the $q$ one-dimensional
representations of ${\mathbb Z}_q$ given by $\sigma_\ell((a^t,b)) = \omega_q^{\ell t}$
for $\ell \in {\mathbb Z}_q$, and $(p-1)/q$ distinct $q$-dimensional representations
$\rho_k$ given by
\[ \rho_k((a^u,b))_{s,t} = \left\{ \begin{array}{ll}
\omega_p^{k a^s b} & t = s + u \bmod q \\
0 & \mbox{otherwise}
\end{array} \right.\enspace ,
\]
for each $0 \leq s,t < q$. Here $k$ ranges over the elements of ${\mathbb Z}_p^* / {\mathbb Z}_q$,
or, to put it differently, $k$ takes values in ${\mathbb Z}_p^*$ but $\rho_k$ and
$\rho_{k'}$ are equivalent if $k$ and $k'$ are in the same coset of $\langle a
\rangle$.
The representations of the affine and $q$-hedral groups are related as follows.
The restriction of the $(p-1)$-dimensional representation $\rho$ of $A_p$ to
$N_q$ is reducible, and is isomorphic to the direct product of the $\rho_k$.
Moreover, if we measure $\rho$ in a \emph{Gel'fand-Tsetlin} basis such
as~\eqref{eq:adaptedbasis} which is \emph{adapted} to the tower of subgroups
\[ A_p > N_q > {\mathbb Z}_p > \{1\} \enspace , \]
then $\rho$ becomes block-diagonal, with $(p-1)/q$ blocks of size $q$, and these
blocks are exactly the representations $\rho_k$ of $N_q$. (See~\cite{MRR03} for
an introduction to adapted bases and their uses in quantum computation.) We
will use this fact in Sections~\ref{sec:info} and~\ref{sec:random} below.
The affine and $q$-hedral groups are \emph{metacyclic} groups, i.e., extensions
of a cyclic group ${\mathbb Z}_p$ by a cyclic group ${\mathbb Z}_q$. In~\cite{Hoyer97}, H{\o}yer
shows how to perform the nonabelian Fourier transform over such groups (up to an
overall phase factor) with a polynomial, i.e., ${\rm polylog}(p)$, number of
elementary quantum operations.
\section{Full reconstructibility}
\label{sec:full-reconstruction}
In this section we show that conjugates of sufficiently large subgroups of the
affine groups are fully reconstructible in polynomial time. For some values of
$p$ and $q$, this allows us to completely solve the Hidden Subgroup Problem for
the $q$-hedral group ${\mathbb Z}_q \ltimes {\mathbb Z}_p$.
\begin{theorem} \label{thm:hcp}
Let $p$ be prime and let $a \in {\mathbb Z}_p^*$ have order $q = (p-1) / {\rm polylog}(p)$.
Then the hidden conjugates of $H_a$ in $A_p$ are fully reconstructible.
\end{theorem}
\begin{proof}
Consider first the maximal non-normal subgroup $H = H_\gamma$ (where $\gamma$ is
a generator of ${\mathbb Z}_p^*$). Carrying out steps 1 through 3 of the Fourier sampling
procedure outlined in the introduction results in a state $\psi_3$ over the
group $G$ which is uniformly supported on a random left coset of the conjugate
$H^b = bHb^{-1}$. Using the procedure of~\cite{Hoyer97}, we now compute the
quantum Fourier transform of this state over $A_p$, in the
basis~\eqref{eq:adaptedbasis}. The associated projection operator is
$$
\pi_{H^b}(\rho)_{j,k} = \frac{1}{p-1} \;\omega_p^{b(j-k)} \enspace,
$$
for $1 \leq j,k < p$. This is a circulant matrix of rank one. More
specifically, every column is some root of unity times the vector
$$
(u_b)_j = \frac{1}{p-1} \;\omega_p^{bj} \enspace,
$$
$1 \leq j < p$. This is also true of $\rho(c) \cdot \pi_{H^b}(\rho)$; since
$\rho(c)$ has one nonzero entry per column, left multiplying by $\rho(c)$ simply
multiplies each column of $\pi_{H^b}(\rho)$ by a phase. Note that in this case
$$
n_\rho = d_\rho |H|/|G| = (p-1)/p = 1-1/p \enspace,
$$
so that upon measurement the $(p-1)$-dimensional representation $\rho$ is
observed with overwhelming probability $1 - 1/p$.
Assuming that we observe $\rho$, we perform another change of basis: namely, we
Fourier transform each column by left-multiplying $\rho(cH)$ by $Q_{\ell,j} =
(1/\sqrt{p-1})\;\omega_{p-1}^{-\ell j}$. In terms of quantum operations, we are
applying the quantum Fourier transform over ${\mathbb Z}_{p-1}$ to the row register,
while leaving the column register unchanged. We can now infer $b$ by measuring
the frequency $\ell$. Specifically, we observe a given value of $\ell$ with
probability
\begin{equation}
P(\ell)
= \left| \frac{1}{p-1} \sum_{j=1}^{p-1} \omega_p^{bj} \omega_{p-1}^{-\ell j}
\right|^2
= \frac{1}{(p-1)^2} \left| \sum_{j=1}^{p-1} {\rm e}^{2 i \theta j} \right|^2
= \frac{1}{(p-1)^2} \frac{\sin^2 (p-1) \theta}{\sin^2 \theta}
\end{equation}
where
\[ \theta = \left( \frac{b}{p} - \frac{\ell}{p-1} \right) \pi \enspace. \]
Now note that for any $b$ there is an $\ell$ such that $|\theta| \leq \pi/(2(p-
1))$. Since
$$
(2x/\pi)^2 \leq \sin^2 x \leq x^2
$$
for $|x| \leq \pi/2$, this gives $P(\ell) \geq (2/\pi)^2$.
Recall that the probability that we observed the $(p-1)$-dimensional
representation $\rho$ in the first place is $n_\rho = 1-1/p$. Thus if we measure
$\rho$, the column, and then $\ell$ and then guess that $b$ minimizes
$|\theta|$, we will be right $\Omega(1)$ of the time. This can be boosted to
high probability, i.e., $1-o(1)$, by repeating the experiment a polynomial
number of times.
Consider now the more general case, when the hidden subgroup is a conjugate of
the subgroup $H_a$ where $a$'s order $q$ is a proper divisor of $p-1$. Recall
that a given conjugate of $H_a$ consists of the elements of the form $(a^t,(1-
a^t)b)$. Then we have
\[
\pi_{H_a^b}(\rho)_{j,k} = \frac{1}{q} \left\{ \begin{array}{ll}
\omega_p^{b(j-k)} & k = a^t j \mbox{ for some } t \\
0 & \mbox{otherwise}
\end{array} \right.
\enspace,
\]
for $1 \leq j,k < p$.
In other words, the nonzero entries are those for which $j$ and $k$ lie in the
same coset of $\langle a \rangle \subset {\mathbb Z}_p^*$. The rank of this projection
operator is thus the number of cosets, which is the index $(p-1)/q$ of $\langle
a \rangle$ in ${\mathbb Z}_p^*$. Since $n_\rho$ is now $q/p$, we again observe $\rho$
with probability
$$
n_\rho \,\textbf{rk}\; \pi_{H_a}(\rho) = (p-1)/p = 1-1/p \enspace.
$$
Following the same procedure as before, we carry out a partial measurement on
the columns of $\rho$, and then Fourier transform the rows. After changing the
variable of summation from $t$ to $-t$ and adding a phase shift of ${\rm e}^{-i
\theta (p-1)}$ inside the $|\cdot|^2$, the probability we observe a frequency
$\ell$, assuming we find ourselves in the $k$th column, is
\begin{equation}
\begin{split}
\label{eq:other}
P(\ell) & =
\left| \frac{1}{\sqrt{q(p-1)}}
\,\sum_{t=0}^{q-1} \omega_p^{b(a^t k \bmod p)} \omega_{p-1}^{-\ell (a^t k
\bmod p)}
\right|^2
\\
& = \frac{1}{q(p-1)} \left|
\sum_{t=0}^{q-1} {\rm e}^{2 i \theta (a^t k \bmod p)}
\right|^2
\enspace.
\end{split}
\end{equation}
Now note that the terms in the sum are of the form ${\rm e}^{i \phi}$ where (assuming
w.l.o.g.\ that $\theta$ is positive)
$$
\phi \in [-\theta (p-1),\theta (p-1)]\enspace.
$$
If we again take $\ell$ so that $|\theta| \leq \pi/(2(p-1))$, then $\phi \in [-
\pi/2,\pi/2]$ and all the terms in the sum have nonnegative real parts. We will
obtain a lower bound on the real part of the sum by showing that a constant
fraction of the terms have $\phi \in (-\pi/3,\pi/3)$, and thus have real part
more than $1/2$. This is the case whenever $a^t k \in (p/6,5p/6)$, so it is
sufficient to prove the following lemma:
\begin{lemma}
Let $a$ have order $q = p/{\rm polylog}(p)$ in ${\mathbb Z}_p^*$, $p$ a prime.
Then at least $(1/3 - o(1)) q$ of the elements in
the coset $\langle a \rangle k$ are in the interval $(p/6,5p/6)$.
\end{lemma}
\noindent
\begin{proof}
We will prove this using \emph{Gauss sums}, which quantify the
interplay between the characters of ${\mathbb Z}_p$ and the characters of
${\mathbb Z}_p^*$. In particular, Gauss sums establish bounds on the
distribution of powers of $a$. Specifically, if $a$ has order $q$
in ${\mathbb Z}_p^*$ then for any integer $k \not\equiv 0 \bmod p$ we have
$$
\sum_{t = 0}^{q - 1} \omega_p^{a^t k} = O(p^{1/2}) = o(p) \enspace .
$$
(See \cite{KonyaginS99} and Appendix~\ref{appendix:gauss-sums}.)
Now suppose $s$ of the elements $x$ in $\langle a \rangle k$ are in the set
$(p/6,5p/6)$, for which ${\rm Re}\, \omega_p^x \geq -1$, and the other $q-s$ elements
are in $[0,p/6] \cup [5p/6,p)$, for which ${\rm Re}\, \omega_p^x \geq 1/2$. Thus we
have
$$
{\rm Re}\, \sum_{t = 0}^{q-1} \omega_p^{a^t k} \geq \,(q/2) - \,(3s/2).
$$
If $s \leq (1/3-\epsilon) q$ for any $\epsilon > 0$ this is $\Theta(q)$, a
contradiction.
\end{proof}
Now that we know that a fraction $1/3-\epsilon$ of the terms in~\eqref{eq:other}
have real part at least $1/2$ and the others have real part at least $0$, we can
take $\epsilon = 1/12$ (say) and write
\[
P(\ell) \geq \frac{1}{q(p-1)} \left( \frac{q}{8} \right)^2 = \frac{1}{64}
\frac{q}{p-1} = \frac{1}{{\rm polylog}(p)} \enspace.
\]
Thus we observe the correct frequency with at least polynomially small
probability; again this can be boosted to high probability by repetition.
\end{proof}
Theorem~\ref{thm:hcp} implies that we can completely solve the Hidden Subgroup
Problem for certain $q$-hedral groups.
\begin{theorem}
\label{thm:hsp}
Let $p$ and $q$ be prime with $q = (p-1)/{\rm polylog}(p)$. Then subgroups of the
$q$-hedral group ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ are fully reconstructible.
\end{theorem}
\begin{proof}
First, note that we can fully reconstruct $H$ if it is non-trivial and normal.
We do this by reconstructing the normal core of $H$,
\[
C(H) = \bigcap_{\gamma \in G} \gamma H \gamma^{-1}
\]
using the techniques of~\cite{HallgrenRT00} (the weak standard method). The
$q$-hedral groups have the special property that no non-normal subgroup contains
a non-trivial normal subgroup;
then $B$ is normal;
in particular, if $H$ is non-normal, then $C(H)$ is the trivial subgroup. Thus
by reconstructing $C(H)$, we either learn $H=C(H)$ or learn that $H$ is either
trivial or non-normal. Furthermore, if $H$ is trivial we will learn this by
checking our reconstruction against the oracle $f$ and finding that it is
incorrect. Therefore, it suffices to consider the non-normal subgroups.
If $q$ is prime, then the non-normal subgroups of ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ are all
conjugate to a single subgroup $K \cong {\mathbb Z}_q$, so the hidden subgroup problem
reduces to the hidden conjugate problem for $K$. While one can construct a
proof similar to that of Theorem~\ref{thm:hcp} directly for the $q$-hedral
groups, it is convenient to embed them in $A_p$ using the isomorphisms $N_q
\cong {\mathbb Z}_q \ltimes {\mathbb Z}_p$ and $H_a \cong K$ and appeal to Theorem~\ref{thm:hcp}.
Now suppose we have an oracle $f: {\mathbb Z}_q \times {\mathbb Z}_p \to S$. We extend this to an
oracle $f'$ on $A_p$ as follows. Choose a generator $\gamma \in {\mathbb Z}_p^*$ and one
of the $q-1$ elements $a \in {\mathbb Z}_p^*$ of order $q$, and let
\[ f': A_p \to S \times \langle a \rangle \]
where
\[ f'((a,b)) = \left( f\left(\Bigr( \Bigl\lfloor \frac{\log a}{(p-1)/q} \Bigr\rfloor, b\Bigr)\right) , a^q
\right) \]
recalling that $\log \gamma^t = t$. The second component of $f'$ serves to
distinguish the cosets of $N_q$ from each other, while the first component maps
each coset of $N_q$ to ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ with the element of ${\mathbb Z}_q$ written
additively, rather than multiplicatively. (This last step is not strictly
necessary---after all, we could have written the elements of $A_p$ in additive
form in the first place---but it can be carried out with Shor's algorithm for
the discrete logarithm~\cite{Shor97}.) This reduces the HCP for $K$ (and
therefore the HSP) on ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ to the HCP for $H_a$ on $A_p$,
completing the proof.
\end{proof}
As an example of Theorem~\ref{thm:hsp}, if $q$ is a \emph{Sophie Germain} prime,
i.e., one for which $p=2q+1$ is also a prime, we can completely solve the HSP
for ${\mathbb Z}_q \ltimes {\mathbb Z}_p$.
\section{Information-theoretic reconstructibility}
\label{sec:info}
In this section, we show that \emph{all} subgroups of the affine and $q$-hedral
groups, regardless of their size, are information-theoretically reconstructible.
We start by considering the hidden conjugate problem for subgroups $H_a =
\langle (a, 0) \rangle$ in $A_p$. Then in Theorem~\ref{thm:infohsp} we show
that we can identify the conjugacy class of a hidden subgroup, and therefore
the subgroup itself. This generalizes the results of Ettinger and
H{\o}yer~\cite{EttingerH98} who show information-theoretic reconstructibility
for the dihedral groups, i.e., the case $q=2$.
\begin{theorem}
\label{thm:infohcp}
Let $p$ be prime and let $a$ be any element of ${\mathbb Z}_p^*$. Then the hidden
conjugates of $H_a$ in $A_p$ are information-theoretically reconstructible.
\end{theorem}
\begin{proof}
Suppose $a$ has order $q$. Recall that $H_a$ and its conjugates $H_a^b$ are
maximal in the subgroup $N_q \cong {\mathbb Z}_q \ltimes {\mathbb Z}_p$. We wish to show that
there is a measurement whose outcomes, given two distinct values of $b$, have
large, i.e., $1/{\rm polylog}(p)$, total variation distance. First, we perform a
series of partial measurements as follows.
\begin{itemize}
\item[(i.)] Measure the name of the representation of $A_p$. If this is not
$\rho$ try again. Otherwise, continue;
\item[(ii.)] Measure the name of the representation $\rho_k$ of $N_q$ inside
$\rho$;
\item[(iii.)] Measure the column of $\rho_k$; and
\item[(iv.)] Perform a POVM with $q$ outcomes, in each of which $s$ is $u$ or
$u+1 \bmod q$ for some $u \in {\mathbb Z}_q$.
\end{itemize}
As in Theorem~\ref{thm:hcp}, we measure the $(p-1)$-dimensional representation
of $A_p$ in a chosen basis. Recall that in the adapted
basis~\eqref{eq:adaptedbasis} the restriction of $\rho$ to $N_q$ is block
diagonal, where the $(p-1)/q$ blocks are the $q$-dimensional representations
$\rho_k$ of $N_q$. Therefore, the projection operator $\pi_{H_a^b}(\rho)$ is
block-diagonal, and each of its blocks is one of the projection operators
$\pi_{H_a^b}(\rho_k)$. Summing $\rho_k$ over $H_a^b = \{(a^t,(1-a^t)b)\}$ gives
$$
\left(\pi_{H_a^b}(\rho_k)\right)_{s,t} = (1/q) \; \omega_p^{k(a^s-a^t)b}
$$
for $0 \leq s,t < q$. This is a matrix of rank 1, where each column (even after
left multiplication by $\rho_k(c)$) is some root of unity times the vector
$(u_k)_s = (1/q) \;\omega_p^{k a^s b}$. Since $n_\rho = q/p$, the probability
that we observe a particular $\rho_k$ is $q/p$. Since $\pi_{H_a^b}(\rho)$ has
$(p-1)/q$ blocks of this kind, it has rank $(p-1)/q$, and the total probability
that we observe $\rho$ is $(p-1)/p=1-1/p$ as before.
Then these four partial measurements determine $k$, remove the effect of the
coset, and determine that $s$ has one of two values, $u$ or $u+1$. Up to an
overall phase we can write this as a two-dimensional vector
\[ \frac{1}{\sqrt{2}}
\left( \! \begin{array}{c}
\omega_p^{k a^u b} \\
\omega_p^{k a^{u+1} b} \end{array} \! \right)\enspace.
\]
We now apply the Hadamard transform
$$
\frac{1}{\sqrt{2}} {\left( \! \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \! \right)}
$$
and measure $s$. The probability we observe that $s=u$ or $u+1$ is then $\cos^2
\theta$ and $\sin^2 \theta$ respectively, where $\theta = (k a^u (a-1) b
\pi)/p$. Now when we observe a $q$-dimensional representation, the $k$ we
observe is uniformly distributed over ${\mathbb Z}_p^* / {\mathbb Z}_q$, and when we perform the
POVM, the $u$ we observe is uniformly distributed over ${\mathbb Z}_q$. It follows that
the coefficient $m = k a^u (u-1)$ is uniformly distributed over ${\mathbb Z}_p^*$. For
any two distinct $b$, $b'$, the total variation distance is then
\[
\frac{1}{2(p-1)} \sum_{m \in {\mathbb Z}_p^*} \left( \left| \cos^2 \frac{\pi m b}{p} -
\cos^2 \frac{\pi m b'}{p} \right| +
\left| \sin^2 \frac{\pi m b}{p} - \sin^2 \frac{\pi m b'}{p} \right| \right)
\enspace .
\]
This we rewrite
\begin{eqnarray*}
& & \frac{1}{p-1} \sum_{m \in {\mathbb Z}_p^*} \left| \cos^2 \frac{\pi m b}{p} - \cos^2
\frac{\pi m b'}{p} \right|\\
& = & \frac{1}{2(p-1)} \sum_{m \in {\mathbb Z}_p} \left| \cos \frac{2 \pi m b}{p} - \cos
\frac{2 \pi m b'}{p} \right| \\
& \geq &\frac{1}{4(p-1)} \sum_{m \in {\mathbb Z}_p} \left( \cos \frac{2 \pi m b}{p} -
\cos \frac{2 \pi m b'}{p} \right)^2 \\
& = & \frac{p}{4(p-1)} > \frac{1}{4} \enspace.
\end{eqnarray*}
(Adding the $m=0$ term contributes zero to the sum in the second line. In the
third line we use the facts that $|x| \leq x^2/2$ for all $|x| \leq 2$, the
average of $\cos^2 x$ is $1/2$, and the two cosines have zero inner product.)
Since the total variation distance between any two distinct conjugates is
bounded below by a constant, we can
distinguish between the $p$ different conjugates with only $O(\log p) =
{\rm poly}(n)$ samples. Thus, hidden conjugates in $A_p$ are information-
theoretically reconstructible, completing the proof.
\end{proof}
\smallskip
By embedding the $q$-hedral groups in $A_p$ as in Theorem~\ref{thm:hsp}, we can
generalize Theorem~\ref{thm:infohcp} to the $q$-hedral groups (note that we do
not require here that $q$ is prime):
\begin{theorem}
\label{thm:infohcpq}
Let $p$ be prime and $q$ a divisor of $p-1$. The subgroups of the $q$-hedral
groups ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ are information-theoretically reconstructible.
\end{theorem}
We now wish to information-theoretically reconstruct all subgroups of the affine
and $q$-hedral groups. We can do this by first reconstructing which conjugacy
class they lie in, and then applying Theorems~\ref{thm:infohcp}
and~\ref{thm:infohcpq}.
\begin{theorem}
\label{thm:infohsp} Let $p$ be prime and $q$ a divisor of $p-1$. The subgroups
of the $q$-hedral groups ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ are information-theoretically
reconstructible. In particular, the subgroups of the affine groups $A_p =
{\mathbb Z}_{p}^* \ltimes {\mathbb Z}_p$ are information-theoretically reconstructible.
\end{theorem}
\begin{proof}
As in Theorem~\ref{thm:hsp}, we can (fully) reconstruct the normal subgroups of
${\mathbb Z}_q \ltimes {\mathbb Z}_p$, so it suffices to consider non-normal subgroups $H$.
Recall that in this case, $H$ is cyclic and $|H|$ is equal to the order of $a$,
where $H = \langle(a,b)\rangle$. Since there is a unique conjugacy class of
subgroups of each order, it suffices to determine $|H|$, at which point the
subgroup $H$ can be determined by Theorem~\ref{thm:infohcpq}.
Let the oracle be $f: {\mathbb Z}_q \ltimes {\mathbb Z}_p \to S$, and let $p_1^{\alpha_1}\ldots
p_k^{\alpha_k}$ be the prime factorization of $q$, in which case $k \leq \sum_i
\alpha_i = O(\log q)$. For each $i \in \{1, \ldots, k\}$ and each $\alpha
\in \{0, \ldots, \alpha_i \}$, we will determine if $p_i^{\alpha} \mid |H|$, and
taking the largest such $\alpha$ for each $i$ gives the prime factorization of
$|H|$.
To do this, for each $i \in [k]$ and $1 \leq \alpha \leq \alpha_i$, let
$\Upsilon_i^\alpha: {\mathbb Z}_{q} \ltimes {\mathbb Z}_p \to {\mathbb Z}_{q/p_i^{\alpha}}$ be the
homomorphism given by
$$
\Upsilon_i^\alpha: (a,b) \mapsto a^{p_i^{\alpha}}\enspace.
$$
Then let
$$
A_i^{\alpha_i} = \ker \Upsilon_i^\alpha = \{ \gamma \in {\mathbb Z}_q \ltimes {\mathbb Z}_p \mid
\gamma^{p_i^{\alpha_i}} = \mathbf{1} \} \enspace,
$$
where $\mathbf{1}$ denotes the identity element of ${\mathbb Z}_q \ltimes {\mathbb Z}_p$.
$A_i^{\alpha_i}$ is the subgroup of ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ consisting of all
elements whose orders are a multiple of $p_i^{\alpha}$. Consider now the
function
$$
f' : {\mathbb Z}_q \ltimes {\mathbb Z}_p \to S \times {\mathbb Z}_{q/p_i^\alpha}
$$
given by
\[
f'(\gamma) = \left( f(\gamma),\Upsilon_i^\alpha(\gamma) \right) \enspace .
\]
Observe that $f'$ is constant (and distinct) on the left cosets of $H \cap
A_i^{\alpha}$ and, furthermore, the subgroup $H \cap A_i^\alpha$ has order
$p^\alpha$ if and only if $p^\alpha$ divides the order of $a$. We may then
determine if $H \cap A_i^\alpha$ has order $p^\alpha$ by assuming that it does,
reconstructing $H$ with Theorem~\ref{thm:infohcpq} using $f'$ as the oracle, and
checking the result against the original oracle $f$. This allows us to determine
the prime factorization of $|H|$ as desired. Therefore, all subgroups of the
$q$-hedral groups ${\mathbb Z}_q \ltimes {\mathbb Z}_p$ are information-theoretically
reconstructible.
\end{proof}
\smallskip As in the dihedral case~\cite{EttingerH98}, we know of no
polynomial-time algorithm which can reconstruct the most likely $b$
from these queries. However, Kuperberg~\cite{Kuperberg03} gives a
quantum algorithm for the HSP in the dihedral group, and more
generally the hidden shift problem, that runs in subexponential
(${\rm e}^{O(\log^{1/2} p)}$) time. Since we can reduce the HSP on
${\mathbb Z}_q \ltimes {\mathbb Z}_p$ to a hidden shift problem by focusing on two
cosets of ${\mathbb Z}_p$, this algorithm applies to the $q$-hedral groups as
well.
\section{Random vs.\ adapted bases}
\label{sec:random}
In Theorems~\ref{thm:infohcp} and~\ref{thm:infohsp}, we measured the high-
dimensional representation $\rho$ in a specific basis which is adapted to the
subgroup structure of $A_p$ and the $q$-hedral groups. In contrast, we show in
this section that if we measure $\rho$ in a \emph{random} basis instead, then
for all but the largest values of $q$ we need an exponential number of
measurements in order to information-theoretically distinguish conjugate
subgroups from each other.
\begin{theorem}
Let $p$ be prime and let $a \in {\mathbb Z}_p^*$ have order $q$ where $q < p^{1-\epsilon}$
for some $\epsilon > 0$. Let $P_b(v)$ be the probability that we observe a basis
vector $v$ in the Fourier basis if the hidden subgroup is $H_a^b$. If we
measure $\rho$ in a random basis, then for any two $b, b'$, with high
probability the $L_1$ distance between these probability distributions is
exponentially small, i.e., there exists $\beta > 0$ such that
\[ \sum_v \left| P_b(v) - P_{b'}(v) \right| < p^{-\beta} \enspace . \]
Thus it takes an exponentially large number of measurements to distinguish the
conjugates $H_a^b$ and $H_a^{b'}$.
\end{theorem}
\begin{proof}
Since we observe the high-dimensional representation $\rho$ with probability $1-
1/p$, it suffices to consider the $L_1$ distance summed over the $d_\rho=p-1$
basis vectors of $\rho$. In fact, we will show that $P_b(v)$ is exponentially
close to the uniform distribution for all $b$.
Write $\pi = \pi_{H_a^b}(\rho)$. Then the probability we observe a given basis
vector $v$, conditioned on observing $\rho$, is
\[ P_b(v) = \frac{1}{\textbf{rk}\; \pi} \abs{\pi \cdot v}^2 \enspace . \]
If $v$ is uniformly random with norm $1$, the expectation of $\abs{\pi \cdot
v}_2^2$ is $(\textbf{rk}\; \pi)/d_\rho$, and so the expectation of $P_b(v)$ is
$1/d_\rho$. We will use the following lemma to show that when $\textbf{rk}\; \pi$ is
sufficiently large, $P_b(v)$ is tightly concentrated around this expectation.
\begin{lemma}
\label{lem:bound}
Let $\pi$ be a projection operator of rank $r$ in a $d$-dimensional space, and
let $v$ be a random $d$-dimensional vector of unit length. Then for all $0 <
\delta < 2$,
\[ \Pr\left[ \,\left| \abs{\pi \cdot v}_2^2 - \frac{r}{d} \right| > \delta
\frac{r}{d} \right]
< 4 {\rm e}^{-r \delta^2 / 48} \enspace . \]
\end{lemma}
\begin{proof}
We use an argument similar to~\cite{GrigniSVV01}. We can think of a random $d$-
dimensional complex vector $v$ as a random $2d$-dimensional real vector of the
same length, and we can think of this in turn as
\[ v_i = \frac{w_i}{\sum_{i=1}^{2d} w_i^2} \]
where the $w_i$ are independent Gaussian variables with zero mean and unit
variance. By choosing a basis in which $\pi$ projects onto the first $r$
(complex) components of $v$, we have
\[ \abs{\pi \cdot v}_2^2 = \frac{\sum_{i=1}^{2r} w_i^2}{\sum_{i=1}^{2d} w_i^2}
= \frac{r}{d} \frac{(1/2r) \sum_{i=1}^{2r} w_i^2}{(1/2d) \sum_{i=1}^{2d} w_i^2}
\enspace .
\]
We now use the following Chernoff bound, which can be derived from the moment
generating function. For any $t$, we have
\[
\Pr\left[ \,\left| \left( \frac{1}{t} \sum_{i=1}^t w_i^2 \right) - 1 \right| >
\epsilon \right]
< 2 \left[ (1+\epsilon)^{1/2} \,{\rm e}^{-\epsilon/2} \right]^t \enspace .
\]
For $|\epsilon| < 1/2$, we have $\ln (1+\epsilon) < \epsilon - \epsilon^2/3$ and this becomes
\begin{equation}
\label{eq:chernoff}
\Pr\left[ \,\left| \left( \frac{1}{t} \sum_{i=1}^t w_i^2 \right) - 1 \right| >
\epsilon \right]
< 2 {\rm e}^{-t \epsilon^2 / 6} \enspace .
\end{equation}
Now, for any $a,b$, if $|a/b - 1| > \delta$ where $\delta < 2$, then either $|a-
1| > \delta/4$ or $|b-1| > \delta/4$. Taking the union bound over these events
where $a = (1/2r) \sum_{i=1}^{2r} w_i^2$ and $b = (1/2d) \sum_{i=1}^{2d} w_i^2$,
setting $\epsilon = \delta/4$ and $t=2r \leq 2d$ in~\eqref{eq:chernoff} gives the
stated bound.
\end{proof}
Setting $d=d_\rho$ and $r = \textbf{rk}\; \pi$, Lemma~\ref{lem:bound} and the union
bound imply that, for any constant $A > \sqrt{48}$, if
\begin{equation}
\label{eq:delta}
\delta = A \sqrt{ \frac{\log d_\rho}{\textbf{rk}\; \pi}}
\end{equation}
then, with high probability, for all $d_\rho$ basis vectors $v$ we have
\[ \abs{ P_b(v) - \frac{1}{d_\rho} } < \frac{\delta}{d_\rho} \enspace . \]
Summing over all $v$, this implies that the $L_1$ distance between $P_b(v)$ and
the uniform distribution is at most $\delta$. Now recall that $\textbf{rk}\; \pi = (p-
1)/q$. If $q < p^{1-\epsilon}$, then $\textbf{rk}\; \pi > p^\epsilon$, and~\eqref{eq:delta}
gives $\delta < p^{-\beta}$ where $\beta = \epsilon/3$, say. Since $P_b(v)$ is
within $\delta$ of the uniform distribution for all $b$, doubling the constant
$A$ and using the triangle inequality completes the proof.
\end{proof}
Several remarks are in order. First, just as for the dihedral group, we can
information-theoretically distinguish conjugate subgroups if we use a random
basis \emph{within} each $q$-dimensional block. The problem is that rather than
having this block-diagonal structure, a random basis cuts across these blocks,
mixing different ``frequencies'' $\rho_k$ and canceling out the useful
information. This is precisely because it is not adapted to the subgroup
structure of $A_p$; it doesn't ``know'' that $\rho$ decomposes into a direct sum
of the $\rho_k$.
Second, it is worth noting that for the values of $q$ for which we have an
algorithm for full (as opposed to information-theoretic) reconstruction, namely
$q=p/{\rm polylog}(p)$, a random basis works as well since the $L_1$ distance
$\delta$ becomes $1/{\rm polylog}(p)$. Based on the strong evidence from
representation theory that some bases are much better for computation than
others, we conjecture that, for some families of groups, adapted bases allow
full reconstruction while random bases do not; but this remains an open
question.
Third, while we focused above on distinguishing conjugate subgroups from each
other, in fact our proof shows that if $q < p^{1-\epsilon}$ a random basis is incapable
of distinguishing $H_a$ from the \emph{trivial} subgroup. In contrast,
Theorems~\ref{thm:infohcp} and~\ref{thm:infohsp} show that an adapted basis
allows us to do this.
\section{Failure of the abelian Fourier transform}
\label{sec:abelian}
In \cite{EttingerH98} the abelian Fourier transform over ${\mathbb Z}_2 \times {\mathbb Z}_p$ is
used in a reconstruction algorithm for the dihedral groups. Using this sort of
``forgetful'' abelian Fourier analysis it is similarly information-theoretically
possible to reconstruct subgroups of the $q$-hedral groups, when $q$ is small
enough.
However, it does not seem possible to reconstruct subgroups of $A_p$ using the
abelian Fourier transform. In particular, we show in this section that if we
think of the affine group as a direct product ${\mathbb Z}_p^* \times {\mathbb Z}_p$ rather than a
semidirect product, then the conjugates of the maximal subgroup become
indistinguishable. This is not surprising, since in an abelian group conjugates
are identical by definition, but it helps illustrate that nonabelian hidden
subgroup problems require nonabelian approaches (most naturally, in
our view, representation theory).
Let us consider the hidden conjugate problem for the maximal subgroup $H$, i.e.,
$H_a$ where $a$ is a generator of ${\mathbb Z}_p^*$. In that case, the characters of
${\mathbb Z}_p^* \times {\mathbb Z}_p$ are simply $\rho_{k,\ell}(a^t,b) = \omega_{p-1}^{kt}
\omega_p^{\ell b}$. Summing these over $H_a = \{ (a^t, (1-a^t)b \}$ shows that
we observe the character $(k,\ell)$ with probability
\begin{align*}
P(k,\ell) &= \frac{1}{p \,(p-1)^2} \left| \sum_{t \in {\mathbb Z}_{p-1}} \omega_{p-
1}^{kt} \omega_p^{\ell (1-a^t) b} \right|^2 \\
&= \frac{1}{p \,(p-1)^2} \left|
\sum_{x \in {\mathbb Z}_p^*} \omega_{p-1}^{k \log_a x} \omega_p^{-\ell x b}
\right|^2
\enspace.
\end{align*}
This is the inner product of a multiplicative character with an additive one,
which is another Gauss sum. In particular, assuming $b \neq 0$, we have
\begin{eqnarray*}
P(0,0) & = & 1/p \\
P(0, \ell \neq 0) & = & 1/ (p\,(p-1)^2) \\
P(k \neq 0, 0) & = & 0 \\
P(k \neq 0, \ell \neq 0) & = & 1/(p-1)^2
\end{eqnarray*}
(see Appendix~\ref{appendix:gauss-sums}). Since these probabilities don't
depend on $b$, the different conjugates $H_a^b$ with $b \neq 0$ are
indistinguishable from each other. Thus it appears essential to use the
nonabelian Fourier transform and the high-dimensional representations of $A_p$.
\section{Hidden shift problems}
\label{sec:shift}
Using the natural action of the affine group on ${\mathbb Z}_p$, we can apply
our algorithm for the hidden conjugate problem studied above to a
natural family of \emph{hidden shift problems}. Specifically, let $M$
be a multiplicative subgroup of ${\mathbb Z}_p^*$ of index $r > 1$, let $S$ be
some set of $r+1$ symbols, and let $f: {\mathbb Z}_p \to S$ be a function for
which
$$
f(x) = f(mx) \Leftrightarrow m \in M
$$
for every $x \in {\mathbb Z}_p$. Observe that $f$ is constant on the
(multiplicative) cosets of $M$ and takes distinct values
on distinct cosets; to put it differently, $f(x)$ is an injective function
of the multiplicative order of $x$ mod $r$.
Furthermore, $f(0) \neq f(x)$ for any nonzero $x$.
The hidden shift problem associated with $f$ is
the problem of determining an unknown element $s \in {\mathbb Z}_p$ given oracle
access to the shifted function
$$
f_s(x) = f(x - s)\enspace.
$$
Such functions have remarkable pseudorandom properties, and have been
proposed as pseudorandom generators for cryptographic purposes, where
$s$ acts as the seed\remove{ or secret key} to generate the sequence (e.g.~\cite{damgard}).
The special case when $f: {\mathbb Z}_p \to {\mathbb C}$ is a \emph{Legendre symbol}, that
is, a multiplicative character of ${\mathbb Z}_p^*$ extended to all of ${\mathbb Z}_p$
by setting $f(0) = 0$, was studied by van Dam, Hallgren, and
Ip~\cite{HallgrenIvD}. They give efficient quantum algorithms for
these hidden shift problems for all characters of ${\mathbb Z}_p^*$. Their
algorithms, however, make explicit use of the complex values taken by
the character, whereas the algorithms we present here depend only on
the symmetries of the underlying function $f$; in particular, in our
case $f$ can be an arbitrary injective function from a multiplicative
character into a set $S$. On the other hand, their algorithms are
efficient for characters of any order, while our algorithms require
that $r$ be at most polylogarithmic in $p$.
Returning to the general problem defined above, let ${\mathcal F}({\mathbb Z}_p, S)$
denote the collection of $S$-valued functions on ${\mathbb Z}_p$. Note that the
affine group $A_p$ acts on the set ${\mathcal F}({\mathbb Z}_p,S)$ by assigning $\alpha \cdot g(x) =
g(\alpha^{-1}(x))$ for each $\alpha \in A_p$ and $g \in F({\mathbb Z}_p,S)$. In particular,
$f_s = (1,s) \cdot f$.
Now note that the isotropy subgroup of $f$, namely the subgroup of $A_p$
that fixes the cosets of $M$, is precisely $H_a = \langle (a,0) \rangle$ where $a
\in {\mathbb Z}_p^*$ has order $q=(p-1)/r$. As we have $f_s = (1,s) \cdot f$, the isotropy
subgroup of $f_s$ is the conjugate subgroup $H_a^s = (1,s) \cdot H_a \cdot
(1,-s)$. Observe now that if we define $F_s : A_p \to ({\mathbb Z}_p)^p$ so that
$F_s(\alpha)$ is the $p$-tuple $(\alpha f_s(0), \alpha f_s(1), \ldots, \alpha f_s(p-1))$ then
\begin{equation}
\label{eqn:shift-symmetry}
F_s(\alpha) = F_s(\beta) \Leftrightarrow \alpha^{-1} \beta \in H_a^s \enspace ,
\end{equation}
i.e., $F_s$ is constant precisely on the left cosets of $H_a^s$.
Evidently, then, the solution to the hidden conjugate problem given by the
oracle $F_s$ determines the solution to the hidden shift problem given
by $f_s$. Unfortunately, the \emph{values} of the oracle $F_s$ are of exponential
size---we cannot afford to evaluate $\alpha f_s(x)$ for all $x \in {\mathbb Z}_p$.
This same symmetries expressed in Equation~\eqref{eqn:shift-symmetry},
however, can be obtained efficiently by
selecting an appropriate subset $R = \{x_1, \ldots, x_m\} \subset {\mathbb Z}_p$ and
considering the oracle that samples $\alpha f_s$ on $R$: that is,
\[ F^R_s(\alpha) = (\alpha f_s(x_1), \ldots, \alpha f_s(x_m)) \enspace . \]
Of course, we have $\alpha f_s = \beta f_s \Rightarrow F^R_s(\alpha) = F^R_s(\beta)$
regardless of $R$; the difficulty is finding a small set $R$ for
which $F^R_s(\alpha) = F^R_s(\beta) \Rightarrow \alpha f_s = \beta f_s$.
We show below that a set of $O(\log p)$ elements selected uniformly at random
from ${\mathbb Z}_p$ has this property with high probability.
Considering that $\alpha f_s(x) = \alpha \cdot (1,s) \cdot f(x)$, it suffices to show that
if $\alpha f \neq \beta f$ then
\[
\Pr_x[\alpha f(x) = \beta f(x)] \leq 1/2\enspace,
\]
where $x$ is selected uniformly at random in ${\mathbb Z}_p$. Note that for
affine functions $\alpha$ and $\beta$ and an element $x \in {\mathbb Z}_p$ for which
$\beta^{-1}(x) \neq 0$,
$$
\alpha f(x) = \beta f(x) \;\Leftrightarrow\; \frac{\alpha^{-1}(x)}{\beta^{-1}(x)} \in M \enspace .
$$
The function $\alpha^{-1}(x)/\beta^{-1}(x)$ is a \emph{fractional linear
transform}, i.e., the ratio of two linear functions; these is the
discrete analog of a M\"{o}bius transformation in the complex
plane. As in the complex case, the fractional linear transform $\gamma(x) /
\delta(x)$ is a bijection on the projective space
${\mathbb Z}_p \cup \{ \infty \}$ unless $\gamma$ and $\delta$ share a root,
or, equivalently, there is a scalar $z \in {\mathbb Z}_p^*$ such that
$\gamma(x) = z\delta(x)$. If $\alpha^{-1}(x) / \beta^{-1}(x)$ is injective, we can
immediately conclude that
$$
\Pr_{x} [ \alpha f(x) = \beta f(x) ] \leq |M|/(p-1) = 1/r \leq 1/2 \enspace.
$$
Otherwise, $\alpha^{-1}(x)/ \beta^{-1}(x) = z$ for some scalar $z$.
Since $\alpha f \neq \beta f$, however, in this case we must have $z \in {\mathbb Z}_p^* \setminus M$.
In particular, $f(zy) \neq f(y)$ for any $y \neq 0$, and so
$$
\Pr_{x} [ \alpha f(x) = \beta f(x) ] = 1/p
$$
since this only occurs at the unique root $x$ of $\alpha^{-1}(x)=0$.
In either case, then, $\alpha f$ and $\beta f$ differ on at least half the elements of ${\mathbb Z}_p$
whenever $\alpha$ and $\beta$ belong to different cosets of $H_a^s$. It follows that if
$R \subset {\mathbb Z}_p$ consists of $m$ elements chosen
independently and uniformly at random from ${\mathbb Z}_p$, we have
$$
\Pr_{R} \left[ \forall x \in R, \alpha f(x) = \beta f(x)\right] \leq 1/2^m
$$
for any $\alpha, \beta \in A_p$ with $\alpha^{-1}\beta \notin H_a$. Taking a union bound over
all pairs of left cosets of $H_a$,
$$
\Pr_{R} \left[ \exists \alpha, \beta \in A_p: \alpha^{-1}\beta \notin H_a, \forall x \in R, \alpha f(x) = \beta
f(x)\right] \leq \left(\frac{p(p-1)}{|H_a|}\right)^2\frac{1}{2^m}\enspace.
$$
Selecting $m = 5 \log p$ ensures that this probability is less than $1/p$.
Since we showed in Section~\ref{sec:full-reconstruction} that we can identify a
hidden conjugate of $H_a$ whenever $H_a$ is of polylogarithmic index in ${\mathbb Z}_p^*$,
and since this index is $(p-1)/q = r$, this provides an efficient solution to the hidden shift
problem so long as $r = {\rm polylog}(p)$.
\section{Closure under extending small groups}
\label{sec:closure}
In this section we show that for any polynomial-size group $K$ and any $H$ for
which we can solve the HSP, we can also solve the HSP for any extension of $K$
by $H$, i.e., any group $G$ with $K \lhd G$ and $G/K \cong H$. (Note that this
is more general than split extensions, i.e., semidirect products $H \ltimes K$.)
This includes the case discussed in~\cite{HallgrenRT00} of Hamiltonian groups,
since all such groups are direct products (and hence extensions) by abelian
groups of the quaternion group $Q_8$~\cite{Rotman94}. It also includes the case
discussed in~\cite{FriedlIMSS02} of groups with commutator subgroups of
polynomial size, such as extra-special $p$-groups, since in that case $K=G'$ and
$H \cong G/G'$ is abelian. Indeed, our proof is an easy generalization of that
in~\cite{FriedlIMSS02}.
\begin{theorem}
\label{thm:semik}
Let $H$ be a group for which hidden subgroups are fully reconstructible, and $K$
a group of polynomial size in $\log |H|$. Then hidden subgroups in any
extension of $K$ by $H$, i.e., any group $G$ with $K \lhd G$ and $G/K \cong H$,
are fully reconstructible.
\end{theorem}
\noindent
\begin{proof}
We assume that $G$ and $K$ are encoded in such a way that multiplication can be
carried out in classical polynomial time. We fix some transversal $t(h)$ of the
left cosets of $K$. First, note that any subgroup $L \subseteq G$ can be
described in terms of i) its intersection $L \cap K$, ii) its projection $L_H =
L/(L \cap K) \subseteq H$, and iii) a representative $\eta(h) \in L \cap (t(h)
\cdot K)$ for each $h \in L_H$. Then each element of $L_H$ is associated with
some left coset of $L \cap K$, i.e., $ L = \bigcup_{h \in L_H} \eta(h) \cdot (L
\cap K)$. Moreover, if $S$ is a set of generators for $L \cap K$ and $T$ is a
set of generators for $L_H$, then $S \cup \eta(T)$ is a set of generators for
$L$.
We can reconstruct $S$ in classical polynomial time simply by querying the
function $h$ on all of $K$. Then $L \cap K$ is the set of all $k$ such that
$f(k) = f(1)$, and we construct $S$ by adding elements of $L \cap K$ to it one
at a time until they generate all of $L \cap K$.
To identify $L_H$, as in~\cite{FriedlIMSS02} we define a new function $f'$ on
$H$ consisting of the unordered collection of the values of $f$ on the
corresponding left coset of $K$:
$$
f'(h) = \{ f(g) \mid g \in t(h) \cdot K \}.
$$
Each query to $f'$ consists of $|K| = {\rm poly}(n)$ queries to $f$. The level sets
of $f'$ are clearly the cosets of $L_H$, so we reconstruct $L_H$ by solving the
HSP on $H$. This yields a set $T$ of generators for $L_H$.
It remains to find a representative $\eta(h)$ in $L \cap (t(h) \cdot K)$ for
each $h \in T$. We simply query $f(g)$ for all $g \in t(h) \cdot K$, and set
$\eta(h)$ to any $g$ such that $f(g) = f(1)$. Since $|T| = O(\log |H|) =
{\rm poly}(n)$ this can be done in polynomial time, completing the proof.
\end{proof}
Unfortunately, we cannot iterate this construction more than a constant number
of times, since doing so would require a superpolynomial number of queries to
$f$ for each query of $f'$. If $K$ has superpolynomial size it is not clear how
to obtain $\eta(h)$, even when $H$ has only two elements. Indeed, this is
precisely the difficulty with the dihedral group.
\section{Conclusion and directions for further work}
We have shown that the ``strong standard method,'' applied with
adapted bases, solves in quantum polynomial time certain nonabelian
Hidden Subgroup Problems that are not solved with any
other known technique, specifically measurements in random
bases or ``forgetful'' abelian approaches.
While we are still very far from an algorithm for HSP in the symmetric group $S_n$ or
for Graph Automorphism, a global understanding of the power of strong Fourier
sampling remains an important goal. Perhaps the next class of groups to try
beyond the affine and $q$-hedral groups are matrix groups such as ${\rm
PSL}_2(p)$, whose maximal subgroups are isomorphic to $A_p$, and which include
one of the infinite families of finite simple groups.
\bigskip {\bf Acknowledgements.} We are grateful to Wim van Dam,
Julia Kempe, Greg Kuperberg,
Frederic Magniez, Martin R\"{o}tteler, and Miklos Santha for helpful
conversations, and to Sally Milius and Tracy Conrad for their support.
Support for this work was provided by the California Institute of
Technology's Institute for Quantum Information (IQI), the Mathematical
Sciences Research Institute (MSRI), the Institute for Advanced Study
(IAS), NSF grants ITR-0220070, ITR-0220264, CCR-0093065,
EIA-0218443, QuBIC-0218563, CCR-0049092,
the Charles Lee Powell Foundation, and the Bell Fund.
|
{
"timestamp": "2005-03-09T20:06:26",
"yymm": "0503",
"arxiv_id": "quant-ph/0503095",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503095"
}
|
\section{Introduction and summary \label{sec:intro}}
One of the most intriguing {experimental} puzzles encountered in
contemporary physics is the evident absence of SUSY partners of
elementary particles in nature. In the context of {field theory}
this means that SUSY, if it exists, must be spontaneously broken.
Witten \cite{Witten} proposed a schematic model which,
incidentally, failed to clarify this breakdown but, nonetheless,
survived and found a number of applications within the
so-called SUSY quantum mechanics (SUSYQM) \cite{CKS}.\par
In the latter formalism one introduces the so-called
superpotential $W(x)$ and defines the two operators
\begin{equation}
{\cal A} = \frac{d}{dx} + W(x) \qquad \bar{\cal A} = - \frac{d}{dx} + W(x)
\end{equation}
with the property that the two related {\em different} (so-called
`SUSY partner') potentials $ V^{(\pm)}- E_0 = W^2 \mp W'$ may
prove {\em both} exactly solvable at the same time. An easy
explanation of this phenomenon lies in the fact that the related
Hamiltonians
\begin{equation}
H^{(\pm)} = - \frac{d^2}{dx^2} + V^{(\pm)}(x) - E_0
\label{eq:Hpm}
\end{equation}
become inter-related, at a convenient auxiliary energy $E = E_0$,
by the factorization rules $H^{(+)}=\bar{\cal A} {\cal A}$ and $H^{(-)}={\cal A}
\bar{\cal A}$. The spectra of $H^{(+)}$ and $H^{(-)}$ are then alike
except possibly for the ground state. In the unbroken SUSY case,
the ground state at vanishing energy is nondegenerate and, in the
present notational set-up, it belongs to $H^{(+)}$. This means
that
\begin{equation}
{\cal A}\, \psi^{(+)}_0(x) = 0 \label{eq:unbroken-susy}
\end{equation}
where $\psi^{(+)}_n(x)$ (resp.\ $\psi^{(-)}_n(x)$), $n=0$, 1,
2,~\ldots, denote the wavefunctions of $H^{(+)}$ (resp.\
$H^{(-)}$). The (double) degeneracy of $\left(\psi^{(+)}_{n+1}(x),
\psi^{(-)}_n(x)\right)$ for $n=0$, 1, 2,~\ldots\ is implied by the
intertwining relationships
\begin{equation}
{\cal A}\, H^{(+)} = H^{(-)} {\cal A} \qquad H^{(+)} \bar{\cal A}
= \bar{\cal A}\, H^{(-)}. \label{eq:intertwining}
\end{equation}
In the conventional setting, the Hamiltonians (\ref{eq:Hpm}) are
assumed self-adjoint.\par
New horizons have been opened by the pioneering letter by Bender
and Boettcher \cite{BB} who noticed, in a slightly different
context, that the latter condition $H = H^\dagger$ might be
relaxed as redundant and replaced by its suitable weakened forms.
For our present purposes, we shall employ their proposal and in equation
(\ref{eq:Hpm}) allow complex potentials that are merely constrained by the
requirement that their real and imaginary parts are spatially symmetric
and antisymmetric, respectively \cite{BBjmp}.\par
It is not too difficult to show that the above SUSYQM
factorization scheme remains unchanged under such a non-Hermitian
generalization~\cite{Andrianov,crea,ptsusy,BMQ1,CQ}. Of course,
the relaxation of the usual condition $H=H^\dagger$ is by far not
a trivial step. Formally, we may put $H^\dagger = {\cal T}H{\cal
T}$ with an antilinear `time-reversal' operator ${\cal T}$
\cite{erratum}. In such a setting, Bender and Boettcher (loc.\
cit., cf.\ also some older studies \cite{BG} or newer developments
\cite{Mali}) merely replaced ${\cal T}$ by its product with parity
${\cal P}$ and conjectured that the above-mentioned and
physically well-motivated weakening of Hermiticity could be most
appropriately characterized as an antilinear `symmetry' or
`$\cal PT$-symmetry', ${\cal PT}H = H{\cal PT}$, of all the
Hamiltonians in question. Equivalently \cite{205} one may speak
about the $\cal P$-pseudo-Hermiticity defined by the relation
\begin{equation}
H^\dagger
={\cal P}\,H\,{\cal P}^{-1}\,.
\label{eq:PTS}
\end{equation}
\par
In this paper we intend to concentrate on
implementing the resulting $\cal PT$-symmetric SUSYQM
factorization scheme in the case of the `simplest' model which remains
`realistic' and `solvable' at the same time. This means that our
`initial' Schr\"{o}dinger equation
\begin{equation}
\left[- \frac{d^2}{d x^2}+V^r(x)+{\rm i}V^i(x)\right]\psi(x)
=E\psi(x)\label{eq:problem}
\end{equation}
(where we dropped the superscript `$(+)$' as temporarily
redundant) will contain just the most trivial infinitely deep
square-well form
\begin{equation}
V^r(x)=\left\{\begin{array}{ll}
+\infty & x<-L\\[0.1cm]
0 & -L<x<L\\[0.1cm]
+\infty& x>L\end{array}\right. \label{eq:V-r}
\end{equation}
for the real part of the potential and the most elementary short-range
one
\begin{equation}
V^i(x)=\left\{\begin{array}{ll}
0 & x<-l\\[0.1cm]
-g & -l<x<0\\[0.1cm]
+g& 0<x<l,\\[0.1cm]
0& x>l
\end{array}\right.
\qquad l<L , \qquad g>0\,
\label{eq:V-i}
\end{equation}
for its imaginary part. As a consequence of (\ref{eq:V-r}), the
wavefunctions will be defined on a finite interval $(-L, L)$ with a
variable length $2L$, on which they satisfy the standard Dirichlet
boundary conditions \cite{ptsqw,Langer}
\begin{equation}
\psi(\pm L)=0 .\label{eq:dirichlet}
\end{equation}
\par
Given the background of the result obtained in \cite{twopoint}, we
derive in section \ref{sec:ptsw}, an elegant trigonometric form of
the standard matching conditions for wavefunctions at the
discontinuities of the potential (subsection \ref{sec:secular})
and discuss the practical semi-numerical determination of the
energies with arbitrary precision (subsection
\ref{sec:graphical}).\par
In section \ref{sec:SUSYQM} we address the key concern of our
present paper, viz., the investigation of the problem in the
context of SUSYQM. Here the non-Hermiticity and discontinuities
create some specific features, which are dealt with in detail.
After deriving the superpotential and the partner potential in
subsection \ref{sec:W}, we construct the eigenfunctions of the
latter and analyze the discontinuities in subsections
\ref{sec:eigenfunctions} and \ref{sec:discontinuities},
respectively.\par
Some physical aspects of our results are finally discussed in
more detail in section~\ref{sec:discussion}.\par
\section{\boldmath Trigonometric secular equation
\label{sec:ptsw}}
\setcounter{equation}{0}
\subsection{\boldmath $\cal PT$-symmetric square well inside a real one
\label{sec:secular}}
Let us denote the four regions $-L < x < -l$, $-l < x < 0$, $0 < x
< l$, $l < x < L$ by $L2$, $L1$, $R1$, $R2$, respectively. We
shall henceforth append these symbols as subscripts to all
quantities pertaining to such regions. The complex potential
$V(x)$, defined in equations (\ref{eq:V-r}) and (\ref{eq:V-i}),
may therefore be rewritten as
\begin{equation}
V_{L2}(x) = 0 \qquad V_{L1}(x) = - {\rm i} g \qquad V_{R1}(x) = {\rm i} g \qquad
V_{R2}(x) = 0. \label{eq:potential}
\end{equation}
\par
The general solution of~(\ref{eq:problem}) satisfying the
conditions (\ref{eq:dirichlet}) can be written as
\begin{equation}
\psi(x) = \left\{\begin{array}{l}
\psi_{L2}(x) = A_L \sin[k(L+x)] \\[0.1cm]
\psi_{L1}(x) = B_L \cosh(\kappa^* x) + {\rm i} \frac{C_L}{\kappa^* l} \sinh(\kappa^*
x) \\[0.1cm]
\psi_{R1}(x) = B_R \cosh(\kappa x) + {\rm i} \frac{C_R}{\kappa l} \sinh(\kappa x)
\\[0.1cm]
\psi_{R2}(x) = A_R \sin[k(L-x)]
\end{array} \right. \label{eq:psi}
\end{equation}
where
\begin{equation}
\kappa=s+{\rm i}t \qquad E=k^2=t^2-s^2 \qquad g=2st. \label{eq:k-kappa}
\end{equation}
Here $s,\ t$ and $k$ are real and, for the sake of definiteness,
are assumed positive. A priori, $A_L$, $B_L$, $C_L$, $A_R$, $B_R$
and $C_R$ are some complex constants.\par
On assuming that $\cal PT$-symmetry is unbroken, we obtain the conditions
\begin{equation}
\psi^*_{L2}(-x) = \psi_{R2}(x) \qquad \psi^*_{L1}(-x) = \psi_{R1}(x) \label{eq:psi-PT}
\end{equation}
from which we get
\begin{equation}
A_L^* = A_R \equiv A \qquad B_L^* = B_R \equiv B \qquad C_L^* = C_R \equiv C.
\label{eq:ABC}
\end{equation}
The derivative of (\ref{eq:psi}), taking (\ref{eq:ABC}) into
account, reads
\begin{equation}
\partial_x \psi(x)=\left\{\begin{array}{l}
\partial_x \psi_{L2}(x) = k A^* \cos[k(L+x)] \\[0.1cm]
\partial_x \psi_{L1}(x) = \kappa^* B^* \sinh(\kappa^*x) + {\rm i} \frac{C^*}{l} \cosh
(\kappa^* x) \\[0.1cm]
\partial_x \psi_{R1}(x) = \kappa B \sinh(\kappa x) + {\rm i} \frac{C}{l}\cosh(\kappa x)
\\[0.1cm]
\partial_x \psi_{R2}(x) = - k A \cos[k(L-x)]
\end{array}\right.. \label{eq:derivative}
\end{equation}
\par
Let us now match the wavefunction and its derivative at $x=0$ and
impose $\cal PT$-symmetry in the neighbourhood of the origin:
\begin{equation}
\psi_{R1}(0) = \psi_{L1}(0) \in \mbox{$\Bbb R$} \qquad \partial_x \psi_{R1}(0) = \partial_x \psi_{L1}(0)
\in {\rm i} \mbox{$\Bbb R$}.
\end{equation}
This leads to
\begin{equation}
B, C \in \mbox{$\Bbb R$}. \label{eq:cond-BC}
\end{equation}
\par
It now remains to match $\psi$ and $\partial_x \psi$ at $x=\pm l$.
Since $\psi$ is $\cal PT$-symmetric, it is enough to impose
matching conditions at $x=l$:
\begin{equation}
\psi_{R2}(l) = \psi_{R1}(l) \qquad \partial_x \psi_{R2}(l) = \partial_x \psi_{R1}(l).
\end{equation}
This yields
\begin{eqnarray}
A\sin [k(L-l)] & = & B \cosh(\kappa l) + {\rm i} \frac{C}{\kappa l} \sinh(\kappa l)
\label{eq:A-BC1} \\[0.1cm]
-k A \cos[k(L-l)] & = & \kappa B \sinh(\kappa l) + {\rm i} \frac{C}{l} \cosh(\kappa l).
\label{eq:A-BC2}
\end{eqnarray}
\par
We conclude that the final form of $\psi$ is
\begin{equation}
\psi(x) = \left\{\begin{array}{l}
\psi_{L2}(x) = A^* \sin[k(L+x)] \\[0.1cm]
\psi_{L1}(x) = B \cosh(\kappa^* x) + {\rm i} \frac{C}{\kappa^* l} \sinh(\kappa^*
x) \\[0.1cm]
\psi_{R1}(x) = B \cosh(\kappa x) + {\rm i} \frac{C}{\kappa l} \sinh(\kappa x)
\\[0.1cm]
\psi_{R2}(x) = A \sin[k(L-x)]
\end{array} \right. \label{eq:psi-bis}
\end{equation}
where the complex constant $A$ is determined by one of the
equations (\ref{eq:A-BC1}) and (\ref{eq:A-BC2}), while the real
constants $B$ and $C$ have to satisfy a condition obtained by
eliminating $A$ between (\ref{eq:A-BC1}) and (\ref{eq:A-BC2}):
\begin{eqnarray}
&& \kappa l B \{k\cos[k(L-l)] \cosh(\kappa l) + \kappa \sin[k(L-l)] \sinh(\kappa l)\}
\nonumber \\
&& \mbox{} + {\rm i} C \{k \cos[k(L-l)] \sinh(\kappa l) + \kappa \sin[k(L-l)]
\cosh(\kappa l)\} = 0. \label{eq:rel-BC}
\end{eqnarray}
We may therefore express both constants $A$ and $C$ in terms of
$B$ as
\begin{eqnarray}
A & = & B\, \frac{\kappa \csc[k(L-l)] \mathop{\rm csch}\nolimits(\kappa l)}{k \cot[k(L-l)] + \kappa
\coth(\kappa l)} \label{eq:A} \\
C & = & {\rm i} \kappa l B\, \frac{k \cot[k(L-l)] \coth(\kappa l) + \kappa}
{k \cot[k(L-l)] + \kappa \coth(\kappa l)}. \label{eq:C}
\end{eqnarray}
\par
Since, from (\ref{eq:cond-BC}), the left-hand side of equation
(\ref{eq:C}) is real, the same should be true for the right-hand
one. The resulting condition can be written as
\begin{eqnarray}
&& k^2 \cot^2[k(L-l)] [\kappa \coth(\kappa l) + \kappa^* \coth(\kappa^* l)]
\nonumber \\
&& \mbox{} + k \cot[k(L-l)] [\kappa^2 + 2 \kappa \kappa^* \coth(\kappa l)
\coth(\kappa^* l) + \kappa^{*2}] \nonumber \\
&& \mbox{} + \kappa \kappa^* [\kappa \coth(\kappa^* l) + \kappa^* \coth(\kappa
l)] = 0. \label{eq:C-real}
\end{eqnarray}
On expressing $k^2$, $\kappa$ and $\kappa^*$ in terms of $s$ and
$t$ through equation (\ref{eq:k-kappa}) and using some elementary
trigonometric identities, condition (\ref{eq:C-real}) is easily
transformed into
\begin{eqnarray}
&& k \sin[2k(L-l)] [s^2 \cosh(2sl) + t^2 \cos(2tl)] \nonumber \\
&& \mbox{} - \cos[2k(L-l)] [s^3 \sinh(2sl) - t^3 \sin(2tl)] \nonumber \\
&& \mbox{} + s t^2 \sinh(2sl) - s^2 t \sin(2tl) = 0 \label{eq:transcendental}
\end{eqnarray}
where $k = \sqrt{t^2 - s^2}$.
\par
\subsection{Graphical and numerical determination of the energies
\label{sec:graphical}}
The transcendental equation (\ref{eq:transcendental}) has to be
complemented by the constraint (\ref{eq:k-kappa}),
\begin{equation}
s t = \frac{1}{2} g .\label{eq:hyperbola}
\end{equation}
The couples of roots $(s_n, t_n)$, $n=0$, 1, 2,~\ldots, of this
pair of equations define all the bound-state energies $E_n$ by the
elementary formula
\begin{equation}
E_n = t_n^2 - s_n^2 \qquad n=0, 1, 2, \ldots. \label{eq:E}
\end{equation}
In practice, the $(s_n, t_n)$ values may be obtained as the
intersection points in the $(s, t)$ plane of the curves
representing the roots of the transcendental equation
(\ref{eq:transcendental}) with the hyperbola
(\ref{eq:hyperbola}).\par
Before proceeding to discuss the graphical and numerical
determination of $E_n$ in general, it is worth reviewing three
interesting limiting cases of equation (\ref{eq:transcendental}).
The first one corresponds to the limit $l \to L$, wherein the
present square well with three matching points reduces to the one
with a single discontinuity. Equation (\ref{eq:transcendental})
then simply becomes
\begin{equation}
s \sinh(2sL) + t \sin(2tL) = 0
\end{equation}
which coincides with equation (9) of \cite{ptsqw} (where $g$ is
denoted by $Z$ and $L=1$).\par
The second limiting case corresponds to $l \to 0$ and gives back
the real square well. Since the constraint (\ref{eq:hyperbola})
then disappears, we are only left with equation
(\ref{eq:transcendental}) acquiring the simple form
\begin{equation}
\sin(2kL) = 0.
\end{equation}
Its solutions are provided by the hyperbolas $t^2 - s^2 =
\left(\frac{n\pi}{2L}\right)^2$, $n=1$, 2,~\ldots, where the $n=0$
value is discarded because no acceptable wavefunction can be
associated with it. We therefore arrive at the well-known
quadratic spectrum $E_n^2 = \left(\frac{n\pi}{2L}\right)^2$,
$n=1$, 2,~\ldots, of the real square well.\par
The existence of the third special limiting regime is connected
with the bounded nature of our imaginary barrier (\ref{eq:V-i}).
In the language of perturbation theory this means \cite{Langer}
that the influence of this barrier on the values of the energies
(\ref{eq:E}) weakens quickly with the growth of the quantum number
$n$. At the higher excitations, as a consequence, the
$n-$dependence of the energies will not deviate too much from the
$l \to 0$ rule $E_n \sim n^2 \gg 1$. In the other words, the
growth of $n$ will imply the growth of $t_n \sim n \gg 1$ and the
decrease and smallness of the roots $s_n = g/(2t_n) \ll 1$. In
this regime, we may imagine that $k = t\,\sqrt{1 - s^2/t^2}= t -
s^2/(2t) + {\cal O}(s^4/t^3)= t-g^2/(8t^3) + {\cal O}(1/n^7)$ so
that the six components of our quantization condition
(\ref{eq:transcendental}), {\it viz.},
\begin{eqnarray}
&& s^2 k \sin[2k(L-l)]\cosh(2sl)
+ t^2 k \sin[2k(L-l)] \cos(2tl) \nonumber \\
&& \mbox{} - s^3 \cos[2k(L-l)] \sinh(2sl)
+ t^3 \cos[2k(L-l)]
\sin(2tl) \nonumber \\
&& \mbox{} + s t^2 \sinh(2sl) - s^2 t \sin(2tl) = 0
\nonumber
\label{eq:appranscend}
\end{eqnarray}
may be characterized by their asymptotic sizes ${\cal O}(1/n) $,
${\cal O}(n^3) $, ${\cal O}(1/n^4) $, ${\cal O}(n^3) $, ${\cal
O}(n^0) $ and ${\cal O}(1/n) $, respectively. Once we omit all the
negligible ${\cal O}(1/n) $ terms and insert $s = g/(2t)$ whenever
necessary, we arrive at the thoroughly simplified approximate
secular equation
\begin{equation}
\sin(2kL) +
\frac{g^2l}{2k^3}
+
{\cal O}\left (\frac{1}{k^4}\right )= 0 .\label{eq:apprcend}
\end{equation}
Its roots are easily found,
\begin{equation}
k=k_n=\frac{\pi\,n}{2L} + (-1)^{n+1}
\frac{2g^2lL^2}{\pi^3 n^3}
+ {\cal O}\left (\frac{1}{n^4}\right ),
\label{eq:aend}
\end{equation}
and give
\begin{equation}
E_n=k_n^2=\left (\frac{\pi\,n}{2L}\right )^2 + (-1)^{n+1}
\frac{2g^2lL}{\pi^2 n^2}
+ {\cal O}\left (\frac{1}{n^3}\right )
\label{eq:nd}
\end{equation}
i.e., a nice and elementary approximate energy formula for all the
highly excited states.
In the general case, the bound-state energies (\ref{eq:E}) of our
model are determined from the simultaneous solutions of equations
(\ref{eq:transcendental}) and (\ref{eq:hyperbola}). Although the
former is transcendental, one of its roots is quite obvious,
namely $s=t$. When we realize that this implies $k=0$ and
substitute the solution into equations (\ref{eq:psi-bis}) --
(\ref{eq:C}), we obtain a vanishing wavefunction. This is in
accordance with an insight provided by the Hermitian limit
$g\rightarrow 0$ or $l \to 0$.\par
The other solutions of~(\ref{eq:transcendental}) can be found
numerically and graphically. As we can see in figure 1 where we
work with re-scaled length units in which $L=1$, they form
semi-ovals in $(s,t)$ plane. We can observe the absence of
robustly real energy levels, i.e., levels remaining real for any
value of $g$, which played their role in~\cite{twopoint}.\par
The locally decreasing character of the semi-oval maxima could
cause a complexification of higher energy pairs while the lower
pairs would remain real. In other words, the semi-oval maxima
might be decreasing faster then the hyperbola
(\ref{eq:hyperbola}). This race in decrease can be judged easily
when we use a hyperbolic coordinate system. As shown in figure 2,
in this setting, the maxima prove to increase monotonically while
the hyperbola is represented by a horizontal straight line.
Consequently, our model preserves a sequential merging of the
energy levels. The critical value $g_c$ of the coupling constant
$g$, for which the two lowest energy levels merge together, is of
high importance. It is the boundary of exact $\cal PT$-symmetry,
which we consider to be physically relevant and assumed in
deriving equation~(\ref{eq:transcendental}). For a higher value of
$g$, the wavefunction $\cal PT$-symmetry would be broken.\par
We found $g_c$ for various values of the parameter $l$. Since
$g_c$ rises rapidly as $l\rightarrow 0$, we present its values in
combination of graph and table (see figure 3 and table 1).
As the parameter $l$ approaches zero, $g_c$ tends to infinity and
the semi-oval maxima run to infinity as well. As explained in
subsection \ref{sec:secular}, equation (\ref{eq:transcendental})
then provides the bound-state energies of the real square well. On
the other hand, for $l \to L=1$, we get back the critical coupling
$g_c \simeq 4.4753$, previously obtained for the square well
in~\cite{ptsqw} and \cite{gezawell}.\par
\section{\boldmath The SUSY partner potential
\label{sec:SUSYQM}} \setcounter{equation}{0}
The purpose of the present section is to construct and study the
SUSY partner $H^{(-)}$ of the square-well Hamiltonian $H^{(+)}$,
defined in equation (\ref{eq:potential}), in the
physically-relevant unbroken $\cal PT$-symmetry regime,
corresponding to $g < g_c$.
\subsection{Determination of the parameters \label{sec:W}}
Identifying $V^{(+)}$ with the square-well potential
(\ref{eq:potential}), i.e., $V^{(+)}_{L2}(x) = 0$,
$V^{(+)}_{L1}(x) = - {\rm i} g$, $V^{(+)}_{R1}(x) = {\rm i} g$,
$V^{(+)}_{R2}(x) = 0$ and $E_0 = k_0^2 = t_0^2 - s_0^2 = -
\kappa_0^2 + {\rm i} g$, we obtain for the superpotential and the
partner potential the results
\begin{equation}
W(x) = \left\{\begin{array}{l}
W_{L2}(x) = k_0 \tan[k_0(x + x_{L2})] \\[0.1cm]
W_{L1}(x) = - \kappa_0^* \tanh[\kappa_0^*(x + x_{L1})] \\[0.1cm]
W_{R1}(x) = - \kappa_0 \tanh[\kappa_0(x - x_{R1})] \\[0.1cm]
W_{R2}(x) = k_0 \tan[k_0(x - x_{R2})]
\end{array} \right.
\end{equation}
and
\begin{equation}
V^{(-)}(x) = \left\{\begin{array}{l}
V^{(-)}_{L2}(x) = 2 k_0^2 \sec^2[k_0(x + x_{L2})] \\[0.1cm]
V^{(-)}_{L1}(x) = - 2 \kappa_0^{*2} \mathop{\rm sech}\nolimits^2[\kappa_0^*(x + x_{L1})] - {\rm i} g
\\[0.1cm]
V^{(-)}_{R1}(x) = - 2 \kappa_0^2 \mathop{\rm sech}\nolimits^2[\kappa_0(x - x_{R1})] + {\rm i} g \\[0.1cm]
V^{(-)}_{R2}(x) = 2 k_0^2 \sec^2[k_0(x - x_{R2})]
\end{array} \right.
\label{eq:partner-0}
\end{equation}
respectively. Here $x_{L2}$, $x_{L1}$, $x_{R1}$ and $x_{R2}$
denote four integration constants.\par
We now choose $x_{L2}$ and $x_{R2}$ as
\begin{equation}
x_{L2} = L + \frac{\pi}{2k_0} \qquad x_{R2} = L - \frac{\pi}{2k_0} \label{eq:integration-2}
\end{equation}
to ensure that $V^{(-)}_{L2}$ and $V^{(-)}_{R2}$ blow up at the
end points $x=-L$ and $x=L$. This is in tune with~\cite{CQ}. We
thus get
\begin{equation}
V^{(-)}_{L2}(x) = 2 k_0^2 \csc^2[k_0(x + L)] \qquad V^{(-)}_{R2}(x) = 2 k_0^2
\csc^2[k_0(x - L)]. \label{eq:partner-bis}
\end{equation}
Observe that for the superpotential, $W_{L2}(x)$ and $W_{R2}(x)$
also blow up at these points:
\begin{equation}
W_{L2}(x) = - k_0 \cot[k_0(x + L)] \qquad W_{R2}(x) = - k_0 \cot[k_0(x - L)].
\end{equation}
\par
Let us next consider the unbroken SUSY condition
(\ref{eq:unbroken-susy}), where according to (\ref{eq:psi-bis})
the ground-state wavefunction of $H^{(+)}$ is given by
\begin{eqnarray}
\psi^{(+)}_{0R2}(x) & = & \psi^{(+)*}_{0L2}(-x) = A^{(+)}_0 \sin[k_0(L - x)] \\
\psi^{(+)}_{0R1}(x) & = & \psi^{(+)*}_{0L1}(-x) = B^{(+)}_0 \cosh(\kappa_0 x) + {\rm i}
\frac{C^{(+)}_0}{\kappa_0 l} \sinh(\kappa_0 x).
\end{eqnarray}
Note that the superscript `$(+)$' is appended to the wavefunction
and the coefficients to signify that we are dealing with
Hamiltonian $H^{(+)}$. It is straightforward to see that equation
(\ref{eq:unbroken-susy}) is automatically satisfied in the regions
$R2$ and $L2$ due to the choice made for the integration constants
$x_{R2}$, $x_{L2}$ in equation (\ref{eq:integration-2}). On the
other hand, in the region $R1$ we find a condition fixing the
value of $x_{R1}$,
\begin{equation}
\tanh(\kappa_0 x_{R1}) = - \frac{{\rm i} C^{(+)}_0}{\kappa_0 l B^{(+)}_0} = \frac{k_0
\cot[k_0(L-l)] \coth(\kappa_0 l) + \kappa_0}{k_0 \cot[k_0(L-l)] + \kappa_0
\coth(\kappa_0 l)} \label{eq:x_R1}
\end{equation}
where in the last step we used equation (\ref{eq:C}). A similar
relation applies in $L1$, thus leading to the result
\begin{equation}
x_{L1} = x_{R1}^*. \label{eq:x_L1}
\end{equation}
\par
Note that in contrast with the real integration constants
$x_{R2}$, $x_{L2}$, the constants $x_{R1}$ and $x_{L1}$ are
complex. Separating both sides of equation (\ref{eq:x_R1}) into a
real and an imaginary part, we obtain the two equations
\begin{eqnarray}
\frac{\sinh X \cosh X}{\cosh^2 X \cos^2 Y + \sinh^2 X \sin^2 Y} & = & \frac{N^r}{D}
\label{eq:x_R1-1} \\
\frac{\sin Y \cos Y}{\cosh^2 X \cos^2 Y + \sinh^2 X \sin^2 Y} & = & \frac{N^i}{D}
\label{eq:x_R1-2}
\end{eqnarray}
where we have used the decompositions $\kappa_0 = s_0 + {\rm
i}t_0$, $x_{R1} = x_{R1}^r + {\rm i} x_{R1}^i$, $\kappa_0 x_{R1} =
X + {\rm i} Y$, implying that
\begin{equation}
X = s_0 x_{R1}^r - t_0 x_{R1}^i \qquad Y = t_0 x_{R1}^r + s_0 x_{R1}^i
\end{equation}
and we have defined
\begin{equation}
N^r = \{- s_0^2 \cos[2k_0(L-l)] + t_0^2\} \sinh(2s_0 l) + k_0 s_0 \sin[2k_0(L-l)]
\cosh(2s_0 l)
\end{equation}
\begin{equation}
N^i = \{s_0^2 - t_0^2 \cos[2k_0(L-l)]\} \sin(2t_0 l) - k_0 t_0 \sin[2k_0(L-l)]
\cos(2t_0 l)
\end{equation}
\begin{eqnarray}
D & = & \{- s_0^2 \cos[2k_0(L-l)] + t_0^2\} \cosh(2s_0 l) + \{s_0^2 - t_0^2
\cos[2k_0(L-l)]\} \cos(2t_0 l) \nonumber \\
&& \mbox{} + k_0 \sin[2k_0(L-l)] [s_0 \sinh(2s_0 l) + t_0 \sin(2t_0 l)].
\end{eqnarray}
Equations (\ref{eq:x_R1-1}) and (\ref{eq:x_R1-2}), when solved
numerically, furnish the values of both the parameters $x_{R1}^r$
and $x_{R1}^i$.\par
One may also observe that the resulting superpotential $W(-x) = - W^*(x)$
and partner potential $V^{(-)}(-x) = V^{(-)*}(x)$ are $\cal
PT$-antisymmetric and $\cal PT$-symmetric, respectively.\par
\subsection{Eigenfunctions in the partner potential \label{sec:eigenfunctions}}
On exploiting the first intertwining relation in
(\ref{eq:intertwining}), the eigenfunctions $\psi^{(-)}_n(x)$,
$n=0$, 1, 2,~\ldots, of $H^{(-)}$ can be obtained by acting with
${\cal A}$ on $\psi^{(+)}_{n+1}(x)$, subject to the preservation of the
boundary and continuity conditions
\begin{eqnarray}
\psi^{(-)}_{nL2}(-L) & = & 0 \qquad \psi^{(-)}_{nR2}(L) = 0 \label{eq:boundary} \\
\psi^{(-)}_{nL2}(-l) & = & \psi^{(-)}_{nL1}(-l) \qquad \partial_x \psi^{(-)}_{nL2}(-l) =
\partial_x \psi^{(-)}_{nL1}(-l) \label{eq:continuity-1} \\
\psi^{(-)}_{nL1}(0) & = & \psi^{(-)}_{nR1}(0) \qquad \partial_x \psi^{(-)}_{nL1}(0) =
\partial_x \psi^{(-)}_{nR1}(0) \label{eq:continuity-2} \\
\psi^{(-)}_{nR1}(l) & = & \psi^{(-)}_{nR2}(l) \qquad \partial_x \psi^{(-)}_{nR1}(l) =
\partial_x \psi^{(-)}_{nR2}(l). \label{eq:continuity-3}
\end{eqnarray}
Application of ${\cal A}$ leads to the forms
\begin{eqnarray}
\psi^{(-)}_{nL2}(x) & = & C^{(-)}_{nL2}\, A^{(+)*}_{n+1} \sin[k_{n+1}(L+x)]\nonumber \\
&& \mbox{} \times \{k_{n+1} \cot[k_{n+1}(L+x)] - k_0 \cot[k_0(L+x)]\}
\label{eq:partner-psi-1} \\
\psi^{(-)}_{nL1}(x) & = & C^{(-)}_{nL1}\, B^{(+)}_{n+1} \sinh(\kappa_{n+1}^* x)
\{\kappa_{n+1}^* - \kappa_0^* \tanh[\kappa_0^*(x + x_{R1}^*)]
\coth(\kappa_{n+1}^* x)\} \nonumber \\
&& \mbox{} + C^{(-)}_{nL1}\, \frac{{\rm i} C^{(+)}_{n+1}}{\kappa_{n+1}^* l}
\sinh(\kappa_{n+1}^* x) \nonumber \\
&& \mbox{} \times \{\kappa_{n+1}^* \coth(\kappa_{n+1}^* x) - \kappa_0^*
\tanh[\kappa_0^*(x + x_{R1}^*)]\} \\
\psi^{(-)}_{nR1}(x) & = & C^{(-)}_{nR1}\, B^{(+)}_{n+1} \sinh(\kappa_{n+1} x)
\{\kappa_{n+1} - \kappa_0 \tanh[\kappa_0(x - x_{R1})]
\coth(\kappa_{n+1} x)\} \nonumber \\
&& \mbox{} + C^{(-)}_{nR1}\, \frac{{\rm i} C^{(+)}_{n+1}}{\kappa_{n+1} l}
\sinh(\kappa_{n+1} x) \nonumber \\
&& \mbox{} \times \{\kappa_{n+1} \coth(\kappa_{n+1} x) - \kappa_0
\tanh[\kappa_0(x - x_{R1})]\} \\
\psi^{(-)}_{nR2}(x) & = & C^{(-)}_{nR2}\, A^{(+)}_{n+1} \sin[k_{n+1}(L-x)]\nonumber \\
&& \mbox{} \times \{- k_{n+1} \cot[k_{n+1}(L-x)] + k_0 \cot[k_0(L-x)]\}
\label{eq:partner-psi-4}
\end{eqnarray}
where $C^{(-)}_{nL2}$, $C^{(-)}_{nL1}$, $C^{(-)}_{nR1}$,
$C^{(-)}_{nR2}$ denote some complex constants and equation
(\ref{eq:x_L1}) has been used. It can be easily checked that the
boundary conditions (\ref{eq:boundary}) are automatically
satisfied by these eigenfunctions. It therefore remains to impose
the continuity conditions (\ref{eq:continuity-1}) --
(\ref{eq:continuity-3}).\par
Let us first match the regions $L1$ and $R1$ at $x=0$. The
continuity conditions (\ref{eq:continuity-2}) yield the two
relations
\begin{equation}
C^{(-)}_{nR1} \left[B^{(+)}_{n+1} \kappa_0 \tanh(\kappa_0 x_{R1}) + \frac{{\rm i}
C^{(+)}_{n+1}}{l}\right] = C^{(-)}_{nL1} \left[- B^{(+)}_{n+1} \kappa_0^* \tanh(
\kappa_0^* x_{R1}^*) + \frac{{\rm i} C^{(+)}_{n+1}}{l}\right] \label{eq:LR-1}
\end{equation}
\begin{eqnarray}
&& C^{(-)}_{nR1} \left\{B^{(+)}_{n+1} [\kappa_{n+1}^2 - \kappa_0^2 \mathop{\rm sech}\nolimits^2(\kappa_0
x_{R1})] + \frac{{\rm i} C^{(+)}_{n+1}}{l} \kappa_0 \tanh(\kappa_0 x_{R1})\right\}
\nonumber \\
&& = C^{(-)}_{nL1} \left\{B^{(+)}_{n+1} [\kappa_{n+1}^{*2} - \kappa_0^{*2}
\mathop{\rm sech}\nolimits^2(\kappa_0^* x_{R1}^*)] - \frac{{\rm i} C^{(+)}_{n+1}}{l} \kappa_0^*
\tanh(\kappa_0^* x_{R1}^*)\right\}. \label{eq:LR-2}
\end{eqnarray}
Since equations (\ref{eq:x_R1}) and (\ref{eq:k-kappa}) provide the
two constraints
\begin{eqnarray}
\kappa_0 \tanh(\kappa_0 x_{R1}) & = & - \kappa_0^* \tanh(\kappa_0^* x_{R1}^*) \\
\kappa_{n+1}^{*2} - \kappa_{n+1}^2 & = & \kappa_0^{*2} - \kappa_0^2 = - 2g
\end{eqnarray}
equations (\ref{eq:LR-1}) and (\ref{eq:LR-2}) are compatible and
lead to the condition
\begin{equation}
C^{(-)}_{nR1} = C^{(-)}_{nL1}.
\end{equation}
\par
Considering next the matching between $R1$ and $R2$ at $x=l$, we
obtain from equation (\ref{eq:continuity-3}) the two conditions
\begin{eqnarray}
&& C^{(-)}_{nR1} \{k_{n+1} \cot[k_{n+1}(L-l)] + \kappa_0 \tanh[\kappa_0(l -
x_{R1})]\} \nonumber \\
&& = C^{(-)}_{nR2} \{k_{n+1} \cot[k_{n+1}(L-l)] - k_0 \cot[k_0(L-l)]\} \label{eq:R12-1}
\end{eqnarray}
\begin{eqnarray}
&& C^{(-)}_{nR1} \biggl(\kappa_{n+1}^2 - \kappa_0^2 + \kappa_0 \tanh[\kappa_0(l -
x_{R1})] \{k_{n+1} \cot[k_{n+1}(L-l)] \nonumber \\
&& \quad\mbox{} + \kappa_0 \tanh[\kappa_0(l - x_{R1})]\}\biggr) \nonumber \\
&& = C^{(-)}_{nR2} \biggl(k_0^2 - k_{n+1}^2 - k_0 \cot[k_0(L-l)] \{k_{n+1}
\cot[k_{n+1}(L-l)] \nonumber \\
&& \quad\mbox{} - k_0 \cot[k_0(L-l)]\}\biggr) \label{eq:R12-2}
\end{eqnarray}
after making use of equations (\ref{eq:A}) and (\ref{eq:C}) to
eliminate $A^{(+)}_{n+1}$, $B^{(+)}_{n+1}$ and $C^{(+)}_{n+1}$.
Equations (\ref{eq:R12-1}) and (\ref{eq:R12-2}) both yield the
same result
\begin{equation}
C^{(-)}_{nR1} = C^{(-)}_{nR2} \label{eq:C-R12}
\end{equation}
due to the two relations
\begin{equation}
\kappa_0 \tanh[\kappa_0(l - x_{R1})] = - k_0 \cot[k_0(L-l)] \label{eq:relation-1}
\end{equation}
and
\begin{equation}
\kappa_{n+1}^2 - \kappa_0^2 = k_0^2 - k_{n+1}^2
\end{equation}
deriving from (\ref{eq:x_R1}) and (\ref{eq:k-kappa}),
respectively.\par
Since a result similar to (\ref{eq:C-R12}) applies at the
interface between regions $L2$ and $L1$, we conclude that the
partner potential eigenfunctions are given by equations
(\ref{eq:partner-psi-1}) -- (\ref{eq:partner-psi-4}) with
\begin{equation}
C^{(-)}_{nL2} = C^{(-)}_{nL1} = C^{(-)}_{nR1} = C^{(-)}_{nR2} \equiv C^{(-)}_n.
\end{equation}
Such eigenfunctions are $\cal PT$-symmetric provided we choose
$C^{(-)}_n$ imaginary:
\begin{equation}
C^{(-)*}_n = - C^{(-)}_n.
\end{equation}
\par
\subsection{Discontinuities in the partner potential \label{sec:discontinuities}}
In subsection \ref{sec:W}, we have constructed the SUSY partner
$V^{(-)}(x)$ of a piece-wise potential with three discontinuities
at $x = -l$, 0 and $l$. We may now ask the following question:
does the former have the same discontinuities as the latter or
could the discontinuity number decrease? We plan to prove here
that the second alternative can be ruled out.\par
{}For such a purpose, we will examine successively under which
conditions $V^{(-)}(x)$ could be continuous at $x=l$ or at $x=0$
and we will show that such restrictions would not be compatible
with some relations deriving from the unbroken-SUSY assumption
(\ref{eq:unbroken-susy}). Observe that we do not have to study
continuity at $x = -l$ separately, since $V^{(-)}(x)$ being $\cal
PT$-symmetric must be simultaneously continuous or discontinuous
at $x = -l$ and $x=l$.\par
Let us start with the point $x=l$. Matching there
$V^{(-)}_{R1}(x)$ and $V^{(-)}_{R2}(x)$, given in equations
(\ref{eq:partner-0}) and (\ref{eq:partner-bis}), respectively,
leads to the relation
\begin{equation}
- 2 \kappa_0^2 \mathop{\rm sech}\nolimits^2[\kappa_0 (l - x_{R1})] + {\rm i} g = 2 k_0^2 \csc^2[k_0(L-l)].
\end{equation}
On using (\ref{eq:relation-1}) and some simple trigonometric
identities, such a relation can be transformed into $k_0^2 = -
\kappa_0^2 + \frac{1}{2} {\rm i} g$, which manifestly contradicts
equation (\ref{eq:k-kappa}). Hence continuity of $V^{(-)}(x)$ at
$x=l$ is ruled out.\par
Consider next the point $x=0$. On equating $V^{(-)}_{R1}(0)$ with
$V^{(-)}_{L1}(0)$ and employing (\ref{eq:partner-0}) and
(\ref{eq:x_L1}), we obtain the condition
\begin{equation}
- 2 \kappa_0^2 \mathop{\rm sech}\nolimits^2(\kappa_0 x_{R1}) + {\rm i} g = - 2 \kappa_0^{*2}
\mathop{\rm sech}\nolimits^2(\kappa_0^* x_{R1}^*) - {\rm i} g.
\end{equation}
Equations (\ref{eq:cond-BC}) and (\ref{eq:x_R1}) then yield the
relation $- \kappa_0^2 + \frac{1}{2} {\rm i} g = - \kappa_0^{*2} -
\frac{1}{2} {\rm i} g$, which contradicts equation
(\ref{eq:k-kappa}) again. Continuity of $V^{(-)}(x)$ at $x=0$ is
therefore excluded too.\par
We conclude that under the simplest assumption of unbroken SUSY
with a factorization energy equal to the ground-state energy of
$H^{(+)}$, the partner potential $V^{(-)}(x)$ has the same three
discontinuities at $x = -l$, 0 and $l$ as $V^{(+)}(x)$.\par
\section{Discussion \label{sec:discussion}}
\setcounter{equation}{0}
Among all the $\cal PT$-symmetric models, field-theoretical
background explains the lasting interest in the purely imaginary
long-range model $V(x)= {\rm i} x^3$ \cite{Bessis,DDT} and its
generalizations $V(x)=x^2({\rm i} x)^\delta$ with the imaginary
part $V^i(x)$ exhibiting, at any $\delta \in [0,2)$, a
characteristic `strongly non-Hermitian' (SNH) long-range growth in
`coordinate' $x \in \mbox{$\Bbb R$}$. Up to the harmonic oscillator at
$\delta=0$, all of the latter SNH $\cal PT$-symmetric models are
only solvable by approximate methods. Still, rigorous proofs exist
showing that their spectra are all real \cite{DDT}.\par
By rigorous means, the reality of the spectrum has also been shown
for many other $\cal PT$-symmetric potentials $V$. Some of them
turn out to be exactly solvable \cite{SI,BR,BQ}, and those for
which $V^i(\pm \infty)=0$ may be called 'weakly non-Hermitian'
(WNH). Their WNH character is reflected not only by a less
explicit influence of the imaginary part of the potential upon the
spectrum, but also by the existence of SUSY partners \cite{BMQ1,
BR, BMQ2} which, in some special cases, may be real and Hermitian
\cite{Andrianov,BR}.\par
In the light of similar observation one might feel tempted to
perceive WNH models as `partially compatible' with our intuitive
expectations. This impression may be further enhanced by noticing
that another exactly solvable model, viz., the typical WNH spiked
form of the $\delta=0$ harmonic oscillator, as described in
\cite{ptho}, proved of particular interest in the SUSYQM context
as well~\cite{crea,BMQ2}.\par
Potentials $V(x)$ with shapes that are piece-wise constant may be
considered equally exceptional. All of these square-well-type
models with forces located inside a finite interval $(-L,L)$ may
be easily classified by the number of their discontinuities.\par
The simplest nontrivial non-Hermitian square-well potential must
have at least one discontinuity (= matching point at $x=0$). While
the real part of this $V$ is just a trivial shift of the energy
scale, it may be kept equal to zero. Then, the non-zero strength
$Z$ of the spatially antisymmetric and purely imaginary $V$ is the
only free (real) parameter of the whole model with SNH features
\cite{CQ,ptsqw}. Its $\cal PT$-symmetry remains unbroken in an
interval of $Z \in (-Z_{crit}, Z_{crit})$ while its ground-state
energy becomes complex beyond $Z_{crit} \approx 4.48$ (in standard
units $\hbar = 2m = 1$~\cite{ptsqw,gezawell}).\par
It is known that some of these features are generic \cite{Langer}.
Quantitatively, their occurrence has also been confirmed for the
twice-constant SNH model $V$ with two discontinuities
\cite{twopoint}. Qualitatively, all of these observations
facilitate the applicability and physical interpretation of the
piece-wise constant models significantly \cite{Batal}, especially
because the numerical values of the maximal allowed couplings
prove to be, in general, quite large. This allows us to guarantee
the (necessary) reality of the energies by keeping simply our
choice of $Z$ safely below this maximum. \par
The family of WNH square-well models may only start at the
piece-wise potential with three discontinuities. In our present
study of such a model it was important to demonstrate the
parallelism of its properties with the exact solutions of the {\em
smooth} complex potentials of similar shapes \cite{crea}.\par
The most obvious parallel lies in the observation that a key
formal feature of the SUSY partners $H^{(\pm)}$ is that they may
remain both non-Hermitian and $\cal PT$-symmetric. Of course, the
parity ${\cal P}$ cannot define the positive-definite norm
\cite{Mali,Langer,srni,Bpriv}. A consistent physical
interpretation of the similar non-Hermitian models was recently
agreed (cf., e.g., \cite{BBJ}) to lie in the existence of {\em a
new} metric-like operator ${\cal P}_{(+)}>0$ which is positive
definite. This Hermitian operator may be assumed to play the role
of the `physical' metric \cite{Geyer}. This means that once our
equation (\ref{eq:PTS}) is satisfied by the old Hamiltonian and by
the new, {\em positive-definite} metric ${\cal P}_{(+)}$, we may
declare the underlying quantum Hamiltonian quasi-Hermitian,
leading to the standard probabilistic interpretation of the theory
(cf.\ the recent discussions of some related subtleties in
\cite{Kretschmer}). Against this background our attention has been
concentrated upon the feasibility of bound-state construction in a
model with a phenomenologically appealing shape of the
potential.\par
A couple of consequences may be expected. Our model may open the
way towards addressing one of the most difficult problems
encountered in $\cal PT$-symmetric quantum mechanics \cite{Bpriv},
viz., the control of a possible instability of the spectrum
reality \cite{twopoint,fragile}. Indeed, due to the
pseudo-Hermiticity property (\ref{eq:PTS}) of our Hamiltonians
$H$, the energies need not be real (i.e., observable) in principle
\cite{205}.\par
Our WNH model may be also characterized by the simplicity of the
bound-state wavefunctions. This allowed us to construct the
superpotential yielding access, rather easily, to the Witten-type
SUSY hierarchy. In this regard the compact form of our
trigonometric secular equation was welcome and particularly
important, especially for any future projects trying to connect
the mathematical $\cal PT$-symmetry with physical
phenomenology.\par
In such a perspective, the most challenging {mathematical}
problems attached to the non-Hermitian models descend from the
reality of their exceptional points \cite{Heiss}. The simplest
solvable models of the square-well type seem to offer a
transparent laboratory for their study since the indeterminate
auxiliary pseudo-metric $\cal P$ coincides with the common
parity.\par
In the context of physics, the phenomenological appeal of all the
piece-wise constant analogues of the purely imaginary cubic force
represented a strong motivation for the systematic constructions
of the positive-definite metric operators of \cite{Geyer}
(cf.\ also \cite{Mali,205,Batal}). In particular, the highly
appealing factorized form ${\cal P}_{(+)}= {\cal CP}> 0$ of these
metric operators has been used and, for physical reasons, the
factor $\cal C$ itself has been called `charge' (cf.\ \cite{BBJ}). For all
the models with relevance in field theory (like $V \sim ix^3$), the
constructions of $\cal C$ were shown feasible by WKB and perturbative
methods~\cite{joness}.\par
In comparison, the solvability of all the simpler models
facilitates the construction of $\cal C$ (called, usually,
quasi-parity in this context \cite{SI,ptho,srni,Quesne}). An
interesting energy-shift interpretation of the quasi-parity (which
is a new symmetry of the Hamiltonian) emerged in the strongly
spiked short-range model considered in~\cite{Omar}.\par
After we return to the square-well models, the quasi-parity or
charge operator $\cal C$ may be constructed in the specific form
which differs sufficiently significantly from the unit operator
just in a finite-dimensional subspace of the Hilbert space
\cite{Langer,twopoint,Batal}. This is one of the most important
merits of this class of models. It seems to open a new inspiration
for a direct physical applicability of non-Hermitian models
whenever their spectrum remains real. \par
\subsection*{Acknowledgements}
The participation of HB, VJ and MZ complied with the Institutional
Research Plan AV0Z10480505. CQ is a Research Director, National
Fund for Scientific Research (FNRS), Belgium. VJ was supported by
the project no. 2388G-6 of FRVS. MZ was supported by the grant
A1048302 of GA AS.\par
\newpage
\section*{Figure captions}
\par
\vspace{1.2cm}
\subsection*{Figure 1: Solutions of~(\ref{eq:transcendental}) form the semi-ovals.
Their intersections with the hyperbola $2st=g$ determine energy levels
$E=k^2=t^2-s^2$ of the system. Here $g=650$ and $l=0.04$.
}
\subsection*{Figure 2: The previous picture (Fig.1) in $[ts,k]$ plane, where {$ k=\sqrt{t^2-s^2} $}.
We set $g=650$ and $l=0.04$ again.}
\subsection*{Figure 3: Fifty values of critical couplings $g_c$,
increasing rapidly as $l$ decreases, $l\rightarrow 0$. }
\par
\vspace{2cm}
\section*{Table captions}
\par
\vspace{1.2cm}
\subsection*{Table 1: Numerical values of $g_c$ in dependence
on the parameter $l$. The table suggests that the critical coupling
grows
faster than $1/l$ for small
$l$.
}
\par
\vspace{2cm}
\section*{Table 1}
\begin{center}\begin{tabular}{c}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}\hline
$l$&1.00 &0.70 &0.50 &0.40 &0.30 &0.20 &0.10 &0.01&0.001 \\
\hline
$g_c\sim$&4.4753 &4.8129 &6.4364 &8.6011 &13.426 &27.273 &95.832
&9895.4&486950 \\
\hline
\end{tabular}\\
\end{tabular}
\end{center}
\par
\newpage
|
{
"timestamp": "2005-03-03T13:59:17",
"yymm": "0503",
"arxiv_id": "quant-ph/0503035",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503035"
}
|
\section{Introduction}
The quantum walk (QW) is an interesting quantum process that is attracting
much attention from the algorithmic point of view \cite{Ambainis04}, but
also because of its intrinsic interest \cite{Kempe03} through its connection
with quantum cellular automata \cite{Meyer96}, and with the physics of the
systems in which it can be implemented. Two different types of QWs have been
introduced, the so--called discrete and continuous QWs. The discrete QW can
be thought of as a quantum version of the classical quantum walk \cite%
{Meyer96,Aharonov93}, whilst the continuous QW is a quantum generalization
of the Markov chain \cite{Fahri98}. In this article we shall deal only with
the discrete QW.
As stated, the discrete QW can be shortly defined as a quantum counterpart
of the random walk. In the random walk on the line, the "walker" moves to
the right or to the left depending on the output of some random process,
e.g., the toss of a coin. In the QW, the classical coin is substituted by a
quantum one, a qubit, and the coin toss is replaced by some unitary
operation acting on the qubit state, e.g, a Hadamard transformation. After
the unitary operation, the qubit state is in a superposition state and thus
there is a finite probability amplitude for the walker to move, in the same
step, to the left and to the right. This leads to the appearance of
interference phenomena in the probability distribution of the walker
localization that makes it very different from its classical counterpart.
The coined QW in one dimension has been studied extensively along the recent
years \cite%
{Nayak,Konno,Carteret,Kendon03,Lopez03,Knight04,Feldman04,Romanelli04,Romanelli05}%
, and some generalizations of the basic process have been recently proposed
\cite%
{Wojcik04,Inui04,Buerschaper04,Ribeiro04,Romanelli04(b),Omar04,Venegas04}.
Regarding physical implementations, there are a number of proposals that
consider quantum systems, i.e., systems whose dynamics can be described only
within the framework of quantum mechanics \cite%
{Travaglione,Dur,Sanders,Zhao,Di04}. Interestingly enough, the
one--dimensional QW has been shown to be implementable by only classical
means, i.e., in setups whose description does not require quantum mechanics
\cite{Hillery,Knight03,Knight03(b),Jeong}; and, in fact, it has been nearly
implemented in an optical cavity \cite{Bouwmeester}, as it is shown in \cite%
{Knight03,Knight03(b)}. Moreover, it has been claimed that the
one--dimensional QW is an interference phenomenon in which entanglement, a
distinctive quantum feature, does not play any role \cite{Knight03} (see
also \cite{Kendon05} for a different view).
Of course, as it is the case for the random walk, the QW can be defined in a
space of arbitrary dimensionality \cite{Mackay02}. In the multidimensional
case, in which the particle "walks" in a $d$--dimensional space, a qubit is
necessary for each spatial dimension or, in other words, a $d$--dimensional
QW requires a qu$d$it. This makes that the unitary transformations, the
analogous to the coin toss, be more complex that in the unidimensional case.
Multidimensional QWs have been studied in some detail in \cite%
{Tregenna03,Inui04(b)} but, to the best of our knowledge, no proposal for
its implementation is available to this day. In this article, we propose a
way for implementing the two--dimensional quantum walk in an optical cavity.
\section{Two--dimensional quantum walk}
Let us briefly introduce the two--dimensional QW, whose implementation is
our main goal. Consider a single particle (the walker) and a qu$d$it with
four states that plays the role of the coin. Notice that the qu$d$it can
correspond to internal states of the particle, although not necessarily. Let
$\mathcal{H}_{P}$ be the Hilbert space of the particle positions on the
plane and
\begin{equation}
\left\{ \left\vert x,y\right\rangle =\left\vert x\right\rangle \left\vert
y\right\rangle ,x,y\in \mathrm{Z}\right\} ,
\end{equation}%
a basis of $\mathcal{H}_{P}$; and let $\mathcal{H}_{C}$ be the
four--dimensional Hilbert space describing coin--qu$d$it, and $\left\{
\left\vert u\right\rangle ,\left\vert d\right\rangle ,\left\vert
r\right\rangle ,\left\vert l\right\rangle \right\} $ a basis of $\mathcal{H}%
_{C}$. The state of the total system belongs to the space $\mathcal{H}=%
\mathcal{H}_{C}\otimes \mathcal{H}_{P}$, and at a given instant of time, say
at iteration $n$, can be expresed as
\begin{equation}
\left\vert \psi \right\rangle _{n}=\sum_{x,y}\left[ r_{x,y}^{\left( n\right)
}\left\vert x,y,r\right\rangle +l_{x,y}^{\left( n\right) }\left\vert
x,y,l\right\rangle +u_{x,y}^{\left( n\right) }\left\vert x,y,u\right\rangle
+d_{x,y}^{\left( n\right) }\left\vert x,y,d\right\rangle \right] ,
\end{equation}%
where the notation is self--explicative.
The dynamics of the system is governed by two physical operations: (i), the
conditional displacement, represented by the operator $\hat{D}$ acting on $%
\mathcal{H}_{P}$
\begin{eqnarray}
\hat{D}\left\vert x,y,r\right\rangle &=&\left\vert x+1,y,r\right\rangle ,\ \
\ \ \hat{D}\left\vert x,y,l\right\rangle =\left\vert x-1,y,l\right\rangle ,
\label{D1} \\
\hat{D}\left\vert x,y,u\right\rangle &=&\left\vert x,y+1,u\right\rangle ,\ \
\ \hat{D}\left\vert x,y,d\right\rangle =\left\vert x,y-1,d\right\rangle ,
\label{D4}
\end{eqnarray}%
i.e., the walker is displaced up, down, rigth or left when the coin is in
the state $\left\vert r\right\rangle $, $\left\vert l\right\rangle $, $%
\left\vert u\right\rangle $, or $\left\vert d\right\rangle $, respectively;
and (ii), the unitary transformation acting on the internal states of the
coin, represented by a unitary operator $\hat{C}_{4}$, which acts on $%
\mathcal{H}_{C}$ and that can be written as a $4\times 4$ matrix. Two
special cases that have been considered in the literature \cite%
{Mackay02,Tregenna03,Inui04(b)} are the Grover coin%
\begin{equation}
\hat{C}_{4,G}=\frac{1}{2}\left(
\begin{array}{cccc}
-1 & 1 & 1 & 1 \\
1 & -1 & 1 & 1 \\
1 & 1 & -1 & 1 \\
1 & 1 & 1 & -1%
\end{array}%
\right) , \label{Grover}
\end{equation}%
and the DFT (discrete Fourier transform) coin%
\begin{equation}
\hat{C}_{4,DFT}=\frac{1}{2}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & i & -1 & -i \\
1 & -1 & 1 & -1 \\
1 & -i & -1 & i%
\end{array}%
\right) . \label{DFT}
\end{equation}
The state of the system after $n$ steps of the walk can be written as
\begin{equation}
\left\vert \psi \right\rangle _{n}=\left( \hat{C}_{4}\hat{D}\right)
^{n}\left\vert \psi \right\rangle _{0},
\end{equation}%
with $\left\vert \psi \right\rangle _{0}$ the initial state of the system.
Finally, the probability distribution for the particle be at position $%
\left( x,y\right) $ after $n$ iterations is given by%
\begin{equation}
P\left( x,y;n\right) =\sum_{c\in \left\{ r,l,u,d\right\} }\left\vert
\left\langle x,y,c\right. \left\vert \psi \right\rangle _{n}\right\vert
^{2}=\sum_{c\in \left\{ r,l,u,d\right\} }P^{c}\left( x,y;n\right) ,
\label{probability}
\end{equation}%
with $P^{c}\left( x,y;n\right) =\left\vert c_{x,y}^{\left( n\right)
}\right\vert ^{2}$ the probability distributions for the particle be at
position $\left( x,y\right) $ and the coin in state $\left\vert
c\right\rangle $, $c\in \left\{ r,l,u,d\right\} $.
\section{Implementation}
In order to implement the two--dimensional QW one needs a walker that can
walk in two orthogonal directions, a plane, and a four--state qu$d$it. Here
we propose an implementation of this process that makes use of classical
resources only, following the same spirit as in \cite{Knight03,Knight03(b)}:
The four states of the coin will correspond to four different spatial paths
that the light field can follow (what, in the notation of \cite{Knight03(b)}%
, borrowed from \cite{Spreeuw01}, corresponds to a four--state position
cebit), and the walker role will be played by the field frequency, again as
in \cite{Knight03,Knight03(b)}, that can be increased or decreased in the
two orthogonal directions corresponding to two orthogonal polarization
states of the light field, say $\mathbf{x}$ and $\mathbf{y}$.
In Fig. 1 a schematic of the first step of the QW is schetched. In Fig. 1(a)
the four parallel light beams, which propagate along the z--axis and are
linearly polarized at $\pi /4$ with respect to the x--axis, first cross an
array of devices that perform the conditional displacement, Eqs. (\ref{D1}%
)--(\ref{D4}): The frequency of the $\mathbf{x}$--polarized ($\mathbf{y}$%
--polarized) light is increased or decreased in beams marked with $r$ or $l$
($u$ or $d$), respectively. Each of these devices can consist, e.g., of a
polarization beam--splitter (that separates the two--polarization components
of the incident beam, the frequency of one of which is suitably increased or
decreased by means of an electrooptic modulator), plus two mirrors and a
second polarization beam--splitter for recombining the two polarization
components back into a single beam after the frequency displacement. After
the implementation of $\hat{D}$, the four beams cross a second device in
which the $\hat{C}_{4}$ operation is implemented. Let us see how this
operation can be done.
In Fig. 2, a schematic of the device performing $\hat{C}_{4}$ is shown. The
four incoming beams suffer five transformations when crossing the $\hat{C}%
_{4}$ device. First, some phase is added to each of the fields, let us call
this operation $\hat{F}_{1}$, which is represented by the operator%
\begin{equation}
\hat{F}_{j}=\left(
\begin{array}{cccc}
e^{i\phi _{j1}} & 0 & 0 & 0 \\
0 & e^{i\phi _{j2}} & 0 & 0 \\
0 & 0 & e^{i\phi _{j3}} & 0 \\
0 & 0 & 0 & e^{i\phi _{j4}}%
\end{array}%
\right) . \label{Fi}
\end{equation}%
with $j=1$. After $\hat{F}_{1}$, beams $r$ and $l$ (and, separately, beams $%
u $ and $d$) are mixed in a beam splitter, let us call this operation $\hat{S%
}_{1}$, which in matrix form reads%
\begin{equation}
\hat{S}_{1}=\left(
\begin{array}{cccc}
\cos \theta _{11} & i\sin \theta _{11} & 0 & 0 \\
i\sin \theta _{11} & \cos \theta _{11} & 0 & 0 \\
0 & 0 & \cos \theta _{12} & i\sin \theta _{12} \\
0 & 0 & i\sin \theta _{12} & \cos \theta _{12}%
\end{array}%
\right) . \label{S1}
\end{equation}%
Then, the third step is similar to the first one, i.e., the phase of the
four beams are increased again. This is represented by the matrix Eq. (\ref%
{Fi}) with $j=2$. In the fourth step, similar to the second one, beams $r$
and $u$ (and, separately, beams $l$ and $d$) are mixed in a beam splitter,
let us call this operation $\hat{S}_{2}$. This is represented by
\begin{equation}
\hat{S}_{2}=\left(
\begin{array}{cccc}
\cos \theta _{21} & 0 & i\sin \theta _{21} & 0 \\
i\sin \theta _{21} & 0 & \cos \theta _{21} & 0 \\
0 & \cos \theta _{22} & 0 & i\sin \theta _{22} \\
0 & i\sin \theta _{22} & 0 & \cos \theta _{22}%
\end{array}%
\right) . \label{S2}
\end{equation}%
The final step is a new dephasing of the beams, represented by Eq. (\ref{Fi}%
) with $j=3$. The global effect of these five operations is given by
\begin{equation}
\hat{C}_{4}=\hat{F}_{3}\cdot \hat{S}_{2}\cdot \hat{F}_{2}\cdot \hat{S}%
_{1}\cdot \hat{F}_{1},
\end{equation}%
whose matrix elements can be writen as
\begin{equation}
\hat{C}_{4}=\left(
\begin{array}{cccc}
c_{11}c_{21}e^{i\alpha _{11}} & is_{11}c_{21}e^{i\alpha _{12}} &
ic_{12}s_{21}e^{i\alpha _{13}} & -s_{12}s_{21}e^{i\alpha _{14}} \\
ic_{11}s_{21}e^{i\alpha _{21}} & -s_{11}s_{21}e^{i\alpha _{22}} &
c_{12}c_{21}e^{i\alpha _{23}} & is_{12}c_{21}e^{i\alpha _{24}} \\
is_{11}c_{22}e^{i\alpha _{31}} & c_{11}c_{22}e^{i\alpha _{32}} &
-s_{11}s_{22}e^{i\alpha _{33}} & ic_{12}s_{22}e^{i\alpha _{34}} \\
-s_{11}s_{22}e^{i\alpha _{41}} & ic_{11}s_{22}e^{i\alpha _{42}} &
is_{12}c_{22}e^{i\alpha _{43}} & c_{12}c_{22}e^{i\alpha _{44}}%
\end{array}%
\right) , \label{Cs}
\end{equation}%
with $s_{ij}=\sin \theta _{ij}$ and $c_{ij}=\cos \theta _{ij}$. The phase
factors appearing in (\ref{Cs}) are related with the phase factors in (\ref%
{Fi}) through
\begin{eqnarray}
\alpha _{11} &=&\phi _{11}+\phi _{21}+\phi _{31},\ \ \ \ \ \alpha _{12}=\phi
_{12}+\phi _{21}+\phi _{31}, \\
\alpha _{13} &=&\phi _{13}+\phi _{23}+\phi _{31},\ \ \ \ \ \alpha _{14}=\phi
_{14}+\phi _{23}+\phi _{31}, \\
\alpha _{21} &=&\phi _{11}+\phi _{21}+\phi _{32},\ \ \ \ \ \alpha _{22}=\phi
_{12}+\phi _{21}+\phi _{32}, \\
\alpha _{23} &=&\phi _{13}+\phi _{23}+\phi _{32},\ \ \ \ \ \alpha _{24}=\phi
_{14}+\phi _{23}+\phi _{32}, \\
\alpha _{31} &=&\phi _{11}+\phi _{22}+\phi _{33},\ \ \ \ \ \alpha _{32}=\phi
_{12}+\phi _{22}+\phi _{33}, \\
\alpha _{33} &=&\phi _{13}+\phi _{24}+\phi _{33},\ \ \ \ \ \alpha _{34}=\phi
_{14}+\phi _{24}+\phi _{33}, \\
\ \alpha _{41} &=&\phi _{11}+\phi _{22}+\phi _{34},\ \ \ \ \ \alpha
_{42}=\phi _{12}+\phi _{22}+\phi _{34}, \\
\alpha _{43} &=&\phi _{13}+\phi _{24}+\phi _{34},\ \ \ \ \ \alpha _{44}=\phi
_{14}+\phi _{24}+\phi _{34},\ \ \ \ \label{alfas}
\end{eqnarray}
Then, the operations performed for constructing $\hat{C}$ provide a class of
possible transformations, and depending on the values of parameters $\theta
_{ij}$ ($i,j=1,2$) and $\phi _{ij}$, through Eqs. (\ref{alfas}), different
transformations are obtained. For example, the Grover coin $\hat{C}_{4G}$,
Eq.(\ref{Grover}), is obtained by taking
\begin{equation}
\theta _{11}=\theta _{12}=\theta _{21}=\theta _{22}=\pi /4, \label{m}
\end{equation}%
for the beam splitters, and
\begin{eqnarray}
\phi _{11} &=&\frac{\pi }{4},\ \ \ \ \notag \\
\phi _{12} &=&\phi _{14}=\phi _{31}=\phi _{34}=0, \notag \\
\phi _{13} &=&-\phi _{21}=-\phi _{22}=\phi _{32}=\phi _{33}=\frac{\pi }{2},\
\ \notag \\
\ \phi _{23} &=&\phi _{24}=\pi ,
\end{eqnarray}%
for the phase filters. With respect to the DFT coin, Eq. (\ref{DFT}), it is
a little bit more complicated: By taking again (\ref{m}) for the beam
splitters and
\begin{eqnarray}
\phi _{11} &=&\phi _{13}=\phi _{22}=\phi _{23}=\phi _{24}=0, \notag \\
\phi _{12} &=&\phi _{14}=-\phi _{21}=\phi _{31}=\phi _{33}=-\frac{\pi }{2},
\notag \\
\phi _{32} &=&\phi _{34}=-\pi ,
\end{eqnarray}%
for the phase filters one obtains%
\begin{equation}
\hat{C}_{4,DFT}^{\prime }=\frac{1}{2}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & i & -i \\
1 & -1 & -i & i%
\end{array}%
\right) , \label{DFTbis}
\end{equation}%
which is very similar to Eq. (\ref{DFT}). In fact, the DFT matrix is
obtained from Eq. (\ref{DFTbis}) by making%
\begin{equation}
\hat{C}_{4,DFT}=\hat{A}\cdot \hat{C}_{4,DFT}^{\prime }\cdot \hat{A}^{-1},
\end{equation}%
with
\begin{equation}
\hat{A}=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1%
\end{array}%
\right) . \label{A}
\end{equation}%
Notice that operator $\hat{A}$ interchanges indexes 2 and 3, what physicaly
means that the light beams $l$ and $u$ must be permuted at the entrance and
at the exit of the scheme in Fig. 1, what can be done by means of a Kepler
telescope.
Up to this point we have seen that a single step of the QW in two dimensions
can be performed by the device represented in Fig. 1. In order to perform $n$
steps, we only need to reinject the output of the device at its entrance.
This is readily achieved by using optical cavities (in Fig. 3 we show a
scheme of the complete setup). In the device, the initial condition is
chosen by fixing the phases and intensities of the four incident beams, and
at the cavity output, the frequency of the emerging field performs the
two--dimensional QW. Of course the output field spectrum must be analyzed,
with polarizers and frequency analizers, in order to extract the
two--dimensional QW: After passing a linear polarizer set to $0%
{{}^o}%
$ ($90%
{{}^o}%
$), from the spectrum of the polarized field one obtains $P\left(
x,0;n\right) $ ($P\left( 0,y;n\right) $), which suitably combined provide $%
P\left( x,y;n\right) $.
Let us note that the use of optical cavities imposses some restrictions (see
\cite{Knight03(b)} for a more detailed discussion on these) as, e.g., the
intracavity field frequencies must resonate with the cavity modes, unless it
be a pulse with a duration shorter than the cavity roundtrip time. Also one
must take care that the optical paths of the different beams be equal and
that the polarization of the light field does not suffer variations along
the roundtrip (what prevents the use of optical fiber cavities). But these
technicalities can be readily solved.
Finally it is worth commenting that the device we are proposing here can
also implement the QW on the line with two coins, as recently proposed in
Ref. \cite{Inui04}. For that purpose, we only need to not distinguish
between the two polarization states of the light, i.e., the walk has to be
performed on a unique dimension, namely, the frequency of the field.
\section{Conclusion}
We have proposed an experimental setup for the implementation of the
two--dimensional QW. Our device consists of classical resources only and has
the advantage that the unitary transformation performed in it is tunable in
the sense that by modifying the parameters of the system, different unitary
transformations can be easily reproduced. The device we are proposing can be
generalized to implement the QW on the circle in either one or the two
dimensions by following the same technical solutions already proposed for
the one--dimensional QW \cite{Knight03(b)}.
The fact that the two--dimensional QW can be implemented by only classical
means suggests, as it was the case for the one--dimensional QW \cite%
{Knight03,Knight03(b)}, that it is a classical process in which nonlocal
entanglement plays no role. Recently \cite{Kendon05} this conclussion has
been discussed and we refer the reader to Ref. \cite{Kendon05} for more
details, as we are not going to discuss this here. Nevertheless, let us
emphasize that in higher dimensional QWs, e.g., the three--dimensional one,
quantum entanglement manifests in the amount of classical resources needed
for the implementation, as the implementation of the three necessary qubits
requires 8 light beams (in general, $n$ qubits would require $2^{n}$ light
beams \cite{Spreeuw01}). In this sense, the two-dimensional QW is the higher
dimensional one that can be implemented classically without a sensible
difference in the resources needed as compared with a \emph{quantum}
implementation.
This work has been financially supported by Spanish Ministerio de Ciencia y
Tecnolog\'{\i}a and European Union FEDER, Project BFM2002-04369-C04-01. We
gratefully acknowledge fruitful discussions with Germ\'{a}n J. de Valc\'{a}%
rcel.
|
{
"timestamp": "2005-03-07T16:18:33",
"yymm": "0503",
"arxiv_id": "quant-ph/0503069",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503069"
}
|
\section{Introduction}
Optical, non imaging detectors are widely used for the detection of weakly
interacting particles. At present the main focus of observation is on
neutrinos and antineutrinos from various sources, but there are also plans to
construct large optical detectors to search for as yet undiscovered particles
such as WIMPs. The detection mechanism is based on the collection of visible
or ultraviolet photons. These are emitted as \v{C}erenkov radiation ({\it
e.g.}, as in Kamiokande~\cite{super-k} and SNO~\cite{sno}) or as
scintillation photons. We will focus our attention in this paper on
scintillator-based, unsegmented detectors.
\subsection{A Brief History of Scintillation Detectors}
The history of scintillator-based detectors is heavily intertwined with that of
neutrino physics. The first neutrino detector ever built, that of Cowan and
Reines in 1953, was a 10.7\,ft$^3$ cylinder filled with a cadmium-doped organic
scintillator and wavelength shifter, which detected
reactor-generated $\bar{\nu_e}$'s by observing the coincidence of $e^+$
annihilation and neutron capture following the inverse beta decay reaction
$\bar{\nu_e}(p, n)e^+$~\cite{cr-scint,cr-discovery,cr-confirmation}.
However, the first large-scale unsegmented liquid scintillator detector was not
built until about 1980. The 100~ton neutrino detector at Artemovsk, Ukraine, a
cylindrical 5.6\,m $\times$ 5.6\,m tank filled with a saturated hydrocarbon
scintillator and fluor, was a direct descendant of Reines and Cowan's original
design. Indeed, it was designed to detect antineutrinos using the same
reactions~\cite{beresnev}. It was buried in a salt mine, 600 meters water
equivalent (m.w.e.) underground, and was later used to study the
interactions of cosmic ray muons with scintillator~\cite{enikeev}.
The 1995 Counting Test Facility (CTF) prototype of the Borexino experiment
further developed the architecture of scintillator-based detectors~\cite{ctf}.
This 4~ton detector was intended primarily as a test bed for technologies of
the full-scale Borexino detector, not as a neutrino detector in its own right.
Nevertheless, it set a record for the lowest detector background achieved
at the time, of 0.03\,counts/(kg\,keV\,yr), in the window 250\,keV to
2.5\,MeV~\cite{ctf-results}. It has as a result produced new upper bounds on
various exotic processes~\cite{ctf-exotic}. Unlike previous scintillation
detectors, it is spherical in design, in order to keep as much
scintillator away from the surface as possible. Liquid scintillator (both
pseudocumene and phenylxylylethane, at different times, again with added
fluors) is contained in a thin spherical nylon balloon, surrounded by 100
inward-facing photomultiplier tubes. This setup is contained in 1000~tons of
ultrapure water in a cylindrical tank. The entire detector is 3400~m.w.e.
underground in the Gran Sasso National Laboratory, in central Italy.
The CTF first established the feasibility of a scintillator-based {\it solar}
neutrino detector with a detailed study of the radioactive contaminants
internal to the scintillator. It was also the first scintillation detector to
introduce an inactive buffer (water) between the active volume of scintillator
and the photomultiplier tubes. As well, it has the capability of position
reconstruction for point-like events.
The CHOOZ detector~\cite{chooz}, built to study oscillations in reactor
antineutrinos from a nuclear power plant by the same name in northern France,
took data in 1997-98. Its layered design incorporated key features of the CTF
and Borexino designs, as well as those of other neutrino detectors such as the
\v{C}erenkov detector SNO~\cite{sno}, and the hybrid \v{C}erenkov/scintillation
light detector LSND~\cite{lsnd}. (The design of the larger hybrid detector
MiniBooNE, built in 1999 in order to confirm or refute results from LSND by
observing 0.5-1\,GeV muon neutrinos produced at the FNAL accelerator,
was based upon the same principles~\cite{miniboone}.)
The interior of the CHOOZ detector featured a
central 5~ton Gd-doped target mass inside a clear roughly egg-shaped Plexiglas
container, surrounded by an undoped 17~ton inactive buffer region contained in
an oblong ``geode,'' and an outer undoped 90~ton volume with its own set of
PMTs, used for vetoing muons from cosmic rays. The detector was placed
at a depth of 300~m.w.e.
The current generation of unsegmented detectors based on organic liquid scintillators -
KamLAND~\cite{kamland}, taking data since 2002, and Borexino~\cite{bx}, soon to
begin operations - retain this sort of layered design, both using the spherical
shape of the CTF. Unlike the detectors described already, Borexino will
observe scintillation light due directly to neutrino scattering from electrons,
and can therefore potentially detect neutrinos with much lower energies (the
threshold $\nu$ energy for the inverse $\beta$ decay is 1.8\,MeV). KamLAND has
observed disappearance of $\bar{\nu_e}$ from reactors using the inverse
$\beta$ decay signature~\cite{kamland-results}, but it is also intended to
observe solar neutrinos directly via
$\nu$-$e$ scattering in the future. The current KamLAND background in the
region below 2\,MeV must be drastically reduced for that goal to be
achieved~\cite{kamland-background}. These detectors are
situated much deeper underground
(Borexino: 3400 m.w.e; KamLAND: 2700 m.w.e.), for further reduction of the
residual muon flux and the production of short-lived cosmogenic isotopes.
Two new experiments with targets of liquified noble gas, also aiming at low
energy solar neutrino detection via the detection of scintillation light,
are currently under development: CLEAN~\cite{clean} and XMASS~\cite{xmass}.
In the case of CLEAN, wavelength-shifter coated windows are offset from the
PMTs by a 5-10\,cm gap which is a thin inactive buffer region.
For these detectors, reliable determination
of the positions of events is even more important, due to the need of rejecting
the higher background rate coming from scintillation events produced in
proximity of the PMTs and container vessel. An additional complication arises because the mean
scattering length of scintillation photons (produced in the ultraviolet range
of the spectrum for noble gases) is much less than the radius of the detector;
scintillation photons propagate from the event origin to the PMTs in a
diffusive mode. Therefore the times of arrival of detected photons provide
less information than in detectors using organic scintillator; these
noble gas detectors will rely heavily upon the
spatial pattern of PMT hits to reconstruct the positions of events.
\subsection{The Necessity of Spatial Event Reconstruction}
Due to the extremely low interaction rates of neutrinos and their antiparticles
(to say nothing of WIMPs and so forth), it is necessary for a detector to
contain a large mass of scintillator with very low levels of internal
radioactive contamination~\cite{bx}. Ultra-pure materials are also used to
screen radioactivity from materials surrounding the detector~\cite{bx,bx-rad}.
Unfortunately, the photosensitive elements used to detect scintillation light
are notorious for being among the main sources of radioactivity in an
ultra-low-background detector.
It is therefore desirable to insert, between the photosensitive elements and
the scintillator, one or more layers of buffer material to suppress radioactive
background. Often the buffers are inactive, {\it i.e.}, not scintillating. An
inactive buffer offers the advantage of minimizing the total trigger rate
caused by the abundant radioactive decays generally produced within the
photosensitive elements~\cite{bx}. On the other hand, if the compositions of
the scintillator and inactive buffer are different, a scintillator containment
system is required to physically separate them~\cite{bx}. The
containment system, being in direct contact with the scintillator,
must satisfy extremely stringent requirements in terms of intrinsic
radiopurity.
For additional background prevention, the outer region of the scintillator
volume can be used as an active buffer. This allows any residual radioactivity
coming from the containment system, or passing through it, to be monitored and
suppressed. A ``fiducial volume'' is commonly defined as a region at
the center of the active volume of the detector in which radioactive background
is expected to be at a minimum. The discrimination between
events belonging to the fiducial and to the non-fiducial regions is performed
by means of software implementation (reconstruction code) of an algorithm
(reconstruction algorithm), which assigns to each single event a reconstructed
position, either inside or outside the fiducial volume. The algorithm also
provides a means of comparing the position of different events and is an
important tool for the identification of several background sources. The
designs of some planned detectors incorporate only a thin inactive buffer
region or none at all, and in these cases, correct assignment of
an event as belonging to the fiducial volume or the buffer region is even more
important. The resolution of detector reconstruction codes are generally
studied with Monte Carlo methods. Event simulations allow close reproductions
of the performance of these codes on real events. Typically, however, the
reconstruction codes are fine tuned by calibrating the detector with the use of
localized sources of radioactivity or light.
What seems lacking from the available literature is a comprehensive discussion
of how the resolutions of detector reconstruction codes are related to some
basic properties of the detector: the linear dimension, the time dispersion of
the photon emission, the scintillator index of refraction, possible processes
of absorption and re-emission and of scattering of the scintillator light, etc.
In this paper we present an analytic study of the resolution for reconstruction
in time and space of scintillation events. The study is restricted, for
simplicity, to the case of events at the center of the detector, simple enough
to be treated, within certain approximation, analytically. Calibrations of
experiments~\cite{ctf-results} and full Monte Carlo studies of the performance
of proposed experiments~\cite{ctf-light} show anyhow that the resolution of the
reconstruction codes depends only in a mild way upon the location of the
scintillation event.
This study also assumes that the optical properties of the media are uniform
throughout the detector, and that the indices of refraction of all materials
between the active scintillator and the photodetectors are approximately the
same.
\section{Likelihood Function Derivation}
The likelihood function is a standard statistical tool used for finding parameters of a physical model. Suppose that a set of $N$ observations is composed of the independent values $\{t_i\}$ and dependent values $\{s_i\}$ ($i = 1, ..., N$). For instance, $\{t_i\}$ could be a list of times at which a radioactive sample is observed, and $\{s_i\}$ a list of observed activities at each time. We wish to model the data using some function $f(s)$ with $n$ free parameters $\vec{a}$. In the example, the function would be a decaying exponential, and the parameters would be the initial activity and the half-life. By definition, the likelihood function over the parameters is a probability distribution of obtaining
the observed data given a specific set of parameters:
\begin{equation}
\mathcal{L}(\vec{a_0};\, \{(t_i, s_i)\})
\; = \; \mathrm{P}(\{(t_i, s_i)\} \; \mathrm{are\; observed}
\;|\; \vec{a} \;=\; \vec{a_0}).
\label{e:base-likelihood}
\end{equation}
The difficult task is to calculate this probability based on the assumption
that the data are correctly described by the model function $f(s)$. Once
this has been done, in order to calculate the most probable value of the
parameters of the model, one simply finds the maximum of the likelihood
function (or, as is usually computationally easier, the minimum of $-\log
\mathcal{L}$) in the $n$-dimensional space defined by the free parameters
$\vec{a}$.
In the case of a scintillator-based detector, the parameters of interest are the position and time of an event in the detector, $\vec{a} = (\vec{x_0}, t_0)$. The observed data are the positions $\{\vec{x_i}\}$ of the
photosensitive elements, usually PMTs (independent values), and the times $\{t_i\}$ at which each element is hit by a photon (dependent values); $i$ ranges from 1 to $N$,
with $N$ being the number of detected photons. For now we assume that at most one photon is detected by each PMT, so all the $\vec{x_i}$'s are distinct, and $N$ is also the number of PMTs that detect a photon.
For conciseness, define the following possible events:
\begin{itemize}
\item{A : detector event occurs at $(\vec{x_0}, t_0)$}
\item{B : detector hit pattern is $\{(\vec{x_i}, t_i)\}$.}
\end{itemize}
Then, Equation~(\ref{e:base-likelihood}) becomes
\begin{equation}
\mathcal{L}(\vec{x_0}, t_0; \{(\vec{x_i}, t_i)\}) \; \equiv \;
\mathrm{P}(\mathrm{B} | \mathrm{A}).
\label{e:bayes}
\end{equation}
\subsection{Factoring the Detector Likelihood Function}
Let us assume that the times at which photons are emitted by the scintillator
are uncorrelated. Then the likelihood function will have one independent
factor for the piece of data provided by each PMT\footnote{
Strictly speaking, this is not exactly true; specifying that $N$
PMTs detected photons causes the PMT hit data to be correlated. For a
reasonably large number of hit PMTs, though, the difference should be
negligible. It would be interesting to compare results derived from
the often-used Poisson and multinomial probabilistic models to the model
put forth here}. Let the total number of working PMTs be $T$, so
that $N$ PMTs (labeled $1, \ldots, N$) have detected a
photon, and $T - N$ PMTs (labeled $N + 1, \ldots, T$) have not. If we
further define
\begin{itemize}
\item{C$_i$ : PMT $i$ is hit}
\item{D$_i$ : PMT $i$ detects a photon}
\item{E$_i$ : PMT $i$ detects a photon at time $t_i$,}
\end{itemize}
then
\begin{eqnarray}
\mathrm{P}(\mathrm{B} | \mathrm{A})\; & = & \;
\prod_{i = 1}^N
\mathrm{P}(\mathrm{E}_i | \mathrm{A}, \, \mathrm{C}_i, \, \mathrm{D}_i)\;
\mathrm{P}(\mathrm{D}_i | \mathrm{A}, \, \mathrm{C}_i)\;
\mathrm{P}(\mathrm{C}_i | \mathrm{A}) \nonumber \\
& & \times\; \prod_{j = N+1}^T \left[
\mathrm{P}(\neg \mathrm{D}_j | \mathrm{A}, \, \mathrm{C}_j)\;
\mathrm{P}(\mathrm{C}_j | \mathrm{A})\; + \;
\mathrm{P}(\neg \mathrm{C}_j | \mathrm{A}) \right]
\label{e:likelihood-factors}
\end{eqnarray}
(where $\neg$ is the logical negation symbol).
Of course, $\mathrm{P}(\mathrm{D}_i | \mathrm{A}, \, \mathrm{C}_i)$ is just
the quantum efficiency $q_i$ of PMT $i$, which is, to a first
approximation, independent of the original event position.
Now define a ``per-PMT'' likelihood function $\mathcal{L}_i$.
\begin{equation}
\mathcal{L}_i(\vec{x_0}, t_0; \vec{x_i}, t_i) \; = \; \left\{
\begin{array}{ll}
q_i\, \mathrm{P}(\mathrm{E}_i | \mathrm{A}, \, \mathrm{C}_i, \, \mathrm{D}_i)\;
\mathrm{P}(\mathrm{C}_i | \mathrm{A}), & i \le N
\\
(1 - q_i) \,
\mathrm{P}(\mathrm{C}_i | \mathrm{A})\; + \;
\mathrm{P}(\neg \mathrm{C}_i | \mathrm{A}), & N < i \le T
\end{array}
\right.
\label{e:per-pmt}
\end{equation}
The total likelihood function is then the product of all per-PMT likelihood
functions. Notice that the per-PMT
likelihood function of a supposedly dead PMT ($q_i = 0$) that does not detect a
photon reduces to 1, so does not influence the total likelihood function, just
as expected.
\subsection{Scintillator Dispersion Time at the Emission Point}
The first factor in the expression for the likelihood function of a PMT that
detects a photon is based solely on timing information of a photon emitted by
the scintillator.
Scintillation photons are emitted as a consequence of the ionization of the scintillator due to interacting particles or radioactive decays. The typical dispersion in the time of emission of organic liquid scintillators is on the order of a few nanoseconds, with a slower component that can reach hundreds of nanoseconds. The emission of photons is uniform over the solid angle. In this discussion we assume that the time of emission of each photon, relative to the time of the event causing scintillation, is an independent random variable $\tau_e$.
Suppose the distribution of the random variable $\tau_e$ is given by some
scintillator response function $p(\tau_e)$. Referring to the left half of
Figure~\ref{f:scintpdf}, one sees that at a specific time $t$, this function
may also be regarded as an outgoing spherical photon probability wave,
integrated over the solid angle 4$\pi$. In fact, the most important factor in
Equation~(\ref{e:likelihood-factors}), the probability $\mathrm{P}(\mathrm{E}_i
| \mathrm{A}, \, \mathrm{C}_i, \, \mathrm{D}_i)$, is equal to it. Let
$\tau_f^i$ be the time of flight from the origin $\vec{x_0}$ of the photon to
the position $\vec{x_i}$ of the $i^{th}$ PMT. Then, with $n$ being the
scintillator index of refraction, we have:
\begin{eqnarray}
\tau_f^i & = & \difrac{\left|\vec{x_i}-\vec{x_0}\right|n}{c} \\
t_i & = & \tau_e + \tau_f^i + t_0.
\end{eqnarray}
As a result,
\begin{equation}
\mathcal{L}_i(\vec{x_0}, t_0; \vec{x_i}, t_i) \; \propto \;
p(t_i - t_0 - \tau_f^i).
\end{equation}
Of course, factors other than the dispersion time of the scintillator may also
affect the probability distribution function of the recorded arrival times
of photons at PMTs. The most important other effects are usually the
effects of scattering in the scintillator and the
finite time resolution of the PMTs themselves. The latter may in general be
incorporated into the distribution $p(\tau_e)$ by convolution with the
scintillator dispersion function. The former requires a bit more care
because scattering effects depend in general upon the light path length
from the event to the PMT; an exact treatment is beyond the scope of this
paper.
\begin{figure}[t!]
\begin{center}
\psfrag{nhat}{$\hat{n}$}
\psfrag{tau}{$\tau_e = t - t_0 - \tau_f^i$}
\psfrag{pdftau}{$p(\tau_e)$}
\psfrag{event}{$(\vec{x_0}, t_0)$}
\psfrag{detector}{$(\vec{x_i}, t_i)$}
\psfrag{da}{$\mathrm{d}A_i$}
\psfrag{domega}{$\mathrm{d}\Omega_i$}
\psfrag{psi}{$\psi_i$}
\epsfig{file=scintpdf.eps,height=3in}
\end{center}
\caption{Geometry of the likelihood function derivation. The concentric
dotted lines, and the graph on the left, represent the probability function
(an expanding spherical wave) of the emission time of a scintillation photon. The rectangle labeled d$A_i$ represents a PMT of infinitesimal size with normal vector $\hat{n}$, subtending a solid angle d$\Omega_i$ as seen from the position of the detector event. The PMT is tilted away from the direction of the event by an angle $\psi_i$. Note that we have
not yet made any assumptions about the geometry of the detector.}
\label{f:scintpdf}
\end{figure}
\subsection{Photon Attenuation}
As photons travel away from their origin, they are attenuated by the
familiar inverse square law. This implies a formula for the probability
$\mathrm{P}(\mathrm{C}_i | \mathrm{A})$ that
a given PMT is hit by a scintillation photon. Suppose a PMT of infinitesimal
area,
at a distance $s_i \equiv \left| \vec{x_i}-\vec{x_0} \right|$ from
the event, subtends a solid angle d$\Omega_i$ as seen from the event location.
Assuming a perfect collection efficiency, it will collect only a fraction
d$\Omega_i / 4\pi$ of all photons emitted. So if $\Gamma$ photons were
emitted, its probability of being struck by at least one of them is
\begin{equation}
\mathrm{P}(\mathrm{C}_i | \mathrm{A}) \; = \;
1 - \left(1 - \frac{\mathrm{d}\Omega_i}{4\pi}\right)^\Gamma
\; \approx \; \Gamma \frac{\mathrm{d}\Omega_i}{4\pi}.
\end{equation}
If the $i^{th}$ PMT has an area d$A_i$ and is tilted away from
the line of sight by an angle $\psi_i$, as shown on the right half of
Figure~\ref{f:scintpdf}, then
$\mathrm{d}\Omega_i = \cos{\psi_i}\, \mathrm{d}A_i / s_i^2$, so
the resulting factor in the likelihood function is given by
\begin{equation}
\mathcal{L}_i(\vec{x_0}, t_0; \vec{x_i}, t_i) \; \propto \;
\Gamma \difrac{\mathrm{d}\Omega_i}{4\pi} \; = \;
\Gamma \difrac{\cos{\psi_i}}{4\pi s_i^2}\, \mathrm{d}A_i.
\end{equation}
As mentioned already, all constant factors in a likelihood function may
be discarded with no effect on the location in parameter space of its
maximum. (To first order, this includes the quantum efficiency $q_i$ of
each PMT.) The per-PMT likelihood function for a PMT detecting a photon may
thus be redefined as
\begin{equation}
\mathcal{L}_i(\vec{x_0}, t_0; \vec{x_i}, t_i) =
p(t_i - t_0 - \tau_f^i) \, \difrac{\cos{\psi_i}}{s_i^2}.
\label{e:likelihood}
\end{equation}
Its logarithm is
\begin{equation}
\log \mathcal{L}_i \; = \; \log p(t_i - t_0 - \tau_f^i) \; + \;
\log \cos \psi_i \;-\; 2\log s_i.
\label{e:log-likelihood}
\end{equation}
\subsection{The PMTs Not Triggered}
For completeness, we now consider the case of a PMT that does not detect
a photon produced by an event in the detector. Its per-PMT likelihood
function, from Equation~(\ref{e:per-pmt}), is given by
\begin{eqnarray}
\mathcal{L}_i(\vec{x_0}, t_0)\, \mathrm{d}^3 \vec{x}\, \mathrm{d}t \;&=&\;
(1 - q_i) \mathrm{P}(\mathrm{C}_i | \mathrm{A})\; + \;
\mathrm{P}(\neg \mathrm{C}_i | \mathrm{A}) \nonumber \\
&=&\; (1 - q_i) \left[1 - \left(1 -
\frac{\mathrm{d}\Omega_i}{4\pi}\right)^\Gamma \right]
\; + \; \left(1 - \frac{\mathrm{d}\Omega_i}{4\pi}\right)^\Gamma
\nonumber \\
&=&\; 1 - q_i + q_i \left(1 - \frac{\mathrm{d}\Omega_i}{4\pi}\right)^\Gamma
\nonumber \\
&\approx&\; 1 - q_i \Gamma \frac{\mathrm{d}\Omega_i}{4\pi}.
\end{eqnarray}
The logarithm of this per-PMT likelihood function is $\approx \; -q_i \Gamma
\mathrm{d}\Omega_i / 4\pi$. This term, containing an infinitesimal, is
negligible in size compared to the terms of Equation~(\ref{e:log-likelihood})
coming from per-PMT likelihood functions for PMTs that have detected a photon.
If PMTs are in fact very small compared to any other relevant dimensions of the
detector, it may therefore be ignored.
\subsection{Specialization to a Spherical Detector}
As written, Equation~(\ref{e:likelihood}) is applicable to any detector with
pointlike PMTs forming the vertices of a convex polyhedron (so that light from an event at any
point inside the detector may reach any one of the PMTs). Let us specialize to
a spherical detector of radius $R$ centered at the origin, having a uniform
distribution of inward-facing PMTs over the surface. As above, we call the
distance from an event to the $i^{th}$ PMT $s_i \equiv \left| \vec{x_i} -
\vec{x_0} \right|$. Let the distance from the center of the detector to the
event be $a \equiv \left| \vec{x_0} \right|$, so we have the geometry of
Figure~\ref{f:circle}.
\begin{figure}[t!]
\begin{center}
\psfrag{psi}{$\psi_i$}
\psfrag{theta}{$\theta_i$}
\psfrag{R}{$R$}
\psfrag{a}{$a$}
\psfrag{s}{$s_i$}
\psfrag{O}{$\mathrm{O}$}
\psfrag{C}{$\mathrm{C}$}
\psfrag{event}{$\mathrm{A} = (\vec{x_0}, t_0)$}
\psfrag{detector}{$\mathrm{B} = (\vec{x_i}, t_i)$}
\epsfig{file=circle.eps,height=2in}
\end{center}
\caption{Geometry of a spherical detector.}
\label{f:circle}
\end{figure}
By dropping a perpendicular from segment OB to point A (shown as
line segment AC), one readily
sees that $s_i \cos{\psi_i} = R - a \cos{\theta_i}$, with $\theta_i$
being the angle between the event and $i^{th}$ PMT
seen from the origin. Hence the likelihood function becomes
\begin{equation}
\mathcal{L}(\vec{x_0}, t_0; \{(\vec{x_i}, t_i)\}) = \prod_{i=1}^N
p\left(t_i - t_0 - \difrac{s_i n}{c}\right) \,
\difrac{R - a \cos{\theta_i}}{s_i^3}
\label{e:sph-likelihood}
\end{equation}
where $s_i$ is given by the Law of Cosines,
\begin{equation}
s_i^2 = R^2 + a^2 - 2 a R \cos{\theta_i}.
\label{e:loc}
\end{equation}
\section{Properties of the Likelihood Function at the Origin}
It may be of interest to examine properties of the likelihood function in the particular case of a hypothetical event occurring at the center of a spherical detector. This allows the general nature of the problem of reconstruction to be understood analytically. For simplicity, let's assume that the distribution of the time emission of the photons is a Gaussian curve with width equal to the characteristic dispersion time of the scintillator:
\begin{equation}
p(\tau_e)=\difrac{e^{-\tau_e^2/2\sigma^2}}{\sqrt{2\pi\sigma^2}}; \;
\log p(\tau_e) = \mbox{\rm const} - \difrac{\tau_e^2}{2 \sigma^2}.
\label{e:gaussian}
\end{equation}
The same equation can also be used for the case when the original scintillation light is absorbed and then re-emitted by scintillation fluors in the immediate proximity of the energy deposition point \cite{ctf-light}. In this case, the dispersion characteristic of the scintillator is effectively broadened by the absorption and re-emission process.
\subsection{Taylor Expansion of the Likelihood Function}
\label{ss:taylor-expansion}
For a point in the detector at a distance $a$ from the center,
in the direction of a particular unit vector $\hat{u}$, the log
likelihood function is
\begin{equation}
\log{\mathcal{L}(a\hat{u},t_0)}= \mbox{\rm const} -\difrac{1}{2\sigma^2}
\sum_{i=1}^{N}\left(t_i-t_0-\difrac{s_i n}{c}\right)^2
+ \sum_{i=1}^{N} \log{\difrac{R - a \cos{\theta_i}}{s_i^3}}
\label{e:spec-likelihood}
\end{equation}
where $s_i$ and $\theta_i$ for each PMT are as shown in figure~\ref{f:circle}.
We assume that the number of hit PMTs $N$ is sufficiently large that we can,
with little error, replace this expression by spatial and temporal averages
over the expected angular and time distributions of the PMT hits. That is
(discarding the constant term),
\begin{equation}
\log{\mathcal{L}(a\hat{u},t_0)} \; \approx \; -\difrac{N}{2\sigma^2}
\left< \left(t-t_0-\difrac{s n}{c}\right)^2 \right>
\; + \; N \left< \log{\difrac{R - a \cos{\theta}}{s^3}} \right>,
\label{e:avg-likelihood}
\end{equation}
where $t$, $s$, $\theta$ are now continuous random variables with the
expected distributions.
We now calculate these averages for a point-like event located in the center
$\vec{x}_0=\vec{0}$ of the detector, occurring at time $t_0 = 0$.
First consider the time average. The time of flight of photons from the
center to each PMT (assuming minimal scattering) is $Rn/c$, where $n$ is the
index of refraction and $c$ is the velocity of light in vacuum.
This means that the distribution curve of $t$ is $p(t-Rn/c)$. From the
properties of a Gaussian distribution, the time averages of time-dependent
quantities are
\begin{eqnarray}
\left< t \right> & = & \difrac{Rn}{c} \\
\left< t^2 \right> & = & \left<t\right>^2 + \sigma_t^2
= \difrac{R^2n^2}{c^2} + \sigma^2.
\end{eqnarray}
Likewise, since all PMTs are equidistant from an event at the center of a
spherical detector, the distribution of PMT hits should be uniform over the
solid angle. Hence the spatial averages over quantities dependent upon the
event-to-PMT angle $\theta$ can be found using Equation~(\ref{e:loc}) and
taking the surface integral over the sphere of PMTs:
\begin{eqnarray}
\left< s \right> & = & \difrac{1}{4\pi} \int \mathrm{d}\phi\,
\mathrm{d}\left(\cos \theta\right)
\sqrt{R^2 + a^2 - 2aR \cos{\theta}}
\; = \; R + \difrac{a^2}{3R} \\
\left< s^2 \right> & = & \difrac{1}{4\pi} \int \mathrm{d}\phi\,
\mathrm{d}\left(\cos \theta\right)
\left(R^2 + a^2 - 2aR\cos{\theta}\right)
\; = \; R^2+a^2
\end{eqnarray}
Finally, we observe that for a point-like event in the center of a uniform
sphere of PMTs, there is no correlation between the expected spatial
distribution of $s$ and temporal distribution of $t$; that is,
$\left< s t \right> = \left< s \right> \left< t \right>$. This and
the above equations allow us to evaluate
\begin{eqnarray}
\left<\left(t-t_0-\frac{s n}{c}\right)^2\right>
& = & \left<t^2 + t_0^2 + \frac{s^2n^2}{c^2}
- 2t t_0 - 2t\frac{s n}{c} + 2t_0\frac{s n}{c}\right> \nonumber \\ & = & \frac{R^2 n^2}{c^2} + t_0^2 + (R^2 + a^2)\frac{n^2}{c^2} - 2\frac{Rn}{c} t_0 \nonumber \\
& & \;\;\;\; -\, 2\frac{Rn^2}{c^2} (R + \frac{a^2}{3R})
+ 2t_0 (R + \frac{a^2}{3R}) \frac{n}{c} \nonumber \\
& = & \, \mbox{\rm const} + t_0^2 + \frac{n^2}{3c^2}a^2
+ \frac{2n}{3cR}a^2 t_0
\end{eqnarray}
where the constant term contains whatever does not depend explicitly on $t_0$ and $a$.
The quantity averaged over in the last term of Equation~(\ref{e:avg-likelihood}), again substituting in
Equation~(\ref{e:loc}), becomes
\begin{eqnarray}
\log{\difrac{R - a \cos \theta_i}{s_i^3}} & = &
\log{\left(\difrac{R - a\cos\theta_i}
{\left( R^2 + a^2 - 2aR \cos\theta_i \right)^{3/2}}\right)}\nonumber \\
&=& -2\log{R} + \difrac{2a}{R}\cos\theta_i
+ \difrac{a^2}{2R^2} \left( 5 \cos^2 \theta_i - 3 \right)
+ ...
\end{eqnarray}
with the last equality above being the expansion into a Taylor series in $a/R$.
By once again averaging the expected distributions in $s$ and $\theta$
over the solid angle, the result, obtained to second order in $a/R$, is
determined to be
\begin{equation}
\left< \log{\difrac{R - a \cos \theta}{s^3}} \right> \approx
\mbox{\rm const} - \difrac{2a^2}{3R^2}.
\end{equation}
The complete likelihood function for an event at the center of a spherical detector, to second order in $a/R$, is thus
\begin{equation}
\log{\mathcal{L}}(a\hat{u}, t_0) \approx
\mbox{\rm const} - N \left[ \difrac{1}{2 \sigma^2}
\left(t_0^2 + \difrac{n^2}{3c^2} a^2 + \difrac{2n}{3cR}a^2 t_0\right)
+ \difrac{2}{3R^2} a^2 \right].
\label{e:likelihood-at-center}
\end{equation}
\subsection{Likelihood Function Maximum and Resolutions}
Solving for the maximum of the likelihood function and requiring $|a| < R$ gives the expected solutions:
\begin{equation}
\left\{
\begin{array}{l}
\difrac{\partial}{\partial t_0}\log \mathcal{L} = 0 \\
\difrac{\partial}{\partial a}\log \mathcal{L} = 0
\end{array}
\right.
\Longleftrightarrow
\left\{
\begin{array}{l}
t_0 = 0 \\
a = 0
\end{array}
\right.
\end{equation}
We next ask about the expected resolution of the detector.
Notice that the information matrix is diagonal because the off-diagonal
terms, $-\partial^2 (\log{\mathcal{L}}) / \partial a \partial t_0$, are
zero when $a = t_0 = 0$. The theoretical resolutions of the detector in
space and time are therefore given by reciprocals of the second derivatives
of the likelihood function:
\begin{equation}
\left\{
\begin{array}{l}
\delta t_0 = \left( -\difrac{\partial^2 \log{\mathcal{L}}}{\partial t_0^2}
\right)^{-1/2} = \difrac{\sigma}{\sqrt{N}} \\
\delta a = \left( -\difrac{\partial^2 \log{\mathcal{L}}}{\partial a^2}
\right)^{-1/2} =
\left( \difrac{Nn^2}{3c^2 \sigma^2} \, + \,
\difrac{4N}{3R^2} \right)^{-1/2}
\end{array}
\right.
\label{e:resolution}
\end{equation}
When the detector dimensions are much larger than the scintillator dispersion time, $R \gg c \sigma / n$, we can approximate $\delta a \approx \sqrt{\difrac{3}{N}} \difrac{c\sigma}{n}$. (It should be noted that this does not take into
account scattering effects, which become increasingly important with
larger detectors.)
Because of the spherical symmetry of the problem,
$\delta a$ can be used as a stand-in for any of the three Cartesian spatial
resolutions $\delta x_0$, $\delta y_0$, $\delta z_0$. One may, for instance,
make the substitution $a^2 = x_0^2 + y_0^2 + z_0^2$ in
Equation~(\ref{e:likelihood-at-center}) and obtain the same results for
the resolution in each Cartesian coordinate.
\subsection{Pattern Matching}
In case of use of a liquified noble gas as scintillator, as in the new generation of solar neutrino detectors \cite{clean,xmass}, Rayleigh scattering of the ultraviolet scintillation photons plays an important role. The photons are scattered intensely by the medium, such that they effectively diffuse out of the medium with a very long dispersion time; then $R \gg c \sigma / n$ is no longer valid. In this case, the information carried by the time of flight method about the original position of the events becomes less reliable. However, it is still possible to reconstruct the original position of the event by taking into account that the density of hits on the PMTs decreases with the inverse of the squared distance from the point where the energy is deposited~\cite{clean-rec}.
Suppose that we have no timing information, so our only information about an event is the pattern of hit PMTs. In this case, the likelihood function simply determines the position of the event. It does not depend on time and cannot be used to reconstruct the time itself. We may set the function $p(\tau_e)$ to be constant and ignore it:
\begin{equation}
\log{\mathcal{L}(a\hat{u})}= \mbox{\rm const}
+ \sum_{i=1}^{N} \log{\difrac{R - a \cos{\theta_i}}{s_i^3}}.
\end{equation}
By the same methods as above, we obtain
\begin{equation}
\log{\mathcal{L}}(a\hat{u}) \approx \mbox{\rm const} - \difrac{2N}{3R^2} a^2
\end{equation}
for the second-order Taylor expansion in $a/R$ of the likelihood function for an event at the detector center. In this case we find
\begin{equation}
\difrac{\partial}{\partial a}\log \mathcal{L} = 0
\Longleftrightarrow
a = 0,
\end{equation}
and for the resolution,
\begin{equation}
\delta a = \left( -\difrac{\partial^2 \log{\mathcal{L}}}{\partial a^2}
\right)^{-1/2} =
\sqrt{\difrac{3}{N}} \difrac{R}{2}.
\label{e:pattern-resolution}
\end{equation}
Recall Equations~(\ref{e:likelihood-at-center}) and~(\ref{e:resolution}) in
the case where timing information {\it is} available:
\begin{eqnarray*}
\log{\mathcal{L}}(a\hat{u}, t_0) & \; \approx \; &
\mbox{\rm const} - \difrac{2N}{3R^2}a^2
- \difrac{N}{\sigma^2}
\left(t_0^2 + \difrac{n^2}{3c^2} a^2 + \difrac{2n}{3cR}a^2 t_0\right) \\
\delta a & \; = \; &
\left( \difrac{Nn^2}{3 c^2 \sigma^2} \, + \,
\difrac{4N}{3R^2} \right)^{-1/2}
\approx \; \sqrt{\difrac{3}{N}} \difrac{c\sigma}{n}.
\end{eqnarray*}
We see that use of timing information improves spatial resolution significantly when the scintillator dispersion time is much less than the travel time for light to cross the detector.
In a liquid noble gas detector, the scintillator time dispersion is very
broad due to the amount of internal Rayleigh scattering of scintillation light.
Nevertheless, use of even the small amount of timing information available has
been shown to improve the spatial resolution by a large
fraction~\cite{neon-darkmatter}.
\subsection{Comparison to Observed Resolutions}
\begin{table}[t!]
\begin{center}
\begin{tabular}{lcrcrcrrr} \hline \hline
Detector & $R$ & $T$ & $n$ & $\sigma$
& $\epsilon$ & $N$ & Pred. & Obs. \\
& [m] & & & [ns] &
[pe] & & \multicolumn{2}{c}{$\delta a$\,[cm]} \\ \hline
\multicolumn{8}{l}{Organic scintillator detectors} \\
CTF, $^{214}$Po $\alpha$~\cite{ctf,ctf-results} & 3.3 & 100 & {\it 1.8} &
{\it 5.1} & 225 & 90 & 12.0 & 12.3 \\
Borexino, 1\,MeV $e^-$ MC~\cite{bx} & 6.5 & 2240 & 1.5 &
{\it 5.1} & 400 & 366 & 8.8 & 8.0 \\ \hline
\multicolumn{8}{l}{Hypothetical
$\ell$Ne detector, 100\,keV $e^-$ MC~\cite{neon-darkmatter}} \\
Spatial data only & 3.0 & 1832 & - & - &
{\it 243} & {\it 243} & 16.7 & 17.0 \\
Timing included & " & " & 1.2 &
{\it 10} & {\it 162} & {\it 155} & 15.0 & 13.6 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of the predicted resolutions of three liquid
scintillator detectors with
the values determined experimentally or by Monte Carlo (MC) methods.
See the text for meanings of the columns and comments on values in
{\it italics}.}
\label{t:resolutions}
\end{table}
Experimentally, the position resolution of a detector can be determined in
several ways. The simplest and most common is the use of a calibration source.
In cases when the detector has not yet been built, Monte Carlo methods are of
course the only method that can be used. The detector resolutions obtained
from experimental results for CTF, and Monte Carlo tests of Borexino
and a hypothetical liquid neon dark matter detector~\cite{neon-darkmatter},
are shown
in the last column of Table~\ref{t:resolutions}. For comparison, the physical
attributes of the detectors and the predicted resolutions $\delta a$ from
Equation~(\ref{e:resolution}) are shown in the other columns of the table. As
above, $R$ is the detector radius, $T$ the total number of PMTs, $n$
the scintillator index of refraction, and $\sigma$ the scintillator dispersion
time. The average number of photoelectrons detected in each event from the
source used is denoted by $\epsilon$.
$N$ is determined in most cases as follows. In detectors using a
time-of-flight position reconstruction method, each PMT can
measure the arrival time only of the first photon to strike it. This
difficulty will be discussed more thoroughly in Section~\ref{s:orderstat}. The
immediate consequence is that $N$ is a measure of the number of hit PMTs rather
than the total number of detected photoelectrons. Basic probability tells us
that given an event in which $\epsilon$ photoelectrons are detected, the
expected number of hit PMTs is
\begin{equation}
\left< N \right> = T \left[1 - \left(\difrac{T - 1}{T}\right)^\epsilon \right].
\label{e:N-from-epsilon}
\end{equation}
Note, however, that for the spatial hit pattern, every photoelectron
contributes to our knowledge, even for multiple hits on a single PMT. This
implies that the term $4N/3R^2$ in the expression for $\delta a$ in
Equation~(\ref{e:resolution}) should in fact include $\epsilon$, not $N$.
In calculating the predicted values of $\delta a$ in Table~\ref{t:resolutions},
we therefore use the modified expression
\begin{equation}
\delta a = \left( \difrac{Nn^2}{3c^2 \sigma^2} \, + \,
\difrac{4\epsilon}{3R^2} \right)^{-1/2}.
\label{e:resolution-modified}
\end{equation}
Some comments on idiosyncracies of the individual detectors are in order. The
value of $n$ of 1.8 tabulated for the CTF is an ``effective index of
refraction.'' In
fact, the CTF volume is partly water ($n = 1.33$) and partly organic
scintillator ($n = 1.5$); this ``effective index'' is an attempt to account
for refraction at the interface between the two fluids. Refraction
causes light to travel a greater distance from event to PMT than it would
through a single medium, so the ``effective $n$'' is higher than that of
either pure fluid. Additionally, note that the observed value
of $\delta a$ for the CTF takes into account only the spread in $x$ and $y$
coordinates; the CTF source had the shape of a cylinder, extended in~$z$.
In the hypothetical liquid neon detector described in
reference~\cite{neon-darkmatter}, events have a prompt component
(relative intensity 2.0) and a delayed component (relative intensity 1.0)
of scintillation light. For the Monte Carlo simulation taking into account
only the spatial pattern of PMT hits (``spatial data only'' row of
Table~\ref{t:resolutions}), both components contribute useful data.
In that case the photoelectron yield is 2428\,pe/MeV, 1.5 times the prompt
light yield of 1619\,pe/MeV (10791.7\,photons/MeV $\times$ 20\% quantum
efficiency $\times$ 75\% geometric coverage) quoted in the reference. For the
position reconstruction calculated from the spatial pattern only, we use
$N = \epsilon_{total} \equiv \epsilon_{prompt} + \epsilon_{delayed}$ in
Equation~(\ref{e:pattern-resolution}).
Calculation of the expected resolution in the liquid Ne detector is trickier
when timing information is included (``timing included'' row of
table~\ref{t:resolutions}). The two terms contributing to $\delta a$ in
Equation~(\ref{e:resolution-modified}) must be evaluated with different values
for $\epsilon$. The term $4\epsilon/3R^2$ comes from the spatial hit pattern
and so uses $\epsilon_{total} = 243$, while the timing-dependent term
$Nn^2/3c^2\sigma^2$ includes only the prompt component of scintillation light,
and thus uses $\epsilon_{prompt} = 162$, with $N = 155$ derived from
Equation~(\ref{e:N-from-epsilon}).
The source of the largest potential errors in the predictions of
Table~\ref{t:resolutions} is the value of the scintillator
dispersion,~$\sigma$. The true scintillator dispersion function of a detector
$p(\tau_e)$ is not actually a Gaussian, so the use of
Equation~(\ref{e:resolution}) is only an approximation. The value of 5.1~ns
used for $\sigma_{CTF}$ is obtained from the fit to CTF data described in
reference~\cite{ctf-light} with the parameters shown in Figure~6 of that paper,
sampled at 1~ns intervals and fit to a Gaussian only. (The same scintillator
dispersion function was used in the Borexino Monte Carlo simulations.)
Nevertheless, the predicted, observed and Monte Carlo values of the position
resolution are in quite good agreement. For the liquid Ne detector,
$\sigma$ was estimated at 10\,ns, based on Figure~7 of
reference~\cite{neon-darkmatter}, as 1/2 the difference between times with
probability values equal to $e^{-0.5}$ times the value at the peak. One
could plausibly estimate this value of $\sigma$ to be anywhere in the range
5.5 to 15\,ns, yielding estimates of $\delta a$ from 12.6 to 15.9\,cm.
This range brackets the Monte Carlo simulation nicely.
\section{Multiple PMT Occupancy and Order Statistics}
\label{s:orderstat}
So far it has largely been assumed that the occupancy of each PMT in the
detector is at most one. If the detector has the capability to measure the
time at which {\it every} photon hits a given PMT, or if the detector
(as with some of the proposed noble gas detectors) has no timing capability
at all, then the assumption may be lifted
with no effect, except that some of the $\vec{x_i}$ (and hence $\theta_i$ and
$s_i$) will be identical in Equation~(\ref{e:sph-likelihood}). For a
detector with timing capabilities, however, it is more
likely that the detector only has the capability to measure the
arrival time of the {\it first} photon to reach each PMT. The probability
function of the first photon to reach a PMT is not the same as that of a random
photon reaching the same PMT; it
is biased toward earlier times. To account for this bias, the scintillator
response function $p(\tau_e)$ must be corrected.
\subsection{Correcting for Timing Bias}
Let the probability function of the first photon to reach a PMT,
out of the $n$ photons reaching that PMT from an event, be represented by
$p_n(\tau_e)$.
This is known as the ``first order statistic.''
Naturally, $p_1(\tau_e) \equiv p(\tau_e)$.
In general, the corrected scintillator response function $p_{corr}$ would then
be some linear combination of the first order statistics,
\begin{equation}
p_{corr}(\tau_e) =
\sum_{n=1}^\infty
p_n(\tau_e) \times
\mathrm{P}(n\, \mathrm{photons\, hit\, the\, PMT}),
\end{equation}
and an {\it a priori} guess would have to be made for the probability that
each possible number of photons had hit the PMT. For simplicity, let us assume
that the number of photons striking each PMT for an event is known (in
Borexino, for instance, this is determined via ADC channels separate from
the timing channels). We can then set $p_{corr}$ equal to the function
$p_n(\tau_e)$.
\begin{figure}[!t]
\begin{center}
\psfrag{tau}{$\tau_e / \sigma$}
\psfrag{poftau}{$p(\tau_e)$}
\psfrag{pnoftau}{$p_n(\tau_e)$}
\psfrag{n2}{$n=2$}
\psfrag{n3}{$n=3$}
\psfrag{n5}{$n=5$}
\psfrag{n10}{$n=10$}
\epsfig{file=orderstats.eps,width=5in}
\end{center}
\caption{
A hypothetical Gaussian scintillator response function $p(\tau_e)$,
and its first order statistics for increasing values of $n = 2, 3, 5, 10$. Note
how as $n$ increases, the corrected response function narrows and shifts toward
earlier times. The time axis is shown in units of the scintillator
dispersion time $\sigma$.}
\label{f:order-statistics}
\end{figure}
It remains only to calculate $p_n(\tau_e)$ given $p(\tau_e)$ and $n$.
Number the emission time of the $n$ photons detected by a given PMT in some
specific but randomly chosen order (for instance, in order of increasing
longitude of their emission directions), $\tau_1, \ldots, \tau_n$.
Also number them in order of
increasing emission time, $s_1, \ldots, s_n$. Then $p_n(\tau_e)$ is
the probability function of the randomly chosen emission time
$\tau_1$ given that $s_1 = \tau_1$:
\begin{eqnarray}
p_n(\tau_e)\, \mathrm{d}\tau_e \;&=&\; \mathrm{P}(\tau_1 \in
[\tau, \tau + \mathrm{d}\tau] | \tau_1 = s_1) \nonumber \\
&=&\; \mathrm{P}(\tau_1 = s_1 | \tau_1 \in [\tau, \tau + \mathrm{d}\tau])
\times \frac{\mathrm{P}(\tau_1 \in [\tau, \tau + \mathrm{d}\tau])}
{\mathrm{P}(\tau_1 = s_1)} \nonumber \\
&=&\; \frac{p(\tau_e)\, \mathrm{d}\tau_e}{(1/n)}\,
\mathrm{P}(\tau_1 = s_1 | \tau_1 \in [\tau, \tau + \mathrm{d}\tau]),
\end{eqnarray}
where the second equality is once again due to Bayes' Theorem. The
probability in the last line above is just the probability that every
other photon has a later arrival time than the randomly selected value $\tau_1$:
\begin{eqnarray}
\mathrm{P}(\tau_1 = s_1 | \tau_1 \in [\tau, \tau + \mathrm{d}\tau])
\;&=&\; \prod_{i = 2}^n \mathrm{P}(\tau_i > \tau_1 | \tau_1 \in
[\tau, \tau + \mathrm{d}\tau]) \nonumber \\
&=&\; \mathrm{P}(\tau_2 > \tau_1 | \tau_1 \in
[\tau, \tau + \mathrm{d}\tau])^{n-1} \nonumber \\
&=&\; \left[ \int_{\tau_e}^\infty p(\tau_e')\, \mathrm{d}\tau_e' \right]^{n-1}.
\end{eqnarray}
Hence (letting $F(\tau_e) \equiv \int_{-\infty}^{\tau_e} p(\tau_e)\,
\mathrm{d}\tau_e$ represent the cumulative distribution function of
$\tau_e$), the first order statistic of $p(\tau_e)$, if $n$ photons are
detected by a given PMT, is
\begin{equation}
p_n(\tau_e) \; = \; n p(\tau_e)\, \left[1 - F(\tau_e) \right]^{n-1}.
\end{equation}
Graphs of the first order statistics of a representative scintillator
response function are shown in Figure~\ref{f:order-statistics} for values of
$n$ equal to 1, 2, 3, 5, and 10. (The specific response function shown is a
Gaussian, Equation~(\ref{e:gaussian}) offset by five units of $\sigma$
from time zero.) Note how as
$n$ increases, the time distribution of the first PMT hit narrows and shifts
toward earlier times.
\subsection{Effects on Detector Resolution}
One may ask about the effect of this correction on the likelihood function
and spatial resolution. Consider again the case of a Gaussian scintillator
time response function. We have
\begin{equation}
\log p_n(\tau_e) \; = \;
\mbox{\rm const} + \log p(\tau_e) + (n - 1)\log [1 - F(\tau_e)].
\end{equation}
Substituting in $F(\tau_e) = (1 + {\rm erf} (\tau_e / \sigma \sqrt{2}))/2$, the
Taylor expansion to second order in $\tau_e$ becomes
\begin{equation}
\log p_n(\tau_e) \; = \; \mbox{\rm const} -
(n - 1)\sqrt{\frac{2}{\pi}}\frac{\tau_e}{\sigma}
- \left(\frac{1}{2} + \frac{n-1}{\pi}\right)\frac{\tau_e^2}{\sigma^2}
+ O(\tau_e^3).
\end{equation}
That is, the first photon detected at each PMT contributes to the log of the
likelihood function in the amount of $-\tau_e^2/2\sigma^2$, but each additional
photon contributes only in the amount of $-\tau_e^2/\pi\sigma^2$ (plus a term
linear in $\tau_e$ which has relatively little effect on the resolution
for a large detector); compare to Equation~(\ref{e:gaussian}). The
resolution is better than if the corrected scintillator response function were
not used, but still poorer than if the time of arrival of every photon could be
measured.
Suppose that the total number of photons detected is $\epsilon$, by $N$ PMTs,
and in particular that the $i^{th}$ PMT sees $n_i$ photons. Denoting the
emission time by $\tau_e^i \equiv t_i-t_0-s_i n / c$, the general
likelihood function is then
\begin{eqnarray}
\log{\mathcal{L}(a\hat{u},t_0)} &\;=\;& \mbox{\rm const}
\,-\, \difrac{1}{\sigma^2}
\sum_{i=1}^{N} \left( \difrac{1}{2} + \difrac{n_i - 1}{\pi} \right)
\left(\tau_e^i\right)^2 \nonumber \\
&\;& -\, \difrac{1}{\sigma}\sqrt{\difrac{2}{\pi}} \sum_{i=1}^N
(n_i - 1)\, \tau_e^i
\,+\, \sum_{j=1}^{\epsilon} \log{\difrac{R - a \cos{\theta_j}}{s_j^3}}.
\end{eqnarray}
Define the excess photon multiplicity as $\delta \equiv (\epsilon - N)/N$. The
likelihood function in the limit of homogeneous PMT coverage as
$N \rightarrow \infty$, for an event at the detector center, becomes
\begin{eqnarray*}
\log{\mathcal{L}(a\hat{u},t_0)} &\;=\;& \mbox{\rm const}
\,-\, \difrac{N}{\sigma^2} \left( \difrac{1}{2} + \difrac{\delta}{\pi} \right)
\left< \left( \tau_e^i \right)^2 \right>\\
&\;& -\, \difrac{N\delta}{\sigma} \sqrt{\difrac{2}{\pi}} \left< \tau_e^i \right>
\,+\, N (\delta + 1)\, \left< \log \difrac{R - a \cos{\theta_j}}{s_j^3}
\right>.
\end{eqnarray*}
Running through calculations analogous to those of Section~\ref{ss:taylor-expansion},we finally obtain the explicit function
\begin{eqnarray}
\log{\mathcal{L}(a\hat{u},t_0)} &\;=\;& \mbox{\rm const}
\,-\, \difrac{N}{\sigma^2} \left( \difrac{1}{2}
+ \difrac{\delta}{\pi} \right)\left( t_0^2 + \difrac{n^2}{3c^2} a^2
+ \difrac{2n}{3cR} a^2 t_0 \right) \nonumber\\
&\;& -\, \difrac{N \delta}{\sigma} \sqrt{\difrac{2}{\pi}}
\left(t_0 + \difrac{n}{3cR} a^2 \right)
\,-\, N (\delta + 1)\difrac{2}{3R^2} a^2 .
\end{eqnarray}
In the limit $c \sigma / R \rightarrow 0$ (that is, for a very large detector
compared to the width of the scintillator response function), it can be shown
that the spatial resolution at the center of a detector, with $N$ and $\delta$
varying while holding $\epsilon$ constant, is proportional to
$\sqrt{\pi(1+\delta)} / \sqrt{\pi+2\delta}$. Hence the resolution of an event
with an average photon multiplicity of $\delta = 0.5$~excess photons per PMT is
6.7\% worse than if PMTs could detect the arrival time of every photon. With
$\delta = 1$~excess photon per PMT (every hit PMT seeing an average of 2
photons), the resolution is 10.5\% worse. In the limit of large $\delta$
(for instance with a high-energy event), the
resolution reaches an asymptote of $\sqrt{\pi/2}$ times (about 25.3\% worse)
that of an ideal detector observing an event of equal energy.
Realistically, construction of an ideal detector, one that measures the time
of arrival for every photon, would be non-trivial. One may on the other hand
ask, given a detector capable of measuring time of
arrival only for the first photon at each PMT, how the use of the statistically
corrected scintillator dispersion function improves the results over the use of
an uncorrected function. This comparison is equivalent to fixing $N$ while
(for the uncorrected dispersion function) setting $\delta$ to zero. In this
case, the use of the corrected dispersion function is an improvement by the
factor $\sqrt{\pi} / \sqrt{\pi + 2\delta}$ (recall that smaller resolutions are
better). For $\delta = 0.5$, the reciprocal of the improvement factor is
1.15, and for
$\delta = 1$, it is 1.28; for large $\delta$, it would theoretically
improve without bound. This analysis even leaves aside the fact that for
events offset from the center of the detector, use of the uncorrected
scintillator dispersion function will produce a statistically biased position
estimate.
\section{Conclusions}
We analyzed the resolution of spherical, optical, non-imaging scintillation based detectors in reconstructing the position of point-like events, limiting the analytic derivation to the case of events near the center of the detector. We found that the fundamental length scale of the resolution given by the time of flight method is proportional to the product of the speed of light in the medium and the dispersion time at the scintillation emission, as in $\delta a \approx \sqrt{\difrac{3}{N}} \difrac{c\sigma}{n}$.
In case the dispersion of the scintillation photons arrival times grows above the ratio of the speed of light to the detector radius, the time of flight method no longer gives relevant information about the point of origin of the event. The position of the event can still be determined by the analysis of the density of hits, and in this case the fundamental resolution is set by the radius of the detector, as in $\delta a = \sqrt{\difrac{3}{N}} \difrac{R}{2}$.
Finally, we made some comments on the need to correct the scintillation
dispersion function in the common case where PMT hit timing information is only
available for the first photon to strike each PMT. In this case, even with a
corrected scintillation dispersion function, the spatial resolution will
be up to 25~percent worse for high-energy events compared to a
similar detector capable of measuring timing information for all photons.
\section{Acknowledgments}
The authors are grateful for the many helpful suggestions and comments of
Kevin Coakley, Dan McKinsey, and Andrea Pocar.
\newpage
|
{
"timestamp": "2005-03-23T21:46:18",
"yymm": "0503",
"arxiv_id": "physics/0503185",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503185"
}
|
\section{Introduction}
Let us recall the notion of a \emph{Poincar\'e embedding}:
\begin{defin}[\rm({Levitt \cite{Levitt}, and \cite[Section 5]{Klein} for a modern exposition})]
\label{def-Pemb}
Let $W$ be a Poincar\'e duality space of dimension $n$ and let
$P$ be a finite CW-complex of dimension $m$.
A \emph{Poincar\'e embedding} of $P$ in $W$ (of \emph{dimension} $n$ and \emph{codimension} $n-m$)
is a commutative diagram of
topological spaces
\begin{equation}
\label{diag-mainsquare} \xymatrix{ \del
T\ar@{->}[r]^i\ar@{->}[d]_k&
P\ar@{->}[d]^f\\
C\ar@{->}[r]_l&W }
\end{equation}
such that \refequ{diag-mainsquare} is a homotopy push-out,
$(P,\del T)$ and $(C,\del T)$ are Poincar\'e duality
pairs\footnote{By abuse of terminology, by the \emph{pair} $(P,\partial T)$
we actually mean the pair $(P',\partial T)$ where $P'$ is the mapping cylinder of $i$,
and similarly for the pair $(C,\partial T)$}
in dimension $n$,
and the map $i$ is $(n-m-1)$-connected.
\end{defin}
The motivating example of a Poincar\'e embedding arises when $W$ is a closed orientable PL-manifold of dimension $n$ and
$f\co P\hookrightarrow W$ is a piecewise
linear embedding of a compact polyhedron $P$ in $W$.
Alternatively we can also take $f$ to be a smooth embedding between smooth compact manifolds.
Then $f(P)$ admits a regular neighborhood, that is
a codimension $0$ compact
submanifold $T\subset W$ that deformation retracts to $P$ (see
\cite[page 33]{RourkeSanderson}.) Let $C:=\overline{W\smallsetminus T}$
be the closure of the complement of $T$ in $W$. Then $C$ and $T$
are both compact manifolds of dimension $n$ with a common boundary
$\del T=\del C$ and $W=T\cup_{\del T} C$. The composition of the inclusion $\del T\hookrightarrow T$
with the retraction $T\quism P$ gives a map $i\co \del T\to P$ and we obtain the pushout \refequ{diag-mainsquare}.
If the polyhedron $P$ is of dimension $m$,
then a general position argument implies that the map $i$ is $(n-m-1)$-connected.
Of course $C$ has the homotopy type of the complement $W\smallsetminus f(P)$.
Thus morally a Poincar\'e embedding is the homotopy generalization of a PL embedding.
Notice that, in Definition \ref{def-Pemb}, $\del T$ is just a topological space and not necessarily
a genuine boundary of a manifold $T$, and $W$ does not need to be a manifold.
Notice also that
by a Poincar\'e embedding we mean \emph{all} of the diagram \refequ{diag-mainsquare} and not only
the map $f$. When such a diagram exists we say that the map $f\co P\to W$ \emph{Poincar\'e embeds}.
The space $C$ in the push-out diagram is called the \emph{complement} of $P$.
A natural question is whether the {homotopy class} of a map $f$ that Poincar\'e embeds
determines the square \refequ{diag-mainsquare} up to homotopy
equivalence and in particular the homotopy type of the complement $C$.
The answer to this question is negative in general as it can be seen
with $W=S^3$ and $P=S^1$. Indeed all PL-embeddings $f\co S^1\hookrightarrow S^3$
are nullhomotopic but the homotopy type of the complement $C\simeq S^3\smallsetminus f(S^1)$
can vary considerably (see for example
\cite[Corollary 11.3]{Lickorish} or \cite{GordonLuecke}.)
This is possible since in general the homotopy class $[f]$ of $f$ does not determine its
isotopy class. On the other hand in the case of a PL-embedding when the codimension is high enough, namely when $n\geq 2m+3$,
then a general position argument implies that $[f]$ determines the isotopy class of $f$. Therefore under
this high codimension hypothesis the homotopy class of a PL-embedding $f$ \emph{does} determine
the homotopy type of the square \refequ{diag-mainsquare}. Similarly under a slightly more restrictive condition
on the codimension, there exists a unique Poincar\'e embedding \refequ{diag-mainsquare} associated to a given homotopy
class $[f]$. See Theorem \ref{thm-unknot} below
for a precise and more general statement for PL-embeddings as
well as a discussion on the corresponding result for Poincar\'e embeddings.
The aim of this paper is to study an algebraic translation of the above question:
can we build algebraic models, such as Sullivan models which
encode rational homotopy type, of the square \refequ{diag-mainsquare} from an algebraic model of
the map $f\,$? In order to be more precise, we first review Sullivan's theory for modeling
rational homotopy types by algebraic models. By a \emph{CDGA},
$A$, we mean a non-negatively graded algebra over the field $\BQ$ of
rational numbers that is commutative in the graded sense and
endowed with a degree $+1$ derivation $d\co A\to A$ such that
$d^2=0$. Sullivan has defined in \cite{Sullivan} a contravariant
functor from topological spaces to CDGA,
$$\Apl\co\mathrm{Top}\to\mathrm{CDGA},$$
mimicking the de Rham complex of differential forms on a smooth
manifold.
By a \emph{CDGA model} of a space $X$, we mean a CDGA,
$A$, linked to $\Apl(X)$ by a chain of CDGA morphisms inducing
isomorphisms in cohomology,
$$
\xymatrix{A&\ar[l]_\simeq A_1\ar[r]^\simeq&\cdots&\ar[l]_\simeq
A_n\ar[r]^-\simeq&\Apl(X).}
$$
The fundamental result of Sullivan's theory is that if $X$ is a
simply-connected space with rational homology of finite type,
then any CDGA model of $X$ determines its rational homotopy type.
There is a similar result for maps and
more generally for finite diagrams. See \cite{FHT-RHT} for a complete
exposition of that theory.
Our first result is the construction, under the high codimension hypothesis
$\dim(W)\geq 2\dim(P)+3$, of an explicit CDGA model
of the Poincar\'e embedding \refequ{diag-mainsquare} out of a CDGA-model of $f$.
To explain this result, we need
some notation which will be made more precise in Section 2.
We denote by $\# V:=\hom(V,\Bk)$ the dual of a $\Bk$-vector space $V$ and by $s^{p}X$
the $p$-th suspension of a graded object $X$, i.e.\ $(s^{p}X)^k=X^{p+k}$. The mapping cone of a cochain map
$f\co M\to N$ is written $N\oplus_fsM$. When $N$ is a CDGA and $M$ is an $N$-DGmodule
this mapping cone
can be endowed with the multiplication
$(n,sm)\cdot(n',sm')=(n\cdot n',s(n\cdot m'\pm n'\cdot m))$.
The differential of the mapping cone does not always satisfy the Leibnitz rule
for this multiplication, but it does under certain conditions on the dimensions
and then the induced structure is called the
\emph{semi-trivial CDGA-structure} on the mapping cone (Definition \ref{def-CDGAMC}).
Our goal is to build
a CDGA model of the homotopy push-out \refequ{diag-mainsquare}, and in particular of the complement $C$,
out of a CDGA model $\phi\co R\to Q$ of $f^*\co \Apl(W)\to\Apl(P)$.
Motivated by Lefschetz duality a first guess for a model of $\Apl(C)$ is the mapping cone
$$
R\oplus_\psi ss^{-n}\#Q
$$
$$\psi\co s^{-n}\#Q\to R\leqno{\rm where}$$
is an $R$-DGmodule map such that $H^n(\psi)$ is an isomorphism.
Unfortunately this naive guess has two flaws:
\begin{enumerate}
\item[(A)] such a map $\psi$ does not necessarily exist, and
\item[(B)] the multiplication on $R\oplus_\psi ss^{-n}\#Q$ does not necessarily define a CDGA
structure because of the possible failure of the Leibnitz rule.
\end{enumerate}
Problem (A) can be addressed by replacing $s^{-n}\#Q$ by a suitable weakly equivalent DG-module $D$, for example a cofibrant one, for
which there exists a map $\psi\co D\to R$ inducing an isomorphism in cohomology in degree $n$. Such a map is called
a \emph{top-degree\ map}\footnote{It was called
a \emph{shriek map} in earlier versions of this paper.} in Definition \ref{def-shriek}.
Problem (B) can be solved by restricting the range of degrees of the graded objects $R$, $Q$, and $D$. This is where
the high codimension hypothesis is needed.
We can now state our first result:
\begin{thm}\label{thm-stableCDGA}%
Consider a Poincar\'e embedding \refequ{diag-mainsquare}
with $P$ and $W$ connected.
If $n\geq2m+3$ and $H^1(f;\BQ)$ is injective
then a model of the commutative CDGA square
$$\BD':=\vcenter{
\xymatrix@1{%
\Apl(W)\ar[r]^{f^*}\ar[d]_{l^*}&
\Apl(P)\ar[d]^{i^*}\\
\Apl(C)\ar[r]_{k^*}&%
\Apl(\del T) }}
$$
can be build explicitly out of any CDGA model of $f^*\co\Apl(W)\to\Apl(P)$.
More precisely, if $n\geq 2m+4$ or if $n\geq2m+3$ and $H^1(f;\BQ)$ is injective, then
the commutative CDGA square $\BD'$ is weakly equivalent to any commutative CDGA square
$$
\BD:=\vcenter{
\xymatrix{%
R\ar[r]^\phi\ar@{^(->}[d]&%
Q\ar@{^(->}[d]\\
R\oplus_{{\psi}}sD\ar[r]_{\phi\oplus\id}&%
Q\oplus_{}sD%
}}$$
where
\begin{enumerate}
\item[\rm(i)] $\phi\co R\to Q$
is a CDGA model of $f^*:\Apl(W)\to\Apl(P)$ with $R^{>n}=0$ and $Q^{>m+2}=0$;
\item[\rm(ii)] $D$ is a $Q$-DGmodule weakly equivalent to $s^{-n}\#Q$ with $D^{>n+1}=0$ and $D^{<n-m}=0$;
\item[\rm(iii)] ${\psi}\co D\to R$ is an $R$-DGmodules map such that $H^n({\psi})$ is an isomorphism
\end{enumerate}
and the mapping cones are endowed with the semi-trivial CDGA structure.
Moreover if $n\geq 2m+3$ and
$H^1(f;\BQ)$ is injective, then $R$, $Q$, $D$, $\phi$, and $\psi$
satisfying (i)-(iii) can be \emph{explicitly} constructed out of any CDGA model of
$f^*\co \Apl(W)\to\Apl(P)$.
\end{thm}
Since CDGA models encode rational homotopy types of simply connected spaces
an immediate corollary of the above theorem is that when $P$ and $W$ are simply connected and
$\dim(W)\geq 2\dim(P)+3$, then the rational homotopy type of the Poincar\'e embedding
\refequ{diag-mainsquare} depends only on the rational homotopy class of $f$.
As a byproduct of this theorem we obtain also a CDGA model $Q\oplus ss^{-n}\#Q$ of
the boundary $\del T$ of a thickening of $P$ under a high codimension hypothesis. This model was already
described in \cite{Lambrechts-thickening} and an analogous model is built in \cite{KahlLVdb}
under weaker hypotheses.
\medbreak
In our first theorem we have supposed that $\dim W\geq2\dim P+3$.
When the connectivity of the embedding is high this condition on the
codimension can be weakened. Indeed in the case of PL-embeddings we
have the following classical result:
\begin{thm}[PL-unknotting, Wall and Hudson]\label{thm-unknot}
Let $P$ be a compact $m$-dimensional polyhedron and let $W$
be a closed $n$-dimensional manifold with $n\geq m+3$. Let $r$ be an integer such that
\begin{equation}\label{equ-unknot}
r\geq2m-n+2.
\end{equation}
Then any two homotopic $r$-connected embeddings $f_0,f_1\co
P\hookrightarrow W$ are isotopic. As a consequence, if $f$ is $r$-connected then the homotopy type
of the square \refequ{diag-mainsquare} depends only on the homotopy class of $f$.
\end{thm}
\begin{proof}
By the uniqueness
part of the Wall's embedding theorem \cite[page 76]{Wall-thick}
$f_0$ and $f_1$ are concordant. Since the codimension
is at least $3$, concordance implies isotopy
\cite{Hudson-conc=>iso}. Therefore $f_0$ is isotopic to $f_1$.
By the uniqueness of a regular neighborhood this implies that the squares \refequ{diag-mainsquare}
for $f_0$ and $f_1$ are homeomorphic.
\end{proof}
The hypothesis that $f$ is $r$-connected with $r$ satisfying the inequality
\refequ{equ-unknot} is called the \emph{unknotting condition}.
The reason for which we have stated Theorem \ref{thm-unknot} in the context of PL-embeddings instead of Poincar\'e
embeddings is that the corresponding result
for Poincar\'e embeddings is known only under a slightly more restrictive condition.
Indeed Klein has proved such an uniqueness result for Poincar\'e embeddings
with an unknotting condition increased by one, i.e.\ $r\geq2m-n+3$ \cite[Theorem 5.4]{Klein}, or
with the sharp unknotting condition \refequ{equ-unknot}
in the metastable range \cite{Klein-compression}. It is still an open question
whether
condition \refequ{equ-unknot} guarantees the uniqueness
of Poincar\'e embeddings in full generality.
We will prove a rational homotopy
theoretical partial version of Theorem \ref{thm-unknot} by establishing
that, under the unknotting condition \refequ{equ-unknot}, the
rational homotopy type of the complement $C$ depends
only on the rational homotopy class of $f$. From Theorem \ref{thm-stableCDGA}
a guess for the model of the complement would be $R\oplus_\psi sD$ with
some assumption on the vanishing of $R$, $Q$, and $D$ in high degrees.
This vanishing assumption can be removed if we truncate the mapping cone $R\oplus_\psi sD$ by a suitable
acyclic module $L$. Moreover only a structure of $R$-DGmodule (instead of $Q$-DGmodule) is needed on
$D$. More precisely we have the following theorem:
\begin{thm}\label{thm-wkstCDGA}
Consider a Poincar\'e embedding \refequ{diag-mainsquare} of codimension
at least $2$ with $P$ and $W$ connected.
Let $r$ be a positive integer such that
$H_*(f;\BQ)$ is $r$-connected, that is $H_i(f;\BQ)$ is an isomorphism for $i<r$ and
an epimorphism for $i=r$.
If
\begin{equation}\label{equ-unknotrht}r\geq2m-n+2.\end{equation}
then a CDGA model of the map $l\co C\to W$ can be build explicitly out of
any CDGA model of $f\co P\to W$.
More precisely, let
\begin{enumerate}
\item[\rm(i)] $\phi\co R\to Q$ be a CDGA model of
$f^*\co\Apl(W)\to\Apl(P)$ with $R$ connected;
\item[\rm(ii)] $D$ be an $R$-DGmodule weakly equivalent to $s^{-n}\#Q$ with
$D^{<n-m}=0$;
\item[\rm(iii)] ${\psi}\co D\to R$ be a top-degree\ map of $R$-DGmodules;
\item[\rm(iv)] $L\subset R\oplus_{{\psi}}sD$ be an acyclic $R$-subDGmodule
with $L^{\leq n-r-2}=0$ and $(R\oplus_{{\psi}}sD)^{\geq n-r}\subset L$.
\end{enumerate}
Then the canonical CDGA map
$$\lambda\co R\to (R\oplus_{{\psi}}sD)/L$$
is a CDGA-model of the map
$$l^*\co\Apl(W)\to\Apl(C).$$
where $\lambda$ is the composition of the inclusion with the projection and the algebra structure
on the truncated mapping cone is induced by the formula
$(r,sd)\cdot(r',sd')=(r\cdot r',s(r\cdot d'\pm r'\cdot d))$.
Moreover under condition \refequ{equ-unknotrht} it is possible to construct explicitly $R$, $Q$, $D$, $L$,
$\phi$, $\psi$ satisfying hypotheses (i)--(iv) out of any CDGA-model of $f^*\co\Apl(W)\to\Apl(P)$.
\end{thm}
\begin{corol}
\label{corol-wkstCDGA} Consider a Poincar\'e embedding \refequ{diag-mainsquare}
of codimension at least $3$ and with $P$ and $W$ simply-connected.
Let $r$ be a positive integer such that $H_*(f;\BQ)$ is $r$-connected. If $r\geq 2m-n+2$
then the rational homotopy type of the complement $C$ depends only on the rational homotopy class of $f$.
\end{corol}
Moreover we will show that the unknotting condition in Theorem \ref{thm-wkstCDGA}
is sharp. More precisely we will construct in Propositions \ref{prop-exsharp1} and \ref{prop-exsharp2}
families of examples for which the unknotting condition \refequ{equ-unknotrht} fails only by a little
but such that the rational cohomology algebra of the complement is not determined
by the rational homotopy class
of the embedding. Note also that our rational result is valid for any
Poincar\'e embeddings satisfying the unknotting condition,
which improves by $1$ the hypothesis under which the ``integral''
homotopy type of the complement
is known to be unique \cite[Corollary B]{Klein-2}.
Unfortunately we were not able to determine the complete rational homotopy type of the square
\refequ{diag-mainsquare} from the rational homotopy class of $f$ under the unknotting condition.
The best result that we can prove in this direction is the determination, under connectivity
hypotheses
on $P$ and $W$ and the extra assumption that
$n\geq m+r+2$, of the modified
square \refequ{diag-mainsquare} where $\del T$ is replaced by the space
$\check{\del T}$ obtained by removing its top cell. See Theorem \ref{thm-wkstCDGAsquare} for
a precise statement.
Our rational models in Theorems \ref{thm-stableCDGA} and \ref{thm-wkstCDGA}
have applications to the construction of the model of blow-ups \cite{LS-stableblowup} and
\cite{LS-unstableblowup}, and of
the configuration space on two points \cite{LS-FM2}.
\medbreak
The above discussion was about CDGA models for the square \refequ{diag-mainsquare}
which determine its rational homotopy type. Instead of CDGA models associated to the functor
$\Apl$
we can associate models to the functor of singular cochains with
coefficients in a field $\Bk$ of arbitrary characteristic,
$S^*(-;\Bk)$. If $Y$ is a space then $S^*(Y;\Bk)$ is a
differential graded algebra (a DGA for short), and if $f\co
X\to Y$ is a continuous map then $S^*(X;\Bk)$ is a differential
graded module (\emph{DGmodule}) over the DGA $S^*(Y;\Bk)$. There
is a notion of models of such DGmodules, and we can build such a model of
the Poincar\'e embedding \refequ{diag-mainsquare} without any restriction on the
codimension or even on the connectivity of $P$. To state the result we use the notion of a
\emph{menorah} as defined in Example \ref{examples-diagram} and
which is essentially a family of maps with same domain.
\begin{thm}\label{thm-DGmodnonconn}%
Consider a Poincar\'e embedding \refequ{diag-mainsquare} with $W$ connected.
Denote the connected components of $P$ by $P_1,\cdots,P_c$ and
set $f_k:=f|P_k$, for $k=1,\cdots,c$. Denote by $C^*$ one of the
functors $S^*(-;\Bk)$ or $\Apl$.
Suppose a quasi-isomorphism of DGA $\rho\co A\quism
C^*(W)$ has been given. Let
$$\set{\phi_k\co R\to Q_k}_{1\leq k\leq c}$$
be a model in $A$-DGMod of the menorah
$$\set{C^*(f_k)\co C^*(W)\to C^*(P_k)}_{1\leq k\leq c}.$$
For $k=1,\cdots,c$, let $D_k$ be an $A$-DGmodule weakly equivalent
to $s^{-n}\#C^*(P_k)$ and let
$
{\psi}_k\co D_k\to R$ be a top-degree\ map of $A$-DGmodules.
Set $D=\oplus_{k=1}^cD_k$, $Q=\oplus_{k=1}^cQ_k$,
$\phi=(\phi_1,\ldots,\phi_c)\co R\to Q$, and
${\psi}=\sum_{k=1}^c{\psi}_k\co D\to R$. Then the two following
commutative
squares are weakly equivalent in $A$-DGMod:
$$\BD:=\vcenter{
\xymatrix@1{%
R\vrule width0pt depth6pt\ar[r]^\phi\ar@{^(->}[d]&%
Q\vrule width0pt depth6pt\ar@{^(->}[d]\\
R\oplus_{{\psi}}sD\ar[r]_{\phi\oplus\id}&%
Q\oplus_{\phi{\psi}}sD%
}}\hbox{\quad\quad and \quad\quad}\BD':=\vcenter{
\xymatrix@1{%
C^*(W)\ar[r]^{f^*}\ar[d]_{l^*}&
C^*(P)\ar[d]^{i^*}\\
C^*(C)\ar[r]_{k^*}&%
C^*(\del T). }}
$$
\end{thm}
This DGmodule model enables us to improve the classical Lefschetz duality
theorem. Indeed this classical result states that the cohomology of the complement,
$H^*(C;\Bk)=H^*(W\smallsetminus f(P);\Bk)$, is determined \emph{as a vector space}
by the algebra map $H^*(f)\co H^*(W)\to H^*(P)$.
Our result gives a way to determine the \emph{$H^*(W)$-module structure} of
$H^*(C)$, and even its algebra structure under the unknotting condition. This is the content
of the following:
\begin{corol}[Improved Lefschetz duality]
\label{corol-HWmodstruct}
Consider a Poincar\'e embedding \refequ{diag-mainsquare} with $W$
connected.
Suppose a quasi-isomorphism of DGA \,$\rho\co A\quism
C^*(W)$ has been given and let $\phi\co R\to Q$ be an $A$-DGmodule model of
$f^*\co C^*(W)\to C^*(P)$. Then we have an
isomorphism of $H^*(W;\Bk)$-modules
$$H^*(C;\Bk)\cong H(s^{-n}\#R\oplus_{s^{-n}\#\phi}s(s^{-n}\#Q)).$$
If moreover
$H_*(f;\Bk)$ is $r$-connected with $r\geq 2m-n+2$
then this isomorphism determines the algebra structure on $H^*(C;\Bk)$.
\end{corol}
Examples of Section 9 will show that the unknotting condition cannot be dropped
when determining the algebra structure in the last corollary.
Christophe Boilley \cite{Boilley}
has constructed examples showing that the $H^*(W)$-module
structure on $H^*(C)$ is neither necessarily given by a trivial extension
nor determined by the map $H^*(f)$ induced in cohomology.
\medbreak
Notice that in all the results of this paper we can replace the Poincar\'e
embedding by the following weaker notion. Let $\Bk$
be a field. A \emph{$\Bk$-Poincar\'e embedding}
is a commutative square \refequ{diag-mainsquare} such that $W$, $(P,\del T)$ and $(C,\del T)$
satisfy Poincar\'e duality in dimension $n$ over $\Bk$,
$m$ is the cohomological dimension of $P$ with coefficients in $\Bk$,
$H^*(i;\Bk)$ is $(n-m-1)$-connected,
and the square \refequ{diag-mainsquare} induces a Mayer-Vietoris long exact sequence in $H^*(-;\Bk)$.
In other words such a $\Bk$-Poincar\'e embedding is a \emph{homological} version of a
Poincar\'e embedding.
As a last remark note that our study is
complementary to the work of Morgan
\cite{Morgan}
who has computed the rational homotopy type of the complement of
divisors $D_i$ with normal crossings in a projective algebraic
variety $W$. In his case the codimension is very low ($D_i$ is of
codimension $2$) but the existence of mixed Hodge structures
\cite{Deligne}
implies that the rational homotopy type of the complement is
determined by the maps induced in cohomology by the inclusion of
divisors. In the case of a single divisor $D$,
Morgan's model for $W\smallsetminus D$ is
expressed in terms of the shriek map
$f^!\co H^{*+2}(D)\to H^*(W)$ which is a special case of our top-degree\ map
(see Example \ref{ex-topdegreeshriek}.)
\medskip
{\bf Plan of the rest of the paper}\qua Section 2 contains notation
and terminology and Section 3 is about diagrams in closed model
categories. We explain in this section what we mean by a model of a
square or a menorah. In Section 4 we define the notion of a
semi-trivial CDGA structure on certain mapping cones and in Section 5
we study the notion of a top-degree\ map and prove their existence and
essential uniqueness. Section 6 is about the DGmodule model of a
Poincar\'e embedding and contains the proofs of Theorem
\ref{thm-DGmodnonconn} and Corollary \ref{corol-HWmodstruct}. Section
7 is about CDGA models of a Poincar\'e embedding in the stable case
and contains the proof of Theorem \ref{thm-stableCDGA}. Section 8
discusses CDGA models of the complement in a Poincar\'e embedding
under the unknotting condition. We prove here Theorem
\ref{thm-wkstCDGA} and its corollaries. We also state and prove
Theorem \ref{thm-wkstCDGAsquare} which exhibits a model of a square
related to \refequ{diag-mainsquare} under a stronger unknotting
condition. Finally Section 9 contains examples of rationally knotted
embeddings and we illustrate by explicit examples the sharpness of the
unknotting condition.
\medskip{\bf Acknowledgements}\qua The authors want to thank Bill Dwyer
for enlightening conversations on closed model structures on
categories of diagrams and John Klein for explaining the proof of
Theorem \ref{thm-unknot}. We thank also the referee for pointing out
that our results could apply to Poincar\'e embeddings. During this
work the first author benefited from the hospitality of the University
of Alberta and of a travel grant from F.N.R.S., and the second author
from the hospitality of the Universit\'e of Louvain. The first author
is Chercheur Qualifi\'e au F.N.R.S.
\section{Notation and terminology} \label{section-toolkitApl}We
denote by $\Bk$ a commutative field. Recall the notions of
\emph{differential graded algebra}, or DGA for short, and of
(left) \emph{graded differential modules} over a DGA $R$, or
$R$-DGmodules for short, as both defined for example in
\cite[Section 3(c)]{FHT-RHT}. We will always suppose that the
DGA are non negatively graded and that the
differentials are of degree $+1$. We denote by $R$-DGMod the
category of $R$-DGmodules.
{\bf Convention on left and right modules}\qua Sometimes in the paper
(in particular in Section \ref{section-DGMod}) it will be important to
distinguish between left and right DGmodules. By an
\emph{$R$-DGmodule} we always mean a {left} $R$-DGmodule, otherwise we
write explicitly \emph{right $R$-DGmodule}. Also by $R$-DGMod we
denote only the category of left $R$-DGmodules. We denote by
$\hom_\Bk$ (resp.\ $\hom_R$) the sets of $\Bk$-modules (resp.\
$R$-modules) morphisms.
We have also a notion of
\emph{commutative differential graded algebra}, or CDGA for short,
which is a DGA such that the multiplication is graded commutative
(\cite[Example 5 in Section 3(b)]{FHT-RHT} where there are called
\emph{commutative cochain algebras}). We denote
by CDGA the corresponding category. A CDGA or more generally a
non-negatively graded vector space, $V$, is called
\emph{connected} if $V^0\cong\Bk$.
The degrees of graded modules and algebras will be written as
superscripts. If $X$ is a graded module or algebra, we will write
$X^{>m}=0$ to express the fact that $X^k=0$ for $k>m$, and
similarly $X^{\geq m}=0$, $X^{<m}=0$, and so on.
The \emph{dual} of a graded $\Bk$-module $M$ will be
denoted by $\#M$ with the grading $ (\#M)^i =
{\mathrm{hom}}(M^{-i},\Bk)$. The duality pairing is defined by
$$
\langle-,-\rangle\co M\otimes \#M \to\Bk,\,x\otimes f\mapsto
\langle x,f\rangle=f(x).
$$
If $(M,d)$ is a differential module then its dual $\#M$ is
equipped with the differential $\delta$ characterized by
$\langle x, \delta(f)\rangle=-(-1)^{|x|}\langle d(x),f\rangle$.
If $M$ is a \emph{right} module over some graded algebra $R$, then
its dual admits a structure of \emph{left} $R$-module
characterized by the formula $ \langle x,a.f \rangle=\langle x.a,f
\rangle$. Similarly if $M$ is a right DGmodule then its dual
becomes a left DGmodule.
The {\em $k$-th suspension} of a graded vector space $M$ is the
graded vector space $s^kM$ defined by $(s^kM)^j\cong M^{k+j}$ and
this isomorphism is denoted by $s^k$. If $M$ is also a left
$R$-module, we transport this structure on the $k$-suspension by
the formula $r.(s^kx)=(-1)^{|r|k}s^k(r.x)$. Also if $M$ is
equipped with a differential $d$, then we define a differential on
$s^kM$ by $d(s^kx)=(-1)^ks^k(dx)$. If $k=1$ we write $sM$ for
$s^1M$.
The {\em mapping cone} of an $R$-DGmodule morphism $f\co X\to Y$
is the $R$-DGmodule $C(f):=(Y\oplus_f sX,d)$ where the
differential is defined by $d(y,sx)=(d_Y(y)+f(x),-sd_X(x))$. If
$f$ is a CDGA morphism, in general there is no natural CDGA
structure on the mapping cone but we will show in Section
\ref{section-MC} that such a CDGA structure exists under favorable
hypotheses.
We will use the functor of (normalized) singular cochains with
coefficients in $\Bk$
$
S^*(-;\Bk)\co\mathrm{ Top}\to \mathrm{DGA}
$
as defined for example in \cite[Chapter 5]{FHT-RHT}. When $\Bk$ is
of characteristic $0$, we have also the de Rham-Sullivan functor
of polynomial forms
$
\Apl\co \mathrm{Top}\to \mathrm{CDGA}
$
as defined in \cite{BG} or \cite[Chapter 10]{FHT-RHT}.
The categories $R$-DGMod and CDGA are closed model categories in the
sense of Quillen for which the weak equivalences are the
quasi-isomorphisms and the fibrations are the surjections (for a
nice review of closed model categories, we refer the reader to
\cite{DwyerSpalinski}). By an \emph{acyclic (co)fibration} we mean
a (co)fibration that is also a weak equivalence. We say that two
objects $X$ and $X'$ in a closed model category are \emph{weakly
equivalent} or that $X$ is a \emph{model} of $X'$ if there exists
a finite chain of weak equivalences joining them,
$$
\xymatrix{X&\ar[l]_\simeq X_1\ar[r]^\simeq&\cdots&\ar[l]_\simeq
X_n\ar[r]^\simeq&X'}.
$$
In that case we will write $X\simeq X'$. Since in Section
\ref{section-diagram} we will consider a closed model structure on
certain categories of diagrams, we can speak of \emph{models} of
that diagrams.
We review quickly the notion of \emph{relative Sullivan algebras}
which is an important class of cofibrations in CDGA. If $V$ is a
non-negatively graded vector space we denote by $\wedge V$ the
free graded commutative algebra generated by $V$ (see \cite[\S
3(b), Example 6]{FHT-RHT}.) A {relative Sullivan algebra}
(\cite[Chapter 14]{FHT-RHT}, or \emph{KS-extension} in the older
terminology of \cite{Halperin-lecturesminmod}) is a CDGA morphism
$\iota\co(A,d_A)\hookrightarrow (A\otimes \wedge V,D)$
where the differential $D$ is an extension of $d_A$ that satisfies
some nilpotence condition (see \cite[Chapter 14]{FHT-RHT} for the
precise definition.) Notice that in this paper we do not assume
that $V^0=0$, following \cite{Halperin-lecturesminmod} but
contrary to \cite{FHT-RHT}. In the special case $A=\Bk$ we get the
notion of a \emph{Sullivan algebra}, $(\wedge V,D)$, which is a
cofibrant object in CDGA. Examples of cofibrant objects in
$R$-DGMod are \emph{semi-free models} as defined in \cite[Chapter
6]{FHT-RHT}. Roughly speaking they are $R$-DGmodules of the form
$(R\otimes V,D)$ where $V$ is a graded vector space and the
differential $D$ satisfies also a nilpotence condition. Finally
remember that every object is fibrant in CDGA and in $R$-DGMod.
To denote that two maps $f_0$ and $f_1$ are homotopic in CDGA or
$R$-DGMod we will write $f_0\sim f_1$, or sometimes $f_0\sim_R
f_1$ to emphasize the underlying DGA. When $P$ and $N$ are
$R$-DGmodules, with $P$ cofibrant, we denote by
$$
[P,N]_R
$$
the set of homotopy classes of $R$-DGmodules from $P$ to $N$.
\section{Diagrams in closed model categories}\label{section-diagram}
In order of being able to speak of models
of objects, maps, commutative squares, and so on, we review in
this section the convenient language of diagrams as described for example in
\cite[Section 10]{DwyerSpalinski}. There will exist a
closed model structure on each of the categories of diagrams that we will
consider. We will finish the section by two useful lemmas to turn
certain homotopy commutative diagrams into commutative ones.
\begin{defin}
Let $\calS$ be a small category and let $\calC$ be any category. A
\emph{diagram in $\calC$ shaped on $\calS$} is a covariant functor
$\BD\co\calS\to\calC$ and we say that $\calS$ is \emph{shaping} the diagram.
A \emph{morphism of diagrams} is a natural transformation between
two diagrams. This defines the category of diagrams $\calC^\calS$.
\end{defin}
We describe now the five main examples of diagrams that we will
consider in this paper. First recall that to each partially
ordered set (or \emph{poset}, for short), $({S},\leq)$, we can
associate a small category $\calS$ whose objects are the elements
of ${S}$ and such that the set of morphisms, ${\mathrm{hom}}_\calS(x,y)$,
between two objects $x$ and $y$ in $\calS$ is a singleton if
$x\leq y$ and is the empty set otherwise.
\begin{examples}$\phantom{999}$
\label{examples-diagram}
\begin{description}
\item[Object] If $\calS$ is the category with only one object and one morphism
(that is the category associated with the poset with only one element) then a diagram
in $\calC$ shaped on $\calS$ is called \emph{an object} of $\calC$.
\item[Map]If $\calS$ is the category associated to the ordered set $\set{0,1}$
then a diagram
in $\calC$ shaped on $\calS$ is just a map between two objects of $\calC$.
Such a diagram is called \emph{a map} of $\calC$.
\item[Commutative square] Let $\calS$ be the category whose objects are the four sets $\emptyset$, $\set{1}$,
$\set{2}$, and $\set{1,2}$,
and whose morphisms are the inclusion maps. A diagram in $\calC$ shaped on $\calS$ is called a
\emph{commutative square} in $\calC$.
\item[Menorah] Let $\calS$ be the category whose objects are
$\emptyset,\set{1},\cdots,\set{n}$, for some positive integer $n$
and where morphisms are inclusions of sets.
Then a diagram in $\calC$ shaped on $\calS$ is just a collection of maps $f_1,\cdots,f_n$ with same domain.
We call such a diagram \emph{a menora
}
and we denote it by $\set{f_i}_{1\leq i\leq
n}$.
\item[Composite] Let $\calS$ be the category corresponding to the ordered
set $\set{0,1,2}$. A diagram shaped on $\calS$ is just two
composable maps $f_0\co X\to Y$ and $f_1\co Y\to Z$. We call such a diagram a \emph{composite}
and we denote it by
$(f_0,f_1)$.
\end{description}
\end{examples}
Each category shaping one of the five diagrams in Example
\ref{examples-diagram} is a \emph{very small category} in the
sense of \cite[Section 10.13]{DwyerSpalinski}. This notion is
useful because of the following:
\begin{prop}\label{prop-CMdiagrams}
Let $\calC$ be a closed model category and let $\calS$ be a very
small category. Then the category $\calC^\calS$ of diagrams in
$\calC$ shaped on $\calS$ admits a closed model structure such
that a map $f\co \BD\to \BD'$ between diagrams is a weak
equivalence (resp.\ a fibration) if and only if for each object $x$
in $\calS$ the map $f(x)\co \BD(x)\to \BD'(x)$ is a weak
equivalence (resp.\ a fibration) in $\calC$.
Moreover if $\hat \BD$ is a cofibrant diagram in $\calC^\calS$ then
for each object $x$ in $\calS$, $\hat \BD(x)$ is a cofibrant
object of $\calC$, and for each morphism $i$ in $\calS$, the map
$\hat \BD(i)$ is a cofibration in $\calC$.
If every object of $\calC$ is fibrant, then the same is true in
$\calC^\calS$.
\end{prop}
\begin{proof} This
model structure is described in \cite[Section
10.13]{DwyerSpalinski}, where the cofibrations in $\calC^\calS$
are also defined (a complete proof of the axioms of Quillen for
this category can be found in \cite[Theorem 5.2.5]{Hovey}). Using
the fact that the initial object $\emptyset$ in $\calC^\calS$ is
the constant diagram with value $\emptyset$ at each object of
$\calS$, it is straightforward to check from the definition of a
cofibration in $\calC^\calS$ (\cite[10.13]{DwyerSpalinski}) that
if $\emptyset\to \hat \BD$ is a cofibration then each object
$\hat \BD(x)$ is cofibrant and each map $\hat \BD(i)$ is a
cofibration. The last statement is obvious.
\end{proof}
In this paper we will always suppose that the closed model
structure on a category of diagrams $\calC^\calS$ is the one
considered in Proposition \ref{prop-CMdiagrams}.
Following the terminology of Section \ref{section-toolkitApl} we can speak of weakly
equivalent diagrams or of a model of a diagram.
\begin{rmk}
\label{rmk-modelmenorah} If a menorah $\{f_k\}_{1\leq k\leq n}$
is a model of another menorah $\{f'_k\}_{1\leq k\leq n}$, then
clearly each map $f_k$ is a model of $f'_k$. It is important to
notice that the converse is \emph{not} true in general. Similarly
if a composite $(f,g)$ is a model of a composite $(f',g')$ then
$f$ is a model of $f'$ and $g$ is a model of $g'$, but again the
converse is not true.
\end{rmk}
The proofs of the following two lemmas are based on standard
techniques of closed model categories and we leave them as
exercises for the reader.
\begin{lemma}\label{lemma-bisurj}%
Let $X$ and $X'$ be two weakly equivalent objects in some closed
model category in which every object is fibrant. Then there exists
a cofibrant object $\hat X$ and acyclic fibrations
$$
\xymatrix{X&\ar@{->>}[l]_\simeq^\beta \hat
X\ar@{->>}[r]^\simeq_{\beta'}&X'}
$$
such that $(\beta,\beta')\co\hat X\to X\times X'$ is also a fibration.
\end{lemma}
\begin{lemma}
\label{lemma-rigidify} Let
$$
\xymatrix{
&\hat A\ar[ld]_f\ar[d]^{\tilde f}\ar[rd]^{f'}\\
X&\ar[l]^\beta \hat X\ar[r]_{\beta'}&X' }
$$
be a homotopy commutative diagram in a closed model category. If
$\hat A$ is a cofibrant object, if $X$ and $X'$ are fibrant, and
if $(\beta,\beta')\co\hat X\to X\times X'$ is a fibration then
there exists a morphism $\hat f\co\hat A\to\hat X$ homotopic to
$\tilde f$ and making the following diagram strictly commute
$$
\xymatrix{
&\hat A\ar[ld]_f\ar[d]^{\hat f}\ar[rd]^{f'}\\
X&\ar[l]^\beta \hat X\ar[r]_{\beta'}&X'. }
$$
\end{lemma}
\section{CDGA structures on mapping cones}\label{section-MC}
The aim of this section is to define a natural extension of the
$R$-DGmodule structure of some mapping cones to CDGA structures,
under certain dimension-connectivity hypotheses.
\begin{defin}
\label{def-semitrivialCGA} Let $R$ be a CDGA and let $f\co X\to
R$ be a morphism of $R$-DGmodules. Consider the mapping cone
$C(f)=R\oplus_f sX$ and define a multiplication
$$
\mu\co C(f)\otimes C(f)\to C(f)
$$
by, for homogeneous elements $r,r'\in R$ and $x,x'\in X$,
\begin{itemize}
\item[(i)] $\mu(r\otimes r')=r.r'$
\item[(ii)] $\mu(r\otimes sx')=(-1)^{\deg(r)}s(r.x')$
\item[(iii)] $\mu(sx\otimes r')=(-1)^{\deg(x).\deg(r')}s(r'.x)$
\item[(iv)] $\mu(sx\otimes sx')=0$.
\end{itemize}
This multiplication defines a commutative graded algebra structure
(not necessarily differential) on $R\oplus_f sX$ that we call the
{\em semi-trivial CGA structure} on the mapping cone.
\end{defin}
This CGA structure on $C(f)$ is compatible with its $R$-module
structure in the sense that the module structure is induced by the
CGA map $R\hookrightarrow R\oplus_f sX$. It is important to notice
that in general the multiplication $\mu$ defined above does not
define a CDGA structure on $C(f)$ because the Leibnitz rule on the
differential of the mapping cone is not necessarily satisfied.
However, we have the following lemmas.
\begin{lemma}\label{lemma-CDGAMC}%
Let $R$ be a CDGA and let $f\co X\to R$ be an $R$-DGmodule
morphism. Suppose that $(sX)^{<k}=0$ and $(R\oplus sX)^{>2k}=0$
for some non negative integer $k$.
Then the mapping
cone $C(f)=R\oplus_f sX$ endowed with its semi-trivial
multiplication is a CDGA and the inclusion map $R\hookrightarrow
R\oplus_f sX$ is a CDGA-morphism.
\end{lemma}
\begin{proof}
This lemma is a special case of the next lemma with $I=0$ and $l=0$.
\end{proof}
\begin{lemma}\label{lemma-CDGAtruncMC}%
Let $R$ be a CDGA, let $f\co X\to R$ be an $R$-DGmodule
morphism, and let $I\subset R\oplus_f sX$ be an $R$-DGsubmodule.
Suppose that
$(sX)^{<k}=0$, $I^{\leq k-l}=0$, and $(R\oplus_f sX)^{\geq
2k-l+1}\subset I$
for non negative
integers $k$ and $l$. Then the semi-trivial multiplication $\mu$ on the
mapping cone $C(f)=R\oplus_fsX$ induces a multiplication on
$C(f)/I$ which endows this quotient with a CDGA-structure, and the
composition
$$
\xymatrix{R\,\ar@{^(->}[r]&R\oplus_fsX\ar[r]^{\pr}&C(f)/I}
$$
is a CDGA morphism.
\end{lemma}
\begin{proof}
We show first that $I$ is an ideal of the CGA $C(f)$ equipped with
its semi-trivial CGA structure. Since $I$ is an $R$-submodule of
$C(f)$ we have that $\mu(R\otimes I)=R.I\subset I$. On the other
hand, for degree reasons $\mu(sX\otimes I)\subset (C(f))^{\geq
2k-l+1}\subset I$. Therefore $\mu(C(f)\otimes I)\subset I$. Thus $I$
is a left ideal, hence a two-sided ideal because $\mu$ is graded
commutative.
This implies that the CGA structure on $C(f)$ induces a CGA
structure on the quotient $C(f)/I$. Denote by $\delta$ the
differential on the mapping cone $C(f)$ and by $\bar\delta$ the
induced differential on the quotient. To prove that
$(C(f)/I,\bar\delta)$ is a CDGA we have only to check the Leibnitz
formula. This will be a consequence of the following relation, for
$c,c'$ homogeneous elements in $R\oplus sX$:
\begin{equation}\label{equ-Leibniztrunc}%
\delta(\mu(c\otimes c'))-\mu(\delta(c)\otimes c')-(-1)^{|c|}
\mu(c\otimes\delta(c'))\,\in\,I.
\end{equation}
To prove \refequ{equ-Leibniztrunc} we study different cases. If
$c,c'\in R$ then the expression in \refequ{equ-Leibniztrunc} is
zero because $R$ is a DGA. If $c\in R$ and $c'\in sX$ then the
expression in \refequ{equ-Leibniztrunc} is zero because $\delta$
is a differential of $R$-DGmodule and the same is true if $c\in
sX$ and $c'\in R$ because $\mu$ is graded commutative. Finally if
$c,c'\in sX$ then the degree of the expression in
\refequ{equ-Leibniztrunc} is at least $2k+1\geq 2k-l+1$, therefore it belongs
to $I$.
This completes the proof that $C(f)/I$ is a CDGA. It is
straightforward to check that the map $R\to C(f)/I$ is a
CDGA-morphism.
\end{proof}
\begin{defin}\label{def-CDGAMC}
The CDGA-structures defined on the mapping cone $R\oplus_f sX$ in
Lemma \ref{lemma-CDGAMC} (respectively on the truncated mapping
cone $(R\oplus_f sX)/I$ in Lemma \ref{lemma-CDGAtruncMC}) is
called the \emph{semi-trivial CDGA structure}.
\end{defin}
Our last lemma gives a sufficient condition for some DGmodule map
between CDGA to be a CDGA morphism.
\begin{lemma}
\label{lemma-DGmodCDGAmap} Let $f\co A\to B$ be a
CDGA-morphism, let
$\xymatrix@1{A\quad\ar@{>->}[r]^-u&A\otimes\wedge X}$ be a relative
Sullivan algebra, and let $\hat f\co A\otimes\wedge X\to B$ be
an $A$-DGmodule morphism extending $f$. If $X^{<k}=0$ and
$B^{\geq2k}=0$ for some non negative integer $k$ then $\hat f$ is a CDGA morphism.
\end{lemma}
\begin{proof}
Since $A\otimes\wedge X$ and $B$ are graded commutative, $\hat f$
is a morphism of $A$-bimodules. The lemma follows from the fact
that for degree reasons $\hat f(A\otimes\wedge^{\geq 2}X)=0$.
\end{proof}
\section{Top-degree\ or shriek map}
The aim of this section is to introduce the simple notion of a \emph{top-degree\ map}
(which was called a \emph{shriek map} in early version of this paper).
A key result will be the existence and essential uniqueness of such top-degree\ maps
(Proposition \ref{prop-existsshriek}.)
We start with the definition and two examples.
\begin{defin}\label{def-shriek}%
Let $R$ be a DGA and assume that $H^*(R)$ is a connected
Poincar\'e duality algebra in dimension $n$.
A \emph{top-degree\ map of $R$-DGmodule}
is an $R$-DGmodule map
$\psi\co D\to R'$
such that $R'$ is weakly equivalent to $R$ and
$H^n(\psi)$ is an isomorphism.
\end{defin}
\begin{example}
\label{ex-topdegreeshriek}
Suppose that $f\co V\hookrightarrow W$ is an embedding of
connected \emph{closed} oriented manifolds of codimension $k$.
Denote by [V] and [W] their homology orientation classes. We have
the classical cohomological shriek map (or Umkehr map, or Gysin
map, see \cite[VI.11.2]{Bredon})
$$
f^!\co s^{-k}H^*(V;\Bk)\to H^*(W;\Bk)
$$
characterized by the equation $f(s^{-k}v)\cap[W]=f_*(v\cap[V])$
(the $k$th-suspension is here only to make $f^!$ a degree
preserving
map.) It is clear that $f^!$ is a map of $H^*(W)$-modules
and that it induces an isomorphism in
degree $n=\dim(W)$. Therefore $f^!$ is a top-degree\ map of
$H^*(W)$-module (here
the differentials are supposed to be $0$).
\end{example}
\begin{example}
\label{ex-topdegreedual}
Let $R$ be a DGA such that $H(R)$ is a connected
Poincar\'e duality algebra in dimension $n$. Let
$\phi\co R\to Q$ be a morphism of \emph{right} $R$-DGmodules
such that $H^0(\phi)$ is an isomorphism. Then $s^{-n}\#R$ is
quasi-isomorphic to $R$ and the map
$$s^{-n}\#\phi\co s^{-n}\#Q\to s^{-n}\#R$$
is a top-degree\ map of (left) $R$-DGmodules.
\end{example}
To prove the existence and uniqueness of top-degree\ maps we need first to study
further sets of homotopy classes of $R$-DGmodules.
For an integer $i$, denote by ${\mathrm{hom}}^i_R(P,N)$ the $\Bk$-module
of $R$-module maps of degree $i$ from $P$ to $N$ and set
$${\mathrm{hom}}^*_R(P,N):=\oplus_{i\in\BZ} {\mathrm{hom}}^i_R(P,N).$$
We can define a degree $+1$ differential $\delta$ on this graded
$\Bk$-module by the formula $\delta(f) = d_N f-(-1)^{|f|}f d_P$.
The following identification is well-known and we omit its proof
(e.g.\ \cite{FHT-gorenstein}):
\begin{lemma}\label{lemma-charsethmtpy}%
Let $R$ be a DGA, let $P$ be a cofibrant $R$-DGmodule, and let $N$
be an $R$-DGmodule. Then we have an isomorphism
$$[P,N]_R\cong H^0({\mathrm{hom}}^*_R(P,N),\delta).$$
\end{lemma}
We have the following important characterization of the set of
homotopy classes into a Poincar\'e duality algebra.
\begin{prop}
\label{prop-PDhmtpyclasses} Let $R$ be a DGA over a field $\Bk$
such that $H^*(R)$ is a connected Poincar\'e duality algebra in
dimension $n$. Let $R'$ be an $R$-DGmodule weakly
equivalent to $R$ and let $P$ be a cofibrant $R$-DGmodule. Then
the map
$$
H^n\co [P,R']_R\to{\mathrm{hom}}_\Bk(H^n(P),H^n(R'))\,,\quad[f]\mapsto
H^n(f).
$$
is an isomorphism of $\Bk$-modules.
\end{prop}
\proof
Without any loss of generality we can suppose that $R'=R$ because
weak equivalences preserve each side of the isomorphism we want to
prove.
Since $H^n(R)\cong\Bk$ there exists a $\Bk$-DGmodule map
$
\epsilon_0\co R\to s^{-n}\Bk
$
inducing an isomorphism in $H^n$. Using the canonical isomorphism
$\# s^nR\cong s^{-n}\#R$ we can interpret $\epsilon_0$ as a
cocycle in $s^{-n}\#R$ and $[\epsilon_0]\not=0$ in
$H^0(s^{-n}\#R)\cong\#H^n(R)$.
Since $R$ is also a \emph{right} $R$-DGmodule, we have a
structure of (left) $R$-DGmodule on $s^{-n}\#R$ (remember our
convention in Section \ref{section-toolkitApl}.) There is a unique
$R$-DGmodule map
$$\epsilon\co R\to s^{-n}\#R$$
sending $1\in R$ to $\epsilon_0$. Thus
$H^*(\epsilon)\co H^*(R)\to s^{-n}\#H^*(R)$
is an $H^*(R)$-module morphism which is an isomorphism in degree
$n$. By Poincar\'e duality of $H^*(R)$ this implies that
$H^*(\epsilon)$ is an isomorphism in every degree. Thus $\epsilon$
is a quasi-isomorphism.
Consider the adjunction isomorphism
$${\mathrm{hom}}_R(P,\#R)\cong{\mathrm{hom}}_\Bk(P,\Bk)\,,\quad\phi\mapsto\hat\phi$$
where $\hat\phi\co P\to\Bk$ is defined by
$\hat\phi(x)=(\phi(x))(1)$ for $x\in P$ and $1$ the unit in $R$.
Combining this isomorphism with Lemma \ref{lemma-charsethmtpy}
we get the following
sequence of isomorphisms
\begin{eqnarray*}
[P,R]_R&\cong& H^0({\mathrm{hom}}_R(P,R))\\
&\stackrel{\epsilon_*}{\cong}&H^0({\mathrm{hom}}_R(P,s^{-n}\#R))\\
&\cong&H^n({\mathrm{hom}}_R(P,\#R))\\
&\cong&H^n({\mathrm{hom}}_\Bk(P,\Bk))\\
&\cong&{\mathrm{hom}}_\Bk(H^n(P),s^n\Bk).
\end{eqnarray*}
Moreover it is straightforward to check that the following diagram
is commutative where the horizontal isomorphism is taken as the
previous sequence of isomorphisms:
$$
\xymatrix{[P,R]_R\ar[r]^-\cong\ar[rd]_{H^n}&{\mathrm{hom}}_\Bk(H^n(P),s^n\Bk)\\
&{\mathrm{hom}}_\Bk(H^n(P),H^n(R)).\ar[u]^\cong_{\epsilon^*_0}}
\eqno{\raise-38pt\hbox{\qed}}
$$
We establishes now the existence and uniqueness (up to homotopy and a scalar multiple)
of top-degree\ maps.
\begin{prop}\label{prop-existsshriek}
Let $R$ be a DGA such that $H^*(R)$ is a connected Poincar\'e
duality algebra in dimension $n$, let $R'$ be an
$R$-DGmodule weakly equivalent to $R$, and let $\hat D$ be a
cofibrant $R$-DGmodule such that $H^n(\hat D)\cong\Bk$. Then there
exists a top-degree\ map of $R$-DGmodules
$$
\psi\co \hat D\to R'.
$$
Moreover if $\psi'\co \hat D\to R'$ is another top-degree\ map
then there exists $u\in\Bk\smallsetminus\{0\}$
such that
$
[\psi]=u.[\psi']
$
in $[\hat D,R']_R$.
\end{prop}
\begin{proof}
By Proposition \ref{prop-PDhmtpyclasses} we have an isomorphism
$$
H^n\co[\hat D, R']_R\iso{\mathrm{hom}}_\Bk(H^n(\hat D),H^n(R')).
$$
Denote by ${\mathrm{iso}}\left(H^n(\hat D),H^n(R')\right)$ the submodule
of ${\mathrm{hom}}_\Bk(H^n(\hat D),H^n(R'))$ consisting of isomorphisms.
Since $H^n(\hat D)\cong\Bk\cong H^n(R)$ there is an obvious
isomorphism
$${\mathrm{iso}}\left(H^n(\hat D),H^n(R')\right)\cong\Bk\smallsetminus\set{0}.
$$
Any homotopy class $\psi\in[\hat D, R']_R$ corresponding to an
element of the non empty set ${\mathrm{iso}}\left(H^n(\hat
D),H^n(R')\right)$ gives a top-degree\ map, which proves the existence
part.
The uniqueness part is based on the same computation and left to the reader.
\end{proof}
We end this section by a lemma on sets of homotopy classes.
\begin{lemma}\label{lemma-stablehmtpyclasses}%
Let $A$ be a DGA, let $D$ be a cofibrant $A$-DGmodule, and let $X$
be an $A$-DGmodule. Suppose that there exist integers $r\geq1$
and $m\geq0$ such that
\begin{itemize}
\item $H^{\leq r-1}(A)=H^0(A)=\Bk$, i.e.\ $A$ is cohomologically
$(r-1)$-connected,
\item $H^{<0}(X)=0$ and $H^{>m}(X)=0$, and
\item $H^{\leq m-r+1}(D)=0$.
\end{itemize}
Then the map
$$H^*\co[D,X]_A\to {\mathrm{hom}}^0_\Bk(H^*(D),H^*(X))\,,\quad[f]\mapsto H^*(f)$$
is an isomorphism of $\Bk$-modules.
If moreover $r=1$ then
$[D,X]_A=0$.
\end{lemma}
\proof
We treat
separately the cases $r=1$ and $r\geq2$. Suppose first that $r=1$.
Then $H^{\leq m}(D)=0=H^{>m}(X)$. By standard obstruction theory
every $A$-DGmodule morphism $f\co D\to X$ is nullhomotopic.
Hence $[D,X]_A=0$. Moreover ${\mathrm{hom}}^0_\Bk(H^*(D),H^*(X))=0$ for
degree reasons. This proves the lemma for $r=1$.
Suppose that $r\geq2$. Using Lemma \ref{lemma-charsethmtpy} one
can prove that the $\Bk$-module $[D,X]_A$ remains unchanged if we
replace $D$, $X$, or $A$ by a cofibrant weakly
equivalent objects (see
\cite[Proposition A.4.(ii)]{FHT-gorenstein}.) Since $H^{\leq
1}(A)=\Bk$, we can replace the DGA $A$ by a minimal free model in
the sense of \cite[Appendix]{HalpLemcatDGA}, therefore we can
suppose that $A^{\leq r-1}=\Bk$. Next by replacing $D$ by a weakly
equivalent minimal semi-free
$A$-DGmodule we can suppose that $D^{\leq m-r+1}=0$. Since $H^{>m}(X)=0$ and
$A$ is connected we can also assume that $X^{>m}=0$.
Then, for degree reasons, the forgetful map
$
\phi^i\co{\mathrm{hom}}^i_A(D,X)\to {\mathrm{hom}}^i_\Bk(D,X)
$
is surjective for $i\geq -1$. Obviously $\phi^i$ is always
injective. Thus in the following commutative diagram, the
horizontal maps are isomorphisms:
$$\xymatrix{
{\mathrm{hom}}_A^1(D,X)\ar[r]^{\cong}_{\phi^1}&{\mathrm{hom}}_\Bk^1(D,X)\\
{\mathrm{hom}}_A^0(D,X)\ar[r]^{\cong}_{\phi^0}\ar[u]_\delta&{\mathrm{hom}}_\Bk^0(D,X)\ar[u]_\delta\\
{\mathrm{hom}}_A^{-1}(D,X)\ar[r]^{\cong}_{\phi^{-1}}\ar[u]_\delta&{\mathrm{hom}}_\Bk^{-1}(D,X)\ar[u]_\delta.}
$$
This implies that $H^0(\phi)\co H^0({\mathrm{hom}}^*_A(D,X),\delta)\to
H^0({\mathrm{hom}}^*_\Bk(D,X),\delta)$ is an isomorphism. We conclude by
using Lemma \ref{lemma-charsethmtpy} and the obvious
identification
$$H^0({\mathrm{hom}}^*_\Bk(D,X),\delta)\cong{\mathrm{hom}}^0_\Bk(H^*(D),H^*(X)).
\eqno{\qed}$$
\section{DGmodule model of a Poincar\'e embedding}\label{section-DGMod}
The aim of this section is to prove Theorem \ref{thm-DGmodnonconn} and
Corollary \ref{corol-HWmodstruct}.
\begin{rmk}
Before proceeding with the proof of Theorem \ref{thm-DGmodnonconn}
we make a comment about the hypothesis on the model of a \emph{menorah}.
Indeed in that theorem we suppose that
$\set{\phi_k}_{1\leq k\leq c}$ is a model of the menorah
$\set{C^*(f_k)}_{1\leq k\leq c}$. As we pointed out in
Remark \ref{rmk-modelmenorah}, when $c\geq 2$ this is a stronger hypothesis
than asking for each $\phi_k$ to be a model of $C^*(f_k)$.
We illustrate this fact by the
following example. Consider the torus $T=S^1\times S^1$ and denote
by $\dot{T}$ this torus with a small open disk removed, so that
$\dot{T}$ is a compact surface of genus $1$ with a circle for
boundary. Let $f\co S^1\hookrightarrow \dot{T}$ be an embedding
such that composed with the inclusion $\dot{T}\subset S^1\times
S^1$ it gives the inclusion of the first factor $S^1$ in
$S^1\times S^1$. Denote by $\dot{T_1}$ and $\dot{T_2}$ two copies
of $\dot{T}$ and let $f_k\co S^1\hookrightarrow\dot{T_k}$ be
the embeddings corresponding to $f$, $k=1,2$. Set
$W=\dot{T_1}\cup_{\del \dot{T}}\dot{T_2}$ which is a closed surface of
genus $2$. It is clear that the complement
$C:=W\smallsetminus(f_1(S^1)\amalg f_2(S^1))$ is connected. Consider
now the obvious automorphism $\phi$ of $W$ permuting $\dot{T_1}$
and $\dot{T_2}$. This automorphism is such that $\phi\circ
f_2=f_1$. By deforming slightly $\phi$ into a diffeotopic
automorphism $\phi'$, we can suppose that $f'_2:=\phi'\circ
f_2$ is an embedding of a circle closed
but disjoint from $f_1(S^1)$. Then $C':=W\smallsetminus(f_1(S^1)\amalg
f'_2(S^1))$ is not connected. Thus $C^*(C)$ and $C^*(C')$ do not
have the same DGmodule model since they have different
cohomologies. On the other hand $C^*(f'_2)$ and $C^*(f_2)$ do
admit the same model since they differ only by the automorphism
$C^*(\phi')$ of $C^*(W)$. The explanation of this apparent
contradiction is in the fact that $\set{C^*(f_1),C^*(f_2)}$ and
$\set{C^*(f_1),C^*(f'_2)}$ do not admit a common model as
\emph{menorah} in the sense of Example \ref{examples-diagram}.
\end{rmk}
The proof of Theorem
\ref{thm-DGmodnonconn} consists of
a series of four lemmas.
Note first that by taking mapping cylinders we can assume
without loss of generality that diagram \refequ{diag-mainsquare}
of Definition \ref{def-Pemb} is a genuine push-out and that each map $i$, $k$, $f$, $l$ is a closed cofibration.
\begin{lemma}\label{lemma-C*MC}
With the same hypotheses as in Theorem \ref{thm-DGmodnonconn}
consider the inclusion map $\iota\co C^*(W,C)\to C^*(W)$. Then
the commutative square $\BD'$ is weakly equivalent in
$C^*(W)$-DGMod to the following commutative square:
$\quad\quad\quad\quad\BD'':=\vcenter{\xymatrix@1{
C^*(\vrule width0pt depth6pt W)_{{}_{{}_{}}}\ar[r]^{f^*}\ar@{^(->}[d]&%
C^*(\vrule width0pt depth6pt P)_{{}_{{}_{}}}\ar@{^(->}[d]\\%
C^*(W)\oplus_{\iota}sC^*(W,C)\ar[r]^{f^*\oplus\id}&%
C^*(P)\oplus_{f^*\iota}sC^*(W,C)%
}}$
\end{lemma}
\begin{proof} Consider the following ladder of short exact sequences in
$C^*(W)$-DGMod
$$
\xymatrix{0\ar[r]&C^*(W,C)\ar[r]^{\iota}\ar[d]_{\simeq}^{f_0^*}&C^*(W)\ar[r]^{l^*}\ar[d]^{f^*}&
C^*(C)\ar[r]\ar[d]^{k^*}&0\\
0\ar[r]&C^*(P,\del T )\ar[r]^{\iota'}&C^*(P)\ar[r]^{i^*}& C^*(\del
T)\ar[r]&0.}
$$
By Mayer-Vietoris $f_0^*$ is a quasi-isomorphism and we have a weak
equivalence
$$ \id\oplus sf_0^*\co
\left(C^*(P)\oplus_{\iota'f_0^*}sC^*(W,C)\right)\quism
\left(C^*(P)\oplus_{\iota'}sC^*(P,\del T)\right).
$$
Thus in diagram $\BD''$ we can replace the right bottom DGmodule
by $C^*(P)\oplus_{\iota'}sC^*(P,\del T)$. To finish the proof apply the five lemma to deduce
that the map $k^*$ is weakly equivalent to the map induced between the mapping cones of $\iota$
and $\iota'$.
\end{proof}
Before stating the next two lemmas we need to introduce further
notation. Let $\del T_k$ be the union of the connected components of $\del T$ that are sent to $P_k$ by $i$. Set
$C_k:=C\cup_{(\del T\smallsetminus\del T_k)}(P\smallsetminus P_k)$, which can
be interpreted as the complement of $P_k$ in $W$ since $W\simeq C_k\cup_{\del T_k}P_k$.
Define also the inclusion maps
$$
\iota_k\co C^*(W,C_k)\hookrightarrow C^*(W).
$$
In the next lemma we build a convenient common model $\hat\phi_k$
of both $\phi_k$ and $f_k^*$.
\begin{lemma}\label{lemma-DGmodphihat}
With the hypotheses of Theorem \ref{thm-DGmodnonconn} there
exists a cofibrant $A$-DGmodule $\hat R$, weak equivalences
$\alpha,\alpha'$, and, for each $k=1,\cdots,c$, an $A$-DGmodule
cofibration
$\xymatrix@1%
{\hat R\quad\ar@{>->}[r]^{\hat\phi_k}&\hat Q_k}%
$ and weak equivalences $\beta_k,\beta'_k$, making the following
diagrams commute
$$
\xymatrix{%
R_{{}_{{}_{}}}\ar[d]_{\phi_k}& \hat
R_{{}_{{}_{}}}\ar[l]_{\alpha}^{\simeq}\ar[r]^{\alpha'}_{\simeq}\ar@{>->}[d]_{\hat\phi_k}&
C^*(W)_{{}_{{}_{}}}\ar[d]^{f^*_k}\\
Q_k& \hat Q_k\ar[l]^{\beta_k}_{\simeq}\ar[r]_{\beta'_k}^{\simeq}&
C^*(P_k), }
$$
and such that $(\alpha,\alpha')\co\hat R\to R\oplus C^*(W)$ and
$(\beta_k,\beta'_k)\co\hat Q_k\to Q_k\oplus C^*(P_k)$ are
surjective.
\end{lemma}
\begin{proof}
Let $\calS$ be the category shaping menorah's.
Apply Lemma \ref{lemma-bisurj} in the category $A$-DGMod$^\calS$
to get a cofibrant menorah ${\set{\hat\phi_k}}_{1\leq k\leq c}$
and weak equivalences
$$
\xymatrix{%
\set{\phi_k}_{1\leq k\leq c}&
{\quad\set{\hat\phi_k}}_{1\leq k\leq c}\quad
\ar@{->>}[l]^-{\set{{(\alpha,\beta_k)}}_k}_{\simeq}
\ar@{->>}[r]_-{\set{{(\alpha',\beta'_k)}}_k}^{\simeq}&
\set{f^*_k}_{1\leq k\leq c}}
$$
with the desired properties. In particular by the second part of
Proposition \ref{prop-CMdiagrams} the maps $\hat\phi_k\co \hat R\to \hat Q_k$
are cofibrations between cofibrant objects.
\end{proof}
\begin{lemma}\label{lemma-DGmodDhat}%
With the hypotheses of Theorem \ref{thm-DGmodnonconn} and with the
notation of Lemma \ref{lemma-DGmodphihat}, there exist for each
$k=1,\cdots,c$, a cofibrant $A$-DGmodule, $\hat D_k$, and weak
equivalences of $A$-DGmodules,
$$
\xymatrix{D_k&\ar[l]^{\simeq}_{\gamma_k}\hat
D_k\ar[r]^-{\gamma'_k}_-{\simeq}& C^*(W,C_k),}
$$
making the following diagram of isomorphisms commute
$$\xymatrix{%
H^n(D_k)\ar[d]_{H^n({\psi}_k)}^\cong&\ar[l]_{H^n(\gamma_k)}^\cong
H^n(\hat
D_k)\ar[r]^{H^n(\gamma'_k)}_\cong&H^n(W,C_k)\ar[d]^{H^n(\iota_k)}_\cong\\
H^n(R)&\ar[l]^{H^n(\alpha)}_\cong H^n(\hat
R)\ar[r]_{H^n(\alpha')}^\cong& H^n(W).}
$$
\end{lemma}
\begin{proof}
Fix $k=1,\cdots,c$. By hypothesis $D_k$ is weakly equivalent as
an $A$-DGmodule to $s^{-n}\#C^*(P_k)$,
by Poincar\'e duality to $C^*(P_k,\del T_k)$, and by Mayer-Vietoris
to $C^*(W,C_k)$. By Lemma \ref{lemma-bisurj}, we can find a
cofibrant $A$-DGmodule, $\hat D_k$, and weak equivalences of
$A$-DGmodules
$$
\xymatrix{D_k&\ar[l]^{\simeq}_{\gamma_k}\hat
D_k\ar[r]^-{\gamma''_k}_-{\simeq}& C^*(W,C_k).}
$$
By Lefschetz duality $H^n(W,C_k)\cong H_0(P_k)\cong\Bk$ and $H^n(\iota_k)$
is an isomorphism.
By definition of a top-degree\ map $H^n({\psi}_k)$ is also an isomorphism.
Thus
the diagram appearing in the statement of the lemma, with
$\gamma''_k$ replacing $\gamma'_k$, is indeed a diagram of
isomorphisms. Since $H^n(\hat D_k)\cong H^n(\hat R)\cong\Bk$, the
two isomorphisms
$$
H^n(\alpha)^{-1}H^n({\psi}_k)H^n(\gamma_k)
\textrm{\,\,\,\,and\,\,\,\,}
H^n(\alpha')^{-1}H^n(\iota_k)H^n(\gamma''_k)
$$
differ only by a multiplicative constant
$u\in\Bk\smallsetminus\set{0}$. Set $\gamma'_k:=u.\gamma''_k$ which is
also a weak equivalence of $A$-DGmodules. Then the diagram of
isomorphisms of the statement commutes.
\end{proof}
Recall the notion of model of a composite from Example
\ref{examples-diagram}.
\begin{lemma}
\label{lemma-DGmodphiphishriek} With the hypotheses of Theorem
\ref{thm-DGmodnonconn}, the composite
$$
\xymatrix{%
D\ar[r]^{{\psi}}&R\ar[r]^\phi&Q}
$$
is an $A$-DGmodule model of the composite
$$
\xymatrix{%
C^*(W,C)\ar[r]^{\iota}&C^*(W)\ar[r]^{f^*}&C^*(P).}
$$
\end{lemma}
\begin{proof}
Consider all the morphisms and DGmodules built in Lemma
\ref{lemma-DGmodphihat} and Lemma \ref{lemma-DGmodDhat}. Fix
$k=1,\cdots,c$. Take a
lifting of $A$-DGmodules $\hat{\psi}_k\co\hat D_k\to\hat R$ of
${\psi_k}\gamma_k$ along the acyclic fibration $\alpha$, so that
\begin{equation}
\label{equ-alphaphi1} \alpha\hat{\psi_k}={\psi}_k\gamma_k
\end{equation}
which is a top-degree\ map. Also $\alpha'\hat{\psi}_k$ and $\iota_k\gamma'_k$ are
top-degree\ maps with values in $C^*(W)$. By Proposition \ref{prop-existsshriek}
there are homotopic up to a multiplicative scalar $u\not=0$ and
Lemma \ref{lemma-DGmodDhat} and \refequ{equ-alphaphi1} imply that $u=1$. Thus
\begin{equation}\label{equ-alphaphi2}%
\alpha'\hat{\psi}_k\simeq_A\iota_k\gamma'_k.
\end{equation}
Set $\hat D:=\oplus_{k=1}^c\hat D_k$,
$\gamma:=\oplus_{k=1}^c\gamma_k$,
$\gamma':=\oplus_{k=1}^c\gamma'_k$, and
$\hat{\psi}:=\sum_{k=1}^c\hat{\psi}_k$. Since the $P_k$'s are
pairwise disjoint, we have an identification
$C^*(W,C)=\oplus_{k=1}^c C^*(W,C_k)$. Equations
$(\ref{equ-alphaphi1})$ and $(\ref{equ-alphaphi2})$ yield to the
following homotopy commutative diagram in $A$-DGMod
$$
\xymatrix{%
D\ar[d]_{{\psi}}& \hat
D\ar[l]_{\gamma}^{\simeq}\ar[r]^-{\gamma'}_-{\simeq}\ar[d]_{\hat{\psi}}&
C^*(W,C)\ar[d]^{\iota}\\
R& \hat R\ar[l]_{\alpha}^{\simeq}\ar[r]^-{\alpha'}_-{\simeq}&
C^*(W). }
$$
Since $(\alpha,\alpha')$ is a fibration we can suppose by Lemma
\ref{lemma-rigidify} that $\hat{\psi}$ has been chosen such that
the above diagram is strictly commutative. Gluing this diagram
with that built in Lemma \ref{lemma-DGmodphihat} we get a
commutative diagram of $A$-DGmodules
$$
\xymatrix{%
D\ar[d]_{{\psi}}& \hat
D\ar[l]_{\gamma}^{\simeq}\ar[r]^-{\gamma'}_-{\simeq}\ar[d]_{\hat{\psi}}&
C^*(W,C)\ar[d]^{\iota}\\
R\ar[d]_{\phi}& \hat
R\ar[l]_{\alpha}^{\simeq}\ar[r]^-{\alpha'}_-{\simeq}\ar[d]_{\hat\phi}&
C^*(W)\ar[d]^{f^*}\\
Q& \hat Q\ar[l]^{\beta}_{\simeq}\ar[r]_-{\beta'}^-{\simeq}& C^*(P)
}
$$
and the lemma is proved.\end{proof}
Collecting the four previous lemmas we achieve the proof of
Theorem \ref{thm-DGmodnonconn} and its corollary.
\begin{proof}[Proof of Theorem \ref{thm-DGmodnonconn}]
Recall diagrams $\BD$ and $\BD'$ defined in Theorem
\ref{thm-DGmodnonconn} and diagram $\BD''$ defined in Lemma
\ref{lemma-C*MC}. Using Lemma \ref{lemma-DGmodphiphishriek} and
taking mapping cones we deduce that the diagrams $\BD$ and
$\BD''$ are weakly equivalent in $A$-DGMod. By Lemma
\ref{lemma-C*MC} diagrams $\BD'$ and $\BD''$ are also weakly
equivalent in $A$-DGMod.
\end{proof}
\begin{proof}[Proof of Corollary \ref{corol-HWmodstruct}]
By Example \ref{ex-topdegreedual} $s^{-n}\#\phi$
is a top-degree\ map of right $A$-DGmodules.
Theorem \ref{thm-DGmodnonconn} implies that
$s^{-n}\#R\oplus_{s^{-n}\#\phi}ss^{-n}\#Q$ is a right $A$-DGmodule model
of $C^*(C)$. Therefore their homologies are isomorphic as right $H^*(W)$-modules
and by commutativity also as left modules.
Since $H^{>m}(P)=0$ and by Lefschetz duality, $H_{<n-m}(W,C)=0$ and $H^i(W)\to H^i(C)$ is
an isomorphism for $i<n-m-1$. Therefore if $x.y$ is
a product in $H^*(C)$ that is not determined by the $H^*(W)$-module structure then
$\deg(x),\deg(y)\geq n-m-1$. Hence $\deg(x.y)\geq 2(n-m-1)\geq n-r$. Since $H^*(f)$ is
$r$-connected we have that $H_{\leq r}(W,P)=0$ and by Lefschetz duality $H^{\geq n-r}(C)=0$.
Therefore $x.y=0$.
\end{proof}
\section{CDGA model of a Poincar\'e embedding in the stable case}
\label{section-stableCDGA}
In this section we give a proof of Theorem \ref{thm-stableCDGA}.
Here is an overview of that proof.
\begin{enumerate}
\item We want to show that the diagrams $\BD$ and $\BD'$ are weakly equivalent as commutative squares of
CDGA. By Theorem \ref{thm-DGmodnonconn} we already know that they are
weakly equivalent in a certain category of DGmodules.
\item
We will build a convenient common CDGA model $\xymatrix@1{\hat
R\quad\ar@{>->}[r]^{\hat\phi}&\hat Q}$ of both $\phi\co R\to
Q$ and $f^*\co\Apl(W)\to\Apl(P)$. We can then consider the
category of ``$\hat\phi$-DGmodules'' whose objects consist of maps
of $\hat R$-DGmodules $M\to N$ such that $N$ is also equipped with
a $\hat Q$-DGmodule compatible with its $\hat R$-DGmodule
structure through the map $\hat\phi$. The morphisms of this
category consist of certain commutative squares that we call
\emph{$\hat\phi$-squares} (see Definition
\ref{def-phihatsq}). In particular the diagrams $\BD$ and $\BD'$
will be $\hat\phi$-squares.
\item A refinement of the arguments of Theorem \ref{thm-DGmodnonconn}
will show that the diagrams $\BD$ and $\BD'$ are weakly equivalent
not only as squares in the category of $\hat R$-DGmodules but also
as $\hat\phi$-squares, which means that the weak equivalences
between the right sides of diagrams $\BD$ and $\BD'$ will be of
$\hat Q$-DGmodules (Lemma \ref{lemma-thetatheta''}.)
\item Using the results of Section \ref{section-MC}
(notably Lemma \ref{lemma-DGmodCDGAmap}), we will
show that this weak equivalence of $\hat\phi$-squares between
$\BD$ and $\BD'$ is indeed a weak equivalence of CDGA squares.
\end{enumerate}
Let's move to the details by establishing a series of lemmas.
Note first that without loss of generality we can assume that \refequ{diag-mainsquare}
is a genuine push-out and that $f$ induces a map of pairs
$f_0\co(P,\del T)\to (W,C)$.
In the next lemma we build a common model $\xymatrix@1{\hat
R\quad\ar@{>->}[r]^{\hat\phi}&\hat Q}$ of both $\phi$ and $f^*$.
\begin{lemma}\label{lemma-CDGAphihat}
With the hypotheses of Theorem \ref{thm-stableCDGA} there exists
a cofibrant CDGA $\hat R$, a relative Sullivan algebra
$\xymatrix@1{\hat R\quad\ar@{>->}[r]^{\hat\phi}&\hat Q}$, and a
commutative diagram of CDGA where horizontal arrows are weak
equivalences
$$
\xymatrix{%
R\ar[d]_{\phi}& \hat
R\ar[l]_{\alpha}^{\simeq}\ar[r]^-{\alpha'}_-{\simeq}\ar[d]_{\hat\phi}&
\Apl(W)\ar[d]^{f^*}\\
Q& \hat Q\ar[l]^{\beta}_{\simeq}\ar[r]_-{\beta'}^-{\simeq}&
\Apl(P), }
$$
and $(\alpha,\alpha')\co\hat R\to R\oplus \Apl(W)$
and $(\beta,\beta')\co\hat Q\to Q\oplus \Apl(P)$ are
surjections.
\end{lemma}
\begin{proof}
This is a consequence of Lemma \ref{lemma-bisurj} in the
the category of maps in CDGA, of the second part of Proposition
\ref{prop-CMdiagrams}, and of the fact that every CDGA cofibration is a retract of a
Sullivan relative algebra. Alternatively the lemma can be proved using standard
techniques of \cite{FHT-RHT}.
\end{proof}
Our next lemma gives a replacement $\bar R$ of $\hat R$ that fibres on
different DGmodules.
\begin{lemma}\label{lemma-Rbar}
With the hypotheses of Theorem \ref{thm-stableCDGA} and the
notation of Lemma \ref{lemma-CDGAphihat} there exists a
factorization of $\hat R$-DGmodules of $(\alpha,\alpha',\hat\phi)$
into an acyclic cofibration $\rho$ followed by a fibration
$(\bar\alpha,\bar\alpha',\bar\phi)$ as follows:
$$
\xymatrix{ \hat
R\,\,\ar@{>->}[r]_{\simeq}^\rho\ar[d]_{(\alpha,\alpha',\hat\phi)}&
\bar
R\ar@{->>}[ld]^{(\bar\alpha,\bar\alpha',\bar\phi)}\\
R\oplus\Apl(W)\oplus\hat Q&}
$$
\end{lemma}
\begin{proof}
The existence of such a factorization is one of the axioms of the
closed model structure on the category $\hat R$-DGMod.
\end{proof}
In the following lemma we give a common model $\hat D$ of both
$D$ and $\Apl(P,\del T)$.
\begin{lemma}\label{lemma-stableDhat}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat} and \ref{lemma-Rbar},
there exists a cofibrant $\hat Q$-DGmodule, $\hat D$, and weak
equivalences of $\hat Q$-DGmodules,
$$
\xymatrix{D&\ar[l]^{\simeq}_{\gamma}\hat
D\ar[r]^-{\gamma'}_-{\simeq}& \Apl(P,\del T),}
$$
making the following diagram of isomorphisms commute
$$\xymatrix{%
H^n(D)\ar[d]_{H^n({\psi})}^\cong &\ar[l]_{H^n(\gamma)}^\cong
H^n(\hat
D)\ar[r]^{H^n(\gamma')}_\cong &H^n(P,\del T)&H^n(W,C)\ar[l]_{H^n(f_0)}^\cong \ar[ld]^{H^n(\iota)}_\cong \\
H^n(R)&\ar[l]^{H^n(\bar\alpha)}_\cong H^n(\bar
R)\ar[r]_{H^n(\bar\alpha')}^\cong & H^n(W).}
$$
Moreover $\hat D$ is also a cofibrant $\hat R$-DGmodule and there
exists an $\hat R$-DGmodule weak equivalence
$$
\gamma''\co\hat D\quism\Apl(W,C)
$$
making the following diagram commute
$$
\xymatrix{\hat
D\ar[r]^-{\gamma''}_-\simeq\ar[rd]_{\gamma'}^\simeq&\Apl(W,C)\ar[d]^{f^*_0}_\simeq\\
&\Apl(P,\del T).}
$$
\end{lemma}
\begin{proof}
The proof of the first part of the lemma
is similar to the proof of Lemma \ref{lemma-DGmodDhat}.
For the second part of the lemma, note that by \cite[Lemma
14.1]{FHT-RHT} $\hat D$ is a cofibrant $\hat R$-DGmodule because
it is a cofibrant $\hat Q$-DGmodule and because
$\hat\phi\co\hat R\to\hat Q$ is a relative Sullivan algebra.
Also $f^*_0$ is a surjective quasi-isomorphism. We take
$\gamma''$ as a lift of $\gamma'$ along the acyclic fibration $f^*_0$.
\end{proof}
\begin{lemma}\label{lemma-CDGAphitildeshriek}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat}--\ref{lemma-stableDhat},
there exists an $\hat R$-DGmodule morphism
$$
\tilde{\psi}\co\hat D\to\bar R
$$
making the following diagram homotopy commute in $\hat R$-DGMod
$$\xymatrix{
D\ar[d]_{{\psi}}
&\ar[l]_{\gamma}\hat
D\ar[r]^-{\gamma''}\ar[d]_{\tilde{\psi}}
&
\Apl(W,C)\ar[d]^{\iota}\\
R&\ar[l]^{\bar\alpha}\bar R\ar[r]_-{\bar\alpha'}&\Apl(W).}$$
\end{lemma}
\begin{proof}
The argument is the same as in the beginning of the
proof of Lemma \ref{lemma-DGmodphiphishriek}.
\end{proof}
We build now a $\hat Q$-DGmodule common model $\chi$ both of
$\phi{\psi}=0$ and of $\iota'\co \Apl(P,\del T)\to\Apl(P)$.
\begin{lemma}\label{lemma-chi}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-CDGAphitildeshriek}, the
composite $\phi{\psi}$ is a $Q$-DGmodule morphism and there exists
a $\hat Q$-DGmodule morphism
$$
\chi\co\hat D\to\hat Q
$$
making the following diagram commute in $\hat Q$-DGMod
$$
\xymatrix{ D\ar[d]_{\phi{\psi}}&\ar[l]_\gamma\hat
D\ar[d]_{\chi}\ar[r]^-{\gamma'}&\Apl(P,\del T)\ar[d]^{\iota'} \\
Q&\ar[l]^{\beta}\hat Q\ar[r]_-{\beta'}&\Apl(P).}
$$
Moreover $\chi\simeq_{\hat R}\bar\phi\tilde{\psi}$.
\end{lemma}
\begin{proof}
Notice that for degree reasons $\phi{\psi}=0$, therefore it is a
morphism of $Q$-DGmodules. Applying Lemma \ref{lemma-stablehmtpyclasses} with $r=1$
we get that $[\hat D,Q]_{\hat Q}=0=[\hat D,\Apl(T)]_{\hat Q}$.
Therefore the diagram of the statement with $0$ replacing $\chi$
is homotopy commutative in $\hat
Q$-DGMod.
Since $(\beta,\beta')$ is a fibration, Lemma \ref{lemma-rigidify}
permits to replace the zero map by a
homotopic $\hat Q$-DGmodule morphism $\chi$
making the diagram strictly commute.
We have also by Lemma \ref{lemma-stablehmtpyclasses} that $ [\hat
D,\hat Q]_{\hat R}=0$, hence $\chi\simeq_{\hat
R}\bar\phi\tilde{\psi}$.
\end{proof}
\begin{lemma}\label{lemma-factorchi}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat}--\ref{lemma-chi}, there
exists a morphism of $\hat R$-DGmodule
$
\bar{\psi}\co\hat D\to\bar R
$
making both of the following diagrams commute
$$\xymatrix{
D\ar[d]_{{\psi}}&\ar[l]_{\gamma}\hat
D\ar[r]^{\gamma''}\ar[d]_{\bar{\psi}}&
\Apl(W,C)\ar[d]^{\iota}&\textrm{\,\,\,\,and\,\,\,\,}
&\hat D\ar[r]^{\bar{\psi}}\ar[rd]_\chi&\bar R\ar[d]^{\bar\phi}\\
R&\ar[l]^{\bar\alpha}\bar R\ar[r]_{\bar\alpha'}&\Apl(W)&&&\hat Q.}
$$
\end{lemma}
\begin{proof}
By Lemmas \ref{lemma-CDGAphitildeshriek} and \ref{lemma-chi}, we
have the following homotopy commutative diagram in $\hat R$-DGMod
$$\xymatrix{&\bar R\ar@{->>}[d]^{(\bar\alpha,\bar\alpha',\bar\phi)}\\
\hat
D\ar[ur]^{\tilde{\psi}}\ar[r]_-{({\psi}\gamma,\iota\gamma'',\chi)}&\quad
R\oplus\Apl(W)\oplus\hat Q. }
$$
Since $(\bar\alpha,\bar\alpha',\bar\phi)$ is a fibration and $\hat
D$ is cofibrant, a standard argument in closed model
categories shows that we can replace $\tilde{\psi}$ by a homotopic
map $\bar{\psi}$ making the diagram strictly commute.
\end{proof}
As we have explained in the overview of the proof, in order to
prove that diagrams $\BD$ and $\BD'$ of Theorem
\ref{thm-stableCDGA} are weakly equivalent in CDGA, we will first
prove that there are weakly equivalent as ``$\hat\phi$-squares''
that we define now. To give a meaning to this assertion we
could define a genuine closed model structure on the category of
$\hat\phi$-squares. Instead of doing so we prefer to introduce the
following \emph{ad hoc} definition of weakly equivalent
$\hat\phi$-squares.
\begin{defin}\label{def-phihatsq}
Let $\hat\phi\co\hat R\to\hat Q$ be a CDGA morphism.
\begin{enumerate}
\item[(i)] By a \emph{$\hat\phi$-square} we mean a commutative square of
$\hat R$-DGmodules
$$
\xymatrix{M\ar[r]^\psi\ar[d]_f&N\ar[d]^g\\
M'\ar[r]_{\psi'}&N'}
$$
such that $N$ and $N'$ have also a structure of $\hat Q$-DGmodule
compatible with their $\hat R$-DGmodule structure through
$\hat\phi$ and such that the right map $g$ is a $\hat
Q$-DGmorphism.
\item[(ii)] A \emph{morphism of $\hat\phi$-squares} is a morphism,
$\Theta$, of commutative squares in $\hat R$-DGmodules between
two
$\hat\phi$-squares
$$\vcenter{\xymatrix@1{%
M\ar[r]^\psi\ar[d]_f &%
N\ar[d]^g\\
M'\ar[r]_{\psi'}&%
N'}}%
\quad\stackrel{\Theta}{\to}\quad%
\vcenter{\xymatrix@1{%
X\ar[r]^\omega\ar[d]_p&%
Y\ar[d]^q\\
X'\ar[r]_{\omega'}&%
Y'}}$$
of the form
$\Theta={\left(\begin{array}{cc}\mu&\nu\\\mu'&\nu'\end{array}\right)}$
such that $\nu$ and $\nu'$ are also morphisms of $\hat
Q$-DGmodules.
\item[(iii)] A morphism $\Theta$ of $\hat\phi$-squares is called a
\emph{fibration} (resp.\ \emph{a weak equivalence}) if each of the
morphisms $\mu,\mu',\nu,\nu'$ is a surjection (resp.\
quasi-isomorphism). A morphism of $\hat\phi$-squares which is
both a fibration and a weak equivalence is called an \emph{acyclic
fibration}.
\end{enumerate}
\end{defin}
Recall the diagrams $\BD$ and $\BD'$ from the statement of Theorem
\ref{thm-stableCDGA}.
\begin{lemma}\label{lemma-phihatsq}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat}--\ref{lemma-factorchi},
Diagrams $\BD$ and $\BD'$ are both commutative squares in CDGA and
$\hat\phi$-squares. The following diagram
$$
\bar \BD:=%
\vcenter{\xymatrix{%
\bar R_{{}_{{}_{}}}\ar[r]^{\bar\phi}\ar@{^(->}[d]&%
\hat Q_{{}_{{}_{}}}\ar@{^(->}[d]\\%
\bar R\oplus_{\bar{\psi}}s\hat D\ar[r]_{\bar\phi\oplus\id}&%
\hat Q\oplus_\chi s\hat D%
}}%
$$
is a $\hat\phi$-square.
\end{lemma}
\begin{proof}
The CDGA structure on the mapping cones of the bottom side of
Diagram $\BD$ are the semi-trivial CDGA structures, which exist by
Lemma \ref{lemma-CDGAMC}. From this it is clear that $\BD$ is a
commutative square of CDGA, as well as $\BD'$. They are also
$\hat\phi$-squares with $\hat R$- and $\hat Q$-DGmodule structures
induced by the maps $\alpha$, $\alpha'$, $\beta$, and $\beta'$.
Using the fact that $\chi$ is a $\hat Q$-DGmodule morphism it is
immediate to check that $\bar \BD$ is a $\hat\phi$-square.
\end{proof}
\begin{lemma}\label{lemma-thetatheta''}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat}--\ref{lemma-phihatsq},
there exist acyclic fibrations of $\hat\phi$-squares
$$
\xymatrix{\BD&%
\ar@{->>}[l]_{\Theta}^{\simeq}\bar\BD\ar@{->>}[r]^{\Theta'}_\simeq&%
\BD'.}
$$
\end{lemma}
\begin{proof}
Using the different maps constructed in our previous series of
lemmas we will describe these two acyclic fibrations
explicitly.
Consider the following commutative square
$$\BD'':=%
\vcenter{\xymatrix@1{%
\Apl(\vrule width0pt depth6pt W)_{{}_{{}_{}}}\ar[r]^{f^*}\ar@{^(->}[d]&%
\Apl(\vrule width0pt depth6pt P)_{{}_{{}_{}}}\ar@{^(->}[d]\\%
\Apl(W)\oplus_\iota s\Apl(W,C)\ar[r]_{f^*\oplus sf^*_0}%
&\Apl(P)\oplus_{\iota'} s\Apl(P,\del T).}}$$
Using the fact that $\iota'\co \Apl(P,\del T)\to \Apl(P)$ is a
morphism of $\Apl(P)$-DGmodules, hence of $\hat Q$-DGmodules, we
see that $\BD''$ is a diagram of $\hat\phi$-squares. Clearly
$\Theta''':=%
\left(%
\begin{array}{cc}%
\id&\id\\%
l^*\oplus 0&i^*\oplus 0%
\end{array}%
\right)\co\BD''\to\BD'%
$ is a surjection, and an argument analogous to that of Lemma
\ref{lemma-C*MC} shows that it is a weak equivalence. Hence
$\Theta'''$ is an acyclic fibration. We have another acyclic
fibration
$\xymatrix@1{\Theta''\co\bar\BD\ar@{->>}[r]^\simeq&\BD''}$
given by
$\Theta'':=%
\left(%
\begin{array}{cc}%
\bar\alpha'&\beta'\\%
\bar\alpha'\oplus s\gamma''&\beta'\oplus s\gamma'%
\end{array}%
\right).%
$ Then $\Theta':=\Theta'''\Theta''$ is one of the required acyclic
fibration. The other one is given by $
\Theta:=%
\left(%
\begin{array}{cc}%
\bar\alpha&\beta\\%
\bar\alpha\oplus s\gamma&\beta\oplus s\gamma%
\end{array}%
\right).%
$
\end{proof}
We sketch now an overview of the end of the proof of the Theorem.
In the next lemma we build an intermediate commutative square,
$\hat\BD$, which is a CDGA model of $\BD'$. Moreover $\hat\BD$ is
also a ``cofibrant $\hat\phi$-square'', therefore by lifting
along the quasi-isomorphisms $\Theta$ and $\Theta'$ we will deduce
that $\hat \BD$ is a model of $\hat\phi$-square of $\BD$.
Finally a degree argument will imply that
this $\hat\phi$-square quasi-isomorphism
$\hat\Theta\co\hat\BD\simeq\BD$ is in fact of CDGA and this
will prove that $\BD$ and $\BD'$ are weakly equivalent CDGA
squares. Let's move to the details.
\begin{lemma}\label{lemma-thetahat''}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat} and
\ref{lemma-phihatsq}, there exists a commutative square in CDGA,
$$\hat\BD:=%
\vcenter{\xymatrix@1{%
\hat {R}_{{}_{{}_{{}_{{}_{}}}}}\quad\ar@{>->}[r]^{\hat\phi}\ar@{>->}[d]_u&%
\hat Q_{{}_{{}_{{}_{{}_{}}}}}\ar@{>->}[d]^v\\
\hat R\otimes\wedge X\quad\ar@{>->}[r]_-{\hat\psi}&%
\hat Q\otimes \wedge X\otimes\wedge Y}} $$
where $\hat\phi$, $\hat\psi$, $u$, and $v$ are cofibrations,
together with a weak equivalence both of CDGA-squares and of
$\hat\phi$-squares
$\hat\Theta'\co\hat\BD\quism \BD'$.
Moreover $X$ and $Y$ can be chosen such that such that
$X^{<n-m-1}=Y^{<n-m-2}=0$. If $H^1(f;\BQ)$ is injective
we can also assume that $Y^{<n-m-1}=0$.
\end{lemma}
\begin{proof}
By taking a {minimal} relative Sullivan algebra of $l^*\alpha'$
we get a commutative diagram of CDGA
$$\xymatrix{
\hat Q\ar[d]_{\beta'}&\ar[l]_{\hat\phi}\hat R\quad\ar[d]_{\alpha
'}\ar@{>->}[r]^u&\hat R\otimes \wedge X\ar[d]^{\lambda'}\\
\Apl(P)&\ar[l]^{f^*}\Apl(W)\ar[r]_{l^*}&\Apl(C).}
$$
Consider the push-out $\hat Q\otimes\wedge X$ of the top line of
the above diagram. By the universal property of the push-out, this
diagram induces a CDGA map
$$\bar\mu'_0\co\hat Q\otimes\wedge X\to\Apl(\del T).$$ The
latter map can be factored into a minimal relative Sullivan
algebra followed by a quasi-isomorphism,
$$\xymatrix{\hat Q\otimes\wedge X\,\,\ar@{>->}[r]^-v&
\hat Q\otimes\wedge X\otimes\wedge Y\ar[r]^-{\mu'}_\simeq&\Apl(\del
T).}
$$
It is immediate to check that the matrix
$\hat\Theta'=\left(\begin{array}{cc}
\alpha'&\beta'\\\lambda'&\mu'\end{array}\right)$ is a weak
equivalence of CDGA-squares and of $\hat\phi$-squares.
We prove now that $X^{<n-m-1}=0$. Since $i$ is $(n-m-1)$-connected
a Mayer-Vietoris argument implies that $H_*(l)$ is $(n-m-1)$-connected. Therefore the same is
true for the map $u$ and by minimality we get that $X^{<n-m-1}=0$.
The model $\hat Q\to \hat Q\otimes\wedge X\otimes\wedge Y$ of $i^*$
is cohomologically $(n-m-1)$-connected and since $X^{<n-m-1}=0$, minimality
implies that $Y^{<n-m-2}=0$.
Assume that $H^1(f)$ is injective. Thus $H^*(f)$ is $1$-connected and $H^*(l)$ is
$(n-m-1)$-connected . By a rational Blackers-Massey argument we deduce that
$\hat Q\otimes\wedge X\to \hat Q\otimes\wedge X\otimes\wedge Y$
is cohomologically $(n-m-1)$-connected. By minimality
we get that $Y^{<n-m-1}=0$.
\end{proof}
\begin{lemma}\label{lemma-bartheta}
With the hypotheses of Theorem \ref{thm-stableCDGA} and with the
notation of Lemma \ref{lemma-thetahat''}, the commutative squares of CDGA
$\hat\BD$ and $\BD$ are weakly equivalent.
\end{lemma}
\begin{proof}
The proof is in two steps:
(i)\qua We will show that the morphism $\hat\Theta'$ constructed in
Lemma \ref{lemma-thetahat''} lifts along the acyclic fibration
$\Theta'$ of Lemma \ref{lemma-thetatheta''} to a weak equivalence
of $\hat\phi$-squares
$\bar\Theta\co\hat\BD\quism\bar\BD$.
(ii)\qua using the map $\Theta$ constructed in Lemma
\ref{lemma-thetatheta''} we will show that the composite
$\hat\Theta:=\Theta\bar\Theta$ is a quasi-isomorphism of CDGA.
(i)\qua The lift will be of the form
$\bar\Theta:=\left(\begin{array}{cc}%
\rho&\id\\%
\bar\lambda&\bar\mu%
\end{array}\right)$, where $\rho$ was defined in Lemma \ref{lemma-Rbar}.
We need only to build the maps $\bar\lambda$ and $\bar\mu$, and
for this we will use the maps $\lambda'$ and $\mu'$ constructed in
the proof of Lemma \ref{lemma-thetahat''}. We have the following
solid commutative diagram of $\hat R$-DGmodules
$$
\xymatrix{ \vrule width0pt depth6pt \hat R\ar@{>->}[d]_u\ar[r]^\rho&\bar R\ar@{^(->}[r]&
\bar R\oplus_{\bar{\psi}}s\hat
D\ar@{->>}[d]_{\simeq}^{l^*\bar\alpha'\oplus 0}\\
\hat R\otimes\wedge
X\ar@{-->}[rru]^{\bar\lambda}\ar[rr]^{\simeq}_{\lambda'}&&\Apl(C).}
$$
Since $u$ is a relative Sullivan algebra, it is an $\hat
R$-DGmodule cofibration. Then, $l^*\bar\alpha\oplus 0$ being an
acyclic fibration, there exists a lift of $\hat R$-DGmodules,
$\bar\lambda$, making both triangles of the diagram commute.
We can define a map of $\hat Q$-DGmodules
$$
\bar\mu_0\co\hat Q\otimes\wedge X\to\hat Q\oplus_\chi s\hat D
$$
by the formula
$$\bar\mu_0(q\otimes\omega)=q.(\bar\phi\oplus\id)(\bar\lambda(1\otimes\omega)),$$
for $q\in\hat Q$ and $\omega\in\wedge X$. It is immediate to check
that $\bar\mu_0$ is a $\hat Q$-DGmodule morphism and that the
following solid diagram of $\hat Q$-DGmodules commutes:
$$\xymatrix{
\hat Q\otimes\wedge X\ar@{>->}[d]\ar[r]^{\bar\mu_0}&\hat
Q\oplus_\chi
s\hat D\ar@{->>}[d]_\simeq^{i^*\beta'\oplus 0}\\
\hat Q\otimes\wedge X\otimes\wedge
Y\ar@{-->}[ru]^{\bar\mu}\ar[r]_{\mu'}^\simeq&\Apl(\del T).}
$$
Therefore there exists a lift, $\bar\mu$, of $\hat Q$-DGmodules
making both triangles of the diagram commute.
It is immediate to check that $\bar\Theta:=\left(\begin{array}{cc}
\rho&\id\\\bar\lambda&\bar\mu\end{array}\right)$ is a weak
equivalence of $\hat\phi$-squares.
(ii)\qua We show now that the composite
$\hat\Theta:=\Theta\bar\Theta\co\hat\BD\to\BD$ is a weak
equivalence in the category of commutative squares of CDGA. We
know already that $\hat\Theta$ is a quasi-isomorphism, since both
$\Theta$ and $\bar\Theta$ are. Recalling the form of $\Theta$ from
the proof of Lemma \ref{lemma-thetatheta''} and of $\bar\Theta$
from the proof of (i), we see that
$$\hat\Theta=
\left(\begin{array}{cc}%
\bar\alpha&\beta\\%
\bar\alpha\oplus s\gamma&\beta\oplus s\gamma%
\end{array}\right)
\left(\begin{array}{cc}%
\rho&\id\\%
\bar\lambda&\bar\mu%
\end{array}\right)=
\left(\begin{array}{cc}%
\alpha&\beta\\%
\lambda&\mu%
\end{array}\right),
$$
where $\alpha$, $\beta$ are CDGA morphisms and $\lambda$ (resp.\
$\mu$) is some $\hat R$-DGmodule (resp.\ $\hat Q$-DGmodule)
morphism.
By the hypotheses of Theorem \ref{thm-stableCDGA}, we
have that $(R\oplus_{{\psi}} sD)^{>n}=0$. Since $n\geq2m+3$, this
implies that $(R\oplus_{{\psi}}sD)^{\geq2(n-m-1)}=0$. Since
$X^{<n-m-1}=0$, Lemma \ref{lemma-DGmodCDGAmap} implies that
$\lambda$ is a CDGA morphism.
Suppose that $H^1(f)$ is injective and $n\geq 2m+3$. A similar argument shows that $\mu$
is a CDGA morphism, which implies that $\hat\Theta$ is a
weak equivalence of squares of CDGA.
Suppose instead that $n\geq 2m+4$. Since $Q$ is connected and
$H^{\geq n}(Q\oplus sD)=0$
there exists an acyclic ideal $L\subset Q\oplus sD$ such that
$\left((Q\oplus sD)/L\right)^{\geq n}=0$. Replace $Q\oplus sD$ in diagram $\BD$
by $(Q\oplus sD)/L$ to get a quasi-)isomorphic CDGA diagram $\tilde\BD$.
Since $Y^{<n-m-2}=0$ and $2(n-m-2)\geq n$,
Lemma \ref{lemma-DGmodCDGAmap} implies that
the composite
$\hat Q\otimes\wedge X\otimes\wedge Y\stackrel{\mu}{\to}Q\oplus sD
\quism(Q\oplus sD)/L$
is a CDGA quasi-isomorphism. Therefore $\hat\BD\simeq\tilde\BD\simeq\BD$ as
CDGA squares.
\end{proof}
Collecting these lemmas we conclude the proof of the first part of the theorem:
\begin{proof}[Proof of Theorem \ref{thm-stableCDGA}]
Lemmas \ref{lemma-thetahat''} and \ref{lemma-bartheta} imply
that the diagrams $\BD$ and $\BD'$ are weakly equivalent CDGA
commutative squares.
We prove now the second part of the theorem.
Suppose given a CDGA model $\phi_0\co R_0\to Q_0$ of $f^*$. Our
goal is to build a model $\phi\co R\to Q$ and a top-degree\ map
${\psi}\co D\to R$ fulfilling hypotheses (i-)-(iii) of Theorem
\ref{thm-stableCDGA}. By replacing
$\phi_0$ by a minimal Sullivan model we can suppose that both
$R_0$ and $Q_0$ are connected. Since $H^{>n-1}(Q_0)=0$ and
$H^{>n}(R_0)$ we can build
another CDGA model of $f^*$ of the form
$\phi_1\co R\to Q_1$
with $R^{>n}=0$, and such that $R$ and $Q_1$ are still connected.
We can factor $\phi_1$ into a minimal relative Sullivan algebra
$\phi_2$ followed by a weak equivalence. This gives another CDGA
model of $f^*$ of the form
$\xymatrix{
\phi_2\co R\quad\ar@{>->}[r]&Q_2:=R\otimes\wedge V}
$
and $V=V^{\geq1}$ because $H^1(f;\BQ)$ is injective.
Let $D_2$ be a minimal semifree model of the $Q_2$-DGmodule
$s^{-n}\#Q_2$. Since $H^{<n-m}(s^{-n}\#Q_2)=0$, minimality implies
that $D_2^{<n-m}=0$. Since $\phi_2$ is a relative Sullivan
algebra, every semifree $Q_2$-DGmodule is also a semifree
$R$-DGmodule. Therefore $D_2$ is also a cofibrant $R$-DGmodule and
Proposition \ref{prop-existsshriek} implies that there exists a
top-degree\ map of $R$-DGmodule
${\psi}_2\co D_2\to R$.
Since $H^{>n}(s^{-n}\#Q_2)=0$
we can replace $D_2$ by a weakly equivalent
$Q_2$-DGmodule, $D$, such that $D^{<n-m}=0$, $D^{>n+1}=0$, and
$D^{\leq n}=D_2^{\leq n}$.
Since $R^{>n}=0$ the map ${\psi}_2$ induces a
top-degree\ map
${\psi}\co D\to R$.
Since $H^{>m}(Q_2)=0$ and $Q_2$ is connected there exists a surjective quasi-isomorphism
of CDGA
$\alpha_2\co Q_2\quism Q$
such that $Q^{>m+2}=0$ and $\ker(\alpha_2)\subset Q^{>m+1}$. For
degree reasons $(\ker\alpha_2).D=0$, therefore the $Q_2$-DGmodule
$D$ inherits a $Q$-DGmodule structure. Set $\phi=\alpha_2\phi_2$.
In summary we have built from $\phi_0$ another CDGA model $\phi$ of $f^*$
and a top-degree\ map of $R$-DGmodule
${\psi}$ satisfying hypotheses (i)--(iii).
\end{proof}
\section{CDGA models of the complement in a Poincar\'e embedding under the unknotting condition}
\label{section-wkstableCDGA}
In this section we give a proof of Theorem \ref{thm-wkstCDGA}
which gives a CDGA model of the complement in a Poincar\'e embedding under the unknotting condition.
We also build a model of a diagram which is almost the Poincar\'e
embedding \refequ{diag-mainsquare} under a slightly stronger unknotting condition
(Theorem \ref{thm-wkstCDGAsquare}).
The proof of Theorem \ref{thm-wkstCDGA} follows the line of the
proof of Theorem \ref{thm-stableCDGA}. In particular we will reuse
many of the lemmas of the previous section. First it is easy to
check that if we replace the hypotheses of Theorem
\ref{thm-stableCDGA} by those of Theorem \ref{thm-wkstCDGA} in
Lemmas \ref{lemma-CDGAphihat} and \ref{lemma-Rbar} then the
conclusions of these lemmas still hold without any change in their
proofs. Since $D$ is supposed to be only an $R$-DGmodule model of $s^{-n}\#Q$ we
replace Lemma \ref{lemma-stableDhat} by the following
\begin{lemma}\label{lemma-wkstDhat}
With the hypotheses of Theorem \ref{thm-wkstCDGA} and with the
notation of Lemmas \ref{lemma-CDGAphihat} and \ref{lemma-Rbar},
there exists a cofibrant $\hat R$-DGmodule, $\hat D$, and weak
equivalences of $\hat R$-DGmodules,
$$
\xymatrix{D&\ar[l]^{\simeq}_{\gamma}\hat
D\ar[r]^-{\gamma''}_-{\simeq}& \Apl(W,C),}
$$
making the following diagram of isomorphisms commute
$$\xymatrix{%
H^n(D)\ar[d]_{H^n({\psi})}^\cong &\ar[l]_{H^n(\gamma)}^\cong
H^n(\hat
D)\ar[r]^{H^n(\gamma'')}_\cong &H^n(W,C) \ar[d]^{H^n(\iota)}_\cong \\
H^n(R)&\ar[l]^{H^n(\bar\alpha)}_\cong H^n(\bar
R)\ar[r]_{H^n(\bar\alpha')}^\cong & H^n(W).}
$$
\end{lemma}
\begin{proof}
It is a special case of Lemma \ref{lemma-DGmodDhat}.
\end{proof}
It can be readily checked that Lemma
\ref{lemma-CDGAphitildeshriek} still holds when we replace the
hypotheses of Theorem \ref{thm-stableCDGA} by those of Theorem
\ref{thm-wkstCDGA}, and the only change in the proof of this lemma
is a replacement of the reference to Lemma \ref{lemma-stableDhat}
to a reference to Lemma \ref{lemma-wkstDhat}.
Moreover by Lemma \ref{lemma-rigidify} we can replace
$\tilde{\psi}$ by $\bar{\psi}$
making the diagram of Lemma
\ref{lemma-CDGAphitildeshriek} strictly commute.
We are now ready for the following
\begin{proof}[Proof of Theorem \ref{thm-wkstCDGA}]
By the same argument as for Theorem \ref{thm-stableCDGA} and using
Lemma \ref{lemma-CDGAphihat}, \ref{lemma-Rbar},
\ref{lemma-wkstDhat}, and \ref{lemma-CDGAphitildeshriek}, we get that
$R\hookrightarrow R\oplus_{{\psi}}sD$ is an $\hat R$-DGmodule
model of $l^*\co\Apl(W)\hookrightarrow\Apl(C)$.
Since $H_*(f)$ is $r$-connected and by Lefschetz duality we have
$H^{\geq n-r}(R\oplus_{{\psi}}sD)=H^{\geq n-r}(C;\BQ)=H_{\leq r}(W,P;\BQ)=0$.
Using the connectivity of $R$ it is easy to build an acyclic subDGmodule
$L\subset R\oplus_{{\psi}}sD$ concentrated in degrees $\geq n-r-1$ and
killing $(R\oplus_{{\psi}}sD)^{\geq n-r}$.
Consider such an acyclic subDGmodule $L$. Set $k=n-m-1$ and $l=n-2m+r-1$.
By Lemma \ref{lemma-CDGAtruncMC}
there is a semi-trivial CDGA structure on
$(R\oplus_{{\psi}}sD)/L$ and the obvious map $R\to
(R\oplus_{{\psi}}sD)/L$ is a CDGA morphism.
Since $L$ is acyclic the map $R\to (R\oplus_{{\psi}}sD)/L$
is also an $\hat R$-DGmodule model $l^*$.
Let $\xymatrix@1{ \hat R\quad\ar@{>->}[r]^-u&\hat R\otimes\wedge
X\ar[r]^{\lambda'}_{\simeq}&\Apl(C)}$
be a CDGA factorization of $l^*\alpha'$
through a relative minimal Sullivan algebra $\hat R\otimes\wedge
X$. By the same argument as in the proof of Lemma
\ref{lemma-thetahat''} we find that $X^{<n-m-1}=0$ because $l^*$
is $(n-m-1)$-connected.
Since $u$ is a model of $\hat R$-DGmodule of $l^*$, the same
argument as in the beginning of the proof of Lemma
\ref{lemma-bartheta} gives a commutative diagram of $\hat
R$-DGmodules
$$
\xymatrix{ \hat R_{{}_{{}_{{}_{}}}}\ar[r]^{\rho}_{\simeq}\ar@{>->}[d]_u&\bar
R_{{}_{{}_{{}_{}}}}\ar@{^(->}[d]\ar[r]^{\bar\alpha}_{\simeq}&
R_{{}_{{}_{{}_{}}}}\ar[d]\\
\hat R\otimes\wedge X\ar[r]^{\bar\lambda}_\simeq
\ar@(d,d)[rr]^{\lambda}_{\simeq}&\bar R\oplus_{\bar{\psi}}s\hat
D\ar[r]^-{\pi(\bar\alpha\oplus
s\gamma)}_-\simeq&(R\oplus_{{\psi}}sD)/L,}
$$
and the composite $\lambda=\pi(\bar\alpha\oplus
s\gamma)\bar\lambda$ is a quasi-isomorphism.
Since $X^{<n-m-1}=0$ and $\left((R\oplus_{{\psi}}sD)/L\right)^{\geq n-r}=0$,
the condition $r\geq2m-n+2$ and Lemma
\ref{lemma-DGmodCDGAmap} imply that $\lambda$ is a CDGA morphism.
Also $\bar\alpha\rho=\alpha$ is a CDGA morphism. Thus $u$ is
a CDGA model of $R\to(R\oplus_{{\psi}}sD)/L$. By
construction $u$ is also a model of $l^*$ and the first part of the theorem is
proved.
The second part of the theorem is proved in a similar way to Theorem \ref{thm-stableCDGA}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{corol-wkstCDGA}]
Since $P$ is simply connected and the codimension is at least $3$,
$\del T$ is simply connected, and since $W$ is also simply-connected, the same is true for $C$ by
Van Kampen theorem. The corollary follows then from the above theorem
and from the fact that a CDGA model of a simply connected
spaces of finite type determines its rational homotopy type.
\end{proof}
In the rest of the section we address the problem of describing a
CDGA model of Diagram \refequ{diag-mainsquare} under some
unknotting condition.
We wish that we could have determined the
rational homotopy type of the entire square
\refequ{diag-mainsquare} from the rational homotopy class of $f$,
but we are only able to determine a slightly less complete square
that we describe now.
Assume that $\del T$ is simply-connected in which case
by Poincar\'e duality in dimension $n-1$ and by
\cite[Proposition 4.1]{Wall-finiteness} we can consider the
space $\del\check T$ obtained by removing the unique top $(n-1)$-cell in
a minimal CW-decomposition of $\del T$. We have then the following
commutative square of topological spaces
\begin{equation}\label{diag-mainsquarepunct}%
\xymatrix{ \del\check T\ar[r]^{\check i}\ar[d]_{\check k}&
P\ar[d]^f\\
C\ar[r]_l&W }
\end{equation}
where $\check i$ and $\check k$ are the
restrictions of $i$ and $k$ to $\del \check T$. Our next theorem
is a description of a CDGA model of \refequ{diag-mainsquarepunct}
under a stronger unknotting condition and two
extra assumptions which are not too restrictive as we explain in
Remark \ref{rmk-extraassumption}.
To state the theorem it is convenient to introduce the following terminology:
if $X$ is an $A$-DGmodule and $l$ is an integer then a \emph{truncation
$A$-subDGmodule of $X$ above degree $l$} is a subDGmodule $L$ such that $L^{\leq l-1}=0$,
$X^{>l}\subset L$ and the projection $\pi\co X\to X/L$ induces an isomorphism
in homology in degrees $\leq l$. Of course $(X/L)^{>l}=0$. It is easy to check that such a truncation
subDGmodule exists when $A$ is connected.
\begin{thm}\label{thm-wkstCDGAsquare}
Consider the diagram \refequ{diag-mainsquarepunct} induced
by a Poincar\'e embedding \refequ{diag-mainsquare} with
$P$ and $W$ connected and $\del T$ simply-connected. Let $r$ be a positive integer
such that $\tilde H_{\leq r-1}(P;\BQ)=\tilde H_{\leq r}(W;\BQ)=0$
and $r\geq 2m-n+2$.
Let $\phi\co R\to Q$ be a CDGA model of
$f^*\co\Apl(W)\to\Apl(P)$ such that $R$ is connected. Let D be
an $R$-DGmodule weakly equivalent to $s^{-n}\#Q$ and such that
$D^{<n-m}=0$. Suppose given a top-degree\ map of $R$-DGmodules
${\psi}\co D\to R$.
Suppose moreover that $n\geq m+r+2$ and that $Q$
is $(r-1)$-connected, that is $Q^{\leq r-1}=Q^0=\BQ$.
Let $I$ be a truncation $R$-subDGmodule of $R$ above degree $n-r-1$,
let $J$ be a truncation $Q$-subDGmodule of $Q$ above degree $m$, and
let $K$ be a truncation $Q$-subDGmodule of $D$ above degree $n-r$.
Then the following two commutative squares are weakly equivalent
in CDGA
$$\BDt:=
\vcenter{\xymatrix{ R\ar[r]^\phi\ar[d]& Q\ar[d]\\
(R\oplus_{{\psi}}sD)/(I\oplus
sK)\ar[r]^{\overline{\phi\oplus\id}}&
(Q\oplus_{\phi{\psi}} sD)/(J \oplus sK)}}
$$
and
$$
\BDt':=%
\vcenter{\xymatrix{%
\Apl(W)\ar[r]^{f^*}\ar[d]_{l^*}&
\Apl(P)\ar[d]^{\check i^*}\\
\Apl(C)\ar[r]^{\check k^*}& \Apl(\del\check T). }}
$$
where, in Diagram $\BDt$, the vertical maps are the composition of
the inclusion with the projection, the bottom map is the one
induced by $\phi\oplus\id_{sD}$, and the CDGA structure on the
truncated mapping cones are the semi-trivial ones.
\end{thm}
\begin{rmk}\label{rmk-extraassumption}
The connectivity hypothesis on $P$ and $W$ are equivalent
to $H_*(f;\BQ)$ is $r$-connected and $\tilde H_{\leq r-1}(P;\BQ)=0$
which is clearly a stronger condition than the unknotting condition \refequ{equ-unknotrht}
because of the high connectivity hypothesis on $P$.
The first extra assumption in the theorem, $n\geq
m+r+2$, is satisfied under the unknotting condition $r\geq 2m-n+2$ as
soon as $m\geq 2r$. On the other hand, if $m<2r$ then by a
rational version of the suspension Freudenthal theorem, $P$ has
the rational homotopy type of a wedge of spheres of dimensions
between $r$ and $2r-1$. Hence this first extra assumption is a
consequence of the unknotting condition when $P$ is not rationally
equivalent to a wedge of spheres. For the second extra assumption
(the $(r-1)$-connectivity of $Q$), since $\tilde H^{\leq r}(P)=0$, one
can always construct an $r$-connected CDGA model $Q$ of $P$, by
taking for example a minimal Sullivan model of any given model of
$P$. Therefore there is no real loss of generality in making this second
assumption.
\end{rmk}
\begin{rmk} It is very likely that under the only unknotting condition \refequ{equ-unknotrht}
one can determine a CDGA model of the
complete Poincar\'e embedding \refequ{diag-mainsquare} but we were
unable to prove this.
\end{rmk}
The rest of the section is devoted to the proof of Theorem
\ref{thm-wkstCDGAsquare} which is a refinement of the proof of
Theorem \ref{thm-stableCDGA}.
Lemmas \ref{lemma-CDGAphihat}--\ref{lemma-CDGAphitildeshriek}
hold with the
hypotheses of Theorem \ref{thm-stableCDGA} replaced by those of
Theorem \ref{thm-wkstCDGAsquare}.
We need the following three lemmas in replacement of Lemma
\ref{lemma-chi}:
\begin{lemma}
\label{lemma-chiwkst} With the hypotheses of Theorem
\ref{thm-wkstCDGAsquare} and with the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-CDGAphitildeshriek}, the
composite $\phi{\psi}$ induces a $Q$-DGmodule map
$
\overline{\phi{\psi}}\co D/K\to Q/J$. There exists a $\hat
Q$-DGmodule morphism $\chi\co \hat D\to\hat Q$
making the following diagram commute in $\hat Q$-DGMod
$$
\xymatrix{
D/K\ar[d]_{\overline{\phi{\psi}}}&\ar[l]_{\pi_0}D&\ar[l]^{\simeq}_{\gamma}\hat
D\ar[r]^-{\gamma'}_-{\simeq}\ar[d]^\chi&\Apl(P,\del T)\ar[d]^{\iota'}\\
Q/J&\ar[l]_{\pi'_0}Q&\ar[l]^{\simeq}_{\beta}\hat
Q\ar[r]^{\beta'}_{\simeq}&\Apl(P)}
$$
where $\pi_0$ and $\pi'_0$ are the canonical projections. Moreover
$\chi\simeq_{\hat R}\bar\phi\tilde{\psi}$.
\end{lemma}
\begin{proof}
Since $n\geq m+r+2$ we have that $\phi{\psi}(K)\subset J$,
hence there is an induced map $\overline{\phi{\psi}}$ between the
quotients. Since $Q$ is $(r-1)$-connected, $(D/K)^{<n-m}=0$,
$(Q/J)^{>m}=0$, and $(n-m)+r>m$, we have
that the $\Bk$-DGmodule map $\overline{\phi{\psi}}$ is a map of
$Q$-DGmodule.
Since $r\geq2m-n+2$ and $D^{<n-m}=0$, we have $H^{\leq m-r+1}(\hat D)=0$.
Also $\tilde H^{\leq r-1}(\hat Q)=H^{>m}(\hat Q)=0$. By Lemma
\ref{lemma-stablehmtpyclasses} we have an isomorphism
$
H^*\co[\hat D,\hat Q]_{\hat
Q}\cong{\mathrm{hom}}^0_{\Bk}(H^*(\hat D),H^*(\hat Q))
$.
Therefore there exists a map
$\chi\co\hat D\to\hat Q$ of $\hat Q$-DGmodules, unique up to
homotopy, such that $H^*(\chi)=H^*(\bar\phi\tilde{\psi})$ where
$\bar\phi$ and $\tilde{\psi}$ were defined in Lemma
\ref{lemma-Rbar} and Lemma \ref{lemma-CDGAphitildeshriek}.
Since $\chi$ induces in cohomology the same map as
$\bar\phi\tilde{\psi}$, Lemma \ref{lemma-CDGAphitildeshriek},
Lemma \ref{lemma-CDGAphihat} and Lemma \ref{lemma-Rbar} imply that
the map $\chi$ makes the diagram of the statement of Lemma
\ref{lemma-chiwkst} commute \emph{in cohomology}. Another
application of Lemma \ref{lemma-stablehmtpyclasses} implies that
this diagram commutes up to a homotopy of $\hat Q$-DGmodules.
Since $(\beta\pi'_0,\beta')$ is surjective we can suppose by Lemma
\ref{lemma-rigidify} that $\chi$ makes the diagram exactly
commute.
Finally we have also $\chi\simeq_{\hat R}\bar\phi\tilde{\psi}$, again by Lemma
\ref{lemma-stablehmtpyclasses}.
\end{proof}
\begin{lemma}\label{lemma-connectivityHconebis}
With the hypotheses of Theorem
\ref{thm-wkstCDGAsquare}, the canonical projection
$$\pi\co R\oplus_{{\psi}} sD\to (R\oplus_{{\psi}} sD)/(I\oplus sK)$$
is a quasi-isomorphism and the
canonical projection
$$\pi'\co Q\oplus_{\phi{\psi}} sD\to (Q\oplus_{\phi{\psi}} sD)/(J\oplus sK)$$
induces an isomorphism in cohomology in all degrees except in
degree $n-1$ where $H^{n-1}(Q\oplus_{\phi{\psi}} sD)\cong\BQ$ and
$H^{n-1}((Q\oplus_{\phi{\psi}} sD)/(J\oplus sK))=0$.
\end{lemma}
\begin{proof}
$L=I\oplus sK$ is a truncation $R$-subDGmodule of $R\oplus_{{\psi}} sD$
above degree $n-r$. Using the fact that $H^n({\psi})$ is an isomorphism
and that $H^i(R)=H^i(sD)=0$ for $n-r\leq i\not =n$,
it comes that $L$ is acyclic, hence $\pi$ is a quasi-isomorphism.
The proof for $\pi'$ is similar after computing that
$H^{\geq n-r}(Q\oplus_{\phi{\psi}} sD)\cong s^{-(n-1)}\BQ$ and using
the assumption $n\geq m+r+2$ to check that $J\oplus sK$ is a differential submodule
of $Q\oplus_{{\psi}}sD$.
\end{proof}
\begin{lemma}\label{lemma-modeldTcheck}
With the hypotheses of Theorem \ref{thm-wkstCDGAsquare} and
with the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-CDGAphitildeshriek} and Lemma
\ref{lemma-chiwkst}, there exists a cofibration of $\hat
Q$-DGmodules
$$\xymatrix{
w\co\hat Q\oplus_\chi s\hat D\,\,\ar@{>->}[r] &(\hat
Q\oplus_\chi s\hat D)\oplus \hat Q\otimes V}
$$
and acyclic fibrations of $\hat Q$-DGmodules $\epsilon$ and
$\epsilon'$ making the following diagram commute
$$\xymatrix{%
Q\oplus_{\phi{\psi}}sD\ar@{->>}[d]_{\pi '}&%
\ar@{->>}[l]_{\alpha\oplus s\gamma}^\simeq {\hat Q\oplus_\chi s\hat
D}_{{}_{{}_{{}_{}}}}\ar@{>->}[d]_w\ar@{->>}[r]^{i^*\alpha'\oplus 0}_\simeq&%
\Apl(\del T)\ar@{->>}[d]^{\check j}\\
(Q\oplus_{\phi{\psi}}sD)/(J\oplus sK)&%
\ar@{->>}[l]_-{\epsilon}^-\simeq (\hat Q\oplus_\chi s\hat
D)\oplus \hat Q\otimes V\ar@{->>}[r]^-{\epsilon'}_-\simeq&%
\Apl(\del \check T). }
$$
\end{lemma}
\begin{proof}
The composite of $\hat Q$-DGmodules
$$
\xymatrix{%
\hat Q\oplus_\chi s\hat D%
\ar@{->>}[r]^{i^*\alpha'\oplus 0}_\simeq&%
\Apl(\del T)\ar@{->>}[r]^{\check j}&%
\Apl(\del \check T)}
$$
can de factored into a \emph{minimal} semi-free extension $w$
followed by a quasi-isomorphism $\epsilon'$. Moreover, since
$\check j(i^*\alpha'\oplus 0)$ is a surjection, so is $\epsilon'$.
Define $\epsilon$ as the extension of $\pi'(\alpha\oplus s\gamma)$ such
that $\epsilon(\hat Q\otimes V)=0$, which is a $\hat Q$-module morphism.
It is clear that $\check j$ is $(n-2)$-connected and by minimality
$V^{<n-2}=0$. Since $r\geq 1$, we have
$((Q\oplus sD)/(J\oplus sK))^{\geq n-1}=0$.
For degree reasons $\epsilon$ is a DGmodule map.
It remains to prove that $\epsilon$ is a quasi-isomorphism. This
is an easy consequence of the fact that $H^{<n-1}(\pi')$ is an
isomorphism and $H^{\geq n-1}((Q\oplus_{\phi{\psi}}sD)/(J\oplus
sK))=0=H^{\geq n-1}(\partial \check T)$.
\end{proof}
Lemma \ref{lemma-factorchi} holds with the hypotheses of Theorem
\ref{thm-wkstCDGAsquare} replacing those of Theorem
\ref{thm-stableCDGA}, without any change in the proof.
To finish the proof of Theorem \ref{thm-wkstCDGAsquare}, we adapt
the four Lemmas \ref{lemma-phihatsq}--\ref{lemma-bartheta} to the
setting of this section. Recall Diagrams $\BDt$ and $\BDt'$ from
the statement of Theorem \ref{thm-wkstCDGAsquare}.
\begin{lemma}\label{lemma-wkphihatsq}
With the hypotheses of Theorem \ref{thm-wkstCDGAsquare} and with
the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-factorchi} and
\ref{lemma-chiwkst}--\ref{lemma-modeldTcheck}, Diagrams $\BDt$ and
$\BDt'$ are both commutative squares in CDGA and
$\hat\phi$-squares, and the following diagram is a
$\hat\phi$-square:
$$
\bar \BDt:=%
\vcenter{\xymatrix@1{%
\bar R\vrule width 0pt depth 6pt\ar[r]^{\bar\phi}\ar@{^(->}[d]&%
\hat Q\vrule width 0pt depth 6pt\ar@{^(->}[d]\\%
\bar R\oplus_{\bar{\psi}}s\hat D\ar[r]_-{w(\bar\phi\oplus\id)}&%
(\hat Q\oplus_\chi s\hat D)\oplus \hat Q\otimes V%
}}%
$$
\end{lemma}
\begin{proof}
We show first that the $\BDt$ is a diagram of CDGA. We have
already shown in the proof of Theorem \ref{thm-wkstCDGA} that
$R\to (R\oplus_{{\psi}}sD)/(I\oplus sK)$ is a CDGA map. The
morphism $\phi{\psi}$ is not a $Q$-DGmodule morphism but, for
degree reasons, the composite $\pi'_0\phi{\psi}\co D\to Q/J$
is. Therefore the truncated mapping cone $(Q\oplus_{\phi{\psi}}
sD)/(J\oplus sK))$ has a natural structure of $Q$-DGmodule. Again
by Lemma \ref{lemma-CDGAtruncMC}, this endows this mapping cone
with a semi trivial CDGA structure and the map $Q\to
(Q\oplus_{\phi{\psi}} sD)/(J\oplus sK)$ is a CDGA map. Moreover,
using the fact that $n\geq m+r+2$ we get that $\phi(I)\subset J$,
therefore $\phi\oplus \id\co R\oplus_{{\psi}}sD\to Q
\oplus_{\phi{\psi}}sD$ induces a map, $\overline{\phi\oplus \id}$,
between the quotients. It is straightforward to check that it is a
CDGA map.
This proves that $\BDt$ is a CDGA square and also a
$\hat\phi$-square where the $\hat R$- and $\hat Q$-DGmodule
structures are induced by the maps $\alpha$ and $\beta$. It is
immediate that $\BDt'$ is a CDGA-square and it is also a
$\hat\phi$-square where the $\hat R$- and $\hat Q$-DGmodule
structures are induced by the maps $\alpha'$ and $\beta'$.
It is immediate that $\bar\BDt$ is a $\hat\phi$-square.
\end{proof}
\begin{lemma}\label{lemma-wkthetatheta''}
With the hypotheses of Theorem \ref{thm-wkstCDGAsquare} and with
the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-factorchi} and
\ref{lemma-chiwkst}--\ref{lemma-wkphihatsq}, there exist acyclic
fibrations of $\hat\phi$-squares
$$
\xymatrix{\BDt&%
\ar@{->>}[l]_{\Theta}^{\simeq}\bar\BDt\ar@{->>}[r]^{\Theta'}_\simeq&%
\BDt'.}
$$
\end{lemma}
\begin{proof}
Set
$\Theta=\left(\begin{array}{cc}\bar\alpha&\beta\\\pi(\alpha\oplus
s\gamma)&\epsilon\end{array}\right)$ and
$\Theta'=\left(\begin{array}{cc}\bar\alpha'&\beta'\\\l^*\bar\alpha'\oplus
0&\epsilon'\end{array}\right)$ where $\epsilon$ and $\epsilon'$
were defined in Lemma \ref{lemma-modeldTcheck}. An argument
analogous to that of Lemma \ref{lemma-thetatheta''} together with
the results of Lemmas \ref{lemma-connectivityHconebis} and
\ref{lemma-chiwkst} finishes the proof.
\end{proof}
\begin{lemma}\label{lemma-wkthetahat''}
With the hypotheses of Theorem \ref{thm-wkstCDGAsquare} and with
the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-factorchi} and
\ref{lemma-chiwkst}--\ref{lemma-wkthetatheta''}, there exists a commutative square in CDGA,
$$\hat\BDt:=%
\vcenter{\xymatrix@1{%
\vrule width0pt depth8pt\hat R_{{}_{{}_{{}_{}}}}\quad\ar@{>->}[r]^{\hat\phi}\ar@{>->}[d]_u&%
\vrule width0pt depth8pt\hat Q_{{}_{{}_{{}_{}}}}\ar@{>->}[d]^v\\
\hat R\otimes\wedge X\,\,\ar@{>->}[r]_-{\hat\psi}&%
\hat Q\otimes \wedge X\otimes\wedge Z}}$$
where $\hat\phi$, $\hat\psi$, $u$, and $v$ are cofibrations, and
there exists a weak equivalence both of CDGA-squares and of
$\hat\phi$-squares
$\hat\Theta'\co\hat\BDt\quism \BDt'$.
Moreover $X$ and $Z$ can be chosen such that such that
$X^{<n-m-1}=Z^{<n-m-1}=0$.
\end{lemma}
\begin{proof}
The proof is completely similar to that of Lemma
\ref{lemma-thetahat''}, replacing $\Apl(\del T)$ by $\Apl(\del
\check T)$, which changes nothing to the $(n-m-1)$-connectivity of
the maps and noticing that since $r$ is positive, $H^1(f)$ is injective.
\end{proof}
\begin{lemma}\label{lemma-wkbartheta}
With the hypotheses of Theorem \ref{thm-wkstCDGAsquare} and with
the notation of Lemmas
\ref{lemma-CDGAphihat}--\ref{lemma-factorchi} and
\ref{lemma-chiwkst}--\ref{lemma-wkthetahat''}, there exists a
quasi-isomorphism of commutative squares in CDGA
$\hat\Theta\co\hat\BDt\quism\BDt$.
\end{lemma}
\begin{proof}
By an completely analogous
argument to that of the first part of the proof of Lemma
\ref{lemma-bartheta}, we get a lifting of $\hat\phi$-squares
$\bar\Theta\co\hat\BDt\quism\bar\BDt$.
It remains then to prove that the composite
$\hat\Theta:=\bar\Theta\hat\Theta'$ is a morphism of squares of
CDGA. This is proved by Lemma \ref{lemma-DGmodCDGAmap} using the
facts that $$((R\oplus_{{\psi}}sD)/(I\oplus sK))^{\geq n-r}=
((Q\oplus_{\phi{\psi}}sD)/(J\oplus sK))^{\geq n-r}=0,
$$
that $X^{<n-m-1}=Z^{<n-m-1}=0$, and that $2(n-m-1)\geq n-r$ by
\refequ{equ-unknotrht}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-wkstCDGA}]
The fact that $\BDt$ is a well defined CDGA square was proved in
the first part of Lemma \ref{lemma-wkphihatsq}. Then by Lemmas
\ref{lemma-wkthetahat''} and \ref{lemma-wkbartheta}, the diagrams
$\BDt$ and $\BDt'$ are weakly equivalent in CDGA.
\end{proof}
\section{Examples of rationally knotted embeddings}
\label{section-examples} The aim of this section is to show by
some examples that the unknotting condition
\refequ{equ-unknotrht}
in Theorem \ref{thm-wkstCDGA} and in the second part of
Corollary \ref{corol-HWmodstruct} is unavoidable and sharp. Recall
that this condition is $r\geq 2m-n+2$ where
\begin{itemize}
\item $m$ is the dimension of the embedded
polyhedron $P$,
\item $n-m$ is the codimension of the embedding,
\item $r$ is the connectivity of the embedding.
\end{itemize}
We will build two families of examples where the unknotting
condition \refequ{equ-unknotrht} is missed by a little and such
that the thesis of Theorem \ref{thm-wkstCDGA} does not hold. The
unknotting condition can be reformulated as $r+(n-m)\geq m+2$ which
can be roughly expressed as $$
\mathrm{connectivity}+\mathrm{codimension}\,\geq\,\mathrm{dimension}+2.
$$
In the first examples that we will build (Proposition
\ref{prop-exsharp1}), the connectivity $r$ is big but the
codimension $n-m$ is not high enough, and in the second family of
examples (Proposition \ref{prop-exsharp2}) the codimension will be
big but the connectivity small. Both of these families of examples
are fairly explicit and are described in the proof of these
propositions.
\begin{prop}
\label{prop-exsharp1} Let $p$ be a positive even integer and let
$n\geq 3p+2$. Set $m=n-p-1$ and $r=2m-n+1$.
Then there exist two $m$-dimensional polyhedra,
$P_0$ and $P_1$, having both the rational homotopy type of the wedge of spheres
$S^{n-2p-1}\vee S^{n-p-1}$, and two
nullhomotopic $r$-connected embeddings $f_0\co P_0\hookrightarrow S^n$ and
$f_1\co P_1\hookrightarrow S^n$, such that the rational
cohomology algebras of the complement of these embedded polyhedra
are not isomorphic:
$$H^*(S^n\smallsetminus f_0(P_0);\BQ)\not\cong
H^*(S^n\smallsetminus f_1(P_1);\BQ).
$$
\end{prop}
\begin{proof}
Set $X_0=S^p\vee S^{2p}$. There exists an obvious PL-embedding
$X_0\subset S^n$. Define $P_0$ as the closure of the complement of
some regular neighborhood of $X_0$ in $S^n$. By Lefschetz duality we have
$\tilde H_*(P_0;\BZ)=\BZ.x_{n-2p-1}\oplus \BZ.y_{n-p-1}$ and by
\cite[Proposition 4.1]{Wall-finiteness}
$P_0$ has the homotopy type of a two-cell CW-complex
$
P_0 \simeq S^{n-2p-1}\cup e^{n-p-1}$. Since $n\geq 3p+2$, we have
that $\pi_{n-p-2}(S^{n-2p-1})\otimes\BQ=0$ and $ P_0\simeq_\BQ S^{n-2p-1}\vee S^{n-p-1}$.
Therefore the rational cohomology algebra $
H^*(S^n\smallsetminus P_0,\BQ)\cong H^*(X_0;\BQ)$
has a trivial multiplication.
On the other hand consider a $(p-1)$-connected and
$2p$-dimensional polyhedron $X_1$ having the homotopy type of the
CW-complex $S^p\cup_{2[\iota,\iota]}e^{2p}$
where $\iota\in\pi_p(S^p)$ represents the identity map and
$[\iota,\iota]$ is the Whitehead bracket. Then
$H^*(X_1;\BQ)\cong \BQ[x]/(x^3)$ with $\deg(x)=p$. By the
embedding theorem of Wall \cite{Wall-thick}, after replacing
$X_1$ by some polyhedron of the same homotopy type, there exists
an embedding $X_1\subset S^n$. Define $P_1$ as the closure of the
complement of a regular neighborhood of $X_1$ in $S^n$. By the
same argument as for $P_0$ we see that $P_1$ has the rational
homotopy type of the same wedge of spheres.
But here the multiplication on the cohomology algebra
$H^*(S^n\smallsetminus P_1;\BQ)\cong H^*(X_1;\BQ)$
is \emph{not} trivial.
Finally it is immediate that both embeddings $P_0\subset S^n$ and
$P_1\subset S^n$ are nullhomotopic and $r$-connected.
\end{proof}
The previous proposition implies that there is no way of getting a
model of the rational homotopy type of the complement
$S^n\smallsetminus f_i(P_i)$ from just a model of the homotopy class of
the embedding $f_i$. Notice that the equation $r=2m-n+1$ is very
close to the unknotting condition \refequ{equ-unknotrht}, showing
that this condition is sharp in Theorem \ref{thm-wkstCDGA}
and Corollary \ref{corol-HWmodstruct}.
Note also that using Spanier-Whitehead duality or the techniques
of \cite{LSV-TAMS}, it can be shown that the two polyhedra $P_0$
and $P_1$ constructed in Proposition \ref{prop-exsharp1} can be
chosen as having the same integral homotopy type. It is even
possible that $P_0$ and $P_1$ might be chosen as being
PL-homeomorphic, but we have no proof of that fact.\medbreak
The
examples of Proposition \ref{prop-exsharp1} show that the
unknotting condition of Theorem \ref{thm-wkstCDGA} is sharp at
least when the codimension $n-m$ is low (even if the connectivity
$r$ is high). In the rest of this section we will build a second
family of examples for which the codimension is high but the
connectivity is low. We prove first a lemma.
\begin{lemma}
\label{lemma-embsusp} Let $i\co X\hookrightarrow S^{n-1}$ be
the inclusion of a polyhedron in a sphere and denote by
$\epsilon\co S^{n-1}\hookrightarrow S^n$ the inclusion of the
equator. Then
$$
S^n\smallsetminus\epsilon(i(X))\simeq \Sigma\left(S^{n-1}\smallsetminus
i(X)\right).
$$
\end{lemma}
\proof
Set $Y=S^{n-1}\smallsetminus i(X)$. It is clear that the complement of
$X$ in $S^n$ has the homotopy type of two disks $D^n$ glued along
$Y\subset S^{n-1}=\partial D^n$. Thus
$$
S^n\smallsetminus i(X)\simeq D^n\cup_Y D^n\simeq \Sigma Y.
\eqno{\qed}$$
\begin{prop}
\label{prop-exsharp2} For $0\leq r\leq 5$ there exists two $r$-connected
homotopic embeddings $f_k\co S^r\times S^7\hookrightarrow
S^{15}$, $k=0,1$, such that the rational cohomology algebras of
their complement are not isomorphic,
$$
H^*(S^{15}\smallsetminus f_0(S^r\times S^7);\BQ)\not\cong
H^*(S^{15}\smallsetminus f_1(S^r\times S^7);\BQ).
$$
\end{prop}
\begin{proof}
We have the standard embeddings $S^r\subset \BR^{r+1}$ and
$S^7\subset \BR^8$, as well as the ``stereographic'' embedding
$\BR^{r+9}\subset (\BR^{r+9}\cup\set{\infty})\cong S^{r+9}$.
Composing those we get an embedding
$$
i\co S^r\times S^7\hookrightarrow
\BR^{r+1}\times\BR^8=\BR^{r+9}\hookrightarrow S^{r+9}.
$$
Since $r+9\leq 14$, we have the inclusion of a subequator
$$
\epsilon\co S^{r+9}\subset S^{15}.
$$
Set $f_0=\epsilon i$. Lemma \ref{lemma-embsusp} implies that
$S^{15}\smallsetminus f_0(S^r\times S^7)$ has the homotopy type of a
suspension. Therefore the multiplication on $H^*(S^{15}\smallsetminus
f_0(S^r\times S^7);\BQ)$ is trivial.
We construct now another embedding $f_1$. Consider the Hopf
fibration
$$
S^7\to S^{15}\stackrel{\pi}{\to}S^8.$$ Consider the inclusion of
$S^r$ in $S^8$ as a subequator. Its complement $S^8\smallsetminus S^r$
has the homotopy type of $S^{7-r}$. Therefore the sphere $S^{15}$
is the union of two polyhedra of the homotopy type of
$\pi^{-1}(S^r)$ and $\pi^{-1}( S^{7-r})$. Since both of the
inclusions $S^r\subset S^8$ and $S^{7-r}\subset S^8$ are
nullhomotopic, the restrictions of the Hopf fibration to these
subspaces are trivial, hence $\pi^{-1}(S^r)\simeq S^r\times S^7$
and $\pi^{-1}(S^{7-r})\simeq S^{7-r}\times S^7$.
This
defines an embedding $f_1\co S^r\times S^7\hookrightarrow
S^{15}$ whose complement has the homotopy type of $S^{7-r}\times
S^7$. Therefore the multiplication on the cohomology algebra $
H^*(S^{15}\smallsetminus f_1(S^r\times S^7);\BQ)$ is not trivial.
Finally it is immediate that the embeddings $f_0$ and $f_1$ are
homotopic since there are both nullhomotopic for
dimension-connectivity reasons.
\end{proof}
Taking $r=0$ in Proposition \ref{prop-exsharp2} gives an example
of two homotopic $0$-connected embeddings of $S^0\times S^7$ in $S^{15}$, of
relatively high codimension, and whose complement do not have the
same rational homotopy type. Again this shows that the unknotting
condition \refequ{equ-unknotrht} is sharp since here $r=2m-n+1$.
Note that $r=0$ is not a positive integer and $P=S^0\times S^7$ is not connected
as it should be in the hypotheses of Theorem
\ref{thm-wkstCDGA}. But if we take $r=1$ we get
two $1$-connected homotopic embeddings of $S^1\times S^7$ into $S^{15}$, and the
unknotting condition is only missed by $2$ in that case.
Examples analogous to those of Proposition \ref{prop-exsharp2} can
be build in other dimensions by replacing the Hopf fibration
$S^7\to S^{15}\to S^8$ by
the Stiefel fibration
$$
S^{2k-1}\to V_2(\BR^{2k+1})\stackrel{\pi'}{\to}S^{2k}
$$
where $V_2(\BR^{2k+1})$ can be seen as the spherical tangent
bundle of $S^{2k}$. Since the Euler characteristic of an
even-dimensional sphere is not zero, it is immediate that
$V_2(\BR^{2k+1})$ has the rational homotopy type of a sphere
$S^{4k-1}$. We leave to the reader the details of the statement
and proof of a proposition analogous to \ref{prop-exsharp2} with
two embeddings of $S^r\times S^{2k-1}$ into
$V_2(\BR^{2k+1})\simeq_\BQ S^{4k-1}$ for which the rational
cohomology algebras of the complements are not isomorphic.
|
{
"timestamp": "2005-03-25T16:47:01",
"yymm": "0503",
"arxiv_id": "math/0503605",
"language": "en",
"url": "https://arxiv.org/abs/math/0503605"
}
|
\section*{Introduction}
Let $M$ be a complex manifold, and $T^*M$ its cotangent bundle endowed with
the canonical symplectic structure. Let $\mathcal{W}_M$ be the sheaf of rings of WKB
operators, that is, microdifferential operators with an extra central
parameter $\tau$. This ring provides a of $T^*M$.
Recall that the order of the operators defines a filtration on $\mathcal{W}_M$ such
that its associated graded ring is isomorphic to $\O_{T^*M}[\tau^{-1},\tau]$.
Then, any filtered sheaf of rings which has $\O_{T^*M}[\tau^{-1},\tau]$ as
graduate ring and which is locally isomorphic to $\mathcal{W}_M$ gives another
deformation quantization of $T^*M$. We call such an object a WKB-algebra.
On a complex symplectic manifold $X$ there may not exist a sheaf of rings of
WKB operators, that is, a sheaf locally isomorphic to $\opb i \mathcal{W}_M$, for any
symplectic local chart $i\colon X\supset U \to T^*M$. However, it is always
defined an algebroid $\mathfrak{W}_X$, which consists, roughly speaking, in
considering the whole family of locally defined sheaves of WKB operators.
This gives a deformation quantization of $X$
(see \cite{Kashiwara1996,Kontsevich2001,Polesello-Schapira,D'Agnolo-Polesello2005}).
Again, the algebroid $\mathfrak{W}_X$ is filtered and its associated graded is the trivial algebroid
$\O_{X}[\tau^{-1},\tau]$. Then we may define a WKB-algebroid to be a filtered
algebroid with the same graded as $\mathfrak{W}_X$, and which is locally equivalent to
$\mathfrak{W}_X$. As before, any of these objects provides a deformation quantization of $X$.
The purpose of this paper is to show that WKB-algebroids are classified by
$H^2(X;k^*_X)$, where $k^*$ is a subgroup of the group of invertible formal
Laurent series in $\opb\tau$.
We refer to \cite{Deligne} for the classification of deformation quantization algebras
on real symplectic manifolds.
The paper is organized as follows: we start by recalling the definition of
WKB operator and that of WKB-algebra on $T^*M$, and by giving their
classification. We then recall the main definitions and properties of filtered
and graded stacks, and those of cohomology with values in a stack. With
these tools at hand, we may define the WKB-algebroids on $X$ and give their
classification.
\medskip
\noindent
{\bf Acknowledgement}
We wish to thank Masaki Kashiwara for useful suggestions.
\section{WKB-algebras}
The relation between Sato's microdifferential operators and WKB
operators\footnote{WKB stands for Wentzel-Kramer-Brillouin.} is classical, and
is discussed e.g.~ in~\cite{Pham,AKKT}. We follow here the presentation
in~\cite{Polesello-Schapira}, and we refer to~\cite{S-K-K,Kashiwara1986,Kashiwara2000}
for the theory of microdifferential operators.
\medskip
Let $M$ be a complex manifold, and denote by $\rho\colon J^1M \to T^*M$ the
projection from the 1-jet bundle to the cotangent bundle.
Let $(t;\tau)$ be the system of homogeneous symplectic coordinates on $T^*\mathbb{C}$, and recall that $J^1 M$ is identified with the affine chart of the projective cotangent bundle $P^*(M\times \mathbb{C})$ given by
$\tau\neq 0$.
Denote by $\mathcal{E}_{M\times\mathbb{C}}$ the sheaf of microdifferential operators on $P^*(M\times\mathbb{C})$.
In a local coordinate system $(x,t)$ on $M\times\mathbb{C}$, consider the subring $\mathcal{E}_{M\times\mathbb{C},\hat t}^{\sqrt v}$ of operators commuting with $\partial_t$.
The ring of WKB operators is defined by
$$
\mathcal{W}_M = \oim\rho (\mathcal{E}_{M\times\mathbb{C},\hat t} |_{J^1M } ).
$$
In a local coordinate system $(x)$ on $M$, with associated symplectic local
coordinates $(x;u)$ on $T^*M$, a WKB operator $P$ of order $m$
defined on a open subset $U$ of $T^*M$ has a total symbol
$$
\sigma(P)=\sum_{j=-\infty}^m p_j(x;u)\tau^{j},
$$
where the $p_j$'s are holomorphic functions on $U$ subject to the estimates
\begin{equation}\label{eq:estmicrod}
\left\{ \begin{array}{l}
\mbox{for any compact subset $K$ of $U$ there exists a constant}\\
\mbox{$C_K>0$ such that for all $j<0$,}
\sup\limits_{K}\vert p_{j}\vert \leq C_K^{-j}(-j)!.
\end{array}\right.
\end{equation}
The product structure on $\mathcal{W}_M$ is given by the Leibniz formula not
involving $\tau$-derivatives. If $Q$ is another WKB operator defined on $U$
of total symbol $\sigma(Q)$, then
$$\sigma(P\circ Q)=\sum_{\alpha\in\mathbb{N}^n} \frac{\tau^{-\vert\alpha\vert}}
{\alpha !} \partial^{\alpha}_u\sigma(P)\partial^{\alpha}_x\sigma(Q).
$$
\begin{remark}
The ring $\mathcal{W}_M$ is a deformation quantization of $T^*M$ in the following
sense. Setting $\hbar=\opb \tau$, the sheaf of formal WKB operators (obtained
by dropping the estimates \eqref{eq:estmicrod}) of degree less than or equal
to 0 is locally isomorphic to $\O_{T^*M}[\![\hbar]\!]$ as $\mathbb{C}_{T^*M}$-modules
(via the total symbol), and it is equipped with an unitary associative product
(the Leibniz rule) which induces a star-product on $\O_{T^*M}[\![\hbar]\!]$.
\end{remark}
Recall that the center of $\mathcal{W}_M$ is the constant sheaf $k_{T^*M}$ with stalk
the subfield $k = \mathcal{W}_{\operatorname{pt}} \subset \mathbb{C}[\![\tau^{-1},\tau]$ of
WKB operators over a point, {\em i.e.} series $\sum_{j}
a_j{\tau}^j$ which satisfie the estimate:
\begin{equation*
\left\{ \begin{array}{l}
\mbox{there exists a constant $C>0$ such that }\\
\mbox{for all $j<0$, }\vert a_{j}\vert \leq C^{-j}(-j)!.
\end{array}\right.
\end{equation*}
The sheaf $\mathcal{W}_M$ is filtered (over $\mathbb{Z}$), and one denotes by
$\mathcal{W}_M(m)$ the sheaf of operators of order less than or equal to $m$.
We denote by
$$
\sigma_m(\cdot)\colon \mathcal{W}_M(m)\to \mathcal{W}_M(m)/\mathcal{W}_M(m-1)
\simeq \mathcal{O}_{T^*M}\cdot\tau^m
$$
the symbol map of order $m$. This function does not depend on the local
coordinate system on $X$. If $\sigma_m(P)$ is not
identically zero, then one says that $P$ has order $m$ and
$\sigma_m(P)$ is called the principal symbol of $P$.
In particular, an element $P$ in $\mathcal{W}_M$ is invertible if and only if
its principal symbol is nowhere vanishing.
Moreover, the principal symbol map induces an isomorphism of graded rings:
$$
\sigma\colon\mathop{\mathcal{G}r}\nolimits(\mathcal{W}_M)\isoto\mathcal{O}_{T^*M}[\opb\tau,\tau].
$$
\medskip
Let $\Omega_M$ be the canonical sheaf on $M$, that is, the sheaf of forms of
top degree. Recall that each locally defined volume form $\theta\in\Omega_M$
gives rise to a local isomorphism $*_{\theta}\colon\mathcal{W}_M^{\mathrm{op}} \isoto \mathcal{W}_M$,
which sends an operator $P$ to its formal adjoint $P^{*_{\theta}}$ with
respect to $\theta$. Twisting $\mathcal{W}_M$ by $\Omega_M$, one then gets a globally
defined isomorphism of rings
$$
\mathcal{W}_M^{\mathrm{op}} \isoto \opb {\pi}\Omega_M\tens\mathcal{W}_M \tens \opb{\pi}
\Omega_M^{\tens -1}
\qquad P\mapsto \theta\tens P^{*_{\theta}}\tens \theta^{\tens -1},
$$
which does not depend on the choice of the volume form.
(Here $\pi\colon T^*M\to M$ denotes the natural projection,
$\Omega_M^{\tens -1}$ the $\O_M$-dual of $\Omega_M$
and the tensor product is over $\opb{\pi}\O_M$.)
This leads to replace the ring $\mathcal{W}_M$ by its twisted version
by half-forms\footnote{Recall
that the sections of $\mathcal{W}^{\sqrt v}_M$ are locally defined by $\theta^{\tens 1/2}\tens
P\tens \theta^{\tens -1/2}$ for a volume form $\theta$ and an operator $P$,
with the equivalence relation $\theta_1^{\tens 1/2}\tens P_1\tens
\theta_1^{\tens -1/2} = \theta_2^{\tens 1/2}\tens P_2\tens
\theta_2^{\tens -1/2}$ if and only if $P_2 = (\theta_1/\theta_2)^{1/2} P_1
(\theta_1/\theta_2)^{-1/2}$.}
$$
\mathcal{W}^{\sqrt v}_M =
\opb {\pi} \Omega_M^{\tens 1/2}\tens\mathcal{W}_M\tens\opb{\pi}\Omega_M^{\tens -1/2}.
$$
The $k$-algebra $\mathcal{W}^{\sqrt v}_M$ is locally isomorphic to $\mathcal{W}_M$ and has the
following properties:
\begin{itemize}
\item[(i)] it is filtered;
\item[(ii)] there is an isomorphism of graded rings
\begin{equation*}
\sigma\colon\mathop{\mathcal{G}r}\nolimits(\mathcal{W}^{\sqrt v}_M)\isoto\mathcal{O}_{T^*M}[\opb\tau,\tau];
\end{equation*}
\item[(iii)] it is endowed with an anti-involution, {\em i.e.} an isomorphism
of rings $$*\colon (\mathcal{W}^{\sqrt v}_M)^{\mathrm{op}}\isoto\mathcal{W}^{\sqrt v}_M \quad \mbox{such that
$*^2=\id$.}$$
\end{itemize}
This suggests the following
\begin{definition}
A WKB-algebra on $T^*M$ is a sheaf of $k$-algebras $\mathcal{A}$ together with
\begin{itemize}
\item[(i)] a filtration $\{F_m\mathcal{A}\}_{m\in\mathbb{Z}}$;
\item[(ii)] an isomorphism of graded rings $\nu\colon\mathop{\mathcal{G}r}\nolimits(\mathcal{A})\isoto
\O_{T^*M}[\opb\tau,\tau]$;
\item[(iii)] an anti-involution $\iota$;
\end{itemize}
such that the triplet $(\mathcal{A},\nu,\iota)$ is locally isomorphic to
$(\mathcal{W}^{\sqrt v}_M, \sigma, *)$.
A morphism of WKB-algebras is a $k$-algebra morphism compatible with the
structures (i), (ii) and (iii).
\end{definition}
By definition, an isomorphism of WKB-algebras $\varphi\colon \mathcal{A}_1\to \mathcal{A}_2$
is a $k$-algebra isomorphism commuting with the anti-involutions, mapping
$F_m\mathcal{A}_1$ to $F_m\mathcal{A}_2$ in such a way that $\nu^2_m(\varphi(P)) =
\nu^1_m(P)$ for all $P\in F_m\mathcal{A}_1$. (Here $\nu^i_m$ denotes the symbol map
$F_m(\mathcal{A}_i)\to F_m(\mathcal{A}_i)/F_{m-1}(\mathcal{A}_i)\simeq
\mathcal{O}_{T^*M}\cdot\tau^m$ of order $m$, for $i=1,2$.)
This translates to WKB operators the notion of equivalence between star-products.
Hence any (formal) WKB-algebra provides a deformation quantization
of $T^*M$. See \cite{BoutetdeMonvel1999,BoutetdeMonvel2002} for similar
definitions in the context of microdifferential and Toeplitz operators.
\begin{example}
Let $f\colon T^*M \to T^*M$ be a symplectic transformation. Then $\mathcal{W}^{\sqrt v}_M$ induces an
anti-involution on $\opb f \mathcal{W}^{\sqrt v}_M$ and a filtration such that the associated graded ring is
isomorphic (via $f$) to $\O_{T^*M}[\opb\tau,\tau]$. By \cite{Polesello-Schapira}, locally there
exists a Quantized Symplectic Transformation over $f$, that is, an isomorphism
$\opb f \mathcal{W}^{\sqrt v}_M\simeq \mathcal{W}^{\sqrt v}_M$of filtered $k$-algebras, commuting with the anti-involutions
and which preserves the graded rings. It follows that $\opb f \mathcal{W}^{\sqrt v}_M$ is a WKB-algebra.
\end{example}
Denote by $\shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_M)$ the group of WKB-algebra automorphisms of
$\mathcal{W}^{\sqrt v}_M$ and set
\begin{equation*}
\begin{split}\mathcal{W}^{\sqrt v, *}_M & = \{P\in\mathcal{W}^{\sqrt v}_M;
\mbox{ $P$ has order 0, $\sigma_0(P)=1$ and $PP^*=1$}\},\\
k^* & = \{s(\tau)\in k;\mbox{ $s(\tau)=1+\sum_{j< 0}a_j{\tau}^j$ and
$s(\tau)s(-\tau) =1$}\}.
\end{split}
\end{equation*}
Note that $\mathcal{W}^{\sqrt v, *}_M$ is a subgroup of the group
$\mathcal{W}^{\sqrt v, \times}_M$ of invertible WKB operators, and that
$k^* = \mathcal{W}^{\sqrt v, *}_{\operatorname{pt}}$.
\begin{lemma}(cf \cite{Polesello-Schapira})\label{lemma:key}
There is an exact sequence of groups on
$T^*M$
\begin{equation}\label{eq:key}
1\to k^*_{T^*M} \to \mathcal{W}^{\sqrt v, *}_M \to[\operatorname{ad}] \shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_M) \to 1,
\end{equation}
where $\operatorname{ad}(P)(Q)=PQP^{-1}$ for any $P\in \mathcal{W}^{\sqrt v, *}_M$ and
$Q\in\mathcal{W}^{\sqrt v}_M$.
\end{lemma}
The set of isomorphism classes of WKB-algebras on $T^*M$ is in bijection with
$H^1(T^*M; \shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_M))$. Hence we get
\begin{corollary}
WKB-algebras on $T^*M$ are classified by the pointed set
$H^1(T^*M;\mathcal{W}^{\sqrt v, *}_M/k^*_{T^*M})$.
\end{corollary}
\section{Filtered and graded stacks}
To define WKB-algebroids, we need to translate the notions of filtration and
graduation from sheaves to stacks. We start here by recalling what a filtered
(resp. graded) category is and how to associate a graded category to a
filtered one. Then we stakify these definitions.
We assume that the reader is familiar with the basic notions from the theory
of stacks which are, roughly speaking, sheaves of categories. (The classical
reference is~\cite{Giraud1971}, and a short presentation is given {\em e.g.}
in~\cite{Kashiwara1996,D'Agnolo-Polesello2003}.)
\medskip
Let $R$ be a commutative ring.
\begin{definition}\label{def:filtered}
A filtered (resp. graded) $R$-category is an
$R$-category\footnote{Recall that an $R$-category is a category whose
sets of morphisms are endowed with an $R$-module structure, so that
composition is bilinear. An $R$-functor is a functor between
$R$-categories which is linear at the level of morphisms.}
$\mathsf{C}$ such that:
\begin{itemize}
\item[$\bullet$] for any objects $P,Q\in\mathsf{C}$, the $R$-module
$\Hom[\mathsf{C}](P,Q)$ is filtered (resp. graded) over $\mathbb{Z}$;
\item[$\bullet$] for any $P,Q,R\in \mathsf{C}$ and any morphisms $f$ in
$F_m\Hom[\mathsf{C}](Q,R)$ (resp. in $G_m\Hom[\mathsf{C}](Q,R)$) and $g$ in
$F_n\Hom[\mathsf{C}](P,Q)$ (resp. in $G_n\Hom[\mathsf{C}](P,Q)$), the composition
$f\circ g$ is in $F_{m+n}\Hom[\mathsf{C}](P,R)$ (resp. in
$G_{m+n}\Hom[\mathsf{C}](P,R)$);
\item[$\bullet$] for each $P\in \mathsf{C}$, the identity morphism $\id_{P}$
is in $F_0\Hom[\mathsf{C}](P,P)$ (resp. in $G_0\Hom[\mathsf{C}](P,P)$).
\end{itemize}
A filtered (resp. graded) $R$-functor is an $R$-functor which respects
the filtrations (resp. graduations) at the level of morphims.
\end{definition}
To any filtered $R$-category $\mathsf{C}$ there is an associated graded $R$-category
$\mathop{\mathrm{Gr}}\nolimits(\mathsf{C})$, whose objects are the same of those of $\mathsf{C}$ and the
morphisms are defined by $\Hom[\mathop{\mathrm{Gr}}\nolimits(\mathsf{C})](P,Q) = \mathop{\mathrm{Gr}}\nolimits(\Hom[\mathsf{C}](P,Q))$ for any objects
$P,Q$. In this way, we get a functor from filtered $R$-categories to graded ones.
Following the presentation in~\cite{D'Agnolo-Polesello2005}, recall that there
is a fully faithful functor from filtered (resp. graded)
$R$-algebras to filtered (resp. graded) $R$-categories, which sends a filtered
(resp. graded) $R$-algebra $A$ the category $\astk A$ with a single
object $\bullet$ and $\Endo(\bullet)=A$ as set of morphisms. Hence, the
functors $\mathop{\mathrm{Gr}}\nolimits$ and $\astk{}$ commutes, that is, for any filtered $R$-algebra
$A$ one has $\mathop{\mathrm{Gr}}\nolimits (\astk A)=\astk{\mathop{\mathrm{Gr}}\nolimits (A)}$.
Note that, if $A$ is a filtered $R$-algebra, the $R$-category $\mathsf{Mod}_F(A)$
of filtered left $A$-modules has a natural filtration: for any filtered
$A$-modules $M$ and $N$, one sets $F_m\Hom[FA](M,N)=\Hom[FA](M,N(m))$,
where $N(m)$ has the same underlying $A$-module as $N$, and the filtration
is given by $F_n N(m) = F_{n+m}N$.
One easily checks that $\mathsf{Mod}_F(A)$ is equivalent to the category
$\catFun[F](\astk A,\mathsf{Mod}_F(R))$ of filtered $R$-functors from
$\astk A$ to $\mathsf{Mod}_F(R)$ and that the Yoneda embedding
$$
\astk A \to \catFun[F]((\astk A)^\mathrm{op}, \mathsf{Mod}_F(R)) \approx
\mathsf{Mod}_F(A^\mathrm{op})
$$
identifies $\astk A$ with the full subcategory of filtered right $A$-modules
which are free of rank one. Everything remains true replacing filtered
algebras and categories by graded ones.
\medskip
Let $X$ be a topological space, and $\mathcal{R}$ a sheaf of commutative rings.
As for categories, there are natural notions of filtered (resp. graded)
$\mathcal{R}$-stack, and of filtered (resp. graded) $\mathcal{R}$-functor between
filtered (resp. graded) $\mathcal{R}$-stacks.
As above, we denote by $\astk{}$ the (faithful and locally full) functor from filtered (resp. graded)
$\mathcal{R}$-algebras to filtered (resp. graded) $\mathcal{R}$-categories, which sends
a filtered (resp. graded) $\mathcal{R}$-algebra $\mathcal{A}$ to the stack $\astk\mathcal{A}$
defined as follows: it is the stack associated with the separated prestack
$X \supset U\mapsto \astk{\mathcal{A}(U)}$.
If $\mathcal{A}$ is a filtered $\mathcal{R}$-algebra, then the stack
$\stkMod[F](\mathcal{A})$ of filtered left $\mathcal{A}$-modules is filtered and equivalent
to the stack of filtered functors $\stkFun[F](\astk \mathcal{A},\stkMod[F](\mathcal{R}))$,
and the Yoneda embedding gives a fully faithful functor
\begin{equation*}
\label{eq:Yoneda}
\astk \mathcal{A} \to \stkFun[F]((\astk \mathcal{A})^\mathrm{op},\stkMod[F](\mathcal{R})) \approx
\stkMod[F](\mathcal{A}^\mathrm{op})
\end{equation*}
into the stack of filtered right $\mathcal{A}$-modules. This identifies $\astk\mathcal{A}$
with the full substack of locally free filtered right $\mathcal{A}$-modules of rank
one. As above, everything remains true replacing filtered algebras and stacks
by graded ones.
Let $\mathfrak{S}$ be a filtered $\mathcal{R}$-stack. We denote by $\mathop{\mathcal{G}r}\nolimits (\mathfrak{S})$ the graded stack
associated to the pre-stack $X \supset U\mapsto \mathop{\mathrm{Gr}}\nolimits (\mathfrak{S}(U))$.
\begin{proposition}
Let $\mathcal{A}$ be a filtered $\mathcal{R}$-algebra and $\mathop{\mathcal{G}r}\nolimits (\mathcal{A})$ its associated graded
ring. Then there is an equivalence of graded
stacks $\mathop{\mathcal{G}r}\nolimits (\mathcal{A}^+)\approx\mathop{\mathcal{G}r}\nolimits (\mathcal{A})^+$.
\end{proposition}
\begin{proof}
Let $\mathcal{L}$ be a locally free right filtered $\mathcal{A}$-module of rank one (that is,
an object of $\mathcal{A}^+$). Its associated graded module $\mathop{\mathcal{G}r}\nolimits (\mathcal{L})$ is a
locally free right graded $\mathop{\mathcal{G}r}\nolimits (\mathcal{A})$-module of rank one (that is, an object
of $\mathop{\mathcal{G}r}\nolimits( \mathcal{A})^+$). Hence the assignement $\mathcal{L} \mapsto \\mathop{\mathcal{G}r}\nolimits (\mathcal{L})$ induces
a functor $\mathop{\mathcal{G}r}\nolimits( \mathcal{A}^+)\to\mathop{\mathcal{G}r}\nolimits( \mathcal{A})^+$ of graded stacks.
Since at each $x\in X$ this reduces to the equality $\mathop{\mathrm{Gr}}\nolimits (\astk {\mathcal{A}_x}) =
\astk{\mathop{\mathrm{Gr}}\nolimits (\mathcal{A}_x)}$, it follows that it is a global equivalence.
\end{proof}
Recall from \cite{Kontsevich2001,D'Agnolo-Polesello2005} that an
$\mathcal{R}$-algebroid stack is an $\mathcal{R}$-stack $\mathfrak{A}$ which is locally non-empty and
locally connected by isomorphisms. Equivalently, for any
$x\in X$ there exist an open subset $U\subset X$ containing $x$ and an
$\mathcal{R}$-algebra $\mathcal{A}$ on $U$ such that $\mathfrak{A}|_U\approx \astk \mathcal{A}$.
\begin{corollary}
Let $\mathfrak{A}$ be a filtered $\mathcal{R}$-stack. If $\mathfrak{A}$ is an $\mathcal{R}$-algebroid stack, then it associated
graded stack $\mathop{\mathcal{G}r}\nolimits(\mathfrak{A})$ is again an $\mathcal{R}$-algebroid stack.
\end{corollary}
\section{WKB-algebroids}
Let $(X,\omega)$ be a complex symplectic manifold. Recall that a local model
for $X$ is an open subset $U$ of the cotangent bundle $T^*M$ of a complex
manifold $M$, equipped with the canonical symplectic form.
Although there may not exist a globally defined WKB-algebra on $X$, that is,
a sheaf locally isomorphic to $\opb i \mathcal{W}_M$ for any symplectic local chart
$i\colon X\supset U \to T^*M$, Polesello-Schapira~\cite{Polesello-Schapira}
defined a canonical stack of WKB-modules on $X$.
Following~\cite{D'Agnolo-Polesello2005}, this result may be restated as:
\begin{theorem}
On any complex symplectic manifold $X$ there exists a canonical $k$-stack
$\mathfrak{W}_X$ which is locally equivalent to $\astk{(\opb i \mathcal{W}^{\sqrt v}_M)}$
for any symplectic local chart $i\colon X\supset U \to T^*M$.
\end{theorem}
By definition, $\mathfrak{W}_X$ is a $k$-algebroid stack. Hence there exists a canonical
WKB-algebra on $X$ if and only if $\mathfrak{W}_X$ has a global object.
\begin{proposition}
The $k$-algebroid stack $\mathfrak{W}_X$ has the following properties:
\begin{itemize}
\item[(i')] it is filtered;
\item[(ii')] there is a natural equivalence of graded stacks
$$\pmb\sigma\colon\mathop{\mathcal{G}r}\nolimits(\mathfrak{W}_X)\approxto \astk {(\O_X[\tau^{-1},\tau])};$$
\item[(iii')] it is endowed with an anti-involution $*$, that is, with a linear
equivalence
$$\pmb*\colon \mathfrak{W}_X^\mathrm{op}\approxto \mathfrak{W}_X$$
and an invertible transformation $\epsilon\colon \pmb*^2 \Rightarrow\id_{\mathfrak{W}_X}$
such that the transformations $\epsilon\id_{\pmb*}\colon \pmb*^3 \Rightarrow \pmb*$ and
$\id_{\pmb*}\epsilon\colon \pmb* \Rightarrow \pmb*^3$ are inverse one to each other.
\end{itemize}
\end{proposition}
We may mimic the definition of WKB-algebra and get the following
\begin{definition}
A WKB-algebroid on $X$ is a $k$-stack $\mathfrak{A}$ endowed with
\begin{itemize}
\item[$\astk {(i)}$] a filtration;
\item[$\astk {(ii)}$] an equivalence of graded stacks
$\pmb\nu\colon\mathop{\mathcal{G}r}\nolimits(\mathfrak{A})\approxto \astk {(\O_X[\tau^{-1},\tau])};$
\item[$\astk {(iii)}$] an anti-involution $\pmb\iota$;
\end{itemize}
such that the triplet $(\mathfrak{A},\pmb\nu,\pmb\iota)$ is locally equivalent to
$(\mathfrak{W}_X,\pmb\sigma,\pmb*)$.
A functor of WKB-algebroids is a $k$-functor compatible with the
structures $\astk {(i)}$, $\astk {(ii)}$ and $\astk {(iii)}$.
\end{definition}
As (formal) WKB-algebras give the deformation quantizations of $T^*M$, we may
say that (formal) WKB-algebroids provide the deformation quantizations of $X$.
\begin{definition}
We call $\mathfrak{W}_X$ the canonical WKB-algebroid on $X$.
\end{definition}
\section{Cohomology with values in a stack}
As for classifying WKB-algebras one uses cohomology with values in a sheaf of
groups, so to classify WKB-algebroids we need a cohomology theory with values
in a stack with group-like properties. In this section we briefly recall the
definition of cohomology with values in a stack and show how to describe it
explicitly by means of the notion of crossed module.
References are made to \cite{Breen1992,Breen1994}.
We assume that the reader is familiar with the notions of monoidal category
and monoidal functor. (The classical reference is \cite{MacLane}.)
Let $X$ be a topological space.
\begin{definition}
\begin{itemize}
\item[(i)] A 2-group\footnote{We follow here the terminology of
Baez-Lauda [{\em Higher-dimensional algebra V: 2-groups}, e-print (2004)
\texttt{arXiv:math.QA/0307200}], which seems to us more
friendly than the classical one of $gr$-category due to Grothendieck.}
is a rigid monoidal groupoid, {\em i.e.} a monoidal category $(\mathsf{G},
\tens,{\bf 1})$ with all the morphisms invertible and such that for any
object $P\in \mathsf{G}$ there exist an object $Q$ and natural
morphisms $P\tens Q\simeq {\bf 1}$ and $Q\tens P\simeq {\bf 1}$.
A functor of 2-groups is a monoidal functor between the underlying
monoidal categories.
\item[(ii)] A pre-stack (resp. stack) of 2-groups on $X$ is a pre-stack (resp. stack)
$\mathfrak{G}$ such that for each open subset $U\subset X$, the category $\mathfrak{G}(U)$ is
2-group and the restriction functors are functors of 2-groups.
\end{itemize}
\end{definition}
If there is no risk of confusion, a stack of 2-groups on $X$ will be simply called a 2-group on $X$.
\begin{example}
Let $\mathcal{G}$ be a sheaf of groups on $X$.
\begin{itemize}
\item[(i)] The discrete stack $\mathcal{G}[0]$ defined by trivially enriching
$\mathcal{G}$ with identity arrows is a 2-group on $X$.
\item[(ii)] Let $\mathcal{G}[1]$ be the stack in groupoids associated to the
separated pre-stack whose category on an open subset $U\subset X$
has a single object $\bullet$ and $\Endo(\bullet)=\mathcal{G}(U)$ as set of
morphisms. Then $\mathcal{G}[1]$ is equivalent to the stack of right
$\mathcal{G}$-torsors and it defines a 2-group on $X$ if and only if $\mathcal{G}$ is
commutative.
\end{itemize}
\end{example}
Let $\mathfrak{G}$ be a pre-stack of 2-groups on $X$.
We define the 0-th cohomology group of $X$ with values in $\mathfrak{G}$ to be
$$ H^0(X;\mathfrak{G})=\ilim[\mathcal{U}] H^0(\mathcal{U};\mathfrak{G}),$$
where $\mathcal{U}$ ranges over open coverings of $X$.
For an open covering $\mathcal{U} = \{U_{i}\}_{i\in I}$, the elements of
$H^0(\mathcal{U};\mathfrak{G})$ are represented by pairs $(\{\mathcal{P}_{i}\}, \{\alpha_{ij}\})$
(the 0-cocycles), where $\mathcal{P}_{i}$ is an object in $\mathfrak{G} (U_{i})$ and
$\alpha_{ij}\colon \mathcal{P}_{j} \isoto \mathcal{P}_{i}$ is an isomorphism on double
intersection $U_{ij} = U_i\cap U_j$, such that
$\alpha_{ij}\circ \alpha_{jk}=\alpha_{ik}$ on triple intersection $U_{ijk}$,
with the relation $(\{\mathcal{P}_{i}\}, \{\alpha_{ij}\})$ is equivalent to
$(\{\mathcal{P}'_{i}\}, \{\alpha'_{ij}\})$
if and only if there exists an isomorphism
$\delta_{i}\colon \mathcal{P}'_{i} \isoto \mathcal{P}_{i}$ compatible with
$\alpha_{ij}$ and $\alpha'_{ij}$ on $U_{ij}$.
Note that, if $\mathfrak{G}$ is a stack of 2-groups, then $H^0(X;\mathfrak{G})$ is
isomorphic to the group of isomorphism classes of objects in $\mathfrak{G}(X)$.
\medskip
Similarly, the 1-st cohomology (pointed) set of $X$ with values in
$\mathfrak{G}$ is defined as
$$H^1(X;\mathfrak{G})=\ilim[\mathcal{U}] H^1(\mathcal{U};\mathfrak{G}),$$
where $\mathcal{U}$ ranges over open coverings of $X$.
For an open covering $\mathcal{U} = \{U_{i}\}_{i\in I}$, the elements of
$H^1(\mathcal{U};\mathfrak{G})$ are given by pairs $(\{\mathcal{P}_{ij}\}, \{\alpha_{ijk}\})$
(the 1-cocycles), where $\mathcal{P}_{ij}$ is an object in $\mathfrak{G} (U_{ij})$ and
$\alpha_{ijk}\colon \mathcal{P}_{ij} \tens \mathcal{P}_{jk} \isoto \mathcal{P}_{ik}$ is an
isomorphism on $U_{ijk}$ such that the diagram on quadruple intersection
$U_{ijkl}$
\begin{equation*}
\xymatrix@C5em{ \mathcal{P}_{ij}\tens \mathcal{P}_{jk}\tens \mathcal{P}_{kl}
\ar[r]^-{\alpha_{ijk}\tens\id_{\mathcal{P}_{kl}}} \ar[d]^{\id_{\mathcal{P}_{ij}}\tens\alpha_{jkl}} &
\mathcal{P}_{ik}\tens \mathcal{P}_{kl}\ar[d]^{\alpha_{ikl}} \\
\mathcal{P}_{ij}\tens \mathcal{P}_{jl} \ar[r]^-{\alpha_{ijl}}
& \mathcal{P}_{il} }
\end{equation*}
commutes. The 1-cocylces $(\{\mathcal{P}_{ij}\}, \{\alpha_{ijk}\})$ and
$(\{\mathcal{P}'_{ij}\}, \{\alpha'_{ijk}\})$ are equivalent if and only if there
exists a pair $(\{\mathcal{Q}_i\}, \{\delta_{ij}\})$, with $\mathcal{Q}_i$ an object of
$\mathfrak{G}(U_i)$ and $\delta_{ij}\colon \mathcal{P}'_{ij} \tens\mathcal{Q}_j \isoto
\mathcal{Q}_i \tens \mathcal{P}_{ij}$ an isomorphism on $U_{ij}$ such that the diagram on
$U_{ijk}$
\begin{equation*}
\xymatrix@C4em@R3em{ \mathcal{P}'_{ij}\tens \mathcal{P}'_{jk}\tens \mathcal{Q}_k
\ar[r]^-{\id_{\mathcal{P}'_{ij}}\tens\delta_{jk}}
\ar[d]^{\alpha'_{ijk}\tens\id_{\mathcal{Q}_{k}}} &
\mathcal{P}'_{ij}\tens\mathcal{Q}_j\tens\mathcal{P}_{jk}
\ar[r]^{\delta_{ij}\tens\id_{\mathcal{P}_{jk}}} &
\mathcal{Q}_i\tens \mathcal{P}_{ij}\tens \mathcal{P}_{jk}
\ar[d]^{\id_{\mathcal{Q}_{i}}\tens\alpha_{ijk}} \\
\mathcal{P}'_{ik}\tens \mathcal{Q}_k \ar[rr]^-{\delta_{ik}}
&& \mathcal{Q}_i\tens \mathcal{P}_{ik} }
\end{equation*}
commutes.
\medskip
In the rest of the section we will give a more explicit description of the cohomology
with values in a stack by means of cocycles with values in a crossed module.
(This was Breen's approach to non abelian cohomology of Giraud~\cite{Giraud1971}.)
\begin{definition}
A crossed module on $X$ is a complex of sheaves of groups
$\mathcal{G}^{-1}\to[d]\mathcal{G}^0$ endowed with a left action of $\mathcal{G}^0$ on $\mathcal{G}^{-1}$
such that for any local sections $g\in\mathcal{G}^0$ and $h,h'\in\mathcal{G}^{-1}$ one has
$$
d({}^gh)=\operatorname{ad}(g)(d(h)) \qquad {}^{d(h')} h=\operatorname{ad}(h')(h).
$$
(Here we use the convention as in \cite{Breen1994}
for which $\mathcal{G}^{i}$ is in $i$-th degree.)
A morphism of crossed modules is a morphism of complexes compatible with the
actions in the natural way.
\end{definition}
Associated to each crossed module $\mathcal{G}^{-1}\to[d]\mathcal{G}^0$ there is
2-group on $X$, which we denote by $[\mathcal{G}^{-1}\to[d]\mathcal{G}^0]$,
defined as follows: it is the stack associated to the separated
pre-stack of 2-groups whose objects on an open subset $U\subset X$ are the
sections $g\in \mathcal{G}^0(U)$ with 2-group law $g\tens g'=gg'$, and whose
morphisms $g\to g'$ are given by sections $h\in \mathcal{G}^{-1}(U)$ such that
$g' = d(h) g$, with the 2-group structure given by $(g_1\to[h_1] g'_1)\tens
(g_2\to[h_2] g'_2) = g_1g_2\to[h_1{}^{g_1}h_2] g'_1g'_2$.
Similarly, each morphism of crossed modules induces a functor of the
corresponding 2-groups.
\begin{remark}
In fact, it is true that any 2-group on $X$ comes from a crossed module.
However, this result is not of practical use. We refer to
\cite{SGA4} for the proof of this fact in the commutative case and to
\cite{Brown-Spencer} for the non commutative case on $X=\operatorname{pt}$.
\end{remark}
\begin{example}
Let $\mathcal{G}$ be a sheaf of groups on $X$.
\begin{itemize}
\item[(i)]The 2-group defined by the
crossed module $1 \to \mathcal{G}$ is identified with $\mathcal{G}[0]$.
\item[(ii)] If moreover $\mathcal{G}$ is commutative, the complex $\mathcal{G} \to 1$
is a crossed module and its associated 2-group is identified with
$\mathcal{G}[1]$.
\end{itemize}
\end{example}
Let $\mathcal{G}^{-1}\to[d]\mathcal{G}^0$ be a crossed module on $X$. Then the cohomology
of $X$ with values in the 2-group $[\mathcal{G}^{-1}\to[d]\mathcal{G}^0]$ admits a very
explicit description, which we recall below. This is usually referred as the
(hyper-)cohomology of $X$ with values in $\mathcal{G}^{-1}\to[d]\mathcal{G}^0$.
By definition, an object $\mathcal{P}$ of $[\mathcal{G}^{-1}\to[d]\mathcal{G}^0]$ on
an open subset $U\subset X$ is described by an open covering $U =
\bigcup\limits\nolimits_i U_i$ and sections $\{g_i\}\in \mathcal{G}^0(U_i)$, subject
to the relation $g_i = d(h_{ij}) g_j$ on double intersections $U_{ij}$, for
given sections $\{h_{ij}\}\in \mathcal{G}^{-1}(U_{ij})$ satisfying $h_{ij}h_{jk}=h_{ik}$
on triple intersections $U_{ijk}$.
Hence, up to a refinement of the open covering $\mathcal{U}=\{U_{i}\}_{i\in I}$ of
$X$, the 0-cocycles on $\mathcal{U}$ with values in $[\mathcal{G}^{-1}\to[d]\mathcal{G}^0]$, may be
described by pairs $(\{g_i\},\{h_{ij}\})$, where $g_i\in \mathcal{G}^0(U_i)$
and $h_{ij}\in \mathcal{G}^{-1}(U_{ij})$ are sections satisfying the relations
\begin{equation*}
\label{nonab1}
\begin{cases}
g_i=d(h_{ij})g_j \quad \text{in }\mathcal{G}^0(U_{ij})\\
h_{ij}h_{jk} = h_{ik} \quad \text{ in } \mathcal{G}^{-1}(U_{ijk}),
\end{cases}
\end{equation*}
and $(\{g_i\},\{h_{ij}\})$ is equivalent to $(\{g'_i\},\{h'_{ij}\})$ if
and only if there exist sections $\{k_i\}\in\mathcal{G}^{-1}(U_i)$ such that the
following relations hold
\begin{equation*}
\label{nonabeq1}
\begin{cases}
g'_i = d(k_i)g_i \\
h'_{ij} k_j = k_i h_{ij} .
\end{cases}
\end{equation*}
\medskip
The same description for 1-cocycles needs some care, since one has to
consider open coverings for any double intersection $U_{ij}$. In other words,
one has to replace coverings by hypercoverings. Indices become thus
very cumbersome, and we will not write them explicitly\footnote{Recall that,
on a paracompact space, usual coverings are cofinal among hypercoverings}.
Hence the 1-cocycles on $\mathcal{U}$ with values in $[\mathcal{G}^{-1}\to[d]\mathcal{G}^0]$, may
be described by pairs $(\{g_{ij}\},\{h_{ijk}\})$, with $g_{ij}\in
\mathcal{G}^0(U_{ij})$ and $h_{ijk}\in \mathcal{G}^{-1}(U_{ijk})$ satisfying the
relations
\begin{equation*}
\label{nonab2}
\begin{cases}
g_{ij}g_{jk}=d(h_{ijk})g_{ik} \quad \text{ in } \mathcal{G}^0(U_{ijk})\\
h_{ijk}h_{ikl} = {}^{g_{ij}}h_{jkl}h_{ijl} \quad \text{in }
\mathcal{G}^{-1}(U_{ijkl}).
\end{cases}
\end{equation*}
Moreover, $(\{g_{ij}\},\{h_{ijk}\})$ is equivalent to
$(\{g'_{ij}\},\{h'_{ijk}\})$ if and only if there exists a pair
$(\{l_i\},\{k_{ij}\})$, with $k_{ij}\in\mathcal{G}^{-1}(U_{ij})$ and
$l_i\in\mathcal{G}^0(U_i)$, such that
\begin{equation*}
\label{nonabeq2}
\begin{cases}
g'_{ij} l_j = d(k_{ij})l_i g_{ij} \\
h'_{ijk} k_{ik} = {}^{g'_{ij}} k_{jk}k_{ij} {}^{l_i} h_{ijk}.
\end{cases}
\end{equation*}
\medskip
Taking the extremal cases $[1\to\mathcal{G}]$ and $[\mathcal{G}\to 1]$, the latter when the
group $\mathcal{G}$ is commutative, one easily recovers from the previous description
the usual definition of the Cech cohomology of $X$ with values in $\mathcal{G}$.
Hence one has the following
\begin{proposition}\label{prop:hyper}
Let $\mathcal{G}$ be a sheaf of groups on $X$. Then there is an isomorphism
(of groups if $i=0$, of pointed sets if $i=1$)
$$
H^i(X;\mathcal{G}[0])\simeq H^{i}(X;\mathcal{G}).
$$
If moreover $\mathcal{G}$ is commutative, then there are isomorphisms of groups (for
$i=0,1$)
$$
H^i(X;\mathcal{G}[1])\simeq H^{i+1}(X;\mathcal{G}).
$$
\end{proposition}
\section{Classification of WKB-algebroids}
Let $(X,\omega)$ be a complex symplectic manifold of dimension $2n$ and $\mathfrak{W}_X$ the
canonical WKB-algebroid on $X$.
Let $\mathfrak{A}$ be another WKB-algebroid. By definition, there exists an open
covering $X=\bigcup\limits\nolimits_i U_i$ such that $\mathfrak{A}|_{U_i}$ is equivalent to
$\mathfrak{W}_X|_{U_i}$ as WKB-algebroids.
Let $\Phi_i \colon \mathfrak{A}|_{U_i} \to \mathfrak{W}_X|_{U_i}$ and
$\Psi_i \colon \mathfrak{W}_X|_{U_i}\to \mathfrak{A}|_{U_i}$ be quasi-inverse to each other.
On double intersections $U_{ij}$ there are WKB-algebroid
equivalences $\Phi_{ij} = \Phi_i\Psi_j \colon \mathfrak{W}_X|_{U_{ij}} \to
\mathfrak{W}_X|_{U_{ij}}$, and on triple intersections $U_{ijk}$ there are
invertible transformations $\alpha_{ijk} \colon \Phi_{ij}\Phi_{jk}
\Rightarrow \Phi_{ik}$ induced by $\Psi_j\Phi_j\Rightarrow\id$.
On quadruple intersections $U_{ijkl}$ the following diagram commutes
\begin{equation}
\label{eq:alpha}
\xymatrix@C5em{
\Phi_{ij}\Phi_{jk}\Phi_{kl} \ar@{=>}[r]^{\alpha_{ijk}\id_{\Phi_{kl}}} \ar@{=>}[d]^{\id_{\Phi_{ij}}\alpha_{jkl}}
& \Phi_{ik}\Phi_{kl} \ar@{=>}[d]^{\alpha_{ikl}} \\
\Phi_{ij}\Phi_{jl} \ar@{=>}[r]^{\alpha_{ijl}} & \Phi_{il} .
}
\end{equation}
It follows that WKB-algebroids are described by 1-cocycles
$(\Phi_{ij},\alpha_{ijk})$ with values in the stack of 2-groups
$\stkAut[\operatorname{WKB}](\mathfrak{W}_X)^\times$ of autoequivalences of $\mathfrak{W}_X$ as
WKB-algebroid. (Here the upper index $\times$ means that all the non-invertible
morphisms have been removed.)
Denote by $\operatorname{WKB}(X)$ the set of equivalence classes of WKB-algebroid
on $X$, pointed by the class of $\mathfrak{W}_X$. Hence one gets an isomorphism of
pointed sets
$$
\operatorname{WKB}(X)\simeq H^1(X;\stkAut[\operatorname{WKB}](\mathfrak{W}_X)^{\times}).
$$
\medskip
Let us briefly recall how to describe more explicitly the 1-cocycle $(\Phi_{ij},\alpha_{ijk})$
attached to a WKB-algebroid $\mathfrak{A}$. We follow \cite{Polesello-Schapira,D'Agnolo-Polesello2005}.
By definition, $\mathfrak{W}_X$ is locally equivalent to
$\astk{(\opb f \mathcal{W}^{\sqrt v}_M)}$ for any symplectic local chart
$f\colon X\supset U \to T^*M$. Hence, up to a refinement of the open covering
$X=\bigcup\limits\nolimits_i U_i$, one may suppose that $\mathfrak{W}_X$ is equivalent on
$U_i$ to $\astk {\mathcal{W}^{\sqrt v}_i{}} = \astk{(\opb{f_i}\mathcal{W}^{\sqrt v}_M)}$, for
a symplectic embedding $f_i \colon U_i \to T^*M$ with $M=\mathbb{C}^n$.
On $U_{ij}$ the functor $\Phi_{ij}\colon \astk{\mathcal{W}^{\sqrt v}_j{}} \to
\astk{\mathcal{W}^{\sqrt v}_i{}}$ is then locally induced by WKB-algebra isomorphisms.
Shrinking again the open covering, we may find an isomorphism of WKB-algebras
$\varphi_{ij} \colon \mathcal{W}^{\sqrt v}_j \to \mathcal{W}^{\sqrt v}_i$ on $U_{ij}$ such
that $\astk{\varphi_{ij}} = \Phi_{ij}|_{U_{ij}}$.
On $U_{ijk}$ we have an invertible transformation
$\alpha_{ijk} \colon \astk{\varphi_{ij}}\astk{\varphi_{jk}} \Rightarrow
\astk{\varphi_{ik}}$, so that there exist a section
$P_{ijk} \in \mathcal{W}^{\sqrt v, *}_i$ such that
$$
\varphi_{ij}\varphi_{jk}= \operatorname{ad}(P_{ijk}) \varphi_{ik}.
$$
Finally, on $U_{ijkl}$ the diagram \eqref{eq:alpha} corresponds to the
equality
$$
P_{ijk} P_{ikl} = \varphi_{ij}(P_{jkl}) P_{ijl}.
$$
The datum of $(\{f_i\},\{\varphi_{ij}\}, \{P_{ijk}\})$ as above is enough
to reconstruct $\mathfrak{A}$ (up to equivalence).
\medskip
In the particular case of $X=T^*M$, one has $f_i=\id$. A direct computation
as above shows that there is an equivalence of 2-groups
\begin{equation}\label{WKB-aut}
\stkAut[\operatorname{WKB}](\astk{(\mathcal{W}^{\sqrt v}_M)})^\times
\approx \left[\mathcal{W}^{\sqrt v, *}_M \to[\operatorname{ad}] \shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_M) \right].
\end{equation}
\bigskip
We are now ready to prove the following
\begin{theorem}\label{th:classification}
There is an isomorphism of pointed sets
$$\operatorname{WKB}(X)\simeq H^2(X;k^*_X).$$
\end{theorem}
\begin{proof}
Consider the natural functor of 2-groups
$$
F\colon k^*_X[1] \longrightarrow\stkAut[\operatorname{WKB}](\mathfrak{W}_X)^{\times}
$$
induced by the functor of pre-stacks which sends the unique object
$\bullet $ to the identity functor $\id_{\mathfrak{W}_X}$.
At any point $p\in X$, we may find a symplectic local chart
$i\colon X\supset U \to T^*M$ around $p$, such that $\mathfrak{W}_X|_U$ is equivalent
to $ \astk{\mathcal{W}^{\sqrt v}_U{}}$ as WKB-algebroid. (Here we set $\mathcal{W}^{\sqrt v}_U =\opb i \mathcal{W}^{\sqrt v}_M$.)
We thus have a chain of equivalences of 2-groups
\begin{equation*}
\begin{split}
\stkAut[\operatorname{WKB}](\mathfrak{W}_X)^\times|_U
& \approx \stkAut[\operatorname{WKB}](\mathfrak{W}_X|_U)^\times\\
& \approx \stkAut[\operatorname{WKB}](\astk{(\mathcal{W}_U^{\sqrt v})})^\times\\
& \approx \left[ \mathcal{W}^{\sqrt v, *}_U \to[\operatorname{ad}] \shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_U) \right], \\
\end{split}
\end{equation*}
(the last one follows from~\eqref{WKB-aut}) and hence the functor $F$
restricts on $U$ to
$$
F|_U\colon k^*_U[1] \longrightarrow \left[\mathcal{W}^{\sqrt v, *}_U\to[\operatorname{ad}]
\shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_U) \right].
$$
By Lemma \ref{lemma:key2}, this is an equivalence of 2-groups so that the
functor $F$ is locally, and hence globally, an equivalence.
We thus get a chain of isomorphisms of pointed sets
\begin{equation*}
H^1(X;\stkAut[\operatorname{WKB}](\mathfrak{W}_X)^{\times}) \simeq H^1(X;k^*_X[1])
\simeq H^2(X;k^*_X),
\end{equation*}
where the latter follows by Proposition \ref{prop:hyper}.
\end{proof}
\begin{lemma}\label{lemma:key2}
Let $M$ be a complex manifold. Then there is an equivalence of 2-groups on
$T^*M$
$$
k^*_{T^*M}[1] \approx \left[\mathcal{W}_M^{\sqrt v, *} \to[\operatorname{ad}]
\shaut[\operatorname{WKB}](\mathcal{W}^{\sqrt v}_M) \right].
$$
\end{lemma}
\begin{proof}
This follows by a direct computation, using \eqref{eq:key}.
\end{proof}
\begin{corollary}
The set $\operatorname{WKB}({X})$ has an abelian group structure.
\end{corollary}
Note that, by much the same proof of Theorem~\ref{th:classification}, one gets
an isomorphism between the group $\operatorname{Pic}_{\operatorname{WKB}}(\mathfrak{W}_{X})$ of isomorphism
classes of autoequivalences\footnote{For a sheaf of rings $\mathcal{A}$, one usually
denotes by $\operatorname{Pic}(\mathcal{A})$ the group of isomorphism classes of invertible
$\mathcal{A}\tens[\mathcal{R}]\mathcal{A}^{\mathrm{op}}$-modules. This is consistent with our notation,
since by Morita theorem $\operatorname{Pic}(\mathcal{A})$ is isomorphic to the group of isomorphism
classes of linear autoequivalences of $\stkMod(\mathcal{A})$.} of $\mathfrak{W}_{X}$ as
WKB-algebroid and $H^1(X;k^*_{X})$.
\begin{remark}
If $X=T^*M$ for a complex manifold $M$, the coboundary map
$$
\delta\colon H^1(T^*M;\mathcal{W}^{\sqrt v, *}_M/k^*_{T^*M}) \to H^2(T^*M;k^*_{T^*M})
$$
associated to the exact sequence~\eqref{eq:key}, may be interpreted as the map
which sends the class $[\mathcal{A}]$ of a WKB-algebra to the class $[\mathcal{A}^+]$ of
the corresponding WKB-algebroid.
We refer to \cite{Deligne,BoutetdeMonvel2002} for similar constructions in the
framework of real manifolds.
\end{remark}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
{
"timestamp": "2005-12-21T18:37:22",
"yymm": "0503",
"arxiv_id": "math/0503400",
"language": "en",
"url": "https://arxiv.org/abs/math/0503400"
}
|
\section{Definitions and Notations}
We briefly recall some well known notions of $CR$ geometry that
will be used in the paper.
Let $N\subset{\mathbb C}} \def\l{\lambda^n$ be a smooth connected real submanifold, and let
$p\in N$. We denote by $T_p(N)$ the tangent space of $N$ at the
point $p$, and by $H_p(N)$ the holomorphic tangent space of $N$ at
the point $p$.
A ($2k+1$)-real submanifold $N\subset{\mathbb C}} \def\l{\lambda^n$, $k\geq1$, is said to
be a \textit{$CR$ submanifold} if ${\sf dim}_{\mathbb C}} \def\l{\lambda H_p(N)$ is constant
along $N$. When this is the case, $H(N)=\cup_p H_p(N)$ is a subbundle of the tangent bundle $T(N)$. If ${\sf dim}_{\mathbb C}} \def\l{\lambda H_p(N)$ is the greatest possible, i.e.\
${\sf dim}_{\mathbb C}} \def\l{\lambda H_p(N) = k$ for every $p$, $N$ is said to be
\textit{maximally complex}.
A $C^\infty$ function $f:N\to{\mathbb C}} \def\l{\lambda$ is said to be a
\textit{$CR$ function} if for a $C^\infty$ extension (and
hence for any) $\widetilde f: U\to {\mathbb C}} \def\l{\lambda$ ($U$ being a neighborhood
of $N$) we have
\begin{equation}\label{1}\left(\oli\partial\widetilde f\right)|_{H(N)}\ =\ 0.\end{equation}
In particular the restriction of a holomorphic function to a $CR$
submanifold is a $CR$ function. It is immediately seen that $f$ is
$CR$ if and only if
\begin{equation}
df\wedge(dz_1\wedge \ldots \wedge dz_n)|_N = 0.
\end{equation}
Similarly $N$ is maximally complex if and only if
$$
(dz_{j_1}\wedge \ldots \wedge dz_{j_{k+1}})|_N = 0,$$
for any $(j_1,\ldots,
j_{k+1})\in\left\{1,\ldots,n\right\}^{k+1}$.
Finally we observe that the boundary $M$ of a complex submanifold
$W$ with ${\sf dim}_{\mathbb C}} \def\l{\lambda W > 1$ is maximally complex. Indeed, for
any $p\in bW=M$, $T_p(bW)$ is a real hyperplane of $T_p(W)=H_p(W)$
and so is $J(T_p(bW))$. Hence $H_p(bW)=T_p(bW)\cap J(T_p(bW))$ is
of real codimension $2$ in $H_p(W)$.
If ${\sf dim}_{\mathbb C}} \def\l{\lambda W=1$ and $bW$ is compact then for any holomorphic
$(1,0)$-form $\omega$ we have
$$\int_{M}\omega\ =\ \iint_W
d\omega\ =\ \iint_W
\partial\omega\ =\ 0,$$
since $\partial\omega|_W \equiv 0$. This
condition for $M$ is called \textit{moments condition} (see
\cite{HL}).
By the same arguments, a ($2n-1$)-real submanifold of ${\mathbb C}} \def\l{\lambda^n$ is maximally complex.
\section{The Local and Semi Global Results}\label{local}
The aim of this section is to prove the local result. Given a
smooth real hypersurface $S$ in ${\mathbb C}} \def\l{\lambda^n$, we denote by $\mathcal L_p(S)$
the Levi form of $S$ at the point $p$. Let $0$ be a point of $M$. We
have the following inclusions of tangent spaces: $$ {\mathbb C}} \def\l{\lambda^n\
\supset\ T_0(S)\ \supset\ H_0(S)\ \supset\ H_0(M);$$ $$
\phantom{{\mathbb C}} \def\l{\lambda^n\ \supset}\ T_0(S)\ \supset\ T_0(M)\ \supset\
H_0(M).$$
\begin{lemma}\label{wk} Let $M$ be a maximally complex submanifold of a
smooth real hypersurface $S$, ${\sf dim}_{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma M=2m+1$, $m\geq1$, $0\in M$. Suppose
that $\mathcal L_0(S)$ has at least $n-m$ eigenvalues of the same
sign. Then $$H_0(S)\not\supset T_0(M).$$ \end{lemma}
\begin{proof} Should the
thesis fail we would have the following chain of inclusions $$ {\mathbb C}} \def\l{\lambda^n
\supset T_0(S) \supset H_0(S) \supset T \supset T_0(M) \supset
H_0(M),$$ where $T$ is the smallest complex space containing
$T_0(M)$ (since $M$ is maximally complex, ${\sf dim}_{\mathbb C}} \def\l{\lambda T=m+1$). Hence, we may choose in a neighborhood of $0$ local complex coordinates $z_k=x_k
+ i y_k$, $k=1,\ldots,m+1$, $w_l=u_l + i v_l$, $l=m+2,\ldots,n$,
in such a way that:
\begin{itemize}
\item $H_0(M) = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k)$, $k=1,\ldots,m$
\item $T_0(M) = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k, \partial/\partial} \def\oli{\overline x_{m+1})$, $k=1,\ldots,m$
\item $T = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k)$, $k=1,\ldots,m+1$
\item $H_0(S) = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k, \partial/\partial} \def\oli{\overline u_l, \partial/\partial} \def\oli{\overline v_l)$, $k=1,\ldots,m+1$,
$l=m+2,\ldots,n-1$, if $m+2\leq n-1$ \\ or
\item $H_0(S) = T$, if $m=n-2$;
\item $T_0(S) = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k, \partial/\partial} \def\oli{\overline u_l, \partial/\partial} \def\oli{\overline v_l, \partial/\partial} \def\oli{\overline u_n)$, $k=1,\ldots,m+1$,
$l=m+2,\ldots,n-1$, if $m+2\leq n-1$ \\ or
\item $T_0(S) = {\sf span} \ (\partial/\partial} \def\oli{\overline x_k, \partial/\partial} \def\oli{\overline y_k, \partial/\partial} \def\oli{\overline u_n)$ $k=1,\ldots,m+1$, if
$m=n-2$.
\end{itemize}
We denote by $z$ the first $m+1$ coordinates, by $\hat z$ the
first $m$, and by $\pi$ the projection on $T$; $\pi$ is obviously
a local embedding of $M$ near $0$, and we set $M_0 = \pi(M)$.\\
Locally at $0$, $S$ is a graph over its tangent space: $$S=\{v_n
= h(u_n,u_j,v_j,x_i,y_i)\}.$$ Observe that the Levi form of $h$
has $n-m$ eigenvalues of the same sign. In order to obtain a similar
description of $M$, we proceed as follows. First, we have $$M_0 =
\{ (\hat z, z_{m+1}): y_{m+1}=\varphi(\hat z,x_{m+1})\}.$$ Then,
we choose $f_j(\hat z,x_{m+1}) = f_j^1(\hat z,x_{m+1}) +
if_j^2(\hat z,x_{m+1})$ (where $f^1_j$ and $f^2_j$ are
real-valued) defined in a neighborhood of $M_0$ in $T$ in such a
way that $$M= \{w_{m+2} = f_{m+2}(\hat z,x_{m+1}), \ldots, w_{n} =
f_{n}(\hat z,x_{m+1}) \}.$$ Observe that the function
$(f_{m+2}(\hat z,x_{m+1}), \ldots, f_{n}(\hat z,x_{m+1}))$ is just
$\pi^{-1}|_{M_0}$, and since $M$ is maximally complex it has to be
a $CR$ map.
By hypothesis, the following equation holds in a neighborhood of
$0$: $$f_n^2(\hat z,x_{m+1}) = h\left(f_n^1(\hat z,x_{m+1}),
f_j^k(\hat z,x_{m+1}),\hat z,x_{m+1}\right).$$ After a computation
of the second derivatives, taking into account that all first
derivatives of $f_j^k$, of $h$ and of $\varphi$ vanish in the
origin, we obtain
$$ \frac{\partial^2 f_n^2}{\partial z_j\partial
\overline z_k}(0)\ =\ \frac{\partial^2 h}{\partial z_j\partial
\overline z_k}(0),$$
i.e.\ the Levi form of $h$ and $f_n^2$ coincide
in $H_0(M)$. By hypothesis $\mathcal L_0(h)$ is strictly positive
definite on a non-zero subspace of $H_0(M)$. We shall obtain a
contradiction by showing that $\mathcal L_0(f_n)$ (and hence $\mathcal L_0(f_n^2)$) vanishes on
$H_0(M)$. Let $\xi\in H_0(M)$. We may assume (up to unitary linear
transformation of coordinates of $H_0(M)$) that $\xi =\partial /
\partial z_1$.
Set $f\doteqdot f_n$. Then, since $f$ is a $CR$ function on $M_0$, we have:
$$\frac{\partial}{\partial \oli z_k}f(\hat z, x_{m+1}) = -\a(\hat
z, x_{m+1}) \frac{\partial}{\partial \oli z_k}\varphi(\hat z,
x_{m+1}),\ \ k=1,\ldots, m $$ and $$ \frac{\partial}{\partial \oli
z_{m+1}}f(\hat z, x_{m+1}) = -i\a(\hat z, x_{m+1}) + \a(\hat z,
x_{m+1}) \frac{\partial}{\partial x_{m+1}}\varphi(\hat z,
x_{m+1}),$$ where $\a(\hat z, x_{m+1})$ is a complex valued function. Differentiating and calculating in $0$ we obtain
\begin{equation} \label{prima}
\frac{\partial^2 f}{\partial z_1 \partial \oli{z_1}}(0) = \a(0) \frac{\partial^2 \varphi}{\partial z_1 \partial \oli{z}_1}(0),
\end{equation}
\begin{equation} \label{seconda}
0 = \frac{\partial f}{\partial x_{m+1}}(0) = i\a(0),
\end{equation}
i.e. $\a(0) = 0$. From (\ref{prima}) we deduce that $\partial^2 f / \partial z_1 \partial \oli z_1 (0) = 0$. Contradiction.
\end{proof}
\begin{lemma}\label{L3} Under the hypothesis of Lemma \ref{wk}, assume that $S$
is the boundary of an unbounded domain $\Omega\subset {\mathbb C}} \def\l{\lambda^n$, $0\in M$
and that the Levi form of $S$ has at least $n-m$ positive
eigenvalues. Then
\begin{enumerate}
\item[\emph{(i)}] there exists an open neighborhood $U$ of $0$ and an $(m+1)$-complex submanifold $W_0\subset U$
with boundary, such that $bW_0=M\cap U$;
\item[\emph{(ii)}] $W_0\subset\Omega\cap U$.
\end{enumerate}
\end{lemma} \begin{proof} To prove the first assertion, observe that to obtain $\mathcal L^M_0(\zeta_0,\oli \zeta_0)$
it suffices to choose a smooth local section $\zeta$ of $H_0(M)$
such that $\zeta(0) = \zeta_0$ and compute the
projection of the bracket $[\zeta,\oli\zeta](0)$ on the real part
of $T_0(M)$. By hypothesis, the intersection of the space
where $\mathcal L_0(S)$ is positive with $H_0(M)$ is non empty; take $\eta_0$ in this intersection. Then $\mathcal L_0^M(\eta_0, \oli\eta_0)\neq 0$. Suppose, by contradiction, that the bracket $[\eta,\oli\eta](0)$ lies in $H_0(M)$, i.e.\ its projection on the real part of the tangent of $M$ is zero. Then, if $\widetilde{\eta}$ is a local smooth extension of the field $\eta$ to $S$, we have $[\widetilde{\eta},\oli{\widetilde{\eta}}](0)= [\eta,\oli\eta](0)\in H_0(M)$. Since $H_0(M)\subset H_0(S)$, this would mean that the Levi form of $S$ in $0$ is zero in $\eta_0$.
Now, we project (generically) $M$ over a ${\mathbb C}} \def\l{\lambda^{m+1}$ in such a way that the
projection $\pi$ is a local embedding near $0$: since the
restriction of $\pi$ to $M$ is a $CR$ function, and since the Levi
form of $M$ has - by the arguments stated above - at least one
positive eigenvalue, it follows that the Levi form of $\pi(M)$ has
at least one positive eigenvalue. Thus, in order to obtain $W_0$,
it is sufficient to apply the Lewy extension theorem \cite{Le} to
the $CR$ function $\pi^{-1}|_M$.
As for the second statement, we observe that the projection by
$\pi$ of the normal vector of $S$ pointing towards $\Omega$ lies into
the domain of ${\mathbb C}} \def\l{\lambda^{m+1}$ where the above extension $W_0$ is
defined. Indeed, the extension result in \cite{Le} gives a holomorphic function in the connected component of (a neighborhood
of $0$ in) ${\mathbb C}} \def\l{\lambda^n \setminus \pi(M)$ for which $\mathcal
L_0(\pi(M))$ has a positive eigenvalue when $\pi(M)$ is oriented
as the boundary of this component. This is precisely the component
towards which the projection of the normal vector of $S$ points
when the orientations of $S$ and $M$ are chosen accordingly. This
fact, combined with Lemma \ref{wk} (which states that any
extension of $M$ must be transverse to $S$) implies that locally
$W_0\subset\Omega\cap U$.
\end{proof}
\begin{corol}[Semi global existence of $W$]\label{L4} Under the same hypothesis of Lemma~\ref{L3}, there exist an open tubular neighborhood $I$ of
$S$ in $\oli \Omega$ and an $(m+1)$-complex submanifold $W_0$ of
\ $\oli\Omega \cap I$, with boundary, such that $S\cap bW_0=M$. \end{corol}
\begin{proof} By Lemma~\ref{L3}, for each point $p\in M$, there exist a
neighborhood $U_p$ of $p$ and a complex manifold
$W_p\subset\oli\Omega\cap U_p$ bounded by $M$. We cover $M$ with
countable many such open sets $U_i$, and consider the union
$W_0=\cup_i W_i$. $W_0$ is contained in the union of the $U_i$'s,
hence we may restrict it to a tubular neighborhood $I_M$ of $M$.
It is easy to extend $I_M$ to a tubular neighborhood $I$ of $S$.
The fact that $W_{i}|_{U_{ij}}=W_{j}|_{U_{ij}}$ if $U_i\cap
U_j=U_{ij}\neq\emptyset$ immediately follows from the construction made in
Lemma~\ref{L3}, in view of the uniqueness of the holomorphic extension of
$CR$ functions. \end{proof}
\begin{ex}\rm\label{E1} Corollary~\ref{L4} could be restated by saying that
if a submanifold $M\subset S$ (${\sf dim}_{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma M \geq 3$) is locally
extendable at each point as a complex manifold, then (one side of) the extension
lies in $\Omega$. This is no longer true, in general, for curves, as
shown in ${\mathbb C}} \def\l{\lambda^n_{(z_1,\ldots,z_{n-1},w)}$, $z_k=x_k+iy_k$, $w=u+iv$, by the following case:
$$S\
=\ \left\{v=u^2+\sum_k\left|z_k\right|^2\right\}, \ \Omega\ =\
\left\{v > u^2+\sum_k\left|z_k\right|^2\right\},$$ $$M\ =\
\left\{y_1=0,\ v=x_1^2,\ u=0,\ z_2=\cdots=z_{n-1}=0\right\}$$ and
$$W\ =\ \left\{w=iz_1^2,\ z_2=\cdots=z_{n-1}=0\right\};$$ we have
that $S\cap W = M$ and $W \subset {\mathbb C}} \def\l{\lambda^n \setminus \Omega$. \end{ex}
\begin{rem}\rm Suppose that $S$ is strongly pseudoconvex and choose, in
${\mathbb C}} \def\l{\lambda^n_{(z_1,\ldots,z_n)}$, a local strogly plurisubharmonic equation $\rho$ for $S$:
$S=\{\rho = 0\}$. Consider the curve
$$\gamma = \{z_j =
\gamma_j(t),\ j=1,\ldots,n, \ t\in (-\e,\e)\}\subset
S.$$
Assume that $\gamma$ is real analytic, so that locally there
exists a complex extension $\widetilde \gamma \supset \gamma$.
Then one side of $\widetilde \gamma$ lies in $\Omega$ if and only if
\begin{equation}\label{curve}
\sum_j Re \frac{\partial \rho}{\partial z_j} \frac{\partial \gamma_j}{\partial t} \neq 0.
\end{equation}
Observe that condition (\ref{curve}), which depends only on $\gamma$ (when $S$ is given), is not satisfied in Example \ref{E1}. Sufficiency of (\ref{curve}) is true when $S$ is \emph{any} real hypersurface: indeed, from a geometric point of view, the condition is equivalent to the transversality of $T(\widetilde \gamma)$ and $H(S)$ (and hence $T(S)$). Pseudoconvexity is required to establish the necessity.
\end{rem}
\section{The Global Result}
In order to make the proof more transparent we first treat the
case when $\Omega$ is an unbounded convex domain with smooth
boundary $b\Omega$. In the next section we will prove the main
theorem in all its generality.
\begin{teorema}\label{MT} Let $M$ be a maximally complex
(connected) $(2m+1)$-real submanifold $(m \geq 1)$ of $b\Omega$. Assume
that $\Omega$ does not contain straight lines and $b\Omega=S$ satisfies the
conditions of Lemma \ref{wk}. Then there exists an $(m+1)$-complex
subvariety $W$ of $\Omega$, with isolated singularities, such that
$bW=M$. \end{teorema}
We observe that under the hypothesis of Theorem \ref{MT}, there
exists a complex strip in a tubular neighborhood with boundary $M$
(see Corollary \ref{L4}). Moreover, since $\Omega$ does not contain
straight lines, we can approximate uniformly from both sides $b\Omega$
by strictly convex domains, see \cite{PT}. It follows that we can
find a real hyperplane $L$ such that, for any translation $L'$ of
$L$, $L'\cap \oli \Omega$ is a compact set. We choose an exhaustive
sequence $L_k$ of such hyperplanes, and we set $\Omega_k$ as the
bounded connected component of $\Omega\setminus L_k$. Then,
approximating from inside, we can choose a strictly convex open
subset $\Omega_k'\subset \Omega$ such that $b\Omega_k' \cap \Omega_k\subset I$,
where $I$ is the tubular neighborhood of Corollary \ref{L4}. It is
easily seen, then, that we are in the situation of the following
\begin{propos}\label{P} Let $D\Subset
B\Subset{\mathbb C}} \def\l{\lambda^n$ ($n\geq4$) be two strictly convex domains. Let
$D_+=D\cap\left\{{\sf Re} \ z_n>0\right\}$, $B_+=B\cap\left\{{\sf
Re} \ z_n>0\right\}$. Then every $(m+1)$-complex subvariety
$(m\geq2)$ with isolated singularities, $A \subset B_+ \setminus
\oli{D}_+\doteqdot C_+$, is the restriction of a complex
subvariety $\widetilde{A}$ of $B_+$ with isolated singularities.
\end{propos}
We treat the cases $m\geq2$ and $m=1$ separately. Indeed all the
main ideas of the proof lie in the case $m\geq2$, while the case
$m=1$ simply adds technical difficulties.
\subsection{$M$ is of dimension at least $5$: $m\geq2$}
Before proving Proposition \ref{P}, we make some considerations
and we prove two lemmata that will be useful.
Let $\varphi$ be a strictly convex function\footnote{In the general case $\varphi$ will be a strongly plurisubharmonic function.} defined in a neighborhood
of $B$ such that $B=\left\{\varphi<0\right\}$. Fixing $\varepsilon>0$
small enough, $B'=\left\{\varphi<-\varepsilon\right\}$ is a strictly convex
domain of $B$ whose boundary $H$ intersects $A$ in a smooth
maximally complex submanifold $N$. A natural way to proceed is to slice $N$ with complex
hyperplanes, in order to apply Harvey-Lawson's theorem. Each slice
of $B'$ is strictly convex, hence strongly pseudoconvex, and so
the holomorphic chain we obtain is contained in $B'$. Thus the set
made up by collecting the chains is contained in $B'$. Analyticity
of this set is the hard part of the proof.
Because of Sard's lemma, for all $ z\in D_+$, there exist a vector
$v$ arbitrarily close to $\partial/\partial} \def\oli{\overline z_n$, and $k\in{\mathbb C}} \def\l{\lambda$ such that $z\in
v_k\doteqdot v^{\perp}+k$ and $A_k\doteqdot v_k\cap N$ is
transversal and compact, and thus smooth.
In a neighborhood of each fixed $z_0\in D_+$, the same vector $v$
realizes the transversality condition. Hence we should now fix our
attention to a neighborhood of the form
$\widehat{U}\doteqdot\bigcup_{k\in U}v_k\cap B_+$, where $v_{k_0}$ is the vector corresponding to $z_0$ and $U\subset{\mathbb C}} \def\l{\lambda$ a neighborhood of $k_0$.
Let
$\pi:\widehat{U}\to{\mathbb C}} \def\l{\lambda^{m}$ be a generic projection: we use
$(w',w)$ as holomorphic coordinates on
$v_{k_{0}}={\mathbb C}} \def\l{\lambda^m\times{\mathbb C}} \def\l{\lambda^{n-m-1}$ (and also for $k$ near to $k_0$).
Let $V_k= {\mathbb C}} \def\l{\lambda^m \setminus \pi(A_k)$, and $V=\cap_k V_k$.
Since $A_{k_{0}}$ has a local extension (given by $v_{k_{0}}\cap
A$), it is maximally complex and so, by Harvey-Lawson's theorem,
there is a holomorphic chain $\widetilde{A}_{k_{0}}$ with
$b\widetilde{A}_{k_{0}}=A_{k_{0}}$, which extends holomorphically
$A_{k_{0}}$.
Our goal is to show that $\widetilde A_U=\cup_k\widetilde A_k$ is
analytic in $\pi^{-1}(V)$. From this, it will follow that
$\widetilde A_U$ is an analytic subvariety of $\widehat{U}$, $\pi$
being a generic projection.
Following an idea of Zaitsev, for $k\in U$,
$w'\in{\mathbb C}} \def\l{\lambda^{m}\setminus\pi(A_k)$ and $\alpha\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon^{n-m-1}$, we
define $$ I^\alpha(w',k)\ \doteqdot\ \int_{(\eta',\eta)\in
A_k}\eta^\alpha\omega_{BM}(\eta'-w'), $$ $\omega_{BM}$ being the
Bochner-Martinelli kernel.
\begin{lemma}[Zaitsev] \label{Zaitsev} Let $F(w',k)$ be the multiple-valued
function which represents $\widetilde A_k$ on
${\mathbb C}} \def\l{\lambda^{m}\setminus\pi(A_k)$; then, if we denote by
$P^\alpha(F(w',k))$ the sum of the $\alpha^\emph{th}$ powers of the
values of $F(w',k)$, the following holds: $$ P^\alpha(F(w',k)) =
I^\alpha(w',k). $$ In particular, $F(w',k)$ is finite. \end{lemma} \begin{proof}
Let $V_0$ be the unbounded component of $V_k$ (where, of course,
$P^\alpha(F(w',k)) = 0$). It is easy to show, following \cite
{HL}, that on $V_0$ also $I^\alpha(F(w',k)) = 0$: in fact, if $w'$
is far enough from $\pi(A_k)$, then $\beta = \eta^\alpha
\omega_{BM}(\eta' - w')$ is a regular $(m,m-1)$-form on some Stein
neighborhood $O$ of $A_k$. So, since in $O$ there exists $\gamma$
such that $\oli\partial\gamma = \beta$, we may write in the
language of currents $$[A_k](\beta) =
[A_k]_{m,m-1}(\oli\partial\gamma) =
\oli\partial[A_k]_{m,m-1}(\gamma) = 0.$$
In fact, since $A_k$ is maximally complex, $[A_k]=[A_k]_{m,m-1} +
[A_k]_{m-1,m}$ and $\oli\partial [A_k]_{m,m-1} = 0$, see
\cite{HL}. Moreover, since $[A_k](\beta)$ is analytic in the
variable $w'$, $[A_k](\beta)=0$ for all $w'\in V_0$.
To conclude our proof, we just need to show that the \lq\lq
jumps\rq\rq\ of the functions $P^\alpha(F(w',k))$ and
$I^\alpha(w',k)$ across the regular part of the common boundary of
two components of $V_k$ are the same.
So, let $z'\in\pi(A_k)$ be a regular point in the common boundary
of $V_1$ and $V_2$. Locally in a neighborhood of $z'$, we can
write $\widetilde A_k$ as a finite union of graphs of holomorphic
functions, whose boundaries $A_k^i$ are either in $A_k$ or empty.
In the first case, the $A_k^i$ are $CR$ graphs over $\pi(A_k)$ in
the neighborhood of $z'$. We may thus consider the jump $j_i$ of
$P^\alpha(F(w',k))$ due to a single function. We remark that the
jump for a function $f$ is $j_i=f(z')^\alpha$. The total jump will
be the sum of them.
To deal with the jump of $I^\alpha(w',k)$ across $z'$, we split
the integration set in the sets $A_k^i$ (thus obtaining the
integrals $I_i^\alpha$) and $A_k\setminus\cup_i A_k^i$
($I_0^\alpha$). Thanks to Plemelj's formulas (see~\cite{HL}) the
jumps of $I_i^\alpha$ are precisely $j_i$. Moreover, since the
form $\eta^\alpha \omega_{BM}(\eta' - z')$ is $C^\infty$
in a neighborhood of $A_k\setminus\cup_i A_k^i$, the jump of
$I_0^\alpha$ is $0$. So $P^\alpha(F(w',k))=I^\alpha(w',k)$.
\end{proof}
\begin{rem}\rm Lemma \ref{Zaitsev} implies, in particular, that the functions
$P^\alpha(F(w',k))$ are continuous in $k$. Indeed, they are
represented as integrals of a fixed form over submanifolds $A_k$
which vary continuously with the parameter $k$.\end{rem}
The functions $P^\alpha(F(w',k))$ and the holomorphic chain
$\widetilde{A}_{k_{0}}$ uniquely determine each other and so,
proving that the union over $k$ of the $\widetilde{A}_{k}$ is an
analytic set is equivalent to proving that the functions
$P^\alpha(F(w',k))$ are holomorphic in the variable $k\in
U\subset{\mathbb C}} \def\l{\lambda$.
\begin{lemma} $P^\alpha(F(w',k))$ is holomorphic in the variable $k\in
U\subset{\mathbb C}} \def\l{\lambda$, for each $\alpha\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon^{n-m-1}$. \end{lemma} \begin{proof} The proof
is very similar to the one of Lewy's main lemma in~\cite{Le}. Let
us fix a point $\left(w',\underline{k}\right)$ such that $w'\notin
A_{\underline{k}}$ (this condition remains true for $k\in
B_\epsilon(\underline{k})$). Consider as domain of $P^\alpha(F)$
the set $\left\{w'\right\}\times B_\epsilon(\underline{k})$. In
view of Morera's theorem, we need to prove that for any simple
curve $\gamma\subset B_\epsilon(\underline{k})$, $$ \int_\gamma
P^\alpha(F(w',k))dk\ =\ 0. $$ Let $\Gamma\subset
B_\epsilon(\underline{k})$ be an open set such that
$b\Gamma=\gamma$. By $\gamma\ast A_k$ ($\Gamma\ast A_k$) we mean
the union of $A_k$ along $\gamma$ (along $\Gamma$). Note that
these sets are submanifolds of $N$ ($\Gamma\ast A_k$ is an open
subset) and $b(\Gamma\ast A_k)=\gamma\ast A_k$. By
Lemma~\ref{Zaitsev} and Stoke's theorem
\begin{eqnarray}
\nonumber\int_\gamma P^\alpha(F(w',k))dk\ &=& \int_\gamma I^\alpha(w',k)dk\ =\\
\nonumber&=&\ \int_\gamma\left(\int_{(\eta',\eta)\in A_k} \eta^\alpha\omega_{BM}(\eta'-w')\right)dk\ =\\
\nonumber&=&\ \iint_{\gamma\ast A_k}\eta^\alpha\omega_{BM}(\eta'-w')\wedge dk\ =\\
\nonumber&=&\ \iint_{\Gamma\ast A_k}d\left(\eta^\alpha\omega_{BM}(\eta'-w')\wedge dk\right)\ =\\
\nonumber&=&\ \iint_{\Gamma\ast A_k}d \eta^\alpha\wedge\omega_{BM}(\eta'-w')\wedge dk\ =\\
\nonumber&=& 0.
\end{eqnarray}
The last equality follows from the fact that since $\eta^\alpha$ is holomorphic, only holomorphic differentials appear in $d\eta^\alpha$. Since all the holomorphic differentials supported by $\Gamma\ast A_k$ already appear in $\omega_{BM}(\eta'-w')\wedge dk$, the integral is zero.
\end{proof}
We may now prove Proposition~\ref{P}.\vspace{0,3cm}
\begin{proof} \textbf{(Proposition~\ref{P}, $m\geq2$)} Up to this point we
have extended the complex manifold $A$ to an analytic set
$$\widetilde{A}_U\doteqdot A\cup\bigcup_{k\in
U}\widetilde{A}_k\subset V_U\doteqdot C_+\cup\bigcup_{k\in
U}\left(v_k\cap B_+\right).$$ The open sets $V_U$ are an open
covering of $B_+$.
Moreover the open sets
$\omega_U\doteqdot\bigcup_{k\in U}(v_k\cap B_+)$ are an open
covering of each compact set $K_\delta\doteqdot \overline
B'\cap\left\{{\sf Re}\, z_n\geq\delta\right\}$. Hence there
exist $\omega_1,\dots,\omega_l$ which cover $K_\delta$ and such
that $\omega_i\cap\omega_{i+1}\cap C_+\neq\emptyset$, for $
i=1,\dots,l-1$ and therefore there exists a countable open cover
$\left\{\omega_i\right\}_{i\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon}$ of $\overline
B'\cap B_+$ such that, for all $i\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon$,
$\omega_i\cap\omega_{i+1}\cap C_+\neq\emptyset$.
So we may
extend $A$ to $C_+\cup\omega_1$ by proceeding as above.
Suppose
now that we have extended $A$ to $C^i\doteqdot
C_+\cup\bigcup^{i}_{j=1}\omega_j$ with an analytic set $A_i$. On
the non-empty intersection $C^i\cap\omega_{i+1}\cap C_+$ $A_i$ and
the extension $\widetilde{A}_{i+1}$ of $A$ to
$C_+\cup\omega_{i+1}$ coincide (as they both coincide with $A$),
hence by analicity they coincide everywhere. Consequently we may
extend $A$ to $C^{i+1}$ by $A_{i+1}\doteqdot
A_i\cup\widetilde{A}_{i+1}$. It follows that, defining $$\widetilde{A}\
\doteqdot \ A\cup\bigcup_{j\in {\mathbb N}} \def\d {\delta} \def\e{\varepsilon}A_j,$$
$\widetilde{A}$ is the
desired extension of $A$ to $B_+$. In order to conclude the proof
we have to show that $\widetilde A$ has isolated singularities.
Let ${\sf Sing} \ (\widetilde A)\subset B'_+$ be the singular
locus of $\widetilde A$.
Recall that $\varphi$ is a strictly convex defining function for $B$. Let us consider the family
$$(\phi_\lambda\ =\ \lambda\varphi+(1-\lambda){\sf Re}\, z_n)_{\lambda\in[0,1]}$$
of strictly convex functions. For $\lambda$ near to $1$,
$\left\{\phi_\lambda=0\right\}$ does not intersect the singular
locus ${\sf Sing} \ (\widetilde A)$. Let $\oli \lambda$ be the
biggest value of $\lambda$ for which $\{\phi_\lambda=0\}\cap {\sf
Sing} \ (\widetilde A)\neq\emptyset$. Then $$\left\{\phi_{\oli
\lambda}<0\right\}\cap B_+\subset B_+$$ is a Stein domain in whose
closure the analytic set ${\sf Sing} \ (\widetilde A)$ is
contained, touching the boundary in a point of strict
convexity. So, by Kontinuit\"atsatz,
$$\{\phi_{\oli
\lambda}=0\}\cap {\sf Sing} \ (\widetilde A)$$
is a set of isolated
points in ${\sf Sing} \ (\widetilde A)$. By repeating the
argument, we conclude that ${\sf Sing} \ (\widetilde A)$ is made
up by isolated points.
\end{proof}
\begin{proof} \textbf{(Theorem~\ref{MT}, $m\geq2$)} Thanks to
Corollary~\ref{L4}, we have a regular submanifold $W_1$ of a
tubular neighborhood $I$, with boundary $M$.
Suppose $0\in M$. The real hyperplanes $H_k\doteqdot T_0(S)+k$, $k\in{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma$, intesect $S$ in a compact set. If the intersection is non-empty, $\oli\Omega$ is divided in two sets. Let $\Omega_k$ be the compact one. We can choose a sequence $H_{k_n}$ such that $\Omega_{k_n}$ is an exaustive sequence for $\oli\Omega$.
We apply proposition~\ref{P} with $B_+=\Omega_{k_n}$, $C_+=I\cap\Omega_{k_n}$, and $A=W_1\cap\Omega_{k_n}$, to obtain an extension of $W_1$ in $\Omega_{k_n}$. Since, by the identity principle, two such extensions coincide in $\Omega_{k_{\min\left\{n,m\right\}}}$, their union is the desired submanifold $W$.
\end{proof}
\subsection{$M$ is of dimension $3$: $m=1$}
We prove now the statement of Proposition~\ref{P} for $m=1$.
Our first step is to show that when we slice transversally $N$ with complex
hyperplanes, we obtain $1$-real submanifolds which satisfy the moments condition.
Again, we fix our attention to a neighborhood of the form
$\widehat{U}\doteqdot\bigcup_{k\in U}v_k\cap B_+$. In $\widehat{U}$,
with coordinates $w_1,\ldots,w_{n-1},k$, we choose an arbitrary
holomorphic $(1,0)$-form which is constant with respect to $k$.
\begin{lemma}\label{omeol}
The function
$$\Phi_\omega(k)\ =\ \int_{A_k}\omega$$
is holomorphic in $U$.
\end{lemma}
\begin{proof}
We use again Morera's theorem. We need to prove that for any simple curve $\gamma\subset U$, $\gamma=b\Gamma$,
$$
\int_\gamma \Phi_\omega(k)dk\ =\ 0.
$$
Applying Stoke's theorem, we have
\begin{eqnarray}
\nonumber\int_\gamma \Phi_\omega(k)dk\ &=& \int_\gamma\left(\int_{A_k}\omega\right)dk\ =\\
\nonumber &=& \iint_{\gamma\ast A_k}\omega\wedge dk\ =\\
\nonumber &=& \iint_{\Gamma\ast A_k}d(\omega\wedge dk)\ =\\
\nonumber &=& \iint_{\Gamma\ast A_k}\partial\omega\wedge dk\ =\\
\nonumber &=& \ 0.
\end{eqnarray}
The last equality is due to the fact that $\Gamma\ast A_k\subset N$ is maximally complex and thus supports only $(2,1)$ and $(1,2)$-forms, while $\partial\omega\wedge dk$ is a $(3,0)$-form.
\end{proof}
Now we can prove Proposition~\ref{P} and Theorem~\ref{MT} also when $m=1$.
We can find a countable covering of $B_+$ made of open subsets $\omega_i=\widehat{U}_i\cap B_+$ in such a way that:
\begin{enumerate}
\item $\omega_0\subset C_+$;
\item if
$$B_l\ =\ \bigcup_{i=1}^l\omega_i,$$
then $\omega_{l+1}\cap B_l\supset v_{l+1}\cap B_+$, where $v_{l+1}$ is a complex hyperplane in $\widehat{U}_{l+1}$.
\end{enumerate}
Now, suppose we have already found $\widetilde A_l$ that extends
$A$ on $B_l$ (observe that in $B_0=\omega_0$, $\widetilde A_0 =A$).
To conclude the proof we have to find $\widetilde A_{l+1}$
extending $A$ on $B_{l+1}$.
Each slice of $N$ in $B_l$ is maximally complex, and so are $v_{l+1}\cap N$ and $v_\epsilon\cap N$, for $v_\epsilon\subset\omega_{l+1}$ sufficiently near to $v_{l+1}$ (because they are in $B_l$ as well).
Now we use Lemma~\ref{omeol} with $\widehat U=\widehat U_{l+1}$. What
we have just observed implies that, for all holomorphic
$(1,0)$-form $\eta$, $\Phi_\eta(k)$ vanishes in an open subset of
$U$ and so is identically zero on $U$. This implies that all
slices in $\omega_{l+1}$ are maximally complex. Again we may apply
Harvey-Lawson's theorem slice by slice and conclude by the methods
of Proposition~\ref{P}.
\subsection{$M$ is of dimension $1$: $m=0$}
We have already observed that if $M$ is one-dimensional the local extension inside $\Omega$ may not exist (see Example~\ref{E1}). Even though there is a local strip in which we have an extension, the methods used to prove Proposition~\ref{P} do not work, since the transversal slices $M$ are either empty or isolated points. Indeed, as the following example shows, that extension result does not hold for $m=0$.
\begin{ex}\rm \label{E2}Using the notation of Proposition~\ref{P}, in
${\mathbb C}} \def\l{\lambda^2$ let $B$ and $D$ be the balls
$$B=\left\{|z_1|^2+|z_2|^2<c\right\},\ \ \
D=\left\{|z_1|^2+|z_2|^2<\e\right\},\ \ \ c>\e>2.$$
Consider the connected irreducible analytic set of codimension one
$$A=\{(z_1,z_2)\in B_+\ :\ z_1z_2=1\}$$ and its restriction $A_C$
to $C_+$. If $A_C$ has two connected components, $A_1$ and $A_2$,
when we try to extend $A_1$ (analytic set of codimension one on
$C_+$) to $B_+$, its restriction to $C_+$ will contain also $A_2$.
So $A_1$ is an analytic set of codimension one on $C_+$ that does
not extend on $B_+$.
So, let us prove that $A_C$ has indeed two connected components. A point of $A$ (of $A_C$) can be written as $z_1=\rho e^{i\theta}$, $z_2=\frac{1}{\rho} e^{-i\theta}$, with $\rho\in{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma^+$ and $\theta\in\left(-\frac\pi2,\frac\pi2\right)$. Hence, points in $A_C$ satisfy
$$
2<\varepsilon<\rho^2+\frac1{\rho^2}<c\ \Rightarrow\ 2<\sqrt{\varepsilon+2}<\rho+\frac1\rho<\sqrt{c+2} .
$$
Since $f(\rho)=\rho+1/\rho$ is monotone decreasing up to $\rho=1$ (where $f(1)=2$), and then monotone increasing, there exist $a$ and $b$ such that the inequalities are satisfied when $a<\rho<b<1$, or when $1<1/b<\rho<1/a$. $A_C$ is thus the union of the two disjoint open sets
$$
\xymatrix{A_1=\left\{ \left(\rho e^{i\theta},\frac1\rho e^{-i\theta}\right)\in {\mathbb C}} \def\l{\lambda^2\ \Big|\ a<\rho<b,\ -\frac\pi2<\theta<\frac\pi2\right\};\\
A_2=\left\{ \left(\rho e^{i\theta},\frac1\rho e^{-i\theta}\right)\in {\mathbb C}} \def\l{\lambda^2\ \Big|\ a<\frac1\rho<b,\ -\frac\pi2<\theta<\frac\pi2\right\}.}$$
\end{ex}
\section{Extension to Pseudoconvex Domains}
We may now prove
\begin{teorema} Let $\Omega$ be an unbounded domain in ${\mathbb C}} \def\l{\lambda^n$ $(n\geq
3)$ with smooth boundary $b\Omega$ and $M$ be a maximally complex
closed $(2m+1)$-real submanifold $(m \geq 1)$ of $b\Omega$. Assume
that
\begin{enumerate}
\item [\emph{(i)}] $b\Omega$ is weakly pseudoconvex and the Levi form $\mathcal
L(b\Omega)$ has at least $n-m$ positive eigenvalues at every point of
$M$;
\item[\emph{(ii)}] $M$ satisfies condition $(\star)$.
\end{enumerate} Then there exists a unique
$(m+1)$-complex analytic subvariety $W$ of $\Omega$, such that $bW =
M$. Moreover the singular
locus of $W$ is discrete and the closure of $W$ in $\oli \Omega
\setminus {\sf Sing} \ W$ is a smooth submanifold with boundary
$M$. \end{teorema}
\begin{proof}
Assume, for the moment, that condition ($\star$) is
replaced by the stronger condition
\begin{itemize}\item[] ${\oli \Omega}^\infty \cap \Sigma_0 =
\emptyset$ where ${\oli \Omega^\infty}$ denotes the projective closure
of $\Omega$.\end{itemize}
The only thing we have to show in order to conclude the proof (by
using the methods of the previous section) is that, up to a
holomorphic change of coordinates and a holomorphic embedding $V:{\mathbb C}} \def\l{\lambda^n\rightarrow {\mathbb C}} \def\l{\lambda^N$, we can choose a sequence of real hyperplanes
$H_k\subset{\mathbb C}} \def\l{\lambda^N$, $k\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon$, which are exhaustive in the following
sense:
\begin{itemize}
\item[1.] $H_k\cap V(S)$ is compact, for all $k\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon$;
\item[2.] one of the two halfspaces in which $H_k$ divides ${\mathbb C}} \def\l{\lambda^N$, say $H_k^+$, intersects $V(\Omega)$ in a relatively compact set;
\item[3.] $\cup_k (H_k^+\cap V(\Omega))=V(\Omega)$.
\end{itemize}
The arguments of Proposition \ref{P}, indeed ---excluded the proof
that the singularities are isolated--- depend only on the fact that
we can cut $M$ by complex hyperplanes, obtaining compact maximally
complex submanifolds. Once we have found $W'\subset V({\mathbb C}} \def\l{\lambda^n)$ ($W'$
is in fact contained in $V({\mathbb C}} \def\l{\lambda^n)$ by analytic continuation, since
it has to coincide with the strip in a neighborhood of $M$), we
set $W= V^{-1}(W')$. Observe that the hypersurfaces $V^{-1}(H_k)$
are an exhaustive sequence for $\Omega$; let $\Omega_k$ be correspondent
sequence of relatively compact subsets. Since $\Omega$ is a domain of
holomorphy, for each $k$ we can choose a strongly pseudoconvex
open subset $\Omega_k'\subset \Omega$ such that $b\Omega_k' \cap \Omega_k \subset
I$, where $I$ is the tubular neighborhood found in Corollary \ref{L4}. So, in each $\Omega_k$ we can suppose that we deal with a
strongly pseudoconvex open set, and thus the proof of the fact
that the singularities are isolated is the same as in Proposition
\ref{P}.
Following~\cite{L2} we divide the proof in two steps.
\emph{Step 1}. $P$ linear. We consider
$\oli\Omega\subset{\mathbb C}} \def\l{\lambda{\mathbb P}^n={\mathbb C}} \def\l{\lambda^n\cup{\mathbb C}} \def\l{\lambda{\mathbb P}^{n-1}_\infty$, which is disjoint
from $\Sigma_0=\left\{P=0\right\}$. So we can consider new coordinates
of ${\mathbb C}} \def\l{\lambda{\mathbb P}^n$ in such a way that $\Sigma_0$ is the ${\mathbb C}} \def\l{\lambda{\mathbb P}^{n-1}$ at
infinity. Now $\Omega$ is a relatively compact open set of
$({\mathbb C}} \def\l{\lambda^n)'={\mathbb C}} \def\l{\lambda{\mathbb P}^n\setminus \Sigma_0$, and
$H_\infty={\mathbb C}} \def\l{\lambda{\mathbb P}^{n-1}_\infty\cap({\mathbb C}} \def\l{\lambda^n)'$ is a complex hyperplane
containing the boundary of $S$. Let $H^{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma_\infty\supset H_\infty$
be a real hyperplane. The intersection between $S$ and a
translated of $H^{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma_\infty$ is either empty or compact. For all
$z\in\Omega$, there exist a real hyperplane $H^{\mathbb R}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma_\infty\not\ni z$,
intersecting $\Omega$, and a small translated $H_{\e_z}$ such that
$z\in H_{\e_z}^+$. Since $\Omega=\cup_z (H_{\e_z}^+\cap\Omega)$, and $\Omega$
is a countable union of compact sets, we may choose an exhaustive
sequence $H_k$.
\emph{Step 2}. $P$ generic. We use the Veronese map $v$ to embed
${\mathbb C}} \def\l{\lambda{\mathbb P}^n$ in a suitable ${\mathbb C}} \def\l{\lambda{\mathbb P}^N$ in such a way that $v(\Sigma_0)=L_0\cap
v({\mathbb C}} \def\l{\lambda{\mathbb P}^n)$, where $L_0$ is a linear subspace. The Veronese map $v$
is defined as follows: let $d$ be the degree of $P$, and let $$N\
=\ {{n+d}\choose{d}}-1. $$ Then $v$ is defined by $$v(z)\ =\
v[z_0:\ldots:z_n]\ =\ [\ldots:w_I:\ldots]_{|I|=d},$$ where
$w_I=z^I$. If $P=\sum_{|I|=d}\alpha_I z^I$, then $v(\Sigma_0)=L_0\cap
v({\mathbb C}} \def\l{\lambda{\mathbb P}^n)$, where $$L_0=\left\{\sum_{|I|=d} \alpha_I w_I = 0
\right\}.$$ Again we can change the coordinates so that $L_0$ is
the ${\mathbb C}} \def\l{\lambda{\mathbb P}^{N-1}$ at infinity. We may now find the exhaustive
sequence $H_k$ as in Step 1.
This achieves the proof in the case when ${\oli\Omega}^\infty \cap \Sigma_0
= \emptyset$. The general case is now an easy consequence.
Indeed, since ${\mathbb C}} \def\l{\lambda{\mathbb P}^n\setminus \Sigma_0$ is Stein, there is a strictly
plurisubharmonic exhaustion function $\psi$. The sets $$\Omega_c\ =\
\left\{\psi<c\right\}$$ are an exhaustive strongly pseudoconvex
family for ${\mathbb C}} \def\l{\lambda{\mathbb P}^n\setminus \Sigma_0$. Thus in view of ($\star$) there exists $\oli c$ such
that $\oli M\subset \Omega_{\oli c}$. $\Omega'\doteqdot\Omega\cap\Omega_{\oli c}$, up to a
regularization of the boundary, is a strongly pseudoconvex open
set verifying ($\star$) in whose boundary lies $M$, and thus $M$
can be extended thanks to what has already been proved.
\end{proof}
\section*{Acknowledgments}
This research was partially supported by the MIUR project \lq\lq Geometric properties of real and complex manifolds\rq\rq.
We wish to thank Giuseppe Tomassini, whose kind help made this work possible.
Useful remarks by the referee helped us to make clearer proofs in the first part of the article and to correct various misprints.
|
{
"timestamp": "2006-07-28T17:00:11",
"yymm": "0503",
"arxiv_id": "math/0503430",
"language": "en",
"url": "https://arxiv.org/abs/math/0503430"
}
|
\section{Collecting the relevant $K$- and $L$-theory.}
\setcounter{altel}{0}
\pagenumbering{arabic}
The material in this first section related to quadratic forms
and L-groups has primarily been extracted from \cite{Giffen;k2} and
\cite{Wall;lfound}, while the facts concerning the algebraic $K$-groups
$K_0$, $K_1$ and $K_2$ have been taken from \cite{Bass} and \cite{Milnor}.
First of all we need the notion of ring with anti-structure.
The word ring will always mean associative ring with identity, written 1.
\begin{defi}
An anti-automorphism $\alpha$ of a ring $R$ is a ring isomorphism
$\alpha\colon R\rightarrow R^\circ$, where $R^\circ$ denotes the opposite ring of
$R$.
A ring with anti-structure $(R,\alpha,u)$,
consists of a ring $R$,
equipped with an anti-automorphism
$\alpha$ of $R$
and a unit $u\in R$ such that
$\alpha(u)u=1$ and $\alpha^2(r)=uru^{-1}$ for every $r\in R$.
\end{defi}
\begin{nitel}{Remark}
Let $(R,\alpha,u)$ be a ring with anti-structure.
\begin{itemize}
\item[$\cdot$]
If $u$ is central in $R$, then $\alpha$ is an anti-involution,
i.e. an anti-automorphism of order at most 2.
\item[$\cdot$]
If $\alpha$ is the identity,
then $R$ must be commutative and $u^2=1$.
The converse is not necessarily true.
\end{itemize}
\end{nitel}
We give a few examples of rings with anti-structure including the most
important ones.
\begin{itemize}
\item[$\cdot$]
$R$ commutative, $\alpha$ the identity, $u=\pm 1$.
\end{itemize}
Let $(R,\alpha,u)$ be a ring with anti-structure.
Then the anti-structure on $R$ can be extended to
\begin{itemize}
\item[$\cdot$]
the group algebra $R[G]$, for every group $G$, by the formula
$$\sum r_ig_i \mapsto \sum \alpha(r_i)g_i^{-1}.$$
\item[$\cdot$]
the ring $M_n(R)$ of $(n\times n)$-matrices over $R$ by
$$A\mapsto A^\alpha,$$
where $(A^\alpha)_{ij}:= \alpha(A_{ji})$.
Thus $A^\alpha$ is the conjugate transpose of $A$.
\item[$\cdot$]
the polynomial ring in one variable $R[T]$ by
$$\sum r_iT^i\mapsto \sum \alpha(r_i)(1-T)^i.$$
\item[$\cdot$]
the polynomial ring in one variable $R[T]$ by
$$\sum r_iT^i\mapsto \sum \alpha(r_i)(-T)^i.$$
\end{itemize}
\begin{defi}\label{defda}
Denote by ${\cal P}(R)$ the category of finitely generated projective
right $R$-modules and $R$-homomorphisms.
The anti-automorphism $\alpha$ enables us to define
a contravariant functor $D_\alpha\colon {\cal P}(R)\rightarrow {\cal P}(R)$
as follows.\hfill\break
For every $P\in \mathop{\rm Obj}\nolimits\,{\cal P}(R)$ we define \hfill\break
$D_\alpha P:= \mathop{\rm Hom}\nolimits_R(P,R)$
equipped with a right $R$-module structure by\hfill\break
$(gr)(p):=\alpha^{-1}(r)g(p)$,
for every $g\in \mathop{\rm Hom}\nolimits_R(P,R)$, $p\in P$ and $r\in R$. \hfill\break
For every $f\in \mathop{\rm Hom}\nolimits_R(P,Q)$ we define\hfill\break
$D_\alpha f\in \mathop{\rm Hom}\nolimits_R(D_\alpha Q,D_\alpha P)$ by
$(D_\alpha f)(h):= h\lower1.0ex\hbox{$\mathchar"2017$} f$ for all $h\in D_\alpha Q $.
\end{defi}
\begin{punt}\label{obsfree}
We make the following observations.
\begin{enumerate}
\item
If $M$ is free with basis $e_1,\ldots,e_m$, then $D_\alpha M$
is free with basis $e_1^*,\ldots,e_m^*$.
Here $e_i^*\in D_\alpha M$ is determined by
$e_i^*(e_j)=\delta_{ij}$ (Kronecker delta).
One calls $e_1^*,\ldots,e_m^*$ the basis dual to
$e_1,\ldots,e_m$.
\item
If $M$ is free with basis $e_1,\ldots,e_m$,
$N$ is free with basis $f_1,\ldots,f_n$ and
$\phi\in \mathop{\rm Hom}\nolimits_R(M,N)$ has $(n\times m)$-matrix
$A$ with respect to these bases,
then $A^\alpha$ is the matrix of $D_\alpha\phi$
with respect to the dual bases.
Just as in the case of square matrices $(A^\alpha)_{ij}=\alpha(A_{ji})$
Note that $(AB)^\alpha=B^\alpha A^\alpha. $
\item
Suppose $e_1,\ldots,e_m$ and $f_1,\ldots,f_m$ are
both bases of $M$ and $X$ is the base-change matrix.
If $A$ is the matrix of $\phi\in \mathop{\rm Hom}\nolimits_R(M,D_\alpha M)$
with respect to $e_1,\ldots,e_m$ and its dual,
then $X^\alpha AX$ is the matrix of $\phi$ with respect to
$f_1,\ldots,f_m$ and its dual.
\end{enumerate}
\end{punt}
\begin{lemma}\label{lemmaeta} \cite[section 1]{Giffen;k2}
The map $\eta_{\alpha,u}\colon 1_{{\cal P}(R)}\rightarrow D_\alpha^2$ defined by \hfill\break
$(\eta_{\alpha,u}P)(p)(g):= u^{-1}\alpha(g(p))$
for every $P\in \mathop{\rm Obj}\nolimits\,{\cal P}(R)$, $p\in P $ and $g\in D_\alpha P $
is a natural equivalence.
\end{lemma}
\begin{proof}
Although the proof is rather straightforward we give some
of the arguments because they might be instructive.
\begin{itemize}
\item[$\cdot$]
$(\eta_{\alpha,u}P)(p)\in D_\alpha^2 P $:
for every $r\in R$ we have
\begin{eqnarray*}
(\eta_{\alpha,u}P)(p)(gr)&=&u^{-1}\alpha(gr(p))\\
&=&u^{-1}\alpha(\alpha^{-1}(r)g(p))\\
&=&u^{-1}\alpha(g(p))r\\
&=&(\eta_{\alpha,u}P)(p)(g)r
\end{eqnarray*}
\item[$\cdot$]
$\eta_{\alpha,u}P\in \mathop{\rm Hom}\nolimits_R(P,D_\alpha^2 P)$:
for every $r\in R$ we have
\begin{eqnarray*}
(\eta_{\alpha,u}P)(pr)(g)&=&u^{-1}\alpha(g(pr))\\
&=&u^{-1}\alpha(g(p)r)\\
&=&u^{-1}\alpha(r)\alpha(g(p))\\
&=&\alpha^{-1}(r)u^{-1}\alpha(g(p))\\
&=&\alpha^{-1}(r)(\eta_{\alpha,u}P)(p)(g)\\
&=&((\eta_{\alpha,u}P)(p)r)(g)
\end{eqnarray*}
\item[$\cdot$]
$\eta_{\alpha,u}$ is natural: for every $\phi\in \mathop{\rm Hom}\nolimits_R(P,Q)$ and
$ h\in D_\alpha Q $ the diagram
$$\diagram{
P&{\buildrel \eta_{\alpha,u}P \over {\hbox to 25pt{\rightarrowfill}}}&
D_\alpha^2P\cr
\mapdown{f}&&\mapdown{D_\alpha^2f}\cr
Q&{\buildrel \eta_{\alpha,u}Q \over {\hbox to 25pt{\rightarrowfill}}}
&D_\alpha^2Q\cr}$$
commutes since
\begin{eqnarray*}
(D_\alpha^2 f)((\eta_{\alpha,u}P)(p))(h)&=&(\eta_{\alpha,u}P)(p)(D_\alpha
f(h))\\
&=&u^{-1}\alpha(h(f(p)))\\
&=&(\eta_{\alpha,u}Q)(f(p))(h)
\end{eqnarray*}
\item[$\cdot$]
$\eta_{\alpha,u}P$ is an isomorphism: there exists a canonical
isomorphism \hfill\break
$ D_{\alpha}(P\oplus Q)\cong D_\alpha P\oplus D_\alpha Q $,
so we may assume that $P$ is free with basis $e_1,\ldots,e_m$ say.
From the definition of $\eta$ we deduce
$(\eta_{\alpha,u}P)(e_i)=e_i^{**}u$.
\end{itemize}
The rest is clear.
\end{proof}
\begin{cor}{}
If $M$ is free with basis $e_1,\ldots,e_m$,
then $\eta_{\alpha,u}(M)\colon M\rightarrow D_\alpha^2 M$ has matrix $uI_m$
with respect to $e_1,\ldots,e_m$ and $e_1^{**},\ldots,e_m^{**}$.
\end{cor}
\begin{nota}
From now on we write
$P^\alpha $ instead of $D_\alpha P $ and $f^\alpha $ instead
of $D_\alpha f$.
\end{nota}
\begin{prop}\label{propadju}
The map $T_{\alpha,u}=T_{\alpha,u}(P,Q)\colon
\mathop{\rm Hom}\nolimits_R(Q,P^\alpha)\rightarrow \mathop{\rm Hom}\nolimits_R(P,Q^\alpha)$
defined by
$$T_{\alpha,u}(f):= f^\alpha\lower1.0ex\hbox{$\mathchar"2017$}\eta_{\alpha,u}P$$
is a natural isomorphism and
$T_{\alpha,u}(P,Q)\lower1.0ex\hbox{$\mathchar"2017$} T_{\alpha,u}(Q,P)=1_{\mathop{\rm Hom}\nolimits_R(P,Q^\alpha)}$.
In other words $T_{\alpha,u}$ defines a self-adjunction of
the functor $D_\alpha$.
\end{prop}
\begin{proof}
As in \cite[Proposition 1.2]{Giffen;k2}
\end{proof}
\begin{lemma}
If $M$ is free with basis $e_1,\ldots,e_m$ and
$\phi\in \mathop{\rm Hom}\nolimits_R(M,M^\alpha)$ has matrix $A$ with respect to this
basis and its dual, then $T_{\alpha,u}(\phi)$
has matrix $A^\alpha u$ with respect to the same bases.
\end{lemma}
\begin{proof}
Immediate by the corollary to definition~\ref{lemmaeta} and the second
observation of~\ref{obsfree}.
\end{proof}
We are now in a position to introduce the notion of quadratic module.
\begin{defi}\label{defnonsing}
In the case that $P=Q$ in proposition~\ref{propadju} we obtain a group
endomorphism $T_{\alpha,u}\colon \mathop{\rm Hom}\nolimits_R(P,P^\alpha)\longrightarrow \mathop{\rm Hom}\nolimits_R(P,P^\alpha)$
satisfying $T_{\alpha,u}^2=1$.\hfill\break
A quadratic, to be precise $(\alpha,u)$-quadratic, $R$-module
is a pair $(P,[\phi])$
consisting of a module $ P\in \mathop{\rm Obj}\nolimits\,{\cal P}(R) $ and the class
$[\phi]\in \mathop{\rm Coker}\nolimits(1-T_{\alpha,u})$ of
an element $\phi\in \mathop{\rm Hom}\nolimits_R(P,P^\alpha)$.\hfill\break
The quadratic module $(P,[\phi])$ is called non-singular
if the image $b_{[\phi]}$ of $[\phi]$ under the
`bilinearization-map' $ b\colon \mathop{\rm Coker}\nolimits(1-T_{\alpha,u})\rightarrow
\mathop{\rm Ker}\nolimits(1-T_{\alpha,u})$, induced by the homomorphism
$1+T_{\alpha,u}\colon \mathop{\rm Hom}\nolimits_R(P,P^\alpha)\rightarrow \mathop{\rm Hom}\nolimits_R(P,P^\alpha)$,
is an isomorphism.
\end{defi}
\begin{nitel}{Remark}
\begin{itemize}
\item[$\cdot$]
If $2$ is invertible in $R$, then $b$ is an isomorphism,
with inverse determined by
$\phi\mapsto [\frac{1}{2}\phi].$
Thus there is a 1-1 correspondence between non-singular quadratic forms
and symmetric non-singular bilinear forms,
i.e. elements of ${\rm Iso}(P,P^\alpha)\cap\mathop{\rm Ker}\nolimits(1-T_{\alpha,u})$.
\item[$\cdot$]
In the literature one denotes by $\mathop{\rm Sesq}\nolimits(P)$ the additive group
of sesquilinear forms on $P$ i.e. biadditive maps
$\phi\colon P\times P \longrightarrow R $ satisfying
$\phi(p_1r_1,p_2r_2)=\alpha^{-1}(r_1)\phi(p_1,p_2)r_2$
for every $p_1,p_2\in P$ and $r_1,r_2\in R$.
In the case that $R$ is commutative and $\alpha$ is the identity,
$\mathop{\rm Sesq}\nolimits(P)$ is
the group of $R$-bilinear maps.
There is a bijective correspondence $\mathop{\rm Sesq}\nolimits(P)\longleftrightarrow
\mathop{\rm Hom}\nolimits_R(P,P^\alpha)$ by associating to an element $\phi\in \mathop{\rm Sesq}\nolimits(P)$
the map $f\in \mathop{\rm Hom}\nolimits_R(P,P^\alpha)$ defined by
$f(p_1)(p_2):= \phi(p_1,p_2)$ for every $p_1,p_2\in P$.
\end{itemize}
\end{nitel}
We proceed to define the various categories of quadratic modules.
Along the way we shall briefly recall the relevant definitions
and facts from algebraic K-theory.\hfill\break
The following categories and functors will all be `categories
with product' as in \cite[Ch.VII, \S1]{Bass}.
\begin{defi}\label{deffunctors}
\begin{itemize}
\item
Let $Q(R,\alpha,u)$ denote the category with \hfill\break
objects: non-singular quadratic (right) $R$-modules,\hfill\break
morphisms: $(P,[\phi])\rightarrow(Q,[\psi])$ are the isomorphisms
$f\colon P\rightarrow Q$ satisfying
$[f^\alpha\psi f]=[\phi]$,\hfill\break
product: $$(P,[\phi])\perp(Q,[\psi]):=
(P\oplus Q, [(\pi_P)^\alpha\phi\pi_P+(\pi_Q)^\alpha\psi\pi_Q]),$$
where $\pi_P\colon P\oplus Q\rightarrow P$ and $\pi_Q\colon P\oplus Q\rightarrow Q$
are the natural projections.
\item
Let $\overline{{\cal P}(R)}$ denote the category with\hfill\break
objects: objects of ${\cal P}(R)$,\hfill\break
morphisms: isomorphisms of ${\cal P}(R)$,\hfill\break
product: product of ${\cal P}(R)$.
\item
Now on one hand we have the forgetful functor
$F\colonQ(R,\alpha,u)\rightarrow\overline{{\cal P}(R)}$, which is of course product
preserving. While on the other hand there is the
so-called hyperbolic functor
$H\colon\overline{{\cal P}(R)}\raQ(R,\alpha,u)$
defined by
$$H(P):=(P\oplus P^\alpha,[\upsilon]),
\quad H(f):= f\oplus(f^\alpha)^{-1},$$ where
$\upsilon\colon P\oplus P^\alpha\rightarrow(P\oplus P^\alpha)^\alpha$
is determined by
$(\upsilon(p,g))(p',g'):= g(p')$.
$H$ is product preserving as well.
The objects $H(P)$ are called hyperbolic.
\item
A product preserving functor $G\colon {\cal C}\rightarrow {\cal D}$ is called
cofinal if for each object $A$ of ${\cal D}$ there exist objects
$B$ of ${\cal D}$ and $C$ of ${\cal C}$,
such that $A\perp B\cong G(C)$.\hfill\break
A subcategory ${\cal C}$ of a category ${\cal D}$ is called cofinal if the
inclusion functor is cofinal.
\end{itemize}
\end{defi}
\begin{lemma} \cite[theorem 3]{Wall;phil}\label{lemmahcof}.
For every $(P,[\phi])\in \mathop{\rm Obj}\nolimits\,Q(R,\alpha,u)$ there exists an isomorphism
$(P,[\phi])\perp (P,-[\phi])\cong H(P)$.
Consequently $H$ is cofinal.
\end{lemma}
\begin{proof}
It is not hard to verify that the morphism
$\xi\colon P\oplus P\rightarrow P\oplus P^\alpha$
defined by
$\xi(p_1,p_2):=
(p_1-b_{[\phi]}^{-1}(\phi(p_1-p_2)),b_{[\phi]}(p_1-p_2))$
does the job.
We refer to {\em loc. cit.} for a detailed proof.
\end{proof}
\begin{defi}\label{defk1}
As usual $\mathop{\rm GL}\nolimits(R)$ denotes the direct limit of the general linear
groups $\mathop{\rm GL}\nolimits_n(R)$ consisting of invertible $n\times n$-matrices
over $R$, with respect to the embeddings
$\mathop{\rm GL}\nolimits_n(R)\hookrightarrow \mathop{\rm GL}\nolimits_{n+1}(R)$ defined by
\[(A)\mapsto\pmatrix{A&0\cr0&1\cr}
\mbox{ \ for all \ } (A)\in \mathop{\rm GL}\nolimits_n(R).\]
A matrix is called elementary if it differs from the identity
matrix at no more than one off-diagonal position. Denote by
$E_n(R)$ resp. $E(R)$ the subgroup of $\mathop{\rm GL}\nolimits_n(R)$ resp. $\mathop{\rm GL}\nolimits(R)$
generated by all elementary matrices.
According to the Whitehead lemma \cite[\S3]{Milnor}
$E(R)$ coincides with the
commutator subgroup of $\mathop{\rm GL}\nolimits(R)$.
By definition $K_1R:= \mathop{\rm GL}\nolimits(R)/E(R)$. We use the additive notation
in the abelian group $K_1R$.
\end{defi}
There is a general procedure for defining the Whitehead group
$K_1{\cal C}$ of a category ${\cal C}$ with product,
but we do not need it for our purposes.
It follows from lemma~\ref{lemmahcof} that the $H(R^n)$ are cofinal in
$Q(R,\alpha,u)$. According to \cite[Ch.VII, \S2.3]{Bass} we may just as well
define $K_1Q(R,\alpha,u)$ as follows under these circumstances.
\begin{defi}
$K_1Q(R,\alpha,u)$ is
the commutator quotient of the direct limit
$$\lim_{\longrightarrow}\,\mathop{\rm Aut}\nolimits(H(R^n))$$
where the limit is taken with respect to the canonical embeddings \hfill\break
$\mathop{\rm Aut}\nolimits(H(R^n))\longrightarrow\mathop{\rm Aut}\nolimits(H(R^n)\perp H(R))\cong\mathop{\rm Aut}\nolimits(H(R^{n+1})).$
\end{defi}
\begin{remark}
Analogously $K_1(R)$ is the Whitehead group of both
${\cal P}(R)$ and $\overline{{\cal P}(R)}$. Since the free modules $R^n$ are cofinal
in both categories, the groups $K_1({\cal P}(R))$ and $K_1(\overline{{\cal P}(R)})$
both coincide with the commutator quotient of the direct limit
\[\lim_{\longrightarrow}\,\mathop{\rm Aut}\nolimits(R^n)\]
where the limit is taken with respect to the canonical embeddings
$\mathop{\rm Aut}\nolimits(R^n)\longrightarrow \mathop{\rm Aut}\nolimits((R^n)\perp(R))\cong\mathop{\rm Aut}\nolimits(R^{n+1})$.
Upon choosing a basis for $R^n$ we may identify $\mathop{\rm Aut}\nolimits(R^n)$ with
$\mathop{\rm GL}\nolimits_n(R)$ and consequently
$K_1{\cal P}(R)\cong K_1(\overline{{\cal P}(R)})\cong K_1R$.
\end{remark}
\begin{punt}\label{defgq}
Let us return to $Q(R,\alpha,u)$.
We choose a basis for $R^n$ and the dual basis for $(R^n)^\alpha$.
Since the matrix of $\upsilon$ with respect to these bases,
takes the form
\[\Sigma_{2n}:=\left(\begin{array}{cc}0&I_n\\0&0\end{array}\right)\]
we may identify $\mathop{\rm Aut}\nolimits(H(R^n))$ with the subgroup of $\mathop{\rm GL}\nolimits_{2n}(R)$
consisting of all matrices
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in \mathop{\rm GL}\nolimits_{2n}(R)
\quad\mbox{\ (here $A,B,C$ and $D$ are $n\times n$-matrices)}\] satisfying
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)^\alpha
\left(\begin{array}{cc}0&I_n\\0&0\end{array}\right)
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)-
\left(\begin{array}{cc}0&I_n\\0&0\end{array}\right)=
X-X^\alpha u\]
for some $(2n\times 2n)$-matrix $X.$
This subgroup of $\mathop{\rm GL}\nolimits_{2n}(R)$ is called the general quadratic group
and is denoted by $\mathop{\rm GQ}\nolimits_{2n}(R)$.
As a consequence $K_1(Q(R,\alpha,u))$ can be identified with the commutator
quotient of the group
\[\mathop{\rm GQ}\nolimits(R):=\lim_{\longrightarrow}\,\mathop{\rm GQ}\nolimits_{2n}(R),\]
where the limit is taken with respect to the embeddings
\[\mathop{\rm GQ}\nolimits_{2n}(R)\hookrightarrow \mathop{\rm GQ}\nolimits_{2(n+1)}(R)\mbox{ defined by}
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\mapsto
\left(\begin{array}{cccc}A&0&B&0\\
0&1&0&0\\
C&0&D&0\\
0&0&0&1
\end{array}\right).\]
\end{punt}
\begin{defi}\label{defantit}
For every $n\in N$ we define
$t_{\alpha,u}\colon \mathop{\rm GL}\nolimits_{2n}(R)\rightarrow \mathop{\rm GL}\nolimits_{2n}(R)$
by
\[t_{\alpha,u}(X)=U_{2n}^{-1}X^\alpha U_{2n}
\mbox{ for every } X\in \mathop{\rm GL}\nolimits_{2n}(R),
\mbox{ here } U_{2n}:=\left(\matrix{0&I_n\cr uI_n&0\cr}\right).\]
Explicitly: for every
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in \mathop{\rm GL}\nolimits_{2n}(R)\]
we have
\[t_{\alpha,u}\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)=
\left(\begin{array}{cc}D^{\alpha^{-1}}&u^{-1}B^\alpha\\
C^\alpha u&A^\alpha\end{array}\right).\]
Note that $D^{\alpha^{-1}}=u^{-1}D^\alpha u$
since $\alpha^2(r)=uru^{-1}$ for every $r\in R.$
Furthermore, $t_{\alpha,u}$ is an anti-involution since
\begin{eqnarray*}
t_{\alpha,u}^2(X)&=&U_{2n}^{-1}(U_{2n}^{-1}X^\alpha U_{2n})^\alpha U_{2n}\\
&=&U_{2n}^{-1}U_{2n}^\alpha X^{\alpha\alpha}(U_{2n}^{-1})^\alpha U_{2n}\\
&=&U_{2n}^{-2}uXu^{-1}U_{2n}^2\\
&=&X
\end{eqnarray*}
\end{defi}
\begin{prop}\label{gqkar}
The following statements are equivalent:
\begin{description}
\item{(a)}
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in
\mathop{\rm GL}\nolimits_{2n}(R)\] belongs to $\mathop{\rm GQ}\nolimits_{2n}(R)$
\item{(b)}
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in
\mathop{\rm GL}\nolimits_{2n}(R)\] and
\[\left(\begin{array}{cc}A^\alpha C&A^\alpha D-1\\
B^\alpha C&B^\alpha D\end{array}\right)=X-X^\alpha u \mbox{ \ for some }X\]
\item{(c)}
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in
\mathop{\rm GL}\nolimits_{2n}(R)\] and
\[\left\{\begin{array}{l}
A^\alpha D+C^\alpha uB=1\\
A^\alpha C+C^\alpha uA=0\\
B^\alpha D+D^\alpha uB=0\\
\mbox{the diagonal entries of $A^\alpha C$ and $B^\alpha D$ belong to}\\
\{x-\alpha(x)u\mid x\in R\}
\end{array}\right.\]
\item{(d)}
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\in \mathop{\rm GL}\nolimits_{2n}(R)\] and
\[\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)^{-1}=
t_{\alpha,u}\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\]
and the diagonal entries of $A^\alpha C$ and $B^\alpha D$ belong to
$\{x-\alpha(x)u\mid x\in R\}$
\item{(e)}
\[\left\{\begin{array}{l}
A^\alpha D+C^\alpha uB=1\\
A^\alpha C+C^\alpha uA=0\\
B^\alpha D+D^\alpha uB=0\\
DA^\alpha +Cu^{-1}B^\alpha=1\\
DC^\alpha +Cu^{-1}D^\alpha=0\\
BA^\alpha +Au^{-1}B^\alpha=0\\
\mbox{the diagonal entries of $A^\alpha C$ and $B^\alpha D$ belong to}\\
\{x-\alpha(x)u\mid x\in R\}
\end{array}\right.\]
\end{description}
\end{prop}
\begin{proof}\hfill\break
{(a)}$\Leftrightarrow$ {(b)}:\hfill\break
Immediate by writing out the condition in ~\ref{defgq}.\hfill\break
{(b)}$\Leftrightarrow$ {(c)}:\hfill\break
From
\[\left(\begin{array}{cc}A^\alpha C&A^\alpha D-1\\
B^\alpha C&B^\alpha D\end{array}\right)=X-X^\alpha u \mbox{ \ for some }X.\]
it follows that
\[\left\{\begin{array}{l}
A^\alpha D-1=-(B^\alpha C)^\alpha u=-C^\alpha uB\\
0=A^\alpha C+(A^\alpha C)^\alpha u=A^\alpha C+C^\alpha uA\\
\mbox{the diagonal entries of $A^\alpha C$ belong to
$\{x-\alpha(x)u\mid x\in R\}$}\\
0=B^\alpha D+(B^\alpha D)^\alpha u=B^\alpha D+D^\alpha uB\\
\mbox{the diagonal entries of $B^\alpha D$ belong to
$\{x-\alpha(x)u\mid x\in R\}$}\\
\end{array}\right .\]
and vice versa.\hfill\break
{(c)}$\Leftrightarrow$ {(d)}:\hfill\break
The identity $A^\alpha D+C^\alpha uB=1$ holds if and only if
$D^{\alpha^{-1}}A+u^{-1}B^\alpha C=1$.
Combined with the other equations of statement {\bf (c)} this reads
\begin{eqnarray*}
\left(\begin{array}{cc}1&0\\0&1\end{array}\right)&=&
\left(\begin{array}{cc}D^{\alpha^{-1}}&u^{-1}B^\alpha\\
C^\alpha u&A^\alpha\end{array}\right)
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\\
&=&\left(t_{\alpha,u}
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\right)
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right).
\end{eqnarray*}
The rest is obvious.\hfill\break
{(d)}$\Leftrightarrow$ {(e)}:\hfill\break
Immediate by writing out the equations
\[\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)
\left(t_{\alpha,u}\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\right)\]
and
\[\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=
\left(t_{\alpha,u}\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right)\right)
\left(\begin{array}{cc}A&B\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&D\end{array}\right).\]
\end{proof}
\begin{punt}
The product preserving functors
$$F\colonQ(R,\alpha,u)\rightarrow\overline{{\cal P}(R)},$$
$$H\colon\overline{{\cal P}(R)}\raQ(R,\alpha,u)$$
and
$$D_\alpha\colon{\cal P}(R)\rightarrow{\cal P}(R)$$
of definition~\ref{deffunctors} and~\ref{defda} induce homomorphisms
$$F_*\colon K_1Q(R,\alpha,u)\rightarrow K_1R,$$
$$H_*\colon K_1R\rightarrow K_1Q(R,\alpha,u)$$
and
$$t=t_\alpha \colon K_1R\rightarrow K_1R.$$
Now $F_*$ is determined by
$$F_*([X])=[X] \mbox{ for every }X\in \mathop{\rm GQ}\nolimits(R),$$
$H_*$ by
\[H_*([X])=\left[\left(\begin{array}{cc}X&0\\0&(X^\alpha)^{-1}\end{array}
\right)\right]\mbox{ for every } X\in \mathop{\rm GL}\nolimits(R).\]
$t$ by
$$t([X])=[X^\alpha] \mbox{ for every } X\in \mathop{\rm GL}\nolimits(R).$$
Note that $t$ is an involution since
$$t^2([X])=[X^{\alpha\alpha}]=[uXu^{-1}]=[X].$$
\end{punt}
\begin{lemma}
$F_*\lower1.0ex\hbox{$\mathchar"2017$} H_*=1-t$.
\end{lemma}
\begin{proof}
For every $X\in \mathop{\rm GL}\nolimits(R)$ we have
$$\pmatrix{X&0\cr 0&(X^\alpha)^{-1}\cr}=
\pmatrix{X(X^\alpha)^{-1}&0\cr 0&1\cr}
\pmatrix{X^\alpha&0\cr 0&(X^\alpha)^{-1}\cr}.$$
But according to \cite[\S2]{Milnor} the class of
$$\pmatrix{X^\alpha&0\cr 0&(X^\alpha)^{-1}\cr}$$
is trivial in $K_1(R)$. In view of the preceding this proves the assertion.
\end{proof}
\begin{defi} \cite{Wall;lfound}
A subgroup ${\cal X}$ of $K_1R$ is called involution invariant if $t({\cal X})={\cal X}$.
For every involution invariant subgroup ${\cal X}$ of $K_1R$ define
$$L_1^{\cal X}(R,\alpha,u):={F_*^{-1}({\cal X})\over H_*({\cal X})}.$$
\end{defi}
\begin{defi}
\begin{itemize}
\item
Let $B(R)$ denote the category with \hfill\break
objects:
$(M,e)$ where $M$ is a free right $R$-module and
$e=[e_1,\ldots,e_{2m}]$ is an equivalence class of bases
of $M$;
two bases being equivalent when the base-change-matrix belongs to $E(R)$,
i.e. it represents $0\in K_1(R)$,\hfill\break
morphisms:
isomorphisms preserving classes,\hfill\break
product:
$(M,e)\perp(N,f):=(M\oplus N,ef)$
where \hfill\break $e=[e_1,\ldots,e_{2m}]$,
$f=[f_1,\ldots,f_{2n}]$ and
$ef=[e_1,\ldots,e_{2m},f_1,\ldots,f_{2n}]$.
\item
Let $BQ(R,\alpha,u)$ denote the category with \hfill\break
objects:
$(M,[\phi],e)$,\hfill\break where
$(M,[\phi])\in \mathop{\rm Obj}\nolimits\,Q(R,\alpha,u)$ and
$(M,e)\in \mathop{\rm Obj}\nolimits\,B(R)$,\hfill\break
morphisms:
isomorphisms preserving both structures,\hfill\break
product:
obvious.
\item
Again there is a product-preserving functor
$H_b\colon B(R)\rightarrow BQ(R,\alpha,u)$ defined by
$$H_b(M,e):=(M\oplus M^\alpha,[\upsilon],ee^*) \quad
H_b(f):= f\oplus(f^\alpha)^{-1}.$$
Here $e^*=[e_1^*,\ldots,e_{2m}^*]$ and $\upsilon$ is as before.
\end{itemize}
\end{defi}
\begin{lemma}\label{hbcofi}
$H_b$ is cofinal.
\end{lemma}
\begin{proof}
Let $(M,\theta,e)$ be an object of $BQ(R,\alpha,u)$.
Lemma~\ref{lemmahcof} supplies a $Q(R,\alpha,u)$-isomorphism
$$\xi\colon(M,\theta,e)\perp(M,-\theta,e)\rightarrow H_b(M,e).$$
Let $\gamma$ be the element $\xi$ determines in $K_1R$.
By choosing the class $f$ of bases of $M\oplus M$
in such a way that
$$\xi\colon (M\oplus M,\theta\perp -\theta,f)\rightarrow H_b(M,e)$$
represents $-\gamma\in K_1R$, we obtain a $BQ(R,\alpha,u)$-isomorphism
\[\xi\perp\xi\colon
(M,\theta,e)\perp(M,-\theta,e)\perp(M\oplus M,\theta -\theta,f)
\rightarrow H_b(M\oplus M,ee).\]
This proves the assertion.
\end{proof}
\begin{defi}
Let $({\cal C},\perp)$ be a category with product.
The Grothendieck group $K_0{\cal C}$ of ${\cal C}$
is defined as the abelian group given by the following presentation:\hfill\break
generators: classes $[A]$ of isomorphic objects $A$ of ${\cal C}$. We
assume that these classes form a set.\hfill\break
relations: $[A]+[B]=[A\perp B]$.
\end{defi}
\begin{punt}\label{k0pres}
Lemma~\ref{hbcofi} implies that
\begin{itemize}
\item[$\cdot$]
each element of $K_0BQ(R,\alpha,u)$ can be written in the form
$[A]-[B]$ where $A\in BQ(R,\alpha,u)$ and $B$ is hyperbolic.
\item[$\cdot$]
the equality $[A]-[B]=[A']-[B']$ holds in $K_0BQ(R,\alpha,u)$
if and only if there exists a hyperbolic object
$C$ such that $A\perp B'\perp C\cong A'\perp B\perp C$.
\end{itemize}
\end{punt}
\begin{defi}
Define $\widetilde{K_0}BQ(R,\alpha,u)$ as the kernel of the rank-map
$$rk\colon\widetilde{K_0}BQ(R,\alpha,u)\rightarrow\Z$$ induced by the map
$$BQ(R,\alpha,u)\rightarrow\Z \mbox{ \ given by \ }
(M,\theta,[e_1,\ldots,e_{2m}])\mapsto 2m.$$
\end{defi}
\begin{defi}
The map $BQ(R,\alpha,u)\rightarrow K_1R$ determined by
$$(M,\theta,e)\mapsto[\hbox{a `matrix' of }
b_\theta\hbox{ with respect to }e\hbox{ and }e^*]$$
induces a homomorphism
$\delta\colon K_0BQ(R,\alpha,u)\rightarrow K_1R$, called discriminant.
\end{defi}
\begin{remark}
$b_\theta $ determines a matrix with respect to $e$ and $e^*$
only up to elementary matrices. It is therefore legitimate to speak about
the class of this `matrix' in $K_1R$.
\end{remark}
\begin{remark}\label{hypnontriv}
Further we ought to mention the fact that $\delta$ is a priori non-trivial
on hyperbolic objects:\hfill\break
given a hyperbolic object $H_b(M,e)=(M\oplus M^\alpha,[\upsilon],ee^*)$ in
$BQ(R,\alpha,u)$, the matrix $\Sigma_{2m}$ of $\upsilon$ actually
(not only up to elementary matrices) takes the form
\[\Sigma_{2m}=\left(\begin{array}{cc}0&I_m\\0&0\end{array}\right)
\mbox{ (no matter what $e$ looks like).}\]
Hence $b_{[\upsilon]}$ has matrix
$U_{2m}=\left(\matrix{0&I_m\cr uI_m&0\cr}\right)$.
The class of this matrix in $K_1R$ is not necessarily trivial.
\end{remark}
\begin{defi} \cite[\S3]{Wall;lfound}\label{deftau}
Define a homomorphism
$\tau\colon K_1(R)\rightarrow \widetilde{K_0}BQ(R,\alpha,u)$ as follows :\hfill\break
Suppose we are given an $x\in K_1(R)$.
Choose $(M,\theta,e)\in BQ(R,\alpha,u)$ and
$\gamma\in\mathop{\rm Aut}\nolimits(M)$ in such a way that the matrix determined by $\gamma$
represents $x$ in $K_1(R)$.
Define $\tau([x]):=[(M,\theta,\gamma(e)]-[M,\theta,e]$ where
$\gamma(e)=[\gamma(e_1),\ldots,\gamma(e_{2m})]$.
It is not hard to check that $\tau$ is a well-defined homomorphism.
\end{defi}
\begin{lemma}
$\delta\lower1.0ex\hbox{$\mathchar"2017$}\tau=1+t$.
\end{lemma}
\begin{proof}
Using the third observation of ~\ref{obsfree} we obtain
$$\delta\lower1.0ex\hbox{$\mathchar"2017$}\tau([A])=[A^\alpha BA]-[B]=[A^\alpha A]=(1+t)([A])\quad
\mbox {for all } \quad A\in\mathop{\rm GL}\nolimits(R),$$
where $B$ is a `matrix' of
$b_\theta$ and $\theta$ is as in the construction of $\tau$.
\end{proof}
\begin{defi}
For every involution invariant subgroup
${\cal X}$ of $K_1R$ define
$$L_0^{\cal X}(R,\alpha,u):={\delta^{-1}({\cal X})\over\tau({\cal X})}$$
here $\delta\colon\widetilde{K_0}BQ(R,\alpha,u)\rightarrow K_1(R)$ is the
restriction of the
discriminant.
\end{defi}
\begin{nota}
Write $L_\varepsilon^s$ instead of $L_\varepsilon^{\{0\}}$
and $L_\varepsilon^h$ instead of $L_\varepsilon^{K_1(R)}$
for $\varepsilon=0,1$.
\end{nota}
\begin{punt}\label{obslgroup}
Let $(R,\alpha,u)$ be a ring with anti-structure and ${\cal X}$ an
involution invariant subgroup of $K_1(R).$
Every element $l$ of $L_0^{\cal X}(R,\alpha,u)$
can be written in the form
$$[M,[\phi],e]-[M',[\phi'],e'],$$
with $rk([M,[\phi],e])=rk([M',[\phi'],e'])=2m$ say.
Let $$\Gamma([M,[\phi],e])\; \hbox{ resp. }\; \Gamma([M',[\phi'],e'])$$
denote the matrix of $\phi$ resp. $\phi'$ with respect
to a basis in the class $e$ resp. $e'$.
Since the quadratic modules
$(M,[\phi])$ and $(M',[\phi'])$ are non-singular,
it follows from definition~\ref{defnonsing} that these matrices
belong to
${\cal N}_{2m}(R),$
where $${\cal N}_{k}(R)
:=\{\Gamma\in M_k(R)\mid \Gamma+\Gamma^\alpha u\in \mathop{\rm GL}\nolimits_k(R)\}.$$
We associate to $l$ the difference
$$[\Gamma([M,[\phi],e])]-[\Gamma([M',[\phi'],e'])]$$
of classes with respect to the following relations:
\begin{itemize}
\item[$\diamond$]
For all
$\Gamma_1,\Gamma_1'\in{\cal N}_{2m_1}(R)$ and
$\Gamma_2,\Gamma_2'\in{\cal N}_{2m_2}(R),$
$$[\Gamma_1]-[\Gamma_1']+[\Gamma_2]-[\Gamma_2']=
[\Gamma_1\perp\Gamma_2]-[\Gamma_1'\perp\Gamma_2'].$$
where $\perp$ is determined by
$$\pmatrix{A&B\cr C&D\cr}\perp\pmatrix{A'&B'\cr C'&D'\cr}=
\pmatrix{A&0&B&0\cr0&A'&0&B'\cr C&0&D&0\cr 0&C'&0&D'\cr}.$$
This follows from the definition of the product in $BQ(R,\alpha,u)$.
\item[$\diamond$]
For all $\Xi\in M_{2m}(R)$
$$[\Gamma]=[\Gamma+\Xi-\Xi^\alpha u]. $$
This is clear in view of definition~\ref{defnonsing} and the
observations of~\ref{obsfree}.
\item[$\diamond$]
For all $\Delta\in \mathop{\rm GL}\nolimits_{2m}(R)$ with $[\Delta]\in{\cal X}$
$$[\Gamma]=[\Delta^\alpha\Gamma\Delta].$$
This is a consequence of definition~\ref{deffunctors} and the
observations of~\ref{obsfree}.
\end{itemize}
Conversely, for all $\Gamma,\Gamma'\in{\cal N}_{2m}(R)$
we associate to $[\Gamma]-[\Gamma']$ the element
$$[R^{2m},[\phi],st]-[R^{2m},[\phi'],st]\in L_0^{\cal X}(R,\alpha,u).$$
Here $st$ denotes the standard basis of $R^{2m}$ and
$\phi$ resp. $\phi'$
is the homomorphism which has matrix $\Gamma$ resp. $\Gamma'$ with
respect to this standard basis.
Thus we have established a bijective correspondence between
elements of $L_0^{\cal X}(R,\alpha,u)$ and
differences of classes of elements of
${\cal N}_{2m}(R)$ under the given relations.
Regarding the first item of \ref{k0pres}
we may thus write every element of
$L_0^{\cal X}(R,\alpha,u)$ as a difference $[\Gamma]-[\Sigma_{2m}]$, with
$\Gamma\in{\cal N}_{2m}(R)$.
Finally, we interpret the second item of \ref{k0pres} as follows.
For all $\Gamma\in{\cal N}_{2m}(R)$ and
$\Gamma'\in{\cal N}_{2m'}(R)$,
$$[\Gamma]-[\Sigma_{2m}]=[\Gamma']-[\Sigma_{2m'}]
\quad\mbox{ in }\quad L_0^{\cal X}(R,\alpha,u)$$
if and only if there exist
$n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$, $\Xi\in M_{2(n+m+m')}$ and $\Delta\in \mathop{\rm GL}\nolimits_{2(n+m+m')}$ such that
$$\Gamma\perp\Sigma_{2(n+m')}=
\Delta^\alpha(\Gamma'\perp\Sigma_{2(n+m)})\Delta +\Xi-\Xi^\alpha u
\quad\mbox{ and }\quad [\Delta]\in{\cal X}.$$
\end{punt}
We conclude this section by stating some definitions and facts from algebraic
$K$- and $L$-theory needed in the sequel.
\begin{thm} \label{exakring}{\rm \cite[Theorem 3]{Wall;lfound}}
Given an abelian group $A$ and an involution $t\colon A\rightarrow A$
the Tate-cohomology groups $H^n(A;t)$ are defined by
$$H^n(A;t):={\mathop{\rm Ker}\nolimits(1-(-1)^nt)\over \mathop{\rm Im}\nolimits(1+(-1)^nt)}. $$
Suppose ${{\cal X}_1}\subset {{\cal X}_2}$ are involution invariant
subgroups of $K_1(R)$,
then there exists an exact sequence
$$
\halign{\quad\hfil$#$\hfil&\hfil$#$\hfil
&\hfil$#$\hfil&
\hfil$#$\hfil&\hfil$#$\hfil
&\hfil$#$\hfil&\hfil$#$\hfil\cr
H^1({{\cal X}_2}/{{\cal X}_1})&\mapright{\tilde{\tau}}&
L_0^{{\cal X}_1}(R,\alpha,u)&
\longrightarrow&L_0^{{\cal X}_2}(R,\alpha,u)&\mapright{\tilde{\delta}}&
H^0({{\cal X}_2}/{{\cal X}_1})\cr
\mapup{}&&&&&&\mapdown{}\cr
L_1^{{\cal X}_2}(R,\alpha,u)&&&&&&L_1^{{\cal X}_1}(R,\alpha,-u)\cr
\mapup{}&&&&&&\mapdown{}\cr
L_1^{{\cal X}_1}(R,\alpha,u)&&&&&&L_1^{{\cal X}_2}(R,\alpha,-u)\cr
\mapup{}&&&&&&\mapdown{}\cr
H^0({{\cal X}_2}/{{\cal X}_1})&\longleftarrow&L_0^{{\cal X}_2}(R,\alpha,-u)&
\longleftarrow&L_0^{{\cal X}_1}(R,\alpha,-u)&\longleftarrow&H^1({{\cal X}_2}/{{\cal X}_1})\cr}
$$
Here $\tilde{\tau}$ resp. $\tilde{\delta}$ is induced by $\tau$ resp.
$\delta$.
\end{thm}
\begin{thm} {\rm \cite{w2}} {\sf Morita invariance.}\hfill\break
If $(R,\alpha,u)$ is a ring with anti-structure and the
matrix ring $M_n(R)$ is equipped with the conjugate transpose anti-structure,
then $L_\varepsilon^*(M_n(R),\alpha,uI_n)$ is isomorphic to
$L_\varepsilon^*(R,\alpha,u)$.
\end{thm}
\begin{thm} {\rm \cite{w2}} {\sf Scaling.}\hfill\break
If $(R,\alpha,u)$ is a ring with anti-structure and $v$ is a unit in $R$,
then $L_\varepsilon^*(R,\alpha,u)$ is isomorphic to
$L_\varepsilon^*(R,\alpha',u')$, where $\alpha'(r):= v\alpha(r)v^{-1}$
and $u':= v\alpha(v^{-1})u$.
\end{thm}
\begin{thm} \label{iadiciso}{\rm \cite[Lemma 5]{Wall;class3}}
Suppose $I$ is a two-sided ideal of $R$ such that
$R$ is complete in the $I$-adic topology.
If $\alpha(I)=I$, then $R/I$ can be equipped with an anti-structure
in an obvious way and the projection $R\rightarrow R/I$ induces an isomorphism
$L_\varepsilon^h(R)\longrightarrow L_\varepsilon^h(R/I)$.
\end{thm}
\begin{defi} \cite{Milnor}
Denote by $e_{ij}(a)\in E_n(R)$ the elementary matrix having the element
$a\in R$ at the $(i,j)$-entry.\hfill\break
For $n\geq 3$ let $\mathop{\rm St}\nolimits_n(R)$ be the group with the following presentation \hfill\break
generators: one generator $x_{ij}(a)$ for every $e_{ij}(a)\in E_n(R)$\hfill\break
relations:\[ x_{ij}(a)x_{ij}(b)=x_{ij}(a+b)\]
\[[x_{ij}(a),x_{kl}(b)]=\left\{\begin{array}{l}
1 \mbox{\hspace{6ex} if } \; i\neq l,\; j\neq k\\
x_{il}(ab) \mbox{\hspace{0.5ex} if }\; j=k,\; i\neq l. \end{array}\right.\]
The Steinberg group of $R$ denoted by $\mathop{\rm St}\nolimits(R)$ is by definition the
direct limit $$\lim_{\longrightarrow}\mathop{\rm St}\nolimits_n(R),$$ where the limit is taken with
respect to the embeddings $\mathop{\rm St}\nolimits_n(R)\hookrightarrow \mathop{\rm St}\nolimits_{n+1}(R)$
coming from the embeddings
$E_n(R)\hookrightarrow E_{n+1}(R)$
of definition~\ref{defk1}.
Since the relations for the $x_{ij}$ in $\mathop{\rm St}\nolimits_n(R)$
also hold for the $e_{ij}$ in $E_n(R)$,
there is a natural homomorphism $\phi\colon \mathop{\rm St}\nolimits_n(R)\rightarrow E_n(R)$,
taking generators $x_{ij}(a)$ to $e_{ij}(a)$,
which in the limit gives rise to a homomorphism
$E(R)\rightarrow \mathop{\rm St}\nolimits(R)$.
The kernel of this last homomorphism
is by definition the $K$-group $K_2R.$
\end{defi}
\begin{lemma} \cite[theorem 5.1]{Milnor}
$K_2R$ is the center of the Steinberg group.
\end{lemma}
\begin{defi}
Denote by $\mathop{\rm GL}\nolimits_{2\infty}(R)$ the direct limit of the groups
$\mathop{\rm GL}\nolimits_{2n}(R)$ with respect to the embeddings
\[\mathop{\rm GL}\nolimits_{2n}(R)\hookrightarrow \mathop{\rm GL}\nolimits_{2(n+1)}(R)\mbox{ defined by }
\left(\matrix{A&B\cr C&D\cr}\right)\mapsto
\left(\matrix{A&0&B&0\cr0&1&0&0\cr C&0&D&0\cr 0&0&0&1\cr}\right)\]
Similarly one defines $E_{2\infty}(R)$ and
correspondingly $\mathop{\rm St}\nolimits_{2\infty}(R).$
\end{defi}
\begin{punt}\label{involstek12} \cite[corollary 1.7]{Giffen;k2}
The anti-involutions $t_{\alpha,u}$ on the $\mathop{\rm GL}\nolimits_{2n}(R)$ give rise
to anti-involutions on the $E_{2n}(R)$ which in turn lift to
anti-involutions of $\mathop{\rm St}\nolimits_{2n}(R).$ See definition~\ref{defantit} for formulas.
These provide for the following commutative diagram with exact rows
and vertical arrows (anti)-involutions:
$$\diagram{
0\longrightarrow&K_2(R)&\longrightarrow&\mathop{\rm St}\nolimits_{2\infty}(R)&\longrightarrow&\mathop{\rm GL}\nolimits_{2\infty}(R)&\longrightarrow&K_1(R)&\lra0\cr
&\mapdown{t_\alpha}&&\mapdown{t_{\alpha,u}}&&\mapdown{t_{\alpha,u}}&
&\mapdown{t_\alpha}&\cr
0\longrightarrow&K_2(R)&\longrightarrow&\mathop{\rm St}\nolimits_{2\infty}(R)&\longrightarrow&\mathop{\rm GL}\nolimits_{2\infty}(R)&\longrightarrow&K_1(R)&\lra0\cr
}$$
\end{punt}
\begin{defi} \label{defgk2i}
Following \cite{Giffen;k2} one can construct a homomorphism
$$G\colon L_0^s(R)\rightarrow H^1(K_2(R);t)$$ as follows: \hfill\break
Let $$l=[\Gamma]-[\Sigma_{2m}]\in L_0^s(R)$$ and
$X=\Gamma+\Gamma^\alpha u$.
Then $$U_{2m}^{-1}X\in E(R)
\mbox{ \ and \ } X^\alpha u=X.$$
Hence $$t_{\alpha,u}(U_{2m}^{-1}X)=
t_{\alpha,u}(X)\cdot t_{\alpha,u}(U_{2m}^{-1})=
U_{2m}^{-1}X^\alpha U_{2m}U_{2m}=U_{2m}^{-1}X^\alpha u=U_{2m}^{-1}X.$$
Now choose a lift $\gamma\in \mathop{\rm St}\nolimits(R)$ of $U_{2m}^{-1}X$
and define $$G(l):=[\gamma^{-1}t_{\alpha,u}\gamma]\in H^1(K_2(R);t).$$
It's not hard to check that $G$ is a well-defined homomorphism.
\end{defi}
\newpage
\section{The Arf invariant.}
\setcounter{altel}{0}
\setcounter{equation}{0}
In this section we define the main object of study in this thesis:
the Arf-groups.
Suppose we are given a ring with anti-structure $(R,\alpha,u)$ and an
involution invariant subgroup ${\cal X}$ of $K_1(R).$
We will analyse the subgroup of $L_0^{\cal X}(R,\alpha,u)$
consisting of all differences of classes of forms whose underlying
bilinear form is standard.
\begin{defi}
Recall the considerations of \ref{obslgroup} and define
$\mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u)$
as the subgroup of $L_0^{\cal X}(R,\alpha,u)$
generated by all elements
$$\plane{A,B}:=\left[\pmatrix{A&I_m\cr 0&B\cr}\right]
-\left[\pmatrix{0&I_m\cr0&0\cr}\right],$$
where $A,B\in\Lambda_m(R):=\{X\in M_m(R)\mid X+X^\alpha u=0\}.$\hfill\break
Note that
$$\pmatrix{A&I_m\cr 0&B\cr}\quad\mbox{ and }\quad\pmatrix{0&I_m\cr 0&0\cr}$$
both belong
to ${\cal N}_{2m}(R).$\hfill\break
Further we define $\Gamma_m(R):=\{X-X^\alpha u\mid X\in M_m(R)\}$.
\end{defi}
\begin{lemma}\label{lemmadiag}
All elements $\plane{A,B}$ of $\mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u)$ can be
written in the form:
\[\plane{A,B}=\sum_{i=1}^m\plane{A_{ii},B_{ii}}.\]
\end{lemma}
\begin{proof}
Recall \ref{obslgroup}.
Since $A+A^\alpha u=B+B^\alpha u=0$ we find
\begin{eqnarray*}
(A,B)&=&\left[\pmatrix{A&I_m\cr 0&B\cr}\right]-
\left[\pmatrix{0&I_m\cr 0&0\cr}\right]\\
&=&\sum_{i=1}^m\left[\pmatrix{A_{ii}&1\cr0&B_{ii}\cr}\right]-
\left[\pmatrix{0&1\cr0&0\cr}\right]\\
&=&\sum_{i=1}^m(A_{ii},B_{ii})
\end{eqnarray*}
\end{proof}
\begin{prop}\label{proparfrel}\hfill\break
Suppose we are given $A,B\in\Lambda_m(R)$ and $A',B'\in\Lambda_{m'}(R).$
Then
$$\plane{A,B}=\plane{A',B'} \mbox{ \ in \ }\quad \mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u),$$
if and only if there exist
$$n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}} \quad \mbox{ \ and \ } \quad
\pmatrix{X&Y\cr Z&T\cr}\in \mathop{\rm GL}\nolimits_{2(n+m+m')}(R) \quad\mbox{ with \ }\quad
\left[\pmatrix{X&Y\cr Z&T\cr}\right]\in{\cal X},$$
such that
$$\begin{array}{l}
A'=X^\alpha AX+X^\alpha Z+Z^\alpha BZ \pmod{\Gamma_{n+m+m'}(R)},\\ \\
B'=Y^\alpha AY+Y^\alpha T+T^\alpha BT \pmod{\Gamma_{n+m+m'}(R)}
\mbox{ \ and }\\ \\
t_{\alpha,u}\pmatrix{X&Y\cr Z&T\cr}=\pmatrix{X&Y\cr Z&T\cr}^{-1}.
\end{array}$$
Here $A,B,A',B'$ are considered to be elements of $M_{n+m+m'}(R)$,
by the embeddings $ M_k(R)\hookrightarrow M_{k+1}(R)$ defined by
$$\pmatrix{C\cr}\lhook\joinrel\longrightarrow\pmatrix{C&0\cr0&0\cr}.$$
\end{prop}
\begin{proof}
Regarding the final assertion of \ref{obslgroup}
it suffices to make the following
statements. Define $k:= n+m+m'$.
\[\left(\begin{array}{cc}X&Y\\Z&T\end{array}\right)^\alpha
\left(\begin{array}{cc}A&I_k\\0&B\end{array}\right)
\left(\begin{array}{cc}X&Y\\Z&T\end{array}\right)
\mbox{ takes the form }
\left(\begin{array}{cc}A'&I_k\\0&B'\end{array}\right)\]
(mod $\Gamma_{2k}(R)$) precisely when the difference
\[\left(\begin{array}{cc}X^\alpha AX+X^\alpha Z+Z^\alpha BZ
&X^\alpha AY+X^\alpha T+Z^\alpha BT\\
Y^\alpha AX+Y^\alpha Z+T^\alpha BT&Y^\alpha AY+Y^\alpha T+T^\alpha BT
\end{array}\right)-
\left(\begin{array}{cc}A'&I_k\\0&B'\end{array}\right)\]
belongs to $\Gamma_{2k}(R).$
From the fact that the matrices $A,B,A',B'$ in this expression belong
to $\Lambda_{2k}(R)$ we deduce:
\[\left\{\begin{array}{l}X^\alpha T+Z^\alpha uY=1\\
X^\alpha Z+Z^\alpha uX=0\\
Y^\alpha T+T^\alpha uY=0\end{array}\right.\]
This is equivalent to
\[\left(t_{\alpha,u}\left(\begin{array}{cc}X&Y\\Z&T\end{array}\right)\right)
\left(\begin{array}{cc}X&Y\\Z&T\end{array}\right)=1.\]
\end{proof}
We will give a presentation for the groups $\mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u)$ in
the next theorem.
Although our definition of the Arf- and $L$-groups
is a priori quite different from the one in \cite{Clauwens;arf},
the presentation is nearly the same.
We refer to \cite{Bak} for a comparison of the various $L$-groups.
Moreover this presentation is not quite the same as the one in
\cite{Clauwens;arf},
because our $u$ is not necessarily central in $R$.
At least not yet.
\begin{thm} \label{thmarfgr}{\rm Compare \cite{Clauwens;arf}}.\hfill\break
As abelian group $Arf^{\cal X}(R,\alpha,u)$ has the following presentation:
\halign{\hfil#&\quad#\hfil&\quad#\hfill\cr
generators:\phantom{1)}
&$\plane{a,b}$&where $a,b\in \Lambda_1(R)$\cr
relations:
1)&$\plane{a,b_1+b_2}=\plane{a,b_1}+\plane{a,b_2}$&
for all $a,b_1,b_2\in\Lambda_1(R)$\cr
2)&$\plane{a_1+a_2,b}=\plane{a_1,b}+\plane{a_2,b}$&
for all $a_1,a_2,b\in\Lambda_1(R)$\cr
3)&$\plane{a,b}=\plane{b,uau^{-1}}$&
for all $a,b\in\Lambda_1(R)$\cr
4)&$\plane{a,b}=0$&for all $a\in\Lambda_1(R),\;\;b\in \Gamma_1(R)$\cr
5)&$\plane{a,\alpha(x)bx}=\plane{xa\alpha^{-1}(x),b}$&
for all $a,b\in\Lambda_1(R),\;\;x\in R$\cr
6)&$\plane{a,b}=\plane{a,ba\alpha^{-1}(b)}.$&for all $a,b\in\Lambda_1(R)$\cr
7)&$\sum_{i=1}^n\plane{(X^\alpha Z)_{ii},(Y^\alpha T)_{ii}}=0$&
if $\pmatrix{X&Y\cr Z&T\cr}\in \mathop{\rm GL}\nolimits_{2n}(R),$\cr
&$t_{\alpha,u}\pmatrix{X&Y\cr Z&T\cr}=\pmatrix{X&Y\cr Z&T\cr}^{-1}$&
and $\left[\pmatrix{X&Y\cr Z&T\cr}\right]\in{\cal X}.$\cr}
\end{thm}
\begin{proof}
$\mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u)$ is generated by the $\plane{a,b}$ because of
lemma~\ref{lemmadiag}.
To prove the relations,
we will now exploit proposition~\ref{proparfrel}.\hfill\break
Let $A,B\in\Lambda_m(R)$.\hfill\break
Choosing
$$\pmatrix{X&Y\cr Z&T\cr}=\pmatrix{I_m&u^{-1}B\cr0&I_m\cr}$$
in proposition~\ref{proparfrel}
yields
\begin{eqnarray*}
\plane{A,B}&=&\plane{A,(u^{-1}B)^\alpha Au^{-1}B+(u^{-1}B)^\alpha+B}\\
&=&\plane{A,B^\alpha uAu^{-1}B+B^\alpha u+B}\\
&=&\plane{A,BAB^{\alpha^{-1}}}.
\end{eqnarray*}
Taking $m=1$ this proves {\em 6}.\hfill\break
Choosing $$\pmatrix{X&Y\cr Z&T\cr}=\pmatrix{I_m&0\cr A&I_m\cr}$$
in proposition~\ref{proparfrel}
yields
$$\plane{A,B}=\plane{A+A+A^\alpha BA,B}=\plane{A^\alpha BA,B}.$$
As a consequence $\plane{a,0}=\plane{0,a}=0$
for all $a\in\Lambda_1(R)$, which proves {\em 4}.\hfill\break
Let $A',B',C',D'\in\Lambda_m(R)$ and $X'\in M_m(R)$.\hfill\break
Choosing $$A=\pmatrix{{A'}&0\cr 0&{C'}\cr},\qquad
B=\pmatrix{{B'}&0\cr 0&{D'}\cr}$$
and $$\pmatrix{X&Y\cr Z&T\cr}=
\pmatrix{I_m&0&0&{X'}\cr0&I_m&-u^{-1}{X'}^\alpha&0\cr
0&0&I_m&0\cr0&0&0&I_m\cr}$$ in
proposition~\ref{proparfrel} yields
\begin{eqnarray*}
\lefteqn{\left(\pmatrix{{A'}&0\cr 0&{C'}\cr},
\pmatrix{{B'}&0\cr 0&{D'}\cr}\right)}\hspace{2ex}\\
&=&\left(\pmatrix{{A'}&0\cr 0&{C'}\cr},\pmatrix{0&-u{X'}\cr {X'}^\alpha&0\cr}
\pmatrix{{A'}&0\cr 0&{C'}\cr}\pmatrix{0&{X'}\cr -u^{-1}{X'}^\alpha&0\cr}+\right.\\
& &\left.\pmatrix{0&-u{X'}\cr {X'}^\alpha&0\cr}+\pmatrix{{B'}&0\cr 0&{D'}\cr}\right)\\
&=&\left(\pmatrix{{A'}&0\cr 0&{C'}\cr},
\pmatrix{u{X'}{C'}u^{-1}{X'}^\alpha&0\cr0&{X'}^\alpha{A'}{X'}\cr}+
\pmatrix{{B'}&0\cr 0&{D'}\cr}\right)
\end{eqnarray*}
Hence
\begin{equation}
\plane{{A'},{B'}}+\plane{{C'},{D'}}=\plane{{A'},{B'}+
u{X'}{C'}u^{-1}{X'}^\alpha}+
\plane{{C'},{X'}^\alpha {A'}{X'}+{D'}}.\label{xeq}
\end{equation}
First choose ${C'}=u^{-1}{B'}u$, ${D'}=0$ and ${X'}=1$ to obtain
$$\plane{{A'},{B'}}=\plane{u^{-1}{B'}u,{A'}},$$ which proves {\em 3}.\hfill\break
Then choose ${A'}={D'}$ and ${X'}=1$ to obtain
$$\plane{{A'},{B'}}+\plane{{C'},{A'}}=\plane{{A'},{B'}+u{C'}u^{-1}}$$
which by {\em 3} is equivalent to
$$\plane{{A'},{B'}}+\plane{{A'},u{C'}u^{-1}}=\plane{{A'},{B'}+u{C'}u^{-1}}.$$
This proves {\em 1}.\hfill\break
Note that {\em 2} follows from and {\em 1} and {\em 3}.\hfill\break
In order to verify {\em 5} we use {\em 1, 2, 3} and {\em 4} to see that
equation~\ref{xeq} comes down to
$$\plane{{A'},u{X'}{C'}u^{-1}{X'}^\alpha}=
\plane{{C'},{X'}^\alpha {A'}{X'}}.$$
But since $\plane{{A'},u{X'}{C'}u^{-1}{X'}^\alpha}=
\plane{{X'}{C'}{X'}^{\alpha^{-1}},{A'}},$
this proves {\em 5}.\hfill\break
Note that all choices for $\pmatrix{X&Y\cr Z&T\cr}$ we have made so far
satisfy the conditions of proposition~\ref{proparfrel}.\hfill\break
Finally suppose $\pmatrix{X&Y\cr Z&T\cr}$ agrees with the conditions
of {\em 7}.
To prove the theorem it suffices to show that
$$(X^\alpha AX+X^\alpha Z+Z^\alpha BZ,Y^\alpha AY+Y^\alpha T+T^\alpha BT)=
(A,B)+(X^\alpha Z,Y^\alpha T)$$
modulo the relations {\em 1} to {\em 6}.
This is accomplished by using the relations for $X,Y.Z$ and $T$
listed in proposition~\ref{gqkar}.
We equate
$$(X^\alpha AX,Y^\alpha AY)=(YX^\alpha AXY^{\alpha^{-1}},A)=
(XY^{\alpha^{-1}},A)$$
and in the same fashion
$$(Z^\alpha BZ,T^\alpha BT)=(ZT^{\alpha^{-1}},B).$$
Further
\begin{eqnarray*}
\lefteqn{(X^\alpha AX,Y^\alpha T)+(X^\alpha Z,Y^\alpha AY)}\\
&=&(A,X^{\alpha^2}Y^\alpha TX^\alpha)+(YX^\alpha ZY^{\alpha^{-1}},A)\\
&=&(u^{-1}X^{\alpha^2}Y^\alpha TX^\alpha u,A)+
(YX^\alpha ZY^{\alpha^{-1}},A)\\
&=&(XY^{\alpha^{-1}},A)
\end{eqnarray*}
and analogously
$$(Z^\alpha BZ,Y^\alpha T)+(X^\alpha Z,T^\alpha BT)=(ZT^{\alpha^{-1}},B).$$
Finally we have
\begin{eqnarray*}
\lefteqn{(X^\alpha AX,T^\alpha BT)+(Z^\alpha BZ,Y^\alpha AY)}\\
&=&(A,X^{\alpha^2}T^\alpha BTX^\alpha)+
(YZ^\alpha BZY^{\alpha^{-1}},A)\\
&=&(A,X^{\alpha^2}T^\alpha BTX^\alpha)+
(A,uYZ^\alpha BZY^{\alpha^{-1}}u^{-1})\\
&=&(A,X^{\alpha^2}T^\alpha BTX^\alpha)+
(A,(1-X^{\alpha^2}T^\alpha)B(1-TX^\alpha))\\
&=&(A,B).
\end{eqnarray*}
This completes the proof.
\end{proof}
\begin{thm}
There is a well-defined homomorphism, called Arf invariant
$$\omega\colon\mathop{\rm Arf}\nolimits^{\cal X}(R,\alpha,u)\rightarrow R/\kappa(R),$$
defined by
$$\plane{A,B}\mapsto\left[Tr(A^\alpha B)\right].$$
Here $\kappa(R)$ denotes the additive subgroup of $R$ generated by
$$\{x+x^2,y+\alpha(y)\mid x,y\in R\}$$
Observe that $xy-yx,2x\in\kappa(R)$ for all $x,y\in R$.
\end{thm}
\begin{proof}
Analogous to the proof of \cite[theorem 2]{Clauwens;arf}.
\end{proof}
\begin{defi}
For every group $G$ we define
$$L^{s,h}(G):= L_0^{s,h}(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G],\alpha,1)$$
and correspondingly
$$\mathop{\rm Arf}\nolimits^{s,h}(G):=\mathop{\rm Arf}\nolimits^{s,h}(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\:_2[G],\alpha,1),$$
where $\alpha$ is determined by $\alpha(g):= g^{-1}$ for all $g\in G$.
Further we define
$$K(G):=
\frac{\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]}{\kappa(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G])}.$$
\end{defi}
\begin{remark}
See also \cite{Clauwens;arf}.\label{remarfgrel}
From the presentation of theorem~\ref{thmarfgr}
we deduce that $\mathop{\rm Arf}\nolimits^{s,h}(G)$
is generated by all
$\plane{g,h}$ with $g,h\in{}_2G:=\{x\in G\mid x^2=1\}$ and that
the following relations hold:
\begin{eqnarray*}
\plane{g,h}&=&\plane{h,g}\\
\plane{g,h}&=&\plane{xgx^{-1},xhx^{-1}}\quad \mbox{ for all }\quad x\in G\\
\plane{g,h}&=&\plane{g,hgh}.
\end{eqnarray*}
The value group $K(G)$ of the Arf invariant $\mathop{\rm Arf}\nolimits^{s,h}(G)\longrightarrow K(G),$
is in fact the $\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$-vectorspace generated by
the quotient set $\cee\!\ell(G):= G/\!\sim$,
where $\sim$ denotes the equivalence relation on
$G$ generated by $g\sim g^{-1}$, $g\sim hgh^{-1}$ and $g\sim g^2$.
\end{remark}
\begin{thm}
The Arf invariant $\mathop{\rm Arf}\nolimits^{s,h}(G)\rightarrow K(G)$ is injective
whenever $G$ is a finite group.
\end{thm}
\begin{proof}
We refer to \cite{Clauwens;arf} for the proof.
\end{proof}
We will revert to these theorems later on.
\begin{lemma}\label{centrel}
Let $a$, $b$ and $c$ be elements of order two in a group $G$ and assume
that $c$ commutes with $a$ and $b$.
Then the relation
$$\plane{a,bc}=\plane{a,b}$$
holds in $\mathop{\rm Arf}\nolimits^{s,h}(G)$.
\end{lemma}
\begin{proof}
$\plane{a,bc}=\plane{a,bcabc}=\plane{a,bab}=\plane{a,b}.$
\end{proof}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$\langle X,S\mid X^{12}=S^2=1,\, SXS=X^5\rangle.$$
So $G$ is a semidirect product of the group of order $2$ and the cyclic
group of order $12$.
\begin{prop}
The elements $\plane{1,1},\,\plane{X^2S,S}$ form a basis for
$\mathop{\rm Arf}\nolimits^{s,h}(G)$.
\end{prop}
\begin{proof}
A little computation yields $\cee\!\ell(G)=\left\{[1],[X]\right\}$.
The Arf invariant is injective and maps $\plane{1,1}$ to $[1]$
and $\plane{X^2S,S}$ to
$[X^2]=[X]$, hence the assertion is true.
\end{proof}
\end{nitel}
The following example is meant to illustrate how tricky
manipulations with the relations in $\mathop{\rm Arf}\nolimits^{s,h}(G)$ can be.
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,S\mid S^2=(XS)^2=Y^{12}=1,\quad SYS=Y^5,
\quad XY=YX\rangle.$$
This group fits into the short exact sequence
$$1\longrightarrow C\times C_{12}\longrightarrow G\longrightarrow C_2\longrightarrow 1,$$
where $C_2$ has generator $S$, $C_{12}$ has generator $Y$ and
$C$ is the infinite cyclic group generated by $X$.
Actually $G$ is a semidirect product of $C_2$ and $C\times C_{12}$.
We show that the elements
$\plane{S,SX^2Y^2}$ and $\plane{SX,SX^3Y^2}$ of $\mathop{\rm Arf}\nolimits^{s,h}(G)$ coincide.
The Arf invariant $\omega$ maps both elements to the class of $X^2Y^2$ in
$K(G)$.
We equate
\begin{eqnarray*}
\plane{S,SX^2Y^2}&=&\plane{S,SX^4Y^4}\\
&=&\plane{S,SX^2Y^{8}}\\
&=&\plane{S,SXY^{4}}\\
&=&\plane{SXY^2SSXY^2,SXY^2SXY^{4}SXY^2}\\
&=&\plane{SX^2Y^4,SX}\\
&=&\plane{SX,SX^2Y^4}\\
&=&\plane{SX,SX^3Y^8}\\
&=&\plane{SX,SX^5Y^4}\\
&=&\plane{SX,SX^3Y^2}.
\end{eqnarray*}
Since the Arf invariant maps both
$\plane{S,SY^2}$ and $\plane{SX,SXY^2}$ to the class of $Y^2$ in $K(G)$,
one might conjecture that these elements are equal too,
but this is false.
\end{nitel}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle Y,S\mid S^2=(YS)^4=(Y^2S)^2=1\rangle.$$
This group is actually an extension of the infinite cyclic group
by the
dihedral group $D_4$:
$$\diagram{
1\longrightarrow C\longrightarrow &G\longrightarrow D_4&\longrightarrow 1\cr
&\begin{array}{l}
S\longmapsto \sigma\\
Y\longmapsto \sigma\tau\vspace{1mm}
\end{array}&\cr}$$
Here $C$ is the infinite cyclic group generated by $Y^2$ and
$D_4$ is the dihedral group with presentation
$$D_4=\langle\sigma,\tau\mid \sigma^2=(\sigma\tau)^2=\tau^4=1\rangle.$$
\begin{prop}
The set
$$\left\{\plane{1,1}\right\}\cup
\left\{\plane{Y^{4i+2}S,S}\mid i>0\right\}$$
constitutes a basis for $\mathop{\rm Arf}\nolimits^{s,h}(G)$.
\end{prop}
\begin{proof}
The elements of order 2 in $G$ are $Y^{2i}S$, $(YS)^2$ and $Y^{2i}S(YS)^2$.
Note that $(YS)^2$ is central in $G$.
So we may use lemma~\ref{centrel} to see that $\mathop{\rm Arf}\nolimits^{s,h}(G)$ is
generated by elements of the form
$\plane{Y^{2i}S,Y^{2j}S}$.
The identities
\begin{eqnarray*}
\plane{Y^{2i}S,Y^{2j}S}&=&
\plane{Y^{-2k}Y^{2i}SY^{2k},Y^{-2k}Y^{2j}SY^{2k}}\\
&=&\plane{Y^{2i-4k}S,Y^{2j-4k}S},\\
\plane{Y^{2i}S,Y^{2}S}&=&
\plane{Y^{2i}S,Y^{2}S(YS)^2}\\
&=&\plane{Y^{2i}S,YSY^{-1}}\\
&=&\plane{Y^{2i-1}SY,S}\\
&=&\plane{Y^{2i-2}S(SY)^2,S}\\
&=&\plane{Y^{2i-2}S,S},\\
\plane{Y^{4i}S,S}&=&
\plane{Y^{2i}SSY^{2i}S,S}\\
&=&\plane{Y^{2i}S,S},\\
\plane{Y^{2i}S,S}&=&
\plane{SY^{2i}SS,S}\\
&=&\plane{Y^{-2i}S,S}
\end{eqnarray*}
show that $\left\{\plane{1,1}\right\}\cup
\left\{\plane{Y^{4i+2}S,S}\mid i>0\right\}$ is a set of
generators for $\mathop{\rm Arf}\nolimits^{s,h}(G)$.
We use the Arf invariant $\mathop{\rm Arf}\nolimits^{s,h}(G) \rightarrow K(G)$
to prove that these elements are independent.
It is easy to verify that $$\cee\!\ell(G)=\left\{[1]\right\}\cup
\left\{[Y^{2i+1}]\mid i>0\right\}$$
by writing down all generating relations in $\cee\!\ell(G)$.\hfill\break
The Arf invariant maps $\plane{1,1}$ to $[1]$ and $\plane{Y^{4i+2}S,S}$ to
$[Y^{4i+2}]=[Y^{2i+1}]$.
This proves the assertion.
\end{proof}
\end{nitel}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,S\mid S^2=(XS)^2=(YS)^4=(Y^2S)^2=1,\quad XY=YX\rangle.$$
This group is actually an extension of the free abelian group $A$ of rank 2
by the dihedral group $D_4$:
$$\diagram{1\longrightarrow&A&\longrightarrow &G&\mapright{\pi}&D_4&\longrightarrow 1\cr}$$
where
$\pi(S):=\sigma$,
$\pi(X)\isdef1$,
$\pi(Y):=\sigma\tau$ and
$A$ is generated by $X$ and $Y^2$.
\begin{prop}
$\mathop{\rm Arf}\nolimits^{s,h}(G)$ is generated by
\begin{eqnarray*}
\left\{\plane{1,1}\right\}&\cup&
\left\{\plane{X^{2i+1}Y^{2j}S,S}\mid i\geq 0\right\}\\&\cup&
\left\{\plane{X^{2i}Y^{4j+2}S,S}\mid j\geq 0\right\}\\&\cup&
\left\{\plane{X^{2i+1}Y^{4j+2}S,XS}\mid j\geq 0\right\}.
\end{eqnarray*}
\end{prop}
\begin{proof}
The elements of order 2 in $G$ are $X^iY^{2j}S$, $(YS)^2$ and
$X^iY^{2j}S(YS)^2$.
Note that $(YS)^2$ is central in $G$ again.
So $\mathop{\rm Arf}\nolimits^{s,h}(G)$ is generated by elements of the form
$\plane{X^{i}Y^{2j}S,X^{k}Y^{2l}S}$.
We may assume that $k,l\in\{0,1\}$ by the identity
$$\plane{X^iY^{2j}S,X^kY^{2l}S}
=\plane{X^{i-2m}Y^{2j-4n}S,X^{k-2m}Y^{2l-4n}S}.$$
We may even assume that $l=0$ by the relation
\begin{eqnarray*}
\plane{X^iY^{2j}S,X^kY^{2}S}&=&
\plane{X^iY^{2j}S,X^kY^{2}S(YS)^2}\\
&=&\plane{X^iY^{2j}S,X^kYSY^{-1}}\\
&=&\plane{X^iY^{2j-1}SY,X^kS}\\
&=&\plane{X^iY^{2j-1}(SY)^3,X^kS}\\
&=&\plane{X^iY^{2j-2}S,X^kS}.
\end{eqnarray*}
When $k=0$ we may assume that $i$ or $j$ is odd:
\begin{eqnarray*}
\plane{X^{2i}Y^{4j}S,S}&=&
\plane{X^iY^{2j}SSX^iY^{2j}S,S}\\
&=&\plane{X^iY^{2j}S,S}
\end{eqnarray*}
In this situation we may assume that one odd exponent is positive:
\begin{eqnarray*}
\plane{X^iY^{2j}S,S}&=&\plane{SX^iY^{2j}SS,S}\\
&=&\plane{X^{-i}Y^{-2j}S,S}
\end{eqnarray*}
When $k=1$ we may assume that $i$ and $j$ are odd:
\begin{eqnarray*}
\plane{X^{2i}Y^{2j}S,XS}&=&
\plane{Y^{2j}S,X^{-2i+1}S}\\
&=&\plane{X^{2i-1}Y^{2j}S,S}\\
\plane{X^{2i+1}Y^{4j}S,XS}&=&\plane{X^{i+1}Y^{2j}SXSX^{i+1}Y^{2j}S,XS}\\
&=&\plane{X^{i+1}Y^{2j}S,XS}
\end{eqnarray*}
And finally, we may assume that j is positive:
\begin{eqnarray*}
\plane{X^{i}Y^{2j}S,XS}&=&
\plane{XSX^{i}Y^{2j}SXS,XS}\\
&=&\plane{X^{-i+2}Y^{-2j}S,XS}
\end{eqnarray*}
This proves the proposition.
\end{proof}
\end{nitel}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,Z\mid X^2=Y^2=Z^2=(XY)^3=(YZ)^{3}=(XZ)^3=1\rangle.$$
This group is known as the affine Weyl group $\widetilde{A_2}$.\hfill\break
Define $U:= XYZY$, $V:= YXZX$ and $W:= ZXYX$.
Then
$UVW=1$ and
$U$, $V$ and $W$ commute.
The subgroup $H$ of $G$ generated by $U$, $V$ and $W$ is normal since,
$$\begin{array}{ll}
XUX=U^{-1}&XVX=W^{-1}\\
YUY=W^{-1}&YVY=V^{-1}\\
ZUZ=V^{-1}&ZWZ=W^{-1}.
\end{array}$$
Further $G/H\cong S_3=\langle x,y\mid x^2=y^2=(xy)^3=1\rangle$.
These groups fit into the short exact sequence
$$1\longrightarrow H\longrightarrow G\rightleftmaps{}{\alpha} S_3\longrightarrow 1,$$
which splits by $\alpha(x):= X$ and $\alpha(y):= Y$.
Thus $G$ is actually a semidirect product of $S_3$ and $H$.
\begin{prop}
The elements
$$\plane{1,1},\quad\plane{X,Y},\quad\plane{Y,Z},\quad\plane{X,Z}\quad
\mbox{ and }\quad\plane{XU^i,X}\quad\mbox{$ i>0 $\, odd}$$
form a basis for $\mathop{\rm Arf}\nolimits^{s,h}(G)$.
\end{prop}
\begin{proof}
We merely sketch the proof.\hfill\break
The elements of order two in $G$ are
$XU^i$, $YV^i$ and $XYXW^i$.
So in $\mathop{\rm Arf}\nolimits^{s,h}(G)$ one has the following types of elements.
\begin{enumerate}
\item
$\plane{XU^i,XU^j}$
\item
$\plane{XU^i,YV^j}$
\item
$\plane{XU^i,XYXW^j}$
\item
$\plane{YV^i,YV^j}$
\item
$\plane{YV^i,XYXW^j}$
\item
$\plane{XYXW^i,XYXW^j}$
\end{enumerate}
We prove that all of these elements are actually of the desired type by
using the relations
\begin{enumerate}
\item[ ]
$XU^i=V^iXV^{-i}=W^iXW^{-i},$
\item[ ]
$YV^i=U^iYU^{-i}=W^iYW^{-i},$
\item[ ]
$XYXYV^iXYX=XU^{-i}.$
\end{enumerate}
\begin{enumerate}
\item
Conjugation by $W^{-j}$ yields $\plane{XU^i,XU^j}=\plane{XU^{i-j},X}$.\hfill\break
And further
\begin{enumerate}
\item[ ]
$\plane{XU^i,X}=\plane{XU^iXXU^i,X}=\plane{XU^{2i},X},$
\item[ ]
$\plane{XU^i,X}=\plane{XXU^iX,X}=\plane{XU^{-i},X}.$
\end{enumerate}
\item
Conjugation by $U^{-j}$ yields $\plane{XU^i,YV^j}=\plane{XU^{i+2j},Y}$.
But because
$$\plane{XU^i,Y}=\plane{U^{-1}WXU^iW^{-1}U,U^{-1}WYW^{-1}U}=
\plane{XU^{i+3},Y},$$
only the elements
\begin{enumerate}
\item[ ]
$\plane{X,Y}$,
\item[ ]
$\plane{XU,Y}=\plane{YZY,Y}=\plane{Y,Z}$ \ and
\item[ ]
$\plane{XU^{-1},Y}=\plane{XYZYX,Y}=\plane{Z,YXYXY}=\plane{X,Z}$ remain.
\end{enumerate}
\item
Conjugation by $X$ yields $\plane{XU^i,XYXW^j}=\plane{XU^{-i},YV^{-j}}$.
\item
Conjugation by $XYX$ yields $\plane{YV^i,YV^j}=\plane{XU^{-i},XU^{-j}}$.
\item
Conjugation by $Y$ yields $\plane{YV^i,XYXW^j}=\plane{YV^{-i},XU^{-j}}$.
\item
Conjugation by $X$ yields $\plane{XYXW^i,XYXW^j}=\plane{YV^{-i},YV^{-j}}$.
\end{enumerate}
We give a list of generating relations in $\cee\!\ell(G)$.
\begin{enumerate}
\item[$\cdot$]
$U^iV^j\sim U^{-i}V^{-j}\sim U^{2i}V^{2j}\sim U^{j-i}V^j\sim U^iV^{i-j}
\sim U^jV^i$
\item[$\cdot$]
$XU^iV^j\sim XV^{j}\sim U^{j}V^{2j}\sim U^{j}V^{-j}$
\item[$\cdot$]
$YU^iV^j\sim YU^{i}\sim U^{2i}V^{i}\sim U^{i}V^{-i}$
\item[$\cdot$]
$XYXU^iV^j\sim YU^{j-i}V^j\sim U^{i-j}V^{j-i}$
\item[$\cdot$]
$YXU^iV^j \sim XYU^{j-i}V^j$
\item[$\cdot$]
$XYU^iV^j \sim XYU^{i+1}V^{j-1}\sim XYU^{i+1}V^{j+2}$
\end{enumerate}
The Arf invariant maps
$$\cases{
\plane{1,1} & to\hspace{3ex} $[1]$\cr
\plane{X,Y} & to\hspace{3ex} $[XY]$\cr
\plane{X,Z} & to\hspace{3ex} $[XZ]=[XYU]$\cr
\plane{Y,Z} & to\hspace{3ex} $[YZ]=[XYU^{-1}]$\cr
\plane{XU^i,X} & to\hspace{3ex} $[U^i]$ \hspace{1ex} $i$ is
positive and odd.\cr}$$
From the list of relations we see that these images are independent,
which proves the proposition
\end{proof}
\end{nitel}
\noindent We will review some of these examples in chapter IV.
\newpage
{\Large {\bf \begin{center}
Chapter II \vspace{4mm}\\
New Invariants for \mbox{\boldmath $L$}-groups.
\end{center}}}
\vspace{6mm}
\setcounter{section}{0}
\section{Extension of the anti-structure to the ring of formal power series.}
\setcounter{altel}{0}
\setcounter{equation}{0}
To construct new invariants we start by extending a given anti-structure
on a ring $R$ to the ring of formal power series $R[[T]]$, in a highly
non-trivial manner. \hfill\break
The fact that the projection $R[[T]]\rightarrow R$ induces an isomorphism of
the associated $L$-groups, enables us to
build new invariants.
\begin{defi}
Suppose we are given a ring with antistructure
$(R,\alpha,u)$.\hfill\break
For every $n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\cup\{\infty\}$ we define
\begin{eqnarray*}
R_n&:=&\cases{
R[T]/(T^{n+1})\,, & the truncated polynomial ring, if $n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$\cr
R[[T]]\,, & the ring of formal power series, if $n=\infty,$\cr}\\
{\cal I}_n&:=& TR_n,
\mbox{ the two-sided ideal of $R_n$ generated by the class of $T$},\\
u_n&:=& u(1+T).
\end{eqnarray*}
Note that the class of $T$ in $R_n$ is also denoted by $T$.
Now we extend the anti-structure on $R$ to an anti-structure on $R_n$ by
the formula
\[
\alpha\left(\sum a_kT^k\right):=
\sum\alpha(a_k)\left({-T\over1+T}\right)^k.
\]
\end{defi}
\begin{lemma}
For every $n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\cup\{\infty\}$
\begin{enumerate}
\item
$(R_n,\alpha,u_n)$ is a ring with antistructure.
\item
${\cal I}_n$ is an involution invariant two-sided ideal of $R_n$, i.e.
$\alpha({\cal I}_n)={\cal I}_n$.
\item
$R_n$ is complete in the ${\cal I}_n$-adic topology.
\item
The projection $R_n\rightarrow R$ splits and $\alpha$ respects this splitting.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is trivial and therefore omitted.
\end{proof}
\newpage
\section{Construction of the invariants $\omega_1^{s,h}$ and $\omega_2$.}
\setcounter{altel}{0}
\setcounter{equation}{0}
\begin{punt}
In algebraic $K$-theory on has functors
$$K_i\colon \quad\mbox{{\sl category of ideals\/}}\rightarrow
\quad\mbox{{\sl category of abelian groups}}$$
for every $i\in {{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$.
The {\sl category of ideals\/} is the category with \hfill\break
objects: pairs $(R,I)$ consisting of a ring $R$ and a two-sided ideal $I$
of $R$\hfill\break
morphisms: $f\colon (R,I)\rightarrow (S,J)$ are the ringhomomorphisms $f\colon R\rightarrow S$
satisfying $f(I)\subseteq J$.\hfill\break
The groups $K_i(R):= K_i(R,R)$ are the ones we already came across
in the first chapter.
For every pair $(R,I)$ there exists a long exact sequence
$$\cdots\rightarrow K_{i+1}(R/I)\rightarrow K_i(R,I)\rightarrow K_i(R)\rightarrow K_i(R/I)\rightarrow\cdots$$
\end{punt}
\begin{punt}
Let $(R,\alpha,u)$ be a ring with anti-structure and
$(R_n,\alpha,u_n)$ the associated extension.
Since the projection $R_n\rightarrow R$ splits, we have
$$K_i(R_n)\cong K_i(R)\oplus K_i(R_n,{\cal I}_n)$$
by the functoriality of the $K_i$.
The involutions $t_\alpha$ on $K_1(R_n)$
and $K_2(R_n)$ induced by $\alpha$ respect this splitting.
Consequently, the Tate cohomology groups split accordingly:
$$H^{0,1}(K_i(R_n))\cong H^{0,1}(K_i(R))\oplus H^{0,1}(K_i(R_n,{\cal I}_n)).$$
\end{punt}
\begin{thm}\label{thminvomega1}
The following periodic sequence is exact.
$$ \halign{\quad\hfil$#$\hfil&\hfil$#$\hfil&\hfil$#$\hfil&
\hfil$#$\hfil&\hfil$#$\hfil&\hfil$#$\hfil&\hfil$#$\hfil\cr
H^1(K_1(R_n,{\cal I}_n))&\buildrel\tilde\tau\over\longrightarrow&L_0^s(R_n,\ol{\phantom{x}},u_n)&
\longrightarrow&L_0^s(R,\ol{\phantom{x}},u)&\buildrel \omega_1^s\over\longrightarrow&H^0(K_1(R_n,{\cal I}_n))\cr
\uparrow&&&&&&\downarrow\cr
L_1^s(R,\ol{\phantom{x}},u)&&&&&&L_1^s(R_n,\ol{\phantom{x}},-u_n)\cr
\uparrow&&&&&&\downarrow\cr
L_1^s(R_n,\ol{\phantom{x}},u_n)&&&&&&L_1^s(R,\ol{\phantom{x}},-u)\cr
\uparrow&&&&&&\downarrow\cr
H^0(K_1(R_n,{\cal I}_n))&\longleftarrow&L_0^s(R,\ol{\phantom{x}},-u)&
\longleftarrow&L_0^s(R_n,\ol{\phantom{x}},-u_n)&\longleftarrow&H^1(K_1(R_n,{\cal I}_n))\cr}$$
Here $\tilde{\tau}$ is induced by the homomorphism $\tau$ of
definition~\ref{deftau}
and $\omega_1^s$ is induced by the discriminant homomorphism.
\end{thm}
\begin{proof}
From theorem~\ref{exakring} of chapter I we obtain the following
commutative diagram with exact rows
$(\varepsilon =0,1)$
$$
\begin{array}{ccccc}
L_{1-\varepsilon}^h(R_n)&\rightarrow H^{1-\varepsilon}(K_1R)\rightarrow
&L_\varepsilon^{K_1(R_n,{\cal I}_n)}(R_n)&\rightarrow L_\varepsilon^h(R_n)\rightarrow
&H^\varepsilon(K_1R)\\
\downarrow&\|&\downarrow&\downarrow&\|\\
L_{1-\varepsilon}^h(R)&\rightarrow H^{1-\varepsilon}(K_1R)\rightarrow&L_\varepsilon^s(R)&\rightarrow L_\varepsilon^h(R)\rightarrow&H^\varepsilon(K_1R)
\end{array}
$$
Theorem~\ref{iadiciso} of chapter I implies that
$L_\varepsilon^h(R_n)\rightarrow L_\varepsilon^h(R)$ is an isomorphism.
Consequently $L_\varepsilon^{K_1(R_n,{\cal I}_n)}(R_n)$ is isomorphic to
$L_\varepsilon^s(R)$
by applying the five lemma to the diagram above.
When we insert this in the sequence of theorem~\ref{exakring} of chapter I
applied to the ring $R_n$
with ${\cal X}_1={0}$ and ${\cal X}_2=K_1(R_n,{\cal I}_n)$,
we obtain the desired periodic exact sequence.
\end{proof}
\begin{defi}\label{defomegaend}
Define $$\omega_1^h\colon L_0^h(R,\alpha,u)\longrightarrow H^0(K_1(R_n,{\cal I}_n);t_\alpha)$$
as the composition of homomorphisms
$$L_0^h(R,\alpha,u)\cong
L_0^h(R_n,\alpha,u_n)\mapright{\tilde{\delta}}
H^0(K_1(R_n);t_\alpha)\longrightarrow H^0(K_1(R_n,{\cal I}_n);t_\alpha),$$
where $\tilde{\delta}$ is induced by the discriminant homomorphism $\delta$.
Notice that $\omega_1^s$ factors through $\omega_1^h$.\hfill\break
Define $d$ as the composition of homomorphisms
$$H^1(K_1(R_n,{\cal I}_n))\mapright{\widetilde{\tau}}
L_0^s(R_n,\ol{\phantom{x}},u_n)\mapright{G}H^1(K_2(R_n,{\cal I}_n)),$$
where $G$ denotes the homomorphism of definition~\ref{defgk2i}
of the first chapter.
\end{defi}
\begin{lemma}\label{expld}
The map $d$ can explicitly be given by
$$d([X])=[\gamma^{-1}t_{\alpha,u}\gamma], \mbox{ for all }X\in GL(R),$$
where $\gamma\in \mathop{\rm St}\nolimits(R)$ is a lift of $(t_{\alpha,u}X)X\in E(R)$.
\end{lemma}
\begin{proof}
Immediate by the definitions of $G$ and $\tau$.
\end{proof}
\begin{thm}
The homomorphism $G$ induces a homomorphism
$$\omega_2\colon \mathop{\rm Ker}\nolimits(\omega_1^s)\rightarrow \mathop{\rm Coker}\nolimits(d).$$
\end{thm}
\begin{proof} This is clear now in view of the exact sequence of
theorem~\ref{thminvomega1} and
definition~\ref{defomegaend}.
\end{proof}
\newpage
\section{Recognition of $\omega_1^h$.}
\setcounter{altel}{0}
\setcounter{equation}{0}
We now proceed to analyse $\omega_1^h$. It will turn out that $\omega_1^h$
is strongly related to the Arf invariant of the first chapter.
\begin{prop}\label{prophk1}
Let $(R,\alpha,u)$ be a ring with anti-structure.
For all $\plane{a,b}\in \mathop{\rm Arf}\nolimits^h(R,\alpha,u)$
\[\omega_1^h(\plane{a,b})=
\left[1+\frac{\alpha(a)bT^2}{1+T}\right]\in H^0(K_1(R_n,{\cal I}_n))\]
\end{prop}
\begin{proof}
We may take
$$\left[\pmatrix{a&1\cr0&b\cr}\right]-\left[\pmatrix{0&1\cr0&0\cr}\right]
\in L_0^h(R_n,\alpha,u_n)$$
as a lift of $\plane{a,b}\in\mathop{\rm Arf}\nolimits^h(R,\alpha,u)\subseteq L_0^h(R,\alpha,u)$.
\begin{eqnarray*}
\omega_1^h(\plane{a,b})
&=&\left[\pmatrix{a&1\cr0&b\cr}+
\pmatrix{\alpha(a) &0\cr1&\alpha(b)\cr}u(1+T)\right]\\
&&-\left[\pmatrix{0&1\cr0&0\cr}+\pmatrix{0&0\cr1&0\cr}u(1+T)\right]\\
&=&\left[\pmatrix{\alpha(a) uT&1\cr u(1+T)&\alpha(b) uT\cr}
\pmatrix{0&1\cr u(1+T)&0\cr}^{-1}\right]\\
&=&\left[\pmatrix{1&\frac{\alpha(a)T}{1+T}\cr\alpha(b)uT&1\cr}\right]\\
&=&\left[\pmatrix{1&\frac{-\alpha(a)T}{1+T}\cr0&1\cr}
\pmatrix{1&\frac{\alpha(a)T}{1+T}\cr\alpha(b)uT&1\cr}
\pmatrix{1&0\cr-\alpha(b)uT&1\cr}\right]\\
&=&\left[1-\frac{\alpha(a)\alpha(b)uT^2}{1+T}\right]\\
&=&\left[1+\frac{\alpha(a) bT^2}{1+T}\right]\\
\end{eqnarray*}
\end{proof}
For the time being we will assume that $R$ is commutative
and write $\ol{\phantom{x}}$ instead of $\alpha$.
\begin{lemma}
$$q\colon H^0(R;\ol{\phantom{x}})\longrightarrow H^0(R;\ol{\phantom{x}})$$
defined by $$[x]\longmapsto [x^2]$$ is a homomorphism.
\end{lemma}
\begin{proof}
$q$ is well-defined:
\begin{itemize}
\item[$\cdot$]
$x^2=\ol{x}^2$ for all $x\in R$, satisfying $x=\ol{x}$.
\item[$\cdot$]
$(x+\ol{x})^2=x^2+x\ol{x}+\ol{x}x+\ol{x}^2=(x^2+x\ol{x})+\ol{(x^2+x\ol{x})}$,
for all $x\in R$.
\end{itemize}
$q$ is a homomorphism, since for all $x,y\in R$ satisfying $x=\ol{x}$,
$y=\ol{y}$
\begin{eqnarray*}
q([x+y])&=&[(x+y)^2]\\
&=&[x^2+xy+yx+y^2]\\
&=&[x^2+xy+\ol{xy}+y^2]\\
&=&[x^2+y^2]\\
&=&q([x])+q([y]).
\end{eqnarray*}
\end{proof}
\begin{defi}
Define $C(R):=\mathop{\rm Coker}\nolimits(1+q)$.
\end{defi}
\begin{prop}
If $n$ is even $(\neq0)$ or $n=\infty$, then
$$\lambda\colon H^0(K_1(R_n,{\cal I}_n);t_\alpha)\longrightarrow C(R)$$ defined below is an
isomorphism.
\end{prop}
\begin{proof}
We denote by $1+{\cal I}_n$ the multiplicative group of units in $R_n$, which
are congruent to $1$ modulo ${\cal I}_n$.
According to \cite[theorem 3.2]{Bass-Murphy}
the homomorphism $(1+{\cal I}_n)\rightarrow K_1(R_n,{\cal I}_n)$ determined by
the composition
$$(1+{\cal I}_n)\subset(R_n)^*=\mathop{\rm GL}\nolimits_1(R_n)\rightarrow K_1(R_n,{\cal I}_n)$$ is an isomorphism.
Since this isomorphism respects the involutions we may and will identify
$H^0(K_1(R_n,{\cal I}_n);t)$ and $H^0(1+{\cal I}_n;\ol{\phantom{x}})$.\hfill\break
Define $Z:=\{f\in1+{\cal I}_n\mid f=\overline{f}\}$
and $B:=\{g\overline{g}\mid g\in1+{\cal I}_n\}$.\hfill\break
If $$f\equiv1+aT+bT^2\pmod{T^3}$$ for certain $a,b\in R$, then
$$\ol{f}\equiv 1-\ol aT+(\ol a+\ol b)T^2\pmod{T^3}$$ and
$$f\ol{f}\equiv 1+(a-\ol a)T+(\ol a-a\ol a+b+\ol b)T^2\pmod{T^3}.$$
So $f\in Z$ implies $a=\overline{b}-b$.
It is easy to verify that the map $Z \rightarrow C(R)$ defined by
$f\mapsto [b\ol b]$
vanishes on $B$ and induces a homomorphism
$$\lambda\colon H^0(1+{\cal I}_n;\ol{\phantom{x}}) \rightarrow C(R).$$
Define $$\mu\colon C(R)\rightarrow H^0(1+{\cal I}_n;\ol{\phantom{x}})$$ by
$$[z]\mapsto [1+zT^2/(1+T)].$$
First note that $1+zT^2/(1+T)\in Z$. We will prove that $\mu$ is
well-defined.
If $[z]=0$ in $C(R)$, there exist $x,y\in R$ with $y=\ol y$, such that
$z=x+\ol x+y+y^2$.\hfill\break
Define
$$f:= 1+zT^2/(1+T) \quad\mbox{ and }\quad g:= 1+yT-(x+y)T^2,$$
then
$$g\ol g=1-(x+\ol x+y+y^2)T^2 \quad \mbox{ and } \quad
fg\ol g\equiv1\pmod{T^3}.$$
So we may assume $f\equiv1 \pmod{T^3}$.\hfill\break
We assert that $[h]=1$ for all $h\in Z$ satisfying $h\equiv1\pmod{T^3}$.\hfill\break
By induction we assume $k>0$ and
$$h\equiv1+aT^{2k+1}+bT^{2k+2}\pmod{T^{2k+3}},$$
for certain $a,b\in R$.
Now
$$\ol h\equiv1-\ol aT^{2k+1}+((2k+1)\ol a+\ol b)T^{2k+2}\pmod{T^{2k+3}}.$$
So $h\in Z$ implies $(2k+1)a=\overline b-b$ and $\overline a=-a.$\hfill\break
Defining $$g:= 1+(b+ka)T^{2k+1}-(k+1)bT^{2k+2},$$
yields
\begin{eqnarray*}
g\ol g&\equiv&1+((ka+b)-(k\ol a+\ol b))T^{2k+1}+\\
& &((2k+1)(k\ol a+\ol b)-(k+1)(b+\ol b))T^{2k+2}\\
&\equiv&1-aT^{2k+1}-bT^{2k+2}\pmod{T^{2k+3}}
\end{eqnarray*}
and
$$hg\overline g\equiv1\pmod{T^{2k+3}}.$$
By induction we find $[h]=1$.\hfill\break
Thus $\mu$ is well-defined.
Finally we prove that $\mu=\lambda^{-1}$:\hfill\break
For all $[z]\in C(R),$
$$\lambda\mu([z])=\lambda(1+zT^2/(1+T))=[z\ol z]=[z^2]=[z].$$
For all $f\isdef1+aT+bT^2+\cdots\in Z,$
$$\mu\lambda([f])=\mu([b\ol b])=\left[1+b\ol bT^2/(1+T)\right],$$
But since
$$f^{-1}(1+b\ol bT^2/(1+T))(1+\ol bT)\ol{(1+\ol bT)}\equiv1\pmod{T^3}$$
we may apply the same argument as before to see that $\mu\lambda([f])=[f].$
\end{proof}
\begin{thm}
The composition of homomorphisms
$$\mathop{\rm Arf}\nolimits^h(R,\ol{\phantom{x}},u)\subseteq
L_0^h(R,\ol{\phantom{x}},u)\mapright{\omega_1^h}H^0(K_1(R_n,{\cal I}_n))\mapright{\lambda}C(R)
\longrightarrow R/\kappa(R),$$
is just the Arf invariant $\mathop{\rm Arf}\nolimits^h(R,\ol{\phantom{x}},u)\rightarrow R/\kappa(R)$
defined in section 2 of the first chapter.
Here $C(R)\rightarrow R/\kappa(R)$ is induced by inclusion.
\end{thm}
\begin{proof}
In view of proposition~\ref{prophk1} we have
\begin{eqnarray*}
\lambda\omega_1^h(\plane{a,b})
&=&\lambda\left(\left[1+\frac{\ol abT^2}{1+T}\right]\right) \\
&=&[\overline{a}b].
\end{eqnarray*}
The rest is clear.
\end{proof}
From now on $R$ is not necessarily commutative.
Let $(R,\ol{\phantom{x}},u)$ be a ring with anti-structure.
We wish to prove that the Arf invariant
$$\mathop{\rm Arf}\nolimits^h(R,\ol{\phantom{x}},u)\rightarrow R/\kappa(R),$$
we dealt with in section 2 of chapter I,
factors through the invariant
$$\omega_1^h\colon \mathop{\rm Arf}\nolimits^h(R,\ol{\phantom{x}},u)\longrightarrow H^0(K_1(R_2,{\cal I}_2)).$$
Here follows an attempt to uncover the connection between
$$R/\kappa(R)$$ and the Tate cohomology group $$H^0(K_1(R_2,{\cal I}_2)),$$
in the non-commutative case.
Let us fix the following notations.
\begin{enumerate}
\item[$\cdot$]
$A$ is the truncated polynomial ring $R_2$.
\item[$\cdot$]
${\cal I}$ is the two-sided ideal of $A$ generated by $T$,
\item[$\cdot$]
$\ol{\phantom{x}}\colon A\rightarrow A$ is the extension of $\ol{\phantom{x}}$ on $R$ to $A$ determined by
$$T\mapsto {-T\over 1+T}=-T+T^2,$$
i.e. $\ol{a+bT+cT^2}=\ol{a}-\ol{b}T+(\ol{b}+\ol{c})T^2$.
\item[$\cdot$]
$1+{\cal I}$ denotes the multiplicative group of units in $A$ which are
congruent to $1$ modulo ${\cal I}$.
\item[$\cdot$]
We write $W=W(A,{\cal I})$ for the subgroup of $1+{\cal I}$ generated by
the set $\{(1+ax)(1+xa)^{-1}\,\mid a\in A, x\in {\cal I}\}$.
According to \cite[theorem 2.1]{Swan}
$W$ is the kernel of
the surjection $1+{\cal I}\rightarrow K_1(A,{\cal I})$.
We will identify $K_1(A,{\cal I})$ and $(1+{\cal I})/W$.
\item[$\cdot$]
For all $r,s\in R$ we define $[r,s]:= rs-sr$.
And $R_{{\rm ab}}:= R/[R,R]$ the quotient of $R$ as an additive group
by the subgroup generated by all $[r,s]$.
This is actually the Hochschild homology group $H_0(R)$.
\end{enumerate}
As we saw in section 2 of chapter II the anti-automorphism $\ol{\phantom{x}}$ of $A$
induces an involution $t$ on the relative $K$-group $K_1(A,{\cal I})$.
We want to investigate the structure of the Tate cohomology groups
$H^0(K_1(A,{\cal I}))\cong H^0((1+{\cal I})/W)$.
We proceed to take a close look at the group $W$.
\begin{lemma}
Every element of $W$ has the form
$$1+\left(\sum_i[u_i,v_i]\right)T+
\left(\sum_k[r_k,s_k]+\sum_{i}u_iv_i[u_i,v_i]+\sum_{i<j}[u_i,v_i][u_j,v_j]
\right)T^2.$$
\end{lemma}
\begin{proof}
Substituting $a=a_0+a_1T$ and $x=x_1T+x_2T^2$ in the expression
$(1+ax)(1+xa)^{-1}$ yields
\begin{eqnarray*}
\lefteqn{(1+(a_0+a_1T)(x_1T+x_2T^2))(1+(x_1T+x_2T^2)(a_0+a_1T))^{-1}}\\
&=&(1+a_0x_1T+(a_0x_2+a_1x_1)T^2)(1+x_1a_0T+(x_1a_1+x_2a_0)T^2)^{-1}\\
&=&(1+a_0x_1T+(a_0x_2+a_1x_1)T^2)
(1-x_1a_0T+((x_1a_0)^2-x_1a_1-x_2a_0)T^2)\\
&=&1+[a_0,x_1]T+([a_0,x_2]+[a_1,x_1]+[x_1,a_0]x_1a_0)T^2.
\end{eqnarray*}
When $a_0=0$ we obtain elements like $$1+[r,s]T^2$$ and modulo such elements
we find expressions of the form
$$1+[u,v]T+uv[u,v]T^2.$$
Note that $$(1+[u,v]T+uv[u,v]T^2)^{-1}=1+[v,u]T+vu[v,u]T^2.$$
Thus $W$ is generated by
$$\left\{1+[u,v]T+uv[u,v]T^2,\,1+[r,s]T^2\,\mid r,s,u,v\in R\right\}.$$
Writing out a product of such elements yields the desired result.
\end{proof}
We also need the Hochschild homology group $H_1(R)$.
We refer to chapter III for the definitions.
The Hochschild homology group $H_1(R)$ and
the cyclic homology group $HC_1(R)$ are defined as:
$$H_1(R):=
\frac{\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)}{\mathop{\rm Im}\nolimits(b\colon R\otimes R\otimes R\rightarrow R)}$$
$$HC_1(R):=
\frac{\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)}{\mathop{\rm Im}\nolimits(b\colon R\otimes R\otimes R\rightarrow R)+\mathop{\rm Im}\nolimits(1-x)}\,,$$
where
$$b(u\otimes v):=[u,v],\quad
b(u\otimes v\otimes w):= uv\otimes w-u\otimes vw+wu\otimes v$$
$$\mbox{ and }\quad x\colon R\otimes R\rightarrow R\otimes R
\mbox{ \ is defined by \ } x(u\otimes v):=-v\otimes u.$$
\begin{punt}
Define $\theta\colon R\otimes R\rightarrow R_{{\rm ab}}$ by
$$\theta(\sum_i u_i\otimes v_i):= \sum_{i<j}[u_i,v_i][u_j,v_j]+
\sum_i u_iv_i[u_i,v_i].$$
$\theta$ is well-defined in the sense that the right-hand side
does not depend on the order of summation in $\sum_i u_i\otimes v_i$.
Observe that
$$\theta(x+y)=\theta(x)+\theta(y)+b(x)\cdot b(y).$$
for all $x,y\in R\otimes R$.
So the restriction of $\theta$ to $\mathop{\rm Ker}\nolimits(b)$ is a homomorphism.
Furthermore it is easy to verify that $\theta$ vanishes on $\mathop{\rm Im}\nolimits(b)$
and $\mathop{\rm Im}\nolimits(1-x)$.
Consequently $\theta$ induces a homomorphism
$\theta'\colon HC_1(R)\rightarrow R_{{\rm ab}}$.
\end{punt}
In view of the preceding it is clear that the sequence
$$\diagram{
HC_1(R)&\mapright{\theta'}& R_{{\rm ab}}&\longrightarrow& K_1(A,{\cal I})&\longrightarrow
& R_{{\rm ab}}&\longrightarrow 0\cr
&&[s]&\longmapsto&[1+sT^2]&&&\cr
&&&&[1+aT+bT^2]&\longmapsto&[a]&\cr}$$
is exact.\hfill\break
The anti-automorphism $\ol{\phantom{x}}\colon R\rightarrow R$ induces
an involution on $ R_{{\rm ab}}$:
$$\ol{[u,v]}=[\ol{v},\ol{u}]$$
$$\left[\ol{\ol{r}}\right]=[uru^{-1}]=[r]\quad\mbox{ in } R_{{\rm ab}}.$$
Furthermore $\mathop{\rm Im}\nolimits(\theta')$ is invariant under this involution.
When we equip $ R_{{\rm ab}}$ on the left-hand side with this involution and $ R_{{\rm ab}}$
on the right-hand side with the involution $[a]\mapsto[-\ol{a}]$,
we obtain the short exact sequence of groups with involutions
$$0\longrightarrow\mathop{\rm Coker}\nolimits(\theta')\longrightarrow K_1(A,{\cal I})\longrightarrow R_{{\rm ab}}\lra0$$
which gives rise to the six-term exact sequence
$$\diagram{
H^0(\mathop{\rm Coker}\nolimits(\theta'))&\longrightarrow&H^0(K_1(A,{\cal I}))&\longrightarrow& H^1(R_{{\rm ab}})\cr
\mapup{\delta}&&&&\mapdown{}\cr
H^0(R_{{\rm ab}})&\longleftarrow&H^1(K_1(A,{\cal I}))&\longleftarrow&H^1(\mathop{\rm Coker}\nolimits(\theta')).\cr}$$
We compute the differential map
$\delta\colon H^0( R_{{\rm ab}})\rightarrow H^0(\mathop{\rm Coker}\nolimits(\theta'))$.
\begin{lemma}\label{lemzeshrand}
If $[a]\in H^0( R_{{\rm ab}})$, i.e. $\ol{a}-a=b(x)$ for some $x\in R\otimes R$,
then $$\delta([a])=[a+a\ol{a}+\theta(x)].$$
\end{lemma}
\begin{proof}
The element $1+aT$ is a lift of $a$ in $K_1(A,{\cal I})$.
And in $K_1(A,{\cal I})$ we have
\begin{eqnarray*}
(1+aT)\ol{(1+aT)}&=&(1+aT)(1-\ol{a}T+\ol{a}T^2)\\
&=&1+(a-\ol{a})T+(\ol{a}-a\ol{a})T^2\\
&=&(1+(a-\ol{a})T+(\ol{a}-a\ol{a})T^2)(1+b(x)T+\theta(x)T^2)\\
&=&1+((a-\ol{a})(\ol{a}-a)+\ol{a}-a\ol{a}+\theta(x))T^2\\
&=&1+((a-\ol{a})(\ol{a}-a)+\ol{a}-a\ol{a}+\theta(x))T^2.
\end{eqnarray*}
But this is the image of
\begin{eqnarray*}
[(a-\ol{a})(\ol{a}-a)+\ol{a}-a\ol{a}+\theta(x)]&=&
[\ol{a}+\ol{a}a+\theta(x)]\\
&=&[a+a\ol{a}+\theta(x)]
\end{eqnarray*}
in $H^0(\mathop{\rm Coker}\nolimits(\theta')).$
\end{proof}
Now we specialize to the case that $R$ is the group ring $\Z[G]$
of an arbitrary group $G$.
\begin{lemma}
$\theta'=0$
\end{lemma}
\begin{proof}
Every cycle of $HC_1(R)$ can be written as
$$\left[\sum_i g_i\otimes h_i\right],$$ by using the relation
$g\otimes h+h\otimes g=0$.
The condition for this element to be a cycle reads $\sum g_ih_i=\sum h_ig_i$.
Such an cycle can be decomposed as a sum of cycles of the form
$$[g\otimes h] \quad\mbox{with} \quad gh=hg$$
or of the form
$$\left[\sum_i^n g_i\otimes h_i\right] \quad \mbox{with}\quad
g_ih_i=\cases{h_{i+1}g_{i+1}& for $i<n$\cr
h_1g_1& for $i=n$\cr}.$$
The homomorphism $\theta'$ is obviously zero on elements of the first type.
As far as the second type is concerned we have the following identities
in $R_{{\rm ab}}$
\begin{eqnarray*}
\theta'\left(\left[\sum g_i\otimes h_i\right]\right)&=&
\sum_{i<j}[g_i,h_i][g_j,h_j]+\sum_ig_ih_i[g_i,h_i]\\
&=&\sum_{i<j}g_ih_ig_jh_j+\sum_{i<j}h_ig_ih_jg_j+\sum_ig_ih_ig_ih_i+\\
& &-\sum_{i<j}h_ig_ig_jh_j-\sum_{i<j}g_ih_ih_jg_j-\sum_ig_ih_ih_ig_i\\
&=&\sum_{i<j}g_ih_ig_jh_j+\sum_{i<j}g_ih_ig_jh_j+\sum_ig_ih_ig_ih_i+\\
& &-\sum_{i<j}g_jh_jh_ig_i-\sum_{i<j}g_ih_ih_jg_j-\sum_ig_ih_ih_ig_i\\
&=&\sum_{i,j}g_ih_ig_jh_j-\sum_{i,j}g_jh_jh_ig_i\\
&=&\left(\sum g_ih_i\right)^2-
\left(\sum g_ih_i\right)\left(\sum h_ig_i\right)\\
&=&0
\end{eqnarray*}
This proves the lemma.
\end{proof}
The next move is to figure out what $\delta\colon
H^0( R_{{\rm ab}})\longrightarrow H^0( R_{{\rm ab}})$ looks like in this case.
Suppose we are given an element $[a]\in H^0( R_{{\rm ab}})$.
Then we may assume that $a=\sum g_i$ by using the fact that
$[g+g^{-1}]=0$ in $H^0( R_{{\rm ab}})$.
The condition for $a$ to be a cycle reads
$$\sum g_i-g_i^{-1}=\sum h_j-h_j',$$
where $h_j\in G$ and $h_j'$ is a conjugate of $h_j$.
From this we conclude that every $g_i$ is conjugated to some $g_j^{-1}$.
Note that $[g+h^{-1}g^{-1}h]=[g+g^{-1}]=0$ in $H^0( R_{{\rm ab}})$.
Thus it suffices to consider the case that
$a=g$ where $g=h^{-1}g^{-1}h$.
We follow lemma~\ref{lemzeshrand}.
Now $g^{-1}-g=[h^{-1},gh]$, so
\begin{eqnarray*}
\delta([g])&=&[g+gg^{-1}+\theta([h^{-1}\otimes gh])]\\
&=&[g+1+h^{-1}gh(g^{-1}-g)]\\
&=&[g+1+g^{-1}(g^{-1}-g)]\\
&=&[g+g^{-2}]\\
&=&[g+g^{2}].
\end{eqnarray*}
As a consequence we have
$$\mathop{\rm Coker}\nolimits(\delta)=
\frac{\{a\in \Z[G]_{{\rm ab}}\mid a=\ol{a}\}}%
{\mathop{\rm Span}\nolimits\{g-h^{-1}gh,g_1+g_1^{-1},g_2+g_2^{2}\mid g_2\sim g_2^{-1}\}}.$$
Our main conclusion is that in the case of a group ring the invariant
$$\omega_1^h\colon \mathop{\rm Arf}\nolimits^h(R,\ol{\phantom{x}},u)\longrightarrow H^0(K_1(A,{\cal I}))$$
factors through an injective homomorphism
$$\mathop{\rm Coker}\nolimits(\delta\colon H^0(R_{{\rm ab}})\rightarrow H^0(R_{{\rm ab}}))\lhook\joinrel\longrightarrow
H^0(K_1(A,{\cal I}))$$
and that there is a homomorphism
$$\mathop{\rm Coker}\nolimits(\delta)\longrightarrow R/\kappa(R).$$
\newpage
\section{Computations on the invariant $\omega_2.$}
\setcounter{altel}{0}
\setcounter{equation}{0}
In order to study the invariant $\omega_2$, we wish to compute
the cokernel of the homomorphism
$$d\colon H^1(K_1(R_n,{\cal I}_n);t_\alpha)\rightarrow H^1(K_2(R_n,{\cal I}_n);t_\alpha).$$
We confine our inquiries to the case where $R$ is commutative,
for then we have the following theorem at our disposal.
\begin{thm}
Let $R$ be a commutative ring with identity and $I$ an ideal
contained in the Jacobson radical of $R$.
Then $K_2(R,I)$ is isomorphic to the abelian group with
presentation:\vspace{1mm}
\halign{#&\quad#\hfill&\quad#\hfill&\quad#\hfill\cr
generators:
&$\denstein{a,b}$& with $a\in I$ or $b\in I$\vspace{1mm}\cr
relations:
&$\denstein{a,b}=-\denstein{b,a}$& if $a\in I$ or $b\in I$\vspace{1mm}\cr
&$\denstein{a,b}+\denstein{a,c}=\denstein{a,b+c-abc}$& if
$a\in I$ or $b,c\in I$\vspace{1mm}\cr
&$\denstein{a,bc}=\denstein{ab,c}+\denstein{ac,b}$& if $a\in I$
or $b\in I$ or $c\in I.$\vspace{1mm}\cr}
\noindent The isomorphism maps $\denstein{a,b}$
to the Dennis-Stein element
$\denstein{a,b}_\circ\in K_2(R,I)$.
\end{thm}
\begin{proof}
See \cite{Maazen-Stienstra,Keune}.
\end{proof}
A little digression seems in order. We refer to \cite[\S9]{Milnor}
and \cite{Dennis-Stein} for more background.\hfill\break
Let $n>2$.\hfill\break
For any unit $r\in R$ one has the elements
$w_{ij}(r):= x_{ij}(r)x_{ji}(-r^{-1})x_{ij}(r)$
and $h_{ij}(r):= w_{ij}(r)w_{ij}(-1)$ in $\mathop{\rm St}\nolimits_{n}(R)$,
where $i$ and $j$ are distinct integers between $1$ and $n$.\hfill\break
Further, for every couple of units $r,s\in R$,
$$h_{ij}(rs)h_{ij}^{-1}(r)h_{ij}^{-1}(s)\in \mathop{\rm St}\nolimits_{n}(R)$$
determines an element $\steinberg{r,s}$ in $K_2(R)$,
which does not depend on $i$ or $j$.\hfill\break
And for all $a,b\in R$ such that $1-ab$ is a unit of $R$,
$$x_{ji}(-b(1-ab)^{-1})x_{ij}(-a)x_{ji}(b)x_{ij}(a(1-ab)^{-1})
h_{ij}^{-1}(1-ab)\in \mathop{\rm St}\nolimits_{n}(R)$$
determines the Dennis-Stein element
$\denstein{a,b}_\circ\in K_2(R)$ which does not depend on $i$ or $j$
either. Note the sign conventions.\hfill\break
In $K_2(R)$ the following relations hold,
whenever the left-hand side is defined.
\begin{eqnarray*}
\steinberg{r_1r_2,s}&=&\steinberg{r_1,s}\steinberg{r_2,s}\\
\steinberg{r,s}&=&\steinberg{s,r}^{-1}\\
\steinberg{r,-r}&=&1\\
\steinberg{r,1-r}&=&1\\
\denstc{a,b}&=&\denstc{b,a}^{-1} \\
\denstc{a,b}\denstc{a,c}&=&\denstc{a,b+c-abc}\\
\denstc{a,bc}&=&\denstc{ab,c}\denstc{ac,b}\\
\denstc{0,a}&=&1 \\
\steinberg{r,s}&=&\denstc{(1-r)s^{-1},s}.
\end{eqnarray*}
Note that we used an additive notation in dealing
with the symbols $\denstein{\;,\;}$
and a multiplicative notation for the corresponding
Dennis-Stein elements $\denstc{\;,\;}$.
Nevertheless we will often omit the ${\scriptstyle \circ}$ .
\begin{prop}\label{propk2inv}
Let $R$ be a commutative ring and $\ol{\phantom{x}}\colon R\rightarrow R$ an involution.
The involution $t$ on $K_2(R)$ induced by $\ol{\phantom{x}}$ satisfies
$$t(\denstc{a,b})=\denstc{\ol b,\ol a}.$$
\end{prop}
\begin{proof}
We will work in $\mathop{\rm St}\nolimits_{2n}(R)$.
We drop the decorations of the anti-involution on the Steinberg group
and simply write $t$.
From definition~\ref{defantit} and \ref{involstek12} of the
first chapter we deduce
$$t(x_{ij}(a))=x_{n+j\,n+i}(\ol a),$$
provided that $i$ and $j$ do not exceed $n$.\hfill\break
Thus $t(w_{12}(r))=w_{n+2\,n+1}(\ol r)$ and
\begin{eqnarray}
t(h_{12}^{-1}(r))&=&w_{n+2\,n+1}^{-1}(\ol r)w_{n+2\,n+1}^{-1}(-1)\nonumber\\
&=&w_{n+2\,n+1}(-\ol r)w_{n+2\,n+1}(1)\\
&=&w_{n+1\,n+2}(\ol r^{-1})w_{n+1\,n+2}(-1)\\
&=&h_{n+1\,n+2}(\ol r^{-1})\nonumber
\end{eqnarray}
In (1) we used the relation $w_{ij}(r)=w_{ij}^{-1}(-r)$ and
(2) follows from the relation $w_{ij}(r)=w_{ji}(-r^{-1})$.
See \cite[lemma 9.5]{Milnor}.
Hence
\begin{eqnarray*}
t(\denstc{a,b})
&=&h_{n+1\,n+2}((1-\ol{ab})^{-1})
x_{n+2\,n+1}(\ol a(1-\ol{ab})^{-1})x_{n+1\,n+2}(\ol b)\cdot\\
&&x_{n+2\,n+1}(-\ol a)x_{n+1\,n+2}(-\ol b(1-\ol{ab})^{-1})\\
&=&h_{n+1\,n+2}((1-\ol{ab})^{-1})
\denstc{-\ol b,-\ol a}h_{n+1\,n+2}(1-\ol{ab})\\
&=&\denstc{-\ol b,-\ol a}\steinberg{(1-\ol{ab})^{-1},1-\ol{ab}}\\
&=&\denstc{\ol b,\ol a}\denstc{-\ol{ab},-1}\steinberg{(1-\ol{ab})^{-1},-1}\\
&=&\denstc{\ol b,\ol a}
\end{eqnarray*}
which proves the assertion.
\end{proof}
To make life more congenial,
we will assume $R$ to carry some additional structure.
In that way $H^1(K_2(R_n,{\cal I}_n))$ becomes fairly accessible for computations
by the techniques of \cite{Clauwens;k}.
The following definition occurs implicitly in \cite{Joyal} and
\cite{Joyal;vec}.
It describes a notion of what one could call `partial $\lambda$-ring'.
\begin{defi}\label{deftheta}
Let $R$ be a commutative ring with identity and $k\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\cup\{\infty\}.$
A $k\lambda$-ring structure on $R$ consists of
operations $\theta^p\colon R\rightarrow R$,
for every prime number $p\leq k$,
which satisfy the following conditions\vspace{1mm}
\halign{#\hfill&\quad#\hfill&\quad#\hfill\cr
1)& $\theta^p(1)=0$&for all \, $p\leq k$\vspace{1mm}\cr
2)& $\theta^p(a+b)=\theta^p(a)+\theta^p(b)
+\sum_{k=1}^{p-1}\frac{1}{p}{p\choose k}a^kb^{p-k}$
&for all \, $p\leq k$\vspace{1mm}\cr
3)& $\theta^p(ab)=\theta^p(a)b^p+\theta^p(b)a^p-p\theta^p(a)\theta^p(b)$
&for all \, $p\leq k$\vspace{1mm}\cr
4)& $\theta^p(\psi^q(a))=\psi^q(\theta^p(a))$
&for all \, $p,q\leq k$\vspace{1mm}\cr}
\noindent here
$\psi^q$ is defined by $\psi^q(a):= a^q-q\theta^q(a).$\hfill\break
We then call $R$ an $k\lambda$-ring.
\end{defi}
\begin{remark}
It is easy to verify that multiplication by $p$ transforms the equations 1
to 4 into
\halign{#\hfill&\quad#\hfill&\quad#\hfill\cr
1')& $\psi^p(1)=1$&for all \, $p\leq k$\vspace{1mm}\cr
2')& $\psi^p(a+b)=\psi^p(a)+\psi^p(b)$
&for all \, $p\leq k$\vspace{1mm}\cr
3')& $\psi^p(ab)=\psi^p(a)\psi^p(b)$
&for all \, $p\leq k$\vspace{1mm}\cr
4')& $\psi^p(\psi^q(a))=\psi^q(\psi^p(a))$
&for all \, $p,q\leq k.$\vspace{1mm}\cr}
Thus the so called Adams operations $\psi_p$ are
ringhomomorphisms, which satisfy the compatibility
conditions 4'.\hfill\break
Conversely, if $R$ is a torsion-free commutative ring equipped with
$\psi_p$ satisfying 1' to 4' such that
$\psi_p(a)\equiv a^p\pmod{pR}$ for all $p\leq k$,
then $R$ becomes a $k\lambda$-ring in the
obvious way and the $\psi_p$ are the associated Adams operations.\hfill\break
As far as the references to \cite{Joyal} and \cite{Joyal;vec} are concerned,
a few remarks are in order.
\begin{itemize}
\item[$\cdot$]
We point out the differences in sign conventions between
the definition in \cite{Joyal;vec} and the one above.
\item[$\cdot$]
Condition 4 in our list is equivalent to what is called
the permutability of $\theta_p$ and $\theta_q$ in \cite{Joyal}.
\end{itemize}
\end{remark}
The terminology is explained by the following theorem.
\begin{thm} {\rm \cite[theorem 3]{Joyal}. }
The notions $\lambda$-ring and $\infty\lambda$-ring coincide.
\end{thm}
\begin{lemma}\label{lringext}
Any structure of $k\lambda$-ring on a ring $R$ admits
a unique extension to the rings $R[T]$ and $R_n$
for all $n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\cup\{\infty\}$, under the condition
that $\theta_p(T)=0$ for all $p\leq k$.
\end{lemma}
\begin{proof}
There exists a unique $k\lambda$-ring structure on the ring of integers $\Z$
defined by $\psi_p:= 1$.\hfill\break
Since the polynomial ring $\Z[T]$ has no torsion and the condition
$\theta_p(T)=0$ implies $\psi_p(T)=T^p$, the formula
$\psi_p(\sum a_iT^i)=\sum a_iT^{ip}$, determines a unique
structure of $k\lambda$-ring on $\Z[T]$.\hfill\break
We now call upon \cite[theorem 3]{Joyal;vec}, which reads as follows.
If $R_1$ and $R_2$ are $k\lambda$-rings, then $R_1\otimes R_2$ can be provided
with a unique structure of $k\lambda$-ring, such that the canonical
maps $R_1\rightarrow R_1\otimes R_2$ and $R_2\rightarrow R_1\otimes R_2$ preserve every $\theta_p$.
Applying this theorem in our situation, proves the assertion for the ring
$R[T]=R\otimes \Z[T]$. \hfill\break
From condition 2 in definition~\ref{deftheta} we deduce that
$f\equiv g\pmod{T^lR[T]}$ implies
$\theta_p(f)\equiv\theta_p(g)\pmod{T^lR[T]}$.
Consequently the $k\lambda$-ring structure on $R[T]$ extends
uniquely to the rings $R_n$ for all $n\in {{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\cup\{\infty\}$.
\end{proof}
\begin{defi}
Let $\delta\colon R\rightarrow\Omega_R$ be the universal derivation on $R$ and
define $\Omega_{R_n,{\cal I}_n}:=\mathop{\rm Ker}\nolimits(\Omega_{R_n}\rightarrow\Omega_R).$\hfill\break
Define recursively
\begin{eqnarray*}
\Omega(R,1)&:=&\Omega_R,\\
\Omega(R,n+1)&:=&\Omega(R,n)\oplus
\frac{R\oplus\Omega_R}{\mathop{\rm Span}\nolimits\{((n+1)a,\delta a)\mid a\in R\}}.
\end{eqnarray*}
Define
\[\widetilde{\Omega}(R,n):=\cases{
\Omega(R,n)\oplus
\frac{{\displaystyle R}}{{\displaystyle(n+1)R}}& if $n$ is odd\vspace{1mm}\cr
\Omega(R,n)& if $n$ is even.\cr}\]
Define
\[\widetilde{K_2}(R_n,{\cal I}_n):=\cases{
K_2(R_n,{\cal I}_n)& if $n$ is odd \vspace{1mm}\cr
\frac{{\displaystyle K_2(R_n,{\cal I}_n)}}{{\displaystyle\mathop{\rm Span}\nolimits\{\denstein{aT^n,T}\mid a\in R\}}}&
if $n$ is even.\cr}\]
\end{defi}
\begin{lemma}\label{beromeg}
As $R$-modules
$$\frac{\Omega_{R_n,{\cal I}_n}}{\delta {\cal I}_n}=\Omega(R,n)\oplus\frac{R}{(n+1)R}$$
\end{lemma}
\begin{proof}
Write $J$ for the ideal of $R[T]$ generated by $T^{n+1}$.
We have
$$\Omega_{R_n}=\frac{\Omega_{R[T]}}{J\Omega_{R[T]}+\delta J}$$
and as $R$-modules
$$\Omega_{R[T]}=(R\otimes_\Z\Omega_{{\displaystyle \Z[T]}})\oplus(R[T]\otimes_R\Omega_R),$$
So
$$\Omega_{R_n,{\cal I}_n}=\underbrace{(R\oplus\Omega_R)\oplus\cdots
\oplus(R\oplus\Omega_R)}_{n \quad{\rm copies}}\oplus\frac{R}{(n+1)R}.$$
Dividing out $\delta {\cal I}_n$ yields the desired result.
\end{proof}
We are now in the position to apply the machinery of \cite{Clauwens;k}
to our situation.
As a matter of fact, the construction in {\em loc. cit.}
yields a homomorphism
\[\nu_n\colon K_2(R_n,{\cal I}_n)\rightarrow\frac{\Omega_{R_n,{\cal I}_n}}{\delta {\cal I}_n},\]
even when $R$ possesses a $(n+1)\lambda$-ring structure.
In view of lemma~\ref{beromeg} we obtain a homomorphism
\[\nu_n\colon K_2(R_n,{\cal I}_n)\rightarrow\Omega(R,n)\oplus\frac{R}{(n+1)R}.\]
Furthermore we obtain a homomorphism
\[\widetilde{\nu_n}\colon\widetilde{K_2}(R_n,{\cal I}_n)\rightarrow
\widetilde{\Omega}(R,n)\]
whenever $R$ is a $n\lambda$-ring ($n>1$).
\begin{thm}\label{thmnuniso}
$\nu_n$ and $\widetilde{\nu_n}$ are isomorphisms.
\end{thm}
\begin{proof}
We refer to {\em loc. cit.} for the definitions of the $\nu_n$.
We proceed by applying induction on $n$.\hfill\break
$n=1$:
$R$ is a $2\lambda$-ring and
$\nu_1\colon K_2(R_1,{\cal I}_1)\rightarrow\Omega_R\oplus\frac{R}{2R}$
is determined by
$$\nu_1\denstein{aT,b}=(a\delta b,[a^2\theta^2(b)]),\quad
\nu_1\denstein{cT,T}=(0,[c]).$$
It is straightforward to check that
$\nu_1^{-1}\colon\Omega_R\oplus\frac{R}{2R}\rightarrow K_2(R_1,{\cal I}_1)$
is well defined by
$$\nu_1^{-1}(a\delta b,[c])=\denstein{aT,b}+\denstein{a^2\theta^2(b)T,T}+
\denstein{cT,T}$$
$n>1$:
Consider the diagram
$$\diagram{
&&&&K_2(R_n,{\cal I}_n^n)&&&&\cr
&&&&\mapdown{\tau}&&&&\cr
0&\rightarrow&\frac{{\displaystyle R\oplus\Omega_R}}%
{{\displaystyle (na,\delta a)}}&\mapright{\iota}&
K_2(R_n,{\cal I}_n)&\mapright{\kappa}&
\Omega(R,n-1)\oplus\frac{{R}}{{(n+1)R}}&\rightarrow&0\cr
&&\mapdown{\pi}&&\mapdown{\chi}&&\mapdown{\pi}&&\cr
0&\rightarrow&\frac{{\displaystyle R}}{{\displaystyle nR}}&\mapright{\iota}&
K_2(R_{n-1},{\cal I}_{n-1})&\mapright{\kappa}&\Omega(R,n-1)&\rightarrow&0\cr}$$
Here $\chi$ and $\tau$ are the obvious maps
and $\mathop{\rm Ker}\nolimits(\chi)=\mathop{\rm Im}\nolimits(\tau)$.
In the top row $\kappa$ is the obvious direct summand of $\nu_n$ and
$\iota([a,b\delta c])=\denstein{aT^{n-1},T}+\denstein{bT^n,c}$.
In the bottom row $\kappa$ is the obvious direct summand of $\nu_{n-1}$ and
$\iota([a])=\denstein{aT^{n-1},T}$.
The maps denoted by $\pi$ are the cononical projections.
We compute
\begin{eqnarray*}
&&\nu_n\denstein{aT^n,b}=[0,a\delta b]\in
\frac{R\oplus\Omega_R}{(na,\delta a)}\\
&&\nu_n\denstein{aT^{n-1},T}=[a,0]\in
\frac{R\oplus\Omega_R}{(na,\delta a)}\\
&&\nu_n\denstein{aT^n,T}=[a]\in\frac{R}{(n+1)R}
\end{eqnarray*}
Therefore the map $\iota$ in the top row is split
by the remaining summand of $\nu_n$;
and since $\pi\nu_n=\nu_{n-1}\chi$ this implies that
the map $\iota$ in the bottom row is split
by the remaining summand of $\nu_{n-1}$.
The bottom row is exact by the induction hypothesis.\hfill\break
Suppose that $x\in K_2(R_n,{\cal I}_n)$ and
$$\kappa(x)=0\in\Omega(R,n-1)\oplus\frac{R}{(n+1)R}\quad (\star).$$
Then there exists a $y\in\frac{R\oplus\Omega_R}{(na,\delta a)}$
such that $\iota(\pi(y))=\chi(x).$
The exactness of the column guarantees the existence of an element
$z\in K_2(R_n,{\cal I}_n^n)$ satisfying $x-\iota(y)=\tau(z)$.
Thus there exists an $r\in R$ such that
$x+\denstein{rT^n,T}\in \mathop{\rm Im}\nolimits(\iota)$.
But $[r]=0\in\frac{R}{(n+1)R}$ because of $(\star)$.
So $x\in\mathop{\rm Im}\nolimits(\iota)$. This proves that $\nu_n$
is an isomorphism.
If $n$ is odd and $n>1$, then the notions $n\lambda$-ring and
$(n+1)\lambda$-ring coincide and the preceding proves that
$\widetilde{\nu_n}$
is an isomorphism.
For $n$ even consider the following diagram.
$$\diagram{
0&\rightarrow&{{\displaystyle R\oplus\Omega_R}\over{\displaystyle(na,\delta a)}}&
\mapright{\widetilde\iota}&
\widetilde{K_2}(R_n,{\cal I}_n)&\mapright{\widetilde\kappa}&\Omega(R,n-1)&\rightarrow&0\cr
&&\mapdown{\pi}&&\mapdown{\widetilde\chi}&&\mapdown{ 1}&&\cr
0&\rightarrow&\frac{{\displaystyle R}}{{\displaystyle nR}}&\mapright{\iota}&
K_2(R_{n-1},{\cal I}_{n-1})&\mapright{\kappa}&\Omega(R,n-1)&\rightarrow&0\cr}$$
and proceed as before.
\end{proof}
\begin{cor}{}
If $R$ possesses a structure of $n\lambda$-ring,
then $H^1(K_2(R_n,{\cal I}_n);t)$ is isomorphic to
$H^1(\widetilde\Omega(R,n);\widetilde{\nu_n}t\widetilde{\nu_n}^{-1})$
\end{cor}
\begin{proof}
Note that
$t\denstein{aT^n,T}=\denstein{\ol{a}T^n,T}$ \,in $K_2(R_n,{\cal I}_n)$
and $\denstein{aT^n,T}$ is an odd torsion element of $K_2(R_n,{\cal I}_n)$
if $n$ is even. But odd torsion elements vanish in $H^1(K_2(R_n,{\cal I}_n))$,
so $H^1(K_2(R_n,{\cal I}_n))\cong H^1(\widetilde{K_2}(R_n,{\cal I}_n))$.
In view of the preceding theorem this yields the desired result.
\end{proof}
This enables us to compute these cohomology groups in the cases where
$\widetilde{\nu_n}$ is manageable.
The next theorem for instance, shows what these groups look
like for $n=1$ and $n=2$.
For all abelian groups $A$ and numbers $k$ we write ${}_kA$ to denote
$\{a\in A\mid ka=0\}.$
\begin{thm}
Let $R$ be a $2\lambda$-ring and $\ol{\phantom{x}}\colon R\rightarrow R$ the identity.Then
\begin{eqnarray*}
H^1(K_2(R_1,{\cal I}_1))
&\cong&\frac{R}{2R}\oplus{}_2(\Omega_R)\\
H^1(K_2(R_2,{\cal I}_2))
&\cong&\{\alpha\in{}_2(\Omega_R)\mid
(1+\phi^2)\alpha\in\delta({}_2R)\}\\
&&\oplus\frac{R}{2R}\oplus
{\Omega_R\over2\Omega_R+\delta R+\mathop{\rm Im}\nolimits(1+\phi^2)}
\end{eqnarray*}
where $\phi^2\colon\Omega_R\rightarrow\Omega_R$
is given by $\phi^2(a\delta b)=\psi^2(a)(b\delta b-\delta\theta^2(b)).$
\end{thm}
\begin{proof}
Again we refer to \cite{Clauwens;k}, for more details on the operations
$\phi^2$.
According to proposition~\ref{propk2inv}
$$t(\denstein{aT,b})=\denstein{b,-aT}=\denstein{aT,b}$$ and
$$t(\denstein{aT,T})=\denstein{-T,aT}=\denstein{aT,T}$$
in $K_2(R_1,{\cal I}_1)$.
So in view of the corollary to theorem~\ref{thmnuniso} we have
$$H^1(K_2(R_1,{\cal I}_1))\cong H^1(\frac{R}{2R}\oplus\Omega_R;1).$$
The isomorphism
$$\widetilde{\nu_2}\colon K_2(R_2,{\cal I}_2)\rightarrow\Omega_R\oplus
{R\oplus\Omega_R\over(2a,\delta a)}$$ is given by
\begin{eqnarray*}
\widetilde{\nu_2}(\denstein{aT,b})&=&(a\delta b,[a^2\theta^2(b),
(a^2-\theta^2(a))\delta\theta^2(b)+
\theta^2(a)b\delta b+\theta^2(b)a\delta a]),\\
\widetilde{\nu_2}(\denstein{aT,T})&=&(0,[a,0]),\\
\widetilde{\nu_2}(\denstein{aT^2,b})&=&(0,[0,a\delta b]).
\end{eqnarray*}
Using proposition~\ref{propk2inv} we compute
$$\widetilde{\nu_2}t\widetilde{\nu_2}^{-1}(\alpha,[b,\gamma])=
(\alpha,[-b,-(1+\phi^2)(\alpha)-\gamma]).$$
Hence
$$\mathop{\rm Ker}\nolimits(1+\widetilde{\nu_2}t\widetilde{\nu_2}^{-1})=
\{(\alpha,[b,\gamma])\mid2\alpha=0\hbox{ and }
[0,(1+\phi^2)(\alpha)]=[0,0]\},$$
$$\mathop{\rm Im}\nolimits(1-\widetilde{\nu_2}t\widetilde{\nu_2}^{-1})=
\{(0,[2b,2\gamma+(1+\phi^2)(\alpha)])\}$$
and the quotient of these groups equals the right-hand-side
of the second isomorphism.
\end{proof}
As far as stability is concerned we have:
\begin{prop}
Let $n\neq 0$ be even.
If $R$ is a $(n+2)\lambda$-ring and $\ol{\phantom{x}}=1$, then
$$H^1(\mathop{\rm Ker}\nolimits(\widetilde{K_2}(R_{n+2},{\cal I}_{n+2})\rightarrow
\widetilde{K_2}(R_n,{\cal I}_n)))\;\cong\;
{{}_{n+2}\mathop{\rm Ker}\nolimits(2\delta)\over{}_{n+2}\mathop{\rm Ker}\nolimits(\delta)}\oplus\frac{R}{2R}.$$
\end{prop}
\begin{proof}
Consider the exact sequence
$$0\rightarrow{R\oplus\Omega_R\over((n+1)a,\delta a)}\oplus
{R\oplus\Omega_R\over((n+2)a,\delta a)}
\stackrel{\widetilde\iota}{\longrightarrow}
\widetilde{K_2}(R_{n+2},{\cal I}_{n+2})\rightarrow
\widetilde{K_2}(R_n,{\cal I}_n)\rightarrow 0,$$
where $\widetilde\iota$ is defined by
$$\widetilde\iota([a,b\delta c],[x,y\delta z])=
\denstein{aT^n,T}+\denstein{bT^{n+1},c}
+\denstein{xT^{n+1},T}+\denstein{yT^{n+2},z}.$$
A splitting $\sigma$ of $\widetilde\iota$ is given by
the appropriate direct summand of $\widetilde\nu_{n+2}$.
The involution $t$ on both $\widetilde{K_2}$-groups
induces the involution $\sigma t\widetilde\iota$ on
$${R\oplus\Omega_R\over((n+1)a,\delta a)}\oplus
{R\oplus\Omega_R\over((n+2)a,\delta a)}.$$
A little computation shows that
$$\sigma t\widetilde\iota([a,\alpha],[b,\beta])=
([a,\alpha],[-b,\delta a-(n+1)\alpha-\beta]).$$
Now $([a,\alpha],[b,\beta])\in \mathop{\rm Ker}\nolimits(1+\sigma t\widetilde\iota)$,
if and only if $([2a,2\alpha],[0,\delta a-(n+1)\alpha])=0$.\hfill\break
Thus putting $n=2m$,
there exist $r,s\in R$ satisfying the relations:
\begin{eqnarray*}
2a&=&(2m+1)r,\\
2\alpha&=&\delta r,\\
(2m+2)s&=&0 \mbox{ \ and}\\
\delta s&=&\delta a-(2m+1)\alpha.
\end{eqnarray*}
Hence
$[a,\alpha]=[a,\delta a-\delta s-m\delta r]%
=[a+(2m+1)mr,\delta (a-s)]=[0,-\delta s]=[-s,0]$
and $2\delta s=(2m+2)s=0$.\hfill\break
Conversely, if $[a,\alpha]=[s,0]$ for some $s\in R$ satisfying
$2\delta s=(2m+2)s=0$,
then $([a,\alpha],[b,\beta])\in \mathop{\rm Ker}\nolimits(1+\sigma t\widetilde\iota)$.\hfill\break
The observation that
$\mathop{\rm Im}\nolimits(1-\sigma t\widetilde\iota)=\{([0,0],[2b,\beta'])\}$
completes the proof.
\end{proof}
The final contribution to the comprehension of the value group of $\omega_2$
comes from the following proposition.
\begin{prop} \label{propdber}
{\rm Compare \cite[theorem 4.1.]{Giffen;k2}}.
Let $(R,\ol{\phantom{x}},u)$ be a commutative ring with antistructure.
If $n$ is even,
$$d\colon H^1(K_1(R_n,{\cal I}_n))\longrightarrow H^1(K_2(R_n,{\cal I}_n))$$ assigns to the class
$[x]$ of the element $x\in 1+{\cal I}_n$ the class $[\{x,-u\}]$.
Recall that we identified $H^1(K_1(R_n,{\cal I}_n))$ and $H^1(1+{\cal I}_n)$.
\end{prop}
\begin{proof}
We will work in $\mathop{\rm GL}\nolimits_{2k}(R_n)$ and $\mathop{\rm St}\nolimits_{2k}(R_n)$.\hfill\break
Suppose $x\in 1+{\cal I}_n$ and $\ol x=x^{-1}$.
Let $X$ be the image of $x$ under the map
$1+{\cal I}_n\longrightarrow \mathop{\rm GL}\nolimits_1(R_n)\lhook\joinrel\longrightarrow \mathop{\rm GL}\nolimits_k(R_n)$.
By definition $t_{\ol{\phantom{x}},u_n}(X)X=\pmatrix{X&0\cr 0&X^{-1}}$ and
$h_{1\,k+1}(x)$ is a lift of this element in $\mathop{\rm St}\nolimits_{2k}(R_n)$.
According to lemma~\ref{expld}
$$d([x])=d([X])=
[h_{1\,k+1}^{-1}(x)\,t_{\ol{\phantom{x}},u_n}(h_{1\,k+1}(x))].$$
But from the definition of $t_{\ol{\phantom{x}},u_n}$ we compute
\begin{eqnarray*}
t_{\ol{\phantom{x}},u_n}(h_{1\,k+1}(x))
&=&t_{\ol{\phantom{x}},u_n}(w_{1\,k+1}(x)w_{1\,k+1}(-1))\\
&=&w_{1\,k+1}(-u_n^{-1})w_{1\,k+1}(u_n^{-1}\ol x)\\
&=&w_{1\,k+1}(-u_n^{-1})w_{1\,k+1}(-1)
w_{1\,k+1}(1)w_{1\,k+1}(u_n^{-1}x^{-1})\\
&=&h_{1\,k+1}(-u_n^{-1})h_{1\,k+1}^{-1}(-u_n^{-1}x^{-1}).
\end{eqnarray*}
Thus
\begin{eqnarray*}
d([x])&=&
[h_{1\,k+1}^{-1}(x)h_{1\,k+1}(-u_n^{-1})h_{1\,k+1}^{-1}(-u_n^{-1}x^{-1})]\\
&=&[\{x,u_n\}]\\
&=&[\{x,-u\}\{x,-(1+T)\}].
\end{eqnarray*}
It remains to show that $\steinberg{x,-(1+T)}$ vanishes
in $H^1(K_2(R_n,{\cal I}_n))$.
First note that $\steinberg{x,-u}$ is a cycle:
\begin{eqnarray*}
t(\steinberg{x,-u})&=&t(\denstein{-u^{-1}(1-x),-u})\\
&=&\denstein{-u^{-1},-u(1-x^{-1})}\\
&=&\steinberg{-u^{-1},x^{-1}}\\
&=&\steinberg{x,-u}^{-1}.
\end{eqnarray*}
Now choose $y\in R_n$ such that $1-x^{-1}=yT$.
So $1-x=-\ol{y}T(1+T)^{-1}$.
We compute
\begin{eqnarray*}
(1-t)(\denstein{T,y})&=&\denstein{T,y}\denstein{-T(1+T)^{-1},\ol y}\\
&=&\denstein{T,y}\denstein{T,-(1+T)^{-1}\ol y}
\denstein{-(1+T)^{-1},\ol{y}T}\\
&=&\denstein{T,y-(1+T)^{-1}\ol y+y\ol{y}T(1+T)^{-1}}\cdot\\
& &\denstein{-(1+T)^{-1},(1+T)(x-1)}\\
&=&\denstein{T,y-(1+T)^{-1}\ol y+y\ol{y}T(1+T)^{-1}}\steinberg{x,-(1+T)}.
\end{eqnarray*}
But since
$$(y-(1+T)^{-1}\ol y+y\ol{y}T(1+T)^{-1})T=1-x^{-1}+1-x+(1-x^{-1})(x-1)=0,$$
we have
$$y-(1+T)^{-1}\ol y+y\ol{y}T(1+T)^{-1}=zT^n \mbox{ \ for some \ } z\in R.$$
For $\steinberg{x,-u}$ is a cycle, so is $\denstein{T,zT^n}$.
What's more $\denstein{T,zT^n}$ is an odd torsion element in $K_2(R_n,{\cal I}_n)$,
because $0=\denstein{T^{n+1},z}=(n+1)\denstein{T,zT^n}$ and $n$ is even.
This finishes the proof.
\end{proof}
\begin{cor}{}
If $\; u=-1$ in the situation of proposition~\ref{propdber},
$ d $ is the zero map.
\end{cor}
\begin{punt}
The composition of homomorphisms
$$\mathop{\rm Arf}\nolimits^s(R,1,-1)\lhook\joinrel\longrightarrow
L_0^s(R,1,-1)\stackrel{\lambda\omega_1^s}{\longrightarrow}
C(R)=\frac{R}{\mathop{\rm Span}\nolimits\{x+x^2\mid x\in R\}}$$
maps $\plane{a,b}$ to $[ab]$.
This surjection splits by the homomorphism $[r]\mapsto\plane{r,1}$.\hfill\break
Writing $\widetilde{\mathop{\rm Arf}\nolimits}(R)$ for the kernel,
we obtain a splitting
$$\mathop{\rm Arf}\nolimits^s(R,1,-1)\cong \widetilde{\mathop{\rm Arf}\nolimits}(R)\oplus
\frac{R}{\mathop{\rm Span}\nolimits\{x+x^2\mid x\in R\}}.$$
$\widetilde{\mathop{\rm Arf}\nolimits}(R)$ is
generated by $\arfred{a,b}:=\plane{a,b}+\plane{ab,1}$, where $a,b\in R$.
The following relations hold in $\widetilde{\mathop{\rm Arf}\nolimits}(R)$:\vspace{1mm}
\halign{#&\quad#\hfil&\quad#\hfill\cr
&$\arfred{a,b_1+b_2}=\arfred{a,b_1}+\arfred{a,b_2}$&\vspace{1mm}\cr
&$\arfred{a,b}=\arfred{b,a}$&\vspace{1mm}\cr
&$\arfred{a,b}=0$&for $a\in 2R$\vspace{1mm}\cr
&$\arfred{ax^2,b}=\arfred{a,bx^2}$&for every $x\in R$\vspace{1mm}\cr
&$\arfred{a,b}=\arfred{a,ab^2}$&\vspace{1mm}\cr
&$\arfred{a,1}=0$&\cr}
\end{punt}
The secondary Arf invariant is by definition the
the restriction of $\omega_2$ to the
$\widetilde{\mathop{\rm Arf}\nolimits}$-part of $\mathop{\rm Ker}\nolimits(\omega_1^s)$:
$$\widetilde{\mathop{\rm Arf}\nolimits}(R)\lhook\joinrel\longrightarrow
\mathop{\rm Ker}\nolimits(\omega_1^s)\stackrel{\omega_2}{\longrightarrow}\mathop{\rm Coker}\nolimits(d)=H^1(K_2(R_2,{\cal I}_2)).$$
The next theorem tells us what this invariant looks like for $n=2$.
\begin{thm}
$\omega_2(\arfred{a,b})=[\denstein{aT^2,b}]\in H^1(K_2(R_2,{\cal I}_2))$.
\end{thm}
\begin{proof}
Let $\arfred{a,b}=\plane{a,b}+\plane{ab,1}$ be represented by
$$\left[\pmatrix{a&0&1&0\cr 0&ab&0&1\cr 0&0&b&0\cr 0&0&0&-1\cr}\right]
-\left[\pmatrix{0&0&1&0\cr 0&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr}\right]\in
L_0^s(R,1,-1).$$
A lift of this element in $L_0^s(R_2,\alpha,-(1+T))$ is given by
$$l:=\left[\pmatrix{a&0&1&0\cr 0&ab&0&1\cr 0&0&b&0\cr 0&0&0&-1\cr}\right]
-\left[\pmatrix{0&0&1&0\cr 0&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr}
\right].$$
To apply the map $G$ of definition~\ref{defomegaend} we choose
$$\gamma:= x_{24}(T^2-T)x_{13}(b(T-T^2))
h_{12}(1+abT^2)x_{31}(-aT)x_{42}(-abT)\in \mathop{\rm St}\nolimits_4(R_2)$$
as a lift of
$$\left(\pmatrix{a&0&1&0\cr 0&ab&0&1\cr 0&0&b&0\cr 0&0&0&-1\cr}
+u_2\pmatrix{a&0&0&0\cr 0&ab&0&0\cr 1&0&b&0\cr 0&1&0&-1\cr}\right)
\pmatrix{0&0&u_2^{-1}&0\cr 0&0&0&u_2^{-1}\cr 1&0&0&0\cr 0&1&0&0\cr}$$
$$=\pmatrix{1&0&b(T-T^2)&0\cr 0&1&0&T^2-T\cr
-aT&0&1&0\cr 0&-abT&0&1\cr}\in E_4(R_2).$$
Using the definition of $t$ and the calculations in the proof of
proposition~\ref{propk2inv} we find
$$t\gamma^{-1}=x_{24}(T-T^2)x_{13}(b(T^2-T))h_{34}(1-abT^2)
x_{31}(aT)x_{42}(abT).$$
A little computation shows that
\begin{eqnarray*}
G(l)&=&[\gamma^{-1}(t\gamma)]\\
&=&[\denstein{abT,T-T^2}\denstein{aT,b(T^2-T)}\\
&&h_{12}(1+abT^2)h_{34}(1-abT^2)
h_{42}(1-abT^2)h_{31}(1+abT^2)]\\
&=&[\denstein{aT^2,b}]\in H^1(K_2(R_2,{\cal I}_2)).
\end{eqnarray*}
But since $\omega_2(\arfred{a,b})=G(l)$ this finishes the proof.
\end{proof}
Taking the (primary) Arf invariant into account we have the following result.
\begin{thm}\label{thmtotw}
Let $R$ be a $2\lambda$-ring.
The invariant
$$\mathop{\rm Arf}\nolimits^s(R,1,-1)\rightarrow\frac{R}{\{x+x^2\}}\oplus
\frac{\Omega_R}{2\Omega_R+\delta R+\{x\delta y+x^2y\delta y\mid x,y\in R\}}$$
maps $\plane{a,b}$ to $([ab],[a\delta b])$.
\end{thm}
\begin{proof}
We compute $\phi^2(a\delta b)$ modulo $2\Omega_R+\delta R$:
\begin{eqnarray*}
\phi^2(a\delta b)&\equiv&\psi^2(a)(b\delta b-\delta\theta^2(b))\\
&\equiv&(a^2-2\theta^2(a))(b\delta b-\delta\theta^2(b))\\
&\equiv&a^2b\delta b-a^2\delta\theta^2(b)\\
&\equiv&a^2b\delta b.
\end{eqnarray*}
Thus
$$2\Omega_R+\delta R+\mathop{\rm Im}\nolimits(1+\phi^2)=
2\Omega_R+\delta R+\{x\delta y+x^2y\delta y\mid x,y\in R\}.$$
In view of the preceding the rest is obvious.
\end{proof}
\noindent
Let $R$ be an arbitrary commutative ring.
We recognize $$\frac{\Omega_R}{2\Omega_R+\delta R}$$
as an instance of a cyclic homology group
{\em viz.} $HC_1(R/2R)$.
The assignment $a\mapsto a\delta a$
determines a well-defined homomorphism
$$q'\colon R\rightarrow \frac{\Omega_R}{\delta R}.$$
Under the assumption that $2R=0$
$$\theta\colon R\rightarrow R \qquad x\mapsto x^2$$
$$\theta'\colon \frac{\Omega_R}{\delta R}\rightarrow \mathop{\rm Coker}\nolimits\,q' \qquad
[a\delta b]\mapsto [a^2b\delta b]$$
are well-defined homomorphisms.
From this point of view
$$\frac{R}{\{x+x^2\}}=\mathop{\rm Coker}\nolimits(1+\theta)$$
and
$$\frac{\Omega_R}{2\Omega_R+\delta R+\{x\delta y+x^2y\delta y\mid x,y\in R\}}
=\mathop{\rm Coker}\nolimits(1+\theta').$$
We are a bit sloppy here in \vspace{1mm} denoting the projection
$\frac{{\displaystyle\Omega_R}}{{\displaystyle\delta R}}\rightarrow \mathop{\rm Coker}\nolimits\,q'$ by 1.
These observations are the motivation for investigating
(operations on) cyclic homology groups.
In the next chapter we will construct the homomorphism
\[\mathop{\rm Arf}\nolimits^s(R,1,-1)\rightarrow \frac{R}{\{x+x^2\mid x\in R\}}\oplus
{\Omega_R\over 2\Omega_R+\delta R+\{(r+r^2\delta s)\delta s\mid
r,s\in R\}}\]
without the assumption that $R$ carries some extra structure.
It turns out that the right generalization in the non-commutative case
involves the notion of quaternionic homology groups.
We will enter into details in the next chapter.
\newpage
\section{Examples.}
\setcounter{altel}{0}
\setcounter{equation}{0}
\begin{nitel}{Example}
Let $R=\Z[X,Y]$ be the polynomial ring in two variables.
\begin{thm}\label{thmzxy}
$$L_0^s(R,1,-1)\cong\frac{R}{\{f+f^2\}}\oplus
\frac{\Omega_R}{2\Omega_R+\delta R+
\{f\delta g+f^2g\delta g\mid f,g\in R\}}.$$
\end{thm}
\begin{proof}
First we claim that $L_0^s(R,1,-1)=\mathop{\rm Arf}\nolimits^s(R,1,-1)$.\hfill\break
\noindent Let $(M,[\phi],e)\in BQ(R,1,-1)$ be given.
Then $b_{[\phi]}(m)(m)=0$ for every $m\in M$.
Choose a basis element $f$ in $M$.
There exists an element $g\in M$ such that $b_{[\phi]}(g)=f^*$.
Thus we obtain a decomposition
$$(M,[\phi],e)\cong(N,[\phi_{\mid N}],[f,g])\perp
(N^\perp,[\phi_{\mid N^\perp}],h),$$
where $N:=\mathop{\rm Span}\nolimits(f,g)$,
$N^{\perp}:=\{m\in M\mid b_{[\phi]}(m)(N)=0\}$
and $h$ is some class of bases.
Given the fact that $K_1(R)\cong\Z/2$ it may be necessary
to interchange the roles of $f$ and $g$ to get the right
class of bases at the right hand side.
In this decomposition the first summand is isomorphic to
$$(R^2,\left[\left(\begin{array}{cc}a&1\\0&b\end{array}\right)\right],
[(1,0),(0,1)])$$ for some $a,b\in R$.
An induction argument proves the claim.\hfill\break
Furthermore $R$ has a structure of $\lambda$-ring by lemma~\ref{lringext}.
Next we claim that
$$\frac{R}{\{f+f^2\}}\oplus
\frac{\Omega_R}{2\Omega_R+\delta R+\{f\delta g+f^2g\delta g\mid f,g\in R\}}
\rightarrow\mathop{\rm Arf}\nolimits^s(R,1,-1)$$
defined by $$\left([x],\sum[a\delta b]\right)\longmapsto
\plane{x,1}+\sum\arfred{a,b}$$
is a well defined inverse of the homomorphism
in theorem~\ref{thmtotw}.
The only non-trivial point on our checklist is: show that
this map respects the relation $$a\delta bc+ab\delta c+ac\delta b=0.$$
This amounts to showing that the relation
$$\arfred{ a,bc}=\arfred{ ab,c}+\arfred{ ac,b}$$
holds in $\widetilde{\mathop{\rm Arf}\nolimits}(R)$.
But this follows immediately from the identity
$$\arfred{ f,g }=
\arfred{ f\frac{\partial g}{\partial x},x}+
\arfred{ f\frac{\partial g}{\partial y},y}
\quad\hbox{for every }f,g\in R.$$
It suffices to prove this for monomials by additivity.
By using the relations in $\widetilde{\mathop{\rm Arf}\nolimits}(R)$ we see that
$$\arfred{X^iY^j,X^kY^l}=
\arfred{X^iY^jkX^{k-1}Y^l,X}+
\arfred{X^iY^jX^klY^{l-1},Y}$$
whenever $k$ or $l$ is even.
By symmetry this is also true when $i$ or $j$ is even.
In the remaining case $i$, $j$, $k$ and $l$ are all odd and
\begin{eqnarray*}
\arfred{X^iY^j,X^kY^l}&=&\arfred{XY,X^{i+k-1}Y^{j+l-1}}\\
&=&\arfred{XY,XYX^{i+k-2}Y^{j+l-2}}\\
&=&\arfred{XY,X^{(i+k-2)/2}Y^{(j+l-2)/2}}.
\end{eqnarray*}
An induction argument finishes the proof.
\end{proof}
\end{nitel}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,S\mid S^2=(XS)^2=(YS)^2=1,\quad XY=YX\rangle.$$
We study $\mathop{\rm Arf}\nolimits^s(G)$ and $\mathop{\rm Arf}\nolimits^h(G)$.
Recall that we are working with the anti-involution determined by
$\ol{g}=g^{-1}$ for all $g\in G$.
Let $H$ be the subgroup of $G$ generated by $X$ and $Y$.
These groups fit into the split short exact sequence
$$1\longrightarrow H\longrightarrow G\longrightarrow C_2\longrightarrow 1,$$
where $C_2$ is the group of order two generated by $S$.
Elements of order two in $G$ have the form $X^iY^jS$ for some $i,j\in \Z$.
Every element $f\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]$ can be decomposed in a unique way as
$f=f_-+f_+S$
with $f_-,f_+\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$.
\begin{prop}\label{propsuf1}
$\mathop{\rm Arf}\nolimits^{s,h}(G)$ is generated by the elements
$$\cases{
\plane{1,1}&\vspace{0.4mm}\cr
\plane{X^{2i}Y^{2j+1}S,S} & with $j\geq 0$\vspace{0.4mm}\cr
\plane{X^{2i+1}Y^{2j}S,S}& with $i\geq 0$\vspace{0.4mm}\cr
\plane{X^{2i+1}Y^{2j+1}S,S}& with $i\geq 0$\vspace{0.4mm}\cr
\plane{X^{2i}Y^{2j+1}S,XS} & with $j\geq 0$\vspace{0.4mm}\cr
\plane{X^{2i+1}Y^{2j+1}S,XS}& with $j\geq 0$\vspace{0.4mm}\cr
\plane{X^{2i+1}Y^{2j+1}S,YS} & with $i\geq 0.$\cr
}$$
\end{prop}
\begin{remark}\label{remsuf1}
We say that an element $f\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$ fulfils condition 1 resp. 2 if
all terms $X^iY^j$ of $f$ satisfy $i\geq 0$ resp. $j\geq 0$.
Using the fact that
for each $h\in \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$ there exist unique $h_0,h_1,h_2,h_3\in
\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$ such that $$h=h_0^2+h_1^2x+h_2^2y+h_3^2xy,$$
we can reformulate
proposition~\ref{propsuf1} as follows.
Every element of $\mathop{\rm Arf}\nolimits^{s,h}(G)$
is of the form
$$\plane{fS,S}+\plane{gS,XS}+\plane{hS,YS},$$
with
\begin{enumerate}
\item[$\cdot$]
$f_1,f_3$ satisfy condition 1, $f_2$ satisfies
condition 2 and $f_0\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$
\item[$\cdot$]
$g_2,g_3$ satisfy condition 2 and $g_0=g_1=0$
\item[$\cdot$]
$h_3$ satisfies condition 1 and $h_0=h_1=h_2=0$.
\end{enumerate}
\end{remark}
\begin{lemma}\label{lemsuf1}
Every element of $\mathop{\rm Arf}\nolimits^{s,h}(G)$ is a sum of elements of the form
$$\plane{X^mY^nS,S},\quad
\plane{X^mY^nS,XS} ,\quad
\plane{X^mY^nS,YS} .$$
\end{lemma}
\begin{proof}
It suffices to prove this for generators
$\plane{X^iY^jS,X^kY^lS}$.
Conjugation by $X$ and $Y$ yields
$$\plane{X^iY^jS,X^kY^lS}=\cases{\plane{X^{i\pm2}Y^jS,X^{k\pm2}Y^lS}&\cr
\plane{X^{i}Y^{j\pm2}S,X^{k}Y^{l\pm2}S}.&\cr}$$
This proves that our generator has the desired form whenever one
of the exponents $i$, $j$, $k$ or $l$ is even.\hfill\break
If all exponents are odd, we have
$$\plane{X^iY^jS,X^kY^lS}=\plane{XYS,X^{k-i+1}Y^{l-j+1}S}$$
where both $k-i+1$
and $l-j+1$ are odd. But since
\begin{eqnarray*}
\plane{XYS,X^{2i+1}Y^{2j+1}S}&=&
\plane{XYS,X^{i+1}Y^{j+1}SXYSX^{i+1}Y^{j+1}S}\\
&=&\plane{XYS,X^{i+1}Y^{j+1}S}
\end{eqnarray*}
and $$\plane{XYS,XYS}=\plane{XYS,1}=\plane{1,1}=\plane{S,S},$$
we can use an induction argument to prove the assertion in this case.
\end{proof}
We turn to the proof of the proposition.
\begin{proof}
By lemma~\ref{lemsuf1} it suffices to prove the claim for the elements
$$\plane{X^mY^nS,S},\quad
\plane{X^mY^nS,XS} ,\quad
\plane{X^mY^nS,YS} .$$
\begin{enumerate}
\item[$\diamond$]
$\plane{X^mY^nS,S}$\hfill\break
We may assume that $m$ or $n$ is odd by using the relations
$$\plane{S,S}=\plane{1,1}$$
$$\plane{X^{2m}Y^{2n}S,S}=\plane{X^mY^nSSX^mY^nS,S}=\plane{X^mY^nS,S}.$$
Further we may assume that the odd exponent is positive since
$$\plane{X^{m}Y^{n}S,S}=\plane{SX^mY^nSS,S}=\plane{X^{-m}Y^{-n}S,S}.$$
\item[$\diamond$]
$\plane{X^mY^nS,XS}$\hfill\break
We may assume that $n$ is odd by
$$\plane{X^{2m}Y^{2n}S,XS}=
\plane{S,X^{-2m+1}Y^{-2n}S}=\plane{X^{2m-1}Y^{2n}S,S}$$
\begin{eqnarray*}
\plane{X^{2m+1}Y^{2n}S,XS}&=&
\plane{X^{m+1}Y^{n}SXSX^{m+1}Y^{n}S,XS}\\
&=&\plane{X^{m+1}Y^{n}S,XS}.
\end{eqnarray*}
And we may assume that $n$ is positive since
$$\plane{X^{m}Y^{n}S,XS}=\plane{XSX^mY^nSXS,XS}=
\plane{X^{-m+2}Y^{-n}S,XS}.$$
\item[$\diamond$]
$\plane{X^mY^nS,YS}$\hfill\break
We may assume that $n$ is odd by
$$\plane{X^{2m}Y^{2n}S,YS}=
\plane{X^{2m}Y^{2n-1}S,S}$$
$$\plane{X^{2m+1}Y^{2n}S,YS}=
\plane{XS,X^{-2m}Y^{-2n+1}S}=\plane{X^{-2m}Y^{-2n+1}S,XS}.$$
We may assume that $m$ is odd by the relation
\begin{eqnarray*}
\plane{X^{2m}Y^{2n+1}S,YS}&=&
\plane{X^{m}Y^{n+1}SYSX^{m}Y^{n+1}S,YS}\\
&=&\plane{X^{m}Y^{n+1}S,YS}.
\end{eqnarray*}
And we may assume that $m$ is positive since
$$\plane{X^{m}Y^{n}S,YS}=\plane{YSX^mY^nSYS,YS}=
\plane{X^{-m}Y^{-n+2}S,YS}.$$
\end{enumerate}
This completes the proof.
\end{proof}
The Arf invariant
$$\mathop{\rm Arf}\nolimits^s(G)\longrightarrow
K(G)=\frac{\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]}{\mathop{\rm Span}\nolimits\{a+\ol{a},b+b^2\mid a,b\in \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]\}}$$
which maps
$$\cases{
\plane{X^iY^jS,X^kY^lS}\vspace{.5mm} & to \ $[X^{i-k}Y^{j-l}]$\cr
\plane{X^iY^jS,1}=\plane{1,1} & to \ $[1]$\cr}$$
splits by
$$\cases{
[X^iY^j] &$\mapsto \plane{X^iY^jS,S}$ \cr
[X^iY^jS] &$\mapsto \plane{1,1}.$ \cr}$$
We write $\widetilde{\mathop{\rm Arf}\nolimits}(G)$ for the remaining summand.
Thus $$\mathop{\rm Arf}\nolimits^s(G)\cong\widetilde{\mathop{\rm Arf}\nolimits}(G)\oplus K(G).$$
Observe that the inclusion
$\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]\lhook\joinrel\longrightarrow\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]$ induces an isomorphism
$$K(H)\lhook\joinrel\surarrow K(G),$$
with inverse $[a]\longmapsto [a_-+a_+\ol{a_+}].$\hfill\break
$\widetilde{\mathop{\rm Arf}\nolimits}(G)$ is
generated by
$$\arfred{a,b}:=\plane{a_+S,b_+S}+\plane{a_+\ol{b_+}S,S},$$
where $a=\ol{a},b=\ol{b}$ in $\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]$.
The following relations hold in $\widetilde{\mathop{\rm Arf}\nolimits}(G)$:\vspace*{1mm}
\halign{#&\quad#\hfil&\quad#\hfill\cr
&$\arfred{a,b}=\arfred{a_+S,b_+S}$&\vspace{1mm}\cr
&$\arfred{a,1}=\arfred{1,a}=0$&\vspace{1mm}\cr
&$\arfred{a,S}=\arfred{S,a}=0$&\vspace{1mm}\cr
&$\arfred{a,b_1+b_2}=\arfred{a,b_1}+\arfred{a,b_2}$&\vspace{1mm}\cr
&$\arfred{a,b}=\arfred{b,a}$&\vspace{1mm}\cr
&$\arfred{\ol{c}ac,b}=\arfred{a,cb\ol{c}}$&for every
$c\in \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]$\vspace{1mm}\cr
&$\arfred{a,b}=\arfred{a,\ol{b}ab}$&\vspace*{1mm}\cr}
\noindent
Now we consider the representation $\rho\colon\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[G]\rightarrow M_2(R)$
of $G$ over the ring $R:=\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$
determined by
\begin{eqnarray*}
X&\longmapsto&\pmatrix{X&0\cr0&X^{-1}\cr}\\
Y&\longmapsto&\pmatrix{Y&0\cr0&Y^{-1}\cr}\\
S&\longmapsto&\pmatrix{0&1\cr1&0\cr}
\end{eqnarray*}
and the diagram
$$\diagram{\mathop{\rm Arf}\nolimits^s(G)&
{\buildrel \psi \over {\hbox to 100pt{\rightarrowfill}}}
&\mathop{\rm Arf}\nolimits^s(R,1,1)\cr
\mapdown{\imath}&&\mapdown{\jmath}\cr
L^s(G)&\mapright{\widetilde{\rho}}\quad
L_0^s(M_2(R),\alpha,1)\quad\mapright{\gamma}&
L_0^s(R,1,1).\cr}$$
Here $\imath$ and $\jmath$ are inclusion maps,\hfill\break
$\widetilde{\rho}$ is induced by $\rho$,\hfill\break
$U:=\pmatrix{0&1\cr1&0\cr}$,\hfill\break
$\alpha(A):= UA^tU$ for all $A\in M_2(R)$,\hfill\break
$\gamma$ is the composition of the `scaling-isomorphism'
$$L_0^s(M_2(R),\alpha,1)\mapright{\cong}L_0^s(M_2(R),{\sf transpose},1)$$
and the `Morita-isomorphism'
$$L_0^s(M_2(R),{\sf transpose},1)\mapright{\cong}L_0^s(R,1,1).$$
\begin{lemma}\label{lempsiar}
$\plane{X^iY^jS,X^kY^lS} \stackrel{\psi}{\longmapsto}
\plane{X^{-i}Y^{-j},X^kY^l}+\plane{X^iY^j,X^{-k}Y^{-l}}$.
\end{lemma}
\begin{proof}
$\imath$ maps
$\plane{X^iY^j,X^kY^l}$
to
$$\left[\pmatrix{X^iY^jS&1\cr 0&X^kY^lS\cr}\right]-
\left[\pmatrix{0&1\cr 0&0\cr}\right],$$
$\widetilde{\rho}$ maps this element to
$$\left[\pmatrix{0&X^iY^j&1&0\cr X^{-i}Y^{-j}&0&0&1\cr
0&0&0&X^kY^l\cr0&0&X^{-k}Y^{-l}&0\cr}\right]-
\left[\pmatrix{0&0&1&0\cr0&0&0&1\cr0&0&0&0\cr0&0&0&0\cr}\right],$$
$\gamma$ maps this element to
$$\left[\pmatrix{X^iY^j&0&0&1\cr 0&X^{-i}Y^{-j}&1&0\cr
0&0&X^kY^l&0\cr 0&0&0&X^{-k}Y^{-l}\cr}\right]-
\left[\pmatrix{0&0&0&1\cr 0&0&1&0\cr 0&0&0&0\cr 0&0&0&0\cr}\right].$$
Now we apply the isometry $\pmatrix{U&0\cr 0&I\cr}$.
Note that this isometry is admissible since its
class in $K_1(R)$ is trivial.
This yields
$$\left[\pmatrix{X^{-i}Y^{-j}&0&1&0\cr 0&X^{i}Y^{j}&0&1\cr
0&0&X^kY^l&0\cr 0&0&0&X^{-k}Y^{-l}\cr}\right]-
\left[\pmatrix{0&0&1&0\cr 0&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr}\right].$$
This element is equal to
$\jmath\left(\plane{X^{-i}Y^{-j},X^kY^l}+\plane{X^iY^j,X^{-k}Y^{-l}}\right).$
\end{proof}
Consequently
$$\psi(\arfred{fS,gS})=
\arfred{\ol{f},g}+\arfred{f,\ol{g}}\in\widetilde{\mathop{\rm Arf}\nolimits}(R)$$
for all $f,g\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$.
We are now in the position to apply the machinery of the previous section
and in particular the secondary Arf invariant
$$\widetilde{\mathop{\rm Arf}\nolimits}(R)\rightarrow
\frac{\Omega_R}{\delta R+\{(a+a^2b)\delta b\mid a,b\in R\}}$$
of theorem~\ref{thmtotw}.
\begin{thm}
The invariant
\begin{eqnarray*}
\mathop{\rm Arf}\nolimits^s(G)&\longrightarrow &
\frac{R}{\mathop{\rm Span}\nolimits\{a+\ol{a},b+b^2\mid a,b\in R\}}\oplus
\frac{\Omega_R}{\delta R+\{(a+a^2b)\delta b\mid a,b\in R\}}\\
\plane{fS,gS}&\longmapsto&([f\ol{g}],[\ol{f}\delta g+f\delta\ol{g}]),
\end{eqnarray*}
is injective and the elements mentioned in proposition~\ref{propsuf1}
constitute a basis for $\mathop{\rm Arf}\nolimits^s(G)$.
\end{thm}
\begin{proof}
By the reformulation of proposition~\ref{propsuf1} in remark~\ref{remsuf1}
it suffices to prove that
$$\plane{f'S,S}+\plane{gS,XS}+\plane{hS,YS}=0\quad \Longrightarrow\quad
f'=g=h=0,$$
whenever $f',g,h\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$ satisfy the conditions mentioned in
remark~\ref{remsuf1}.
Suppose $\xi:=\plane{f'S,S}+\plane{gS,XS}+\plane{hS,YS}=0$.
Define $f:= f'+gX^{-1}+hY^{-1}$.
Then
$$\xi=\plane{fS,S}+\arfred{gS,XS}+\arfred{hS,YS}$$
and $f$ still fulfils the condition of
remark~\ref{remsuf1}.
The image
$$([f],[(gX^{-1}+\ol{g}X)X^{-1}\delta X +(hY^{-1}+\ol{h}Y)Y^{-1}\delta Y])$$
of $\xi$ vanishes in
$$\frac{R}{\mathop{\rm Span}\nolimits\{a+\ol{a},b+b^2\mid a,b\in R\}}\oplus
\frac{\Omega_R}{\delta R+\{(a+a^2b)\delta b\mid a,b\in R\}}.$$
We will exploit the following facts to show that $f=g=h=0$.
\begin{enumerate}
\item[$\cdot$]
For each $h\in R$ there are unique $h_0,h_1,h_2,h_3\in R$ such that\hfill\break
$h=h_0^2+h_1^2X+h_2^2Y+h_3^2XY$.
\item[$\cdot$]
If $h\in R$ is symmetric, i.e. $\ol h=h$ and the constant term of $h$
is zero, then $h=p+\ol p$ for some $p\in R$.
\item[$\cdot$]
If $h\in R$ is symmetric,
then $h_0^2$, $h_1^2X$, $h_2^2Y$ and $h_3^2XY$ are symmetric.
\end{enumerate}
The fact that $[f]=0$ guarantees the existence of $a,b\in R$ such that
$$f=a+a^2+b+\ol{b}.$$
This implies: $f_0=0$ and $a_0^2+a^2$ is symmetric.
So $a_0+a=a_0+a_0^2+a_1^2X+a_2^2Y+a_3^2XY$ is symmetric as well.
By applying induction on
$$\max\{|i|+|j|\, \mid X^iY^j \mbox{ is a term of } a+a^2\}$$
we conclude that $a+a^2$ is symmetric.
Hence $f_1^2X+f_2^2Y+f_3^2XY$ is symmetric, but the conditions on
$f_1,f_2,f_3$ make this impossible unless $f=0$.\hfill\break
Since
$[(gX^{-1}+\ol{g}X)X^{-1}\delta X +(hY^{-1}+\ol{h}Y)Y^{-1}\delta Y]=0$
there exist $a,b,c\in R$ such that
$$(gX^{-1}+\ol{g}X)X^{-1}\delta X +(hY^{-1}+\ol{h}Y)Y^{-1}\delta Y
=(a+a^2)X^{-1}\delta X +(b+b^2)Y^{-1}\delta Y +\delta c$$
Since
\begin{eqnarray*}
\delta c&=&\delta(c_0^2+c_1^2X+c_2^2Y+c_3^2XY)\\
&=&c_1^2XX^{-1}\delta X +c_2^2YY^{-1}\delta Y
+c_3^2XYX^{-1}\delta X +c_3^2XYY^{-1}\delta Y ,
\end{eqnarray*}
we may assume that $c_0=0$ and
it follows that
$$gX^{-1}+\ol{g}X=a+a^2+c_1^2X+c_3^2XY,$$
$$hY^{-1}+\ol{h}Y=b+b^2+c_2^2Y+c_3^2XY.$$
Substituting $g=g_2^2Y+g_3^2XY$ and $h=h_3^2XY$ gives us the identities
$$g_2^2X^{-1}Y+g_3^2Y+\ol{g_2^2X^{-1}Y+g_3^2Y}=a+a^2+c_1^2X+c_3^2XY,$$
$$h_3^2X+\ol{h_3^2X}=b+b^2+c_2^2Y+c_3^2XY.$$
From these equations we deduce that
$a_0=a$ and $b_0=b$, thus $a+a^2=b+b^2=0$.
Hence $c_1=c_2=c_3=0$.
But then the restrictions on $g_2$, $g_3$ and $h_3$
imply $g_2=g_3=h_3=0$.
This finishes the proof.
\end{proof}
\end{nitel}
\newpage
{\Large {\bf \begin{center}
Chapter III \vspace{4mm}\\
Hochschild, cyclic and quaternionic homology.
\end{center}}}
\vspace{6mm}
\setcounter{section}{0}
\section{Definitions and notations.}\label{defhomolo}
\setcounter{altel}{0}
\setcounter{equation}{0}
In the fourth section of the previous chapter we explained why we are
interested in constructing certain operations on cyclic homology groups.
We start by summing up
the definitions of the various homologies we need.
We refer to \cite{LQ,Loday} for more details.
Let $k$ denote a commutative ring with identity.
\begin{defi}
A simplicial $k$-module is a series of $k$-modules $\{M_n\mid n\in {{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\},$
endowed with $k$-module homomorphisms
\[d_i\colon M_n\rightarrow M_{n-1}\quad\mbox{ for all }\quad i\in\{0,1,\ldots,n\}\]
\[s_i\colon M_n\rightarrow M_{n+1}\quad\mbox{ for all }\quad i\in\{0,1,\ldots,n\},\]
satisfying
\begin{eqnarray*}
d_id_j&=&d_{j-1}d_i \quad\mbox{ if } i<j\\
d_is_j&=&\cases{
s_{j-1}d_i & if $i<j$ \cr
1 & if $j\leq i\leq j+1$\cr
s_jd_{i-1} & if $i>j+1$ \cr}\\
s_is_j&=&s_{j+1}s_i \quad\mbox{ if } i\leq j.
\end{eqnarray*}
\end{defi}
\begin{defi}
A cyclic $k$-module is a simplicial $k$-module
$\{M_n \mid n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\}$ equipped with homomorphisms
\[x\colon M_n\rightarrow M_n\] satisfying
\begin{eqnarray*}
x^{n+1}&=&1\\
d_ix&=&-xd_{i-1}\quad\mbox{ for all }\quad i\in\{1,\ldots,n\}\\
d_0x&=&(-1)^nd_n\\
s_ix&=&-xs_{i-1}\quad\mbox{ for all }\quad i\in\{1,\ldots,n\}\\
s_0x&=&(-1)^{n+1}x^2s_n.
\end{eqnarray*}
\end{defi}
\begin{defi}
A quaternionic $k$-module consists of a simplicial
$k$-module $\{M_n\mid n\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\}$ and homomorphisms
\[\left\{\begin{array}{l}x\colon M_n\rightarrow M_n\\
y\colon M_n\rightarrow M_n\end{array}\right.\]
satisfying
\[\begin{array}{lcll}
x^{n+1}&=&y^2&\\
xyx&=&y&\\
d_ix&=&-xd_{i-1}&\mbox{ for all }\quad i\in\{1,\ldots,n\}\\
s_ix&=&-xs_{i-1}&\mbox{ for all }\quad i\in\{1,\ldots,n\}\\
d_iy&=&(-1)^nyd_{n-i}&\mbox{ for all }\quad i\in\{0,\ldots,n\}\\
s_iy&=&(-1)^{n+1}ys_{n-i}&\mbox{ for all }\quad i\in\{0,\ldots,n\}.
\end{array}\]
\end{defi}
\begin{defi}
A quaternionic $k$-module is called a dihedral $k$-module when
$y^2=1$.
\end{defi}
\begin{exa}\label{kanex1}
Let $R$ be a $k$-algebra.
We write $R^{n+1}$ as an abbreviation for the (n+1)-fold tensor product
$R\otimes_kR\otimes_k\cdots\otimes_kR.$
The $k$-modules\[M_n:= R^{n+1}\]
and the homomorphisms $d_i$ and $s_i$ determined by
\begin{eqnarray*}
d_i(a_0\otimes\cdots\otimes a_n)&:=&\cases{
a_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_n&for $0\leq i<n$\cr
a_na_0\otimes a_1\otimes\cdots\otimes a_{n-1}&for $i=n$\cr}\\
s_i(a_0\otimes\cdots\otimes a_n)&:=&
a_0\otimes\cdots\otimes a_i\te1\otimes a_{i+1}\otimes\cdots\otimes a_n
\quad\mbox{ for all }0\leq i\leq n
\end{eqnarray*}
constitute a simplicial $k$-module.
The homomorphisms $x\colon R^{n+1}\rightarrow R^{n+1}$ determined by
\[x(a_0\otimes\cdots\otimes a_n):=
(-1)^na_n\otimes a_0\otimes\cdots\otimes a_{n-1}\]
make this simplicial module into a cyclic module.\hfill\break
If in addition $R$ is equipped with an anti-involution of $k$-algebras
$\ol{\phantom{x}}\colon R\rightarrow R$, it even becomes a dihedral module by defining
\[y(a_0\otimes\cdots\otimes a_n):=
(-1)^{\frac{1}{2}n(n+1)}
(\overline{a_0}\otimes\overline{a_n}\otimes\cdots\otimes\overline{a_1}).\]
\end{exa}
\begin{exa}\label{kanex2}
More general, given a $k$-algebra $R$ and a $R$-bimodule $P$
we can turn
\[M_n:= P\otimes_kR^n\]
into a simplicial $k$-module through the homomorphisms
\begin{eqnarray*}
d_i(p\otimes r_1\otimes\cdots\otimes r_n)&:=&\cases{
pr_1\otimes r_2\otimes\cdots\otimes r_n& for $i=0$\cr
p\otimes r_1\otimes\cdots\otimes r_ir_{i+1}\otimes\cdots\otimes r_n& for $0<i<n$\cr
r_np\otimes r_1\otimes\cdots\otimes r_{n-1}& for $i=n$\cr}\\
s_i(p\otimes r_1\otimes\cdots\otimes r_n)&:=&
p\otimes r_1\otimes\cdots\otimes r_i\te1\otimes r_{i+1}\otimes\cdots\otimes r_n \\
&&\mbox{ for } 0\leq i\leq n
\end{eqnarray*}
\end{exa}
\begin{defi}\label{defh}
For every simplicial $k$-module $M_*$ one constructs the chain complex
${\cal B}(M_*)$ called Hochschild complex as follows:
$$\diagram{\cdots\mapright{b}M_{n+1}\mapright{b}M_n\mapright{b}M_{n-1}
\mapright{b}\cdots\mapright{b}M_0}$$
where $$ b:=\sum_{i=0}^n(-1)^id_i.$$
The Hochschild-homology of $M_*$ is by definition the homology
of this chain complex.\hfill\break
In case $M_*$ is the simplicial $k$-module of example~\ref{kanex1}
we denote this chain complex by $(R^*,b)$ and its homology by $H_*(R).$
\end{defi}
\begin{defi}\label{defhc}
If $M_*$ is a cyclic $k$-module one can build a double complex
${\cal C}(M_*)$:
$$\diagram{\vdots&&\vdots&&\vdots&&\vdots&&\cr
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\cr
M_n&\mapleft{1-x}&M_n&\mapleft{L}&M_n&
\mapleft{1-x}&M_n&\mapleft{}&\cdots\cr
\mapdown{b}&&\mapdown{-b'}&&\mapdown{b}&&\mapdown{-b'}&&\cr
M_{n-1}&\mapleft{1-x}&M_{n-1}&\mapleft{L}&M_{n-1}&
\mapleft{1-x}&M_{n-1}&\longleftarrow&\cdots\cr
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\cr
\vdots&&\vdots&&\vdots&&\vdots&& \cr}$$
where \begin{eqnarray*}b&:=&\sum_{i=0}^n(-1)^id_i\\
b'&:=&\sum_{i=0}^{n-1}(-1)^id_i\\
L&:=&\sum_{i=0}^nx^i \end{eqnarray*}
The cyclic homology $HC_n(M_*)$ of $M_*$
is by definition the n-th homology of the total
complex $\mathop{\rm Tot}\nolimits{\cal C}(M_*)$ associated to ${\cal C}(M_*)$, i.e.
\[HC_n(M_*):= H_n(\mathop{\rm Tot}\nolimits{\cal C}(M_*)).\]
In the case that $M_*$ is the cyclic module
of example~\ref{kanex1} we
denote this cyclic homology by \[HC_n(R).\]
\end{defi}
\begin{defi}\label{defhq}
If $M_*$ is a quaternionic module one can build a double complex
${\cal D}(M_*)$ as follows:
\halign{\hfil$#$\hfil&\quad\hfil$#$\hfil&\quad\hfil$#$\hfil&\quad
\hfil$#$\hfil&\quad\hfil$#$\hfil&\quad\hfil$#$\hfil&\quad
\hfil$#$\hfil&\quad\hfil$#$\hfil&\quad\hfill$#$\hfil&\quad\hfil$#$\hfil\cr
\vdots&&\vdots&&\vdots&&\vdots&&\vdots&\cr
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow&\cr
M_n&\stackrel{\alpha}{\leftarrow}&M_n\oplus M_n&\stackrel{\beta}{\leftarrow}
&M_n\oplus M_n&
\stackrel{\gamma}{\leftarrow}&M_n&\stackrel{N}{\leftarrow}
&M_n&\leftarrow\cdots\cr
\mapdown{b}&&\mapdown{ -\widetilde{B}}&
&\mapdown{ \widehat{B}}&&\mapdown{-b'}&&\mapdown{b}&\cr
M_{n-1}&\stackrel{\alpha}{\leftarrow}&M_{n-1}
\oplus M_{n-1}&\stackrel{\beta}{\leftarrow}
&M_{n-1}\oplus M_{n-1}&
\stackrel{\gamma}{\leftarrow}&M_{n-1}&\stackrel{N}{\leftarrow}
&M_{n-1}&\leftarrow\cdots\cr
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow&\cr
\vdots&&\vdots&&\vdots&&\vdots&&\vdots&\cr}
where \begin{eqnarray*}b&:=&\sum_{i=0}^n(-1)^id_i\\
b'&:=&\sum_{i=0}^{n-1}(-1)^id_i\\
\widetilde{B}&:=& \pmatrix{b'&0\cr0&b\cr}\\
\widehat{B}&:=& \pmatrix{b&0\cr0&b'\cr}\\
L&:=&\sum_{i=0}^nx^i\\
N&:=&\sum_{i=0}^3Ly^i\\
\alpha&:=&\pmatrix{1-x&1-y\cr}\\
\beta&:=& \pmatrix{L&1+yx\cr-1-y&x-1\cr}\\
\gamma&:=&\pmatrix{1-x\cr yx-1\cr}
\end{eqnarray*}
The quaternionic homology $HQ_n(M_*)$ of $M_*$
is by definition the n-th homology of the total
complex $\mathop{\rm Tot}\nolimits{\cal D}(M_*)$ associated to ${\cal D}(M_*)$ i.e.
\[HQ_n(M_*):= H_n(\mathop{\rm Tot}\nolimits{\cal D}(M_*)).\]
In the case that $M_*$ is the quaternionic module of example~\ref{kanex1} we
denote this quaternionic homology by \[HQ_n(R).\]
\end{defi}
\newpage
\section{Reduced power operations.}\label{sechomoperaties}
\setcounter{altel}{0}
\setcounter{equation}{0}
In this section we will construct operations on various
low dimensional homology groups. These operations will be used
later on to define new Arf invariants.
We feel that the material in this section is interesting
in its own right.
\begin{nota}
Let $p$ be a fixed prime number for the rest of this section.
For every $n\in {{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$, $I_n$ denotes the set $\{1,2,\ldots,n\}$.
$I_n$ will act as a set of indices.
The symmetric group of degree $p$, $S_p$ acts on the
$p$-fold cartesian product $I_n^p$ of $I_n$ by
$$\tau(i_1,\ldots,i_p):=
(i_{\tau(1)},\ldots,i_{\tau(p)}) \mbox{ \ for all \ }(i_1,\ldots,i_n)\in
I_n^p,\tau\in S_p.$$
Consider the permutation $\sigma:=(1\,2\cdots p)^{-1}$.
Define $\Delta_n:=\{\gamma\in I_n^p\mid \sigma\gamma=\gamma\}$.
Let $\Gamma_n$ denote a set of representatives for the $\sigma$-orbits
of the free action of $\sigma$ on $I_n^p-\Delta_n$.
\end{nota}
Let $R$ be an associative ring with identity.
Now recall the definitions of the Hochschild homology group $H_0(R)$ and
the cyclic homology group $HC_0(R)$.
Observe that both groups are equal to $\mathop{\rm Coker}\nolimits(b)$, where
$b\colon R\otimes R\rightarrow R$
is defined by $$b(r_1\otimes r_2)=r_1r_2-r_2r_1.$$
For all $r\in R$ we denote by $[r]$ the class of $r$ in $H_0(R)$.
\begin{prop}\label{propthh0}
$\theta_p\colon H_0(R)\rightarrow H_0(R/pR)$ defined by
$$\theta_p([r]):=[r^p],$$
is a well-defined homomorphism.
\end{prop}
\begin{proof}
For all maps $\alpha\colon I_n\rightarrow R$ and elements
$\gamma=(i_1,\ldots,i_p)\in I_n^p$, we will write $\gamma(\alpha)$ instead of
$\alpha_{i_1}\alpha_{i_2}\cdots \alpha_{i_p}$.
We assert that
$$
\sum_{k=1}^p\sigma^k\gamma(\alpha)=p\gamma(\alpha)-
b\left(\sum_{l=1}^{p-1}\alpha_{i_1}\cdots \alpha_{i_l}\otimes
\alpha_{i_{l+1}}\cdots \alpha_{i_p}\right).
$$
This is easily verified by writing everything out.
For all $\alpha\colon I_2\rightarrow R$,
the following identity holds in $H_0(R/pR)$:
\begin{eqnarray*}
[(\alpha_{1}+\alpha_{2})^p]
&=&[\alpha_{1}^p+\alpha_{2}^p+
\sum_{\gamma\in I_2^p-\Delta_2}\gamma(\alpha)]\\
&=&[\alpha_{1}^p+\alpha_{2}^p+
\sum_{\gamma\in \Gamma_2}\sum_{k=\;1}^p\sigma^k\gamma(\alpha)]\\
&=&[\alpha_{1}^p+\alpha_{2}^p] \\
&=&[\alpha_{1}^p]+[\alpha_{2}^p].
\end{eqnarray*}
So it suffices to show that
$[(b(\alpha_{1}\otimes \alpha_{2}))^p]=0$ in $H_0(R/pR)$.
Now then:
\begin{eqnarray*}
[(b(\alpha_{1}\otimes \alpha_{2}))^p]&=&
[(\alpha_{1}\alpha_{2}-\alpha_{2}\alpha_{1})^p]\\
&=&[(\alpha_{1}\alpha_{2})^p+(-1)^p(\alpha_{2}\alpha_{1})^p]\\
&=&[\alpha_{1}\alpha_{2}(\alpha_{1}\alpha_{2})^{p-1}-
\alpha_{2}(\alpha_{1}\alpha_{2})^{p-1}\alpha_{1}]\\
&=&[b(\alpha_{1}\otimes \alpha_{2}(\alpha_{1}\alpha_{2})^{p-1})]\\
&=&0
\end{eqnarray*}
This proves the proposition.
\end{proof}
Recall the definitions of the Hochschild homology group $H_1(R)$ and
the cyclic homology group $HC_1(R)$:
$$H_1(R):=
\frac{\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)}{\mathop{\rm Im}\nolimits(b\colon R\otimes R\otimes R\rightarrow R)}$$
$$HC_1(R):=
\frac{\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)}{\mathop{\rm Im}\nolimits(b\colon R\otimes R\otimes R\rightarrow R)+\mathop{\rm Im}\nolimits(1-x)}\,,$$
where
$$b(r_1\otimes r_2)=r_1r_2-r_2r_1,$$
$$b(r_1\otimes r_2\otimes r_3)=r_1r_2\otimes r_3-r_1\otimes r_2r_3+r_3r_1\otimes r_2,$$
$$x\colon R\otimes R\rightarrow R\otimes R
\mbox{ \ is defined by \ } x(r_1\otimes r_2)=-r_2\otimes r_1.$$
For all $\xi\in\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)$, we denote
by $[\xi]$ the class of
$\xi$ in $H_1(R)$ as well as in $HC_1(R)$.\hfill\break
Let $\alpha,\beta\colon I_n\rightarrow R$ be set-theoretic maps.
For every $p$-tuple $\gamma=(i_1,\ldots,i_p)\in I_n^p$ we write
$$\gamma(\alpha,\beta)$$ instead of
$$\alpha_{i_1}\beta_{i_1}%
\alpha_{i_2}\beta_{i_2}\cdots\alpha_{i_{p-1}}\beta_{i_{p-1}}\otimes
\alpha_{i_p}\beta_{i_p}\in R\otimes R.$$
\begin{thm}\label{thmoperaties}
The map $\theta_p\colon H_1(R)\rightarrow HC_1(R/pR)$ determined by
\[
\left[\sum_{i\in I_n}\alpha_i\otimes\beta_i\right]\mapsto
\left[\sum_{i\in I_n}(\alpha_i\beta_i)^{p-1}\alpha_i\otimes\beta_i+
\sum_{\gamma\in\Gamma_n}\sum_{t=1}^{p-1}\left(t\sigma^t\gamma(\alpha,\beta)-
t\sigma^t\gamma(\beta,\alpha)\right)\right]
\]
is a well-defined homomorphism.
\end{thm}
\begin{remark}
In the case that $p=2$ this reads
$\theta_2\colon H_1(R)\rightarrow HC_1(R/2R)$
$$\left[\sum_{i=1}^n\alpha_i\otimes\beta_i\right]\mapsto
\left[\sum_{i=1}^n\alpha_i\beta_i\alpha_i\otimes\beta_i+
\sum_{i<j}\left(\alpha_i\beta_i\otimes \alpha_j\beta_j+
\beta_i\alpha_i\otimes \beta_j\alpha_j\right)\right].$$
\end{remark}
We will prove this theorem with the help of a series of lemmas.
\begin{lemma}\label{lemmacor}
Let $m>1$. For all $r_1,r_2,\ldots,r_m\in R$:
\begin{eqnarray*}
\sum_{i=1}^mr_{i+1}r_{i+2}\cdots r_mr_1r_2\cdots r_{i-1}\otimes r_i&=&
(1-x)(r_1\cdots r_m\te1)\\
&+&b\left(\sum_{i=1}^{m-2}r_{i+2}\cdots r_m\otimes
r_1\cdots r_i\otimes r_{i+1}\right)\\
&+&b(1\otimes r_1\cdots r_{m-1}\otimes r_m)\\
&-&b(1\otimes r_1\cdots r_m\te1)
\end{eqnarray*}
\end{lemma}
\begin{proof}
Simply a matter of writing everything out.
\end{proof}
\begin{cor}{}
For all $\alpha,\beta\colon I_n\rightarrow R$ and $\gamma\in I_n^p$
$$\left[\sum_{t=1}^{p-1}(t\sigma^t\gamma(\alpha,\beta)-
t\sigma^{t+1}\gamma(\alpha,\beta))\right]=
\left[\sum_{t=1}^p\sigma^t\gamma(\alpha,\beta)\right]=0.$$
\end{cor}
\begin{cor}{}
$\theta_p$ does not depend on the choice of $\Gamma_n$.
\end{cor}
\begin{lemma}\label{lemmatens}
Let ${\cal F}(R\times R)$ be the free abelian monoid on the set $R\times R$ and
$\otimes\colon {\cal F}(R\times R)\rightarrow R\otimes R$ be the canonical morphism.
There is a bijective correspondence
between homomorphisms on $\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R)$ and morphisms
on $\mathop{\rm Ker}\nolimits(b\otimes\colon {\cal F}(R\times R)\rightarrow R)$ which kill all elements of the
form \vspace{1mm}\hfill\break
$\begin{array}{ll}(u,0)&u\in R\\
(0,u)&u\in R\\
(u,v+w)+(u,-v)+(u,-w)&u,v,w\in R\\
(u+v,w)+(-u,w)+(-v,w)&u,v,w\in R.
\end{array}$
\end{lemma}
\begin{proof}
To a homomorphism $f$ on $\mathop{\rm Ker}\nolimits(b)$, we associate the morphism $f\otimes$ on
$\mathop{\rm Ker}\nolimits(b\otimes)$. It is clear that this morphism meets all requirements.
Conversely suppose $f$ is a morphism on $\mathop{\rm Ker}\nolimits(b\otimes)$ as in the statement
above.
Define the homomorphism $g$ on $\mathop{\rm Ker}\nolimits(b)$ as follows:
If $\xi=\sum_i\alpha_i\otimes\beta_i$ belongs to
$\mathop{\rm Ker}\nolimits(b)$, we choose
$\eta=\sum_i(\alpha_i,\beta_i)$ as a lift of $\xi$ in $\mathop{\rm Ker}\nolimits(b\otimes)$,
and define $g(\xi):= f(\eta)$.
Let us verify that this is well-defined.
Suppose $\tilde{\eta}=\sum_i(\tilde{\alpha}_i,\tilde{\beta}_i)$
is another lift of $\xi$ in $\mathop{\rm Ker}\nolimits(b\otimes)$.
Consider the difference $\eta-\tilde{\eta}$ in the free abelian group
${\cal F}\!g(R\times R)$.
By definition of the tensor-product, this takes the form:
\begin{eqnarray*}
&&\sum_{k_1}\left\{(u_{k_1}+v_{k_1},w_{k_1})-
(u_{k_1},w_{k_1})-(v_{k_1},w_{k_1})\right\}+\\
&&\sum_{k_2}\left\{(u_{k_2},w_{k_2})+(v_{k_2},w_{k_2})-
(u_{k_2}+v_{k_2},w_{k_2})\right\}+\\
&&\sum_{k_3}\left\{(u_{k_3},v_{k_3}+w_{k_3})-
(u_{k_3},v_{k_3})-(u_{k_3},w_{k_3})\right\}+\\
&&\sum_{k_4}\left\{(u_{k_4},v_{k_4})+(u_{k_4},w_{k_4})-
(u_{k_4},v_{k_4}+w_{k_4})\right\}
\end{eqnarray*}
for certain $u_{k_i},v_{k_i},w_{k_i}\in R.$
As a consequence we have in ${\cal F}(R\times R)$:
\begin{eqnarray*}
\eta&+&\sum_{k_1}\left\{(u_{k_1},w_{k_1})+
(-u_{k_1},w_{k_1})+(0,w_{k_1})\right\}+\\
& &\sum_{k_1}\left\{(v_{k_1},w_{k_1})+
(-v_{k_1},w_{k_1})+(0,w_{k_1})\right\}+\\
& &\sum_{k_2}\left\{(u_{k_2}+v_{k_2},w_{k_2})+
(-u_{k_2},w_{k_2})+(-v_{k_2},w_{k_2})\right\}+\\
& &\sum_{k_2}\left\{2(0,w_{k_2})\right\}+\\
& &\sum_{k_3}\left\{(u_{k_3},v_{k_3})+
(u_{k_3},-v_{k_3})+(u_{k_3},0)\right\}+\\
& &\sum_{k_3}\left\{(u_{k_3},w_{k_3})+(u_{k_3},-w_{k_3})+
(u_{k_3},0)\right\}+\\
& &\sum_{k_4}\left\{(u_{k_4},v_{k_4}+w_{k_4})+
(u_{k_4},-v_{k_4})+(u_{k_4},-w_{k_4})\right\}+\\
& &\sum_{k_4}\left\{2(u_{k_4},0)\right\}=\\
\tilde{\eta}&+&\sum_{k_1}\left\{(u_{k_1}+v_{k_1},w_{k_1})+
(-u_{k_1},w_{k_1})+(-v_{k_1},w_{k_1})\right\}+\\
& &\sum_{k_1}\left\{2(0,w_{k_1})\right\}+\\
& &\sum_{k_2}\left\{(u_{k_2},w_{k_2})+
(-u_{k_2},w_{k_2})+(0,w_{k_2})\right\}+\\
& &\sum_{k_2}\left\{(v_{k_2},w_{k_2})+
(-v_{k_2},w_{k_2})+(0,w_{k_2})\right\}+\\
& &\sum_{k_3}\left\{(u_{k_3},v_{k_3}+w_{k_3})+
(u_{k_3},-v_{k_3})+(u_{k_3},-w_{k_3})\right\}+\\
& &\sum_{k_3}\left\{2(u_{k_3},0)\right\}+\\
& &\sum_{k_4}\left\{(u_{k_4},v_{k_4})+
(u_{k_4},-v_{k_4})+(u_{k_4},0)\right\}+\\
& &\sum_{k_4}\left\{(u_{k_4},w_{k_4})+(u_{k_4},-w_{k_4})+
(u_{k_4},0)\right\}.
\end{eqnarray*}
This implies $f(\eta)=f(\tilde{\eta})$. Hence $g$ is well-defined.
The rest is obvious.
\end{proof}
We want to apply this lemma to the map
$$\tilde{\theta}_p\colon\mathop{\rm Ker}\nolimits(b\otimes)\rightarrow HC_1(R/pR)$$
defined by
$$
\sum_{i\in I_n}(\alpha_i,\beta_i)\mapsto
\left[\sum_{i\in I_n}(\alpha_i\beta_i)^{p-1}\alpha_i\otimes\beta_i+
\sum_{\gamma\in\Gamma_n}\sum_{t=1}^{p-1}\left(t\sigma^t\gamma(\alpha,\beta)-
t\sigma^t\gamma(\beta,\alpha)\right)\right]
$$
But first we need another lemma to show that $\tilde{\theta}_p$
is well-defined in the sense that the formula on the right-hand side
defines a cycle in $HC_1(R/pR)$.
\begin{lemma}
For all $\alpha,\beta\colon I_n\rightarrow R$ with
$\sum_{i\in I_n}(\alpha_i,\beta_i)\in\mathop{\rm Ker}\nolimits(b\otimes)$
$$b\left(\sum_{i\in I_n}(\alpha_i\beta_i)^{p-1}\alpha_i\otimes\beta_i+
\sum_{\gamma\in\Gamma_n}\sum_{t=1}^p\left(t\sigma^t\gamma(\alpha,\beta)-
t\sigma^t\gamma(\beta,\alpha)\right)\right)=0.
$$
\end{lemma}
\begin{proof}
Writing $\ol{\gamma}(\alpha,\beta)$ instead of
$\alpha_{i_1}\beta_{i_1}\cdots \alpha_{i_p}\beta_{i_p}$,
for every $\gamma=(i_1,\ldots,i_p)\in I_n^p$, the expression becomes
\begin{eqnarray*}
\lefteqn{\sum_{\gamma\in\Delta_n}(\ol{\gamma}(\alpha,\beta)-
\ol{\gamma}(\beta,\alpha))+}\hspace{2ex}\\
& &\hspace*{-9ex}\sum_{\gamma\in\Gamma_n}\sum_{t=1}^{p-1}
(t\ol{\sigma^t\gamma}(\alpha,\beta)-
t\ol{\sigma^{t+1}\gamma}(\alpha,\beta)-
t\ol{\sigma^t\gamma}(\beta,\alpha)+
t\ol{\sigma^{t+1}\gamma}(\beta.\alpha))\\
&=&\sum_{\gamma\in\Delta_n}(\ol{\gamma}(\alpha,\beta)-\ol{\gamma}(\beta,\alpha))+
\sum_{\gamma\in\Gamma_n}\sum_{t=1}^p(\ol{\sigma^t\gamma}(\alpha,\beta)-
\ol{\sigma^t\gamma}(\beta,\alpha))\\
&=&\sum_{\gamma\in\Delta_n}(\ol{\gamma}(\alpha,\beta)-\ol{\gamma}(\beta,\alpha))+
\sum_{\gamma\in I_n^p-\Delta_n}(\ol{\gamma}(\alpha,\beta)-\ol{\gamma}(\beta,\alpha))\\
&=&\sum_{\gamma\in I_n^p}(\ol{\gamma}(\alpha,\beta)-\ol{\gamma}(\beta,\alpha))\\
&=&\left(\sum_{i\in I_n}\alpha_i\beta_i\right)^p-
\left(\sum_{i\in I_n}\beta_i\alpha_i\right)^p\\
&=&0.
\end{eqnarray*}
This proves the assertion.
\end{proof}
We proceed by showing that $\tilde{\theta}_p$ is a morphism on
$\mathop{\rm Ker}\nolimits(b\otimes)$.
\begin{punt}
Suppose we are given $\alpha,\beta\colon I_n\rightarrow R$ and
$\alpha',\beta'\colon I_{n'}\rightarrow R$, such that
$$\eta=\sum_{i\in I_n}(\alpha_i,\beta_i) \quad\mbox{ and }\quad
\eta'=\sum_{i\in I_{n'}}(\alpha_i',\beta_i')$$
are in $\mathop{\rm Ker}\nolimits(b\otimes)$.
Let's say
$$r:=\sum_{i\in I_n}\alpha_i\beta_i=\sum_{i\in I_n}\beta_i\alpha_i
\quad\mbox{ and }\quad
r':=\sum_{i\in I_{n'}}\alpha_i'\beta_i'=
\sum_{i\in I_{n'}}\beta_i'\alpha_i'.$$
We identify the disjoint union $I_n\vee I_{n'}$ and $I_{n+n'}$.
Define $\tilde{\alpha}\colon I_{n+n'}\rightarrow R$ by
$$\tilde{\alpha}(i):=\cases{\alpha(i)& if $i\in I_n$\cr
\alpha'(i)&if $i\in I_{n'}$\cr}$$
and define $\tilde{\beta}$ in the same way.
The map $I_{n+n'}\rightarrow I_2$ defined by
$$i\mapsto\cases{1& if $i\in I_n$\cr 2& if $i\in I_{n'}$\cr}$$
induces a map $\pi\colon I_{n+n'}^p\rightarrow I_2^p$ which preserves
the $\sigma$-action.
Therefore
$$\Gamma_{n+n'}=\Gamma_n\cup \Gamma_{n'}\cup
\bigcup_{\lambda\in\Gamma_2}\pi^{-1}(\lambda).$$
Using this terminology we equate
\begin{eqnarray*}
\lefteqn{\tilde{\theta}_p(\eta+\eta')-\tilde{\theta}_p(\eta)
-\tilde{\theta}_p(\eta')}\\
&=&\left[\sum_{t=1}^{p-1}\sum_{\lambda\in\Gamma_2}%
\sum_{\gamma\in\pi^{-1}(\lambda)}%
(t\sigma^t\gamma(\tilde{\alpha},\tilde{\beta})-
t\sigma^t\gamma(\tilde{\beta},\tilde{\alpha}))\right]\\
&=&\left[\sum_{t=1}^{p-1}\sum_{\lambda\in\Gamma_2}%
(t\sigma^t\lambda(\rho)-t\sigma^t\lambda(\rho))\right]\\
&=&0,
\end{eqnarray*}
where $\rho\colon I_2\rightarrow R$ is defined by $\rho(1)=r$ and $\rho(2)=r'$.\hfill\break
And $\lambda(\rho)=\rho_{i_1}\cdots\rho_{i_{p-1}}\otimes \rho_{i_p}$ if
$\lambda=(i_1,\ldots,i_p)\in I_2^p$.
\end{punt}
\begin{punt}
Now it is time to apply lemma~\ref{lemmatens} and show that
$\tilde{\theta}_p$ induces a homomorphism $\theta_p'$ on
$\mathop{\rm Ker}\nolimits(b\colon R\otimes R\rightarrow R):$
\begin{itemize}
\item[$\diamond$]
It is clear that
$\tilde{\theta}_p(u,0)=\tilde{\theta}_p(0,u)=0$, for all $u\in R$.
\item[$\diamond$]
$\tilde{\theta}_p((u,v+w)+(u,-v)+(u,-w))=0,$
for all $u,v,w\in R$:\hfill\break
Define
$\alpha,\beta\colon I_3\rightarrow R$ by
$$\begin{array}{lll}
\alpha(1):= u &\alpha(2):= u& \alpha(3):= u\\
\beta(1):= v+w &\beta(2):= -v& \beta(3):= -w.
\end{array}$$
The map $I_3\rightarrow I_2$ defined by $1\mapsto1,\;\;2\mapsto2,\;\;3\mapsto2$
induces a map $\pi\colon I_3^p\rightarrow I_2^p$ which preserves the
$\sigma$-action.
Define
$\alpha',\beta'\colon I_2\rightarrow R$ by
$$\begin{array}{ll}
\alpha'(1):= u&\alpha'(2):= u\\
\beta'(1):= -v&\beta'(2):= -w.
\end{array}$$
And finally we define
$$\gamma_1:= u\beta_1'u\beta_2'\cdots u\otimes \beta_{i_p}'\qquad
\gamma_2:= \beta_1'u\beta_2'u\cdots \beta_{i_p}'\otimes u$$
for all $\gamma=(i_1,\ldots,i_p)\in I_2^p.$
$$\tilde{\theta}_p((u,v+w)+(u,-v)+(u,-w))=
\tilde{\theta}_p\left(\sum_{i\in I_3}(\alpha_i,\beta_i)\right).$$
\begin{eqnarray*}
\lefteqn{(u(v+w))^{p-1}u\otimes(v+w)-(uv)^{p-1}u\otimes v-(uw)^{p-1}u\otimes w}
\hspace{10ex}\\
&=&-\sum_{\gamma\in I_2}\gamma_1+\sum_{\gamma\in\Delta_2}\gamma_1\\
&=&-\sum_{\gamma\in I_2-\Delta_2}\gamma_1
\phantom{xxxxxxxxxxxxxxxxxxxxxxxxx}\\
&=&-\sum_{\gamma\in\Gamma_2}\sum_{t=1}^p(\sigma^t\gamma)_1
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{\sum_{\gamma\in\Gamma_3}\sum_{t=1}^{p-1}
(t\sigma^t\gamma(\alpha,\beta)-t\sigma^t\gamma(\beta,\alpha))}\hspace{10ex}\\
&=&\sum_{\lambda\in\Gamma_2}\sum_{t=1}^{p-1}
\sum_{\gamma\in\pi^{-1}(\lambda)}
(t\sigma^t\gamma(\alpha,\beta)-t\sigma^t\gamma(\beta,\alpha))\\
&&+\sum_{\gamma\in\Gamma_2}\sum_{t=1}^{p-1}
(t\sigma^t\gamma(\alpha',\beta')-t\sigma^t\gamma(\beta',\alpha'))
\end{eqnarray*}
But for all $\lambda\in \Gamma_2$ we have
\begin{eqnarray*}
\lefteqn{\left[\sum_{\gamma\in\pi^{-1}(\lambda)}
(t\sigma^t\gamma(\alpha,\beta)-t\sigma^t\gamma(\beta,\alpha))\right]}\\
&=&\pm\left[t((u(v+w))^{p-1}\otimes u(v+w)-
((v+w)u)^{p-1}\otimes (v+w)u)\right]\\
&=&0,
\end{eqnarray*}
since $[(ab)^{k}\otimes ab-(ba)^{k}\otimes ba]=0$ in $HC_1(R)$. Further
\begin{eqnarray*}
\lefteqn{\left[\sum_{\gamma\in\Gamma_2}\sum_{t=1}^{p-1}
(t\sigma^t\gamma(\alpha',\beta')-t\sigma^t\gamma(\beta',\alpha'))\right]}\\
&=&\left[\sum_{\gamma\in\Gamma_2}\sum_{t=1}^{p-1}
(t(\sigma^{t+1}\gamma)_2-t(\sigma^t\gamma)_2)\right]\\
&=&\left[-\sum_{\gamma\in\Gamma_2}\sum_{t=1}^{p}
(\sigma^{t}\gamma)_2\right]
\end{eqnarray*}
Conclusion:
\begin{eqnarray*}
\tilde{\theta}_p((u,v+w)+(u,-v)+(u,-w))&=&
\left[-\sum_{\gamma\in\Gamma_2}\sum_{t=1}^p
((\sigma^t\gamma)_1+(\sigma^t\gamma)_2)\right]\\
&=&\left[-\sum_{\gamma\in\Gamma_2}\sum_{t=1}^p
\sigma^t\gamma(\beta',\alpha')\right]\\
&=&0
\end{eqnarray*}
according to the corollary following lemma~\ref{lemmacor}.
\item[$\diamond$]
In a similar way one can prove that
$\tilde{\theta}_p((u+v,w)+(-u,w)+(-v,w))=0,$
for all $u,v,w\in R$.
\end{itemize}
Thus we obtain a homomorphism $\theta_p'\colon\mathop{\rm Ker}\nolimits(b)\longrightarrow HC_1(R/pR)$.
\end{punt}
\begin{prop}\label{prophc}
$\theta_p'(u\otimes v+v\otimes u)=[(uv)^{p-1}\otimes uv]\mbox{ for all }
u,v\in R.$
\end{prop}
\begin{proof}
Define $\alpha,\beta\colon I_2\rightarrow R$ by
$\alpha(1):= u,\;\;\alpha(2):= v,$
$\beta(1):= v,\;\;\beta(2):= u.$
We equate
\begin{eqnarray*}
\lefteqn{\theta_p'(u\otimes v+v\otimes u)}\\
&=&\left[(uv)^{p-1}u\otimes v+(vu)^{p-1}v\otimes u+\sum_{\gamma\in\Gamma_2}
\sum_{t=1}^{p-1}t\sigma^t\gamma(\alpha,\beta)-
t\sigma^t\gamma(\beta,\alpha)\right]
\end{eqnarray*}
The permutation $I_2\rightarrow I_2$ determined by
$1\mapsto2,\;\;2\mapsto1,$ induces a
permutation $\pi\colon I_2^p\rightarrow I_2^p$ which preserves the $\sigma$-action.
Since $\pi\gamma(\alpha,\beta)=\gamma(\beta,\alpha)$ for every
$\gamma\in \Gamma_2,$ the term involving the double sum in the equation
above vanishes.\hfill\break
Adding this to the fact that
$[(uv)^{p-1}u\otimes v+(vu)^{p-1}v\otimes u]=[(uv)^{p-1}\otimes uv]$
proves the proposition.
\end{proof}
\begin{punt}
To finish the proof of theorem~\ref{thmoperaties}
it only remains to show that
$$\theta_p'(uv\otimes w-u\otimes vw+wu\otimes v)=0 \mbox{ \ for all \ } u,v,w\in R.$$
For this purpose we define
$\alpha,\beta\colon I_3\rightarrow R$ by
$$\begin{array}{lll}
\alpha(1):= uv&\alpha(2):= vw&\alpha(3):= wu\\
\beta(1):= w&\beta(2):= u&\beta(3):= v
\end{array}$$
We use proposition~\ref{prophc} to equate
\begin{eqnarray*}
\lefteqn{\theta_p'(uv\otimes w-u\otimes vw+wu\otimes v)}\\
&=&\theta_p'((uv\otimes w+vw\otimes u+wu\otimes v)-(u\otimes vw+vw\otimes u))\\
&=&[(uvw)^{p-1}uv\otimes w+(vwu)^{p-1}vw\otimes u+(wuv)^{p-1}wu\otimes v\\
& &+\sum_{\gamma\in \Gamma_3}\sum_{t=1}^{p-1}
(t\sigma^t\gamma(\alpha,\beta)-t\sigma^t\gamma(\beta,\alpha))-
(uvw)^{p-1}\otimes uvw].
\end{eqnarray*}
The permutation $I_3\rightarrow I_3$ defined by
$1\mapsto3,\;\;2\mapsto1,\;\;3\mapsto2,$ induces a permutation $\pi$
of $I_3^p$ which respects the $\sigma$-action.
Since $\pi\gamma(\alpha,\beta)=\gamma(\beta,\alpha)$ for every
$\gamma\in \Gamma_3,$ the term involving the double sum in the equation
above vanishes.
And because
$$[(uvw)^{p-1}uv\otimes w+(vwu)^{p-1}vw\otimes u+(wuv)^{p-1}wu\otimes v
-(uvw)^{p-1}\otimes uvw]=0,$$
we are done.
\end{punt}
This completes the proof of theorem~\ref{thmoperaties}.
\begin{punt}
Let $B\colon HC_0(R)\rightarrow H_1(R)$ denote the homomorphism determined by
$[r]\mapsto [r\te1+1\otimes r]=[1\otimes r]$.
The composition of $B$ and $\theta_p\colon H_1(R)\rightarrow HC_1(R/pR)$ yields
a homomorphism $q\colon HC_0(R)\rightarrow HC_1(R/pR)$, which,
as a consequence of proposition~\ref{prophc},
maps $[r]$ to $[r^{p-1}\otimes r].$
\end{punt}
\begin{thm}
The homomorphism
$\theta_p\colon H_1(R)\rightarrow HC_1(R/pR)$ induces a homomorphism
$HC_1(R)\rightarrow \mathop{\rm Coker}\nolimits(q).$
\end{thm}
\begin{proof}
This is an immediate consequence of proposition~\ref{prophc}.
\end{proof}
\begin{punt}\label{altdefhq}
Now recall the definition~\ref{defhq}
of the quaternionic
homology group $HQ_1(R)$.
There is an isomorphism
\[HQ_1(R)\lhook\joinrel\surarrow
{\mathop{\rm Ker}\nolimits((b\;\;1-y)\colon(R\otimes R)\oplus R\rightarrow R)\over
\mathop{\rm Span}\nolimits\left\{\begin{array}{c}(r\otimes s+s\otimes r,-rs-\ol{rs}),\\
(u\otimes v+\ol{u}\otimes\ol{v},vu-uv),\\
(0,2(w+\ol{w})),\\
(xy\otimes z-x\otimes yz+zx\otimes y,0)\end{array}\right\}}\]
defined by
$$[\varpi,a,b]\mapsto[\varpi,0,b+a+\ol{a}].$$
Here
$$b\colon R\otimes R\rightarrow R
\mbox{ \ is defined by \ }
b(r_1\otimes r_2)=r_1r_2-r_2r_1,$$
$$y\colon R\rightarrow R \mbox{ \ is defined by \ }y(r)=\ol{r}.$$
\end{punt}
\begin{punt}\label{punteentensx}
The correspondence $x\mapsto[1\otimes x,0]$ obviously defines a
homomorphism $\nu_R\colon R\rightarrow HQ_1(R)$.
\end{punt}
\begin{lemma}\label{lemmaxtensx}
The map $x\mapsto[x\otimes x,0]$
determines a well-defined homomorphism $\mu_R\colon R\rightarrow\mathop{\rm Coker}\nolimits\,\nu_R$.
\end{lemma}
\begin{proof}
For all $x,y\in R$,\hfill\break
$\mu_R(x+y)-\mu_R(x)-\mu_R(y)=[x\otimes y+y\otimes x,0]=[\nu_R(xy)]=0$.
The rest is obvious.
\end{proof}
\begin{thm}\label{thmhqoperatie}
There exists a well-defined homomorphism
$$\vartheta=\vartheta_R\colon
HQ_1(R)\rightarrow\mathop{\rm Coker}\nolimits\,(\mu_{R/2R})={HQ_1(R/2R)\over\mathop{\rm Span}\nolimits\{[x\otimes x,0]\mid
x\in R\}}$$
defined by:
\begin{eqnarray*}
\left[
\sum_{i\in I_n}\alpha_i\otimes \beta_i, c\right]&\mapsto&
\left[\sum_{i\in I_n}\alpha_i\beta_i\alpha_i\otimes \beta_i
+\sum_{i\in I_n}\alpha_i\beta_i\otimes \beta_i\alpha_i\right.\\
& &\hspace*{.5ex}+\left.
\sum_{i<j}(\alpha_i\beta_i+\beta_i\alpha_i)\otimes
(\alpha_j\beta_j+\beta_j\alpha_j)+c\otimes\ol{c},c^2\right]
\end{eqnarray*}
\end{thm}
The proof will come about in a few steps.
\begin{remark}
\begin{eqnarray*}
\vartheta\left(\left[\sum_{i\in I_n}\alpha_i\otimes \beta_i, c\right]\right)
&=&\left[\sum_{i\in I_n}\alpha_i\beta_i\alpha_i\otimes \beta_i
+\left(\sum_{i\in I_n}\alpha_i\beta_i\right)\otimes
\left(\sum_{i\in I_n}\beta_i\alpha_i\right)\right.\\
& &+\left.\sum_{\gamma\in\Gamma_2}(\gamma(\alpha,\beta)-
\gamma(\beta,\alpha))+c\otimes\ol{c},c^2\right].
\end{eqnarray*}
$\sum_{\gamma\in\Gamma_2}[\gamma(\alpha,\beta)-
\sigma\gamma(\alpha,\beta)]=0$,
since $[r\otimes s+s\otimes r,0]=0$ in $\mathop{\rm Coker}\nolimits(\mu_{R/2R})$.
Thus it is clear that $\vartheta$ does not depend on the way the sum
$\sum_{i\in I_n}\alpha_i\otimes \beta_i$, is ordered.
\end{remark}
\begin{lemma}
If $(\sum_{i\in I_n}\alpha_i\otimes \beta_i,c)\in\mathop{\rm Ker}\nolimits(b\;\;1-y)$, then
\begin{eqnarray*}
&&\left(\sum_{i\in I_n}\alpha_i\beta_i\alpha_i\otimes \beta_i+
\alpha_i\beta_i\otimes \beta_i\alpha_i+\right.\\
&&\qquad
\left.\sum_{i<j}(\alpha_i\beta_i+\beta_i\alpha_i)\otimes(\alpha_j\beta_j+
\beta_j\alpha_j)+c\otimes\ol{c},c^2\right)
\end{eqnarray*}
is a cycle in $HQ_1(R/2R)$.
\end{lemma}
\begin{proof}
The image of this expression under the homomorphism $(b\;\;1-y)$ equals
\begin{eqnarray*}
&&\sum_{i\in I_n}(\alpha_i\beta_i)^2+(\beta_i\alpha_i)^2+
\alpha_i\beta_i\beta_i\alpha_i+\beta_i\alpha_i\alpha_i\beta_i+\\
&&\sum_{i<j}(\alpha_i\beta_i+\beta_i\alpha_i)
(\alpha_j\beta_j+\beta_j\alpha_j)+
(\alpha_j\beta_j+\beta_j\alpha_j)(\alpha_i\beta_i+\beta_i\alpha_i)+\\
&&c\ol{c}+\ol{c}c+c^2+\ol{c}^2=\\
&&\sum_{i,j\in I_n}(\alpha_i\beta_i+\beta_i\alpha_i)(\alpha_j\beta_j+
\beta_j\alpha_j)+c\ol{c}+\ol{c}c+c^2+\ol{c}^2=\\
&&(c+\ol{c})^2+c\ol{c}+\ol{c}c+c^2+\ol{c}^2=0
\end{eqnarray*}
\end{proof}
\begin{lemma}\label{lemmaextens}
As before ${\cal F}(R\times R)$ denotes the free abelian monoid on the set
$R\times R$ and
$\otimes\colon {\cal F}(R\times R)\rightarrow R\otimes R$ is the canonical mapping.
Compare lemma~\ref{lemmatens}.
There is a bijective correspondence between homomorphisms on
$$\mathop{\rm Ker}\nolimits((b\;\;1-y)\colon(R\otimes R)\oplus R\rightarrow R)$$
and morphisms on
$$\mathop{\rm Ker}\nolimits\left((b\;\;1-y)\pmatrix{\otimes&0\cr0&1\cr}\right)=
\mathop{\rm Ker}\nolimits((b\!\otimes\;\;1-y)\colon {\cal F}(R\times R)\oplus R\rightarrow R),$$
which kill all elements of the form\vspace{1mm}\hfill\break
$\begin{array}{ll}((u,0),0)&u\in R\\
((0,u),0)&u\in R\\
((u,v+w)+(u,-v)+(u,-w),0)&u,v,w\in R\\
((u+v,w)+(-u,w)+(-v,w),0)&u,v,w\in R.
\end{array}$
\end{lemma}
\begin{proof}
Modulo a few minor adjustments the proof of lemma~\ref{lemmatens} will do.
\end{proof}
We apply this lemma to the map
$\tilde{\vartheta}\colon\mathop{\rm Ker}\nolimits(b\!\otimes\,\;1-y)\rightarrow \mathop{\rm Coker}\nolimits(\mu_{R/2R})$
defined by
\begin{eqnarray*}
\tilde{\vartheta}\left(\sum_{i\in I_n}(\alpha_i,\beta_i),c\right)&=&
\left[\sum_{i\in I_n}\alpha_i\beta_i\alpha_i\otimes \beta_i+
\alpha_i\beta_i\otimes \beta_i\alpha_i+\right.\\
& &\left.
\;\sum_{i<j}(\alpha_i\beta_i+\beta_i\alpha_i)\otimes
(\alpha_j\beta_j+\beta_j\alpha_j)+c\otimes\ol{c},c^2\right].
\end{eqnarray*}
\begin{lemma}
$\tilde{\vartheta}$ is a morphism on $\mathop{\rm Ker}\nolimits(b\!\otimes\;\;1-y)$.
\end{lemma}
\begin{proof}
If $$\eta:=\left(\sum_{i\in I_n}(\alpha_i,\beta_i),c\right)
\quad\mbox{ and }\quad
\eta':=\left(\sum_{i\in I_{n'}}(\alpha_i',\beta_i'),c'\right)$$
are in $\mathop{\rm Ker}\nolimits(b\!\otimes\;\;1-y)$, then
\begin{eqnarray*}
\lefteqn{\tilde{\vartheta}(\eta+\eta')-\tilde{\vartheta}(\eta)-
\tilde{\vartheta}(\eta')}\hspace{2ex}\\
&=&\left[\left(\sum_{i\in I_n}\alpha_i\beta_i+\beta_i\alpha_i\right)\otimes
\left(\sum_{i\in I_{n'}}\alpha_i'\beta_i'+\beta_i'\alpha_i'\right)+\right.\\
& &\left.(c+c')\otimes(\ol{c}+\ol{c'})+c\otimes\ol{c}+c'\otimes\ol{c'},
(c+c')^2+c^2+{c'}^2\right]\\
&=&\left[(c+\ol{c})\otimes(c'+\ol{c'})+c\otimes\ol{c'}+c'\otimes\ol{c},cc'+c'c\right]\\
&=&\left[c\otimes c'+\ol{c}\otimes c'+c'\otimes\ol{c}+\ol{c}\otimes\ol{c'},cc'+c'c\right]\\
&=&0
\end{eqnarray*}
which proves the lemma.
\end{proof}
The next step is to prove that $\tilde{\vartheta}$ induces a homomorphism
$\vartheta'$ on
$\mathop{\rm Ker}\nolimits(b\;\;1-y)$.
\begin{punt}
We use lemma~\ref{lemmaextens}:
\begin{enumerate}
\item[$\diamond$]
It is clear that
$\tilde{\vartheta}((u,0),0)=\tilde{\vartheta}((0,u),0)=0$
for all $u\in R$.
\item[$\diamond$]
For all $u,v,w\in R$,
\begin{eqnarray*}
\lefteqn{\tilde{\vartheta}((u,v+w)+(u,-v)+(u,-w),0)}\hspace{4ex}\\
&=&[u(v+w)u\otimes(v+w)+uvu\otimes v+uwu\otimes w+\\
& &\;u(v+w)\otimes(v+w)u+uv\otimes vu+uw\otimes wu+\\
& &\;(u(v+w)+(v+w)u)\otimes(uv+vu)+\\
& &\;(u(v+w)+(v+w)u)\otimes(uw+wu)+\\
& &\;(uv+vu)\otimes(uw+wu),0]\\
&=&[uvu\otimes w+uwu\otimes v+uv\otimes wu+uw\otimes vu+\\
& &\;(uv+vu)\otimes(uw+wu)]\\
&=&0.
\end{eqnarray*}
\item[$\diamond$]
In the same fashion one proves that
$\tilde{\vartheta}((u+v,w)+(-u,w)+(-v,w),0)=0$
for all $u,v,w\in R$.
\end{enumerate}
\end{punt}
Finally we use \ref{altdefhq} to verify that $\vartheta'$ induces
the promised homomorphism $\vartheta$.
\begin{punt}
For all $r,s,u,v,w,x,y,z\in R$ we have
\begin{eqnarray*}
\lefteqn{\vartheta'(r\otimes s+s\otimes r,rs+\ol{rs})}\hspace{5ex}\\
&=&[rsr\otimes s+rs\otimes sr+srs\otimes r+sr\otimes rs+\\
& &\;(rs+sr)\otimes(rs+sr)+(rs+\ol{rs})\otimes(rs+\ol{rs}),(rs+\ol{rs})^2]\\
&=&[0,rsrs+rs\ol{rs}+\ol{rs}rs+\ol{rs}\ol{rs}]\\
&=&0,\\
\lefteqn{\vartheta'(u\otimes v+\ol{u}\otimes\ol{v},vu-uv)}\hspace{5ex}\\
&=&[uvu\otimes v+\ol{u}\ol{v}\ol{u}\otimes\ol{v}+uv\otimes vu+
\ol{u}\ol{v}\otimes\ol{v}\ol{u}+\\
& &\;(uv+vu)\otimes(\ol{u}\ol{v}+\ol{v}\ol{u})+(uv+vu)\otimes(\ol{uv+vu}),
(uv+vu)^2]\\
&=&[uvu\otimes v+\ol{u}\ol{v}\ol{u}\otimes\ol{v}+uv\otimes vu+
\ol{u}\ol{v}\otimes\ol{v}\ol{u},(uv+vu)^2]\\
&=&0,\\
\lefteqn{\vartheta'(xy\otimes z+x\otimes yz+zx\otimes y,0)}\hspace{5ex}\\
&=&[xyzxy\otimes z+xyzx\otimes yz+zxyzx\otimes y+\\
& &\;xyz\otimes zxy+xyz\otimes yzx+zxy\otimes yzx+\\
& &\;(xyz+zxy)\otimes(xyz+yzx)+\\
& &\;(xyz+zxy)\otimes(zxy+yzx)+\\
& &\;(xyz+yzx)\otimes(zxy+yzx),0]\\
&=&0,\\
\lefteqn{\vartheta'(0,2(w+w'))=0.}\hspace*{5ex}
\end{eqnarray*}
\end{punt}
This step completes the proof of theorem~\ref{thmhqoperatie}.
\newpage
\section{Morita invariance.}
\setcounter{altel}{0}
\setcounter{equation}{0}
\begin{thm}\label{thmmorita}
Let $A$ be the ring of $m\times m$-matrices over the $k$-algebra $R$.
The trace-maps $\mathop{\rm Tr}\nolimits\colon A^n\rightarrow R^n$ determined by
\[\mathop{\rm Tr}\nolimits(X_1\otimes X_2\otimes\cdots\otimes X_n):=
\sum_{i_1,\ldots,i_n}\left(X_1\right)_{i_1i_2}
\otimes\left(X_2\right)_{i_2i_3}\otimes\cdots\otimes\left(X_n\right)_{i_ni_1}\]
yield a chain equivalence between the Hochschild complexes
$(A^*,b)$ and $(R^*,b)$.
A chain inverse is given by the maps $\iota\colon R^n\rightarrow A^n$ defined by
\[\iota(r_1\otimes r_2\otimes\cdots\otimes r_n):= E_{11}(r_1)\otimes
\cdots \otimes E_{11}(r_n)\]
Where $E_{ij}(r)$
denotes the $m\times m$-matrix with $r$ in the
$(i,j)$-entry and zeros in all other entries.
\end{thm}
\begin{proof}
It's easy to check that $\mathop{\rm Tr}\nolimits$ and $\iota$ are chain maps.
We immediately see that $\mathop{\rm Tr}\nolimits\lower1.0ex\hbox{$\mathchar"2017$}\iota=1$.
We will show that $\iota\lower1.0ex\hbox{$\mathchar"2017$}\mathop{\rm Tr}\nolimits\simeq1$ simply by giving a chain homotopy.
For that purpose we proceed to introduce the following definitions:\hfill\break
Define
$$\gamma\colon A^{n+1}\rightarrow A^{n+1}$$ by
$$\gamma(X_0\otimes X_1\otimes\cdots\otimes X_n):=
(-1)^{n+1}\sum_{i=1}^m E_{i1}(1)\otimes E_{1i}(1)X_nX_0\otimes X_1\otimes\cdots\otimes
X_{n-1},$$
$$s\colon A^n\rightarrow A^{n+1}$$
by
$$s(X_1\otimes\cdots\otimes X_n):= X_1\otimes\cdots\otimes X_n\otimes 1$$
and finally
$$\chi_n\colon A^n\rightarrow A^{n+1}$$
by
$$\chi_n:= (-1)^{n+1}\sum_{k=1}^n\gamma^ks.$$
The following relations are valid:
\setcounter{equation}{0}
\begin{eqnarray}
\sum_{i=1}^mE_{i1}(1)E_{1i}(1)&=&1\\
d_0\gamma&\stackrel{1}{=}&(-1)^{n+1}d_n \nonumber\\
d_0\gamma^k&=&(-1)^{n+1}d_n\gamma^{k-1} \mbox{ \ if \ } k>0\\
d_i\gamma&=&-\gamma d_{i-1} \quad\mbox{ \ if \ } 1\leq i<n\\
d_i\gamma^k&=&=(-1)^k\gamma^kd_{i-k}\mbox{ \ if \ }k\leq i<n\\
d_1\gamma^2&\stackrel{3}{=}&-\gamma d_0\gamma \\
&\stackrel{2}{=}&(-1)^n\gamma d_n \nonumber\\
&=&(-1)^n\gamma d_{n-1} \nonumber\\
d_i\gamma^k&\stackrel{4}{=}&(-1)^{i-1}\gamma^{i-1}d_1\gamma^{k-i+1}\\
&\stackrel{5}{=}&(-1)^{n+i-1}\gamma^id_{n-1}\gamma^{k-i-1} \nonumber\\
&=&(-1)^{n+k}\gamma^{k-1}d_{n+i-k}\mbox{ \ if \ }0<i<k\nonumber\\
\gamma sd_n&=&\gamma d_ns\\
E_{1i}(1)XE_{j1}(1)&=&E_{11}(X_{ij})\\
d_n\gamma^ns&\stackrel{8}{=}&\iota\lower1.0ex\hbox{$\mathchar"2017$}\mathop{\rm Tr}\nolimits
\end{eqnarray}
Now we are in the position to prove that
$$b\chi_n+\chi_{n-1}b=1-\iota\mathop{\rm Tr}\nolimits:$$
\begin{eqnarray*}
b\chi_n
&=&(-1)^{n+1}\sum_{k=1}^n\sum_{i=0}^n(-1)^id_i\gamma^ks\\
&=&(-1)^{n+1}(\sum_{k=1}^n(d_0\gamma^ks+(-1)^nd_n\gamma^ks)+
\sum_{k=1}^n\sum_{i=1}^{n-1}(-1)^id_i\gamma^ks)\\
&\stackrel{2}{=}&1-d_n\gamma^ns+\\
& &(-1)^{n+1}(\sum_{k=1}^{n-1}\sum_{i=k}^{n-1}(-1)^id_i\gamma^ks+
\sum_{k=2}^n\sum_{i=1}^{k-1}(-1)^id_i\gamma^ks)\\
&\stackrel{4\,6}{=}&1-\iota\mathop{\rm Tr}\nolimits+\\
& &(-1)^{n+1}(\sum_{k=1}^{n-1}\sum_{i=k}^{n-1}(-1)^{i+k}\gamma^kd_{i-k}s+
\sum_{k=2}^{n}\sum_{i=1}^{k-1}(-1)^{n+i+k}\gamma^{k-1}d_{n+i-k}s)\\
&=&1-\iota\mathop{\rm Tr}\nolimits+\\
& &(-1)^{n+1}(\sum_{k=1}^{n-1}\sum_{m=0}^{n-k-1}(-1)^m\gamma^kd_ms+
\sum_{k=2}^{n}\sum_{m=n-k+1}^{n-1}(-1)^m\gamma^{k-1}d_ms)\\
&=&1-\iota\mathop{\rm Tr}\nolimits+
(-1)^{n+1}(\sum_{k=1}^{n-1}\sum_{m=0}^{n-1}(-1)^m\gamma^kd_ms)\\
&\stackrel{7}{=}&1-\iota\mathop{\rm Tr}\nolimits+
(-1)^{n+1}(\sum_{k=1}^{n-1}\sum_{m=0}^{n-1}(-1)^m\gamma^ksd_m)\\
&=&1-\iota\mathop{\rm Tr}\nolimits-\chi_{n-1}b.
\end{eqnarray*}
\end{proof}
Let $\ol{\phantom{x}}\colon R\rightarrow R$ be an anti-involution of $k$-algebras.
We extend this anti-involution to an anti-involution $\ol{\phantom{x}}\colon A\rightarrow A$
by defining $(\overline{X})_{ij}=\overline{X_{ji}}$ for every $X\in A$.
According to example~\ref{kanex1}
we may regard
both $R^*$ and $A^*$ as quaternionic modules.
\begin{thm}
The map $\mathop{\rm Tr}\nolimits$ induces isomorphisms
$$H_*(A)\mapright{\mathop{\rm Tr}\nolimits} H_*(R)$$
$$HC_*(A)\mapright{\mathop{\rm Tr}\nolimits} HC_*(R)$$
$$HQ_*(A)\mapright{\mathop{\rm Tr}\nolimits} HQ_*(R)$$
\end{thm}
\begin{proof}
It is clear from the definitions that both $\iota$ and $\mathop{\rm Tr}\nolimits$ preserve
$x$ and $y$.
\end{proof}
\begin{thm}\label{thmmorhq}
The following diagrams commute
\begin{itemize}
\item[$\diamond$]
$$\diagram{HC_0(A)&\mapright{B}&H_1(A)\cr
\mapdown{\mathop{\rm Tr}\nolimits}&&\mapdown{\mathop{\rm Tr}\nolimits}\cr
HC_0(R)&\mapright{B}&H_1(R)\cr}$$
\item[$\diamond$]
$$\diagram{H_0(A)&\mapright{\theta_p}&H_0(A/pA)\cr
\mapdown{\mathop{\rm Tr}\nolimits}&&\mapdown{\mathop{\rm Tr}\nolimits}\cr
H_0(R)&\mapright{\theta_p}&H_0(R/pR)\cr}$$
\item[$\diamond$]
$$\diagram{
HC_1(A)&\mapright{\theta_p}&HC_1(A/pA)/\mathop{\rm Im}\nolimits(q)\cr
\mapdown{\mathop{\rm Tr}\nolimits}&&\mapdown{\mathop{\rm Tr}\nolimits}\cr
HC_1(R)&\mapright{\theta_p}&HC_1(R/pR)/\mathop{\rm Im}\nolimits(q)\cr}$$
\item[$\diamond$]
$$\diagram{
HQ_1(A)&\mapright{\vartheta_A}&\mathop{\rm Coker}\nolimits(\mu_{(A/2A)})\cr
\mapdown{\mathop{\rm Tr}\nolimits}&&\mapdown{\mathop{\rm Tr}\nolimits}\cr
HQ_1(R)&\mapright{\vartheta_R}&\mathop{\rm Coker}\nolimits(\mu_{(R/2R)})\cr}$$
\end{itemize}
\end{thm}
\begin{proof}
A little examination of the definitions shows that
$B$,
$\theta_p$
and $\vartheta$ commute with
$\iota$.
\end{proof}
\newpage
\section{Generalized Arf invariants.}
\setcounter{altel}{0}
\setcounter{equation}{0}
Let $(R,\alpha,u)$ be a ring with anti-structure with $u=\pm 1$.
Thus $u$ is central and $\alpha$ is an anti-involution.
\begin{thm}
The map
\[\Upsilon\colon\mathop{\rm Arf}\nolimits^h(R,\alpha,u)\rightarrow\mathop{\rm Coker}\nolimits(1+\vartheta_R)\]
determined by
\[\plane{a,b}\mapsto [a\otimes b,ab]\]
is a well-defined homomorphism.
\end{thm}
We are a bit sloppy here in denoting the projection
$HQ_1(R)\rightarrow\mathop{\rm Coker}\nolimits(\mu_{R/2R})$ by $1$.
\begin{proof}
Recall the presentation of $\mathop{\rm Arf}\nolimits^h(R,\alpha,u)$ from
theorem~\ref{thmarfgr}.\hfill\break
For all $a,b\in \Lambda_1(R)$ the element $(a\otimes b,ab)$
is a cycle in $HQ_1(R/2R)$:
$$(b\;\;1-y)(a\otimes b,ab)=
ab+ba+ab+ba=0.$$
Next we will check that $\Upsilon$ respects all the relations of the
aforementioned presentation.
\begin{enumerate}
\item
obvious
\item
obvious
\item
$[a\otimes b+b\otimes a,ab+ba]=0$
\item
$[a\otimes(x+\alpha(x)),a(x+\alpha(x)]=[a\otimes x+\alpha(x)\otimes a,ax+xa]=0$
\item
$[a\otimes \alpha(x) bx+xa\alpha(x)\otimes b,a\alpha(x) bx+xa\alpha(x) b]=$\hfill\break
$[a\alpha(x) b\otimes x+xa\otimes \alpha(x) b,a\alpha(x) bx+xa\alpha(x) b]=$\hfill\break
$[a\alpha(x) b\otimes x+bxa\otimes \alpha(x),a\alpha(x) bx+xa\alpha(x) b]=0$
\item
$\vartheta([a\otimes b,ab])=[aba\otimes b+ab\otimes ba+ab\otimes ba,abab]=[aba\otimes b,abab]$
\item
Suppose
$$\pmatrix{X&Y\cr Z&T\cr}\in \mathop{\rm GL}\nolimits_{2n}(R)
\mbox{ \ satisfies \ }
t_{\alpha,u}\left(\pmatrix{X&Y\cr Z&T\cr}\right)=
\pmatrix{X&Y \cr Z&T \cr}^{-1}.$$
Then using the relations for $X$, $Y$, $Z$ and $T$, we equate
\begin{eqnarray*}
\lefteqn{(1+\vartheta)
[X^\alpha\otimes T+Z\otimes Y^\alpha,X^\alpha T]}\hspace{2ex}\\
&=&[X^\alpha\otimes T+Z\otimes Y^\alpha+\\
& &X^\alpha TX^\alpha\otimes T+
ZY^\alpha Z\otimes Y^\alpha+X^\alpha T\otimes TX^\alpha+ZY^\alpha\otimes Y^\alpha Z+\\
& &(X^\alpha T+TX^\alpha)\otimes(ZY^\alpha+Y^\alpha Z)+X^\alpha T\otimes T^\alpha X,
X^\alpha T+(X^\alpha T)^2]\\
&=&[X^\alpha ZY^\alpha\otimes T+TX^\alpha Z\otimes Y^\alpha,X^\alpha ZY^\alpha T]\\
&=&[X^\alpha Z\otimes Y^\alpha T,X^\alpha ZY^\alpha T].
\end{eqnarray*}
Now theorem~\ref{thmmorhq} finishes the job.
\end{enumerate}
This finishes the proof.
\end{proof}
\begin{remark}
In the case that $R$ is commutative and $\alpha$ is the identity we have
\begin{eqnarray*}
\mathop{\rm Coker}\nolimits(1+\vartheta_R)&=&\frac{R}{\mathop{\rm Span}\nolimits\{x+x^2\mid x\in R\}}\oplus\\
& &{\Omega_R\over 2\Omega_R+\delta R+\{(r+r^2\delta s)\delta s\mid
r,s\in R\}}\\
&=&\mathop{\rm Coker}\nolimits(1+\theta_2\colon H_0(R)\rightarrow H_0(R/2R))\oplus\\
& &\mathop{\rm Coker}\nolimits(1+\theta_2\colon H_1(R)\rightarrow HC_1(R/2R)).
\end{eqnarray*}
This can be verified by a little examination of \ref{altdefhq} and
the definitions of $\theta_2$ and $\vartheta$ in
proposition~\ref{propthh0},
theorem~\ref{thmoperaties} and
theorem~\ref{thmhqoperatie}.
The projection of
$$\Upsilon\colon\mathop{\rm Arf}\nolimits^h(R,1,-1)\rightarrow\mathop{\rm Coker}\nolimits(1+\theta_2)\oplus\mathop{\rm Coker}\nolimits(1+\theta_2)$$
on the first summand is just the old primary Arf invariant.
The secondary Arf invariant
$$\mathop{\rm Arf}\nolimits^s(R,1,-1)\longrightarrow {\Omega_R\over 2\Omega_R+\delta R+
\{(r+r^2\delta s)\delta s\mid r,s\in R\}}$$
factors through the projection of $\Upsilon$ on the second summand.
\end{remark}
\newpage
{\Large {\bf \begin{center}
Chapter IV \vspace{4mm}\\
Applications to group rings.
\end{center}}}
\vspace{6mm}
\setcounter{section}{0}
\section{Quaternionic homology of group rings.}\label{secquahom}
\setcounter{altel}{0}
\setcounter{equation}{0}
The following exposition is based upon the work of J.-L. Loday in
\cite{Loday}.\hfill\break
Let $k$ be a commutative ring with identity, $G$ a group and $k[G]$ the
group algebra of $G$ over $k$.
By providing $k[G]$ with the anti-involution $\ol{\phantom{x}}$ determined by
$\ol{g}=g^{-1}$ for all $g\in G$, \hspace{1ex}
$k[G]\otimes_k k[G]^n$ becomes a quaternionic module by means of
example~\ref{kanex1} of chapter III.
\begin{nota}
Denote by $\Gamma$ the set of conjugacy classes of $G$ and by $C\colon G\rightarrow
\Gamma$ the map which assigns to $g\in G$ its conjugacy class $C(g)$.
Further we choose a section $S\colon\Gamma\rightarrow G$ of $C$ such that
$S(C(g^{-1}))=(S(C(g))^{-1}$ for every $g\in G$ with $C(g)\neq C(g^{-1})$.
Finally,
for every set $V$ endowed with a right $G$-action
we supply the free $k$-module $k[V]$
with a $k[G]$-bimodule structure by letting $G$ act trivially from the
left-hand side on $V$.
\end{nota}
\begin{punt}
For every $z\in G$, the right action $C(z)\times G\rightarrow C(z)$ of $G$ on $C(z)$
defined by
$(x,g)\mapsto g^{-1}xg$ for all
$x\in C(z)$ and $g\in G$,
makes $k[C(z)]$ into a $k[G]$-bimodule.
\end{punt}
\begin{lemma}
The map $$\phi\colon k[G]\otimes_k k[G]^n\rightarrow\bigoplus_{z\in\mathop{\rm Im}\nolimits S}
k[C(z)]\otimes_k k[G]^n$$
determined by
$$\phi(g\otimes g_1\otimes\cdots\otimes g_n):=
\cases{g_1\cdots g_ng\otimes g_1\otimes\cdots\otimes
g_n & if $gg_1\cdots g_n\in C(z)$\cr
0&otherwise,\cr}$$
is an isomorphism of simplicial modules with inverse determined by
$$h\otimes g_1\otimes\cdots\otimes g_n\mapsto(g_1\cdots g_n)^{-1}h\otimes g_1\otimes\cdots\otimes
g_n\mbox{ for all }h\in C(z).$$
\end{lemma}
\begin{proof}
See \cite{Loday}.
\end{proof}
\begin{defi}
We say that $z\in G$ is of type \newline
$\begin{array}{lll}
\mbox{ \ \ } &1&\mbox{ if }\;z=z^{-1},\\
&2&\mbox{ if }\;z^{-1}\in C(z)\mbox{ \ and \ } z\neq z^{-1},\\
&3&\mbox{ if }\;z^{-1}\not\in C(z).
\end{array}$\hfill\break
For each $i\in\{1,2,3\}$ let $S_i$ denote the subset of $\mathop{\rm Im}\nolimits S$
consisting of all elements of type $i$.
Notice that $\mathop{\rm Im}\nolimits S$ is the disjoint union of the $S_i$.
Now $S$ was chosen in such a way that $z\in S_3\Leftrightarrow z^{-1}\in
S_3$, This allows us to write $S_3$ as a disjoint union of sets $S_3^+$
and $S_3^-$ such that $z\in S_3^+\Leftrightarrow z^{-1}\in S_3^-.$
\end{defi}
\begin{defi}\label{defxeny}
The simplicial module $k[C(z)\cup C(z^{-1})]\otimes_k k[G]^n$
becomes a quaternionic module
by defining
$$x(g\otimes g_1\otimes\cdots\otimes g_n):=
(-1)^n(g_1\cdots g_n)^{-1}gg_1\cdots g_n
\otimes(g_1\cdots g_n)^{-1}g\otimes g_1\otimes\cdots\otimes g_{n-1}$$
and
$$y(g\otimes g_1\otimes\cdots\otimes g_n):=
(-1)^{\frac{n(n+1)}{2}}(g_1\cdots g_n)^{-1}g^{-1}g_1\cdots g_n
\otimes g_n^{-1}\otimes\cdots\otimes g_1^{-1}$$
\end{defi}
\begin{thm}\label{thmophak}
$$\phi\colon k[G]\otimes_k k[G]^n\rightarrow\bigoplus_{z\in S_1\cup S_2\cup S_3^+}
k[C(z)\cup C(z^{-1})]\otimes_k k[G]^n$$
is an isomorphism of quaternionic modules.
\end{thm}
\begin{proof}
This is easy to check. The maps
$x$ and $y$ were defined so as to make $\phi$ respect the quaternionic
structure.
\end{proof}
\begin{punt}\label{koleq}
For every group $G$ we define
$$d_i\colon k[G]^{n+1}\rightarrow k[G]^n$$
by
$$\begin{array}{rcl}
d_i(g_0\otimes g_1\otimes\cdots\otimes g_n)&:=&
g_0\otimes\cdots\otimes g_ig_{i+1}\otimes\cdots\otimes g_n \quad\mbox{ if }\quad
0\leq i<n\\
d_n(g_0\otimes g_1\otimes\cdots\otimes g_n)&:=&
g_0\otimes\cdots\otimes g_{n-1},
\end{array}
$$
$$d\colon k[G]^{n+1}\rightarrow k[G]^n \quad\mbox{ \ by \ }\quad
d:=\sum_{i=0}^n(-1)^id_i$$
and
$$d'\colon k[G]^{n+1}\rightarrow k[G]^n\quad\mbox{ \ by \ }\quad
d':=\sum_{i=0}^{n-1}(-1)^id_i.$$
Now the map
$$s\colon k[G]^{n+1}\rightarrow k[G]^{n+2}\mbox{ \ determined by \ }
s(g_0\otimes\cdots\otimes g_n)\isdef1\otimes g_0\otimes\cdots\otimes g_n$$
satisfies $sd+ds=sd'+d's=1$ and therefore
provides for a
chain contraction of both the chain complexes
$$(k[G]^{*+1},d)\quad\mbox{ \ and \ }\quad (k[G]^{*+1},d').$$
Now let $G$ be a group and $H$ a subgroup of $G$.
Choose a set-theoretic section $\beta\colon H\backslash G\rightarrow G$, of the
canonical projection $\pi\colon G\rightarrow H\backslash G$,
satisfying $\beta(H)=1$ and define $\gamma:=\beta\lower1.0ex\hbox{$\mathchar"2017$}\pi$.\hfill\break
In what follows we will give homotopy-inverse maps of the inclusion-induced
maps
$$j_*\colon(k[H]^{*+1},d)\rightarrow (k[G]^{*+1},d)$$
$$j_*'\colon(k[H]^{*+1},d')\rightarrow (k[G]^{*+1},d')$$
and appropriate chain homotopies.\hfill\break
The chain map $p_*$ determined by
$$p_n\colon k[G]^{n+1}\rightarrow k[H]^{n+1}$$
$$p_n(g_0\otimes\cdots\otimes g_n):=$$
$$g_0\gamma(g_0)^{-1}\otimes\gamma(g_0)g_1\gamma(g_0g_1)^{-1}\otimes\cdots\otimes
\gamma(g_0g_1\cdots g_{n-1})g_n\gamma(g_0\cdots g_n)^{-1}$$
is a chain inverse to $j_*$, through the homotopies
$$h_n\colon k[H]^{n+1}\rightarrow k[H]^{n+2} \quad\mbox{\ defined by \ }\quad
h_n\isdef0$$ and
$$\ol{h}_n\colon k[G]^{n+1}\rightarrow k[G]^{n+2}\quad\mbox{\ defined by \ }\quad
\ol{h}_n:= s(j_np_n-1).$$
Thus \begin{eqnarray*}
p_nj_n-1&=&dh_n+h_{n-1}d\\
j_np_n-1&=&d\ol{h}_n+\ol{h}_{n-1}d.\end{eqnarray*}
Analogously we define
$$p_n'\colon k[G]^{n+1}\rightarrow k[H]^{n+1}\quad
\mbox{ \ by \ }\quad p_n':= 0,$$
$$h_n'\colon k[H]^{n+1}\rightarrow k[H]^{n+2}
\quad\mbox{ \ by \ }\quad h_n':= -s$$
and
$$\ol{h'}_n\colon k[G]^{n+1}\rightarrow k[G]^{n+2}\quad \mbox{ \ by \ }\quad
\ol{h'}_n:= -s.$$
Then again $p_n'$ determines a chain map and
\begin{eqnarray*}
p_n'j_n'-1&=&d'h_n'+h_{n-1}'d'\\
j_n'p_n'-1&=&d'\ol{h'}_n+\ol{h'}_{n-1}d'.
\end{eqnarray*}
\end{punt}
\begin{defi}\label{defgzstreep}
For all $z\in G$ one defines the subgroups $G_z$ and $\ol{\gz}$ of $G$ by
$$G_z:=\{g\in G\mid gz=zg\}$$
and
$$\ol{\gz}:=\left\{g\in G\mid g^{-1}zg\in\{z,z^{-1}\}\right\}.$$
\end{defi}
Notice that
\begin{itemize}
\item[$\cdot$]
$G_z=G_{z^{-1}}$.
\item[$\cdot$]
the correspondences $G_z\backslash G\rightarrow C(z)$ and
$G_z\backslash\ol{\gz}\rightarrow\{z,z^{-1}\}$ determined by
$G_z a\mapsto a^{-1}za$ are bijective.
\item[$\cdot$]
$\ol{\gz}$ acts from the right on $\{z,z^{-1}\}$ by conjugation
and this makes $k[z,z^{-1}]$ into a $k[\ol{\gz}]$-bimodule.
\end{itemize}
\begin{thm}\label{thmdc}
For all $z\in G$ the inclusion $\ol{\gz}\subseteq G$ induces a morphism
$$k[z,z^{-1}]\otimes_kk[\ol{\gz}]^n\longrightarrow k[C(z)\cup C(z^{-1})]\otimes_kk[G]^n$$
$$a\otimes g_1\otimes\cdots\otimes g_n\mapsto a\otimes g_1\otimes\cdots\otimes g_n$$
of quaternionic modules.
\end{thm}
\begin{proof}
We distinguish between three cases and keep \ref{koleq} and
definition~\ref{defgzstreep} in mind.
\begin{enumerate}
\item
For all $z$ of type 1 we have $G_z=\ol{\gz}$ and the inclusion $G_z\subseteq G$
induces a morphism
of quaternionic modules
$$\diagram{
k[z]\otimes_kk[G_z]^n\cr
\isodown{}\cr
k\otimes_{k[G_z]}k[G_z]^{n+1}\cr
\mapdown{}\cr
k\otimes_{k[G_z]}k[G]^{n+1}\cr
\isodown{}\cr
k[C(z)]\otimes_kk[G]^n\cr}$$
mapping
$$z\otimes g_1\otimes\cdots\otimes g_n\mbox{ \ to \ }z\otimes g_1\otimes\cdots\otimes g_n.$$
Formulas for $x$ and $y$ can be found in definition~\ref{defxeny}.
\item
For all $z$ of type 2 we have $C(z)=C(z^{-1})$ and the
inclusion $\ol{\gz}\subseteq G$
induces a morphism of quaternionic modules
$$\diagram{
k[z,z^{-1}]\otimes_kk[\ol{\gz}]^n\cr
\isodown{}\cr
k\otimes_{k[G_z]}k[\ol{\gz}]^{n+1}\cr
\mapdown{}\cr
k\otimes_{k[G_z]}k[G]^{n+1}\cr
\isodown{}\cr
k[C(z)]\otimes_kk[G]^n\cr}$$
mapping
$$a\otimes g_1\otimes\cdots\otimes g_n\mbox{ \ to \ }a\otimes g_1\otimes\cdots\otimes g_n.$$
Formulas for $x$ and $y$ can be found in definition~\ref{defxeny}.
\item
For all $z$ of type 3 we have $G_z=G_{z^{-1}}=\ol{\gz}$ and the inclusion
$G_z\subseteq G$
induces a morphism of quaternionic modules
$$\diagram{
k[z,z^{-1}]\otimes_kk[G_z]^n\cr
\isodown{}\cr
(k\otimes_{k[G_z]}k[G_z]^{n+1})\oplus (k\otimes_{k[G_{z^{-1}}]}k[G_{z^{-1}}]^{n+1})\cr
\mapdown{}\cr
(k\otimes_{k[G_z]}k[G]^{n+1})\oplus (k\otimes_{k[G_{z^{-1}}]}k[G]^{n+1})\cr
\isodown{}\cr
k[C(z)\cup C(z^{-1})]\otimes_kk[G]^n\cr}$$
mapping
$$a\otimes g_1\otimes\cdots\otimes g_n\mbox{ \ to \ }a\otimes g_1\otimes\cdots\otimes g_n.$$
Formulas for $x$ and $y$ can be derived from definition~\ref{defxeny}.
\end{enumerate}
In all cases this morphism induces a chain map of the associated
quaternionic double complexes.
\end{proof}
By applying $k\otimes_{k[G_z]}-$ in the various situations of \ref{koleq}
that occur here, we see that these maps are
chain equivalences on the columns by Shapiro's lemma.
Further \ref{koleq} enables us to compute explicit
chain inverses and chain homotopies.
To obtain the inverse homomorphism on the level of quaternionic homology
we use the following lemma.
\begin{lemma}\label{lemmadubbelcomplexiso}
Suppose $j\colon{\cal C}\rightarrow\ol{\cee}$ is a chain map of double complexes
$$\diagram{C_{20}& & & & & &\ol{C}_{20}&&&&\cr
\mapdown{d_{20}^v}&&\vdots&&& &\mapdown{\ol{d}_{20}^v}&&\vdots&&\cr
C_{10}&\mapleft{d_{11}^h}&C_{11}&\cdots&&\mapright{j}
&\ol{C}_{10}&\mapleft{\ol{d}_{11}^h}&\ol{C}_{11}&\cdots&\cr
\mapdown{d_{10}^v}&&\mapdown{d_{11}^v}&&&
&\mapdown{\ol{d}_{10}^v}&&\mapdown{\ol{d}_{11}^v}&&\cr
C_{00}&\mapleft{d_{01}^h}&C_{01}&\mapleft{d_{02}^h}&C_{02}&&
\ol{C}_{00}&\mapleft{\ol{d}_{01}^h}&\ol{C}_{01}&\mapleft{\ol{d}_{02}^h}&\ol{C}_{02}\cr}
$$
which is a chain equivalence on the columns.
Let
$p_{*\,k}$ be a chain inverse of $j_{*\,k}$ and
\begin{eqnarray*}
p_{m\,k}j_{m\,k}-1&=&d_{m+1\,k}^vh_{m\,k}+h_{m-1\,k}d_{m\,k}^v\\
j_{m\,k}p_{m\,k}-1&=&\ol{d}_{m+1\,k}^v\ol{h}_{m\,k}+\ol{h}_{m-1\,k}\ol{d}_{m\,k}^v.
\end{eqnarray*}
Then $$\tau\colon H_1(\mathop{\rm Tot}\nolimits\ol{\cee})\rightarrow H_1(\mathop{\rm Tot}\nolimits{\cal C})$$
defined by $$[a,b]\mapsto
[p_{10}a+p_{10}\ol{d}_{11}^h\ol{h}_{01}b+h_{00}d_{01}^hp_{01}b,p_{01}b]$$
for all $(a,b)\in\mathop{\rm Ker}\nolimits(\ol{d}_{10}^v\;\; \ol{d}_{01}^h)$,
is the inverse of $$j_*\colon H_1(\mathop{\rm Tot}\nolimits{\cal C})\rightarrow H_1(\mathop{\rm Tot}\nolimits\ol{\cee}).$$
\end{lemma}
\begin{proof}
The map $j_*$ is an isomorphism
since $j$ is an equivalence on the columns.
By definition of double complex:
\begin{eqnarray*}
d_{m-1\,k}^hd_{m\,k}^v+d_{m\,k-1}^vd_{m\,k}^h&=&0
\quad\mbox{ \ for all \ }m,k\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\\
\ol{d}_{m-1\,k}^h\ol{d}_{m\,k}^v+\ol{d}_{m\,k-1}^v\ol{d}_{m\,k}^h&=&0
\quad\mbox{ \ for all \ }m,k\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}
\end{eqnarray*}
Now suppose $a\in\ol{C}_{10}$ and $b\in\ol{C}_{01}$ satisfy
$$\ol{d}_{10}^va+\ol{d}_{01}^hb=0.$$
Then
\begin{eqnarray*}
\lefteqn{d_{10}^v(p_{10}a+p_{10}\ol{d}_{11}^h\ol{h}_{01}b+h_{00}d_{01}^hp_{01}b)+
d_{01}^hp_{01}b }\hspace{4em} \\
&=&-p_{00}\ol{d}_{01}^hb-p_{00}\ol{d}_{01}^h\ol{d}_{11}^v\ol{h}_{01}b+
p_{00}j_{00}d_{01}^hp_{01}b\\
&=&-p_{00}\ol{d}_{01}^hj_{01}p_{01}b+p_{00}\ol{d}_{01}^hj_{01}p_{01}b\\
&=&0
\end{eqnarray*}
proves that $\tau([a,b])\in H_1(\mathop{\rm Tot}\nolimits{\cal C}).$
Further we equate
\begin{eqnarray*}
\lefteqn{j_*\tau([a,b])-[a,b]}\hspace{2em}\\
&=&[j_{10}p_{10}(a+\ol{d}_{11}^h\ol{h}_{01}b)+
j_{10}h_{00}d_{01}^hp_{01}b-a,(j_{01}p_{01}-1)b]\\
&=&[(j_{10}p_{10}-1)(a+\ol{d}_{11}^h\ol{h}_{01}b)+
j_{10}h_{00}d_{01}^hp_{01}b,0]\\
&=&[j_{10}p_{10}(j_{10}p_{10}-1)(a+\ol{d}_{11}^h\ol{h}_{01}b)+
j_{10}p_{10}j_{10}h_{00}d_{01}^hp_{01}b,0].
\end{eqnarray*}
To obtain this last identity we used the fact that
$$(j_{10}p_{10}-1)(a+\ol{d}_{11}^h\ol{h}_{01}b)+
j_{10}h_{00}d_{01}^hp_{01}b\in\mathop{\rm Ker}\nolimits(\ol{d}_{10}^v)$$
and
$$(j_{10}p_{10}-1)\mathop{\rm Ker}\nolimits(\ol{d}_{10}^v)\subseteq\mathop{\rm Im}\nolimits(\ol{d}_{20}^v).$$
To continue the computation we define
$c:= j_{10}p_{10}(j_{10}p_{10}-1)(a+\ol{d}_{11}^h\ol{h}_{01}b).$
\begin{eqnarray*}
\lefteqn{[c+j_{10}p_{10}j_{10}h_{00}d_{01}^hp_{01}b,0]}\\
&=&[c+j_{10}h_{00}p_{00}j_{00}d_{01}^hp_{01}b,0]\\
&=&[c+j_{10}h_{00}p_{00}\ol{d}_{01}^hj_{01}p_{01}b,0]\\
&=&[c+j_{10}h_{00}p_{00}\ol{d}_{01}^h(\ol{d}_{11}^v\ol{h}_{01}+1)b,0]\\
&=&[c-j_{10}h_{00}p_{00}\ol{d}_{10}^va-
j_{10}h_{00}p_{00}\ol{d}_{10}^v\ol{d}_{11}^h\ol{h}_{01}b,0]\\
&=&[c-j_{10}h_{00}d_{10}^vp_{10}(a+\ol{d}_{11}^h\ol{h}_{01}b),0]\\
&=&[c-j_{10}p_{10}(j_{10}p_{10}-1)(a+\ol{d}_{11}^h\ol{h}_{01}b),0]\\
&=&0.
\end{eqnarray*}
Thus we find $j_*\tau=1$ and since $j_*$ is already
an isomorphism, this proves the lemma.
\end{proof}
\begin{punt}\label{puntranden}
We apply lemma~\ref{lemmadubbelcomplexiso} to the situation of
theorem~\ref{thmdc}:
Write ${\cal D}_1$ for the double complex
$${\cal D}\left(k[z,z^{-1}]\otimes k[\ol{\gz}]^*\right),$$
${\cal D}_2$ for the double complex
$${\cal D}\left(k[C(z)\cup C(z^{-1})]\otimes k[G]^*\right)$$
and $$j\colon {\cal D}_1\rightarrow{\cal D}_2$$ for the
chain map induced by the morphism of theorem~\ref{thmdc}.
See definition~\ref{defhq} of chapter III for the definition of ${\cal D}$.
Picture ${\cal D}_1:$
$$\diagram{
k[z,z^{-1}]\otimes k[\ol{\gz}]^2&&&\cr
\mapdown{d_{20}^v}&&&\cr
k[z,z^{-1}]\otimes k[\ol{\gz}]&\mapleft{d_{11}^h}&
(k[z,z^{-1}]\otimes k[\ol{\gz}])\oplus&\vspace{-1.0ex}\cr
&&\hspace{1em}(k[z,z^{-1}]\otimes k[\ol{\gz}])&\cr
\mapdown{d_{10}^v}&&\mapdown{d_{11}^v}&\cr
k[z,z^{-1}]&\mapleft{d_{01}^h}&k[z,z^{-1}]\oplus k[z,z^{-1}]&\mapleft{d_{02}^h}
k[z,z^{-1}]\oplus k[z,z^{-1}]\cr}$$
and
${\cal D}_2:$
$$\diagram{
k[C(z)\cup C(z^{-1})]\otimes k[G]^2&&\cr
\mapdown{\ol{d}_{20}^v}&&\cr
k[C(z)\cup C(z^{-1})]\otimes k[G]&\mapleft{\ol{d}_{11}^h}&
(k[C(z)\cup C(z^{-1})]\otimes k[G])\oplus\vspace{-1.0ex}\cr
&&\hspace{1em}(k[C(z)\cup C(z^{-1})]\otimes k[G])\cr
\mapdown{\ol{d}_{10}^v}&&\mapdown{\ol{d}_{11}^v}\cr
k[C(z)\cup C(z^{-1})]&\mapleft{\ol{d}_{01}^h}&k[C(z)\cup C(z^{-1})]\oplus
k[C(z)\cup C(z^{-1})]\mapleft{\ol{d}_{02}^h}\cr}$$
We use \ref{koleq}, definition~\ref{defxeny}, theorem~\ref{thmdc} and
lemma~\ref{lemmadubbelcomplexiso} to obtain the following formulas.
\begin{eqnarray*}
d_{10}^v,\ol{d}_{10}^v&\colon&a\otimes g\mapsto g^{-1}ag-a\\
d_{20}^v,\ol{d}_{20}^v&\colon&a\otimes g_1\otimes g_2\mapsto g_1^{-1}ag_1\otimes g_2-a\otimes
g_1g_2+a\otimes g_1\\
d_{11}^v,\ol{d}_{11}^v&\colon&(a\otimes g_1,0)\mapsto (-g_1^{-1}ag_1,0)\\
&&(0,b\otimes g_2)\mapsto (0,b-g_2^{-1}bg_2)\\
d_{01}^h, \ol{d}_{01}^h&\colon&(a,0)\mapsto 0\\
&&(0,b)\mapsto b-b^{-1}\\
d_{11}^h,\ol{d}_{11}^h&\colon&(a\otimes g_1,0)\mapsto a\otimes g_1+
g_1^{-1}ag_1\otimes g_1^{-1}a\\
&&(0,b\otimes g_2)\mapsto b\otimes g_2+g_2^{-1}b^{-1}g_2\otimes g_2^{-1}\\
d_{02}^h,\ol{d}_{02}^h&\colon&(a,0)\mapsto(a,-a-a^{-1})\\
&&(0,b)\mapsto(b+b^{-1},0)\\
p_{10}&\colon&g_1^{-1}ag_1\otimes g_2\mapsto
\gamma(g_1)g_1^{-1}ag_1\gamma(g_1)^{-1}\otimes\gamma(g_1)g_2\gamma(g_1g_2)^{-1}\\
p_{01}&\colon&(g_1^{-1}ag_1,0)\mapsto 0\\
&&(0,g_2^{-1}bg_2)\mapsto(0,\gamma(g_2)g_2^{-1}bg_2\gamma(g_2)^{-1})\\
h_{00}=0&&\\
\ol{h}_{01}&\colon&(g_1^{-1}ag_1,0)\mapsto(a\otimes g_1,0)\\
&&(0,g_2^{-1}bg_2)\mapsto(0,b\otimes g_2-b\otimes g_2\gamma(g_2)^{-1})
\end{eqnarray*}
\end{punt}
\begin{thm}\label{thminvketen}
The inverse
$$\tau\colon H_1(\mathop{\rm Tot}\nolimits({\cal D}_2))\longrightarrow H_1(\mathop{\rm Tot}\nolimits({\cal D}_1))$$
of $j_*$ is \underline{determined} by
$(x,y)\longmapsto$
$$(\gamma(g_1)g_1^{-1}ag_1\gamma(g_1)^{-1}\otimes\gamma(g_1)g_2\gamma(g_1g_2)^{-1}
+b\otimes b,(0,\gamma(g_4)g_4^{-1}cg_4\gamma(g_4)^{-1})),$$
where
\begin{eqnarray*}
x&=&g_1^{-1}ag_1\otimes g_2\in k[C(z)\cup C(z^{-1})]\otimes k[G]\\
y&=&(g_3^{-1}bg_3,g_4^{-1}cg_4)\in k[C(z)\cup C(z^{-1})]\oplus
k[C(z)\cup C(z^{-1})].
\end{eqnarray*}
\end{thm}
\begin{proof}
Under the given conditions we have
\begin{eqnarray*}
\lefteqn{p_{10}\ol{d}_{11}^h\ol{h}_{01}(g_3^{-1}bg_3,g_4^{-1}cg_4)}\\
&=&p_{10}\ol{d}_{11}^h(b\otimes g_3,c\otimes g_4-c\otimes g_4\gamma(g_4)^{-1})\\
&=&p_{10}(b\otimes g_3+g_3^{-1}bg_3\otimes g_3^{-1}b+c\otimes g_4+
g_4^{-1}c^{-1}g_4\otimes g_4^{-1}\\
&&-c\otimes g_4\gamma(g_4)^{-1}-\gamma(g_4)g_4^{-1}c^{-1}g_4\gamma(g_4)^{-1}\otimes
\gamma(g_4)g_4^{-1})\\
&=&b\otimes g_3\gamma(g_3)^{-1}+\gamma(g_3)g_3^{-1}bg_3\gamma(g_3)^{-1}\otimes
\gamma(g_3)g_3^{-1}b.
\end{eqnarray*}
Applying the first relation of the list of theorem~\ref{thmrelaties} yields
$$[b\otimes g_3\gamma(g_3)^{-1}+\gamma(g_3)g_3^{-1}bg_3\gamma(g_3)^{-1}\otimes
\gamma(g_3)g_3^{-1}b]=[b\otimes b].$$
Using the formula for $\tau$ in lemma~\ref{lemmadubbelcomplexiso}
yields the desired result.
\end{proof}
\begin{thm}\label{thmrelaties}
For every $g,g_1,g_2\in\ol{\gz}$ and $a\in\{z,z^{-1}\}$,
the following relations are valid in $H_1(\mathop{\rm Tot}\nolimits({\cal D}_1))$.
\begin{enumerate}
\item
$[g_1^{-1}ag_1\otimes g_2+a\otimes(g_1-g_1g_2),0,0]=0,$
\item
$[0,z+z^{-1},0]=0,$
\item
$[0,a,a+a^{-1}]=0$ and $[0,0,2(z+z^{-1})]=0,$
\item
$[z\otimes z,0,z+z^{-1}]=0,$
\item
$[z\otimes g-z^{-1}\otimes g,0,z-g^{-1}zg]=0,$
\item
$[z\otimes (g_1+g_2-g_1g_2),0,\epsilon(g_1,g_2)]=0,$ where
$$\epsilon(g_1,g_2):=
\cases{z-z^{-1}&if $g_1,g_2\not\inG_z$\cr 0&otherwise\cr}.$$
\end{enumerate}
\end{thm}
\begin{proof}
\begin{enumerate}
\item[{\em 1}]
follows immediately from the definition of $d_{20}^v.$
\item[{\em 2}]
is clear since $d_{02}^h(0,z)=(z+z^{-1},0).$
\item[{\em 3}]
$d_{02}^h(a,0)=(a,-a-a^{-1})$ and {\em 2} imply that
$[0,0,2(z+z^{-1})]=0$. The rest is obvious.
\item[{\em 4}]
Using the definitions of $d_{11}^h$ and $d_{11}^v$ we find
\begin{eqnarray*}
0&=&[z\otimes g+g^{-1}zg\otimes g^{-1}z,-g^{-1}zg,0]\\
&=&[z\otimes g+z\otimes(z-g),0,z+z^{-1}] \mbox{ \ by {\em 1} and {\em 3} \ }\\
&=&[z\otimes z,0,z+z^{-1}].
\end{eqnarray*}
\item[{\em 5}]
Using the definitions of $d_{11}^h$ and $d_{11}^v$ we equate
\begin{eqnarray*}
0&=&[z\otimes g+g^{-1}z^{-1} g\otimes g^{-1},0,z-g^{-1}zg]\\
&=&[z\otimes g+z^{-1}\otimes(1-g),0,z-g^{-1}zg] \mbox{ \ by {\em 1} \ }\\
&=&[z\otimes g-z^{-1}\otimes g,0,z-g^{-1}zg].
\end{eqnarray*}
Note that $[z\te1,0,0]=0$ by taking $g_1=g_2=1$ in {\em 1}.
\item[{\em 6}]
If $g_1\inG_z$, then {\em 6} follows from {\em 1}. \hfill\break
If $g_1\not\inG_z$, then {\em 6} follows from {\em 1} and {\em 5}.
\end{enumerate}
This completes the list of relations.
\end{proof}
\begin{nota}
For every group $J$ we denote by
$J_{{\rm ab}}$ the commutator quotient of $J$,
i.e. $J_{{\rm ab}}=J/[J,J]$, and by $J_\#$ the quotient group
$J_{{\rm ab}}/(J_{{\rm ab}})^2.$
\end{nota}
\begin{thm}\label{thmdelenuitrek}
Let $k=\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\:_2$.
\begin{enumerate}
\item
For every $z$ of type {\rm 1} the map
$$\eta\colon
H_1(\mathop{\rm Tot}\nolimits({\cal D}_1))\mapright{}\left((G_z)_\#/<z>\right)\times C_2$$
determined by
$$\left[\sum_i z\otimes g_i,n_1z,n_2z\right]\mapsto
\left(\left[\prod_ig_i\right],t^{n_2}\right)$$
for all $n_1,n_2\in k$ and $g_i\inG_z$, where
$C_2$ denotes the cyclic group of order two generated by $t$,
is an isomorphism.
\item
For every $z$ of type {\rm 2} the map
$$\eta\colon
H_1(\mathop{\rm Tot}\nolimits({\cal D}_1))\mapright{}\frac{(\ol{\gz}\times_{C_2}C_4)_\#}{<[z,t^2]>}$$
determined by
$$\left[\sum_ia_i\otimes g_i,\rho(n_1)z+\rho(n_2)z^{-1},
\rho(n_3)z+\rho(n_4)z^{-1}\right]\mapsto\left[\prod_ig_i,t^n\right]$$
for all $n_1,n_2,n_3,n_4\in \Z$, $a_i\in\{z,z^{-1}\}$ and $g_i\in\ol{\gz}$,
satisfying the cycle condition $\sum\rho(w(g_i))=\rho(n_3-n_4)$,
is an isomorphism.
Here
$$\rho \mbox{ \ is the canonical map \ } \Z\rightarrow k,$$
$$n:= \sum w(g_i)+2\left(n_1+n_2+n_4+\sum w'(a_i)w(g_i)\right),$$
$$w'(z)\isdef0,\,\,w'(z^{-1})\isdef1,$$
$$w(g):= w'(g^{-1}zg) \mbox{ \ for all \ } g\in\ol{\gz}. $$
And $\ol{\gz}\times_{C_2}C_4$ is the pull-back of the diagram
$$\diagram{&&C_4\cr&&\mapdown{\pi_1}\cr\ol{\gz}&\mapright{\pi_2}&C_2\cr}$$
Here $C_4$ denotes the cyclic group of order four generated by $t$,
$\pi_1$ is the non-trivial map and $\pi_2(g):= t^{w(g)}$ for
every $g\in \ol{\gz}$.
\item
For every $z$ of type {\rm 3} the map
$$\eta\colon H_1(\mathop{\rm Tot}\nolimits({\cal D}_1))\mapright{}(G_z)_\#$$
determined by
$$\left[\sum_i a_i\otimes g_i,n_1z+n_2z^{-1},n_3z+n_4z^{-1}\right]\mapsto
\left[\prod_ig_iz^{n_1+n_2+n_3}\right]$$
for all $n_1,n_2,n_3,n_4\in k$, $a_i\in\{z,z^{-1}\}$ and $g_i\inG_z$,
satisfying the cycle condition $n_3=n_4$,
is an isomorphism.
\end{enumerate}
\end{thm}
\begin{proof}
We will not enter into all the details of the proof; it is not difficult
but rather tedious.
\begin{enumerate}
\item[{\em 1}]
The data in ~\ref{puntranden} make it is easy to verify that the map
on $$\mathop{\rm Ker}\nolimits(d_{10}^v\;\;d_{01}^h)=(k[z]\otimes k[G_z])\oplus k[z]\oplus k[z]$$
determined by the expression in the definition of $\eta$
is a homomorphism which
vanishes on $\mathop{\rm Im}\nolimits(d_{02}^h)$, $\mathop{\rm Im}\nolimits(d_{20}^v)$
and $\mathop{\rm Im}\nolimits(d_{11}^h\;\;d_{11}^v).$\hfill\break
Theorem~\ref{thmrelaties} enables us to check that
the inverse of $\eta$ is determined by
$$([g],t^n)\mapsto [z\otimes g,0,\ol{n}z]$$
for every $g\inG_z$, $n\in\Z$.
\item[{\em 2}]
Again $\eta$ is a well-defined homomorphism.
The inverse homomorphism is determined by
$$[g,t^n]\mapsto [z\otimes g,0,\rho(\mathop{\rm ent\/}\nolimits((n+1)/2))z+\rho(\mathop{\rm ent\/}\nolimits(n/2))z^{-1}]$$
for all
$g\in\ol{\gz}$ and $n\in\Z$ satisfying $\rho(w(g))=\rho(n)$.
\item[{\em 3}]
The homomorphism $\eta^{-1}$ maps $[g]$ to $[z\otimes g,0,0]$ for all $g\inG_z$.
\end{enumerate}
Here $\mathop{\rm ent\/}\nolimits$ denotes the entier function.
\end{proof}
\begin{nota}
Write $$\Sigma(G)=
\bigoplus_{z\in S_1}\left(\left((G_z)_\#/<z>\right)\times C_2\right)
\oplus
\bigoplus_{z\in S_2}\frac{(\ol{\gz}\times_{C_2}C_4)_\#}{<[z,t^2]>}
\oplus
\bigoplus_{z\in S_3^+} (G_z)_\#$$
\end{nota}
\begin{thm}\label{thmhqiso}
We have an isomorphism $\Psi\colon HQ_1(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\;_2[G])\longrightarrow\Sigma(G)$.
\end{thm}
\begin{proof}
By theorem~\ref{thmophak}, theorem~\ref{thminvketen} and
theorem~\ref{thmdelenuitrek}.
\end{proof}
\newpage
\section{Managing Coker$(1+\vartheta).$}
\setcounter{altel}{0}
\setcounter{equation}{0}
Before we start with our reflections on
$\mathop{\rm Coker}\nolimits(1+\vartheta_{{\displaystyle \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\;_2[G]}})$
recall the theorems~\ref{thmophak}, \ref{thminvketen},
\ref{thmdelenuitrek} and \ref{thmhqiso} of the previous section.
\begin{lemma}
The isomorphism $\Psi$ induces an isomorphism
$$\Psi_1\colon\mathop{\rm Coker}\nolimits(\nu)\lhook\joinrel\surarrow\mathop{\rm Coker}\nolimits(\Psi\lower1.0ex\hbox{$\mathchar"2017$}\nu)$$
and
$$\diagram{\mathop{\rm Coker}\nolimits(\Psi\lower1.0ex\hbox{$\mathchar"2017$}\nu)
&=&\bigoplus_{z\in S_1}\left(\left((G_z)_\#/<z>\right)\times C_2\right)
\oplus\cr
&&\bigoplus_{z\in S_2}\left(\ol{\gz}\right)_\#/<z>
\oplus\hfill\cr
&&\bigoplus_{z\in S_3^+} (G_z)_\#/<z>\hfill\cr}
$$
\end{lemma}
\begin{proof}
To determine $\mathop{\rm Coker}\nolimits(\Psi\lower1.0ex\hbox{$\mathchar"2017$}\nu)$
we compute $$\Psi(\nu(x))=\Psi([1\otimes x,0,0])$$ for $x\in\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\;_2[G]$.
We may assume that $x\in G$.
There exist $z\in\mathop{\rm Im}\nolimits\,S$ and $g\in G$ such that $x=g^{-1}zg.$
Now
$$\phi(1\otimes g^{-1}zg)=g^{-1}zg\otimes g^{-1}zg,$$
\begin{eqnarray*}
\tau([g^{-1}zg\otimes g^{-1}zg,0,0])&=&
[\gamma(g)g^{-1}zg\gamma(g)^{-1}\otimes
\gamma(g)g^{-1}zg\gamma(zg)^{-1},0,0]\vspace{.5mm}\\
&=&[\gamma(g)g^{-1}zg\gamma(g)^{-1}\otimes\gamma(g)g^{-1}zg\gamma(g)^{-1},0,0]
\vspace{.5mm}\\
&=&\cases{[z\otimes z,0,0]& if $z$ is of \vspace{.5mm}type 1\cr
[z^{\pm1}\otimes z^{\pm1},0,0]& if $z$ is of \vspace{.5mm}type 2\cr
[z\otimes z,0,0]& if $z$ is of type 3\cr}
\end{eqnarray*}
applying the isomorphism $\eta$ we find
$$\Psi([1\otimes g^{-1}zg,0,0])=
\cases{([z],1)=1\in((G_z)_\#/<z>)\times C_2
&if $z$ is \vspace{1mm} of type 1\cr
[z,1]\in(\ol{\gz}\times_{C_2}C_4)_\#/<[z,t^2]>
&if $z$ \vspace{1mm}is of type 2\cr
[z]\in(G_z)_\#&if $z$ is of type 3\cr}$$
The rest is clear now.
\end{proof}
\begin{defi}
Let $G$ be a group. Define \[\tilde{F}(z):=\cases{
{\displaystyle {(G_z)_\#\over
<\{x\in G\mid x=z \,\vee\, x^2=z\}>}}\times C_2,&
if $z$ is of\vspace{1mm} type $1$ \cr
{\displaystyle {(\ol{G_z})_\# \over
<\{x\in G\mid x=z\,\vee\, x^2=z\}>}}, & if $z$ is of \vspace{1mm}type $2$\cr
{\displaystyle {(G_z)_\#\over<\{x\in G\mid x=z\,\vee\, x^2=z\}>}},
&if $z$ is of type $3$.\cr
}\]
\end{defi}
\begin{lemma}
The isomorphism $\Psi_1$ induces an isomorphism
$$\Psi_2\colon\mathop{\rm Coker}\nolimits(\mu)\lhook\joinrel\surarrow\mathop{\rm Coker}\nolimits(\Psi_1\lower1.0ex\hbox{$\mathchar"2017$}\mu)$$ and
$$
\mathop{\rm Coker}\nolimits(\Psi_1\lower1.0ex\hbox{$\mathchar"2017$}\mu)=
\bigoplus_{z\in S_1\cup S_2\cup S_3^+}\tilde{F}(z)$$
\end{lemma}
\begin{proof}
To determine $\mathop{\rm Coker}\nolimits(\Psi_1\lower1.0ex\hbox{$\mathchar"2017$}\mu)$
we compute $\Psi_1([x\otimes x,0,0])$.
Again we may assume that $x\in G$.
There exist $z\in\mathop{\rm Im}\nolimits\,S$ and $g\in G$ such that $x^2=g^{-1}zg.$
Observe that $(gxg^{-1})^2=z$.
Now $$\phi(x\otimes x)=x^2\otimes x=g^{-1}zg\otimes x.$$
Notice that $\gamma(gx)=\gamma(g)$ since $gxg^{-1}\inG_z$.
\begin{eqnarray*}
\tau([g^{-1}zg\otimes x,0,0])&=&
[\gamma(g)g^{-1}zg\gamma(g)^{-1}\otimes
\gamma(g)x\gamma(gx)^{-1},0,0]\vspace{.5mm}\\
&=&[\gamma(g)g^{-1}zg\gamma(g)^{-1}\otimes
\gamma(g)x\gamma(g)^{-1},0,0]\vspace{.5mm}\\
&=&\!\cases{
[z\otimes \gamma(g)g^{-1}gxg^{-1}g\gamma(g)^{-1},0,0]
&if $z$ is of\vspace{.5mm} type 1\cr
[z^{\pm1}\otimes \gamma(g)g^{-1}gxg^{-1}g\gamma(g)^{-1},0,0]
&if $z$ is of\vspace{.5mm} type 2\cr
[z\otimes \gamma(g)g^{-1}gxg^{-1}g\gamma(g)^{-1},0,0]
&if $z$ is of type 3\cr}
\end{eqnarray*}
applying the isomorphism $\eta$ we find
$$\Psi_1([x\otimes x,0,0])=
\cases{([gxg^{-1}],1)\in((G_z)_\#/<[z]>)\times C_2
&if $z$ is of\vspace{.5mm} type 1\cr
[gxg^{-1}]\in(\ol{\gz})_\#/<[z]>
&if $z$ is of\vspace{.5mm} type 2\cr
[gxg^{-1}]\in(G_z)_\#/<[z]>
&if $z$ is of type 3\cr}$$
This proves the claim.
\end{proof}
The isomorphism $\Psi_2$ induces an isomorphism
$$\Psi_3\colon\mathop{\rm Coker}\nolimits(1+\vartheta)\lhook\joinrel\surarrow
\mathop{\rm Coker}\nolimits(\Psi_2(1+\vartheta)\Psi^{-1}):$$
$$\diagram{
HQ_1(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\:_2[G])&\mapright{1+\vartheta}&\mathop{\rm Coker}\nolimits(\mu)
&\longrightarrow&\mathop{\rm Coker}\nolimits(1+\vartheta)\cr
\mapdown{\Psi}&&\mapdown{\Psi_2}&&\mapdown{\Psi_3}\cr
{\cal S}&\longrightarrow&\mathop{\rm Coker}\nolimits(\Psi_1\lower1.0ex\hbox{$\mathchar"2017$}\mu)&
\longrightarrow&\mathop{\rm Coker}\nolimits(\Psi_2(1+\vartheta)\Psi^{-1})\cr}$$
\begin{lemma}\label{identlem}
$\mathop{\rm Coker}\nolimits(\Psi_2(1+\vartheta)\Psi^{-1})$ arises from $\mathop{\rm Coker}\nolimits(\Psi_1\lower1.0ex\hbox{$\mathchar"2017$}\mu)$
by imposing the following identifications.
For all $z$ of type
\begin{enumerate}
\item[1]
identify $$([g],t^i)\in\tilde{F}(z) \mbox{ \ and \ }
([g],t^i)\in\tilde{F}(1)$$
\item[2]
identify $$[g]\in\tilde{F}(z) \mbox{ \ and \ }
\cases{
([g],t^{w(g)})\in\tilde{F}(z^2)& if $z^2$ is\vspace{.5mm} of type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 2\cr}$$
\item[3]
identify $$[g]\in\tilde{F}(z) \mbox{ \ and \ }
\cases{
([g],1)\in\tilde{F}(z^2)& if $z^2$ is of\vspace{.5mm} type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of \vspace{.5mm}type 2\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 3.\cr}$$
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
Let $([g],t^i)\in((G_z)_\#/<z>)\times C_2.$
$$\Psi^{-1}([g],t^i)=[g^{-1}z\otimes g,0,iz]$$
since
\begin{eqnarray*}
\phi(g^{-1}z\otimes g)&=&(z\otimes g) \quad\mbox{and}\quad
\phi(iz)=iz,\\
\tau([z\otimes g,0,iz])&=&[z\otimes g\gamma(g)^{-1},0,iz]\;=\;[z\otimes g,0,iz],\\
\eta([z\otimes g,0,iz])&=&([g],t^i).
\end{eqnarray*}
$\vartheta([g^{-1}z\otimes g,0,iz])=[g^{-1}z^2\otimes g+z\otimes z+iz\otimes z,0,iz^2]=
[g^{-1}\otimes g,0,i]$.
$$\Psi_2(g^{-1}\otimes g,0,i])=([g],t^i)\in\tilde{F}(z^2)=\tilde{F}(1)$$
since
\begin{eqnarray*}
\phi(g^{-1}\otimes g)&=&(1\otimes g) \quad\mbox{and}\quad \phi(i)\;=\;i,\\
\tau([1\otimes g,0,iz])&=&[1\otimes g\gamma(g)^{-1},0,i]\;=\;[1\otimes g,0,i],\\
\eta([1\otimes g,0,i])&=&([g],t^i).
\end{eqnarray*}
\item
Let $[g,t^i]\in(\ol{\gz}\times_{C_2}C_4)_\#/<[z,t^2]>$.
$$\Psi^{-1}([g,t^i])=[g^{-1}z\otimes g,0,y],$$
where $y=\mathop{\rm ent\/}\nolimits((i+1)/2)z+\mathop{\rm ent\/}\nolimits(i/2)z^{-1}$,
since
\begin{eqnarray*}
\phi(g^{-1}z\otimes g)&=&(z\otimes g)\quad\mbox{ and}\quad \phi(y)\;=\;y,\\
\tau([z\otimes g,0,y])&=&
[z\otimes g\gamma(g)^{-1},0,y]\;=\;[z\otimes g,0,y],\\
\eta([z\otimes g,0,y])&=&[g,t^{w(g)+2\mathop{\rm ent\/}\nolimits(i/2)}]\;=\;[g,t^i].
\end{eqnarray*}
$\vartheta([g^{-1}z\otimes g,0,y])=
[g^{-1}z^2\otimes g,0,\mathop{\rm ent\/}\nolimits((i+1)/2)z^2+\mathop{\rm ent\/}\nolimits(i/2)z^{-2}]$.\hfill\break
Note that
$[z\otimes z^{\pm1},0,0]=[z^{\pm1}\otimes z,0,0]=0$ in $\mathop{\rm Coker}\nolimits(\mu)$.\hfill\break
Define $y':=\mathop{\rm ent\/}\nolimits((i+1)/2)z^2+\mathop{\rm ent\/}\nolimits(i/2)z^{-2}$.
$$\Psi_2([g^{-1}z^2\otimes g,0,y'])=
\cases{
([g],t^i)\in\tilde{F}(z^2)& if $z^2$ is of type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 2\cr}$$
since
\begin{eqnarray*}
\phi(g^{-1}z^2\otimes g)&=&(z^2\otimes g) \quad\mbox{ and }\quad \phi(y')\;=\;y',\\
\tau([z^2\otimes g,0,y'])&=&[z^2\otimes g\gamma(g)^{-1},0,y']\;=\;
[z^2\otimes g,0,y'],\\
\eta([z^2\otimes g,0,y'])&=&
\cases{
([g],t^i)\in\tilde{F}(z^2)& if $z^2$ is of type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 2\cr}
\end{eqnarray*}
\item
Let $[g]\in(G_z)_\#$.
$$\Psi^{-1}([g])=[g^{-1}z\otimes g,0,0]$$
since
\begin{eqnarray*}
\phi(g^{-1}z\otimes g)&=&(z\otimes g),\\
\tau([z\otimes g,0,0])&=&
[z\otimes g\gamma(g)^{-1},0,0]\;=\;[z\otimes g,0,0],\\
\eta([z\otimes g,0,0])&=&[g].
\end{eqnarray*}
$\vartheta([g^{-1}z\otimes g,0,0])=[g^{-1}z^2\otimes g,0,0]$.
$$\Psi_2([g^{-1}z^2\otimes g,0,0])=
\cases{
([g],1)\in\tilde{F}(z^2)& if $z^2$ is of type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 2\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 3\cr}$$
since
\begin{eqnarray*}
\phi(g^{-1}z^2\otimes g)&=&(z^2\otimes g),\\
\tau([z^2\otimes g,0,0])&=&[z^2\otimes g\gamma(g)^{-1},0,0]\;=\;
[z^2\otimes g,0,0],\\
\eta([z^2\otimes g,0,0])&=&
\cases{
([g],1)\in\tilde{F}(z^2)& if $z^2$ is of type 1\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 2\cr
[g]\in\tilde{F}(z^2)& if $z^2$ is of type 3\cr}
\end{eqnarray*}
\end{enumerate}
This completes the proof.
\end{proof}
\begin{defi}
Let $G$ be a group. For every $z\in G$ we define $\sqrt{z}$ as
the subgroup of $(G_z)_\#$ resp. $(\ol{G_z})_\#$ generated by the set
$$\{g\in G\mid g^{2^k}=z \mbox{ for some } k\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}\}.$$
\end{defi}
\begin{defi}
Define
\[{\cal J}(G):=\lim_{\stackler{z}{\longrightarrow}} F(z),\]
where \[F(z):=\cases{
{\displaystyle {(G_z)_\#\over\sqrt{z}}}\times C_2
& if $ z$ is of\vspace{1mm} type $1$\cr
{\displaystyle {(\ol{G_z})_\# \over\sqrt{z}}}
& if $z$ is of\vspace{1mm} type $2$\cr
{\displaystyle {(G_z)_\#\over\sqrt{z}}}
& if $z$ is of type $3$\cr}\]
and the limit is taken with respect to the homomorphisms
\begin{enumerate}
\item[$\cdot$]
$F(z)\longrightarrow F(x^{-1}zx)$ for every $x\in G$
defined by
$$\cases{
([g],t^i)\mapsto([x^{-1}gx],t^i)& for all $z$ of\vspace{1mm} type $1$\cr
[g]\mapsto [x^{-1}gx]& for all $z$ of type $2$ and $3$\cr}$$
\item[$\cdot$]
$F(z)\rightarrow F(z^2)$ defined by
$$\cases{
([g],t^i)\mapsto([g],t^i)& for all $z$ of\vspace{1mm} type $1$\cr
[g]\mapsto\cases{([g],t^{w(g)})& if $z^2$ is of type $1$\cr
[g] & if $z^2$ is of type $2$\cr}&
for all $z$ of\vspace{1mm} type $2$\cr
[g]\mapsto\cases{([g],1)& if $z^2$ is of type $1$\cr
[g] & if $z^2$ is of type $2$\cr
[g] & if $z^2$ is of type $3$\cr}&
for all $z$ of type $3$\cr}$$
\item[$\cdot$]
$F(z)\rightarrow F(z^{-1})$ defined by
$[g]\mapsto[g]$
for all $z$ of type $3$.
\end{enumerate}
\end{defi}
\begin{remark}\label{remjgstruct}
$${\cal J}(G)\cong\bigoplus_{{\displaystyle c\in\cee\!\ell(G)}} {\cal L}(c),$$
where
$${\cal L}(c):= \lim_{\stackler{z\in c}{{\displaystyle \longrightarrow}}} F(z).$$
\end{remark}
\begin{thm}
\[\mathop{\rm Coker}\nolimits(1+\vartheta)\cong{\cal J}(G)\]
\end{thm}
\begin{proof}
Obvious in view of lemma~\ref{identlem}.
\end{proof}
\begin{prop}\label{propupsarf}
Suppose $\plane{g,h}$ is an element of $\mathop{\rm Arf}\nolimits^h(G)$.
The invariant $\Upsilon$ of {\rm chapter III} maps $\plane{g,h }$ to
\[\cases{[1,t]\in {\cal L}([1]) & if $gh$ is of type $1$\cr
[h]\in {\cal L}([gh]) & if $gh$ is of type $2$\cr}\]
Note that $gh$ is never of type $3$.
\end{prop}
\begin{proof}
Define $z:= gh$.
Note that $g^2=h^2=1$ and $hzh=z^{-1}$.
By definition $\Upsilon(\plane{g,h})=[g\otimes h,gh]\in\mathop{\rm Coker}\nolimits(1+\vartheta).$
By the definitions of $\phi$, $\tau$ and $\eta$:
\begin{eqnarray*}
\phi(gh)&=&gh, \\
\phi(g\otimes h)&=&hg\otimes h,\\
\tau([hg\otimes h,0,gh])&=&\tau([hzh\otimes h,0,z])\\
&=&[\gamma(h)hzh\gamma(h)^{-1}\otimes \gamma(h)h,0,z]\\
&=&[z^{-1}\otimes h,0,z]\\
\eta([z^{-1}\otimes h,0,z])&=&\cases{
([h],t)\in\tilde{F}(z)& if $z$ is of type 1\cr
[h]\in\tilde{F}(z)& if $z$ is of type 2\cr}
\end{eqnarray*}
Hence
$$\Psi_3([g\otimes h,gh])=\cases{
([h],t)=([1],t)& if $gh$ is of type 1\cr
[h]& if $gh$ is of type 2\cr}$$
\end{proof}
\begin{lemma}
For all $z\in G$
$$\mathop{\rm Ker}\nolimits(\ol{\gz}\rightarrow(\ol{\gz})_\#/\sqrt{z})\subset G_z.$$
\end{lemma}
\begin{proof}
Every commutator is a product of squares:
$xyx^{-1}y^{-1}=x^2(x^{-1}y)^2y^{-2}$.
Every square of an element in $\ol{\gz}$ belongs to $G_z$.
If $y^{{2^k}}=z$, then $y\inG_z$.
\end{proof}
\begin{lemma}\label{lemmantgh}
$\Upsilon(\plane{g,h})$ is never trivial in ${\cal J}(G)$.
\end{lemma}
\begin{proof}
Define $z:= (gh)^{2^k}$, with $k$ large.
If $z$ is of type 1, the statement is true by
proposition~\ref{propupsarf}.
If $z$ is of type 2, then $g\in\ol{\gz}\setminusG_z$. Therefore
$[g]\in {\cal L}([z])$
cannot be trivial.
\end{proof}
Now we review one of the examples we encountered in section 5 of chapter II.
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,S\mid S^2=(XS)^2=(YS)^2=1,\quad XY=YX\rangle.$$
To compute $\mathop{\rm Arf}\nolimits^h(G)$ we determine
$${\cal J}(G)=\bigoplus_{{\displaystyle c\in\cee\!\ell(G)}} {\cal L}(c).$$
It is immediately clear from the presentation of $G$ that $\sqrt{1}=G$ and
for all $z\in H$ we have $\ol{\gz}=G$.
(Recall that $H$ is the subgroup generated by $X$ and $Y$.)
Therefore
$$\left(\ol{\gz}\right)_\#=G/G^2=G/\langle X^2,Y^2\rangle.$$
A little examination shows that
$$\cee\!\ell(G)=
\left\{[1]\right\}\cup
\left\{[X^{2i}Y^{2j+1}],\,[X^{2k+1}Y^{2l}],\,[X^{2m+1}Y^{2n+1}]
\quad\mid j,k,m\geq 0\right\}$$
and
$${\cal L}(c)=\cases{C_2& if $c=[1]$\cr
G/\langle X^2,Y\rangle \cong C_2\times C_2 & if $c=[X^{2i}Y^{2j+1}]$\cr
G/\langle X,Y^2\rangle \cong C_2\times C_2 & if $c=[X^{2k+1}Y^{2l}]$\cr
G/\langle X^2,XY\rangle\cong C_2\times C_2 & if $c=[X^{2m+1}Y^{2n+1}]$.\cr}$$
Proposition~\ref{propsuf1} of chapter II says that the elements
$$\cases{
\plane{1,1}&\cr
\plane{X^{2i}Y^{2j+1}S,S} & for $j\geq 0$\cr
\plane{X^{2i+1}Y^{2j}S,S}& for $i\geq 0$\cr
\plane{X^{2i+1}Y^{2j+1}S,S}& for $i\geq 0$\cr
\plane{X^{2i+1}Y^{2j+1}S,XS}& for $j\geq 0$\cr
\plane{X^{2i+1}Y^{2j+1}S,YS} & for $i\geq 0.$\cr
\plane{X^{2i}Y^{2j+1}S,XS} & for $j\geq 0$\cr
}$$
generate $\mathop{\rm Arf}\nolimits^h(G)$.
But since
\begin{eqnarray*}
\Upsilon(\plane{1,1}) &=&t \in C_2\\
\Upsilon(\plane{X^{2i}Y^{2j+1}S,S}) &=&[S] \in{\cal L}([X^{2i}Y^{2j+1}])\\
\Upsilon(\plane{X^{2i+1}Y^{2j}S,S}) &=&[S] \in{\cal L}([X^{2i+1}Y^{2j}])\\
\Upsilon(\plane{X^{2i+1}Y^{2j+1}S,S}) &=&[S] \in{\cal L}([X^{2i+1}Y^{2j+1}])\\
\Upsilon(\plane{X^{2i+1}Y^{2j+1}S,XS})&=&[XS]\in{\cal L}([X^{2i}Y^{2j+1}])\\
\Upsilon(\plane{X^{2i+1}Y^{2j+1}S,YS})&=&[YS]\in{\cal L}([X^{2i+1}Y^{2j}])\\
\Upsilon(\plane{X^{2i}Y^{2j+1}S,XS}) &=&[XS]\in{\cal L}([X^{2i-1}Y^{2j+1}])
\end{eqnarray*}
we may conclude that these elements constitute a basis for $\mathop{\rm Arf}\nolimits^h(G)$.
\end{nitel}
We revert to one of the examples of chapter I.
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$G:=\langle X,Y,S\mid S^2=(XS)^2=(YS)^4=(Y^2S)^2=1,\quad XY=YX\rangle.$$
\begin{prop}
\begin{eqnarray*}
\left\{\plane{1,1}\right\}&\cup&
\left\{\plane{X^{2i+1}Y^{2j}S,S}\mid i\geq 0\right\}\\
&\cup&\left\{\plane{X^{2i}Y^{4j+2}S,S}\mid j\geq 0\right\}\\
&\cup&\left\{\plane{X^{2i+1}Y^{4j+2}S,XS}\mid j\geq 0\right\}
\end{eqnarray*}
is a basis for $\mathop{\rm Arf}\nolimits^{s,h}(G)$.
\end{prop}
\begin{proof}
We know already that these elements generate $\mathop{\rm Arf}\nolimits^h(G)$.
To prove independence we use our invariant $\Upsilon$.
We proceed to compute the summands ${\cal L}(c)$ of value group ${\cal J}(G)$.
It is not hard to verify that
$$\cee\!\ell(G)=
\left\{[1]\right\}\cup
\left\{[X^{2i+1}Y^{2j}]\,\mid i\geq 0\right\}\cup
\left\{[X^{i}Y^{2j+1}]\,\mid j\geq 0\right\}.$$
We omit the proof.
$${\cal L}(c)=\cases{
C_2
& if $c=[1]$\cr
G/\langle X,Y^2,(YS)^2\rangle \cong C_2\times C_2
& if $c=[X^{2i+1}Y^{2j}]$\cr
G/\langle X^2,Y,(YS)^2\rangle \cong C_2\times C_2
& if $c=[X^{2i}Y^{2j+1}]$ \cr
G/\langle X^2,XY,(YS)^2\rangle\cong C_2\times C_2
& if $c=[X^{2i+1}Y^{2j+1}]$.\cr}$$
Note that the class of $S$ is non-trivial in any ${\cal L}(c)$.
Further, the classes of $X$, $S$ and $XS$ in ${\cal L}([X^{2i}Y^{2j+1}])$ as well
as in ${\cal L}([X^{2i+1}Y^{2j+1}])$ are distinct.
Now we can use the list of images
\begin{eqnarray*}
\Upsilon(\plane{1,1}) &=&t \in C_2\\
\Upsilon(\plane{X^{2i+1}Y^{2j}S,S}) &=&[S] \in{\cal L}([X^{2i+1}Y^{2j}])\\
\Upsilon(\plane{X^{4i}Y^{4j+2}S,S}) &=&[S] \in{\cal L}([X^{2i}Y^{2j+1}])\\
\Upsilon(\plane{X^{4i+2}Y^{4j+2}S,S}) &=&[S] \in{\cal L}([X^{2i+1}Y^{2j+1}])\\
\Upsilon(\plane{X^{4i+1}Y^{4j+2}S,XS})&=&[XS]\in{\cal L}([X^{2i}Y^{2j+1}])\\
\Upsilon(\plane{X^{4i+3}Y^{4j+2}S,XS})&=&[XS]\in{\cal L}([X^{2i+1}Y^{2j+1}])
\end{eqnarray*}
to see that the assertion is true.
\end{proof}
\end{nitel}
\begin{nitel}{Example}
Let $G$ be the group with presentation
$$\langle X,Y,Z,S\mid X,Y,Z \mbox{ commute },
S^2=(XS)^2=(YS)^2=(ZS)^2=1\rangle.$$
Let $c\in\cee\!\ell(G)$ be the class of $XYZ$.
The invariant $\Upsilon$ maps
$$\xi:=\plane{XYS,SZ}+\plane{XZS,SY}+\plane{YZS,SX}+\plane{XYZS,S}
\in\mathop{\rm Arf}\nolimits^h(G)$$
to the class $[SZSYSXS]=[1]\in {\cal L}(c)=G/\langle X^2,Y^2,Z^2,XYZ\rangle.$
But it is not clear at all whether $\xi$ is trivial in $\mathop{\rm Arf}\nolimits^h(G)$.
\end{nitel}
\newpage
\section{Groups with two ends.}
\setcounter{altel}{0}
\setcounter{equation}{0}
We wish to prove that our invariant $\Upsilon$ is injective for all groups
having two ends.
For that purpose theorem~\ref{chargp2e} gives a suitable characterization
of these groups.
\begin{nota}
Throughout this section
\[\begin{array}{ll}
G & \mbox{ denotes a group,}\\
E & \mbox{ denotes a finite group,}\\
C & \mbox{ denotes the infinite cyclic group,}\\
C_m & \mbox{ denotes the cyclic group of order $m$,}\\
D & \mbox{ denotes the infinite dihedral group }\\
& \mbox{ with presentation }<S,T\mid S^2=(ST)^2=1>,\\
D_{m}& \mbox{ denotes the dihedral group of order $2m$}\\
& \mbox{ with presentation }
<\sigma,\tau\mid \sigma^2=(\sigma\tau)^2=\tau^m=1>.
\end{array}\]
\end{nota}
\begin{thm} {\rm \cite{Wall;gpth} }
The following statements are equivalent;
\begin{enumerate}
\item $G$ has two ends.
\item $G$ has an infinite cyclic subgroup of finite index.
\item $G$ has an infinite cyclic normal subgroup of finite index.
\end{enumerate}
\end{thm}
\begin{proof}
We refer to {\em loc. cit.} for a proof.
\end{proof}
\begin{defi}
A group extension of $C$ by $E$ is a short exact sequence of groups and
homomorphisms
\[1\rightarrow C\rightarrow G\rightarrow E\ra1\]
The extension is called central if the image of $C$ is central in $G$.
\end{defi}
\begin{thm} \label{chargp2e}
\begin{enumerate}
\item
$1\rightarrow C\rightarrow G\rightarrow E\ra1$ is a central extension if and only if $G$ fits into a
pull-back diagram
\begin{eqnarray*}G&\rightarrow&E\\\downarrow&&\downarrow\{{\bf C}\llap{\vrule height6pt width0.5pt depth0pt\kern.45em}}&\rightarrow&C_m\end{eqnarray*}
\item
$1\rightarrow C\rightarrow G\rightarrow E\ra1$ is a non-central extension if and only if $G$
fits into a pull-back diagram
\begin{eqnarray*}G&\rightarrow&E\\\downarrow&&\downarrow\{\cal D}&\rightarrow&D_m\end{eqnarray*}
\end{enumerate}
\end{thm}
\begin{proof}
In the sequel we will regard $C$ as a subgroup of $G$.
\begin{itemize}
\item[{\em 1.}] ``$\Rightarrow$''
Suppose $1\rightarrow C\rightarrow G\mapright{\pi}E\ra1$
is a central extension.
Define the so-called transfer homomorphism $\phi\colon G\rightarrow C$ as follows:
choose a set theoretic section $\alpha\colon E\rightarrow G$
of the projection $\pi\colon G\rightarrow E$ such that $\alpha(1)=1$ and define
$$\phi(g):=\prod_{e\in E}\alpha(e)g\alpha(e\pi(g))^{-1}
\mbox{ \ for all \ } g\in G.$$
Note that
\begin{itemize}
\item[$\diamond$]
$\alpha(e)g\alpha(e\pi(g))^{-1}\in\mathop{\rm Ker}\nolimits\pi=C$ for all $e\in E$ and $g\in G$.
\item[$\diamond$]
$\phi$ does not depend on the choice of $\alpha$:\hfill\break
If $\alpha'$ is another section of $\pi$ we have
$\alpha(e)\alpha'(e)^{-1}\in C$ for every $e\in E$.
Hence
\begin{eqnarray*}
\prod_{e\in E}\alpha(e)g\alpha(e\pi(g))^{-1}&=&
\prod_{e\in E}\alpha'(e)g\alpha'(e\pi(g))^{-1}\cdot\\
&&\prod_{e\in E}\alpha(e)\alpha'(e)^{-1}\cdot\\
&&\prod_{e\in E}\alpha'(e\pi(g))\alpha(e\pi(g))^{-1}\\
&=&\prod_{e\in E}\alpha'(e)g\alpha'(e\pi(g))^{-1}
\end{eqnarray*}
\item[$\diamond$]
$\phi$ is a homomorphism:
\begin{eqnarray*}
\phi(g_1g_2)&=&\prod_{e\in E}\alpha(e)g_1g_2\alpha(e\pi(g_1g_2))^{-1}\\
&=&\prod_{e\in E}\alpha(e)g_1\alpha(e\pi(g_1))^{-1}\cdot
\alpha(e\pi(g_1))g_2\alpha(e\pi(g_1)\pi(g_2))^{-1}\\
&=&\phi(g_1)\phi(g_2)
\end{eqnarray*}
\item[$\diamond$]
$\phi(c)=c^{|E|}$ for every $c\in C$.
Here $|E|$ denotes the cardinality of $E$.
\end{itemize}
Now it is easy to verify that $G$ fits into the pull-back diagram
$$\diagram{G&\mapright{\pi}&E\cr\mapdown{\phi'}
&&\mapdown{p\phi'\alpha}\cr C&\mapright{p}&C_m\cr}$$
where
$m:=|E|/[C:\mathop{\rm Im}\nolimits\phi]$,\hfill\break
$[C:\mathop{\rm Im}\nolimits\phi]$ is the index of $\mathop{\rm Im}\nolimits\phi$ in $C$,\hfill\break
$\phi':=\epsilon\lower1.0ex\hbox{$\mathchar"2017$}\phi$,\hfill\break
$\epsilon$ is an isomorphism $\mathop{\rm Im}\nolimits\phi\rightarrow C$ and \hfill\break
$p\colon C\rightarrow C_m$ is the canonical projection.\hfill\break
Note that $p\phi'\alpha$ does not depend on $\alpha$.
\item[$\phantom{2.}$] ``$\Leftarrow$''
Suppose
$$\diagram{G&\mapright{\pi}&E\cr\mapdown{}
&&\mapdown{}\cr C&\mapright{p}&C_m\cr}$$
is a pull-back diagram, then
$$\diagram{1\longrightarrow&C\longrightarrow & G &\mapright{\pi}E\lra1\hfil\cr
&c\mapsto&(c^m,1)& \hfil\cr
& &(c,e) &\mapsto e \hfil\cr}$$
is a central extension.
\item[{\em 2.}] ``$\Rightarrow$''
Suppose $1\rightarrow C\rightarrow G\mapright{\pi}E\ra1$ is a non-central extension.
Choose a set theoretic section $\alpha\colon E\rightarrow G$ as before.
The homomorphism $$w\colon E\rightarrow \mathop{\rm Aut}\nolimits(C)\cong C_2$$ defined by
$w(e)(c):=\alpha(e)c\alpha(e)^{-1}$ for all $c\in C$ and $e\in E$,
does not depend on the choice of $\alpha$.
Let $\phi\colon\mathop{\rm Ker}\nolimits(w\pi)\rightarrow C$ be the transfer homomorphism associated to
the central extension
$$1\rightarrow C\rightarrow \mathop{\rm Ker}\nolimits(w\pi)\mapright{\pi}\mathop{\rm Ker}\nolimits(w)\ra1.$$
Choose an element $u\in G\setminus\mathop{\rm Ker}\nolimits(w\pi)$ and
define $$\psi\colon G\rightarrow D$$
$$\psi(g):=\cases{\phi(g)&if $g\in\mathop{\rm Ker}\nolimits(w\pi)$\cr
\phi(gu^{-1})S&if $g\in G\setminus\mathop{\rm Ker}\nolimits(w\pi)$\cr}.$$
For every $g\in\mathop{\rm Ker}\nolimits(w\pi)$ we equate
\begin{eqnarray*}
\phi(ugu^{-1})^{-1}&=&u^{-1}\phi(ugu^{-1})u\\
&=&\prod_{e\in\mathop{\rm Ker}\nolimits(w)}u^{-1}\alpha(e)ugu^{-1}\alpha(e\pi(ugu^{-1}))^{-1}u\\
&=&\prod_{e\in\mathop{\rm Ker}\nolimits(w)}u^{-1}\alpha(e)u\alpha(\pi(u)^{-1}e\pi(u))^{-1}\cdot\\
&&\prod_{e\in\mathop{\rm Ker}\nolimits(w)}\alpha(\pi(u)^{-1}e\pi(u))g
\alpha(\pi(u)^{-1}e\pi(u)\pi(g))^{-1}\cdot\\
&&\prod_{e\in\mathop{\rm Ker}\nolimits(w)}\alpha(\pi(u)^{-1}e\pi(u)\pi(g))u^{-1}
\alpha(e\pi(ugu^{-1}))^{-1}u\\
&=&\phi(g).
\end{eqnarray*}
In particular $\phi(u^2)=1$ and $\psi$ is a homomorphism.\hfill\break
Again it is easy to verify that $G$ fits into the pull-back diagram
$$\diagram{G&\mapright{\pi}&E\cr\mapdown{\psi'}
&&\mapdown{p\psi'\alpha}\cr D&\mapright{p}&D_{m}\cr}$$
where
$2m:=|E|/[D:\mathop{\rm Im}\nolimits\psi]$.\hfill\break
Note that $m\cdot[D:\mathop{\rm Im}\nolimits\psi]=|\mathop{\rm Ker}\nolimits(w)|$ and
$|E|=2|\mathop{\rm Ker}\nolimits(w)|$.\hfill\break
$\psi':=\epsilon\lower1.0ex\hbox{$\mathchar"2017$}\psi$,\hfill\break
$\epsilon$ is an isomorphism $\mathop{\rm Im}\nolimits\psi\rightarrow D$ and \hfill\break
$p\colon D\rightarrow D_m$ is the canonical projection.\hfill\break
Note that $p\psi'\alpha$ does not depend on $\alpha$.
\item[$\phantom{2.}$] ``$\Leftarrow$''
If
$$\diagram{G&\mapright{\pi}&E\cr\mapdown{}
&&\mapdown{}\cr D&\mapright{p}&D_{m}\cr}$$
is a pull-back diagram, then
$$\diagram{1\longrightarrow&C\longrightarrow & G &\mapright{\pi}E\lra1\hfil\cr
&c\mapsto&(c^m,1)& \hfil\cr
& &(d,e) &\mapsto e \hfil\cr}$$
is obviously a non-central extension.
\end{itemize}
This completes the proof.
\end{proof}
\newpage
\section{$\Upsilon$ for groups with two ends.}
\setcounter{altel}{0}
\setcounter{equation}{0}
This section is devoted to the following theorem.
\begin{thm}\label{thmin2end}
The invariant $\Upsilon\colon\mathop{\rm Arf}\nolimits^h(G)\rightarrow{\cal J}(G)$ is injective for all
groups $G$ having two ends.
\end{thm}
\begin{lemma}\label{lem2powz}
For all $k\in{{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$ the relation
$$\plane{a,b}=\plane{a,a(ab)^{2^k}}=\plane{b,b(ab)^{2^k}}$$
holds in $\mathop{\rm Arf}\nolimits^h(G)$.
\end{lemma}
\begin{proof}
By the relations mentioned in remark~\ref{remarfgrel} of chapter I,
we have
$\plane{a,b}=\plane{a,bab}=\plane{a,a(ab)^2}$ and $\plane{a,b}=\plane{b,a}$.
The rest is obvious.
\end{proof}
\begin{lemma}\label{lemmaeindord}
If $\plane{a,az}$ and $\plane{b,bz}$ are elements of $\mathop{\rm Arf}\nolimits^h(G)$ and
$abz^i$ has finite order, for some $i\in\Z$,
then $\plane{a,az}=\plane{b,bz}$.
\end{lemma}
\begin{proof}
First we consider the case that $i=0$.
Let us write $x=ab$ and let's say that the order
of $x$ equals $2^km$ with $m$ odd. \hfill\break
Note that $a^2=b^2=(az)^2=1 $ and $xz=zx$.
Using the relations of remark~\ref{remarfgrel} of chapter I we equate
\begin{eqnarray*}
\plane{b,bz}&=&\plane{ax,axz}\\
&=&\plane{ax^m,ax^mz}\\
&=&\plane{ax^m,ax^mz^{2^k}}\\
&=&\plane{ax^m,ax^m(ax^maz)^{2^k}}\\
&=&\plane{ax^m,az}\\
&=&\plane{az,ax^m}\\
&=&\plane{az,az(azax^m)^{2^k}}\\
&=&\plane{az,az(aza)^{2^k}}\\
&=&\plane{az,a}\\
&=&\plane{a,az}
\end{eqnarray*}
The case $i\neq 0$ can be reduced to the previous case:
$$\plane{b,bz}=\plane{z^{-j}bz^j,z^{-j}bzz^j}=\plane{bz^{2j},bz^{2j+1}}$$
$$\plane{b,bz}=\plane{b,bbzb}=\plane{b,bz^{-1}}=\plane{bz^{2j},bz^{2j-1}}
=\plane{bz^{2j-1},bz^{2j}},$$
thus $\;\plane{b,bz}=\plane{bz^i,bz^{i+1}}$.
\end{proof}
\begin{prop}
In the case where $G$ fits into a pull-back diagram
$$\diagram{G&\longrightarrow&E\cr\mapdown{}&&\mapdown{}\cr C&\longrightarrow&C_m\cr}$$
$\Upsilon$ is injective.
\end{prop}
\begin{proof}
Let $x\in \mathop{\rm Ker}\nolimits(\Upsilon)$.
The relations in $\mathop{\rm Arf}\nolimits^h(G)$ listed in remark~\ref{remarfgrel} of chapter I
and remark~\ref{remjgstruct} on the structure of ${\cal J}(G)$ allow us to assume,
without loss of generality,
that $$x=\sum\plane{a_i,a_iz}.$$
Every product of two elements of order two, is of finite order,
since all elements of order two in $G$
take the form $(1,e)$.
So we may use lemma~\ref{lemmaeindord} to see that $x=0$ or $x=\plane{a,az}$.
But according to
lemma~\ref{lemmantgh} $\Upsilon(\plane{a,az})$ is non-trivial,
so the second case does not occur.
\end{proof}
It remains to show that $\Upsilon$ is injective for groups $G$ which fit
into a pull-back diagram
$$\diagram{G&\mapright{\pi}&E\cr
\mapdown{\psi}&&\mapdown{\hat p}\cr
D&\mapright{p}&D_{2m}\cr}$$
\underline{Intermezzo}.
\begin{defi}
Let $G$ be a group.
Suppose we have 2-primary elements $a,b\in G$ which satisfy
$[a,b^2]=[a^2,b]=1$.
Here $[x,y]$ denotes the commutator $xyx^{-1}y^{-1}$.
Denote by $H$ the subgroup of $G$ generated by $a$ and $b$. The matrix
$$\pmatrix{a&1\cr 0&b\cr}+\pmatrix{a&1\cr 0&b\cr}^\alpha=
\pmatrix{a+a^{-1}&1\cr 1&b+b^{-1}\cr}=
\pmatrix{a(1+a^{-2})&1\cr 1&b(1+b^{-2})\cr}$$
is invertible, since $1+a^{-2}$ and $1+b^{-2}$ are nilpotent and central
in $\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$.
It is therefore legitimate to define
$$\wp(a,b):=\left[\pmatrix{a&1\cr 0&b\cr}\right]-
\left[\pmatrix{0&1\cr0&0\cr}\right]\in L^h(H).$$
We call such elements of $L^h(H)$ as well as their images in $L^h(G)$
pseudo-arfian.
\end{defi}
Notice that $\wp(a,b)$ is not necessarily an element of
$\mathop{\rm Arf}\nolimits^h(H)$ or $\mathop{\rm Arf}\nolimits^h(G)$.
However, applying theorem~\ref{iadiciso} of chapter I to the ring
$\;\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[H]$ and its nilpotent ideal
$(a^2+1,b^2+1)$ yields an isomorphism
$$L^h(H)\lhook\joinrel\surarrow L^h(H/<a^2,b^2>),$$ which maps $\wp(a,b)\in L^h(H)$
to the Arf-element
$\plane{a,b}\in \mathop{\rm Arf}\nolimits^h(H/<a^2,b^2>):$
$$\diagram{&&\wp(a,b)&\longmapsto&\plane{a,b}\cr
\mathop{\rm Arf}\nolimits^h(G)&\longleftarrow&\mathop{\rm Arf}\nolimits^h(H)&\lhook\joinrel\longrightarrow&\mathop{\rm Arf}\nolimits^h(H/<a^2,b^2>)\cr
\bigcap&&\bigcap&&\bigcap\cr
L^h(G)&\longleftarrow&L^h(H)&\lhook\joinrel\surarrow&L^h(H/<a^2,b^2>)\cr}$$
\begin{punt}\label{defpseu}
Let $G$ be a group and $g,z\in G$.
Assume $g^{-1}zg=z^{-1}$ and $g$ is of finite order, say $2^rr_0$
with $r_0$ odd. Define $H$ as the subgroup of $G$
generated by $z$ and $h:= g^{r_0}$.\hfill\break
Since $h^{-1}zh=z^{-1}$, i.e. $h^2=(hz)^2$
we obtain a pseudo-arf element $\wp(h,hz)\in L^h(H).$
The question is whether this element depends on $h$.
\end{punt}
\begin{thm}\label{thmwinjend}
Let $E$ be a finite group.
The invariant $\omega_1^h$ of chapter II induces an isomorphism
$$L^h(E)\longrightarrow\bigoplus k/\{x+x^2\mid x\in k\}$$
Here the summation runs through all representations $\rho\colon
\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E]\rightarrow M_n(k)$ which take the form
$\rho(e)^{-1}=P^{-1}\rho(e)^tP$ for all $e\in E$,
for some invertible matrix $P$ and $t$ means matrix transpose.
What's more, the image of $\wp(h,hz)$ under this isomorphism
is the element which has $[\mathop{\rm Tr}\nolimits(\rho(z))]$ at the place with index $\rho$.
In particular $\wp(h,hz)$ does not depend on $h$.
\end{thm}
\begin{proof}
Define $R:=\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E]/\mathop{\rm rad}\nolimits(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E])$ where $\mathop{\rm rad}\nolimits$ means
Jacobson radical.
For every ring $A$, we denote by $\widetilde{A}$ the truncated polynomial
ring $A[T]/(T^3)$ and in this context
$(T)$ is the two-sided ideal of $\widetilde{A}$ generated by $T$.
Consider the following diagram.
$$\diagram{
L^h(E)&&\cr
\isodown{1}&&\cr
L^h(R)&{\buildrel \omega_1^h \over {\hbox to 50pt{\rightarrowfill}}}
&H^0(K_1(\widetilde{R}))\cr
\isodown{2}&&\isodown{6}\cr
L^h\left(\prod D_i\right)&&
H^0\left(\bigoplus K_1(\widetilde{D_i})\right)\cr
\isodown{3}&&\isodown{7}\cr
\bigoplus_j L^h(D_j)&&
\bigoplus_j H^0\left( K_1(\widetilde{D_j})\right)\cr
\isodown{4}&&\isodown{}\cr
\bigoplus_j L^h(k_j)&&
H^0\left(\bigoplus( k_j^*\oplus1+T\widetilde{k_j})\right)\cr
\isodown{5}&&\isodown{}\cr
\bigoplus_j\mathop{\rm Arf}\nolimits^h(k_j)&&
H^0\left(\bigoplus_j 1+T\widetilde{k_j}\right)\cr
\hfill\searrow& &\swarrow\hfill\cr
&\bigoplus_j\mathop{\rm Coker}\nolimits(1+\sigma_j)&\cr
&\mapdown{\cong}&\cr
&\bigoplus_j \Z/2&\cr}$$
We elucidate the diagram.
\begin{enumerate}
\item
It follows from theorem~\ref{iadiciso} of chapter I
that $L^h(E)$ and $L^h(R)$ are isomorphic,
because $\mathop{\rm rad}\nolimits(\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E])$ is a nilpotent ideal.
\item
The ring $R$ is artinian and $\mathop{\rm rad}\nolimits(R)=0$, so we can apply the
Wedderburn-Artin theorem. In our case this reads:
$R$ is isomorphic to a direct product of full matrix rings over finite
fields of characteristic two.
Explicitly, $$R\cong \prod D_i;$$
here $D_i:= M_{n_i}(k_i)$ is the ring of $(n_i\times n_i)$-matrices
over the finite field $k_i$ and char$(k_i)=2$. \hfill\break
Denote by $\rho_i$ the composition
$\,\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E]\rightarrow R\rightarrow \bigoplus D_i\rightarrow D_i.$
\item
Let $\,\ol{\phantom{x}}\,$ denote the (anti-) involutions on
$R$ and $\prod D_i$ induced
by the involution on $\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E]$.
Before we decompose $\prod D_i$ as a product of rings with involution,
we fix some notations concerning finite fields of characteristic two.
If $k$ is such a field, the group of automorphisms of $k$ is cyclic and
generated by the Frobenius automorphism $\sigma\colon k\rightarrow k$ which
assigns to an element $x$ of $k$ its square.
The field trace $\mathop{\rm Tr}\nolimits\colon k\longrightarrow\mkern-15mu\rightarrow \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$ induces an isomorphism
$\mathop{\rm Coker}\nolimits(1+\sigma)\rightarrow\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$.
If the degree of $k$ over $\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$ is even there
exists a unique automorphism of order two, which we denote by
$\hat{\sigma}$.\hfill\break
Now, for a factor $D=M_n(k)$ of $\prod D_i$ we have three possible cases:
\begin{itemize}
\item
$D$ is invariant under the involution, i.e. $D=\ol{D}$, and the restriction
of $\,\ol{\phantom{x}}\,$ to $k$ is $\hat{\sigma}$.
\item
$D=\ol{D}$ and the restriction of $\,\ol{\phantom{x}}\,$ to $k$ is the identity.
Since the composition of the (anti-) involution $\,\ol{\phantom{x}}\,$ with matrix
transpose is a $k$-linear automorphism of $D$, this composition takes the
form $X\mapsto PXP^{-1}$, for some invertible matrix $P$.
Further we may assume that $P$ is symmetric,
since this automorphism is of order two.
Thus for all $X\in D$ we have $\ol{X}=P^{-1}X^tP$.
\item
$\ol{D}\neq D$. So $D\times\ol{D}$ is a factor of $\prod D_i$.
If $D\times D^{\circ}$ is endowed with the involution $(x,y)\mapsto (y,x)$,
the map $D\times\ol{D}\rightarrow D\times D^{\circ}$ defined by
$(x,y)\mapsto(x,\ol y)$
is an isomorphism of rings with involution.
Recall that ${\scriptstyle \circ}$ means opposite multiplication.
\end{itemize}
Thus we obtain a decomposition of $\prod D_i$ in which
three different types of factors occur.
The $L$-groups split accordingly. See e.g. \cite{w2}.
We assert that only the groups $L^h(D)$,
where $D$ is of the second type, survive.\hfill\break
In the first case we have
$$L^h(D,\ol{\phantom{x}},1)\cong L^h(k,\hat{\sigma},1)$$
by Morita invariance.
But $L^h(k,\hat{\sigma},1)=0$ by \cite[\S6]{Wall;lfound}.\hfill\break
For the third case we will show that quadratic
modules $(M,\theta)$ over the ring $D\times D^{\circ}$,
with involution $\alpha(x,y)=(y,x)$, are in fact hyperbolic.
Note that there is no need to worry about bases, because we are working in
$L^h$.
Define $\lambda:=(1,0)$,
$M_1:= \lambda M$ and $M_2:=(1+\lambda)M$. So $M=M_1\oplus M_2$.
Since $b_\theta\colon M\rightarrow M^\alpha$ is an isomorphism and for all $m,n\in M$
$$b_\theta((1+\lambda)m)((1+\lambda)n)=\lambda(1+\lambda)b_\theta(m)(n)=0,$$
the restriction of $b_\theta$ to $M_2$ yields an isomorphism
$M_2\rightarrow M_1^\alpha$.
Now it is easy to verify that the map
$$(M,\theta)\longrightarrow H(M_1)=(M_1\oplus M_1^\alpha,\upsilon)$$
defined by
$$ m\mapsto(\lambda m,b_\theta((1+\lambda)m))$$
is an isometry.
In Walls terminology \cite{Wall;lfound} this says
that $(D\times\{0\})M$ is a subkernel of $M$.
This proves our assertion.\hfill\break
We use the index $j$ to refer to summands of the second type.
\item
$L^h(D_j,\ol{\phantom{x}},1)$ is isomorphic to $L^h(k_j,1,1)$ by
Morita invariance.
\item
Since the field trace $\mathop{\rm Tr}\nolimits\colon k\longrightarrow\mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2$ is surjective we can choose
an element $a$ in $k$ such that $\mathop{\rm Tr}\nolimits(a)=1$.
The Arf invariant $$\omega_1^h\colon \mathop{\rm Arf}\nolimits^h(k,1,1)\rightarrow \mathop{\rm Coker}\nolimits(1+\sigma)\cong \Z/2$$
maps $\plane{a,1}$ to the non-trivial element in $\Z/2$.
Combining this with the fact that $L^h(k,1,1)\cong\Z/2$,
see \cite[\S6]{Wall;lfound},
we find $$L^h(k,1,1)=\mathop{\rm Arf}\nolimits^h(k,1,1)\cong\Z/2.$$
\item
It is almost immediately clear from the definition of $K_1$,
that
for all rings $A,B$ one has $K_1(A\times B)\cong K_1(A)\oplus K_1(B)$.
\item
The argument here is roughly the same as the one on the `$L$-side'
of the diagram (item 3).
By Morita theory we have $K_1(\widetilde{D})\cong K_1(\widetilde{k})$.
Alternatively one can see this directly by looking at the
definition of $K_1$.
Since the projection $\widetilde{k}\rightarrow k$ splits we have
$$K_1(\widetilde{k})\cong K_1(k)\oplus K_1(\widetilde{k},(T)).$$
It is well-known that $K_1(k)=k^*$, the group of units in $k$
and we already saw that
$K_1(\widetilde{k},(T))=1+T\widetilde{k}$.
To decompose $\bigoplus_i K_1(\widetilde{D_i})$ into invariant parts,
we consider the same three possibilities:
\begin{itemize}
\item
$D$ is invariant under the involution and the restriction
of $\,\ol{\phantom{x}}\,$ to $k$ is $\hat{\sigma}$.
In this case
$$H^0(K_1(\widetilde{D}))=H^0(k^*\oplus (1+T\widetilde{k}))
=H^0(k^*)\oplus H^0(1+T\widetilde{k}).$$
But $H^0(k^*)$ vanishes, because $k^*$ has odd order.
And in the third section of chapter II we computed
$H^0(1+T\widetilde{k})=C(k)$, but this also disappears since
$H^0(k;\hat{\sigma})=0$.
\item
$D=\ol{D}$ and the restriction of $\ol{\phantom{x}}$ to $k$ is the identity.
By the same arguments as in the previous case we obtain
$H^0(K_1(\widetilde{D}))=C(k)$, but now $C(k)$ is precisely $\mathop{\rm Coker}\nolimits(1+\sigma)$.
\item
$D\neq\ol{D}$.
Here the involution interchanges the summands
$K_1(\widetilde{D})$ and $K_1(\widetilde{\ol{D}})$, so $H^0$ clearly dies.
\end{itemize}
Thus only the summands of the second type survive.
\end{enumerate}
This completes the proof of the first part of what the theorem asserts.
To prove the second part
let $\rho\colon \mbox {F{\llap {I \kern.13em}}}\hspace{-.4em}\,_2[E]\rightarrow D=M_n(k)$ be a representation of the special
kind, i.e. $D$ is of the second type, and assume that
$\rho(h)=H$ and $\rho(z)=Z$.
Then
$$\omega_1^h\left(\wp(h,hz)\right)=
\omega_1^h\left(\left[\pmatrix{h&1\cr0&hz\cr}\right]-
\left[\pmatrix{0&1\cr0&0\cr}\right]\right).$$
Now
$$\pmatrix{h&1\cr0&hz\cr}+(1+T)\pmatrix{h&1\cr0&hz\cr}^\alpha=
\pmatrix{h+(1+T)h^{-1}&1\cr(1+T)&hz+(1+T)(hz)^{-1}\cr}$$
and
$$\pmatrix{0&1\cr0&0\cr}+(1+T)\pmatrix{0&1\cr0&0\cr}^\alpha=
\pmatrix{0&1\cr(1+T)&0\cr},$$
so
\begin{eqnarray*}
\lefteqn{\omega_1^h\left(\wp(h,hz)\right)=}\\
&&\left[\pmatrix{h+(1+T)h^{-1}&1\cr(1+T)&hz+(1+T)(hz)^{-1}\cr}
\pmatrix{0&1\cr(1+T)&0\cr}^{-1}\right]=\\
&&\left[\pmatrix{1&h(1+T)^{-1}+h^{-1}\cr hz+(1+T)(hz)^{-1}&1\cr}\right]=\\
&&\left[\pmatrix{1+h^2z(1+T)^{-1}+h^{-2}z(1+T)&h(1+T)^{-1}+h^{-1}\cr
0&1\cr}\right]=\\
&&\left[\pmatrix{1+h^2z(1+T)^{-1}+h^{-2}z(1+T)&0\cr
0&1\cr}\right]=\\
&&\left[\left(1+h^2z(1+T)^{-1}+h^{-2}z(1+T)\right)\right]\in
H^0(K_1(\widetilde{R})).
\end{eqnarray*}
The image of this element in $H^0(1+T\widetilde{k})$ equals
\begin{eqnarray*}
\lefteqn{\left[\det\left(1+H^2Z(1+T+T^2)+H^{-2}Z(1+T)\right)\right]} \\
&=&\left[\det\left(1+H^2ZT^2\right)\right]
\left[\det\left(1+(H^2+H^{-2})Z(1+T)\right)\right]\\
&= &\left[\det\left(1+H^2ZT^2\right)\right] \\
&=&[1+\mathop{\rm Tr}\nolimits(H^2Z)T^2]\\
&=&[1+\mathop{\rm Tr}\nolimits(Z)T^2]
\end{eqnarray*}
Here $$\det\left(1+(H^2+H^{-2})Z(1+T)\right)=1 \quad\mbox{ and }\quad
\mathop{\rm Tr}\nolimits(H^2Z)=\mathop{\rm Tr}\nolimits(Z)$$
by lemma~\ref{lemmanilp},
because $(H^2+H^{-2})Z$ and $1+H^2$ are nilpotent.
Finally, the image of $[1+\mathop{\rm Tr}\nolimits(Z)T^2]$ in $\mathop{\rm Coker}\nolimits(1+\sigma)$ equals
$[\mathop{\rm Tr}\nolimits(Z)]=[\mathop{\rm Tr}\nolimits(\rho(z))]$ according to the computations in chapter II.
\end{proof}
\begin{lemma}\label{lemmanilp}
If $V$ is a finite dimensional $k$-vectorspace, $N\colon V\rightarrow V$ is a
nilpotent linear map and $s$ is an indeterminate,
then $$\mathop{\rm Tr}\nolimits(N)=0 \mbox{ \ and \ } \det(1+sN)=1.$$
\end{lemma}
\begin{proof}
Suppose $N^n=0$.
We apply induction on $n$. \hfill\break
If $n=1$ the matter is clear.\hfill\break
If $n>1$ consider the diagram
$$\diagram{0&\longrightarrow& NV &\longrightarrow&V &\longrightarrow&V/NV &\longrightarrow&0\cr
& &\mapdown{N}& &\mapdown{N}& &\mapdown{0}& & \cr
0&\longrightarrow& NV &\longrightarrow&V &\longrightarrow&V/NV &\longrightarrow&0\cr}$$
The first vertical map in this diagram has nilpotency degree $n-1$
and $N\colon V\rightarrow V$ takes the form $\pmatrix{*&*\cr0&0\cr}$.
This proves the assertions.
\end{proof}
\underline{End intermezzo}.
\begin{thm}\label{thm2stuks}
If $\plane{a,b}+\plane{c,d}\in\mathop{\rm Ker}\nolimits(\Upsilon)$, then
$\plane{a,b}=\plane{c,d}$ in $\mathop{\rm Arf}\nolimits^h(G)$.
\end{thm}
\begin{proof}
Note that $\plane{a,b}+\plane{c,d}\in\mathop{\rm Ker}\nolimits(\Upsilon)$ if and only if
$$[ab]=[cd]\in \cee\!\ell(G)\quad\mbox{ and }\quad
[bd]=1\in{\cal L}([ab]).$$
Again the relations in $\mathop{\rm Arf}\nolimits^h(G)$ and the structure of ${\cal J}(G)$
allow us to assume that $ab=cd$.
Elements of order two in $G$ either have the form
$(1,e)$ with $e^2=1$ and $\hat p(e)=1$ or the form
$(ST^i,e)$ with $e^2=1$ and $\hat p(e)=p(ST^i).$
Thus we may assume that
$$\plane{a,b}+\plane{c,d}=
((\Delta,e),(\Delta T^i,ez))+((\Delta T^j,ex),(\Delta T^{i+j},exz)),$$
where
\begin{itemize}
\item[$\cdot$]
$\Delta=ST^\nu$: if $\Delta=1$ we are through by lemma~\ref{lemmaeindord}
\item[$\cdot$]
$e,x,z\in E$ satisfy $e^2=(ex)^2=(ez)^2=1$ and $xz=zx$
\item[$\cdot$]
$[(T^j,x)]=1\in {\cal L}([(T^i,z)])$.
\end{itemize}
Lemma~\ref{lem2powz} permits us to
replace $(T^i,z)$ by any power-of-two power
of $(T^i,z)$. Hence we may assume that $z$ has odd order, let's say order
$l_0=2l-1$.
\begin{nitel}{Case 1$\colon$ $i\neq0$}\hfill\break
Write $m=2^\mu m_0$ and $i=2^\tau i_0$ with $m_0$ and $i_0$ odd.
Since $(T^i,z)\in G$ and $z$ has order $l_0$ in $E$, we have $m|il_0$, i.e.
$\mu\leq\tau$ and $m_0|i_0l_0$.
If $\mu<\tau$, then
$$((\Delta,e),(\Delta T^i,ez))=((\Delta,e),(\Delta T^{i/2},ez^l)),$$
by lemma~\ref{lem2powz},
where
\begin{itemize}
\item[$\cdot$]
$(T^{i/2},z^l)\in G$, \ because $il\equiv i/2\pmod{m}$
\item[$\cdot$]
$(T^{i/2},z^l)^2=(T^i,z).$
\end{itemize}
So we may assume that $\mu=\tau$.\hfill\break
Further, conjugation by a suitable power of $(T^j,x)$ allows us
to replace $(T^j,x)$ by any odd power of $(T^j,x)$.
Thus we may assume that $x$ has order a power of 2, let's say $2^k$.
\begin{itemize}
\item[$\diamond$]
If necessary we conjugate by $(T^m,1)$ to achieve that $0\leq j<2m$.
\item[$\diamond$]
If $m<j<2m$, then conjugation by $(T^{j-m},x)$ yields
$$((\Delta T^j,ex),(\Delta T^{i+j},exz))=
((\Delta T^{2m-j},ex^{-1}),(\Delta T^{i+2m-j},ex^{-1}z)),$$
so we may assume that $0\leq j\leq m$. \hfill\break
It is important to note that these changes do not affect the order of $x$.
\item[$\diamond$]
If $j=0$, then lemma~\ref{lemmaeindord} gives the desired result.
\item[$\diamond$]
If $j=m$, then conjugation by $(T^{(m+i)/2},z^l)$ yields
\begin{eqnarray*}
((\Delta T^m,ex),(\Delta T^{i+m},exz))&=&
((\Delta T^{-i},exz^{-2l}),(\Delta ,exz^{1-2l}))\\
&=&((\Delta ,e),(\Delta T^i,ez))
\end{eqnarray*}
The second identity follows from lemma~\ref{lemmaeindord}.
Note that $(T^{(m+i)/2},z^l)\in G$ if and only if
$(i+m)/2\equiv il\pmod{m}$. But this condition is satisfied
because
$$(i+m)/2=2^\mu(i_0+m_0)/2\equiv 2^\mu i_0l=il \pmod{2^\mu m_0}.$$
This finishes the proof in the case that $j=m$.
\item[$\diamond$]
If $0<j<m$, write $j=2^\nu j_0$ with $j_0$ odd.\hfill\break
We know that $(T^j,x)\in G$ and $x$ has order $2^k$ in $E$,
hence $m|j2^k$, i.e. $\mu\leq k+\nu$ and $m_0|j_0$.
Taking the fact that $j<m$ into account this implies $\nu<\mu$.
\begin{itemize}
\item[$\cdot$]
Choose $r\in {{\bf N}\llap{\vrule height6.5pt width0.5pt depth0pt\kern.8em}}$ such that $r>k+\nu$ and $l_0|2^r-1$.
\item[$\cdot$]
Define $w:= z^{l^{\mu-\nu}}$.
\item[$\cdot$]
Choose an $\epsilon$ which satisfies the congruence
$$j\epsilon+il^{\mu-\nu}\equiv j+i_02^\nu\pmod{m}.$$
This is possible:\hfill\break
mod $m_0$ it reads
\begin{eqnarray*}
i_02^\nu2^{\mu-\nu} l^{\mu-\nu}&\equiv&i_02^\nu\\
i_02^\nu((2l)^{\mu-\nu}-1)&\equiv&0\\
i_02^\nu((l_0+1)^{\mu-\nu}-1)&\equiv&0,
\end{eqnarray*}
but since $m_0|i_0l_0$, this is automatically true;\hfill\break
mod $2^\mu$ it reads
$$2^\nu j_0(\epsilon-1)\equiv 2^\nu i_0\pmod{2^\mu}$$
which is equivalent to
$$j_0(\epsilon-1)\equiv i_0\pmod{2^{\mu-\nu}},$$
but since $j_0$ is odd this is solvable.
\item[$\cdot$]
$ex(exex^\epsilon w)^{2^{r-\nu}}=
ex(x^\epsilon w)^{2^{r-\nu}}=
exz^{2^{r-\mu}}.$
\item[$\cdot$]
Define $\tilde{j}:= j-(2^r-1)2^\nu i_0$ and
$\tilde{x}:= x^\epsilon$.
\end{itemize}
These definitions and facts support the following computation.
\begin{eqnarray*}
((\Delta T^j,ex),(\Delta T^{i+j},exz))&=&\\
\mbox{ \ ($r-\mu$ times lemma~\ref{lem2powz})}&=&
((\Delta T^j,ex),(\Delta T^{i2^{r-\mu}+j},exz^{2^{r-\mu}}))\\
\mbox{ \ ($r-\nu$ times lemma~\ref{lem2powz})}&=&
((\Delta T^j,ex),(\Delta T^{i_02^\nu+j},e\tilde{x}w))\\
\mbox{ \ ($r$ times lemma~\ref{lem2powz})}&=&
((\Delta T^{j+2^\nu i_0-2^{r+\nu}i_0},e\tilde{x}),
(\Delta T^{i_02^\nu+j},e\tilde{x}w))\\
&=&((\Delta T^{\tilde{j}},e\tilde{x}),
(\Delta T^{i_02^{r+\nu}+\tilde{j}},e\tilde{x}w))\\
\mbox{ \ ($\mu-\nu$ times lemma~\ref{lem2powz})}&=&
((\Delta T^{\tilde{j}},e\tilde{x}),(\Delta T^{i2^r+\tilde{j}},e\tilde{x}w))\\
\mbox{ \ ($r$ times lemma~\ref{lem2powz})}&=&
((\Delta T^{\tilde{j}},e\tilde{x}),(\Delta T^{i+\tilde{j}},e\tilde{x}z))
\end{eqnarray*}
Observe that $\tilde{j}$ is a multiple of $2^{\nu+1}m_0$.
Thus we replace the old $(T^j,x)$ by a new one.
Apply one of the preceding steps if $j\geq m$ or $j\leq0$.
\end{itemize}
Repeat this process until $\mu=\nu$, which implies $m|j$.
This completes the proof in this first case.
We did not need the fact that
$[(T^j,x)]=1\in {\cal L}([(T^i,z)])$!
This means that the primary Arf invariant is already good enough to detect
the Arf-elements in this case.
\end{nitel}
\begin{nitel}{Case 2$\colon$ $i=0$}\hfill\break
Our purpose is to show that
$$((\Delta ,e),\Delta ,ez))=((\Delta T^j,ex),(\Delta T^j,exz))\in\mathop{\rm Arf}\nolimits^h(G).$$
We apply induction on $j$, as follows.
\begin{itemize}
\item[$\diamond$] If $j<0$ or $j>2m$,\hfill\break
we conjugate by a suitable power of $(T^m,1)$,
to attain $0\leq j\leq 2m$.
\item[$\diamond$] If $m<j\leq2m$,\hfill\break
we conjugate by $(\Delta T^m,e)$, to achieve $0\leq j\leq m$.
\item[$\diamond$] If $j=0$,\hfill\break
lemma~\ref{lemmaeindord} does the job.
\item[$\diamond$] If $j\not|\,m$,\hfill\break
we define $d:=\gcd(j,m)$.
Obviously $(T^d,x^{n_0})\in\ol{G_{(1,z)}}$ for some $n_0\in \Z$.
Now conjugating by $(T^d,x^{n_0})$ allows us to replace $j$ by $j-2d$.
Notice that $j-2d>0$.
\item[$\diamond$] If $j|m$,\hfill\break
there are two possibilities.
If there exists $(T^c,y)\in\ol{G_{(1,z)}}$ with $0<c<j$, then
we conjugate by $(T^c,y)$ to replace $j$ by $j-2c$. We have
$$-j+2\leq j-2c\leq j-2.$$
Conjugating by $(\Delta ,e)$,
if necessary, yields $$0\leq j-2c\leq j-2.$$
If there is not such a $c$, then the elements of
$\ol{G_{(1,z)}}$ either have the
form $(T^{jv},\cdot\cdot)$ or
$(\Delta T^{jv},\cdot\cdot)$.
Since any element of $$\mathop{\rm Ker}\nolimits(\ol{G_{(1,z)}}\rightarrow F(1,z))$$
is a product of
squares and $2$-power roots of $(1,z)$, so is $(T^j,x)$.
A little examination reveals that this can only happen when there exist
$2$-power roots $y_1$ and $y_2$ of $z$ such that
$(\Delta ,y_1),(\Delta T^j,y_2)\in G$\hfill\break
Consider the pull-back diagram
$$\diagram{G&\longrightarrow&E\cr
\mapdown{ }&&\mapdown{\hat p}\cr
D&\longrightarrow&D_{m}\cr}$$
and define
$$\begin{array}{ll}
F_1:= \hat{p}^{-1}(<\sigma\tau^\nu>)&\\
j_1\colon F_1\rightarrow G \mbox{ \ by \ }
&j_1(f):=\cases{(1,f)& if $\hat p(f)=1$\cr (\Delta,f)&otherwise\cr}\\
F_2:= \hat{p}^{-1}(<\sigma\tau^{\nu+j}>)&\\
j_2\colon F_2\rightarrow G \mbox{ \ by \ }
&j_2(f):=\cases{(1,f)& if $\hat p(f)=1$\cr (\Delta T^j,f)&otherwise\cr}\\
F_0:= F_1\cap F_2=\mathop{\rm Ker}\nolimits(\hat p)&\vspace{1mm}\\
E_0:=\mathop{\rm Ker}\nolimits(w\colon E\rightarrow \mathop{\rm Aut}\nolimits(C))&
\end{array}$$
Now $z\in F_0$, $e\in F_1$, $ex\in F_2$ and in the diagram
$$\diagram{F_0&\subset&E_0\cr\bigcap&&\bigcap\cr F_1&\subset&E\cr}$$
$[F_1:F_0]=[E:E_0]=2$ and $[E:F_1]=[E_0:F_0]=m$.\hfill\break
We know there exist $y_1\in F_1\setminus F_0$ such that $y_1z=zy_1$.
Then we have $ey_1\in F_0$ and $ey_1z(ey_1)^{-1}=z^{-1}$, so
~\ref{defpseu} guarantees the existence of a pseudo-arf element
$\wp(f_1,f_1z)\in L^h(F_0)$.
Analogous, the existence $y_2\in F_2\setminus F_0$, satisfying
$y_2z=zy_2$, yields a pseudo-arf element $\wp(f_2,f_2z)\in L^h(F_0)$.
through the element $exy_2\in F_0$.\hfill\break
But these pseudo-arf elements must coincide by theorem~\ref{thmwinjend}.
$$\diagram{
&\wp(f_2,f_2z)&&(ex,exz)\cr
\wp(f_1,f_1z)&L^h(F_0)&{\hbox to 30pt{\rightarrowfill}}&L^h(F_2)\cr
&\mapdown{}&
\begin{picture}(20,20)
\put(-8,15){\vector(3,-2){32}}
\end{picture}
&\mapdown{j_{2*}}\cr
(e,ez)&L^h(F_1)&
{\buildrel j_{1*}\over {\hbox to 30pt{\rightarrowfill}}} &L^h(G)\cr
}$$
Therefore we may conclude that
$$j_{1*}((e,ez))=j_{2*}((ex,exz))\in L^h(G).$$
\end{itemize}
\end{nitel}
This completes the proof of theorem~\ref{thm2stuks}.
\end{proof}
\begin{nitel}{proof of theorem~\ref{thmin2end}}
Suppose we have an element of $\mathop{\rm Arf}\nolimits^h(G)$ which is killed by $\Upsilon$.
As before we may assume that it has the form $\sum\plane{a_i,a_iz}$.
We apply induction on the number of terms occuring in the expression
for our element in $\mathop{\rm Arf}\nolimits^h(G)$.
Recall that we are dealing with terms like
$((\Delta ,e),(\Delta T^{i},ez))$ and
$((\Delta T^{j},ex),(\Delta T^{i+j},exz))$.
If there are less than three terms theorem~\ref{thm2stuks} does the job.
Thus assume that the number of terms exceeds two.
If a term $((1,\cdot\cdot),\cdots)$ appears,
lemma~\ref{lemmaeindord} enables us
to cancel two terms.
Otherwise there are two cases:
\begin{itemize}
\item[$\cdot$]
$i\neq0$\hfill\break
We can cancel terms by the first case of theorem~\ref{thm2stuks}
without having any information on $(T^j,x)$.
\item[$\cdot$]
$i=0$\hfill\break
The following terms occur:
\begin{eqnarray*}
&&((\Delta ,e),(\Delta T^{i},ez))\\
&&((\Delta T^{j_1},ex_1),(\Delta T^{i+j_1},ex_1z))\\
&&((\Delta T^{j_2},ex_2),(\Delta T^{i+j_2},ex_2z))
\end{eqnarray*}
Now define $j=\gcd(j_1,j_2)$, say $j=a_1j_1+a_2j_2$.
We conjugate the second and third term by a suitable power of
$$(T^j,x_1^{a_1}x_2^{a_2})$$ to obtain
$$((\Delta ,e\tilde{x}),(\Delta ,e\tilde{x}z))$$
or
$$((\Delta T^{j},e\tilde{x}),(\Delta T^{i+j},e\tilde{x}z)).$$
Applying lemma~\ref{lemmaeindord} once more, we can cancel terms.
\end{itemize}
We see that two terms cancel in all cases.
\end{nitel}
|
{
"timestamp": "2005-03-24T14:18:00",
"yymm": "0503",
"arxiv_id": "math/0503538",
"language": "en",
"url": "https://arxiv.org/abs/math/0503538"
}
|
\section{Introduction.}\nin
Let $X$ be a (connected and reduced) complex space. We
recall that $X$ is said to be {\it strongly} $q$-{\it pseudoconvex} in the sense of
Andreotti-Grauert~\cite{AG} if there exists a compact subset $K$ and a smooth function
$\varphi:X\to{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma$, $\varphi\ge 0$, which is strongly $q$-plurisubharmonic on $X\smi
K$ and such that:
\begin{itemize}
\item[a)] $0=\min\limits_X\,\varphi<\min\limits_K\,\varphi$;
\item[b)] for every $c>\max\limits_K\,\varphi$ the subset
$$
B_c=\{x\in X:\varphi(x)<c\}
$$
is relatively compact in $X$.
\end{itemize}
If $K=\ES$, $X$ is said to be $q$-{\it complete}. We remark that, for a space, being $1$-complete is equivalent to being Stein.
Replacing the condition b) by
\begin{itemize}
\item[b')] for every $0<\varepsilon<\min\limits_K\,\varphi$
and $c>\max\limits_K\,\varphi$ the subset
$$
B_{\e,c}=\{x\in X:\e<\varphi(x)<c\}
$$
is relatively compact in $X$,
\end{itemize}
we obtain the notion of $q$-{\it corona} (see~\cite{AG},~\cite{AT}).
A $q$-corona is said to be {\it
complete} whenever $K=\ES$.
The extension problem for analytic objects defined on $q$-coronae was studied by many authors (see e.g. \cite{FG}, \cite{Se}, \cite{Si}, \cite{SiT}, \cite{T70}). In this paper we deal with the larger class of the
semi $q$-coronae which are defined as follows. Consider a strongly $q$-pseudoconvex space (or,
more generally, a $q$-corona) $X$, and a smooth function $\varphi:X\to{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma$ displaying the
$q$-pseudoconvexity of $X$. Let $B_{\varepsilon,c}\!\subset\!} \def\nsbs{\!\not\subset\! X$ and let $h:X\to{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma$ be a
pluriharmonic function (i.e.\ locally the real part of a holomorphic function)
such that $K\cap\{h=0\}=\ES$. A connected component of $B_{\varepsilon,c}\smi\{h=0\}$ is, by definition, a {\it
semi} $q$-{\it corona}.
Another type of semi $q$-corona is obtained by replacing the zero set of $h$ with the
intersection of $X$ with a Levi flat hypersurface. More precisely, consider a closed
strongly $q$-pseudoconvex subspace $X$ of an open subset of ${\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^n$ and the $q$-corona
$C=B_{\varepsilon,c}=B_c\smi{\overline B}_\varepsilon$. Let $H$ be a Levi flat
hypersurface of a neighbourhood $U$ of ${\overline B}_c$ such that $H\cap K=\ES$. The
connected components $C_m$ of $C\smi H$ are called semi $q$-coronae.
In both cases the semi $q$-coronae are differences $A_c\smi{\overline A}_\varepsilon$
where $A_c$, $A_\varepsilon$ are strongly $q$-pseudoconvex spaces. Indeed, the function
$\psi=-\log h^2$ (respectively $\psi=-\log \delta_H(z)$, where $\delta_H(z)$ is the
distance of $z$ from $H$) is plurisubharmonic in $W\smi\{h=0\}$ (respectively
$W\smi H$) where $W$ is a neighbourhood of $B_c\cap \{h=0\}$ (respectively $B_c\cap H$). Let $\chi:{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma\to{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma$ be an
increasing convex function such that $\chi\circ\varphi>\psi$ on a neighbourhood of
$B_c\smi W$. The function $\Phi=\sup\,(\chi\circ\varphi,\psi)+\varphi$ is an exhaustion function
for $B_c\smi \{h=0\}$ (for $B_c\smi H$) and it is strongly
$q$-plurisubharmonic in $B_c\smi(\{h=0\}\cup K)$ (in
$B_c\smi H\cup K)$.
The interest for
domains whose boundary contains a ``Levi flat part'' originated from an extension theorem for
CR-functions proved in \cite{LT} (see also \cite{La}, \cite{LaP}, \cite{St}).
Using cohomological techniques developped in \cite{AG}, \cite{AT}, \cite{BS}, \cite{C} we prove that, under
appropriate regularity conditions, holomorphic functions defined on a complete semi
$1$-corona \lq\lq fill in the holes\rq\rq\ (Corollaries~\ref{cD} and~\ref{oE}). Meanwhile we also
obtain more general extension theorems for sections of coherent sheaves
(Theorems~\ref{cC} and~\ref{oCw}). As an application, we finally obtain an
extension theorem for divisors (Theorems~\ref{divis} and~\ref{divis2}) and for analytic sets
of codimension one (Theorem~\ref{ansets}).
We remark that this approach fails in the case when the objects to be extended are not
sections of a sheaf defined on the whole $B_c$. In particular, this applies for analytic
sets of higher codimension. This is closely related with the general, definitely more
difficult, problem of extending analytic objects assigned on some semi $q$-corona when
the subsets $B_c$ are not relatively compact in $X$ i.e.\ when $X$ is a genuine $q$-corona. It is worth noticing that a similar extension theorem for complex submanifold of higher codimension has been recently obtained in~\cite{DS} by different methods based on Harvey-Lawson's theorem~\cite{HL}.\vspace{0,3cm}
We wish to thank Mauro Nacinovich and Viorel V\^aj\^aitu for their kind help and suggestions.
\section{Cohomology and extension of sections.}\nin
\subsection{Closed $q$-coronae}
Let $X$ be a strictly $q$-pseudoconvex space (respectively $X\subset{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^n$ be a strictly $q$-pseudoconvex open set) and $H=\left\{h=0\right\}$ (respectively $H$ Levi-flat), and $C=B_{\varepsilon,c}=B_c\smi{\overline B}_\varepsilon$ a $q$-corona.
We can suppose that $B_c \smi H$ has two connected components, $B_+$ and $B_-$, and define $C_+=B_+\cap C$, $C_-=B_-\cap C$.
If $\mathcal F\in{\rm Coh}(B_c)$, we define $p(\mathcal F)=\inf\limits_{x\in
B_c}\,{\rm depth}({\mathcal F}_x)$, the depth of $\mathcal F$ on $B_c$. If $\mathcal F=\mathcal O$, the structure sheaf of $X$, we define $p(B_c)=p(\mathcal O)$.
\begin{teorema}\label{Ac}
Let $\mathcal F\in {\rm Coh}(B_c)$. Then the image of the homomorphism
$$
H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{B}_+,\mathcal F)\oplus H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{C},\mathcal F)\longrightarrow H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+,\mathcal F)
$$
(all closures are taken in $B_c$), defined by $(\xi\oplus\eta)\mapsto\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+}-\eta_{|\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+}$ has finite codimension provided that $q-1\le
r\le p(\mathcal F)-q-2$ .
\end{teorema}
\par\noindent{\bf Proof.\ }
Consider the Mayer-Vietoris sequence applied to the closed sets $\overline} \def\rar{\rightarrow} \def\tms{\times{B}_+$ and $\overline} \def\rar{\rightarrow} \def\tms{\times{C}$
\begin{eqnarray}\label{suc1}
\cdots &\to& H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{B}_+\cup \overline} \def\rar{\rightarrow} \def\tms{\times{C},\mathcal F)\to H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{B}_+,\mathcal F)\oplus H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{C},\mathcal
F)\stackrel{\d}{\to}\\ &\stackrel{\d}{\to}& H^r(\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+,\mathcal F)\to H^{r+1}(\overline} \def\rar{\rightarrow} \def\tms{\times{B}_+\cup \overline} \def\rar{\rightarrow} \def\tms{\times{C},\mathcal
F)\to\cdots\nonumber
\end{eqnarray}
$\d(a\oplus b)=a_{|\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+}-b_{|\overline} \def\rar{\rightarrow} \def\tms{\times{C}_+}$.
$\overline} \def\rar{\rightarrow} \def\tms{\times B_+\cup \overline} \def\rar{\rightarrow} \def\tms{\times C=B_c\smi U$ where $U=B_-\cap B_\varepsilon$. $U$ is $q$-complete, so the groups of compact support cohomology $H^{r}_c(U,\mathcal F)$ are zero for $q\leq r\leq p(\mathcal{F})-q$.
From the exact sequence of compact support cohomology
\begin{eqnarray}
\cdots &\to& H^r_c(U,\mathcal F)\to H^r(B_c,\mathcal F)\to\\
&\to& H^r(B_c\smi U,\mathcal F)\to H^{r+1}_c(U,\mathcal F)\to\cdots\nonumber
\end{eqnarray}
it follows that
\begin{equation}\label{isomBc-U}
H^r(B_c,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(B_c\smi U,\mathcal F),
\end{equation}
for $q\leq r \leq p(\mathcal{F})-q-1$.
Since $B_c$ is $q$-pseudoconvex,
$$\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(B_c,\mathcal F)<\IN
$$
for $q\le r$ \cite[Th\'eor\`eme 11]{AG}, and so
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r( B_c\smi U,\mathcal F)<\IN
$$
for $q\le r\le p(\mathcal F)-q-1$.
From (\ref{suc1}) we see that $\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda H^r( B_c\smi U,\mathcal F)=\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda H^r(\overline} \def\rar{\rightarrow} \def\tms{\times B_+\cup\overline} \def\rar{\rightarrow} \def\tms{\times C ,\mathcal F)$ is greater than or equal to the codimension of the homomorphism $\delta$.
\ $\Box$\par\vskip.6truecm
\begin{corol}\label{cB}
Under the same assumption of Theorem~\ref{Ac}, if $K\cap H=\ES$,
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)<\IN
$$
for $q\le r\le p(\mathcal F)-q-2$.
\end{corol}
\par\noindent{\bf Proof.\ }
Since $K\cap H=\ES$, $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ is a $q$-pseudoconvex space, and by virtue of \cite[Th\'eor\`eme 11]{AG} we have
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)<\IN
$$
for $r\ge q$. On the other hand, $\overline} \def\rar{\rightarrow} \def\tms{\times C$ is
a $q$-corona, thus we obtain
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)<\IN
$$
for $q\le r\le p(\mathcal F)-q-1$ in view of \cite[Theorem 3]{AT}. By Theorem~\ref{Ac} we then get that for $q\le r\le p(\mathcal
F)-q-1$ the vector space $H^r(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)\oplus H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)$ has finite dimension and for $q-1\leq r\leq p(\mathcal F)-q-2$ its image in $H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)$ has finite codimension. Thus
$H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)$ has finite dimension for $q\le r\le p(\mathcal F)-q-2$.
\ $\Box$\par\vskip.6truecm
\begin{teorema}\label{cC}
If $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ is a $q$-complete space, then
$$
H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)
$$
for $q\le r\le p(\mathcal F)-q-2$ and the homomorphism
\begin{equation}\label{eqA}
H^{q-1}(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)\oplus H^{q-1}(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)\longrightarrow H^{q-1}(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)
\end{equation}
is surjective for $p(\mathcal F)\geq2q+1$.
If $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ is a $1$-complete space and $p(\mathcal F)\ge 3$, the homomorphism
$$
H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)\longrightarrow H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)
$$
is surjective.
\end{teorema}
\par\noindent{\bf Proof.\ }
Since by hypothesis $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ is a $q$-complete space, $H^r(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)=\{0\}$ for
$q\le r$ \cite[Th\'eor\`eme 5]{AG}. From (\ref{isomBc-U}) it follows that $H^r(\overline} \def\rar{\rightarrow} \def\tms{\times B_+\cup \overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)=\{0\}$ for
$q\le r\le p(\mathcal F)-q-1$. Thus, the Mayer-Vietoris sequence (\ref{suc1}) implies that $H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)$ for $q \le r \le p(\mathcal F)-q-2$ and that the homomorphism (\ref{eqA}) is surjective if $p(\mathcal F)\geq 2q+1$.
In particular, if $q=1$ and $p(\mathcal F)\ge 3$ the homomorphism$$
H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)\oplus H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)\longrightarrow H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)
$$
is surjective i.e.\ every section $\s\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C_+,\mathcal F)$ is a difference $\s_1-\s_2$ of two sections $\s_1\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)$, $\s_2\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)$. Since $B_\varepsilon$ is Stein, the cohomology group with compact supports $H^1_k(B_\varepsilon,\mathcal F)$
is zero, and so the Mayer-Vietoris compact support cohomology sequence implies that the restriction homomorphism
$$
H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_c,\mathcal F)\longrightarrow H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_c\smi B_\varepsilon,\mathcal
F)=H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)
$$
is surjective, hence $\s_2\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times C,\mathcal F)$ is restriction of $\widetilde{\s}_2\in H^0(B_c,\mathcal F)$. So $\s$ is restriction to $\overline} \def\rar{\rightarrow} \def\tms{\times C_+$ of $(\s_1-\widetilde{\s}_{2|\overline} \def\rar{\rightarrow} \def\tms{\times B_+})\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal F)$, and the restriction homomorphism is surjective.
\ $\Box$\par\vskip.6truecm
\begin{corol}\label{cD}
Let $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ be a $1$-complete space and $p(B_c)\ge 3$. Then every holomorphic function on $\overline} \def\rar{\rightarrow} \def\tms{\times C_+$ extends holomorphically on $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$.
\end{corol}
\subsection{Open $q$-coronae}
Most of the Theorems and Corollaries of the previous section still hold in the open case and their proofs are very similar. First we give the proof of the extension results using directly Theorem~\ref{cC}. We have to assume that $H$ is the zero set of a pluriharmonic function $h$ and we define $B_c$, $C$, $B_+$, $B_-$, $C_+$ and $C_-$ as we did before.
Let us suppose $B_+$ is $1$-complete and $p(\mathcal F)\geq 3$. Let $s\in H^0(C_+,\mathcal F)$. For all $\epsilon>0$, we consider the closed semi $1$-corona
$$
\overline} \def\rar{\rightarrow} \def\tms{\times C_\epsilon=\overline} \def\rar{\rightarrow} \def\tms{\times{B_{\varepsilon+\epsilon,c}\cap\{h>\epsilon\}}\subset C_+
$$
Let $\s_\epsilon=s_{|\overline} \def\rar{\rightarrow} \def\tms{\times C_\epsilon}$. By Theorem~\ref{cC} (applied to $
\overline} \def\rar{\rightarrow} \def\tms{\times C_\epsilon$, $H_\epsilon=\{h=\epsilon\}$), we obtain that $\s_\epsilon$ extends to a section $\widetilde\s_\epsilon\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon,\mathcal F)$, where $\overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon=\overline} \def\rar{\rightarrow} \def\tms{\times{B_+\cap\{h>\epsilon\}}$. Since $B_+=\cup_\epsilon \overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon$, if for all $\epsilon_2>\epsilon_1>0$,
\begin{eqnarray}\label{*}
\widetilde\s_{\epsilon_1|_{\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}}}=\widetilde\s_{\epsilon_2}
\end{eqnarray}
the sections $\widetilde\s_{\epsilon}$ can be glued toghether to a section $\s\in H^0(B_+,\mathcal F)$ extending $s$.
Let $\epsilon_1,\epsilon_2$, $\epsilon_2>\epsilon_1>0$, be fixed. We have to show that~(\ref{*}) holds. By definition,
$$
\left(\widetilde\s_{\epsilon_1|_{\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}}}-\widetilde\s_{\epsilon_2}\right)_{|\overline} \def\rar{\rightarrow} \def\tms{\times C_{\epsilon_2}}=s-s=0.
$$
Thus, the support of $\widetilde\s_{\epsilon_1|_{\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}}}-\widetilde\s_{\epsilon_2}$, $S$, is an analytic set contained in $\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}\smi C_{\epsilon_2}$. Let us consider the family $(\phi_\lambda=\lambda(\varphi-\epsilon_2)+(1-\lambda)(h-\epsilon_2))_{\lambda\in[0,1]}$ of strictly plurisubharmonic functions. Let $\overline} \def\rar{\rightarrow} \def\tms{\times \lambda$ be the smallest value of $\lambda$ for which $\{\phi_\lambda=0\}\cap S\neq\ES$. Then $\{\phi_{\overline} \def\rar{\rightarrow} \def\tms{\times \lambda}<0\}\cap B_+\subset B_+$ is a Stein domain in which the analytic set $S$ intersects the boundary; so the maximum principle for plurisubharmonic functions and the strict plurisubharmonicity of $\phi_{\overline} \def\rar{\rightarrow} \def\tms{\times \lambda}$ toghether imply that $\{\phi_{\overline} \def\rar{\rightarrow} \def\tms{\times \lambda}=0\}\cap S$ is a set of isolated points in $S$. By repeating the argument, we show that $S$ has no components of positive dimension. Hence $\widetilde\s_{\epsilon_1|_{\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}}}-\widetilde\s_{\epsilon_2}$ is zero outside a set of isolated points. Since $p(\mathcal F)\geq3$, the only section of $\mathcal F$ with compact support is the zero-section \cite[Th\'eor\`eme 3.6 (a), p.\ 46]{BS}, and so $\widetilde\s_{\epsilon_1|_{\overline} \def\rar{\rightarrow} \def\tms{\times B_{\epsilon_2}}}-\widetilde\s_{\epsilon_2}$ is zero.
Hence, there exists a section $\s\in H^0(B_+,\mathcal F)$ such that $\s_{|C_+}=s$. Thus we have proved the following
\begin{teorema}\label{oCw}
If a $B_+$ is $1$-complete space, $\mathcal F$ a coherent sheaf on $B_+$ with $p(\mathcal F)\ge 3$, the homomorphism
$$
H^0(B_+,\mathcal F)\longrightarrow H^0(C_+,\mathcal F)
$$
is surjective.
\end{teorema}
In particular,
\begin{corol}\label{oE}
If $B_+$ is a $1$-complete space and $p(B_c)\ge 3$, every holomorphic function on $C_+$
can be holomorphically extended on $B_+$.
\end{corol}
\begin{teorema}\label{oA}
Let $\emph{Sing}(B_c)=\ES$. Let $\mathcal F\in {\rm Coh}(B_c)$. Then the image of the homomorphism
$$
H^r(B_+,\mathcal F)\oplus H^r(C,\mathcal F)\longrightarrow H^r(C_+,\mathcal F)
$$
defined by $(\xi,\eta)\mapsto\xi_{|C_+}-\eta_{|C_+}$ has finite codimension for $q-1\le
r\le p(\mathcal F)-q-2$. For $q=1$ the thesis holds true also dropping the assumption $\emph{Sing}(B_c)=\ES$.
\end{teorema}
\par\noindent{\bf Proof.\ }
Consider the Mayer-Vietoris sequence applied to the open sets $B_+$ and $C$
\begin{eqnarray}\label{1open}
\cdots &\to& H^r(B_+\cup C,\mathcal F)\to H^r(B_+,\mathcal F)\oplus H^r(C,\mathcal
F)\stackrel{\d}{\to}\\ &\stackrel{\d}{\to}& H^r(C_+,\mathcal F)\to H^{r+1}(B_+\cup C,\mathcal
F)\to\cdots,\nonumber
\end{eqnarray}
$\d(a\oplus b)=a_{|C_+}-b_{|C_+}$. $B_+\cup C=B_c\smi K_0$ where $K_0=\overline} \def\rar{\rightarrow} \def\tms{\times{B}_-\cap\overline} \def\rar{\rightarrow} \def\tms{\times{B}_\varepsilon$. $K_0$ has a $q$-complete neighbourhoods system and so the local cohomology groups $H^r_{K_0}(B_c,\mathcal F)$ are zero for $q\leq r\le p(\mathcal F)-q$ \cite{C} (in the general case for $q=1$, see \cite[Lemme 2.3, p.\ 29]{BS}).
Then, from the local cohomology exact sequence
\begin{eqnarray}
\cdots &\to& H^r_{K_0}(B_c,\mathcal F)\to H^r(B_c,\mathcal F)\to\\
&\to& H^r(B_c\smi K_0,\mathcal F)\to H^{r+1}_{K_0}(B_c,\mathcal F)\to\cdots\nonumber
\end{eqnarray}
follows that
\begin{equation}\label{eq3}
H^r(B_c,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(B_c\smi K_0,\mathcal F),
\end{equation}
for $q\le r\le p(\mathcal F)-q-1$.
Since $B_c$ is $q$-pseudoconvex,
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(C,\mathcal F)<\IN
$$
for $q\le r$ \cite[Th\'eor\`eme 11]{AG}, and so
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(B_c\smi K_0,\mathcal F)<\IN
$$
for $q\le r\le p(\mathcal F)-q-1$.
From (\ref{1open}) we see that $\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda H^r(B_c\smi K_0,\mathcal F)=\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda H^r(B_+\cup C,\mathcal F)$ is greater than or equal to the codimension of the homomorphism $\d$.
\ $\Box$\par\vskip.6truecm
\begin{corol}\label{oB}
Under the same assumption of Theorem~\ref{oA}, if $K\cap H=\ES$,
$$
\dim_{\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda\,H^r(C_+,\mathcal F)<\IN
$$
for $q\le r\le p(\mathcal F)-q-2$.
\end{corol}
\par\noindent{\bf Proof.\ }
The proof is similar to that of Corollary~\ref{cB}.
\ $\Box$\par\vskip.6truecm
\begin{teorema}\label{oC}
Suppose that $\emph{Sing}(B_c)=\ES$ and $B_+$ is a $q$-complete space, then
$$
H^r(C,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(C_+,\mathcal F)
$$
for $q\le r\le p(\mathcal F)-q-2$ and the homomorphism
\begin{eqnarray}\label{Aopen}
H^{q-1}(B_+,\mathcal F)\oplus H^{q-1}(C,\mathcal F)\longrightarrow H^{q-1}(C_+,\mathcal F)
\end{eqnarray}
is surjective if $p(\mathcal F)\geq 2q+1$. If $q=1$, both results hold true for an arbitrary complex space $B_c$.
\end{teorema}
\par\noindent{\bf Proof.\ }
The proof is similar to that of Theorem~\ref{cC}.
\ $\Box$\par\vskip.6truecm
\subsection{Corollaries of the extension theorems.}
From now on, unless otherwise stated, by $B$, $B_+$, $B_\varepsilon$, $C$ and $C_+$ we denote both the open sets and their closures, and we suppose that $H=\{h=0\}$, $h$ pluriharmonic.
\subsubsection{}Let $f\in H^0(C_+,\mathcal O^*)$. In the hypothesis of Corollaries~\ref{cD} and~\ref{oE}, both $f$ and $1/f$ extend holomorphically on $B_+$ Hence:
\begin{corol}\label{O*sur}
If $B_+$ is a $1$-complete space and $p(B_c)\ge 3$, the restriction homomorphism
$$
H^0(B_+,\mathcal O^*)\longrightarrow H^0(C_+,\mathcal O^*)
$$
is surjective.
\end{corol}
\subsubsection{}In Theorems~\ref{cC} and~\ref{oCw} we have estabilished the isomorphism
$$
H^r(C,\mathcal F)\stackrel{\sim}{\rightarrow} H^r(C_+,\mathcal F).
$$
In some special cases this leads to vanishing-cohomology theorems for $C_+$. An example is provided by a $q$-corona $C$ which is contained in an affine variety. In such a situation, we have that $H^r(C,\mathcal F)=\left\{0\right\}$, for $q\leq r\leq p(\mathcal F)-q-2$ \cite{AT}, and consequently $H^r(C_+,\mathcal F)=\left\{0\right\}$ in the same range of $r$.
\subsubsection{}Let $X$ be a Stein space. Let $H=\left\{h=0\right\}\subset X$ be the zero set of a pluriharmonic function, and let $S$ be a real hypersurface of $X$ with boundary, such that $S\cap H=b S=b A$, where $A$ is an open set in $H$. Let $D\subset X$ be the relatively compact domain bounded by $S\cup A$. In \cite{LT} it is proved that, for $X={\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^n$, $CR$-functions on $S$ extend holomorphically to $D$. As a corollary of the previous theorems, we can obtain a similar result for section of a coherent sheaf on an arbitrary Stein space $X$.
Let us consider the connected component $Y$ of $X\smi H$ containing $D$, the closure $\overline} \def\rar{\rightarrow} \def\tms{\times D$ of $D$ in $Y$, and let be $F=Y\smi D$ and $S_Y=S\cap Y$. For every coherent sheaf $\mathcal F$ on $X$, with $p(\mathcal F)\geq3$ we have the Mayer-Vietoris exact sequence
$$
\cdots\ \to\ H^0(\overline} \def\rar{\rightarrow} \def\tms{\times D,\mathcal F)\oplus H^0(F,\mathcal F)\ \to\ H^0(S_Y,\mathcal F)\ \to\ H^1(Y, \mathcal F) \ \to\ \cdots
$$
Since $Y$ is Stein, $H^1(Y, \mathcal F)$ is zero, and every section $\s$ on $S_Y$ is a difference $s_1-s_2$, where $s_1\in H^0(\overline} \def\rar{\rightarrow} \def\tms{\times D,\mathcal F)$ and $s_2\in H^0(F,\mathcal F)$. By choosing an $\varepsilon$ big enough so that $S$ is contained in the ball $B_\e(x_0)$ of radius $\e$ of $X$ centered in $x_0$, we can apply Theorem~\ref{oCw} to the semi $1$-corona $C_+=Y\smi(B_\e\cap Y)$, to extend $s_{2|_{C_+}}$ to a section $\tilde s_2$ defined on $Y$. In order to conclude that $s_1-\tilde s_{2|_{\overline} \def\rar{\rightarrow} \def\tms{\times D}}$ extends the section $\s$, we have to prove that $s_{2|_F}-\tilde s_{2|_F}=0$.
As before, we consider the set $\Sigma=\left\{s_{2|_F}-\tilde s_{2|_F}\neq0\right\}\subset B_\e\cap Y$ and conclude that $\Sigma$ is a set of isolated points. Since $p(\mathcal F)\geq3$, $\mathcal F$ has no non zero section with compact support \cite[Th\'eor\`eme 3.6 (a), p.\ 46]{BS}. Thus $\Sigma=\ES$ and we have obtained the following:
\begin{corol}\label{Lupac1}
Let $X$ be a Stein space. Let $H=\left\{h=0\right\}\subset X$ be the zero set of a pluriharmonic function, and $S$ be a real hypersurface of $X$ with boundary, such that $S\cap H=b S=b A$, where $A$ is an open set in $H$. Let $D\subset X$ be the relatively compact domain bounded by $S\cup A$ and $\mathcal F$ be a coherent sheaf with $p(\mathcal F)\geq3$. All sections of $\mathcal F$ on $S$ extend (uniquely) to $D$.
\end{corol}
We can go further:
\begin{corol}\label{Lupac} Let $X$ be a Stein manifold, $\mathcal F$ a coherent sheaf on $X$ such that $p(\mathcal F)\geq3$ and $D$ be a bounded domain and $K$ a compact subset of $b D$ such that $b D\smi K$ is smooth. Assume that $K$ is $\mathcal O(D)$-convex, i.e.
$$ K=\left\{z\in\overline} \def\rar{\rightarrow} \def\tms{\times D\ :\ |f(z)|\leq\max_K |f|\right\}. $$
Then every section of $\mathcal F$ on $b D\smi K$ extends to $D$.
\end{corol}
\par\noindent{\bf Proof.\ } We recall that since $U$ is an open subset of a Stein manifold there exists
an envelope of holomorphy $\widetilde U$ of $U$ (cfr. \cite{DG}) $\widetilde U$ is a Stein
domain $\pi_U:\widetilde U\to X$ over $X$ and there exists and open embedding $j:U\to\widetilde U$ such that $\pi_U\circ j=id_U$ and $J^*:\mathcal O(\widetilde U)\to\mathcal O(U)$ is an isomorphism. In particular $\pi_U^*\mathcal F$ is a coherent sheaf with
the same depth as $\mathcal F$, which extends ${\mathcal F}_{|U}$.
Let us fix an arbitrary point $x\in D$. We need to show that any given section $\s\in H^0(b D\smi K,\mathcal F)$ extends to a neighbourhood of $x$. Since $x\not\in K=\widehat{K}$, there exists an holomorphic function $f$, defined on a neighbourhood $U$ of $\overline} \def\rar{\rightarrow} \def\tms{\times D$, such that $|f(x)|>\max_K |f(z)|$.
Then $\s$ extends to a section $\widetilde\s\in H^0(\pi^{-1}(D\smi K),\mathcal F)$. Let $\widetilde f$ be the
holomorphic extension of $f$ to $\widetilde U$. The hypersurface
$$
H=\left\{z\in\widetilde U:\vert \widetilde
f(z)\vert=\max\limits_K\vert\widetilde f\vert\right\}
$$
is the zero-set of a pluriharmonic function and, by construction, $$x\in \widetilde D_+=\left\{z\in\widetilde U:\vert \widetilde
f(z)\vert>\max\limits_K\vert\widetilde f\vert\right\}.$$
Now we are in the situation of Corollary~\ref{Lupac1} so $\widetilde\s$ extends to a section on $\widetilde D_+$. Since
$x\in\widetilde D_+$, this ends the proof.
\ $\Box$\par\vskip.6truecm
\section{Extension of divisors and analytic sets of codimension one.}
First of all, we give an example in dimension $n=2$ of a regular complex curve of $C_+$ which does not extend on $B_+$. Hence, not every divisor on $C_+$ extends to a divisor on $B_+$.\vspace{0.25cm}\\
\nin\textbf{Example}. Using the same notation as before, let $B_c$ be the ball $\left\{|z_1|^2+|z_2|^2<c\right\}$, $c>2$, in ${\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^2$, and $H$ be the hyperplane $\left\{x_2=0\right\}$ ($z_j=x_j+iy_j$). Let $2<\e<c$, $C=B_c\smi\overline} \def\rar{\rightarrow} \def\tms{\times B_\e$, $B_+=B_c\cap\left\{x_2>0\right\}$, $C_+=C\cap\left\{x_2>0\right\}$.
Consider the connected irreducible analytic set of codimension one
$$A=\{(z_1,z_2)\in B_+\ :\ z_1z_2=1\}$$
and its restriction $A_C$ to $C_+$. If $A_C$ has two connected components, $A_1$ and $A_2$, if we try to extend $A_1$ (analytic set of codimension one on $C_+$) to $B_+$, its restriction to $C_+$ will contain also $A_2$. So $A_1$ is an analytic set of codimension one on $C_+$ that does not extend on $B_+$.
So, let us prove that $A_C$ has indeed two connected components. A point of $A$ (of $A_C$) can be written as $z_1=\rho e^{i\theta}$, $z_2=\frac{1}{\rho} e^{-i\theta}$, with $\rho\in{\mathbb {R}}} \def\a {\alpha} \def\b {\beta}\def\g{\gamma^+$ and $\theta\in\left(-\frac\pi2,\frac\pi2\right)$. Hence, points in $A_C$ satisfy
$$
2<\varepsilon<\rho^2+\frac1{\rho^2}<c\ \Rightarrow\ 2<\sqrt{\varepsilon+2}<\rho+\frac1\rho<\sqrt{c+2} .
$$
Since $f(\rho)=\rho+1/\rho$ is monotone decreasing up to $\rho=1$ (where $f(1)=2$), and then monotone increasing, there exist $a$ and $b$ such that the inequalities are satisfied when $a<\rho<b<1$, or when $1<1/b<\rho<1/a$. $A_C$ is thus the union of the two disjoint open sets
$$
\xymatrix{A_1=\left\{ \left(\rho e^{i\theta},\frac1\rho e^{-i\theta}\right)\in {\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^2\ \Big|\ a<\rho<b,\ -\frac\pi2<\theta<\frac\pi2\right\};\\
A_2=\left\{ \left(\rho e^{i\theta},\frac1\rho e^{-i\theta}\right)\in {\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^2\ \Big|\ a<\frac1\rho<b,\ -\frac\pi2<\theta<\frac\pi2\right\}.}$$\vspace{0.25cm}
The aim of this section is to prove an extension theorem for divisors, i.e.\ to prove that, under certain hypothesis, the homomorphism
\begin{eqnarray}\label{DivSurg}
H^0(B_+,\mathcal D)\to H^0(C_+,\mathcal D)
\end{eqnarray}
is surjective.
In order to get this result, we observe that from the exact sequence
\begin{eqnarray}\label{esattaD}
0\to\mathcal O^*\to\mathcal M^*\to\mathcal D\to 0
\end{eqnarray}
we get the commutative diagram (horizontal lines are exact)
$
\xymatrix{H^0(B_+,\mathcal M^*)\ar[r]\ar[d]_{\alpha} & H^0(B_+,\mathcal D)\ar[r]\ar[d]_{\beta} & H^1(B_+,\mathcal O^*)\ar[r]\ar[d]_{\gamma} & H^1(B_+,\mathcal M^*)\ar[d]_{\delta} \\ H^0(C_+,\mathcal M^*)\ar[r] & H^0(C_+,\mathcal D)\ar[r] & H^1(C_+,\mathcal O^*)\ar[r] & H^1(C_+,\mathcal M^*)}
$$
Thus, in view of the \lq\lq five lemma\rq\rq, in order to conclude that $\beta$ is surjective it is sufficient to show that $\alpha$ and $\gamma$ are surjective, and $\delta$ is injective.
\begin{Lemma}\label{alphaS}
If $\emph{Sing}(B_+)=\ES$, $B_c$ is $1$-complete and $p(B_c)\geq 3$, then $\alpha$ is surjective.
\end{Lemma}
\par\noindent{\bf Proof.\ }
Let $f$ be a meromorphic invertible function on $C_+$. Since $C_+$ is an open set of the Stein manifold $B_+$, $f=f_1 f_2^{-1}$, $f_1,f_2\in H^0(C_+,\mathcal O)$. By Corollary~\ref{cD} (\ref{oE}), $f_1$ and $f_2$ extend to holomorphic functions on $B_+$ and consequently $f$ extends on $B_+$ as well.
\ $\Box$\par\vskip.6truecm
\begin{Lemma}\label{gammaS}
Assume that the restriction $H^2(B_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)\to H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$ is surjective. If $B_c$ is $1$-complete and $p(B_c)\geq4$, then $\gamma$ is surjective.
\end{Lemma}
We remark that if $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)=\{0\}$ the first condition is satisfied.\vspace{0.5cm}
\par\noindent{\bf Proof.\ }
From the exact sequence
\begin{eqnarray}
0\to{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta\to\mathcal O\to\mathcal O^*\to 0
\end{eqnarray}
we get the commutative diagram (horizontal lines are exact)
$$
\xymatrix{H^1(B_+,\mathcal O)\ar[r]\ar[d]_{f_2} & H^1(B_+,\mathcal O^*)\ar[r]\ar[d]_{\gamma} & H^2(B_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)\ar[r]\ar[d]_{f_4} & H^2(B_+,\mathcal O)\ar[d]_{f_5} \\ H^1(C_+,\mathcal O)\ar[r] & H^1(C_+,\mathcal O^*)\ar[r] & H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)\ar[r] & H^2(C_+,\mathcal O)}
$$
where $H^1(B_+,\mathcal O)=H^2(B_+,\mathcal O)=\left\{0\right\}$ because $B_+$ is Stein, and $f_4$ is surjective by hypothesis. Thus in order to prove that $\gamma$ is surjective by the \lq\lq five lemma\rq\rq\ it is sufficient to show that $f_2$ is surjective, i.e.\ that $H^1(C_+,\mathcal O)=\left\{0\right\}$.
Since $p(B_c)\geq4$, by Theorem~\ref{cC} (\ref{oC}) it follows that
\begin{eqnarray}\label{C=C+}
H^1(C,\mathcal O)\stackrel{\sim}{\longrightarrow} H^1(C_+,\mathcal O).
\end{eqnarray}
Consider the local, respectively compact support, cohomology exact sequence
$$
\xymatrix{H^1_{\overline} \def\rar{\rightarrow} \def\tms{\times B_\varepsilon}(B_c,\mathcal O)\ar[r] & H^1(B_c,\mathcal O)\ar[r] & H^1(C,\mathcal O)\ar[r] & H^2_{\overline} \def\rar{\rightarrow} \def\tms{\times B_\varepsilon}(B_c,\mathcal O) \\ H^1_k(B_\varepsilon,\mathcal O)\ar[r] & H^1(B_c,\mathcal O)\ar[r] & H^1(C,\mathcal O)\ar[r] & H^2_k(B_\varepsilon,\mathcal O)}
$$
Since $B_c$ is Stein, $H^1(B_c,\mathcal O)=\left\{0\right\}$ and $H^r_k(B_\e,\mathcal O)=H^r_{\overline} \def\rar{\rightarrow} \def\tms{\times B_\e}(B_c,\mathcal O)=\left\{0\right\}$ for $1\leq r\leq p(B_\e)-1$~\cite{C}. In particular, since $p(B_\e)\geq p(B_c)\geq4$, it follows that
\begin{eqnarray}\label{Bc=C}
\{0\}=H^1(B_c,\mathcal O)\stackrel{\sim}{\longrightarrow} H^1(C,\mathcal O).
\end{eqnarray}
(\ref{C=C+}) and (\ref{Bc=C}) give
$$
\{0\}=H^1(B_c,\mathcal O)\stackrel{\sim}{\longrightarrow}H^1(C,\mathcal O)\stackrel{\sim}{\longrightarrow} H^1(C_+,\mathcal O).
$$and this proves the lemma.
\ $\Box$\par\vskip.6truecm
In the case $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)=\{0\}$ we remark that from the proof of Lemma~\ref{gammaS} it follows that the sequence
$$\{0\}\longrightarrow H^1(C_+,\mathcal{O}^*)\longrightarrow\{0\}$$
is exact, that is $H^1(C_+,\mathcal{O}^*)=\{0\}$. Hence, the commutative diagram relative to (\ref{esattaD}) becomes (horizontal lines are exact)
\begin{eqnarray}\label{last}
\xymatrix{H^0(B_+,\mathcal M^*)\ar[r]\ar[d]_{\alpha} & H^0(B_+,\mathcal D)\ar[r]\ar[d]_{\beta} & H^1(B_+,\mathcal O^*)\ar[d]_{\gamma} \\ H^0(C_+,\mathcal M^*)\ar[r] & H^0(C_+,\mathcal D)\ar[r] & \{0\}}
\end{eqnarray}
and it is then easy to see that a divisor on $C_+$ can be extended to a divisor on $B_+$.
Thus we have proved the following:
\begin{teorema}\label{divis}
Let $B_c$ be $1$-complete, $p(B_c)\geq4$, and $C_+$ satisfy the topological condition $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)=\{0\}$. Then, if $\emph{Sing}(B_+)=\ES$, all divisors on $C_+$ extend (uniquely) to divisors on $B_+$.
\end{teorema}
\begin{corol}\label{corxi}
Let $B_c$ be $1$-complete, $p(B_c)\geq4$, $\emph{Sing}(B_+)=\ES$, and $\xi$ be a divisor on $C_+$ with zero Chern class in $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$. Then $\xi$ extends (uniquely) to a divisor on $B_+$.
\end{corol}
\par\noindent{\bf Proof.\ }
Use diagram~(\ref{last}).
\ $\Box$\par\vskip.6truecm
\begin{teorema}\label{ansets}
Assume that $H^2(C_+,{\mathbb Q}} \def\o{\omega}\def\p{\partial}\def\r{\varrho)=\{0\}$. If $\emph{Sing}(B_+)=\ES$, $B_c$ is $1$-complete and $p(B_c)\geq4$, then all analytic sets of codimension $1$ on $C_+$ extend to analytic sets on $B_+$.
\end{teorema}
\par\noindent{\bf Proof.\ }
Let $A$ be an analytic set of codimension $1$ on $C_+$. Since $B_+$ is a Stein manifold, $C_+$ is locally factorial, and so there exists a divisor $\xi$ on $C_+$ with support $A$. Since
$H^2(C_+,{\mathbb Q}} \def\o{\omega}\def\p{\partial}\def\r{\varrho)=\{0\}$, there exists $n\in{\mathbb N}} \def\d {\delta} \def\e{\varepsilon$ such that $n c_2(\xi)=0\in H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$. Hence $n\xi$ has zero Chern class in $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$, and so, by Corollary~\ref{corxi} $n\xi$ can be extended to a divisor $\widetilde{n\xi}$ on $B_+$. The support of $\widetilde{n\xi}$ is an analytic set $\widetilde A$ which extends to $B_+$ the support $A$ of $n\xi$.
\ $\Box$\par\vskip.6truecm
In Theorem~\ref{divis} the condition $H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)=\{0\}$ can be relaxed and replaced by the weaker one: the restriction map $H^2(B_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)\to H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$ is surjective. We need the following
\begin{Lemma}\label{deltaI}
$\delta$ is injective.
\end{Lemma}
\par\noindent{\bf Proof.\ }
First we prove lemma for $C_+$ closed. Let $\xi\in H^1(\overline} \def\rar{\rightarrow} \def\tms{\times B_+,\mathcal M^*)$ be such that $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times C_+}=0$. Consider the set
$$A=\{\eta\in[0,\varepsilon]\ :\ \xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi\overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta}}=0\}.$$
If we prove that $0\in A$, we are done, because $0=\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_0}=\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_+}=\xi$. Obviously $\eta_0\in A$ implies $\forall\eta>\eta_0$, $\eta\in A$.
$A\neq\ES$. Since $C_+=B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_\e$ and $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times C_+}=0$, $\varepsilon\in A$.
$A$ is closed. If $\eta_n\in A$, for all $n$, and $\eta_n\searrow\eta_\infty$, $\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta_\infty}=\cup_n (\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta_n})$, hence $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta_n}}=0$ for all $n$ implies $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta_\infty}}=0$, i.e.\ $\eta_\infty\in A$.
$A$ is open. Suppose $0<\eta_0\in A$. We denote $C_{\eta_0}=\overline} \def\rar{\rightarrow} \def\tms{\times B_+\smi \overline} \def\rar{\rightarrow} \def\tms{\times B_{\eta_0}$. Let $\mathcal A$ be the family of open covering $\{U_i\}_{i\in I}$ of $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ such that:
\begin{itemize}
\item[$\alpha$)] $U_i$ is isomorphically equivalent to an holomorphy domain in ${\mathbb C}} \def\en{\varepsilon_n} \def\l{\lambda^n$;
\item[$\beta$)] If $U_i\cap b B_{\eta_0}\neq\ES$, the restriction homomorphism
$$
H^0(U_i,\mathcal O)\rightarrow H^0(U_i\cap C_{\eta_0},\mathcal O)$$
is bijective;
\item[$\gamma$)] $U_i\cap U_j$ is simply connected.
\end{itemize}
$\mathcal A$ is not empty and it is cofinal in the set of open coverings of $\overline} \def\rar{\rightarrow} \def\tms{\times B_+$ \cite[Lemma 2, p.\ 222]{AG}. Let $\mathcal U=\{U_i\}_{i\in I}\in \mathcal A$, and $\{f_{ij}\}\in Z^1(\mathcal U,\mathcal M^*)$ be a representative of $\xi$. Let $W_i=U_i\cap C_{\eta_0}$. Since $\eta_0\in A$, if $W_i\cap W_j\neq\ES$, $f_{ij|W_i\cap W_j}=f_i f_j^{-1}$ ($f_\nu\in H^0(W_\nu,\mathcal M^*)$). By $\alpha$), $f_\nu=p_\nu q_\nu^{-1}$, $p_\nu, q_\nu\in H^0(W_\nu,\mathcal O)$. By $\beta$), both $p_\nu$ and $q_\nu$ can be holomorphically extended on $U_\nu$, with $\widetilde p_\nu$ and $\widetilde q_\nu$. Hence we have $f_{ij}=\widetilde p_i \widetilde q_i^{-1}(\widetilde p_j \widetilde q_j^{-1})^{-1}$ on $U_i\cap U_j$ (which is simply connected, so that there is no polidromy). So $\xi=0$ in an open neighborhood $U$ of $C_{\eta_0}$ and, by compactness, there exists $\epsilon'>0$ such that $C_{\eta_0-\epsilon'}\subset U$. So $\eta_0-\epsilon'\in A$ and consequently $A$ is open.
Thus $A=[0,\varepsilon]$, and the lemma is proved if $C_+$ is closed.
If $C_+$ is open, we consider $C_+$ as a union of the closed semi $1$-coronae
$$
\overline} \def\rar{\rightarrow} \def\tms{\times C_\epsilon=\overline} \def\rar{\rightarrow} \def\tms{\times{B_{\varepsilon+\epsilon',c}\cap\{h>\epsilon'\}}\subset C_+.
$$
Let $\xi\in H^1(B_+,\mathcal M^*)$ be such that $\xi_{|C_+}=0$. Then $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times C_\epsilon'}=0$, for all $\epsilon'>0$. Consequently from what we have already proved $\xi_{|\overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon'}=0$, where $\overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon=\overline} \def\rar{\rightarrow} \def\tms{\times{B_+\cap\{h>\epsilon'\}}$. Since $\cup_\epsilon' \overline} \def\rar{\rightarrow} \def\tms{\times B_\epsilon'=B_+$, $\xi=0$ and the lemma is proved.
\ $\Box$\par\vskip.6truecm
Lemma~\ref{alphaS}, Lemma~\ref{gammaS} and Lemma~\ref{deltaI} lead to the following generalization of Theorem~\ref{divis}:
\begin{teorema}\label{divis2}
Assume that the restriction $H^2(B_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)\to H^2(C_+,{\mathbb Z}} \def\s{\sigma}\def\t{\theta}\def\z{\zeta)$ is surjective. If $\emph{Sing}(B_+)=\ES$, $B_c$ is $1$-complete and $p(B_c)\geq4$, then all divisors on $C_+$ extend to divisors on $B_+$.
\end{teorema}
|
{
"timestamp": "2005-03-23T14:12:59",
"yymm": "0503",
"arxiv_id": "math/0503490",
"language": "en",
"url": "https://arxiv.org/abs/math/0503490"
}
|
\section*{Introduction}
Throughout the paper $k$ is a fixed algebraically closed field.
All considered categories are additive $k$-categories and all
functors are $k$-functors.
One of the aims of the representation theory of finite-dimensional
algebras is a description of indecomposable modules and
homomorphism spaces between them. A guiding example is that of
special biserial algebras, for which a full description of the
indecomposable modules and the Auslander--Reiten sequences was
given by Wald and Waschb\"usch~\cite{WalWas} (see
also~\cite{BuRi}). Homomorphism spaces between indecomposable
modules were also investigated (see for example~\cite{CB1}).
Another class of algebras whose representation theory is described
is formed by clannish algebras (or more generally, clan problems)
introduced by Crawley-Boevey~\cite{CB4} (see also~\cites{Bond,
De}). Homomorphism spaces and Auslander--Reiten sequences for this
class of problems were studied by Gei\ss~\cite{Ge} (see
also~\cite{GePe} for a description of the Auslander-Reiten
components).
According to Drozd's Tame and Wild Theorem~\cite{Dr} (see
also~\cite{CB2}) one may hope to obtain classifications like these
above only for so called tame algebras. First examples of tame
algebras are provided by the representation-finite algebras, for
which there are only finitely many isomorphism classes of
indecomposable modules. The representation theory of the
representation-finite algebras has been intensively studied (see for
example~\cites{BaGaRoSa, Bong, BongGa, BrGa}) and seems to be
well-understood. One knows that an algebra is representation-finite
if and only if its infinite radical vanishes.
The first level in the hierarchy of representation-infinite algebras
is occupied by the domestic algebras, for which in each dimension
all but finitely many indecomposable modules can be parameterized by
finitely many lines (see also~\cite{CB3} for a different
characterization of the domestic algebras). Schr\"oer's
work~\cite{Sc} on the infinite radical of special biserial algebras
gives hope to characterize the domestic algebras in terms of the
infinite radial. In~\cite{BobDrSk} (continued by~\cites{Bob1,
BobSk2}), we initiated the study of a new class of domestic
algebras, which may be seen as a test class for this
characterization. The results obtained so far concern the
Auslander--Reiten theory. In order to deal with the infinite radical
one needs to have a more precise knowledge about indecomposable
modules and homomorphisms spaces between them. In this paper we make
a first step in this direction, namely we give a description of the
indecomposable modules. This description resembles the description
obtained for clans, thus one may hope that the corresponding results
about homomorphisms can be also transferred.
The paper is organized as follows. In Section~\ref{mainres} we
present the main result of the paper, in Section~\ref{sectvect} we
recall necessary information about vector space categories, and in
final Section~\ref{sectproof} we prove the main theorem. The paper
was written during the author held a one year post-doc position at
the University of Bern. Author gratefully acknowledges the support
from the Schweizerischer Nationalfonds and the Polish Scientific
Grant KBN No.~1 P03A 018 27.
\section{Strings, the corresponding modules and the main result}
\label{mainres}
In this section we first introduce notation, which is necessary to
formulate the main result of the paper given at the end of the
section.
\subsection{}
In the paper, by $\mathbb{Z}$ (respectively, $\mathbb{N}_0$, $\mathbb{N}$) we denote
the set of (nonnegative, positive) integers. If $m$ and $n$ are
integers, then by $[m, n]$ we denote the set of all integers $l$
such that $m \leq l \leq n$. For a sequence $f : [1, n] \to \mathbb{N}$,
$n \in \mathbb{N}_0$, of positive integers we denote $n$ by $|f|$. We
identify finite subsets of $\mathbb{N}$ with the corresponding
increasing sequences of positive integers. In particular, if $F$
is a finite subset of $\mathbb{N}$ and $i \in [1, |F|]$, then $F_i$
denotes the $i$-th element of $F$ with respect to the usual order
of integers.
\subsection{}
By a quiver $Q$ we mean an oriented graph, i.e., a set of vertices
$Q_0$, a set of arrows $Q_1$ and two maps $s_Q, t_Q : Q_1 \to Q_0$,
which assign to an arrow $\alpha$ in $Q$ its starting and
terminating vertex, respectively. If $\alpha \in Q_1$, $s_Q (\alpha)
= x$ and $t_Q (\alpha) = y$, then we write $\alpha : x \to y$. By a
path in $Q$ we mean a sequence $\rho = \alpha_1 \cdots \alpha_n$ of
arrows in $Q$ such that $t_Q (\alpha_{i + 1}) = s_Q (\alpha_i)$ for
all $i \in [1, n - 1]$. The number $n$ is called the length of
$\rho$ and denoted $|\rho|$. We write $s_Q (\rho)$ for $s_Q
(\alpha_n)$ and $t_Q (\rho)$ for $t_Q (\alpha_1)$, and we say that
$\rho$ starts at $s_Q (\rho)$ and terminates at $t_Q (\rho)$. For
each vertex $x$ of $Q$ we denote also by $x$ the path of length $0$
at vertex $x$ ($s_Q (x) = x = t_Q (x)$). For paths $\rho = \alpha_1
\cdots \alpha_n$ and $\rho' = \alpha_1' \cdots \alpha_m'$ in $Q$
such that $s_Q (\rho) = t_Q (\rho')$, we denote by $\rho \rho'$ the
path $\alpha_1 \cdots \alpha_n \alpha_1' \cdots \alpha_m'$. In
particular, $\rho s_Q (\rho) = \rho = t_Q (\rho) \rho$.
\subsection{}
By a defining system we mean a quadruple $(p, q, S, T)$, where $p$
and $q$ are sequences of positive integers such that $|q| = |p|$
and $\sum_{i = 1}^{|p|} p_i \geq 2$, and $S = (S_i)_{i = 1}^{|p|}$
and $T = (T_i)_{i = 1}^{|p|}$ are families of finite subsets of
$\mathbb{N}$ such that for each $i \in [1, |p|]$ hold: $T_i \subseteq
S_i \subseteq [2, p_i + |T_i|]$, if $j \in S_i$ then $j + 1 \not
\in S_i$, and $p_i + |T_i| \not \in T_i$. We write $T_{i, j}$
instead of $(T_i)_j$ for $i \in [1, |p|]$ and $j \in [1, |T_i|]$.
Throughout the rest of the section $(p, q, S, T)$ is a fixed
defining system.
\subsection{}
We define a quiver $Q$ by
\begin{align*}
Q_0 & = \{ x_{i, j} \mid i \in [1, |p|], \, j \in [0, p_i + |T_i|]
\} \\
& \cup \{ y_{i, j} \mid i \in [1, |p|], \, j \in [1, q_i - 1] \}
\\
& \cup \{ z_{i, j} \mid i \in [1, |p|], \, j \in S_i \} \\
\intertext{and} %
Q_1 & = \{ \alpha_{i, j} : x_{i, j} \to x_{i, j - 1} \mid i \in
[1, |p|], \, j \in [1, p_i + |T_i|] \} \\
& \cup \{ \beta_{i, j} : y_{i, j} \to y_{i, j - 1} \mid i \in [1,
|p|], \, j \in [1, q_i] \} \\
& \cup \{ \gamma_{i, j} : z_{i, j} \to x_{i, j} \mid i \in [1,
|p|], \, j \in S_i \} \\
& \cup \{ \xi_{i, j} : x_{i, p_i + j} \to z_{i, T_{i, j}} \mid i
\in [1, |p|], \, j \in [1, |T_i|] \},
\end{align*}
where $y_{i, 0} = x_{i + 1, 0}$ (with $x_{|p| + 1, 0} = x_{1, 0}$)
and $y_{i, q_i} = x_{i, p_i}$ for $i \in [1, |p|]$.
Let $A$ be the path algebra of $Q$ bounded by relations
\begin{gather*}
\alpha_{i, j - 1} \alpha_{i, j} \gamma_{i, j}, \, i \in [1, |p|],
\, j \in S_i, \\
\beta_{i, q_i} \alpha_{i, p_i + 1}, \, i \in [1, |p|] \text{ such
that } |T_i| > 0, \\
\xi_{i, j - 1} \alpha_{i, p_i + j}, \, i \in [1, |p|], \, j \in
[2, |T_i|], \\
\intertext{and} %
\alpha_{i, T_{i, j}} \gamma_{i, T_{i, j}} \xi_{i, j} - \alpha_{i,
T_{i, j}} \alpha_{i, T_{i, j} + 1} \cdots \alpha_{i, p_i + j - 1}
\alpha_{i, p_i + j}, \, i \in [1, |p|], \, j \in [1, |T_i|].
\end{gather*}
Recall that by~\cite{Bob1}*{Theorem~1.1} the class of algebras
defined in the above way coincides with the class of admissible
algebras with formal two-ray modules introduced in~\cite{BobSk2}.
In order to clarify a bit the above definitions we give a simple
example. If $p = (6, 3)$, $q = (2, 2)$, $S = (\{ 2, 4, 6, 8 \}, \{ 2
\})$ and $T = (\{ 4, 6 \}, \varnothing)$, then $A$ is the path
algebra of the quiver
\[
\xymatrix{%
\bullet \save*+!R{\scriptstyle z_{1, 8}} \restore
\ar[rd]_{\gamma_{1, 8}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 8}} \restore
\ar[d]^{\alpha_{1, 8}} \ar[ld]_{\xi_{1, 2}} \\ %
\bullet \save*+!R{\scriptstyle z_{1, 6}} \restore
\ar[rd]_{\gamma_{1, 6}} & \bullet \save*+!L{\scriptstyle x_{1, 7}}
\restore \ar[d]^{\alpha_{1, 7}} \ar[ldddd]_(.4){\xi_{1, 1}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 6}} \restore
\ar[d]^(.7){\alpha_{1, 6}} \ar[rrdd]^{\beta_{1, 2}} & & & & \bullet
\save*+!R{\scriptstyle x_{2, 3}} \restore
\ar[dd]_(.7){\alpha_{2, 3}} \ar[lldddd]_{\beta_{2, 2}} \\ %
\bullet \save*+!R{\scriptstyle z_{1, 4}} \restore
\ar[rd]_(.3){\gamma_{1, 4}} & \bullet \save*+!L{\scriptstyle x_{1,
5}} \restore \ar[d]^{\alpha_{1, 5}} & & & & & \bullet
\save*+!L{\scriptstyle z_{2, 2}} \restore \ar[ld]^{\gamma_{2, 2}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 4}} \restore
\ar[d]^{\alpha_{1, 4}} & & \bullet \save*+!R{\scriptstyle y_{1, 1}}
\restore \ar[rrdddd]_{\beta_{1, 1}} & & \bullet
\save*+!R{\scriptstyle x_{2, 2}} \restore \ar[dd]_{\alpha_{2, 2}} \\ %
\bullet \save*+!R{\scriptstyle z_{1, 2}} \restore
\ar[rd]_{\gamma_{1, 2}} & \bullet \save*+!L{\scriptstyle x_{1, 3}}
\restore \ar[d]^{\alpha_{1, 3}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 2}} \restore
\ar[d]^{\alpha_{1, 2}} & & \bullet \save*+!R{\scriptstyle y_{2, 1}}
\restore \ar[lldd]^{\beta_{2, 1}} & & \bullet \save*+!R{\scriptstyle
x_{2, 1}} \restore \ar[dd]_(.3){\alpha_{2,
1}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 1}} \restore
\ar[d]^(.3){\alpha_{1, 1}} \\ %
& \bullet \save*+!L{\scriptstyle x_{1, 0}} \restore & & & & \bullet
\save*+!R{\scriptstyle x_{2, 0}} \restore}
\]
bounded by relations
\begin{gather*}
\alpha_{1, 1} \alpha_{1, 2} \gamma_{1, 2}, \; \alpha_{1, 3}
\alpha_{1, 4} \gamma_{1, 4}, \; \alpha_{1, 5} \alpha_{1, 6}
\gamma_{1, 6}, \; \alpha_{1, 7} \alpha_{1, 8} \gamma_{1, 8}, \;
\alpha_{2, 1} \alpha_{2, 2} \gamma_{2, 2}, \; \beta_{1, 2}
\alpha_{1, 7},
\\ %
\xi_{1, 1} \alpha_{1, 8}, \; \alpha_{1, 2} \alpha_{1, 3} \alpha_{1,
4} \alpha_{1, 5} \alpha_{1, 6} \alpha_{1, 7} - \alpha_{1, 2}
\gamma_{1, 2} \xi_{1, 1}, \; \alpha_{1, 6} \alpha_{1, 7} \alpha_{1,
8} - \alpha_{1, 6} \gamma_{1, 6} \xi_{1, 2}.
\end{gather*}
\subsection{}
Let
\[
Q_1' = \{ \alpha_{i, j} : x_{i, j} \to x_{i, j - 1} \mid i \in [1,
|p|], \, j \in [1, p_i + |T_i|] \}
\]
and $Q_1'' = Q_1 \setminus Q_1'$. Let $Q^*$ be the quiver with
same set of vertices and arrows as $Q$, but with the arrows from
$Q_1''$ reversed, i.e., $Q_0^* = Q_0$, $Q_1^* = Q_1$ and
\[
s_{Q^*} (\alpha) =
\begin{cases}
s_Q (\alpha) & \alpha \in Q_1', \\
t_Q (\alpha) & \alpha \in Q_1'',
\end{cases} \text{ and }
t_{Q^*} (\alpha) =
\begin{cases}
t_Q (\alpha) & \alpha \in Q_1', \\
s_Q (\alpha) & \alpha \in Q_1''.
\end{cases}
\]
By a string in $Q$ we mean a path in $Q^*$ which does not contain
a subpath $\alpha_{i, T_{i, j}} \alpha_{i, T_{i, j} + 1} \cdots
\alpha_{i, p_i + j}$ for $i \in [1, |p|]$ and $j \in [1, |T_i|]$.
For formal reasons we also introduce the empty string denoted by
$\varnothing$. By convention the length of $\varnothing$ is $-1$,
the maps $s_{Q^*}$ and $t_{Q^*}$ are not defined for $\varnothing$
and it cannot be composed with other strings. If $C$ is a string
and $C = C' C''$ for strings $C'$ and $C''$, then $C'$ is called a
terminating substring of $C$ and $C''$ is called a starting
substring of $C$.
If $C = c_1 \cdots c_n$ is a string and $x \in Q_0$, then we put
\begin{align*}
J_C^x & = \{ i \in [0, n - 1] \mid t_{Q^*} (c_{i + 1}) = x \} \\
\intertext{and} %
I_C^x & =
\begin{cases}
J_C^x \cup \{ n \} & s_{Q^*} (c_n) = x, \\
J_C^x & s_{Q^*} (c_n) \neq x.
\end{cases}
\end{align*}
In particular, $J_y^x = \varnothing$ for all $y \in Q_0$, $I_x^x =
\{ 0 \}$, and $I_y^x = \varnothing$ if $y \neq x$.
\subsection{}
For each vertex $x$ of $Q$ we denote by $\omega_x$ (respectively
$\mu_x$) the longest string terminating at $x$ and consisting only
of elements of $Q_1'$ ($Q_1''$). Similarly, by $\pi_x$
(respectively $\nu_x$) we denote the longest string starting at
$x$ and consisting only of elements of $Q_1'$ ($Q_1'')$.
Let
\begin{align*}
Q_0' = \{ x_{i, j} \mid i \in [1, |p|], \, j \in S_i \}, \\
\intertext{and} %
Q_0'' = \{ x_{i, j} \mid i \in [1, |p|], \, j \in T_i \}.
\end{align*}
For $x \in Q_0'$, $x = x_{i, j}$, we denote $\alpha_{i, j}$ by
$\alpha_x$ and $\gamma_{i, j}$ by $\gamma_x$.
Let $x \in Q_0''$, $x = x_{i, T_{i, j}}$. We put
\[
B_x = \alpha_{i, T_{i, j} + 1} \cdots \alpha_{i, j} \xi_{i, j}
\gamma_{i, T_{i, j}}.
\]
For a string $C$ terminating at $x$ we denote by $p_C$ the maximal
integer $p \geq 0$ such that $B_x^p$ is a terminating substring of
$C$, where $B_x^p$ denotes the $p$-fold composition of $B_x$ with
itself (with the convention that $B_x^0 = x$). If $x \in Q_0'
\setminus Q_0''$ then we set $B_x = x$ and $p_C = 0$ for each
string $C$ terminating at $x$.
\subsection{} \label{sectord}
For a given vertex $x$ of $Q$ we introduce a linear order in the
set of all strings terminating at $x$. Let $C$ and $C'$ be two
strings terminating at $x$ and let $C_0$ be the longest string
which is both a terminating substring of $C$ and a terminating
substring of $C'$. Then $C < C'$ if and only if either $C = C_0
\beta D$ for $\beta \in Q_1''$ and a string $D$ or $C' = C_0
\alpha D'$ for $\alpha \in Q_1'$ and a string $D'$. Note that the
maximal string terminating at $x$ is $\omega_x$ and the minimal
one is $\mu_x$.
If $C \neq \omega_x$ is a string terminating at $x$, then there
exists a direct successor $C_+$ of $C$, which can be described in
the following way. If there exists $\alpha \in Q_1'$ such that $C
\alpha$ is a string, then $C_+ = C \alpha \mu_{s_Q (\alpha)}$.
Otherwise, there exist a string $C'$ and $\beta \in Q_1''$ such
that $C = C' \beta \omega_{t_Q (\beta)}$. In this case $C_+ = C'$.
We also put $(\omega_x)_+ = \varnothing$.
Similarly, we may define a string ${}_+ C$, which is a direct
successor of $C$ with respect to the appropriate order in the set
of all strings starting at $s_{Q^*} (C)$. Since this order will
play no role in the sequel, we only give a description of ${}_+
C$. If there exists $\beta \in Q_1''$ such that $\beta C$ is a
string, then ${}_+ C = \pi_{s_Q (\beta)} \beta C$. Otherwise,
${}_+ C = C''$, if $C = \nu_{t_Q (\alpha)} \alpha C''$ for $\alpha
\in Q_1'$ and a string $C''$, or ${}_+ C = \varnothing$ if $C =
\nu_x$.
Let $C$ be a string such that $|C_+| + |{}_+ C| \geq |C|$ (this is
equivalent to saying that $C \neq \nu_x \omega_x$ for a vertex $x$
of $Q$). Then we define ${}_+ C_+$ by
\[
{}_+ C_+ =
\begin{cases}
{}_+ (C_+) & C_+ \neq \varnothing, \\
({}_+ C)_+ & {}_+ C \neq \varnothing.
\end{cases}
\]
One easily verifies that the above definition is correct and ${}_+
C_+ \neq \varnothing$. We also put ${}_+ (\nu_x \omega_x)_+ =
\varnothing$ for $x \in Q_0$.
\subsection{}
Let $\mathcal{S}$ be the set of all strings in $Q$. For $x \in Q_0'$ we
denote by $\mathcal{S}_x$ the set of all strings $C$ terminating at $x$
such that $\alpha_x C'$ is a string, where $C = B_x^{p_C} C'$
($\mathcal{S}_x$ is the set of all strings terminating at $x$ if $x \in
Q_0' \setminus Q_0''$). Let $\mathcal{P}_x$ be the set all pairs $(C,
C')$ of $C, C' \in \mathcal{S}_x$ such that $C < C'$ and, if $x \in
Q_0''$, $C' < B_x C$. Finally, we put
\[
\mathcal{B}' = \{ B_x \mid x \in Q_0'' \} \text{ and } \mathcal{B} = \{ B_0 \}
\cup \mathcal{B}',
\]
where
\[
B_0 = \alpha_{1, 1} \cdots \alpha_{1, p_1} \beta_{1, q_1} \cdots
\beta_{1, 1} \cdots \alpha_{|p|, 1} \cdots \alpha_{|p|, p_{|p|}}
\beta_{|p|, q_{|p|}} \cdots \beta_{|p|, 1}.
\]
\subsection{}
Let $B = b_1 \cdots b_n \in \mathcal{B}$, $\lambda \in k^*$ and $m \in
\mathbb{N}$. We define a representation $R (B, \lambda, m)$ of $Q$ as
follows:
\begin{align*}
R (B, \lambda, m)_y & = \bigoplus_{j \in [1, m]} \bigoplus_{i \in
J_B^y} k v_i^{(j)} \\
\intertext{and} %
R (B, \lambda, m)_\alpha (v_i^{(j)}) & =
\begin{cases}
v_{i - 1}^{(j)} & \alpha \in Q_1', \, \alpha = b_i, \, i \in [1, n
- 1],
\\ %
v_{i + 1}^{(j)} & \alpha \in Q_1'', \, \alpha = b_{i + 1}, \, i
\in [0, n - 2],
\\ %
\lambda v_0^{(j)} + v_0^{(j + 1)} & \alpha = b_n, \, i = n - 1,
\\ %
& \qquad j \in [1, m - 1],
\\ %
\lambda v_0^{(m)} & \alpha = b_n, \, i = n - 1, \, j = m, \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
We also put $R (B, \lambda, 0) = 0$.
\subsection{}
Let $x \in Q_0''$, $B = b_1 \cdots b_n = B_x$ and $m \in \mathbb{N}$. We
define a representation $Q (B, m)$ of $Q$ as follows:
\begin{align*}
Q (B, m)_y & =
\begin{cases}
k v' \oplus \bigoplus_{j \in [1, m]} \bigoplus_{i \in J_C^y} k
v_i^{(j)} & y = t_Q (\alpha_x), \\
\bigoplus_{j \in [1, m]} \bigoplus_{i \in J_C^y} k v_i^{(j)} &
\text{otherwise},
\end{cases} \\
Q (B, m)_\alpha (v_i^{(j)}) & =
\begin{cases}
v' & \alpha = \alpha_x, \, i = 0, \, j = 1,
\\ %
v_{i - 1}^{(j)} & \alpha \in Q_1', \, \alpha = b_i, \, i \in [1, n
- 1],
\\ %
v_{i + 1}^{(j)} & \alpha \in Q_1'', \, \alpha = b_{i + 1}, \, i
\in [0, n - 2],
\\ %
v_0^{(j)} + v_0^{(j + 1)} & \alpha = b_n, \, i = n - 1, \, j \in [1, m - 1],
\\ %
v_0^{(m)} & \alpha = b_n, \, i = n - 1, \, j = m,
\\ %
0 & \text{otherwise},
\end{cases}
\intertext{and} %
Q (B, m)_\alpha (v') & = 0.
\end{align*}
\subsection{}
Let $C = c_1 \cdots c_n \in \mathcal{S}$. We define a representation $M
(C)$ of $Q$ as follows:
\begin{align*}
M (C)_y & = \bigoplus_{i \in I_C^y} k v_i
\\ %
\intertext{and} %
M (C)_\alpha (v_i) & =
\begin{cases}
v_{i - 1} & \alpha \in Q_1', \, \alpha = c_i, \, i \in [1, n],
\\ %
v_{i + 1} & \alpha \in Q_1'', \, \alpha = c_{i + 1}, \, i \in [0,
n - 1],
\\ %
0 & \text{otherwise}.
\end{cases}
\end{align*}
In particular, $M (x)$ is the simple representation of $Q$ at $x$.
We also put $M (\varnothing) = 0$.
\subsection{}
Let $x \in Q_0'$ and $C = c_1 \cdots c_n \in \mathcal{S}_x$. We define a
representation $N (C)$ of $Q$ as follows:
\begin{align*}
N (C)_y & =
\begin{cases}
k v' \oplus \bigoplus_{i \in I_C^y} k v_i & y = t_Q (\alpha_x),
\\ %
k v'' \oplus \bigoplus_{i \in I_C^y} k v_i & y = s_Q (\gamma_x),
\\ %
\bigoplus_{i \in I_C^y} k v_i & \text{otherwise},
\end{cases}
\\ %
N (C)_\alpha (v_i) & =
\begin{cases}
v' & \alpha = \alpha_x, \, i = p |B_x|, \, p \in [0, p_C],
\\ %
v_{i - 1} & \alpha \in Q_1', \, \alpha = c_i, \, i \in [1, n],
\\ %
v_{i + 1} & \alpha \in Q_1'', \, \alpha = c_{i + 1}, \, i \in [0,
n - 1],
\\ %
0 & \text{otherwise},
\end{cases}
\\ %
N (C)_\alpha (v') & = 0,
\\ %
\intertext{and} %
N (C)_\alpha (v'') & =
\begin{cases}
v_0 & \alpha = \gamma_x,
\\ %
0 & \text{otherwise}.
\end{cases}
\end{align*}
We also put $N (\varnothing) = M (s_Q (\gamma_x))$ (more
precisely, we should write $N_x (\varnothing)$, but we omit the
vertex if it causes no confusion).
\subsection{}
Let $x \in Q_0''$ and $C = c_1 \cdots c_n \in \mathcal{S}_x$ be such
that $p_C > 0$. We define a representation $L (C)$ of $Q$ as
follows:
\begin{align*}
L (C)_y & =
\begin{cases}
k v' \oplus \bigoplus_{i \in I_C^y} k v_i & y = t_Q (\alpha_x),
\\ %
\bigoplus_{i \in I_C^y} k v_i & \text{otherwise},
\end{cases}
\\ %
L (C)_\alpha (v_i) & =
\begin{cases}
v' & \alpha = \alpha_x, \, i = p |B_x|, \, p \in [0, p_C],
\\ %
v_{i - 1} & \alpha \in Q_1', \, \alpha = c_i, \, i \in [1, n],
\\ %
v_{i + 1} & \alpha \in Q_1'', \, \alpha = c_i, \, i \in [0, n -
1],
\\ %
0 & \text{otherwise},
\end{cases}
\\ %
\intertext{and} %
L (C)_\alpha (v') & = 0.
\end{align*}
\subsection{}
Let $x \in Q_0'$, and $(C = c_1 \cdots c_n, C' = c_1' \cdots c_m')
\in \mathcal{P}_x$. We define a representation $N (C, C')$ of $Q$ as
follows:
\begin{align*}
N (C, C')_y & =
\begin{cases}
k v' \oplus \bigoplus_{i \in I_C^y} k v_i \oplus \bigoplus_{i \in
I_{C'}^y} k v_i' & y = t_Q (\alpha_x),
\\ %
k v'' \oplus \bigoplus_{i \in I_C^y} k v_i \oplus \bigoplus_{i \in
I_{C'}^y} k v_i' & y = s_Q (\gamma_x),
\\ %
\bigoplus_{i \in I_C^y} k v_i \oplus \bigoplus_{i \in I_{C'}^y} k
v_i' & \text{otherwise},
\end{cases}
\\ %
N (C, C')_\alpha (v_i) & =
\begin{cases}
v' & \alpha = \alpha_x, \, i = p |B_x|, \, p \in [0, p_C],
\\ %
v_{i - 1} & \alpha \in Q_1', \, \alpha = c_i, \, i \in [1, n],
\\ %
v_{i + 1} & \alpha \in Q_1'', \, \alpha = c_{i + 1}, \, i \in [0,
n - 1],
\\ %
0 & \text{otherwise},
\end{cases}
\\ %
N (C, C')_\alpha (v_i') & =
\begin{cases}
v' & \alpha = \alpha_x, \, i = p |B_x|, \, p \in [0, p_{C'}],
\\ %
v_{i - 1}' & \alpha \in Q_1', \, \alpha = c_i', \, i \in [1, m],
\\ %
v_{i + 1}' & \alpha \in Q_1'', \, \alpha = c_{i + 1}', \, i \in
[0, m - 1],
\\ %
0 & \text{otherwise},
\end{cases} \\ %
N (C, C')_\alpha (v') & = 0,
\\ %
\intertext{and} %
N (C, C')_\alpha (v'') & =
\begin{cases}
v_0 & \alpha = \gamma_x, \\ %
0 & \text{otherwise}.
\end{cases}
\end{align*}
We also put $N (C, \varnothing) = M (\gamma_x C)$, $N (C, C) = N
(C) \oplus M (C)$ and, if $x \in Q_1''$, $N (C, B_x C) = L (B_x C)
\oplus M (\gamma_x C)$.
\subsection{} \label{maintheo}
Let
\begin{multline*}
\mathcal{S}' = \mathcal{S} \setminus (\{ \nu_x \omega_x \mid x \in Q_0 \} \cup
\{ C \mid C \in \mathcal{S}_x, \, x \in Q_0' \} \\ %
\cup \{ \alpha_x C \mid C \in \mathcal{S}_x, \, x \in Q_0' \} \cup \{
\gamma_x C \mid C \in \mathcal{S}_x, \, x \in Q_0'' \}).
\end{multline*}
Observe, that $\nu_x \omega_x \in \mathcal{S}$ for all $x \in Q_0$, and
$\alpha_x C \in \mathcal{S}$ for all $x \in Q_0' \setminus Q_0''$ and $C
\in \mathcal{S}_x$. Moreover, if $x \in Q_0''$ and $C \in \mathcal{S}_x$, then
$\gamma_x C \in \mathcal{S}$, but $\alpha_x C \in \mathcal{S}$ if and only if
$\omega_x$ is not a terminating substring of $C$.
The following theorem is the main result of the paper.
\begin{theo*}
Let $(p, q, S, T)$ be a defining system and let $A$ be the
corresponding algebra.
\begin{enumerate}
\item
Representations
\begin{align*}
& R (B, \lambda, m), \, B \in \mathcal{B}, \, \lambda \in k^*, \, m \in
\mathbb{N}, \\ %
& Q (B, m), \, B \in \mathcal{B}', \, m \in \mathbb{N}, \\ %
& M (C), \, C \in \mathcal{S}, \\ %
& N (C), \, C \in \mathcal{S}_x, \, x \in Q_0', \\ %
& L (B_x C), \, C \in \mathcal{S}_x, \, x \in Q_0'', \\ %
& N (C, C'), (C, C') \in \mathcal{P}_x, \, x \in Q_0',
\end{align*}
form a complete set of pairwise nonisomorphic indecomposable
modules over $A$.
\item
Sequences
\begin{align*}
& 0 \to R (B, \lambda, m) \to R (B, \lambda, m + 1) \oplus R (B,
\lambda, m - 1) \to R (B, \lambda, m)
\\ %
& \qquad \to 0, \, (B, \lambda, m) \in \mathcal{B} \times k^* \times
\mathbb{N}, \, B = B_0 \text{ or } \lambda \neq 1,
\\ %
& 0 \to R (B, 1, m) \to Q (B, m + 1) \oplus R (B, 1, m - 1) \to Q
(B, m) \to 0,
\\ %
& \qquad B \in \mathcal{B}', \, m \in \mathbb{N},
\\ %
& 0 \to Q (B, m) \to R (B, m) \oplus Q (B, m - 1) \to R (B, 1, m -
1) \to 0,
\\ %
& \qquad B \in \mathcal{B}', \, m \in \mathbb{N}, \, m > 1,
\\ %
& 0 \to M (C) \to M (C_+) \oplus M ({}_+ C) \to M ({}_+ C_+) \to
0, \, C \in \mathcal{S}',
\\ %
& 0 \to M (C) \to M (C_+) \oplus N (\mu_x, {}_+ C) \to N (\mu_x,
{}_+ C_+) \to 0,
\\ %
& \qquad C = \alpha_x C', \, C' \in \mathcal{S}_x, \, x \in Q_0',
\\ %
& 0 \to M (C) \to N (C, C_+) \to N (C_+) \to 0, \, C \in \mathcal{S}_x,
\, x \in Q_0',
\\ %
& 0 \to M (\gamma_x C) \to N (C_+, B_x C) \to L (B_x C_+) \to 0,
\, C \in \mathcal{S}_x, \, x \in Q_0'',
\\ %
& 0 \to N (C) \to N (C, C_+) \to M (C_+) \to 0, \, C \in \mathcal{S}_x,
\, x \in Q_0', \, C \neq \omega_x,
\\ %
& 0 \to L (B_x C) \to N (C_+, B_x C) \to M (\gamma_x C_+) \to 0,
\, C \in \mathcal{S}_x, \, x \in Q_0'',
\\ %
& 0 \to N (C, C') \to N (C, C_+') \oplus N (C_+, C') \to N (C_+,
C_+') \to 0,
\\ %
& \qquad (C, C') \in \mathcal{P}_x, \, x \in Q_0',
\end{align*}
form a complete list of Auslander--Reiten sequences in $\mod A$.
\end{enumerate}
\end{theo*}
We finish this section with some remarks concerning the above
theorem. First of all, if $x \in Q_0'$ then $\omega_x \in \mathcal{S}_x$
if and only if $x \not \in Q_0''$. If $x \in Q_0'$, $C \in
\mathcal{S}_x$ and $\alpha_x C \in \mathcal{S}$, then ${}_+ (\alpha_x C) = C$
and ${}_+ (\alpha_x C)_+ = C_+$. Moreover, if $C \neq \omega_x$,
then $(\alpha_x C)_+ = \alpha_x C_+$. Finally, if $x \in Q_0'
\setminus Q_0''$, then $(\alpha_x \omega)_+ = \varnothing$.
\section{Vector space categories} \label{sectvect}
In this section we describe vector space categories and subspace
categories needed in the proof of our main result.
\subsection{}
Following~\cite{Si}*{Section~17.1} (see
also~\cite{Ri2}*{Section~2.4}) by a vector space category we mean
a pair $\mathbb{K} = (\mathcal{K}, {|-|})$, where $\mathcal{K}$ is a Krull--Schmidt
category and ${|-|} : \mathcal{K} \to \mod k$ is a faithful functor. For
a vector space category $\mathbb{K}$ we consider the subspace category
$\mathcal{U} (\mathbb{K})$ of $\mathbb{K}$. The objects of $\mathcal{U} (\mathbb{K})$ are
triples $V = (V_0, V_1, \gamma_V)$ with $V_0 \in \mathcal{K}$, $V_1 \in
\mod k$ and $\gamma_V : V_1 \to |V_0|$ a $k$-linear map. If $V =
(V_0, V_1, \gamma_V)$ and $W = (W_0, W_1, \gamma_W)$ are two
objects of $\mathcal{U} (\mathbb{K})$, then a morphism $f : V \to W$ in $\mathcal{U}
(\mathbb{K})$ is a pair $f = (f_0, f_1)$, where $f_0 : V_0 \to W_0$ is a
morphism in $\mathcal{K}$, $f_1 : V_1 \to W_1$ is a $k$-linear map and
the condition $|f_0| \gamma_V = \gamma_W f_1$ is satisfied. By
$\overline{0}$ we denote the triple $(0, k, 0)$ in $\mathcal{U} (\mathbb{K})$.
\subsection{}
An ordered set $I$ is called semi-admissible, if the order is linear
and for each element of $I$ which is not maximal there exists a
direct successor. If in addition, there exist a minimal and a
maximal elements in $I$, then we call $I$ admissible. If $I$ is a
semi-admissible ordered set and $\gamma \in I$ is not maximal in
$I$, then by $\gamma_+$ we denote the direct successor of $\gamma$
in $I$.
If $I_1$ and $I_2$ are two semi-admissible ordered sets, then we
introduce the order in $I_1 \times I_2$ by saying that $(x_1, y_1)
\leq (x_2, y_2)$ if either $x_1 < x_2$ or $x_1 = x_2$ and $y_1 \leq
y_2$, for $x_1, x_2 \in I_1$ and $y_1, y_2 \in I_2$. If $(x, y) \in
I_1 \times I_2$, then we put $(x, y)^+ = (x_+, y)$. If in addition
$I_1$ and $I_2$ are disjoint, then by $I_1 + I_2$ we denote the
ordered set $I_1 \cup I_2$ with the elements of $I_1$ smaller than
the elements of $I_2$.
If $I$ is an admissible ordered set, then we denote by $I_-$ the
set $\{ * \} + I$, where $* \not \in I$. Note that in this case $*
= \min I_-$ and $*_+ = \min I$. Similarly, we put $I_+ = I + \{ *
\}$ (thus in this case $* = \max I_+ = (\max I)_+$). Finally, we
denote by $I'$ the ordered set $I \setminus \{ \max I \}$.
\subsection{}
Let $I_1$, \ldots, $I_{r + 1}$, $r \in \mathbb{N}_0$, be a family of
admissible ordered sets. Let $\mathcal{K}$ be the Krull--Schmidt
category, whose indecomposable objects are
\begin{itemize}
\item
$X_\gamma$, $\gamma \in I_p'$, $p \in [1, r + 1]$,
\item
$X_{\max I_p}'$, $X_{\max I_p}''$, $p \in [1, r]$,
\end{itemize}
and all indecomposable objects of $\mathcal{K}$ are one-dimensional, i.e.,
for each indecomposable object $X$ of $\mathcal{K}$, $\dim_k |X| = 1$. If
$U$ and $V$ are indecomposable objects of $\mathcal{K}$, then
$\Hom_{\mathcal{K}} (U, V) \neq 0$ if and only if one of the following
conditions holds:
\begin{itemize}
\item
$U = X_{\gamma'}$, $V = X_{\gamma''}$, $\gamma' \in I_p'$,
$\gamma'' \in I_q'$, $(p, \gamma') \leq (q, \gamma'')$,
\item
$U = X_\gamma$, $V = X_{\max I_q}'$, $\gamma \in I_p'$, $p \leq
q$,
\item
$U = X_\gamma$, $V = X_{\max I_q}''$, $\gamma \in I_p'$, $p \leq
q$,
\item
$U = X_{\max I_p}'$, $V = X_\gamma$, $\gamma \in I_q'$, $p < q$,
\item
$U = X_{\max I_p}'$, $V = X_{\max I_q}'$, $p \leq q$,
\item
$U = X_{\max I_p}'$, $V = X_{\max I_q}''$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_\gamma$, $\gamma \in I_q'$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_{\max I_q}'$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_{\max I_q}''$, $p \leq q$.
\end{itemize}
By $\mathbb{K}_{I_1, \ldots, I_{r + 1}}$ we denote the vector space
category $(\mathcal{K}, {|-|})$, where ${|-|} : \mathcal{K} \to \mod k$ is the
forgetful functor.
\subsection{}
Let $I$ be an admissible ordered set. Let $\mathcal{L}$ be the
Krull--Schmidt category, whose indecomposable objects are
\begin{itemize}
\item
$X_\gamma$, $\gamma \in I$,
\item
$Y_\gamma$, $\gamma \in I$,
\end{itemize}
and all indecomposable objects of $\mathcal{L}$ are one-dimensional. If
$U$ and $V$ are indecomposable objects of $\mathcal{L}$, then
$\Hom_{\mathcal{L}} (U, V) \neq 0$ if and only if one of the following
conditions holds:
\begin{itemize}
\item
$U = X_{\gamma'}$, $V = X_{\gamma''}$, $\gamma' \leq \gamma''$,
\item
$U = X_{\gamma'}$, $V = Y_{\gamma''}$, $\gamma' \leq \gamma''$,
\item
$U = Y_{\gamma'}$, $V = Y_{\gamma''}$, $\gamma' \leq \gamma''$.
\end{itemize}
By $\mathbb{L}_I$ we denote the vector space category $(\mathcal{L}, {|-|})$,
where ${|-|} : \mathcal{L} \to \mod k$ is the forgetful functor.
\subsection{} \label{subspaceone}
We have the following description of the indecomposable objects and
the Auslander--Reiten sequences in $\mathcal{U} (\mathbb{L}_I)$. For definitions
of the relevant objects and the proof we refer
to~\cite{BobDrSk}*{Section~3}.
\begin{prop*}
Let $I$ be an admissible ordered set.
\begin{enumerate}
\item
Objects
\begin{align*}
& M_{\min I_-, \gamma} = X_\gamma, \, \gamma \in I, \,
\\ %
& M_{\gamma', \gamma''} = \overline{Y_{\gamma'} X_{\gamma''}}, \,
\gamma', \gamma'' \in I, \, \gamma' < \gamma'',
\\ %
& M_{\gamma, \max I_+} = \overline{Y_\gamma}, \, \gamma \in I,
\\ %
& M_{\gamma, \gamma}' = Y_\gamma, \, \gamma \in I,
\\ %
& M_{\gamma, \gamma}'' = \overline{X_\gamma}, \, \gamma \in I,
\\ %
& M_{\max I_+, \max I_+}'' = \overline{0},
\end{align*}
form a complete set of pairwise nonisomorphic indecomposable
objects in $\mathcal{U} (\mathbb{L}_I)$.
\item
Sequences
\begin{align*}
& 0 \to M_{\gamma', \gamma''} \to M_{\gamma'_+, \gamma''} \oplus
M_{\gamma', \gamma''_+} \to M_{\gamma'_+, \gamma''_+} \to 0, \,
\gamma', \gamma'' \in I_-, \, \gamma' < \gamma'',
\\ %
& 0 \to M_{\gamma, \gamma}' \to M_{\gamma, \gamma_+} \to
M_{\gamma_+, \gamma_+}'' \to 0, \, \gamma \in I,
\\ %
& 0 \to M_{\gamma, \gamma}'' \to M_{\gamma, \gamma_+} \to
M_{\gamma_+, \gamma_+}' \to 0, \, \gamma \in I',
\end{align*}
form a complete list of Auslander--Reiten sequences in $\mathcal{U}
(\mathbb{L}_I)$, where
\begin{align*}
& M_{\gamma, \gamma} = M_{\gamma, \gamma}' \oplus M_{\gamma,
\gamma}'', \, \gamma \in I,
\\ %
& M_{\min I_-, \max I_+} = 0.
\end{align*}
\end{enumerate}
\end{prop*}
\subsection{}
Let $I_0$, \ldots, $I_{r + 1}$, $r \in \mathbb{N}$, be a family of
admissible ordered sets. Let $\mathcal{L}$ be the Krull--Schmidt
category, whose indecomposable objects are
\begin{itemize}
\item
$X_\gamma$, $\gamma \in I_p'$, $p \in [0, r + 1]$,
\item
$X_{\max I_p}'$, $X_{\max I_p}''$, $p \in [0, r]$,
\item
$Y_\gamma$, $\gamma \in I_0'$,
\item
$Z$.
\end{itemize}
If $U$ is an indecomposable object of $\mathcal{L}$, then
\[
\dim_k |U| =
\begin{cases}
2 & U = X_{\min I_1}, \\ %
1 & \text{otherwise}.
\end{cases}
\]
If $U$ and $V$ are indecomposable objects of $\mathcal{L}$, then $\dim_k
\Hom_{\mathcal{L}} (U, V) \leq 2$, $\Hom_{\mathcal{L}} (U, V) \neq 0$ if and
only if one of the following conditions holds:
\begin{itemize}
\item
$U = X_{\gamma'}$, $V = X_{\gamma''}$, $\gamma' \in I_p'$,
$\gamma'' \in I_q'$, $(p, \gamma') \leq (q, \gamma'')$,
\item
$U = X_\gamma$, $V = X_{\max I_q}'$, $\gamma \in I_p'$, $p \leq
q$,
\item
$U = X_\gamma$, $V = X_{\max I_q}''$, $\gamma \in I_p'$, $p \leq
q$,
\item
$U = X_{\gamma'}$, $V = Y_{\gamma''}$, $\gamma', \gamma'' \in
I_0'$, $\gamma' \leq \gamma''$,
\item
$U = X_\gamma$, $V = Z$, $\gamma \in I_0'$,
\item
$U = X_{\max I_p}'$, $V = X_\gamma$, $\gamma \in I_q'$, $p < q$,
\item
$U = X_{\max I_p}'$, $V = X_{\max I_q}'$, $p \leq q$,
\item
$U = X_{\max I_p}'$, $V = X_{\max I_q}''$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_\gamma$, $\gamma \in I_q'$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_{\max I_q}'$, $p < q$,
\item
$U = X_{\max I_p}''$, $V = X_{\max I_q}''$, $p \leq q$,
\item
$U = X_{\max I_0}''$, $V = Z$,
\item
$U = Y_\gamma$, $V = X_{\min I_1}$,
\item
$U = Y_{\gamma'}$, $V = Y_{\gamma''}$, $\gamma' \leq \gamma''$,
\item
$U = Y_\gamma$, $V = Z$,
\item
$U = Z$, $V = Z$,
\end{itemize}
and $\dim_k \Hom_{\mathcal{L}} (U, V) = 2$ if and only if $U = X_\gamma$,
$\gamma \in I_0'$, $V = X_{\min I_1}$. By $\mathbb{L}_{I_0, \ldots, I_{r +
1}}$ we denote the vector space category $(\mathcal{L}, {|-|})$, where
${|-|} : \mathcal{L} \to \mod k$ is the forgetful functor. We refer the
reader to~\cite{BobSk1}*{Section~1} for pictures presenting vector
space categories of the above type, and in particular explaining how
the forgetful functor ${|-|}$ is defined on $\Hom_{\mathcal{L}} (X_\gamma,
X_{\min I_1})$ for $\gamma \in I_0'$.
\subsection{} \label{propLIr}
We describe the indecomposable objects and the Auslander--Rei\-ten
sequences in $\mathcal{U} (\mathbb{L}_{I_0, \ldots, I_{r + 1}})$. We refer
to~\cite{BobSk1} for definitions of the objects listed in the below
proposition and its proof.
\begin{prop*}
Let $I_0$, \ldots, $I_{r + 1}$, $r \in \mathbb{N}$, be admissible
ordered sets. Put
\[
I_p'' =
\begin{cases}
I_1' \setminus \{ \min I_1 \} & p = 1, \\ %
I_p' & p \in [2, r + 1],
\end{cases}
\]
\begin{enumerate}
\item
Objects
\begin{align*}
& M_{(-1, \max I_0), (0, \gamma)} = X_\gamma, \, \gamma \in I_0',
\\ %
& M_{(0, \gamma'), (0, \gamma'')} = \overline{Y_{\gamma'} X_{\gamma''}},
\, \gamma', \gamma'' \in I_0', \, \gamma' < \gamma'',
\\ %
& M_{(n - 1, \max I_0), (n, \gamma)} = \overline{Y_\gamma X_{\max I_0}'
X_{\max I_0}'' X_{\min I_1}^{2 n - 1}}^{2 n}, \, \gamma \in I_0',
\, n \in \mathbb{N},
\\ %
& M_{(n, \gamma), (n, \max I_0)} = \overline{Y_\gamma X_{\max I_0}'
X_{\max I_0}'' X_{\min I_1}^{2 n}}^{2 n + 1}, \, \gamma \in I_0',
\, n \in \mathbb{N}_0,
\\ %
& M_{(n, \gamma''), (n + 1, \gamma')} = \overline{Y_{\gamma''}
Y_{\gamma'} X_{\max I_0}' X_{\max I_0}'' X_{\min I_1}^{2 n}}^{2 n
+ 2},
\\ %
& \qquad \gamma', \gamma'' \in I_0', \gamma' < \gamma'', \, n \in
\mathbb{N}_0,
\\ %
& M_{(n, \gamma'), (n, \gamma'')} = \overline{Y_{\gamma'} Y_{\gamma''}
X_{\max I_0}' X_{\max I_0}'' X_{\min I_1}^{2 n - 1}}^{2 n + 1},
\\ %
& \qquad \gamma', \gamma'' \in I_0', \gamma' < \gamma'', \, n \in
\mathbb{N},
\\ %
& M_{(n, \gamma), (n, \gamma)}' = \overline{Y_\gamma X_{\min I_1}^n}^n,
\gamma \in I_0', \, n \in \mathbb{N}_0,
\\ %
& M_{(n, \max I_0), (n, \max I_0)}' = \overline{X_{\min I_1}^{n + 1}}^n,
\, n \in \mathbb{N}_0,
\\ %
& M_{(0, \gamma), (0, \gamma)}'' = \overline{X_\gamma}, \, \gamma \in
I_0',
\\ %
& M_{(n, \max I_0), (n, \max I_0)}'' = \overline{X_{\max I_0}' X_{\max
I_0}'' X_{\min I_1}^n}^{n + 1}, \, n \in \mathbb{N}_0,
\\ %
& M_{(n, \gamma), (n, \gamma)}'' = \overline{Y_\gamma X_{\max I_0}'
X_{\max I_0}'' X_{\min I_1}^{n - 1}}^{n + 1}, \, \gamma \in I_0',
\, n \in \mathbb{N},
\\ %
& M_{(n - 1, \max I_0), (n, \max I_0)}' = \overline{X_{\max I_0}'
X_{\min I_1}^n}^n, \, n \in \mathbb{N}_0,
\\ %
& M_{(n, \gamma), (n + 1, \gamma)}' = \overline{Y_\gamma X_{\max I_0}'
X_{\min I_1}^n}^{n + 1}, \, \gamma \in I_0', \, n \in \mathbb{N}_0,
\\ %
& M_{(n - 1, \max I_0), (n, \max I_0)}'' = \overline{X_{\max I_0}''
X_{\min I_1}^n}^n, \, n \in \mathbb{N}_0,
\\ %
& M_{(n, \gamma), (n + 1, \gamma)}'' = \overline{Y_\gamma X_{\max I_0}''
X_{\min I_1}^n}^{n + 1}, \, \gamma \in I_0', \, n \in \mathbb{N}_0,
\\ %
& R_n^\lambda = \overline{X_{\min I_1}^n}^n (\lambda), \, \lambda \in
k^*, \, \lambda \neq 1, \, n \in \mathbb{N},
\\ %
& R_{2 n - 1}^1 = \overline{X_{\max I_0}'' X_{\min I_1}^{n - 1}}^n, \, n
\in \mathbb{N},
\\ %
& R_{2 n}^1 = \overline{X_{\min I_1}^n}^n (1), \, n \geq 1,
\\ %
& R_{2 n - 1, 0}^\infty = \overline{X_{\max I_0}' X_{\min I_1}^{n -
1}}^n, \, n \in \mathbb{N},
\\ %
& R_{2 n - 1, 1}^\infty = \overline{X_{\min I_1}^{n - 1} Z}^{n - 1}, \,
n \in \mathbb{N},
\\ %
& R_{2 n, 0}^\infty = \overline{X_{\min I_1}^n}^n (\infty), \, n \in
\mathbb{N},
\\ %
& R_{2 n, 1}^\infty = \overline{X_{\max I_0}' X_{\min I_1}^{n - 1} Z}^n,
\, n \in \mathbb{N},
\\ %
& S_{p, (n - 1, \max I_0), (m - 1, \max I_0)} = \overline{X_{\min I_1}^n
X_{\max I_p}' X_{\max I_p}'' X_{\min I_1}^m}^{n + m}, \\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0, \, n < m,
\\ %
& S_{p, (n, \gamma), (m - 1, \max I_0)} = \overline{Y_\gamma X_{\min
I_1}^n X_{\max I_p}' X_{\max I_p}'' X_{\min I_1}^m}^{n + m + 1},
\\ %
& \qquad p \in [1, r], \, \gamma \in I_0', \, n, m \in \mathbb{N}_0, \,
n < m,
\\ %
& S_{p, (n - 1, \max I_0), (m, \gamma)} = \overline{X_{\min I_1}^n
X_{\max I_p}' X_{\max I_p}'' X_{\min I_1}^m Y_\gamma}^{n + m + 1},
\\ %
& \qquad p \in [1, r], \, \gamma \in I_0', \, n, m \in \mathbb{N}_0, \,
n \leq m,
\\ %
& S_{p, (n, \gamma'), (m, \gamma'')} = \overline{Y_{\gamma''} X_{\min
I_1}^n X_{\max I_p}' X_{\max I_p}'' X_{\min I_1}^m
Y_{\gamma'}}^{n + m + 2}, \\ %
& \qquad p \in [1, r], \, \gamma, \gamma' \in I_0', \, n, m \in
\mathbb{N}_0, \, (n, \gamma') < (m, \gamma''),
\\ %
& S_{p, (n - 1, \max I_0), (n - 1, \max I_0)}' = \overline{X_{\min
I_1}^n X_{\max I_p}'}^n, \, p \in [1, r], \, n \in \mathbb{N}_0,
\\ %
& S_{p, (n, \gamma), (n, \gamma)}' = \overline{Y_\gamma X_{\min I_1}^n
X_{\max I_p}'}^{n + 1}, \, p \in [1, r], \, \gamma \in I_0', \, n
\in \mathbb{N}_0,
\\ %
& S_{p, (n - 1, \max I_0), (n - 1, \max I_0)}'' = \overline{X_{\min
I_1}^n X_{\max I_p}''}^n, \, p \in [1, r], \, n \in \mathbb{N}_0,
\\ %
& S_{p, (n, \gamma), (n, \gamma)}'' = \overline{Y_\gamma X_{\min I_1}^n
X_{\max I_p}''}^{n + 1}, \, p \in [1, r], \, \gamma \in I_0', \, n
\in \mathbb{N}_0,
\\ %
& T_{p, \gamma, (m - 1, \max I_0)} = \overline{X_{\min I_1}^m
X_\gamma}^m, \, p \in [1, r + 1], \, \gamma \in I_p'', \, m \in
\mathbb{N}_0,
\\ %
& T_{p, \gamma', (m, \gamma'')} = \overline{Y_{\gamma''} X_{\min I_1}^m
X_{\gamma'}}^{m + 1}, \, p \in [1, r + 1], \, \gamma' \in I_p'',
\, \gamma'' \in I_0', \, m \in \mathbb{N}_0,
\\ %
& T_{r + 1, \max I_{r + 1}, (m - 1, \max I_0)} = \overline{X_{\min
I_1}^m}^m (0), \, m \in \mathbb{N},
\\ %
& T_{r + 1, \max I_{r + 1}, (m, \gamma)} = \overline{Y_\gamma X_{\min
I_1}^m}^{m + 1}, \, \gamma \in I_0', \, m \in \mathbb{N}_0,
\\ %
& U_{p, 2 n, (m - 1, \max I_0)} = \overline{X_{\min I_1}^m X_{\max I_p}'
X_{\max I_p}'' X_{\min I_1}^n}^{n + m + 1},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0,
\\ %
& U_{p, 2 n, (m, \gamma)} = \overline{Y_\gamma X_{\min I_1}^m X_{\max
I_p}' X_{\max I_p}'' X_{\min I_1}^n}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, \gamma \in I_0', \, n, m \in \mathbb{N}_0,
\\ %
& U_{p, 2 n + 1, (m - 1, \max I_0)} = \overline{X_{\min I_1}^m X_{\max
I_p}' X_{\max I_p}'' X_{\min I_1}^n Z}^{n + m + 1},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0,
\\ %
& U_{p, 2 n + 1, (m, \gamma)} = \overline{Y_\gamma X_{\min I_1}^m
X_{\max I_p}' X_{\max I_p}'' X_{\min I_1}^n Z}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, \gamma \in I_0', \, n, m \in \mathbb{N}_0,
\\ %
& V_{p, 2 n, \gamma} = \overline{X_{\min I_1}^n X_\gamma}^{n + 1}, \, p
\in [1, r + 1], \, \gamma \in I_p'', \, n \in \mathbb{N}_0,
\\ %
& V_{p, 2 n + 1, \gamma} = \overline{X_{\min I_1}^n X_\gamma Z}^{n + 1},
\, p \in [1, r + 1], \, \gamma \in I_p'', \, n \in \mathbb{N}_0,
\\ %
& V_{r + 1, 2 n, \max I_{r + 1}} = \overline{X_{\min I_1}^n}^{n + 1}, \,
n \in \mathbb{N}_0,
\\ %
& V_{r + 1, 2 n + 1, \max I_{r + 1}} = \overline{X_{\min I_1}^n Z}^{n +
1}, \, n \in \mathbb{N}_0,
\\ %
& W_{p, 2 n, 2 m} = \overline{X_{\min I_1}^n X_{\max I_p}' X_{\max
I_p}'' X_{\min I_1}^m}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0, \, m < n,
\\ %
& W_{p, 2 n + 1, 2 m} = \overline{Z X_{\min I_1}^n X_{\max I_p}' X_{\max
I_p}'' X_{\min I_1}^m}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0, \, m \leq n,
\\ %
& W_{p, 2 n, 2 m + 1} = \overline{X_{\min I_1}^n X_{\max I_p}' X_{\max
I_p}'' X_{\min I_1}^m Z}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0, \, m < n,
\\ %
& W_{p, 2 n + 1, 2 m + 1} = \overline{Z X_{\min I_1}^n X_{\max I_p}'
X_{\max I_p}'' X_{\min I_1}^m Z}^{n + m + 2},
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}_0, \, m < n,
\\ %
& W_{p, 2 n, 2 n}' = \overline{X_{\min I_1}^n X_{\max I_p}'}^{n + 1}, \,
p \in [1, r], \, n \in \mathbb{N}_0,
\\ %
& W_{p, 2 n + 1, 2 n + 1}' = \overline{X_{\min I_1}^n X_{\max I_p}'
Z}^{n + 1}, \, p \in [1, r], \, n \in \mathbb{N}_0,
\\ %
& W_{p, 2 n, 2 n}'' = \overline{X_{\min I_1}^n X_{\max I_p}''}^{n + 1},
\, p \in [1, r], \, n \in \mathbb{N}_0,
\\ %
& W_{p, 2 n + 1, 2 n + 1}'' = \overline{X_{\min I_1}^n X_{\max I_p}''
Z}^{n + 1}, \, p \in [1, r], \, n \in \mathbb{N}_0,
\end{align*}
form a complete list of indecomposable objects in $\mathcal{U}
(\mathbb{L}_{I_0, \ldots, I_{r + 1}})$.
\item
Sequences
\begin{align*}
& 0 \to M_{\gamma', \gamma''} \to M_{\gamma'_+, \gamma''} \oplus
M_{\gamma', \gamma''_+} \to M_{\gamma'_+, \gamma''_+} \to 0,
\\ %
& \qquad \gamma', \gamma'' \in \mathbb{Z} \times I_0, \, (-1, \max I_0)
\leq \gamma' < \gamma'' < (\gamma')^+,
\\ %
& 0 \to M_{\gamma, \gamma}' \to M_{\gamma, \gamma_+} \to
M_{\gamma_+, \gamma_+}'' \to 0, \, \gamma \in \mathbb{N}_0 \times I_0,
\\ %
& 0 \to M_{\gamma, \gamma}'' \to M_{\gamma, \gamma_+} \to
M_{\gamma_+, \gamma_+}' \to 0, \, \gamma \in \mathbb{N}_0 \times I_0,
\\ %
& 0 \to M_{\gamma, \gamma^+}' \to M_{\gamma_+, \gamma^+} \to
M_{\gamma_+, \gamma_+^+}'' \to 0, \, \gamma \in \mathbb{Z} \times I_0,
\, (-1, \max I_0) \leq \gamma,
\\ %
& 0 \to M_{\gamma, \gamma^+}'' \to M_{\gamma_+, \gamma^+} \to
M_{\gamma_+, \gamma_+^+}' \to 0, \, \gamma \in \mathbb{Z} \times I_0, \,
(-1, \max I_0) \leq \gamma,
\\ %
& 0 \to R_n^\lambda \to R_{n + 1}^\lambda \oplus R_{n - 1}^\lambda
\to R_n^\lambda \to 0, \, \lambda \in k^*, \, \lambda \neq 1, \, n
\in \mathbb{N},
\\ %
& 0 \to R_{n + 1}^1 \to R_{n + 2}^1 \oplus R_{n - 1}^1 \to R_n^1
\to 0, \, n \in \mathbb{N},
\\ %
& 0 \to R_{n, i}^\infty \to R_{n + 1, i}^\infty \oplus R_{n - 1, i
+ 1}^\infty \to R_{n, i + 1}^\infty \to 0, \, i \in \mathbb{Z}_2, \, n
\in \mathbb{N},
\\ %
& 0 \to S_{p, \gamma', \gamma''} \to S_{p, \gamma'_+, \gamma''}
\oplus S_{p, \gamma', \gamma''_+} \to S_{p, \gamma'_+, \gamma''_+}
\to 0,
\\ %
& \qquad p \in [1, r], \, \gamma', \gamma'' \in \mathbb{Z} \times I_0,
\, (-1, \max I_0) \leq \gamma' < \gamma'',
\\ %
& 0 \to S_{p, \gamma, \gamma}' \to S_{p, \gamma, \gamma_+} \to
S_{p, \gamma_+, \gamma_+}'' \to 0,
\\ %
& \qquad p \in [1, r], \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max
I_0) \leq \gamma,
\\ %
& 0 \to S_{p, \gamma, \gamma}'' \to S_{p, \gamma, \gamma_+} \to
S_{p, \gamma_+, \gamma_+}' \to 0,
\\ %
& \qquad p \in [1, r], \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max
I_0) \leq \gamma,
\\ %
& 0 \to T_{r + 1, \max I_{r + 1}, (0, \max I_0')} \to T_{r + 1,
\max I_{r + 1}, (0 \max I_0)}
\\ %
& \qquad \to T_{1, \min I_1'', (0, \max I_0)} \to 0,
\\ %
& 0 \to T_{p, \gamma', \gamma''} \to T_{p, \gamma'_+, \gamma''}
\oplus T_{p, \gamma', \gamma''_+} \to T_{p, \gamma'_+, \gamma''_+}
\to 0, \, p \in [1, r + 1],
\\ %
& \qquad \gamma' \in (I_p'')_-,\, \gamma'' \in \mathbb{Z} \times I_0, \,
(-1, \max I_0) \leq \gamma'',
\\ %
& 0 \to U_{p, n, \gamma} \to U_{p, n, \gamma_+} \oplus U_{p, n -
1, \gamma} \to U_{p, n - 1, \gamma_+} \to 0,
\\ %
& \qquad p \in [1, r], \, n \in \mathbb{N}, \, \gamma \in \mathbb{N}_0 \times
I_0,
\\ %
& 0 \to V_{p, n, \gamma} \to V_{p, n, \gamma_+} \oplus V_{p, n -
1, \gamma} \to V_{p, n - 1, \gamma_+} \to 0,
\\ %
& \qquad p \in [1, r + 1], \, n \in \mathbb{N}, \, \gamma \in (I_p'')_-,
\\ %
& 0 \to W_{p, n, m} \to W_{p, n - 1, m} \oplus W_{p, n, m - 1} \to
W_{p, n - 1, m - 1} \to 0,
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}, m < n,
\\ %
& 0 \to W_{p, n, n}' \to W_{p, n, n - 1} \to W_{p, n - 1, n - 1}''
\to 0, \, p \in [1, r], \, n \in \mathbb{N},
\\ %
& 0 \to W_{p, n, n}'' \to W_{p, n, n - 1} \to W_{p, n - 1, n - 1}'
\to 0, \, p \in [1, r], \, n \in \mathbb{N},
\end{align*}
form a complete list of Auslander--Reiten sequences in the
category $\mathcal{U} (\mathbb{L}_{I_0, \ldots, I_{r + 1}})$, where
\begin{align*}
& M_{\gamma, \gamma} = M_{\gamma, \gamma}' \oplus M_{\gamma,
\gamma}'', \, \gamma \in \mathbb{N}_0 \times I_0,
\\ %
& M_{\gamma, \gamma^+} = M_{\gamma, \gamma^+}' \oplus M_{\gamma,
\gamma^+}'', \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max I_0) \leq
\gamma,
\\ %
& R_0^\lambda = 0, \, \lambda \in k^*,
\\ %
& R_{0, i}^\infty = 0, \, i \in \mathbb{Z}_2,
\\ %
& S_{\gamma, \gamma} = S_{\gamma, \gamma}' \oplus S_{\gamma,
\gamma}'', \, \gamma \in \mathbb{N}_0 \times I_0,
\\
& T_{1, \min (I_1'')_-, \gamma} = T_{r + 1, \max I_{r + 1},
\gamma^+}, \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max I_0) \leq
\gamma,
\\ %
& T_{p, \min (I_p'')_-, \gamma} = U_{p - 1, 0, \gamma}, \, p \in
[2, r + 1], \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max I_0) \leq
\gamma,
\\ %
& T_{p, \max I_p, \gamma} = S_{p, (-1, \max I_0), \gamma}, \, p
\in [1, r], \, \gamma \in \mathbb{Z} \times I_0, \, (-1, \max I_0) \leq
\gamma,
\\ %
& T_{r + 1, \max I_{r + 1}, (-1, \max I_0)} = 0,
\\ %
& V_{1, n, \min (I_1'')_-} = V_{r + 1, n + 2, \max I_{r + 1}}, \,
n \in \mathbb{N}_0,
\\ %
& V_{p, n, \min (I_p'')_-} = W_{p - 1, n, 0}, \, p \in [2, r + 1],
\, n \in \mathbb{N}_0,
\\ %
& V_{p, n, \max I_p} = U_{p, n, (-1, \max I_0)}, \, p \in [1, r],
\, n \in \mathbb{N}_0,
\\ %
& W_{p, n, n} = W_{p, n, n}' \oplus W_{p, n, n}'', \, p \in [1,
r], \, n \in \mathbb{N}_0.
\end{align*}
\end{enumerate}
\end{prop*}
\section{Proof of the main result} \label{sectproof}
In this section we present the proof of the main theorem of the
paper.
\subsection{}
Let $A$ be an algebra and let $R$ be an $A$-module. By $A [R]$ we
denote the one-point extension of $A$ by $R$ defined as
\[
\begin{bmatrix}
A & R \\ 0 & k
\end{bmatrix}.
\]
The category of $A [R]$-modules is equivalent to the category of
triples $(V_0, V_1, \gamma_V)$, with $V_0 \in \mod A$, $V_1 \in
\mod k$ and $\gamma_V : V_1 \to \Hom_A (R, V_0)$ is a $k$-linear
map (see \cite{Ri2}*{2.5(8)}).
Let $\Hom (R, \mod A)$ be the vector space category $(\mathcal{K},
{|-|})$, where $\mathcal{K} = \mod A / \Ker \Hom_A (R, -)$ and ${|-|} :
\mathcal{K} \to \mod k$ is the functor induced by $\Hom_A (R, -)$. It
follows from the above remark that we may view the objects of
$\mathcal{U} (\Hom (R, \mod A))$ as objects of $\mod A [R]$.
Consequently, if $X$ is an indecomposable $A [R]$-module then
either $X \in \mod A$ or $X \in \mathcal{U} (\Hom (R, \mod A))$.
Moreover, each Auslander--Reiten sequence in $\mod A [R]$ is
either of the form
\begin{multline*}
0 \to (X, \Hom_A (R, X), \Hom_A (R, \Id_X))
\\ %
\to (Y, \Hom_A (R, X), \Hom_A (R, f)) \to (Z, 0, 0) \to 0
\end{multline*}
for an Auslander--Reiten sequence $0 \to X \xrightarrow{f} Y \to Z
\to 0$ in $\mod A$, or comes from an Auslander--Reiten sequence in
$\mathcal{U} (\Hom (R, \mod A))$.
\subsection{}
From now on we assume that $(p, q, S, T)$ is a fixed defining
system. We also use notation introduced in Section~\ref{mainres}.
A vertex $x$ of $Q$ is called admissible if one of the following
possibilities holds:
\begin{itemize}
\item
$x = x_{i, j}$, $i \in [1, |p|]$, $j \in [2, p_i + |T_i|]$, $j -
1, j, j + 1 \not \in S_i$,
\item
$x = z_{i, j}$, $i \in [1, |p|]$, $j \in S_i \cap [T_{i, |T_i|} +
2, p_i + |T_i|]$.
\end{itemize}
For an admissible vertex $x$ of $Q$ we define a defining system
$(p, q, S^x, T^x)$ by:
\begin{itemize}
\item
if $x = x_{i_0, j_0}$, then
\[
S_i^x =
\begin{cases}
S_{i_0} \cup \{ j_0 \} & i = i_0,
\\ %
S_i & i \neq i_0,
\end{cases}
\text{ and } T^x = T,
\]
\item
if $x = z_{i_0, j_0}$, then
\[
S^x = S \text{ and } T_i^x =
\begin{cases}
T_{i_0} \cup \{ j_0 \} & i = i_0,
\\ %
T_i & i \neq i_0.
\end{cases}
\]
\end{itemize}
A defining system $(p, q, S, T)$ if called fundamental if $S_i =
\varnothing = T_i$ for all $i$. The following observation allows
us to perform inductive proofs: each defining system is an
iterated extension of a fundamental one by admissible vertices.
\subsection{} \label{sectlemm}
For $x \in Q_0$, $x = x_{i, j}$, let $X_x = M (\mu_x)$, $I_x = M
(\omega_x)$ and $R_x = M (\alpha_x \mu_x)$. Similarly, if $x =
z_{i, j}$, then $X_x = M (\gamma_y \mu_y)$ and $R_x = N (\mu_y,
\omega_y)$, where $y = x_{i, j}$.
For a vertex $x$ of $Q$ let $\mathcal{C}_x$ denote the set of all strings
terminating at $x$ ordered by the relation introduced
in~\ref{sectord}. Recall that $\mathcal{C}_x' = \mathcal{C}_x \setminus \{
\omega_x \}$. We prove the main theorem inductively together with
the following series of lemmas.
\begin{lemm} \label{lemmR}
Let $x$ be an admissible vertex of $Q$.
\begin{enumerate}
\item
If $x = x_{i_0, j_0}$, then the assignment
\begin{align*}
X_C & \mapsto M (\alpha_x C), \, C \in \mathcal{C}_x,
\\ %
Y_C & \mapsto M (C), \, C \in \mathcal{C}_x,
\end{align*}
induces an equivalence between $\mathbb{L}_{\mathcal{C}_x}$ and $\Hom (R_x,
\mod A)$.
\item
If $x = z_{i_0, j_0}$, let $\{ j_1 < \cdots < j_r \} = S_{i_0}
\cap [j_0 + 1, p_{i_0} + |T_{i_0}|]$ and $j_{r + 1} = p_{i_0} +
|T_{i_0}| + 1$. The assignment
\begin{align*}
X_C & \mapsto N (C, \omega_{x_{i_0, j_p}}), \, C \in
\mathcal{C}_{x_{i_0, j_p}}', \, p \in [0, r],
\\ %
X_{j_p} & \mapsto M (\gamma_{i_0, j_p} \omega_{x_{i_0, j_p}}), \,
p \in [0, r],
\\ %
X_j & \mapsto M (\omega_{x_{i_0, j}}), \, j \in [j_p + 1, \ldots,
j_{p + 1} - 1], \, p \in [0, r],
\\ %
X_{\omega_{x_{i_0, j_p}}}' & \mapsto M (\omega_{x_{i_0, j_p}}), \,
p \in [0, r],
\\ %
X_{\omega_{x_{i_0, j_p}}}'' & \mapsto N (\omega_{x_{i_0, j_p}}),
\, p \in [0, r],
\\ %
Y_C & \mapsto M (\gamma_{i_0, j_0} C), \, C \in \mathcal{C}_{x_{i_0,
j_0}}',
\\ %
Z & \mapsto M (x),
\end{align*}
induces an equivalence between
\[
\mathbb{L}_{\mathcal{C}_{x_{i_0, j_0}}, [j_0, j_1 - 1] + \mathcal{C}_{x_{i_0, j_1}},
\ldots, [j_{r - 1}, j_r - 1] + \mathcal{C}_{x_{i_0, j_r}}, [j_r, j_{r +
1}]} \text{ and } \Hom (R_x, \mod A).
\]
\end{enumerate}
\end{lemm}
\begin{lemm} \label{lemmX}
Let $x$ be an admissible vertex of $Q$. The assignment
\[
X_C \mapsto M (C), \, C \in \mathcal{C}_x,
\]
induces an equivalence between $\mathbb{K}_{\mathcal{C}_x}$ and $\Hom (X_x,
\mod A)$.
\end{lemm}
\begin{lemm} \label{lemmI}
Let $x = x_{i_0, j_0}$ be such that $j_0 \in [T_{i_0, |T_{i_0}|} +
1, p_{i_0} + |T_{i_0}|] \setminus S_{i_0}$. Let $\{ j_1 < \cdots <
j_r \} = S_{i_0} \cap [j_0 + 1, p_{i_0} + |T_{i_0}|]$ and $j_{r +
1} = p_{i_0} + |T_{i_0}| + 1$. The assignment
\begin{align*}
X_C & \mapsto N (C, \omega_{x_{i_0, j_p}}), \, C \in
\mathcal{C}_{x_{i_0, j_p}}', \, p \in [1, r],
\\ %
X_{j_0} & \mapsto M (\omega_{x_{i_0, j_0}}),
\\ %
X_{j_p} & \mapsto M (\gamma_{i_0, j_p} \omega_{x_{i_0, j_p}}), \,
p \in [1, r],
\\ %
X_j & \mapsto M (\omega_{x_{i_0, j}}), \, j \in [j_p + 1, j_{p
+ 1} - 1], \, p \in [0, r],
\\ %
X_{\omega_{x_{i_0, j_p}}}' & \mapsto M (\omega_{x_{i_0, j_p}}), \,
p \in [1, r],
\\ %
X_{\omega_{x_{i_0, j_p}}}'' & \mapsto N (\omega_{x_{i_0, j_p}}),
\, p \in [1, r],
\end{align*}
induces an equivalence between
\[
\mathbb{K}_{[j_0, j_1 - 1] + \mathcal{C}_{x_{i_0, j_1}}, \ldots, [j_{r - 1},
j_r - 1] + \mathcal{C}_{x_{i_0, j_r}}, [j_r, j_{r + 1} - 1]} \text{ and
} \Hom (I_x, \mod A).
\]
\end{lemm}
\subsection{}
If $(p, q, S, T)$ is a fundamental defining system, then
Theorem~\ref{maintheo} and Lemmas~\ref{sectlemm} are easy
exercises in the representation theory of a hereditary algebra of
type $\tilde{\mathbb{A}}_{p, q}$.
From now on we assume that we Theorem~\ref{maintheo} and
Lemmas~\ref{sectlemm} have been proved for $(p, q, S, T)$. Let $x$
be an admissible vertex of $Q$. We will show that
Theorem~\ref{maintheo} and Lemmas~\ref{sectlemm} hold for $(p, q,
S^x, T^x)$.
By $Q^x$ (respectively, $A^x$) we will denote the quiver (algebra)
associated with $(p, q, S^x, T^x)$. We also define $R_{x'}^x$,
$X_{x'}^x$ and $I_{x'}^x$ in the analogous way as the
corresponding modules for $(p, q, S, T)$.
\subsection{}
Assume first that $x = x_{i_0, j_0}$. Let $\gamma = \gamma_{i_0,
j_0}$ be the new arrow of $Q^x$ and $z = z_{i_0, j_0}$ be the new
vertex of $Q^x$. Theorem~\ref{maintheo} for $(p, q, S^x, T^x)$
follows from the induction hypothesis (Theorem~\ref{maintheo} and
Lemma~\ref{lemmR} for $(p, q, S, T)$), Proposition~\ref{subspaceone}
and the following isomorphisms
\begin{align*}
& \overline{M (\alpha_x C)} \simeq N (C), \, C \in \mathcal{C}_x,
\\ %
& \overline{M (C)} \simeq M (\gamma C), \, C \in \mathcal{C}_x,
\\ %
& \overline{M (C') M (\alpha_x C'')} \simeq N (C', C''), \, C', C'' \in
\mathcal{C}_x, \, C' < C'',
\\ %
& \overline{0} \simeq M (z).
\end{align*}
\subsection{}
Now we prove Lemma~\ref{lemmR} for $(p, q, S^x, T^x)$. Let $x'$ be
an admissible vertex of $Q^x$. Then either $x'$ is an admissible
vertex of $Q$ or $x' = z$. In the first case there are still two
possibilities: either $x' = x_{i, j}$ or $x' = z_{i, j}$.
Consider first the case $x' = x_{i, j}$. Then either $i \neq i_0$ or
$i = i_0$ and $|j - j_0| > 1$, hence it is easily seen that in this
case we also have $R_{x'}^x = R_{x'}$ and $\Hom (R_{x'}^x, \mod A^x)
= \Hom (R_{x'}, \mod A)$, thus the claim follows.
Let now $x' = z_{i, j}$ for $(i, j) \neq (i_0, j_0)$. In this case
also $R_{x'}^x = R_{x'}$. Moreover, if $i \neq i_0$ or $i = i_0$ and
$j_0 < j$, then $\Hom (R_{x'}^x, \mod A^x) = \Hom (R_{x'}, \mod A)$.
If $j < j_0$, then the claim about $\Hom (R_{x'}^x, \mod A^x)$
follows by observing that its indecomposable objects are the
indecomposable objects of $\Hom (R_{x'}, \mod A)$ and
\[
\overline{M (C) M (\alpha_x \omega_x)}, \, C \in \mathcal{C}_x', \, \overline{M
(\alpha_x \omega_x)}, \, \overline{M (\omega_x)}.
\]
Finally, let $x' = z$. Then $R_{x'}^x = \overline{I_x X_x}$ and the
claim follows from Lemmas~\ref{lemmX} and \ref{lemmI}.
\subsection{}
In order to show Lemma~\ref{lemmX} we have to consider the cases
analogous to the ones considered above. If $x' = x_{i, j}$ or $x' =
z_{i, j}$, $x' \neq z$, then $X_{x'}^x = X_{x'}$ and $\Hom
(X_{x'}^x, \mod A^x) = \Hom (X_{x'}, \mod A)$. Thus it remains to
consider the case $x' = z$. In this case $X_{x'}^x = \overline{X_x}$ and
the description of $\Hom (X_{x'}^x, \mod A^x)$ follows easily from
the description of $\Hom (X_x, \mod A)$.
\subsection{}
It remains to show Lemma~\ref{lemmI}. Let $x' = x_{i, j}$ be the
vertex of $Q^x$ satisfying the hypothesis of Lemma~\ref{lemmI}. We
have $I_{x'}^x = I_{x'}$. If $i \neq i_0$ or $i = i_0$ and $j_0 <
j$, then also $\Hom (I_{x'}^x, \mod A^x) = \Hom (I_{x'}, \mod A)$.
If $j < j_0$, then we have observed that indecomposable objects of
$\Hom (I_{x'}^x, \mod A^x)$ are the indecomposable objects of $\Hom
(I_{x'}, \mod A)$ and
\[
\overline{M (C) M (\alpha_x \omega_x)}, \, C \in \mathcal{C}_x', \, \overline{M
(\alpha_x \omega_x)}, \, \overline{M (\omega_x)}.
\]
\subsection{}
Assume now that $x = z_{i_0, j_0}$. Let $\{ j_1 < \cdots < j_r \}
= S_{i_0} \cap [j + 1, p_{i_0} + |T_{i_0}|]$ and $j_{r + 1} =
p_{i_0} + |T_{i_0}| + 1$. Put
\begin{align*}
\mathcal{C} & = \mathcal{C}_{x_{i_0, j_0}}, &
\mathcal{C}_p & = \mathcal{C}_{x_{i_0, j_p}}, \, p \in [1, r],
\\ %
\gamma & = \gamma_{i_0, j_0}, &
\gamma_p & = \gamma_{i_0, j_p}, \, p \in [1, r],
\\ %
\omega & = \omega_{x_{i_0, j_0}}, &
\omega_j & = \omega_{x_{i_0, j}}, \, j \in [j + 1, p_{i_0} +
|T_{i_0}|],
\\ %
\alpha & = \alpha_{i_0, p_{i_0} + |T_{i_0}| + 1}, &
\xi & = \xi_{i_0, |T_{i_0}| + 1},
\\ %
\intertext{and} %
B & = \omega \alpha \xi \gamma, &
B_j & = \omega_j \alpha \xi \gamma, \, j \in [j + 1, p_{i_0} +
|T_{i_0}|].
\end{align*}
Finally, let $z = x_{i_0, j_{r + 1}}$.
\subsection{}
In this case Theorem~\ref{maintheo} for $(p, q, S^x, T^x)$ follows
from the induction hypothesis, Proposition~\ref{propLIr} and the
following isomorphisms
\begin{align*}
& \overline{M (\gamma C') N (C'', \omega)} \simeq N (C'', B C'), \, C',
C'' \in \mathcal{C}', \, C' < C'',
\\ %
& \overline{M (\gamma C) M (\omega) N (\omega) M (\gamma \omega)^{2 n -
1}}^{2 n} \simeq N (B^n C, B^n \omega), \, C \in \mathcal{C}', \, n \in
\mathbb{N},
\\ %
& \overline{M (\gamma C) M (\omega) N (\omega) M (\gamma \omega)^{2 n}}^{2
n + 1} \simeq N (B^n \omega, B^{n + 1} C), \, C \in \mathcal{C}', \, n \in
\mathbb{N},
\\ %
& \overline{M (\gamma C'') M (\gamma C') M (\omega) N (\omega) M (\gamma
\omega)^{2 n}}^{2 n + 2} \simeq N (B^{n + 1} C', B^{n + 1} C''),
\\ %
& \qquad C', C'' \in \mathcal{C}', \, C' < C'', \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma C') M (\gamma C'') M (\omega) N (\omega) M (\gamma
\omega)^{2 n - 1}}^{2 n + 1} \simeq N (B^n C'', B^{n + 1} C'),
\\ %
& \qquad C', C'' \in \mathcal{C}', \, C' < C'', \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^n}^n \simeq M (\gamma B^n C),
\, C \in \mathcal{C}', \, \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^{n + 1}}^n \simeq M (\gamma B^n \omega), \,
n \in \mathbb{N},
\\ %
& \overline{N (C, \omega)} \simeq L (B C), \, C \in \mathcal{C}',
\\ %
& \overline{M (\omega) N (\omega) M (\gamma \omega)^n}^{n + 1} \simeq L
(B^{n + 1} \omega), \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma C) M (\omega) N (\omega) M (\gamma \omega)^{n -
1}}^{n + 1} \simeq L (B^{n + 1} C), C \in \mathcal{C}', \, n \in \mathbb{N},
\\ %
& \overline{M (\omega) M (\gamma \omega)^n}^n \simeq M (B^n \omega), \, n
\in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\omega) M (\gamma \omega)^n}^{n + 1} \simeq M
(B^{n + 1} C), \, C \in \mathcal{C}', \, n \in \mathbb{N}_0,
\\ %
& \overline{N (\omega) M (\gamma \omega)^n}^n \simeq N (B^n \omega), \, n
\in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\omega) M (\gamma \omega)^n}^{n + 1} \simeq N
(B^{n + 1} C), \, C \in \mathcal{C}', \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^n}^n (\lambda) \simeq R (B, \lambda, n), \,
n \in \mathbb{N}, \, \lambda \in k^*,
\\ %
& \overline{N (\omega) M (\gamma \omega)^n}^{n + 1} \simeq Q (B^{n + 1}),
\, n \in \mathbb{N}_0,
\\ %
& \overline{M (\omega) M (\gamma \omega)^n}^{n + 1} \simeq M (B^n \omega
\alpha), \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^n M (x)}^n \simeq M (\gamma B^{n - 1} \omega
\alpha \xi), \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^n}^n (\infty) \simeq M (\gamma B^{n - 1}
\omega \alpha), \, n \in \mathbb{N},
\\ %
& \overline{M (\omega) M (\gamma \omega)^n M (x)}^{n + 1} \simeq M (B^n
\omega \alpha \xi), \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p}) M (\gamma \omega)^m}^{m}
\simeq N (\omega_{j_p}, B_{j_p} B^{m - 1} \omega), \, p \in [1, r],
\, m \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m}^{m + n} \simeq N (B_{j_p} B^{n - 1} \omega,
B_{j_p} B^{m -
1} \omega),\\ %
& \qquad \, p \in [1, r], \, n, m \in \mathbb{N}, \, n < m,
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^n M (\omega_{j_p}) N
(\omega_{j_p}) M (\gamma \omega)^m}^{m + n + 1}
\\ %
& \qquad \simeq N (B_{j_p} B^n C, B_{j_p} B^{m - 1} \omega), \, p
\in [1, r], \, C \in \mathcal{C}', \, n, m \in \mathbb{N}_0, \, n < m,
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p}) M (\gamma \omega)^m M
(\gamma C)}^{m + 1} \simeq N (\omega_{j_p}, B_{j_p} B^m C),
\\ %
& \qquad p \in [1, r], \, C \in \mathcal{C}', \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m M (\gamma C)}^{m + n + 1}
\\ %
& \qquad \simeq N (B_{j_p} B^{n - 1} \omega, B_{j_p} B^m C), \, p
\in [1, r], \, C \in \mathcal{C}', \, n, m \in \mathbb{N}, \, n \leq m,
\\ %
& \overline{M (\gamma C') M (\gamma \omega)^n M (\omega_{j_p}) N
(\omega_{j_p}) M (\gamma \omega)^m M (\gamma C'')}^{m + n + 1}
\\ %
& \qquad \simeq N (B_{j_p} B^n C', B_{j_p} B^m C''),
\\ %
& \qquad p \in [1, r], \, C', C'' \in \mathcal{C}', \, n, m \in \mathbb{N}_0, \,
(n, C') < (m, C''),
\\ %
& \overline{M (\gamma \omega)^n M (\omega_j)}^n \simeq M (B_j B^{n - 1}
\omega), \, j \in [j_0 + 1, j_{r + 1} - 1], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^n M (\omega_j)}^{n + 1} \simeq
M (B_j B^n C), \, j \in [j_0 + 1, j_{r + 1} - 1], \, n \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^n N (\omega_{j_p})}^n \simeq N (B_{j_p} B^{n
- 1} \omega), \, p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^n N (\omega_{j_p})}^{n + 1}
\simeq N (B_{j_p} B^n C), \, p \in [1, r], \, C \in \mathcal{C}', \, n \in
\mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^m M (\gamma_p \omega_{j_p})}^m \simeq M
(\gamma_p B_{j_p} B^{m - 1} \omega), \, p \in [1, r], \, m \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^m N (C, \omega_{j_p})}^m \simeq N (C,
B_{j_p} B^{m - 1} \omega), \, C \in \mathcal{C}_p', \, p \in [1, r], \, m
\in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m M (\gamma_p \omega_{j_p})}^{m
+ 1} \simeq M (\gamma_p B_{j_p} B^m C),
\\ %
& \qquad C \in \mathcal{C}', \, p \in [1, r], \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma C') M (\gamma \omega)^m N (C'', \omega_{j_p})}^{m +
1} \simeq N (C'', B_{j_p} B^m C'),
\\ %
& \qquad C' \in \mathcal{C}', \, C'' \in \mathcal{C}_p', \, p \in [1, r], \, m
\in \mathbb{N}_0,
\\ %
& \overline{M (\gamma \omega)^m}^m (0) \simeq M (\xi \gamma B^{m - 1}
\omega), \, m \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m}^{m + 1} \simeq M (\xi \gamma
B^m C), \, C \in \mathcal{C}', \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p})}^1 \simeq N (\omega_{j_p},
\omega_{j_p} \alpha), \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^m M (\omega_{j_p}) N (\omega_{j_p})}^{m + 1}
\simeq N (B_{j_p} B^{m - 1} \omega, \omega_{j_p} \alpha), \, p \in
[1, r], \, m \in \mathbb{N},
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p}) M (\gamma \omega)^n}^{n + 1}
\simeq N (\omega_{j_p}, B_{j_p} B^{n - 1} \omega \alpha), \, p \in
[1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^m M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^n}^{n + m + 1} \simeq N (B_{j_p} B^{m - 1} \omega,
B_{j_p} B^{n - 1} \omega \alpha),
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m M (\omega_{j_p}) N
(\omega_{j_p})}^{m + 2} \simeq N (B_{j_p} B^m C, \omega_{j_p}
\alpha),
\\ %
& \qquad C \in \mathcal{C}', \, p \in [1, r], \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m M (\omega_{j_p}) N
(\omega_{j_p}) M (\gamma \omega)^n}^{n + m + 2}
\\ %
& \qquad \simeq N (B_{j_p} B^m C, B_{j_p} B^{n - 1} \omega \alpha),
\, C \in \mathcal{C}', \, p \in [1, r], \, n \in \mathbb{N}, \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p}) M (x)}^1 \simeq N
(\omega_{j_p}, \omega_{j_p} \alpha \xi), \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^m M (\omega_{j_p}) N (\omega_{j_p}) M
(x)}^{m + 1} \simeq N (B_{j_p} B^{m - 1} \omega, \omega_{j_p} \alpha
\xi),
\\ %
& \qquad p \in [1, r], \, m \in \mathbb{N},
\\ %
& \overline{M (\omega_{j_p}) N (\omega_{j_p}) M (\gamma \omega)^n M
(x)}^{n + 1} \simeq N (\omega_{j_p}, B_{j_p} B^{n - 1} \omega \alpha
\xi),
\\ %
& \qquad p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^m M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^n M (x)}^{n + m + 1}
\\ %
& \qquad \simeq N (B_{j_p} B^{m - 1} \omega, B_{j_p} B^{n - 1}
\omega \alpha \xi), \, p \in [1, r], \, n, m \in \mathbb{N},
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m M (\omega_{j_p}) N
(\omega_{j_p}) M (x)}^{m + 2} \simeq N (B_{j_p} B^m C, \omega_{j_p}
\alpha \xi),
\\ %
& \qquad C \in \mathcal{C}', \, p \in [1, r], \, m \in \mathbb{N}_0,
\\ %
& \overline{M (\gamma C) M (\gamma \omega)^m M (\omega_{j_p}) N
(\omega_{j_p}) M (\gamma \omega)^n M (x)}^{n + m + 2}
\\ %
& \qquad \simeq N (B_{j_p} B^m C, B_{j_p} B^{n - 1} \omega \alpha
\xi), \, C \in \mathcal{C}', \, p \in [1, r], \, n \in \mathbb{N}, \, m \in
\mathbb{N}_0,
\\ %
& \overline{M (\gamma_p \omega_{j_p})}^1 \simeq M (\gamma_p \omega_{j_p}
\alpha), \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p})}^{n + 1} \simeq
M (\gamma_p B_{j_p} B^{n - 1} \omega \alpha), \, p \in [1, r], \, n
\in \mathbb{N},
\\ %
& \overline{M (\omega_j)}^1 \simeq M (\omega_j \alpha), \, j \in [j_0 + 1,
j_{r + 1} - 1],
\\ %
& \overline{M (\gamma \omega)^n M (\omega_j)}^{n + 1} \simeq M (B_j B^{n -
1} \omega \alpha), \, j \in [j_0 + 1, j_{r + 1} - 1], \, n \in \mathbb{N},
\\ %
& \overline{N (C, \omega_{j_p})}^1 \simeq N (C, \omega_{j_p} \alpha), \, C
\in \mathcal{C}_p, \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^n N (C, \omega_{j_p})}^{n + 1} \simeq N (C,
B_{j_p} B^{n - 1} \omega \alpha), \, C \in \mathcal{C}_p, \, p \in [1, r],
\, n \in \mathbb{N},
\\ %
& \overline{M (\gamma_p \omega_{j_p}) M (x)}^1 \simeq M (\gamma_p
\omega_{j_p} \alpha \xi), \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p}) M (x)}^{n + 1}
\simeq M (\gamma_p B_{j_p} B^{n - 1} \omega \alpha \xi), \, p \in
[1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\omega_j) M (x)}^1 \simeq M (\omega_j \alpha \xi), \, j \in
[j_0 + 1, j_{r + 1} - 1],
\\ %
& \overline{M (\gamma \omega)^n M (\omega_j) M (x)}^{n + 1} \simeq M (B_j
B^{n - 1} \omega \alpha \xi),
\\ %
& \qquad j \in [j_0 + 1, j_{r + 1} - 1], \, n \in \mathbb{N},
\\ %
& \overline{N (C, \omega_{j_p}) M (x)}^1 \simeq N (C, \omega_{j_p} \alpha
\xi), \, C \in \mathcal{C}_p, \, p \in [1, r],
\\ %
& \overline{M (\gamma \omega)^n N (C, \omega_{j_p}) M (x)}^{n + 1} \simeq
N (C, B_{j_p} B^{n - 1} \omega \alpha \xi),
\\ %
& \qquad C \in \mathcal{C}_p, \, p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{0}^1 \simeq M (z),
\\ %
& \overline{M (\gamma \omega)^n}^{n + 1} \simeq M (\xi \gamma B^{n - 1}
\omega \alpha), \, n \in \mathbb{N},
\\ %
& \overline{M (x)}^1 \simeq M (\xi),
\\ %
& \overline{M (\gamma \omega)^n M (x)}^{n + 1} \simeq M (\xi \gamma B^{n
-1} \omega \alpha \xi), \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p})}^{n + 2}
\simeq N (B_{j_p} B^{n - 1} \omega \alpha, \omega_{j_p} \alpha), \,
p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m}^{n + m + 2} \simeq N (B_{j_p} B^{n - 1} \omega
\alpha, B_{j_p} B^{m - 1} \omega \alpha),
\\ %
& \qquad p \in [1, r], \, n, m \in \mathbb{N}, \, m < n,
\\ %
& \overline{M (x) M (\omega_{j_p}) N (\omega_{j_p})}^{2} \simeq N
(\omega_{j_p} \alpha \xi, \omega_{j_p} \alpha), \, p \in [1, r],
\\ %
& \overline{M (x) M (\gamma \omega)^n M (\omega_{j_p}) N
(\omega_{j_p})}^{n + 2} \simeq N (B_{j_p} B^{n - 1} \omega \alpha
\xi, \omega_{j_p} \alpha),
\\ %
& \qquad p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (x) M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m}^{n + m + 2}
\\ %
& \qquad \simeq N (B_{j_p} B^{n - 1} \omega \alpha \xi, B_{j_p} B^{m
- 1} \omega \alpha), \, p \in [1, r], \, n, m \in \mathbb{N}, \, m \leq n,
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(x)}^{n + 2} \simeq N (B_{j_p} B^{n - 1} \omega \alpha, \omega_{j_p}
\alpha \xi),
\\ %
& \qquad p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m M (x)}^{n + m + 2}
\\ %
& \qquad \simeq N (B_{j_p} B^{n - 1} \omega \alpha, B_{j_p} B^{m -
1} \omega \alpha \xi), \, p \in [1, r], \, n, m \in \mathbb{N}, \, m < n,
\\ %
& \overline{M (x) M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(x)}^{n + 2} \simeq N (B_{j_p} B^{n - 1} \omega \alpha \xi,
\omega_{j_p} \alpha \xi),
\\ %
& \qquad p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{M (x) M (\gamma \omega)^n M (\omega_{j_p}) N (\omega_{j_p}) M
(\gamma \omega)^m M (x)}^{n + m + 2}
\\ %
& \qquad \simeq N (B_{j_p} B^{n - 1} \omega \alpha \xi, B_{j_p} B^{m
- 1} \omega \alpha \xi), \, p \in [1, r], \, n, m \in \mathbb{N}, \, m <
n,
\\ %
& \overline{N (\omega_{j_p})}^1 \simeq N (\omega_{j_p} \alpha), \, p \in
[1, r],
\\ %
& \overline{N (\gamma \omega)^n M (\omega_{j_p})}^1 \simeq N (B_{j_p} B^{n
- 1} \omega \alpha), \, p \in [1, r], \, n \in \mathbb{N},
\\ %
& \overline{N (\omega_{j_p}) M (x)}^1 \simeq N (\omega_{j_p} \alpha \xi),
\, p \in [1, r],
\\ %
& \overline{N (\gamma \omega)^n M (\omega_{j_p}) M (x)}^1 \simeq N
(B_{j_p} B^{n - 1} \omega \alpha \xi), \, p \in [1, r], \, n \in
\mathbb{N},
\end{align*}
\subsection{}
We now prove Lemma~\ref{lemmR}. Let $x'$ be an admissible index of
$Q^x$. Let first $x' = x_{i, j}$, $x' \neq z$. In this case
$R_{x'}^x = R_{x'}$ . If $i \neq i_0$ or $i = i_0$ and $j < j_0$
then also $\Hom (R_{x'}^x, \mod A^x) = \Hom (R_{x'}, \mod A)$. If $i
= i_0$ and $j_0 < j$, then the indecomposable objects of $\Hom
(R_{x'}^x, \mod A^x)$ are the indecomposable objects of $\Hom
(R_{x'}, \mod A)$ and
\begin{gather*}
\overline{M (\gamma \omega)^n M (\omega_{j - 1})}^n, \, \overline{M (\gamma
\omega)^n M (\omega_j)}^n, \, n \in \mathbb{N},
\\ %
\overline{M (\gamma C) M (\gamma \omega)^n M (\omega_{j - 1})}^{n + 1},
\, \overline{M (\gamma C) M (\gamma \omega)^n M (\omega_j)}^{n + 1}, C
\in \mathcal{C}', n \in \mathbb{N}_0,
\\ %
\overline{M (\gamma \omega)^n M (\omega_{j - 1})}^{n + 1}, \, \overline{M
(\gamma \omega)^n M (\omega_j)}^{n + 1}, \, n \in \mathbb{N}_0,
\\ %
\intertext{and}
\overline{M (\gamma \omega)^n M (\omega_{j - 1}) M (x)}^{n + 1}, \,
\overline{M (\gamma \omega)^n M (\omega_j) M (x)}^{n + 1}, \, n \in
\mathbb{N}_0.
\end{gather*}
Assume now that $x' = z_{i, j}$. If $i \neq i_0$, then again
$R_{x'}^x = R_{x'}$ and $\Hom (R_{x'}^x, \mod A^x) = \Hom (R_{x'},
\mod A)$. If $i = i_0$ then $j = j_p$ for $p \in [1, r]$. Moreover,
$R_{x'}^x = \overline{R_{x'}}$ and the indecomposable objects of $\Hom
(R_{x'}^x, \mod A^x)$ are
\begin{gather*}
M (\gamma_p C), \, C \in \mathcal{C}_p, \, M (x'),
\\ %
\overline{N (C, \omega_{j_q})}, \, \overline{M (\gamma_q \omega_{j_q})}, \,
\overline{N (\omega_{j_q})}, C \in \mathcal{C}_{j_q}', \, \, q \in [p, r],
\\ %
\overline{M (\omega_l)}, \, l \in [j_p, \ldots, j_{r + 1} - 1], \,
\overline{0},
\\ %
\overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p})}^n, \, \overline{M
(\gamma \omega)^{n - 1} M (\omega_{j_q}) N (\omega_{j_q})}^n, \, n
\in \mathbb{N}, \, q \in [p, r],
\\ %
\overline{M (\gamma C) M (\gamma \omega)^n M (\gamma_p \omega_{j_p})}^{n
+ 1}, \, C \in \mathcal{C}', \, n \in \mathbb{N}_0,
\\ %
\overline{M (\gamma C) M (\gamma \omega)^n M (\omega_{j_q}) N
(\omega_{j_q})}^{n + 2}, \, C \in \mathcal{C}_{j_q}', \, n \in \mathbb{N}_0,
\, q \in [p, r],
\\ %
\overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p})}^{n + 1}, \,
\overline{M (\gamma \omega)^{n - 1} M (\gamma_p \omega_{j_p}) M (x)}^n,
\, n \in \mathbb{N},
\\ %
\overline{M (\gamma \omega)^n M (\omega_{j_q}) N (\omega_{j_q})}^{n +
2}, \, n \in \mathbb{N}_0, \, q \in [p, r],
\\ %
\intertext{and}
\overline{M (\gamma \omega)^n M (\omega_{j_q}) N (\omega_{j_q}) M
(x)}^{n + 2}, \, n \in \mathbb{N}_0, \, q \in [p, r].
\end{gather*}
Finally, assume that $x' = z$ (it is possible, if $p_{i_0} +
|T_{i_0}| \not \in S_{i_0}$). In this case $R_{x'}^x = \overline{X_x M
(x_{i_0, p_{i_0} + |T_{i_0}|})}$. It follows that the indecomposable
objects of $\Hom (R_{x'}^x, \mod A^x)$ are
\begin{gather*}
\overline{M (C) M (x_{i_0, p_{i_0} + |T_{i_0}|})}, \, \overline{M (C)}, \, C
\in \mathcal{C}_x,
\\ %
\intertext{and} \overline{M (x_{i_0, p_{i_0} + |T_{i_0}|})}, \, \overline{0}.
\end{gather*}
\subsection{}
Now we indicate how to prove Lemma~\ref{lemmX}. Let $x'$ be an
admissible index of $Q^x$. If $x' = x_{i, j}$, $x' \neq z$, then
$X_{x'}^x = X_{x'}$. If in addition, $i \neq i_0$ or $i = i_0$ and
$j < j_0$, then $\Hom (X_{x'}^x, \mod A^x) = \Hom (X_{x'}, \mod A)$.
Let $i = i_0$ and $j_0 < j$. Then the indecomposable objects of
$\Hom (X_{x'}^x, \mod A^x)$ are the indecomposable objects of $\Hom
(X_{x'}, \mod A)$ and
\begin{gather*}
\overline{M (\gamma \omega)^{n + 1} M (\omega_j)}^{n + 1}, \, \overline{M
(\gamma C) M (\gamma \omega)^n M (\omega_j)}^{n + 1}, C \in
\mathcal{C}', n \in \mathbb{N}_0,
\\ %
\intertext{and}
\overline{M (\gamma \omega)^n M (\omega_j)}^{n + 1}, \, \overline{M (\gamma
\omega)^n M (\omega_j) M (x)}^{n + 1}, \, n \in \mathbb{N}_0.
\end{gather*}
Assume now that $x' = z_{i, j}$. Again $X_{x'}^x = X_{x'}$ and if $i
\neq i_0$ then $\Hom (X_{x'}^x, \mod A^x) = \Hom (X_{x'}, \mod A)$.
Let $i = i_0$. Then $j = j_p$ for $p \in [1, r]$. The indecomposable
objects of $\Hom (X_{x'}^x, \mod A^x)$ are the indecomposable
objects of $\Hom (X_{x'}, \mod A)$ and
\begin{gather*}
\overline{M (\gamma \omega)^{n + 1} M (\gamma_p \omega_{j_p})}^{n + 1},
\, \overline{M (\gamma C) M (\gamma \omega)^n M (\gamma_p
\omega_{j_p})}^{n + 1}, \, C \in \mathcal{C}', \, n \in \mathbb{N}_0,
\\ %
\\ %
\intertext{and}
\overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p})}^{n + 1}, \,
\overline{M (\gamma \omega)^n M (\gamma_p \omega_{j_p}) M (x)}^{n + 1},
\, n \in \mathbb{N}_0.
\end{gather*}
Finally, let $x' = z$. In this case $X_{x'}^x = \overline{X_x}$ and the
indecomposable objects of $\Hom (X_{x'}, \mod A^x)$ are
\[
\overline{M (C)}, \, C \in \mathcal{C}_x, \text{ and } \overline{0}.
\]
\subsection{}
It remains to give the proof of Lemma~\ref{lemmI}. Let $x' = x_{i,
j}$ be the vertex of $Q^x$ satisfying the hypothesis of
Lemma~\ref{lemmI}. If $i \neq i_0$ then $I_{x'}^x = I_{x'}$ and
$\Hom (I_{x'}^x, \mod A^x) = \Hom (I_{x'}, \mod A)$. Assume now that
$i = i_0$. Then $j \in [j_0 + 1, j_{r + 1}$. Let $p = \min \{ q \in
[1, r + 1] \mid j \leq j_q \}$. First consider the case $j \neq j_{r
+ 1}$. Then $I_{x'}^x = \overline{I_{x'}}$ and the indecomposable objects
of $\Hom (I_{x'}^x, \mod A^x)$ are
\begin{gather*}
\overline{N (C, \omega_{j_q})}, \, \overline{M (\gamma_q \omega_{j_q})}, \,
\overline{N (\omega_{j_q})}, \, C \in \mathcal{C}_{j_q}', \, q \in [p, r], \,
\\ %
\overline{M (\omega_l)}, \, l \in [j, \ldots, j_{r + 1} - 1], \, \overline{0},
\\ %
\overline{M (\gamma \omega)^{n - 1} M (\omega_{j_q}) N
(\omega_{j_q})}^n, \, n \in \mathbb{N}, \, q \in [p, r],
\\ %
\overline{M (\gamma C) M (\gamma \omega)^n M (\omega_{j_q}) N
(\omega_{j_q})}^{n + 2}, \, C \in \mathcal{C}_{j_q}', \, n \in \mathbb{N}_0,
\, q \in [p, r],
\\ %
\overline{M (\gamma \omega)^n M (\omega_{j_q}) N (\omega_{j_q})}^{n +
2}, \, n \in \mathbb{N}_0, \, q \in [p, r],
\\ %
\intertext{and}
\overline{M (x) M (\gamma \omega)^n M (\omega_{j_q}) N
(\omega_{j_q})}^{n + 2}, \, n \in \mathbb{N}_0, \, q \in [p, r].
\end{gather*}
If $j = j_{r + 1}$ then the claim is clear.
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|
{
"timestamp": "2005-08-02T10:23:19",
"yymm": "0503",
"arxiv_id": "math/0503513",
"language": "en",
"url": "https://arxiv.org/abs/math/0503513"
}
|
\section{Introduction}
\label{sec:intro}
The known meson spectrum contains three pseudoscalars [$I^G (J^{PC}) = 1^- (0^{-+}) $], all with masses below $2\,$GeV \cite{pdg}: $\pi(140)$; $\pi(1300)$; and $\pi(1800)$. The lightest of these, the pion [$\pi(140)$], is much studied and well understood as QCD's Goldstone mode. It is the basic degree of freedom in chiral effective theories, and a veracious explanation of its properties requires an approach to possess a valid realisation of chiral symmetry and its dynamical breaking.
The $\pi(1300)$ is broad, with a width of $200$ to $600\,$MeV. In the framework of constituent-quark models it is usually interpreted as the pion's first radial excitation. Namely, the $\pi(1300)$ is pictured as: an $I^G (J^{P}) L = 1^- (0^-)S$ $Q \bar Q$ meson, where $Q$ denotes a constituent-quark; and the first radially excited state of the $\pi(140)$ on a $Q \bar Q$ $n\, ^1\!S_0$ trajectory, where $n$ is the ``principal quantum number'' \cite{anisovich,norbury}.
At first sight it might appear natural to interpret the $\pi(1800)$ as the third state on the $n\, ^1\!S_0$ trajectory. However, in comparison with $\pi(1300)$, the $\pi(1800)$ is narrow, with a width of $207\pm 13\,$MeV, and has a decay pattern that may be consistent with its interpretation as a \emph{hybrid} meson in constituent-quark models \cite{page}. This picture has the constituent-quarks' spins aligned to produce $S_{Q\bar Q} = 1$ with $J=0$ obtained by coupling $S_{Q\bar Q}$ to a spin-$1$ excitation of the confinement potential.
It is legitimate to ask for a unified theoretical understanding of these states and, indeed, the entire trajectory of pseudoscalar mesons. This is a topical question; e.g., Refs.\,\cite{shakin,ji,barnes,yudichev,metsch2,bakker,krassnigg1,andreastunl,%
lc03,andreasrapid}, and it is easy to identify at least one reason why. In the context of a constituent-quark model Hamiltonian a subset, if not all, of the pseudoscalar mesons form a $Q\bar Q$ $n\, ^1\!S_0$ trajectory. In this framework the support possessed at long-range by the bound state's wave function grows with increasing $n$. Hence the properties of radially excited states become increasingly sensitive to the manner by which confinement is expressed in the potential. As we have already noted, in this same context a definition and representation of hybrid mesons requires that explicit excitation of the confinement potential be included as an additional degree of freedom. Seen from this perspective one may anticipate that the properties of all the heavier pseudoscalar mesons are likely to be sensitive to the long-range part of the interaction between light-quarks in QCD, whether they be radial excitations or hybrid mesons. This suggests that the study of their properties can provide a map of what might be called the confinement potential between light-quarks. (NB. The information obtained thereby is complementary to that gathered in studies of axial-vector mesons \cite{ericaxial,a1b1,burdenpichowsky,jarecke,peteraxial}, which in constituent-quark models are interpreted as orbital excitations of the $\pi$- and $\rho$-mesons.)
It is not possible to accurately describe pseudoscalar mesons using a framework that fails to respect the axial-vector Ward-Takahashi identity. For example, chiral symmetry and its dynamical breakdown force the leptonic decay constant of every pseudoscalar meson, \emph{except} the Goldstone mode, to vanish in the chiral limit \cite{yudichev,krassnigg1,andreastunl,lc03,andreasrapid}. Herein we therefore employ QCD's Dyson-Schwinger equations (DSEs) (modern applications are reviewed in Refs.\,\cite{bastirev,reinhardrev,pieterrev}) for which a systematic, Poincar\'e covariant and symmetry preserving treatment of quark-antiquark bound states has been established \cite{bender,detmold,mandarvertex}. To provide exemplars we will focus primarily on the $\pi(140)$ and the next-lightest pseudoscalar state. Nonetheless, the exact results will apply to all elements on the pseudoscalar meson trajectory.
It is noteworthy that in Poincar\'e covariant quantum field theory all bound states with given quantum numbers; e.g., ($I^G$, $J^{PC}$), are described by the same homogeneous Bethe-Salpeter equation (BSE). This is kindred to the statement that all interpolating fields with the same quantum numbers are on-shell equivalent, a fact which is apparent in numerical simulations of lattice-QCD; e.g., Ref.\,\cite{hedditch}. Hence a given homogeneous BSE yields the mass and Bethe-Salpeter amplitude of every bound state in the channel specified by ($I^G$, $J^{PC}$).
In a confining theory a given $J^P$ trajectory will likely contain a countable infinity of bound states. The lowest mass member of the trajectory is conventionally described as the ground state. All other members may reasonably be described as excited states. The radial excitation of a state with a given $J^P$ preserves this total-momentum\,$+\,$parity assignment. However, it may be distinguished from the ground state by the pointwise behaviour of its Bethe-Salpeter amplitude, which when analysed appropriately exhibits a finite number of zeros. As in quantum mechanics, the number of zeros can be associated with a principal quantum number $n$. Studies of pseudoscalar mesons show that the ground state amplitude has no zeros and can therefore be associated with $n=0$. The amplitude of the next highest mass pseudoscalar possesses one zero and is therefore identified with $n=1$; e.g., \cite{yudichev,krassnigg1,andreastunl,lc03,andreasrapid}. In simple models, this pattern continues \cite{metsch2,bakker}.
It may be that hybrid mesons, if they exist, can likewise be identified through the pointwise behaviour of their Bethe-Salpeter amplitudes. For example, a solution of the pseudoscalar BSE, heavier than the first radial excitation, whose Bethe-Salpeter amplitude exhibits both: a pattern of zeros which does not match that associated with radial excitations; and relationships between component functions in the Bethe-Salpeter amplitude different from those present in the lower mass solutions, would appear a reasonable hybrid candidate.
Of course, Bethe-Salpeter amplitudes are not themselves observable and the experimental categorisation of ground, and excited and putative hybrid states proceeds via analysis of their decay patterns. Notwithstanding this, the order in those decay patterns is determined in large part by the Bethe-Salpeter amplitudes' pointwise behaviour. We therefore anticipate that a natural distinction between straightforward radial excitations and hybrids may be possible without recourse to a constituent-quark model basis.
In Sec.\,\ref{gapbse} we recapitulate on aspects of the DSEs and truncation scheme that are relevant to our study. The Abelian anomaly features in Sec.\,\ref{exact}, wherein exact results are derived regarding the coupling of pseudoscalar mesons to two photons. We outline a renormalisation-group-improved model of the quark-antiquark scattering kernel in Sec.\,\ref{model}. It is used in that section to illustrate the exact results and explore effects of the model's realisation of light-quark confinement on, e.g., bound state charge radii. Section~\ref{epilogue} is an epilogue.
\section{BETHE-SALPETER AND GAP EQUATIONS}
\label{gapbse}
A Poincar\'e covariant and symmetry preserving treatment of quark-antiquark bound states can be based on the homogeneous Bethe-Salpeter equation (BSE) \cite{fn:Eucl}
\begin{equation}
\label{bse1}
[\Gamma(k;P)]_{tu} = \int^\Lambda_q [\chi(q;P)]_{sr}\, K_{rs}^{tu}(q,k;P)\,,
\end{equation}
where: $k$ is the relative and $P$ the total momentum of the constituents; $r$,\ldots,\,$u$ represent colour, Dirac and flavour indices;
\begin{equation}
\label{definechi}
\chi(q;P)= S(q_+) \Gamma(q;P) S(q_-)\,,
\end{equation}
$q_\pm = q\pm P/2$; and $\int^\Lambda_q$ represents a Poincar\'e invariant regularisation of the integral, with $\Lambda$ the regularisation mass-scale \cite{mrt98,mr97}. (We shall subsequently describe regularisation explicitly.) In Eq.\,(\ref{bse1}), $S$ is the renormalised dressed-quark propagator and $K$ is the fully amputated dressed-quark-antiquark scattering kernel; namely, it is the sum of all diagrams that cannot be disconnected by cutting two fermion lines. The product $(SS) K$ is a renormalisation point invariant. Hence, when the kernel is expressed completely in terms of renormalised Schwinger functions, the homogeneous BSE's solution is independent of the regularisation mass-scale, which may be removed; viz., $\Lambda \to \infty$.
In a given channel the homogeneous BSE only has solutions for particular, separated values of $P^2$: $P^2=-m_n^2$, where $m_n$ is a bound state's mass, whereat $\Gamma_n(k;P)$ is that bound state's Bethe-Salpeter amplitude. In the flavour nonsinglet pseudoscalar channel the lowest mass solution is associated with the $\pi(140)$. It is precisely QCD's Goldstone mode \cite{mrt98}, and we denote it by a value of $n=0$. The homogeneous BSE next possesses a $J^{PC}=0^{-+}$ solution when $P^2$ assumes a value associated with the mass of the $\pi(1300)$. We label this state by $n=1$. In the study of this meson in Ref.\,\cite{andreastunl} the Tchebychev moments of the Lorentz scalar functions that appear in the matrix-valued Bethe-Salpeter amplitude each exhibit a single zero. It can therefore be described as a radially excited state. (NB. Hereafter the subscript $n$ is merely a counter labelling states of increasing mass: $m_0<m_1<m_2<\ldots$, etc.)
The pattern of isolated solutions continues so that in principle one may obtain the mass and amplitude of every pseudoscalar meson from Eq.\,(\ref{bse1}). Herein we will exploit this in comparing properties of the two lowest-mass flavour-nonsinglet $J^{PC}=0^{-+}$ mesons just described.
The dressed-quark propagator appearing in the BSE's kernel is determined by the renormalised gap equation
\begin{eqnarray}
S(p)^{-1} & =& Z_2 \,(i\gamma\cdot p + m^{\rm bm}) + \Sigma(p)\,, \label{gendse} \\
\Sigma(p) & = & Z_1 \int^\Lambda_q\! g^2 D_{\mu\nu}(p-q) \frac{\lambda^a}{2}\gamma_\mu S(q) \Gamma^a_\nu(q,p) , \label{gensigma}
\end{eqnarray}
wherein: $D_{\mu\nu}$ is the dressed-gluon propagator, $\Gamma_\nu(q,p)$ is the dressed-quark-gluon vertex, and $m^{\rm bm}$ is the $\Lambda$-dependent current-quark bare mass. The quark-gluon-vertex and quark wave function renormalisation constants, $Z_{1,2}(\zeta^2,\Lambda^2)$, depend on the gauge parameter, the renormalisation point, $\zeta$, and the regularisation mass-scale. A Poincar\'e invariant regularisation of the integral is essential and, since pseudoscalar mesons are our focus, we employ a Pauli-Villars scheme. That is implemented in Eq.\,(\ref{gendse}) by considering the quarks as minimally anticoupled ($g^{PV}=ig$) to additional massive gluons ($m_g^{PV}=\Lambda$). This effects a tempering of the integrand, which is expressed via a modification of the gluon propagator's ultraviolet behaviour:
\begin{equation}
\frac{1}{(p-q)^2} \to \frac{1}{(p-q)^2} - \frac{1}{(p-q)^2+\Lambda^2}\,,
\end{equation}
and regulates the integral's superficial linear divergence.
The gap equation's solution has the form
\begin{eqnarray}
S(p)^{-1} & = & i \gamma\cdot p \, A(p^2,\zeta^2) + B(p^2,\zeta^2) \,,\\
& =& \frac{1}{Z(p^2,\zeta^2)}\left[ i\gamma\cdot p + M(p^2)\right] .
\label{sinvp}
\end{eqnarray}
It is obtained from Eq.\,(\ref{gendse}) augmented by the renormalisation condition
\begin{equation}
\label{renormS} \left.S(p)^{-1}\right|_{p^2=\zeta^2} = i\gamma\cdot p +
m(\zeta)\,,
\end{equation}
where $m(\zeta)$ is the renormalised (running) current-quark mass:
\begin{equation}
Z_2(\zeta^2,\Lambda^2) \, m^{\rm bm}(\Lambda) = Z_4(\zeta^2,\Lambda^2) \, m(\zeta)\,,
\end{equation}
with $Z_4$ the Lagrangian mass renormalisation constant. At one-loop order in perturbative QCD
\begin{equation}
m(\zeta) = \frac{\hat m}{(\ln \zeta/\Lambda_{\rm QCD})^{\gamma_m}}\,,
\end{equation}
with $\gamma_m= 12/(33-2 N_f)$, where $N_f$ is the number of active current-quark flavours, and $\hat m$ is the renormalisation-point-invariant current-quark mass. The chiral limit is unambiguously defined by setting $\hat m = 0$ \cite{mrt98,mr97,langfeld}, which is equivalent to the requirement
\begin{equation}
\label{limchiral}
Z_2(\zeta^2,\Lambda^2) \, m^{\rm bm}(\Lambda) \equiv 0 \,,\; \forall \Lambda \gg \zeta \,.
\end{equation}
The behaviour and features of the solution of QCD's gap equation are reviewed in Refs.\,\cite{bastirev,reinhardrev,pieterrev}. It is a longstanding prediction of DSE studies that the dressed-quark propagator is strongly dressed at infrared length-scales, namely, $p^2\lesssim 2\,$GeV$^2$ and that this is materially important in explaining a wide range of hadron properties \cite{pieterrev}. Indeed, an enhancement of the mass function, $M(p^2)$, is central to the appearance of a constituent-quark mass-scale and an existential prerequisite for Goldstone modes. The DSE results have been confirmed in numerical simulations of lattice-regularised QCD \cite{bowman} and the conditions have been explored under which pointwise agreement between DSE results and lattice simulations may be obtained \cite{bhagwat,maris,bhagwat2}.
The $I^G (J^{PC}) = 1^- (0^{-+})$ trajectory contains the pion, whose properties are fundamentally governed by the phenomenon of dynamical chiral symmetry breaking (DCSB). One expression of the chiral properties of QCD is the axial-vector Ward-Takahashi identity
\begin{eqnarray}
\nonumber
P_\mu \Gamma_{5\mu}^j(k;P) & =& S^{-1}(k_+) i \gamma_5\frac{\tau^j}{2}
+ i \gamma_5\frac{\tau^j}{2} S^{-1}(k_-)\\
&& - \, 2i\,m(\zeta) \,\Gamma_5^j(k;P) ,
\label{avwtim}
\end{eqnarray}
which we have here written for two quark flavours, each with the same current-quark mass: $\{\tau^i:i=1,2,3\}$ are flavour Pauli matrices. In Eq.\,(\ref{avwtim}), $\Gamma_{5\mu}^j(k;P)$ is the axial-vector vertex:
\begin{eqnarray}
\nonumber
\left[\Gamma^j_{5\mu}(k;P)\right]_{tu}
& = & Z_2 \left[\gamma_5\gamma_\mu \frac{\tau^j}{2} \right]_{tu}\\
%
&+& \int^\Lambda_q
[\chi^j_{5\mu}(q;P)]_{sr} K_{tu}^{rs}(q,k;P)\,,
\label{avbse}
\end{eqnarray}
and $\Gamma_5^j(k;P)$ is the pseudoscalar vertex
\begin{eqnarray}
\nonumber
\left[\Gamma_{5}(k;P)\right]_{tu}
& = & Z_4 \left[\gamma_5 \frac{\tau^j}{2}\right]_{tu}\\
%
&+& \int^\Lambda_q
[\chi^j_{5}(q;P)]_{sr} K_{tu}^{rs}(q,k;P)\,.
\label{psbse}
\end{eqnarray}
The quark propagator, axial-vector and pseudoscalar vertices are all expressed via integral equations; i.e., DSEs. Equation~(\ref{avwtim}) is an exact statement about chiral symmetry and the pattern by which it is broken. Hence it must always be satisfied. Since that cannot credibly be achieved through fine tuning, the distinct kernels of Eqs.\,(\ref{gendse}), (\ref{gensigma}), (\ref{avbse}), (\ref{psbse}) must be intimately related. Any theoretical tool employed in calculating properties of the pseudoscalar and pseudovector channels must preserve that relationship if the results are to be both quantitatively and qualitatively reliable.
While a weak coupling expansion of the DSEs yields perturbation theory and satisfies this constraint, that truncation scheme is not useful in the study of bound states nor of other intrinsically nonperturbative phenomena; such as confinement and DCSB. Fortunately at least one nonperturbative, systematic and symmetry preserving scheme exists. (References\,\cite{detmold,mandarvertex} give details.) This entails that the full implications of Eq.\,(\ref{avwtim}) can be elucidated and illustrated.
Unless there is a reason for the residue to vanish, every isovector pseudoscalar meson appears as a pole contribution to the axial-vector and pseudoscalar vertices \cite{mrt98}:
\begin{eqnarray}
\nonumber
\left. \Gamma_{5 \mu}^j(k;P)\right|_{P^2+m_{\pi_n}^2 \approx 0}&=& \frac{f_{\pi_n} \, P_\mu}{P^2 +
m_{\pi_n}^2} \Gamma_{\pi_n}^j(k;P) \\
& & + \; \Gamma_{5 \mu}^{j\,{\rm reg}}(k;P) \,, \label{genavv} \\
\nonumber
\left. i\Gamma_{5 }^j(k;P)\right|_{P^2+m_{\pi_n}^2 \approx 0}
&=& \frac{\rho_{\pi_n}(\zeta) }{P^2 +
m_{\pi_n}^2} \Gamma_{\pi_n}^j(k;P)\\
& & + \; i\Gamma_{5 }^{j\,{\rm reg}}(k;P) \,; \label{genpv}
\end{eqnarray}
viz., each vertex may be expressed as a simple pole plus terms regular in the neighbourhood of this pole, with $\Gamma_{\pi_n}^j(k;P)$ representing the bound state's canonically normalised Bethe-Salpeter amplitude:
\begin{eqnarray}
\nonumber
\lefteqn{
\Gamma_{\pi_n}^j(k;P) = \tau^j \gamma_5 \left[ i E_{\pi_n}(k;P) + \gamma\cdot P F_{\pi_n}(k;P) \right. }\\
&+& \left.
\gamma\cdot k \,k \cdot P\, G_{\pi_n}(k;P) +
\sigma_{\mu\nu}\,k_\mu P_\nu \,H_{\pi_n}(k;P) \right] \! ; \label{genpibsa} \end{eqnarray}
and
\begin{eqnarray}
\label{fpin} f_{\pi_n} \,\delta^{ij} \, P_\mu &=& Z_2\,{\rm tr} \int^\Lambda_q
\sfrac{1}{2} \tau^i \gamma_5\gamma_\mu\, \chi^j_{\pi_n}(q;P) \,, \\
\label{cpres} i \rho_{\pi_n}\!(\zeta)\, \delta^{ij} &=& Z_4\,{\rm tr}
\int^\Lambda_q \sfrac{1}{2} \tau^i \gamma_5 \, \chi^j_{\pi_n}(q;P)\,.
\end{eqnarray}
The residues expressed in Eqs.\,(\ref{fpin}) and (\ref{cpres}), are gauge invariant and cutoff independent.
For an elementary pseudoscalar meson, $F_{\pi_n}(k;P)\equiv 0 \equiv G_{\pi_n}(k;P) \equiv H_{\pi_n}(k;P)$ in Eq.\,(\ref{genpibsa}). The first two of these functions can be described as characterising the pseudoscalar meson's pseudovector components; and the last, its pseudotensor component. The associated Dirac structures necessarily occur in a Poincar\'e covariant bound state description: they signal the presence of quark orbital angular momentum.
Equation\,(\ref{avwtim}) combined with Eqs.\,(\ref{genavv}) -- (\ref{cpres}) yields \cite{mrt98,mr97}
\begin{equation}
\label{gmorgen} f_{\pi_n} m_{\pi_n}^2 = 2 \, m(\zeta) \,
\rho_{\pi_n}(\zeta)\,;
\end{equation}
i.e., an identity valid: for every flavour nonsinglet $0^-$ meson; and irrespective of the magnitude of the current-quark mass \cite{mishasvy}. In the chiral limit additional information about the ground state pseudoscalar ($n=0$) is available; namely, an array of quark-level Goldberger-Treiman relations \cite{mrt98}
\begin{eqnarray}
\label{bwti} f_{\pi_0}^0 E_{\pi_0}(k;0) &= & B(k^2)\,, \\
F_R(k;0) + 2 \, f_{\pi_0}^0 F_{\pi_0}(k;0) & = & A(k^2)\,,\label{fwti}\\
G_R(k;0) + 2 \,f_{\pi_0}^0 G_{\pi_0}(k;0) & = & 2 A^\prime(k^2)\,,\label{gwti}\\
H_R(k;0) + 2 \,f_{\pi_0}^0 H_{\pi_0}(k;0) & = & 0\,, \label{hwti}
\end{eqnarray}
where $F_R$, $G_R$, $H_R$ are, respectively, the coefficient functions of $\gamma_5 \gamma_\mu$, $\gamma\cdot k k_\mu$, $\sigma_{\mu\nu} k_\nu$ in $\Gamma_{5 \mu}^{j\,{\rm reg}}(k;P)$ and
\begin{equation}
f_{\pi_n}^0 := \lim_{\hat m \to 0}\, f_{\pi_n} .
\end{equation}
Equations~(\ref{bwti}) -- (\ref{hwti}) are a pointwise consequence of DCSB and a pointwise expression of Goldstone's theorem. They can be used to show
\begin{equation}
\rho_{\pi_0}^0(\zeta) := \lim_{\hat m \to 0}\,\rho(\zeta) = -\frac{1}{f^0_{\pi_0} } \langle \bar q q \rangle^0_\zeta\,,
\end{equation}
wherein
\begin{equation}
\label{qbq0} \,-\,\langle \bar q q \rangle_\zeta^0 = \lim_{\Lambda\to \infty}
Z_4(\zeta^2,\Lambda^2)\, N_c \, {\rm tr}_{\rm D}\int^\Lambda_q\!
S^{0}(q,\zeta)\,,
\end{equation}
is the vacuum quark condensate \cite{langfeld}. It is now plain from Eq.\,(\ref{gmorgen}) that in the neighbourhood of $\hat m = 0$
\begin{equation}
(f_{\pi_0}^0)^2 m_{\pi_0}^2 = -\, 2 \, m(\zeta) \, \langle \bar q q \rangle_\zeta^0\,;
\end{equation}
viz., the Gell-Mann--Oakes--Renner relation is a corollary of Eq.\,(\ref{gmorgen}).
\section{Two photon coupling of Pseudoscalar Mesons: Exact Results}
\label{exact}
\subsection{Abelian anomaly}
To be concrete we will begin by considering the two-photon coupling as expressed via the renormalised triangle diagrams:
\begin{eqnarray}
\nonumber T^3_{5\mu\nu\rho}(k_1,k_2) &=& {\rm tr}\int_\ell^M {\cal S}(\ell_{0+}) \, \Gamma^3_{5\rho}(\ell_{0+},\ell_{-0}) \, {\cal S}(\ell_{-0}) \\%
\nonumber & \times& \, i{\cal Q}\Gamma_\mu(\ell_{-0},\ell) \, {\cal S}(\ell) \, i {\cal Q}\Gamma_\nu(\ell,\ell_{0+})\,,\\
&& \label{Tmnr}\\
\nonumber T^3_{5\mu\nu}(k_1,k_2) &=& {\rm tr}\int_\ell^M {\cal S}(\ell_{0+}) \, \Gamma^3_{5}(\ell_{0+},\ell_{-0}) \, {\cal S}(\ell_{-0}) \\
\nonumber &\times& \, i{\cal Q}\Gamma_\mu(\ell_{-0},\ell) \, {\cal S}(\ell) \, i {\cal Q}\Gamma_\nu(\ell,\ell_{0+})\,,\\
&& \label{Pmnr}
\end{eqnarray}
where $\ell_{\alpha\beta}=\ell+\alpha k_1+\beta k_2$, the electric charge matrix ${\cal Q}={\rm diag}[e_u,e_d]=e\,{\rm diag}[2/3,-1/3]$, ${\cal S}= {\rm diag}[S_u,S_d]$ and
\begin{equation}
\left[\Gamma_{\mu}(k;P)\right]_{tu} = Z_2 \left[\gamma_\mu \right]_{tu}\\
+ \int^\Lambda_q
[\chi^j_{\mu}(q;P)]_{sr} K_{tu}^{rs}(q,k;P)
\end{equation}
is the renormalised dressed-quark-photon vertex.
The bare axial-vector--vector--vector vertex exhibits a superficial linear divergence and, as with all other Schwinger functions, it must be rigorously defined via a Poincar\'e invariant regularisation scheme. In this case an appropriate Pauli-Villars prescription corresponds to minimally anticoupling the photon to additional flavoured quarks with a large mass $m^{PV}=M$. To elucidate, we introduce
\begin{eqnarray}
\nonumber \lefteqn{
\tilde T^3_{5\mu\nu\rho}(k_1,k_2;\hat m) := {\rm tr}\int_\ell {\cal S}_{\hat m}(\ell_{0+}) \, \Gamma^{3\,\hat m}_{5\rho}(\ell_{0+},\ell_{-0}) }\\
\nonumber
&& \times \, {\cal S}_{\hat m}(\ell_{-0})\, i {\cal Q} \Gamma^{\hat m}_\mu(\ell_{-0},\ell) \, {\cal S}_{\hat m}(\ell) \, i {\cal Q}\Gamma^{\hat m}_\nu(\ell,\ell_{0+})\,,\\ \label{PVregd}
\end{eqnarray}
wherein the current-quark-mass dependence is explicit, so that Eq.\,(\ref{Tmnr}) can rigorously be written as
\begin{equation}
T^3_{5\mu\nu\rho}(k_1,k_2;\hat m) = \tilde T^3_{5\mu\nu\rho}(k_1,k_2;\hat m) - \tilde T^3_{5\mu\nu\rho}(k_1,k_2;M)\,,
\end{equation}
with $M \to \infty$ as the last step in the calculation.
\begin{widetext}
\begin{figure}[h]
\begin{center}
\hspace*{0em}\includegraphics[width=0.99\textwidth]{Fig1.eps}
\parbox{\textwidth}{\caption{\label{figAVWTI} This axial-vector Ward-Takahashi identity is an analogue of Eq.\,(\protect\ref{avwti0}). It is valid if, and only if: the dressed-quark propagator, $S$, is obtained from Eq.\,(\ref{rainbowdse}); the axial-vector vertex, $\Gamma_{5\mu}$, is obtained from Eq.\,(\protect\ref{avbse}) with the kernel constructed from $S$ and Eq.\,(\protect\ref{ladderK}); the pseudoscalar vertex is constructed analogously; and the unamputated renormalised quark-antiquark scattering matrix: $G= (SS) + (SS)K(SS) + (SS)K(SS)K(SS)+ [\ldots]$, is constructed from the elements just described.}}
\end{center}
\end{figure}
\end{widetext}
The dressed-quark propagators in Eqs.\,(\ref{Tmnr}) -- (\ref{PVregd}) are understood to be calculated using the rainbow-truncation gap equation, which is defined by Eq.\,(\ref{gendse}) with
\begin{equation}
\Sigma(p)=\int^\Lambda_q\! {\cal G}((p-q)^2) D_{\mu\nu}^{\rm free}(p-q) \frac{\lambda^a}{2}\gamma_\mu S(q) \frac{\lambda^a}{2}\gamma_\nu , \label{rainbowdse}
\end{equation}
wherein $D_{\mu\nu}^{\rm free}(\ell)$ is the free gauge boson propagator \cite{fn:landau} and ${\cal G}(\ell^2)$ will subsequently be specified. The remaining element, the axial-vector vertex, is obtained from the ladder Bethe-Salpeter equation, whose kernel (see Eq.\,(\ref{bse1}), for example) is defined by the dressed-quark propagators just specified and
\begin{eqnarray}
\nonumber \lefteqn{
K^{tu}_{rs}(q,k;P) = }\\
&& \!\!\! - \,{\cal G}((k-q)^2) \, D_{\mu\nu}^{\rm free}(k-q)\,\left[\gamma_\mu \frac{\lambda^a}{2}\right]_{ts} \, \left[\gamma_\nu \frac{\lambda^a}{2}\right]_{ru} \!\!\!. \label{ladderK}
\end{eqnarray}
In what follows it is important that the rainbow-ladder truncation is the first term in the systematic and symmetry preserving truncation scheme described in Refs.\,\cite{bender,detmold,mandarvertex} and, furthermore, that with the choice
\begin{equation}
\label{calGuv}
{\cal G}(\ell^2) = 4\pi \alpha_S(\ell^2)\,,\; \ell^2\gg \Lambda_{\rm QCD}^2\,,
\end{equation}
the rainbow-ladder truncation is guaranteed to express the one-loop renormalisation group properties of QCD.
The axial-vector Ward-Takahashi identity depicted in Fig.\,\ref{figAVWTI} is an analogue of
\begin{eqnarray}
\nonumber \lefteqn{P_\mu {\cal S}(k_+) \,\Gamma_{5\mu}^j(k;P)\, {\cal S}(k_-) = i \gamma_5\frac{\tau^j}{2} \, {\cal S}(k_-) }\\
\nonumber
&+& {\cal S}(k_+)\, i \gamma_5\frac{\tau^j}{2} - {\cal S}(k_+) \{{\cal M}(\zeta)\, , \,i\Gamma_5^j(k;P)\} {\cal S}(k_-)\,.\\
\label{avwti0}
\end{eqnarray}
It can be derived following the method in Refs.\,\cite{bicudo,marisbicudo} if, and only if, every dressed-quark propagator that appears is obtained from the rainbow DSE and the accompanying dressed vertices are determined from the ladder Bethe-Salpeter equation, both of which have just been defined.
Using the identity in Fig.\,\ref{figAVWTI} it can be shown \cite{lc03} that
\begin{equation}
\label{anomaly}
P_\rho T^3_{5\mu\nu\rho}(k_1,k_2) + 2 i m(\zeta) \, T^3_{5\mu\nu}(k_1,k_2) = \frac{\alpha}{2 \pi} \varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,,
\end{equation}
where $\alpha= e^2/(4\pi)$. This is an explicit demonstration that the triangle-diagram representation of the axial-vector--two-photon coupling calculated in the rainbow-ladder truncation is a necessary and sufficient pairing to preserve the Abelian anomaly.
In general the coupling of an axial-vector current to two photons is described by a six-point Schwinger function, to which Eq.\,(\ref{Tmnr}) is an approximation. The same is true of the pseudoscalar--two-photon coupling and its connection with Eq.\,(\ref{Pmnr}). Equation~(\ref{anomaly}) is valid for any and all values of $P^2=(k_1+k_2)^2$. It is an exact statement of a divergence relation between these two six-point Schwinger functions, which is preserved by the truncation we will subsequently employ in illustrative quantitative studies. Before providing those illustrations, however, we derive corollaries of Eq.\,(\ref{anomaly}) that have important implications for the properties of pseudoscalar bound states.
If one inserts Eqs.\,(\ref{genavv}) and (\ref{genpv}) into Eq.\,(\ref{anomaly}) and uses Eq.\,(\ref{gmorgen}), one finds that in the neighbourhood of each electric-charge-neutral pseudoscalar-meson bound-state pole
\begin{eqnarray}
\nonumber \lefteqn{P_\rho T_{5\mu\nu\rho}^{3\,{\rm reg}}(k_1,k_2) + 2 i m(\zeta) \, T_{5\mu\nu}^{3\,{\rm reg}}(k_1,k_2)}\\
& & + f_{\pi_n} \,T^{\pi_n^0}_{\mu\nu}(k_1,k_2) = \frac{\alpha}{2 \pi} i\varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,.
\label{reganomaly}
\end{eqnarray}
In this equation, $T^{3\,{\rm reg}}(k_1,k_2)$ are nonresonant or \emph{continuum} contributions to the relevant Schwinger functions, whose form is concretely illustrated herein upon substitution of $\Gamma_{5 \mu}^{j\,{\rm reg}}(k;P)$ and $\Gamma_{5}^{j\,{\rm reg}}(k;P)$ into Eqs.\,(\ref{Tmnr}) and (\ref{Pmnr}), respectively. Moreover, $T^{\pi_n^0}$ is the six-point Schwinger function describing the bound state contribution, which in rainbow-ladder truncation is realised as
\begin{eqnarray}
\nonumber T^{\pi_n^0}_{\mu\nu}(k_1,k_2) &=& {\rm tr}\int_\ell^{M\to\infty} \!\! {\cal S}(\ell_{0+}) \, \Gamma_{\pi_n^0}(\ell_{-\frac{1}{2}\frac{1}{2}};P) \, {\cal S}(\ell_{-0}) \\%
&\times& \, i{\cal Q}\Gamma_\mu(\ell_{-0},\ell) \, {\cal S}(\ell) \, i {\cal Q}\Gamma_\nu(\ell,\ell_{0+}).
\label{Tpingg}
\end{eqnarray}
This Schwinger function describes the direct coupling of a pseudoscalar meson to two photons. The support properties of the bound state Bethe-Salpeter amplitude guarantee that the renormalised Schwinger function is finite so that the regularising parameter can be removed; i.e., $M\to \infty$, in general and in our truncation, Eq.\,(\ref{Tpingg}).
We note that owing to the $O(4)$ (Euclidean Lorentz) transformation properties of each term on the l.h.s.\ in Eq.\,(\ref{anomaly}), one may write
\begin{eqnarray}
P_\rho T_{5\mu\nu\rho}^{3\,{\rm reg}}(k_1,k_2) & = & \frac{\alpha}{\pi} i\varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,A^{3\,{\rm reg}}(k_1,k_2) \,,\; \\
T_{5\mu\nu}^{3\,{\rm reg}}(k_1,k_2) & = & \frac{\alpha}{\pi} i\varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,P^{3\,{\rm reg}}(k_1,k_2)\,,\; \\
T^{\pi_n^0}_{\mu\nu}(k_1,k_2) & = & \frac{\alpha}{\pi} i\varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,G^{\pi_n^0}(k_1,k_2)\,, \; \label{TGdef}
\end{eqnarray}
so that Eq.\,(\ref{anomaly}) can be compactly expressed as
\begin{equation}
\label{reganomaly0}
A^{3\,{\rm reg}}(k_1,k_2) + 2 i m(\zeta) P^{3\,{\rm reg}}(k_1,k_2) + f_{\pi_n} G^{\pi_n^0}(k_1,k_2) = \frac{1}{2}.
\end{equation}
It has been proven \cite{andreasrapid} that in the chiral limit
\begin{equation}
\label{fpizero}
f_{\pi_n}^0 \equiv 0\; \forall n\geq 1.
\end{equation}
Hence it follows from Eq.\,(\ref{reganomaly}) that in this limit all pseudoscalar mesons, \emph{except} the Goldstone mode, decouple from the divergence of the axial-vector--two-photon vertex. (This is true unless $G^{\pi_n^0}(k_1,k_2)$ diverges in the chiral limit, which is not the case, as we will see.)
In the chiral limit the pole associated with the ground state pion appears at $P^2=0$ and thus
\begin{eqnarray}
\nonumber
\lefteqn{\left. P_\rho T_{5\mu\nu\rho}^{3}(k_1,k_2)\right|_{P^2\neq 0}}\\
&& = \left.P_\rho T_{5\mu\nu\rho}^{3\,{\rm reg}}(k_1,k_2)\right|_{P^2\neq 0} = \frac{\alpha}{2 \pi} i\varepsilon_{\mu\nu\rho\sigma} k_{1\rho} k_{2\sigma}\,;
\end{eqnarray}
namely, outside the neighbourhood of the ground state pole the regular (or continuum) part of the divergence of the axial-vector vertex saturates the anomaly in the divergence of the axial-vector--two-photon coupling.
On the other hand, in the neighbourhood of $P^2=0$
\begin{eqnarray}
\left. A^{3\,{\rm reg}}(k_1,k_2) \right|_{ P^2\simeq 0} + f_{\pi_0} \,G^{\pi_0}(k_1,k_2)
& =& \frac{1}{2}\,; \label{anomalypion}
\end{eqnarray}
i.e., on this domain the contribution to the axial-vector--two-photon coupling from the regular part of the divergence of the axial-vector vertex combines with the direct $\pi_0^0 \gamma \gamma$ vertex to fulfill the anomaly. This fact was illustrated in Ref.\,\cite{mrpion} by direct calculation: Eqs.\,(\ref{bwti}) -- (\ref{hwti}) are an essential part of that demonstration.
If one defines
\begin{equation}
\label{TpiG}
{\cal T}_{\pi_n^0}(P^2,Q^2) = \left. G^{\pi_n^0}(k_1,k_2) \right|_{k_1^2=Q^2=k_2^2},
\end{equation}
in which case $ P^2= 2(k_1\cdot k_2+Q^2)$, then the physical width of the neutral ground state pion is determined by
\begin{equation}
g_{\pi_0^0 \gamma\gamma}:= {\cal T}_{\pi_0^0}(-m_{\pi_0^0}^2,0) ;
\end{equation}
viz., the second term on the l.h.s.\ of Eq.\,(\ref{anomalypion}) evaluated at the on-shell points. This result is not useful unless one has a means of estimating the contribution from the first term; viz., $A^{3\,{\rm reg}}(k_1,k_2)$. However, that is readily done. A consideration \cite{mrt98} of the structure of the regular piece in Eq.\,(\ref{genavv}) indicates that the impact of this continuum term on the $\pi_0^0 \gamma\gamma$ coupling is modulated by the magnitude of the pion's mass, which is small for realistic $u$ and $d$ current-quark masses and vanishes in the chiral limit. One therefore expects this term to contribute very little and anticipates from Eq.\,(\ref{anomalypion}) that
\begin{equation}
\label{anomalycouple}
g_{\pi_0^0 \gamma\gamma} = \frac{1}{2} \frac{1}{f_{\pi_0}}
\end{equation}
is a good approximation. This is verified in explicit calculations; e.g., in Ref.\,\cite{maristandypi0}, which evaluates the triangle diagrams described herein, the first term on the l.h.s.\ modifies the result in Eq.\,(\ref{anomalycouple}) by less than 2\%.
There is no reason to expect an analogous result for pseudoscalar mesons other than the $\pi(140)$; i.e., the states which we denote by $n\geq 1$. Indeed, as all known such pseudoscalar mesons have experimentally determined masses that are greater than $1\,$GeV, the reasoning used above suggests that the presence of the continuum terms, $A^{3\,{\rm reg}}(k_1,k_2)$ and $P^{3\,{\rm reg}}(k_1,k_2)$, must materially impact upon the value of $g_{\pi_n^0 \gamma\gamma}$. This will subsequently be illustrated using the rainbow-ladder truncation.
\subsection{Asymptotic behaviour of transition form factor}
\label{exactUV}
We have stated that the rainbow-ladder truncation preserves the one-loop renormalisation group properties of QCD. It follows that Eq.\,(\ref{Tpingg}) should reproduce the leading large-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_n(P) \gamma^\ast(Q)$ transition form factor inferred from perturbative QCD. The QCD analysis has been performed for the ground state pion ($n=0$) with the result \cite{uvQQ}
\begin{equation}
\label{TpiuvQCD}
{\cal T}_{\pi_0^0}(P^2=-m_{\pi_0}^2,Q^2) \stackrel{Q^2\gg \Lambda_{\rm QCD}^2}{=} \frac{4\pi^2}{3} \frac{f_{\pi_0}}{Q^2}\,,
\end{equation}
and Ref.\,\cite{kekez} verified that this is indeed the result contained in Eq.\,(\ref{Tpingg}). However, it is useful for our purposes to recapitulate on that derivation.
Consider Eq.\,(\ref{Tpingg}): the integral is finite and hence a shift in the integration variable is permitted,
\begin{eqnarray}
\nonumber \lefteqn{T^{\pi_n^0}_{\mu\nu}(k_1,k_2) = {\rm tr}\int_\ell^{M\to \infty} \!\! \chi_{\pi_n^0}(\ell;P) }\\%
\nonumber &\times& i{\cal Q} \Gamma_\mu(\ell_{-P},\ell_{K}) \, {\cal S}(\ell_{K}) \, i {\cal Q}\Gamma_\nu(\ell_{K},\ell_{P}),\\
\label{TpinggN}
\end{eqnarray}
where $\ell_P:= \ell_{\frac{1}{2}\frac{1}{2}}= \ell + P/2$ and $\ell_K:= \ell_{\frac{1}{2}-\frac{1}{2}}=: \ell + K$. We assume that $k_1^2=Q^2=k_2^2$ with $Q^2\gg \Lambda_{\rm QCD}^2$ and, because we do not restrict ourselves to ground state pseudoscalar mesons, assume besides that for the given $n$ under consideration $Q^2 \gg m_{\pi_n}^2$. On this domain $K\cdot P\equiv 0$, $K^2=Q^2$, and it is valid at leading $(1/Q^2)$-order in Eq.\,(\ref{TpinggN}) to write \cite{cdrcroat,pctcroat}
\begin{eqnarray}
\label{expand}
\nonumber && i{\cal Q} \Gamma_\mu(\ell_{-P},\ell_{K}) \, {\cal S}(\ell_{K}) \, i {\cal Q}\Gamma_\nu(\ell_{K},\ell_{P}) \\
& = & Z_2 \, i{\cal Q} \gamma_\mu \, \frac{-i\gamma\cdot \ell_{K}}{\ell_{K}^2} \, i{\cal Q} \gamma_\nu
\end{eqnarray}
so that
\begin{eqnarray}
\nonumber \lefteqn{ T^{\pi_n^0}_{\mu\nu}(k_1,k_2) }\\
\nonumber &=& \frac{4 \pi \alpha}{3}\, i\varepsilon_{\mu\nu\rho\sigma}\, {\rm tr} \, Z_2 \int_\ell^{M} \!\! \sfrac{1}{2} \tau^3\, \gamma_5 \gamma_\sigma \, \chi_{\pi_n^0}(\ell;P) \, \frac{(\ell_{K})_\rho}{\ell_{K}^2}.\\
\label{TpinggNa}
\end{eqnarray}
Since we are concerned with $J^{PC} = 0^{-+}$ states, it follows that
\begin{eqnarray}
\nonumber \lefteqn{ T^{\pi_n^0}_{\mu\nu}(k_1,k_2) = \frac{4 \pi \alpha}{3}\, i\varepsilon_{\mu\nu\rho\sigma}}\\
& & \times \left[K_\rho {\cal I}_\sigma(K,P) - K_\alpha {\cal J}_{\rho\sigma\alpha}(K,P)\right],
\label{Tanswer}
\end{eqnarray}
where Eq.\,(\ref{TpinggNa}) yields
\begin{eqnarray}
\nonumber
\lefteqn{{\cal I}_\sigma(K,P) }\\
\nonumber
&= & {\rm tr} \, Z_2 \int_\ell^{M} \!\! \sfrac{1}{2} \tau^3\, \gamma_5 \gamma_\sigma \, \chi_{\pi_n^0}(\ell;P) \,(\ell^2 + K^2) \, \Delta(\ell,K) \\ \label{Ires}\\
\nonumber
\lefteqn{K_\alpha {\cal J}_{\rho\sigma\alpha}(K,P)}\\
\nonumber
&=& {\rm tr} \, Z_2 \int_\ell^{M} \!\! \sfrac{1}{2} \tau^3\, \gamma_5 \gamma_\sigma \, \chi_{\pi_n^0}(\ell;P) \, 2 \, \ell_\rho\, \ell\cdot K \, \Delta(\ell,K)\\
\label{Jres}
\end{eqnarray}
with $\Delta(l,K) = 1/[(\ell^2+K^2)^2 - 4 (\ell\cdot K)^2]$.
As we show in the Appendix, on the large-$Q^2$ domain, that part of ${\cal I}_\sigma(K,P)$ which contributes to $T^{\pi_n^0}_{\mu\nu}(k_1,k_2)$ is
\begin{equation}
{\cal I}_\sigma(K,P) = P_\sigma
\left\{ \frac{f_{\pi_n}}{Q^2} + F^{(2)}_{\cal I}(P^2) \frac{\ln^{\gamma} Q^2/\omega_{\pi_n}^2}{Q^4} \right\}, \label{Iuv}
\end{equation}
$P^2= -m_{\pi_n}^2$, where $\gamma$ is an anomalous dimension and $\omega_{\pi_n}$ is a mass-scale associated with the momentum space width of the meson's Bethe-Salpeter wave function. Similar reasoning exposes the leading contribution to Eq.\,(\ref{Tanswer}) from Eq.\,(\ref{Jres}):
\begin{equation}
K_\alpha {\cal J}_{\rho\sigma\alpha}(K,P) = K_\rho P_\sigma \, F^{(2)}_{\cal J}(P^2)\frac{\ln^{\gamma} Q^2/\omega_{\pi_n}^2}{Q^4} \,,
\end{equation}
$P^2= -m_{\pi_n}^2$. Combining these results one arrives at
\begin{eqnarray}
\nonumber \lefteqn{
T^{\pi_n^0}_{\mu\nu}(k_1,k_2) \stackrel{Q^2\to \infty}{=} \frac{4 \pi \alpha }{3} i\varepsilon_{\mu\nu\rho\sigma}\, k_{1\rho} k_{2\sigma} }\\
&\times & \left[\frac{f_{\pi_n}}{Q^2} + F^{(2)}_{n }(P^2)\frac{\ln^{\gamma} Q^2/\omega_{\pi_n}^2}{Q^4} \right].\label{enduv}
\end{eqnarray}
We emphasise that the coefficient of the leading $1/Q^2$-term in Eq.\,(\ref{enduv}) is exact and model-independent.
That is not true of the subleading $1/Q^4$ term. Furthermore, with a given \textit{Ansatz} for ${\cal G}(k^2)$ in Eqs.\,(\ref{rainbowdse}) and (\ref{ladderK}), Eq.\,(\ref{expand}) is not sufficient to accurately determine the value of the coefficient of the $1/Q^4$ term or the anomalous dimension because, for example, momentum-dependent dressing of the quark-photon vertex can contribute at this order. Nevertheless, our analysis highlights the existence of a nonzero subleading $1/Q^4$ contribution whose strength is sensitive to features of the dynamics. These observations were made previously for the ground state ($n=0$) pion \cite{yeh}.
We can now return to one of the stated reasons for this analysis: Eq.\,(\ref{enduv}) inserted in Eq.\,(\ref{TGdef}) and combined with Eq.\,(\ref{TpiG}) reproduces the leading order result obtained in perturbative QCD, Eq.\,(\ref{TpiuvQCD}). In fact, it provides more. The perturbative result was only derived for the ground state pseudoscalar meson. Our analysis shows that for each meson on the pseudoscalar trajectory, identified herein by a value of $n$, QCD predicts
\begin{eqnarray}
\nonumber
\lefteqn{{\cal T}_{\pi_n^0}(-m_{\pi_n}^2,Q^2) \stackrel{Q^2\gg \Lambda_{\rm QCD}^2}{=} \frac{4\pi^2}{3}}\\
& \times & \left[ \frac{f_{\pi_n}}{Q^2} + F_n^{(2)}(-m_{\pi_n}^2)
\frac{\ln^{\gamma} Q^2/\omega_{\pi_n}^2}{Q^4}
\right] .
\label{UVnot0}
\end{eqnarray}
It is now apparent from Eq.\,(\ref{fpizero}) that $\forall n\geq 1$
\begin{eqnarray}
\nonumber \lefteqn{\lim_{\hat m\to 0} {\cal T}_{\pi_n^0}(-m_{\pi_n}^2,Q^2) }\\
&& \stackrel{Q^2\gg \Lambda_{\rm QCD}^2}{=} \frac{4\pi^2}{3}\left. F^{(2)}_{n }(-m_{\pi_n}^2)\frac{\ln^{\gamma} Q^2/\omega_{\pi_n}^2}{Q^4}\right|_{\hat m=0} \,;
\label{UVchiralnot0}
\end{eqnarray}
namely, in the chiral limit the leading-order power-law in the transition form factor for excited state pseudoscalar mesons is O$(1/Q^4)$. This result is model-independent.
Furthermore, while we cannot determine the QCD value of the coefficient $F_n^{(2)}(-m_{\pi_n}^2)$ in the present truncation, in general that coefficient is \emph{not} proportional to $f_{\pi_n}$, or some power thereof, for any value of $n$. We will see this clearly in the $n\geq 1$ transition form factor for which, if that were the case, the $1/Q^4$-term would be absent in the chiral limit. For all pseudoscalar states there are mass-scales other than $f_\pi$ that are nonzero even in the chiral limit when chiral symmetry is dynamically broken.
\section{Couplings of Pseudoscalar Mesons: Model Results}
\label{model}
\subsection{Rainbow-ladder truncation}
\label{sec:rl}
In order to illustrate the results presented above and calculate other observables it is necessary to specify ${\cal G}(k^2)$ in Eqs.\,(\ref{rainbowdse}) and (\ref{ladderK}). We choose
\begin{equation}
\label{calG}
\frac{{\cal G}(s)}{s} = \frac{4\pi^2}{\omega^6} \, D\, s\, {\rm e}^{-s/\omega^2}+ \frac{8\pi^2 \gamma_m}{\ln\left[ \tau + \left(1+s/\Lambda_{\rm QCD}^2\right)^2\right]} \, {\cal F}(s)\,,
\end{equation}
with ${\cal F}(s)= [1-\exp(-s/[4 m_t^2])]/s$, $m_t=0.5\,$GeV, $\ln(\tau+1)=2$, $\gamma_m=12/25$ and $\Lambda_{\rm QCD} = \Lambda^{(4)}_{\overline{MS}} = 0.234\,$GeV.
This form expresses the interaction as a sum of two terms. The second guarantees Eq.\,(\ref{calGuv}) and therefore ensures that perturbative behaviour is correctly realised at short range; namely, as written, for $(k-q)^2 \sim k^2 \sim q^2 \gtrsim 1 - 2\,$GeV$^2$, $K$ is precisely as prescribed by QCD. On the other hand, the first term in ${\cal G}(k^2)$ is a model for the long-range behaviour of the interaction. It is a finite width representation of the form introduced in Ref.\,\cite{mn83}, which has been rendered as an integrable regularisation of $1/k^4$ \cite{mm97}. This interpretation, when combined with the result that in a heavy-quark--heavy-antiquark BSE the renormalisation-group-improved ladder truncation is exact \cite{mandarvertex}, is consistent with ${\cal G}(k^2)$ leading to a Richardson-like potential \cite{richardson} between static sources.
The active parameters in Eq.\,(\ref{calG}) are $D$ and $\omega$, which together determine the integrated infrared strength of the rainbow-ladder kernel, but they are not independent. In fitting a selection of ground state observables \cite{mt99}, a change in one is compensated by altering the other; e.g., on the domain $\omega\in[0.3,0.5]\,$GeV, the fitted observables are approximately constant along the trajectory
\begin{equation}
\omega D = (0.72 \, {\rm GeV})^3 =: m_g^3\,.
\end{equation}
(NB. The value of $m_g$ is typical of the mass-scale associated with nonperturbative gluon dynamics.) Herein, unless otherwise stated, we use
\begin{equation}
\label{omegavalue}
\omega= 0.35\,{\rm GeV.}
\end{equation}
Equation~(\ref{calG}) defines a renormalisation-group-im\-proved rainbow-ladder truncation. This form, introduced in Refs.\,\cite{mr97,mt99}, has been employed extensively in the calculation of properties of ground state pseudoscalar and vector mesons \cite{fn:jain}. These applications are reviewed in Ref.\,\cite{pieterrev}, from which it is apparent that the model describes a basket of thirty-one hadron observables with a rms error between calculation and experiment of $15$\%.
The calculation of observables is now straightforward. The kernel of the gap equation, Eq.\,(\ref{rainbowdse}), is completely specified. Thus a solution follows immediately upon fixing the current-quark mass: this sets the \emph{boundary condition}, Eq.\,(\ref{renormS}). We focus on the $u$-$d$ sector and assume isospin symmetry:
\begin{equation}
\hat m_u=\hat m_d= \hat m\,.
\end{equation}
With a result for the dressed-quark propagator in hand, the kernel of Bethe-Salpeter equations is also complete. The solutions of these equations yield: the bound state Bethe-Salpeter amplitudes; the axial-vector and pseudoscalar vertices; and the dressed-quark-photon vertex, all of which appear above. At this point one has every element necessary for the calculation of an amplitude such as Eq.\,(\ref{TpinggN}) and therewith experimental observables. The numerical procedures are described in Refs.\,\cite{mr97,mt99,mt00,krassnigg1}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig2.eps}
\caption{\label{fig:Tpiggnmm} Small-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_n(P) \gamma^\ast(Q)$ transition form factor, defined in Eq.\,(\protect\ref{TpiG}), calculated with the current-quark mass in Eq.\,(\protect\ref{qmass}). The ground state's two-photon coupling suggested by Eq.\,(\protect\ref{anomalycouple}) is marked by ``$\times$''. }
\end{center}
\end{figure}
\subsection{Two photon couplings of pseudoscalar mesons}
Figure~\ref{fig:Tpiggnmm} depicts the small-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_n(P) \gamma^\ast(Q)$ transition form factor defined in Eq.\,(\protect\ref{TpiG}), calculated for the two lowest-mass $0^{-+}$ states with
\begin{equation}
\label{qmass}
m(\zeta_0) := \frac{\hat m}{(\ln\zeta_0/\Lambda_{\rm QCD})^{\gamma_m}} = 5.5\,{\rm MeV}\,,\; \zeta_0= 1\,{\rm GeV}\,.
\end{equation}
(Recall that in this model the $n=1$ state is a radial excitation.) It is notable that while ${\cal T}_{\pi_0^0}(-m_{\pi_0}^2,Q^2)>0$,
\begin{equation}
{\cal T}_{\pi_1^0}(-m_{\pi_1}^2,Q^2)<0 \,, \; Q^2\geq -m_{\pi_1}^2/4 ;
\end{equation}
viz., it is negative on the entire kinematically accessible domain. Moreover, for nonzero current-quark mass we expect the sign of this form factor to duplicate the pattern set by the leptonic decay constant, which is $(-1)^n$ \cite{andreasrapid}. NB.\ On the depicted domain and with the resolution in this figure there is no perceptible difference between these curves and those obtained in the chiral limit. That is not true for larger $Q^2$, as will become apparent.
The coupling constants for decay into two real photons are presented in Table~\ref{table:couplings}, as are the associated decay widths, calculated using
\begin{equation}
\label{ggwidth}
\Gamma_{\pi^0_n \gamma\gamma} = \alpha_{\rm em}^2\, \frac{m_{\pi_n}^3}{16\pi^3} \, g^2_{\pi_n \gamma\gamma}.
\end{equation}
It is evident from Table~\ref{table:couplings} that Eq.\,(\protect\ref{anomalycouple}) is truly a good approximation for the $\pi(140)$.
\begin{table}[t]
\caption{\label{table:couplings} Results for a range of properties of the two lowest mass $0^{-+}$ mesons. Note that for $n=0$, Eq.\,(\protect\ref{anomalycouple}) yields: chiral limit, $5.68\,$GeV$^{-1}$; massive, Eq.\,(\protect\ref{qmass}), $5.41\,$GeV$^{-1}$. Decay widths: calculated from Eqs.\,(\protect\ref{ggwidth}); value known experimentally \cite{pdg}: $\Gamma_{\pi_0 \gamma\gamma}=7.84\pm 0.56\,$eV. Also \protect\cite{pdg}: $m_{\pi_0} = 0.14\,$GeV; $m_{\pi_1} = 1.3\pm 0.1\,$GeV. [NB.\ Our best estimate is $\Gamma_{\pi_1^0 \gamma\gamma} \approx 240$eV, for reasons presented in connection with Eq.\,(\protect\ref{Gpiggbest}).]}
\begin{ruledtabular}
\begin{tabular*}
{\hsize} {l@{\extracolsep{0ptplus1fil}}
l@{\extracolsep{0ptplus1fil}}|l@{\extracolsep{0ptplus1fil}}
l@{\extracolsep{0ptplus1fil}}l@{\extracolsep{0ptplus1fil}}l@{\extracolsep{0ptplus1fil}}}
& & $m_n\,$ & $f_n\,$ & $g_{{\pi_n}\gamma\gamma}$ & $\Gamma_{\pi_n^0 \gamma\gamma}$ \\
& & (GeV) & (GeV) & (GeV)$^{-1}$ & (eV) \\\hline
$\pi_0$ & $\hat m =0$ & $0.0$ & $\;\;\;0.088$ & $\;\;\;5.31$ & \\
& $\hat m$, Eq.\,(\protect\ref{qmass})~ & $0.14$ & $\;\;\;0.092$ & $\;\;\;5.25$ & $\;\;7.9$ \\
$\pi_1$ & $\hat m =0$ & $1.04$ & $\;\;\;0.0$ & $-0.71$ \\
& $\hat m$, Eq.\,(\protect\ref{qmass})~ & $1.06$ & $-0.0016$ & $-0.70$ & $63.0$\\
\end{tabular*}
\end{ruledtabular}
\end{table}
The result for $g_{\pi_1 \gamma\gamma}$ is, however, striking. This coupling is negative because the $\pi_1$'s Bethe-Salpeter amplitude has a significant domain of negative support \cite{andreasrapid}; and while its magnitude is material, $\sim 0.13\,g_{\pi_0 \gamma\gamma}$, it is finite even in the chiral limit. The last fact demonstrates that the $\pi_1 \gamma\gamma$ coupling is not inversely proportional to $f_{\pi_1}$ cf.\ Eq.\,(\ref{anomalycouple}). This confirms that the excited state decouples from the axial-vector--two-photon vertex in the chiral limit, as described in connection with Eq.\,(\ref{reganomaly0}). Consequently, the evolution with $P^2$ of the regular (or continuum) part of the divergence of the axial-vector--two-photon vertex is smooth; i.e.,
\begin{equation}
\left. A^{3\,{\rm reg}}(k_1,k_2) \right|_{ P^2\simeq -m_{\pi_1}^2} \approx \left. A^{3\,{\rm reg}}(k_1,k_2) \right|_{ P^2= -m_{\pi_1}^2} \,,
\end{equation}
and in addition
\begin{equation}
\left[A^{3\,{\rm reg}}(k_1,k_2) + 2 i m(\zeta) P^{3\,{\rm reg}}(k_1,k_2) \right]_{ P^2= -m_{\pi_1}^2} \approx \frac{1}{2}\,,
\end{equation}
with exact equality for $\hat m =0$.
In Fig.\,\ref{fig:UV01m} we depict the large-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_n(P) \gamma^\ast(Q)$ transition form factor obtained with the nonzero current-quark mass in Eq.\,(\ref{qmass}), for the two lowest mass pseudoscalars. The ultraviolet behaviour anticipated for the ground state from perturbative QCD, Eq.\,(\ref{TpiuvQCD}), is evident. This is a numerical verification of the argument associated with Eqs.\,(\ref{TpinggN}) -- (\ref{UVchiralnot0}); viz., that the truncation we employ preserves leading-order QCD results. The analogous result for the first excited state, indicated by Eq.\,(\ref{UVnot0}), is also conspicuous.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig3.eps}
\caption{\label{fig:UV01m} Calculated large-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_n(P) \gamma^\ast(Q)$ transition form factor, Eq.\,(\protect\ref{TpiG}): \textit{diamonds} -- ground state, $n=0$; and \textit{circles} -- first excited state, $n=1$. The \textit{solid-lines} are Eq.\,(\protect\ref{TpiuvQCD}) with either $f_{\pi_0}$ or $f_{\pi_1}$ from Table~\protect\ref{table:couplings}, as appropriate.}
\end{center}
\end{figure}
For the ground state the behaviour of the transition form factor in the chiral limit is not markedly different from that found with $\hat m$ in Eq.\,(\ref{qmass}) and illustrated in Fig.\,\ref{fig:UV01m}. As evident in Fig.\,\ref{fig:UV1chiral}, that is not the case for $\gamma^\ast(Q) \pi_1(P) \gamma^\ast(Q)$ in the chiral limit. While the form factor is initially negative, as may be anticipated from Fig.\,\ref{fig:Tpiggnmm}, it is positive for $Q^2 \gtrsim 8\,$GeV$^2$ and the asymptotic behaviour indicated in Eq.\,(\ref{UVchiralnot0}) is exhibited for $Q^2\gtrsim 50\,$GeV$^2$. With the model's parameter value specified in Eq.\,(\ref{omegavalue}), we find
\begin{equation}
\label{Fnvalue}
\left. F_1^{(2)}(-m_{\pi_1}^2) \, \ln^{\gamma} Q^2/\omega_{\pi_1}^2\right|_{\hat m=0} \approx (0.22\,{\rm GeV})^3.
\end{equation}
This mass-scale is commensurate with that set by the vacuum quark condensate. The magnitude of $F_1^{(2)}$ depends on the model parameter. So, too, does the precise location of the boundary between the domains on which the transition form factor has negative and positive support. However, qualitative features, such as the existence of these domains, are robust.
It is noteworthy that while $f_{\pi_1}\equiv 0$ algebraically in the chiral limit, in practice there is always a numerical error. Hence, as is plain from Eq.\,(\ref{UVnot0}), there will inevitably be a value of $Q^2$ beyond which the erroneous nonzero value of $f_{\pi_1}$, produced by the numerical error, will come to dominate the chiral-limit transition form factor. To obtain the value in Eq.\,(\ref{Fnvalue}) we estimated the magnitude of this pollution and subtracted it. For this reason, within the accuracy of our numerical analysis, we cannot provide reliable information on the $\ln Q^2$-modification. The figure hints, however, at the presence in our model of such a modification to the $1/Q^4$-behaviour.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig4.eps}
\caption{\label{fig:UV1chiral} Large-$Q^2$ behaviour of the $\gamma^\ast(Q) \pi_1(P) \gamma^\ast(Q)$ transition form factor, Eq.\,(\protect\ref{TpiG}): \textit{Diamonds} -- the result obtained with $\hat m$ in Eq.\,(\protect\ref{qmass}); \textit{Circles} -- our chiral limit calculation ($\hat m = 0$); \textit{Solid line} -- the curve $\frac{4\pi^2}{3} (0.22\,{\rm GeV})^3/Q^4$.}
\end{center}
\end{figure}
\subsection{Charge radii}
At leading order in the truncation scheme we are using, and in the isospin symmetric limit, the elastic electromagnetic form factor of a pseudoscalar meson is described by
\begin{eqnarray}
\nonumber
\lefteqn{
e\, (p_1+p_2) \, F_{\pi_n}(Q^2) := e \, \Lambda_{\mu}(p_1,p_2)}\\
\nonumber
& = & {\rm tr} \int_\ell %
\chi_{\pi_n}(\ell_{0,\frac{1}{2}})\,
i {\cal Q} \Gamma_{\mu}(\ell_{-\frac{1}{2}\frac{1}{2}},\ell_{\frac{1}{2}-\frac{1}{2}})\\
&& \times \,
\chi_{\pi_n}(\ell_{\frac{1}{2}0};-p_2) \,
{\cal S} (\ell_{\frac{1}{2}\frac{1}{2}})^{-1}\,, \label{piem}
\end{eqnarray}
with $Q=p_1-p_2$. Each element that appears in the integrand is fully renormalised and the integral is finite. The expression automatically satisfies \cite{mt00,cdrpion}
\begin{equation}
(p_1-p_2)_\mu \, \Lambda_{\mu}(p_1,p_2) = 0\,
\end{equation}
and guarantees
\begin{equation}
F_{\pi_n}(Q^2=0) = 1\,.
\end{equation}
In Ref.\,\cite{mt00} the model described in Sec.\,\ref{sec:rl} was employed to calculate the electromagnetic form factor of the pion using Eq.\,(\ref{piem}). The prediction was subsequently verified in a JLab experiment performed at intermediate $Q^2$ \cite{volmer}.
We have calculated the charge radii of the two lowest mass pseudoscalars using the standard definition:
\begin{equation}
\label{usualradius}
r_{\pi_n}^2 = - 6 \, F_{\pi_n}^\prime(Q^2=0)\,.
\end{equation}
Our results appear in Fig.\,\ref{fig:emradii}. As promised in association with Eq.\,(\ref{omegavalue}), the ground state's properties are almost insensitive to the model's mass-scale, $\omega$: in formulating the model, a path appeared in the $(D,\omega)$ parameter space along which vacuum and ground state properties vary little. The orthogonality of the excited states with respect to the ground state means there is no reason to expect such insensitivity in properties of the excited states. And, indeed, one observes that the charge radius of the first excited state changes rapidly with increasing $\omega$, with the ratio $r_{\pi_1}/r_{\pi_0}$ varying from $0.9$ -- $1.2$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig5.eps}
\caption{\label{fig:emradii} Evolution of ground and first excited state pseudoscalar mesons' electromagnetic charge radii with the model's scale parameter $\omega$. \textit{Dotted line}: $r_\pi=0.66\,$fm, which indicates the experimental value of the ground state's radius. We must estimate the derivative in Eq.\,(\protect\ref{usualradius}) numerically. That is the primary source of the numerical error depicted in the figure, which corresponds to a relative error $\lesssim 1$\% for $n=0$ and $\lesssim 3$\% for $n=1$.}
\end{center}
\end{figure}
This outcome can readily be interpreted. The length-scale $r_a := 1/\omega$ measures the range of strong attraction in our model: magnifying $r_a$ increases the range of strong attraction. In Sec.\,\ref{sec:intro} we argued that the properties of radial excitations should be sensitive to the nature of the interaction between light quarks at long-range. It is now apparent that this is true. Moreover, decreasing $\omega$ has the effect of increasing the active range of the confining piece of the interaction in Eq.\,(\ref{calG}). This effectively strengthens the confinement force. That compresses the bound state, as one observes in Fig.\,\ref{fig:emradii}: $r_{\pi_1}$ decreases rapidly with decreasing $\omega$ (increasing $r_a$).
A similar result for the evolution of the mass was observed in Ref.\,\cite{andreasrapid}; namely, the mass of the first excited state dropped rapidly with increasing $r_a$. On the domain illustrated in Fig.\,\ref{fig:emradii}, the mass of the ground state obtained with nonzero current-quark mass varied by only 3\% while that of the first excited state changed by 14\%. It is natural to expect that an increase in the strength of the confinement force should increase the magnitude of the binding energy and hence reduce the mass, and that is precisely what occurs. (NB.\ Independent of the parameters, the ground state mass is identically zero in the chiral limit because the truncation is symmetry preserving. Dynamical chiral symmetry breaking, which has many consequences, is another reason why properties of the ground state pseudoscalar meson do not respond rapidly to modest parameter changes.)
It is natural to suppose $r_{\pi_1}>r_{\pi_0}$; namely, that a radial excitation is larger than the associated ground state. However, our calculations illustrate that with the ground state pseudoscalar meson's properties constrained by Goldstone's theorem and its pointwise consequences, Eqs.\,(\ref{bwti}) -- (\ref{hwti}), it is possible for a confining interaction to compress the excited state with the consequence that $r_{\pi_1}<r_{\pi_0}$. An analysis of the $\omega$-dependence of $m_{\pi_1}$ indicates that a value of $1.3\,$GeV may be obtained with $\omega \approx 0.48$ \cite{menu}. However, quantitative difficulties connected with the behaviour of the dressed-quark propagator in the complex-$\ell^2$ plane \cite{jarecke,pichowskycomplex} currently prevent us from studying the excited state directly with $\omega > 0.4$ in Eq.\,(\ref{calG}). Hence, we cannot make a firm prediction for $r_{\pi_1}$. However, our results suggest $1.1 < r_{\pi_1}/r_{\pi_0}<1.6$, with a linear extrapolation giving
\begin{equation}
r_{\pi_1}\simeq 1.4\,r_{\pi_0}\,.
\end{equation}
Naturally, we have also studied the evolution of $g_{\pi_n \gamma \gamma}$ with $\omega$. On the domain illustrated in Fig.\,\ref{fig:emradii}, $g_{\pi_0 \gamma \gamma}$ varies by no more than 1\%, whereas $ g_{\pi_1 \gamma \gamma}(\omega=0.3)=-0.55$ and $ g_{\pi_1 \gamma \gamma}(\omega=0.4)=-0.80$, which is a variation over a range of $\sim 40$\%. Following the reasoning above, and taking account of the variation in $m_{\pi_1}$, we conclude that it is likely $\Gamma_{\pi_1\gamma\gamma} > 150\,$eV $\gtrsim 20\,\Gamma_{\pi_0\gamma\gamma}$. Our best estimate is $200 <\Gamma_{\pi_1\gamma\gamma} ({\rm eV}) <300$ and linear extrapolation gives
\begin{equation}
\label{Gpiggbest}
\Gamma_{\pi_1\gamma\gamma} \simeq 240\,{\rm eV}.
\end{equation}
\section{Epilogue}
\label{epilogue}
The strong interaction spectrum exhibits trajectories of mesons with the same spin\,$+$\,parity, $J^P$. One may distinguish between the states on these trajectories by introducing an integer label $n$, with $n=0$ denoting the lowest-mass state, $n=1$ the next-lightest state, etc. The Bethe-Salpeter equation (BSE) yields the mass and amplitude of every bound state in a given channel specified by $J^{P}$. Hence it provides a practical tool for the Poincar\'e covariant study of mesons on these trajectories.
In applying the Bethe-Salpeter equation to a study of pseudoscalar mesons we made use of the fact that at least one nonperturbative and symmetry preserving Dyson-Schwinger equation (DSE) truncation scheme exists. This fact supports a proof that, in the chiral limit, excited state $0^-$ mesons do not couple to the axial-vector current; viz., $f_{\pi_n}\equiv 0$ $\forall n \geq 1$.
We demonstrated that the leading-order (rainbow-ladder) term in the DSE truncation scheme, when consistently implemented, is necessary and sufficient to express the Abelian anomaly. It can therefore be used to illustrate the anomaly's observable consequences. We capitalised on this to show that even though excited state pseudoscalar mesons decouple from the axial-vector current in the chiral limit, they nevertheless couple to two photons. (NB.\ The strength of this coupling is materially affected by the continuum contribution to the Abelian anomaly.) Hence the Primakov process, as employed for example in \emph{PrimEx} at JLab \cite{primex}, may be used as a tool for their production and study.
A renormalisation-group-improved rainbow-ladder trun\-cation is guaranteed to express the one-loop renormalisation group properties of QCD. We exploited this and thereby determined the leading power-law behaviour of the $\gamma^\ast \pi_n \gamma^\ast$ transition form factor. When the current-quark mass is nonzero then, for all $n$, this form factor behaves as $(4\pi^2/3) (f_{\pi_n}/Q^2)$ at deep spacelike momenta. For all but the Goldstone mode this leading order contribution vanishes in the chiral limit. In that case, however, the form factor remains nonzero and the ultraviolet behaviour is $\simeq (4\pi^2/3) (-\langle \bar q q \rangle/Q^4)$. Although only exposed starkly in the chiral limit for excited states, this subleading power-law contribution to the $\gamma^\ast \pi_n \gamma^\ast$ transition form factor is always present and in general its coefficient is not simply related to $f_{\pi_n}$.
As one might rationally expect, the properties of excited ($n\geq 1$) states are sensitive to the pointwise behaviour of what might be called the confinement potential between light-quarks. We illustrated this by laying out the evolution of the charge radii of the $n=0,1$ pseudoscalar mesons. As it is shielded by Goldstone's theorem, the ground state's radius can be insensitive to details of the long-range part of the interaction. However, that is not true of $r_{\pi_1}$, the radius of the first excited state, which is orthogonal to the vacuum. An increase in the length-scale that characterises the range of the confining potential reduces $r_{\pi_1}$. This result states that increasing the confinement force compresses the excited state: indeed, it is possible to obtain $r_{\pi_1} < r_{\pi_0}$. However, our current best estimate is $r_{\pi_1} \simeq 1.4\, r_{\pi_0}$.
A detailed exploration of the properties of collections of mesons on particular $J^P$ trajectories offers the hope of exposing features of the long-range part of the interaction between light-quarks. In principle, this interaction can be quite different to that between heavy-quarks. The pseudoscalar trajectory is of particular interest because its lowest mass entry is QCD's Goldstone mode. Chiral current conservation places constraints on some properties of every member of this trajectory, whose study may therefore provide information about the interplay between confinement and dynamical chiral symmetry breaking.
\bigskip
\centerline{Acknowledgments}\medskip
We acknowledge profitable interactions with S.\,J.~Brodsky, R.\,J.\ Holt and P.\,C.\ Tandy.
This work was supported by: Austrian Research Foundation \textit{FWF,
Erwin-Schr\"odinger-Stipendium} no.\ J2233-N08; Department of Energy,
Office of Nuclear Physics, contract nos.\ W-31-109-ENG-38 and DE-FG02-00ER41135; National Science Foundation contract no.\ INT-0129236; the \textit{A.\,v.\
Humboldt-Stiftung} via a \textit{F.\,W.\ Bessel Forschungspreis}; and benefited from the facilities of the ANL Computing Resource Center and the NSF Terascale Computing System at the Pittsburgh Supercomputing Center.
\nopagebreak
|
{
"timestamp": "2005-03-16T01:27:42",
"yymm": "0503",
"arxiv_id": "nucl-th/0503043",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503043"
}
|
\section{Main Theorem}
\label{intro}
Vector bundles over the projective space
${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ are one of the main subjects in both (algebraic) geometry
and commutative algebra. The most fundamental result in this area is
the theorem due to Grothendieck which asserts that any
holomorphic vector bundle over ${\mathbf{P}}}% \P == \mathbb{P^1_{\mathbb{K}}}% \P == \mathbb{P$ splits into a direct
sum of line bundles.
When $n\geq 2$, vector bundles over ${\mathbf{P}}}% \P == \mathbb{P^n_{{\mathbb{K}}}% \P == \mathbb{P}$ do not necessarily
split. Indeed, the tangent bundle is indecomposable.
In these cases, some sufficient conditions for vector bundles to
split have been established.
The following is one of such criterions, which we call
``Restriction criterion''.
\begin{theorem}[Horrocks]
\label{rest}
Let ${\mathbb{K}}}% \P == \mathbb{P$ be an algebraically closed field,
$n$ be an integer greater than or equal to 3,
and let $E$ be a locally free sheaf on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ of $\rank\ r\ (\ge 1)$.
Then
$E$
splits
into a direct sum of line bundles if and only if
there exists a hyperplane $H \subset {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ such that
$E|_H$ splits into a direct sum of line bundles.
\end{theorem}
In other words,
the splitting of a vector bundle can be characterized
by using a hyperplane section.
However, vector bundles, or equivalently locally free sheaves, form a
small class among all coherent sheaves. There are some
important wider classes of coherent sheaves, e.g.,
reflexive sheaves or torsion free sheaves.
The purpose of this article is to generalize the
``Restriction criterion'' to
one for
reflexive sheaves, and we also show that it fails in the class of
torsion free sheaves.
Our main theorem is as follows.
\begin{theorem}\label{main}
Let ${\mathbb{K}}}% \P == \mathbb{P$ be an algebraically closed field,
$n$ be an integer greater than or equal to 3,
and let $E$ be a reflexive sheaf on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ of $\rank\ r\ (\ge 1)$. Then
$E$
splits
into a direct sum of line bundles if and only if
there exists a hyperplane $H \subset {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ such that
$E|_H$ splits into a direct sum of line bundles.
\end{theorem}
We give two proofs for Theorem \ref{main}. The first proof
is basically parallel to that of Theorem \ref{rest}, in which
we also establish a general principle that the
structure of a reflexive sheaf can be recovered from
its hyperplane section (Theorem \ref{sp}).
The second proof is based on a cohomological characterization
for a coherent sheaf to be locally free. By using it,
the proof is reduced to Theorem \ref{rest}.
The organization of this paper is as follows.
In \S\ref{pre}, we recall some basic results on reflexive sheaves
from \cite{H2}.
In \S\ref{pf}, we give the first proof of the main theorem.
In \S\ref{app}, we give the second proof by using a cohomological
characterization for a coherent sheaf to be locally free.
To each hyperplane arrangement in a vector space,
we can associate a reflexive sheaf over the projective space.
The splitting of this reflexive sheaf defines an important
class of arrangements, namely, free arrangements.
As an application of our main theorem,
we give a criterion for an arrangement
to be free in \S\ref{arr}, which has been also obtained in
\cite{Y}.
\textbf{Acknowledgement.}
The authors learned results of \S\ref{app} from Professor
F.-O. Schreyer. They are grateful to him.
The authors also thank to Takeshi Abe and Florin Ambro for many
helpful comments and pointing out mistakes in our draft.
The second author was supported by the JSPS Research Fellowship for
Young Scientists.
\section{Preliminaries}\label{pre}
In this section, we fix the notation and prepare some results for the proof
of Theorem
{\rmfamily \ref{main}}. We use the terms ``vector bundle''
and ``locally free sheaf''
interchangeably. The term ``variety'' means a integral
scheme of finite type over a field.
Let $X$ be a smooth variety of dimension $n$ over a
field ${\mathbb{K}}}% \P == \mathbb{P$,
where $n \ge 1$ and ${\mathbb{K}}}% \P == \mathbb{P$ is an
algebraically closed field.
For a coherent sheaf $E$ on $X$ we denote by
$\Sing (E)$ the non-free locus of $E$, i.e.,
$\Sing(E):=\{ x \in X|
E_x\ \mbox{is not a free}\
\mathcal{O}_{x,X} \mbox{-module}\}$. The dual of a coherent sheaf
$E$ (on $X$) is denoted by $E^*$.
In this article, we employ homological algebra to investigate
properties of
a coherent sheaf on a smooth variety $X$.
Let us
review some definitions and results. For a coherent sheaf $E$ on
$X$ over ${\mathbb{K}}}% \P == \mathbb{P$ and for a point $x \in X$
(denoted by $\depth_{\mathcal{O}_X} (E_x))$ as the length of
a maximal $E_x$-regular sequence in $\mathcal{M}_x$, where
$\mathcal{M}_x$ is the unique maximal ideal of a local ring
$\mathcal{O}_{x, X}$.
Moreover, we define the projective dimension of an
$\mathcal{O}_{x,X}$-module $E_x$ (denoted by
$\pd_{\mathcal{O}_{x,X}} (E_x))$
as the length of a minimal free resolution of $E_x$ as an
$\mathcal{O}_{x,X}$-module.
It is known that every module which is finitely generated over a regular
local ring has finite projective
dimension. These two quantities
are related by the famous Auslander-Buchsbaum formula as follows.
\[
\depth_{\mathcal{O}_{x,X}}(E_x)+ \pd_{\mathcal{O}_{x,X}}
(E_x)=\dim
\mathcal{O}_{x,X}.
\]
Hence it follows easily that a coherent sheaf $E$ on $X$ is locally
free if and only if
$\depth_{\mathcal{O}_{x,X}} (E_x)= \dim \mathcal{O}_{x,X}$ for
all
$x \in X$. For details and proofs, see \cite{M}.
The projective dimension can also be characterized as follows
(for example, see \cite{OSS} Chapter II).
\begin{lemma}
\label{chara}
Let $X$ be a smooth variety and
$E$ be a coherent sheaf on $X$. Then $\pd_{\mathcal{O}_{x,X}}(E_x)\leq q$
if
and only if for
all $i>q$ we have
$$
\mathcal{E}xt^i_{\mathcal{O}_X}(E, \mathcal{O}_X)_x=0.
$$
\end{lemma}
In particular, $E$ is locally free if and only if
$
\mathcal{E}xt^i_{\mathcal{O}_X}(E, \mathcal{O}_X)=0
$
for all $i>0$.
Next, let us review definitions and results on reflexive sheaves on
${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. Reflexive sheaves form a category between torsion free sheaves
and
vector bundles.
\begin{define}
We say a coherent sheaf $E$ on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ is reflexive if the canonical
morphism
$E \rightarrow E^{**}$ is an isomorphism.
\end{define}
In this article, we use the following results on reflexive sheaves. For the
proofs and details, see \cite{H2}.
\begin{prop}[\cite{H2}, Proposition 1.3]\label{depth}
A coherent sheaf $E$ on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ is reflexive if and only if $E$ is
torsion
free and
$\depth_{\mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}}(E_x) \ge 2$ for all points $x \in
{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$
such that
$\dim \mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P} \ge 2$.
\end{prop}
\begin{cor}[\cite{H2}, Corollary 1.4]
$\codim_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P} \Sing (E) \ge 3$ for a reflexive sheaf $E$ on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$.
\end{cor}
\begin{prop}[\cite{H2}, Proposition 1.6]\label{123}
For a coherent sheaf $E$ on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$, the following are equivalent.
\begin{itemize}
\item[1. ] $E$ is reflexive.
\item[2. ] $E$ is torsion free and normal.
\item[3. ] $E$ is torsion free and for each open set
$U \subset {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ and each closed set $Z$ in $U$
satistying $\codim_{U}(Z) \ge 2$,
we have $E|_U \simeq j_* (E|_{U\setminus Z})$, where
$j: U \setminus Z \rightarrow Z$ is an open immersion.
\end{itemize}
\end{prop}
\section{The first proof of Theorem \ref{main}}\label{pf}
Let us prove Theorem {\rmfamily \ref{main}}.
It suffices to show the ``if'' part of the statement.
First, let us assume that $\dim (\Sing(E)) \ge 1$. Then any hyperplane $H
\subset {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ intersects
$\Sing(E)$. Take a point $x \in H \cap \Sing(E)
\neq \emptyset$. Note that $\depth_{\mathcal{O}_{x, {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}}
(E_x) \le \dim \mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}-1$.
Since the equation $h \in \mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}$
which defines $H$ at $x$
is a regular element for the reflexive $\mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}$-module
$E_x$, it follows that $\depth_{\mathcal{O}_{x,H}} (E|_H)_x < \dim
\mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}-1=
\dim \mathcal{O}_{x,H}$. From Auslander-Buchsbaum formula,
we conclude that $E|_H$ can not even be locally free.
Hence we may assume that $\dim (\Sing(E)) =0$.
The next lemma is a generalization of Theorem 2.5 in
\cite{H2}.
\begin{lemma}\label{h1}
Let $E$ be a reflexive sheaf on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$ ($n\geq 3$)
with $\dim (\Sing(E)) =0$.
Suppose the restriction $E|_H$ to a hyperplane $H$ splits into a direct
sum of
line bundles. Then
$$
H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E(k))=0, \mbox{ for all } k\in{\mathbb{Z}}}% \Z == \mathbb{Z.
$$
\end{lemma}
{\bf Proof of Lemma \ref{h1}}.
We use the long exact sequence associated with the
short exact sequence
\[
0 \rightarrow E(k-1) \rightarrow E(k) \rightarrow E(k)|_H \rightarrow 0.
\]
Because $E(k)|_H$ is a direct sum of line bundles, it follows that
$H^1(H, E(k)|_H)=0$. So
we have surjections
\begin{equation}\label{surj}
H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P, E(k-1)) \twoheadrightarrow H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E(k)),\ \forall k \in
{\mathbb{Z}}}% \Z == \mathbb{Z.
\end{equation}
To see that these cohomology groups are equal to zero,
let us consider the spectral
sequence of
local and global Ext functors:
\[
E_2^{p,q}=H^p({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,\mathcal{E}xt_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^q(E,\omega)) \Rightarrow
E^{p+q}=\Ext_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^{p+q}(E,\omega)
\]
where $\omega$ is the dualizing sheaf of ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. The assumption
$\dim(\Sing (E))=0$ implies $\dim
(\Supp(\mathcal{E}xt_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^q(E,\omega))) =0$ for all $q>0$. Thus it
follows
that $E_2^{p,q}=0$ unless
$p =0$ or $q = 0$. Moreover,
Proposition \ref{depth} implies
$\depth_{\mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}}(E_x) \ge 2$. From
Auslander-Buchsbaum formula, we have
$\pd_{\mathcal{O}_{x,{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}}E_x < n-1$ for all $x \in {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$.
It follows that
$\mathcal{E}xt_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^q(E,\omega)=0$ for $\forall q\geq n-1$.
Hence
we have
$E_2^{p,q}=0$ for $q \ge n-1$. Considering the convergence of this spectral
sequence,
we obtain the surjection
\begin{equation}\label{surj2}
H^{n-1}({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,\mathcal{H}om_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(E,\omega)) \simeq
H^{n-1}({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E^* \otimes \omega) \twoheadrightarrow
\Ext^{n-1}_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(E,\omega).
\end{equation}
Since $\Ext^{n-1}_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(E(k),\omega)$ is the Serre dual to
$H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P, E(k))$, they have the same dimension.
From (\ref{surj2}),
we have
\begin{equation}\label{ineq}
\dim H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E(k))\le
\dim H^{n-1}({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P, E^*(-k)\otimes \omega)
\end{equation}
for all $k \in {\mathbb{Z}}}% \Z == \mathbb{Z$. The right hand side of (\ref{ineq})
vanishes for $k\ll 0$. Then together with the surjectivity (\ref{surj}),
we conclude that
$H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E(k))=0, \mbox{ for all } k\in{\mathbb{Z}}}% \Z == \mathbb{Z$.
\hfill$\square$
Now, let us put
\[
E|_H \simeq \ \oplus_{i=1}^r \mathcal{O}_H(a_i)
\]
and
$F :=\oplus_{i=1}^r \mathcal{O}_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(a_i)$. Noting that
$\Ext_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^1(F,E(-1)) \simeq H^1({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P, E(-a_i-1))=0$,
Theorem \ref{main} follows from the following theorem,
which asserts that, roughly speaking, the structure of a reflexive sheaf
can be recovered from its restriction to a hyperplane.
\begin{theorem}\label{sp}
Let $E$ and $F$ be
reflexive sheaves on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P\ ( n \ge 2)$ and $H$ be a hyperplane in
${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. Suppose
$E|_H \cong F|_H$ and $\Ext_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^1(F,E(-1))=0$. Then
$E\cong F$.
\end{theorem}
{\bf Proof of Theorem \ref{sp}}.
We want to extend the
isomorphism
$\varphi:F|_H \rightarrow E|_H$ to one over ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. That is possible
since there
is an exact sequence
\begin{eqnarray*}
0 &\rightarrow& \Hom_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(F,E(-1)) \rightarrow \Hom_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(F,E)
\rightarrow \Hom_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(F,E|_H)\\
&\rightarrow& \Ext_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}^1(F,E(-1)) =0,
\end{eqnarray*}
and every morphism $F|_H \rightarrow E|_H$ has a canonical extension to a
morphism $F
\rightarrow E|_H$. Let us fix
an extended morphism $f:F \rightarrow E$ which satisfies $f|_H =\varphi$.
Now, let us
consider the morphism $\det f : \det F \rightarrow \det E$. This
is a monomorphism because $f$ is already a monomorphism.
Since $E|_H \simeq F|_H$, ranks and first Chern classes of $E$ and $F$ are
the same. Henceforth
we can see that $\det f$ is a multiplication
of some constant element in ${\mathbb{K}}}% \P == \mathbb{P$. Note that this constant is not zero. For
$\det f$ is not zero on $H$. Thus at each point $x \in {\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P \setminus
(\Sing
(E) \cup \Sing(F))$, the
morphism $f_x$ is an isomorphism because at these points $f_x$
are the endomorphism of a direct sum of local rings of the same rank.
Since $\codim_{{\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P}(\Sing(E) \cup \Sing(F)) > 2$ and both of
$E$ and $F$ are reflexive, the third condition of Proposition \ref{123}
implies
that
$f$ is also an isomorphism on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. \hfill$\square$
\begin{rem}
In Theorem {\rmfamily \ref{main}}, we can not omit the assumption that
$E$ is
reflexive, i.e.,
``Restriction criterion'' fails for torsion free sheaves.
For example, consider the ideal sheaf $I_p$ on ${\mathbf{P}}}% \P == \mathbb{P^3_{{\mathbb{K}}}% \P == \mathbb{P}$ which
corresponds to a closed point $p \in {\mathbf{P}}}% \P == \mathbb{P^3_{{\mathbb{K}}}% \P == \mathbb{P}$. Note that
$I_p$ is not reflexive. Indeed, let us put
$U={\mathbf{P}}}% \P == \mathbb{P^3_{{\mathbb{K}}}% \P == \mathbb{P} \setminus \{p\}$ and $j:U \rightarrow {\mathbf{P}}}% \P == \mathbb{P^3_{{\mathbb{K}}}% \P == \mathbb{P}$
be an open immersion. It is easy to see that
$I_p|_U \simeq \mathcal{O}_U$. If $I_p$ is reflexive, then according to
Proposition \ref{123},
$j_* (I_p|_U) \simeq I_p$ must hold. However, clearly this is not ture.
Hence $I_p$ is not reflexive.
Now,
if we cut $I_p$ by a plane $H$ which does not contain $p$,
then it is easily seen that
$I_p|_H \simeq \mathcal{O}_H$. However, of course, $I_p$ is not a line
bundle on ${\mathbf{P}}}% \P == \mathbb{P^3$.
\end{rem}
\section{The second proof}\label{app}
Instead of Theorem {\rmfamily \ref{sp}}, we can use the following result,
which is the generalization of the famous Horrocks' splitting criterion
(For example,
see \cite{OSS}). Combining this criterion with usual cohomological
arguments and Lemma \ref{h1},
we can give the second proof of Theorem {\rmfamily \ref{main}}. However, it
seems
that this theorem is not so familiar. Hence let us show the result with a
complete proof.
\begin{theorem}\label{gspc}
Let ${\mathbb{K}}}% \P == \mathbb{P$ be an algebraically closed field, $n$ be a integer greater
than or equal to 2,
and let $E$ be a coherent sheaf on ${\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P$. Then $E$ splits into a
direct sum of line bundles if and only if $H^i({\mathbf{P}}}% \P == \mathbb{P^n_{\mathbb{K}}}% \P == \mathbb{P,E(k))=0$ for all $k
\in {\mathbb{Z}}}% \Z == \mathbb{Z,\ i=1,\cdots,n-1$ and $H^0({\mathbf{P}}}% \P == \mathbb{P^n_{{\mathbb{K}}}% \P == \mathbb{P}, E(k))=0$ for all $k \ll 0$.
\end{theorem}
\begin{rem}
Note that when $E$ is torsion free, then $H^0({\mathbf{P}}}% \P == \mathbb{P^n_{{\mathbb{K}}}% \P == \mathbb{P}, E(k))=0$ for all
$k \ll 0$. This follows from the fact that all torsion free sheaves
can be embedded into a direct sum of line bundles on ${\mathbf{P}}}% \P == \mathbb{P^n_{{\mathbb{K}}}% \P == \mathbb{P}$. So in the
theorem, the condition $H^0({\mathbf{P}}}% \P == \mathbb{P^n_{{\mathbb{K}}}% \P == \mathbb{P}, E(k))=0$
is automatically satisfied for torsion free sheaves.
\end{rem}
When $E$ is a vector bundle, Theorem \rmfamily \ref{gspc} is just the
splitting criterion of
Horrocks. Thus for the proof of this theorem, it suffices to show the
following lemma.
\begin{lemma}\label{schreyer}
Let $X$ be a nonsingular projective variety over an algebraically
closed field ${\mathbb{K}}}% \P == \mathbb{P$ of dimension $n>1$, $L$ be an ample line bundle on $X$,
and let $E$ be a
coherent sheaf on $X$.Then
$E$ is locally free if and only if $H^i(X, E(k))=0$ for all $k \ll 0$ and
$i=0,1,\cdots,n-1$, where $E(k)=E\otimes L^k$.
\end{lemma}
{\bf Proof of Lemma \ref{schreyer}}.
From Serre duality, the ``only if'' part follows immediately. Let us show
the
``if'' part of the statement.
Recall that $E$ is locally free on $X$ if and only
if
$\mathcal{E}xt^i_X(E, \mathcal{O}_X)=0$
for all $i >0$,
see \S\ref{pre}.
Consider the spectral sequence
\[
E_2^{p,q}(k)=H^p(X,\mathcal{E}xt^q_X(E(k),\omega)) \Rightarrow
E^{p+q}(k)=\Ext^{p+q}_X
(E(k),\omega),
\]
where $k \in {\mathbb{Z}}}% \Z == \mathbb{Z$ and $\omega$ is the dualizing sheaf on $X$.
By Serre
duality, $H^i(X,E(k))^* \simeq \Ext^{n-i}_X(E(k), \omega)$ for
$i=0,1,\cdots, n$.
So for each $i > 0,\ E^i(k)= \Ext^i_X(E(k),\omega) = 0$ for sufficiently
small
$k \in {\mathbb{Z}}}% \Z == \mathbb{Z$. Now let us
assume that there exists an integer $i>0$ such that
$\mathcal{E}xt^i_X(E,\mathcal{O}_X) \neq 0$,
and we show that this leads to a contradiction.
It is easy to see that
\[
E_2^{0,i}(k)=H^0(X, \mathcal{E}xt_X^i(E, \omega) \otimes
\mathcal{O}_X(-k)) \neq 0,\ \mbox{for }\forall k\ll 0.
\]
On the other hand, for
$p>0$,
$$
E_2^{p,q}(k)=H^p(X,\mathcal{E}xt^q_X(E,\omega)\otimes
\mathcal{O}_X(-k))
=0,\ \mbox{for }\forall k\ll 0.
$$
From the definition of spectral sequence,
$$
\Ext ^i_X(E(k), \omega)=E_2^{0,i}(k)\neq 0,
$$
for $\forall k\ll 0$.
This contradicts the
assumption that
for each $i>0,\ E^i(k) = 0$ for sufficiently small $k \in {\mathbb{Z}}}% \Z == \mathbb{Z$. Hence we
can see that
$\mathcal{E}xt^i_X(E, \mathcal{O}_X)=0$ for all
$i>0$, so $E$ is a locally free sheaf. \hfill$\square$
\section{Application to hyperplane arrangements}\label{arr}
In this section, we describe an application of our main theorem
to the theory of hyperplane arrangements. As mentioned in
\S \ref{intro}, each hyperplane arrangement determines a
reflexive sheaf. We start with a
more general setting. To every divisor $D$ in a complex manifold $M$
we can associate a reflexive sheaf as follows.
\begin{define}
\label{log}
A vector field $\delta$ on an open set $U\subset M$ is
said to be logarithmic tangent to $D$ if for a local defining
equation $h$ of $D\cap U$ on $U$, $\delta h\in (h)$.
The sheaf associated with logarithmic vector fields
is denoted by $\Der_M(-\log D)$.
\end{define}
In the definition above, a vector field $\delta$ is identified with
a derivation $\delta:\mathcal{O}_M\longrightarrow \mathcal{O}_M$, and
$\Der_M(-\log D)$ can be considered as a subsheaf of the tangent sheaf.
The sheaf of logarithmic vector fields $\Der_M(-\log D)$ is
not necessarily locally free, but in \cite{Slog}, K. Saito proved
the following.
\begin{theorem}[\cite{Slog}]
$\Der_M(-\log D)$ is a reflexive sheaf.
\end{theorem}
From now on,
we restrict ourselves to the case where $D$
is a hyperplane arrangement.
Let $V$ be an $\ell$-dimensional linear space over
${\mathbb{K}}}% \P == \mathbb{P$ and
$S:={\mathbb{K}}}% \P == \mathbb{P[V^*]$ be the algebra of polynomial functions on $V$ that is
naturally isomorphic to ${\mathbb{K}}}% \P == \mathbb{P[z_1, z_2, \cdots, z_\ell]$ for
any choice of basis $(z_1, \cdots, z_\ell)$ of $V^*$.
A (central) hyperplane arrangement $\mathcal{A}$ is a finite
collection of codimension one
linear subspaces in $V$.
For each hyperplane $H$ of $\mathcal{A}$, fix
a nonzero linear form $\alpha_H\in V^*$ vanishing on $H$ and
put $Q:=\prod_{H\in\mathcal{A}}\alpha_H$.
The characteristic polynomial of $\mathcal{A}$ is defined as
$$
\chi(\mathcal{A}, t)=\sum_{X\in L_\mathcal{A}}\mu(X)t^{\dim X},
$$
where $L_\mathcal{A}$ is a lattice which consists of the intersections of
elements of $\mathcal{A}$, ordered by reverse inclusion,
$\hat{0}:=V$ is the unique minimal element of $L_\mathcal{A}$ and
$\mu:L_\mathcal{A}\longrightarrow{\mathbb{Z}}}% \Z == \mathbb{Z$ is the M\"obius function
defined as follows:
\begin{eqnarray*}
\mu(\hat{0})&=&1,\\
\mu(X)&=&-\sum_{Y<X}\mu(Y),\ \mbox{if}\ \hat{0}<X.
\end{eqnarray*}
The characteristic polynomial is one of the most important
concepts in the theory of hyperplane arrangements. Actually
there are a lot of combinatorial or geometric interpretations
of characteristic polynimial.
For details, see \cite{OT}.
Denote by $\Der_V:={\mathbb{K}}}% \P == \mathbb{P[V^*]\otimes V$ the $S$-module of
all polynomial vector fields on $V$. The following definition
was given by G. Ziegler.
\begin{define}[\cite{Z}]
For a given arrangement $\mathcal{A}$ and a map
$m:\mathcal{A}\longrightarrow{\mathbb{Z}}}% \Z == \mathbb{Z_{\geq 0}$, we define modules
of logarithmic vector fields with multiplicity $m$ by
$$
D(\mathcal{A}, m)=\{\delta\in\Der_V\ |\ \delta \alpha_H \in S
\alpha^{m(H)},\
\forall H\in\mathcal{A}\}
$$
When the multiplicity $m$ is the constant map $\underbar{{\rm 1}}(H)\equiv
1\
(\forall H\in\mathcal{A})$, $D(\mathcal{A}, \underbar{{\rm 1}})$ is simply
denoted by $D(\mathcal{A})$.
\end{define}
It is known that the graded $S$-module $D(\mathcal{A}, m)$ is a reflexive
module of rank $l= \dim V$.
\begin{define}
\begin{itemize}
\item[(1)]
An arrangement with a multiplicity $(\mathcal{A}, m)$ is called
free with exponents $(e_1, \cdots, e_\ell)$ if $D(\mathcal{A}, m)$ is
a free $S$-module, with a homogeneous basis
$\delta_1, \cdots, \delta_\ell$ such that
$$
\deg \delta_i=e_i.
$$
Note that a vector field
$$
\delta=\sum_i f_i\frac{\partial}{\partial x_i}
$$
is said to be homogeneous if coefficients $f_1, \cdots, f_\ell$ are
all homogeneous with the same degree
and put $\deg \delta :=\deg f_i$.
\item[(2)]
An arrangement $\mathcal{A}$ is called free if
$(\mathcal{A}, \underbar{{\rm 1}})$ is free, i.e.,
$D(\mathcal{A})$ is a free $S$-module.
\end{itemize}
\end{define}
Since $D(\mathcal{A})$ contains the Euler vector field
$\theta_E:=\sum_{i=1}^\ell x_i\frac{\partial}{\partial x_i}$,
the exponents $(e_1, \cdots, e_\ell)$ of a free arrangement
$\mathcal{A}$ contains $1$.
H. Terao proved that the freeness of $\mathcal{A}$
implies a remarkable behavior of the characteristic polynomial.
\begin{theorem}[\cite{Tfact}]
\label{factor}
Suppose $\mathcal{A}$ is a free arrangement with the exponents
$(e_1, \cdots, e_\ell)$, then
$$
\chi(\mathcal{A}, t)=\prod_{i=1}^\ell(t-e_i).
$$
\end{theorem}
As we will see later, in Corollary \ref{free},
the freeness is equivalent to the splitting of a reflexive sheaf, and
exponents are corresponding to the splitting type. On the other hand,
the left hand side of the Theorem \ref{factor}
is obtained from the intersection poset, thus
determined by the combinatorial structure.
This theorem connects two regions in mathematics:
combinatorics of arrangements and geometry of reflexive sheaves.
It enables us to study combinatorics of arrangements via a
geometric method.
For example, in \cite{Y} characteristic polynomials for some
arrangements are computed by using this interpretation.
In \cite{Z}, Ziegler studied the relation between
the freeness and the freeness with a multiplicity.
Fixing a hyperplane $H_0\in\mathcal{A}$, let us define
an arrangement
$$
\mathcal{A}^{H_0}:=\{ H_0\cap K\ |\ K\in\mathcal{A},\ K\neq H_0\},
$$
over $H$ and the natural multiplicity
$$
\underline{m}(X):=\sharp\{ K\in\mathcal{A}\ |\ K\cap H_0=X\}
$$
for $X\in\mathcal{A}^{H_0}$.
\begin{theorem}[\cite{Z}]
\label{thm:zie}
If $\mathcal{A}$ is a free arrangement with
exponents $(1, e_2, \cdots, e_\ell)$, then the restricted
arrangement with natural multiplicity
$(\mathcal{A}^{H_0}, \underbar{m})$ is also free
with exponents $(e_2, \cdots, e_\ell)$.
\end{theorem}
More precisely,
let $\alpha=\alpha_{H_0}$ be a defining equation of $H_0$ and define
$$
D_0(\mathcal{A}):=\{ \delta\in D(\mathcal{A})\ |\
\delta\alpha=0\}.
$$
It is easily seen that $D(\mathcal{A})$ has a direct sum
decomposition into graded $S$-modules
$$
D(\mathcal{A})=S\cdot \theta_E\oplus D_0(\mathcal{A}).
$$
Ziegler proved that if $\delta_1=\theta_E, \delta_2, \cdots, \delta_\ell$
is a basis of $D(\mathcal{A})$ with
$\delta_2, \cdots, \delta_\ell \in D_0(\mathcal{A})$, then
$\delta_2|_{H_0}, \cdots, \delta_\ell|_{H_0}$ form a basis
of $D(\mathcal{A}^{H_0}, \underbar{m})$.
Recall that a graded $S$-module $M=\oplus_{k\in{\mathbb{Z}}}% \Z == \mathbb{Z}M_k$ determines
a coherent sheaf $\tilde{M}$ over ${\mathbf{P}}}% \P == \mathbb{P^{\ell-1}=\Proj S$. Conversely
for any coherent sheaf $\mathcal{F}$ over ${\mathbf{P}}}% \P == \mathbb{P^{\ell-1}$,
$\Gamma_*(\mathcal{F}):=\bigoplus_{k\in{\mathbb{Z}}}% \Z == \mathbb{Z}\Gamma({\mathbf{P}}}% \P == \mathbb{P^{\ell-1},
\mathcal{F}(k))$
defines the graded $S$-module associated with $\mathcal{F}$.
We have the natural $S$-homomorphism
$\alpha:M\rightarrow \Gamma_*(\tilde{M})$, which is neither
injective nor surjective in general. In the case of $M=D(\mathcal{A})$,
however,
we have the following lemma.
\begin{lemma}
\label{graded}
$
\alpha: D(\mathcal{A})\stackrel{\cong}{\longrightarrow}
\Gamma_*\left({\mathbf{P}}}% \P == \mathbb{P^{\ell-1}, \widetilde{D(\mathcal{A})}\right)$
is isomorphic.
\end{lemma}
{\bf Proof of Lemma \ref{graded}}. We prove the surjectivity.
Since $\bigcup_{i=1}^\ell D(z_i)={\mathbf{P}}}% \P == \mathbb{P^{\ell-1}$, any element
in $\Gamma({\mathbf{P}}}% \P == \mathbb{P^{\ell-1}, \widetilde{D(\mathcal{A})}(k))$ can be
expressed as
$$
\delta=
\frac{\delta_1}{z_1^{d_1}}=
\frac{\delta_2}{z_2^{d_2}}=\cdots =
\frac{\delta_\ell}{z_\ell^{d_\ell}},
$$
where $\delta_i\in D(\mathcal{A})_{d_i+k}$. From the facts that
$\delta_i$ is an element of a $S$-free module $\Der_V$ and
$S$ is UFD,
it is easily seen that $\delta$ is also a polynomial
vector field, so contained in $\Der_V$.
Let $\alpha_H$ be a defining linear form
of $H\in\mathcal{A}$, and we may choose $i$ such that
$\alpha_H$ and $z_i$
are linearly independent. Then the right hand side of
$$
z_i^{d_i}\cdot \delta\alpha_H=\delta_i\alpha_H
$$
is divisible by $\alpha_H$, so is the left.
Hence $\delta\alpha_H$ is also divisible by
$\alpha_H$, and we can conclude that $\delta\in D(\mathcal{A})$.
\hfill$\square$
The above lemma enable us to connect freeness and splitting.
\begin{cor}
\label{free}
$\mathcal{A}$ is free with exponents $(e_1, \cdots, e_\ell)$ if
and only if
$$
\widetilde{D(\mathcal{A})}=
\mathcal{O}_{{\mathbf{P}}}% \P == \mathbb{P^{\ell-1}}(-e_1)\oplus\cdots\oplus\mathcal{O}_{{\mathbf{P}}}% \P == \mathbb{P^{\ell-1}}(-e_\ell)
$$
\end{cor}
Now, the following theorem, which has been proved and
played an important role in the proof of Edelman and Reiner conjecture in
\cite{Y},
is naturally proved from Theorem \ref{main}.
\begin{theorem}[\cite{Y}]
\label{thm:y}
$\mathcal{A}$ is free if and only if
there exists a hyperplane $H_0\in\mathcal{A}$ such that
\begin{itemize}
\item[(a)] $(\mathcal{A}^{H_0}, \underbar{m})$ is free, and
\item[(b)] $\mathcal{A}_x:=\{H\in\mathcal{A}\ |\ H\ni x\}$ is
free for all $x\in H_0\setminus \{0\}$.
\end{itemize}
\end{theorem}
{\bf Proof of Theorem \ref{thm:y}}.
Let us denote by ${\mathbf{P}}}% \P == \mathbb{P(V)$ the
projective space of one-dimensional subspaces in a vector space $V$.
Recall that $D_0(\mathcal{A})$ is a graded reflexive $S$-module.
So it determines a reflexive sheaf $\widetilde{D_0(\mathcal{A})}$ over
${\mathbf{P}}}% \P == \mathbb{P(V)$. As is mentioned in \cite{MS}, the local structure of
$\widetilde{D_0(\mathcal{A})}$ is determined by the local structure
of $\mathcal{A}$, i.e.,
$$
\widetilde{D_0(\mathcal{A})}_{\bar{x}}=
\widetilde{D_0(\mathcal{A}_x)}_{\bar{x}},
$$
for $\bar{x}\in{\mathbf{P}}}% \P == \mathbb{P(V)$. Using Theorem \ref{thm:zie} locally,
condition (b) in Theorem \ref{thm:y} implies that
$$
\widetilde{D_0(\mathcal{A})}_{\bar{x}}|_{{\mathbf{P}}}% \P == \mathbb{P(H_0)}
=
\widetilde{D(\mathcal{A}^{H_0}, \underbar{m})}_{\bar{x}}.
$$
Now condition (a) in Theorem \ref{thm:zie} means that
$\widetilde{D_0(\mathcal{A})}|_{{\mathbf{P}}}% \P == \mathbb{P(H_0)}$ splits into
a direct sum of line bundles.
From Theorem \ref{main}, we may conclude that
$\widetilde{D_0(\mathcal{A})}$ is also splitting.
Hence
$$
\bigoplus_{k\in{\mathbb{Z}}}% \Z == \mathbb{Z}\Gamma\left({\mathbf{P}}}% \P == \mathbb{P(V), \widetilde{D_0(\mathcal{A})}(k)\right)
=D_0(\mathcal{A})
$$
is a free module over $S$.
Thus $\mathcal{A}$ is a free arrangement.
\hfill$\square$
|
{
"timestamp": "2005-03-30T16:08:12",
"yymm": "0503",
"arxiv_id": "math/0503710",
"language": "en",
"url": "https://arxiv.org/abs/math/0503710"
}
|
\section{Introduction}
The geometric approach to the theory of linear
dynamical systems has provided deep insights
and elegant solutions to many control problems,
such as the disturbance decoupling problem, the block
decoupling problem, and the model matching problem
(see~\cite{wonham} and the references therein).
The concept of $(A,B)$-invariant subspace
(or controlled invariant subspace, see~\cite{BasMar91})
has played a significant role in the development of this approach.
It is natural to try to apply the same kind of methods to discrete
event systems. Several mathematical models have been proposed, see
in particular~\cite{CassLafoOlsd} for a survey of the following
approaches. Ramadge and Wonham~\cite{ramadge87a} initiated the logical,
language-theoretic approach, in which the precise ordering of the events
is of interest and time does not play an explicit role.
This theory addresses the synthesis of controllers in
order to satisfy some qualitative specifications on the admissible orderings
of the events. Another approach is the max-plus algebra based control approach
initiated by Cohen et al.~\cite{cohen85a}, in which in addition to
the ordering, the timing of the events plays an essential role. A third
approach is the perturbation analysis of Cassandras
and Ho~\cite{CassaHo83}, which deals with stochastic timed
discrete event systems.
The max-plus semiring is the set $\R\cup\{-\infty\}$, equipped with
$\max$ as addition and the usual sum as multiplication.
Linear dynamical systems with coefficients in the max-plus
semiring turn out to be useful for modeling and analyzing
many discrete event dynamic systems subject to synchronization
constraints (see~\cite{bcoq}). Among these, we can mention
some manufacturing systems (Cohen et al.~\cite{cohen85a}),
computer networks (Le Boudec and Thiran~\cite{leboudec}) and
transportation networks (Olsder et al.~\cite{OlsSubGett98},
Braker~\cite{braker91,braker}, and de Vries et al.~\cite{deVDeSdeM98}).
Many results from linear system theory have been extended to
systems with coefficients in the max-plus semiring, such as
the connection between spectral theory and stability questions
(see~\cite{cohen89a}) or transfer series methods (see~\cite{bcoq}).
Several interesting control problems have also been studied by,
for example, Boimond et al.~\cite{BoiCott99,BoiHar00},
Cottenceau et al.~\cite{CottHar03} and Lhommeau~\cite{Lhommeau}.
In contrast to the approach presented here, which is based on state space
representation, their approach uses transfer series and residuation methods
and therefore deals with different types of specifications.
This motivates the attempt to extend the geometric approach, and in
particular the concept of $(A,B)$-invariant subspace, to the
theory of linear dynamical systems over the max-plus
semiring, a question which is raised in~\cite{ccggq99}.
The same kind of generalization, which was initiated by Hautus, Conte and
Perdon, has been widely studied for linear dynamical systems over rings
(see~\cite{hautus82,hautus84,conte94,conte95,assan,AssLafPer}).
In this paper we will see that the extension of the
geometric approach to linear systems over the max-plus
semiring presents similar difficulties to those
encountered in dealing with coefficients in a ring
rather than coefficients in a field. The $(A,B)$-invariance problem has been studied
in the framework of formal series over some complete idempotent
semirings by Klimann~\cite{klimann99}.
To illustrate one of the possible applications of the results presented in
this paper, we apply the methods presented here to the study of transportation
networks which evolve according to a timetable. Max-plus linear models for
transportation networks have been studied by several authors, see for
example~\cite{OlsSubGett98,braker91,braker,deVDeSdeM98}. Let us
consider the simple railway network given in Figure~\ref{figure1}, which
has been borrowed from~\cite{deVDeSdeM98}.
\begin{figure}
\begin{center}
\input figure1V2
\end{center}
\caption{A simple transportation network}
\label{figure1}
\end{figure}
In this network, we assume that in the initial state there is a train
running along each of the tracks which connect the following stations:
$P$ with $Q$, $Q$ with $P$, $Q$ with $Q$ via $R$ and finally $Q$ with
$Q$ via $S$. In Figure~\ref{figure1}, these tracks are denoted by
$d_1$, $d_2$, $d_3$ and $d_4$ respectively. The traveling time on
track $d_i$ is given by $t_i$, for $i=1,\ldots ,4$. We will assume that the following
conditions are satisfied. A first condition is that at station $Q$ the
trains coming from stations $P$ and $S$ have to ensure a connection to the
train which leaves for destination $R$ and vice versa. The second
condition is that a train cannot leave before its scheduled departure
time which is given by a timetable. If we assume that a train leaves
as soon as all the previous conditions have been satisfied, then the
evolution of the transportation network can be described by a max-plus
linear dynamical system where the scheduled departure times can be
seen as controls (see Section~\ref{aplicacionSec}). We will see that
the tools presented in this paper can be used to analyze this kind of network.
For example, it is possible to determine whether there exists a
timetable that satisfies such conditions as the following.
A first condition could be that the time between two consecutive
departures of trains in the same direction be less than a certain
given bound. As a second condition we could require that the time
that passengers have to wait to make some connections be less
than another given bound. Of course, more general specifications could be
analyzed. We show how to compute a
timetable which satisfies these requirements when it exists.
For instance, suppose that in the railway network given in
Figure~\ref{figure1} we want the time between two
consecutive departures of trains in the same direction to be less
than $15$ time units and the maximal time that passengers
have to wait to make any connection to be less than $4$ time units.
In Section~\ref{aplicacionSec} we show that this is possible
and give a timetable which satisfies these requirements.
This paper is organized as follows. In Section~\ref{geomABinvSec},
after a short introduction to max-plus type semirings, we introduce
the concept of geometrically $(A,B)$-invariant semimodule and generalize
the Wonham fixed point algorithm (which is used to compute the maximal
$(A,B)$-invariant subspace contained in a given space, see~\cite{wonham})
to max-plus algebra. In Section~\ref{volumeSec} we introduce the concept of volume of
a semimodule and study its properties. In Section~\ref{finitevolumeSec}
we use volume arguments to show that the fixed point algorithm introduced
in Section~\ref{geomABinvSec} converges in a finite number of steps for an
important class of semimodules. In Section~\ref{algABinvSec} we consider the
concept of algebraically $(A,B)$-invariant semimodule and give a method to
decide whether a finitely generated semimodule is algebraically
$(A,B)$-invariant. Finally, in Section~\ref{aplicacionSec} we apply the
methods given in this paper to the study of transportation networks
which evolve according to a timetable.
Let us finally mention that some of the results presented here were
announced in~\cite{gk03} and considered in~\cite{katz}.
\medskip\noindent{\em Acknowledgment.}\/
The author would like to thank S. Gaubert for many helpful
suggestions and comments on preliminary versions of this manuscript
and J.-J. Loiseau for useful references. He would also like to thank
J. E. Cury and the anonymous reviewers who helped to improve
this paper.
\section{Geometrically $(A,B)$-invariant semimodules}\label{geomABinvSec}
Let us first recall some definitions and results. A {\em monoid}
is a set equipped with an associative internal composition law
which has a (two sided) neutral element. A {\em semiring} is a
set ${\mathcal{S}} $ equipped with two internal composition laws $\oplus $
and $\otimes $, called addition and multiplication respectively,
such that ${\mathcal{S}}$ is a commutative monoid for addition, $\mathcal{S}$ is a
monoid for multiplication, multiplication distributes over addition,
and the neutral element for addition is absorbing for multiplication.
We will sometimes denote by $({\mathcal{S}},\oplus,\otimes,\varepsilon,e)$
the semiring ${\mathcal{S}}$, where $\varepsilon$ and $e$ represent the
neutral elements for addition and for multiplication respectively. We
say that a semiring ${\mathcal{S}}$ is {\em idempotent} if $x\oplus x=x$
for all $x\in \mathcal{S}$. In this paper, we are mostly interested in
some variants of the max-plus semiring $\R_{\max} $, which is the set
$\R\cup\{-\infty\}$ equipped with $\oplus =\max $ and $\otimes =+$
(see~\cite{pin95} for an overview). Some of these variants can be
obtained by noting that a semiring $M_{\max }$, whose set of elements
is $M\cup\{-\infty\}$ and laws are $\oplus =\max $ and $\otimes =+ $, is
associated with a submonoid $(M,+)$ of $(\mathbb{R},+)$. Symmetrically,
we can consider the semiring $M_{\min }$ with the set
of elements $M\cup\{+\infty\}$ and laws $\oplus =\min $
and $\otimes =+$.
For instance, taking $M=\mathbb{Z}$ we get the semiring $\Z_{\max} =(\Z\cup\{-\infty\},\max,+)$,
which is the main semiring we are going to work
with, and taking $M=\mathbb{N}$ we get the semiring $\N_{\min} =(\N\cup\{+\infty\},\min,+)$,
which is known as the {\em tropical semiring} (see~\cite{pin95}).
Recall that an idempotent semiring
$({\mathcal{S}},\oplus,\otimes)$ is equipped with the {\em natural order}:
$x\preceq y \iff x\oplus y=y$ (see for example~\cite{bcoq}).
Sometimes it is useful to add a maximal element for the natural order
to the semirings $M_{\max }$ and $M_{\min }$, obtaining in
this way the {\em complete} semirings
${\overline{M}}_{\max}=(M\cup\{\pm\infty\},\max,+)$ and
${\overline{M}}_{\min}=(M\cup\{\pm\infty\},\min,+)$, respectively.
Note that, in the semirings ${\overline{M}}_{\max}$
and ${\overline{M}}_{\min}$, the value of
$(-\infty)+(+\infty)=(+\infty)+(-\infty)$ is determined
by the fact that the neutral element for addition
is absorbing for multiplication. Then, we know that
$(-\infty)+(+\infty)=(+\infty)+(-\infty)=-\infty$ in
${\overline{M}}_{\max}$ and
$(-\infty)+(+\infty)=(+\infty)+(-\infty)=+\infty$ in
${\overline{M}}_{\min}$.
We next introduce the concept of semimodules which is the analogous
over semirings of vector spaces (we refer the reader to~\cite{GargKumar95}
and~\cite{gaubert98n} for more details on semimodules). A (left) {\em semimodule}
over a semiring $({\mathcal{S}},\oplus,\otimes,\varepsilon_{{\mathcal{S}}},e)$
is a commutative monoid $({\mathcal{X}},\hat{\oplus })$,
with neutral element $\varepsilon_{{\mathcal{X}}}$, equipped with a map
${\mathcal{S}}\times {\mathcal{X}}\to {\mathcal{X}}$, $(\lambda,x) \to \lambda \cdot x$
(left action), which satisfies:
\begin{eqnarray*}
(\lambda \otimes \mu)\cdot x= \lambda \cdot(\mu \cdot x)\; , \\
\lambda \cdot (x\; \hat{\oplus }\;y) =\lambda \cdot x \; \hat{\oplus }\; \lambda \cdot y\;,\\
(\lambda \oplus \mu)\cdot x = \lambda \cdot x \; \hat{ \oplus }\; \mu \cdot x \; , \\
\varepsilon_{\mathcal{S}}\cdot x = \varepsilon_{\mathcal{X}}\; , \\
\lambda \cdot \varepsilon_{\mathcal{X}} = \varepsilon_{\mathcal{X}} \; , \\
e \cdot x =x \; ,
\end{eqnarray*}
for all $x,y\in {\mathcal{X}}$ and $\lambda,\mu\in {\mathcal{S}}$.
We will usually use concatenation to denote both the
multiplication of ${\mathcal{S}}$ and the left action,
and we will denote by $\varepsilon$ both the zero
element $\varepsilon_{{\mathcal{S}}}$ of ${\mathcal{S}}$ and the zero
element $\varepsilon_{{\mathcal{X}}}$ of ${\mathcal{X}}$. A {\em subsemimodule}
of ${\mathcal{X}}$ is a subset ${\mathcal{Z}}\subset {\mathcal{X}}$ such that
$\lambda x \hat{\oplus } \mu y \in {\mathcal{Z}}$,
for all $x,y\in {\mathcal{Z}}$ and $\lambda,\mu\in {\mathcal{S}}$.
In this paper, we will mostly consider subsemimodules
of the {\em free semimodule} ${\mathcal{S}}^n$, which is the set of
$n$-dimensional vectors over ${\mathcal{S}}$, equipped with
the internal law $(x\hat{\oplus }y)_i=x_i\oplus y_i$ and the
left action $(\lambda\cdot x)_i=\lambda \otimes x_i$.
If $G\subset {\mathcal{X}}$, we will denote by $\mbox{\rm span}\, G $
the subsemimodule of ${\mathcal{X}}$ generated by $G$,
that is, the set of all $x\in {\mathcal{X}}$ for which there
exists a finite number of elements $u_1,\ldots ,u_k$
of $G$ and a finite number of scalars
$\lambda_1,\ldots ,\lambda_k\in {\mathcal{S}}$, such that
$x=\hat{\bigoplus}_{i=1,\ldots , k}\lambda_i u_i$.
Finally, if $C\in {\mathcal{S}}^{n\times r}$, we will denote by $\mbox{\rm Im}\, C$ the
subsemimodule of ${\mathcal{S}}^n$ generated by the columns of $C$.
Let $({\mathcal{S}},\oplus ,\otimes)$ denote a semiring. By a {\em system with
coefficients in ${\mathcal{S}}$}, or a {\em system over ${\mathcal{S}}$}, we mean a
linear dynamical system whose evolution is
determined by a set of equations of the form
\begin{equation}\label{dynamicsystem}
x(k)=Ax(k-1)\oplus Bu(k)\; ,
\end{equation}
where $A\in \mathcal{S}^{n\times n}$, $B\in \mathcal{S}^{n\times q}$, and
$x(k)\in \mathcal{S}^{n\times 1}$, $u(k)\in \mathcal{S}^{q\times 1}$, $k=1,2,\ldots $
are the sequences of state and control vectors respectively.
We are interested in studying the following problem:
Given a certain specification for the state space of
system~\eqref{dynamicsystem}, which we suppose is given by a
semimodule $\mathcal{K} \subset \mathcal{S}^n$, we want to compute the maximal
set of initial states $\mathcal{K}^*$ for which there exists a sequence
of control vectors which makes the state of system~\eqref{dynamicsystem}
stay in $\mathcal{K}$ forever, that is, such that $x(k)\in \mathcal{K}$ for all $k\geq 0$.
To treat this problem it is convenient to make the following definition.
\begin{definition}\label{ABinvariante2}
Given the matrices $A\in{\mathcal{S}}^{n\times n}$ and $B\in {\mathcal{S}}^{n\times q}$,
we say that a semimodule ${\mathcal{X}} \subset {\mathcal{S}}^n$ is
{\rm (geometrically) $(A,B)$-invariant} if for all $x\in \mathcal{X}$ there exists
$u\in {\mathcal{S}}^q$ such that $Ax \oplus Bu$ belongs to $\mathcal{X}$.
\end{definition}
The proof of the following lemma is identical to the case of linear dynamical systems over rings.
We include it for completeness.
\begin{lemma}\label{obs1}
If ${\mathcal{K}} \subset {\mathcal{S}}^n$ is a semimodule, then ${\mathcal{K}}^*$ is the maximal
(geometrically) $(A,B)$-invariant semimodule contained in ${\mathcal{K}}$.
\end{lemma}
\begin{proof}
In the first place, note that a semimodule $\mathcal{X} \subset {\mathcal{S}}^n$ is (geometrically)
$(A,B)$-invariant if and only if for each $x\in {\mathcal{X}}$ there exists a sequence of control
vectors such that the trajectory of the dynamical system~\eqref{dynamicsystem},
associated with this control sequence and the initial condition $x(0)=x$,
is completely contained in ${\mathcal{X}}$. Therefore, any (geometrically) $(A,B)$-invariant
semimodule contained in ${\mathcal{K}}$ is also contained in $\mathcal{K}^*$. In the second place,
note that $\mathcal{K}^*$ is a subsemimodule of $\mathcal{S}^n$ since system~\eqref{dynamicsystem}
is linear and $\mathcal{K}$ is a semimodule. Then, to prove the lemma,
it only remains to show that $\mathcal{K}^*$ is (geometrically) $(A,B)$-invariant.
Let $x$ be an arbitrary element of $\mathcal{K}^*$. We must see that there is a control
$u(1)\in \mathcal{S}^q$ such that $x(1)=Ax \oplus Bu(1)$ belongs to $\mathcal{K}^*$.
Since $x\in \mathcal{K}^*$, we know that there exists a sequence of control
vectors $u(k)$, $k=1, 2, \ldots $, such that the trajectory
$x(0)$, $x(1)$, $x(2)$, $\ldots$ of system~\eqref{dynamicsystem},
associated with this control sequence and the initial condition $x(0)=x$,
is completely contained in $\mathcal{K}$. Therefore, $x(1)\in \mathcal{K}^*$
since there exists a sequence of control vectors
($u'(k)=u(k+1)$, $k=1, 2, \ldots $) which makes the state of
system~\eqref{dynamicsystem} stay in $\mathcal{K}$ forever when the initial
state is $x(1)$.
\end{proof}
To tackle the previous problem in the case of max-plus type semirings,
we generalize the classical fixed point algorithm which is
used to compute the maximal $(A,B)$-invariant subspace contained
in a given space (see~\cite{wonham}). With this purpose in mind,
we set ${\mathcal{B}}=\mbox{\rm Im}\, B$ and consider the self-map $\varphi$
of the set of subsemimodules of ${\mathcal{S}}^n$, given by:
\begin{equation}\label{definicionphi}
\varphi({\mathcal{X}})={\mathcal{X}} \cap A^{-1}({\mathcal{X}} \ominus {\mathcal{B}})
\enspace ,
\end{equation}
where $A^{-1}({\mathcal{Y}})=\set{u\in {\mathcal{S}}^n}{Au\in{\mathcal{Y}}}$ and
${\mathcal{Z}}\ominus {\mathcal{Y}}=\set{u\in {\mathcal{S}}^n}{\exists y\in{\mathcal{Y}}, u\oplus y\in {\mathcal{Z}}}$
for all ${\mathcal{Z}},{\mathcal{Y}}\subset{\mathcal{S}}^n$.
\begin{remark}\label{ObsComputo}
Note that when ${\mathcal{S}}=\Z_{\max}$ or ${\mathcal{S}}=\N_{\min}$, if the semimodule ${\mathcal{X}}$ is finitely
generated, then the semimodule $\varphi({\mathcal{X}})$ is also finitely generated. In fact,
given the sets of generators of some finitely generated semimodules ${\mathcal{Z}}$ and ${\mathcal{Y}}$,
the semimodules ${\mathcal{Y}} \ominus {\mathcal{Z}}$, $A^{-1}({\mathcal{Y}})$ and ${\mathcal{Y}} \cap {\mathcal{Z}}$
can be expressed as the images by suitable matrices of the sets of solutions
of appropriate max-plus linear systems of the form $Dx=Cx$
(see~\cite{gaubert98n} for details). Therefore, their sets of generators
can be explicitly computed using a general elimination algorithm
due to Butkovi\v{c} and Heged\"{u}s~\cite{butkovicH} and Gaubert~\cite{gaubert92a}.
Then, when ${\mathcal{X}}$ is finitely generated, the set of generators of
$\varphi({\mathcal{X}})$ can also be computed using this algorithm. More generally, if ${\mathcal{X}}$
belongs to the class of rational semimodules (this class, which extends the
notion of finitely generated semimodule, turns out to be useful in the geometric
approach to discrete event systems, see~\cite{gk02a}),
then $\varphi({\mathcal{X}})$ is also a rational semimodule and can be computed by
Theorem~3.5 of~\cite{gk02a}.
\end{remark}
\begin{lemma}\label{obs2}\
A semimodule ${\mathcal{X}} \subset {\mathcal{S}}^n$ is (geometrically) $(A,B)$-invariant
if and only if ${\mathcal{X}}=\varphi({\mathcal{X}})$.
\end{lemma}
\begin{proof}
Since
\begin{eqnarray*}
A^{-1}({\mathcal{X}} \ominus {\mathcal{B}}) & = &\set{x\in {\mathcal{S}}^n}{Ax\in{\mathcal{X}} \ominus {\mathcal{B}}}= \\
& = & \set{x\in {\mathcal{S}}^n}{\exists b\in {\mathcal{B}}, Ax\oplus b\in {\mathcal{X}}}= \\
& = & \set{x\in {\mathcal{S}}^n}{\exists u\in {\mathcal{S}}^q, Ax\oplus Bu\in {\mathcal{X}}} \; ,
\end{eqnarray*}
we see that $A^{-1}({\mathcal{X}} \ominus {\mathcal{B}})$ is the set of initial
states $x(0)$ of the dynamical system~\eqref{dynamicsystem}
for which there exists a control $u(1)$ which makes the new
state of the system, that is $x(1)=Ax(0)\oplus Bu(1)$, belong
to ${\mathcal{X}}$. Then, it readily follows from Definition~\ref{ABinvariante2}
that a semimodule ${\mathcal{X}} \subset {\mathcal{S}}^n$ is (geometrically) $(A,B)$-invariant
if and only if ${\mathcal{X}}\subset A^{-1}({\mathcal{X}} \ominus {\mathcal{B}})$. Therefore, a semimodule
${\mathcal{X}} \subset {\mathcal{S}}^n$ is (geometrically) $(A,B)$-invariant if and only if
${\mathcal{X}}=\varphi({\mathcal{X}})$, that is, (geometrically) $(A,B)$-invariant semimodules
are precisely the fixed points of the map $\varphi$ defined
by~\eqref{definicionphi}.
\end{proof}
Inspired by the algorithm in the classical case,
we define the following sequence of semimodules:
\begin{equation}\label{algoABinv}
{\mathcal{X}}_1={\mathcal{K}}\; , \quad {\mathcal{X}}_{r+1}=\varphi({\mathcal{X}}_r)\;, \quad \forall r\in \mathbb{N}.
\end{equation}
Then we have the following lemma.
\begin{lemma}\label{lemaalgoAB}
Let ${\mathcal{K}} \subset {\mathcal{S}}^n$ be an arbitrary semimodule.
Then the sequence of semimodules $\{{\mathcal{X}}_r\}_{r\in \mathbb{N}}$
defined by~\eqref{algoABinv} is decreasing, i.e.
${\mathcal{X}}_{r+1}\subset {\mathcal{X}}_r$ for all $r\in \mathbb{N}$. Moreover,
if we define ${\mathcal{X}}_{\omega}=\cap_{r\in \mathbb{N}} {\mathcal{X}}_r$, then every
(geometrically) $(A,B)$-invariant semimodule
contained in ${\mathcal{K}}$ is also contained in ${\mathcal{X}}_{\omega}$.
In particular, it follows that ${\mathcal{K}}^* \subset {\mathcal{X}}_{\omega}$.
\end{lemma}
\begin{proof}
The fact that the sequence of semimodules $\{{\mathcal{X}}_r\}_{r\in \mathbb{N}}$
is decreasing is a consequence of the definition of the map $\varphi$:
\[
{\mathcal{X}}_{r+1}=\varphi({\mathcal{X}}_r)={\mathcal{X}}_r \cap A^{-1}({\mathcal{X}}_r \ominus {\mathcal{B}})\subset {\mathcal{X}}_r,
\]
for all $r\in \mathbb{N}$.
To prove the second part of Lemma~\ref{lemaalgoAB},
firstly it is convenient to notice that $\varphi$
satisfies the following property:
\[
\forall {\mathcal{Z}},{\mathcal{Y}}\subset{\mathcal{S}}^n\;,\enspace
{\mathcal{Z}}\subset {\mathcal{Y}} \Rightarrow \varphi({\mathcal{Z}})\subset \varphi({\mathcal{Y}})\;,
\]
that is, $\varphi $ is monotonic when the set of
subsemimodules of ${\mathcal{S}}^n$ is equipped with the order:
${\mathcal{Z}} \leq {\mathcal{Y}}$ if and only if ${\mathcal{Z}}\subset {\mathcal{Y}}$.
Now let ${\mathcal{X}} \subset {\mathcal{K}}$ be an arbitrary (geometrically)
$(A,B)$-invariant semimodule. We will prove by induction
on $r$ that ${\mathcal{X}}\subset {\mathcal{X}}_r$ for all $r\in \mathbb{N}$,
and therefore that ${\mathcal{X}}\subset \cap_{r\in \mathbb{N}} {\mathcal{X}}_r={\mathcal{X}}_{\omega}$.
In the first place, we know that ${\mathcal{X}} \subset {\mathcal{K}} ={\mathcal{X}}_1$.
Since ${\mathcal{X}}$ is a (geometrically) $(A,B)$-invariant semimodule,
thanks to Lemma~\ref{obs2}, it follows that ${\mathcal{X}} =\varphi({\mathcal{X}})$.
If we now assume that ${\mathcal{X}} \subset {\mathcal{X}}_t$, then we have:
\[
{\mathcal{X}} = \varphi({\mathcal{X}})\subset \varphi({\mathcal{X}}_t)={\mathcal{X}}_{t+1}\;.
\]
Therefore, ${\mathcal{X}}\subset {\mathcal{X}}_r$ for all $r\in \mathbb{N}$, as we wanted to show.
\end{proof}
Note that if the sequence $\{{\mathcal{X}}_r\}_{r\in \mathbb{N}}$
stabilizes\footnote{Throughout this paper, we will use the word
``stabilize'' to mean ``converge in a finite number of steps''.},
that is, if there exists $k\in \mathbb{N}$ such that ${\mathcal{X}}_{k+1}={\mathcal{X}}_k$,
then our problem will be solved. Indeed, if there exists $k\in \mathbb{N}$
such that ${\mathcal{X}}_k={\mathcal{X}}_{k+1}=\varphi({\mathcal{X}}_k)$ then,
thanks to Lemma~\ref{obs2}, we know that ${\mathcal{X}}_k$ is a
(geometrically) $(A,B)$-invariant semimodule which is contained in ${\mathcal{K}}$
(since ${\mathcal{X}}_1={\mathcal{K}}$ and by Lemma~\ref{lemaalgoAB}
the sequence $\{{\mathcal{X}}_r\}_{r\in \mathbb{N}}$ is decreasing). Therefore
${\mathcal{X}}_k\subset {\mathcal{K}}^*$, and as by Lemma~\ref{lemaalgoAB} we know
that ${\mathcal{K}}^*\subset {\mathcal{X}}_k$, it follows finally that ${\mathcal{K}}^*={\mathcal{X}}_k$.
\begin{example}\label{ejemplo1}
Let ${\mathcal{S}}=\Z_{\max}$. Let us consider the matrices
\[
A=
\begin{pmatrix} -\infty & 0 \\ 0 & -\infty \end{pmatrix}
\enspace \mbox{ and } \enspace
B=\begin{pmatrix} 0 \\ 0\end{pmatrix}\; ,
\]
and the semimodule ${\mathcal{K}}=\set{(x,y)^T\in \Z_{\max}^2}{y\geq x+1}$.
Let us compute, in this particular case,
the sequence of semimodules $\{{\mathcal{X}}_r\}_{r\in \mathbb{N}}$ defined
by~\eqref{algoABinv}. By definition we know that
${\mathcal{X}}_1={\mathcal{K}}=\set{(x,y)^T\in \Z_{\max}^2}{y\geq x+1}$. Since
there exists $\lambda \in \Z_{\max}$ such that
$\max(y,\lambda)\geq\max(x,\lambda)+1$ (that is,
there exists $(\lambda,\lambda)^T\in \mathcal{B}$ such that
$(x,y)^T\oplus(\lambda,\lambda)^T\in\mathcal{X}_1$)
if and only if $y\geq x+1$ (that is, $(x,y)^T\in \mathcal{X}_1$),
we get $\mathcal{X}_1 \ominus \mathcal{B}=\mathcal{X}_1$. Therefore,
\begin{eqnarray*}
A^{-1}(\mathcal{X}_1 \ominus \mathcal{B}) & = & A^{-1}(\mathcal{X}_1) \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{A(x,y)^T\in\mathcal{X}_1} \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{(y,x)^T\in\mathcal{X}_1} \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{x\geq y+1 }\;,
\end{eqnarray*}
and thus
\begin{eqnarray*}
{\mathcal{X}}_2 & = & {\mathcal{X}}_1 \cap A^{-1}({\mathcal{X}}_1 \ominus {\mathcal{B}}) \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{y\geq x+1} \cap
\set{(x,y)^T\in \Z_{\max}^2}{x\geq y+1 } \\
& = & \{ (-\infty ,-\infty )^T\} \; .
\end{eqnarray*}
Then, since by Lemma~\ref{lemaalgoAB} the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ is decreasing, it follows that
$\mathcal{X}_k=\{ (-\infty,-\infty)^T\}$ for all $k\geq 2$. Therefore,
the maximal (geometrically) $(A,B)$-invariant semimodule contained
in $\mathcal{K}$ is trivial: $\mathcal{K}^*=\mathcal{X}_\omega=\{ (-\infty,-\infty)^T\} $.
\end{example}
In the case of the theory of linear dynamical systems over a field,
the sequence $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ always converges in at most
$n$ steps, since it is a decreasing sequence of subspaces
of a vector space of dimension $n$. However, one of the problems in
the max-plus case, which is reminiscent of difficulties of the theory
of linear dynamical systems over rings
(see~\cite{assan,AssLafPer,conte94,conte95,hautus82,hautus84}),
is that the sequence $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ may not stabilize
(see Example~\ref{ejemplo2} below). This difficulty comes from
the fact that the semimodule $\Z_{\max}^n$ is not Artinian,
that is, there are infinite decreasing sequences of subsemimodules
of $\Z_{\max}^n$. In the case of linear dynamical systems over rings,
the convergence of the sequence $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ in a
finite number of steps is not guaranteed either, and
although there exists a procedure for finding $\mathcal{K}^*$
when $\mathcal{S}$ is a Principal Ideal Domain (see~\cite{conte94}),
in general the computation of $\mathcal{K}^*$ remains a difficult problem.
\begin{example}\label{ejemplo2}
Let $\mathcal{S}=\Z_{\max}$. Let us consider the matrices
\[
A=
\begin{pmatrix}
-1 & -\infty \\
-\infty & 0
\end{pmatrix}
\enspace \mbox{ and } \enspace
B=
\begin{pmatrix}
0 \\
0
\end{pmatrix}\; ,
\]
and the semimodule $\mathcal{K}=\set{(x,y)^T\in \Z_{\max}^2}{y\leq x-1}$.
Note that $\mathcal{K}=\mbox{\rm Im}\, K$, where
\[
K=
\begin{pmatrix}
0 & 0 \\
-1 & -\infty
\end{pmatrix}\;.
\]
Next we show that in this case the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv} is given by:
\begin{equation}\label{suceje2}
\mathcal{X}_r=\set{(x,y)^T\in \Z_{\max}^2}{y\leq x-r}=
\mbox{\rm Im}\,
\begin{pmatrix}
0 & 0 \\
-r & -\infty
\end{pmatrix}
\;,
\end{equation}
for all $r\in \mathbb{N}$. We prove~\eqref{suceje2}
by induction on $r$. Let us note, in the first place,
that~\eqref{suceje2} is satisfied by definition when $r=1$.
Assume now that~\eqref{suceje2} holds for $r=k$, that is:
\[
\mathcal{X}_k=\set{(x,y)^T\in \Z_{\max}^2}{y\leq x-k}=
\mbox{\rm Im}\,
\begin{pmatrix}
0 & 0 \\
-k & -\infty
\end{pmatrix}\;.
\]
Let us note that $\mathcal{X}_k \ominus \mathcal{B} =\mathcal{X}_k$, since
there exists $\lambda \in \Z_{\max}$ such that
$\max(y,\lambda)\leq\max(x,\lambda)-k$ (that is,
there exists $(\lambda,\lambda)^T\in \mathcal{B}$ such that
$(x,y)^T\oplus(\lambda,\lambda)^T\in\mathcal{X}_k$)
if and only if $y\leq x-k$ (that is, $(x,y)^T\in \mathcal{X}_k$).
Therefore,
\begin{eqnarray*}
A^{-1}(\mathcal{X}_k \ominus \mathcal{B}) & = & A^{-1}(\mathcal{X}_k) \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{A(x,y)^T\in\mathcal{X}_k} \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{(x-1,y)^T\in\mathcal{X}_k} \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{y\leq x-1-k }\;,
\end{eqnarray*}
and thus
\begin{eqnarray*}
\mathcal{X}_{k+1} & = & \mathcal{X}_k \cap A^{-1}(\mathcal{X}_k \ominus \mathcal{B}) \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{y\leq x-k} \cap
\set{(x,y)^T\in \Z_{\max}^2}{y\leq x-1-k } \\
& = & \set{(x,y)^T\in \Z_{\max}^2}{y\leq x-(1+k) }\;,
\end{eqnarray*}
which shows that~\eqref{suceje2} holds for all $r\in \mathbb{N}$.
We see in this way that the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ is strictly decreasing and
therefore does not stabilize. Let us finally note that the semimodule
$\mathcal{X}_{\omega}=\cap_{r\in \mathbb{N}} \mathcal{X}_r=\set{(x,y)^T\in \Z_{\max}^2}{y=-\infty}$
is $A$-invariant, that is, $A(\mathcal{X}_{\omega})\subset\mathcal{X}_{\omega}$.
Then, $\mathcal{X}_{\omega}$ is in particular (geometrically)
$(A,B)$-invariant and therefore
$\mathcal{K}^*= \mathcal{X}_{\omega}=\set{(x,y)^T\in \Z_{\max}^2}{y=-\infty}$.
\end{example}
An open problem is to determine whether it is always the case
that $\mathcal{K}^*= \mathcal{X}_{\omega}$.
It is worth mentioning that this equality does not necessarily hold in
the case of linear dynamical systems over rings.
\begin{remark}
Even when $\mathcal{S}$ is a
Principal Ideal Domain, it could be necessary to compute more than once
(but a finite number of times) the limit $\mathcal{X}_{\omega}$ of sequences
defined as in~\eqref{algoABinv}. To be more precise, in such a case
$\mathcal{X}_1$ is defined as $\mathcal{K}$ in the first step and, if it is necessary
(that is, when $\mathcal{X}_{\omega}$ is not a geometrically $(A,B)$-invariant module),
in the next steps $\mathcal{X}_1$ is defined as the smallest {\em closed} submodule
containing the previous limit $\mathcal{X}_{\omega}$ (see~\cite{conte94} for details).
\end{remark}
Sufficient conditions for the stabilization of the
sequence $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv},
and therefore for the equality $\mathcal{K}^*= \mathcal{X}_{\omega}$
to hold true, will be given in Section~\ref{finitevolumeSec}
in the case $\mathcal{S}=\Z_{\max}$. Note that Example~\ref{ejemplo2} shows that
even in the case of the tropical semiring $\N_{\min}$ the sequence of
semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ may not stabilize (indeed all the
computations in Example~\ref{ejemplo2} are valid when we restrict
ourselves to the semiring $\mathbb{N}_{\max }^{-}=(\mathbb{N}^{-}\cup\{-\infty\},\max ,+)$,
which is clearly isomorphic to $\N_{\min}$). However, more general sufficient
conditions for the equality $\mathcal{K}^*= \mathcal{X}_{\omega}$ to hold true can be
given in the case of the tropical semiring using compactness arguments.
With this aim, let us consider the topology of $\N_{\min} $ defined by the
metric:
\[
d(x,y)=|\exp(-x)-\exp(-y)| \; ,
\]
for all $x,y\in \N_{\min} $. Note that $\N_{\min} $ is compact equipped
with this topology and therefore $\N_{\min}^n$ is also compact
equipped with the product topology. As a matter of fact, given
a sequence $\{x_r\}_{r\in \mathbb{N}}$ of elements of $\N_{\min} $, if the value $+\infty $ appears in
$\{x_r\}_{r\in \mathbb{N}}$ an infinite number of times or if the set of finite values (that is, in $\mathbb{N} $)
of $\{x_r\}_{r\in \mathbb{N}}$ is unbounded (in the usual sense), then $+\infty $ is an
accumulation point of $\{x_r\}_{r\in \mathbb{N}}$. Otherwise, some finite element $x_k$ of
$\{x_r\}_{r\in \mathbb{N}}$ must appear in this sequence an infinite number of times and then $x_k$
is an accumulation point of $\{x_r\}_{r\in \mathbb{N}}$. Now we have the following lemma.
\begin{lemma}\label{compact}
Finitely generated subsemimodules of $\N_{\min}^n$ are compact.
\end{lemma}
\begin{proof}
Firstly, let us notice that $\N_{\min} $ is a {\em topological semiring},
that is, for all sequences $\{x_r\}_{r\in \mathbb{N}}$ and $\{y_r\}_{r\in \mathbb{N}}$
of elements of $\N_{\min} $ the following equalities are satisfied:
\[
\lim_{r\rightarrow \infty}\left(x_r\oplus y_r\right) =
\left(\lim_{r\rightarrow \infty}x_r\right) \oplus
\left(\lim_{r\rightarrow \infty}y_r\right) \; ,
\]
and
\[
\lim_{r\rightarrow \infty}\left(x_r\otimes y_r\right) =
\left(\lim_{r\rightarrow \infty}x_r\right) \otimes
\left(\lim_{r\rightarrow \infty}y_r\right) \; .
\]
Let us now see that a finitely generated semimodule
$\mathcal{X}\subset \N_{\min}^n$ is compact. Indeed, since $\mathcal{X}$ is
finitely generated there exists a matrix $Q\in \N_{\min}^{n\times p}$,
for some $p\in \mathbb{N}$, such that $\mathcal{X} =\mbox{\rm Im}\, Q$. Let $\{Qy_r\}_{r\in \mathbb{N}}$
be an arbitrary sequence of elements of $\mathcal{X}$. To prove that $\mathcal{X}$ is
compact, we must show that $\{Qy_r\}_{r\in \mathbb{N}}$ has a subsequence
which converges to an element of $\mathcal{X}$. Since $\N_{\min}^p$ is compact, we know that
there exists a subsequence $\{y_{r_k}\}_{k\in \mathbb{N}}$ of $\{y_r\}_{r\in \mathbb{N}}$
and an element $y\in \N_{\min}^p$ such that $\lim_{k\rightarrow \infty}y_{r_k}=y$.
Then, using the fact that $\N_{\min}$ is a topological semiring, it follows that
\[
\lim_{k\rightarrow \infty}\left( Qy_{r_k}\right) =
Q\left( \lim_{k\rightarrow \infty}y_{r_k}\right) = Qy\in \mathcal{X} \; .
\]
Therefore, $\mathcal{X}$ is compact.
\end{proof}
The following theorem shows that in the case of $\N_{\min}$ the equality
$\mathcal{K}^*= \mathcal{X}_{\omega}$ holds when $\mathcal{K}$ is finitely generated.
\begin{theorem}
Let $\mathcal{K} \subset \N_{\min}^n$ be a finitely generated semimodule. Then, for all
matrices $A\in \N_{\min}^{n\times n}$ and $B\in \N_{\min}^{n\times q}$, the maximal
(geometrically) $(A,B)$-invariant semimodule $\mathcal{K}^*$ contained in $\mathcal{K}$
is given by $\mathcal{X}_{\omega}=\cap_{r\in \mathbb{N}} \mathcal{X}_r$, where the sequence
of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ is defined by~\eqref{algoABinv}.
\end{theorem}
\begin{proof}
By Lemmas~\ref{obs1} and~\ref{lemaalgoAB}, to prove the theorem, it suffices to show
that $\mathcal{X}_{\omega}$ is a (geometrically) $(A,B)$-invariant semimodule,
which is equivalent to showing that $\mathcal{X}_{\omega}=\varphi(\mathcal{X}_{\omega})$
by Lemma~\ref{obs2}.
Since $\mathcal{X}_{\omega}\subset \mathcal{X}_r$ for all $r\in \mathbb{N}$, it follows that
$\varphi(\mathcal{X}_{\omega})\subset \varphi(\mathcal{X}_r)=\mathcal{X}_{r+1}$ for all
$r\in \mathbb{N}$. Therefore, $\varphi(\mathcal{X}_{\omega})\subset \cap_{r\in \mathbb{N}} \mathcal{X}_r =
\mathcal{X}_{\omega}$.
Let us now see that $\mathcal{X}_{\omega}\subset \varphi(\mathcal{X}_{\omega})$.
Let $x$ be an arbitrary element of $\mathcal{X}_{\omega}$. Then, since
$x\in \varphi(\mathcal{X}_r)=\mathcal{X}_{r+1}$ for all $r\in \mathbb{N}$, we know that
there exists a sequence $\{b_r\}_{r\in \mathbb{N}}\subset \mathcal{B}$ such that
$Ax\oplus b_r$ belongs to $\mathcal{X}_r$ for all $r\in \mathbb{N}$.
As $\mathcal{B}$ is compact by Lemma~\ref{compact}, there exists $b\in \mathcal{B}$
and a subsequence $\{b_{r_k}\}_{k\in \mathbb{N}}$ of $\{b_r\}_{r\in \mathbb{N}}$ such
that $\lim_{k\rightarrow \infty}b_{r_k} =b$. Now, since by
Lemma~\ref{lemaalgoAB} the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ is decreasing,
it follows that $Ax\oplus b_{r_j}\in \mathcal{X}_{r_k}$ for all $j\geq k$.
Therefore, $Ax\oplus b\in \mathcal{X}_{r_k}$ for all $k\in \mathbb{N}$ (recall that the
semimodules $\mathcal{X}_r$ are all finitely generated and then,
by Lemma~\ref{compact}, in particular closed). Then, $Ax\oplus b$
belongs to $\mathcal{X}_{\omega}$, from which we see that $x\in \varphi(\mathcal{X}_{\omega})$.
Therefore, $\mathcal{X}_{\omega} \subset \varphi(\mathcal{X}_{\omega})$.
\end{proof}
\section{Volume}\label{volumeSec}
In the next section we will give sufficient conditions on
the semimodule $\mathcal{K}$, when $\mathcal{S}=\Z_{\max}$, to assure that the sequence
of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined
by~\eqref{algoABinv} stabilizes. For this purpose it is
convenient to introduce first the notion of volume of a
subsemimodule of $\Z_{\max}^n$ and study its properties.
\begin{definition}\label{defvolumen}
Let $\mathcal{K}\subset \Z_{\max}^n$ be a semimodule. We call the {\rm volume} of
$\mathcal{K}$, represented by $\mbox{\rm vol}\,(\mathcal{K})$, the cardinality of the set
$\set{x\in \mathcal{K}}{x_1\oplus \cdots \oplus x_n=0}$, that is,
$\mbox{\rm vol}\,(\mathcal{K})=\mbox{\rm card}\,\left(\set{x\in \mathcal{K}}{x_1\oplus \cdots \oplus x_n=0}\right)$.
Also, if $K\in \Z_{\max}^{n\times p}$, we represent by $\mbox{\rm vol}\,(K)$
the volume of the semimodule $\mathcal{K}=\mbox{\rm Im}\, K$, that is,
$\mbox{\rm vol}\,(K)=\mbox{\rm vol}\,(\mbox{\rm Im}\, K)$.
\end{definition}
Before stating the following results, which provide some properties
of the volume, it is convenient to introduce the following notation:
if $\mathcal{X} \subset \Z_{\max}^n$, then we define
$\tilde{\mathcal{X}}=\set{x\in \mathcal{X}}{x_1\oplus \cdots \oplus x_n=0}$.
\begin{remark}\label{obsvolproy}
Let us consider the max-plus parallelism relation
$\sim$ on $\Z_{\max}^n$ defined by: $x\sim y$ if and only if
$x=\lambda y$ for some $\lambda \in \mathbb{R}$ (that is,
$x_i=\lambda+y_i$ for all $1\leq i\leq n$, in the usual algebra).
We denote by $\mathcal{K}/\sim$ the quotient of a semimodule $\mathcal{K}\subset \Z_{\max}^n$
by this relation and by $[x]$ the equivalence class of $x\in \Z_{\max}^n$.
Then, since the function $f:\tilde{\mathcal{K}}\mapsto (\mathcal{K}/\sim)-[\varepsilon]$ defined by
$f(x)=[x]$ is a bijection, it follows that the volume of $\mathcal{K}$ is equal to $\mbox{\rm card}\,(\mathcal{K}/\sim)-1$,
that is, the cardinality of the set of {\em nontrivial lines} (i.e. the equivalence classes of
nonzero elements) contained in $\mathcal{K}$.
The {\em max-plus projective space} is the quotient of $\R_{\max}^n$ by the parallelism relation.
\end{remark}
\begin{lemma}\label{lemapropvol}
Let $A\in \Z_{\max}^{r\times n}$, $B\in \Z_{\max}^{n\times p}$
and $C\in \Z_{\max}^{p\times q}$ be matrices
and $\mathcal{Z},\mathcal{Y}\subset \Z_{\max}^n$ be semimodules. Then we have:
\begin{enumerate}
\item \label{p1} $\mathcal{Y} \subset \mathcal{Z} \Rightarrow \mbox{\rm vol}\,(\mathcal{Y}) \leq \mbox{\rm vol}\,(\mathcal{Z}) \;$,
\item \label{p2} if $\mbox{\rm vol}\,(\mathcal{Y})<\infty$,
then $\mathcal{Y} \varsubsetneq \mathcal{Z} \Rightarrow \mbox{\rm vol}\,(\mathcal{Y}) <\mbox{\rm vol}\,(\mathcal{Z}) \;$,
\item \label{p3} $\mbox{\rm vol}\,(A\mathcal{Y}) \leq \mbox{\rm vol}\,(A)$ and then
$\mbox{\rm vol}\,(AB) \leq \mbox{\rm vol}\,(A)\;$,
\item \label{p4} $\mbox{\rm vol}\,( A\mathcal{Y}) \leq \mbox{\rm vol}\,(\mathcal{Y})$ and then
$\mbox{\rm vol}\,(AB) \leq \mbox{\rm vol}\,(B)\;$,
\item \label{p5} $\mbox{\rm vol}\,(ABC) \leq \mbox{\rm vol}\,(B)\;$,
\item \label{p6} if $P\in \Z_{\max}^{n\times n}$ and $Q\in \Z_{\max}^{p\times p}$
are invertible\footnote{A matrix $P$ is invertible if there exists
a matrix $P^{-1}$ such that $PP^{-1}=P^{-1}P=I$, where $I$ is the max-plus identity
matrix. In the max-plus semiring, this means that the columns of $P$ are equal,
up to a permutation, to the columns of $I$ multiplied by non-zero scalars.},
then $\mbox{\rm vol}\,(PBQ) =\mbox{\rm vol}\,(B)\;$,
\item \label{p7} $\mbox{\rm vol}\,(A) =\mbox{\rm vol}\,( A^{T})\; $.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{p1}. This property is a consequence of the definition of volume:
$\mathcal{Y} \subset \mathcal{Z} \Rightarrow \tilde{\mathcal{Y}} \subset \tilde{\mathcal{Z}}
\Rightarrow
\mbox{\rm card}\, (\tilde{\mathcal{Y}})\leq \mbox{\rm card}\, (\tilde{\mathcal{Z}}) \Rightarrow
\mbox{\rm vol}\,(\mathcal{Y})\leq \mbox{\rm vol}\,(\mathcal{Z})$.
\ref{p2}. In the first place, we will show that the following simple
property is satisfied: for all semimodules $\mathcal{Y},\mathcal{Z}\subset \Z_{\max}^n$,
\begin{eqnarray}\label{tonta}
\mathcal{Y} \varsubsetneq \mathcal{Z} \Rightarrow \tilde{\mathcal{Y}} \varsubsetneq \tilde{\mathcal{Z}}\; .
\end{eqnarray}
As a matter of fact, assume that $\mathcal{Y} \varsubsetneq \mathcal{Z}$. Then, there exists
$x\in \mathcal{Z} - \mathcal{Y}$. Therefore, we know that
$x \neq(-\infty,\ldots ,-\infty)^T$ and we can define the
vector $\tilde{x}=\left( x_1\oplus \cdots \oplus x_n\right)^{-1}x$
(that is, $\tilde{x}_i=x_i-\max\{x_1,\ldots ,x_n\}$ for all
$1\leq i \leq n$, in the usual algebra). Now, it follows that
$\tilde{x}\in \tilde{\mathcal{Z}}- \tilde{\mathcal{Y}}$ and thus
$\tilde{\mathcal{Y}} \varsubsetneq \tilde{\mathcal{Z}}$.
This proves property~\eqref{tonta}.
Now, using property~\eqref{tonta} and the fact that $\mbox{\rm vol}\,(\mathcal{Y})<\infty$,
we get:
$\mathcal{Y} \varsubsetneq \mathcal{Z} \Rightarrow
\tilde{\mathcal{Y}} \varsubsetneq \tilde{\mathcal{Z}} \Rightarrow
\mbox{\rm card}\, (\tilde{\mathcal{Y}}) < \mbox{\rm card}\, (\tilde{\mathcal{Z}}) \Rightarrow
\mbox{\rm vol}\,(\mathcal{Y}) < \mbox{\rm vol}\,(\mathcal{Z})$.
\ref{p3}. Since $A\mathcal{Y}\subset \mbox{\rm Im}\, A$, applying Statement~\ref{p1},
we have: $\mbox{\rm vol}\,(A\mathcal{Y}) \leq \mbox{\rm vol}\,(\mbox{\rm Im}\, A)=\mbox{\rm vol}\,(A)$.
\ref{p4}. From the definition of the set $\tilde{\mathcal{Y}}$ it follows
that for each $y\in \mathcal{Y}- \{ (-\infty,\ldots ,-\infty )^T\}$
there exists $\tilde{y}\in \tilde{\mathcal{Y}}$ and $\lambda \in {\mathbb{Z}}$ such that
$y=\lambda \tilde{y}$ (it suffices to take
$\lambda = y_1\oplus \cdots \oplus y_n$ and $\tilde{y}=\lambda^{-1}y$).
Therefore,
\[
A\mathcal{Y} - \{ (-\infty,\ldots ,-\infty )^T\} \subset
\set{\lambda A\tilde{y}}{\tilde{y}\in \tilde{\mathcal{Y}}, \lambda \in \mathbb{Z}}\;,
\]
and then we get:
\begin{eqnarray*}
&\mbox{\rm vol}\,(A\mathcal{Y})=\mbox{\rm card}\, ( \set{x\in A\mathcal{Y}}{x_1\oplus \cdots \oplus x_r=0} ) \\
& \leq \mbox{\rm card}\, (\set{x=\lambda A\tilde{y}}{\tilde{y}\in \tilde{\mathcal{Y}}, \lambda \in \mathbb{Z}, x_1\oplus \cdots \oplus x_r=0} ) \\
& \leq \mbox{\rm card}\, (\set{A\tilde{y}}{\tilde{y}\in \tilde{\mathcal{Y}}}) \leq
\mbox{\rm card}\,(\tilde{\mathcal{Y}})=\mbox{\rm vol}\,(\mathcal{Y})\;.
\end{eqnarray*}
\ref{p5}. Applying Statements~\ref{p3} and~\ref{p4} we get:
$\mbox{\rm vol}\,(ABC)\leq \mbox{\rm vol}\,(AB)\leq \mbox{\rm vol}\,(B)$.
\ref{p6}. From Statement~\ref{p5} we obtain:
$\mbox{\rm vol}\,(B) =\mbox{\rm vol}\,(P^{-1}PBQQ^{-1})\leq \mbox{\rm vol}\,(PBQ)\leq \mbox{\rm vol}\,(B)$.
Therefore, $\mbox{\rm vol}\,(B) =\mbox{\rm vol}\,(PBQ)$.
\ref{p7}. Let us note, in the first place,
that we can define in a completely analogous way
the volume of a subsemimodule of $\Z_{\min}^n$. Then,
since the function $x\rightarrow -x$ is an
isomorphism from $\Z_{\max}$ to $\Z_{\min}$, it is clear that
$\mbox{\rm vol}\,(\mathcal{Z})=\mbox{\rm vol}\,(-\mathcal{Z})$ for every subsemimodule
$\mathcal{Z}\subset \Z_{\max}^n$. Let us now consider the matrix
$A^{\sharp}=-A^T$ and the semimodule
$\mathcal{Y}=\mbox{\rm Im}\,(A^{\sharp})\subset \Z_{\min}^n$. Since
$\mathcal{Y}=-\mbox{\rm Im}\,(A^T)$, we know that $\mbox{\rm vol}\,(A^T)=\mbox{\rm vol}\,(\mathcal{Y})$.
Now, using elements of residuation theory (we refer the reader to~\cite{BlythJan72}
for an extensive presentation of this theory), it can be shown
(see for example~\cite{bcoq} or~\cite{gaubert01a})
that the following two properties hold:
\begin{eqnarray*}
A(A^{\sharp}(Ax)) & = & Ax\;,\enspace \forall x\in \Z_{\max}^n\;, \mbox{ and }\\
A^{\sharp}(A(A^{\sharp}y)) & = & A^{\sharp}y\;,\enspace \forall y\in \Z_{\min}^r \;,
\end{eqnarray*}
where the products by $A$ are performed in $\overline{\Z}_{\max}$ and the products
by $A^{\sharp}$ are performed in $\overline{\Z}_{\min}$. Therefore,
the function $f:\mbox{\rm Im}\,(A)\mapsto \mbox{\rm Im}\,(A^{\sharp})$ defined by
$f(y)=A^{\sharp}y$ is a bijection with inverse $g(x)=Ax$.
Then, the function $F$ from $\mbox{\rm Im}\,(A)/\sim$ to $\mbox{\rm Im}\,(A^{\sharp})/\sim$
defined by $F([y])=[A^{\sharp}y]$, where $[x]$ denotes
the equivalence class of $x$ by the parallelism relation $\sim$,
is also a bijection. Now, using Remark~\ref{obsvolproy},
we obtain:
$\mbox{\rm vol}\,(A)=\mbox{\rm card}\,(\mbox{\rm Im}\,(A)/\sim)-1=\mbox{\rm card}\,(\mbox{\rm Im}\,(A^{\sharp})/\sim)-1=\mbox{\rm vol}\,(A^{\sharp})=\mbox{\rm vol}\,(\mathcal{Y})$,
and then $\mbox{\rm vol}\,(A)=\mbox{\rm vol}\,(\mathcal{Y})=\mbox{\rm vol}\,(A^T)$.
\end{proof}
\section{Specifications with finite volume}\label{finitevolumeSec}
In the next theorem we give a condition on the specification $\mathcal{K}$,
when $\mathcal{S}=\Z_{\max}$, ensuring that the sequence of semimodules defined
by~\eqref{algoABinv} stabilizes.
\begin{theorem}\label{th-inv}
Let $\mathcal{K}\subset\Z_{\max}^n$ be a semimodule with finite volume.
Then, for all $A\in\Z_{\max}^{n\times n}$ and $B\in \Z_{\max}^{n\times p}$,
the maximal (geometrically) $(A,B)$-invariant semimodule $\mathcal{K}^*$
contained in $\mathcal{K}$ is finitely generated. Moreover,
if we define the sequence of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$
by~\eqref{algoABinv}, then $\mathcal{K}^*=\mathcal{X}_k$ for some $k\leq \mbox{\rm vol}\,(\mathcal{K})+1$.
\end{theorem}
\begin{proof}
First of all, let us note that every semimodule $\mathcal{Y}\subset\Z_{\max}^n$
with finite volume is necessarily finitely generated. Indeed,
this property is a consequence of the fact that
$\mathcal{Y}=\mbox{\rm span}\,(\tilde{\mathcal{Y}})$. Now, as $\mathcal{K}^*\subset\mathcal{K}$, applying
Statement~\ref{p1} of Lemma~\ref{lemapropvol} it follows that
$\mbox{\rm vol}\,(\mathcal{K}^*)\leq \mbox{\rm vol}\,(\mathcal{K})<\infty$, and then $\mathcal{K}^*$ is finitely generated.
Let us now see that the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv}
must stabilize in at most $\mbox{\rm vol}\,(\mathcal{K})+1$ steps.
Indeed, by Lemma~\ref{lemaalgoAB}
we know that the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ is decreasing. Then,
using Statement~\ref{p1} of Lemma~\ref{lemapropvol},
we see that $\{\mbox{\rm vol}\,(\mathcal{X}_r)\}_{r\in \mathbb{N}}$ is a decreasing
sequence of nonnegative integers. Therefore,
there exists $k\leq \mbox{\rm vol}\,(\mathcal{X}_1)+1=\mbox{\rm vol}\,(\mathcal{K})+1$ such that
$\mbox{\rm vol}\,(\mathcal{X}_{k+1})=\mbox{\rm vol}\,(\mathcal{X}_k)$. Then,
as $\mathcal{X}_{k+1}\subset \mathcal{X}_k\subset \mathcal{K}$ by Lemma~\ref{lemaalgoAB},
we know that $\mbox{\rm vol}\,(\mathcal{X}_{k+1})=\mbox{\rm vol}\,(\mathcal{X}_k)\leq \mbox{\rm vol}\,(\mathcal{K})<\infty$
(once again, by Statement~\ref{p1} of Lemma~\ref{lemapropvol}).
Finally, applying Statement~\ref{p2} of Lemma~\ref{lemapropvol}
to the semimodules $\mathcal{X}_{k+1}$ and $\mathcal{X}_k$, it follows that
$\mathcal{X}_{k+1}=\mathcal{X}_k$, from which we conclude that $\mathcal{K}^*=\mathcal{X}_k$.
\end{proof}
An important particular case of Theorem~\ref{th-inv}
is the one in which the semimodule $\mathcal{K}$ is
generated by a finite number of vectors whose
entries are all finite. In this case it is possible
to bound the volume of $\mathcal{K}$ by means of the additive
version of Hilbert's projective metric:
for all $x\in\mathbb{Z}^n$, define
\[
\|x\|_H=\max\set{x_i}{1\leq i \leq n}-\min\set{x_i}{1\leq i \leq n} \enspace,
\]
and for all $K\in \mathbb{Z}^{n\times s}$, define
\[
\Delta_H(K)= \max\set{\|K_{\cdot i}\|_H}{1\leq i\leq s} \enspace,
\]
where $K_{\cdot i}$ denotes the $i$-th column of the matrix $K$.
Then we have the following corollary.
\begin{corollary}\label{corvolfin}
Let $\mathcal{K}=\mbox{\rm Im}\, K$, where $K\in \Z_{\max}^{n\times s}$
is a matrix whose entries are all finite.
Then, for all $A\in\Z_{\max}^{n\times n}$ and $B\in \Z_{\max}^{n\times p}$,
the maximal (geometrically) $(A,B)$-invariant semimodule $\mathcal{K}^*$
contained in $\mathcal{K}$ is finitely generated and, if we define the
sequence of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ by~\eqref{algoABinv},
there exists some $k\leq (\Delta_H(K)+1)^n-\Delta_H(K)^n+1$
such that $\mathcal{K}^*=\mathcal{X}_k$.
\end{corollary}
\begin{proof}
By Theorem~\ref{th-inv}, to prove the corollary,
it suffices to show that
\begin{equation}\label{projective}
\mbox{\rm vol}\,(\mathcal{K})\leq (\Delta_H(K)+1)^n-\Delta_H(K)^n \enspace ,
\end{equation}
where the power $n$ is in the usual algebra.
Since the additive version of Hilbert's projective metric
$\|\cdot\|_H$ satisfies the following properties:
\begin{eqnarray*}
\|\lambda x\|_H & = & \|x\|_H \;, \\
\|x\oplus y\|_H & \leq &\|x\|_H\oplus \|y\|_H \;,
\end{eqnarray*}
for all $x,y\in \mathbb{Z}^n$ and $\lambda \in \mathbb{Z}$,
it follows that $\|x\|_H\leq\Delta_H(K)$ for all
$x\in \mathcal{K}- \{(-\infty,\ldots ,-\infty)^T\}$
and therefore $\mathcal{K}$ is contained in the semimodule
\[
\mathcal{Y}=\set{x\in \mathbb{Z}^n}{\|x\|_H\leq\Delta_H(K)}\cup \{(-\infty,\ldots ,-\infty)^T\}
\]
(note that the only vector in $\mathcal{K}$ with at least one entry
equal to $-\infty$ is $(-\infty,\ldots ,-\infty)^T$).
Then, by Statement~\ref{p1} of Lemma~\ref{lemapropvol},
to prove~\eqref{projective} it suffices
to show that $\mbox{\rm vol}\,(\mathcal{Y})=(\Delta_H(K)+1)^n-\Delta_H(K)^n$.
With this aim, we must compute the number of elements of the set:
\begin{eqnarray*}
\tilde{\mathcal{Y}} & = & \set{x\in \mathcal{Y}}{x_1\oplus \cdots \oplus x_n=0} \\
& = & \set{x\in \mathbb{Z}^n}{\|x\|_H\leq\Delta_H(K),x_1\oplus \cdots \oplus x_n=0}\;,
\end{eqnarray*}
that is, the number of vectors $x$ in $\mathbb{Z}^n$ with entries
between $-\Delta_H(K)$ and zero (since $\max_ix_i=x_1\oplus \cdots \oplus x_n=0$ and
$\Delta_H(K)\geq \|x\|_H =\max_ix_i-\min_ix_i=-\min_ix_i$) and with at least one entry
equal to zero (since $\max_ix_i=0$).
We know that there are ${n \choose r} \Delta_H(K)^{n-r}$
elements in the set $\tilde{\mathcal{Y}}$ with exactly $r$ entries equal to zero.
To be more precise, there exist ${n \choose r}$
different ways of choosing the $r$ entries which will
have the value zero, and there exist $\Delta_H(K)^{n-r}$
different ways of assigning values to the $n-r$
remaining entries among the $\Delta_H(K)$ possible values.
Therefore, the number of elements of the set $\tilde{\mathcal{Y}}$ is:
\[
\sum_{r=1}^{r=n} {n \choose r}\Delta_H(K)^{n-r} =
(\Delta_H(K)+1)^n-\Delta_H(K)^n \;,
\]
and then $\mbox{\rm vol}\,(\mathcal{Y})=(\Delta_H(K)+1)^n-\Delta_H(K)^n$.
\end{proof}
Note that in the proof of Corollary~\ref{corvolfin} we showed,
in particular, that for each matrix $K\in \Z_{\max}^{n\times s}$
whose entries are all finite, the volume $\mbox{\rm vol}\,(K)$ is bounded
by $(\Delta_H(K)+1)^n-\Delta_H(K)^n$ (this is inequality~\eqref{projective}).
We next show that this bound is tight. Indeed,
let us consider the semimodule
\[
\mathcal{Y}=\set{x\in \mathbb{Z}^n}{\|x\|_H\leq M}\cup \{(-\infty,\ldots ,-\infty)^T\}\;,
\]
where $M\in \mathbb{N}$. Note that in the proof of Corollary~\ref{corvolfin}
we proved that $\mathcal{Y}$ has volume $(M+1)^n-M^n$. Now,
if we define the matrix $K\in \Z_{\max}^{n\times n}$ by
$K_{ij}=M$ if $i=j$ and $K_{ij}=0$ otherwise,
it follows that $\mathcal{Y}=\mbox{\rm Im}\,(K)$ and $\Delta_H(K)=M$.
Therefore, there exist matrices $K\in \Z_{\max}^{n\times s}$
(whose entries are all finite) which have volume equal to
$(\Delta_H(K)+1)^n-\Delta_H(K)^n$.
Theorem~\ref{th-inv} is useful in many practical problems because
in such problems the specification $\mathcal{K}$ frequently has finite volume.
This is often the case when $\mathcal{K}$ models certain stability conditions,
as for example, ``bounded delay'' requirements. To be more precise, let us
assume that system~\eqref{dynamicsystem} is the dater representation
of a timed event graph (we refer the reader to~\cite{bcoq} for more
details on the modeling of timed event graphs). Then, a typical
case of semimodule $\mathcal{K}$ which arises in applications is:
\begin{equation}\label{semiacot}
\mathcal{K} =\set{x\in \Z_{\max}^n}{x_i-x_j \leq d_{ij}, \forall 1\leq i,j\leq n}\; ,
\end{equation}
where $D=(d_{ij})$ is a matrix with entries in $\mathbb{Z}\cup \{+\infty\}$.
Note that the state vector $x(k)$, representing the dates of the
firings numbered $k$, belongs to $\mathcal{K}$ if and only if $x(k)_i-x(k)_j \leq d_{ij}$,
for all $1\leq i,j\leq n$, which means that the delay between the
$k$-th firing of the transition labeled $j$ and the $k$-th
firing of the transition labeled $i$ should not exceed $d_{ij}$.
Note also that in practice we usually can assume that $D$ only has finite
entries, since we can replace $+\infty$ by a sufficiently large constant.
We next show that in such a case, the semimodule $\mathcal{K}$ defined by~\eqref{semiacot}
has finite volume. Let us first recall that a
directed graph $\mathcal{G} (A)$, called the {\em precedence graph} of $A$,
is associated with a matrix $A=(a_{ij})\in \R_{\max}^{n\times n}$. This graph
is defined as follows: there exists a directed arc of
{\em weight} $a_{ji}$ from node $i$ to node $j$
if and only if $a_{ji}\not = -\infty$. A matrix whose
precedence graph is strongly connected is called
{\em irreducible}. The spectral radius $\rho_{\max }(A)$
of $A$ is defined by:
\[
\rho_{\max }(A)=\bigoplus_{k=1}^{n}\mbox{tr}(A^k)^\frac{1}{k}=
\max_{1\leq k\leq n} \max_{i_1,\ldots ,i_k} \frac{a_{i_1i_2}+\cdots
+a_{i_ki_1}}{k} \; ,
\]
that is, the maximal circuit mean of $\mathcal{G} (A)$.
Before stating the following lemma, which shows in particular that the
semimodule~\eqref{semiacot} has finite volume
when $D$ only has finite entries, let us note that
\begin{equation}\label{semiacot2}
\mathcal{K} =\set{x\in \Z_{\max}^n}{Ex \leq x}\; ,
\end{equation}
where $E=(-D)^T$. Then we have:
\begin{lemma}\label{lemaHstar}
If the matrix $E$ is irreducible, then the semimodule
$\mathcal{K}$ defined by~\eqref{semiacot2} has finite volume.
Moreover, if $E$ has spectral radius strictly greater
than the unit (that is, 0), then $\mathcal{K}$ reduces to the
null vector.
\end{lemma}
\begin{proof}
In the first place, let us see that
$\mathcal{K}=\mbox{\rm Im}\,(E^*)\cap \Z_{\max}^n$, where
\[
E^*=\bigoplus_{r=0}^{\infty }E^r=I\oplus E\oplus E^2\oplus \cdots
\]
(note that the matrix $E^*$ can have entries equal to $+\infty$,
so that $E^*$ should be thought of as a map from $\overline{\Z}_{\max}^n$ to $\overline{\Z}_{\max}^n$).
Indeed, we have:
\begin{eqnarray*}
& x\in \mathcal{K} \Rightarrow Ex\leq x,x\in \Z_{\max}^n \Rightarrow \\
& E^{r}x\leq x , \forall r\in \mathbb{N} , x\in \Z_{\max}^n
\Rightarrow E^*x\leq x , x\in \Z_{\max}^n \Rightarrow \\
& E^*x = x , x\in \Z_{\max}^n \Rightarrow x\in \mbox{\rm Im}\,(E^*)\cap \Z_{\max}^n \; ,
\end{eqnarray*}
and
\begin{eqnarray*}
& x\in \mbox{\rm Im}\,(E^*)\cap \Z_{\max}^n \Rightarrow \\
& x=E^*y, \mbox { for some } y\in \overline{\Z}_{\max}^n , x\in \Z_{\max}^n \Rightarrow \\
& Ex\leq E^*x=E^*E^*y=E^*y=x , x\in \Z_{\max}^n \Rightarrow x\in \mathcal{K} \; .
\end{eqnarray*}
When $E$ has spectral radius less than or equal to the unit, we know that:
\[
E^*=I\oplus E\oplus \cdots \oplus E^{n-1}\; ,
\]
since $E^r\leq I\oplus E\oplus \cdots \oplus E^{n-1}$
for all $r\geq n$ (see for example Theorem~3.20 of~\cite{bcoq}).
Moreover, since $E$ is irreducible, we know that all the entries
of $E^*$ are finite. Indeed, this follows from the fact that $E^k_{ij}$,
for $i\not =j$, is the maximal weight of all paths of length $k$ running from
$j$ to $i$ in the precedence graph of $E$. Then,
the proof of Corollary~\ref{corvolfin} shows that $\mathcal{K}$ has finite volume.
When $E$ has spectral radius strictly greater than the unit, since
$E$ is irreducible, all the entries of $E^*$ are equal to
$+\infty$ (once again by the interpretation of the entries of
the matrix $E^k$ in terms of the weight of paths in the precedence
graph of $E$). Therefore, the only vector in $\mathcal{K}=\mbox{\rm Im}\, (E^*)\cap \Z_{\max}^n$
is the null vector.
\end{proof}
We end this section with an example showing that in Theorem~\ref{th-inv},
the bound $\mbox{\rm vol}\,(\mathcal{K})+1$ on the number of steps needed to stabilize
the sequence of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv},
cannot be improved.
\begin{example}
Let us consider the matrices
\[
A=
\begin{pmatrix} 1 & -\infty \\ -\infty & 0 \end{pmatrix}
\enspace \mbox{ and } \enspace
B=\begin{pmatrix} 0 \\ 0\end{pmatrix}\; ,
\]
and the semimodule $\mathcal{K}=\set{(x,y)^T\in \Z_{\max}^2}{x+1\leq y\leq x+l}$,
where $l\in \mathbb{N}$. Then, in this case we have:
\[
\tilde{\mathcal{K}}=\set{(x,y)^T\in \mathcal{K}}{x\oplus y=0}=\{(-1,0)^T,\ldots ,(-l,0)^T\}\; ,
\]
from which we get $\mbox{\rm vol}\,(\mathcal{K})=l$.
Therefore, we are able to apply Theorem~\ref{th-inv}.
In fact, $\mathcal{K}=\mbox{\rm Im}\, K$ where
\[
K=\begin{pmatrix} 0 & 0 \\ 1 & l \end{pmatrix}\;,
\]
so we are also in a position to apply Corollary~\ref{corvolfin}.
By Theorem~\ref{th-inv} we know that the sequence of semimodules
$\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv} must stabilize
in at most $\mbox{\rm vol}\,(\mathcal{K})+1=l+1$ steps.
Let us check this fact in this particular case.
In the first place, note that
$\mathcal{K}\subset \set{(x,y)^T\in \Z_{\max}^2}{x+1\leq y}$, so that
$\mathcal{X}_r\subset\mathcal{K}\subset \set{(x,y)^T\in \Z_{\max}^2}{x+1\leq y}$
for all $r\in \mathbb{N}$.
Then, it is easy to show (applying a straightforward variant
of the computation of $\mathcal{X}_r\ominus \mathcal{B}$ done in
Example~\ref{ejemplo2}) that $\mathcal{X}_r\ominus \mathcal{B}=\mathcal{X}_r$
for all $r\in \mathbb{N}$. In this way we get:
\begin{eqnarray*}
\mathcal{X}_1 & = & \left\{(x,y)^T\in \Z_{\max}^2\mid x+1\leq y\leq x+l\right\}
\;,\\
\mathcal{X}_2 & = &\mathcal{X}_1 \cap A^{-1}(\mathcal{X}_1 \ominus \mathcal{B})=\mathcal{X}_1 \cap A^{-1}(\mathcal{X}_1) \\
& = &\left\{(x,y)^T\in \Z_{\max}^2\mid x+1\leq y\leq x+l\right\}\cap
\\ & & \enspace \;
\left\{(x,y)^T\in \Z_{\max}^2\mid x+2\leq y\leq x+l+1\right\} \\
& = &\left\{(x,y)^T\in \Z_{\max}^2\mid x+2\leq y\leq x+l\right\}
\varsubsetneq \mathcal{X}_1\;,\\
& \vdots & \\
\mathcal{X}_l & = & \mathcal{X}_{l-1} \cap A^{-1}(\mathcal{X}_{l-1} \ominus \mathcal{B})=\mathcal{X}_{l-1} \cap A^{-1}(\mathcal{X}_{l-1}) \\
& = & \left\{(x,y)^T\in \Z_{\max}^2\mid x+l-1\leq y\leq x+l\right\}\cap
\\ & & \enspace \;
\left\{(x,y)^T\in \Z_{\max}^2\mid x+l\leq y\leq x+l+1\right\} \\
& = & \left\{(x,y)^T\in \Z_{\max}^2\mid x+l\leq y\leq x+l\right\} \\
& = & \left\{(x,y)^T\in \Z_{\max}^2\mid y= x+l\right\}
\varsubsetneq \mathcal{X}_{l-1} \;,\\
\mathcal{X}_{l+1} & = &\mathcal{X}_l \cap A^{-1}(\mathcal{X}_l \ominus \mathcal{B})=\mathcal{X}_l \cap A^{-1}(\mathcal{X}_l) \\
& = & \left\{(x,y)^T\in \Z_{\max}^2\mid y = x+l\right\}\cap
\left\{(x,y)^T\in \Z_{\max}^2\mid y= x+l+1\right\} \\
& = & \left\{(-\infty ,-\infty)^T\right\} \varsubsetneq \mathcal{X}_l\;.
\end{eqnarray*}
Then, since by Lemma~\ref{lemaalgoAB} we know that
\[
\left\{(-\infty ,-\infty)^T\right\} \subset \mathcal{X}_{l+2} \subset
\mathcal{X}_{l+1}=\left\{(-\infty ,-\infty)^T\right\}\; ,
\]
it is clear that $\mathcal{X}_{l+2}=\mathcal{X}_{l+1}$, and therefore
\[
\mathcal{K}^*=\mathcal{X}_{l+1}=\left\{(-\infty ,-\infty)^T\right\} \; .
\]
In this way we see that in this particular case the sequence
of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ stabilizes in exactly
$\mbox{\rm vol}\,(\mathcal{K})+1=l+1$ steps.
\end{example}
\section{Algebraically $(A,B)$-invariant semimodules}\label{algABinvSec}
This section deals with another fundamental problem in the geometric
approach to the theory of linear dynamical systems:
the computation of a linear feedback. Let us once again consider the dynamical
system~\eqref{dynamicsystem}. Let us assume that we already know the maximal
(geometrically) $(A,B)$-invariant semimodule $\mathcal{K}^*$ contained in a given
semimodule $\mathcal{K}\subset \mathcal{S}^n$. From a dynamical point of view, this means
that the trajectories of system~\eqref{dynamicsystem} starting in $\mathcal{K}^*$
can be kept inside $\mathcal{K}^*$ by a suitable choice of the control. Our new problem
is to determine whether this control can be generated by using a state feedback.
In other words, we want to determine whether there exists a linear feedback
$u(k)=Fx(k-1)$, where $F\in \mathcal{S}^{q\times n}$, which makes $\mathcal{K}^*$ invariant
with respect to the resulting closed loop system:
\begin{equation}\label{sisaut}
x(k)=(A\oplus BF)x(k-1)\; ,
\end{equation}
that is, such that every trajectory of the closed loop
system~\eqref{sisaut} is completely contained in $\mathcal{K}^*$ when its initial
state is in $\mathcal{K}^*$. If a linear feedback with this property exists, we will
say that $\mathcal{K}^*$ is an algebraically $(A,B)$-invariant semimodule. Some
authors call this notion $(A+BF)$-invariance (see~\cite{assan}) or the
feedback property (see~\cite{hautus82,conte95,conte94}).
\begin{definition}\label{defABFinv}
Given the matrices $A\in\mathcal{S}^{n\times n}$ and $B\in \mathcal{S}^{n\times q}$, we say
that a semimodule $\mathcal{X} \subset \mathcal{S}^n$ is {\rm algebraically
$(A,B)$-invariant} if there exists $F\in \mathcal{S}^{q\times n}$ such that
\[
(A\oplus BF) \mathcal{X} \subset \mathcal{X} \enspace .
\]
\end{definition}
Obviously, every algebraically $(A,B)$-invariant semimodule is
also geometrically $(A,B)$-invariant. Nevertheless, when $\mathcal{S}=\Z_{\max}$
it is not clear whether a geometrically $(A,B)$-invariant semimodule
is algebraically $(A,B)$-invariant.
Once again, this problem is reminiscent of difficulties of
the theory of linear dynamical systems over rings
(see~\cite{hautus82,hautus84,conte94,conte95,assan,AssLafPer}).
Indeed, in the case of linear dynamical systems with
coefficients in a field, the class of geometrically
$(A,B)$-invariant spaces coincides with the class of
algebraically $(A,B)$-invariant spaces (see~\cite{wonham}).
This property makes the (geometrically) $(A,B)$-invariant
spaces very useful in the classical theory.
However, this crucial feature is no longer true for
linear dynamical systems with coefficients in a ring,
that is, there exist geometrically $(A,B)$-invariant
modules which are not algebraically $(A,B)$-invariant
(see~\cite{hautus82}, in particular Example~2.3).
The following example shows that this is also the case for
linear dynamical systems over the tropical
semiring $\N_{\min}=(\N\cup\{+\infty\},\min,+)$.
\begin{remark}
In the case of rings, a necessary and sufficient condition for $\mathcal{K}^*$
to be algebraically $(A,B)$-invariant can be given in the form of a factorization
condition on the transfer function, assuming that the system is
reachable and injective (see~\cite{hautus82}). When $\mathcal{S}$ is a
Principal Ideal Domain, it can be shown that $\mathcal{K}^*$ is
algebraically $(A,B)$-invariant if and only if it is a direct
summand (see~\cite{hautus82,conte95,conte94}).
\end{remark}
\begin{example}
Let $\mathcal{S}=\N_{\min}$. Let us consider the matrices
\[
A=
\begin{pmatrix} 1 & +\infty \\ 1 & 0 \end{pmatrix}
\enspace \mbox{ and } \enspace
B=\begin{pmatrix} 1 \\ 1\end{pmatrix}\; ,
\]
and the semimodule $\mathcal{K}=\left\{(x,y)^T\in \N_{\min}^2\mid x\leq y \right\}$.
In the first place, let us compute the maximal geometrically
$(A,B)$-invariant semimodule $\mathcal{K}^*$ contained in $\mathcal{K}$. With this aim,
we will compute the sequence of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$
defined by~\eqref{algoABinv}. We have:
\begin{eqnarray*}
\mathcal{X}_1 & = & \mathcal{K}=\left\{(x,y)^T\in \N_{\min}^2\mid x\leq y \right\}
\;,\\
\mathcal{X}_2 & = & \mathcal{X}_1 \cap A^{-1}(\mathcal{X}_1 \ominus \mathcal{B}) \\
& = & \left\{(x,y)^T\in \N_{\min}^2\mid x\leq y \right\}\cap
\left\{(x,y)^T\in \N_{\min}^2\mid 1\leq y\right\} \\
& = & \left\{(x,y)^T\in \N_{\min}^2\mid x\leq y, 1\leq y\right\}
\;, \\
\mathcal{X}_3 & = & \mathcal{X}_2 \cap A^{-1}(\mathcal{X}_2 \ominus \mathcal{B})= \\
& = & \left\{(x,y)^T\in \N_{\min}^2\mid x\leq y,1\leq y\right\}\cap
\left\{(x,y)^T\in \N_{\min}^2\mid 1\leq y\right\} \\
& = & \mathcal{X}_2 \;.
\end{eqnarray*}
Then, we get $\mathcal{K}^*=\mathcal{X}_2=\left\{(x,y)^T\in \N_{\min}^2\mid x\leq y,1\leq y\right\}$.
Indeed, it is easy to check that a trajectory which starts at a point of
$\mathcal{K}^*=\mathcal{K}- \left\{(0,0)^T\right\}$ can be kept inside $\mathcal{K}$ with
the sequence of controls identically equal to $(1,1)^T$, and that a
trajectory which starts at the point $(0,0)^T$ cannot be kept inside $\mathcal{K}$
(since for all controls in $\mathcal{B}$ the next state of the system is always
$(1,0)^T$, which does not belong to $\mathcal{K}$).
Let us now see that $\mathcal{K}^*$ is not an algebraically $(A,B)$-invariant
semimodule. With this aim, we will show that a trajectory which starts at the
point $(1,1)^T\in\mathcal{K}^*$ cannot be kept inside $\mathcal{K}^*$ when a linear state
feedback is applied. Let $F\in \N_{\min}^{1\times2}$ be an arbitrary feedback.
Then, since $F (1,1)^T\geq 1$, we know that $BF(1,1)^T=(\alpha,\alpha)^T$,
where $\alpha \geq 2$. Therefore,
$$(A\oplus BF)\begin{pmatrix} 1 \\ 1\end{pmatrix} =
\begin{pmatrix} 2 \\ 1\end{pmatrix}\oplus \begin{pmatrix} \alpha \\ \alpha \end{pmatrix}=
\begin{pmatrix} 2 \\ 1\end{pmatrix}\not \in \mathcal{K}^*\; ,$$
which shows that $\mathcal{K}^*$ is not an algebraically $(A,B)$-invariant semimodule.
\end{example}
We next show how we can decide, using the existing results
on max-plus linear system of equations, whether a finitely generated
subsemimodule of $\Z_{\max}^n$ is algebraically $(A,B)$-invariant.
This method also computes a linear feedback with the required
property when it exists. Let $A\in\Z_{\max}^{n\times n}$, $B\in \Z_{\max}^{n\times q}$,
and let $\mathcal{X}$ be a finitely generated subsemimodule of
$\Z_{\max}^n$, so that there exists $Q\in \Z_{\max}^{n\times r}$,
for some $r\in \mathbb{N}$, such that $\mathcal{X}=\mbox{\rm Im}\, Q$. Then,
from Definition~\ref{defABFinv} it readily follows
that $\mathcal{X}$ is an algebraically $(A,B)$-invariant semimodule
if and only if there exist matrices
$F\in \Z_{\max}^{q\times n}$ and $G\in \Z_{\max}^{r\times r}$
such that:
\begin{equation}\label{sistnohom}
(A\oplus BF)Q=QG\; .
\end{equation}
As~\eqref{sistnohom} is a two sided max-plus linear system of equations,
we know that its set of solutions $(F,G)$ is a finitely
generated max-plus convex set,
which can be explicitly computed by the general elimination methods
(see~\cite{butkovicH,gaubert92a,gaubert98n,maxplus97}).
In this way we see that we can effectively decide whether a finitely
generated subsemimodule of $\Z_{\max}^n$ is algebraically $(A,B)$-invariant.
\begin{remark}\label{obssisequa}
The elimination algorithm shows that the set of solutions of a homogeneous
max-plus linear system of the form $Dx=Cx$, where $D, C$ are matrices of
suitable dimensions, is a finitely generated semimodule. This algorithm relies
on the fact that hyperplanes of $\R_{\max}^n$ (that is, the set of solutions of
an equation of the form $dx=cx$, where $d,c\in \R_{\max}^n$ are row vectors)
are finitely generated. It is worth mentioning that the resulting
naive algorithm has an a priori doubly exponential complexity. However, the
doubly exponential bound is pessimistic. It is possible to incorporate
in this algorithm the elimination of redundant generators which reduces its
execution time. In fact, we are currently working on this subject and we
believe that improvements are possible, since we have shown by direct
arguments that the number of generators of the set of solutions is at
most simply exponential. This will be the subject of a further work.
\end{remark}
Let us note that to decide whether $\mathcal{X}=\mbox{\rm Im}\, Q$ is an algebraically
$(A,B)$-invariant semimodule it suffices to know whether the system of
equations~\eqref{sistnohom} has at least one solution. Taking this into account,
it is worth mentioning that there are algorithms to compute a single solution
(with finite entries) of homogeneous max-plus linear systems which seem to be
more efficient in practice than the elimination methods (see~\cite{bcg99,walkup}).
Indeed, it is known that the problem of the existence of a solution (with finite entries)
of a homogeneous max-plus linear system can be reduced to the problem of the existence
of a sub-fixed point of a min-max function (for more background on min-max functions
we refer the reader to~\cite{cras,gg} and the references therein). To be more precise,
observe that $Dx=Cx$ is equivalent to
$x\leq \min \left\{ D\backslash Cx, C\backslash Dx\right\}$, where
$D\backslash Cx=\sup \set{y\in \overline{\R}_{\max}^n}{Dy\leq Cx}$ ($C\backslash Dx$
is defined analogously). Since $D\backslash Cx$ can be computed as
$(-D^T)(Cx)$, where the product by $-D^T$ is performed in $\overline{\R}_{\min}$ (see~\cite{bcoq}),
it follows that $f(x)=\min \left\{ D\backslash Cx, C\backslash Dx\right\}$ is a min-max
function. Then, there is $x\in \mathbb{R}^n$ such that $x\leq f(x)$
(that is, a sub-fixed point of $f$) if and only if all the entries of the
{\em cycle time vector} of $f$, which is defined as
$\chi(f)=\lim_{k\rightarrow \infty} f^k(x)/k$, are nonnegative
(see~\cite{cras,gg}). The cycle time vector $\chi(f)$, and, if it exists, a solution of
$x\leq f(x)$ can be efficiently computed via the min-max Howard algorithm (we refer the
reader to~\cite{cras,gg} for a detailed presentation of this algorithm).
Although the min-max Howard algorithm behaves remarkably well in practice,
its complexity is not yet well understood (\cite{cras,gg}).
To be able to apply this algorithm to solve our problem, firstly we need to
add one unknown $t$ to system~\eqref{sistnohom} in order to obtain a homogeneous max-plus
linear system of equations:
\begin{equation}\label{sisthom}
(At\oplus BF)Q=QG\; .
\end{equation}
Then, as system~\eqref{sistnohom} has at least one solution if and only if
system~\eqref{sisthom} has at least one solution with $t\not = -\infty$,
the semimodule $\mathcal{X}=\mbox{\rm Im}\, Q$ is algebraically $(A,B)$-invariant if and only if
system~\eqref{sisthom} has at least one solution with $t\not = -\infty$
(note that if $(t,F,G)$ is a solution of~\eqref{sisthom} with $t\not = -\infty$,
then $t^{-1}F=(-t)F$ is the feedback we are looking for).
Therefore, as $(t,F,G)$ is a solution of~\eqref{sisthom} if and only if
\begin{eqnarray}\label{subpunfijo}
t & \leq & (AQ)\backslash (QG)\; , \nonumber \\
F & \leq & B\backslash (QG)/Q \; , \label{sistdes}\\
G & \leq & Q\backslash ((At\oplus BF)Q) \; , \nonumber
\end{eqnarray}
where $D\backslash C$ is defined as $\sup \set{E\in \overline{\Z}_{\max}^{p\times r}}{DE\leq C}$
for all $D\in \Z_{\max}^{n\times p}$ and $C\in \Z_{\max}^{n\times r}$ (the function $/$ is
defined in an analogous way), if we can find a sub-fixed point of the min-max function
defined by the right hand side of~\eqref{sistdes}, then the semimodule $\mathcal{X}=\mbox{\rm Im}\, Q$ is
algebraically $(A,B)$-invariant.
\section{Application to transportation networks with a timetable}\label{aplicacionSec}
Let us consider the railway network given in Figure~\ref{figure1}.
Firstly, we will recall how the evolution of this kind of
transportation network can be described by max-plus linear
dynamical systems of the form of~\eqref{dynamicsystem} (we refer the reader
to~\cite{bcoq,OlsSubGett98,braker91,deVDeSdeM98} for details on max-plus models for
transportation networks). We are interested in the departure times of the trains from the stations.
Let us assume that in the initial state there is a train running along
each of the following tracks: the one connecting $P$ with $Q$, the one connecting $Q$ with $P$,
the one connecting $Q$ with $Q$ via $R$, and finally the one connecting $Q$ with $Q$ via $S$.
We call these tracks directions $d_1$, $d_2$, $d_3$ and $d_4$ respectively, as it is shown in
Figure~\ref{figure1}. In general, we can have $n$ different directions. The traveling time in
direction $d_i$ (to which the time needed for passengers to leave and board the train is added)
will be denoted by $t_i$. For our example these times are given in Figure~\ref{figure1}.
Let $x_i(k)$ denote the $k$-th departure time of the train which leaves in direction $d_i$.
As we explained in the introduction, a train cannot leave before a number of
conditions have been satisfied.
A first condition is that the train must have arrived at the station.
For instance, let us assume that the train which leaves in
direction $d_i$ is the one which comes from direction $d_{r(i)}$
(in Figure~\ref{figure1} we have: $r(1)=2$, $r(2)=4$, $r(3)=3$, and $r(4)=1$).
Then, the following condition must be satisfied:
\begin{equation}\label{condicion1}
t_{r(i)}+x_{r(i)}(k-1)\leq x_i(k) \;.
\end{equation}
A second constraint follows from the demand that trains must connect.
This gives rise to the following condition
\begin{equation}\label{condicion2}
t_j+x_j(k-1)\leq x_i(k) \;,\;\forall j\in C(i)\; ,
\end{equation}
where $C(i)$ is the set of indexes of all the directions of the trains which
have to provide a connection with the train which leaves in direction $d_i$
(in the case of the network given in Figure~\ref{figure1} we have: $C(1)=\emptyset$,
$C(2)=\left\{3\right\}$, $C(3)=\left\{1,4\right\}$, and $C(4)=\left\{3\right\}$).
Finally, the last condition is that a train cannot leave before its
scheduled departure time. This yields
\begin{equation}\label{condicion3}
u_i(k)\leq x_i(k) \;,
\end{equation}
where $u_i(k)$ denotes the scheduled departure time for the $k$-th
train in direction $d_i$. Now, if we assume that a train leaves as soon as
all the previous conditions have been satisfied, in max-plus notation
conditions~\eqref{condicion1}, \eqref{condicion2} and~\eqref{condicion3}
lead to
\begin{equation}\label{ecuacion1}
x_i(k)=\bigoplus_{j\in C(i)} t_j x_j(k-1)\oplus t_{r(i)} x_{r(i)}(k-1)\oplus u_i(k) \;.
\end{equation}
Therefore, if we define the matrix $A=(a_{ij})\in \Z_{\max}^{n\times n}$ by:
\[
a_{ij}= \left\{
\begin{array}{ll}
t_j & \mbox{ if } j \in C(i) \cup \{r(i)\} , \\
-\infty & \mbox{ otherwise,}
\end{array}
\right.
\]
then~\eqref{ecuacion1} can be written in matrix form as
\begin{equation}\label{ecuacion2}
x(k)=A x(k-1) \oplus u(k) \;,
\end{equation}
where $x(k)=(x_1(k),\ldots ,x_n(k))^T$ and $u(k)=(u_1(k),\ldots ,u_n(k))^T$, which is a system of
the form of~\eqref{dynamicsystem}. In the particular case of the railway network shown in
Figure~\ref{figure1} we have
\[
A=
\begin{pmatrix}
-\infty & 17 & -\infty & -\infty \\
-\infty & -\infty & 11 & 9 \\
14 & -\infty & 11 & 9 \\
14 & -\infty & 11 & -\infty
\end{pmatrix}\; .
\]
Suppose now that we want to decide whether there exists a timetable
such that the time between two consecutive train departures
in the same direction is less than a certain given bound or such
that the time that passengers have to wait to make some connections
is less than another given bound. To be able to model this kind of
requirement it is convenient to introduce the extended state vector
$\overline{x}(k)=(x_1(k),\ldots ,x_n(k),x_1(k-1),\ldots ,x_n(k-1))^T$.
Then~\eqref{ecuacion2} can be rewritten as
$\overline{x}(k)=\overline{A}\overline{x}(k-1)\oplus \overline{B}u(k)$,
where
\[
\overline{A}=
\begin{pmatrix}
A & \varepsilon \\
I & \varepsilon
\end{pmatrix}\; \mbox{ and }\;
\overline{B}=
\begin{pmatrix}
I \\
\varepsilon
\end{pmatrix}
\]
(here $I,\varepsilon \in \Z_{\max}^{n\times n}$ denote the max-plus
identity and zero matrices, respectively).
Assume that we want the time between two consecutive train departures
in direction $d_i$ to be less than $L_i$ time units. This can be
expressed as $\overline{x}_i(k)-\overline{x}_{i+n}(k)\leq L_i$,
or equivalently as $\overline{x}_i(k)-L_i\leq \overline{x}_{i+n}(k)$.
For simplicity we will take the same bound $L$ for all the directions,
although everything that follows can be done with different bounds.
Then the previous condition can be written in matrix form as
\begin{equation}\label{especificacion1}
\begin{pmatrix}
\varepsilon & \varepsilon \\
(-L) I & \varepsilon
\end{pmatrix}
\overline{x}(k) \leq \overline{x}(k)\; ,\; \forall k\in \mathbb{N} \; .
\end{equation}
Suppose now that we want passengers coming from direction
$d_i$ not to have to wait more than $M_{ij}$ time units for the
departure of the train which leaves in direction $d_j$. This can be
expressed as $\overline{x}_j(k)-a_{ji}-\overline{x}_{i+n}(k)\leq M_{ij}$,
which is equivalent to
$\overline{x}_j(k)-a_{ji}-M_{ij}\leq \overline{x}_{i+n}(k)$. Once again,
if for simplicity we take the same bound $M$ for all the possible
connections, the previous condition can be written in matrix form as
\begin{equation}\label{especificacion2}
\begin{pmatrix}
\varepsilon & \varepsilon \\
(-M) S & \varepsilon
\end{pmatrix}
\overline{x}(k) \leq \overline{x}(k)\; , \; \forall k\in \mathbb{N} \; ,
\end{equation}
where the matrix $S=(s_{ij})\in \Z_{\max}^{n\times n}$ is defined
by: $s_{ij}=-a_{ji}$ if $a_{ji}\not = -\infty$ and $s_{ij}= -\infty$
otherwise. Finally, in order to have {\em realistic} initial states for the extended state
vector, we can consider the obvious physical constraints $x(k-1)\leq x(k)$ and
$Ax(k-1)\leq x(k)$, which lead to the following condition:
\begin{equation}\label{especificacion3}
\begin{pmatrix}
\varepsilon & I\oplus A \\
\varepsilon & \varepsilon
\end{pmatrix}
\overline{x}(k) \leq \overline{x}(k)\; , \; \forall k\in \mathbb{N} \; .
\end{equation}
Therefore, to get the desired behavior of the network,
the timetable $u(k)$ should be such that the extended state
vector satisfies conditions~\eqref{especificacion1},
\eqref{especificacion2} and~\eqref{especificacion3}, that is,
such that $E\overline{x}(k) \leq \overline{x}(k)$ for all
$k\in \mathbb{N}$,
where
\[
E=
\begin{pmatrix}
\varepsilon & I\oplus A \\
(-M) S\oplus (-L) I & \varepsilon
\end{pmatrix}
\; .
\]
For instance, let us take $L=15$ and $M=4$ in the case of the
railway network shown in Figure~\ref{figure1}. Then
$E\overline{x}(k) \leq \overline{x}(k)$ is equivalent to
$\overline{x}(k)\in \mbox{\rm Im}\, E^*$ (see the proof of
Lemma~\ref{lemaHstar}), where
\[
E^*=
\begin{pmatrix}
0 & 2 & -2 & -2 & 12 & 17 & 13 & 11 \\
-5 & 0 & -4 & -4 & 10 & 12 & 11 & 9 \\
-1 & 1 & 0 & -3 & 14 & 16 & 12 & 10 \\
-1 & 1 & -3 & 0 & 14 & 16 & 12 & 10 \\
-15 & -13 & -17 & -17 & 0 & 2 & -2 & -4 \\
-20 & -15 & -19 & -19 & -5 & 0 & -4 & -6 \\
-16 & -14 & -15 & -15 & -1 & 1 & 0 & -5 \\
-14 & -12 & -13 & -15 & 1 & 3 & -1 & 0
\end{pmatrix}\; .
\]
Therefore, our problem is to determine the maximal geometrically
$(\overline{A},\overline{B})$-invariant semimodule contained
in $\mathcal{K}=\mbox{\rm Im}\, E^*$. With this aim we compute the sequence
of semimodules $\{\mathcal{X}_r\}_{r\in \mathbb{N}}$ defined by~\eqref{algoABinv}
following the method described in Remark~\ref{ObsComputo}
(which has been implemented with scilab, see~\cite{toolbox}).
Since the entries of $E^*$ are all finite, from Corollary~\ref{corvolfin}
we know that this sequence must stabilize. In fact, we have:
$\mathcal{X}_5=\mathcal{X}_4\varsubsetneq \mathcal{X}_3\varsubsetneq \mathcal{X}_2\varsubsetneq \mathcal{X}_1=\mathcal{K}$.
Then, the maximal geometrically $(\overline{A},\overline{B})$-invariant
semimodule $\mathcal{K}^*$ contained in $\mathcal{K}$ is $\mathcal{X}_4$,
which is generated by the columns of the following matrix
\[
\begin{pmatrix}
17 & 17 & 17 & 18 & 17 \\
15 & 15 & 14 & 15 & 15 \\
18 & 18 & 17 & 18 & 18 \\
19 & 19 & 18 & 19 & 19 \\
4 & 2 & 2 & 3 & 2 \\
0 & 0 & 0 & 0 & 0 \\
4 & 4 & 3 & 4 & 4 \\
5 & 5 & 4 & 5 & 2
\end{pmatrix}\; .
\]
Consequently, it is possible to obtain the desired
behavior of the network with a suitable choice of
the timetable $u(k)$ when the initial state belongs
to $\mathcal{K}^*$. To be able to compute these
timetables we use the method described at the end of
Section~\ref{algABinvSec} to decide whether $\mathcal{K}^*=\mathcal{K}_4$ is
an algebraically $(\overline{A},\overline{B})$-invariant semimodule
(that is, we apply the min-max Howard algorithm to find a state feedback).
In this way we can see that $\mathcal{K}^*$ is algebraically
$(\overline{A},\overline{B})$-invariant and one possible state
feedback is given by
\[
\overline{F}=
\begin{pmatrix}
14 & 14 & 14 & 13 & 14 & 14 & 14 & 14 \\
11 & 14 & 11 & 10 & 14 & 14 & 14 & 14 \\
14 & 14 & 14 & 13 & 14 & 14 & 14 & 14 \\
14 & 14 & 14 & 14 & 14 & 14 & 14 & 14
\end{pmatrix}\; .
\]
For instance, let us consider the evolution of the railway
network when the initial state is
$\overline{x}(0)=(17,15,18,19,4,0,4,5)^T \in \mathcal{K}^*$ and
the control $\overline{F}$ is applied. In this case we obtain the
following trajectory $x(k)$ of the system
\[
\begin{pmatrix}
4 \\
0 \\
4 \\
5
\end{pmatrix},\;
\begin{pmatrix}
17 \\
15 \\
18 \\
19
\end{pmatrix},\;
\begin{pmatrix}
32 \\
29 \\
32 \\
33
\end{pmatrix},\;
\begin{pmatrix}
46 \\
43 \\
46\\
47
\end{pmatrix},\;
\begin{pmatrix}
60 \\
57 \\
60 \\
61
\end{pmatrix},\;
\begin{pmatrix}
74 \\
71 \\
74 \\
75
\end{pmatrix},\;
\ldots
\]
which clearly satisfies the constraints imposed on the network.
However, if no control is applied, we get the following trajectory starting
from the same initial state
\[
\begin{pmatrix}
4 \\
0 \\
4 \\
5
\end{pmatrix},\;
\begin{pmatrix}
17 \\
15 \\
18 \\
19
\end{pmatrix},\;
\begin{pmatrix}
32 \\
29 \\
31 \\
31
\end{pmatrix},\;
\begin{pmatrix}
46 \\
42 \\
46\\
46
\end{pmatrix},\;
\begin{pmatrix}
59 \\
57 \\
60 \\
60
\end{pmatrix},\;
\begin{pmatrix}
74 \\
71 \\
73 \\
73
\end{pmatrix},\;
\ldots
\]
which does not satisfy the constraints imposed on the network, since for
example the passengers coming from station $S$ on the third train (which
leaves from station $Q$ in direction $d_4$ at time $31$) will have to wait $6$
time units for the next departure of a train in direction $d_3$ toward
station $R$ (which will take place at time $46$).
If we want to obtain the desired behavior of the network with a
periodic timetable, that is with a timetable $u(k)$ of the form
$u(k)=\lambda^k u$, where $\lambda \in \Z_{\max}$ and $u\in \Z_{\max}^n$,
then what we can do is to see if the matrix
$\overline{A}\oplus \overline{B} \overline{F}$ has an eigenvector in $\mathcal{K}^*$.
In this case it can be shown that
$\overline{x}(0)=(17,14,17,18,3,0,3,4)^T \in \mathcal{K}^*$
is an eigenvector of $\overline{A}\oplus \overline{B} \overline{F}$
corresponding to the eigenvalue $\lambda =14$, that is,
the following equality is satisfied:
\[
(\overline{A}\oplus \overline{B} \overline{F})\overline{x}(0)=
14\overline{x}(0)\; .
\]
Therefore, the periodic timetable
\[
u(k)=\overline{F}\overline{x}(k-1)=14^{(k-1)}\overline{F}\overline{x}(0)=
14^{(k+1)}\begin{pmatrix}
3 \\
0 \\
3 \\
4
\end{pmatrix}
\]
leads to the desired behavior of the network when the initial state
is $\overline{x}(0)$. In other words, one train should leave in each
direction every $14$ time units but the $k$-th departure time of the
trains in direction $d_1$ and $d_3$, respectively in direction $d_4$,
should be scheduled $3$ time units, respectively $4$ time units,
after the $k$-th scheduled departure time of the train in direction $d_2$.
Let us finally mention that the computations of the examples presented
in this paper have been checked using the max-plus toolbox of scilab
(see~\cite{toolbox}).
\section{Conclusion}
In this paper, the classical concept of $(A,B)$-invariant space is extended to linear
dynamical systems over the max-plus semiring. This extension presents similar
difficulties to those encountered in dealing with coefficients in a ring rather than
coefficients in a field. On the one hand, we show that the classical algorithm for the
computation of the maximal $(A,B)$-invariant subspace contained in a given space, which is
generalized to the max-plus algebra framework, need not converge in a finite number of steps.
However, sufficient conditions for the convergence of this algorithm are given. In particular,
it is shown that these conditions are satisfied by a class of semimodules of practical interest.
On the other hand, the existence (which is not guaranteed) and the computation of linear state
feedbacks are also discussed in the case of finitely generated semimodules.
Finally, we show that this approach is capable of providing solutions
to some control problems by considering its application to the study of transportation
networks which evolve according to a timetable.
\bibliographystyle{alpha}
|
{
"timestamp": "2007-04-06T21:04:51",
"yymm": "0503",
"arxiv_id": "math/0503448",
"language": "en",
"url": "https://arxiv.org/abs/math/0503448"
}
|
\section{Introduction}
In a typical Bell experiment, two or more entangled particles are distributed to separate observers. Each observer measures on his particle one from a set of possible observables and obtains some outcome. One of the most striking features of quantum mechanics is that the resulting joint outcome probabilities can violate a Bell inequality \cite{bel64}, indicating that quantum mechanics is not, in Bell's terminology, locally causal. This prediction has been confirmed, up to some loopholes, in numerous laboratory experiments \cite{asp99,tw01}. The implications of nonlocality for our fundamental description of nature \cite{bel87,cm89} have long been discussed; more recently, nonlocality has also acquired a significance in quantum information science \cite{eke91,ags03,bhk04,asw02,bra03,bzp04,blm05}. From this perspective, being able to decide whether a joint probability distribution can be reproduced with classical randomness only, or whether entanglement is necessary, is an important issue.
For a given number of observers, measurement settings, and measurement outcomes, the set of joint probabilities accessible to locally causal theories is a convex polytope \cite{ww01b}. It is therefore completely characterized by a finite number of linear inequalities that these probabilities must satisfy --- that is, by a finite number of Bell inequalities. Each of these inequalities corresponds to a \emph{facet} of the local polytope. Note, however, that not every Bell inequality represents a facet. Facet inequalities are the ones which characterize precisely the border between the local and the nonlocal region. They form a minimal and complete set of Bell inequalities.
In the simple situation where they are only two observers, two measurement choices, and two outcomes per measurement, all the facet inequalities are known \cite{fro81,fin82}: up to permutation of the outcomes, they correspond to the Clauser-Horne-Shimony-Holt (CHSH) inequality \cite{chs69}. Beyond this, little is known. It is in principle possible to obtain all the facet inequalities of an arbitrary Bell polytope using specific algorithms. In practice this only allows one to extend the range of solved cases to a few more observers, measurements or outcomes \cite{ps01,cg04}, as these algorithms are excessively time-consuming.
The problem of listing all facet inequalities has in fact been demonstrated to be NP-complete \cite{aii04}; it is therefore unlikely that it could be solved in full generality. Discouraging as this result may seem, it nevertheless leaves open several possibilities. First, complete sets of facet inequalities may be obtained for particular classes of Bell polytopes or for simplified versions of them. For instance, in the case where ``full correlation functions" are considered instead of complete joint probability distributions, all facet inequalities are known for Bell scenarios consisting of an arbitrary number of parties with two measurement choices and two outcomes \cite{ww01,zb02}. Second, in more complicated situations it may still be possible to obtain partial lists of facets. For instance, families of facet inequalities are known for arbitrary number of measurements \cite{aii04} or outcomes \cite{mas03}.
Further progress in the derivation of Bell inequalities would certainly benefit from a better characterization of the general properties of Bell polytopes. This is the motivation behind the present article. The question that we will investigate is how, and to what extent, the facial structure of a Bell polytope determines the facial structure of more complex polytopes. More specifically consider a bipartite Bell experiment characterized by the probability $p_{k_1k_2|j_1j_2}$ for the first observer to obtain outcome $k_1$ and for the second one to obtain outcome $k_2$, given that the first observer measures $j_1$ and the second one $j_2$. Suppose that each observer chooses one from two dichotomic observables, that is, $k_1,k_2\in \{1,2\}$ and $j_1,j_2\in \{1,2\}$. A necessary condition for this experiment to be reproducible by a local model is that the joint probabilities satisfy the CHSH inequality
\begin{eqnarray}\label{chsh}
&p_{11|11}+p_{11|12}+p_{11|21}-p_{11|22}&\nonumber \\
+&p_{22|11}+p_{22|12}+p_{22|21}-p_{22|22}&\geq 0\,.
\end{eqnarray}
Although this inequality is defined for the specific Bell scenario that we have just described, it also constrains the set of local joint probabilities involving more observers, measurements, and outcomes. Indeed, as was noted by Peres \cite{per99} there are obvious ways to extend Bell inequalities to more complex situations, or to \emph{lift} them following the terminology of polytope theory. As an illustration, let us consider the following three possible extensions of our CHSH scenario.
\emph{(i) More observers.} Consider a tripartite Bell experiment with joint probability distribution $p_{k_1k_2k_3|j_1j_2j_3}$, where $k_1,k_2,k_3\in \{1,2\}$ and $j_1,j_2,j_3\in \{1,2\}$. A necessary condition for this tripartite distribution to be local is that the probabilities $\widetilde p_{k_1k_2|j_1j_2}$ for the first two observers to measure $j_1$ and $j_2$ and to obtain outcomes $k_1$ and $k_2$ \emph{conditional} on the third observer measuring $j_3=1$ and obtaining $k_3=1$ satisfy the CHSH inequality. These conditional probabilities are given by $\widetilde p_{k_1k_2|j_1j_2}=p_{k_1k_21|j_1j_21}/p_{1_3|1_3}$, where the marginal $p_{1_3|1_3}=\sum_{k_1,k_2}p_{k_1k_21|j_1j_21}$ is independent of $j_1$ and $j_2$ by nosignaling\footnote{See Section \ref{dim}.}. Inserting these probabilities in (\ref{chsh}) and multiplying both side by $p_{1_3|1_3}$ leads to
\begin{eqnarray}\label{chshmo}
&p_{111|111}+p_{111|121}+p_{111|211}-p_{111|221}&\nonumber \\
+&p_{221|111}+p_{221|121}+p_{221|211}-p_{221|221}&\geq 0\,,
\end{eqnarray}
a natural extension of the CHSH inequality to three parties.
\emph{(ii) More measurements.} Consider our original bipartite Bell scenario, but assume that the second observer may choose between three different measurement settings $j_2\in\{1,2,3\}$. Clearly, a necessary condition for the corresponding joint distribution to be reproducible by a local model is that, when restricted to the probabilities involving $j_2\in\{1,2\}$, it satisfies the CHSH inequality. Therefore, inequality (\ref{chsh}) is, as such, a valid Bell inequality for this three-measurement scenario.
\emph{(iii) More outcomes.} Suppose now that the measurement apparatus of the second observer may output one out of three distinct values $k_2\in\{1,2,3\}$. Merging the outcomes $k_2=2$ and $k_2=3$, we obtain an effective two-outcomes distribution with probabilities $\widetilde p_{k_11|j_1j_2}=p_{k_11|j_1j_2}$ and $\widetilde p_{k_12|j_1j_2}=p_{k_12|j_1j_2}+p_{k_13|j_1j_2}$. The existence of a local model for the original distribution obviously implies a model for the coarse-grained one. Expressing the fact that the $\widetilde p_{k_1k_2|j_1j_2}$ should satisfy (\ref{chsh}), we thus deduce the following lifting
\begin{eqnarray}
&p_{11|11}+p_{11|12}+p_{11|21}-p_{11|22}&\nonumber \\
+&p_{22|11}+p_{22|12}+p_{22|21}-p_{22|22}&\nonumber \\
+&p_{23|11}+p_{23|12}+p_{23|21}-p_{23|22}&\geq 0
\end{eqnarray}
of the CHSH inequality to three outcomes.
These three examples can be combined and used sequentially to lift the CHSH inequality to an arbitrary number of observers, measurements, and outcomes. It is also straightforward to generalize them to other Bell inequalities than the CHSH one. How strong are the constraints on the joint probabilities obtained in this way? We will show that if the original inequality describes a facet of the original polytope, then the lifted one is also a facet of the more complex polytope. This implies, for instance, that the CHSH inequality is a facet of every Bell polytope since it is a facet of the simplest one.
This article is organized as follows. Section II introduces the concepts and notations that will be used in the remainder of the paper. In particular, we briefly review the definition of Bell polytopes and elementary notions of polytope theory. In Section III, we derive some basic properties of Bell polytopes that are necessary to prove our main results concerning the lifting of facet inequalities. These results are presented in Section IV. We conclude with a discussion and some open questions in Section V.
\section{Definitions}
\subsection{Bell scenario}
Consider $n$ systems and assume that on each system $i$ a measurement $j\in\{1,\ldots,m_i\}$ is made, yielding an outcome $k\in\{1,\ldots, v_{ij}\}$. Note that the number of possible measurements $m_i$ may be different for each system $i$, and that the number of possible outcomes $v_{ij}$ may be different for each measurement $j$ on system $i$. Such a Bell scenario is thus characterized by the triple $(n,m,v)$ where $m=(m_1,\ldots,m_n)$ specifies the number of possible measurements per system, and where the table $v=\big[(v_{11},\ldots,v_{1m_1});\ldots;(v_{n1},\ldots,v_{nm_n})\big]$ specifies the number of possible outcomes per measurement on each system. When notations such as $(n,2,v)$ are used, it should be understood that $m_i=2$ for all $i$.
The joint probability of obtaining the outcomes $(k_1,\ldots,k_n)$ given the measurement settings $(j_1,\ldots,j_n)$ will be denoted $p_{k_1\ldots k_n|j_1 \ldots j_n}$. We will view these $t=\prod_{i=1}^{n}\left(\sum_{j=1}^{m_i}v_{ij}\right)$ probabilities as forming the components of a vector $p$ in $\mathbb{R}^t$.
For a given observer $i\in\{1,\ldots,n\}$, measurement $j\in\{1,\ldots,m_i\}$ and outcome $k\in\{1,\ldots,v_{ij}\}$, we will often be interested in the subset of the components of $p$ that have the indices $k_i$ and $j_i$ corresponding to observer $i$ fixed, and equal, respectively, to $k$ and $j$. In other words, we will be interested in the variables $p_{k_1\ldots k_{i-1}k\,k_{i+1}\ldots k_n|j_1\ldots j_{i-1}j\,j_{i+1}\ldots j_n}$. The restriction of $p$ to these components will be denoted $p(i,j,k)$.
\subsection{Bell polytopes}
The set $\mathcal{B}\subseteq \mathbb{R}^t$ of correlations reproducible within a locally causal model is the set of correlations $p$ satisfying
\begin{equation*}
p_{k_1\ldots k_n|j_1\ldots j_n}=\int\!\mathrm{d} \mu\, q(\mu) P(k_1|j_1,\mu)\ldots P(k_n|j_n,\mu)\,,
\end{equation*}
where $q(\mu)\geq 0$, $\int\!\mathrm{d}\mu\, q(\mu)=1$, and $P(k_i|j_i,\mu)$ is the probability of obtaining the measurement outcome $k_i$ given the setting $j_i$ and the hidden-variable $\mu$ \cite{bel64,bel87}.
From this definition it is easily deduced (see \cite{ww01b} for instance) that $p$ is generated by specifying probabilities for every assignment of one of the possible outcomes to each of the measurement settings. More precisely, let the table $\lambda=\big[(\lambda_{11},\ldots,\lambda_{1m_1});\ldots;(\lambda_{n1},\ldots,\lambda_{nm_n})\big]$ assign to each measurement $j$ on system $i$ the outcome $\lambda_{ij}$. The (finite) set of all such possible assigmenents will be denoted $\Lambda$. Let
\begin{equation}\label{defdetvect}
p^\lambda_{k_1\ldots k_n|j_1\ldots j_n}=\left\{\begin{array}{ll}1 &\text{if }\lambda_{1j_1}=k_1,\ldots,\lambda_{nj_n}=k_n\\ 0&\mbox{otherwise}\end{array}\right.
\end{equation}
be the deterministic vector corresponding to the assignment $\lambda$. Then
\begin{equation}\label{localpolya}
\mathcal{B}=\{p\in\mathbb{R}^t\mid p=\sum_{\lambda\in\Lambda} q_\lambda\, p^\lambda,\, q_\lambda\geq0,\, \sum_{\lambda\in\Lambda} q_\lambda=1\}\,.
\end{equation}
The set $\mathcal{B}$ of local correlations is thus the convex hull of a finite number of points, i.e., it is a polytope. The deterministic vectors $\{p^\lambda|\lambda\in\Lambda\}$ form the extreme points of this polytope.
\subsection{Notions of polytope theory}\label{polrev}
We review in this section some elementary notions of polytope theory. For more detailed introductions, see \cite{nw88,sch89,zie95}.
The points $p_1,\ldots,p_n$ in $\mathbb{R}^t$ are said to be affinely independent if the unique solution to $\sum_i \mu_ip_i=0$, $\sum_i\mu_i=0$ is $\mu_i=0$ for all $i$, or equivalently, if the points $p_2-p_1,\ldots,p_n-p_1$ are linearly independent. They are affinely dependent otherwise. The affine hull of a set of points is the set of all their affine combinations. An affine set has dimension $D$, if the maximum number of affinely independent points it contains is $D+1$.
Let $\mathcal{B}\subseteq \mathbb{R}^t$ be a polytope defined as in (\ref{localpolya}). Let $(b,b_0)\in\mathbb{R}^{t+1}$ define the inequality $b\cdot p\geq b_0$. If this inequality is satisfied for all $p\in\mathcal{B}$, it is called a valid inequality for the polytope $\mathcal{B}$, or a Bell inequality in the context of Bell polytopes. Note that to check whether an inequality is a valid inequality, it is sufficient, by convexity, to check whether it is satisfied by the extreme points $\{p^\lambda|\lambda\in\Lambda\}$. Given the valid inequality $b\cdot p\geq b_0$, the set $F=\{p\in \mathcal{B}\mid b\cdot p=b_0\}$ is called a face of $\mathcal{B}$ and the inequality is said to support $F$. If $F\neq\emptyset$ and $F\neq\mathcal{B}$, it is a proper face. The dimension of $F$ is the dimension of its affine hull. Proper faces clearly satisfy $\dim F\leq \dim \mathcal{B}-1$. Proper faces of maximal dimension are called facets. An inequality $b\cdot p\geq b_0$ thus supports a facet of $\mathcal{B}$ if and only if $\dim \mathcal{B}$ affinely independent of $\mathcal{B}$ satisfy it with equality.
A fundamental result in polyhedral theory, known as Minkowski-Weyl's theorem, states that a polytope represented as the convex hull of a finite number of points, as in (\ref{localpolya}), can equivalently be represented as the intersection of finitely many half-spaces:
\begin{equation}\label{localpolyb}
\mathcal{B}=\{p\in\mathbb{R}^t\mid b^i\cdot p\geq b^i_0,\, \mbox{for all } i\in I\}\,,
\end{equation}
where $\{b^i\cdot p\geq b^i_0,\,i\in I\}$ is a finite set of inequalities. The inequalities supporting facets of $\mathcal{B}$ provide a minimal set of such inequalities\footnote{Note that if $\mathcal{B}\subseteq \mathbb{R}^t$ is not full dimensional, that is if $\dim\mathcal{B}<t$, then \emph{equality constraints} describing the affine hull of $\mathcal{B}$ must also be included in the above description.}. In particular, any valid inequality for $\mathcal{B}$ can be derived from the facet inequalities.
Given a Bell scenario $(n,m,v)$, the task of finding all the Bell inequalities is thus the problem of finding all the facets of the convex polytope $\mathcal{B}(n,m,v)$ defined by (\ref{defdetvect}) and (\ref{localpolya}). This connection between the search for optimal Bell inequalities and polyhedral geometry was observed by different authors \cite{fro81,gm84,pit89,per99}. For discussions on the complexity of this facet enumeration task see \cite{pit91,aii04}. For the instances for which this problem has been partially or completely solved, see \cite{fro81,fin82,ps01,ww01,zb02,mas03,sli03,cg04,aii04,lpz04}.
\section{Basic properties of Bell polytopes}\label{bp}
\subsection{Affine hull}\label{dim}
Local correlations $p\in\mathcal{B}$ satisfy the following equality constraints:\\
\emph{The normalization conditions}
\begin{equation}\label{multinorma}
\sum_{k_1\ldots k_n} p_{k_1 \ldots k_n|j_1 \ldots j_n}=1
\end{equation}
for all $j_1,\ldots,j_n$;\\
\emph{and the nosignaling conditions}
\begin{equation}\label{multinosig}
\sum_{k_{i}}p_{k_1\ldots k_{i}\ldots k_n|j_1\ldots j_{i}\ldots j_n}=\sum_{k_{i}}p_{k_1\ldots k_{i}\ldots k_n|j_1\ldots j'_{i}\ldots j_n}
\end{equation}
for all $i$, $k_1,\ldots k_{i-1},k_{i+1},\ldots,k_n$ and $j_1,\ldots j_{i-1},j_{i},j'_{i},\linebreak[4]j_{i+1},\ldots,j_n$.
The nosignaling conditions imply that for each subset $\{i_1,\ldots,i_q\}$ of size $q$ of the observers, the $q$-marginals $p_{k_{i_1}\ldots k_{i_q}|j_{i_1}\ldots j_{i_q}}=\sum_{k_{i_{q+1}}}\ldots\sum_{k_{i_n}}p_{k_1\ldots k_n|j_1\ldots j_n}$ are well-defined, that is, are independent of the precise value of the measurement settings $j_{i_{q+1}}\ldots j_{i_n}$.
The two conditions (\ref{multinorma}) and (\ref{multinosig}) also imply that the polytope $\mathcal{B}$ is not full dimensional in $\mathbb{R}^t$, i.e., it is contained in an affine subspace. The following theorem generalizes results given in \cite{mas03} and \cite{aii04}.
\begin{theorem}\label{localdim}
The constraints (\ref{multinorma}) and (\ref{multinosig}) fully determine the affine hull of $\mathcal{B}$ and
\begin{equation}
\dim\mathcal{B}=\prod_{i=1}^n\left(\sum_{j=1}^{m_i}\left(v_{ij}-1\right)+1\right)-1\,.
\end{equation}
\end{theorem}
\noindent\emph{Proof.}
Consider the marginals $p_{k_{i_1}\ldots k_{i_q}|j_{i_1}\ldots j_{i_q}}$ as defined above for all possible subsets $\{i_1,\ldots,i_q\}$ of size $q$, and for all $q=1,\ldots,n$. Of these marginals retain only the ones such that $k_{i}\neq 1$ for all $i$ $\in\{i_1,\ldots,i_q\}$. These probabilities define in total $D=\prod_{i=1}^n\Big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\Big)-1$ numbers. It is straightforward to check that their knowledge is sufficient to reconstruct, using the normalization and nosignaling conditions, the original $p_{k_1 \ldots k_n|j_1 \ldots j_n}$. This implies that the affine subspace defined by (\ref{multinorma}) and (\ref{multinosig}) is of dimension $\leq D$.
Let us now show that $\dim\mathcal{B}\geq D$, or equivalently that $\mathcal{B}$ contains $D+1$ affinely independent points. For this, note that the definition (\ref{defdetvect}) implies that an extreme point $p^\lambda$ can be written as the product $p^\lambda_{k_1\ldots k_n|j_1\ldots j_n}=p^\lambda_{k_1|j_1}\ldots p^\lambda_{k_n|j_n}$,
where $p^\lambda_{k_i|j_i}$ is a vector of length $\sum_{j=1}^{m_i}v_{ij}$ such that
\begin{equation}\label{defdetvect3}
p^\lambda_{k_i|j_i}=\left\{\begin{array}{ll}1 &\text{if } \lambda_{ij_i}=k_i\\ 0&\mbox{otherwise .}\end{array}\right.
\end{equation}
For fixed $i$, consider, for each $j_i'\in\{1,\ldots,m_i\}$ and for each $k_i'\in\{2,\ldots,v_{ij_i'}\}$, the points $p^\lambda_{k_i|j_i}$ defined by $\lambda_{ij_i}=1$ for all $j_i\neq j_i'$ and $\lambda_{ij'_i}=k'_i$. In addition, consider the vector $p^\lambda_{k_i|j_i}$ defined by $\lambda_{ij_i}=1$ for all $j_i$. These $\sum_{j=1}^{m_i}(v_{ij}-1)+1$ points are linearly independent. The products $p^\lambda_{k_1\ldots k_n|j_1\ldots j_n}=p^\lambda_{k_1|j_1}\ldots p^\lambda_{k_n|j_n}$ of all these points thus define $\prod_{i=1}^{n}\Big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\Big)=D+1$ linearly independent extreme points of $\mathcal{B}$, which are therefore also affinely independent.\hfill$\square$
Since $\mathcal{B}$ is not full dimensional, it follows that there is no unique way to write down a valid inequality for $\mathcal{B}$. More specifically, the inequalities $b\cdot p\geq b_0$ and $(b+\mu c)\cdot p\geq (b+\mu c_0)$, where $\mu \in \mathbb{R}$ and where $c\cdot p=c_0$ is a linear combination of the equalities (\ref{multinorma}) and (\ref{multinosig}), impose the same constraints on $\mathcal{B}$. In particular, it is always possible to use the normalization conditions to rewrite an inequality such that its lower bound is $0$, that is, in the form $b\cdot p\geq 0$. This fact will be used later on.
\subsection{Trivial facets and nontrivial polytopes}\label{trivfac}
In addition to the normalization and nosignaling conditions, $\mathcal{B}$ also satisfy the following \emph{positivity conditions}:
\begin{equation}\label{multipos}
p_{k_1 \ldots k_n|j_1 \ldots j_n}\geq 0
\end{equation}
for all $k_1,\ldots,k_n$ and $j_1,\ldots,j_n$.
\begin{theorem}
The positivity conditions support facets of $\mathcal{B}$.
\end{theorem}
\noindent\emph{Proof.}
Without loss of generality, suppose that $p_{k_1 \ldots k_n|j_1 \ldots j_n}\geq 0$ is such that the $k_1,\ldots,k_n$ are all different than $1$. Then, in the proof of Theorem 1, we enumerated $\dim \mathcal{B}+1$ affinely independent points, $\dim \mathcal{B}$ of which satisfy $p_{k_1 \ldots k_n|j_1 \ldots j_n}=0$.\hfill$\square$
The normalization, nosignaling, and positivity conditions are obviously not only satisfied by local probabilities, but also by all nosignaling nonlocal ones, and in particular by quantum ones. The only useful constraints that separate the local region from the nonlocal thus correspond to the facets of $\mathcal{B}$ that are not of the form (\ref{multipos}).
Let us also note that when determining the facets of a Bell polytope, we can always assume that $n$, $m_i$ and $v_{ij}$ are all $\geq 2$ because otherwise all the corresponding facets are trivial or belong to simpler polytopes. Indeed,
\begin{enumerate}
\item[(i)] the only facet inequalities of one-partite polytopes are the positivity constraints,
\item[(ii)] all the facet inequalities of a polytope where $m_i=1$ for some party $i$ are equivalent to the facet inequalities of the polytope obtained by discarding that party,
\item[(iii)] a polytope with $v_{ij}=1$ for some measurement $j$ of party $i$ is equivalent to the polytope obtained by discarding that measurement choice.
\end{enumerate}
Point (i) is easily established. To show (ii), assume that $\mathcal{B}$ is a polytope such that for party $i$ the only measurement choice is $j\in\{1\}$. A valid inequality for $\mathcal{B}$ can thus be written as
\begin{equation}\label{trivmeas1}
\sum_k b_k\cdot p(i,j,k) \geq 0\,,
\end{equation}
where, without loss of generality, the right-hand side is equal to zero. It then follows that for all $k\in\{1,\ldots,v_{ij}\}$ the following inequalities
\begin{equation}\label{trivmeas2}
b_k\cdot p(i,j,k)\geq 0
\end{equation}
are also valid for $\mathcal{B}$. Indeed, for each extreme point $p^\lambda$, either the assignment $\lambda$ is such that $\lambda_{ij}=k$ and (\ref{trivmeas1}) and (\ref{trivmeas2}) impose the same constraints on $p^\lambda$, or $\lambda_{ij}\neq k$ and (\ref{trivmeas2}) gives the trivial inequality $0\geq 0$. Every extreme point satisfying (\ref{trivmeas1}) thus also satisfies (\ref{trivmeas2}). Note further that every extreme point satisfying (\ref{trivmeas1}) with equality also satisfies (\ref{trivmeas2}) with equality. This implies that the face supported by (\ref{trivmeas1}) cannot be --- unless (\ref{trivmeas1}) is itself equivalent to one of the inequalities (\ref{trivmeas2}) --- a facet of $\mathcal{B}$, because it lies in the intersection of the faces supported by (\ref{trivmeas2}) and is therefore of dimension $<\dim\mathcal{B}-1$. We can thus assume that all facet inequalities of $\mathcal B$ are of the form (\ref{trivmeas2}). It will be shown in Section \ref{morobs}, that all these facet inequalities are equivalent to facet inequalities of the polytope obtained by discarding party $i$. Finally, point (iii) follows immediately when we notice that a polytope with $v_{ij}=1$ for some measurement $j$ of party $i$ and the polytope obtained by discarding that measurement have the same dimension and have their extreme points in one-to-one correspondence.
\subsection{A useful lemma}
As we have reminded earlier an inequality defines a facet of a polytope $\mathcal{B}$ if and only if it is satisfied by $\dim{B}$ affinely independent points of $\mathcal{B}$. To prove the results of the next section concerning the lifting of facet inequalities, we will then need to count the number of affinely independent points that a facet contains. The following lemma will be our main tool to achieve this task.
\begin{lemma}\label{lemma}
Let the inequality $b\cdot p\geq b_0$ support a facet of $\mathcal{B}(n,m,v)$. Let $i'\in\{1,\ldots,n\}$, $j'\in\{1,\ldots,m_{i'}\}$ and $k'\in\{1,\ldots,v_{i'j'}\}$. Then there are at exactly $r$ extreme points $p^\lambda$ of $\mathcal{B}$ such that $b\cdot p^\lambda=b_0$, $\lambda_{i'j'}=k'$, and such that the $r$ restrictions $p^\lambda(i',j',k')$ are affinely independent, where
\begin{enumerate}
\item[(i)] $r=\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)-1$, if $b\cdot p\geq b_0$ is equivalent to an inequality of the form\linebreak[4] $c\cdot p(i',j',k')\geq 0$;
\item[(ii)] $r=\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)$, otherwise.
\end{enumerate}
\end{lemma}
\noindent\emph{Proof.}
Let $\{p^\delta\,|\,\delta\in \Delta\subseteq\Lambda\}$ be $\dim\mathcal{B}$ affinely independent extreme points which belong to the facet supported by $b\cdot p\geq b_0$. Among these, let $\{p^\gamma\,|\,\gamma\in \Gamma\subseteq\Delta\}$ be the extreme points satisfying $\gamma_{i'j'}=k'$ and such that their restrictions $\{p^\gamma(i',j',k')\,|\gamma\in \Gamma\}$ are affinely independent.
Consider the polytope $\mathcal{B}^{n-1}$ obtained from $\mathcal{B}$ by discarding party $i'$. The components of $p\in\mathcal{B}^{n-1}$ are thus of the form $p_{k_1\ldots k_{i'-1}k_{i'+1}\ldots k_n|j_1\ldots j_{i'-1}j_{i'+1}\ldots j_n}$. Given that $p^\gamma(i',j',k')$ corresponds to the components of $p^\gamma$ where the indices associated to the ${i'}^\mathrm{th}$ party are fixed and satisfy $k_{i'}=k'$, $j_{i'}=j'$, given that $\gamma_{i'j'}=k'$, and given definition (\ref{defdetvect}), it follows that each $p^\gamma(i',j',k')$ can be identified with an extreme point of the $(n-1)$-partite polytope $\mathcal{B}^{n-1}$ (and conversely, each extreme point of $\mathcal{B}^{n-1}$ can be identified with the restriction $p^\gamma(i',j',k')$ of some extreme point $p^\gamma\in\mathcal{B}$ satisfying $\gamma_{i'j'}=k'$). Thus no more than $\dim\mathcal{B}^{n-1}$ of the $p^\gamma(i',j',k')$ can be affinely independent, and $r\leq\dim\mathcal{B}^{n-1}+1=\prod_{i\neq i'}\big(\sum_{i=1}^{m_i}(v_{ij}-1)+1\big)$. Alternatively, one could have deduced the same result starting from the fact that the $p^\gamma$ satisfy the implicit equalities (\ref{multinorma}) and (\ref{multinosig}), and counting the number of constraints that these equalities impose on the $p^\gamma(i',j',k')$.
Suppose that $r<\dim\mathcal{B}^{n-1}+1$. Then the $\{p^\gamma\,|\,\gamma\in \Gamma\}$ satisfy at least one constraint
\begin{equation}\label{dmn}
c\cdot p(i',j',k')=0
\end{equation}
linearly independent from the implicit equalities of $\mathcal{B}$. Following the remark at the end of Section \ref{dim}, we have not lost generality by taking the right-hand side of (\ref{dmn}) equal to zero. Note that the constraint (\ref{dmn}) is in fact satisfied by all $\{p^\delta\,|\,\delta\in \Delta\}$. Indeed, either $\delta_{i'j'}\neq k'$ and (\ref{dmn}) gives the trivial equation $0=0$, or $p^\delta(i',j',k')$ is affinely dependent from the $p^\gamma(i',j',k')$, which satisfy (\ref{dmn}).
As the $\{p^\delta\,|\,\delta\in \Delta\}$ form a set of $\dim\mathcal{B}$ independent extreme points, they can satisfy at most one constraint linearly independent from the implicit equalities of $\mathcal{B}$, i.e., there can only be one constraint of the form (\ref{dmn}). Thus at most $r=\dim\mathcal{B}^{n-1}=\prod_{i\neq i'}\big(\sum_{i=1}^{m_i}(v_{ij}-1)+1\big)-1$. Furthermore, as the $\{p^\delta\,|\,\delta\in \Delta\}$ already satisfy the equality $b\cdot p=b_0$, this can only be the case if (\ref{dmn}) is equivalent to $b\cdot p=b_0$, that is if $b\cdot p\geq b_0$ is equivalent either to $c\cdot p(i',j',k')\geq 0$ or $(-c)\cdot p(i',j',k')\geq 0$. \hfill$\square$
\section{Lifting Bell inequalities}\label{sectlifting}
We now move on to study the liftings of Bell inequalities that we have presented in the introduction and their natural generalizations. We will prove that these liftings are facet-preserving. It was already shown in \cite{aii04} that a Bell inequality that supports a facet of $\mathcal{B}(2,m,2)$ also supports a facet of $\mathcal{B}(2,m',2)$ for all $m'\geq m$. Furthermore, in \cite{kvk98} liftings of ``partial constraint satisfaction polytopes" (polytopes encountered in certain optimization problems) were considered. Although such liftings were studied independently from any potential relation to Bell inequalities, it turns out that partial constraint satisfaction polytopes over a complete bipartite graph are bipartite Bell polytopes (in particular, the ``4-cycle inequality" introduced in \cite{kvk98} corresponds to the CHSH inequality). The results presented in \cite{kvk98} then imply that an inequality that supports a facet of $\mathcal{B}(2,m,v)$ also supports a facet of $\mathcal{B}(2,m',v')$ for all $m'\geq m$, $v'\geq v$. It is in fact these results that inspired the ones that are presented here.
In the next three subsections, we will see that the lifting of an arbitrary inequality to a situation involving, respectively, one more observer, one more measurement outcome, and one more measurement setting are facet-preserving. Combined together these results imply that a Bell inequality that supports a facet of a Bell polytope $\mathcal{B}(n,m,v)$, also supports, when lifted in the appropriate way, a facet of any higher dimensional polytope $\mathcal{B}(n',m',v')$ with $n'\geq n$, $m'\geq m$, $v'\geq v$.
\subsection{One more observer}\label{morobs}
Consider a polytope $\mathcal{B}\equiv\mathcal{B}(n,m,v)$, where the $n$ parties are labeled $\{1,\ldots,i'-1,i'+1\ldots,n+1\}$ for some value $i'$. Let the inequality
\begin{equation}\label{origineqpart}
b\cdot p\geq 0
\end{equation}
be valid for $\mathcal{B}$. Note that we have taken, without loss of generality, the right-hand side of (\ref{origineqpart}) to be equal to $0$. Let us extend the polytope $\mathcal{B}$ by inserting an additional observer in position $i'$. The resulting $(n+1)$-partite polytope will be denoted $\mathcal{B}^{n+1}$.
Given a point $p\in\mathcal{B}^{n+1}$, remember that $p(i',j',k')$ represents the probabilities of $p$ for which the indices corresponding to the measurement setting and the outcome of party $i'$ are fixed, and are equal, respectively, to $j'$ and $k'$. Therefore $p(i',j',k')/p_{k'_{i'}|j'_{i'}}$, where $p_{k'_{i'}|j'_{i'}}$ denotes the marginal probability for observer $i'$ to measure $j'$ and obtain $k'$, is the joint outcome probability distribution for the $n$ observers $\{1,\ldots,i'-1,i'+1,\ldots n+1\}$ conditional on party $i'$ measuring $j'$ and obtaining $k'$. Either this conditional probability is equal to zero, or it corresponds to a point of $\mathcal{B}$. In both cases, it satisfies (\ref{origineqpart}). It thus follows immediately that the following inequality
\begin{equation}\label{liftineqpart}
b\cdot p(i',j',k')\geq 0
\end{equation}
is valid for $\mathcal{B}^{n+1}$. Further, this lifting is facet-preserving.
\begin{theorem}\label{liftparttheo}
The inequality (\ref{origineqpart}) supports a facet of $\mathcal{B}$ if and only if (\ref{liftineqpart}) supports a facet of $\mathcal{B}^{n+1}$.
\end{theorem}
\noindent\emph{Proof.}
As we have noted in the proof of Lemma \ref{lemma}, the restriction $p^\lambda(i',j',k')$ of an extreme point $p^\lambda$ of $\mathcal{B}^{n+1}$ satisfying $\lambda_{i'j'}=k'$ can be identified with an extreme point of $\mathcal{B}$, and conversely. Moreover, it is clear that if $p^\lambda(i',j',k')$ satisfy (\ref{liftineqpart}) with equality the corresponding extreme point of $\mathcal{B}$ satisfy (\ref{origineqpart}) with equality, and the other way around.
Assume that (\ref{liftineqpart}) supports a facet of $\mathcal{B}^{n+1}$. Then it follows from Lemma \ref{lemma} that they are $\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)-1=\dim \mathcal{B}$ extreme points of $\mathcal{B}^{n+1}$ that satisfy (\ref{liftineqpart}) with equality, such that $\lambda_{i'j'}=k'$ and for which the restrictions $p^\lambda(i',j',k')$ are affinely independent. By the above remark, these extreme points define $\dim \mathcal{B}$ affinely independent extreme points of $\mathcal{B}$ that satisfy (\ref{origineqpart}) with equality, hence this inequality supports a facet of $\mathcal{B}$.
To prove the converse statement, suppose now that (\ref{origineqpart}) defines a facet of $\mathcal{B}$, that is, there exist $\dim \mathcal{B}$ affinely independent extreme points of $\mathcal{B}$ that satisfy it with equality. By the above remark, there thus exist $\dim\mathcal{B}$ extreme points of $\mathcal{B}^{n+1}$ that satisfy (\ref{liftineqpart}) with equality, such that $\lambda_{i'j'}=k'$ and for which the restrictions $p^\lambda(i',j',k')$ are affinely independent. To show that (\ref{liftineqpart}) defines a facet of $\mathcal{B}^{n+1}$, it thus remain to find $\dim\mathcal{B}^{n+1}-\dim\mathcal{B}$ affinely independent points satisfying it with equality. For this, consider\footnote{We use the fact that $v_{ij}\geq 2$, following the remark at the end of Section \ref{trivfac}.} the extreme points of $\mathcal{B}^{n+1}$ with $\lambda_{i'j'}\neq k'$. They form an affine subspace of dimension $\dim\mathcal{B}^{n+1}-\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)=\dim\mathcal{B}^{n+1}-\dim\mathcal{B}-1$ since they can be identified with the extreme points of the polytope involving one outcome less than $\mathcal{B}^{n+1}$ for the measurement $j'$. Moreover, because they verify $p^\lambda(i',j',k')=0$, they satisfy (\ref{liftineqpart}) with equality, and are affinely independent from the extreme points for which $\lambda_{i'j'}=k'$.
\hfill$\square$
We thus have just shown that any facet inequality of an $n$-partite polytope can be extended to a facet inequality for a situation involving $n+1$ parties. This result can be used sequentially so that facets of $n$-party polytopes are lifted to $(n+k)$-partite polytopes. For instance, the positivity conditions (\ref{multipos}) can be viewed as the successive lifting of $1$-party inequalities.
The result holds in the other direction as well, since any facet inequality of the form (\ref{liftineqpart}) is the lifting of an $n$-partite inequality. When studying Bell polytopes, it is thus in general sufficient to consider \emph{genuinely $n$-partite inequalities}, that is, inequalities that cannot be written in a form that involves only probabilities associated with one specific measurement setting $j'$ and one specific outcome $k'$ for some party $i'$. Note that we can extend this definition to exclude also all inequalities such as (\ref{trivmeas1}) that involve only probabilities associated to one measurement setting (but possibly several outcomes corresponding to this measurement). Indeed, we have noted at the end of section \ref{trivfac} that such inequalities cannot be stronger than inequalities of the form (\ref{liftineqpart}).
\subsection{One more measurement outcome}
Consider a polytope $\mathcal{B}\equiv\mathcal{B}(n,m,v)$, where for measurement $j'$ of party $i'$ the $v_{i'j'}$ outcomes are labeled $\{1,\ldots,k'-1,k'+1,\ldots,v_{i'j'}+1\}$ for some $k'$. Let
\begin{equation}\label{origineqpart2}
b\cdot p\geq b_0
\end{equation}
be a genuinely $n$-partite inequality valid for $\mathcal{B}$. Let us consider the polytope $\mathcal{B}^{v+1}$ obtained from $\mathcal{B}$ by allowing an extra outcome $k'$ for the measurement $j'$ of party $i'$. To lift the inequality $b\cdot p\geq b_0$ to the polytope $\mathcal{B}^{v+1}$, we can merge the additional outcome $k'$ with some other outcome $k^*\in\{1,\ldots,k'-1,k'+1,\ldots,v_{i'j'}+1\}$, and insert the resulting probability distribution in (\ref{origineqpart}). This results in the inequality
\begin{equation}\label{liftineqout}
b\cdot p+b(i',j',k^*)\cdot p(i',j',k')\geq b_0\,.
\end{equation}
\begin{theorem}\label{theoliftout}
If the genuinely $n$-partite inequality (\ref{origineqpart}) supports a facet of $\mathcal{B}$, then (\ref{liftineqout}) supports a facet of $\mathcal{B}^{v+1}$.
\end{theorem}
\noindent\emph{Proof.}
The dimension of $\mathcal{B}^{v+1}$ equals $\dim\mathcal{B}+\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)$. The extreme points of $\mathcal{B}$ that belong to the facet $b\cdot p\geq b_0$ provide $\dim\mathcal{B}$ affinely independent points satisfying (\ref{liftineqout}) with equality. By Lemma \ref{lemma}, there exist $\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)$ extreme points $p^\lambda$ with $\lambda_{i'j'}=k^*$ that saturate (\ref{origineqpart}), and thus (\ref{liftineqout}), and for which the $p^\lambda(i',j',k^*)$ are affinely independent. Replace $k^*$ by $k'$ in these extreme points. These new extreme points still satisfy (\ref{liftineqout}) with equality and are affinely independent with all the previous ones, since they are the unique extreme points with $p^\lambda(i',j',k')\neq 0$. In total, we thus enumerated $\dim\mathcal{B}^{v+1}=\dim\mathcal{B}+\prod_{i\neq i'}\big(\sum_{j=1}^{m_i}(v_{ij}-1)+1\big)$ affinely independent point satisfying (\ref{liftineqout}) with equality.
\hfill$\square$
\subsection{One more measurement setting}
Consider a polytope $\mathcal{B}\equiv\mathcal{B}(n,m,v)$, where for party $i'$ the $m_{i'}$ measurements are labeled $\{1,\ldots,j'-1,j'+1,\ldots,m_{i'}+1\}$ for some $j'$. Let the polytope $\mathcal{B}^{m+1}$ be the polytope obtained from $\mathcal{B}$ by allowing the additional measurement setting $j'$ for party $i'$. An inequality $b\cdot p\geq b_0$ valid for $\mathcal{B}$ is also clearly valid for $\mathcal{B}^{m+1}$. Moreover, the following stronger result holds.
\begin{theorem}
Let $b\cdot p\geq b_0$ be a genuinely $n$-partite inequality supporting a facet of $\mathcal{B}$. Then it is also support a facet of $\mathcal{B}^{m+1}$.
\end{theorem}
\noindent\emph{Proof.}
Consider the polytope $\widetilde{\mathcal{B}}^{m+1}$ defined as $\mathcal{B}^{m+1}$ but such that for the measurement $j'$ of party $i'$ is associated a single possible outcome, i.e., $v_{i'j'}=1$. The inequality $b\cdot p\geq b_0$ is a valid genuinely $n$-partite inequality for $\widetilde{\mathcal{B}}^{m+1}$. Further, since $\widetilde{\mathcal{B}}^{m+1}$ and $\mathcal{B}$ have the same dimension, it is also facet defining for $\widetilde{\mathcal{B}}^{m+1}$. Following the procedure to lift an inequality to more outcomes delineated in the previous subsection, this inequality can be lifted from $\widetilde{\mathcal{B}}^{m+1}$ to $\mathcal{B}^{m+1}$. Since $b\cdot p\geq b_0$ does not involve components associated with the measurement $j'$ of party $i'$, this results in the inequality $b\cdot p\geq b_0$ itself. By Theorem \ref{theoliftout}, this inequality is facet defining for $\mathcal{B}^{m+1}$.
\hfill$\square$
\section{Conclusion}
We have shown that the facial structure of Bell polytopes is organized in a hierarchical way, with all the facets of a given polytope inducing, through their respective liftings, facets of more complex polytopes. Instead of considering the entire set of facets of a Bell polytope, it is thus in general sufficient to characterize the ones that do not belong to simpler polytopes. It would be interesting to investigate whether this fact could be exploited to improve the efficiency of the algorithms used to list facet inequalities or to simplify analytical derivations of Bell inequalities.
Note that for certain polytopes, the complete set of facet inequalities is constituted entirely by inequalities lifted from more elementary polytopes. For instance for Bell scenarios involving two observers, the first having a choice between two dichotomic measurements and the second one between an arbitrary number of them, all the facet-defining inequalities correspond to liftings of the CHSH inequality \cite{sli03,cg04}. A natural extension of the results reported in this article would then be to investigate more generally when inequalities lifted from simpler polytopes describe complete sets of facets. Progress along this line would allow one to narrow down the class of Bell scenarios that have to be considered to find new Bell inequalities.
Following this approach, all the polytopes for which the only facets correspond to liftings of the CHSH inequality have recently been characterized \cite{sp}.
Finally, let us note that while the facet-preserving liftings that we have considered are interesting because they throw light on the structure of Bell polytopes, the inequalities obtained in this way are not essentially different from the original ones, they are merely re-expressions of these inequalities adapted to more general scenarios. However, it is also in principle possible to consider more complicated generalizations of Bell inequalities that alter significantly their intrinsic structure. For instance, the family of Bell inequalities introduced in \cite{cgl02} can be understood as being generated by successive nontrivial liftings of the CHSH inequality. Studying such liftings, as well as the other possible extensions of our results, seems a promising path towards a more accurate characterization of the constraints that separate the set of local joint probabilities from the set of nonlocal ones.
\acknowledgments
I would like to thank Jean-Paul Doignon and Serge Massar for helpful discussions. This work is supported by the David and Alice Van Buuren fellowship of the Belgian American Educational Foundation and by the National Science Foundation under Grant No. EIA-0086038.
|
{
"timestamp": "2005-06-18T04:21:07",
"yymm": "0503",
"arxiv_id": "quant-ph/0503179",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503179"
}
|
\section{Introduction}
\indent\indent We study the long time behaviour of solutions to a
class Korteweg-de Vries-type equations, with an additional term
$b(t,x)u$. These equations, from now on called the bKdV, are of
the form
\begin{align}
\partial_t u=-\partial_x\left(\partial_x^2 u+f(u)-b(t,x)u\right),
\label{Eqn:KdvGeneralizedWithPotential}
\end{align}
where $b(t,x)$ is a real valued function and $f$ is a
nonlinearity. In this paper we consider a restricted class of
nonlinearities. In particular, for monomial nonlinearities, we
give a result only for $f(u)=u^3$, corresponding to the modified
KdV (mKdV). When $b=0$, Equation
\eqref{Eqn:KdvGeneralizedWithPotential} reduces to the generalized
Korteweg-de Vries equation (GKdV)
\begin{equation}
\partial_t u=-\partial_x(\partial_x^2 u+f(u)). \label{Eqn:GKdV}
\end{equation}
A remarkable property of the GKdV is the existence of spatially
localized solitary (or travelling) waves, i.e. solutions of the
form $u=Q_c(x-a-c t)$, where $a\in{\mathbb R}$ and $c$ in some interval
$I$. When $f(u)=u^p$ and $p\ge 2$, solitary waves are explicitly
computed to be
\begin{equation*}
Q_c(x)=c^\frac{1}{p-1}Q(c^\frac{1}{2}x),
\end{equation*}
where
\begin{equation*}
Q(x)=\left(\frac{p+1}{2}\right)^\frac{1}{p-1} \left(\cosh\left(\frac{p-1}{2}
x\right)\rb^2.
\end{equation*}
It is generally believed that an arbitrary, say $\Hs{1}$, solution
to equation \eqref{Eqn:GKdV} eventually breaks up into a
collection of solitary waves and radiation. A discussion of this
phenomenon for the generalized KdV appears in Bona \cite{BoSo94}.
For the general, but integrable case see Deift and Zhou \cite{DeZh93}.
The mKdV equation is fundamental in many areas of applied
mathematics ranging from traffic flow to plasma physics (see
\cite{Ku1985,ChRa1987,LuMa1997,Na2002}) and arises from an
approximation of a more complicated systems. The effects of higher
order processes can often be collected into a term of the form
$b(t,x) u$. Our main result stated at the end of the next
section gives, for long time, an explicit, leading order
description of a solution to the bKdV initially close to a
solitary wave solution of the GKdV.
We assume that the coefficient $b$ and nonlinearity $f$ are such
that \eqref{Eqn:KdvGeneralizedWithPotential} has global solutions
for $\Hs{1}$ data and that \eqref{Eqn:KdvGeneralizedWithPotential}
with $b=0$ possesses solitary wave solutions. Precise conditions
will be formulated in the next section. Here we mention that the
literature regarding well-posedness of the KdV ($b=0$,
$f(u)=u^2$) is extensive and well developed. The Miura transform
(see \cite{Mi1968}) then gives well-posedness results for the
mKdV. Bona and Smith \cite{BoSm1975} proved global wellposedness
of the KdV in $\Hs{2}$. See also \cite{Ka1983}. Kenig, Ponce, and
Vega \cite{KePoVe1996} have proved local wellposedness in $\Hs{s}$
for $s\ge -\frac{3}{4}$ and similar results are available for the
generalized KdV ($b=0$, monomial nonlinearity $f(u)=u^p$ with
$p=2,3,4$)\cite{KePoVe1993}. In particular, local well-posedness
for the mKdV in $\Hs{s}$ with $s\ge\frac{1}{4}$ and global
well-posedness for $s\ge 1$ are known. More recently, results
extending local wellposedness in negative index Sobolev spaces to
global wellposedness have been proven \cite{CoSt1999,CoKe2001}.
There is little literature on global well-posedness of the bKdV in
energy space, however, under a smallness assumption on the
coefficient $b$, Dejak and Sigal \cite{DeSi2004} proved global
well-posedness in $\Hs{1}$ of the bKdV with $f(u)=u^p$, $p=2,3,4$.
They used results of \cite{KePoVe1993}, and perturbation and
energy arguments.
Soliton solutions of the KdV equation are known to be orbitally
stable. Although the linearized analysis of Jeffrey and Kakutani
\cite{JeKa1970} suggested orbital stability, the first nonlinear
stability result was given by Benjamin \cite{Be1972}. He assumed
smooth solutions and used Lyapunov stability and spectral theory
to prove his results. Bona \cite{Bo1975} later corrected and
improved Benjamin's result to solutions in $\Hs{2}$. Weinstein
\cite{We1985} used variation methods, avoiding the use of an
explicit spectral respresentaion, and extended the orbital
stability result to the GKdV. More recently, Grillakis, Satah,
and Strauss \cite{GrShSt87} extended the Lyapunov method to
abstract Hamiltonian systems with symmetry. Numerical simulations
of soliton dynamics for the KdV were performed by Bona et al. See
\cite{BoDo86,BoDo91,BoDo95,BoDo96}.
For nonlinear Schr\"{o}dinger and Hartree equations, long-time
dynamics of solitary waves were studied by Bronski and Gerrard
\cite{BrJe2000}, Fr\"{o}hlich, Tsai and Yau \cite{FrTs2003},
Keraani \cite{Ke02}, and Fr\"{o}hlich, Gustafson, Jonsson, and
Sigal \cite{FrGu2003, FrGuJoSi2005}. For related results and
techniques for the nonlinear Schr\"{o}dinger equations see also
\cite{BuPe1992,BuSu2003,GaSi2004,RoScSoPreprint,RoScSoPreprintII,TsYa2002I,TsYa2002II,TsYa2002III,SoWe90}.
In our approach we use the fact that the bKdV is a
(non-autonomous, if $b$ depends on time) Hamiltonian system. As
in the case of the nonlinear Schr\"{o}dinger equation (see
\cite{FrGu2003}), we construct a Hamiltonian reduction of this
original, infinite dimensional dynamical system to a two
dimensional dynamical system on a manifold of soliton
configurations. The analysis of the general bKdV immediately runs
into the problem that the natural symplectic form $\omega$ is not
defined on the tangent space of the soliton manifold. In this
paper we prove the main theorem in the cases where the symplectic
form is well defined on the tangent space. One such case is when
the nonlinearity is $f(u)=u^3$. For the general case see
\cite{DeSi2004}. We remark here that the dynamics for the special
case considered here include the higher order correction terms for
the scaling parameter $c$, which cannot be included in the general
case.
\vspace{4mm}\noindent{\bf\large Acknowledgements} \vspace{4mm}\\
We are grateful to I.M. Sigal for useful discussions.
\section{Preliminaries, Assumptions, Main Results}
\label{Section:AssumptionsAndMainResults}The bKdV can be written
in Hamiltonian form as
\begin{eqnarray}
\partial_t u=\dxH_b'(u),
\label{Eqn:KdVVariationalForm}
\end{eqnarray}
where $H_b'$ is the $\Lp{2}$ function
corresponding to the Fr\'{e}chet derivative
$\partialH_b$ in the $\Lp{2}$ pairing. Here the
Hamiltonian $H_b$ is
\begin{eqnarray*}
H_b(u):=\intR{ \frac{1}{2}(\partial_x
u)^2-F(u)+\frac{1}{2} b(t,x) u^2},
\end{eqnarray*}
where the function $F$ is the antiderivative of $f$ with $F(0)=0$.
The operator $\partial_x$ is the anti-self-adjoint operator (symplectic
operator) generating the Poisson bracket
\begin{equation*}
\{G_1, G_2\}:=\frac{1}{2}\intR{G_1'(u)\partial_x G_2'(u)-G_2'(u)\partial_x
G_1'(u)},
\end{equation*}
defined for any $G_1$, $G_2$ such that $G_1', G_2'\in
\Hs{\frac{1}{2}}$.
The corresponding symplectic form is
\begin{equation*}
\omega(v_1,\, v_2):=\frac{1}{2}\intR{v_1(x)\partial_x^{-1}
v_2(x)-v_2(x)\partial_x^{-1} v_1(x)},
\end{equation*}
defined for any $v_1, v_2\in \Lp{1}$. Here the operator
$\partial_x^{-1}$ is defined as
\begin{equation*}
\partial_x^{-1} v(x):=\int_{-\infty}^x v(y)\, dy.
\end{equation*}
Note that $\partial_x^{-1}\cdot \partial_x=I$ and, on the space
$\{u\in\Lp{2}\,|\, \intR{u}=0\}$, $\partial_x^{-1}$ is formally
anti-self-adjoint with inverse $\partial_x$. Hence, if
$\intR{v_1(x)}=0$, then $\omega(v_1,\, v_2)=\intR{v_1(x)\partial_x^{-1}
v_2(x)}$.
Note that if $b$ depends on time $t$, then equation
\eqref{Eqn:KdVVariationalForm} is non-autonomous. It is, however,
in the form of a conservation law, and hence the integral of the
solution $u$ is conserved provided $u$ and its derivatives decay
to zero at infinity:
\begin{equation*}
\dd{t}\intR{u}=0.
\end{equation*}
There are also conserved quantities associated to symmetries of
\eqref{Eqn:KdvGeneralizedWithPotential} when $b=0$. The simplest
such corresponds to time translation invariance and is the
Hamiltonian itself. This is also true if $b$ is non-zero but
time independent. If the potential $b=0$, then
\eqref{Eqn:KdvGeneralizedWithPotential} is also spatially
translation invariant. Noether's theorem then implies that the
flow preserves the momentum
\begin{eqnarray*}
P(u):=\frac{1}{2}\LpNorm{2}{u}^2.
\end{eqnarray*}
In general, when $b\ne 0$ the temporal and spatial translation
symmetries are broken, and hence, the Hamiltonian and momentum are
no longer conserved. Instead, one has the relations
\begin{align}
\dd{t}H_b(u)&=\frac{1}{2}\intR{(\partial_tb) u^2},
\label{Eqn:ConservationHamiltonian}\\
\dd{t}P(u)&=\frac{1}{2}\intR{b'u^2},
\label{Eqn:ConservationMomentum}
\end{align}
where $b'(t,x):=\partial_x b(t,x)$. For later use, we also state the
relation
\begin{eqnarray}
\dd{t}\frac{1}{2}\intR{b u^2}=\intR{\frac{1}{2} u^2\partial_tb+b'\left( u
f(u)-\frac{3}{2}(\partial_x u)^2-F(u)\right)-b'' u \partial_x u}.
\label{Eqn:ConservationPotentialMomentum}
\end{eqnarray}
Assuming (\ref{Eqn:KdvGeneralizedWithPotential}) is well-posed in
$\Hs{2}$, the above equalities are obtained after multiple
integration by parts. Then, by density of $\Hs{2}$ in $\Hs{1}$,
the equalities continue to hold for solutions in $\Hs{1}$. To
avoid these technical details, we assume the Hamiltonian flow on
$\Hs{1}$ enjoys (\ref{Eqn:ConservationHamiltonian}),
(\ref{Eqn:ConservationMomentum}) and
(\ref{Eqn:ConservationPotentialMomentum}).
Consider the GKdV, i.e. equation \eqref{Eqn:GKdV}. Under certain
conditions on $f$, this equation has travelling wave solutions of
the form $Q_c(x-c t)$, where $Q_c$ a positive $\Hs{2}$ function.
Substituting $u=Q_c(x-ct)$ into the GKdV gives the scalar field
equation
\begin{equation}
-\partial_x^2Q_c+cQ_c-f(Q_c)=0. \label{Eqn:ScalarFieldEquation}
\end{equation}
Existence of solutions to this equation has been studied by
numerous authors. See \cite{St1977, BeLi1983}. In particular,
Berestyki and Lions \cite{BeLi1983} give sufficient and necessary
conditions for a positive and smooth solution $Q_c$ to exist. We
assume $g:=-c u+f(u)$ satisfies the following conditions:
\begin{enumerate}
\item $g$ is locally Lipschitz and $g(0)=0$, \item
$x^*:=\inf\{x>0\,|\,\int_0^x g(y)\,dy\}$ exists with $x^*>0$ and
$g(x^*)>0$, and \item $\lim_{s\rightarrow 0}\frac{g(s)}{s}\le
-m<0$.
\end{enumerate}
Then, as shown by Berestycki and Lions,
\eqref{Eqn:ScalarFieldEquation} has a unique (modulo translations)
solution $Q_c\in C^2$ for $c$ in some interval, which is positive,
even (when centred at the origin), and with $Q_c$, $\partial_xQ_c$, and
$\partial_x^2Q_c$ exponentially decaying to zero at infinity ($\partial_xQ_c<0$
for $x>0$). Furthermore, if $f$ is $C^2$, then the implicit
function theorem implies that $Q_c$ is $C^2$ with respect to the
parameter $c$ on some interval $I_0\subset\R_+$. We assume that
$x^m\partial_c^nQ_c\in\Lp{1}$ for $n=1,2$, $m=0,1,2$ so that integrals
containing $\partial_c^nQ_c$ are continuous and differentiable with
respect to $c$.
We also make the assumption that
\begin{equation}
\intR{\partial_cQ_c}=0 \label{Eqn:IntVcAssump}
\end{equation}
for all $c\in I$. This implies that
\begin{eqnarray}
\int_{-\infty}^x \partial_cQ_c(z)\, dz, \int_{-\infty}^x \partial_c^2Q_c(z)\,
dz\in\Lp{2}. \label{Eqn:LIIAssumptions}
\end{eqnarray}
To see this use the isometry property of the Fourier transform and
the decay properties of $\partial_cQ_c$. The above requirements of $Q_c$
are implicit assumptions on the nonlinearity $f$ and are true when
$f(u)=u^3$. Assumption \eqref{Eqn:IntVcAssump} is a very important
and restrictive requirement; it does not hold when $f(x)=x^p$ and
$p\ne 3$. For the case where \eqref{Eqn:IntVcAssump} does not
hold see \cite{DeSi2004}.
The solitary waves $Q_c$ are orbitally stable if $\delta'(c)>0$,
where $\delta(c)=P(Q_c)$. See Weinstein \cite{We1985} the first
proof for general nonlinearities. Moreover, in \cite{GrShSt87},
Grillakis, Shatah and Strauss proved that $\delta'(c)>0$ is a
necessary and sufficient condition for $Q_c$ to be orbitally
stable. In this paper, we assume that $Q_c$ is stable for all $c$
in some compact interval $I\subset I_0$, or equivalently that
$\delta'(c)>0$ on $I$. For $f(u)=u^p$, we have
$\delta'(c)=\frac{5-p}{4(p-1)}\LpNorm{2}{Q_{c=1}}^2$, which
implies the well known stability criterion $p<5$ corresponding to
subcritical power nonlinearities.
The scalar field equation \eqref{Eqn:ScalarFieldEquation} for the
solitary wave can be viewed as an Euler-Lagrange equation for the
extremals of the Hamiltonian $H_{b=0}$
subject to constant momentum $P(u)$. Moreover, $Q_c$ is a stable
solitary wave if and only if it is a minimizer of
$H_{b=0}$ subject to constant momentum $P$.
Thus, if $c$ is the Lagrange multiplier associated to the momentum
constraint, then $Q_c$ is an extremal of
\begin{align}
\Lambda_{ca}(u)&:=H_{b=0}(u)+c P(u)
\label{DefinitionLS}\\&=\intR{ \frac{1}{2}(\partial_x u )^2+\frac{1}{2}c
u^2-F(u)},\nonumber
\end{align}
and hence $\Lambda_{ca}'(Q_c)=0$.
The functional $\Lambda_{ca}$ is translationally invariant.
Therefore, $Q_{ca}(x):=Q_c(x-a)$ is also an extremal of
$\Lambda_{ca}$, and $Q_c(x-c t-a)$ is a solitary wave solution
of (\ref{Eqn:KdvGeneralizedWithPotential}) with $b=0$. All such
solutions form the two dimensional $C^\infty$ manifold of solitary
waves
\begin{equation*}
M_s:=\{Q_{ca}\,|\,c\in I, a\in {\mathbb R}\},
\end{equation*}
with tangent space $T_{\Qca}M_s$ spanned by the vectors
\begin{eqnarray}
\zeta^{tr}_{c a}:=\partial_aQ_{ca}=-\partial_xQ_{ca}\ \mbox{and}\ \zeta^n_{c a}:=\partial_cQ_{ca},
\label{Eqn:DefinitionOfTangentVectors}
\end{eqnarray}
which we call the translation and normalization vectors. Notice
that the two tangent vectors are orthogonal in $\Lp{2}$.
In addition to the requirement on $b$ that
(\ref{Eqn:KdvGeneralizedWithPotential}) is globally wellposed, we
assume the potential $b$ is bounded, twice differentiable, and
small in the sense that
\begin{align}
|\partial_t^n\partial_x^mb|\le\epsilon_a\epsilon_t^n\epsilon_x^m, \label{Eqn:AssumptionOnPotential}
\end{align}
for $n=0,1$, $m=0,1,2$, and $n+m\le 2$. The positive constants
$\epsilon_a$, $\epsilon_x$, and $\epsilon_t$ are amplitude, length, and time scales of
the function $b$. We assume all are less than or equal to one.
Lastly, we make some explicit assumptions on the local
nonlinearity $f$. We require the nonlinearity to be $k$ times
continuously differentiable, with $f^{(k)}$ bounded for some $k\ge
3$ and $f(0)=f'(0)=0$. These assumptions ensure the Hamiltonian
is finite on the space $\Hs{1}$ and, since $Q_c$ decays
exponentially (see \cite{BeLi1983}), both $f(Q_c)$ and $f'(Q_c)$
have exponential decay.
We are ready to state our main result. Recall that $I_0\subset
{\mathbb R}_+$ is an interval where $Q_c$ is twice continuously
differentiable.
\begin{thm}
Let the above assumptions hold and assume $\delta'(c)>0$ for all
$c$ in a compact set $I\subset I_0$. Assume $\epsilon_a\le 1$. Then, if
$\epsilon_x\le 1$, $\epsilon_0$ and $\epsilon_t$ are small enough, there is a
positive constant $C$ such that the solution to
(\ref{Eqn:KdvGeneralizedWithPotential}) with an initial condition
$u_0$ satisfying $\inf_{Q_{ca}\in M_s}\HsNorm{1}{u_0-Q_{c a}}\le
\epsilon_0$ can be written as
\begin{eqnarray*}
u(x,t)=Q_{c(t)}(x-a(t))+\xi(x,t),
\end{eqnarray*}
where $\HsNorm{1}{\xi(t)}=\O{
\epsilon_0+(\epsilon_a\epsilon_x\epsilon_0)^\frac{1}{2}+\epsilon_x+\epsilon_t}$ for all times
$t \le C(\epsilon_a \epsilon_x)^{-1}$. Moreover, during this time interval the
parameters $a(t)$ and $c(t)$ satisfy the equations
\begin{eqnarray*}
\left(\begin{array}{c}
\dot{a} \\
\dot{c}
\end{array}\right)&=&
\left(\begin{array}{c}
c-b(a)\\
0
\end{array}\right)+
b'(a)\frac{\delta(c)}{\delta'(c)}\left(\begin{array}{c}
0\\
1
\end{array}\right)+\O{(\epsilon_0+\epsilon_x+\epsilon_t)^2+(\epsilon_a\epsilon_x\epsilon_0)^\frac{1}{2}(\epsilon_x+\epsilon_t+\epsilon_0)},
\end{eqnarray*}
where $c$ is assumed to lie in the compact set $I$.
\label{MainThm}
\end{thm}
\begin{proof}[Sketch of Proof and Paper Organization]
To realize the Hamiltonian reduction we decompose functions in a
neighbourhood of the soliton manifold $M_s$ as
\begin{equation*}
u=Q_{ca}+\xi
\end{equation*}
with $\xi$ symplectically orthogonal to $T_{\Qca}M_s$, i.e.
$\xi\bot\partial_x^{-1}T_{\Qca}M_s$. We show that there is an $\epsilon_0>0$
such that if the solution $u$ satisifes the estimate
$\inf_{Q_{ca}}\HsNorm{1}{u-Q_{ca}}<\epsilon_0$, then there are unique
$C^1$ functions $a(u)$ and $c(u)$ such that $u=Q_{c(u) a(u)}+\xi$
with $\xi\bot\partial_x^{-1}T_{\Qca}M_s$.
With the knowledge that the symplectic decomposition exists, we
substitute $u=Q_{ca}+\xi$ into the bKdV
\eqref{Eqn:KdvGeneralizedWithPotential} and split the resulting
equation according to the decomposition
\begin{equation*}
\Lp{2}=\partial_x^{-1}T_{\Qca}M_s\oplus\left(\partial_x^{-1}T_{\Qca}M_s\right)^{\bot}
\end{equation*}
to obtain equations for the parameters $c$ and $a$, and an
equation for the (infinite dimensional) fluctuation $\xi$. In
Section \ref{Section:Projection} we isolate the leading order
terms in the equations for $a$ and $c$ and estimate the remainder,
including all terms containing $\xi$. In Sections
\ref{Section:HessianAndItsProperties} and
\ref{Section:Positivity}, we establish spectral properties and a
lower bound of the Hessian $\Lambda_{ca}''$ on the space
$\left(\partial_x^{-1}T_{\Qca}M_s\right)^{\bot}$.
The proof that $\HsNorm{1}{\xi}$ is sufficiently small is the final
ingredient in the proof of the main theorem. The remaining
sections concentrate on proving this crucial result. We employ a
Lyapunov method and in Section \ref{Section:LyapDeriv} we
construct the Lyapunov function $\Gamma_c$ and prove an
estimate on its time derivative. This estimate is later time
maximized over an interval $[0,T]$, and integrated to obtain an
upper bound on $\Gamma_c$ involving the time $T$ and
the norms of $\xi$. We combine this upper bound with the lower
bound on $\Gamma_c$ following from the results of
Section \ref{Section:Positivity}, and obtain an inequality
involving $\HsNorm{1}{\xi}$. In Section \ref{Section:BoundOnFluct}
we solve the inequality to find an upper bound on $\HsNorm{1}{\xi}$
provided $\HsNorm{1}{\xi(0)}$ is small enough. We substitute this
bound into the bound appearing in the dynamical equation for $a$
and $c$, and take $\epsilon_a\epsilon_x$ and $\epsilon_0$ small enough so that
all intermediate results hold to complete the proof.
\end{proof}
\section{Modulation of Solutions}
\label{Section:Decomposition} As stated in the previous section,
we begin the proof by decomposing the solution of
\eqref{Eqn:KdvGeneralizedWithPotential} into a modulated solitary
wave and a fluctuation $\xi$:
\begin{eqnarray}
u(x,t)=Q_{c(t)a(t)}(x)+\xi(x,t),
\label{EquationWithUErrorQDecomposition}
\end{eqnarray}
with $a$, $c$, and $\xi$ fixed by the orthogonality condition
\begin{align}
\xi\bot \dx^{-1}T_{\Qca}M_s, \label{Cond:Orthogonality}
\end{align}
where
\begin{align*}
\dx^{-1}:g\mapsto \int_{-\infty}^x g(z)\, dz.
\end{align*}
Note that $\dx^{-1}T_{\Qca}M_s$ is a subset of $\Lp{2}$ (see
\eqref{Eqn:LIIAssumptions}).
The existence and uniqueness of parameters $a$ and $c$ such that
$\xi=u-Q_{ca}$ satisfies \eqref{Cond:Orthogonality} follows from the
next lemma concerning a restriction of $\dx^{-1}$ and the implicit
function theorem.
The restriction $K$ of $\dx^{-1}$ to the tangent space $T_{\Qca}M_s$ is
defined by the equation $K P_T=P_T\dx^{-1} P_T$, where
$P_T$ is the orthogonal projection onto $T_{\Qca}M_s$. In the
natural basis $\{\zeta^{tr}_{c a},\zeta^n_{c a}\}$ of the tangent space $T_{\Qca}M_s$, the
matrix representation of $K$ is $N^{-1}{\Omega_{c a}}$, where
\begin{eqnarray*}
N&:=\left(\begin{array}{cc}
\LpNorm{2}{\zeta^{tr}_{c a}}^2 & 0 \\
0 & \LpNorm{2}{\zeta^n_{c a}}^2
\end{array}\right)
\end{eqnarray*}
and
\begin{eqnarray}
{\Omega_{c a}}&:=
\left(\begin{array}{cc}
\ip{\zeta^{tr}_{c a}}{\dx^{-1}\zeta^{tr}_{c a}} & \ip{\zeta^n_{c a}}{\dx^{-1}\zeta^{tr}_{c a}} \\
\ip{\zeta^{tr}_{c a}}{\dx^{-1}\zeta^n_{c a}} & \ip{\zeta^n_{c a}}{\dx^{-1}\zeta^n_{c a}}
\end{array}\right).
\label{Eqn:DefinitionOfOiN}
\end{eqnarray}
Recall that $\delta(c)=\frac{1}{2}\LpNorm{2}{Q_c}^2$.
\begin{lemma}
\label{Lemma:InvertibilityOfOiOmegaBound} If $\delta'(c)> 0$ on
the compact set $I\subset\R_+$, then the matrix ${\Omega_{c a}}$ is
invertible for all $c\in I$, and
\begin{align} {\Omega_{c a}^{-1}}=\frac{1}{\delta'(c)}\left(\begin{array}{cc}
0 & 1\\
-1 & 0\\
\end{array}\right).
\label{Eqn:LeadingOrderExpressionOfSymplecticInverse}
\end{align}
Clearly, $\|{\Omega_{c a}^{-1}}\|\le \InfI{\delta'}^{-1}$, where
$\InfI{\delta'}:=\inf_I \delta'(c)$.
\end{lemma}
\begin{proof}
The lemma follows from the relations $\ip{\zeta^{tr}_{c a}}{\dx^{-1}\zeta^{tr}_{c a}}=0$,
$\ip{\zeta^n_{c a}}{\dx^{-1}\zeta^n_{c a}}=0$ and
$\ip{\zeta^{tr}_{c a}}{\dx^{-1}\zeta^n_{c a}}=\ip{\zeta^n_{c a}}{Q_c}=\delta'(c)$.
\end{proof}
Given $\varepsilon>0$, define the tubular neighbourhood
$U_{\varepsilon}:=\{u\in\Lp{2}\,|\,\inf_{(c,\,a)\in
I\times{\mathbb R}}\LpNorm{2}{u-Q_{ca}}<\varepsilon\}$ of the solitary wave
manifold $M_s$ in $\Lp{2}$.
\begin{prop}
\label{Prop:ExistenceOfDecomposition} Let $I\subset\R_+$ be a
compact interval such that $c\mapstoQ_{ca}$ is $C^1(I)$. Then there
exists a positive number
$\varepsilon_0=\varepsilon_0(I)=\O{\InfI{\delta'}^2}$ dependent on
$I$ and unique $C^1$ functions $
a:U_{\varepsilon_0}\rightarrow\R_+$ and
$c:U_{\varepsilon_0}\rightarrow I$, such that
\begin{equation*}
\ip{Q_{c(u)a(u)}-u}{\dx^{-1}\zeta^{tr}_{c(u)a(u)}}=0\ \mbox{and}\
\ip{Q_{c(u)a(u)}-u}{\dx^{-1}\zeta_{c(u)a(u)}^n}=0
\end{equation*}
for all $u\in U_{\varepsilon_0}$. Moreover, there is a positive
real number $C=C(I)$ such that
\begin{equation}
\HsNorm{1}{u-Q_{c(u) a(u)}}\le C\inf_{Q_{ca}\in
M_s}\HsNorm{1}{u-Q_{ca}} \label{Ineq:InitialConditionIFT}
\end{equation}
for all $u\in U_{\varepsilon_0}\cap\Hs{1}$.
\end{prop}
\begin{proof}
Let $\mu:=(\mu^1,\mu^2)^T\in \R_+\times I$ and define $G:\R_+\times
I\times\Hs{1}\rightarrow{\mathbb R}^2$ as
\begin{eqnarray*}
G:(\mu,u)\mapsto\left(\begin{array}{c}
\ip{Q_{ca}-u}{{\Omega_{c a}}\zeta^{tr}_{c a}}\\
\ip{Q_{ca}-u}{{\Omega_{c a}}\zeta^n_{c a}}
\end{array}\right),
\end{eqnarray*}
where $a=\mu^1$ and $c=\mu^2$. The proposition is equivalent to
solving $G(g(u),u)=0$ for a $C^1$ function $g$. Let $\mu_0=(a\,
c)^T$. If $G$ is $C^1$, $G(\mu_0,Q_{ca})$=0, and $\partial_\mu
F(\mu_0,Q_{ca})$ is invertible, then the implicit function theorem
asserts the existence of an open ball $B_{\varepsilon_0}(Q_{ca})$ of
radius $\varepsilon_0$ with centre $Q_{ca}$, and a unique function
$g_{Q_{ca}}:B_\delta(Q_{ca})\rightarrow\R_+\times I$, such that
$G(g_{Q_{ca}}(u),u)=0$ for all $u\in B_{\varepsilon_0}(Q_{ca})$. The
first two conditions are trivial, and the third follows from Lemma
\ref{Lemma:InvertibilityOfOiOmegaBound} since $\partial_\mu
G(\mu_0,Q_{ca})={\Omega_{c a}}$. The radius of the balls
$B_\varepsilon(Q_{ca})$ depend on the parameters $c$ and $a$. To
obtain an estimate of the radius, and to show that we can take
$\varepsilon$ independent of the parameters $c$ and $a$, we give a
proof of the existence of the above function $g_{Q_{ca}}$ for our
special case using the contraction mapping principle.
We wish to solve $G(\mu,u)=0$ for $\mu:=(\mu^1,\mu^2)^T$ with $u$
close to $Q_{ca}$ in $\Lp{2}$. Expand $G(\mu,u)$ in $\mu$ about
$\mu_0=(a\, c)^T$: $G(\mu,u)=G(\mu_0,u)+\partial_\mu
G(\mu_0,u)(\mu-\mu_0)+R(\mu,u)$, with
$R(\mu,u)=\O{\|\mu-\mu_0\|^2}$ ($G$ is $C^2)$. Thus, we must
solve $\mu=\mu_0-[\partial_\mu G(\mu_0,u)]^{-1}\left( G(\mu_0,u)+R(\mu,u)
\right)$ for $\mu$. Clearly, since $\partial_\mu G(\mu_0,u)={\Omega_{c a}}$,
$\mu$ must be a fixed point of
\begin{equation*}
H_{u \mu_0}(\mu):=\mu_0-{\Omega_{c a}^{-1}}[G(\mu_0,u)+R(\mu,u)].
\end{equation*}
We now show that $H_{u \mu_0}$ is a strict contraction, and hence
has a fixed point. By the mean value theorem
\begin{equation*}
\|H_{u \mu_0}(\mu_2)-H_{u \mu_0}(\mu_1)\|\le \sup\|\partial_\mu H_{u
\mu_0}\|\|\mu_2-\mu_1\|,
\end{equation*}
where the supremum is taken over all allowed parameter values.
Furthermore, we have
\begin{align*}
\partial_\mu H_{u \mu_0}(\mu)&=-{\Omega_{c a}^{-1}}[\partial_\mu G(\mu,u)-\partial_u
G(\mu_0,u)]\\
&=-{\Omega_{c a}^{-1}}[\partial_\mu G(\mu,u)-\partial_\mu G(\mu,Q_{ca})+\partial_\mu
G(\mu,Q_{ca})-\partial_\mu G(\mu_0,Q_{ca})+\partial_\mu G(\mu_0,Q_{ca})-\partial_u
G(\mu_0,u)]
\end{align*}
Using the mean value theorem again, we compute that
\begin{equation*}
\|\partial_\mu G(\mu,u)-\partial_\mu G(\mu_0, u)\|\le C_1\delta+
C_2\varepsilon
\end{equation*}
for some constants $C_1$ and $C_2$ if $\|\mu-\mu_0\|<\delta$ and
$\LpNorm{2}{u-Q_{ca}}<\varepsilon$. Combining all the estimates
gives
\begin{equation*}
\|H_{u \mu_0}(\mu_2)-H_{u \mu_0}(\mu_1)\|\le
\sup\|{\Omega_{c a}^{-1}}\|\left( C_1\delta+C_2\varepsilon
\right)\|\mu_2-\mu_1\|.
\end{equation*}
Thus, if $\delta=\frac{1}{4}(C_1 \sup\|{\Omega_{c a}^{-1}}\|)^{-1}$ and
$\varepsilon=\frac{1}{4}(C_2 \sup\|{\Omega_{c a}^{-1}}\|)^{-1}$, then
$H_{u \mu_0}$ is a contraction.
We now choose $\delta$ and $\varepsilon$ so that $H_{u \mu_0}$
maps $B_\delta(\mu_0)$ to $B_\delta(\mu_0)$. We have that
\begin{equation*}
\|H_{u \mu_0}-\mu_0\|\le \|{\Omega_{c a}^{-1}} \left( G(\mu_0, u)+R(\mu,u)
\right)\|\le \sup\|{\Omega_{c a}^{-1}}\|\left(
\|G(\mu_0,u)-G(\mu_0,Q_{ca})\|+\O{\delta^2} \right).
\end{equation*}
By the mean value theorem $\|G(\mu_0,u)-G(\mu_0,Q_{ca})\|\le
C_3\varepsilon$. Thus, if we take
$\delta=\O{\sup\|{\Omega_{c a}^{-1}}\|^{-1}}$ so that $\O{\delta^2}\le
\frac{1}{4}\left(\sup\|{\Omega_{c a}^{-1}}\|\right)^{-1}\delta$, then
\begin{equation*}
\|H_{u \mu_0}-\mu_0\|\le
C_3\sup\|{\Omega_{c a}^{-1}}\|\varepsilon+\frac{1}{4}\delta.
\end{equation*}
We now take $\varepsilon<\frac{1}{4}\left( C_3\sup\|{\Omega_{c a}^{-1}}\|
\right)^{-1}\delta$ to obtain $\|H_{u \mu_0}-\mu_0\|\le
\frac{1}{2}\delta$. To complete the argument, take $\delta$ to be
the smaller of $\frac{1}{4}\left( C_1\sup\|{\Omega_{c a}^{-1}}\|\right)^{-1}$
and the above choice, and then $\varepsilon$ to be the smaller of
$\frac{1}{4}(C_2\sup\|{\Omega_{c a}^{-1}}\|)^{-1}$ and $\delta(4
C_3\sup\|{\Omega_{c a}^{-1}}\|)^{-1}$. Using the bound on
$\|{\Omega_{c a}^{-1}}\|$ we find that
\begin{equation*}
\varepsilon=\O{\InfI{\delta'}^2}
\end{equation*}
if $\sup\|{\Omega_{c a}^{-1}}\|\ge 1$, or equivalently, when
$\InfI{\delta'}$ is sufficiently small.
The above argument shows that there exists balls $\{
B_{\varepsilon}(Q_{ca})\, |\, a\in\R_+, c\in I\}$ with radius
$\varepsilon$ dependent only on the compact set $I$. Then,
defining $U_{\varepsilon_0}=\bigcup\{ B_{\varepsilon_0}(Q_{ca})\,
|\, a\in\R_+, c\in I\}$ and pasting the $C^1$ functions $g_{Q_{ca}}$
together, into a $C^1$ function $g_{
I}:U_{\varepsilon_0}\rightarrow \R_+\times I$, proves existence of
the required $C^1$ functions $a(u)$ and $c(u)$. Uniqueness follows
from the uniqueness of the functions $g_{Q_{ca}}$.
Let $u\in U_{\varepsilon}$, $c\in I$, and $a\in {\mathbb R}$, and consider
the equation
\begin{equation*}
u-Q_{c(u)a(u)}=u-Q_{ca}+Q_{ca}-Q_{c(u)a(u)}.
\end{equation*}
Clearly, inequality \eqref{Ineq:InitialConditionIFT} will follow
if $\HsNorm{1}{Q_{ca}-Q_{c(u)a(u)}}\le C\HsNorm{1}{u-Q_{ca}}$ for some
positive constant $C$. Since the derivatives $\partial_cQ_{ca}$ and
$\partial_aQ_{ca}$ are uniformly bounded in $\Hs{1}$ over $I\times{\mathbb R}$, the
mean value theorem gives that $\HsNorm{1}{Q_{ca}-Q_{c(u)a(u)}}\le
C\|(c,a)^T-(c(u),a(u))^T\|$, where the constant $C$ does not
depend on $c$, $a$. The relations $g_{ I}(Q_{ca})=(c,a)^T$ and $g_{
I}(u)=(c(u),a(u))^T$ then imply $\HsNorm{1}{Q_{ca}-Q_{c(u)a(u)}}\le
C\|g_{ I}(Q_{ca})-g_{ I}(u)\|$. Again, we appeal to the mean value
theorem and use the properties of ${\Omega_{c a}}$ and that $\partial_u g_{
I}=\partial_\mu G^{-1}\partial_u G$ is uniformly bounded in the parameters $c$
and $a$ to obtain \eqref{Ineq:InitialConditionIFT}.
\end{proof}
\section{Evolution Equations for Parameters $\xi$, $a$ and $c$}
\label{Section:Projection} In Section \ref{Section:Decomposition}
we proved that if $u$ remains close enough to the solitary wave
manifold $M_s$, then we can write a solution $u$ to
\eqref{Eqn:KdvGeneralizedWithPotential} uniquely as a sum of a
modulated solitary wave $Q_{ca}$ and a fluctuation $\xi$ satisfying
the orthogonality condition \eqref{Cond:Orthogonality}. Thus, as
$u$ evolves according to the initial value problem
\eqref{Eqn:KdvGeneralizedWithPotential}, the parameters $a(t)$ and
$c(t)$ trace out a path in ${\mathbb R}^2$. The goal of this section is to
derive the dynamical equations for the parameters $a$ and $c$, and
the fluctuation $\xi$. We obtain such equations by substituting the
decomposition $u=Q_{ca}+\xi$ into
\eqref{Eqn:KdvGeneralizedWithPotential} and then projecting the
resulting equation onto appropriate directions, with the intent of
using the orthogonality condition on $\xi$.
From now on, $u$ is the solution of
\eqref{Eqn:KdvGeneralizedWithPotential} with initial condition
$u_0$ satisfying $\epsilon_0:=\inf_{Q_{ca}\in
M_s}\HsNorm{1}{u_0-Q_{ca}}<\varepsilon_0$, and $T_0=T_0(u_0)$ is the
maximal time such that $u(t)\in U_\varepsilon$ for $0\le t\le
T_0$. Then for $0\le t\le T_0$, $u$ can be decomposed as in
\eqref{EquationWithUErrorQDecomposition} and
\eqref{Cond:Orthogonality}.
\begin{prop}
\label{Prop:EvolutionEquationAndBoundForAandC}
Assume
$\delta'(c)\ne 0$. Say $u=Q_{ca}+\xi$ is a solution to
(\ref{Eqn:KdvGeneralizedWithPotential}), where $\xi$ satisfies
(\ref{Cond:Orthogonality}). Then, if $\HsNorm{1}{\xi}$ is small
enough, $\epsilon_x\le 1$, and $c\in I$,
\begin{align}
\label{Eqn:DynamicalEquationForCAndA} \left(\begin{array}{c}
\dot{a} \\
\dot{c}
\end{array}\right)&=
\left(\begin{array}{c}
c-b(t,a)\\
0
\end{array}\right)+
b'(t,a)\frac{\delta(c)}{\delta'(c)}\left(\begin{array}{c}
0\\
1
\end{array}\right)+Z(a,\,c,\,\xi),
\end{align}
where
$Z(a,\,c,\,\xi)=\O{\epsilon_a\epsilon_x^2+\epsilon_a\epsilon_x\HsNorm{1}{\xi}+\HsNorm{1}{\xi}^2}$.
\end{prop}
\begin{proof}
Recall that the solitary wave $Q_{ca}$ is an extremal of the
functional $\Lambda_{ca}$. To use this fact we rearrange
definition \eqref{DefinitionLS} of $\Lambda_{ca}$ to write the
Hamiltonian $H_b$ as
\begin{equation*}
H_b(u)=\Lambda_{ca}(u)-cP(u)+\frac{1}{2}\intR{b
u^2(x)},
\end{equation*}
where for notational simplicity we have suppressed the space and
time dependency of $b$. Substituting $Q_{ca}+\xi$ for $u$ in
\eqref{Eqn:KdVVariationalForm} and using the above expression for
$H_b$ gives the equation
\begin{equation*}
\dot{a}\zeta^{tr}_{c a}+\dot{c}\zeta^n_{c a}+\dot{\xi}=\dx\Lambda_{ca}'(Q_{ca}+\xi)-c\dx[Q_{ca}+\xi]+\dx[(Q_{ca}+\xi)b],
\end{equation*}
where dots indicate time differentiation. Let
$\mathcal{L}_{Q}:=\Lambda_{ca}''(Q_{ca})$,
\begin{eqnarray*}
\delta b:=b(t,x)-b(t,a)
\end{eqnarray*}
and
\begin{eqnarray*}
\delta^2b:=b(t,x)-b(t,a)-b'(t,a)(x-a).
\end{eqnarray*}
Taylor expanding $\Lambda_{ca}'(Q_{ca}+\xi)$ to linear order in
$\xi$, using that $Q_{ca}$ is an extremal of $\Lambda_{ca}$ and
the relation $\zeta^{tr}_{c a}=-\dxQ_{ca}$ gives that
\begin{align}
\dot{\xi}=\dx\left[(\mathcal{L}_{Q}+\delta b+b(a)-c)\xi\right]&+\dx\NpA{\xi}-[\dot{a}-c+b(a)]\zeta^{tr}_{c a}-\dot{c}\zeta^n_{c a}\nonumber\\&+b'(a)\dx[(x-a)Q_{ca}]+\dx[\RBQ_{ca}].
\label{Eqn:KdVEquationForXiAndParameters}
\end{align}
The nonlinear terms have been collected into $\NpA{\xi}$ given by
(\ref{Eqn:NpA}) in Appendix
\ref{Appendix:EstimateNonlinearRemainders}.
Define the vectors $\zeta_1:=\zeta^{tr}_{c a}$ and $\zeta_2:=\zeta^n_{c a}$. Projecting
(\ref{Eqn:KdVEquationForXiAndParameters}) onto $\dx^{-1}\zeta_i$
for $i=1,2$ and using the antisymmetry of $\dx$ gives the
two equations
\begin{align}
[\dot{a}-c+b(a)]\left[
\ip{\zeta^{tr}_{c a}}{\dx^{-1}\zeta_i}+\ip{\xi}{\zeta_i}\right]&+\dot{c}\ip{\zeta^n_{c a}}{\dx^{-1}\zeta_i}+\ip{\dot{\xi}}{\dx^{-1}\zeta_i}-\dot{a}\ip{\xi}{\zeta_i}=-b'(t,a)\ip{(x-a)Q_{ca}}{\zeta_i}\nonumber\\
&-\ip{\RBQ_{ca}}{\zeta_i}-\ip{\delta b\xi}{\zeta_i}-\ip{\NpA{\xi}}{\zeta_i}-\ip{\mathcal{L}_{Q}\xi}{\zeta_i}.
\label{Eqn:KdVEquationForXiAndParametersSecond}
\end{align}
We can replace the term containing $\dot{\xi}$ since the time
derivative of the orthogonality condition
$\ip{\xi}{\dx^{-1}\zeta_i}=0$ implies
$\ip{\dot{\xi}}{\dx^{-1}\zeta_i}=\dot{a}\ip{\xi}{\zeta_i}-\dot{c}\ip{\xi}{\partial_c\dx^{-1}\zeta_i}$.
Note that we have used the relation $\partial_a\zeta_i=-\partial_x\zeta_i$.
Thus, in matrix form,
(\ref{Eqn:KdVEquationForXiAndParametersSecond}) becomes
\begin{align}
(I+B){\Omega_{c a}}\left(\begin{array}{c}\dot{a}-c+b(t,a)\\
\dot{c}\end{array}\right)=X+Y, \label{Eqn:ApproximateDynamicalSystem}
\end{align}
where
\begin{align*}
X&:=-b'(t,a)\delta(c)\left(\begin{array}{c}1\\0\end{array}\right)-\left(\begin{array}{c}\ip{\RBQ_{ca}}{\zeta^{tr}_{c a}}\\ \ip{\RBQ_{ca}}{\zeta^n_{c a}}\end{array}\right),\\
Y&:=-\left(\begin{array}{c}\ip{\delta b\xi}{\zeta^{tr}_{c a}}+\ip{\NpA{\xi}}{\zeta^{tr}_{c a}}+\ip{\mathcal{L}_{Q}\xi}{\zeta^{tr}_{c a}}\\
\\
\ip{\delta b\xi}{\zeta^n_{c a}}+\ip{\NpA{\xi}}{\zeta^n_{c a}}+\ip{\mathcal{L}_{Q}\xi}{\zeta^n_{c a}}\end{array}\right),
\end{align*}
and
\begin{eqnarray*}
B:=\left(\begin{array}{cc}\ip{\xi}{\zeta^{tr}_{c a}} & \ip{\xi}{\zeta^n_{c a}}\\\ip{\xi}{\zeta^n_{c a}} &
-\ip{\xi}{\partial_c\dx^{-1}\zeta^n_{c a}}\end{array}\right){\Omega_{c a}^{-1}}.
\end{eqnarray*}
We have explicitly computed $\ip{(x-a)Q_{ca}}{\zeta_i}$ to obtain
the above expression for $X$.
We now estimate the error terms and solve for $\dot{a}$ and
$\dot{c}$. The assumption on the potential implies the bounds
\begin{align}
|\delta b|\le \epsilon_a\epsilon_x (x-a)\ \mbox{and}\ |\delta^2b|\le \epsilon_a\epsilon_x^2
(x-a)^2.\label{Eqn:SizeDV}
\end{align}
Thus, H\"{o}lder's inequality and exponential decay of $Q_{ca}$
imply
\begin{align}
X&=-b'(t,a)\delta(c) \left(\begin{array}{c}
1\label{Eqn:EstimateOnXFirst}\\
0
\end{array}\right)+\O{\epsilon_a\epsilon_x^2}\\
&=\O{\epsilon_a\epsilon_x}. \nonumbe
\end{align}
Similarly, exponential decay of $\zeta^{tr}_{c a}$ and $\zeta^n_{c a}$ implies
$\ip{\delta b\xi}{\zeta_i}=\O{\epsilon_a\epsilon_x\HsNorm{1}{\xi}}$. The linear term
$\ip{\mathcal{L}_{Q}\xi}{\zeta_i}$ is zero since $\mathcal{L}_{Q}\zeta^{tr}_{c a}=0$, $\mathcal{L}_{Q}\zeta^n_{c a}=-Q_{ca}$
and $\xi\bot \dx^{-1}\zeta^{tr}_{c a}=-Q_{ca}$. Lastly, $\ip{\NpA{\xi}}{\zeta_i}\le
C\HsNorm{1}{\xi}^2$ by the first estimate in Lemma
\ref{Appendix:EstimateNonlinearRemainders}.\ref{Lemma:NonlinearEstimates}.
Combining the above estimates gives the bound
\begin{align*}
\|Y\|=\O{\epsilon_a\epsilon_x\HsNorm{1}{\xi}+\HsNorm{1}{\xi}^2}.
\end{align*}
By the second inclusion of (\ref{Eqn:LIIAssumptions}),
$\partial_c\dx^{-1}\zeta^n_{c a}\in \Lp{2}$. H\"{o}lder's inequality then implies
$\|B\|=\O{\HsNorm{1}{\xi}}$. Thus, if $\HsNorm{1}{\xi}$ is
sufficiently small, say so that $\|B\|\le \frac{1}{2}$, then $I+B$
is invertible and $\|\left( I+B\right)^{-1}\|\le 2$. Acting on equation
(\ref{Eqn:ApproximateDynamicalSystem}) by
$(I+B)^{-1}=I-B(I+B)^{-1}$ and then ${\Omega_{c a}^{-1}}$ gives the
equation
\begin{align*}\left(
\begin{array}{c}\dot{a}-c+V(a)\\ \dot{c}\end{array}\right)={\Omega_{c a}^{-1}}
[X+B(I-B)^{-1} X+(I-B)^{-1} Y].
\end{align*}
Using the above estimates of $\|B\|$, $\|(I-B)^{-1}\|$, $\|X\|$,
and $\|Y\|$ implies
\begin{align*}\left(
\begin{array}{c}\dot{a}-c+V(a)\\ \dot{c}\end{array}\right)={\Omega_{c a}^{-1}}
X+\O{\epsilon_a\epsilon_x\HsNorm{1}{\xi}+\HsNorm{1}{\xi}^2}.
\end{align*}
Replacing $X$ by (\ref{Eqn:EstimateOnXFirst}) completes the proof.
\end{proof}
\section{The Lyapunov Functional}
\label{Section:LyapDeriv} In the last section we derived dynamical
equations for the modulation parameters. These equations contain
the $\Hs{1}$ norm of the fluctuation. In this section we begin to
prove a bound on $\xi$. Recall that the latter bound is needed to
ensure that $u$ remains close to the manifold of solitary waves
$M_s$ for long time.
We employ a Lyapunov argument with Lyapunov function
\begin{align}
\Gamma_c(t):=\Lambda_{ca}(Q_{ca}+\xi)-\Lambda_{ca}(Q_{ca})+b'(a)\ip{(x-a)Q_{ca}}{\xi}.
\label{equ:LSDiffDef}
\end{align}
Remark: if $f(u)=u^3$, the last term in the Lyapunov functional is
not needed; however, apart from computational complexity, there is
no disadvantage in using the above function for this special case
as well.
\begin{lemma}
\label{Lemma:AlmostConservationOfLyapunov} Say $u=Q_{ca}+\xi$ is a
solution to (\ref{Eqn:KdvGeneralizedWithPotential}), where $\xi$
satisfies (\ref{Cond:Orthogonality}). Say $\epsilon_a\le 1$. If
$\delta'(c)>0$, and $\epsilon_x$ and $\HsNorm{1}{\xi}$ are less than 1,
with $\HsNorm{1}{\xi}$ small enough, then
\begin{align}
\dd{t}
\Gamma_c(t)&=\O{\epsilon_a^2\epsilon_x^3+\left(\epsilon_a\epsilon_x\epsilon_t+\epsilon_a\epsilon_x^2\right)\HsNorm{1}{\xi}+\epsilon_a\epsilon_x\HsNorm{1}{\xi}^2+\HsNorm{1}{\xi}^4}.
\label{TimeDerivativeLiapunovFunctional}
\end{align}
\end{lemma}
\begin{proof}
Suppressing explicit dependence on $x$ and $t$, we have by
definition
\begin{align*}
\Lambda_{ca}(u):=H_b(u)-\frac{1}{2}\intR{
u^2 b}+cP(u).
\end{align*}
Thus, relations (\ref{Eqn:ConservationHamiltonian}),
(\ref{Eqn:ConservationMomentum}) and
(\ref{Eqn:ConservationPotentialMomentum}) imply that the time
derivative of $\Lambda_{ca}$ along the solution $u$ is
\begin{align*}
\dd{t}\Lambda_{ca}(u)=\intR{\frac{1}{2}\dot{c} u^2
+b'\left[\frac{1}{2}c u^2- u f(u)+\frac{3}{2}(\partial_x
u)^2+F(u)\right]+b''\, u \partial_x u}.
\end{align*}
Substituting $Q_{ca}+\xi$ for $u$, manipulating the result using
antisymmetry of $\partial_x$, and collecting appropriate terms into
$b'(a)\ip{\mathcal{L}_{Q}\xi}{\dx((x-a)Q_{ca})}$,
$\ip{\NpA{\xi}}{\partial_x[\delta b(Q_{ca}+\xi)]}$, and
$\ip{\Lambda_{ca}'(Q_{ca})}{\partial_x(\delta b(Q_{ca}+\xi))}$ gives the
relation
\begin{align*}
\dd{t}[\Lambda_{ca}(Q_{ca}+\xi)-\Lambda_{ca}(Q_{ca})]=&b'(a)\ip{\mathcal{L}_{Q}\xi}{\dx((x-a)Q_{ca})}+\dot{c}\ip{Q_{ca}}{\xi}+\ip{\mathcal{L}_{Q}\xi}{\partial_x\left(\RBQ_{ca}\right)}+\dot{c}\frac{1}{2}\LpNorm{2}{\xi}^2\\
&+c\frac{1}{2}\ip{b'\xi}{\xi}+\frac{3}{2}\ip{b'\partial_x\xi}{\partial_x\xi}-\ip{f'(Q_{ca})\xi}{\partial_x(\delta b\xi)}\\
&+\ip{\NpA{\xi}}{\partial_x[\delta b(Q_{ca}+\xi)]}+\ip{b''\xi}{\partial_x\xi}+\ip{\Lambda_{ca}'(Q_{ca})}{\partial_x[\delta b(Q_{ca}+\xi)]}.
\end{align*}
The last term is zero because $\Lambda_{ca}'(Q_{ca})=0$ and
since $\xi\botQ_{ca}$, the quantity $\dot{c}\ip{\xi}{Q_{ca}}$ is also
zero. We use Lemma \ref{Lemma:NonlinearEstimates}, assumptions
(\ref{Eqn:AssumptionOnPotential}) on the potential, estimates
(\ref{Eqn:SizeDV}), and
\begin{align*}
|\delta b'|&\le\epsilon_a\epsilon_x^2 x\
\end{align*}
to estimate the size of the time derivative. We also use that
$Q_{ca}$, $\partial_xQ_{ca}$, $\partial_x^2Q_{ca}$ and $f'(Q_{ca})$ are exponentially
decaying. When $\epsilon_x\le 1$, higher order terms like
$\ip{b''\xi}{\partial_x\xi}$ are bounded above by lower order terms like
$\ip{b'\xi}{\xi}$. Similarly, if $\HsNorm{1}{\xi}\le 1$, then
$\epsilon_a\epsilon_x\HsNorm{1}{\xi}^2\le\epsilon_a\epsilon_x\HsNorm{1}{\xi}$. This procedure
gives the estimate
\begin{align*}
\dd{t}[\Lambda_{ca}(Q_{ca}+\xi)-\Lambda_{ca}(Q_{ca})]=&b'(a)\ip{\xi}{\mathcal{L}_{Q}\dx((x-a)Q_{ca})}+\ip{\NpA{\xi}}{\delta b\partial_x\xi}\\
&+\O{|\dot{c}|\HsNorm{1}{\xi}^2+\epsilon_a\epsilon_x^2\HsNorm{1}{\xi}+\epsilon_a\epsilon_x\HsNorm{1}{\xi}^2}.
\end{align*}
Applying the chain rule to the integrand of
\begin{equation*}
\intR{\partial_x\left[\left(
F(Q_{ca}+\xi)-F(Q_{ca})-f(Q_{ca})\xi-\frac{1}{2}f'(Q_{ca})\xi^2\right)\delta b\right]}=0
\end{equation*}
and using the definition of $\NpA{\xi}$ gives that
\begin{align*}
\ip{\NpA{\xi}}{\delta b\partial_x\xi}=&\ip{\NpA{\xi}+\frac{1}{2}f''(Q_c)\xi^2}{\delta b\partial_xQ_c}\\
&-\intR{\left(
F(Q_{ca}+\xi)-F(Q_{ca})-f(Q_{ca})\xi-\frac{1}{2}f'(Q_{ca})\xi^2\right) b'}.
\end{align*}
The second estimate and the proof of the third estimate of
Lemma \ref{Lemma:NonlinearEstimates} of Appendix \ref{Appendix:EstimateNonlinearRemainders} then imply the bound
$\ip{\NpA{\xi}}{\delta b\partial_x\xi}=\O{\epsilon_a\epsilon_x\HsNorm{1}{\xi}^3}$. Thus, since
$\epsilon_a\epsilon_x\HsNorm{1}{\xi}^3\le \epsilon_a\epsilon_x\HsNorm{1}{\xi}^2$ when
$\HsNorm{1}{\xi}\le 1$, we have
\begin{multline}
\dd{t}[\Lambda_{ca}(Q_{ca}+\xi)-\Lambda_{ca}(Q_{ca})]=b'(a)\ip{\xi}{\mathcal{L}_{Q}\dx((x-a)Q_{ca})}+\O{|\dot{c}|\HsNorm{1}{\xi}^2+\epsilon_a\epsilon_x^2\HsNorm{1}{\xi}+\epsilon_a\epsilon_x\HsNorm{1}{\xi}^2}.
\label{Eqn:PropAlmostLiapunovConservationDLS}
\end{multline}
When $f(u)=u^3$, $\ip{\xi}{\mathcal{L}_{Q}\dx((x-a)Q_{ca})}=0$ since
$\zeta^n_{c a}=\dx[(x-a)Q_{ca}]$. In this special case the above
estimate is sufficient for our purposes, but in general, we need
to use the corrected Lyapunov functional. When $\xi\in
C({\mathbb R},\,\Hs{1})\cap C^1({\mathbb R},\, \Hs{-2})$, $b'(a)\ip{\xi}{(x-a)Q_{ca}}$
is continuously differentiable with respect to time;
\begin{align*}
\dd{t}\left[ b'(a)\ip{\xi}{(x-a)Q_{ca}}
\right]=&\partial_tb'\ip{\xi}{(x-a)Q_{ca}}+b'(a)\ip{\dot{\xi}}{(x-a)Q_{ca}}+\dot{c}b'(a)\ip{\xi}{(x-a)\zeta^n_{c a}}\\
&+\dot{a}b'(a)\ip{\xi}{(x-a)\zeta^{tr}_{c a}}+\dot{a}b''(a)\ip{\xi}{(x-a)Q_{ca}},
\end{align*}
where $\ip{\xi}{Q_{ca}}=0$ has been used to simplify the derivative.
Substituting for $\partial_t\xi$ using
(\ref{Eqn:KdVEquationForXiAndParameters}) gives
\begin{align*}
\dd{t}[b'(a)\ip{\xi}{(x-a)Q_{ca}}]=&-b'(a)\ip{\xi}{\mathcal{L}_{Q}\dx((x-a)Q_{ca})}-[\dot{a}-c+b(a)]b'(a)\frac{1}{2}\LpNorm{2}{Q_{ca}}^2+\partial_tb'\ip{\xi}{(x-a)Q_{ca}}\\
&+[\dot{a}-c+b(a)]b'(a)\ip{\partial_x\xi}{(x-a)Q_{ca}}+[\dot{a}-c+b(a)] b''(a)\ip{\xi}{(x-a)Q_{ca}}\\
&+\dot{c} b'(a)\ip{\xi}{(x-a)\zeta^n_{c a}}-b'(a)\ip{\xi}{\delta b\partial_x((x-a)Q_{ca})}-b'(a)\ip{\NpA{\xi}}{\partial_x((x-a)Q_{ca})}\\
&-b'(a)\ip{\RBQ_{ca}}{\partial_x((x-a)Q_{ca})}+[c-b(a)]b''(a)\ip{\xi}{(x-a)Q_{ca}}.
\end{align*}
We estimate using the same assumptions used to derive
(\ref{Eqn:PropAlmostLiapunovConservationDLS}). If
$\HsNorm{1}{\xi}$ and $\epsilon_x$ are less than 1, then
\begin{align*}
\dd{t}[b'(a)\ip{\xi}{(x-a)Q_{ca}}]=&-b'(a)\ip{\xi}{\mathcal{L}_{Q}\dx((x-a)Q_{ca})}+\O{ |\dot{a}-c+b(a)|\epsilon_a\epsilon_x+|\dot{c}|\epsilon_a\epsilon_x\HsNorm{1}{\xi}}\\
&+\O{\epsilon_a^2\epsilon_x^3+((1+\epsilon_a)\epsilon_x^2+\epsilon_x\epsilon_t)\epsilon_a\HsNorm{1}{\xi}+\epsilon_a\epsilon_x\HsNorm{1}{\xi}^2}.
\end{align*}
Adding the above expression to
(\ref{Eqn:PropAlmostLiapunovConservationDLS}) gives an upper bound
containing $|\dot{c}|$ and $|\dot{a}-c+b(a)|$. Replacing these
quantities using the bounds
\begin{equation*}
|\dot{c}|=\O{\epsilon_a\epsilon_x+\epsilon_a\epsilon_x\HsNorm{1}{\xi}+\HsNorm{1}{\xi}^2}\\
\end{equation*}
and
\begin{equation*}
|\dot{a}-c+b(a)|=\O{\epsilon_a\epsilon_x^2+\epsilon_a\epsilon_x\HsNorm{1}{\xi}+\HsNorm{1}{\xi}^2}\\
\end{equation*}
from Proposition \ref{Prop:EvolutionEquationAndBoundForAandC}, and
bounding higher order terms by lower order terms gives
(\ref{TimeDerivativeLiapunovFunctional}). To use the above bounds
on $|\dot{c}|$ and $|\dot{a}-c+b(a)|$ we must assume
$\HsNorm{1}{\xi}$ is small enough so that Proposition
\ref{Prop:EvolutionEquationAndBoundForAandC} holds.
\end{proof}
\section{Spectral Properties of the Hessian $\mathcal{L}_{Q}$}
\label{Section:HessianAndItsProperties} The Hessian
$\partial^2\Lambda_{ca}$ at $Q_{ca}$ in the $\Lp{2}$ pairing is
computed to be the unbounded operator
\begin{align}
\mathcal{L}_{Q}&:=-\partial_x^2+c-f'(Q_{ca}), \label{Eqn:Hessian}
\end{align}
defined on $\Lp{2}$ with domain $\Hs{2}$. We extend this operator
to the corresponding complex spaces.
\begin{prop} \label{Prop:Spectrum
The self-adjoint operator $\mathcal{L}_{Q}$ has the following properties.
{\begin{enumerate}
\item $\mathcal{L}_{Q}\zeta^{tr}_{c a}=0$ and $\mathcal{L}_{Q}\zeta^n_{c a}=-Q_{ca}$.
\item All eigenvalues of $\mathcal{L}_{Q}$ are simple, and
$\Null{\mathcal{L}_{Q}}=\Span{\zeta^{tr}_{c a}}$.
\item $\mathcal{L}_{Q}$ has exactly one negative eigenvalue.
\item The essential spectrum is $[c,\infty)\subset\R_+$.
\item $\mathcal{L}_{Q}$ has a finite number of eigenvalues in $(-\infty,
c)$.
\end{enumerate}}
\end{prop}
\begin{proof}
Recall that the vectors $\zeta^{tr}_{c a}:=-\partial_xQ_{ca}$ and $\zeta^n_{c a}:=\partial_cQ_{ca}$ are in
the Sobolev space $\Hs{2}$. Thus, relations $\mathcal{L}_{Q}\zeta^{tr}_{c a}=0$ and
$\mathcal{L}_{Q}\zeta^n_{c a}=-Q_{ca}$ make sense, and are obtained by differentiating
$\Lambda_{ca}'(Q_{ca})=0$ with respect to $a$ and $c$. The
first relation above proves that $\zeta^{tr}_{c a}$ is a null vector.
Say $\zeta,\eta\in \Hs{2}$ are linearly independent eigenvectors
of $\mathcal{L}_{Q}$ with the same eigenvalue. Then, since $\mathcal{L}_{Q}$ is a second
order linear differential operator without a first order
derivative, the Wronskian
\begin{eqnarray*}
W(\eta,\zeta)=\zeta\partial_x \eta-\eta\partial_x\zeta
\end{eqnarray*}
is a non-zero constant. With $\eta$ and $\zeta$ both in $\Hs{2}$
however, the limit $\lim_{x\rightarrow \infty} W(\eta,\zeta)$ is
zero. This contradicts the non vanishing of the Wronskian, and
hence all eigenvalues of $\mathcal{L}_{Q}$ are simple and, in particular,
$\Null{\mathcal{L}_{Q}}=\Span{\zeta^{tr}_{c a}}$.
Next we prove that the operator $\mathcal{L}_{Q}$ has exactly one negative
eigenvalue using Sturm-Liouville theory on an infinite interval.
Recall that the solitary wave $Q_{ca}(x)$ is a differentiable
function, symmetric about $x=a$ and monotonically decreasing if
$x>a$. This implies that the null vector $\zeta^{tr}_{c a}$, or equivalently,
the derivative of $Q_{ca}$ with respect to $x$, has exactly one root
at $x=a$. Therefore, by Sturm-Liouville theory, zero is the second
eigenvalue and there is exactly one negative eigenvalue.
We use standard methods to compute the essential spectrum. Since
the function $f'(Q_{ca}(x))$ is continuous and decays to zero at
infinity, the bottom of the essential spectrum begins at
$\lim_{x\rightarrow \infty} (c-f'(Q_{ca}(x)))=c$ and extends to
infinity: $\sigma_{ess}(\mathcal{L}_{Q})=[c,\infty)$. Furthermore, the bottom
of the essential spectrum is not an accumulation point of the
discrete spectrum since $f'(Q_{ca}(x))$ decays faster than $x^{-2}$
at infinity. Hence, there is at most a finite number of
eigenvalues in the interval $(-\infty,c)$. For details see
\cite{ReSiI, ReSiIV, GuSi2003}.
\end{proof}
\section{Strict Positivity of the Hessian}
\label{Section:Positivity} In this section we prove strict
positivity of the Hessian $\mathcal{L}_{Q}$ on the orthogonal complement to
the 2-dimensional space $\dx^{-1}T_{\Qca}M_s=\Span{Q_{ca},\,\dx^{-1}\zeta^n_{c a}}$.
This result is a crucial ingredient needed to prove the bound on
the fluctuation $\xi$.
\begin{prop}
\label{Prop:Positivity} Assume $\delta'(c)>0$ on $I\subset\R_+$. If
$\xi\bot\dx^{-1}T_{\Qca}M_s$, then there is a positive constant $\rho$
such that $\ip{\mathcal{L}_{Q}\xi}{\xi}\ge\rho\HsNorm{1}{\xi}^2$.
\end{prop}
\begin{proof}
Define $X:=\{\xi\in\Hs{1}\ |\ \xi\bot\dx^{-1}T_{\Qca}M_s,\
\LpNorm{2}{\xi}=1\}$. By the max-min principle, $\inf_{X\cap\Hs{2}}
\ip{\mathcal{L}_{Q}\xi}{\xi}$ is attained or is equal to $\inf
\sigma_{ess}(\mathcal{L}_{Q})=c$. If the later holds the proof is complete.
In the former case, let $\eta$ be the minimizer.
We claim the set of vectors $\{\zeta^{tr}_{c a},\zeta^n_{c a},\eta\}$ is an linearly
independent set. If they were dependent, then, since $\zeta^{tr}_{c a}$ and
$\zeta^n_{c a}$ are orthogonal, there are non-zero constants $\alpha$ and
$\beta$ such that $\eta=\alpha\zeta^{tr}_{c a}+\beta\zeta^n_{c a}$. Projecting this
equation onto $\dx^{-1}\zeta^{tr}_{c a}$ and $\dx^{-1}\zeta^n_{c a}$ gives the equations
$\beta\delta'(c)=0$ and $\alpha\delta'(c)=0$. Thus, the
assumption $\delta'(c)>0$ implies $\eta=0$. A contradiction since
the zero function does not lie in the set $X$. Note that in
deriving $\alpha\delta'(c)=0$ we have used that $\dx^{-1}$ is
antisymmetric on the span of $\zeta^n_{c a}$ since $\dx^{-1}\zeta^n_{c a}\in \Lp{2}$.
By the min-max principle, if
\begin{align*}
\lambda_3&:=\inf \left\{ \max \left\{\ip{\mathcal{L}_{Q}\xi}{\xi}\, |\, \xi\in
V,\, \LpNorm{2}{\xi}=1 \right\} \, |\, V\subset \Hs{2},\,
\mbox{dim}\,
V=3 \right\}\\
&\le \max \left\{ \ip{\mathcal{L}_{Q}\xi}{\xi} \,|\,
\xi\in\Span{\zeta^{tr}_{c a},\,\zeta^n_{c a},\,\eta}\right\}
\end{align*}
is below the essential spectrum, then it is the third eigenvalue
counting multiplicity. Let $\xi=\alpha\eta+\beta\zeta^{tr}_{c a}+\gamma\zeta^n_{c a}$
where $\alpha$, $\beta$ and $\gamma$ are arbitrary apart from
satisfying $\LpNorm{2}{\xi}=1$. Thus, since the third eigenvalue of
$\mathcal{L}_{Q}$ is positive (see Section
\ref{Section:HessianAndItsProperties}),
\begin{align*}
0<\ip{\mathcal{L}_{Q}\xi}{\xi}=\alpha^2\ip{\mathcal{L}_{Q}\eta}{\eta}-\gamma^2\delta'(c)\le\alpha^2\ip{\mathcal{L}_{Q}\eta}{\eta},
\end{align*}
and hence $\ip{\mathcal{L}_{Q}\eta}{\eta}>0$. The function
$\sigma(c)=\ip{\mathcal{L}_{Q}\eta}{\eta}$ is continuous since both
$\dx^{-1}\zeta^{tr}_{c a}$ and $\dx^{-1}\zeta^n_{c a}$ are continuous in $\Lp{2}$ as
functions of $c$. Set $\varrho=\inf_I \sigma(c)$.
We now improve the result to an $\Hs{1}$ norm. If we define the
constant $K(I):=\sup_I \SupNorm{c-f'(Q_{ca})}$, then
$\ip{\mathcal{L}_{Q}\xi}{\xi}\ge\LpNorm{2}{\partial_x\xi}^2-K(I)\LpNorm{2}{\xi}^2$.
Adding to this bound the factor $\frac{K+1}{\varrho}$ of the lower
bound $\ip{\mathcal{L}_{Q}\xi}{\xi}\ge \varrho\LpNorm{2}{\xi}^2$ derived above
completes the proof.
\end{proof}
\section{Bound on the Fluctuation}
\label{Section:BoundOnFluct} We are now ready to prove the bound
on the fluctuation.
\begin{prop}
\label{Prop:FluctuationBound} Say $\epsilon_a\le 1$. Then, for small
enough $\epsilon_x\le 1$ and initial fluctuation $\HsNorm{1}{\xi(0)}\le
1$, there exists a constant $C$ such that the bound
\begin{align*}
\HsNorm{1}{\xi(t)}=\O{\epsilon_0+\left(\epsilon_a\epsilon_x\right)^\frac{1}{2}\epsilon_0^\frac{1}{2}+\epsilon_x+\epsilon_t}
\end{align*}
holds for all times $t\le T=C\left(\epsilon_a\epsilon_x\right)^{-1}$.
\end{prop}
\begin{proof}
Lemma \ref{Lemma:AlmostConservationOfLyapunov} implies
\begin{align*}
\left|\dd{t} \Gamma_c(t)\right|\le
C\left(\epsilon_a^2\epsilon_x^3+\left(\epsilon_a\epsilon_x\epsilon_t+\epsilon_a\epsilon_x^2\right)\HTNorm{\xi}+\epsilon_a\epsilon_x\HTNorm{\xi}^2+\HTNorm{\xi}^4\right)
\end{align*}
for some constant $C>0$ where $\HTNorm{\xi}:=\sup_{0\le t\le
T}\HsNorm{1}{\xi}$. Integrating over $[0,T]$ gives an upper bound
on $\Gamma_c(T)$. A lower bound is obtained by
expanding $\Lambda_{ca}(Q_{ca}+\xi)$ to quadratic order then
using Proposition \ref{Prop:Positivity}, the third estimate of
Lemma \ref{Lemma:NonlinearEstimates} and
$V'(a)\ip{\xi}{(x-a)Q_{ca}}=\O{\epsilon_a\epsilon_x\HsNorm{1}{\xi}}$. We obtain,
after setting all non-essential constants to one,
\begin{align*}
\HTNorm{\xi}^2-\HTNorm{\xi}^3-\epsilon_a\epsilon_x\HTNorm{\xi}\le
\Gamma_c(T)\le
|\Gamma_c(0)|+\left(\epsilon_a^2\epsilon_x^3+\left(\epsilon_a\epsilon_x\epsilon_t+\epsilon_a\epsilon_x^2\right)\HTNorm{\xi}+\epsilon_a\epsilon_x\HTNorm{\xi}^2+\HTNorm{\xi}^4\right)
T
\end{align*}
for all $T>0$. Take $T=\O{\left(\epsilon_a\epsilon_x\right)^{-1}}$. Then, under the
smallness assumption $\HsNorm{1}{\xi}\ll(\epsilon_a\epsilon_x)^\frac{1}{2}$,
\begin{align*}
\HsNorm{1}{\xi}=\O{|\Gamma_c(0)|^\frac{1}{2}+\epsilon_x+\epsilon_t}.
\end{align*}
The initial value of the Lyapunov functional
$\Gamma_c(0)$ can be bounded by the $\Hs{1}$ norm of
the initial fluctuation $\HsNorm{1}{\xi(0)}\le C\epsilon_0$ (recall
that $\epsilon_0:=\inf_{Q_{ca}\in M_s}\HsNorm{1}{u_0-Q_{ca}}$. Indeed,
Taylor expanding $\Lambda_{ca}(Q_{ca}+\xi)$ to second order in
$\xi$ and using the third estimate in Lemma
\ref{Lemma:NonlinearEstimates} gives
$|\Gamma_c(0)|=\O{\epsilon_0^2+\epsilon_a\epsilon_x\epsilon_0}$ if
$\epsilon_0\ll1$. To complete the proof we take $\epsilon_x$ and
$\epsilon_0$ small enough so that $\HsNorm{1}{\xi(t)}$ is
sufficiently small for Lemma
\ref{Lemma:AlmostConservationOfLyapunov} to hold.
\end{proof}
We now prove the main theorem.
\begin{proof}[Proof of Theorem \ref{MainThm}]
By our choice $\epsilon_0<\varepsilon_0$, there is a (maximal)
time $T_0$ such that the solution $u$ in
\eqref{Eqn:KdvGeneralizedWithPotential} is in $U_{\varepsilon_0}$
for time $t\le T_0$. Hence decomposition
\eqref{EquationWithUErrorQDecomposition} with
\eqref{Cond:Orthogonality}, and Proposition
\ref{Prop:FluctuationBound} are valid for the solution $u$ over
this time and imply the statements of the main theorem. In
particular $\HsNorm{1}{\xi(t)}=
\O{\epsilon_0+\left(\epsilon_a\epsilon_x\right)^\frac{1}{2}\epsilon_0^\frac{1}{2}+\epsilon_x+\epsilon_t}$
for times $t\le \min\{T_0, T\}$. Taking
$\epsilon_0+\left(\epsilon_a\epsilon_x\right)^\frac{1}{2}\epsilon_0^\frac{1}{2}+\epsilon_x+\epsilon_t\ll\varepsilon_0$,
we must have $t\le T$ by maximality of the time $T_0$.
\end{proof}
|
{
"timestamp": "2005-03-08T18:12:31",
"yymm": "0503",
"arxiv_id": "math-ph/0503016",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503016"
}
|
\section{Introduction}
Human altruistic behavior is a long-standing problem in
evolutionary theory, as first realized by Darwin himself:
\begin{quotation}
He who was ready to sacrifice his life (\dots) rather than betray his comrades,
would often leave no offspring to inherit his noble nature\dots Therefore,
it seems scarcely possible (\dots) that the number of men gifted with such virtues
(\dots) would be increased by natural selection, that is, by the survival of
the fittest.
\cite{Darwin-Descent}
\end{quotation}
At the crux of the problem lies the fact that Darwin developed his
theory assuming that natural selection acts exclusively on individuals
only. On this grounds, he could not possibly understand altruistic behavior
in humans, i.e., acts that decrease the actor's fitness while increasing that
of others. Reluctantly, he had to call for selection at group level:
\begin{quotation}
A man who was not impelled by any deep, instinctive feeling, to sacrifice
his life for the good of others, yet was roused to such actions by a sense
of glory, would by his example excite the same wish for glory in other men,
and would strengthen by exercise the noble feeling of admiration. He might
thus do far more good to his tribe than by begetting offsprings with a
tendency to inherit his own high character.
\cite{Darwin-Descent}
\end{quotation}
In fact, human behavior is unique in nature. Indeed, altruism or
cooperative behavior exists in other species, but it can be
understood in terms of genetic relatedness (kin selection,
introduced by Hamilton \cite{Hamilton:1964}) or of repeated
interactions (as proposed by Trivers \cite{Trivers:1971}).
However, human cooperation extends to genetically unrelated
individuals and to large groups, characteristics that cannot
be understood within those schemes. Subsequently, a number of
theories based on group and/or cultural evolution have been put
forward in order to explain altruism (see
\cite{Hammerstein} for a review).
\section{The Ultimatum game}
In order to address quantitatively the issues above, behavioral
researchers use evolutionary game theory \cite{Gintisbook,Camerer} to
design experiments that try to find the influence of different
factors.
In this paper, we analyze this problem in the context of a
specific set of such experiments, related to the so called
Ultimatum game \cite{Guth:1982,Henrich}.
In the Ultimatum game,
under conditions of anonymity, two players are shown a sum
of money, say 100 \EUR{}. One of the players, the ``proposer'',
is instructed to offer any amount, from 1 \EUR{} to 100 \EUR{},
to the other, the ``responder''. The proposer can make only one
offer, which the responder can accept or reject. If the offer is
accepted, the money is shared accordingly; if rejected, both
players receive nothing. Since the game is played only once
(no repeated interactions) and anonymously (no reputation gain;
for more on explanations of altruism relying on reputation
see \cite{Nowak-Sigmund}),
a self-interested responder will accept any amount of money
offered. Therefore, self-interested proposers will offer the
minimum possible amount, 1 \EUR{}, which will be accepted.
Notwithstanding,
in actual Ultimatum game experiments with human subjects,
average offers do not even approximate the self-interested prediction.
Generally speaking, proposers offer respondents very substantial
amounts (50 \% being a typical modal offer) and respondents
frequently reject offers below 30 \% \cite{Fehr2003}. Most of the
experiments have been carried out with university students in
western countries, showing a large degree of individual variability
but a striking uniformity between groups in average behavior.
A large study in 15 small-scale societies \cite{Henrich}
found that, in all
cases, respondents or proposers behave in a reciprocal manner.
Furthermore, the behavioral variability across groups was much
larger than previously observed: while mean offers in the case
of university students are in the range 43\%-48\%, in the
cross-cultural study they ranged from 26\% to 58\%.
The fact that indirect reciprocity is excluded by the anonymity
condition and that interactions
are one-shot (i.e., repeated interaction does not apply)
allows one to interpret rejections in terms of the so-called
strong reciprocity \cite{Gintis2000,Fehr2002}.
This amounts to considering that
these behaviors are truly altruistic, i.e., that
they are costly for the individual performing them in so far as
they do not result in direct or indirect benefit. As a consequence,
we return to our evolutionary puzzle: The negative effects of
altruistic acts must decrease the altruist's fitness as compared to
that of the recipients of the benefit, ultimately leading to
the extinction of altruists. Indeed, standard evolutionary game
theory arguments applied to the Ultimatum game lead to the expectation
that in a mixed population, punishers (individuals
who reject low offers) have less chance to survive
than rational
players (indivuals who accept any offer) and eventually disappear.
In the remainder of the paper, we will show that this conclusion
depends on the dynamics, and that different dynamics leads to
the survival of punishers through fluctuations.
\section{The model}
We consider a population of $N$ players (agents) of the Ultimatum game
with a fixed sum of money $M$ per game.
Random pairs of players are chosen, of which one is the proposer
and another one is the respondent. In its simplest version,
we will assume that
players are capable of other-regarding behavior (empathy); consequently,
in order to optimize their gain,
proposers offer the minimum amount of money
that they would accept. Every agent has her own, fixed
acceptance threshold, $1\leq t_i\leq M$ ($t_i$ are always integer
numbers for simplicity). Agents have only one strategy:
respondents reject any offer
smaller than their own acceptance threshold, and
accept offers otherwise.
Money
shared as a consequence of accepted offers accumulates to the
capital of each of the involved players. As our main aim is to
study selection acting on modified descendants, hereafter we interpret this
capital as `fitness'
(here used in a loose, Darwinian sense, not in the more
restrictive one of reproductive rate).
After $s$ games,
the agent with the overall minimum fitness is
removed (randomly picked if there are several)
and a new agent is introduced by duplicating that
with the maximum fitness, i.e., with the same threshold and the
same fitness (again randomly picked if there are
several). Mutation is introduced in the duplication process by
allowing changes of $\pm 1$ in the acceptance threshold of the
newly generated player with probability 1/3 each. Agents
have no memory (i.e., interactions are one-shot) and no information
about other agents (i.e., no reputation gains are possible).
We stress that the model is dramatically simplified; however, we
have studied more complicated versions (including separate acceptance
and offer thresholds) and the results are similar to the ones we
discuss below. Another factor we have considered is smaller mutation
rates, again without qualitative changes in the result. Therefore,
for the sake of brevity we concentrate here on the simple model
summarized above and refer the reader to \cite{Cuesta-Sanchez}
for a more detailed analysis including those other versions.
\section{Results}
Figure \ref{figure1} shows the typical outcome of simulations of our model.
As we can see, the mean acceptance threshold rapidly evolves towards
values around 40\%, while the whole
distribution of thresholds converges to a peaked function, with
the range of acceptance thresholds for the agents covering about a
10\% of the available ones.
These are values compatible with the experimental results discussed
above. The
mean acceptance threshold fluctuates
during the length of the simulation, never reaching a stationary value
for the durations we have explored. The width of the peak fluctuates
as well, but in a much smaller scale than the position.
The fluctuations are larger for smaller values of $s$, and when $s$
becomes of the order of $N$ or larger, the evolution of the mean
acceptance threshold is very smooth. This is a crucial point and will
be discussed in more detail below.
Importantly, the typical evolution
we are describing does not depend on the initial condition. In particular,
a population consisting solely of self-interested agents, i.e., all
initial thresholds are set to $t_i=1$, evolves in the same fashion.
Indeed, the distributions shown in the left panel of Figure
\ref{figure1} have been obtained with such an initial condition,
and it can be clearly observed that self-interested agents disappear
in the early stages of the evolution.
The number of players and the value $M$ of the capital at stake in every
game are not important either, and increasing $M$ only leads
to a higher resolution of the threshold distribution function.
\begin{figure}
\label{figure1}
\includegraphics[height=.2\textheight]{granada1.eps}
\hspace*{5mm}
\includegraphics[height=.2\textheight]{granada2.eps}
\caption{Left: mean acceptance threshold as a function of simulation
time. Initial condition is that all agents have $t_i=1$.
Right: acceptance threshold distribution after $10^8$ games.
Initial condition is that all agents have uniformly distributed, random $t_i$.
In both cases, $s$ is as indicated from the plot.}
\end{figure}
\section{Discussion}
As we mentioned in the preceding section,
we have observed that taking very large values for $s$ or, strictly
speaking, considering the limit $s/N\to\infty$, does lead to different
results. In this respect, let us recall
previous studies of the Ultimatum game by Page and
Nowak \cite{PageNowak00,PageNowak02}.
The model introduced in those works has a dynamics completely
different from ours: following standard evolutionary game theory,
every player plays every other one in both roles (proponent and
respondent), and afterwards players reproduce with probability
proportional to their payoff (which is fitness in the reproductive
sense). Simulations and adaptive dynamics equations show then that the
population ends up composed by players with fair (50\%) thresholds.
This is different from our observations, in which we hardly ever
reach an equilibrium (only for large $s$) and even then equilibria
set up at values different from the fair share. The reason for this
difference is that the Page-Nowak model dynamics describes the
$s/N\to\infty$ limit of our model, in which between death-reproduction
events the time average gain all players obtain is
the mean payoff with high accuracy.
We thus see that our model is more general
because it has one free parameter, $s$, that allows selecting different
regimes whereas the Page-Nowak dynamics is only one limiting case.
Those different regimes are what we have described as fluctuation dominated
(when $s/N$ is finite and not too large) and the regime analyzed by
Page and Nowak (when $s/N\to\infty$).
This amounts to saying that by varying $s$ we can
study regimes far from the standard evolutionary game theory
limit. As a result, we find a variability of outcomes for the
acceptance threshold consistent with the observations in real
human societies %
\cite{Henrich,Fehr2003}.
In fact, fluctuations due to the finite number of games are at the
heart of our results. Among the results summarized above, the
evolution of a population entirely
formed by self-interested players into a diversified population with a
large majority of altruists is the most relevant and surprising one.
We will now argue that the underlying reason for this is precisely
the presence of
fluctuations in our model. For the sake of definiteness, let us
consider the case $s=1$ (agent replacement takes place after every game)
although the discussion applies to larger (but finite) values of $s$ as
well. After one or more games, a mutation event will take place
and a ``weak altruistic punisher'' (an agent with $t_i=2$) will appear
in the population,
with a fitness inherited from its ancestor. For this new agent to be
removed at the next iteration so that the population reverts to its
uniform state, our model rules imply that this agent has to have
the lowest fitness, that is the only one with that value of fitness,
{\em and also} that it does not play as a proposer in
the next game (if playing as a responder the agent will earn nothing
because of her threshold). In any other event this altruistic punisher
will survive at
least one cycle, in which an additional one can appear by mutation.
It is thus clear that fluctuations indeed help altruists to take
over: As soon as a few altruists are present in the population, it is
easy to see analytically that they will survive and proliferate even
in the limit $s/N\to\infty$.
\section{The Stag-Hunt game}
This far, we have shown that considering that players play a finite
number of games between death-birth events in the Ultimatum game leads
to results unexpected from standard evolutionary game theory arguments.
Hence, the question arises as to whether this is a consequence of the
many strategies available in the Ultimatum game (as many as possible
values for $t_i$, 100 with our choice for the parameters) or, on the
contrary, it is a general phenomenon. To show that the latter is the
case, we have considered a completely different, much simpler kind of
game: the so-called Stag-Hunt game \cite{Gintisbook,Camerer,Henrich}.
In this game, two hunters cooperate in hunting for stag, which is the
most profitable option;
however, hunting a stag is impossible unless both work together, and
they have the option of hunting for rabbit, less profitable, but
with sure earnings. This is reflected in the following payoff matrix
(C stands for cooperation in hunting stag, D stands for defection and
hunting rabbit alone):
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
\mbox{ } & C & D \\ \hline
\hline
C & 6 & 0 \\ \hline
D & 5 & 1 \\ \hline
\end{tabular}
\end{center}
This game belongs in the class of coordination games: In the
language of game theory, it has two
Nash equilibria, (C,C) and (D,D), and the players would like
to coordinate in choosing the first one (so called payoff-dominant).
However, the second one is a safer choice
because it has the largest guaranteed minimum payoff
(so called risk-dominant).
We have been working on the evolutionary dynamics of this game and,
specifically, on the equilibrium selection problem
\cite{todos}. For this example, we have chosen the dynamics given
by the Moran process \cite{Nowak2004}, in which after $s$ games
an agent is duplicated with probability proportional to the fitness
accumulated during the $s$ games, and another one is killed
randomly. With such a simple dynamics, it is an elementary
exercise to show that, in the limit $s/N\to\infty$, the whole
population becomes C (resp.\ D) strategists if the initial density
of C strategists is larger (resp.\ smaller) than 1/2. As Fig.\
\ref{figure2} shows, simulation results for finite
$s$ are largely different from that analytical prediction:
\begin{figure}
\label{figure2}
\includegraphics[height=.2\textheight]{altruism_100.eps}
\hspace*{5mm}
\includegraphics[height=.2\textheight]{altruism_1000.eps}
\caption{Fraction of games that end up with a cooperator-only
population vs density of cooperators in the initial state
for $N=100$ (left) and $N=1000$ (right) agents playing
the Stag-Hunt game. Results are obtained from simulations of the Stag-Hunt
game with the Moran dynamics, and for every initial density the final
density is averaged over 100 games. Values of $s$ are as indicated in the plot.}
\end{figure}
Indeed, we see that for cooperators to prevail in the final state, an initial
density larger than 1/2 is needed. In particular, for $s=1$, all agents become
defectors except for initial densities close to 1 in the case $N=100$ (left
panel), and for
all initial densities for $N=1000$ (right panel) or larger (not shown). The
plots also show that larger populations lead to better statistics (meaning that
curves are smoother and less noisy; it is evident that $\tilde{x}$ has a smaller
variance for larger populations), and the
trend upon increasing $N$ is that the curves become step functions (as should
be for an infinite population). Importantly, the effect, namely that the basin of
attraction of the (D,D) equilibrium is enlarged for finite $s$, persists even
in the infinite population limit. In addition, it is also robust upon changes
in the dynamics: we have verified that choosing the agent to be eliminated with
probability inversely proportional to the agent's fitness leads to qualitatively
similar result. We are thus faced with another clear-cut manifestation of the
relevance of taking the limit of infinite games before the dynamics occurs or,
on the contrary, sticking to a finite number of games. Once again, we stress that
the setup is completely different from the Ultimatum game and, as a consequence,
we claim that this kind of phenomena is generic and should be observed in
many other problems.
\section{Conclusions}
In this paper, we have shown that altruistic-like behavior, specifically,
altruistic punishment, may arise by means of exclusive individual selection
even in the absence of repeated interactions and reputation gains. Our
conclusion is important in so far as it is generally believed that
some kind of group selection is needed to understand the observed human
behavior. The reason for that is that game theoretical arguments
apparently show that altruists are at disadvantage with respect to
selfish individual. In this respect, another relevant conclusion
of the present work is that
perspectives and approaches alternative to standard evolutionary
game theory may be needed in order to understand paradoxical
features such as the appearance of altruistic punishment.
As additional evidence supporting this claim,
we have briefly discussed, in the context of the much simpler problem
of the stag-hunt game, that
equilibrium selection is indeed dramatically modified by taking
into account a finite number of games. Therefore, we conclude that
the dynamics postulated for a particular application of evolutionary
game theory must be closely related to the specific problem as the
outcome can be completely different depending on the dynamics.
\begin{theacknowledgments}
AS thanks the organizers of the 8th Granada Seminar, specially Joaqu\'\i n
Marro, for the opportunity to present these results and to discuss with
the Seminar attendees.
We acknowledge financial support from Ministerio de Ciencia y Tecnolog\'\i a
(Spain) through grants BFM2003-07749-C05-01 (AS) and BFM2003-0180 (JAC).
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
{
"timestamp": "2005-03-16T20:03:02",
"yymm": "0503",
"arxiv_id": "q-bio/0503024",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503024"
}
|
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\begin{document}
\author{{\sc Partha Sarathi Chakraborty} and
{\sc Arupkumar Pal}}
\title{Equivariant spectral triples
for $SU_q(\ell+1)$ and the odd dimensional
quantum spheres}
\maketitle
\begin{abstract}
We formulate the notion of equivariance of an operator
with respect to a covariant representation of a $C^*$-dynamical system.
We then use a combinatorial technique used by the authors earlier
in characterizing spectral triples for $SU_q(2)$ to investigate
equivariant spectral triples for two classes of spaces:
the quantum groups $SU_q(\ell+1)$ for $\ell>1$, and
the odd dimensional quantum spheres $S_q^{2\ell+1}$
of Vaksman \& Soibelman. In the former case,
a precise characterization of the sign and the singular
values of an
equivariant Dirac operator acting on the $L_2$ space is
obtained. Using this, we then exhibit
equivariant Dirac operators with nontrivial sign
on direct sums of multiple copies of the $L_2$ space.
In the latter case, viewing $S_q^{2\ell+1}$ as a homogeneous space
for $SU_q(\ell+1)$, we give a complete characterization
of equivariant Dirac operators, and also produce an
optimal family of spectral triples with nontrivial
$K$-homology class.
\end{abstract}
{\bf AMS Subject Classification No.:} {\large 58}B{\large 34}, {\large
46}L{\large 87}, {\large
19}K{\large 33}\\
{\bf Keywords.} Spectral triples, noncommutative geometry,
quantum group.
\newsection{Introduction}
Groups have always played
a very crucial role in the study of geometry of a space, mainly as
objects that govern the symmetry of the space. One would
expect the same in noncommutative geometry also.
Moreover, since one now deals with a larger class of
spaces, mainly noncommutative ones, it is natural to expect that one
would require a larger class, Hopf algebras or the quantum groups, to
play a similar role. In the classical case, groups which govern
symmetry are themselves nice geometric objects. Here we want to look
at quantum groups from the same angle.
In a previous paper~(\cite{c-p1}), the authors treated the case of the quantum $SU(2)$
group and found a family of
spectral triples acting on its $L_2$-space
that are equivariant with respect to its natural (co)action.
This family is optimal, in the sense
that given any nontrivial equivariant Dirac operator $D$ acting
on the $L_2$ space, there exists a Dirac operator $\widetilde{D}$
belonging to this family such that
$\mbox{sign\,}D$ is a compact perturbation of $\mbox{sign\,}\widetilde{D}$
and there exist reals $a$ and $b$ such that
\[
|D| \leq a + b|\widetilde{D}|.
\]
A generic triple from this family,
that is also a generator of the $K$-homology group,
was analysed by Connes in \cite{co3}
where he used the general theory developed by him and Moscovici
(\cite{c-m}) to make elaborate
computations and finally ended up with a local index formula.
One beautiful and somewhat surprising observation
in his paper was that the description of the cocycle
given by the difference between the character of the
triple and the cocycle for which index formula was given
involved the Dedekind eta function.
This gave further impetus to the construction of
spectral triples for quantum groups and their homogeneous
spaces (\cite{d-s}, \cite{d-l-p-s}, \cite{d-l-s-s-v},
\cite{h-l}, \cite{kr}, \cite{s-d-l-s-v}).
It should perhaps be pointed out here
that the construction by
Kr\"{a}hmer~(\cite{kr}) is algebraic in nature and does not
address the crucial analytic issues involved in the
definition of a spectral triple.
The construction by Hawkins \& Landi~(\cite{h-l}) on the other
hand does not deal with equivariance; and more crucially,
they restrict themselves to the construction of bounded
Kasparov modules. But in Noncommutative geometry,
spectral triples or the unbounded Kasparov modules
are key ingredients, as they work as a looking glass allowing
one to distinguish between continuous and smooth functions.
Our aim in the present paper is to look for higher
dimensional counterparts of the spectral triples
found in \cite{c-p1}.
We first formulate precisely what
one means by an equivariant spectral triple in a general
set up (this is already implicit in \cite{c-p1})
and then study equivariant Dirac operators
for two classes of spaces, both of which can
be thaught of as higher dimensional analogues
of $SU_q(2)$ which was worked out earlier.
First, we analyse equivariant Dirac operators
acting on the $L_2$-spaces of the groups $SU_q(\ell+1)$.
We derive a precise expression for the singular
values of an equivariant Dirac operator, and show that
a Dirac operator with these singular values will
have the correct summability property.
We also show that for $\ell>1$,
an equivariant Dirac operator acting on $L_2(G)$
have to have trivial sign.
Thus for $\ell>1$, one would be forced to bring in multiplicity
when looking for equivariant Dirac operators
with nontrivial sign.
Using this observation, we then exhibit a
family of equivariant Dirac operators
acting on direct sums of multiple copies of
the $L_2$ space and having nontrivial sign.
Whether these Dirac operators
have nontrivial $K$-homology class is still not known.
In the last section, we take up the odd dimensional
quantum spheres $S_q^{2\ell+1}$. In this case, the outcome
turns out to be more satisfactory. After characterizing
the sign and the singular values of Dirac operators on
$L_2(S_q^{2\ell+1})$ equivariant under the action of
the group $SU_q(\ell+1)$, we produce, just like
in the $SU_q(2)$ case, an optimum family of
nontrivial equivariant Dirac operators that are
$(2\ell+1)$-summable.
The paper is organised as follows.
In the next section, we will recall from~\cite{c-p0} the
combinatorial method that was earlier used implicitly
in \cite{c-p1} and \cite{c-p2}.
In section~3, we formulate the notion of equivariance.
This has been done using the quantum group at
the function algebra level rather than passing on
to the quantum universal envelopping algebra level.
In section~4, we briefly recall the quantum group
$SU_q(\ell+1)$ and its representation theory.
In particular, we describe a nice basis for
the $L_2$ space and study the Clebsch-Gordon
coefficients. These are used in section~5 to describe the
action by left multiplication on the $L_2$ space explicitly.
In section~6, we write down the conditions
coming from the boundedness of commutators with $D$.
In sections~7 and 8, we analyze the equivariant Dirac operators
for $SU_q(\ell+1)$. First we give a precise characterization of
the singular values in section~7,
and then a characterization of the sign in section~8.
In section~9, we deal with the odd dimensional quantum spheres.
\newsection{The general scheme}
Let us recall the combinatorial set up from~\cite{c-p0}.
Suppose $\mathcal{H}$ is a Hilbert space,
and $D$ is a self-adjoint operator on $\mathcal{H}$ with compact resolvent.
Then $D$ admits a spectral resolution $\sum_{\gamma\in\Gamma} d_\gamma
P_\gamma$, where the $d_\gamma$'s are all distinct and each $P_\gamma$
is a finite dimensional projection. Assume now onward that all the
$d_\gamma$'s are nonzero. Let $c$ be a positive real. Let
us define a graph $\mathcal{G}_c$ as follows: take the vertex set $V$ to
be $\Gamma$. Connect two vertices $\gamma$ and $\gamma'$ by an edge
if $|d_\gamma-d_{\gamma'}|<c$. Let $V^+=\{\gamma\in V: d_\gamma>0\}$ and
$V^-=\{\gamma\in V: d_\gamma<0\}$. This will give us a partition of
$V$.
This partition has the following important property:
there does not exist infinite number of disjoint
paths each going from a point in $V^+$
to a point in $V^-$. Here disjoint paths mean paths
for which the set of vertices of one does not
intersect the set of vertices
of the other.
This is easy to see, because
if there is a path from $\gamma$ to $\delta$
and $d_\gamma>0$, $d_\delta<0$, then for some $\alpha$ on the path,
one must have $d_\alpha\in[-c,c]$.
Since the paths are disjoint, it
would contradict the compact resolvent condition.
We will call such a partition a sign-determining partition.
We will use this knowledge about the graph.
We start with an equivariant operator that is self-adjoint
and has discrete spectrum. Equivariance will give us
an idea about the spectral resolution
$\sum_{\gamma\in\Gamma}d_\gamma P_\gamma$.
Next we use the action of the algebra elements on the basis
elements of $\mathcal{H}$ and the boundedness of their
commutators with $D$.
This gives certain growth restrictions
on the $d_\gamma$'s. These will give us some information
about the edges in the graph. We exploit this knowledge
to characterize those partitions $(V_1,V_2)$ of the vertex set
that are sign-determining, i.\ e.\ do not admit any infinite ladder.
The sign of the operator $D$ must be of the form
$\sum_{\gamma\in V_1}P_\gamma-\sum_{\gamma\in V_2}P_\gamma$
where $(V_1,V_2)$ is a sign-determining partition.
Of course, for a given $c$, the graph
$\mathcal{G}_c$ may have no edges, or too few edges (if the singular values
of $D$ happen to grow too fast), in which case, we will
be left with too many sign-determining
partitions.
Fortunately, the operators we are interested in are meant to be the
Dirac operators of some commutative/noncommutative manifold. Therefore
the singular values of $D$ will grow at the rate of $O(n^{1/d})$ for
some $d\geq 1$. So one can choose a large enough $c$ and work with
the graph $\mathcal{G}_c$.
In other words, we would like to characterize
those partitions that are sign-determining
for all sufficiently large values of $c$.
\newsection{Equivariance}
Suppose $G$ is a compact group, quantum or classical,
and $\mathcal{A}$ is a unital $C^*$-algebra. Assume that
$G$ has an action on $\mathcal{A}$ given by
$\tau:\mathcal{A}\rightarrow\mathcal{A}\otimes C(G)$,
so that $(\mbox{id}\otimes\Delta)\tau=(\tau\otimes\mbox{id})\tau$,
$\Delta$ being the coproduct.
In other words, we have a $C^*$-dynamical system $(\mathcal{A},G,\tau)$.
Our goal is to study spectral triples for $\mathcal{A}$
equivariant under this action.
Let us first say what we mean by `equivariant' here.
A covariant representation $(\pi,u)$
of $(\mathcal{A},G,\tau)$ consists of
a unital *-representation $\pi:\mathcal{A}\rightarrow\mathcal{L}(\mathcal{H})$,
a unitary representation $u$ of $G$ on $\mathcal{H}$, i.e.\
a unitary element of the multiplier algebra $M(\mathcal{K}(\mathcal{H})\otimes C(G))$
such that they obey the condition
$(\pi\otimes\mbox{id})\tau(a)=u(\pi(a)\otimes I)u^*$ for all $a\in\mathcal{A}$.
\begin{dfn}\rm
Suppose $(\mathcal{A}, G,\tau)$ is a $C^*$-dynamical system.
An operator $D$ acting on a Hilbert space $\mathcal{H}$
is said to be \textbf{equivariant} with respect to a covariant
representation $(\pi,u)$ of the system if
$D\otimes I$ commutes with $u$.
\end{dfn}
Since the operator $D$ is self-adjoint with compact resolvent, it will
admit a spectral resolution $\sum_\lambda d_\lambda P_\lambda$, where
the $d_\lambda$'s are distinct and each $P_\lambda$ is finite
dimensional. Also, $D$ has been assumed to be equivariant --- so that
the $P_\lambda$'s commute with $u$ (to be precise, the
$(P_\lambda\otimes I)$'s do), i.e.\ $u$ keeps each $P_\lambda\mathcal{H}$
invariant. As $G$ is compact, each $P_\lambda\mathcal{H}$ will decompose
further as $\oplus_\mu P_{\lambda\mu}\mathcal{H}$ such that the restriction
of $u$ to each $P_{\lambda\mu}$ is irreducible. In other words, one
can now write $D$ in the form $\sum_{\gamma\in\Gamma}d_\gamma
P_\gamma$ for some index set $\Gamma$ and a family of finite
dimensional projections $P_\gamma$ such that each $P_\gamma$ commutes
with $u$ and the restriction of $u$ to each $P_\gamma$ is irreducible.
In this paper, we will deal with two cases,
the group in question in both cases will be $G=SU_q(\ell+1)$.
The $C^*$-algebra $\mathcal{A}$ on which the group
acts will be $C(SU_q(\ell+1))$ in one case and
$C(S_q^{2\ell+1})$ in the other.
Let us discuss the first case a little here.
The action $\tau$ here will be the natural action coming from the
coproduct, $\mathcal{H}$ is $L_2(G)$, $\pi$ is the
representation of $\mathcal{A}=C(SU_q(\ell+1))$
on $\mathcal{H}$ by left multiplication, and $u$ is the right regular
representation. Structure of the regular representation of a compact
(quantum) group along with the remarks made above tell us the
following. Let $\Lambda$ be the set of unitary irreducible
representation-types for $G$. Then $\mathcal{H}$ decomposes as
$\oplus_{\lambda\in\Lambda}\mathcal{H}_\lambda$, where the restriction of $u$
to $\mathcal{H}_\lambda$ is equivalent to $\mbox{dim}\,\lambda$ copies of the
irreducible $\lambda$, and also that $D$ respects this decomposition.
Further, restriction of $D$ to $\mathcal{H}_\lambda$ is of the form
$\sum_{\mu}d_{\lambda\mu}P_{\lambda\mu}$, $u$ commutes with each of
these $P_{\lambda\mu}$'s, and the restriction of $u$ to
$P_{\lambda\mu}\mathcal{H}$ is equivalent to $\lambda$. Let $N_\lambda$ be
any set with $|N_\lambda|=\mbox{dim}\,\lambda$. One can then choose
an orthonormal basis $\{e^\lambda_{ij}:i,j\in N_\lambda\}$ such that
the spaces $P_{\lambda\mu}\mathcal{H}$ are precisely
$\mbox{span}\,\{e^\lambda_{ij}:j\in N_\lambda\}$ for distinct values
of $i\in N_\lambda$. Since $D$ is of the form $\sum_\lambda\sum_\mu
d_{\lambda\mu}P_{\lambda\mu}$, in this system of bases, $D$ will look
like $e^\lambda_{ij}\mapsto d(\lambda,i)e^\lambda_{ij}$. In what
follows, we will make a special choice of $N_\lambda$, which will make
the combinatorial analysis very convenient.
\newsection{Preliminaries on $SU_q(\ell+1)$}
Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $\ell$.
let $(\!(a_{ij})\!)$ be the associated Cartan matrix,
$q$ be a real number lying in the interval $(0,1)$
and let $q_i=q^{(\alpha_i,\alpha_i)/2}$, where $\alpha_i$'s are the simple roots
of $\mathfrak{g}$.
Then the quantised universal envelopping algebra (QUEA)
$U_q(\mathfrak{g})$ is the algebra
generated by $E_i$, $F_i$, $K_i$ and $K_i^{-1}$, $i=1,\ldots,\ell$, satisfying the
following relations
\begin{displaymath}
K_iK_j=K_jK_i,\quad K_iK_i^{-1}=K_i^{-1}K_i=1,
\end{displaymath}
\begin{displaymath}
K_iE_jK_i^{-1}=q_i^{\frac{1}{2} a_{ij}}E_j,\quad
K_iF_jK_i^{-1}=q_i^{-\frac{1}{2} a_{ij}}F_j,
\end{displaymath}
\begin{displaymath}
E_iF_j-F_jE_i=\delta_{ij}\frac{K_i^2-K_i^{-2}}{q_i-q_i^{-1}},
\end{displaymath}
\begin{displaymath}
\sum_{r=0}^{1-a_{ij}}(-1)^r{{1-a_{ij}}\choose r}_{q_i}
E_i^{1-a_{ij}-r}E_jE_i^r =0 \quad\forall\, i\neq j,
\end{displaymath}
\begin{displaymath}
\sum_{r=0}^{1-a_{ij}}(-1)^r{{1-a_{ij}}\choose r}_{q_i}
F_i^{1-a_{ij}-r}F_jF_i^r =0\quad \forall\, i\neq j,
\end{displaymath}
where ${n\choose r}_q$ denote the $q$-binomial coefficients.
Hopf *-structure comes from the following maps:
\[
\Delta(K_i)=K_i\otimes K_i,\quad \Delta(K_i^{-1})=K_i^{-1}\otimes K_i^{-1},
\]
\[
\Delta(E_i)=E_i\otimes K_i + K_i^{-1}\otimes E_i,\quad
\Delta(F_i)=F_i\otimes K_i + K_i^{-1}\otimes F_i,
\]
\[
\epsilon(K_i)=1,\quad \epsilon(E_i)=0=\epsilon(F_i),
\]
\[
S((K_i)=K_i^{-1},\quad S(E_i)=-q_iE_i,\quad S(F_i)=-q_i^{-1}F_i,
\]
\[
K_i^*=K_i,\quad E_i^*=-q_i^{-1}F_i,\quad F_i^*=-q_iE_i.
\]
In the type A case, the associated Cartan matrix is given by
\[
a_{ij}=\cases{2& if $i=j$,\cr
-1 & if $i=j\pm1$,\cr
0 & otherwise,}
\]
and $(\alpha_i,\alpha_i)=2$ so that $q_i=q$ for all $i$.
The QUEA in this case is denoted by $u_q(su(\ell+1))$.
Take the collection of matrix entries of all finite-dimensional
unitarizable $u_q(su(\ell+1))$-modules. The algebra generated by these
gets a natural Hopf*-structure as the dual of $u_q(su(\ell+1))$. One
can also put a natural $C^*$-norm on this. Upon completion with
respect to this norm, one gets a unital $C^*$-algebra that plays the
role of the algebra of continuous functions on $SU_q(\ell+1)$. For a
detailed account of this, refer to chapter~3, \cite{ko-so}. In
\cite{w}, Woronowicz gave a different description of this
$C^*$-algebra. which was later shown by Rosso (\cite{r}) to be
equivalent to the earlier one.
For remainder of this article, we will take $G$ to be $SU_q(\ell+1)$
and $\mathcal{A}$ will be the $C^*$-algebra of continuous functions on $G$.
\paragraph{Gelfand-Tsetlin tableaux.}
Irreducible unitary representations of the group
$SU_q(\ell+1)$ are indexed by
Young tableaux $\lambda=(\lambda_1,\ldots,\lambda_{\ell+1})$,
where $\lambda_i$'s are nonnegative integers,
$\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_{\ell+1}$
(Theorem~1.5, \cite{w}).
Write $\mathcal{H}_\lambda$ for the Hilbert space where
the irreducible $\lambda$ acts.
There are various ways of indexing the basis elements
of $\mathcal{H}_\lambda$. The one we will use is due to Gelfand
and Tsetlin.
According to their prescription, basis elements for
$\mathcal{H}_\lambda$ are parametrized by arrays of the form
\[
\mathbf{r}=\left(\matrix{r_{11}&r_{12} &\cdots&r_{1,\ell}&r_{1,\ell+1}\cr
r_{21}&r_{22}&\cdots &r_{2,\ell}&\cr
&\cdots&&&\cr
r_{\ell,1}&r_{\ell,2}&&&\cr
r_{\ell+1,1}&&&&}\right),
\]
where $r_{ij}$'s are integers satisfying
$r_{1j}=\lambda_j$ for $j=1,\ldots,\ell+1$,
$r_{ij}\geq r_{i+1,j}\geq r_{i,j+1}\geq 0$ for all $i$, $j$.
Such arrays are known as Gelfand-Tsetlin tableaux, to be abreviated
as GT tableaux for the rest of this section.
For a GT tableaux $\mathbf{r}$, the symbol $\mathbf{r}_{i\cdot}$ will denote its
$i$\raisebox{.4ex}{th} row.
It is well-known that two representations indexed respectively
by $\lambda$ and $\lambda'$ are equivalent if and only if
$\lambda_j-\lambda_j^\prime$ is independent of $j$ (\cite{w}).
Thus one gets an equivalence relation on the set of Young tableaux
$\{ \lambda=(\lambda_1,\ldots,\lambda_{\ell+1}):
\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_{\ell+1}, \lambda_j\in\mathbb{N}\}$.
This, in turn, induces an equivalence relation on the set of
all GT tableaux $\Gamma=\{\mathbf{r}: r_{ij}\in\mathbb{N},
r_{ij}\geq r_{i+1,j}\geq r_{i,j+1}\}$: one says $\mathbf{r}$ and $\mathbf{s}$
are equivalent if $r_{ij}-s_{ij}$ is independent of $i$ and $j$.
By $\Gamma$ we will mean the above set modulo this equivalence.
We will denote by $u^\lambda$ the irreducible unitary indexed by $\lambda$,
$\{e(\lambda,\mathbf{r}):\mathbf{r}_{1\cdot}=\lambda\}$ will denote an orthonormal basis
for $\mathcal{H}_\lambda$ and $u^\lambda_{\mathbf{r}\mathbf{s}}$ will stand for the matrix entries
of $u^\lambda$ in this basis. The symbol ${1\!\!1}$ will denote the Young tableaux
$(1,0,\ldots,0)$. We will often omit the symbol ${1\!\!1}$
and just write $u$ in order to denote $u^{1\!\!1}$.
Notice that any GT tableaux $\mathbf{r}$ with first row ${1\!\!1}$
must be, for some $i\in\{1,2,\ldots,\ell+1\}$, of the form $(r_{ab})$, where
\[
r_{ab}=\cases{1 &if $1\leq a\leq i$ and $b=1$,\cr
0 &otherwise.}
\]
Thus such a GT tableaux is uniquely determined by the integer $i$.
We will write just $i$ for this GT tableaux $\mathbf{r}$.
Thus for example, a typical matrix entry of $u^{1\!\!1}$ will be
written simply as $u_{ij}$.
Let $\mathbf{r}=(r_{ab})$ be a GT tableaux.
Let
$H_{ab}(\mathbf{r}):=r_{a+1,b}-r_{a,b+1}$ and
$V_{ab}(\mathbf{r}):=r_{ab}-r_{a+1,b}$.
An element $\mathbf{r}$ of $\Gamma$ is completely
specified by the following differences
\[
\mathbf{D}(\mathbf{r})=\left(\matrix{V_{11}(\mathbf{r})&H_{11}(\mathbf{r})
&H_{12}(\mathbf{r})&\cdots&H_{1,\ell-1}(\mathbf{r})&H_{1,\ell}(\mathbf{r})\cr
V_{21}(\mathbf{r})&H_{21}(\mathbf{r})&H_{22}(\mathbf{r})&\cdots&H_{2,\ell-1}(\mathbf{r})&\cr
&\cdots&&&&\cr
V_{\ell,1}(\mathbf{r})&H_{\ell,1}(\mathbf{r})&&&&}\right).
\]
The differences satisfy the following inequalities
\begin{equation}\label{ineq}
\sum_{k=0}^b H_{a-k,k+1}(\mathbf{r})\leq V_{a+1,1}(\mathbf{r})
+\sum_{k=0}^b H_{a-k+1,k+1}(\mathbf{r}),\quad
1\leq a\leq \ell,\;\;0\leq b\leq a-1.
\end{equation}
Conversely, if one has an array of the form
\[
\left(\matrix{V_{11}&H_{11}&H_{12}&\cdots&H_{1,\ell-1}&H_{1,\ell}\cr
V_{21}&H_{21}&H_{22}&\cdots&H_{2,\ell-1}&\cr
&\cdots&&&&\cr
V_{\ell,1}&H_{\ell,1}&&&&}\right),
\]
where $V_{ij}$'s and $H_{ij}$'s are in $\mathbb{N}$ and obey
the inequalities~(\ref{ineq}), then the above array is of the form
$\mathbf{D}(\mathbf{r})$ for some GT tableaux $\mathbf{r}$. Thus the quantities
$V_{a1}$ and $H_{ab}$ give a coordinate system for elements in $\Gamma$.
The following diagram explains this new coordinate system.
The hollow circles stand for the $r_{ij}$'s.
The entries are decreasing along the direction of the arrows,
and the $V_{ij}$'s and the $H_{ij}$'s are the difference
between the two endpoints of the corresponding arrows.\\
\hspace*{100pt}
\def\scriptstyle{\scriptstyle}
\xymatrix@C=35pt@R=35pt{
& & j\ar@{.>}[r] &&\\
& \circ\ar@{->}[r]\ar@{->}[d]_{V_{11}} & \circ\ar@{->}[r]
& \circ\ar@{->}[r] &\circ\\
i\ar@{.>}[d] & \circ\ar@{->}[r]\ar@{->}[d]_{V_{21}}\ar@{->}[ur]_{H_{11}} &
\circ\ar@{->}[r]\ar@{->}[ur]_{H_{12}} & \circ\ar@{->}[ur]_{H_{13}} & \\
& \circ\ar@{->}[r]\ar@{->}[d]_{V_{31}}\ar@{->}[ur]_{H_{21}} &
\circ\ar@{->}[ur]_{H_{22}} &\\
& \circ\ar@{->}[ur]_{H_{31}} & }\\
\paragraph{Clebsch-Gordon coefficients.}
Look at the representation $u^{1\!\!1}\otimes u^\lambda$
acting on $\mathcal{H}_{1\!\!1}\otimes\mathcal{H}_\lambda$.
The representation decomposes as a direct sum
$\oplus_\mu u^\mu$, i.e.\ one has a corresponding
decomposition $\oplus_\mu\mathcal{H}_\mu$ of $\mathcal{H}_{1\!\!1}\otimes\mathcal{H}_\lambda$.
Thus one has two orthonormal bases
$\{e^\mu_\mathbf{s}\}$ and $\{e^{1\!\!1}_i\otimes e^\lambda_\mathbf{r}\}$.
The Clebsch-Gordon coefficient $C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{s})$
is defined to be the inner product
$\langle e^\mu_\mathbf{s}, e^{1\!\!1}_i\otimes e^\lambda_\mathbf{r}\rangle$.
Since ${1\!\!1}$, $\lambda$ and $\mu$ are just the first rows of
$i$, $\mathbf{r}$ and $\mathbf{s}$ respectively, we will often denote
the above quantity just by $C_q(i,\mathbf{r},\mathbf{s})$.
Next, we will compute the quantities $C_q(i,\mathbf{r},\mathbf{s})$. We will
use the calculations given in (\cite{k-s}, pp.\ 220), keeping in mind
that for our case (i.e.\ for $SU_q(\ell+1)$), the top right entry of
the GT tableaux is zero.
Let $M=(m_1,m_2,\ldots,m_i)\in\mathbb{N}^i$ be such that $1\leq m_j\leq \ell+2-j$.
Denote by $M(\mathbf{r})$ the tableaux $\mathbf{s}$ defined by
\begin{equation}\label{movenotation}
s_{jk}=\cases{r_{jk}+1 & if $k=m_j$, $1\leq j\leq i$,\cr
r_{jk} & otherwise.}
\end{equation}
With this notation, observe now that
$C_q(i,\mathbf{r},\mathbf{s})$ will be zero unless $\mathbf{s}$ is
$M(\mathbf{r})$ for some $M\in\mathbb{N}^i$.
(One has to keep in mind though that not all tableaux of the form $M(\mathbf{r})$
is a valid GT tableaux)
From (\cite{k-s}, pp.\ 220), we have
\begin{equation}\label{cgc1}
C_q(i,\mathbf{r},M(\mathbf{r}))=\prod_{a=1}^{i-1}
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{a\cdot} \cr
(1,\mathbf{0}) &\mathbf{r}_{a+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{a\cdot}+e_{m_a}\cr
\mathbf{r}_{a+1\cdot}+e_{m_{a+1}}
\end{array}\right.\right\rangle
\times
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{i\cdot} \cr
(0,\mathbf{0}) &\mathbf{r}_{i+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{i\cdot}+e_{m_i}\cr
\mathbf{r}_{i+1\cdot}
\end{array}\right.\right\rangle,
\end{equation}
where $e_k$ stands for a vector (in the appropriate space) whose
$k$\raisebox{.4ex}{th} coordinate is 1 and the rest are all zero, and
\begin{eqnarray}
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{a\cdot} \cr
(1,\mathbf{0}) &\mathbf{r}_{a+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{a\cdot}+e_j\cr
\mathbf{r}_{a+1\cdot}+e_k
\end{array}\right.\right\rangle^2
&=&
q^{-r_{aj}+r_{a+1,k} - k+j}
\times
\prod_{{i=1}\atop{i\neq j}}^{\ell+2-a}
\frac{[r_{a,i}-r_{a+1,k}-i+k]_q }{[r_{a,i}-r_{a,j}-i+j]_q} \nonumber \\
&& \times
\prod_{{i=1}\atop{i\neq k}}^{\ell+1-a}
\frac{[r_{a+1,i}-r_{a,j}-i+j-1]_q }{[r_{a+1,i}-r_{a+1,k}-i+k-1]_q},\label{corrected_1}\\
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{a\cdot} \cr
(0,\mathbf{0}) &\mathbf{r}_{a+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{a\cdot}+e_j\cr
\mathbf{r}_{a+1\cdot}
\end{array}\right.\right\rangle^2
&=& q^{\left(1-j+\sum_{i=1}^{\ell+1-a}r_{a+1,i} -
\sum_{{i=1}\atop{i\neq j}}^{\ell+2-a}r_{a,i}\right)} \nonumber \\
&&
\times \left(
\frac{\prod_{i=1}^{\ell+1-a}[r_{a+1,i}-r_{aj}-i+j-1]_q }
{\prod_{{i=1}\atop{i\neq j}}^{\ell+2-a}[r_{a,i}-r_{aj}-i+j]_q }\right),
\label{corrected_2}
\end{eqnarray}
where for an integer $n$, $[n]_q$ denotes the $q$-number $(q^n-q^{-n})/(q-q^{-1})$.
After some lengthy but straightforward computations,
we get the following two relations:
\begin{equation}
\left|
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{a\cdot} \cr
(1,\mathbf{0}) &\mathbf{r}_{a+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{a\cdot}+e_j\cr
\mathbf{r}_{a+1\cdot}+e_k
\end{array}\right.\right\rangle
\right| = A'q^A,
\end{equation}
\begin{equation}
\left|
\left\langle \begin{array}{ll}
(1,\mathbf{0}) &\mathbf{r}_{a\cdot} \cr
(0,\mathbf{0}) &\mathbf{r}_{a+1\cdot}
\end{array}\left| \begin{array}{l}
\mathbf{r}_{a\cdot}+e_j\cr
\mathbf{r}_{a+1\cdot}
\end{array}\right.\right\rangle
\right| = B'q^B,
\end{equation}
where
\begin{eqnarray}
A&=&\cases{\displaystyle{\sum_{j\wedge k < b < j\vee k}(r_{a+1,b}-r_{a,b})}
+(r_{a+1,j\wedge k}-r_{a,j\vee k}) & if $j\neq k$,\cr
0 & if $j=k$.} \cr
&=& \sum_{j\wedge k \leq b < j\vee k}(r_{a+1,b}-r_{a,b+1})
+2 \sum_{k < b < j}(r_{a,b}-r_{a+1,b}) \cr
&=& \sum_{j\wedge k \leq b < j\vee k}H_{ab}(\mathbf{r})
+ 2 \sum_{k < b < j}V_{ab}(\mathbf{r}).\label{cgc2}\\
B &=& \sum_{j \leq b < \ell+2-a}H_{ab}(\mathbf{r}),\label{cgc3}
\end{eqnarray}
and $A'$ and $B'$ both lie between two positive constants
independent of $\mathbf{r}$, $a$, $j$ and $k$
(Here and elsewhere in this paper, an empty summation
would always mean zero).
Combining these, one gets
\begin{equation} \label{cgc4}
C_q(i,\mathbf{r}, M(\mathbf{r}))=P\cdot q^{C(i,\mathbf{r},M)},
\end{equation}
where
\begin{equation} \label{cgc5}
C(i,\mathbf{r},M)=\sum_{a=1}^{i-1}\left(
\sum_{m_a\wedge m_{a+1} \leq b < m_a\vee m_{a+1}}H_{ab}(\mathbf{r})
+2 \sum_{m_{a+1} < b < m_a}V_{ab}(\mathbf{r})\right)
+\sum_{m_i \leq b < \ell+2-i}H_{ib}(\mathbf{r}),
\end{equation}
and $P$ lies between two positive constants
that are independent of $i$, $\mathbf{r}$ and $M$.
\begin{rmrk}\rm
The formulae (\ref{corrected_1}) and (\ref{corrected_2})
are obtained from equations~(45) and (46), page 220, \cite{k-s}
by replacing $q$ with $q^{-1}$. Equation~(45) is a special
case of the more general formula (48), page 221, \cite{k-s}.
However, there is a small error in equation~(48) there.
The correct form can be found in equations~(3.1, 3.2a, 3.2b)
in \cite{a-s}. That correction has been incorporated in
equations~(\ref{corrected_1}) and (\ref{corrected_2}) here.
\end{rmrk}
\newsection{Left multiplication operators}
The matrix entries $u^\lambda_{\mathbf{r}\mathbf{s}}$ form a complete orthogonal set
of vectors in $L_2(G)$. Write $e^\lambda_{\mathbf{r}\mathbf{s}}$ for
$\|u^\lambda_{\mathbf{r}\mathbf{s}}\|^{-1}u^\lambda_{\mathbf{r}\mathbf{s}}$.
Then the $e^\lambda_{\mathbf{r}\mathbf{s}}$'s form a complete orthonormal basis
for $L_2(G)$. Let $\pi$ denote the representation of $\mathcal{A}$ on
$L_2(G)$ by left multiplications. We will now derive an expression for
$\pi(u_{ij})e^\lambda_{\mathbf{r}\mathbf{s}}$.
From the definition of matrix entries and that of the CG coefficients,
one gets
\begin{equation} \label{cb1}
u^\rho e(\rho,\mathbf{t})=\sum_\mathbf{s} u^\rho_{\mathbf{s}\mathbf{t}}e(\rho,\mathbf{s}),
\end{equation}
\begin{equation} \label{cb2}
e(\mu,\mathbf{n})=\sum_{j,\mathbf{s}}C_q(j,\mathbf{s},\mathbf{n})e({1\!\!1},j)\otimes e(\lambda,\mathbf{s}).
\end{equation}
Apply $u\otimes u^\lambda$ on both sides and note that
$u\otimes u^\lambda$ acts on $e(\mu,\mathbf{n})$ as $u^\mu$:
\begin{equation} \label{cb3}
\sum_\mathbf{m} u^\mu_{\mathbf{m}\mathbf{n}}e(\mu,\mathbf{m})=
\sum_{j,\mathbf{s}}\sum_{i,\mathbf{r}}C_q(j,\mathbf{s},\mathbf{n})
u_{ij}u^\lambda_{\mathbf{r}\mathbf{s}}e({1\!\!1},i)\otimes e(\lambda,\mathbf{r}).
\end{equation}
Next, use (\ref{cb2}) to expand $e(\mu,\mathbf{m})$ on the left hand side to get
\begin{equation}
\sum_{i,\mathbf{r},\mathbf{m}} u^\mu_{\mathbf{m}\mathbf{n}}
C_q(i,\mathbf{r},\mathbf{m})e({1\!\!1},i)\otimes e(\lambda,\mathbf{r})
=
\sum_{j,\mathbf{s}}\sum_{i,\mathbf{r}}C_q(j,\mathbf{s},\mathbf{n})
u_{ij}u^\lambda_{\mathbf{r}\mathbf{s}}e({1\!\!1},i)\otimes
e(\lambda,\mathbf{r}).
\end{equation}
Equating coefficients, one gets
\begin{equation}
\sum_{\mathbf{m}} C_q(i,\mathbf{r},\mathbf{m})u^\mu_{\mathbf{m}\mathbf{n}}
=
\sum_{j,\mathbf{s}}C_q(j,\mathbf{s},\mathbf{n})
u_{ij}u^\lambda_{\mathbf{r}\mathbf{s}}.
\end{equation}
Now using orthogonality of the matrix
$(\!(C_q({1\!\!1},\lambda,\mu;j,\mathbf{s},\mathbf{n}))\!)_{(\mu,\mathbf{n}),(j,\mathbf{s})}$,
we obtain
\begin{equation}\label{alg_left_mult}
u_{ij}u^\lambda_{\mathbf{r}\mathbf{s}}
= \sum_{\mu,\mathbf{m},\mathbf{n}}
C_q(i,\mathbf{r},\mathbf{m})C_q(j,\mathbf{s},\mathbf{n})u^\mu_{\mathbf{m}\mathbf{n}}.
\end{equation}
From (\cite{k-s}, pp.\ 441), one has
$\|u^\lambda_{\mathbf{r}\mathbf{s}}\|=d_\lambda^{-\frac{1}{2}}q^{-\psi(\mathbf{r})}$,
where
\[
\psi(\mathbf{r})=-\frac{\ell}{2}\sum_{j=1}^{\ell+1}r_{1j}
+ \sum_{i=2}^{\ell+1}\sum_{j=1}^{\ell+2-i}r_{ij},
\qquad
d_\lambda=\sum_{\mathbf{r}:\mathbf{r}_1=\lambda} q^{2\psi(\mathbf{r})}
\]
Therefore
\begin{equation}\label{left_mult}
\pi(u_{ij})e^\lambda_{\mathbf{r}\mathbf{s}}
= \sum_{\mu,\mathbf{m},\mathbf{n}}
C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{m})C_q({1\!\!1},\lambda,\mu;j,\mathbf{s},\mathbf{n})
d_\lambda^\frac{1}{2} d_\mu^{-\frac{1}{2}}q^{\psi(\mathbf{r})-\psi(\mathbf{m})}
e^\mu_{\mathbf{m}\mathbf{n}}.
\end{equation}
Write
\begin{equation}
\kappa(\mathbf{r},\mathbf{m})=
d_\lambda^\frac{1}{2} d_\mu^{-\frac{1}{2}}q^{\psi(\mathbf{r})-\psi(\mathbf{m})}.
\end{equation}
\begin{lmma}\label{krmbound}
There exist constants $K_2>K_1>0$ such that
$K_1< \kappa(\mathbf{r}, M(\mathbf{r}))<K_2$ for all $\mathbf{r}$.
\end{lmma}
\noindent{\it Proof\/}:
Observe that (\cite{ch-pr}, pp-365)
\[
d_\lambda=\prod_{1\leq i\leq j\leq\ell+1}
\frac{[\lambda_i-\lambda_j+j-i]_q}{[j-i]_q}.
\]
Therefore one gets
\[
\frac{d_\lambda}{d_{\lambda+e_k}}=
\prod_{j:k<j}\frac{[\lambda_k-\lambda_j+j-k]_q}{[\lambda_k-\lambda_j+j-k+1]_q}
\times
\prod_{i:i<k}\frac{[\lambda_i-\lambda_k+k-i]_q}{[\lambda_i-\lambda_k+k-i-1]_q}.
\]
There are $\ell$ terms in the above product, and each term
lies between two positive quantities that depend just on $q$.
Next, we have
\[
\psi(\mathbf{r})=-\frac{\ell}{2}\sum_{j=1}^{\ell+1}r_{1j}
+ \sum_{i=2}^{\ell+1}\sum_{j=1}^{\ell+2-i}r_{ij}.
\]
It follows from this that $\psi(\mathbf{r})-\psi(\mathbf{m})$
is bounded.
Therefore the result follows.
\qed
\newsection{Boundedness of commutators
Let $D$ be an equivariant Dirac operator acting on $L_2(G)$.
It follows from the discussion in section~3
that $D$ must be of the form
\begin{equation}
e^\lambda_{\mathbf{r}\mathbf{s}}
\mapsto
d(\mathbf{r})e^\lambda_{\mathbf{r}\mathbf{s}},
\end{equation}
(Here, for a Young tableaux $\lambda$, $N_\lambda$
is the set of all GT tableaux, modulo the appropriate equivalence
relation, with top row $\lambda$).
Then we have
\begin{equation}\label{bdd_comm}
[D,\pi(u_{ij})]e^\lambda_{\mathbf{r}\mathbf{s}}=
\sum (d(\mathbf{m})-d(\mathbf{r}))C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{m})
C_q({1\!\!1},\lambda,\mu;j,\mathbf{s},\mathbf{n})
\kappa(\mathbf{r},\mathbf{m})e^\mu_{\mathbf{m}\mathbf{n}}.
\end{equation}
Therefore the condition for boundedness of commutators reads
as follows:
\begin{equation} \label{eqbdd1}
|(d(\mathbf{m})-d(\mathbf{r}))C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{m})
C_q({1\!\!1},\lambda,\mu;j,\mathbf{s},\mathbf{n})
\kappa(\mathbf{r},\mathbf{m})|<c,
\end{equation}
where $c$ is independent of $i$, $j$, $\lambda$, $\mu$, $\mathbf{r}$, $\mathbf{s}$, $\mathbf{m}$ and $\mathbf{n}$.
Using lemma~\ref{krmbound}, we get
\begin{equation}\label{eqbdd2}
|(d(\mathbf{m})-d(\mathbf{r}))C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{m})
C_q({1\!\!1},\lambda,\mu;j,\mathbf{s},\mathbf{n})|<c.
\end{equation}
Choosing $j$, $\mathbf{s}$ and $\mathbf{n}$ suitably, one can ensure that
(\ref{eqbdd2}) implies the following:
\begin{equation}\label{eqbdd3}
|(d(\mathbf{m})-d(\mathbf{r}))C_q({1\!\!1},\lambda,\mu;i,\mathbf{r},\mathbf{m})|<c.
\end{equation}
It follows from~(\ref{bdd_comm}) that this condition is also sufficient for
the boundedness of the commutators $[D, u_{ij}]$.
From (\ref{cgc4}), one gets
\begin{equation} \label{eqbdd4}
|d(\mathbf{r})-d(M(\mathbf{r}))|
\leq c q^{-C(i,\mathbf{r},M)}.
\end{equation}
Let us next form a graph $\mathcal{G}_c$ as described
in section~1 by connecting
two elements $\mathbf{r}$ and $\mathbf{r}'$ if
$|d(\mathbf{r})-d(\mathbf{r}')|<c$.
We will assume
the existence of a partition
$(\Gamma^+,\Gamma^-)$ that does not admit any infinite ladder.
For any subset $F$ of $\Gamma$, we will denote by $F^\pm$ the sets
$F\cap \Gamma^\pm$.
Our next job is to study this graph in more detail
using the boundedness conditions above.
Let us start with a few definitions and notations.
By an \textbf{elementary move}, we will mean a map $M$
from some subset of
$\Gamma$ to $\Gamma$ such that $\gamma$ and $M(\gamma)$
are connected by an edge.
A \textbf{move} will mean a composition of a finite number of
elementary moves.
If $M_1$ and $M_2$ are two moves, $M_1M_2$ and $M_2M_1$ will
in general be different.
For a family of moves $M_1, M_2,\ldots, M_r$,
we will denote by
$\sum_{{j=1}}^{r}M_j$
the move $M_1M_2\ldots M_r$,
and by
$\sum_{j=1}^{r}M_{r+1-j}$
the move $M_r\ldots M_2M_1$.
For a nonnegative integer $n$ and a move $M$, we will denote
by $nM$ the move obtained by applying $M$ successively $n$ times.
Of special interest to us will be moves of the
form $M:\mathbf{r}\mapsto\mathbf{s}$, where
$\mathbf{s}$ is given by (\ref{movenotation}).
We will use the vector $(m_1,\ldots, m_{k})$
to denote $M$. The following families of moves will
be particularly useful to us:
\[
M_{ik}=(i,i-1,\ldots,i-k+1)\in\mathbb{N}^k,\quad
N_{ik}=(\underbrace{i+1,\ldots,i+1}_{\mbox{$k$}},
i,i,\ldots,i)\in\mathbb{N}^{\ell+2-i}.
\]
For describing a path in our graph, we will
often use phrases like `apply the move
$\sum_{{j=1}}^{k}M_j$
to go from $\mathbf{r}$ to $\mathbf{s}$'. This will refer
to the path given by
\[
\Bigl(\mathbf{r},\, M_k(\mathbf{r}),
M_{k-1}M_k(\mathbf{r}),\,\ldots,\,M_1M_2\ldots
M_k(\mathbf{r})=\mathbf{s}\Bigr).
\]
The following lemma will be very useful in the next
two sections.
\begin{lmma}\label{freemove}
Let $N_{jk}$ and $M_{ik}$ be the moves defined above. Then
\begin{enumerate}
\item $|d(\mathbf{r})-d(N_{j0}(\mathbf{r}))|\leq c$,
\item $|d(\mathbf{r})-d(M_{ik}(\mathbf{r}))|\leq
cq^{-\sum_{a=1}^{k-1}H_{a,i+1-a}-\sum_{b=i}^{\ell}H_{k,b+k-1}}$. In
particular, if $H_{a,i+1-a}(\mathbf{r})=0$ for $1\leq a\leq k-1$ and
$H_{k,b+k-1}(\mathbf{r})=0$ for $i\leq b\leq \ell$, then
$|d(\mathbf{r})-d(M_{ik}(\mathbf{r}))|\leq c$.
\end{enumerate}
\end{lmma}
\noindent{\it Proof\/}:
Direct consequence of~(\ref{eqbdd4}).
\qed
\newsection{Characterization of $|D|$}
In this section and the next, we will use lemma~\ref{freemove}
to prove a characterization theorem for the sign of the
operator $D$. Along the way,
we will also give a very precise description
of the singular values of $D$.
The main ingredients in the proof are
the finiteness of exactly one of the sets $F^+$ and $F^-$
for appropriately chosen subsets $F$ of $\Gamma$.
General form of the argument for proving this will be
as follows:
for a carefully chosen coordinate $C$ (in the present case, $C$
would be one of the $V_{a1}$'s or $H_{ab}$'s), a sweepout argument
will show that any $\gamma$ can be connected by a path,
throughout which $C(\cdot)$ remains constant, to another point $\gamma'$
for which $C(\gamma')=C(\gamma)$ and all other coordinates of $\gamma'$
are zero.
This would help connect any two points $\gamma$ and $\delta$ by a path
such that $C(\cdot)$ would lie between $C(\gamma)$ and $C(\delta)$
on the path. This would finally result in the finiteness
of at least one (and hence exactly one) of $C(F^+)$ and $C(F^-)$.
Next, assuming one of these, say $C(F^-)$ is finite,
one shows that for any other coordinate $C'$,
$C'(F^-)$ is also finite.
This is done as follows. If $C'(F^-)$ is infinite, one chooses
elements $y_n\in F^-$ with
$C'(y_n)<C'(y_{n+1})$ for all $n$.
Now starting at each $y_n$, produce paths
keeping the $C'$-coordinate constant and taking the
$C$-coordinate above the plane $C(\cdot)=K$, where $C(F^-)\subseteq [-K,K]$.
This will produce an infinite ladder.
The argument is explained in the following diagram.\\[3ex]
\hspace*{60pt}
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}\\[2ex]
Our next job is to define an important class of subsets of $\Gamma$.
Observe that lemma~\ref{freemove}
tells us that for any $\mathbf{r}$ and any $j$, the points
$\mathbf{r}$ and $N_{j0}(\mathbf{r})$ are connected by an edge,
whenever $N_{j0}(\mathbf{r})$ is a GT tableaux.
Let $\mathbf{r}$ be an element of $\Gamma$. Define the
\textbf{free plane passing through $\mathbf{r}$} to be the minimal
subset of $\Gamma$ that contains $\mathbf{r}$
and is closed under application of the moves $N_{j0}$.
We will denote this set by $\mathscr{F}_\mathbf{r}$.
The following is an easy consequence of this definition.
\begin{lmma} \label{freecriterion}
Let $\mathbf{r}$ and $\mathbf{s}$ be two GT tableaux. Then
$\mathbf{s}\in \mathscr{F}_\mathbf{r}$ if and only if
$V_{a,1}(\mathbf{r})=V_{a,1}(\mathbf{s})$ for all $a$ and for each $b$, the difference
$H_{a,b}(\mathbf{r})-H_{a,b}(\mathbf{s})$ is independent of $a$.
\end{lmma}
\begin{crlre} \label{freedisjt}
Let $\mathbf{r},\mathbf{s}\in\Gamma$. Then either $\mathscr{F}_\mathbf{r}=\mathscr{F}_\mathbf{s}$ or
$\mathscr{F}_\mathbf{r}\cap \mathscr{F}_\mathbf{s}=\phi$.
\end{crlre}
Let $\mathbf{r}\in\Gamma$.
For $1\leq j\leq \ell+1$, define
$a_j$ to be an integer such that $H_{a_j,j}(\mathbf{r})=\min_i H_{ij}(\mathbf{r})$.
Note three things here:\\
1. definition of $a_j$ depends on $\mathbf{r}$,\\
2. for a given $j$ and given $\mathbf{r}$, $a_j$ need not be unique, and\\
3. if $\mathbf{s}\in\mathscr{F}_\mathbf{r}$, then for each $j$, the set of
$k$'s for which $H_{kj}(\mathbf{s})=\min_i H_{ij}(\mathbf{s})$ is same
as the set of all $k$'s for which $H_{kj}(\mathbf{r})=\min_i H_{ij}(\mathbf{r})$.
Therefore, the $a_j$'s can be chosen in a manner such that
they remain the same for all elements lying on a given free plane.
\begin{lmma}\label{sweep1}
Let $\mathbf{s}\in \mathscr{F}_\mathbf{r}$. Let $\mathbf{s}'$ be another GT tableaux
given by
\[
V_{a1}(\mathbf{s}')=V_{a1}(\mathbf{s}) \mbox{ and }
H_{a1}(\mathbf{s}')=H_{a1}(\mathbf{s}) \mbox{ for all }a,\quad
H_{a_b,b}(\mathbf{s}')=0 \mbox{ for all }b>1,
\]
where the $a_j$'s are as defined above.
Then there is a path in $\mathscr{F}_\mathbf{r}$ from $\mathbf{s}$ to $\mathbf{s}'$
such that $H_{11}(\cdot)$ remains constant throughout this path.
\end{lmma}
\noindent{\it Proof\/}:
Apply the move
$\sum_{{b=2}}^{\ell}
\left(\sum_{j=2}^{\ell+2-b} H_{a_j,j}(\mathbf{s})\right)N_{\ell+3-b,0}$.\qed
The following diagram will help explain the steps involved
in the above proof in the case where $\mathbf{r}$ is the constant
tableaux.\\[2ex]
\def\scriptstyle{\scriptstyle}
\xymatrix@C=.6pt@R=.6pt{
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \odot\ar@{.}[d] && \cdot&& \cdot&\\
0 & a & & b & & c && d &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot\ar@{.}[u]\ar@{.}[d] && \cdot&\\
0 & a & & b & & c &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \odot\ar@{.}[u]&\\
0 & a && b & \\
\cdot\ar@{}[r] && \cdot&\\
0 &a & \\
\cdot&}
\hspace{-2em} \xymatrix@C=20pt@R=12pt{&\\&\\ \ar@{->}[r]^{bN_{30}}&\\}\hspace{.3em}
\xymatrix@C=.6pt@R=.6pt{
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \odot\ar@{.}[d]&& \cdot&\\
0 & a & & 0 & & b+c && d &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \odot\ar@{.}[u]&\\
0 & a & & 0 & & b+c &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot&\\
0 & a && 0 & \\
\cdot\ar@{}[r] && \cdot&\\
0 &a & \\
\cdot&}
\hspace{-2em} \xymatrix@C=20pt@R=12pt{&\\&\\ \ar@{->}[r]^{(b+c)N_{40}}&\\}\hspace{.3em}
\xymatrix@C=.6pt@R=.6pt{
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \cdot& \odot&\\
0 & a & & 0 & & 0 && b+c+d &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \cdot&\\
0 & a & & 0 & & 0 &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot&\\
0 & a && 0 & \\
\cdot\ar@{}[r] && \cdot&\\
0 &a & \\
\cdot&}\\[2ex]
\hspace*{12em} \xymatrix@C=20pt@R=12pt{&\\&\\ \ar@{->}[r]^{(b+c+d)N_{50}}&\\}\hspace{.3em}
\xymatrix@C=.6pt@R=.6pt{
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \cdot&& \cdot&\\
0 & a & & 0 & & 0 && 0 &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot && \cdot&\\
0 & a & & 0 & & 0 &\\
\cdot\ar@{}[r] & & \cdot\ar@{}[r] && \cdot&\\
0 & a && 0 & \\
\cdot\ar@{}[r] && \cdot&\\
0 &a & \\
\cdot&}\\
A dotted line joining two circled dots signifies a move that
increases the $r_{ij}$'s lying on the dotted line by one.
Where there is one circled dot and no dotted line, it means
one applies the move that raises the $r_{ij}$ corresponding to
the circled dot by one.
\begin{ppsn}\label{signfree1}
Let $\mathbf{r}$ be a GT tableaux.
Then either $\mathscr{F}_{\mathbf{r}}^+$ is finite or $\mathscr{F}_{\mathbf{r}}^-$ is finite.
\end{ppsn}
\noindent{\it Proof\/}:
Suppose, if possible, both $H_{11}(\mathscr{F}_{\mathbf{r}}^+)$ and $H_{11}(\mathscr{F}_{\mathbf{r}}^-)$
are infinite. Then there exist two sequences of elements
$\mathbf{r}_n$ and $\mathbf{s}_n$ with $\mathbf{r}_n\in \mathscr{F}_\mathbf{r}^+$
and $\mathbf{s}_n\in \mathscr{F}_\mathbf{r}^-$,
such that
\[
H_{11}(\mathbf{r}_1)<H_{11}(\mathbf{s}_1)<H_{11}(\mathbf{r}_2)<H_{11}(\mathbf{s}_2)<\cdots.
\]
Now starting from $\mathbf{r}_n$, employ the forgoing lemma
to reach a point $\mathbf{r}'_n\in\mathscr{F}_{\mathbf{r}}$ for which
\[
V_{a1}(\mathbf{r}'_n)=V_{a1}(\mathbf{r}_n) \mbox{ and }
H_{a1}(\mathbf{r}'_n)=H_{a1}(\mathbf{r}_n) \mbox{ for all }a,\quad
H_{a_b,b}(\mathbf{r}'_n)=0 \mbox{ for all }b>1.
\]
Similarly, start at $\mathbf{s}_n$ and go to
a point $\mathbf{s}'_n\in\mathscr{F}_{\mathbf{r}}$ for which
\[
V_{a1}(\mathbf{s}'_n)=V_{a1}(\mathbf{s}_n) \mbox{ and }
H_{a1}(\mathbf{s}'_n)=H_{a1}(\mathbf{s}_n) \mbox{ for all }a,\quad
H_{a_b,b}(\mathbf{s}'_n)=0 \mbox{ for all }b>1.
\]
Now use the move $N_{10}$ to get to $\mathbf{s}'_n$ from $\mathbf{r}'_n$.
The paths thus constructed are all disjoint, because
for the path from $\mathbf{r}_n$ to $\mathbf{s}_n$, the
$H_{11}$ coordinate lies between
$H_{11}(\mathbf{r}_n)$ and $H_{11}(\mathbf{s}_n)$.
This means $(\mathscr{F}_\mathbf{r}^+, \mathscr{F}_\mathbf{r}^-)$ admits an infinite ladder.
So one of the sets $H_{11}(\mathscr{F}_\mathbf{r}^+)$ and $H_{11}(\mathscr{F}_\mathbf{r}^-)$
must be finite. Let us assume that $H_{11}(\mathscr{F}_\mathbf{r}^-)$ is finite.
Let us next show that for any $b>1$, $H_{ab}(\mathscr{F}_\mathbf{r}^-)$ is finite.
Let $K$ be an integer such that $H_{11}(\mathbf{s})<K$ for all $\mathbf{s}\in \mathscr{F}_\mathbf{r}^-$.
If $H_{ab}(\mathscr{F}_\mathbf{r}^-)$ was infinite, there would exist elements
$\mathbf{r}_n\in \mathscr{F}_\mathbf{r}^-$ such that
\[
H_{ab}(\mathbf{r}_1)<H_{ab}(\mathbf{r}_2)<\cdots.
\]
Now start at $\mathbf{r}_n$ and employ the move $N_{10}$ successively
$K$ times to reach a point in
$\mathscr{F}_\mathbf{r}^+=\mathscr{F}_\mathbf{r}\backslash\mathscr{F}_\mathbf{r}^-$.
These paths will all be disjoint, as throughout the path,
$H_{ab}$ remains fixed.
Since the coordinates $(H_{11},H_{12},\ldots,H_{1,\ell})$
completely specify a point in $\mathscr{F}_\mathbf{r}$, it follows that
$\mathscr{F}_\mathbf{r}^-$ is finite.
\qed
Next we need a set that can be used for a proper
indexing of the free planes.
Such a set will be called a complementary axis.
\begin{dfn}\rm\rm
A subset $\mathscr{C} $ of $\Gamma$ is called a \textbf{complementary axis} if
\begin{enumerate}
\item $\cup_{\mathbf{r}\in \mathscr{C} }\mathscr{F}_\mathbf{r} =\Gamma$,
\item if $\mathbf{r},\mathbf{s}\in \mathscr{C} $, and $\mathbf{r}\neq \mathbf{s}$, then
$\mathscr{F}_\mathbf{r}$ and $\mathscr{F}_\mathbf{s}$ are disjoint.
\end{enumerate}
\end{dfn}
Let us next give a choice of a complementary axis.
\begin{thm} \label{compl}
Define
\[
\mathscr{C} =\{\mathbf{r}\in \Gamma: \Pi_{a=1}^{\ell+1-b} H_{ab}(\mathbf{r})=0
\mbox{ for } 1\leq b\leq \ell\}.
\]
The set $\mathscr{C} $ defined above is a complementary axis.
\end{thm}
\noindent{\it Proof\/}:
Let $\mathbf{s}\in\Gamma$.
A sweepout argument almost identical to that used in
lemma~\ref{sweep1} (application of the move
$\sum_{{b=1}}^\ell
\left(\sum_{j=1}^{\ell+1-b} H_{a_j,j}(\mathbf{s})\right)N_{\ell+2-b,0}$ )
will connect $\mathbf{s}$ to another element $\mathbf{s}'$
for which $H_{a_b,b}(\mathbf{s}')=0$ for $1\leq b\leq\ell$
by a path that lies entirely on $\mathscr{F}_\mathbf{s}$.
Clearly, $\mathbf{s}'\in\mathscr{C}$. Since $\mathbf{s}'\in\mathscr{F}_\mathbf{s}$,
by corollary~\ref{freedisjt}, $\mathbf{s}\in\mathscr{F}_{\mathbf{s}'}$.
It remains to show that if $\mathbf{r}$ and $\mathbf{s}$ are two distinct elements of
$\mathscr{C}$, then $\mathbf{s}\not\in\mathscr{F}_\mathbf{r}$.
Since $\mathbf{r}\neq\mathbf{s}$, there exist two integers $a$ and $b$,
$1\leq b\leq \ell$ and $1\leq a\leq \ell+2-b$, such that
$H_{ab}(\mathbf{r})\neq H_{ab}(\mathbf{s})$.
Observe that $H_{1\ell}(\cdot)$ must be zero for both,
as they are members of $\mathscr{C}$. So $b$ can not be $\ell$ here.
Next we will produce two integers $i$ and $j$ such that
the differences $H_{ib}(\mathbf{r})-H_{ib}(\mathbf{s})$
and $H_{jb}(\mathbf{r})-H_{jb}(\mathbf{s})$ are distinct.
If there is an integer $k$ for which
$H_{kb}(\mathbf{r})=H_{kb}(\mathbf{s})=0$, then take $i=a$, $j=k$.
If not, there would exist two integers $i$ and $j$ such that
$H_{ib}(\mathbf{r})=0$, $H_{ib}(\mathbf{s})>0$
and
$H_{jb}(\mathbf{r})>0$, $H_{jb}(\mathbf{s})=0$.
Take these $i$ and $j$.
Since $H_{ib}(\mathbf{r})-H_{ib}(\mathbf{s})$
and $H_{jb}(\mathbf{r})-H_{jb}(\mathbf{s})$ are distinct,
by lemma~\ref{freecriterion}, $\mathbf{r}$ and $\mathbf{s}$ can not lie
on the same free plane.
\qed
\begin{lmma} \label{sweep2}
Let $\mathbf{r}$ be a GT tableaux. Let $\mathbf{s}$ be the GT tableaux
defined by the prescription
\[
V_{a1}(\mathbf{s})=V_{a1}(\mathbf{r})\mbox{ for all }a,\quad
H_{ab}(\mathbf{s})=H_{ab}(\mathbf{r}) \mbox{ for all }a\geq 2,\mbox{ for all }b,\quad
H_{1,b}(\mathbf{s})=0 \mbox{ for all }b.
\]
Then there is a path from $\mathbf{r}$ to $\mathbf{s}$ such that
$V_{a1}(\cdot)$ remains constant throughout the path.
\end{lmma}
\noindent{\it Proof\/}:
Apply the move
$\displaystyle{\sum_{{b=1}}^\ell} H_{1,b}(\mathbf{r})M_{b+1,1}$.\qed
The above lemma is actually the first step in the following
slightly more general sweepout algorithm.
\begin{lmma} \label{sweep3}
Let $\mathbf{r}$ be a GT tableaux. Let $\mathbf{s}$ be the GT tableaux
defined by the prescription
\[
V_{11}(\mathbf{s})=V_{11}(\mathbf{r}),\quad
V_{a1}(\mathbf{s})=0\mbox{ for all }a>1,\quad
H_{ab}(\mathbf{s})=0 \mbox{ for all }a,b.
\]
Then there is a path from $\mathbf{r}$ to $\mathbf{s}$ such that
$V_{11}(\cdot)$ remains constant throughout the path.
\end{lmma}
\noindent{\it Proof\/}:
Apply successively the moves
\[
\sum_{{b=1}}^\ell H_{1,b}(\mathbf{r})M_{b+1,1},\quad
\sum_{{b=1}}^{\ell-1} H_{2,b}(\mathbf{r})M_{b+2,2},\quad
\ldots,\quad
H_{\ell,1}(\mathbf{r})M_{\ell+1,\ell},
\]
followed by
\begin{equation}\label{movseq}
V_{21}(\mathbf{r})M_{33},\quad (V_{21}(\mathbf{r})+V_{31}(\mathbf{r}))M_{44},\quad
\ldots,\quad \left(\sum_{a=2}^\ell V_{a1}(\mathbf{r})\right)M_{\ell+1,\ell+1}.
\end{equation}
\qed
The following diagram
will help explain the procedure described above in a simple case.\\
\def\scriptstyle{\scriptstyle}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \cdot\ar@{}[r] & \odot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \cdot\ar@{}[ur]|\star & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \\
\cdot\ar@{}[ur]|\star& }
\hspace{-1em} \xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{41}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \odot\ar@{}[r] & \cdot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \cdot\ar@{}[ur]|0 & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \\
\cdot\ar@{}[ur]|\star& }
\hspace{-1em} \xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{31}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \odot\ar@{}[r] & \cdot\ar@{}[r] & \cdot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|0 & \cdot\ar@{}[ur]|0 & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \\
\cdot\ar@{}[ur]|\star& \\
}
\hspace{-1em} \xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{21}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \cdot\ar@{}[r] & \odot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \cdot\ar@{}[ur]|0 & \odot\ar@{.}[ur]|0 & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|\star & \\
\cdot\ar@{}[ur]|\star& \\
}\\[2ex]
\hspace*{60pt}
\xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{42}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \odot\ar@{}[r] & \cdot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \odot\ar@{.}[ur]|0 & \cdot\ar@{}[ur]|0 & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|\star& \cdot\ar@{}[ur]|0 & \\
\cdot\ar@{}[ur]|\star& \\
}
\hspace{-1em} \xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{32}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \cdot\ar@{}[r] & \odot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \cdot\ar@{}[ur]|0 & \cdot\ar@{.}[ur]|0 && \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \odot\ar@{.}[ur]|0 &&& \\
\cdot\ar@{}[ur]|\star& &&&\\
}
\hspace{-1em} \xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{43}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \odot\ar@{}[r] & \cdot&\\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \cdot\ar@{.}[ur]|0 & \cdot\ar@{}[ur]|0 & \\
\odot\ar@{}[d]|\star\ar@{.}[ur]|0& \cdot\ar@{}[ur]|0 & \\
\cdot\ar@{}[ur]|0& \\
}\\[2ex]
\hspace*{60pt}
\xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{33}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \cdot\ar@{}[r] & \odot&\\
\cdot\ar@{}[d]|0\ar@{}[ur]|0& \cdot\ar@{}[ur]|0 & \cdot\ar@{.}[ur]|0 & \\
\cdot\ar@{}[d]|\star\ar@{}[ur]|0& \cdot\ar@{.}[ur]|0 & \\
\odot\ar@{.}[ur]|0& \\
}
\xymatrix@C=20pt@R=12pt{&\\ \ar@{->}[r]^{M_{44}}&\\}\hspace{.5em}
\xymatrix@C=10pt@R=8pt{
\cdot\ar@{}[r]\ar@{}[d]|\star & \cdot\ar@{}[r] & \cdot\ar@{}[r] & \cdot&\\
\cdot\ar@{}[d]|0\ar@{}[ur]|0& \cdot\ar@{}[ur]|0 & \cdot\ar@{}[ur]|0 & \\
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\cdot\ar@{}[ur]|0& \\
}
\begin{crlre}\label{growth5}
$|d(\mathbf{r})|=O(r_{11})$.
\end{crlre}
\noindent{\it Proof\/}:
If one employs the sequence of moves
\[
V_{11}(\mathbf{r})M_{22},\quad (V_{11}(\mathbf{r})+V_{21}(\mathbf{r}))M_{33},\quad
\ldots,\quad \left(\sum_{a=1}^\ell V_{a1}(\mathbf{r})\right)M_{\ell+1,\ell+1}
\]
instead of the sequence given in (\ref{movseq}),
one would reach the constant (or zero) tableaux.
Total length of this path from $\mathbf{r}$ to the zero tableaux
is
\[
\sum_{a=1}^\ell\sum_{b=1}^{\ell+1-a}H_{ab}(\mathbf{r})
+ \sum_{b=1}^\ell\sum_{a=1}^b V_{a1}(\mathbf{r}),
\]
which can easily be shown to be bounded by $\ell r_{11}$.
\qed
\begin{thm}\label{singular}
Let $\widetilde{D}$ be the following operator:
\begin{equation}
\widetilde{D}: e^\lambda_{\mathbf{r},\mathbf{s}}\mapsto r_{11}e^\lambda_{\mathbf{r},\mathbf{s}}
\end{equation}
Then $(\mathcal{A},\mathcal{H},\widetilde{D})$ is an equivariant $\ell(\ell+2)$-summable odd spectral triple.
Moreover, if $D$ is any equivariant Dirac operator
acting on the $L_2$ space of $SU_q(\ell+1)$, then
there exist positive reals $a$ and $b$ such that
$|D| \leq a + b\widetilde{D}$.
In particular,
$D$ cannot be $p$-summable for $p<\ell(\ell+2)$.
\end{thm}
\noindent{\it Proof\/}:
Boundedness of commutators with algebra elements
follow from the observation that
$|d(\mathbf{r})-d(M(\mathbf{r})|\leq 1$ and hence
equation~(\ref{eqbdd3}) is satisfied.
Observe that the number of Young tableux
$\lambda=(\lambda_1,\ldots,\lambda_\ell,\lambda_{\ell+1})$
with
$n=\lambda_1\geq \lambda_2\geq \ldots \lambda_\ell\geq \lambda_{\ell+1}=0$
is
\[
\sum_{i_1=0}^{n}\sum_{i_2=0}^{i_1}\ldots\sum_{i_{\ell-1}=0}^{i_{\ell-2}}1
=
\mbox{polynomial in $n$ of degree $\ell-1$}.
\]
Thus the number of such Young tableaux is $O(n^{\ell-1})$.
Next, let
$\lambda:n=\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_{\ell}\geq 0$
be an Young tableaux, and let $V_\lambda$ be the space
carrying the irreducible representation parametrized by
$\lambda$. Then
\begin{eqnarray*}
\mbox{dim}\, V_\lambda &=&
\prod_{1\leq i<j\leq \ell+1}
\frac{(\lambda_{i}-\lambda_{i+1})+
\ldots (\lambda_{j-1}-\lambda_{j})+j-i}{j-i}\\
&=& \prod_{1\leq i<j\leq \ell+1}
\frac{\lambda_{i}-\lambda_{j}+j-i}{j-i}\\
&\leq& (n+1)^{\frac{\ell(\ell+1)}{2}}.
\end{eqnarray*}
Thus the dimension of an irreducible representation corresponding to
a Young tableaux
\[
n=\lambda_1\geq \lambda_2\geq \ldots \lambda_\ell\geq \lambda_{\ell+1}=0
\]
is $O(n^{\frac{1}{2}\ell(\ell+1)})$.
Using the two observations above, one can now show
that the summability
of $\widetilde{D}$ is $\ell(\ell+2)$.
Optimality of $\widetilde{D}$ follows from
corollary~\ref{growth5}.\qed
One should note, however, that the $\widetilde{D}$ defined
above has trivial sign, and consequently trivial $K$-homology class.
\begin{lmma}\label{optimality1}
Let $D$ be an equivariant Dirac operator
on $L_2(G)\otimes\mathbb{C}^m$. Then
there are positive reals $a,b$ such that
$|D|\leq a+b|\widetilde{D}\otimes I|$.
\end{lmma}
\noindent{\it Proof\/}:
Let $D$ be an equivariant Dirac operator on $L_2(G)\otimes\mathbb{C}^m$.
Then $D$ must be of the form
$e_{\mathbf{r},\mathbf{s}}\otimes v\mapsto e_{\mathbf{r},\mathbf{s}}\otimes T(\mathbf{r})v$
where $T(\mathbf{r})$ are self-adjoint operators acting on $\mathbb{C}^m$.
The growth conditions coming out of the boundedness of
the commutators will now be exactly as in~(\ref{eqbdd4}), with
the scalars $d(\cdot)$ replaced by operators $T(\cdot)$ and
absolute value replaced by operator norm.
If we now form a graph by joining two vertices
$\mathbf{r}$ and $\mathbf{s}$ whenever $\|T(\mathbf{r})-T(\mathbf{s})\|\leq c$,
then exactly as in the proof of corollary~\ref{growth5},
one can show that any point $\mathbf{r}$ can be connected to
the zero tableaux by a path of length $O(r_{11})$.
This implies that there are positive reals $a$ and $b$ such
that $|T(\mathbf{r})|\leq a+br_{11}$.
The assertion in the lemma now follows from this.\qed
\newsection{Characterization of $\mbox{sign}\,D$}
We continue our analysis of the growth
conditions on the $d(\mathbf{r})$'s in this section
in order to come up with a complete characterization
of the sign of $D$.
\begin{lmma}\label{about_v11}
The sets $V_{11}(\Gamma^+)$ and $V_{11}(\Gamma^-)$ can not both be infinite.
\end{lmma}
\noindent{\it Proof\/}:
If both the sets are infinite, then one can
choose two sequences of points $\mathbf{r}_n$ and $\mathbf{s}_n$
such that $\mathbf{r}_n\in \Gamma^+$, $\mathbf{s}_n\in \Gamma^-$ and
\[
V_{11}(\mathbf{r}_1)<V_{11}(\mathbf{s}_1)< V_{11}(\mathbf{r}_2)< V_{11}(\mathbf{s}_2)<\ldots.
\]
Start at $\mathbf{r}_n$ and use lemma~\ref{sweep3} above to reach a point
$\mathbf{r}'_n$ for which $V_{11}(\mathbf{r}'_n)=V_{11}(\mathbf{r}_n)$ and all other coordinates
are zero through a path where the $V_{11}$ coordinate remains constant.
Similarly, from $\mathbf{s}_n$, go to a point $\mathbf{s}'_n$ for which
$V_{11}(\mathbf{s}'_n)=V_{11}(\mathbf{s}_n)$ and all other coordinates
are zero. Now apply the move
$(V_{11}(\mathbf{s}_n)-V_{11}(\mathbf{r}_n))M_{11}$ to go from $\mathbf{r}'_n$ to $\mathbf{s}'_n$.
This will give us a path $p_n$ from $\mathbf{r}_n$ to $\mathbf{s}_n$
on which $V_{11}(\cdot)$ remains between $V_{11}(\mathbf{r}_n)$
and $V_{11}(\mathbf{s}_n)$. Therefore all the paths $p_n$ are disjoint.
Thus $(\Gamma^+,\Gamma^-)$ admits an infinite ladder.
So at least one of $V_{11}(\Gamma^+)$ and $V_{11}(\Gamma^-)$
must be finite.\qed
\begin{lmma}
Let $C$ be any of the coordinates
$V_{a1}$ or $H_{ab}$ where $a>1$.
If $V_{11}(\Gamma^-)$ is finite, then $C(\Gamma^-)$ is also finite.
\end{lmma}
\noindent{\it Proof\/}:
Assume $K$ is a positive integer such that $V_{11}(\Gamma^-)\subseteq [0,K]$.
Now suppose, if possible, that $C(\Gamma^-)$ is infinite.
Let $\mathbf{r}_n$ be a sequence of points in $\Gamma^-$ such that
\[
C(\mathbf{r}_1)<C(\mathbf{r}_2)<\ldots.
\]
Start at $\mathbf{r}_n$, and use lemma~\ref{sweep2} to reach a point
$\mathbf{r}'_n$ and then apply $M_{11}$ for $K+1$ times to get to a point
$\mathbf{s}_n$ for which $V_{11}(\mathbf{s}_n)>K$.
Throughout this path, $C(\cdot)$ is constant, so that the paths are all
disjoint.
Since $V_{11}(\mathbf{s}_n)>K$, we have $\mathbf{s}_n\in \Gamma^+$.
Thus this gives us an infinite ladder for $(\Gamma^+,\Gamma^-)$,
which is impossible.\qed
\begin{lmma}
Suppose $H_{1\ell}(F)$ is bounded.
If $V_{11}(\Gamma^-)$ is finite, then $F^-$ is finite.
\end{lmma}
\noindent{\it Proof\/}:
The previous lemma, along with the assumption here
tells us that the sets $V_{a1}(F^-$) and $H_{a,\ell+1-a}(F^-)$
are all bounded for $1\leq a\leq\ell$.
Since for an $\mathbf{r}\in V$, one has
$r_{11}=\sum_{a=1}^\ell V_{a1}(\mathbf{r})+\sum_{a=1}^\ell H_{a,\ell+1-a}(\mathbf{r})$,
the set
$\{r_{11}:\mathbf{r}\in F^-\}$ is bounded.
It follows that $F^-$ is finite.\qed
\begin{crlre}\label{signcomp}
If $V_{11}(\Gamma^-)$ is finite, then $\mathscr{C} ^-$ is finite.
\end{crlre}
\noindent{\it Proof\/}:
Follows from the observation that $H_{1\ell}(\mathbf{r})=0$
for all $\mathbf{r}\in \mathscr{C} $.\qed\\
A similar argument will tell us that
if $V_{11}(\Gamma^+)$ is finite, then $\mathscr{C} ^+$ is finite.
Thus from lemma~\ref{about_v11}, it follows that
either $\mathscr{C}^+$ or $\mathscr{C}^-$ is finite.
\begin{thm}\label{eqsign}
Let $D$ be an equivariant Dirac operator on $L_2(SU_q(\ell+1))$.
Then $\mbox{sign\,} D$ must be of the form $2P-I$
or $I-2P$ where $P$ is, up to a compact perturbation, the projection
onto the closed span of
$\{e^\lambda_{\mathbf{r},\mathbf{s}}: \mathbf{r}\in \mathscr{F}_{\mathbf{r}_i} \mbox{ for some }i\}$,
with $\mathbf{r}_1,\ldots,\mathbf{r}_k$ being a finite collection of GT-tableaux.
\end{thm}
\noindent{\it Proof\/}:
Let
$\mathscr{C} '=\{\mathbf{r}\in \mathscr{C} : \mathscr{F}_\mathbf{r}^+ \neq\phi\neq \mathscr{F}_\mathbf{r}^-\}$.
Let us first show that $\mathscr{C} '$ is finite, i.e.\ except for
finitely many $\mathbf{r}$'s in $\mathscr{C} $, one has either
$\mathscr{F}_\mathbf{r}\subseteq \Gamma^+$ or $\mathscr{F}_\mathbf{r}\subseteq \Gamma^-$.
It follows from the argument used in the proof of theorem~\ref{compl}
that any two points on a free plane can be connected by a
path lying entirely on the plane. If $\mathscr{C}'$ is infinite,
one can easily produce an infinite ladder using this fact.
Thus there are only finitely many free
planes $\mathscr{F}_\mathbf{r}$ for which
both $\mathscr{F}_\mathbf{r}^+$ and $\mathscr{F}_\mathbf{r}^-$ are nonempty.
Since we already know that for every $\mathbf{r}$,
either $\mathscr{F}_\mathbf{r}^+$ or $\mathscr{F}_\mathbf{r}^-$ is finite,
it follows that by applying a compact perturbation,
one can ensure that for every $\mathbf{r}$, exactly one of the sets
$\mathscr{F}_\mathbf{r}^+$ and $\mathscr{F}_\mathbf{r}^-$ is empty.
This, along with the observations that
$\mathscr{C}\cap\mathscr{F}_\mathbf{r}=\{\mathbf{r}\}$ and that
either $\mathscr{C}^+$ or $\mathscr{C}^-$ is finite
gives us the required conclusion.\qed
As a consequence of this sign characterization, we now get
the following theorem.
\begin{thm}
Let $\ell>1$. Let $D$ be an equivariant
Dirac operator acting on $L_2(G)$. Then
$D$ must have trivial sign.
\end{thm}
\noindent{\it Proof\/}:
We will show that if $P$ is as in the earlier theorem,
then the commutators $[P,\pi(u_{ij})]$ can not all be compact.
Let us first prove it
in the case when $P$ is the projection onto the span of
$\{e_{\mathbf{r}\mathbf{s}}: \mathbf{r}\in\mathscr{F}_0\}$, where $\mathscr{F}_0$
is the free plane passing through
the constant tableaux.
We have
\[
[P,\pi(u_{ij})]e_{\mathbf{r}\mathbf{s}}=\cases{
P\pi(u_{ij})e_{\mathbf{r}\mathbf{s}} & if $\mathbf{r}\not\in \mathscr{F}_0$,\cr
(P-I)\pi(u_{ij})e_{\mathbf{r}\mathbf{s}} & if $\mathbf{r}\in \mathscr{F}_0$}.
\]
Recall (section~5) the expression for $\pi(u_{ij})e_{\mathbf{r}\mathbf{s}}$:
\[
\pi(u_{ij})e_{\mathbf{r}\mathbf{s}}
=\sum_{{R\in\mathbb{N}^i, S\in\mathbb{N}^j}\atop{R(1)=S(1)}}
C_q(i,\mathbf{r},R(\mathbf{r}))C_q(j,\mathbf{s},S(\mathbf{s}))k(\mathbf{r},R(\mathbf{r}))
e_{R(\mathbf{r})S(\mathbf{s})}.
\]
Hence for $\mathbf{r}\in \mathscr{F}_0$,
\begin{eqnarray*}
[P,\pi(u_{ij})]e_{\mathbf{r}\mathbf{s}} &=& (P-I)\pi(u_{ij})e_{\mathbf{r}\mathbf{s}}\\
&=& -\sum_{{R\in\mathbb{N}^i, S\in\mathbb{N}^j}\atop{R(1)=S(1),R\neq N_{i0}}}
C_q(i,\mathbf{r},R(\mathbf{r}))C_q(j,\mathbf{s}, S(\mathbf{s}))k(\mathbf{r},R(\mathbf{r}))
e_{R(\mathbf{r}),S(\mathbf{s})}.
\end{eqnarray*}
In particular, for $i=j=1$, one gets
\[
[P,\pi(u_{11})]e_{\mathbf{r}\mathbf{s}} =
-\sum_{k=1}^\ell
C_q(1,\mathbf{r},M_{k1}(\mathbf{r}))C_q(1,\mathbf{s}, M_{k1}(\mathbf{s}))k(\mathbf{r},M_{k1}(\mathbf{r}))
e_{M_{k1}(\mathbf{r}),M_{k1}(\mathbf{s})}.
\]
Now suppose $\mathbf{r}\in\mathscr{F}_0$ satisfies
\begin{equation} \label{choices}
r_{1,\ell}=0=r_{2,\ell}=r_{1,\ell+1}.
\end{equation}
Then
\[
\langle e_{M_{\ell 1}(\mathbf{r}),M_{\ell 1}(\mathbf{r})},
[P,\pi(u_{11})]e_{\mathbf{r}\bldr}\rangle =
- C_q(1,\mathbf{r}, M_{\ell 1}(\mathbf{r}))^2k(\mathbf{r},M_{\ell 1}(\mathbf{r})).
\]
It follows from~(\ref{cgc4}) and (\ref{cgc5})
that $C_q(1,\mathbf{r}, M_{\ell 1}(\mathbf{r}))$ is bounded away from zero,
so long as $\mathbf{r}$ obeys (\ref{choices}).
We have also seen (lemma~\ref{krmbound}) that
$k(\mathbf{r},M_{\ell 1}(\mathbf{r}))$ is bounded away from zero.
Now it is easy to see that if $\ell>1$, then there are infinitely many choices
of $\mathbf{r}$ satisfying (\ref{choices}) such that they all lie in $\mathscr{F}_0$.
Therefore $[P,\pi(u_{11})]$
is not compact.
For more general $P$ (as in the previous theorem),
the idea would be similar, but this time
one has to get hold of a positive integer $n$ such that
for any $\mathbf{r}\in\cup_{i=1}^k\mathscr{F}_{\mathbf{r}_i}$,
$nM_{\ell 1}(\mathbf{r})\not\in \cup_{i=1}^k\mathscr{F}_{\mathbf{r}_i}$,
and then compute
$\langle e_{nM_{\ell 1}(\mathbf{r}),nM_{\ell 1}(\mathbf{r})},
(P-I)\pi(u_{11})^n e_{\mathbf{r} \mathbf{r}}\rangle$.
\qed
As mentioned in the introduction, the above theorem
in particular says that in order to get equivariant
Dirac operators with nontrivial
sign for for $\ell>1$,
one needs to bring in multiplicities. We will see below
that if one takes the tensor product of
$L_2(G)$ with a suitable space, it is possible to produce
such operators.
\begin{thm}
Let $\widetilde{D}$ be as in theorem~\ref{singular} and let
$N_i$ be the following operators on $L_2(G)$:
\[
N_i e_{\mathbf{r},\mathbf{s}}=f_i(\mathbf{r}) e_{\mathbf{r},\mathbf{s}},
\]
where $f_i(\mathbf{r})=\min\{H_{ai}(\mathbf{r}):1\leq a\leq \ell+1-i\}$.
Let $\gamma_1,\gamma_2,\ldots,\gamma_{\ell+1}$ be
$\ell+1$ spin matrices acting on $\mathbb{C}^m$. Define
an operator $D$ on $L_2(G)\otimes\mathbb{C}^m$ as follows:
\[
D = \sum_{i=1}^\ell N_i\otimes \gamma_i
+ \widetilde{D}\otimes \gamma_{\ell+1}.
\]
Then $(L_2(G)\otimes\mathbb{C}^m,\pi\otimes I, D)$ is
an equivariant
$\ell(\ell+2)$-summable spectral triple.
Moreover, the operator $D$ is optimal,
in the following sense:
given any equivariant Dirac operator
$D'$ on $L_2(G)\otimes\mathbb{C}^m$
there are positive reals $a,b$ such that
$|D'|\leq a+b|D|$.
\end{thm}
\noindent{\it Proof\/}:
Compact resolvent condition and summability
of $D$ follow from
the fact that the operator $|D|$ is given by
$|D| e_{\mathbf{r},\mathbf{s}}=\lambda_\mathbf{r} e_{\mathbf{r},\mathbf{s}}$,
where the singular values $\lambda_\mathbf{r}$ obey the inequality
\[
r_{11}\leq \lambda_\mathbf{r} \leq K r_{11}
\]
for some constant $K$ that depends only on $\ell$.
Boundedness of commutators follow from the boundedness
of commutators of the $N_i$'s and $\widetilde{D}$ with
the algebra elements, which is clear from
condition~(\ref{eqbdd4}).
Observe that $\widetilde{D}\otimes I\leq |D|$.
Therefore optimality follows from lemma~\ref{optimality1}.
\qed
\begin{rmrk}\rm
Let $\widehat{V}_{i1}$ and $\widehat{H}_{ij}$ denote the following
operators on $L_2(G)$:
\[
\widehat{V}_{i1}e_{\mathbf{r},\mathbf{s}}=V_{i1}(\mathbf{r})e_{\mathbf{r},\mathbf{s}},
\quad
\widehat{H}_{ij}e_{\mathbf{r},\mathbf{s}}=H_{ij}(\mathbf{r})e_{\mathbf{r},\mathbf{s}},
\quad
i+j\leq \ell+1.
\]
Suppose now that $\gamma_1,\gamma_2,\ldots,\gamma_{\ell(\ell+3)/2}$
be spin matrices acting on some space $\mathbb{C}^m$, and
$D_k$ for $1\leq k\leq \frac{\ell(\ell+3)}{2}$ are the operators
$\widehat{V}_{i1}$ and $\widehat{H}_{ij}$ in some order.
Now define $D$ on $L_2(G)\otimes\mathbb{C}^m$ to be the operator
\[
D=\sum D_k\otimes \gamma_k.
\]
Then this operator $D$ also enjoys all the features
described in the above theorem.
\end{rmrk}
\section{The odd dimensional quantum spheres}
In this section, we will use the combinatorial
technique and the calculations done in the earlier
sections to investigate equivariant Dirac operators
for all the odd dimensional quantum spheres $S_q^{2\ell+1}$
of Vaksman \& Soibelman~(\cite{v-s}).
In what follows, we will write $G$ for $SU_q(\ell+1)$
and $H$ for $SU_q(\ell)$.
The $C^*$-algebra $C(S_q^{2\ell+1})$ of the quantum
sphere $S_q^{2\ell+1}$
is the universal $C^*$-algebra generated by
elements
$z_1, z_2,\ldots, z_{\ell+1}$
satisfying the following relations (see~\cite{h-s}):
\begin{eqnarray*}
z_i z_j & =& qz_j z_i,\qquad 1\leq j<i\leq \ell+1,\\
z_i z_j^* & =& q z_j^* z_i ,\qquad 1\leq i\neq j\leq \ell+1,\\
z_i z_i^* - z_i^* z_i +
(1-q^{2})\sum_{k>i} z_k z_k^* &=& 0,\qquad \hspace{2em}1\leq i\leq \ell+1,\\
\sum_{i=1}^{\ell+1} z_i z_i^* &=& 1.
\end{eqnarray*}
Just like their classical counterparts,
these spheres can be viewed as quotient spaces
of the quantum groups $SU_q(\ell+1)$, i.\ e.\
\begin{equation}
C(S_q^{2\ell+1}) \cong C(G\verb1\1H) =
\{a\in C(G): (\phi\otimes id)\Delta (a)=I\otimes a\},
\end{equation}
where $\phi$ is a $C^*$-homomorphism
from $C(G)$ onto $C(H)$ that preserves the comultiplication,
that is, it satisfies
$\Delta\phi=(\phi\otimes\phi)\Delta$, where the $\Delta$
on the right hand side is the comultiplication for
$G$ and the $\Delta$ on the left hand side stands for the
comultiplication for $H$.
(For a formulation of quotient spaces etc.\ in the context
of compact quantum groups, see~\cite{po})
The group $G$ has a canonical right action
$\tau:C(G\verb1\1H)\rightarrow C(G\verb1\1H)\otimes C(G)$
coming from the comultiplication $\Delta$
(i.\ e.\ $\tau$ is just the restriction of $\Delta$ to $C(G\verb1\1H)$).
Let $\rho$ denote the restriction of the Haar state on $C(G)$
to $C(G\verb1\1H)$.
Then clearly one has
$(\rho\otimes id)\tau (a) = \rho(a)I$,
which means $\rho$ is the invariant state for $C(G\verb1\1H)$.
This also means that $L_2(G\verb1\1H)=L_2(\rho)$ is just the
closure of $C(G\verb1\1H)$ in $L_2(G)$.
\begin{ppsn}
Assume $\ell>1$.
The right regular representation $u$ of $G$ keeps
$L_2(G\verb1\1H)$ invariant, and the restriction of $u$ to
$L_2(G\verb1\1H)$ decomposes as a direct sum of exactly one copy
of each of the irreducibles given by the young tableaux
$\lambda_{n,k}:=(n+k, k,k,\ldots, k,0)$, with $n,k\in\mathbb{N}$.
\end{ppsn}
\noindent{\it Proof\/}:
Write $\sigma$ for the composition $h_H\circ\phi$
where $h_H$ is the Haar state for $H$.
From the description of $C(G\verb1\1H)$ above, it follows
that
\begin{eqnarray*}
C(G\verb1\1H) &=& \{a\in C(G): (\sigma \otimes id)\Delta (a)=a\}\\
& =&\{(\sigma \otimes id)\Delta (a): a\in C(G)\}.
\end{eqnarray*}
Now
the map $a\mapsto \sigma\ast a:=(\sigma\otimes id)\Delta(a)$
on $C(G)$ extends to a bounded linear operator $L_\sigma$ on $L_2(G)$
(lemma~3.1, \cite{pa}), and it is easy to see that
$L_\sigma^2=L_\sigma$. It follows then that
$L_2(G\verb1\1H)=\ker(L_\sigma -I)=\mbox{ran}\,L_\sigma$.
From the discussion preceeding theorem~3.3, \cite{pa},
it now follows that $u$ keeps $L_2(G\verb1\1H)$ invariant and in fact
the restriction of $u$ to $L_2(G\verb1\1H)$ is the representation
induced by the trivial repersentation of $H$.
From the analogue of Frobenius reciprocity theorem for
compact quantum groups (theorem~3.3, \cite{pa}) it now
follows that the multiplicity of any irreducible $u^\lambda$
in it would be same as the multiplicity of the trivial
representation of $H$ in the restriction of $u^\lambda$ to $H$.
But from the representation theory of $SU_q(\ell+1)$,
we know that the restriction of $u^\lambda$ to $SU_q(\ell)$
decomposes into a direct sum of one copy of
each irreducible $\mu:(\mu_1\geq \mu_2\geq \ldots \geq\mu_\ell)$
of $SU_q(\ell)$ for which
\begin{equation}\label{induced}
\lambda_1\geq \mu_1 \geq \lambda_2\geq \mu_2\geq \ldots
\geq\lambda_\ell \geq \mu_\ell \geq 0.
\end{equation}
Now the trivial representation of $SU_q(\ell)$ is indexed
by Young tableaux of the form
$\mu:(k,k,\ldots,k)$ where $k\in\mathbb{N}$.
But such a $\mu$ will obey the restriction~\ref{induced} above
if and only if $\lambda$ is of the form
$(n+k,k,k,\ldots,k,0)$.
\qed
\begin{rmrk}\rm
For the case $\ell=1$, the restriction of the irreducible
$(n,0)$ to the trivial subgroup decomposes into $n+1$ copies
of the trivial representation. Therefore, in this case,
$L_2(S_q^3)$ decomposes into a direct sum of $n+1$ copies of
each representation $(n,0)$.
\end{rmrk}
Next, we will make an explicit choice of $\phi$
that would help us make use of the calculations
already done in the initial sections for analyzing
Dirac operators acting on $L_2(G\verb1\1H)$.
More specifically, we will choose our $\phi$ in such
a manner that $L_2(G\verb1\1H)$ turns out to be
the span of certain rows of the $e_{\mathbf{r},\mathbf{s}}$'s.
Let $u^{1\!\!1}$ denote the fundamental unitary for $G$,
i.\ e.\ the irreducible unitary representation corresponding to the
Young tableaux ${1\!\!1}=(1,0,\ldots,0)$.
Similarly write $v^{1\!\!1}$ for the fundamental unitary for $H$.
Fix some bases for the corresponding representation spaces.
Then $C(G)$ is the $C^*$-algebra generated by the matrix
entries $\{u^{1\!\!1}_{ij}\}$ and $C(H)$ is the
$C^*$-algebra generated by the matrix
entries $\{v^{1\!\!1}_{ij}\}$.
Now define $\phi$ by
\begin{equation}
\phi(u^{1\!\!1}_{ij})=\cases{ I & if $i=j=1$,\cr
v^{1\!\!1}_{i-1,j-1} & if $2\leq i,j\leq \ell+1$,\cr
0 & otherwise.}
\end{equation}
Then $C(G\verb1\1H)$ is the $C^*$-subalgebra of
$C(G)$ generated by the entries $u_{1,j}$ for $1\leq j\leq \ell+1$
(one recovers the relations for the generators of $C(S_q^{2\ell+1})$
if one sets $z_i=q^{-i+1}u^*_{1,i})$.
\begin{ppsn}
Let $\Gamma_0$ be the set of all GT tableaux $\mathbf{r}^{nk}$
given by
\[
r^{nk}_{ij}=\cases{ n+k & if $i=j=1$,\cr
0 & if $i=1$, $j=\ell+1$,\cr
k & otherwise,}
\]
for some $n,k \in \mathbb{N}$.
Let $\Gamma_0^{nk}$ be the set of all GT tableaux with
top row $(n+k,k,\ldots,k,0)$.
Then the family of vectors
\[
\{e_{\mathbf{r}^{nk},\mathbf{s}}: n,k\in\mathbb{N},\, \mathbf{s}\in\Gamma_0^{nk}\}
\]
form a complete
orthonormal basis for $L_2(G\verb1\1H)$.
\end{ppsn}
\noindent{\it Proof\/}:
Let $A$ be the linear span of the elements
$\{u_{\mathbf{r}^{n,k},\mathbf{s}}: n,k\in\mathbb{N}, \mathbf{s}\in\Gamma_0^{n,k}\}$.
Clearly the closure of $A$ in $L_2(G)$ is the closed
linear span of $\{e_{\mathbf{r}^{nk},\mathbf{s}}: n,k\in\mathbb{N},\, \mathbf{s}\in\Gamma_0^{nk}\}$.
It is also immdiate that the restriction of the
right regular representation to the above subspace
is a direct sum of one copy of each of the irreducibles
$(n+k,k,k,\ldots,k,0)$.
We will next show that for any $a\in A$, $u_{1j}a$ and $u_{1j}^*$ a are also
in $A$.
Take $a=u_{\mathbf{r}^{n,k},\mathbf{s}}$. Use equation~(\ref{alg_left_mult}) to get
\begin{eqnarray}
u_{1,j}u_{\mathbf{r}^{n,k},\mathbf{s}} &=&
\sum_{M, M'}
C_q(1,\mathbf{r}^{n,k},M(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M'(\mathbf{s}))
u_{M(\mathbf{r}^{n,k}),M'(\mathbf{s})}\cr
&=& \sum_{M'}
C_q(1,\mathbf{r}^{n,k},M_{11}(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M'(\mathbf{s}))
u_{M_{11}(\mathbf{r}^{n,k}),M'(\mathbf{s})} \cr
&& +
\sum_{M''}
C_q(1,\mathbf{r}^{n,k},M_{\ell+1,1}(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M''(\mathbf{s}))
u_{M_{\ell+1,1}(\mathbf{r}^{n,k}),M''(\mathbf{s})} \cr
&=&\sum_{M'}
C_q(1,\mathbf{r}^{n,k},\mathbf{r}^{n+1,k})C_q(j,\mathbf{s},M'(\mathbf{s}))
u_{\mathbf{r}^{n+1,k},M'(\mathbf{s})}\cr
&& +
\sum_{M''}
C_q(1,\mathbf{r}^{n,k},\mathbf{r}^{n,k-1}))C_q(j,\mathbf{s},M''(\mathbf{s}))
u_{\mathbf{r}^{n,k-1},M''(\mathbf{s})},
\end{eqnarray}
where the first sum is over all moves $M'\in\mathbb{N}^{j}$
whose first coordinate is 1 and the second
sum is over all moves $M''\in\mathbb{N}^{j}$
whose first coordinate is $\ell+1$.
Thus $u_{1j}a\in A$.
Next, note that if
$\langle u_{1j}^* e_{\mathbf{r}^{n,k},\mathbf{s}}, e_{\mathbf{r}',\mathbf{s}'}\rangle\neq 0$,
then one must have
$\mathbf{r}'=\mathbf{r}^{n-1,k}$ or $\mathbf{r}'=\mathbf{r}^{n,k+1}$.
Therefore it follows that $u_{1j}^* u_{\mathbf{r}^{n,k},\mathbf{s}}$
is a linear combination of the $u_{\mathbf{r}^{n-1,k},\mathbf{s}}$
$u_{\mathbf{r}^{n,k+1},\mathbf{s}}$'s, and hence belongs to $A$.
Since $A$ contains the element $u_{\mathbf{0},\mathbf{0}}=1$,
it contains $u_{1j}$ and $u_{ij}^*$. Thus $A$ contains the $*$-algebra
$B$ generated by the $u_{1j}$'s.
But by the previous theorem, restriction of the right regular representation
to the $L_2$ closure $L_2(G\verb1\1H)$ of $B$ also decomposes
as a direct sum of one copy
of each of the irreducibles $(n+k,k,\ldots,k,0)$.
So it follows that $L_2(G\verb1\1H)$ is equal to
the subspace stated in the theorem.
\qed
A self-adjoint operator with compact resolvent on $L_2(G\verb1\1H)$
that commutes with the restriction of $u$ there would be
of the form
\[
e_{\mathbf{r},\mathbf{s}}\mapsto d(\mathbf{r})e_{\mathbf{r},\mathbf{s}},\quad \mathbf{r}\in\Gamma_0.
\]
Next, let us look at the growth restrictions coming from the
boundedness of commutators.
In this case, one has the boundedness of only the operators
$[D,\pi(u_{ij})]$. Which means, in effect, one will now have
the condition~(\ref{eqbdd4}) only for $i=1$ and $\mathbf{r}\in\Gamma_0$:
\begin{equation}\label{eqbdd_sph1}
|d(\mathbf{r})-d(M(\mathbf{r}))|\leq c q^{-C(1,\mathbf{r},M)}.
\end{equation}
Observe that only allowed moves here are the moves
$M=M_{1,1}\equiv(1)$ and $M=M_{\ell+1,1}\equiv(\ell+1)$.
Looking at the corresponding quantity
$C(1,\mathbf{r},M)$, we find that there are two conditions:
\begin{eqnarray}
|d(\mathbf{r}^{nk})-d(\mathbf{r}^{n,k-1})| &\leq & c,\label{eqbdd_sph2}\\
|d(\mathbf{r}^{nk})-d(\mathbf{r}^{n+1,k})| &\leq &
cq^{-\sum_{j=1}^{\ell}H_{1j}(\mathbf{r}^{nk})}
=cq^{-k}.\label{eqbdd_sph3}
\end{eqnarray}
As in the earlier sections, we can now
form a graph by taking $\Gamma_0$ to be the set of vertices,
and by joining two vertices $\mathbf{r}$ and $\mathbf{s}$ by an edge if
$|d(\mathbf{r})-d(\mathbf{s})|\leq c$.
\begin{lmma}
Let $\mathscr{F}_n=\{\mathbf{r}^{n,k}:k\in\mathbb{N}\}$, $n\in\mathbb{N}$.
Then any two points in $\mathscr{F}_n$ are connected
by a path lying entirely in $\mathscr{F}_n$.
If $n<n'$, then any point in $\mathscr{F}_n$ is connected to
any point in $\mathscr{F}_{n'}$ by a path such that
$n\leq V_{1,1}(\mathbf{r}) \leq n'$
for every vertex $\mathbf{r}$ lying on that path.
\end{lmma}
\noindent{\it Proof\/}:
Take two points $\mathbf{r}^{n,j}$ and $\mathbf{r}^{n,k}$
in $\mathscr{F}_n$. Assume $j<k$.
From the condition (\ref{eqbdd_sph2}), it follows
that any point $\mathbf{r}$ is connected to $M_{\ell+1,1}(\mathbf{r})$
by an edge. Therefore the first conclusion follows
from the observation that if we start at $\mathbf{r}^{n,k}$
and apply the move $M_{\ell+1,1}$ successively $k-j$ number of times,
we reach the point $\mathbf{r}^{n,j}$, and the vertices on this path
are the points $\mathbf{r}^{n,i}$ for $i=j, j+1,\ldots,k$.
Observe also that throughout this path, $V_{1,1}(\mathbf{r})$
remains $n$.
For the second part, take a point $\mathbf{r}^{n,k}$ in $\mathscr{F}_n$
and a point $\mathbf{r}^{n',j}$ in $\mathscr{F}_{n'}$.
From what we have done above, there is a path
from $\mathbf{r}^{n,k}$ to $\mathbf{r}^{n,0}$ throughout which
$V_{1,1}(\mathbf{r})=n$.
Similarly there is a path
from $\mathbf{r}^{n',j}$ to $\mathbf{r}^{n',0}$ throughout which
$V_{1,1}(\mathbf{r})=n'$.
Next, note from (\ref{eqbdd_sph3}) that for $p\in\mathbb{N}$,
the points
$\mathbf{r}^{p,0}$ and $\mathbf{r}^{p+1,0}$ are connected by an edge
and
$V_{1,1}(\mathbf{r}^{p,0})=p$, $V_{1,1}(\mathbf{r}^{p+1,0})=p+1$.
So start at $\mathbf{r}^{n,0}$ and reach successively the
points
$\mathbf{r}^{n+1,0}$, $\mathbf{r}^{n+2,0}$ and so on to
eventually reach the point $\mathbf{r}^{n',0}$;
also the coordinate $V_{1,1}(\cdot)$ remains between $n$ and $n'$
on this path.\qed
\begin{thm}\label{eqsign_sphere}
Let $D$ be an equivariant Dirac operator on $L_2(G\verb1\1H)$.
Then
\begin{enumerate}
\item
$D$ must be of the form
\[
e_{\mathbf{r},\mathbf{s}}\mapsto d(\mathbf{r})e_{\mathbf{r},\mathbf{s}},\quad \mathbf{r}\in\Gamma,
\]
where the singular values obey $|d(\mathbf{r})|=O(r_{11})$, and
\item
$\mbox{sign\,} D$ must be of the form $2P-I$
or $I-2P$ where $P$ is, up to a compact perturbation, the projection
onto the closed span of
$\{e_{\mathbf{r}^{nk},\mathbf{s}}: n\in F, k\in\mathbb{N}, \mathbf{s}\in \Gamma_0^{nk}\}$,
for some finite subset $F$ of $\mathbb{N}$.
\end{enumerate}
\end{thm}
\noindent{\it Proof\/}:
Start with an equivariant self-adjoint operator
$D$ with compact resolvent, so that it is indeed of the form
$e_{\mathbf{r},\mathbf{s}}\mapsto d(\mathbf{r})e_{\mathbf{r},\mathbf{s}}$.
By applying a compact perturbation if necessary,
make sure that $d(\mathbf{r})\neq 0$ for all $\mathbf{r}\in\Gamma_0$.
We have seen during the proof of the previous lemma that
for any $n$ and $k$ in $\mathbb{N}$, the vertices
$\mathbf{r}^{nk}$ and $\mathbf{r}^{n,k+1}$ are connected by an edge,
and for any $n\in\mathbb{N}$, the vertices
$\mathbf{r}^{n,0}$ and $\mathbf{r}^{n+1,0}$ is connected by an edge.
Thus any vertex $\mathbf{r}^{nk}$ can be reached from the vertex
$\mathbf{r}^{00}$ by a path of length $n+k$. Therefore one gets the
first assertion.
Next, define
\begin{eqnarray*}
\Gamma_0^+ &=& \{\mathbf{r}\in\Gamma_0: d(\mathbf{r})>0\},\\
\Gamma_0^- &=& \{\mathbf{r}\in\Gamma_0: d(\mathbf{r})<0\},\\
\mathscr{F}_n^+ &=& \mathscr{F}_n\cap \Gamma_0^+,\\
\mathscr{F}_n^- &=& \mathscr{F}_n\cap \Gamma_0^-.
\end{eqnarray*}
Observe that for the path produced in the proof
of the forgoing lemma to connect two
points $\mathbf{r}^{n,k}$ and $\mathbf{r}^{n,j}$ in $\mathscr{F}_n$,
the coordinate $H_{1,\ell}(\cdot)$ remains between $j$ and $k$.
Now suppose for some $n$,
both $\mathscr{F}_n^+$ and $\mathscr{F}_n^-$ are infinite.
Then there are points
\[
0\leq k_1 < k_2 < \ldots
\]
such that $\mathbf{r}^{nk}$ is in $\mathscr{F}_n^+$ for $k=k_{2j}$
and $\mathbf{r}^{nk}$ is in $\mathscr{F}_n^-$ for $k=k_{2j+1}$.
Using the above observation, we can then produce
an infinite ladder by joining
each $\mathbf{r}^{n,k_{2j-1}}$ to $\mathbf{r}^{n,k_{2j}}$.
Thus for each $n\in\mathbb{N}$, exactly one of the sets
$\mathscr{F}_n^+$ and $\mathscr{F}_n^-$ is finite.
Also, note that by the first part of the previous lemma,
the set of all $n\in\mathbb{N}$ for which
both $\mathscr{F}_n^+$ and $\mathscr{F}_n^-$ are nonempty is finite.
Therefore by applying a compact perturbation, we can ensure that
for every $n$, either $\mathscr{F}_n^+=\mathscr{F}_n$ or
$\mathscr{F}_n^-=\mathscr{F}_n$.
Finally, if there are infinitely many $n$'s for which
$\mathscr{F}_n^+=\mathscr{F}_n$
and infinitely many $n$'s for which $\mathscr{F}_n^-=\mathscr{F}_n$,
then one can choose a sequence of integers
\[
0\leq n_1 < n_2 <\ldots
\]
such that
$\mathscr{F}_n^+=\mathscr{F}_n$ for $n=n_{2j}$
and
$\mathscr{F}_n^-=\mathscr{F}_n$ for $n=n_{2j+1}$.
Now use the second part of the previous lemma
to join each $\mathbf{r}^{n_{2j-1},0}$ to $\mathbf{r}^{n_{2j},0}$
to produce an infinite ladder.
Thus there is a finite subset $F$ of $\mathbb{N}$ such that
exactly one of the following is true:
\[
\mathscr{F}_n=\cases{\mathscr{F}_n^+ & if $n\in F$,\cr
\mathscr{F}_n^- & if $n\not\in F$,}
\qquad
\mbox{or }
\qquad
\mathscr{F}_n=\cases{\mathscr{F}_n^- & if $n\in F$,\cr
\mathscr{F}_n^+ & if $n\not\in F$.}
\]
This is precisely what the second part
of the theorem says.\qed
Next, take the operator $D:e_{\mathbf{r},\mathbf{s}}\mapsto d(\mathbf{r})e_{\mathbf{r},\mathbf{s}}$
on $L_2(G\verb1\1H)$ where the $d(\mathbf{r})$'s are given by:
\begin{equation}\label{eq_sphere1}
d(\mathbf{r}^{nk})=\cases{-k & if $n=0$,\cr
n+k & if $n>0$.}
\end{equation}
\begin{thm}\label{generic_d_sph}
The operator $D$ is an equivariant $(2\ell+1)$-summable
Dirac operator acting on $L_2(G\verb1\1H)$, that gives a
nondegenerate pairing with $K_1(C(G\verb1\1H))$.
The operator $D$ is optimal, i.\ e.\
if $D_0$ is any equivariant Dirac operator on $L_2(G\verb1\1H)$,
then there are positive reals $a$ and $b$ such that
\[
|D_0|\leq a+b|D|.
\]
\end{thm}
\noindent{\it Proof\/}:
Recall from equation~(\ref{left_mult}) that the elements
$u_{1,j}$ act on the basis
elements $e_{\mathbf{r}^{n,k},\mathbf{s}}$ as follows:
\begin{eqnarray}\label{l2_repn_sph}
u_{1,j}e_{\mathbf{r}^{n,k},\mathbf{s}} &=&
\sum_{M, M'}
C_q(1,\mathbf{r}^{n,k},M(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M'(\mathbf{s}))
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{M(\mathbf{r}^{n,k}),M'(\mathbf{s})}\cr
&=& \sum_{M'}
C_q(1,\mathbf{r}^{n,k},M_{11}(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M'(\mathbf{s}))
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{M_{11}(\mathbf{r}^{n,k}),M'(\mathbf{s})} \cr
&& +
\sum_{M''}
C_q(1,\mathbf{r}^{n,k},M_{\ell+1,1}(\mathbf{r}^{n,k}))C_q(j,\mathbf{s},M''(\mathbf{s}))
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{M_{\ell+1,1}(\mathbf{r}^{n,k}),M''(\mathbf{s})} \cr
&=&\sum_{M'}
C_q(1,\mathbf{r}^{n,k},\mathbf{r}^{n+1,k})C_q(j,\mathbf{s},M'(\mathbf{s}))
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{\mathbf{r}^{n+1,k},M'(\mathbf{s})}\cr
&& +
\sum_{M''}
C_q(1,\mathbf{r}^{n,k},\mathbf{r}^{n,k-1}))C_q(j,\mathbf{s},M''(\mathbf{s}))
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{\mathbf{r}^{n,k-1},M''(\mathbf{s})},
\end{eqnarray}
where the first sum is over all moves $M'\in\mathbb{N}^{j}$
whose first coordinate is 1 and the second
sum is over all moves $M''\in\mathbb{N}^{j}$
whose first coordinate is $\ell+1$.
If we now plug in the values of the Clebsch-Gordon coefficients
from equations~(\ref{cgc4}) and~(\ref{cgc5}), we get
\begin{eqnarray}
u_{1,j}e_{\mathbf{r}^{n,k},\mathbf{s}} &=&
\sum_{M'}
P'_1 P'_2 q^{k+C(j,\mathbf{s},M')}
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{\mathbf{r}^{n+1,k},M'(\mathbf{s})}\cr
&& +
\sum_{M''}
P''_1 P''_2 q^{C(j,\mathbf{s},M'')}
\kappa(\mathbf{r}^{n,k},\mathbf{s})
e_{\mathbf{r}^{n,k-1},M''(\mathbf{s})},
\end{eqnarray}
where $P'_i$, $P''_j$ and $k(\mathbf{r}^{n,k},\mathbf{s})$
all lie between two fixed positive numbers.
Boundedness of the commutators $[D,u_{1,j}]$
now follow directly.
For summability, notice that the eigenspace
of $|D|$ corresponding to the eigenvalue $n\in\mathbb{N}$
is the span of
\[
\{e_{\mathbf{r}^{k,n-k},\mathbf{s}}: 0\leq k\leq n, \mathbf{s}\in\Gamma_0^{k,n-k}\}.
\]
Now just count the number of elements in the above set
to get summability.
Next, we will compute the pairing of the $K$-homology class
of this $D$ with a generator of the $K_1$ group.
Write $\omega_q:=q^{-\ell}u_{1,\ell+1}$.
From the commutation relations, it follows that
this element has spectrum
\[
\{z\in\mathbb{C}: |z|=0 \mbox{ or }q^n\mbox{ for some }n\in\mathbb{N}\}.
\]
Then the element $\gamma_q:=\chi_{\{1\}}(\omega_q^*\omega_q)(\omega_q-I)+I$
is unitary.
We will show that the index of the operator
$Q\gamma_q Q$ (viewed as an operator on $QL_2(G\verb1\1H)$)
is $1$, where $Q=\frac{I-\mbox{sign\,} D}{2}$, i.\ e.\ it
is the projection onto the closed
linear span of
$\{e_{\mathbf{r}^{0,k},\mathbf{s}}:k\in\mathbb{N}, \mathbf{s}\in\Gamma_0^{0,k}\}$.
What we will actually do is
compute the index of the
operator $Q\gamma_0 Q$ and appeal to continuity of the index.
From equation~(\ref{l2_repn_sph}), we get
\begin{eqnarray}\label{for_q=0_1}
\lefteqn{u_{1,\ell+1}e_{\mathbf{r}^{0,k},\mathbf{s}}}\cr
&=&
C_q(1,\mathbf{r}^{0,k},M_{11}(\mathbf{r}^{0,k}))C_q(\ell+1,\mathbf{s},N_{1,0}(\mathbf{s}))
\kappa(\mathbf{r}^{0,k},M_{11}(\mathbf{r}^{0,k}))e_{\mathbf{r}^{1,k},N_{1,0}(\mathbf{s})}\cr
&& + C_q(1,\mathbf{r}^{0,k},M_{\ell+1,1}(\mathbf{r}^{0,k}))
C_q(\ell+1,\mathbf{s},M_{\ell+1,\ell+1}(\mathbf{s}))
\kappa(\mathbf{r}^{0,k},M_{\ell+1,1}(\mathbf{r}^{0,k}))
e_{\mathbf{r}^{0,k-1},M_{\ell+1,\ell+1}(\mathbf{s})}.\cr
&&
\end{eqnarray}
Use the formula~(\ref{cgc1}) for Clebsch-Gordon coefficients
to get
\begin{eqnarray}
C_q(1,\mathbf{r}^{0,k},M_{11}(\mathbf{r}^{0,k}))
&=& q^{k}(1+ o(q)),\\
C_q(1,\mathbf{r}^{0,k},M_{\ell+1,1}(\mathbf{r}^{0,k})
&=& 1+ o(q),\\
C_q(\ell+1,\mathbf{s},N_{1,0}(\mathbf{s}))
&=& 1+ o(q),\\
C_q(\ell+1,\mathbf{s},M_{\ell+1,\ell+1}(\mathbf{s}))
&=& q^{s_{\ell+1,1}+\ell}(1+ o_4(q)),
\end{eqnarray}
where $o(q)$ signifies a function of $q$ that is
continuous at $q=0$ and
$o(0)=0$.
We also have
\begin{eqnarray}
\kappa(\mathbf{r}^{0,k},M_{11}(\mathbf{r}^{0,k}))
&=& q^\ell (1+o(q)),\\
\kappa(\mathbf{r}^{0,k},M_{\ell+1,1}(\mathbf{r}^{0,k}))
&=& 1+o(q),
\end{eqnarray}
where $o(q)$ is as earlier.
Plugging these values in~(\ref{for_q=0_1})
we get
\begin{equation}
\omega_q e_{\mathbf{r}^{0,k},\mathbf{s}}
= q^{k}(1+o(q))e_{\mathbf{r}^{1,k},N_{1,0}(\mathbf{s})}
+ q^{s_{\ell+1,1}}(1+o(q))e_{\mathbf{r}^{0,k-1},M_{\ell+1,\ell+1}(\mathbf{s})}
\end{equation}
Putting $q=0$,
we get
\begin{equation}
\omega_0 e_{\mathbf{r}^{0,k},\mathbf{s}}
= \cases{
e_{\mathbf{r}^{0,k-1},M_{\ell+1,\ell+1}(\mathbf{s})}
& if $k>0$ and $s_{\ell+1,1}=0$,\cr
e_{\mathbf{r}^{1,0},N_{1,0}(\mathbf{s})} & if $k=0$,\cr
0 & otherwise.}
\end{equation}
Thus $\omega_0^*\omega_0$ is the projection onto the span
of
$\{e_{\mathbf{r}^{0,k},\mathbf{s}^k}: k\in\mathbb{N}\}$
where
$\mathbf{s}^k$ is the GT tableaux given by
\[
s^k_{ij}=\cases{0 & if $i=\ell+2-j$,\cr
k & otherwise,}
\]
which is uniquely determined by the conditions
$s_{\ell+1,1}=0$ and that $\mathbf{s}\in\Gamma_0^{0,k}$.
Therefore the operator $\gamma_0$ is given by
\[
\gamma_0 e_{\mathbf{r}^{0,k},\mathbf{s}} =
e_{\mathbf{r}^{0,k},\mathbf{s}} - \chi_{\{\mathbf{s}=\mathbf{s}^k\}}e_{\mathbf{r}^{0,k},\mathbf{s}}
+ \chi_{\{\mathbf{s}=\mathbf{s}^k\}}
e_{\mathbf{r}^{0,k-1},\mathbf{s}^{k-1}}.
\]
It now follows that the index of $Q\gamma_0 Q$ is $1$.
Optimality follows from part~1 of the previous theorem.
\qed
|
{
"timestamp": "2005-04-21T17:52:13",
"yymm": "0503",
"arxiv_id": "math/0503689",
"language": "en",
"url": "https://arxiv.org/abs/math/0503689"
}
|
\section{Introduction}
The Heider balance \cite{h46,hei2,hara,dor1,wt} is a final state of personal relations
between members of a society, reached when these relations evolve according to
some dynamical rules. The relations are assumed to be symmetric, and they can be friendly
or hostile. The underlying psycho-sociological mechanism of the rules is an attempt of
the society members to remove a cognitive dissonance, which we feel when two of our friends
hate each other or our friend likes our enemy. As a result of the process, the society
is split into two groups, with friendly relations within the groups and hostile
relations between the groups. As a special case, the size of one group is zero,
i.e. all hostile relations are removed. HB is the final state if each member
interacts with each other; in the frames of the graph theory, where the problem is formulated,
the case is represented by a fully connected graph.
Recently a continuous dynamics has been introduced to describe the time evolution of the
relations \cite{my1}. In this approach, the relations between nodes $i$ and $j$were
represented by matrix elements $r(i,j)$, which were real numbers, friendly ($r(i,j)>0$) or
hostile ($r(i,j)<0)$. As a consequence of the
continuity, we observed a polarization of opinions: the absolute values of the matrix
elements $r(i,j)$ increase. Here we continue this discussion, but the condition of maximal
connectivity is relaxed, as it could be unrealistic in large societies.
The purpose of first part of this work is to demonstrate, that even if HB is not
present, the above mentioned polarization remains true. In Section II we present
new numerical results for a society of $N=100$ members, represented by Barab\'asi-Albert (BA)
network \cite{ab}. Although this size of considered social structure is rather small, it is
sufficient to observe some characteristics which are different than those in the exponential
networks. In second part (Section III) we compare the results of our equations of motion with
some examples, established in the literature of the subject. The Section is closed by final
conclusions.
\section{Calculations for Barab\'asi-Albert networks}
The time evolution of $r(i,j)$ is determined by the equation of motion \cite{my1}
\begin{equation}
\frac{dr(i,j)}{dt}=\Big\{1-\Big(\frac{r(i,j)}{R}\Big)^2\Big\}\sum_k r(i,k) r(k,j)
\end{equation}
where $R$ is a sociologically justified limitation on the absolute value of $r(i,j)$ \cite{my1}.
Here $R=5.0$. Initial values of $r(i,j)$ are random numbers, uniformly distributed in the range
$(-0.5,0.5)$.
The equation is solved numerically with the Runge-Kutta IV method with variable length of timestep
\cite{RK4}, simultaneously
for all pairs $(i,j)$ of linked nodes. The method of construction of BA networks was described
in \cite{MK}. The connectivity parameter is selected to be $M=7$, because in this case the
probability $p(M)$ of HB has a clear minimum for BA networks of $N=100$ nodes, and
$p(M=7)\approx 0.5$ (see Fig. 1). This choice of $M$ is motivated by our aim to falsify the result on the
polarization of opinions. This polarization was demonstrated \cite{my1} to be a consequence of HB;
therefore, the question here is if it appears also when HB is not present. An example of
time evolution of such a network is shown in Fig. 2.
Our result is that the polarization is present in all investigated cases. As time
increases,
the distribution of $r(i,j)$ gets wider and finally it reaches a stable shape, with two large peaks
at $r(i,j)\approx\pm R$ and one smaller peak at the centre, where $r(i,j)\approx 0$.
In Fig. 3, we show a series of histograms of $r(,j)$ in subsequent times (A-E).
Particular networks differ quantitatively with respect to the heights of the peaks, but these
differences are small.
We note here that when some links are absent, the definition of HB should be somewhat relaxed,
because some other links, which do not enter to any triad $(i,j,k)$, will not evolve at all.
Therefore we should admit that some negative relations survive within a given group. We classify a
final state of the graph as HB if there are no chains of friendly relations between the subgroups.
On the other hand, more than two mutually hostile subgroups can appear. These facts were recognized
already in literature \cite{hara,wt}. Surprisingly enough, subgroups of $1<N<97$ nodes are never
found in our BA networks. On the contrary, in the exponential networks groups of all sizes
were detected. In Figs. 4 and 5 we show diagrams for BA networks and exponential networks, respectively.
Each point at these diagrams marks the value of $r(i,j)$ and the size of the subgroup which
contains nodes $(i,j)$. Links between different subgroups are omitted. We see that for BA
networks (Fig.4), the lowest value of $N$ is 97. The remaining three nodes are linked with all
other nodes by hostile relations.
\section{Examples}
In Ref. \cite{my1}, an example of polarization of opinions on the lustration law in Poland in 1999
was brought up. The presented statistical data \cite{cbos} displayed two maxima at negative and
positive opinions and a lower value at the centre of the plot. In our simulations performed for
fully connected graphs \cite{my1}, the obtained value for the center was zero. However, it is clear
that in any group larger than, say, 50 persons some interpersonal relations will be absent. Taking this
into account, we can claim than the statistical data of \cite{cbos} should be compared to the
results discussed here rather than to those for a fully connected graph. Here we reproduce a
peak of the histogram at its centre, on the contrary to the results in \cite{my1}. This fact
allows to speak on a qualitative accordance of the results of our calculations with the statistical
data of \cite{cbos}.
Next example is the set of data of the attendance of 18 'Southern women' in local meetings in
Natchez, Missouri, USA in 1935 \cite{free}. These data were used to compare 21 methods of finding
social groups. The results were analysed with respect to their consensus, and ranked with consensus
index from 0.543 (much worse than all others) to 0.968. To apply our dynamics, we use the
correlation function $<p(i,j)>-<p(i)><p(j)>$ as initial values of $r(i,j)$. Our method produced
the division (1-9) against (10-18), what gives the index value 0.968. As a by-product, the method
can provide the time dynamics of the relations till HB and, once HB is reached, the leadership within
the cliques \cite{bl}. We should add that actually, we have no data on the possible friendship or
hostility between these women, then the interpretation of these results should be done with care.
Last example is the set of data about a real conflict in the Zachary karate club \cite{za,bonet,gir}.
The input data are taken from \cite{wbpg}. All initial values of the matrix elements are reduced
by a constant $\epsilon$ to evade the case of overwhelming friendship. The obtained splitting of
the group is exactly as observed by Zachary: (1-8,11-14,17,18,20,22) against
(9,10,15,16,19,21,23-34). These results were checked not to vary for $\epsilon$ between 1.0
and 3.0. The status of all group members can be obtained with the same method as in the previous
example.
To conclude, the essence of Eq. (1) is the nonlinear coupling between links $r(i,j)$, which produces the positive
feedback between
the actual values of the relations and their time evolution. We should add that the idea of such
a feedback is not entirely new.
It is present, for example, in Boltzmann-like nonlinear master equations applied to behavioral
models \cite{hlb}. On the contrary, it is absent in later works on formal theory of social influence
\cite{cons}. On the other hand, the theories of status \cite{bl}
are close to the method of transition matrix, known in non-equilibrium statistical mechanics
\cite{re}.
\bigskip
|
{
"timestamp": "2005-03-11T17:49:04",
"yymm": "0503",
"arxiv_id": "physics/0503085",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503085"
}
|
\section{Introduction \label{In}}
\bigskip
Although there is a long history of theoretical work on the solution of
the Coulomb problem in three-particle scattering
\cite{alt:78a,berthold:90a,kievsky:96a,kievsky:01a,chen:01a,alt:02a,
suslov:04a}, the work of Refs.~\cite{kievsky:96a,kievsky:01a}
pioneered the effort on fully converged numerical calculations for
proton-deuteron $(pd)$ elastic scattering including
the Coulomb repulsion between protons together with realistic nuclear
interactions. In their work the authors use the charge-dependent AV18
potential together with the Urbana IX three-nucleon force and
proceed to solve the three-particle Schr\"odinger equation using the Kohn
variational principle (KVP); the wave function satisfies appropriate Coulomb
distorted asymptotic boundary conditions and is expanded at short distances
in a pair correlated hyperspherical harmonics basis set. The results
presented were fully converged vis-\`a-vis the size of the basis set and the
angular momentum states included in the calculation.
In parallel a benchmark calculation was performed~\cite{kievsky:01b}
where results obtained variationally
were compared with those obtained from the solution of coordinate-space
Faddeev equations for the AV14 potential
at energies below three-body breakup threshold.
In a recent publication~\cite{deltuva:05a} the momentum-space solution of the
Alt-Grassberger-Sandhas (AGS) equation~\cite{alt:67a} for two protons
and a neutron was successfully applied, not only to $pd$ elastic
scattering but also to radiative $pd$ capture and two-body
electromagnetic disintegration of ${}^3\mathrm{He}$.
The treatment of the Coulomb interaction is based on the ideas
proposed by Taylor~\cite{taylor:74a} for two charged particle scattering and
extended in Ref.~\cite{alt:78a} for three-particle scattering with
two charged particles alone. The Coulomb potential is screened, standard
scattering theory for short-range potentials is used, and the obtained
results are corrected for the unscreened limit using the renormalization
prescription~\cite{taylor:74a,alt:78a}. The results presented in
Ref.~\cite{deltuva:05a} are converged vis-\`a-vis the screening radius $R$
and the number of included two-body and three-body angular momentum states.
Although in Ref.~\cite{deltuva:05a} the hadron dynamics is based
on the purely nucleonic charge-dependent (CD) Bonn potential
and its realistic coupled-channel extension
CD Bonn + $\Delta$, allowing for single virtual
$\Delta$-isobar excitation,
other realistic potential models may be used easily as well.
Motivated by recent experimental efforts in the measurements of $pd$ elastic
observables~\cite{sagara:94a,exp1,exp2},
in the present paper we present benchmark
results for a number of $pd$ elastic scattering observables,
both below and above three-body breakup threshold, using
the charge-dependent AV18~\cite{wiringa:95a} two-nucleon potential
and no three-nucleon force.
In Sec.~\ref{Methods} we make a short
description of the methods we use, in Sec.~\ref{sec:results} we present
the results, and in Sec.~\ref{sec:conclusions} the conclusions.
\section{The Methods \label{Methods}}
In this section we briefly introduce both methods and provide the basic
framework for a general understanding of the technical procedures;
further details may be found in the appropriate references.
We choose to describe the method based on KVP using its traditional notation,
which we attempt to carry over to the discussion of the integral equation
approach in Sec.~\ref{sec:IE} and \ref{sec:results}.
Therefore the presentation of the integral equation approach will not be
in the notation used in Ref.~\cite{deltuva:05a}.
\subsection{The Kohn variational principle}
\def{\bf x}{{\bf x}}
\def{\bf y}{{\bf y}}
\def{\rho}{{\bf r}}
\def{\alpha}{{\alpha}}
\def{\rho}{{\rho}}
\def{\alpha\alpha'}{{\alpha\alpha'}}
\def{kk'}{{kk'}}
\def\rightarrow{\rightarrow}
\def{\hbar^2\over M_N}{{\hbar^2\over M_N}}
The KVP can be used to describe nucleon-deuteron $(Nd)$ elastic scattering.
Below the three-body breakup threshold the
collision matrix is unitary and the problem can be formulated
in terms of the real reactance matrix ($K$--matrix).
Above the three-body breakup threshold the elastic part of the
collision matrix is no longer unitary and the formulation in terms of the
$S$-matrix, the complex form of the KVP, is convenient.
Referring to Refs.~\cite{kievsky:01a,KRV99,VKR00} for details,
a brief description of the method is given below.
The scattering wave function (w.f.) $\Psi$ is
written as sum of two terms
\begin{equation}
\Psi=\Psi_C+\Psi_A \
\label{eq:psi}
\end{equation}
which carry the appropriate asymptotic boundary conditions.
The first term, $\Psi_C$, describes the
system when the three--nucleons are close to each other. For large
interparticle separations and energies below the
three-body breakup threshold it goes to zero,
whereas for higher energies it must
reproduce a three outgoing particle state. It
is written as a sum of three Faddeev--like amplitudes
corresponding to the three cyclic permutations of the particle indices.
Each amplitude $\Psi_C({\bf x}_i,{\bf y}_i)$, where ${\bf x}_i,{\bf y}_i$ are
the Jacobi coordinates corresponding to the $i$-th permutation, has
total angular momentum $JJ_z$ and total isospin $TT_z$ and is
decomposed into channels using $LS$ coupling, namely
\begin{eqnarray}
\Psi_C({\bf x}_i,{\bf y}_i) &=& \sum_{\alpha=1}^{N_c} \phi_\alpha(x_i,y_i)
{\cal Y}_\alpha (jk,i) \\
{\cal Y}_\alpha (jk,i) &=&
\Bigl\{\bigl[ Y_{\ell_\alpha}(\hat x_i) Y_{L_\alpha}(\hat y_i)
\bigr]_{\Lambda_\alpha} \bigl [ s_\alpha^{jk} s_\alpha^i \bigr ]
_{S_\alpha}
\Bigr \}_{J J_z} \; \bigl [ t_\alpha^{jk} t_\alpha^i \bigr ]_{T T_z},
\end{eqnarray}
where $x_i,y_i$ are the moduli of the Jacobi coordinates and
${\cal Y}_\alpha$ is the angular-spin-isospin function for each channel.
The maximum number of channels considered in the expansion is $N_c$.
The two-dimensional amplitude $\phi_\alpha$ is expanded in terms of the
pair correlated hyperspherical harmonic basis \cite{KVR93,KVR94}
\begin{equation}
\phi_\alpha(x_i,y_i) = \rho^{-5/2} f_\alpha(x_i)
\left[ \sum_K u^\alpha_K(\rho) {}^{(2)}P^{\ell_\alpha,L_\alpha}_K(\phi_i)
\right] \ ,
\label{eq:PHH}
\end{equation}
where the hyperspherical variables, the hyperradius $\rho$ and
the hyperangle $\phi_i$, are defined by the relations
$x_i=\rho\cos{\phi}_i$ and $y_i=\rho\sin{\phi}_i$. The factor
${}^{(2)}P^{\ell,L}_K(\phi)$ is a hyperspherical polynomial and
$f_\alpha(x_i)$ is a pair correlation function introduced
to accelerate the convergence of the expansion. For small values
of the interparticle distance $f_\alpha(x_i)$ is regulated by the
$NN$ interaction whereas for large separations
$f_\alpha(x_i)\rightarrow 1$.
The second term, $\Psi_A$, in the variational wave function of
Eq.(\ref{eq:psi})
describes the asymptotic motion of a deuteron relative to the third
nucleon. It can also be written as a sum
of three amplitudes with the generic one having the form
\begin{equation}
\Omega^\lambda_{LSJ}({\bf x}_i,{\bf y}_i) = \sum_{l_{\alpha}=0,2} w_{l_{\alpha}}(x_i)
{\cal R}^\lambda_L (y_i)
\left\{\left[ [Y_{l_{\alpha}}({\hat x}_i) s_{\alpha}^{jk}]_1 s^i \right]_S
Y_L({\hat y}_i) \right\}_{JJ_z}
[t_{\alpha}^{jk}t^i]_{TT_z}\ , \label{eq:omega}
\end{equation}
where $w_{l_{\alpha}}(x_i)$ is the deuteron w.f. radial component in the
state $l_{\alpha} =0,2$.
In addition, $s_{\alpha}^{jk}=1,t_{\alpha}^{jk}=0$ and $L$ is the relative nucleon-deuteron
angular momentum. The superscript $\lambda$ indicates
the regular ($\lambda\equiv R$) or the irregular ($\lambda\equiv I$)
solution. In the $pd$ $(nd)$ case, the functions
${\cal R}^\lambda$ are related to
the regular or irregular Coulomb (spherical Bessel) functions.
The functions $\Omega^\lambda$ can be combined to form a general
asymptotic state ${}^{(2S+1)}L_J$
\begin{equation}
\Omega^+_{LSJ}({\bf x}_i,{\bf y}_i) = \Omega^0_{LSJ}({\bf x}_i,{\bf y}_i)+
\sum_{L'S'}{}^J{\cal L}^{SS'}_{LL'}\Omega^1_{L'S'J}({\bf x}_i,{\bf y}_i) \ ,
\end{equation}
where
\begin{eqnarray}
\Omega^0_{LSJ}({\bf x}_i,{\bf y}_i) =& u_{00}\Omega^R_{LSJ}({\bf x}_i,{\bf y}_i)+
u_{01}\Omega^I_{LSJ}({\bf x}_i,{\bf y}_i) \ , \\
\Omega^1_{LSJ}({\bf x}_i,{\bf y}_i) =& u_{10}\Omega^R_{LSJ}({\bf x}_i,{\bf y}_i)+
u_{11}\Omega^I_{LSJ}({\bf x}_i,{\bf y}_i) \ .
\end{eqnarray}
The matrix elements $u_{ij}$ can be selected according to the
four different choices of the matrix ${\cal L}=$ $K$-matrix,
$K^{-1}$-matrix, $S$-matrix or $T$-matrix. A general
three-nucleon scattering w.f. for an incident
state with relative angular momentum $L$,
spin $S$ and total angular momentum $J$ is
\begin{equation}
\Psi^+_{LSJ}=\sum_{i=1,3}\left[ \Psi_C({\bf x}_i,{\bf y}_i)+\Omega^+_{LSJ}({\bf x}_i,{\bf y}_i)
\right] \ ,
\end{equation}
and its complex conjugate is $\Psi^-_{LSJ}$. A variational estimate of the
trial parameters in the w.f. $\Psi^+_{LSJ}$ can
be obtained by requiring, in accordance with
the generalized KVP, that the functional
\begin{equation}
[{}^J{\cal L}^{SS'}_{LL'}]= {}^J{\cal L}^{SS'}_{LL'}-{\frac{2}{{\rm det}(u)}}
\langle\Psi^-_{LSJ}|H-E|\Psi^+_{L'S'J}\rangle \ ,
\label{eq:kohn}
\end{equation}
be stationary. Below the three-body breakup threshold,
due to the unitarity of the
$S$-matrix, the four forms for the ${\cal L}$-matrix are equivalent.
However, it was shown that when the
complex form of the principle is used, there is a considerable
reduction of numerical instabilities~\cite{kiev97}.
Above the three-body breakup threshold it is convenient to formulate
the variational principle
in terms of the $S$--matrix. Accordingly, we get the following functional:
\begin{equation}
[{}^J{S}^{SS'}_{LL'}]= {}^J{S}^{SS'}_{LL'}+{i}
\langle\Psi^-_{LSJ}|H-E|\Psi^+_{L'S'J}\rangle \ .
\label{eq:ckohn}
\end{equation}
The variation of the functional with respect to the hyperradial
functions $u^\alpha_K(\rho)$
leads to the following set of coupled equations:
\begin{equation}
\sum_{\alpha',k'}
\Bigl[ A^{\alpha\alpha'}_{kk'} ({\rho} ){d^2\over d{\rho}^2}+ B^{\alpha\alpha'}_{kk'} ({\rho} ){d\over d{\rho}}
+ C^{\alpha\alpha'}_{kk'} ({\rho} )+
{M_N\over\hbar^2} E\; N^{\alpha\alpha'}_{kk'} ({\rho} )\Bigr ]
u^{\alpha'}_{k'}({\rho})= D^\lambda_{\alpha k}(\rho) \ .
\label{eq:siste}
\end{equation}
For each asymptotic state $^{(2S+1)}L_J$ two different inhomogeneous terms
are constructed corresponding to the asymptotic $\Omega^\lambda_{LSJ}$
functions with $\lambda\equiv 0,1$. Accordingly, two sets of solutions
are obtained and
combined to minimize the functional (\ref{eq:ckohn}) with respect to
the $S$-matrix elements. This is the first order solution, the second order
estimate of the $S$-matrix is obtained after replacing the first order
solution in Eq.(\ref{eq:ckohn}).
In order to solve the above system of equations appropriate
boundary conditions must be specified for the hyperradial functions.
For energies below the three-body breakup threshold
they go to zero when $\rho\rightarrow\infty$, whereas
for higher energy they asymptotically describe the breakup configuration.
The boundary conditions to be applied in this case have
been discussed in Refs.~\cite{kievsky:01a,VKR00,KVR97}
and are briefly illustrated below.
To simplify the notation let us label the basis
elements with the index $\mu\equiv[{\alpha},K]$,
and introduce the completely antisymmetric correlated
spin-isospin-hyperspherical basis element ${\cal Q}_\mu(\rho,\Omega)$
as linear combinations of the products
\begin{equation}
\label{eq:bco}
\sum_{i=1}^3
f_{\alpha}(x_i)\; {}^{(2)}P^{\ell_\alpha,L_\alpha}_K(\phi_i)
{\cal Y}_\alpha(jk,i) \ ,
\end{equation}
which depend on $\rho$ through the correlation factor.
In terms of the ${\cal Q}_\mu(\rho,\Omega)$ the internal part is written as
\begin{equation}
\Psi_C= \rho^{-5/2}\sum_{\mu=1}^{N_m}
\omega_{\mu}(\rho) {\cal Q}_\mu(\rho,\Omega) \ ,
\end{equation}
with $N_m$ the total number of basis functions considered.
The hyperradial functions $u_\mu(\rho)$ and $\omega_{\mu}(\rho)$
are related by an unitary transformation imposing that
the ``uncorrelated'' basis elements ${\cal Q}^0_\mu(\Omega)$,
obtained by setting all the correlation functions $f_{\alpha}(x_i)=1$,
form an orthogonal basis.
Explicitly, the matrix elements of the norm $N$ behave as
\begin{equation}
N_{\mu\mu'}(\rho)= \int d\Omega\;
{\cal Q}_\mu(\rho,\Omega)^\dag
{\cal Q}_{\mu'} (\rho,\Omega) \rightarrow N^{(0)}_{\mu\mu'}
+{ N^{(3)}_{\mu\mu'}\over \rho^3}+{\cal O}(1/\rho^5)
\ ,\qquad {\rm for\ }\rho\rightarrow\infty\ , \label{eq:n}
\end{equation}
where, in particular,
\begin{equation}
N^{(0)}_{\mu\mu'}= \int d\Omega\;
{\cal Q}^0_\mu (\Omega)^\dag
{\cal Q}^0_{\mu'} (\Omega)\ . \label{eq:n1}
\end{equation}
is diagonal with diagonal elements ${\cal N}_\mu$
either $1$ or $0$. Therefore, some correlated elements
have the property: ${\cal Q}_\mu(\rho,\Omega) \rightarrow 0$ as $\rho\rightarrow\infty$.
In the following we arrange the new basis in such a way that for values
of the index $\mu\le\overline{N}_m$ the eigenvalues of the
norm are ${\cal N}_\mu=1$ and
for $\overline{N}_m+1 \le \mu \le N_m$ they are ${\cal N}_\mu=0$.
For $\rho\rightarrow\infty$, neglecting terms going to zero faster
than $\rho^{-2}$, the asymptotic expression of the set of Eqs.(\ref{eq:siste})
rotated using the unitary transformation defined above,
reduces to the form
\begin{equation}
\label{eq:c0}
\sum_{\mu'} \biggl\{
-{\hbar^2\over M_N} \left( {d^2\over d\rho^2} -{{\cal K}_\mu({\cal K}_\mu +1)\over\rho^2}
+ Q^2 \right ){\cal N}_\mu
\delta_{\mu,\mu'} +
{2\;Q\; \chi_{\mu\mu'}\over \rho} \;
+{\cal O}({1\over\rho^3})\biggr\}\omega_{\mu'}(\rho) = 0 \ ,
\end{equation}
where $E=\hbar^2 Q^2/M_N$ and ${\cal K}_\mu= G_\mu+3/2$. Here $G_\mu$ is
the grand-angular quantum number defined as $G_\mu=l_\alpha+L_\alpha + 2 K$
and the matrix $\chi$ is defined as
\begin{equation}
\label{eq:c}
{ \chi}_{\mu\mu'}= \int d\Omega\;
{\cal Q}^0_{\mu} (\Omega)^\dag
\; \hat \chi \;
{\cal Q}^0_{\mu'} (\Omega)
\ .
\end{equation}
The dimensionless operator $\hat\chi$ originates from the Coulomb interaction
as
\begin{equation}
\hat \chi = {M_N\over 2\hbar^2 Q}
\sum_{i=1}^3 {e^2\over \cos\phi_i} {1+\tau_{j,z} \over 2}
{1+\tau_{k,z} \over 2} \ .
\label{eq:chi}
\end{equation}
It should be noticed that $\chi_{\mu\mu'}=0$ if $\mu,\mu'>{\overline{N}_m}$.
In practice,
the functions $\omega_\mu(\rho)$ are chosen to be regular at the origin, i.e.
$\omega_\mu(0)=0$ and, in accordance with the equations to be satisfied for
$\rho\rightarrow\infty$, to have the following behavior
($\mu\le\overline{N}_m$)
\begin{equation}
\label{eq:asy2}
\omega_\mu(\rho) \rightarrow
- \sum_{\mu'=1}^{\overline{N}_m}
\left ( e^{-i {\hat \chi} \ln 2 Q\rho} \right)_{\mu\mu'}\;
b_{\mu'} \; e^{i Q\rho} \ ,
\end{equation}
where $ b_{\mu'}$ are unknown coefficients. This form corresponds
to the asymptotic behavior of three outgoing particles
interacting through the Coulomb potential~\cite{merkuriev2}.
In the case of $nd$ scattering ($\chi\equiv 0$)
the outgoing solutions evolve as outgoing Hankel functions
$H^{(1)}(Q\rho)$ ($\omega_\mu(\rho)\rightarrow -b_\mu e^{iQ\rho}$).
For values of the index $\mu > \overline{N}_m$ the eigenvalues of the
norm are ${\cal N}_\mu=0$ and the leading terms
in Eq.(\ref{eq:c0}) vanish. So, the asymptotic behavior of these
$\omega_\mu$ functions is governed by the next order terms.
However, for $\mu > \overline{N}_m$, it is verified that
$\omega_\mu{\cal Q}_\mu\rightarrow 0$ as $\rho\rightarrow\infty$.
In order to solve the system of Eqs.(\ref{eq:siste})
the hyperradial functions are expanded in terms of Laguerre
polynomials plus an auxiliary function
\begin{equation} \label{eq:M}
\omega_\mu(\rho)=\rho^{5/2}\sum_{m=0}^M A^m_{\mu}
L^{(5)}_m(z)\exp(-{z\over 2}) +A^{M+1}_{\mu} \overline \omega_{\mu}(\rho) \ ,
\end{equation}
where $z=\gamma\rho$ and $\gamma$ is a nonlinear parameter.
The linear parameters $A^m_{\mu}$ $(m=0,....,M+1)$
are determined by the variational procedure.
The inclusion of the auxiliary functions $\overline \omega_{\mu}(\rho)$
defined in Eq.(\ref{eq:M}) is useful for reproducing the oscillatory
behavior shown by the hyperradial functions for $\rho\gtrsim 30$ fm.
Otherwise
a rather large number $M$ of polynomials should be included in
the expansion. A convenient choice is to take them
as the regular solutions of a one dimensional
differential equation corresponding to the $\mu$-th equation of
the system whose asymptotic behavior is the one of Eq.(\ref{eq:c0}).
In the cases considered here
the solutions obtained for the $S$-matrix stabilize
for values of the matching radius $\rho_0>100$ fm.
\subsection{The integral equation approach \label{sec:IE}}
The integral equation to be solved is the AGS equation~\cite{alt:67a}
for three-particle scattering where each pair of nucleons
interacts through the strong potential $v$ and the Coulomb potential
$w$ acts only between charged nucleons. The work in
Ref.~\cite{deltuva:05a} follows the seminal work of
Refs.~\cite{taylor:74a,alt:78a} in the sense that the treatment of the
Coulomb interaction is based
on screening, followed by the use of standard scattering theory for
short-range potentials and renormalization of the obtained results in order
to correct for the unscreened limit. Nevertheless there are important
differences relative to Ref.~\cite{alt:02a} that are paramount to the fast
convergence of the calculation in terms of screening radius R and the
effective use of realistic interactions:
{\bf a)} We work with a screened Coulomb potential
\begin{gather}
w_R(r) = w(r) \; e^{-(r/R)^n}
\end{gather}
where $w (r) = \frac{\alpha}{r}$ is the true Coulomb
potential, $\alpha$ being the fine structure constant and $n$ a power
controlling the smoothness of the screening. We prefer to work with a
sharper screening than the Yukawa screening $(n=1)$ of Ref.~\cite{alt:02a}
because we want to ensure that the screened Coulomb potential $w_R$
approximates well the true Coulomb one $w$ for distances $r<R$ and
simultaneously vanishes rapidly for $r>R$, providing a comparatively rapid
convergence of the partial wave expansion. In contrast, the sharp cutoff
$(n \to \infty)$ yields unpleasant oscillatory behavior in momentum space
representation, leading to convergence problems. We find values
$3 \le n \le 6$ to provide a sufficient smoothness and fast convergence;
$n = 4$ is used for the calculations of this paper.
{\bf b)} Although the choice of the screened potential improves the
partial wave convergence, the practical implementation of the solution of
AGS equation still places a technical difficulty, i.e., the calculation of
the AGS operators for nuclear plus screened Coulomb potentials requires
two-nucleon partial waves with pair orbital angular momentum considerably
higher than required for the hadronic potential alone. In this context the
perturbation theory for higher two-nucleon partial waves developed in
Ref.~\cite{deltuva:03b} is a very efficient and reliable technical tool for
treating the screened Coulomb interaction in high partial waves.
As a result of these two technical implementations, the
method~\cite{deltuva:03a} that was developed before for solving
three-particle AGS equations without Coulomb could be successfully used in
the presence of screened Coulomb. Using the usual three-body notation, the
full multichannel transition matrix reads
\begin{subequations}\label{eq:a+b}
\begin{gather} \label{eq:Uba}
\begin{align}
U^{(R)}_{\beta \alpha}(Z) = {} & \bar{\delta}_{\beta \alpha} G_0^{-1}(Z)
+ \sum_{\sigma} \bar{\delta}_{\beta \sigma} T^{(R)}_\sigma (Z) G_0(Z)
U^{(R)}_{\sigma \alpha}(Z),
\end{align}
\end{gather}
\noindent where the superscript $(R)$ denotes the dependence on the screening
radius $R$ of the Coulomb potential, $G_0(Z) = (Z - H_0)^{-1}$ the free
resolvent, $\bar{\delta}_{\beta \alpha} = 1- \delta_{\beta\alpha}$, and
\begin{gather} \label{eq:TR}
\begin{align}
T^{(R)}_\alpha (Z) = {}& (v_\alpha + w_{\alpha R}) +
(v_\alpha + w_{\alpha R}) G_0(Z) T^{(R)}_\alpha (Z).
\end{align}
\end{gather}
\end{subequations}
The two-particle transition matrix $T^{(R)}_\alpha (Z)$ results from
the nuclear interaction $v_{\alpha}$ between hadrons plus the screened
Coulomb $w_{\alpha R}$ between charged nucleons ($w_{\alpha R} = 0$ otherwise).
As expected the full multichannel transition matrix
$U^{(R)}_{\beta \alpha}(Z)$ must contain the pure Coulomb transition
matrix $T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} (Z)$ derived from the screened Coulomb
$W^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}$ between the spectator proton and the center of mass
(c.m.) of the remaining neutron-proton $(np)$ pair in channel $\alpha$
\begin{gather} \label{eq:Tcm}
T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} (Z) = W^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} +
W^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} G^{(R)}_{\alpha} (Z) T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} (Z),
\end{gather}
where $W^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} = 0$ for $n(pp) \; \;\alpha$ channels
and $G^{(R)}_{\alpha}$ the channel resolvent
\begin{gather} \label{eq:GRa}
G^{(R)}_\alpha (Z) = (Z - H_0 - v_\alpha - w_{\alpha R})^{-1}.
\end{gather}
In a system of two charged particles and a neutral one, when
$w_{\alpha R} = 0$, $\; W^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R} \ne 0 $ and vice versa.
As demonstrated in Refs.~\cite{alt:78a,deltuva:05a} the split of
the multichannel transition matrix
\begin{gather} \label{eq:GR3}
U^{(R)}_{\beta \alpha}(Z) = \delta_{\beta\alpha}
T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z) + [ U^{(R)}_{\beta \alpha}(Z) -
\delta_{\beta\alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z)]
\end{gather}
into a long-range part $ \delta_{\beta \alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z) $
and a Coulomb distorted short-range part
$[U^{(R)}_{\beta\alpha}(Z) - \delta_{\beta\alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z)]$
is extremely convenient to recover the unscreened Coulomb limit.
According to Refs.~\cite{alt:78a,deltuva:05a} the full $pd$ transition
amplitude $ \langle \phi_\beta (\mbf{q}_f) \nu_{\beta_f} | U_{\beta \alpha}
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle $ for the initial
and final channel states with relative $pd$ momentum $\mbf{q}_i$ and
$\mbf{q}_f$, $q_f = q_i$, energy $E_{\alpha}(q_i)$, and discrete
quantum numbers $\nu_{\alpha_i}$ and $\nu_{\beta_f}$,
is obtained via the renormalization of the on-shell
$U^{(R)}_{\beta \alpha}(Z)$ with $Z = E_{\alpha}(q_i) + i0$
in the infinite $R$ limit. For the screened
Coulomb transition matrix $T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z)$,
contained in $ U^{(R)}_{\beta \alpha}(Z)$, that limit can be carried out
analytically, yielding the proper Coulomb transition amplitude
$\langle \phi_\beta (\mbf{q}_f) \nu_{\beta_f} |T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha C}
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle $ \cite{alt:78a,taylor:74a},
while the Coulomb distorted short-range part requires the explicit use of a
renormalization factor,
\begin{gather} \label{eq:UC2}
\begin{split}
\langle \phi_\beta (\mbf{q}_f) & \nu_{\beta_f} | U_{\beta \alpha}
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle \\ = {}&
\delta_{\beta \alpha}
\langle \phi_\beta (\mbf{q}_f) \nu_{\beta_f} |T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha C}
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle \\ & +
\lim_{R \to \infty} \{ \mathcal{Z}_R^{-\frac12}(q_f)
\langle \phi_\beta (\mbf{q}_f) \nu_{\beta_f} |
[ U^{(R)}_{\beta \alpha}(E_\alpha(q_i) + i0) \\ & -
\delta_{\beta\alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(E_\alpha(q_i) + i0)]
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle
\mathcal{Z}_R^{-\frac12}(q_i) \}.
\end{split}
\end{gather}
The renormalization factor
\begin{subequations}
\begin{gather}\label{eq:zrq}
\mathcal{Z}_R(q) = e^{-2i \phi_R(q)},
\end{gather}
contains a phase $\phi_R(q)$ which, though independent of the
$pd$ relative orbital momentum $L$ in the infinite $R$ limit,
is given by \cite{taylor:74a}
\begin{gather} \label{eq:phiRl}
\phi_R(q) = \sigma_L(q) -\eta_{LR}(q),
\end{gather}
\end{subequations}
where $\eta_{LR}(q)$ is the diverging screened Coulomb phase
shift corresponding to standard boundary conditions, and $\sigma_L(q)$
the proper Coulomb phase referring to logarithmically distorted
Coulomb boundary conditions. The limit of the Coulomb distorted
short-range part of the multichannel transition matrix
$[ U^{(R)}_{\beta\alpha}(Z) - \delta_{\beta\alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z)]$
has to be performed
numerically but, due to its short-range nature, the limit is reached
with sufficient accuracy at finite screening radii $R$. Furthermore,
due to the choice of screening and perturbation technique to deal with
high angular momentum states, $[ U^{(R)}_{\beta \alpha}(Z) -
\delta_{\beta\alpha} T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(Z)]$ is calculated through the
numerical solution of Eqs.~(\ref{eq:a+b}) and
(\ref{eq:Tcm}), using partial-wave expansion.
In actual calculations we use the isospin formulation and, therefore, the
nucleons are considered identical. Instead of Eq.~\eqref{eq:Uba} we
solve a symmetrized AGS equation
\begin{gather} \label{eq:UR}
U^{(R)}(Z) = P G_0^{-1}(Z) + P T^{(R)}_{\alpha}(Z) G_0(Z) U^{(R)}(Z),
\end{gather}
$P$ being the sum of the two cyclic three-particle permutation operators,
and use a properly symmetrized $pd$ transition amplitude
\begin{gather}
\begin{split}\label{eq:Uasym}
\langle \phi_\alpha (\mbf{q}_f) & \nu_{\alpha_f} |
U |\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle \\ = {} &
\langle \phi_\alpha (\mbf{q}_f) \nu_{\alpha_f} |
T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha C}|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle
\\ & + \lim_{R \to \infty}
\{ \mathcal{Z}_R^{-\frac12}(q_f) \langle \phi_\alpha (\mbf{q}_f) \nu_{\alpha_f}|
[ U^{(R)}(E_\alpha(q_i) + i0) \\ & -
T^{\mathrm{c\!\:\!.m\!\:\!.}}_{\alpha R}(E_\alpha(q_i) + i0)]
|\phi_\alpha (\mbf{q}_i) \nu_{\alpha_i} \rangle \mathcal{Z}_R^{-\frac12}(q_i) \}
\end{split}
\end{gather}
for the calculation of observables.
For further technical details we refer to Ref.~\cite{deltuva:05a}.
\section{Results \label{sec:results}}
\bigskip
In this section we compare numerical calculations for a number of
elastic observables performed using the KVP and the integral equation
approach. Three different lab energies have been considered: $3$, $10$, and
$65$ MeV. The Coulomb effects are expected to be
sizable in most of the observables at the first two energies.
The two methods use a different scheme to construct the
scattering states with total angular momentum and parity $J^\pi$.
In the KVP the $LS$ coupling is used and channels are ordered by
increasing values of $\ell_\alpha + L_\alpha$. The expansion of
the scattering state is truncated at values
$\ell_\alpha + L_\alpha=L_{max}+2$, where $L_{max}$ is the maximum value
of $L$ corresponding to the asymptotic states ${}^{(2S+1)}L_J$.
In the integral equation approach the $jj$ scheme has been used. The
channels have been ordered for increasing values of the two-body angular
momentum $j$ and for the strong interaction
the maximum value $j_{max}=5$ has been considered
for the first two energies whereas at $65$ MeV the value $j_{max}=6$
has been used; the screened Coulomb interaction is taken into
account up to $j_{max}=25$ as described in Ref.~\cite{deltuva:05a}.
Both numerical calculations presented here are
converged relative to the number of three-body partial waves. In addition,
the variational calculations are converged relative to the size of
the hyperspherical basis set and, in the integral equation approach, the
results are converged with respect to the screening radius $R$.
In Figs.~\ref{fig:obs1} and \ref{fig:obs2}
we compare the differential cross section and vector and tensor
analyzing powers for $pd$ elastic scattering at the three selected
energies, $3$, $10$ and $65$ MeV proton lab energies.
In Fig.~\ref{fig:stc} a selection of spin transfer coefficients at
$65$ MeV is shown. In the figures, two different
curves are shown corresponding to calculations using the KVP
(thin solid line) and integral equation approach (dotted line).
By inspection of the figures one may conclude that the agreement is excellent
because the numerical calculations agree to better than 1\%.
In fact the curves are practically one on top of the other, the exceptions
being the maximum of $T_{21}$ and some spin transfer coefficients
at $65$ MeV in which a small disagreement is observed.
Nevertheless, it is important to mention that in all cases the difference
between the two curves is smaller than the experimental accuracy
for the corresponding data sets. Likewise the agreement between the
two calculations largely exceeds the agreement of any of them with
the data as shown in Refs.~\cite{kievsky:01a,deltuva:05a}.
The present results can be used to study Coulomb effects by comparing
$nd$ to $pd$ calculations.
In Fig.~\ref{fig:obs3} we analyze the evolution of the Coulomb effects
for the differential cross section, the nucleon analyzing
power $A_y$ and two tensor analyzing powers, $T_{20}$ and $T_{21}$ at
$3$, $10$ and $65$ MeV proton lab energies.
In order to reduce the number of curves in the figure for the sake of clarity
we present results obtained using the integral equation approach. The results
obtained using the KVP for $nd$ scattering agree at the same level
already shown for the $pd$ case in the previous figures.
In Fig.~\ref{fig:obs3} the thin solid line denotes the $pd$ calculation
whereas the dotted line denotes the corresponding $nd$ calculation.
The latter agrees well with
the results of other existing $nd$ calculations~\cite{witala:pc}.
From the figure we observe that Coulomb effects are appreciable at
$3$ and $10$ MeV but are considerably reduced at $65$ MeV.
A more exhaustive analysis on Coulomb effects can be found
in Refs.~\cite{kievsky:01a,KRV01,deltuva:05a}.
In addition to the benchmark comparison using AV18 potential
we also give one result for the Malfliet-Tjon (MT) I-III potential,
in order to resolve an existing problem.
Reference \cite{suslov:04a} reports a disagreement between
$pd$ phase shifts results for MT I-III potential
calculated using the first technique of this paper,
the KVP \cite{kievsky:01a},
and the configuration-space Faddeev equations \cite{suslov:04a}.
The calculation based on the second technique of this paper,
the momentum-space integral equations \cite{deltuva:05a},
clearly confirms the results of Ref.~\cite{kievsky:01a}.
A detailed comparison of $pd$ and $nd$ phase shift results for
MT I-III potential is given in Table~\ref{tab:MT}.
In the following we discuss some of the limitations inherent to the
two methods used to describe $pd$ elastic scattering.
The KVP, as presented here, reduces the scattering problem to the solution
of a linear set of equations in which the matrix elements of
the Hamiltonian have to be computed between basis states; increasing the
energy, appreciable contributions from states with high values of
$\ell_\alpha + L_\alpha$ appear. In order to take into account these
contributions, a very large basis has to be used with the consequence
that numerical instabilities start to appear.
In the integral equation approach at very low energies
convergence in terms of screening radius requires
$R > 30 \:\mathrm{fm}$, which in turn increases the number of two-body partial waves
that are needed for convergence. The interplay of these two requirements
makes the integral equation solution unstable at those very low energies.
An interesting heuristic argument to understand the size of the screening
radius needed for convergence is the wave length $\lambda$ corresponding to
the on-shell momentum. At 3 MeV, 10 MeV, and 65 MeV proton lab energy,
for which a screening radius of 20 fm, 10 fm, and 7 fm is needed
for convergence,
$\lambda$ is $24.8\:\mathrm{fm}$, $13.6\:\mathrm{fm}$, and $5.3\:\mathrm{fm}$, respectively.
It appears that for the calculation of $pd$ elastic scattering observables
the screening has to be only so large that one wave
length can be accommodated in the Coulomb tail outside the range of the
hadronic interaction; seeing proper Coulomb over one wave length appears
enough to provide, with the additional help of renormalization, the true
Coulomb characteristics of scattering despite screening.
\section{Conclusions \label{sec:conclusions}}
\bigskip
In the present paper two methods devised to describe elastic
$pd$ scattering are compared for a wide range of energies. One of
the methods, the KVP, was developed a few years ago and
used to study how realistic potential models,
including two-body and three-body forces, describe the
elastic observables measured for that reaction. On the other
hand, numerical accurate results have been recently obtained solving
the AGS equation for $pd$ scattering using a screened Coulomb potential
corrected for the unscreened limit using a renormalization
prescription. As has been briefly described in the present paper,
both methods are substantially different. It is satisfactory to observe
that both methods produce essentially the same results for a large
variety of elastic observables using a realistic two-nucleon potential.
We stress the fact that the selection of observables here presented is only
part of the observables compared. In all cases, similar patterns have
been obtained.
In addition, by comparing the $pd$ calculations to the corresponding $nd$
calculations, Coulomb effects have been estimated. As expected these
effects are sizable at low energies but at the highest analyzed
energy, $65$ MeV, they are small, except at forward scattering angles.
From these considerations it is
possible to identify on a firm basis which $pd$ observables may or may
not be analyzed by calculations in which the Coulomb interaction
has been neglected.
We can conclude that at present it is possible to describe $pd$ elastic
scattering, including the Coulomb repulsion, using standard techniques
as the Faddeev equations in configuration and momentum space or
variational principles. Moreover, in Ref.~\cite{KVM04} the treatment
of other terms of the $NN$ electromagnetic potential as the magnetic
moment interaction has been discussed.
\begin{acknowledgments}
The authors are grateful to H.~Wita{\l}a for the comparison of $nd$ results.
A.D. is supported by the FCT grant SFRH/BPD/14801/2003,
A.C.F. in part by the grant POCTI/FNU/37280/2001,
and P.U.S. in part by the DFG grant Sa 247/25.
\end{acknowledgments}
\bibliographystyle{prsty}
|
{
"timestamp": "2005-03-04T18:22:53",
"yymm": "0503",
"arxiv_id": "nucl-th/0503015",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503015"
}
|
\section{Introduction}
\label{sec:introduction}
Half wave plate (HWP) retarders are used extensively for polarimetric
measurements. The technique is used across a broad range of
electro-magnetic frequencies because it provides an effective way to
discriminate against systematic errors. The modulation efficiency of
a HWP\ that is constructed from a single birefringent plate can reach
100\% for a set of discrete electro-magnetic frequencies but away from
these frequencies the efficiency drops rapidly. To overcome this
limitation it has been proposed to stack several birefringent plates
with specific relative angles of their optical
axes~\cite{pancharatnam55,title81,title75}. Such a construction has
been called achromatic HWP\ (AHWP) because it has a broader
frequency range over which the polarimetric efficiency is high
compared to a HWP\ that is made from a single plate. The efficiency
of an AHWP\ depends on the number of plates in the stack and on their
relative angles. The concept has been demonstrated experimentally in
the optical and IR bands~\cite{tinbergenbook}. Murray
et~al.~\cite{murray97} have described briefly measurements of an
AHWP\ made of 5 quartz plates for wavelengths between 350 and 850
$\mu$m and an AHWP\ made of 3 quartz plates for wavelengths between 1
and 2~mm. No detailed information is given about the measurements, the
analysis, tests for systematic errors, or about the optimization of
the AHWP\ in respect to the relative angles between the plates. A
2-element achromatic waveguide polarizers for operation at $\sim$1~cm
is mentioned by Leitch et al.~\cite{leitch02b}.
In this paper we present the construction of a sapphire AHWP\ and
measurements of its properties at a wavelength of 2.7~mm (110~GHz).
We also present an analysis of the design of a three-plate sapphire
AHWP\ for a wavelength of 2~mm. There is currently interest in
an AHWP\ that is suitable for the mm-wave band because of the
increase in experimental efforts to measure the polarization of the
cosmic microwave background radiation. Several experiments will use
HWP s and increasing the bandwidth where the efficiency is high will
increase the signal-to-noise ratio of the experiment.
\section{Experimental Setup}
A top-view sketch of the experimental setup is shown in
Figure~\ref{fig:exp_setup}. We used a Gunn
oscillator~\cite{spacek_source} at 110~GHz and a diode
detector~\cite{spacek_detector} as a source and detector of radiation,
respectively. Both source and detector had conical horns
that provided beams of 12~degrees full width at
half maximum. They emitted and were sensitive to linearly polarized
radiation with a -15~dB maximum level of cross polarization at
$\sim$10 degrees from peak gain. The source and detector were aligned
by maximizing the signal received by the detector as a function of its
orientation relative to the fixed orientation of the source.
\begin{figure
\centerline{\rotatebox{0}
{\scalebox{1.0}{\includegraphics{setup_nov8.eps} } } }
\caption{A top view sketch of the experimental setup.}
\label{fig:exp_setup}
\end{figure}
We used wire grid polarizers to increase the level of linear
polarization of the light emitted by the source and detected by the
detector. The grids, which were made by Buckbee-Meers, were measured
to have a modulation efficiency of 97\%~\cite{johnson_thesis}.
The source, detector, and polarizers were housed in metallic boxes
that were lined inside (outside) with Emerson and Cuming Eccosorb
LS-16 (LS-14). One side of the boxes was open.
Between the boxes were placed two 1.25~cm thick plates of Emerson and
Cuming Eccosorb MF-124 which served as collimators. They had
19~mm diameter knife-edged holes which faced the source. The knife
edges were covered with 0.07~mm thick aluminum tape. The HWP\ was
installed between the collimators in a 5~cm diameter Newport mount
that could rotate around the $x$ axis with a resolution of
1~degree. The mount was held by a cylindrical leg that gave it another
degree of freedom for rotation around the $z$ axis. The beam filled
the central 15\% area of the HWP. Its angular extent when it reached
the detector was 2~degrees.
The entire experiment was mounted on a metallic optics bench.
Aluminum sheet metal lined with egg-crate Eccosorb-CV3
enclosed the experiment from three sides. Egg-crate Eccosorb was also
placed both in front and above the source and detector boxes, as
shown in Figure~\ref{fig:exp_setup}.
\section{Achromatic Half Wave Plate}
We used a stack of three sapphire a-cut plates to construct the
AHWP. Each of the fine ground plates had a thickness of
$2.32\pm0.05$~mm, which made each a HWP\ for a frequency of 193~GHz.
The three plates were mounted together with a front and back
anti-reflection coating made of 0.35~mm thick polished Herasil. The
orientation of the second plate was rotated by 50.5~deg with respect
to the orientation of the aligned first and third plates. We had an
angular accuracy of $\pm 1$~degree in assembling the stack and an
accuracy of $\pm 1.5$~degrees in orienting the stack-mount normal to
the incoming beam. The ordinary and extraordinary axes of any of the
plates were known to within 0.5~degree.
We compared the performance of the AHWP\ to the performance of a
`chromatic' plate, a single a-cut plate of sapphire with a thickness
of 2.32~mm. The chromatic plate was stacked with the same layers of
anti-reflection coating as the AHWP.
We used a frequency of 110~GHz to make the measurements because at
this frequency the difference between the modulation efficiency of
the AHWP\ and of the single plate are nearly maximized
thereby providing a clear demonstration
of the achromaticity of the stack.
\section{Measurements, Analysis, and Results}
To quantify the efficiency of the plates we measured the detected
intensity as a function of their rotation angle $\alpha$ about the $x$
axis. Data were taken every 10~degree in angle and are shown in
Figure~\ref{fig:data}. Error bars are the standard deviation of 5
repeat measurements of the efficiency. A repeat measurement consisted
of assembling all individual pieces into a stack, mounting the stack,
and taking data. No changes in other elements in the experiment were
made between repeat measurements.
\begin{figure
\centerline{\rotatebox{-90}
{\scalebox{.7}{\includegraphics{data_hannes_dec30.ps}}}}
\caption{Measurements (points) and theoretical
predictions (dash) of the signal detected as a function of
rotation angle of the plates for the chromatic plate (blue diamond)
and for the AHWP\ (red triangles). Error bars are the standard
deviations of 5 repeated measurements. The theoretical predictions
have no free parameters.}
\label{fig:data}
\end{figure}
A constant offset of about 0.7~mV was measured when the aperture of
the detector box was blocked and was subtracted from the data. This
level was constant with rotation of the plates, between different
independent measurements of a given stack, and between measurements
with different stacks. The data was then fit with the following
model
\begin{equation}
D = \sum_{i=0}^{8} A_{i}\cos(i\alpha + \phi_{i}).
\label{eqn:hwp_model}
\end{equation}
The output of the fitting were the 9 amplitudes and 8 phases, where
$\phi_{0}$ was set to zero. The modulation efficiency was defined as
\begin{equation}
\epsilon = { A_{4} \over A_{0} }.
\label{eqn:efficiency}
\end{equation}
The value of $\epsilon$ did not change when we fit the data only up to
the fourth harmonic (5 amplitudes and 4 phases). The quality of the
fit however degraded from a reduced $\chi^2$ of 0.27 and 0.9 for the
achromatic and chromatic plates, respectively, with 8 harmonics to 5.8
and 2.6, respectively, with 4 harmonics.
Predictions about the efficiency of the plates were calculated using
the technique of Mueller matrices. The intensity of the light incident
on the detector was generated by multiplying an incident Stokes vector
representing 100\% $Q$ polarized light by Mueller matrices that
simulated the response of the two anti-reflection layers, the plates,
and a 100\% $Q$ polarized detector. An overall normalization was taken
from a measurement of the power detected in the absence of a HWP\ in
the light path. The phase was taken from the known orientation of the
plates. Normal incidence was assumed throughout. A prediction for the
detected intensity was calculated as a function of $\alpha$ in steps
of 1~degree, fitted by the model given in
Equation~\ref{eqn:hwp_model}, and a predicted efficiency was
calculated using Equation~\ref{eqn:efficiency}. The predicted response
of the plates as a function of angle is shown in
Figure~\ref{fig:data}. The prediction shown is not a fit to the
data. There are no free parameters in this prediction.
Figure~\ref{fig:efficiency} shows the predicted efficiency of the
chromatic and achromatic plates as a function of frequency and our
measured values of $43 \pm 4 \%$ and $96 \pm 1.5 \%$, respectively.
The predicted values are 43.5\% and 100\%, respectively. Uncertainty
in the predicted values of the efficiency, due to uncertainty
in the indices of sapphire~\cite{lamb96}, is 1.5\% for the single plate
and negligible for the AHWP. The errors on the measurements of
the modulation efficiency were calculated by summing the statistical
and an estimate of the systematic errors in quadrature.
\begin{figure
\centerline{\rotatebox{90}
{\scalebox{0.7}{\includegraphics{efficiency_hannes_march10.eps}}}}
\caption{The predicted modulation
efficiency as a function of frequency of the AHWP\ (red broad) and of
a single plate (blue narrow) and the measured
efficiency of both plates.}
\label{fig:efficiency}
\end{figure}
We also measured the efficiency for angles of incidence that are not
normal by tilting the AHWP\ about the $z$ axis between angles of zero
and 15 degrees. We found no change in the efficiency as a function of
angle within statistical errors.
Spurious signals generated by reflections can be a source of
systematic errors. We checked the level of signal detected by the
detector when either of the collimators were blocked with metal or
with a piece of Eccosorb MF124. The level was 0.7 mV for all cases and
did not change as a function of the rotation angle of the plates in
their mount.
The experiment was repeated for various distances of the plates from
the source. The efficiency of the single plate varied in a sinusoidal
manner with position with an amplitude of 1.9 \% and a period of 1.4
mm. This period is also half the wavelength of the source and we
hypothesize that reflections in the setup cause the small variation in
efficiency. We have also observed that the shape of the deviations
between the theoretical prediction for the detected signal and the one
measured vary as a function of the position of the plate. The data
shown in Figure~\ref{fig:data} is representative of the magnitude of
such deviations. For the achromatic plate the peak-to-peak changes in
efficiency as a function of distance were smaller than the quoted
statistical error.
\section{Discussion}
There is good agreement between each of the no-free-parameters
predictions shown in Fig.~\ref{fig:data} and the data. Both the
predicted overall modulation amplitude and the relative phase shift
are reproduced by the measurements. The measured modulation
efficiencies are close to the predicted values.
AHWP's can be constructed with various combinations of birefringent
plates each giving a different degree of
achromaticity. Title~\cite{title75} showed that with 3 plates of the
same material an AHWP\ should have the first and last plates aligned
and most of our discussion is restricted to such a
stack. Figure~\ref{fig:breadth_ripple} shows the efficiency of an
AHWP\ made of three sapphire plates as a function of frequency and
for three different orientations of the second plate. Each of the
plates is a HWP\ at odd harmonics of 50~GHz, suitable for a
cosmic microwave background polarization experiment - EBEX - that
we are currently constructing. EBEX will operate at 150, 250, 350 and
450~GHz. An orientation angle of 58~degrees gives close to a constant
modulation efficiency over a band of $\sim$40~GHz. A plate
orientation angle of 47~degrees gives a band of $\sim$60~GHz at the
expense of variations of the efficiency within that band. It is
therefore interesting to quantify the {\it average} modulation
efficiency as a function of bandwidth and as a function of rotation
angle of the second plate. The results are shown in
Figure~\ref{fig:contour} for a top-hat frequency response and they
demonstrate several features. The maximum average efficiency decreases
as a function of bandwidth but with a proper choice of angle average
efficiencies that are larger than 95\% are achievable with up to
60~GHz of bandwidth. The angular precision required for the
orientation of the second plate is rather coarse. The efficiency for
60~GHz of bandwidth is larger than 95\% for any angle between 47 and
56~degrees. Even smaller accuracy is required for narrower bandwidths.
\begin{figure
\centerline{\rotatebox{90}
{\scalebox{.6}{\includegraphics{deviation_zoom.ps}}}}
\caption{Predicted modulation
efficiency of an AHWP\ as a function of frequency near 150~GHz for
rotation angles of 47 (dash dot, green), 53 (solid, red) and 58 (dash,
blue)~degrees of the second plate. Each sapphire plate in the stack is
a HWP\ for a frequency of 50~GHz.}
\label{fig:breadth_ripple}
\end{figure}
A stack of 5 plates can give high modulation efficiency over an even
broader range of frequencies compared to a 3-stack; see
Figures~\ref{fig:5stack} and~\ref{fig:5stack_int}. With an assumption
of a top-hat frequency response of the instrument we calculate that
for the balloon-borne EBEX the penalty in increased absorption and
emission from the thicker stack of sapphire plates would be smaller
than the increase in signal and therefore a properly designed 5-stack
would increase the signal-to-noise ratio of the experiment.
\begin{figure
\centerline{\rotatebox{90}
{\scalebox{.9}{\includegraphics{contour2.ps}}}}
\caption{The average modulation efficiency
(color scale and contours) as a function of the orientation of the
second plate and the spectral width of a top hat band centered on
150~GHz (for example, a width of 60~GHz means $150 \pm
30$~GHz).}
\label{fig:contour}
\end{figure}
Interest in mm-wave AHWP\ has increased recently because of the
scientific interest in the polarization of the cosmic microwave
background radiation. Several experiments including our own EBEX
are proposing to use HWP's as means to modulate the
incident polarization~\cite{oxley04,church03}. The results presented
in this paper provide reassurance that these experiments can rely on
an AHWP\ and that the efficiency of such a plate is constant for
a relatively broad range of incidence angles.
\begin{figure
\centerline{\rotatebox{0}
{\scalebox{1.0}{\includegraphics{5and3hwp_new.eps}}}}
\caption{The modulation efficiency of an
AHWP\ made of a stack of 5 plates compared to the modulation
efficiency of an AHWP\ made of a 3-stack. The 5-stack has orientation
angles of 28.8, 94.5, 28.8 and 2~degrees for the plates after the
first, respectively. For the 3-stack the second plate is at
57.5~degrees. Each of the plates is sapphire and is optimized for
50~GHz.}
\label{fig:5stack}
\end{figure}
\begin{figure
\centerline{\rotatebox{90}
{\scalebox{1.0}{\includegraphics{5stack_bj_dec30.ps}}}}
\caption{The average modulation
efficiency (color scale and contours) for an AHWP\ made of a
5-stack. The efficiency is given as a function of the orientation of
the second and fourth plates (relative to the first) and the spectral
width of a top hat band centered on 150~GHz. The relative angles of
the third and fifth plates are 94.5 and 2~degrees, respectively.}
\label{fig:5stack_int}
\end{figure}
\newpage
|
{
"timestamp": "2005-03-15T09:30:31",
"yymm": "0503",
"arxiv_id": "physics/0503122",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503122"
}
|
\section{Introduction}
The Casimir effect concerns the Lamb shifts in the frequency of radiation
modes due to the interaction between photon modes and electrical currents.
The photon mode Lagrangian is discussed in Sec.\ref{LCMD}. Mode
frequency shifts induce changes in the free energy which in the zero
temperature limit\cite{Casimir:1948,Casimir::1948} reduce to changes in the zero point
energy\cite{Bordag:2001,Milton:2001}
\begin{math} \delta E_0=(\hbar /2)\sum_{a} \delta \Omega_a \end{math}.
The Lamb frequency shifts are usually small and can be understood from a
perturbation theory viewpoint.
Such damping is discussed in Sec.\ref{OCD}. The conventional Casimir effect
theory thereby considers Feynman diagram corrections to the free energy
containing one photon loop\cite{Dzyaloshinski:1960,Dzyaloshinski:1961}.
In Sec.\ref{TS} it is shown how a one loop instability can
arise if the coupling between a photon oscillation mode
and the surrounding currents is too strong.
If the damping functions and frequency shifts are also oscillating
functions of time, then (over and above single photon absorption and
emission processes) there is the absorption and emission of
{\em photon pairs}\cite{Dodonov:1998}. The photon pair processes constitute a
dynamical Casimir effect\cite{Dalvit:1999,Dodonov:1999}. Frequency modulations
tend to heat up the cavity. In Sec.\ref{DCE}, the noise temperature description
is discussed. In Sec.\ref{PFM}, the heating of a cavity mode by periodic frequency
modulation is explored. In an unstable regime, the temperature of (say)
a microwave cavity mode grows {\em exponentially}. The implied
purely theoretical {\em microwave oven} would be much more hot than
that which could be observed in experimental reality. Nonlinear
higher loop photon processes producing dynamic microwave intensity
stability are discussed in Sec.\ref{MS}.
\section{Lagrangian Circuit Mode Description \label{LCMD}}
Our purpose in this section is to provide a Lagrangian description
of a single microwave cavity mode which follows from the action
principle formulation of electrodynamics\cite{Widom:1987}. For this purpose
we employ the Coulomb gauge, \begin{math} div{\bf A}_{mode}=0 \end{math},
for the vector potential. The vector potential representing the
cavity mode may be written
\begin{equation}
{\bf A}_{mode}({\bf r},t)=\Phi(t){\bf K}({\bf r}).
\label{ML1}
\end{equation}
The mode electromagnetic fields are then given by
\begin{eqnarray}
{\bf E}_{mode}({\bf r},t)
=-\frac{1}{c}\left[\frac{{\bf A}_{mode}({\bf r},t)}{\partial t}\right]
=-\frac{\dot{\Phi }(t)}{c}{\bf K}({\bf r}),
\nonumber \\
{\bf B}_{mode}({\bf r},t)
= curl{\bf A}_{mode}({\bf r},t)=\Phi(t)\ curl{\bf K}({\bf r}).
\label{ML2}
\end{eqnarray}
The Lagrangian
\begin{equation}
L_{field}=\frac{1}{8\pi }\int_{cavity}
\left[\left|{\bf E}_{mode}({\bf r},t)\right|^2-
\left|{\bf B}_{mode}({\bf r},t)\right|^2\right]d^3{\bf r}
\label{ML3}
\end{equation}
describes the mode in terms of a simple oscillator circuit.
The capacitance \begin{math} C \end{math} and inductance
\begin{math} \Lambda \end{math} of the circuit are defined,
respectively, by
\begin{eqnarray}
C=\frac{1}{4\pi }\int_{cavity} \left|{\bf K}({\bf r})\right|^2 d^3{\bf r},
\nonumber \\
\frac{1}{\Lambda }=
\frac{1}{4\pi }\int_{cavity} \left|curl{\bf K}({\bf r})\right|^2 d^3{\bf r}.
\label{ML4}
\end{eqnarray}
The circuit electromagnetic field Lagrangian follows from
Eqs.(\ref{ML2}), (\ref{ML3}) and (\ref{ML4}). It is of the
simple \begin{math} \Lambda C \end{math} oscillator form
\begin{equation}
L_{field}(\dot{\Phi },\Phi )=\frac{C}{2c^2}\dot{\Phi }^2
-\frac{1}{2\Lambda }\Phi^2,
\label{ML5}
\end{equation}
wherein the bare circuit frequency obeys
\begin{equation}
\Omega_\infty ^2=\frac{c^2}{\Lambda C}\ .
\label{ML6}
\end{equation}
The interactions between cavity wall currents and an electromagnetic
mode are conventionally described by
\begin{eqnarray}
L_{int}=\frac{1}{c}\int {\bf J}
\cdot {\bf A}_{mode }d^3{\bf r},
\nonumber \\
L_{int}= \frac{1}{c}I\Phi ,
\nonumber \\
I(t) = \int {\bf J}({\bf r},t)\cdot {\bf K}({\bf r})d^3{\bf r},
\label{ML7}
\end{eqnarray}
where the current \begin{math} I \end{math} drives the oscillator circuit.
In total, the circuit mode Lagrangian follows from
Eqs.(\ref{ML5}) and (\ref{ML7}) as
\begin{equation}
L=\frac{C}{2c^2}\dot{\Phi }^2
-\frac{1}{2\Lambda }\Phi^2+\frac{1}{c}I\Phi +L^\prime
\label{ML8}
\end{equation}
wherein \begin{math} L^\prime \end{math} describes all of the
other degrees of freedom which couple into the mode coordinate.
Maxwell's equations for a single microwave mode then takes the form
\begin{eqnarray}
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\Phi}}\right)
=\left(\frac{\partial L}{\partial \Phi}\right),
\nonumber \\
C\left(\ddot{\Phi }+\Omega_\infty ^2\Phi \right)=cI.
\label{ML9}
\end{eqnarray}
The damping of the oscillator will first be discussed from a classical
electrical engineering viewpoint and only later from a fully quantum
electrodynamic viewpoint.
\section{Oscillator Circuit Damping \label{OCD}}
From an electrical engineering viewpoint, let us consider a small external
current source \begin{math} \delta I_{ext} \end{math} which drives the mode
coordinate \begin{math} \delta \Phi \end{math}. Eq.(\ref{ML9}) now reads
\begin{equation}
\frac{C}{c^2}\delta \ddot{\Phi }+\frac{1}{\Lambda }\delta \Phi =
\frac{1}{c}\delta I=\frac{1}{c}\left(\delta I_{ext}+\delta I_{ind}\right)
\label{CD1}
\end{equation}
were \begin{math} \delta I_{ind} \end{math} is the current induced by the
coordinate response \begin{math} \delta \Phi \end{math}. In the complex
frequency \begin{math} \zeta \end{math} domain we have in (the upper half
\begin{math} {\Im }m\ \zeta >0 \end{math} plane)
\begin{eqnarray}
\delta I_{ext}(t) = {\Re }e\left\{\delta I_{ext,\zeta}e^{-i\zeta t}\right\}
\nonumber \\
\delta \Phi (t) = {\Re }e\left\{\delta I_{ext,\zeta}
{\cal D}(\zeta )e^{-i\zeta t}\right\}.
\label{CD2}
\end{eqnarray}
The induced current is determined by the ``surface admittance''
\begin{math} Y(\zeta ) \end{math} of the cavity walls;
In detail
\begin{eqnarray}
\delta I_{ind}(t) = -\frac{1}{c}\int_0^\infty
{\cal G}(t^\prime )\delta \dot{\Phi }(t-t^\prime ) dt^\prime ,
\nonumber \\
Y(\zeta ) = \int_0^\infty e^{i\zeta t}{\cal G}(t)dt,
\label{CD3}
\end{eqnarray}
so that
\begin{eqnarray}
\left\{-\frac{C}{c^2}\zeta ^2+\frac{1}{\Lambda }
-\frac{i\zeta }{c^2}Y(\zeta )\right\}{\cal D}(\zeta )
=\frac{1}{c}\ ,
\nonumber \\
-i\zeta \varepsilon (\zeta )C=-i\zeta C+Y(\zeta ),
\label{CD4}
\end{eqnarray}
wherein the effective frequency dependent capacitance
\begin{math} \varepsilon (\zeta )C \end{math}
determines the mode dielectric
response function \begin{math} \varepsilon (\zeta ) \end{math}.
The retarded propagator for the mode in the frequency domain obeys
\begin{equation}
{\cal D}(\zeta )=\frac{\Lambda }{c}
\left[\frac{\Omega_\infty ^2}{\Omega_\infty ^2-\zeta ^2-\Pi(\zeta )}\right]
\label{CD5}
\end{equation}
wherein the ``self energy'' \begin{math} \Pi(\zeta ) \end{math} is
determined by the induced current admittance via
\begin{equation}
{\Pi (\zeta )}=\frac{i\zeta Y(\zeta )}{C}\ .
\label{CD6}
\end{equation}
The self energy describes both frequency shift and damping properties of the
mode.
Causality dictates that all engineering response functions obey analytic
dispersion relations (\begin{math} {\Im }m\ \zeta >0 \end{math}) of the form
\begin{eqnarray}
{\cal D}(\zeta )=\frac{2}{\pi }\int_0^\infty
\frac{\omega {\Im }m{\cal D}(\omega+i0^+)d\omega }{\omega^2 -\zeta ^2}\ ,
\nonumber \\
\Pi (\zeta )=\frac{2}{\pi }\int_0^\infty
\frac{\omega {\Im }m\Pi (\omega+i0^+)d\omega }{\omega^2 -\zeta ^2}\ .
\label{CD7}
\end{eqnarray}
The damping rate for the oscillation is determined by
\begin{equation}
{\Im }m\Pi (\omega+i0^+)=\omega {\Re}e\Gamma (\omega +i0^+)
=\frac{\omega {\Re}e Y(\omega +i0^+)}{C}\ .
\label{CD8}
\end{equation}
The shifted frequency,
\begin{equation}
\Omega_0^2=\Omega_\infty ^2-\Pi(0),
\label{CD9}
\end{equation}
obeys the dispersion relation sum rule
The shifted frequency is related to the damping rate via the sum rule
\begin{equation}
\Omega_\infty ^2=\Omega_0^2+\frac{2}{\pi }
\int_0^\infty {\Re}e\Gamma (\omega +i0^+)d\omega ,
\label{CD10}
\end{equation}
which follows from Eqs.(\ref{CD6}) - (\ref{CD9}).
Finally, the the quality factor \begin{math} Q \end{math} for the mode
frequency \begin{math} \Omega_0 \end{math} is well defined as
\begin{equation}
\frac{\Omega_0}{Q}={\Re}e\Gamma (\Omega_0 +i0^+)
\label{CD11}
\end{equation}
if and only if the mode is under damped by a large margin; e.g.
\begin{math} Q>>1 \end{math}. On the other hand,
if the damping is sufficiently strong, then the mode can go unstable.
Let us consider this physical effect in more detail.
\section{Thermodynamic Stability \label{TS}}
If the mode where uncoupled to the damping current, then then the free energy
of the oscillator would be
\begin{eqnarray}
f_\infty (T)= -k_BT
\ln \left[\sum_{N=0}^\infty e^{-(N+1/2)\hbar \Omega_\infty /k_BT}\right],
\nonumber \\
f_\infty (T)=k_BT
\ln \left[\sinh \left(\frac{\hbar \Omega_\infty}{2k_BT}\right)\right].
\label{TS1}
\end{eqnarray}
The damping effects give rise to Lamb shifted frequencies and a Casimir-Lifshitz
renormalization in the free energy; It is
\begin{eqnarray}
f(T) = f_\infty (T)+f_1(T),
\nonumber \\
f_1(T) = \left(\frac{k_BT}{2}\right)\sum_{n=-\infty}^\infty
\ln\left[1-\left(\frac{\Pi (i|\omega_n|)}{\Omega_\infty^2+\omega_n^2}\right)\right],
\nonumber \\
\hbar \omega_n = 2\pi nk_BT .
\nonumber \\
\Pi (i|\omega_n|) = \frac{2}{\pi }\int_0^\infty
\frac{\omega {\Im }m\Pi (\omega+i0^+)d\omega }{\omega^2 +\omega_n^2}\ .
\label{TS2}
\end{eqnarray}
A sufficient condition for the validity of Eqs.(\ref{TS2}) is that the mode oscillator
obeys a linear equation of motion. From Eqs.(\ref{TS1}) and (\ref{TS2}) we deduce the
following thermodynamic stability\cite{Widom:2004}
\medskip
\par \noindent
{\bf Theorem 1:} {\it The Casimir free energy shift of an oscillator mode is stable if
and only if \begin{math} \Pi (0)<\Omega_\infty ^2 \end{math}. If
\begin{math} \Pi (0)>\Omega_\infty ^2 \end{math}, then the one loop
free energy in {Eq.{\rm (\ref{TS2})}} becomes complex yielding finite lifetime effects.}
\medskip
\par \noindent
Thermodynamic stability can be restored if the one goes beyond the one loop
approximation in the effective Lagrangian, e.g. the oscillator can shift its
minimum form zero to \begin{math} \Phi_0 \end{math}. For such a thermodynamic
instability in which \begin{math} \omega_0^2=\Pi(0)-\Omega_\infty ^2>0 \end{math},
the effective Lagrangian may be taken as
\begin{equation}
L_{effective}=\frac{C}{2c^2}\dot{\Phi }^2+
\frac{C\omega_0^2}{4\Phi_0^2 c^2}\left(\Phi^2-\Phi_0^2\right)^2.
\label{TS3}
\end{equation}
The stability is restored via a stabilizing term representing four photon
interactions. Such a Lagrangian can appear for modes whose surrounding
walls are at least in part ferromagnetic.
A high quality photon oscillator mode is only weakly damped so that
the one loop perturbation approximation is virtually exact. On the
other hand {\em dynamical instabilities} may still require higher order
photon interaction terms to understand the ultimate stabilities in
laboratory systems.
\section{Dynamical Casimir Effects \label{DCE}}
Suppose that the dielectric response function
\begin{math} \varepsilon (\zeta ) \end{math} of the mode
in Eq.(\ref{CD4}) is made to vary time; i.e.
\begin{equation}
\varepsilon (\zeta )\ \Rightarrow\ \varepsilon(\zeta ,t)
\ \ {\rm equivalently}
\ \ \Pi (\zeta )\ \Rightarrow\ \Pi(\zeta ,t).
\label{DCE1}
\end{equation}
If the resulting differential equation for the
\begin{math} \Phi =\Re e \{\phi \}\end{math} signal obeys
to a sufficient degree of accuracy
\begin{eqnarray}
\ddot{\phi}(t)+\Omega^2 (t)\phi (t)=0,
\nonumber \\
\Omega (t\to \pm \infty)=\Omega_0,
\label{DCE2}
\end{eqnarray}
then there exists a solution of the form
\begin{eqnarray}
\phi (t\to \infty )=e^{i\Omega_0 t}+\rho e^{-i\Omega_0 t},
\nonumber \\
\phi (t\to -\infty )=\sigma e^{i\Omega_0 t},
\nonumber \\
|\rho |^2+|\sigma |^2=1.
\label{DCE3}
\end{eqnarray}
From a quantum mechanical viewpoint, the time variation
\begin{math} e^{i\Omega_0 t} \end{math} may represent a
photon moving backward in time and
\begin{math} e^{-i\Omega_0 t} \end{math}
may represent photon moving forward in time. In Eq.(\ref{DCE3}),
the reflection amplitude for a photon moving backward in time
to bounce forward in time is given by
\begin{math} \rho \end{math}. A backward in time moving photon
reflected forward in time appears in the laboratory to be a pair
of photons being created.
\begin{figure}[bp]
\scalebox {0.8}{\includegraphics{cdfig1}}
\caption{If $T_i$ represents the initial cavity mode temperature and $T^*$
represents the noise temperature of the pair radiated photons, then the final
temperature $T_f$ of the of the cavity mode is enhanced (over
and above $T^*)$ via the initial photon population. The resulting radiation
enhancement is plotted for photons with energy $E_\gamma =\hbar \Omega_0$.}
\label{Fig1}
\end{figure}
The probability of such a photon pair creation event defines
a {\em photon pair creation noise temperature} \begin{math} T^* \end{math}
induced by the time varying frequency via
\begin{equation}
R=|\rho |^2=e^{-\hbar \Omega_0/k_BT^*}.
\label{DCE4}
\end{equation}
The mean number \begin{math} \bar{N} \end{math} of photons which
would be radiated from the vacuum by a time varying frequency
modulation \begin{math} \Omega (t) \end{math} obeys a formal Planck
law
\begin{equation}
\bar{N}=\frac{R}{1-R}=\frac{1}{e^{\hbar \Omega_0/k_BT^*}-1}\ .
\label{DCE5}
\end{equation}
Suppose (for example) that a microwave cavity is initially in thermal
equilibrium at temperature \begin{math} T_i \end{math}. The mean
number of initial microwave photons in a given normal mode is then given by
\begin{equation}
N_i=\frac{1}{e^{\hbar \Omega_0/k_BT_i}-1}\ .
\label{DCE6}
\end{equation}
After a sequence of frequency modulation pulses the mean number of final
photons in the cavity mode is
\begin{equation}
N_f=(2\bar{N}+1)N_i+\bar{N}=
N_i\coth\left(\frac{\hbar \Omega_0}{2k_BT^*}\right)
+\frac{1}{e^{\hbar \Omega_0/k_BT^*}-1}\ .
\label{DCE7}
\end{equation}
Note that the existence of an {\em initial} number of photons
\begin{math} N_i \end{math} in the cavity mode makes larger the
final number of of photons
\begin{equation}
N_f=\frac{1}{e^{\hbar \Omega_0/k_BT_f}-1}
\label{DCE6f}
\end{equation}
via the {\em induced} radiation of additional
photon pairs. If the microwave frequency large margin inequality
\begin{equation}
\hbar \Omega_0\ll k_BT^*
\label{DCE8}
\end{equation}
holds true, then Eqs.(\ref{DCE5}) - (\ref{DCE8}) imply an
approximate law for the {\em final} cavity mode noise temperature
is given by
\begin{equation}
T_f \approx T^* \coth\left(\frac{\hbar \Omega_0}{2k_BT_i}\right).
\label{DCE9}
\end{equation}
The resulting enhancement \begin{math} (T_f/T^*) \end{math} is
plotted in Fig.\ref{Fig1}. The dynamical Casimir effect for frequency
modulation pulses is thereby described in terms of the amount of
heat that raises the temperature \begin{math} T_i\to T_f \end{math}
of the microwave cavity.
\section{Periodic Frequency Modulations\label{PFM}}
For periodic modulations in the frequency one must examine\cite{Wilhelm:2003}
the differential equation
\begin{eqnarray}
\ddot{\phi}(t)+\Omega^2 (t)\phi (t)=0,
\nonumber \\
\Omega^2 (t)=\Omega_0^2+\nu^2(t),
\nonumber \\
\nu(t+\tau)=\nu (t).
\label{PFM1}
\end{eqnarray}
From a mathematical viewpoint, Eq.(\ref{PFM1}) has been well
studied. If \begin{math} \nu(t) \end{math} can be represented as
a non-overlapping pulse sequence of the form
\begin{equation}
\nu (t)=\sum_{n=-\infty}^\infty \varpi(t-n\tau ),
\label{PFM2}
\end{equation}
then the transmission problem for a single pulse,
\begin{equation}
\ddot{\phi}_1(t)+\{\Omega_0^2+ \varpi^2(t)\}\phi_1 (t)=0,
\label{PMF3}
\end{equation}
yields a complete solution to the general problem. In
particular we examine the two photon creation problem as in
Eq.(\ref{DCE3}); i.e.
\begin{eqnarray}
\phi_1(t\to \infty )=e^{i\Omega_0 t}+\rho_1 e^{-i\Omega_0 t},
\nonumber \\
\phi_1 (t\to -\infty )=\sigma_1 e^{i\Omega_0 t},
\nonumber \\
|\rho_1 |^2+|\sigma_1 |^2=R_1+P_1=1,
\nonumber \\
\sigma_1=\sqrt{P_1}\ e^{-i\Theta_1}.
\label{PMF4}
\end{eqnarray}
Employing the characteristic function
\begin{equation}
\mu (\Omega_0)=\frac{\cos(\Omega_0 \tau +\Theta_1(\Omega_0))}
{\sqrt{P_1(\Omega_0)}},
\label{PMF5}
\end{equation}
one may study the stability problem for the dynamic Casimir effect.
For {\em periodic} frequency modulations there are two cases of interest:
\par \noindent
Case I: {\em Stable Motions \begin{math} -1< \mu (\Omega_0)<+1 \end{math}}
\begin{eqnarray}
\mu(\Omega_0)=\cos(\Omega \tau )
\nonumber \\
\phi_\pm (t+\tau )=e^{\pm i\Omega t}\phi_\pm (t).
\label{PMF6}
\end{eqnarray}
Case II: {\em Unstable Motions
\begin{math} \mu (\Omega_0)>+1 \end{math} {\rm or}
\begin{math}\mu (\Omega_0)<-1 \end{math}}
\begin{eqnarray}
\mu(\Omega_0)=\cosh(\gamma \tau )
\ \ {\rm or}\ \ \mu(\Omega_0)=-\cosh(\gamma \tau )
\nonumber \\
\phi_\pm (t+\tau )=e^{\pm \gamma t}\phi_\pm (t).
\label{PMF7}
\end{eqnarray}
In the unstable regime, \begin{math} 2\gamma \end{math} represents
the number of cavity photons being produced per unit time. If
the cavity mode has a high quality factor \begin{math} Q\gg 1 \end{math},
then photons are also absorbed at a rate
\begin{math} (\Omega_0/Q) \end{math}. The net photon production
rate in this approximation would then be
\begin{equation}
\Gamma_1\simeq \left(2\gamma -\frac{\Omega_0}{Q}\right),
\label{PMF8}
\end{equation}
and the theoretical noise temperature after \begin{math} n_p \end{math}
pulses would be
\begin{equation}
k_BT_1^* \approx \hbar \Omega_0 \exp(n_p \tau \Gamma_1 ).
\label{PMF9}
\end{equation}
As an example, let us suppose a sequence of rectangular pulse sequences
of the form
\begin{eqnarray}
\Omega (t)=\Omega_0 \ \ \ {\rm if}
\ \ \ t_0+n\tau < t < t_0+(n+1/2)\tau ,
\nonumber \\
\Omega (t)=(1+\alpha )\Omega_0 \ \ \ {\rm if}
\ \ \ t_0+(n+1/2)\tau < t < t_0+(n+1)\tau ,
\label{PMF10}
\end{eqnarray}
wherein \begin{math} n=1,2,\ldots ,n_p \end{math}. The estimate
\begin{equation}
\exp(n_p \tau \Gamma_1 )\sim \exp(n_p\alpha /2)
\ \ \ {\rm for} \ \ \ 1\gg \alpha \gg (\Omega_0\tau)/Q
\label{PMF11}
\end{equation}
is not unreasonable.
The exponential temperature {\em instability} for high quality
cavity modes, i.e. \begin{math} \Gamma_1 > 0 \end{math} in
Eqs.(\ref{PMF8}) - (\ref{PMF11}), would be sufficient for large
\begin{math} n_p \end{math} to {\it melt} the cavity. No microwave
oven works that efficiently even if the dynamic Casimir effect were employed
for exactly that purpose. The one loop photon approximation is evidently at
fault and higher loops (non-linear processes) must be invoked for the noise
temperature of the mode to be theoretically stable as would be laboratory
microwave cavities.
\section{Microwave Intensity Stability\label{MS}}
The stability of the microwave cavity is due to the fact that the modulation
is induced by a {\em pump} which supplies the energy of the induced cavity
radiation. One may define a {\em pump coordinate} \begin{math} \eta \end{math} which
in general is a quantum mechanical operator. In principle, one might mechanically
vibrate a wall in the cavity in which case \begin{math} \eta \end{math} would be
proportional to a mechanical displacement. In practice, changing the frequency
by electronic means may well be more efficient. Be that as it may, let us define
the coordinate so that
\begin{equation}
\left<\eta (t)\right>=\frac{\nu^2(t)}{\Omega_0^2}\ ,
\label{MS1}
\end{equation}
wherein the quantities on the right hand side of Eq.(\ref{MS1}) are given
in Eq.(\ref{PFM1}).
If the quantum pump coordinate exhibits stationary fluctuations
\begin{equation}
\Delta \eta =\eta -\left<\eta \right>
\label{MS2}
\end{equation}
with quantum noise
\begin{equation}
\frac{1}{2}\left< \Delta \eta (t) \Delta \eta (t^\prime )+
\Delta \eta (t^\prime )\Delta \eta (t)\right>=
\int_{-\infty}^\infty \bar{S}_\eta (\omega )
e^{-i\omega (t-t^\prime )}d\omega ,
\label{MS3}
\end{equation}
then two photon absorption and two photon emission processes are described
by the additional noise Hamiltonian
\begin{equation}
\Delta H=\frac{1}{4}\hbar \Omega_0
\left(a^\dagger a^\dagger+a a\right)\Delta \eta .
\label{MS4}
\end{equation}
The usual mode photon creation and destruction operators
are \begin{math} a^\dagger \end{math} and
\begin{math} a \end{math}, respectively.
When the Hamiltonian in Eq.(\ref{MS4}) is taken to second order in
perturbation theory, the resulting energies involve four boson processes
and thereby introduces multi-photon loop processes.
With the pump coordinate positive and negative frequency spectral functions
\begin{eqnarray}
\left< \Delta \eta (t) \Delta \eta (t^\prime )\right>
=\int_{-\infty }^\infty S_\eta ^+ (\omega )
e^{-i\omega (t-t^\prime )}d\omega ,
\nonumber \\
\left<\Delta \eta (t^\prime )\Delta \eta (t) \right>
=\int_{-\infty }^\infty S_\eta ^+ (\omega )
e^{-i\omega (t-t^\prime )}d\omega ,
\label{MS5}
\end{eqnarray}
the two photon Fermi golden rule transition rates which follow from
Eqs.(\ref{MS4}) and (\ref{MS5}) read
\begin{eqnarray}
\Gamma^+ (n\to n-2) = \frac{\pi \Omega_0^2}{8}
S_\eta ^+ (\omega =2\Omega_0)n(n-1),
\nonumber \\
\Gamma^- (n-2\to n) = \frac{\pi \Omega_0^2}{8}
S_\eta ^- (\omega =2\Omega_0)n(n-1).
\label{MS6}
\end{eqnarray}
The pump coordinate also has a noise temperature
\begin{math} T_\eta \end{math} may be defined via
\begin{equation}
S_\eta ^- (2\Omega_0)=
e^{-2\hbar \Omega_0/k_BT_\eta }S_\eta ^+ (2\Omega_0).
\label{MS7}
\end{equation}
If there a many photons in the mode, then the net
rate of photon absorption is given by
\begin{equation}
\Gamma_{absorption} \simeq
\left(\frac{\pi \Omega_0^2\bar{S}_\eta (\omega = 2\Omega_0) }{2}\right)
\tanh\left(\frac{\hbar \Omega_0}{k_BT_\eta }\right)n^2 .
\label{MS8}
\end{equation}
On the other hand the frequency modulation produces photons
at a rate
\begin{equation}
\Gamma_{emmision} \simeq 2\gamma n
\ \ \ {\rm where} \ \ \ n \gg 1\ ,
\label{MS9}
\end{equation}
and \begin{math} \gamma \end{math} is defined in Eq.(\ref{PMF7}). We may
now state the central result of this section:
\medskip
\par \noindent
{\bf Theorem 2:} {\it If the pump coordinate pushes the cavity mode into
a modulation dynamic Casimir instability, then the quantum noise will
stabilize the cavity mode according to the equation}
\begin{eqnarray}
\frac{dn}{dt} = 2(\gamma n - \tilde{\gamma } n^2),
\nonumber \\
\tilde{\gamma} ={\pi \Omega_0^2\bar{S}_\eta (\omega = 2\Omega_0) }
\tanh \left(\frac{\hbar \Omega_0}{k_BT_\eta }\right).
\label{MS10}
\end{eqnarray}
\medskip
\par \noindent
The cavity photon occupation number will then saturate according to
\begin{equation}
\bar{n}_{saturate}=
\frac{\gamma }{\pi \Omega_0^2 \bar{S}_\eta (\omega = 2\Omega_0) }
\coth \left(\frac{\hbar \Omega_0}{k_BT_\eta }\right).
\label{MS11}
\end{equation}
More simply, with the response function
\begin{equation}
\chi(\zeta )=\frac{i}{\hbar}\int_0^\infty
\left<\left[\eta (t),\eta (0)\right]\right>e^{i\zeta t}dt,
\label{MS12}
\end{equation}
the fluctuation dissipation theorem
\begin{equation}
\bar{S}_\eta (\omega)=\left(\frac{\hbar }{2\pi}\right)
\coth\left(\frac{\hbar \omega }{2k_BT_\eta }\right){\Im m}\chi(\omega +i0^+).
\label{MS13}
\end{equation}
together with Eqs.(\ref{MS11})
and (\ref{MS12}) reads
\begin{equation}
\bar{n}_{saturate}=
\frac{2\gamma }{ \Omega_0^2 [\hbar {\Im m}\chi(2\Omega_0 +i0^+)]}.
\label{MS14}
\end{equation}
The relation time \begin{math} \tau^\dagger \end{math} for the parameter
\begin{math} \eta \end{math} may be conventionally
defined\cite{Martin:1968} by
\begin{equation}
\chi(0)\tau^\dagger =
\lim_{\omega \to 0}\frac{{\Im} m\chi(\omega +i0^+)}{\omega }
\label{MS15}
\end{equation}
so that
\begin{equation}
\bar{n}_{saturate}\approx
\frac{\gamma }{ \Omega_0^3 \tau^\dagger \hbar \chi(0)}.
\label{MS16}
\end{equation}
Eq.(\ref{MS16}) is our final answer for the number of final photons
at saturation.
\section{A Numerical Example \label{ANE}}
In order to make our final answer less abstract, let us consider a
proposed\cite{Braggio:2004} experiment. In that proposal, the parameter
\begin{math} \eta \end{math} describes the metallic conductivity in a
semiconductor plate due to a laser beam inducing particle hole pairs.
If we let \begin{math} \tau_R \end{math} represent the recombination
time taken to annihilate a particle hole pair in the semiconductor
and let \begin{math} \omega_L \end{math} represent the laser frequency,
then we estimate that
\begin{equation}
\frac{1}{\tau^\dagger}\sim\frac{\hbar \omega_L\chi(0)}{\tau_R}
\label{ANE1}
\end{equation}
which implies
\begin{equation}
\bar{n}_{saturate}\sim
\left(\frac{\gamma }{ \Omega_0}\right)
\left(\frac{1}{\Omega_0 \tau_R}\right)\left(\frac{\omega_L}{\Omega_0}\right).
\label{ANE2}
\end{equation}
The following estimates are reasonable for the proposal\cite{Braggio:2004}:
\begin{eqnarray}
\left(\frac{\gamma }{ \Omega_0}\right)\sim 0.05,
\nonumber \\
\left(\frac{1}{\Omega_0 \tau_R}\right)\sim 10,
\nonumber \\
\left(\frac{\omega_L}{\Omega_0}\right)\sim 2\times 10^5,
\nonumber \\
\bar{n}_{saturate}\sim 10^5 \ {\rm microwave\ photons.}
\label{ANE3}
\end{eqnarray}
\section{Conclusion\label{Conc}}
We have explored the concept of induced instabilities in both the static
and dynamic Casimir effects. For the static case, large quantum
electrodynamic collective Lamb shifts in condensed matter can
induce a phase transition requiring a new equilibrium position
of the microwave oscillator coordinates. In particular, when at the quadratic
level and oscillator goes unstable, quartic terms can be invoked to make the
system stable. For the dynamic case, even if the frequency shifts are small,
perfect periodicity in modulation pulses can build up to exponentially large
proportions again leading to an instability. Again dynamic quartic terms
can stabilize the cavity modes. The basic principle involved is that
the shifted frequencies themselves must undergo fluctuations. Given the noise
fluctuations in the pump coordinate, the final saturation temperature
of the microwave cavity can be computed from Eq.(\ref{MS16}).
\vskip 0.5cm
|
{
"timestamp": "2005-03-01T19:02:52",
"yymm": "0503",
"arxiv_id": "quant-ph/0503016",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503016"
}
|
\section{Introduction}
In this note, we study graphs without cycles of prescribed even
lengths. For a finite or infinite set ${\cal C}$ of cycles, define
$\mbox{ex}(n,{\cal C})$ to be the maximum possible number of edges
in an $n$-vertex graph which does not contain any of the cycles in
${\cal C}$. The asymptotic behaviour of the function
$\mbox{ex}(n,{\cal C})$ is particularly interesting when at least
one of the cycles in ${\cal C}$ is of even length, and was
initiated by Erd\H{os} \cite{Erd}. In general, it is the lower
bounds for $\mbox{ex}(n,{\cal C})$ -- that is, the construction of
dense graphs without certain even cycles -- which are hard to come
by. The best known lower bounds are based on finite geometries,
such as polarity graphs of generalized polygons~\cite{LUW2}, and
the algebraic constructions given by Lazebnik, Ustimenko and
Woldar~\cite{LUW1} and Ramanujan graphs of Lubotsky, Phillips and
Sarnak~\cite{LPS}; see also~\cite{LUW3}. In the direction of upper
bounds, the first major result is known as the even circuit
theorem, due to Bondy and Simonovits \cite{BS}, who proved that
$\mbox{ex}(n,\{C_{2k}\}) \leq 100kn^{1+\frac{1}{k}}$. A more
extensive study of $\mbox{ex}(n,{\cal C})$ was carried out by
Erd\H{o}s and Simonovits~\cite{ES}. Our point of departure is the
study of $\mbox{ex}(n,{\cal C})$ when ${\cal C}$ consists only of
the even cycles of length at most $2k$. The main result of this
article is the following:
\begin{theorem}
\label{thm:main}
Let $k \geq 2$ be an integer. Then, for all $n$,
\[ \mbox{ex}(n,\{C_4,C_6,\dots,C_{2k}\}) \; \; \leq \; \; \textstyle{\frac{1}{2}}n^{1 + \frac{1}{k}} + 2^{k^2}n.\]
Furthermore, when $k \in \{2,3,5\}$, the $n$-vertex
polarity graphs of generalized $(k + 1)$-gons in {\rm \cite{LUW2}}
have $\frac{1}{2}n^{1 + 1/k} + O(n)$
edges and no even cycles of length at most $2k$.
\end{theorem}
For the statement about the number of edges in the polarity
graphs, see~\cite{LUW2}, page 9. Theorem \ref{thm:main} extends
the Moore bound (see \cite{Big}) up to an additive term, and a
more recent result of Alon, Hoory, and Linial~\cite{AHL}, who
proved that an $n$-vertex graph without cycles of length at most
$2k$ has at most $\frac{1}{2}(n^{1 + 1/k} + n)$ edges (see
Proposition \ref{prop:AHL}). In other words, we do not require
that the odd cycles be forbidden, and the same bound still holds,
but with a weaker additive linear term. Our result is also best
possible in the following sense: if we forbid only the $2k$-cycle
in our graphs, then the upper bounds in Theorem \ref{thm:main} no
longer hold -- it was shown recently, in \cite{FNV}, that
$\mbox{ex}(n,\{C_6\}) > 0.534n^{4/3}$ and $\mbox{ex}(n,\{C_{10}\})
> 0.598n^{6/5}$ as $n$ tends to infinity.
\bigskip
\section{Local Structure}
Let $G$ be a graph with no even cycles of length less than or
equal to $2k$. We write $P[u,v]$ to indicate that a path $P
\subset G$ has end vertices $u$ and $v$, and we order the vertices
of $P$ from $u$ to $v$. Let $\prec$ denote this ordering along $P$.
A {\it vine} on a path $P$ is a graph consisting of
the union of $P$ together with paths $Q[u_i,v_i]$
which are internally disjoint from $P$ for $i = 1,2,\dots,r$,
and where $u \preceq u_1 \prec v_1 \preceq u_2 \prec v_2 \preceq \dots
\preceq u_r \prec v_r \preceq v$.
A $uv$-path of shortest length is called a {\it $uv$-geodesic}.
A {\it $\theta$-graph} consists of three internally
disjoint paths with the same pair of endpoints.
\medskip
\begin{lemma}\label{theta}
Any $\theta$-graph contains an even cycle.
\end{lemma}
\begin{proof} If $P, Q$ and $R$ are the internally disjoint paths
in the $\theta$-graph with the same pair of endpoints, then
$|P \cup Q| + |Q \cup R| + |P \cup R| = 2|P| + 2|Q| + 2|R|$,
which is even. Therefore one of the cycles
$P \cup Q$, $Q \cup R$ or $P \cup R$ must have
even length.
\end{proof}
\medskip
\begin{lemma}
\label{lem:short}
Let $P^*$ be a $uv$-geodesic of length at most $k$. Then the union $H$
of all $uv$-paths of length at most $k$ is a vine on $P^{*}$ and $P^{*}$ is the unique $uv$-geodesic.
\end{lemma}
\begin{proof}
Suppose, for a contradiction, that $H$ is not a vine on $P^*$.
Let $x \prec v$ be a vertex of $P^*$ at a maximum distance from $u$ on $P^*$
such that the union of all $ux$-paths in $H$
is a vine on $P^*[u,x]$. By the maximality of $x$, there is a
$uv$-path $P$ of length at most $k$ such that $x$ has degree three in
$P \cup P^*$. If $P$ has minimum possible length, then
$P[x,y] \cup P^*[x,y]$ is the only cycle in $P \cup P^*$
for some $y \succ x$ on $P^*$. By the maximality of $x$, the union of all $uy$-paths
in $H$ is not a vine. Therefore there must be a $uv$-path $Q$ of
length at most $k$ such that $Q \cup P \cup P^*$ is not a vine on $P^*$.
If $Q$ has minimum possible length, then $P \cup Q$ and $P^* \cup Q$
each have exactly one cycle. It follows that there is a path $Q[w,z] \subset Q$ such that
\[ Q[u,x] = P^*[u,x] \; \; \mbox{ and } \; \; Q[x,w] \cup Q[z,v] \subset P[x,v] \cup P^*[x,v]\]
and $Q[w,z]$ is internally disjoint from $P \cup P^*$. Since $P
\cup P^* \cup Q$ is not a vine, $w \in P[x,y] \cup P^*[x,y]$ and
$w \neq y$. If $z \in P^*[y,v]$, then $P^*[x,z] \cup P[x,z] \cup
Q[w,z]$ is a $\theta$-graph (see Figure 1).
\medskip
\SetLabels
\R(.41*.00)$P^*[x,y]$\\
\R(.20*.44)$P^*$\\
\R(.35*.29)$x$\\
\R(.01*.29)$u$\\
\R(1.00*.29)$v$\\
\R(.82*.29)$z$\\
\R(.73*.29)$y$\\
\R(.90*.92)$Q[w,z]$\\
\R(.45*.65)$w$\\
\R(.65*.60)$P[x,y]$\\
\endSetLabels
\begin{center}
\centerline{\AffixLabels{\includegraphics[width=3in]{paths.eps}}}
\end{center}
\begin{center}
{\sf Figure 1 : A $\theta$-graph in $Q \cup P \cup P^*$.}
\end{center}
\medskip
The cycles in this $\theta$ graph are $P[w,z] \cup Q[w,z] \subset P \cup Q$
and $P[x,y] \cup P^*[x,y] \subset P \cup P^*$ and
$P^*[x,z] \cup Q[x,z] \subset P^* \cup Q$. Each
of these cycles has length at most $2k$, since the paths
$P,Q$ and $P^*$ each have length at most $k$.
By Lemma \ref{theta}, one of these cycles has even length, which is a contradiction.
A similar argument works when $z \not \in P^*[y,v]$.
Therefore $H$ is a vine on $P^*$.
\bigskip
To complete the proof, we must show that $P^{*}$ is the unique
$uv$-geodesic. By definition, $H$ consists of the union of $P^{*}$
and paths $P_i = P_i[u_i,v_i]$ for $i \in [r]$, and let $P^{*}_i =
P^{*}[u_i,v_i]$. Since each cycle $P^{*}_i \cup P_i$ is of length
at most $2k$, each cycle in the vine has odd length. Now suppose
$P$ is another $uv$-geodesic. Then $P_i \subset P$ for some $i$.
Since $P_i \cup P_i^{*}$ is an odd cycle, we may assume $|P_i| <
|P^{*}_i|$. By replacing $P^{*}_i$ with $P_i$ on $P^{*}$, we
obtain a $uv$-path of length $|P^{*}| - |P^*_i| + |P_i| <
|P^{*}|$, which contradicts the fact that $P^{*}$ is a
$uv$-geodesic. So $P^{*}$ is the unique $uv$-geodesic.
\end{proof}
\bigskip
Henceforth, the paths
in the vine on $P^{*}$ will be denoted $P_i = P_i[u_i,v_i]$,
and $P^{*}[u_i,v_i] = P^*_i$, for $i \in [r]$. Let $\mathcal{P}_k(u,v)$ denote
the set of all $uv$-paths of length $k$, and define the map
\[
f: \mathcal{P}_k(u,v) \rightarrow 2^{[r]} \; \; \mbox{ by } \; \;
f(P) = \set{i \in [r] \; \mid \; P_{i}[u_i,v_i] \subset P }.
\]
Then $f(P)$ records the set of integers $i$ for which the path $P
\in \mathcal{P}_k(u,v)$ uses the path $P_i[u_i,v_i]$ in the vine on $P^{*}$
instead of $P^{*}[u_i,v_i]$. Let ${\cal F}$ be the image of
$\mathcal{P}_k(u,v)$ under $f$.
\medskip
\begin{lemma}
\label{lem:paths}
The map $f$ is an injection, and the family ${\cal F}$
is an antichain of sets of size at most $k - |P^*|$ in
the partially ordered set of all subsets of $[r]$.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:short}, each $P \in \mathcal{P}_k(u,v)$ is the union of
some (possibly none) of the paths $P_{i}$ together with internally
disjoint subpaths of $P^{*}$. Therefore the set $f(P)$ uniquely
determines $P$, and $f$ is an injection. If two sets in ${\cal F}$
are comparable, say $f(P) \subset f(Q)$, then $|Q| > |P|$ and $Q
\not \in \mathcal{P}_k(u,v)$, which is a contradiction. So ${\cal F}$ is an
antichain. Finally, any path $P \in \mathcal{P}_k(u,v)$ has length at least
$|P^{*}| + |f(P)|$, by Lemma \ref{lem:short}, so all sets in
${\cal F}$ have size at most $k - |P^{*}|$.
\end{proof}
\begin{theorem}\label{thm:max}
Let $G$ be a graph containing no even cycles of length at most $2k$.
Then
\[ |\mathcal{P}_k(u,v)| \; \leq \; \max\left( {r \choose m} : r \leq k \; \mbox{and}\; m = \min
\set{\Big\lfloor \frac{r}{2} \Big\rfloor,k-r}\right).\] The equality is achieved
when $r = |P^{*}|$ and the vine on $P^{*}$ comprises $|P^{*}|$
triangles.
\end{theorem}
\begin{proof}
The family ${\cal F}$ is an antichain, by Lemma \ref{lem:paths}.
By Sperner's Theorem and the LYM inequality \cite{Eng}, this means
that $|{\cal F}| \leq {r \choose m}$ where $m = \min
\set{\floor{\frac{r}{2}},k-|P^*|}$.
\end{proof}
\bigskip
A \emph{non-returning} walk of length $r$ in $G$ is a walk whose
consecutive edges are distinct. Let $\mathcal{W}_r$ be the set of
non-returning $r$-walks (for $r = 0$, $\mathcal{W}_0$ consists of single
vertices). The final result required for the proof of Theorem
\ref{thm:main} is the following lower bound on the number of
non-returning walks, by Alon, Hoory and Linial~\cite{AHL}, which
gives the best known upper bound on
$\mbox{ex}(n,\{C_3,C_4,\dots,C_{2k}\})$:
\begin{proposition}\label{prop:AHL}
Let $G$ be an $n$-vertex graph of average degree $d \geq 2$. Then
$|\mathcal{W}_r| \geq \; nd(d-1)^{r-1}$. Moreover, if $G$ has average degree $d \geq 2$ and no cycles of length at most
$2k$, then $d(d - 1)^{k-1} \leq n$.
\end{proposition}
In \cite{AHL}, the number $\mathcal{W}_r/nd$ is denoted $N_{r-1}$ and shown
to be less than $(d-1)^{r-1}$. The second statement of the
Proposition is an immediate consequence of the main theorem there.
\bigskip
\section{Proof of Theorem \ref{thm:main}}
Let $G$ be a counterexample to Theorem \ref{thm:main} with minimal
number of vertices $n$ and average degree $d$. Then $d >
n^{\frac{1}{k}} + 2^{k^2}$, and $G$ has minimum degree at least
$\lfloor d/2 \rfloor + 1$, otherwise we remove a vertex of lower
degree, keeping the average degree non-increasing, to obtain a
smaller counterexample than $G$. We may also assume $n > 2^{k^2}$.
Now let $v$ be a vertex of $G$ of maximum degree, $\Delta$. Pick a breadth-first search tree $T$
rooted at $v$, and let $T_r$ be the set of vertices of $G$ at
distance at most $r$ from $v$. Then no vertex of $T_r$ is joined
to two vertices in $T_{r-1}$, and the set of edges in $T_{r-1}
\backslash T_{r-2}$ form a matching, for all $r \leq k$. So every
vertex of $T$ has degree at least $\delta - 2$, where $\delta$ is
the minimum degree in $G$, from which we deduce
\[ 1+\Delta +\Delta(\delta-2)+ \dots + \Delta(\delta -2)^{k-1} \; \leq \; |V(T)| \; \leq \; n.\]
Since $\delta > \lfloor d/2 \rfloor$ and
$d > n^{\frac{1}{k}} + 4$, we find $\Delta < 2^{k-1}n^{\frac{1}{k}}$.
\bigskip
Now let $\mathcal{P}_r$ be the set of paths of length $r$ in $G$, and let
$\mathcal{Q}_r = \mathcal{W}_r - \mathcal{P}_r$ be the set of non-returning walks with $r$
edges which are not paths. There are at least $\delta - k$
extensions of a given path of length $r$ in $G$, for any $r < k$.
Therefore
\begin{equation}\label{eq:walkbound}
|\mathcal{P}_k| \geq (\delta - k)^{k - \ell}|\mathcal{P}_{\ell}| \; \; \mbox{ and }
\; \; |\mathcal{Q}_k| \leq \Delta^{k-1} k n < k2^{(k-1)^2}
n^{\frac{2k-1}{k}}.
\end{equation}
By Lemma \ref{lem:short}, for any pair $(u,v)$ of distinct
vertices, joined by at least two paths of length $k$, there is a
$uv$-geodesic of length $\ell < k$. By Theorem \ref{thm:max},
$|\mathcal{P}_k(u,v)| < 2^k$, so the number of ordered pairs of vertices
joined by exactly one $k$-path is at least
\begin{eqnarray*}
|\mathcal{P}_k| - 2^{k} \sum_{\ell = 1}^{k-1}|\mathcal{P}_{\ell}| &\geq& |\mathcal{P}_k|
\brac{ 1 - \frac{2^{k}}{\delta-k-1}}\\
&=& \left(\; |\mathcal{W}_k| - |\mathcal{Q}_k| \;\right) \cdot \brac{ 1 - \frac{2^{k}}{\delta-k-1}}\\
&>& \left(nd(d-1)^{k-1} - k2^{(k-1)^2} n^{\frac{2k-1}{k}}\right)
\cdot \brac{ 1 - \frac{2^{k}}{\delta-k-1}}.
\end{eqnarray*}
In the last line, we used (\ref{eq:walkbound}) and Proposition
\ref{prop:AHL}. There are $n(n - 1)$ (ordered) pairs of distinct
vertices which could be joined by a unique path of length $k$, so
the expression above is less than $n^2$. Using $\delta-k-1 \geq
\frac{d}{4}$ and substituting $d = n^{\frac{1}{k}} + 2^{k^2}$
into the last line, we get
\begin{eqnarray*}
n^2 &>& \left(n(n^{\frac{1}{k}} + 2^{k^2})(n^{\frac{1}{k}} +
2^{k^2} - 1)^{k-1} - k2^{(k-1)^2} n^{\frac{2k-1}{k}}\right)
\left(1 - \frac{2^{k+2}}{n^{\frac{1}{k}} + 2^{k^2}}\right) \\ \\
&=& \left(n^{\frac{2k - 1}{k}}(n^{\frac{1}{k}} + 2^{k^2})(1 +
n^{-\frac{1}{k}}(2^{k^2} - 1))^{k-1} -
k2^{(k - 1)^2}n^{\frac{2k - 1}{k}}\right)\left(1 - \frac{2^{k+2}}{n^{\frac{1}{k}} + 2^{k^2}}\right) \\ \\
&>& \left(n^{\frac{2k - 1}{k}}(n^{\frac{1}{k}} + 2^{k^2})(1 +
n^{-\frac{1}{k}}(k - 1)(2^{k^2} - 1)) -
k2^{(k - 1)^2}n^{\frac{2k-1}{k}}\right)\left(1 - \frac{2^{k+2}}{n^{\frac{1}{k}} + 2^{k^2}}\right) \\ \\
&>& n^2 \left(1 + \frac{2^{k^2}}{n^{\frac{1}{k}} + 2^{k^2}}\right)
\left(1 - \frac{2^{k + 2}}{n^{\frac{1}{k}} + 2^{k^2}}\right) \; \; > \; \; n^2\\
\end{eqnarray*}
which gives a contradiction. We must thus have $d <
n^{\frac{1}{k}} + 2^{k^2}$. \vrule height10pt width5pt depth1pt
\bigskip
\section{Concluding Remarks}
If $G$ is $d$-regular, then picking a breadth first search tree
as in the calculation of the maximum degree we obtain
\[ 1+d+d(d-2)+ \dots +d(d-2)^{k-1} \leq n.\]
So in this case we have $d < n^{\frac{1}{k}} + 2$.
The main points at which the large linear term
is introduced in the proof of Theorem \ref{thm:main}
is in the estimate of the maximum degree
and the upper bound on $|{\cal Q}_k|$. We believe it
should be possible to circumvent these bounds to obtain
a linear term of the form $cn$, for some absolute constant $c$.
Finally, we note that the analogous extremal problem when some of
the short odd cycles are forbidden seems to be very difficult.
For example, it is known that
\[ \frac{1}{2\sqrt{2}} \; \leq \; \liminf_{n \rightarrow \infty} \frac{\mbox{ex}(n,\{C_3,C_4\})}{n^{3/2}} \; \leq \;
\limsup_{n \rightarrow \infty}
\frac{\mbox{ex}(n,\{C_3,C_4\})}{n^{3/2}} \; \leq\; \frac{1}{2},\]
but the asymptotic value of $\mbox{ex}(n,\{C_3,C_4\})$ remains an
open question (posed by Erd\H{o}s).
\bigskip
\textbf{Acknowledgements.}
The first author would like to thank Terence Tao for supervising him during
his undergraduate thesis, which led to this work.
|
{
"timestamp": "2005-03-28T02:39:06",
"yymm": "0503",
"arxiv_id": "math/0503623",
"language": "en",
"url": "https://arxiv.org/abs/math/0503623"
}
|
\section{Introduction}
Many aspects of life as a responsible citizen in society
involve having an understanding of the probability
of one type of event in comparison to others. Yet event probabilities are
often expressed using unfamiliar or varied terminology (\textit{i.e.,}
negative exponents, such as $10^{-4}$ or $10^{-5}$, one part in a thousand, etc.) with the
result that,
for the ordinary person, the comparison of event probabilities and the drawing of valid conclusions are made more difficult.
\section{Proposed Remedy}
As a remedy for this, I propose the Improbability Scale, or $IS$, defined as:
{
\mathversion{bold}
\begin{equation}
IS = - \log_{10} (p)
\label{mp_master}
\end{equation}
}
where $p$ is the probability of the event.
$IS$ takes on the value of 0 for absolutely certain events and proceeds upwards for events with greater and greater {\it im}probability.
Table~I lists some events and their $IS$ values.
\begin{center}
\bigskip
\begin{tabular}{|l|c|}
\hline
\hfil \textbf{Event} \hfil & \textbf{IS} \\ \hline
Rolling a 7 on the next roll of a pair of dice~\citep{twodie} & $0.8$ \\ \hline
Space Shuttle major failure on next launch - current experience~\citep{shuttle} & $2.3$ \\ \hline
One's birthday occuring tomorrow within a given year~\citep{birthday} & $2.6$ \\ \hline
Space Shuttle major failure on next launch - near term goal~\citep{shuttle} & $4.0$ \\ \hline
Being struck by lightning within a given year~\citep{lightning} & $5.4$ \\ \hline
Drawing a royal flush on the next deal of five cards~\citep{royalflush} & $5.8$ \\ \hline
Space shuttle major failure on next launch - eventual goal~\citep{shuttle} & $6.0$ \\ \hline
Winning the jackpot in the next Powerball Lottery~\citep{powerball} & $8.1$ \\ \hline
A core-collapse Supernova occurring within a given year close & \\
enough to Earth (8 parsecs) to cause significant biological effects~\citep{supernova} & $8.8$ \\ \hline
\end{tabular}
\\
\bigskip
Table I \\
Some Events and their $IS$ values\\
\bigskip
\end{center}
Because Improbability Scale values are typically small numbers between $0$ and $10$, they are easily remembered---particularly in the case of personally meaningful events. The public can use the $IS$ values for such events to ``customize'' its understanding of the Improbability Scale. When a new or less familiar event is presented, the public can use the event{\tt '}s $IS$ value to put its improbability into proper perspective and, by implication, to draw conclusions about the event{\tt '}s {\it probability} as well.
\section{Examples of the Utility of the Improbability Scale}
A standout example of how the Improbability Scale could have served better to communicate the risks of a technological endeavor may be found in an October 2000 speech~\citep{shuttle} on the topic of {\it NASA in the 21st Century} given by then NASA Administrator Daniel Goldin. The speech was given to a Laboratory audience at the Applied Physics Laboratory Colloquium~\cite{APL} of The Johns Hopkins University and was also reported on {\bf Space.com} by Leonard David to a readership more characteristic of the interested general public. There, Goldin is reported as saying: \begin{quote}\begin{center}``We want to take the probability of a major failure of today{\tt '}s space shuttle from one part
in 200 to one part in 10,000,
and eventually to one part in 1,000,000 with about the same reliability of today{\tt '}s commercial aircraft.''\end{center}\end{quote} No doubt for the experienced Laboratory audience the implications of a risk assessment of ``one part in 200'' were well understood. For the interested general public with little or no context in which to place that assessment, the same is not clear.
However, with context provided by the Improbability Scale and a knowledge of the $IS$ for familiar events, such as that for {\it certainty} equaling $0$ and that for tomorrow being one's birthday equaling $2.6$, the public would have almost certainly understood the implications of a risk assessment that stated: \begin{quote}\begin{center}``On the Improbability Scale, a major failure \\of today{\tt '}s Space Shuttle has a rank of $2.3$.''\end{center}\end{quote}
Another example of the utility of the Improbability Scale relates to the $IS$ for several independent events occuring together. The $IS$ for the combined occurrence is the sum of the $IS$ values for the individual events. This simple combination rule makes it easy for the general public to use its knowledge of the $IS$ values for familiar events to understand the improbability of a new or less familiar event.
Knowing that the $IS$ for one{\tt '}s birthday occuring tomorrow within a given year is $2.6$ and that the $IS$ for being struck by lightning within a given year is $5.4$, one has an immediate understanding of just how improbable an $IS$ $8.0$ event is---namely, it is as improbable as getting struck by lightning on one's birthday. One can then apply that understanding to even mundane matters, such as when one learns that winning the Powerball Lottery jackpot on the next drawing~\citep{powerball} has an $IS$ of $8.1$.
\section{Conclusion}
I suggest that researchers quote the Improbability Scale values when writing for the general public. Widespread adoption of this way of characterizing events will enhance the public's understanding of the predictions of science and help in obtaining the public's support
for actions related to those predictions in, for example, such cases as natural disasters and technological failures.
\section{Acknowledgements}
I am grateful to Robert Cousins, Department of Physics and Astronomy, UCLA for a number of discussions. I thank Mariano Zimmler, Division of Engineering and Applied Sciences, Harvard University for a careful reading of the manuscript. Fermilab is operated under DOE contract DE-AC02-76CH03000.
\newpage
\section{References}
|
{
"timestamp": "2005-04-01T01:56:05",
"yymm": "0503",
"arxiv_id": "physics/0503229",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503229"
}
|
\section{Motivation}
Short-term financial data usually exhibit similar properties called `stylized facts' like, e.g., leptokurtosis,
dependence of simultaneous extremes, radial asymmetry, vola\-tility clustering, etc., especially if the
log-price changes (called the `log-returns') of stocks, stock indices, and foreign exchange rates are
considered. Particularly, high-frequency data usually are non-stationary, have jumps, and are strongly
dependent. Cf., e.g., Bouchaud, Cont, and Potters, 1998, Breymann, Dias, and Embrechts, 2003, Eberlein and
Keller, 1995, Embrechts, Frey, and McNeil, 2004 (Section 4.1.1), Engle, 1982, Fama, 1965, Junker and May, 2002, Mandelbrot, 1963, and Mikosch, 2003 (Chapter 1).
Figure 1 contains QQ-plots of $\text{GARCH}(1,1)$ residuals of daily log-returns of the NASDAQ and the
S\&P 500 indices from 1993-01-01 to 2000-06-30. It is clearly indicated that the normal distribution hypothesis is not appropriate for the loss parts of the distributions whereas the Gaussian law seems to be acceptable for the profit parts. Hence the probability of extreme losses is higher than suggested by the normal distribution assumption.
\begin{center}
\includegraphics[scale=.34]{NASDAQ_QQ-Plot}
\includegraphics[scale=.34]{SP500_QQ-Plot}\\[.25cm]
\end{center}
{\bf Fig. 1:} QQ-plots of NASDAQ (left hand) and S\&P 500 (right hand) $\text{GARCH}(1,1)$ residuals from
1993-01-01 to 2000-06-30 ($n=1892$).\\[.25cm]
The next picture shows the joint distribution of the GARCH residuals considered above.
\begin{center}
\includegraphics[scale=.35]{NASDAQ_vs_SP500_-_emp_ohne_Konturen}\\[.25cm]
\end{center}
{\bf Fig. 2:} NASDAQ vs. S\&P 500 $\text{GARCH}(1,1)$ residuals from 1993-01-01 to 2000-06-30 ($n=1892$).\\[.25cm]
Except for one element all extremes occur simultaneously. The effect of simultaneous extremes can be observed
more precisely in the following picture. It shows the total numbers of S\&P 500 stocks whose absolute values of
daily log-returns exceeded $10\%$ for each trading day during 1980-01-02 to 2003-11-26. On the 19th October
1987 (i.e. the `Black Monday') there occurred 239 extremes. This is suppressed for the sake of transparency.
\begin{center}
\includegraphics[scale=.35]{outlier_profile}\\[.25cm]
{\bf Fig. 3:} Number of extremes in the S\&P 500 during 1980-01-02 to 2003-11-26.\\[.25cm]
\end{center}
The latter figure shows the concomitance of extremes. If extremes would occur independently then the number of
extremal events (no matter if losses or profits) should be small and all but constant over time. Obviously,
this is not the case. In contrast one can see the October Crash of 1987 and several extremes which occur
permanently since the beginning of the bear market in 2000. Hence there is an increasing tendency of
simultaneous losses which is probably due to globalization effects and relaxed market regulation. The phenomenon of simultaneous extremes is often denoted by `asymptotic dependence' or `tail dependence'.
The traditional class of elliptically symmetric distributions (Cambanis, Huang, and Simons, 1981, Fang, Kotz,
and Ng, 1990, and Kelker, 1970) is often proposed for the modeling of financial data (cf., e.g., Bingham and
Kiesel, 2002). But elliptical distributions suffer from the pro\-perty of radial symmetry. The pictures above
show that financial data are not always symmetrically distributed. For this reason the authors will bear on the
assumption of gene\-ralized elliptically distributed (Frahm, 2004) log-returns. This allows for the modeling of tail dependence and radial asymmetry.
The quintessence of modern portfolio theory is that the portfolio diversification effect depends essentially on
the covariances. But the parameters for portfolio optimization, i.e. the mean vector and the covariance matrix,
have to be estimated. Especially for portfolio risk minimization a reliable estimate of the covariance matrix
is necessary (Chopra and Ziemba, 1993). For covariance matrix estimation generally one should use as much
available data as possible. But since daily log-returns and all the more high-frequency data are not normally
distributed, standard estimators like the sample covariance matrix may be highly inefficient leading to
erroneous implications (see, e.g., Oja, 2003 and Visuri, 2001). This is because the sample covariance matrix is
very sensitive to outliers. The smaller the distribution's tail index (Hult and Lindskog, 2002), i.e. the
heavier the tails of the log-return distributions the higher the estimator's variance. So the quality of the
parameter estimates depends essentially on the true multivariate distribution of log-returns.
In the following it is shown how the linear dependence structure of generalized elliptical random vectors can be
estimated robustly. More precisely, it is shown that Tyler's (1987) robust M-estimator for the dispersion matrix
$\Sigma$ of elliptically distributed random vectors remains completely robust for generalized elliptically
distributed random vectors. This estimator is not disturbed neither by asymmetries nor by outliers and all the
available data points can be used for estimation purposes. Further, the impact of high-dimensional (financial)
data on statistical inference will be discussed. This is done by referring to a branch of statistical physics
called `Random Matrix Theory' (Hiai and Petz, 2000 and Mehta, 1990). Random matrix theory (RMT) is concerned
with the distribution of eigenvalues of high-dimensional randomly generated matrices. If each component of a sample is independent and identically distributed then the distribution of the eigenvalues of the sample covariance matrix converges to a specified law which does not depend on the specific distribution of the sample components. The circumstances under which this result of RMT can be properly adopted to generalized elliptically distributed data will be examined.
\section{Generalized Elliptical Distributions}
It is well known that an elliptically distributed random vector $X$ can be represented stochastically by
$X\! =_{\mathrm{d}}\! \mu +\mathcal{R}\Lambda U^{\left( k\right)}$, where $\mu\in\mathbb{R}^{d}$, $\Lambda\in\mathbb{R}^{d\times k}$
with $\mathrm{r}(\Lambda)=k$, $U^{\left( k\right) }$ is a $k$-dimensional random vector uniformly distributed on the
unit hypersphere $\mathcal{S}^{k-1}$, and $\mathcal{R}$ is a nonnegative random variable stochastically
independent of $U^{\left( k\right) }$. The positive semi-definite matrix $\Sigma := \Lambda\Lambda^{\mathrm{T}}$ characterizes the linear dependence structure of $X$ and is referred to as the `dispersion matrix'.
\begin{definition}[Generalized elliptical distribution]
The $d$-dimensional random vector $X$ is said to be `generalized elliptically distributed' if and only if
\begin{equation*}
X\overset{\mathrm{d}}{=}\mu +\mathcal{R}\Lambda U^{\left( k\right) }.
\end{equation*}
where $U^{\left( k\right) }$ is a $k$-dimensional random vector uniformly distributed on $\mathcal{S}^{k-1}$,
$\mathcal{R}$ is a random variable, $\mu \in \mathbb{R}^{d}$, and $\Lambda \in \mathbb{R}^{d\times k}$.
\end{definition}
Note that the definition of generalized elliptical distributions preserves all the ordinary components of
elliptically symmetric distributions (i.e. $\mu$, $\Sigma$, and $\mathcal{R}$). But in contrast the generating
variate $\mathcal{R}$ may be negative and even more it may depend on $U^{\left( k\right) }$. It is worth to point out that the class of generalized elliptical distributions contains the class of skew-elliptical distributions (Branco and Dey, 2001, and Frahm, 2004, Section 3.2).
The next figure shows once again the joint distribution of the GARCH residuals of the NASDAQ and
S\&P 500 log-returns from 1993-01-01 to 2000-06-30 from Figure 2. The right hand of Figure 4 contains
simulated GARCH residuals on the basis of a generalized $t$-distribution. More precisely, the generating variate
$\mathcal{R}$ corres\-ponds to $\sqrt{\nu \cdot \chi _{2}^{2}/\chi _{\nu }^{2}}\,$ but the number of degrees of
freedom $\nu$ depends on $U^{(2)}$, i.e. $\nu = 4 + 996\cdot\left(\delta(\Lambda u / \|\Lambda
u\|_{2},v\right))^{3}$ $(\|u\|_{2}=1)$. Here $\delta$ is a function that measures the distance between $\Lambda
u / \|\Lambda u\|_{2}$ and the reference vector $v=\left(-\cos \left( \pi /4\right) ,-\sin \left( \pi /4\right)
\right)$, $\delta(u,v) := \angle(u,v)/\pi = \arccos(u^{\mathrm{T}} v)/\pi$. Hence, random vectors which are close to the
reference vector (i.e. close to the `perfect loss scenario') are supposed to be $t$-distributed with $\nu=4$
degrees of freedom whereas random vectors which are opposite are assumed to be nearly Gaussian ($\nu=1000$)
distributed. This is consistent with the phenomenon observed in Figure 1. The pseudo-correlation coefficient is
set to $0.78$.
\begin{center}
\includegraphics[scale=.34]{emp}
\includegraphics[scale=.34]{sim}\\[.25cm]
\end{center}
{\bf Fig. 4:} Observed $\text{GARCH}(1,1)$ residuals of NASDAQ and S\&P 500 (left hand) and simulated
generalized $t$-distributed random noise ($n=1892$) (right hand).\\[.25cm]
\section{Robust Covariance Matrix Estimation}
It is well-known that the sample covariance matrix corresponds both to the moment estimator and to the
ML-estimator for the dispersion matrix $\Sigma$ of normally distributed data. But given any other elliptical
distribution family the dispersion matrix usually does not correspond to the covariance matrix. Generally,
robust covariance matrix estimation means to estimate the dispersion matrix, that is the covariance matrix up
to a scaling constant. There are many applications like, e.g., principal components analysis, canonical
correlation analysis, linear discriminant ana\-lysis, and multivariate regression where only the dispersion
matrix is demanded (Oja, 2003). Particularly, by Tobin's two-fund separation theorem (Tobin, 1958) the
optimal portfolio of risky assets does not depend on the scale of the covariance matrix. Thus in the following
we will loosely speak of `covariance matrix estimation' rather than of estimating the dispersion matrix for the
sake of simplicity.
As mentioned before the true linear dependence structure of elliptically distributed data can not be estimated efficiently by the sample covariance matrix, generally. Especially, if the data stem from a regularly varying random
vector the smaller the tail index, i.e. the heavier the tails the larger the estimator's variance. But in the following it is shown that there exists a completely robust alternative to the sample covariance matrix.
Let $X$ be a $d$-dimensional generalized elliptically distributed random vector where $\mu$ is supposed to be
known, $\Lambda \in \mathbb{R}^{d\times k}$ with $\mathrm{r}(\Lambda)=d$, and $P(\mathcal{R}=0)=0$. Further, let the unit
random vector generated by $\Lambda$ be defined as
\begin{equation*}
S := \frac{\Lambda U^{\left( k\right) }}{ {\big |\!|}\Lambda U^{\left( k\right) }{\big |\!|}_{2}}.
\end{equation*}
Due to the stochastic representation of $X$ the following relations hold,
\begin{equation*}
\frac{X-\mu}{{\big |\!|}X-\mu{\big |\!|}_{2}}\overset{\mathrm{d}}{=}%
\frac{\mathcal{R}\Lambda U^{\left( k\right) }}{
{\big |\!|}\mathcal{R}\Lambda U^{\left( k\right) }{\big |\!|}_{2}}\overset{\mathrm{a.s.}}{=}%
\pm\frac{\Lambda U^{\left( k\right) }}{ {\big |\!|}\Lambda U^{\left( k\right) }{\big |\!|}_{2}}=\pm S,
\end{equation*}
where $\pm :=\mathrm{sgn}(\mathcal{R})$. The random vector $\pm S$ does not depend on the absolute value of
$\mathcal{R}$. So it is completely robust against extreme outcomes of the generating variate. But the sign of
$\mathcal{R}$ still remains and this may depend on $U^{\left( k\right) }$, anymore. Suppose for the moment that
$\pm$ is known for each realization of $\mathcal{R}$. Then the dispersion matrix of $X$ can be estimated
robustly via maximum-likelihood estimation using the density function of $S$ which is only a function of
$\Lambda$. This is given by the next theorem.
\begin{theorem}
The spectral density function of the unit random vector generated by $\Lambda \in \mathbb{R}^{d\times k}$ corresponds
to
\begin{equation*}\label{spectral_density}
s\longmapsto \psi \left( s\right) =\frac{\Gamma \left( \frac{d}{2}\right) }{2\pi ^{d/2}}\cdot \sqrt{\det
(\Sigma ^{-1})}\cdot \sqrt{s^{\mathrm{T}}\Sigma ^{-1}s}^{\,-d},\qquad \forall \ s\in \mathcal{S}^{d-1},
\end{equation*}
where $\Sigma :=\Lambda \Lambda ^{\mathrm{T}}$.
\end{theorem}
\begin{proof}
See, e.g., Frahm, 2004, pp. 59-60.\hfill \medskip
\end{proof}
Since $\psi$ is a symmetric density function the sign of $\mathcal{R}$ does not matter at all. Hence the
ML-estimation approach works even if the data are skew-elliptically distributed, for instance.
The desired `spectral estimator' is given by the fixed-point equation (Frahm, 2004, Section 4.2.2)
\begin{equation*}
\widehat{\Sigma}_{\mathrm{S}}=\frac{d}{n}\cdot \sum_{j=1}^{n}\frac{s_{j}s_{j}^{\mathrm{T}}}{s_{j}^{\mathrm{T}}\widehat{\Sigma}_{\mathrm{S}}^{-1}s_{j}},
\end{equation*}
where $s_{j}:=\left(x_{j}-\mu\right)/\left({\big |\!|}x_{j}-\mu {\big |\!|}_{2}\right)$
for $j=1,...,n$. Since the solution of the fixed-point equation is only unique up to a scaling constant in the
following it is implicitly required that the upper left element of $\widehat{\Sigma}_{\mathrm{S}}$ corresponds
to $1$.
The spectral estimator $\widehat{\Sigma}_{\mathrm{S}}$ cor\-responds to Tyler's robust M-estimator (Tyler,
1983 and Tyler, 1987) for elliptical distributions, i.e.
\begin{equation*}
\widehat{\Sigma}_{\mathrm{S}}=\frac{d}{n}\cdot \sum_{j=1}^{n}\frac{\left( x_{j}-\mu \right) \left(
x_{j}-\mu \right) ^{\mathrm{T}}}{\left( x_{j}-\mu \right) ^{\mathrm{T}}\widehat{\Sigma}_{\mathrm{S}}^{-1}\left( x_{j}-\mu \right) }.
\end{equation*}
Hence Tyler's M-estimator remains completely robust within the class of generalized elliptical distributions.
The following figure shows the sample covariance matrix (left hand) of a sample with $n=1000$ observations and
$d=500$ dimensions drawn from a multivariate $t$-distribution with $\nu=4$ degrees of freedom. Note
that the tail index of the multivariate $t$-distribution corresponds to $\nu$. Each cell of the plots represents
a matrix element where the blue colored cells symbolize small numbers and the red colored cells indicate large
numbers. The true dispersion matrix is given in the middle whereas the spectral estimate is given by the right
hand.
\begin{center}
\includegraphics[height=4.5cm,width=4.5cm]{nu4momest.eps}\quad
\includegraphics[height=4.5cm,width=4.5cm]{true.eps}\quad
\includegraphics[height=4.5cm,width=4.5cm]{nu4specest.eps}\\[.25cm]
\end{center}
{\bf Fig. 5:} Sample covariance matrix (left hand), true covariance matrix (middle), and spectral estimate
(right hand) of multivariate $t$-distributed realizations ($n=1000,\,d=500,\,\nu=4$).\\[.25cm]
\section{Random Matrix Theory}\label{RMT}
RMT is concerned with the distribution of the eigenvalues of high-dimensional randomly gene\-rated matrices. A
random matrix is simply a matrix of random variables. We will consider only symmetric random matrices. Thus the
corresponding eigenvalues are always real. The empirical distribution function of eigenvalues is defined as
follows.
\begin{definition}[Empirical distribution function of eigenvalues]
Let $\widehat{\Sigma}$ be a $d\times d$ symmetric random matrix with eigenvalues
$\widehat{\lambda}_{1},\widehat{\lambda}_{2},\ldots ,\widehat{\lambda}_{d}\,$. Then the function
\begin{equation*}
\lambda \longmapsto \widehat{W}_{d}\left( \lambda \right) :=\frac{1}{d}\cdot
\sum_{i=1}^{d}1\!\!1_{\widehat{\lambda}_{i}\leq \,\lambda }
\end{equation*}
is called the `empirical distribution function of the eigenvalues' of $\,\widehat{\Sigma}$.
\end{definition}
Note that each eigenvalue of a random matrix in fact is random but per se not a random variable since there is
no single-valued mapping $\widehat{\Sigma}\mapsto\widehat{\lambda}_{i}$ $\left( i\in \left\{ 1,\ldots ,d\right\}
\right)$ but rather $\widehat{\Sigma}\mapsto\lambda (\widehat{\Sigma})$ where $\lambda (\widehat{\Sigma})$
denotes the set of all eigenvalues of $\widehat{\Sigma}$. This can be simply fixed by assuming that the
eigenvalues $\widehat{\lambda}_{1},\widehat{\lambda}_{2},\ldots ,\widehat{\lambda}_{d}$ are sorted either in an
increasing or decreasing order.
\begin{theorem}[Mar\v{c}enko and Pastur, 1967]\label{MP_law}
Let $U_{1}^{\left( d\right) },U_{2}^{\left( d\right) },\ldots ,U_{n}^{\left( d\right) }$ $\left( n=1,2,\ldots
\right)$ be sequences of independent random vectors uniformly distributed on the unit hypersphere
$\mathcal{S}^{d-1}$ and consider the random matrix
\begin{equation*}
\widehat{\Sigma}_{\mathrm{MP}}:=\frac{d}{n}\cdot\sum_{j=1}^{n}U_{j}^{\left( d\right) }U_{j}^{\left( d\right)
\mathrm{T}},
\end{equation*}%
where its empirical distribution function of the eigenvalues is denoted by $%
\widehat{W}_{d}\,$. Suppose that $n\rightarrow \infty $,$\ d\rightarrow \infty $, $n/d\rightarrow q<\infty $.
Then
\begin{equation*}
\widehat{W}_{d}\overset{\mathrm{p}}{\longrightarrow }F_{\mathrm{MP}}\left(\cdot\,;q\right),
\end{equation*}
at all points where $F_{\mathrm{MP}}$ is continuous. More precisely, $\lambda \mapsto F_{\mathrm{MP}}\left(
\lambda \,;q\right) =F_{\mathrm{MP}}^{\mathrm{Dir}}\left( \lambda \,;q\right)
+F_{\mathrm{MP}}^{\mathrm{Leb}}\left( \lambda \,;q\right) $ where the Dirac part is given by
\begin{equation*}
\lambda \longmapsto F_{\mathrm{MP}}^{\mathrm{Dir}}\left( \lambda \,;q\right) =\left\{
\begin{array}{lll}
1-q, & & \lambda \geq 0,\,0\leq q<1, \\
\rule{0cm}{0.5cm}0, & & \text{else},%
\end{array}%
\right.
\end{equation*}%
and the Lebesgue part $\lambda \mapsto F_{\mathrm{MP}}^{\mathrm{Leb}}\left(
\lambda \,;q\right) =\int_{-\infty }^{\lambda }f_{\mathrm{MP}}^{\mathrm{Leb}%
}\left( x\,;q\right) dx$ is determined by the density function%
\begin{equation*}
\lambda \longmapsto f_{\mathrm{MP}}^{\mathrm{Leb}}\left( \lambda \,;q\right) =\left\{
\begin{array}{lll}
\frac{q}{2\pi}\cdot \frac{\sqrt{\left( \lambda _{\max }-\lambda \right) \left( \lambda -\lambda _{\min }\right)
}}{\lambda }, & & \lambda _{\min }< \lambda < \lambda _{\max }, \\
\rule{0cm}{0.5cm}0, & & \text{else},%
\end{array}%
\right.
\end{equation*}%
where%
\begin{equation*}
\lambda _{\min ,\max }:=\left( 1\pm \frac{1}{\sqrt{q}}\right) ^{2}.
\end{equation*}
\end{theorem}
\begin{proof}
Mar\v{c}enko and Pastur, 1967.\hfill \medskip
\end{proof}
In the following $\widehat{\Sigma}_{\mathrm{MP}}$ will be called `Mar\v{c}enko-Pastur operator'. The next
corollary states that the Mar\v{c}enko-Pastur law $F_{\mathrm{MP}}$ holds not only for the empirical
distribution function of eigenvalues of the Mar\v{c}enko-Pastur operator but also for that obtained by the
sample covariance matrix if the data are standard normally distributed and independent.
\begin{corollary}
Let $X,X_{1},X_{2},\ldots ,X_{n}$ $\left( n=1,2,\ldots \right)$ be sequences of independent and standard normally
distributed random vectors with uncorrelated components. Then the empirical distribution function of the eigenvalues of
\begin{equation*}
\frac{1}{n}\cdot\sum_{j=1}^{n}X_{j}X_{j}^{\mathrm{T}}
\end{equation*}
converges in probability to the Mar\v{c}enko-Pastur law stated in Theorem \ref{MP_law}.
\end{corollary}
\begin{proof}
Due to the strong law of large numbers $\chi _{d}^{2}/d\overset{\mathrm{a.s.}}{\rightarrow }1$
$(d\rightarrow\infty)$ and thus
\begin{equation*}
\widehat{\Sigma}_{\mathrm{MP}} \sim \frac{d}{n}\cdot \sum_{j=1}^{n}\frac{\chi _{d,j}^{2}}{d}\cdot U_{j}^{\left(
d\right) }U_{j}^{\left( d\right) \mathrm{T}} \overset{\mathrm{d}}{=}
\frac{1}{n}\cdot\sum_{j=1}^{n}X_{j}X_{j}^{\mathrm{T}}.
\end{equation*}
\rule{.5cm}{0cm}\hfill\medskip
\end{proof}
Moreover, the Mar\v{c}enko-Pastur law holds even if $X$ is an arbitrary random vector with standardized i.i.d.
components provided the second moment is finite (Yin, 1986). More precisely, consider the random vector $X$ with
$E(X)=\mu$ and $Var(X)=\sigma^2 I_{d}$ where the components of $X$ are supposed to be stochastically
independent. Then the Mar\v{c}enko-Pastur law can be applied on the empirical distribution function of the
eigenvalues of
\begin{equation*}
\frac{1}{n}\cdot\sum_{j=1}^{n}\left(\frac{X_{j}-\widehat{\mu}}{\widehat{\sigma}}\right)
\left(\frac{X_{j}-\widehat{\mu}}{\widehat{\sigma}}\right)^{\mathrm{T}}= \widehat{\Sigma}/\widehat{\sigma}^2,
\end{equation*}
where $\widehat{\Sigma}$ denotes the sample covariance matrix and
\begin{equation*}
\widehat{\sigma}^2:=\frac{\mathrm{tr}(\widehat{\Sigma})}{d}=\frac{1}{d}\cdot
\sum_{i=1}^{d}\widehat{\lambda}_{i}=:\overline{\lambda}.
\end{equation*}
Hence, the Mar\v{c}enko-Pastur law can be applied virtually ever on the empirical distribution function of
$\widehat{\lambda}_{1}/\overline{\lambda},...,\widehat{\lambda}_{d}/\overline{\lambda}$ where the estimated
eigenvalues are given by the sample covariance matrix provided the sample elements, i.e. the realized random
vectors consist of stochastically independent components. But within the class of elliptical distributions this
holds only for uncorrelated normally distributed data. Hence linear independence and stochastical independence
are not equivalent for genera\-lized elliptically distributed data. This is because even if there is no linear dependence between the components of an elliptically distributed random vector another sort of nonlinear dependence caused by the gene\-rating variate $\mathcal{R}$ remains, generally.
For instance, consider the unit random vector
$U^{(2)}=(U_{1},U_{2})$. Then
\begin{equation*}
U_{2}\overset{\mathrm{a.s.}}{=}\pm \sqrt{1-U_{1}^{2}},
\end{equation*}%
i.e. $U_{2}$ depends strongly on $U_{1}$ though indeed the elements of $U^{(2)}$ are uncorrelated.
Tail dependent random variables cannot be stochastically independent. Especially, if the random components of an elliptically distributed random vector are heavy tailed, i.e. if the generating variate is regularly varying then they possess the property of tail dependence (Schmidt, 2002). In that case the eigenspectrum generated by the sample covariance matrix may lead to erroneous implications.
For instance, consider a sample (with sample size $n=1000$) of $500$-dimensional random vectors where each vector element is standardized $t$-distributed with $\nu=5$ degrees of freedom and stochastically independent of each other. Here the eigenspectrum obtained by the sample covariance matrix indeed is consistent with the Mar\v{c}enko-Pastur law (upper left part of Figure 6). But if the data stem from a multivariate $t$-distribution possessing the same parameters and each vector component is uncorrelated then the eigenspectrum obtained by the sample covariance matrix does not correspond to the Mar\v{c}enko-Pastur law (upper right part of Figure 6). Actually, there are $24$ eigenva\-lues exceeding the Mar\v{c}enko-Pastur upper bound $\lambda _{\max}=(1+1/\sqrt{2}\,)^{2}=2.91$ and the largest eigenvalue corresponds to $10.33$. But fortunately
the eigenspectra obtained by the spectral estimator are consistent with the Mar\v{c}enko-Pastur law as
indicated by the lower part of Figure 6.
\begin{center}
\includegraphics[scale=.34]{MP1mom}
\includegraphics[scale=.34]{MP2mom}\\[.25cm]
\includegraphics[scale=.34]{MP1spec}
\includegraphics[scale=.34]{MP2spec}\\[.25cm]
\end{center}
{\bf Fig. 6:} Eigenspectra of univariate (left part) and multivariate (right part) uncorrelated $t$-distributed
data ($n=1000,\,d=500,\,\nu=5$) obtained by the sample covariance matrix (upper part) and by the spectral
estimator (lower part).\\[.25cm]
Tyler (1987) shows that the spectral estimator converges strongly to the true dispersion matrix $\Sigma $. That means %
\begin{equation*}
\frac{s_{j}s_{j}^{\mathrm{T}}}{s_{j}^{\mathrm{T}}%
\widehat{\Sigma }^{-1}s_{j}}\longrightarrow \frac{%
s_{j}s_{j}^{\mathrm{T}}}{s_{j}^{\mathrm{T}}\Sigma ^{-1}s_{j}},\qquad n\longrightarrow \infty ,\ d\text{ const.,}
\end{equation*}%
for $j=1,2,\ldots$ and $P$-almost all realizations. Consequently, if $\Sigma =I_{d}$ (up to a scaling constant) then%
\begin{equation*}
\frac{s_{j}s_{j}^{\mathrm{T}}}{s_{j}^{\mathrm{T}}%
\widehat{\Sigma }^{-1}s_{j}}\longrightarrow s_{j}s_{j}^{\mathrm{T}} \equiv u_{j}^{\left(d\right)}u_{j}^{\left(d\right)\mathrm{T}},
\end{equation*}%
as $n\rightarrow\infty$ and $d$ constant. Hence the spectral estimator and the Mar\v{c}enko-Pastur operator are
asymptotically equivalent provided $\Sigma =\sigma^{2}I_{d}$. The authors believe that the strong
convergence holds even for $n\rightarrow \infty $, $d\rightarrow \infty $, $n/d\rightarrow q>1$ for $P$-almost
all realizations where the spectral estimate exists. The proof of this conjecture is due to a forthcoming work.
Note that for $q\leq 1$ the spectral estimate does not exist at all. Further, Tyler (1987) shows that the
spectral estimate exists (a.s.) if $n>d\left(d-1\right)$, i.e. $q>d-1$. Indeed, this is a sufficient condition
for the existency of the spectral estimator. But in practice the spectral estimator seems to exist in most cases
when $n$ is already slightly larger than $d$.
We conclude that testing high-dimensional data for the null hypothesis $\Sigma =\sigma^{2}I_{d}$ by means of the sample covariance matrix may lead to wrong conclusions provided the data are generalized elliptically distributed. In contrast, the spectral estimator seems to be a robust alternative for applying the results of RMT in the context of generalized elliptical distributions.
\section{Financial Applications}
\subsection{Portfolio Risk Minimization}
In this section it is supposed that $n/d\rightarrow \infty$, i.e. from the viewpoint of RMT we
study low-dimensional problems. Let $R=(R_{1},R_{2},...,R_{d})$ be an elliptically distributed random vector of
short-term (e.g. daily) log-returns. If the fourth order cross moments of the log-returns are
finite then the elements of the sample covariance matrix are multivariate normally distributed, asymptotically.
The asymptotic covariance of each element is given by (see, e.g., Praag and Wesselman, 1989)
\begin{equation*}
\mathrm{ACov}\left(\hat{\sigma}_{ij},\hat{\sigma}_{kl}\right) =\left( 1+\kappa \right) \cdot \left( \sigma
_{ik}\sigma _{jl}+\sigma _{il}\sigma _{jk}\right) +\kappa\cdot\sigma _{ij}\sigma _{kl},
\end{equation*}
where $\Sigma=[\sigma_{ij}]$ denotes the true covariance matrix of $R$ and
\begin{equation*}
\kappa :=\frac{1}{3}\cdot \frac{E\left( R_{i}^{4}\right) }{E^{2}\!\left( R_{i}^{2}\right) }-1
\end{equation*}
is called the `kurtosis parameter'. Note that the kurtosis parameter does not depend on $i\in\{1,...,d\}$. It
is well-known that in the case of normality $\kappa =0$. A distribution with positive (or even infinite)
$\kappa $ is called `leptokurtic'. Particularly, regularly varying distributions are leptokurtic.
It is well-known that the portfolio which minimizes the portfolio return variance (the so called `global minimum
variance portfolio') is given by the vector of portfolio weights
\begin{equation*}\label{GMVP}
w := \frac{\Sigma ^{-1}\text{$\underline{1}$}}{\text{$\underline{1}$}^{\mathrm{T}}\Sigma
^{-1}\text{$\underline{1}$}}.
\end{equation*}
Now, suppose for the sake of simplicity that $R$ is spherically distributed, i.e. that $\mu = 0$ and $\Sigma$ is proportional to the identity matrix. Since the weights of the global minimum variance portfolio do not depend on the scale of $\Sigma$ we may assume $\Sigma = I_{d}$ w.l.o.g. Then the asymptotic covariances of the sample covariance matrix elements are simply given by
\begin{equation*}
\mathrm{ACov}\left(\hat{\sigma}_{ij},\hat{\sigma}_{kl}\right) =\left\{
\begin{array}{rcl}
2+3\kappa , & & i=j=k=l, \\ \rule{0cm}{0.5cm}\kappa , & & i=j,\, k=l,\, i\neq k, \\ \rule{0cm}{0.5cm}1+\kappa
, & & i=k,\, j=l,\, i\neq j, \\ \rule{0in}{0.5cm}0, & & \text{else}.
\end{array}\right.
\end{equation*}
For instance suppose that the random vector $R$ is multivariate $t$-distributed with $\nu>4$ de\-grees of
freedom. Then the kurtosis parameter corresponds to $\kappa =2/(\nu -4)$ (see, e.g., Frahm, 2004, p. 91).
Hence, the smaller $\nu$ the larger the asymptotic variances and covariances and these quantities tend to
infinity for $\nu \searrow 4$. Further, if $\nu\leq 4$ the sample covariance matrix even is no longer
multivariate nor\-mally distributed, asymptotically.
In contrast, the asymptotic covariance of each element of the spectral estimator (Frahm, 2004, p.
76) is given by
\begin{equation*}
\mathrm{ACov}\left(\hat{\sigma}_{\mathrm{S},ij},\hat{\sigma}_{\mathrm{S},kl}\right) =\left\{
\begin{array}{rcl}
4\cdot\frac{d+2}{d} , & & i=j=k=l, \\ \rule{0cm}{0.5cm}2\cdot\frac{d+2}{d} , & & i=j,\, k=l,\, i\neq k, \\
\rule{0cm}{0.5cm}\frac{d+2}{d} , & & i=k,\, j=l,\, i\neq j, \\ \rule{0in}{0.5cm}0, & & \text{else}.
\end{array}\right.
\end{equation*}
Note that the same holds even if $R$ is not $t$-distributed but only generalized elliptically distributed since
$\widehat{\Sigma}_{\mathrm{S}}$ does not depend on the generating variate of $R$. Particularly, the spectral
estimator is not disturbed by the tail index of $R$.
Now one may ask when the sample covariance matrix is dominated (in a component-wise manner) by the spectral
estimator provided the data are multivariate $t$-distributed. Regarding the main diagonal entries of the
covariance matrix estimate this is given by
\begin{equation*}
4\cdot \frac{d+2}{d}<2\cdot \frac{\nu -1}{\nu -4},
\end{equation*}
i.e. if $\nu <4 + 3d/(d+4)$ the variance of the spectral estimator's main diagonal elements is smaller than
the variance of the corresponding main diagonal elements of the sample covariance matrix, asymptotically. Concerning its off diagonal entries we obtain
\begin{equation*}
\frac{d+2}{d}<\frac{\nu -2}{\nu -4},
\end{equation*}
i.e. $\nu < 4+d$. It is worth to note that several empirical studies indicate that the tail indices of daily log-returns generally lie between $4$ and $7$ (see, e.g., Embrechts, Frey, and McNeil, 2004, p. 81 and Junker and May, 2002).
In the following the daily log-returns from 1980-01-02 to 2003-10-06 of 285 S\&P 500 stocks are analyzed for studying the robustness of the spectral estimator vs. the sample covariance matrix. The considered stocks belong to the `survivors' of the S\&P 500 composite at the last quarter of 2003. The sample size corresponds to $n=6000$. The total sample period is partitioned into $10$ sub-periods each containing $600$ daily log-returns. Further, each sub-period is divided into `even' and `odd' days, i.e. there is a sub-sample containing the 1st, 3rd, \ldots, 599th log-returns and another sub-sample with the 2nd, 4th, \ldots, 600th log-returns. Hence each sub-sample contains $300$ daily log-returns of $285$ stocks. Both the sample covariance matrix and the spectral estimator are used for estimating the relative eigenspectrum of the true covariance matrix, i.e. $\lambda_{1}/\sum_{i=1}^{d}\lambda_{i},\ldots ,\lambda_{d}/\sum_{i=1}^{d}\lambda_{i}$ for each even and odd sub-sample, separately. If the covariance matrix estimator is robust against outliers then the estimated eigenspectra of each sub-sample should be similar since even if the true eigenspectrum changes dynamically over time this must affect both the even and the odd days, equally. The eigenspectrum obtained in the even sub-sample can be compared with the eigenspectrum given by the odd sub-sample simply by the differences of the ordered (relative) eigenvalues.
\begin{center}
\includegraphics[scale=.21]{even_oddMnew
\includegraphics[scale=.21]{even_oddSnew}\\[.25cm]
\end{center}
{\bf Fig. 7:} Eigenvalue differences for each ordered eigenvalue given by the sample covariance matrix (left hand) and by the spectral estimate (right hand).\\[.25cm]
On the left hand of Figure 7 we see the eigenvalue differences for each $10$ sub-periods caused by the sample covariance matrix. Similarly, the right hand of Figure 7 shows the eigenvalue differences given by the spectral estimate. Figure 7 indicates that the spectral estimator leads to more robust estimates of the eigenspectra of financial data. But note that - concerning the overall eigenspectrum - the sample covariance matrix performs well up to the 4th sub-period. This is the period which contains the famous October Crash of $1987$. In contrast, the spectral estimator is not affected by extreme values.
\begin{center}
\includegraphics[scale=.21]{5even_oddM}
\includegraphics[scale=.21]{5even_oddS}\\[.25cm]
\end{center}
{\bf Fig. 8:} Eigenvalue differences for the largest $5$ eigenvalues given by the sample covariance matrix (left hand) and by the spectral estimate (right hand).\\[.25cm]
Figure 8 focuses on the differences of the $5$ largest eigenvalues. It shows that the sample covariance matrix particularly fails for estimating the largest eigenvalue. Once again this phenomenon is caused by the Black Monday which belongs to the even sub-sample of the 4th sub-period. Note that the largest eigenvalue of the even sub-sample exceeds the largest eigenvalue of the odd sub-sample by almost $12$ percentage points. We conclude that although the sample covariance matrix works quite good for the most time it is not appropriate for measuring the linear dependence structure of financial data. This is due to a few but extreme fluctuations on financial markets.
\subsection{Principal Components Analysis}
Now, consider a $d$-dimensional vector $R=(R_{1},...,R_{d})$ of long-term (e.g. yearly) i.i.d. log-returns. Due to the central limit theorem each vector component of $R$ is approximately normal distributed provided the covariance matrix of the short-term (e.g. daily) log-returns exists and is finite. Since the sum of i.i.d. elliptical random vectors is always elliptically distributed, too (see, e.g., Hult and Lindskog, 2002) one may take for granted that the vector components of $R$ are jointly normally distributed, approximately. But this is not true if the number of dimensions $d$ is large relative to the sample size $n$.
For instance, consider a $d$-dimensional random vector $X$ which is multivariate $t$-distributed with $\nu>2$ degrees of freedom, location vector $\mu = 0$, and dispersion matrix $\Sigma = (\nu -2)/\nu\cdot I_{d}$. Due to the multivariate central limit theorem one could believe that
\begin{equation*}
Y := \frac{1}{\sqrt{n}}\cdot\sum_{j=1}^{n} X_{j}\overset{\cdot}{\sim}N_{d}\left( 0,I_{d}\right),
\end{equation*}
where $X_{1},\ldots,X_{n}$ are independent copies of $X$. But indeed $Y^{\text{T}}Y \overset{\cdot}{\sim}\chi_{d}^{2}$ holds only if $q:=n/d$ is large rather than $n$ being large (cf. Frahm, 2004, Section 6.2). Thus the quantity $q$ can be interpreted as `effective sample size'.
In the following it is assumed that $R$ is elliptically distributed with location vector $\mu$ and dispersion matrix $\Sigma$. Let $\Sigma = \mathcal{O}\mathcal{D}\mathcal{O}^{\text{T}}$ be a spectral decomposition of $\Sigma$. Then
\begin{equation*}
R\overset{\mathrm{d}}{=}\mu +\mathcal{O}\sqrt{\mathcal{D}}\,Y,
\end{equation*}
where $Y$ spherically distributed with $\Sigma = I_{d}$.
We assume that the elements of $\mathcal{D}$, i.e. the eigenvalues of $\Sigma$ are given in a descending order
and that the first $k$ eigenvalues are large whereas the residual ones are small. The elements of $Y$ are called `principal components' of $R$. Since $\mathcal{O}$ is orthonormal the distribution of $\sqrt{\mathcal{D}}\,Y$ remains up to a rotation in $\mathbb{R}^{d}$. The direction of each principal component is given by the corresponding column of $\mathcal{O}$.
Hence the first $k$ eigenvalues correspond to the variances (up to a scaling constant) of the `driving risk factors' contained in the first part of $Y$, i.e. $\left( Y_{1},\ldots,Y_{k}\right)$. For the purpose of dimension reduction $k$ shall not be too large. Because the $d-k$ residual risk factors contained in $\left( Y_{k+1},\ldots
,Y_{d}\right) $ are supposed to have (relatively) small variances they can be interpreted as the components of the
idiosyncratic risks of each firm, i.e.
\begin{equation*}
\varepsilon _{i}:=\sum_{j=k+1}^{d}\sqrt{\lambda_{j}}\,\mathcal{O}_{ij}Y_{j},\qquad i=1,\ldots ,d,
\end{equation*}
where $\lambda_{j}:=\mathcal{D}_{jj}$.
Thus we obtain the following principal components model for long-term log-returns,
\begin{equation*}
R_{i}\overset{\mathrm{d}}{=}\mu_{i}+\beta _{i1}Y_{1}+\ldots +\beta _{ik}Y_{k}+\varepsilon _{i},\qquad
i=1,\ldots ,d,
\end{equation*}
where the driving risk factors $Y_{1},...,Y_{k}$ are uncorrelated. Further, each noise term
$\varepsilon_{i}$ $(i=1,...,d)$ is uncorrelated to $Y_{1},...,Y_{k}$, too. But note that $\varepsilon_{1},\ldots ,\varepsilon_{d}$ are correlated, generally. The `Betas' are given by $\beta_{ij} = \sqrt{\lambda_{j}}\,\mathcal{O}_{ij}$ for $i=1,\ldots , d$ and $j=1,\ldots ,k$.
The purpose of principal components analysis is to reduce the complexity caused by the number of dimensions.
This can be done successfully only if there is indeed a number of principal components accountable for the most
part of the distribution. Additionally, the covariance matrix estimator which is used for extracting the
principal components should be robust against outliers.
For example, let the daily log-returns be multivariate $t$-distributed with $\nu$ degrees of freedom and suppose
that $d=500$ and $n=1000$. Note that due to the central limit theorem the normality assumption concerning the long-term log-returns makes sense whenever $\nu >2$. The black lines in Figure 9 show the true proportion of the total variation for a set of $500$ eigenvalues. We see that the largest $20\%$ of the eigenvalues accounts for $%
80\%$ of the overall variance. This is known in economics as `80/20 rule' or `Pareto's principle'. The estimated
eigenvalue proportions obtained by the sample covariance matrix are represented by the red lines whereas the
corres\-ponding estimates based on the spectral estimator are given by the green lines. Each line is an average
over $100$ concentration curves drawn from samples of the corresponding multivariate $t$-distribution.
If the data have a small tail index as given by the lower right of Figure 9 then the sample covariance matrix
tends to underestimate the number of driving risk factors, essentially. This is similar to the phenomenon
observed in Figure 6 where the number of large eigenvalues is overestimated. In contrast, the concentration
curves obtained by the spectral estimator are robust against heavy tails. This holds even if the long-term log-returns are not asymptotically normal distributed.
\begin{center}
\includegraphics[scale=.34]{PCA2}
\includegraphics[scale=.34]{PCA3}\\[.25cm]
\includegraphics[scale=.34]{PCA1}
\includegraphics[scale=.34]{PCA4}\\[.25cm]
\end{center}
{\bf Fig. 9:} True proportion of the total variation (black line) and proportions obtained by the sample
covariance matrix (red lines) and by the spectral estimator (green lines). The samples are drawn from a
multivariate $t$-distribution with $\nu =\infty$ (i.e. the multivariate normal distribution, upper left),
$\nu=10$ (upper right), $\nu =5$ (lower left), and $\nu =2$ (lower right).\\[.25cm]
In the simulated example of Figure 9 it is assumed that the small eigenvalues are equal. This is equivalent to
the assumption that the residual risk factors are spherically distributed, i.e. that they contain no
more information about the linear dependence structure of $R$. But even if the true eigenvalues are equal
the corresponding estimates will not share this property because of estimation errors. Yet it is important to know whether the residual risk factors have structural information or the differences between the eigenvalue estimates are only caused by random noise. This is not an easy task, especially if the data are not normally distributed and the number of dimensions is large which is the issue of the next section.
\subsection{Signal-Noise Separation}
In the previous section it was mentioned that the central limit theorem fails in the context of high-dimensional data, i.e. if $n/d$ is small. Hence, now we leave the field of classical multivariate analysis and get to the domain of RMT.
Let $\Sigma =\mathcal{ODO}^{\mathrm{T}}\in \mathbb{R}^{d\times d}$ be a spectral decomposition where $\mathcal{D}$ shall be a diagonal matrix containing a `bulk' of small and equal eigenvalues and some large (but
not necessarily equal) eigenvalues. For the sake of simplicity suppose%
\begin{equation*}
\mathcal{D}=\left[
\begin{array}{cc}
cI_{k} & 0 \\
\rule{0cm}{.5cm} 0 & bI_{d-k}%
\end{array}%
\right] \qquad c>b>0,
\end{equation*}%
where $d-k$ is large. Hence $\Sigma$ has two different characteristic manifolds. The `major' one is determined
by the first $k$ column vectors of $\mathcal{O}$ (the `signal part' of $\Sigma$) whereas the `minor' one is
given by the $d-k$ residual column vectors of $\mathcal{O}$ (the `noise part' of $\Sigma$). We are interested in
separating signal from noise that is to say estimating $k$, properly.
For instance, assume that $n=1000$, $d=500$, and that a sample consists of normally distributed random vectors
with covariance matrix $\Sigma$, where $b=1$, $c=5$, and $k=100$. By using the sample covariance matrix and
normalizing the eigenvalues one obtains exemplarily the histogram of eigenvalues given on the left hand of
Figure 10. As might be expected the Mar\v{c}enko-Pastur law is not valid due to the two different regimes of
eigenvalues. In contrast, when focusing on the smallest $400$ eigenvalues, i.e. on the noise part of
$\widehat{\Sigma}$ the Mar\v{c}enko-Pastur law becomes valid as we see on the right hand of Figure 10.
\begin{center}
\includegraphics[scale=.34]{SNS1}
\includegraphics[scale=.34]{SNS2}\\[.25cm]
\end{center}
{\bf Fig. 10:} Histogram of all $d=500$ eigenvalues (left hand) and of the noise part (right hand) consisting of
the $d-k=400$ smallest eigenvalues. The Mar\v{c}enko-Pastur law is represented by the green lines.\\[.25cm]
Thus separating signal from noise means sorting out the largest eigenvalues successively until the residual
eigenspectrum is consistent with the Mar\v{c}enko-Pastur law. This is given, e.g., when there are no more
eigenvalues exceeding the Mar\v{c}enko-Pastur upper bound $\lambda _{\max}$. In our case-study this is given
for $397$ eigenvalues (see the figure below), i.e. $\widehat{k}=103$.
\begin{center}
\includegraphics[scale=.35]{SNS3}\\[.25cm]
{\bf Fig. 11:} Histogram of the remaining $397$ eigenvalues after signal-noise separation.\\[.25cm]
\end{center}
As it was shown in Section \ref{RMT} this approach is promising only if the data are not regularly varying. Hence
for financial data not the sample covariance matrix but the spectral estimator is proposed for a proper signal-noise
separation.
\section{Conclusions}
Due to the stylized facts of empirical finance the Gaussian distribution hypothesis is not appropriate for the modeling of financial data. For that reason the authors rely on the broad class of generalized elliptical distributions. This class allows for tail dependence and radial asymmetry. Although the sample covariance matrix works quite good with financial data for the most time it is not appropriate for measuring their linear dependence structure. This is due to a few but extreme fluctuations on financial markets.
It is shown that there exists a completely robust ML-estimator (the `spectral estimator') for the dispersion matrix of generalized elliptical distributions. This estimator corresponds to Tyler's M-estimator for elliptical distributions. Further, it is shown that the Mar\v{c}enko-Pastur law fails if the sample covariance matrix is considered as random matrix in the context of elliptically or even generalized elliptically distributed data. This is due to the fact that stochastical independence implies linear independence but conversely uncorrelated random variables are not necessarily independent. In contrast, the Mar\v{c}enko-Pastur law remains valid if the data are uncorrelated and the spectral estimator is considered as random matrix.
The robustness property of the spectral estimator can be demonstrated for several financial applications like, e.g., portfolio risk minimization, principal components analy\-sis, and signal-noise separation. If the data are heavy tailed the principal components analy\-sis tends to underestimate the number of driving risk factors if the sample covariance matrix is used for extracting the eigenspectrum. This means that the contribution of the largest eigenvalues to the total variation of the data is overestimated, systemati\-cally. Consequently, in the context of signal-noise separation the largest eigenvalues are overestimated by the sample covariance matrix. This can be fixed simply by using the spectral estimator, instead.
|
{
"timestamp": "2005-03-01T11:59:44",
"yymm": "0503",
"arxiv_id": "physics/0503007",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503007"
}
|
\section{Introduction}
Quantum states which differ only by a overall phase cannot be distinguished by measurements
in quantum mechanics. Hence phases were thought to be unimportant
until Berry made an important and interesting observation regarding the behavior of pure quantum
systems in a slowly changing environment \cite{2}. The adiabatic theorem makes sure that, if a system is initially in an eigenstate of the
instantaneous Hamiltonian, it remains so. When the environment (more precisely, the Hamiltonian) returns to it's initial state
after undergoing slow changes, the system acquires a measurable phase, apart from the well known dynamical phase,
which is purely of geometric origin \cite{2}.
Simon\cite{3} showed this to be a consequence of parallel transport in a curved space appropriate to the quantum system.
Berry's phase was reconsidered
by Aharanov and Anandan, who shifted the emphasis from changes in the environment, to the motion of the pure quantum system
itself and found that the for all the changes in the environment, the same geometric phase is obtained which is uniquely
associated with the motion of the pure quantum system and hence enabled them to generalize Berry's phase to non-adiabatic motions
\cite{5}.
For a spin half particle subjected to a magnetic field $\bf B$, the non adiabatic cyclic Aharanov-Anandan
phase is just the
solid angle determined by the path in the projective Hilbert space \cite{5}.
Yet another interesting discovery in the fundamentals of quantum physics was the observation that by accessing a
large Hilbert space spanned by the linear combination of quantum states and by intelligently manipulating them, some of
the problems intractable for classical computers can be solved efficiently \cite{rf,preskill}.
This idea of quantum computation
using coherent quantum mechanical systems has excited a number of research groups \cite{bou,chuangbook}.
Various physical systems including nuclear magnetic resonance (NMR) are being examined to built a suitable physical
device which would perform quantum information processing and quantum simulations \cite{dg,cory98,na,ernst,jo}.
Also, the quantum correlation inherently present in the
entangled quantum states was found to be useful for quantum computation, communication and cryptography \cite{preskill}.
Geometric quantum computing is a way of manipulating quantum states using quantum gates based on
geometric phase shifts \cite{4,9}.
This approach is particularly useful because of the built-in fault tolerance, which arises due to the fact that geometric phases
depend only upon some global geometric features and it is robust against certain errors and dephasing \cite{4,9,10,11}.
In nuclear magnetic resonance (NMR), the acquisition of geometric phase by a spin was first verified by Pines et.al.\cite{6} in adiabatic regime by
subjecting a nuclear spin to an effective magnetic field which slowly sweeps a cone.
A similar approach was adopted by Jones et.al. to demonstrate the construction of controlled phase shift gates in a
two-qubit system using adiabatic geometric phase \cite{4,9}. Pines et.al. also studied the
geometric phase in non-adiabatic regime, namely the Aharanov-Anandan phase by NMR \cite{7}.
They used a system of two dipolar coupled identical proton spins
which form a three level system. A two level subsystem was made to undergo a cyclic
evolution in the Hilbert space by applying a time-dependent magnetic field, while geometric phase was observed in the
modulation of the coherence of the other two-level subsystem \cite{7}.
Recently, non-adiabatic geometric phase has also been observed for mixed states by NMR using evolution in
tilted Hamiltonian frame \cite{mix}.
In the work reported here, we have adopted a scheme similar to that of Pines et.al. \cite{7} to demonstrate
construction of controlled phase shift gates in a two-qubit system using non-adiabatic geometric phase by NMR. The scheme is easily
scalable to higher qubit systems. The geometric controlled phase gates were used to implement Deutsch-Jozsa (DJ) algorithm
\cite{deu} and Grover's search algorithm \cite{grover} in the two-qubit system. To the best of our knowledge, this is the first implementation of
quantum algorithms using geometric phase.
\section{non-adiabatic geometric phase gate}
Consider a two qubit system, which has four eigenstates $\vert 00\rangle$, $\vert 01\rangle$, $\vert 10\rangle$
and $\vert 11\rangle$. The two-state subsystem of $\vert 10\rangle$ and $\vert 11\rangle$ can be taken through
a circuit enclosing a solid angle $\Omega$ \cite{7}. If the other dynamical phases are canceled during the process, these
two states gain a non-adiabatic phase purely due to geometric topology. Since the operation is done selectively
with the states where first qubit is in state $\vert 1 \rangle$, this acts as a controlled phase gate where the
second qubit gains a phase only when the first qubit is $\vert 1 \rangle$ \cite{bou,chuangbook,4} .
The transport of the selected states through a closed circuit can be accomplished by selective excitation.
Such selective excitation can be performed by pulses having a small bandwidth which excite a selected transition in the
spectrum and leave the others unaffected \cite{cory98,freeman,free,kd,pram1,rd1}. In the following we consider geometric phase
acquired by two different paths in a Bloch sphere, respectively known as slice circuit and triangular circuit \cite{7}.
\subsection{Geometric phase acquired by a slice circuit}
In a slice circuit, the state vector cuts a slice out of the Bloch sphere, Figure 1(a).
The slice circuit can be achieved by two pulses A.B=$(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$.
$(\pi)_{\theta+\pi+\phi}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$, where the pulses are applied from left to
right. Here
$(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$ denotes a selective $\pi$-pulse on
$\vert 10\rangle \leftrightarrow \vert 11\rangle$ transition with phase $\theta$. The first
$(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$ pulse rotates the
polarization vector of the subsystem through $\pi$ about the an axis with azimuthal angle $\theta$ in the
x-y plane (Fig. 1).
The vector is brought back to its original position completing a closed circuit
by the second $\pi$-pulse about the axis in the x-y plane with azimuthal angle $(\theta+\pi+\phi)$.
The resulting path encloses a solid angle of 2$\phi$,
The operator of the two pulses can be calculated as,
\begin{eqnarray}
{\mathrm A.B}=&&(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}
(\pi)_{\theta+\pi+\phi}^{\vert 10\rangle \leftrightarrow \vert 11\rangle} \nonumber \\
=&&exp[-i(I_x^{\vert 10\rangle \leftrightarrow \vert 11\rangle} cos(\theta)+
I_y^{\vert 10\rangle \leftrightarrow \vert 11\rangle} sin(\theta))\pi] \nonumber \\
&&exp[-i(I_x^{\vert 10\rangle \leftrightarrow \vert 11\rangle} cos(\theta +\pi +\phi)+
I_y^{\vert 10\rangle \leftrightarrow \vert 11\rangle} sin(\theta +\pi+\phi))\pi] \nonumber \\
=&&\pmatrix{1 & 0 & 0 & 0 \cr 0& 1& 0& 0& \cr 0& 0& 0& -sin\theta-icos\theta \cr 0& 0& sin\theta-icos\theta & 0 } \times \nonumber \\
&&\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& 0& sin(\theta+\phi)+icos(\theta+\phi)
\cr 0& 0& -sin(\theta+\phi)+icos(\theta+\phi)& 0} \nonumber \\
&&=\pmatrix{1 & 0 & 0 & 0 \cr 0& 1& 0& 0& \cr 0& 0& e^{i\phi}& 0 \cr 0& 0& 0 & e^{-i\phi}},
\end{eqnarray}
where $I_x^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$ and
$I_y^{\vert 10\rangle \leftrightarrow \vert 11\rangle}$ are the fictitious spin-1/2 operators \cite{vega} for the
two-state subsystem of $\vert 10\rangle$ and $\vert 11\rangle$, given by;
\begin{eqnarray}
I_x^{\vert 10\rangle \leftrightarrow \vert 11\rangle}=\frac{1}{2}\pmatrix{0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 1 \cr 0 & 0 & 1 & 0} ~~~~~{\mathrm and}~~~~~
I_y^{\vert 10\rangle \leftrightarrow \vert 11\rangle}=\frac{1}{2}\pmatrix{0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & -i \cr 0 & 0 & i & 0}.
\end{eqnarray}
Note that the combined operator of the two pulses in Eq.[1] attributes a non-adiabatic geometric phase
proportional to the solid angle of
the circuit traversed. However, the phase it attributed only to the
two states where first qubit is in state $\vert 1\rangle$. Since the selective excitation does not perturb the
other transitions, the subsystem $\vert 00\rangle \leftrightarrow \vert 01\rangle$ do not gain any phase,
This is analogous to the controlled phase gate where
the second qubit acquires a phase controlled by the state of first qubit \cite{chuangbook,4}.
To demonstrate the operation of a controlled geometric phase gate, we have taken the two qubit system of
carbon-13 labeled chloroform ($^{13}$CHCl$_3$), where the two nuclear spins $^{13}$C and $^1$H forms the two-qubit
system. The sample of $^{13}$CHCl$_3$ was dissolved in the solvent of CDCl$_3$, and experiments were performed
at room temperature at a magnetic field of B$_0$=11.2 Tesla. At this high-field the resonance frequency of proton is
500 MHz and that of carbon is 125 MHz. The indirect spin-spin coupling (the J-coupling)
between the two qubits is 210 Hz. Starting from equilibrium,
the $\vert 00\rangle$ pseudopure state was prepared by spatial averaging method using the pulse sequence \cite{mix},
\begin{eqnarray}
(\pi/3)^2_x-G_z-(\pi/4)^2_x-\frac{1}{2J}-(\pi/4)^2_{-y}-G_z,
\end{eqnarray}
where the pulses were applied on the second qubit, denoted by $2$ in superscript, which in our case is the proton
spin. After creation of pps, a pseudo-Hadamard gate \cite{djjo,grojo} was applied on the first qubit, which in our case was $^{13}$C. The pseudo-Hadamard gate
was implemented by a $(\pi/2)^1_y$ where the superscript denotes the qubit and the subscript denotes the phase of the pulse \cite{djjo,grojo}.
This gate creates a an uniform superposition of the first qubit $\vert 00\rangle +\vert 10\rangle$. The operation of the
controlled phase gate would now transform the state into $\vert 00\rangle + e^{i\phi}\vert 10\rangle$.
For the slice circuit, the proton dynamical phase would vanish since the applied field is always orthogonal to the
polarization vector, generating parallel transport \cite{7}. However, the carbon coherence would undergo evolution due to the
internal Hamiltonian during the pulses. Hence the pulse sequence of the gate was incorporated into a Hahn-echo
\cite{echo,ernstbook} sequence of the
form $\tau-(\pi)_x-\tau$, where the pulse sequence of the gate were applied during the second $\tau$ period, as given in
figure 1(b). The
intermediate $(\pi)$-pulse refocuses inhomogeneity of the B$_0$ field, the chemical shift of carbon and its J-coupling to the proton.
However, to restore the state of the first qubit altered by the $(\pi)$-pulse,
the pulse sequence of Eq. [1] has to be supplemented by adding
a $(\pi)^1_{-x}$ pulse (figure 1(b)), yielding the sequence:
\begin{eqnarray}
&&(\pi)^1_{x}.(\pi)_{\theta}^{\vert 00\rangle \leftrightarrow \vert 01\rangle}.
(\pi)_{\theta+\pi+\phi}^{\vert 00\rangle \leftrightarrow \vert 01\rangle}.(\pi)^1_{-x} \nonumber \\
&&=\pmatrix{0 & 0 & i & 0 \cr 0 & 0 & 0 & i \cr i & 0 & 0 & 0 \cr 0 & i & 0 & 0}.
\pmatrix{e^{i\phi} & 0 & 0 & 0 \cr 0& e^{-i\phi} & 0& 0& \cr 0& 0& 1 & 0 \cr 0& 0& 0 & 1}.
\pmatrix{0 & 0 & -i & 0 \cr 0 & 0 & 0 & -i \cr -i & 0 & 0 & 0 \cr 0 & -i & 0 & 0} \nonumber \\
&&=\pmatrix{1 & 0 & 0 & 0 \cr 0& 1& 0& 0& \cr 0& 0& e^{i\phi}& 0 \cr 0& 0& 0 & e^{-i\phi}},
\end{eqnarray}
where the selective pulses were applied on the $\vert 00\rangle \leftrightarrow \vert 01\rangle$ transition to
achieve the exact form of controlled phase gate.
The selective excitation was obtained with Gaussian shaped pulses of 13.2 ms duration. The non-adiabatic geometric phase
was observed in the phase of $\vert 00\rangle \leftrightarrow \vert 10\rangle$ coherence. We have observed the geometric phase
for the slice circuit with various solid angles $(2\phi)$, each time varying the phase $\phi$ of the second selective $(\pi)$-pulse.
The corresponding spectra are given in figure 2(c), where the $\vert 00\rangle \leftrightarrow \vert 10\rangle$ shows a
phase change of $e^{i\phi}$. For $\phi=0$, there is no phase change and the peak
is absorptive. With increase of $\phi$, the phase of the peak changes and it becomes dispersive for $\phi=\pi/2$,
and subsequently, a negative absorptive for $\phi=\pi$.
The three small lines in the spectra comes from the naturally abundant
$^{13}$C signal of CDCl$_3$, which provide a reference.
Since all dynamical phases due to evolution under chemical shift and J-couplings were refocused,
the solvent $^{13}$C signal is absorptive in all the spectra. However, solute $^{13}$C signal gains phase because it is coupled to
the protons, one of whose transition is taken through a closed circuit.
This result thus provides a graphic display of geometric phase by non-adiabatic evolution. To accurately read the phase angle
of each spectrum in Fig. 2(c), a zero-order phase correction was applied to the spectra in Fig. 2(c), till the observed peak became
absorptive. The change of phase of $\vert 00\rangle \leftrightarrow \vert 10\rangle$ coherence due to geometric phase
is plotted against the solid-angle (2$\phi$), in figure 3. The graph in figure 3 shows the high fidelity of the
experimental implementation of the slice circuit in this case.
\subsection{Geometric phase acquired by a triangular circuit}
In the triangular circuit, the state vector traverses a triangular path on the Bloch sphere figure 1(c) \cite{7}.
The solid angle enclosed by the triangular circuit of figure 1(c) is $\phi$.
The controlled phase shift gate can be implemented by the non-adiabatic phase acquired when the appropriate
sub-system goes through this circuit.
The pulse sequence for the circuit and the corresponding operator can be calculated,
similar to that of the sliced circuit, as
\begin{eqnarray}
A.C.B=&&(\pi/2)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{\theta}.
(\phi)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{z}.
(\pi/2)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{\theta+\pi-\phi} \nonumber \\
=&&\pmatrix{1 & 0 & 0 & 0 \cr 0& 1& 0& 0& \cr 0& 0& \frac{1}{\sqrt{2}}& \frac{-sin\theta-icos\theta}{\sqrt{2}}
\cr 0& 0& \frac{sin\theta-icos\theta}{\frac{1}{\sqrt{2}}} & \frac{1}{\sqrt{2}} } \times
\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& e^{-i\phi/2}& 0 \cr 0 &0& 0& e^{i\phi/2}} \times \nonumber \\
&&\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& \frac{1}{\sqrt{2}}& \frac{sin(\theta-\phi)+icos(\theta-\phi)}{\sqrt{2}} \cr
0& 0& \frac{-sin(\theta-\phi)+icos(\theta-\phi)}{\sqrt{2}}& \frac{1}{\sqrt{2}}} \nonumber \\
&&=\pmatrix{1 & 0 & 0 & 0 \cr 0& 1& 0& 0& \cr 0& 0& e^{-i\phi/2}& 0 \cr 0& 0& 0 & e^{i\phi/2}},
\end{eqnarray}
The intermediate $(\phi)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{z}$ pulse can be applied by the
composite z-pulse sequence $(\pi/2)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{y}
(\phi)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{-x}(\pi/2)^{\vert 10\rangle \leftrightarrow \vert 11\rangle}_{-y}$
\cite{lev,ranajmr}.
In the experiments, we have chosen $\theta=3\pi/2$. The state of $\vert 00\rangle +\vert 10\rangle$ was prepared and
then the pulse sequence of figure 1(d) was applied. Similar to the slice circuit, the sequence was incorporated
in a Hahn-echo and the pulses were applied on the $\vert 00\rangle \leftrightarrow \vert 01\rangle$ transition.
The operator of Eq.[5] transforms $\vert 00\rangle +\vert 10\rangle$ to $\vert 00\rangle +e^{-i\phi/2}\vert 10\rangle$.
The phase of the $\vert 00\rangle \leftrightarrow \vert 10\rangle$ was observed for various $\phi$, by changing the angle
of the z-pulse and the phase of the last pulse in Eq.[5]. The spectra are given in figure 4. Once again, the peak changes
from absorptive to dispersive and then to a negative absorptive in correspondence with the change of $\phi$.
However, there are two major
differences between the spectra of figure 2(c) and 4(c). Note that after the phase gate, the state of the system is
$\vert 00\rangle +e^{i\phi}\vert 10\rangle$ for slice circuit and $\vert 00\rangle +e^{-i\phi/2}\vert 10\rangle$
for triangular circuit. This is because the solid angle of the slice circuit if 2$\phi$, whereas that of the
triangular circuit is $\phi$. Hence, in the slice circuit the coherences become a negative
absorptive for $\phi=\pi$,
whereas in the triangle circuit the same observation is obtained for $\phi=2\pi$. Moreover, the phase of the pulses
corresponding to the triangle circuit is chosen such that the sign of phase is opposite to that of the
slice circuit. This difference is clearly reflected in the sign of the coherences between figure 2(c) and 4(c).
A plot of the absolute value of observed phase change
against solid angle is given in figure 5, whose high fidelity validate the use of such gates for quantum computing.
\section{Deutsch-Jozsa algorithm}
Deutsch-Jozsa (DJ) algorithm provides a demonstration of the advantage of quantum superpositions over
classical computing \cite{deu}.
The DJ algorithm determines the type of an unknown function when it is either constant or balanced. In the simplest
case, $f(x)$ maps a single bit to a single bit. The function is called constant if $f(x)$ is
independent of $x$ and it is balanced if $f(x)$ is zero for one value of $x$ and unity for the other
value. For N qubit system, $f(x_1,x_2,...x_N)$ is constant if it is independent of $x_i$ and balanced if it is zero
for half the values of $x_i$ and unity for the other half. Classically it requires ($2^{N-1}+1$)
function calls to check if $f(x_1,x_2,...x_N)$ is constant or balanced.
However the DJ algorithm would require only a single function call \cite{deu}. The Cleve version of DJ algorithm
implemented by using a unitary transformation by the propagator
$U_f$ while adding an extra qubit, is given by \cite{cleve},
\begin{eqnarray}
\vert x_1,x_2,...x_N\rangle \vert x_{N+1}\rangle \stackrel{U_f}{\longrightarrow}
\vert x_1,x_2,...x_N\rangle \vert x_{N+1}\oplus f(x_1,x_2,...x_N)\rangle
\end {eqnarray}
The four possible functions for the single-bit DJ algorithm are $f_{00}$, $f_{11}$,
$f_{10}$ and $f_{01}$. $f_{00}(x)=0$ for $x=$0 or 1, $f_{11}(x)=1$ for $x=$0 or 1,
$f_{10}(x)=$1 or 0 corresponding to $x=$0 or 1, while $f_{01}(x)=$0 or 1 corresponding to $x=$0 or 1.
The unitary transformations corresponding to the four possible propagators $U_f$ are
\begin{eqnarray}
U_{f_{00}}=\pmatrix{1&0&0&0\cr 0&1&0&0\cr 0&0&1&0\cr 0&0&0&1},~~~~~~~~~~
U_{f_{11}}=\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0}, \nonumber \\
\nonumber \\
U_{f_{10}}=\pmatrix{1&0&0&0\cr 0&1&0&0\cr 0&0&0&1\cr 0&0&1&0},~~~~~~~~~~
U_{f_{01}}=\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&1&0\cr 0&0&0&1}.
\end{eqnarray}
For higher qubits the functions are easy to evaluate using Eq.[6].
DJ-algorithm has been demonstrated using dynamic phase by several research groups \cite{djjo,djchu,ka1,pram,ranajcp}.
The quantum circuit for single-bit Cleve version of DJ algorithm is given in figure 6(a) \cite{djchu}.
The algorithm starts with $\vert 00\rangle$ pseudopure state. The pair of pseudo-Hadamard gates
$(\pi/2)^1_y(\pi/2)^2_{-y}$ create superposition of the form
$[(\vert 0\rangle + \vert 1\rangle)/\sqrt{2}][(\vert 0\rangle - \vert 1\rangle)/\sqrt{2}]$.
Then the operator $U_f$ is applied. When the function is constant, i.e. $f(0)=f(1)$, the input qubit is in the state
$(\vert 0\rangle + \vert 1\rangle)/\sqrt{2}$, else
the function is balanced in which case it is in the state $(\vert 0\rangle - \vert 1\rangle)/\sqrt{2}$.
Thus, the answer is stored in the
relative phase between the two states of the input qubit. A final pair of pseudo-Hadamard gates
$(\pi/2)^1_{-y}(\pi/2)^2_{y}$ converts the superposition back into the eigenstates.
The work qubit comes back to state $\vert 0\rangle$, where as the input qubit becomes $\vert 0\rangle$ or $\vert 1\rangle$
corresponding to the function being constant or balanced.
The operator of $U_{f_{00}}$ is identity matrix and corresponds to no operation. The operator of $U_{f_{11}}$
can be achieved by a $(\pi)_x$ pulse on the second qubit. In this experiment, unlike section II, we label proton as
the first qubit and carbon as the second qubit, and consequently the $(\pi)_x$ pulse was applied on the carbon.
The $U_{f_{10}}$ operator is a controlled-NOT gate which flips the second qubit when the first qubit is $\vert 1\rangle$.
This gate can be achieved by a controlled phase gate sandwiched between two pseudo-Hadamard gates on the second qubit [],
$U_{f_{10}}=h-C_{11}(\pi)-h^{-1}$, where the controlled phase gate is of the form,
\begin{eqnarray}
C_{11}(\phi)=\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& 1& 0 \cr 0& 0& 0& e^{i\phi}}.
\end{eqnarray}
This precise form of controlled phase gate can be achieved by a recursive use of the phase gates demonstrated
in section II. Since the gate A.B given in Eq.[1] attributes a phase $e^{i\phi}$ to the state $\vert 10\rangle$ and $e^{-i\phi}$
to the state $\vert 11\rangle$, we denote this gate as $C_{10}(\phi).C_{11}(-\phi)$, where
\begin{eqnarray}
A.B=[C_{10}(\phi).C_{11}(-\phi)]=\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& e^{i\phi}& 0 \cr 0& 0& 0& e^{-i\phi}}.
\end{eqnarray}
The phase gate $C_{11}(\phi)$ can be constructed by a suitable combination of these gates,
\begin{eqnarray}
&&[C_{00}(-\phi/4).C_{10}(\phi/4)] \times
[C_{01}(-\phi/4).C_{11}(\phi/4)] \times [C_{10}(-\phi/2).C_{11}(\phi/2)] \nonumber \\
&&= \pmatrix{e^{-i\phi/4}& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& e^{i\phi/4}& 0 \cr 0& 0& 0& 1} \times
\pmatrix{1& 0& 0& 0 \cr 0& e^{-i\phi/4}& 0& 0 \cr 0& 0& 1& 0 \cr 0& 0& 0& e^{i\phi/4}} \times
\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& e^{-i\phi/2}& 0 \cr 0& 0& 0& e^{i\phi/2}} \nonumber \\
&&=\pmatrix{e^{-i\phi/4}& 0& 0& 0 \cr 0& e^{-i\phi/4}& 0& 0 \cr 0& 0& e^{-i\phi/4}& 0 \cr 0& 0& 0& e^{i3\phi/4}}
=e^{-i\phi/4}\pmatrix{1& 0& 0& 0 \cr 0& 1& 0& 0 \cr 0& 0& 1& 0 \cr 0& 0& 0& e^{i\phi}}=e^{-i\phi/4}C_{11}(\phi).
\end{eqnarray}
Note that if performed in fault-tolerant manner by using non-adiabatic geometric phase,
the first gate requires a rotation of the transition
$\vert 00\rangle \leftrightarrow \vert 10\rangle$ through a closed circuit. We have used the slice circuit, where it requires
a sequence of two $\pi$-pulses, $(\pi)_{\theta}^{\vert 00\rangle \leftrightarrow \vert 10\rangle}
(\pi)_{\theta+\pi-\phi/4}^{\vert 00\rangle \leftrightarrow \vert 10\rangle}$. Similarly, the second phase gate of Eq.[7] can be
achieved by the pulse sequence $(\pi)_{\theta}^{\vert 01\rangle \leftrightarrow \vert 11\rangle}
(\pi)_{\theta+\pi-\phi/4}^{\vert 01\rangle \leftrightarrow \vert 11\rangle}$. Note that these two sequence is require
pulsing of both the transitions of first qubit, $\vert 00\rangle \leftrightarrow \vert 10\rangle$ for the first gate and
$\vert 01\rangle \leftrightarrow \vert 11\rangle$ for the second.
Hence, they can be performed simultaneously by a couple spin-selective pulses $(\pi)_{\theta}^1(\pi)_{\theta+\pi-\phi/4}^1$,
where the pulses are applied on the first qubit (denoted by superscript). Thus,
\begin{eqnarray}
C_{11}(\phi)=(\pi)_{\theta}^1.(\pi)_{\theta+\pi-\phi/4}^1.(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}.
(\pi)_{\theta+\pi-\phi/2}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}.
\end{eqnarray}
In this case $\phi=\pi$, and we have chosen $\theta=3\pi/2$.
The last two pulses are however transition selective pulses, which were incorporated into a refocusing sequence,
$\tau-(\pi/2)^1_x-\tau-(\pi/2)^2_x-\tau-(\pi/2)^1_x-\tau-(\pi/2)^2_x$, where the selective pulses were applied in the
last $\tau$ period, and the pulses were applied on the $\vert 00\rangle \leftrightarrow \vert 01\rangle$ transition.
It may be noted that the triangular circuit
could have also used for the same purpose. The pseudo-Hadamard pulses on second qubit were achieved by
$h=(\pi/2)^2_y$ and $h^{-1}=(\pi/2)^2_{-y}$ pulses.
The operator of $U_{f_{01}}$ can be implemented in the similar manner by $h-C_{00}(\pi)-h^{-1}$, where
$C_{00}(\phi)$ can be implemented by
\begin{eqnarray}
C_{00}(\phi)=(\pi)_{\theta}^1.(\pi)_{\theta+\pi+\phi/4}^1.(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}
.(\pi)_{\theta+\pi+\phi/2}^{\vert 00\rangle \leftrightarrow \vert 01\rangle}.
\end{eqnarray}
The equilibrium spectrum of the two qubits are given in figure 6(b).
After creating the superposition from pps,
applying the various $U_f$, and applying the last set of $(\pi/2)$ pulses,
the spectra of proton and carbon were recorded in two different experiments by selective $(\pi/2)$ pulses after a gradient. The
spectra corresponding to various functions are given in figure 6(c), (e), (g) and (i).
The intensities of the peaks in the spectra provide a measure of the diagonal elements of the density matrix.
The complete tomographed \cite{chutomo,rd} density matrices in each case is given in figures 6(d), (f), (h) and (j).
When $U_{f_{00}}$ and $U_{f_{11}}$ are implemented, the
final state is $\vert 00\rangle$, and since the state of input qubit is $\vert 0\rangle$, the corresponding functions
$f_{00}$ and $f_{11}$ are inferred to be constant. Whereas in the case of $U_{f_{01}}$ and $U_{f_{10}}$, the final state of the system in
$\vert 10\rangle$. The state of input qubit being $\vert 1\rangle$, the corresponding functions $f_{01}$ and $f_{10}$ are
balanced. Theoretically, it is expected that the density matrices will have only the populations
corresponding to the final pure states. There were however errors due to r.f. inhomogeneity and relaxation.
The deviation from the expected results are within 13$\%$.
\section{Grover's search algorithm}
Grover's search algorithm can search an unsorted database of size N in $O(\sqrt{N})$ steps while a classical search would require
$O(N)$ steps \cite{grover}. Grover's search algorithm has been earlier demonstrated by several
workers by NMR, all using dynamic phase \cite{grojo,grochu,ap,ranacpl,ranajcp}.
The quantum circuit for implementing Grover's search algorithm on two qubit system is given in figure 7(a).
The algorithm starts from a $\vert 00\rangle$ pseudopure state. A uniform superposition of all states are created by
the initial Hadamard gates $(H)$. Then the sign of the searched state $``x"$ is inverted by the oracle through the operator
\begin{eqnarray}
U_x=I-2\vert x\rangle \langle x \vert,
\end{eqnarray}
where $U_x$ is a controlled phase shift gate $C_{x}(\pi)$. $C_{11}(\pi)$ and $C_{00}(\pi)$ gates were implemented by the
pulse sequences given in Eq.[11] and [12] respectively. The oracle for the other two states $\vert 01\rangle$ and $\vert 10\rangle$
were implemented by the sequences,
\begin{eqnarray}
C_{01}(\phi)=(\pi)_{\theta}^1.(\pi)_{\theta+\pi+\phi/4}^1.(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}
.(\pi)_{\theta+\pi-\phi/2}^{\vert 00\rangle \leftrightarrow \vert 01\rangle}, \nonumber \\
C_{10}(\phi)=(\pi)_{\theta}^1.(\pi)_{\theta+\pi-\phi/4}^1.(\pi)_{\theta}^{\vert 10\rangle \leftrightarrow \vert 11\rangle}.
(\pi)_{\theta+\pi+\phi/2}^{\vert 10\rangle \leftrightarrow \vert 11\rangle},
\end{eqnarray}
where $\phi=\pi$, as required in our case.
An inversion about mean is performed on all the states by a diffusion operator $HU_{00}H$ \cite{grover}, where
\begin{eqnarray}
U_{00}=I-2\vert 00\rangle \langle 00 \vert,
\end{eqnarray}
where $U_{00}$ is nothing but $C_{00}(\pi)$, and was implemented by the pulse sequence of Eq.[12].
For an N-sized database the algorithm requires $O(\sqrt{N})$ iterations of $U_x HU_{00}H$ \cite{grover}. For a 2-qubit system
with four states, only one iteration is required \cite{grojo,grochu}.
We have created a $\vert 00\rangle$ pseudopure state using Eq.[3] and applied the quantum circuit of figure 7(a), for
$\vert x \rangle=\vert 00 \rangle$, $\vert 01 \rangle$, $\vert 10 \rangle$ and $\vert 11 \rangle$. Finally,
the spectra of proton and carbon were recorded individually in two different experiments by selective $(\pi/2)$
pulses after a gradient.
The complete tomographed density matrices in each case is given in figures 7(d), (f), (h) and (j).
In each case, the searched state $\vert x\rangle$ was found to be with highest probability.
Ideally in a two-qubit system, probability should exist only in the searched state, and there should be no coherences.
Experimentally however, other states were also found with low probability, and some coherences were found in the
off-diagonal elements of the density matrix. These errors are mainly due to relaxation and imperfection of pulses caused by r.f. inhomogeneity.
Imperfection of r.f. pulses can cause imperfect refocusing of dynamic phase.
However, it was found that setting the duration of selective pulses to multiples of
(2/J) yielded better results. We have used 13.2ms (6/J) duration Gaussian shaped pulses. The maximum errors in the diagonal elements
are within 10$\%$ and that in the off-diagonal elemenst are within 15$\%$.
\section{conclusion}
A technique of using non-adiabatic geometric phase for quantum computing by NMR is demonstrated.
The technique uses selective excitation of subsystems, and is easily scalable to higher qubit systems provided
the spectrum is well resolved. Since the non-adiabatic geometric phase does not depend on the details of the
path traversed, it is insusceptible to certain errors yielding inherently fault-tolerant quantum computation \cite{fcomp1,fcomp2}.
The controlled geometric phase gates were also used to implement DJ-algorithm and
Grover's search algorithm in a two-qubit system. Implementation of fault-tolerant
controlled phase gates using adiabatic geometric phase demands that the evolution should be 'adiabatic', which requires
long experimental time. To avoid decoherence, use of non-adiabatic geometric phase might be utile.
\section{acknowledgment}
The authors thank K.V. Ramanathan for useful discussions. The use of DRX-500 NMR spectrometer funded by the Department of
Science and Technology (DST), New Delhi, at the Sophisticated Instruments Facility,
Indian Institute of Science, Bangalore, is gratefully acknowledged.
AK acknowledges ``DAE-BRNS" for senior scientist support and DST for a research grant for
"Quantum Computing by NMR".
$^*$DAE/BRNS Senior Scientist, e-mail: anilnmr@physics.iisc.ernet.in
|
{
"timestamp": "2005-03-03T07:41:01",
"yymm": "0503",
"arxiv_id": "quant-ph/0503032",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503032"
}
|
\section{Introduction, Notations}
\medskip
As is well known, the CSM converts the description of resonances by
non-integrable Gamow states into one by square integrable states while
leaving the discrete spectrum unchanged \cite{ABC}. Cuts describing the
continuum are rotated, however, but this may be advantageous, since they are
thus disentangled when their thresholds differ from one another. (We are not
interested, in this paper, in the case of channels with identical thresholds.)
It is then expected that the continuum corresponding to such rotated cuts
makes a much smoother contribution to the calculation of collision amplitudes,
level densities, strength functions and sum rules \cite{Kato1} \cite{Kato2},
since narrow resonant processes have been assumed to be peeled out explicitly
by the CSM. The CSM Hamiltonian, unfortunately, is not hermitian any more,
and it is not obvious that a resolution of the identity in terms of its bound
states, resonances and presumably damped continuum is possible. For the one
channel case, convincing arguments have been advanced a long time ago
\cite{Berg1} to prove that this resolution exists. More recently \cite{us}, a
detailed investigation of the case of two channels, coupled by straightforward
potentials, generated a contour integration of the usual Green's function
which provided the identity resolution. The task was made reasonably easy by
the small complication of the Riemann surface in that case. The purpose of the
present paper is to capitalize on the methods used for that two channel case
and attempt a generalization to any finite number of channels, despite the
more complicated nature of the relevant Riemann surface. We shall assume,
naturally, that there already exists, derived from single poles and usual
cuts, a resolution of the identity for the initial Hamiltonian, before its
modification by complex scaling. Our problematics would be meaningless
otherwise.
\medskip
Several earlier studies, in particular by \cite{KFFGR} \cite{ML}, have been
concerned with a description of resonances with square integrable states,
without complex scaling. They did not restrict to the consideration of just
simple poles of the $S$-matrix and investigated how one might, as rigorously
as possible, define initial wave packets for the description of decaying
states; the non purely exponential nature of their decays received a detailed
attention, via the analysis of their time dependent evolutions. The present
paper, however, will be content with a Gamow definition of resonances, by
means of simple poles; our aim is just to generate a resolution of the
identity, with time independent states extending to asymptotic regions. For
earlier searches of a complete basis of states, including resonances, but
within a compact interaction volume, we may refer to the review by \cite{BRT}
of $R$-matrix methods and their extensions. See also \cite{R} and in particular
the comparison of ``class B'' and ``class D'' theories.
\medskip
In this paper, we shall again assume that all potentials $V_{in}(r)$ driving
the channels and their couplings are local and so short ranged, Gaussian-like
for instance, that the $2N$ Jost solutions of the $N$ coupled equation system,
\begin{equation}
-\psi_{ij}''(k_j,r)+\sum_{n=1}^N
\left[ e^{2i\theta} V_{in}(e^{i\theta}r)+
\left(\frac{\ell_i(\ell_i+1)}{r^2}-k_i^2\right) \delta_{in} \right]
\psi_{nj}(k_j,r)=0,\ \ i,j=1,...,N,
\label{coupl}
\end{equation}
exist and are analytical in the whole complex domain of all the momenta $k_j.$
The radius $r$ runs from $0$ to $+\infty,$ obviously, and the number $N$ of
channels is taken as finite. As an additional technicality we also assume,
naturally, that the products $V_{in}\, \psi_{nj}$ do not diverge for
$r \rightarrow 0$ when singular solutions of Eqs.(\ref{coupl}) are considered.
\medskip
We select the threshold of the lowest channel as the origin of the complex
energy plane, hence $E \equiv k_1^2.$ The other channels with their
physical thresholds $E_j^*,$ which are real and positive numbers, now
define channel momenta according to, $E_j=k_j^2=E-e^{2i\theta}E_j^*.$ Notice
that, given a real number $E_j^*$ defining a physical threshold, the usual
complex scaling where $p^2$ becomes $e^{-2i\theta} p^2$ and $r$ becomes
$e^{i\theta} r$ does not change $E_j^*$ and rotates the corresponding cut by
angle $-2\theta.$ But here, we have a slightly different representation,
because the Hamiltonian has been multiplied by $e^{2i\theta}.$ Hence kinetic
operators in our Hamiltonian $H,$ see Eqs.(\ref{coupl}), are just $-d^2/dr^2,$
every cut rotates back into being ``horizontal''and starts from
$e^{2i\theta} E^*.$ For time dependent studies. it will make sense to scale
time, conjugate of energy, by a factor $e^{-2i\theta}.$ This will prevent
those resonant wave packets, the energies of which have a positive imaginary
part as eigenvalues of $H,$ from exploding when $t \rightarrow + \infty.$
\medskip
Also in this paper no rearrangement is allowed, channels are defined by
just internal excitations of the projectile and/or the target, hence all
reduced masses are equal. Finally we exclude from this paper the consideration
of abnormal thresholds; we shall only discuss the case of ``square root
thresholds''. This is generic enough.
\medskip
It is understood here and from now on that a first subscript, such as $i$ or
$n,$ denotes the component of each wave $\psi$ in channel $i$ or $n,$ then
that any superscript, $\pm,$ or second subscript, $j,$ denotes the boundary
condition which defines $\psi.$ For a Jost solution $f^{\pm}_{.j},$ the
boundary condition that we choose is ``asymptotic flux
$e^{\pm i(k_jr-\ell_j \pi/2)}$ in channel $j$ and no asymptotic flux in the
other channels''. It is well known that for $r \rightarrow 0,$ the components
of such Jost solutions are proportional to $(k_ir)^{-\ell_i}(2\ell_i-1)!!.$
For a regular solution $\varphi_{.j},$ the boundary condition that we choose
sits at $r=0$ and reads,
``$\lim_{r \rightarrow 0}\, (k_i r)^{-\ell_i-1}\, \varphi_{ij}(r) = 0\ \,
\forall i \ne j,$ while, for $i=j,$ then
$\lim_{r \rightarrow 0}\, (k_j r)^{-\ell_j-1}\, \varphi_{jj}(r) =
1/(2\ell_j+1)!!.$
\medskip
Following Newton \cite{Newt}, it is convenient, given $E$ and $r,$ to set the
column vectors $\varphi_{.j}$ into a matrix ${\bf \Phi}(E,r)$ of regular
solutions and the Jost solutions $f^+_{.j}$ (resp. $f^-_{.j}$) into a similar
matrix ${\bf f}^+(E,r)$ (resp. ${\bf f}^-$).
It is also convenient to notice that ${\bf \Phi}$, viewed as a function
of the $k_j$'s as if these were independent momenta, is even under any
reversal of a $k_j$ into $-k_j.$ Such is not the case for ${\bf f}^+;$
analytic continuations in either energy or momenta planes can introduce
one (or several) $f^-_{.j}$'s into ${\bf f}^+.$
\medskip
For our oncoming argument we must use the Wronskian matrix with matrix
elements the Wronskians ${\cal W}\left(f^+_{.m},\varphi_{.n}\right)$ of the
Jost solutions $f^+_{.m}$ with the regular ones $\varphi_{.n}.$ This, for
$s$ waves, is the transposed of ${\bf f}^+$ at $r=0,$
\begin{equation}
{\bf W}(E)=\tilde{{\bf f}}^+(E,0),
\label{witch}
\end{equation}
and for other angular momenta is only a slight modification of
$\tilde{{\bf f}}^+(E,0).$ (Rather than just $\tilde{{\bf f}}^+(E,0)$ one
must use limits of products $(k_ir)^{\ell_i} f_{ij}^+/(2\ell_i-1)!!$
at $r=0,$ explicitly, but we will disregard this technicality.)
The Green's function ${\bf G}$ is then found as,
\begin{equation}
{\bf G}(E,r,r')={\bf \Phi}(E,r)\, [{\bf W}(E)]^{-1}\, \tilde {\bf f}^+(E,r')
\ \ {\rm if}\ r < r',\ \ \
{\bf G}(E,r,r')={\bf f}^+(E,r)\, [\tilde {\bf W}(E)]^{-1}\, \tilde {\bf \Phi}
(E,r')\ \ {\rm if}\ r > r'.
\label{Green}
\end{equation}
Here each tilde $\tilde{ }$ means transposition; we refer to \cite{Newt}
or to Appendix A of \cite{us} for the derivation of such formulae for
${\bf G}.$ Despite different formulae whether $r > r'$ or $r < r',$ and the
lack of hermiticity, ${\bf G}$ is symmetric, namely
${\bf G}(r,r')={\bf G}(r',r).$
\medskip
It will be noticed that the CSM, as we describe it by the system of
Eqs.(\ref{coupl}), locates thresholds on a segment of the complex $E$ plane
with slope $2\theta,$ extending from $E=0$ to $e^{2i\theta}E_N^*,$ and
that the channel cuts are rotated back into being ``horizontal''. Conversely,
bound states lie on a negative semiaxis rotated by $2\theta$ and resonances
are rotated by $2\theta$ as well. This slight change of representation changes
nothing to the physics, obviously. For trivial technical reasons \cite{us},
we normalize energy units so that $E_N^*=4.$ Also we shall use a short
notation, $k \equiv k_1$ and $K \equiv k_N.$ We show in Figure 1 the cut
energy plane in an illustrative, four channel situation when $\theta=\pi/6,$
$E_2^*=1.5$ and $E_3^*=3.5.$
\medskip
Equipped with this slightly unwieldy formalism, we can now investigate whether
there exists a representation, and an integration contour, such that the
traditional integral, ${\cal I}=\int dE\, {\bf G}(E,r,r'),$ calculated in two
different ways, generates a resolution of the identity. This question of a
representation and a contour is the subject of Section II, the main part of
our argument. Additional considerations on the two ways of calculating this
integral make the subject of Section III. A discussion and conclusion are
proposed in Section IV.
\begin{figure}[htb] \centering
\mbox{ \epsfysize=110mm
\epsffile{figcc1.eps}
}
\caption{$E$-plane. Physical cuts for a four channel case when $\theta=\pi/6,$
$E^*_2=1.5,$ $E^*_3=3.5$ and $E^*_4=4.$ Lowest channel, heavy full lines,
highest channel, heavy dashed lines, intermediate channels, lighter full
lines. The dotted segment with slope $\pi/3$ is the locus of thresholds
(big dots) in this representation.}
\end{figure}
\section{Representations and contours}
\subsection{Energy plane}
\medskip
From Fig. 1 it is intuitive that one could start, for instance, from
$+ \infty$ along the lower rim of the lowest channel cut, return to the origin,
$E=0,$ proceed to $+ \infty$ again on the upper rim, then join there the lower
rim of the second cut, return to the threshold of this second cut, go to
$e^{2i\theta}E^*_2+ \infty$ along the upper rim, join the third cut lower rim
at infinity, etc., until arriving at $e^{2i\theta}E^*_N + \infty$ along the
upper rim of the highest channel. Then the contour would be closing at infinity
by means of an almost complete circle, counterclockwise, terminating at the
starting point, namely at $+ \infty$ on the lower rim of the lowest channel.
\medskip
Along such a contour, it would be necessary to investigate the behaviors of the
ingredients ${\bf f}^+,$ ${\bf W}$ and ${\bf \Phi}$ of ${\bf G}.$ Furthermore,
information is needed about the singularities of ${\bf G}$ inside the contour;
indeed, residues of simple poles are essential for a calculation of
$\int dE\, {\bf G}(E)$ by Cauchy's theorem; one also needs reasons why no
singularities higher than simple poles occur.
\medskip
The representation discussed in the next subsection makes easier the needed
investigation, for it opens two of the cuts and limits the discussion to
situations where all momenta have semipositive imaginary parts,
$\Im k_j \ge 0.$
\subsection{Pseudomomentum plane}
\medskip
A generalization from \cite{us}, where there were two channels only, the
present ``$P$ representation'' consists in joining the upper rim of the
lowest cut and the lower rim of the highest cut, and in opening both cuts,
by {\it rational} formulae,
\begin{equation}
k=P+Q^2/P,\ \ K=P-Q^2/P,
\end{equation}
where $Q=e^{i\theta}$ makes a short notation for our scaling of energies such
that $E^*_N=4$ and $k^2-K^2=4Q^2.$ Trivially, $P$ is the average $(k+K)/2$ of
$k$ and $K.$ The point is, despite an obvious failure to open additional cuts,
$P$ also give the ``dominant'' part of any other momentum when
$\Im P \rightarrow + \infty.$
Indeed, when $|P|$ is large, say $|P| >> 2,$ then an asymptotic value can be
defined for $k_j,\, j \ne 1,\, j \ne N,$ according to the rule,
\begin{equation}
k_j \equiv (k^2-Q^2 E^*_j)^{\frac{1}{2}} =
(P^2+2Q^2-Q^2 E^*_j+Q^4/P^2)^{\frac{1}{2}} = P + Q^2(1-E^*_j/2)/P +
{\cal O}(P^{-2}).
\end{equation}
Thus the semicircle at infinity in the upper $P$ plane corresponds to
$\Im k_j >0,\, \forall j.$ This is of critical value for the zoology
of our Jost functions and it is expected that this semicircle properly
closes the integration contour under design.
\medskip
Set now $P=x+iy$ and short notations $c=\cos2\theta$ and $s=\sin2\theta.$
A trivial calculation separates the real and imaginary parts
of the (complex) energies driving each channel,
\begin{equation}
(x^2+y^2)^2\, \Re(k_j^2)=
[(x^2+y^2+s)(x+y)+(x-y)c] \, [(x^2+y^2-s)(x-y)+(x+y)c]
- E_j^* (x^2+y^2)^2 c,
\label{rea}
\end{equation}
and
\begin{equation}
(x^2+y^2)^2\, \Im(k_j^2)=2[(x^2+y^2) x + x c + y s]\, [(x^2+y^2) y + x s - y c]
- E_j^* (x^2+y^2)^2 s.
\label{ima}
\end{equation}
and it is trivial to recover the images, in this new representation, of the
cuts displayed in Fig. 1. Polar coordinates, with $P=p e^{i\eta},$
can be also be used to decribe the $j$-th cut from Eq.(\ref{ima}) by,
\begin{equation}
p^2 \sin2\eta + \frac{\sin(4\theta-2\eta)}{p^2}=(E^*_j-2)\, \sin2\theta.
\label{polar}
\end{equation}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=110mm
\epsffile{figcc2alt.eps}
}
\caption{$P$ plane. Cuts for the same four channel case, $\theta=\pi/6,$
$E^*_2=1.5,$ $E^*_3=3.5$ and $E^*_4=4.$ Opened cut for lowest channel,
heavy full line. Opened cut for highest channel, heavy dashed line.
Intermediate channel cuts, not open, lighter full lines. The dotted segment
is the locus of thresholds (big dots) in this $P$ representation.}
\end{figure}
Results are shown in Figure 2 for the same special case as Fig. 1.
As in \cite{us}, the lowest channel is represented by the heavy,
shoulder shaped line, that starts from $-\infty$ on the real P axis,
bends up, then backs into the origin $P=0,$ where it terminates with a slope
$2 \theta.$ Along the curve, $k$ is real and runs from $-\infty$ to $+\infty,$
covering both rims of the initial cut. The threshold $k=0$ is represented by
$P=iQ=e^{i(\theta+\frac{1}{2}\pi)}.$ Partner points where
$k \leftrightarrow -k$ obtain under the symmetric transformation
$P \leftrightarrow -Q^2/P.$ In the same way, for the highest channel, $K$ runs
with real values along the heavy dashed line, from $-\infty$ at $P=0$ to
$+\infty$ at the end of the positive $\Re P$ semiaxis, via $K=0$ for $P=Q.$
The transform, $P \leftrightarrow Q^2/P,$ makes partners with opposite values
of $K.$
\medskip
The other cuts remain cuts. Their thresholds lie on the image, shown as a
dotted line again, of the segment already pointed out at the stage of Fig. 1.
Because both $\Re (k_j^2)$ and $\Im (k_j^2)$ vanish for such points, it is
easy to eliminate $E_j^*$ between the right hand sides of
Eqs.(\ref{rea},\ref{ima}) and obtain the condition for such a locus,
\begin{equation}
x^2+y^2=1,
\end{equation}
a very simple result indeed. With $|P|=1,$ the positions of the thresholds are
easy to obtain. The special cases $j=1$ and $j=N$ give the argument
$\eta \equiv ArgP$ as $\eta=\theta+\pi/2$ and $\eta=\theta,$ respectively.
This was already known from \cite{us}. The function
$\sin2\eta+\sin(4\theta-2\eta),$ see Eq.(\ref{polar}), decreases monotonically
when $\eta$ increases from $\theta$ to $\theta+\pi/2,$ hence a unique solution
for each $E_j^*,$ and an obvious symmetry about $\theta+\pi/4$ corresponding
to the symmetry about $E_j^*=2.$ Then each intermediate cut generates, from
Eq.(\ref{ima}), an image which joins its threshold to the origin $P=0,$ while
$k_j,$ a real number along this image, runs from $0$ to $\pm \infty,$
according to the rim. The image lies between the heavy full and dashed lines,
and, being pinched between them at $P=0,$ also reaches the origin with slope
$2 \theta.$ While the pinching makes numerics slightly difficult, it is easy
to verify analytically from Eqs.(\ref{rea},\ref{ima}) that {\it
infinitesimally away from both rims of such an intermediate cut, but inside
the wedge created by the heavy line curves, $\Im k_j$ remains positive}.
\begin{figure}[htb] \centering
\mbox{ \epsfysize=110mm
\epsffile{figcc3alt.eps}
}
\caption{$P$ plane. Again $\theta=\pi/6.$ Cut for the channel defined by
$E^*_2=1.5.$ The center line, between dots, is the cut. Cut then continued
for negative energies in the channel. Additional lines, lower rim (leftmost
curve) and upper rim (rightmost curve), respectively. Both rims extended
below threshold. Heavy line bar, connection between extended rims.}
\end{figure}
To illustrate our full control of the various $\Im k_j$'s provided by this $P$
representation, whether inside the wedge or near the positive infinity
semicircle, we show in Figure 3 the cut corresponding to $E_2^*,$ and its
continuation beyond threshold. By ``beyond'', we mean still canceling
$\Im k_2^2,$ while $\Re k_2^2$ becomes more and more negative. This allows
reaching the ``semicircle''. Simultaneously, we generate rims of the cut, and
beyond again below threshold. To generate rims, we use Eq.(\ref{ima}), or as
well Eq.(\ref{polar}), with $E_2^*$ replaced by $E_2^*-0.2$ and $E_2^*+0.2$
for the lower and upper rim, respectively. (The choice $\pm 0.2$ was made for
graphical convenience, but we tested much smaller intervals, naturally.) The
dots represent $P=0,$ where the channel energy is infinite, and the threshold,
where it vanishes by definition. Like the cut, the rims are pinched by the
wedge.
Then we show in Figure 4 the trajectory of $k_3$ when $P$ follows this
cut from $P=0,$ to the threshold and beyond. Notice that, $E_2$ being real
along the line, then the imaginary part of $E_3=E_2+e^{2i\theta}(E_2^*-E_3^*)$
is obviously negative. This does not prevent a choice of $k_3$ with
$\Im k_3>0,$ generating the leftmost trajectory in Fig. 4.
Simultaneously, we show the trajectories of $k_2$ from both rims of the same
cut. The left hand side (when seen in Fig. 3) rim induces
$\Re k_2 \rightarrow -\infty$ when $P \rightarrow 0,$ with an
infinitesimally positive $\Im k_2.$ Conversely the right hand side rim
induces $\Re k_2 \rightarrow +\infty$ when $P \rightarrow 0,$ with still
an infinitesimally positive $\Im k_2.$ When we go from either rim towards the
upper semicircle at infinity, this induces $\Im k_2 \rightarrow + \infty,$
as expected. The rims can be connected by any small path, see the bar
above the threshold in Fig. 3, and the values of $k_2$ along the rims can be
smoothly matched, see the curved bar in Fig. 4, the trajectory of
$k_2$ when $P$ follows the bar in Fig. 3. Generalizations to every $k_j$ in
every part of the wedge are trivial.
\begin{figure}[htb] \centering
\mbox{ \epsfysize=110mm
\epsffile{figcc4alt.eps}
}
\caption{$k_2,k_3$ planes. Still $\theta=\pi/6,$ $E_2^*=1.5$ and $E_3^*=3.5.$
Leftmost curve, trajectory of $k_3$ when $P$ follows the central line of
Fig. 3. Intermediate curve, trajectory of $k_2$ for extended lower
rim, see leftmost curve in Fig. 3. Rightmost curve, trajectory of $k_2$
induced by extended upper rim, see rightmost curve in Fig. 3. Heavy line
curved bar, connection trajectory for $k_2$ when $P$ turns around the
threshold, below it.}
\end{figure}
\subsection{Contour}
\medskip
To synthetize this Section, the $P$ representation defines a physical sheet
similar to the physical sheet of the energy plane. The region of interest is
that region above the two curves which open the cuts for the lowest and the
highest channels, while cuts remain for the intermediate channels. All momenta
inside the wedge, and all the way to the upper semicircle at infinity, can
be defined with positive imaginary parts. A contour can be found, following
all cuts and closing at infinity in the upper plane.
\medskip
The intuition which was present in the $E$ representation can be
substantiated in the $P$ plane. Start from $- \infty$ on the real axis, follow
the ``opener curve'' which corresponds to the lowest channel, all the way
to $P=0.$ From there, follow the lower rim of the cut corresponding to the
second channel, back to its threshold, then turn around the threshold to
follow its upper rim, down to $P=0.$ In turn, follow the lower rim of each
intermediate channel, then its upper rim. After bouncing $N-1$ times at
$P=0,$ follow the ``opener curve'' corresponding to the upper channel, until
$P \rightarrow + \infty$ on the real axis. Then close the contour by means of
the upper semicircle at infinity. In the next Section, we shall investigate
what happens to the integral, ${\cal I}=\int dE\, {\bf G}(E,r,r'),$ when
considered along this contour in the $P$ plane.
\section{Three contributions to the Green's function integral}
\subsection{Upper semicircle}
\medskip
At infinity in this upper $P$ plane, the integration weight,
$dE=2\left(P-Q^4/P^3\right)\,dP,$ boils down to $2P\,dP.$ All the $N$
distinct Jost solutions boil down to
$\exp\left[i(Pr-\frac{1}{2}\ell_j\pi)\right]$ in their
respective ``flux channel $j$'', while vanishing in the other channels. At
the same time, the $N$ distinct regular solutions similarly boil down to
$\sin\left(Pr-\frac{1}{2}\ell_j\pi\right)/P$ in their respective flux channel
and vanish in the other channels. The Wronskian matrix boils down to the
$N$-dimensional unit matrix.
\medskip
Assume now $r > r',$ for instance, and thus consider the second of
Eqs.(\ref{Green}). The product
${\bf f}^+\, [\tilde {\bf W}]^{-1}\, \tilde {\bf \Phi}$ boils down to a
diagonal matrix. Its $j$-th diagonal element reads,
\begin{equation}
\int_{sc}2\ dP\ e^{i(Pr-\ell_j\pi/2)}\,
\sin\left(Pr'-\ell_j \frac{\pi}{2}\right),
\end{equation}
and can be easily calculated by reducing the semicircle back to the real $P$
axis. The result does not depend on $j,$
\begin{equation}
-i \int_{\infty}^{-\infty}\, dP\, e^{i(Pr-\frac{1}{2}\ell_j\pi)}\,
\left[ \exp\left(iPr'-i \ell_j \frac{\pi}{2}\right) -
\exp\left(i \ell_j \frac{\pi}{2}-iPr'\right) \right] =
2 i \pi [\delta(r+r') - \delta(r-r')].
\end{equation}
It is trivial to verify that the same result is obtained if $r<r'.$
Furthermore the term $\delta(r+r')$ cancels out in the space of regular radial
waves. Hence the contribution ${\cal I}_{sc}$ of the semicircle makes nothing
but the multichannel identity, multiplied by $(-2i\pi).$ Notice that,
differing from \cite{Kato1}, this identity is not multiplied by a factor
depending on $\theta,$ since for us the ends of the semicircle, $-\infty$ and
$+\infty,$ both lie on the real $P$ axis.
\subsection{Continuum}
It makes no difference here whether we consider the contribution of one of the
``opener line'' or that of one of the intermediate cuts. For in both cases
we group partner terms. Such partners either come from a transform
$P \leftrightarrow \pm Q^2/P$ or from opposite rims of the intermediate cut
under consideration.
What is important to notice is that momenta retain their finite and positive
imaginary parts and do not change when we compare two partner points, except
that momentum specific to the opener line or the cut. For that momentum, which
is real, ``partnership'' means $k_j \leftrightarrow -k_j,$ with still an
infinitesimal positive imaginary part. Keeping in mind
that ${\bf \Phi}$ is even under such a momentum flip, the contribution of
such a continuum thus reads, if $r > r'$ for instance,
\begin{equation}
{\cal I}_j=\int_0^{\infty} 2k_j\, dk_j\, {\bf D}_j(E,r)\,
\tilde {\bf \Phi}(E,r'),
\label{cut}
\end{equation}
where ${\bf D}_j(E,r)$ represents the following difference between partners,
\begin{equation}
{\bf D}_j(E,r)={\bf f}^+(E, r)\, [\tilde {\bf W}( E)]^{-1} -
{\bf f}^+(-k_j,r)\, [\tilde {\bf W}(-k_j)]^{-1},
\label{discont}
\end{equation}
a discontinuity across the cut. The notation used here takes advantage of
the fact that $dE=2k_j dk_j,$ and that $k_j$ is a convenient label along
the line or the cut.
The first term, ${\bf f}^+(E,r)\, [\tilde {\bf W}(E)]^{-1},$
in the right hand side of Eq.(\ref{discont}) clearly comes from the upper rim.
The notation that we use for the second term,
${\bf f}^+(-k_j,r)\, [\tilde {\bf W}(-k_j)]^{-1},$ indicates that, because of
analytic continuation in the physical sheet around the threshold, one Jost
solution $f^-_{.j}$ now makes the $j$-th column of ${\bf f}$ and that of
$\tilde {\bf W}.$ {\it All other columns are unchanged, and this strong
similarity reduces the difference ${\bf D}_j$ to be a rank one dyadic.}
An elementary proof of this dyadic result was given in Appendix C of \cite{us}.
Nothing changes in the argument if $r < r'.$
\medskip
As a consequence of the dyadic nature of ${\bf D}_j,$ and of the symmetry
${\bf G}(E,r,r')={\bf G}(E,r',r),$ hence of the same symmetry for
discontinuities across cuts, there exists as a column vector a solution
$\phi_{.j}$ of Eqs.(\ref{coupl}) that is able to represent symmetrically
both ${\bf D}_j(E, r)\, \tilde {\bf \Phi}(E,r')$
and
${\bf \Phi}(E,r)\, \tilde {\bf D}_j(E,r')$
in a self dual way as an outer product,
\begin{equation}
{\cal I}_j=\int_0^{\infty} 2k_j\, dk_j\, \frac{\phi_{.j}(E,r)\,
{\tilde \phi}_{.j}(E,r')}{{\cal D}(E)}.
\label{continuum}
\end{equation}
This solution belongs to the set of regular solutions, naturally, because of
the regularity of ${\bf G}$ at both $r=0,$ and $r'=0,$ illustrated by the
presence of ${\bf \Phi}$ in Eqs.(\ref{Green}).
The exact natures of this $\phi_{.j}$ and of the ``normalizing''
denominator ${\cal D}$ are discussed in the Appendix.
\medskip
At this stage, the full integral along the full contour thus gives the
sum of the multichannel identity and ``pseudoprojectors on the continuum'',
one pseudoprojector for each channel,
\begin{equation}
\frac{i}{2\pi}\int dE\, {\bf G}(E,r,r')=
\left[\matrix{\delta(r-r') & 0 & ... & 0 \cr
0 & \delta(r-r')& ... & 0 \cr
. & . & . & . \cr
0 & 0 & ... & \delta(r-r')}\right] +
\frac{i}{\pi}\sum_{j=1}^N
\int_0^{\infty} k_j\, dk_j\, \frac {\phi_{.j}(E,r)\, {\tilde \phi}_{.j}(E,r')}
{{\cal D}(E)}.
\label{resola}
\end{equation}
The next subsection shows what happens if the same integral is evaluated
by means of the Cauchy theorem.
\subsection{Residues at poles}
\medskip
We assumed that, before complex scaling, namely for $\theta=0,$ there existed
an identity resolution in terms of unscaled bound states and unscaled
scattering states. In other words we assumed that the corresponding, unscaled
${\bf G}(E)$ shows only isolated, simple poles, besides the physical cuts.
Such poles can be on the real $E$ axis of the physical sheet, describing
bound states, or away from this axis, then describing resonances or
antiresonances. The point is, now, that the CSM cannot change the nature of
such poles \cite{ABC}. Within our description by Eqs.(\ref{coupl}), the CSM
just rotates such poles by $2\theta$ in the energy representation,
along circular arcs, concentric around $E=0.$ In the $P$ representation,
the images of such arcs are also concentric arcs, with angular extension
$\theta$ only. This is trivially seen from the equation which, for each
initial position $\varepsilon$ of a pole, defines those values of $P$ which
represent $e^{2i\theta}\varepsilon,$
\begin{equation}
\left(P+e^{2i\theta}/P\right)^2=e^{2i\theta}\, \varepsilon.
\end{equation}
Indeed, $\theta$ disappears from this equation if one sets $P=e^{i\theta}P_0,$
where $P_0$ solves for the initial position $\varepsilon.$ It can be concluded
that only simple poles will be found when a finite $\theta$ is used
for our CSM. Notice, incidentally, that for $\varepsilon$ real and negative
(bound states), the $P$ representation will align poles along the axis with
polar angle $\theta+\pi/2,$ further than the circle with radius $1$ that we
found as the locus of thresholds. There will be no such alignment for
resonances.
\medskip
For the calculation of ${\cal I}$ by Cauchy's theorem, poles are not due to
either ${\bf f}$ or ${\bf \Phi},$ since these, as functions of $E$ or $P,$ are
regular. Only the divergence of ${\bf W}^{-1}$ can create poles. The
situations of interest are those when the roots of the determinant,
$\det {\bf W},$ are located inside the integration contour. We know that
such is the case for the bound states. Depending upon $\theta,$ some
resonances may also rotate into the domain. It is already known that only
simple, isolated poles occur. The only question to solve is, what is the
residue of ${\bf G}$ at such a pole.
\medskip
Residues of ${\bf G}$ at its poles will now be obtained from derivatives
$d/dE.$ That is equivalent to a calculation in the $P$ representation, anyhow,
and slighly easier. We shall use short notations in which the dependence of
${\bf \Phi},$ ${\bf f}^+,$ ${\bf W},$ upon $r,$ and/or $r'$ and/or $E$ will be
most often understood. However, at those energies $E_{\nu}$
where a pole occurs, we use an explicit subscript $\nu$ to specify that such
quantities ${\bf \Phi},$ ... , ${\bf W}$ are evaluated at $E_{\nu}.$
\medskip
Poles occur because of ${\bf W}^{-1}.$ Hence, we must only find the residue,
\begin{equation}
{\cal R}_{\nu}=\lim_{E \rightarrow E_{\nu}}\ (E-E_{\nu})\ {\bf W}^{-1}(E)\, ,
\label{residue1}
\end{equation}
and form the matrix product,
${\bf \Phi}_{\nu}\,{\cal R}_{\nu}\,\tilde {\bf f}^+_\nu$ and its
transpose ${\bf f}^+_\nu\,\tilde {\cal R}_{\nu}\,\tilde {\bf \Phi}_{\nu}.$
\medskip
At a (simple!) root $E_{\nu}$ of $\det{\bf W}(E),$ there is necessarily one,
and just one, null right eigenvector $\Lambda_{\nu}$ of ${\bf W}.$ Similarly
there is one, and just one, null left eigenvector $\Lambda'_{\nu}.$ We write
them as columns and normalize them by the condition,
\begin{equation}
\tilde \Lambda'_{\nu}\, \Lambda_{\nu}=1.
\label{biorth}
\end{equation}
Then the divergent part of ${\bf W}^{-1}$ in a neighborhood of $E_{\nu}$ is
nothing but the truncation,
\begin{equation}
{\bf W}^{-1}_{tr}=\frac{\Lambda_{\nu}\, \tilde \Lambda_{\nu}'}
{\tilde \Lambda_{\nu}'\, {\bf W}(E)\, \Lambda_{\nu}}\, ,
\label{trunc}
\end{equation}
where there is an explicit dependence on $E$ in the denominator. This
denominator, a number, vanishes at $E=E_{\nu}.$ As a matrix element of
${\bf W}$ it is nothing but the Wronskian of the following two waves,
$F \equiv {\bf f}^+\, \Lambda_{\nu}'$
and
$\xi \equiv {\Phi}\, \Lambda_{\nu}.$
The former, $F,$ is irregular, the latter, $\xi,$ is regular. While
$\Lambda_{\nu}$ and $\Lambda'_{\nu}$ do not depend on $E,$ since they were
defined at $E=E_{\nu},$ both $F$ and $\xi$ depend on $E,$ via ${\bf f}^+$ and
${\bf \Phi}.$ When their Wronskian vanishes, $F$ and $\xi$ become proportional
to each other, and there exits a number $c$ such that $F_{\nu}=c\, \xi_{\nu}.$
This special wave is both a mixture of regular solutions and a mixture of Jost
solutions, with positive imaginary parts in the momenta driving all Jost
solutions. Therefore it decreases exponentially in all channels when
$r \rightarrow \infty$ and it is square integrable as well as regular. As
expected it represents either a bound state or a regularized resonance.
\medskip
According to Eqs.(\ref{residue1},\ref{trunc}), the residue under study comes
from just the reciprocal of the derivative of the Wronskian of $F$ and $\xi,$
\begin{equation}
{\cal R}_{\nu} = \frac{\Lambda_{\nu}\, \tilde \Lambda_{\nu}'}
{
d \left[\tilde \Lambda_{\nu}'\, {\bf W}(E)\, \Lambda_{\nu}\right]/{dE}
\, |_{E=E_{\nu}}
}\, .
\label{residue2}
\end{equation}
In short, we must calculate the derivative of a Wronskian with respect to
the energy, $d \left[\tilde \Lambda_{\nu}'\, {\bf W}(E)\, \Lambda_{\nu}\right]
/dE.$ To help manipulations with Wronskians,
define an operator matrix ${\bf U}$ with matrix elements the CSM potentials,
completed by the centrifugal barriers and the thresholds,
\begin{equation}
{\bf U}_{ij}=e^{2i\theta}U_{ij}\left( e^{i\theta} r \right) + \delta_{ij}
\left[e^{2i\theta}E_j^*+\frac{\ell_j(\ell_j+1)}{r^2} \right].
\end{equation}
Then elementary, but slightly tedious manipulations, which are already
described in \cite{Newt} or in Appendix B of \cite{us},
give the remarquably simple result,
\begin{equation}
d \left[\tilde \Lambda_{\nu}'\, {\bf W}(E)\, \Lambda_{\nu}\right] /
dE\, |_{E=E_{\nu}} = -\, c \,
\int_0^\infty dr\, \tilde \xi(E_{\nu},r)\, \xi(E_{\nu},r).
\label{Euclid}
\end{equation}
Then the constant $c$ cancels out between this and the numerators of
${\bf f}^+_\nu\,\tilde {\cal R}_{\nu}\,\tilde {\bf \Phi}_{\nu}$
and
${\bf \Phi}_{\nu}\,{\cal R}_{\nu}\,\tilde {\bf f}^+_\nu,$
which make the same, symmetric formula anyway, whether $r > r'$ or $r < r',$
since $F_{\nu}=c\, \xi_{\nu}.$
\medskip
Summing upon all such residues obtained at roots $E_{\nu}$ of
$\det {\bf W}$ above the ``opener'' curves in the $P$ upper half-plane, the
contour integral reads,
\begin{equation}
{\cal I}(r,r') = -\, 2\, i \, \pi\, \sum_{\nu} \
\frac{ {\bf \Phi}(E_{\nu},r)\ \Lambda_{\nu}\
\tilde \Lambda_{\nu}\ \tilde {\bf \Phi}(E_{\nu},r') }
{\int_0^{\infty}dr''\,
\tilde \Lambda_{\nu}\ \tilde {\bf \Phi}(E_{\nu},r'')\
{\bf \Phi}(E_{\nu},r'')\ \Lambda_{\nu} } \, .
\label{residue3}
\end{equation}
Here we state again that the column vector $\Lambda_{\nu}$ is the null,
right-hand side eigenvector of ${\bf W}(E_{\nu}),$ namely
${\bf W}(E_{\nu})\, \Lambda_{\nu}=0,$ then the
column vector ${\bf \Phi}(E_{\nu})\, \Lambda_{\nu}$ of wave functions is the
wave function of the bound state or resonance, and the denominator plays the
role of a ``Euclidean-like square norm''. This denominator is non vanishing;
this corresponds to the hypothesis of single, isolated poles. All these are
labeled by $\nu,$ a discrete index, or as well by $P_{\nu},$
an isolated root of ${\bf W}$ if viewed as a function of $P.$
\subsection{Completeness}
\medskip
Since the three contributions ${\cal I}_{sc},$ $\sum_j{\cal I}_j$ and
${\cal I}$ are obviously related by ${\cal I}_{sc} + \sum_j{\cal I}_j =
{\cal I},$ it is trivial to equate $\frac{i}{2\pi}{\cal I}_{sc},$ the
multichannel identity, with the difference between $\frac{i}{2\pi}{\cal I},$
the pseudoprojector on both bound states and resonances, and $\frac{i}{2\pi}
\sum_j {\cal I}_j,$ the latter term making the pseudoprojector upon the
continuum for all channels. Naturally, in practical calculations, a cutoff and
some amount of discretization will be necessary to integrate such continuum
terms, but the $P$ representation provides a suitable frame for testing the
convergence of such a resolution for sum rules, level densities and similar
observables. Notice that, because of the use of complex, self dual bras
and kets in the resolution, such cutoff and discretization manipulations
may generate spurious imaginary parts for the expectation values of hermitian
observables. For a discussion and possible interpretation of imaginary
parts in individual matrix elements, we refer to \cite{Berg2}. But, when
summed upon all discrete and integral terms provided by the resolution,
such imaginary parts must add up to a negligible, spurious noise compared to
the real parts. This requested cancellation makes one more criterion to
validate numerical operations.
\section{Discussion And Conclusion}
\medskip
Once again we used the ABC theorems \cite{ABC} to locate the discrete
spectrum at trivially rotated positions deduced from the discrete spectrum
of an initial, hermitian Hamiltonian. The topological similitude provided
by the CSM rotation warrants that, as long as there are no double poles or
higher singularities with the initial Hamiltonian, the same will be true with
the CSM Hamiltonian.
\medskip
Then it was not very difficult to find a representation which allows a suitable
contour integration of the Green's function. There was still a slightly
complicated Riemann surface to handle, for the number of cuts was reduced to
$N-2$ only \cite{Weid}, but we took great care, including a few numerical,
illustrative examples, to show that all cuts in the new representation are
well understood, all thresholds are easily located, all complex momenta to be
used for proofs have positive imaginary parts in a physical domain of a
suitable sheet, and in general that all technicalities are sound.
\medskip
This proof of the CSM completeness for $N$ channels is restricted to a finite
number of well separated channels, normal square root threshold singularities,
in a purely inelastic situation, without rearrangement, and with short ranged
forces. The case of long range forces makes a more difficult question, indeed
\cite{atomists1} \cite{atomists2}. But our restrictions still allow a large
class of practical problems, and for instance in nuclear physics, a very large
number of collective resonances can be described by the coupled channel
equations that we studied.
\bigskip \noindent
Acknowledgment: B.G.G. thanks the Hokkaido University for its hospitality
during part of this work.
|
{
"timestamp": "2005-03-17T18:30:57",
"yymm": "0503",
"arxiv_id": "nucl-th/0503049",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503049"
}
|
\section{Introduction}\label{s:intro}
Analysis of complex high dimensional data is an exploding area of
research, with applications in diverse fields, such as machine
learning, statistical data analysis, bio-informatics, meteorology,
chemistry and physics. In the first three application fields, the
underlying assumption is that the data is sampled from some
unknown probability distribution, typically without any notion of
time or correlation between consecutive samples. Important tasks
are dimensionality reduction, e.g., the representation of the high
dimensional data with only a few coordinates, and the study of the
geometry and statistics of the data, its possible decomposition
into clusters, etc \cite{Hastie}. In addition, there are many
problems concerning supervised learning, in which additional
information, such as a discrete class $g(\mb{x})\in\{g_1,...,g_k\}$ or
a continuous function value $f(\mb{x})$ is given to some of the data
points. In this paper we are concerned only with the unsupervised
case, although some of the methods and ideas presented can be
applied to the supervised or semi-supervised case as well
\cite{PNAS1}.
In the later three above-mentioned application fields the data is
typically sampled from a complex biological, chemical or physical
{\em dynamical} system, in which there is an inherent notion of
time. Many of these systems involve multiple time and length
scales, and in many interesting cases there is a separation of
time scales, that is, there are only a few "slow" time scales at
which the system performs conformational changes from one
meta-stable state to another, with many additional fast time
scales at which the system performs local fluctuations within
these meta-stable states. In the case of macromolecules the slow
time scale is that of a conformational change, while the fast time
scales are governed by the chaotic rotations and vibrations of the
individual chemical bonds between the different atoms of the
molecule, as well as the random fluctuations due to the frequent
collisions with the surrounding solvent water molecules. In the
more general case of interacting particle systems, the fast time
scales are those of density fluctuations around the mean density
profiles, while the slow time scales correspond to the time
evolution of these mean density profiles.
Although on the fine time and length scales the full description
of such systems requires a high dimensional space, e.g. the
locations (and velocities) of all the different particles, these
systems typically have an intrinsic low dimensionality on coarser
length and time scales. Thus, the coarse time evolution of the
high dimensional system can be described by only a few dynamically
relevant variables, typically called reaction coordinates. Important tasks in such systems are
the reduction of the dimensionality at these coarser scales (known
as homogenization), and the efficient representation of the
complicated linear or non-linear operators that govern their
(coarse grained) time evolution. Additional goals are the
identification of the meta-stable states, the characterization of
the transitions between them and the efficient computation of mean
exit times, potentials of mean force and effective diffusion
coefficients \cite{GKS,Huisinga,Huisinga03,Elber}.
In this paper, following \cite{Lafon}, we consider a family of
diffusion maps for the analysis of these problems. Given a large
dataset, we construct a family of random walk processes based on
isotropic and anisotropic diffusion kernels and study their first
few eigenvalues and eigenvectors (principal components). The key
point in our analysis is that these eigenvectors and eigenvalues
capture important geometrical and statistical information
regarding the structure of the underlying datasets.
It is interesting to note that similar approaches have been
suggested in various different fields. In graph theory, the first
few eigenvectors of the normalized graph Laplacian have been used
for spectral clustering \cite{Weiss99,Weiss}, approximations to
the optimal normalized-cut problem \cite{Malik} and dimensionality
reduction \cite{Belkin,Saerens}, to name just a few. Similar
constructions have also been used for the clustering and
identification of meta-stable states for datasets sampled from
dynamical systems \cite{Huisinga}. However, it seems that the
connection of these computed eigenvectors to the underlying
geometry and probability density of the dataset has not been fully
explored.
In this paper, we consider the connection of these eigenvalues and
eigenvectors to the underlying geometry and probability density
distribution of the dataset. To this end, we assume that the data
is sampled from some (unknown) probability distribution, and view
the eigenvectors computed on the finite dataset as discrete
approximations of corresponding eigenfunctions of suitably defined
continuum operators in an infinite population setting. As the
number of samples goes to infinity, the discrete random walk on
the set converges to a diffusion process defined on the
$n$-dimensional space but with a non-uniform probability density.
By explicitly studying the asymptotic form of the
Chapman-Kolmogorov equations in this setting (e.g., the
infinitesimal generators), we find that for data sampled from a
general probability distribution, written in Boltzmann form as
$p(\mb{x})= e^{-U(\mb{x})}$, the eigenfunctions and eigenvalues of the
standard normalized graph Laplacian construction correspond to a
diffusion process with a potential $2 U(\mb{x})$ (instead of a single
$U(\mb{x})$). Therefore, a subset of the first few eigenfunctions are
indeed well suited for spectral clustering of data that contains
only a few well separated clusters, corresponding to deep wells in
the potential $U(\mb{x})$.
Motivated by the well known connection between diffusion processes
and Schr\"odinger operators \cite{Bernstein}, we propose a
different novel non-isotropic construction of a random walk on the
graph, that in the asymptotic limit of infinite data recovers the
eigenvalues and eigenfunctions of a diffusion process with the
same potential $U(\mb{x})$. This normalization, therefore, is most
suited for the study of the long time behavior of complex
dynamical systems that evolve in time according to a stochastic
differential equation. For example, in the case of a dynamical
system driven by a bistable potential with two wells, (e.g. with
one slow time scale for the transition between the wells and many
fast time scales) the second eigenfunction can serve as a
parametrization of the reaction coordinate between the two states,
much in analogy to its use for the construction of an
approximation to the optimal normalized cut for graph
segmentation. For the analysis of dynamical systems, we also
propose to use a subset of the first few eigenfunctions as
reaction coordinates for the design of fast simulations. The main
idea is that once a parametrization of dynamically meaningful
reaction coordinates is known, and lifting and projection
operators between the original space and the diffusion map are
available, detailed simulations can be initialized at different
locations on the reaction path and run only for short times, to
estimate the transition probabilities to different nearby
locations in the reaction coordinate space, thus efficiently
constructing a potential of mean field and an efficient diffusion
coefficient on the reaction path \cite{Yannis}.
Finally, we describe yet another random walk construction that in
the limit of infinite data recovers the Laplace-Beltrami (heat)
operator on the manifold on which the data resides, regardless of
the possibly non-uniform sampling of points on the manifold. This
normalization is therefore best suited for learning the geometry
of the dataset, as it separates the geometry of the manifold from
the statistics on it.
Our analysis thus reveals the intimate connection between the
eigenvalues and eigenfunctions of different random walks on the
finite graph to the underlying geometry and probability
distribution $p=e^{-U}$ from which the dataset was sampled. These
findings lead to a better understanding of the advantages and
limitations of diffusion maps as a tool to solve different tasks
in the analysis of high dimensional data.
\section{Problem Setup}\label{s:setup}
Consider a finite dataset $\{\mb{x}_i\}_{i=1}^N \in \mathbb{R}^n$. We
consider two different possible scenarios for the origin of the
data. In the first scenario, the data is not necessarily derived
from a dynamical system, but rather it is randomly sampled from
some arbitrary probability distribution $p(\mb{x})$. In this case we
define an associated potential
\begin{equation}
U(\mb{x}) = - \log p(\mb{x})
\end{equation}
so that $p= e^{-U}$.
In the second scenario, we assume that the data is sampled from a
dynamical system in equilibrium. We further assume that the
dynamical system, defined by its state $\mb{\mb{x}}(t)\in\mathbb{R}^n$
at time $t$, satisfies the following generic stochastic
differential equation (SDE)
\begin{equation}
\dot{\mb{x}} = -\nabla U(\mb{x}) + \sqrt{2} \dot{\mb{w}}
\label{SDE}
\end{equation}
where a dot on a variable means differentiation with respect to
time, $U$ is the free energy at $\mb{x}$ (which, with some abuse of
nomenclature, we will also call the potential at $\mb{x}$), and
$\mb{w}(t)$ is an $n$-dimensional Brownian motion process. In this
case there is an explicit notion of time, and the transition
probability density $p(\mb{x},t|\mb{y},s)$ of finding the system at
location $\mb{x}$ at time $t$, given an initial location $\mb{y}$ at time
$s$ ($t > s$), satisfies the forward Fokker-Planck equation (FPE)
\cite{Schuss,Gardiner}
\begin{equation}
\frac{\partial p}{\partial t} = \nabla \cdot \left(\nabla p + p \nabla U(\mb{x})\right)
\label{FPE}
\end{equation}
with initial condition
\begin{equation}
\lim_{t\to s^+} p(\mb{x},t|\mb{y},s) = \delta(\mb{x} - \mb{y})
\end{equation}
Similarly, the backward Fokker-Planck equation for the density
$p(\mb{x},t | \mb{y}, s)$, in the backward variables $\mb{y},s$ ($s<t$) is
\begin{equation}
-\frac{\partial p}{\partial s } = \Delta p - \nabla p \cdot \nabla U(\mb{y})
\label{backward_FPE}
\end{equation}
where differentiations in (\ref{backward_FPE}) are with respect to
the variable $\mb{y}$, and the Laplacian $\Delta$ is a negative
operator, defined as $\Delta u = \nabla \cdot (\nabla u)$.
As time $t\to\infty$ the steady state solution of (\ref{FPE}) is
given by the equilibrium Boltzmann probability density,
\begin{equation}
\mu(\mb{x})d\mb{x} = \Pr\{\mb{x}\}d\mb{x} = \frac{\exp(-U(\mb{x}))}{Z}d\mb{x}
\label{mu_x}
\end{equation}
where $Z$ is a normalization constant (known as the partition
function in statistical physics), given by
\begin{equation}
Z = \int_{\mathbb{R}^n} \exp(-U(\mb{x})) d\mb{x}
\end{equation}
In what follows we assume that the potential $U(\mb{x})$ is shifted by
the suitable constant (which does not change the SDE (\ref{SDE})),
so that $Z=1$. Also, we use the notation $\mu(\mb{x}) = \Pr\{\mb{x}\} =
p(\mb{x})=e^{-U(\mb{x})}$ interchangeably to denote the (invariant)
probability measure on the space.
Note that in both scenarios, the steady state probability density,
given by (\ref{mu_x}) is identical. Therefore, for the purpose of
our initial analysis, which does not directly take into account the
possible time dependence of the data, it is only the features of
the underlying potential $U(\mb{x})$ that come into play.
The Langevin equation (\ref{SDE}) or the corresponding
Fokker-Planck equation (\ref{FPE}) are commonly used to describe
mechanical, physical, chemical, or biological systems driven by
noise. The study of their behavior, and specifically the decay to
equilibrium has been the subject of much theoretical research
\cite{Risken}. In general, the solution of the Fokker-Planck
equation (\ref{FPE}) can be written in terms of an eigenfunction
expansion
\begin{equation}
p(\mb{x},t) = \sum_{j=0}^\infty a_j e^{-\lambda_j t} \varphi_j(\mb{x})
\end{equation}
where $-\lambda_j$ are the eigenvalues of the FP operator, with
$\lambda_0 = 0 < \lambda_1\leq \lambda_2\leq\ldots$,
$\varphi_j(\mb{x})$ are their corresponding eigenfunctions, and the
coefficients $a_j$ depend on the initial conditions. Obviously,
the long term behavior of the system is governed only by the first
few eigenfunctions $\varphi_0,\varphi_1,\ldots,\varphi_k$, where
$k$ is typically small and depends on the time scale of interest.
In low dimensions, e.g. $n\leq 3$ for example, it is possible to
calculate approximations to these eigenfunctions via numerical
solutions of the relevant partial differential equations. In high
dimensions, however, this approach is in general infeasible and
one typically resorts to simulations of trajectories of the
corresponding SDE (\ref{SDE}). In this case, there is a need to
employ statistical methods to analyze the simulated trajectories,
identify the slow variables, the meta-stable states, the reaction
pathways connecting them and the mean transition times between
them.
\section{Diffusion Maps}
\subsection{Finite Data}\label{s:discrete}
Let $\{\mb{x}_i\}_{i=1}^N$, denote the $N$ samples, either merged from
many different simulations of the stochastic equation (\ref{SDE}),
or simply given without an underlying dynamical system. In
\cite{Lafon}, Coifman and Lafon suggested the following method,
based on the definition of a Markov chain on the data, for the
analysis of the geometry of general datasets:
For a fixed value of $\varepsilon$ (a metaparameter of the algorithm),
define an isotropic diffusion kernel,
\begin{equation}
k_\varepsilon(\mb{x},\mb{y}) =
\exp\left(-\frac{\norm{\mb{x}-\mb{y}}^2}{2\varepsilon}\right) \label{k_epsilon}
\end{equation}
Assume that the transition probability between points $\mb{x}_i$ and
$\mb{x}_j$ is proportional to $k_\varepsilon(\mb{x}_i,\mb{x}_j)$, and construct an
$N\times N$ Markov matrix, as follows
\begin{equation}
M(i,j) = \frac{k_\varepsilon(\mb{x}_i,\mb{x}_j)}{p_\varepsilon(\mb{x}_j)}
\label{M_discrete}
\end{equation}
where $p_\varepsilon$ is the required normalization constant, given by
\begin{equation}
p_\varepsilon(\mb{x}_j) = \sum_i k_\varepsilon(\mb{x}_i,\mb{x}_j)
\label{p_ve_discrete}
\end{equation}
For large enough values of $\varepsilon$ the Markov matrix $M$ is fully
connected (in the numerical sense) and therefore has an eigenvalue
$\lambda_0=1$ with multiplicity one and a sequence of additional
$n-1$ non-increasing eigenvalues $\lambda_j < 1$, with
corresponding eigenvectors $\varphi_j$.
The diffusion map at time $m$ is defined as the mapping from $\mb{x}$
to the vector
\[
\Phi_m(\mb{x}) = \left(\lambda_0^m
\varphi_0(\mb{x}),\lambda_1^m\varphi_1(\mb{x}),\ldots,\lambda_k^m
\varphi_k(\mb{x})\right) \] for some small value of $k$. In
\cite{Lafon}, it was demonstrated that this mapping gives a low
dimensional parametrization of the geometry and density of the
data. In the field of data analysis, this construction is known as
the {\em normalized graph Laplacian}. In \cite{Malik}, Shi and
Malik suggested using the first non-trivial eigenvector to compute
an approximation to the optimal normalized cut of a graph, while
the first few eigenvectors were suggested by Weiss et al.
\cite{Weiss99,Weiss} for clustering. Similar constructions,
falling under the general term of kernel methods have been used in
the machine learning community for classification and regression
\cite{Kernel}. In this paper we elucidate the connection between
this construction and the underlying potential $U(\mb{x})$.
\subsection{The Continuum Diffusion Process}\label{sec: continuum diffusion process}
To analyze the eigenvalues and eigenvectors of the normalized
graph Laplacian, we consider them as a finite approximation of a
suitably defined diffusion operator acting on the continuum
probability space from which the data was sampled. We thus
consider the limit of the above Markov chain process as the number
of samples approaches infinity. For a finite value of $\varepsilon$, the
Markov chain in discrete time and space converges to a Markov
process in discrete time but continuous space. Then, in the limit
$\varepsilon\to0$, this jump process converges to a diffusion process on
$\mathbb{R}^n$, whose local transition probability depends on the
non-uniform probability measure $\mu(\mb{x}) = e^{-U(\mb{x})}$.
We first consider the case of a fixed $\varepsilon > 0$, and take $N\to\infty$. Using the
similarity of (\ref{k_epsilon}) to the diffusion kernel, we view
$\varepsilon$ as a measure of time and consider a discrete jump process at
time intervals $\Delta t= \varepsilon$, with a transition probability
between points $\mb{y}$ and $\mb{x}$ proportional to $k_\varepsilon(\mb{x},\mb{y})$.
However, since the density of points is not uniform but rather
given by the measure $\mu(\mb{x})$, we define an associated
normalization factor $p_\varepsilon(\mb{y})$ as follows,
\begin{equation}
p_\varepsilon(\mb{y}) = \int k_\varepsilon(\mb{x},\mb{y}) \mu(\mb{x}) d\mb{x}
\label{p_ve}
\end{equation}
and a forward transition probability
\begin{equation}
M_f(\mb{x}|\mb{y}) = \Pr(\mb{x}(t+\varepsilon) = \mb{x}\,|\mb{x}(t)=\mb{y}) =
\frac{k_\varepsilon(\mb{x},\mb{y})}{p_\varepsilon(\mb{y})}
\label{M_f}
\end{equation}
Equations (\ref{p_ve}) and (\ref{M_f}) are the continuous
analogues of the discrete equations (\ref{p_ve_discrete}) and
(\ref{M_discrete}). For future use, we also define a symmetric
kernel $M_s(\mb{x},\mb{y})$ as follows,
\begin{equation}
M_s(\mb{x},\mb{y}) = \frac{k_\varepsilon(\mb{x},\mb{y})}{\sqrt{p_\varepsilon(\mb{x})p_\varepsilon(\mb{y})}}
\label{M_s}
\end{equation}
Note that $p_\varepsilon(\mb{x})$ is an estimate of the local probability
density at $\mb{x}$, computed by averaging the density in a
neighborhood of radius $O(\varepsilon^{1/2})$ around $\mb{x}$. Indeed, as
$\varepsilon\to 0$, we have that
\begin{equation}
p_\varepsilon(\mb{x}) = p(\mb{x}) + \frac\varepsilon{2} \Delta p(\mb{x}) + O(\varepsilon^{3/2})
\end{equation}
We now define forward, backward and symmetric Chapman-Kolmogorov operators
on functions defined on this probability space, as follows,
\begin{equation}
T_f[\varphi](\mb{x}) = \int M_f(\mb{x}|\mb{y}) \varphi(\mb{y}) d\mu(\mb{y})
\end{equation}
\begin{equation}
T_b[\varphi](\mb{x}) = \int M_f(\mb{y}|\mb{x}) \varphi(\mb{y}) d\mu(\mb{y})
\end{equation}
and
\begin{equation}
T_s[\varphi](\mb{x}) = \int M_s(\mb{x},\mb{y}) \varphi(\mb{y}) d\mu(\mb{y})
\end{equation}
If $\varphi(\mb{x})$ is the probability of finding the system at
location $\mb{x}$ at time $t=0$, then $T_f[\varphi]$ is the evolution
of this probability to time $t=\varepsilon$. Similarly, if $\psi(\mb{z})$
is some function on the space, then $T_b[\psi](\mb{x})$ is the mean
(average) value of that function at time $\varepsilon$ for a random walk
that started at $\mb{x}$, and so $T_b^m[\psi](\mb{x})$ is the average
value of the function at time $t=m\varepsilon$.
By definition, the operators $T_f$ and $T_b$ are adjoint under the
inner product with weight $\mu$, while the operator $T_s$ is self
adjoint under this inner product,
\begin{equation}
\langle T_f \varphi , \psi \rangle_{\mu} = \langle \varphi, T_b \psi
\rangle_{\mu},
\quad \quad
\langle T_s \varphi , \psi \rangle_{\mu} = \langle \varphi, T_s \psi
\rangle_{\mu}
\end{equation}
Moreover, since $T_s$ is obtained via conjugation of the kernel
$M_f$ with $\sqrt{p_\varepsilon(\mb{x})}$ all three operators share the same
eigenvalues. The corresponding eigenfunctions can be found via
conjugation by $\sqrt{p_\varepsilon}$. For example, if $T_s\varphi_s =
\lambda \varphi_s$, then the corresponding eigenfunctions for
$T_f$ and $T_b$ are $\varphi_f = \sqrt{p_\varepsilon} \varphi_s$ and
$\varphi_b = \varphi_s/\sqrt{p_\varepsilon}$, respectively. Since
$\sqrt{p_\varepsilon}$ is the first eigenfunction with $\lambda_0 = 1$ of
$T_s$, the steady state of the forward operator is simply
$p_\varepsilon(\mb{x})$, while for the backward operator it is the uniform
density, $\varphi_b=1$.
Obviously, the eigenvalues and eigenvectors of the discrete Markov
chain described in the previous section are discrete
approximations to the eigenvalues and eigenfunctions of these
continuum operators. Rigorous mathematical proofs of this
convergence as $N\to\infty$ under various assumptions have been
recently obtained by several authors \cite{BelkinC,Hein}.
Therefore, for a better understanding of the finite sample case,
we are interested in the properties of the eigenvalues and
eigenfunctions of either one of the operators $T_f,T_b$ or $T_s$,
and how these relate to the measure $\mu(\mb{x})$ and to the
corresponding potential $U(\mb{x})$. To this end, we look for
functions $\varphi(\mb{x})$ such that
\begin{equation}
T_j\varphi = \int M_j(\mb{x},\mb{y}) \varphi(\mb{y}) \Pr\{\mb{y}\} d\mb{y} = \lambda
\varphi(\mb{x})
\label{Tlambda}
\end{equation}
where $j \in\{f,b,s\}$.
While in the case of a finite amount of data, $\varepsilon$ must remain
finite so as to have enough neighbors in a ball of radius
$O(\varepsilon^{1/2})$ near each point $\mb{x}$, as the number of samples goes
to infinity, we can take smaller and smaller values of $\varepsilon$.
Therefore, it is instructive to look at the limit $\varepsilon \to 0$. In
this case, the transition probability densities of the continuous
in space but discrete in time Markov chain converge to those of a
diffusion process, whose time evolution is described by a
differential equation
\[
\frac{\partial \varphi}{\partial t} = {\cal H}_f \varphi
\]
where ${\cal H}_f$ is the infinitesimal generator or propagator of
the forward operator, defined as
\[
{\cal H}_f = \lim_{\varepsilon \to 0}\frac{I - T_f}{\varepsilon}
\]
As shown in the Appendix, by computing the asymptotic expansion of
the corresponding integrals in the limit $\varepsilon\to0$, we obtain that
\begin{equation}
{\cal H}_f \varphi = \Delta \varphi
- \varphi \left(e^U\Delta e^{-U}\right)
\label{H_f}
\end{equation}
Similarly, the inifinitesimal operator of the backward operator is
given by
\begin{equation}
{\cal H}_b \psi = \lim_{\varepsilon \to 0} \frac{T_b-I}{\varepsilon}\psi = \Delta
\psi - 2
\nabla \psi \cdot \nabla U
\label{H_b}
\end{equation}
As expected, $\psi_0=1$ is the eigenfunction with $\lambda_0=0$ of
the backward infinitesimal operator, while $\varphi_0=e^{-U}$ is
the eigenfunction of the forward one.
A few important remarks are due at this point. First, note that
the backward operator (\ref{H_b}) has the same functional form as
the backward FPE (\ref{backward_FPE}), but with a potential $2
U(\mb{x})$ instead of $U(\mb{x})$. The forward operator (\ref{H_f}) has a
different functional form from the forward FPE (\ref{FPE})
corresponding to the stochastic differential equation (\ref{SDE}).
This should come as no surprise, since (\ref{H_f}) is the
differential operator of an isotropic diffusion process on a space
with non-uniform probability measure $\mu(\mb{x})$, which is obviously
different from the standard anisotropic diffusion in a space with
a uniform measure, as described by the SDE (\ref{SDE})
\cite{Gardiner}.
Interestingly, however, the form of the forward operator is
the same as the Schr\"{o}dinger operator of quantum physics
\cite{Singh}, e.g.
\begin{equation}
{\cal H}\varphi = \Delta \varphi - \varphi V(\mb{x}) \label{schrodinger}
\label{QM}
\end{equation}
where in our case the potential $V(\mb{x})$ has the following specific
form
\begin{equation}
V(\mb{x}) =\left(\nabla U(\mb{x})\right)^2 - \Delta U(\mb{x}).
\label{Vx}
\end{equation}
Therefore, in the limit $N \to \infty, \varepsilon\to 0$, the
eigenfunctions of the diffusion map are the same as those of the
Schr\"odinger operator (\ref{schrodinger}) with a potential
(\ref{Vx}). The properties of the first few of these
eigenfunctions have been extensively studied theoretically for
simple potentials $V(\mb{x})$ \cite{Singh}.
In order to see why the forward operator ${\cal H}_f$ also
corresponds to a potential $2U(\mb{x})$ instead of $U(\mb{x})$, we recall
that there is a well known correspondence \cite{Bernstein},
between the Schr\"{o}dinger equation with a sypersymmetric
potential of the specific form (\ref{Vx}) and a diffusion process
described by a Fokker-Planck equation of the standard form
(\ref{FPE}). The transformation
\begin{equation}
p(\mb{x},t) = \psi(\mb{x},t) e^{-U(\mb{x})/2}
\label{transformation}
\end{equation}
applied to the original FPE (\ref{FPE}) yields the Schr\"odinger
equation with imaginary time
\begin{equation}
- \frac{\partial \psi}{\partial t} = \Delta \psi - \psi\left(\frac{(\nabla
U)^2}4 - \frac{\Delta U}2\right)\label{eq:Schrodinger imaginary time}
\end{equation}
Comparing (\ref{eq:Schrodinger imaginary time}) with (\ref{Vx}),
we conclude that the eigenvalues of the operator (\ref{H_f}) are
the same as those of a Fokker-Planck equation with a potential $2
U(\mb{x})$. Therefore, in general, for data sampled from the SDE
(\ref{SDE}), there is no direct correspondence between the
eigenvalues and eigenfunctions of the normalized graph Laplacian
and those of the corresponding Fokker-Planck equation (\ref{FPE}).
However, when the original potential $U(\mb{x})$ has two metastable
states separated by a large barrier, corresponding to two well
separated clusters, so does $2U(\mb{x})$. Therefore, the first
non-trivial eigenvalue is governed by the mean passage time
between the two barriers, and the first non-trivial eigenfunction
gives a parametrization of the path between them (see also the
analysis of the next section).
We note that in \cite{Horn}, Horn and Gottlieb suggested a
clustering algorithm based on the Schr\"{o}dinger operator
(\ref{QM}), where they constructed an approximate eigenfunction
$\psi(\mb{x}) = p_\varepsilon(\mb{x})$ as in our eq. \ref{p_ve_discrete}), and
computed its corresponding potential $V(\mb{x})$ from eq. (\ref{QM}).
The clusters were then defined by the minima of the potential $V$.
Employing the same asymptotic analysis of this paper, one can show
that in the appropriate limit, the computed potential $V$ is given
by (\ref{Vx}). This asymptotic analysis and the connection between
the quantum operator and a diffusion process thus provides a
mathematical explanation for the success of their method. Indeed,
when $U$ has a deep parabolic minima at a point $\mb{x}$,
corresponding to a well defined cluster, so does $V$.
\section{Anisotropic Diffusion Maps}\label{ref: anisotropic diffusion maps}
In the previous section we showed that asymptotically, the
eigenvalues and eigenfunctions of the normalized graph Laplacian
operator correspond to the Fokker-Planck equation with a potential
$2U(\mb{x})$ instead of the single $U(\mb{x})$. In this section we present
a different normalization that yields infinitesimal generators
corresponding to the potential $U(\mb{x})$ without the additional
factor of two.
In fact, following \cite{Lafon}
we consider in more generality a whole family of
different normalizations and their corresponding diffusions, and
we show that, in addition to containing the graph Laplacian
normalization used in the previous section, this collection of
diffusions includes at least two other Laplacians of interest: the
Laplace-Beltrami operator, which captures the Riemannian geometry
of the data set, and the backward Fokker-Planck operator of
equation (\ref{backward_FPE}).
Instead of applying the graph Laplacian normalization to the
isotropic kernel $k_\varepsilon(\mb{x},\mb{y})$, we first appropriately
renormalize the kernel into an anisotropic one to obtain a new
weighted graph, and then apply the graph Laplacian normalization
to this graph. More precisely, we proceed as follows: start with a
Gaussian kernel $k_\varepsilon(\mb{x},\mb{y})$ and let $\alpha>0$ be a
parameter indexing our family of diffusions. Define an estimate
for the local density as
\[
p_\varepsilon(\mb{x})=\int k_\varepsilon(\mb{x},\mb{y}) \Pr\{\mb{y}\}d\mb{y}
\]
and consider the family of kernels
\[
k^{(\alpha)}_\varepsilon(\mb{x},\mb{y})=\frac{k_\varepsilon(\mb{x},\mb{y})}{p_\varepsilon^\alpha(\mb{x})p_\varepsilon^\alpha(\mb{y})}
\]
We now apply the graph Laplacian normalization by computing the
normalization factor
\[
d_\varepsilon^{(\alpha)}(\mb{y})=\int
k^{(\alpha)}_\varepsilon(\mb{x},\mb{y})\Pr\{\mb{x}\}d\mb{x}
\]
and forming a forward transition probability kernel
\[
M_f^{(\alpha)}(\mb{x}|\mb{y})=\Pr\{\mb{x}(t+\varepsilon)=\mb{x}|\mb{x}(t)=\mb{y}\}=\frac{k_\varepsilon^{(\alpha)}(\mb{x},\mb{y})}{d_\varepsilon^{(\alpha)}(\mb{y})}
\]
Similar to the analysis of section \ref{sec: continuum diffusion
process}, we can construct the corresponding forward, symmetric
and backward diffusion kernels. It can be shown (see appendix
\ref{infinitesimal computations}) that the forward and backward
infinitesimal generators of this diffusion are
\begin{eqnarray}
\mathcal H_b^{(\alpha)}\psi &=& \Delta \psi -
2(1-\alpha)\nabla\phi\cdot \nabla U \\
\mathcal H_f^{(\alpha)} \varphi&=&\Delta
\varphi-2\alpha \nabla \varphi \cdot \nabla U + (2\alpha-1)
\varphi \left((\nabla U)^2 - \Delta U\right)
\end{eqnarray}
We mention three interesting cases:
\begin{itemize}
\item For $\alpha=0$, this construction yields the classical
normalized graph Laplacian with the infinitesimal operator given
by equation (\ref{H_f})
\[
\mathcal H_f \varphi=\Delta \varphi-\left(e^{U}\Delta
e^{-U}\right)\varphi
\]
\item For $\alpha=1$, the backward generator gives the
Laplace-Beltrami operator:
\begin{equation}
\mathcal H_b\psi=\Delta \psi
\end{equation}
In other words, this diffusion captures only the geometry of the
data, in which the density $e^{-U}$ plays absolutely no role.
Therefore, this normalization separates the geometry of the
underlying manifold from the statistics on it.
\item For $\alpha=\frac 1 2$, the infinitesimal operator of the
forward and backward operators coincide and are given by
\begin{equation}
\mathcal H_f \varphi= \mathcal H_b \varphi = \Delta \varphi-
\nabla \varphi \cdot \nabla U
\end{equation}
which is exactly the backward FPE (\ref{backward_FPE}), with a
potential $U(\mb{x})$.
\end{itemize}
Therefore, the last case with $\alpha=1/2$ provides a consistent
method to approximate the eigenvalues and eigenfunctions
corresponding to the stochastic differential equation (\ref{SDE}).
This is done by constructing a graph Laplacian with an
appropriately anisotropic weighted graph.
As explained in \cite{Lafon,Saerens,new}, the Euclidian distance
between any two points after the diffusion map embedding into
$\mathbb{R}^k$ is almost equal to their diffusion distance on the
original dataset. Thus, for dynamical systems with only one or two
slow time scales, and many fast time scales, only a small number
of diffusion map coordinates need be retained for the coarse
grained representation of the data at medium to long times, at
which the fast coordinates have equilibrated. Therefore, the
diffusion map can be considered as an empirical method to perform
data-driven or equation-free homogenization. In particular, since
this observation yields a computational method for the
approximation of the top eigenfunctions and eigenvalues, this
method can be applied towards the design of fast and efficient
simulations that can be initialized on arbitrary points on the
diffusion map. This application will be described in a separate
publication \cite{new}.
\section{Examples}
In this section we present the potential strength of the diffusion
map method by analyzing, both analytically and numerically a few
toy examples, with simple potentials $U(\mb{x})$. More complicated
high dimensional examples of stochastic dynamical systems are
analyzed in \cite{new}, while other applications such as the
analysis of images for which we typically have no underlying
probability model appear in \cite{Lafon}.
\subsection{Parabolic potential in 1-D}
We start with the simplest case of a parabolic potential in one
dimension, which in the context of the SDE (\ref{SDE}),
corresponds to the well known Ornstein-Uhlenbeck process. We thus
consider a potential $U(x) = x^2 /2 \tau$, with a corresponding
normalized density $p = e^{-U}/\sqrt{2\pi\tau}$.
The normalization factor $p_\varepsilon$ can be computed explicitly
\[
p_\varepsilon(y) = \int \frac{e^{-(x-y)^2/2\varepsilon}}{\sqrt{2\pi\varepsilon}}
\frac{e^{-x^2/2\tau}}{\sqrt{2\pi\tau}}dx = \frac{1}{\sqrt{2\pi(\tau + \varepsilon)}}
e^{-y^2/2(\tau+\varepsilon)}
\]
where, for convenience, we multiplied the kernel $k_\varepsilon(x,y)$ by a
normalization factor $1/\sqrt{2\pi\varepsilon}$. Therefore, the
eigenvalue/eigenfunction problem for the symmetric operator $T_s$
with a finite $\varepsilon$ reads
\[
T_s\varphi = \int
\frac{\exp\left(-\frac{(x-y)^2}{2\varepsilon}\right)}{\sqrt{2\pi\varepsilon}}
\exp\left(\frac{x^2+y^2}{4(\varepsilon+\tau)}\right)
\exp\left(-\frac{y^2}{2\tau}\right) \sqrt{\frac{\tau+\varepsilon}{\tau}}
\varphi(y)dy = \lambda \varphi(x)
\]
The first eigenfunction, with eigenvalue $\lambda_0=1$ is given by
\[
\varphi_0(x) = C \sqrt{p_\varepsilon(x)} = C \exp\left(-\frac{x^2}{4(\varepsilon + \tau)}\right)
\]
The second eigenfunction, with eigenvalue $\lambda_1 = \tau/(\tau
+ \varepsilon) < 1$ is, up to normalization
\[
\varphi_1(x) = x \exp\left(-\frac{x^2}{4(\varepsilon + \tau)}\right)
\]
In general, the sequence of eigenfunctions and eigenvalues is
characterized by the following lemma:
\noindent {\bf \em Lemma:} The eigenvalues of the operator $T_s$
are $\lambda_k = \left(\tau/(\tau+\varepsilon)\right)^k$, with the
corresponding eigenvectors given by
\begin{equation}
\varphi_k(x) = p_k(x)
\exp\left(-\frac{x^2}{4(\tau+\varepsilon)}\right)
\end{equation}
where $p_k$ is a polynomial of degree $k$ (even or odd depending
on $k$).
In the limit $\varepsilon\to 0$, we obtain the eigenfunctions of the
corresponding infinitesimal generator. For the specific potential
$U(x)=x^2/2\tau$, the eigenfunction problem for the backward
generator reads
\begin{equation}
\psi_{xx} - 2 \frac{x}{\tau} \psi_x = - \lambda \psi
\end{equation}
The solutions of this eigenfunction problem are, up to scaling of
$x$, the well known Hermite polynomials, which by the
correspondence of this operator to the Schr\"{o}dinger
eigenvector/eigenvalue problem, are also the eigenfunctions of the
quantum harmonic oscillator (after multiplication by the
appropriate Gaussian) \cite{Singh}.
Note that plotting the second vs. the first eigenfunctions (with
the convention that the zeroth eigenfunction is the constant one,
which we typically ignore), is the same as plotting $x^2+1$ vs
$x$, e.g. a parabola. Therefore, we expect that for a large enough
and yet finite data-set sampled from this potential, the plot of
the corresponding discrete eigenfunctions should lay on a
parabolic curve (see next section for a numerical example).
\subsection{Multi-Dimensional Parabolic Potential}
We now consider a harmonic potential in $n$-dimensions, of the
form
\begin{equation}
U(\mb{x}) = \sum_j \frac{x_j^2}{2\tau_j}
\end{equation}
where, in addition, we assume $\tau_1\gg \tau_2 > \tau_3 >\ldots >
\tau_n$, so that $x_1$ is a slow variable in the context of the
SDE (\ref{SDE}).
We note that for this specific potential, the probability density
has a separable structure, $p(\mb{x}) = p_1(x_1)\ldots p_n(x_n)$, and
so does the kernel $k_\varepsilon(\mb{x},\mb{y})$, and consequently, also the
symmetric kernel $M_s(\mb{x},\mb{y})$. Therefore, there is an
outer-product structure to the eigenvalues and eigenfunctions. For
example, in two dimensions the eigenfunctions and eigenvalues are
given by
\begin{equation}
\varphi_{i,j}(x_1,x_2) = \varphi_{1,i}(x_1)\varphi_{2,j}(x_2)\quad
\mbox{and}\quad\lambda_{i,j} = \mu_1^i \mu_2^j
\end{equation}
where $\mu_1 = \tau_1/(\tau_1+\varepsilon)$ and $\mu_2 = \tau_2/(\tau_2
+\varepsilon)$. Since by assumption $\tau_1 \gg \tau_2$, then upon
ordering of the eigenfunctions by decreasing eigenvalue, the first
non-trivial eigenfunctions are
$\varphi_{1,0},\varphi_{2,0},\ldots$, which depend only on the
slow variable $x_1$. Note that indeed, regardless of the value of
$\varepsilon$, as long as $\tau_2 > 2 \tau_1$, we have that $\lambda_1^2 >
\lambda_2$. Therefore, in this example the first few coordinates
of the diffusion map give a (redundant) parametrization of the
slow variable $x_1$ in the system.
In figure \ref{f:u1} we present numerical results corresponding to
a 2-dimensional potential with $\tau_1=1,\tau_2=1/25$. In the
upper left some 3500 points sampled from the distribution
$p=e^{-U}$ are shown. In the lower right and left panels, the
first two non-trivial backward eigenfunctions $\psi_1$ and
$\psi_2$ are plotted vs. the slow variable $x_1$. Note that except
at the edges, where the statistical sampling is poor, the first
eigenfunction is linear in $x_1$, while the second one is
quadratic in $x_1$. In the upper right panel we plot $\psi_2$ vs.
$\psi_1$ and note that they indeed lie on a parabolic curve, as
predicted by the analysis of the previous section.
\begin{figure}[t]
\mbox{
\begin{minipage}[t] {\textwidth}
\begin{center}
\begin{tabular}{c}
\psfig{figure=u1.eps,width=8.0cm}\\
\end{tabular}
\end{center}
\end{minipage}
}
\\
\caption{The anisotropic diffusion map on a harmonic potential in
2-D. } \label{f:u1}
\end{figure}
\subsection{A potential with two minima}
We now consider a double well potential $U(x)$ with two minima,
one at $x_L$ and one at $x_R$. For simplicity of the analysis, we
assume a symmetric potential around $(x_L+x_R)/2$, with
$U(x_L)=U(x_R) = 0$ (see figure \ref{f:u2}). In the context of
data clustering, this can be viewed as approximately a mixture of
two Gaussian clouds, while in the context of stochastic dynamical
systems, this potential defines two meta-stable states.
We first consider an approximation to the quantity $p_\varepsilon(x)$,
given by eq. (\ref{p_ve}). For $x$ near $x_L$, $U(x) \approx
(x-x_L)^2/\tau_L$, while for $x$ near $x_R$, $U(x) \approx
(x-x_R)^2/\tau_R$. Therefore,
\begin{equation}
e^{-U(y)} \approx e^{-(y-x_L)^2/2\tau_L} + e^{-(y-x_R)^2/2\tau_R}
\end{equation}
and
\begin{eqnarray}
p_\varepsilon(x) &\approx& \frac{1}{\sqrt{2}} \left(\frac{\sqrt{\tau_L}}{\sqrt{\tau_L+\varepsilon}}e^{-(x-x_L)^2/2(\tau_L + \varepsilon)}
+\frac{\sqrt{\tau_R}}{\sqrt{\tau_R+\varepsilon}}e^{-(x-x_R)^2/2(\tau_R+
\varepsilon)}
\right) \nonumber \\
&=& \frac1{\sqrt{2}} \left[\varphi_L(x) + \varphi_R(x)\right]
\end{eqnarray}
where $\varphi_L$ and $\varphi_R$ are the first forward
eigenfunctions corresponding to a single well potential centered
at $x_L$ or at $x_R$, respectively. As is well known both in the
theory of quantum physics and in the theory of the Fokker-Planck
equation, an approximate expression for the next eigenfunction is
\[
\varphi_1(x) = \frac1{\sqrt{2}} \left[\varphi_L(x) -
\varphi_R(x)\right]
\]
Therefore, the first non-trivial eigenfunction of the backward
operator is given by
\[
\psi_1(x) = \frac{\varphi_L(x) - \varphi_R(x)}{\varphi_L(x) +
\varphi_R(x)}
\]
This eigenfunction is roughly $+1$ in one well and $-1$ in the
other well, with a sharp transition between the two values near
the barrier between the two wells. Therefore, this eigenfunction
is indeed suited for clustering. Moreover, in the context of a
mixture of two Gaussian clouds, clustering according to the sign
of $\psi_1(x)$ is asymptotically equivalent to the optimal Bayes
classifier.
\noindent {\bf Example:} Consider the following potential in two
dimensions,
\begin{equation}
U(x,y) = \frac1{4}\,x^4-\frac{25}{12}x^3+\frac9{2}x^2 + 25
\frac{y^2}2
\end{equation}
In the $x$ direction, this potential has a double well shape with
two minima, one at $x=0$ and one at $x=4$, separated by a
potential barrier with a maximum at $x=2.25$.
In figure \ref{f:u2} we show some numerical results of the
diffusion map on some 1200 points sub-sampled from a stochastic
simulation with this potential which generated about 40,000
points. On the upper right panel we see the potential $U(x,0)$,
showing the two wells. In the upper left, a scatter plot of all
the points, color coded according to the value of the local
estimated density $p_\varepsilon$, (with $\varepsilon=0.25$) is shown, where the
two clusters are easily observed. In the lower left panel, the
first non-trivial eigenfunction is plotted vs. the first
coordinate $x$. Note that even though there is quite a bit of
variation in the $y$-variable inside each of the wells, the first
eigenfunction $\psi_1$ is essentially a function of only $x$,
regardless of the value of $y$. In the lower right we plot the
first three backward eigenfunctions. Note that they all lie on a
curve, indicating that the long time asymptotics are governed by
the passage time between the two wells and not by the local
fluctuations inside them.
\begin{figure}[t]
\mbox{
\begin{minipage}[t] {\textwidth}
\begin{center}
\begin{tabular}{c}
\psfig{figure=u2.eps,width=8.0cm}\\
\end{tabular}
\end{center}
\end{minipage}
}
\\
\caption{Numerical results for a double well potential in 2-D. }
\label{f:u2}
\end{figure}
\subsection{Potential with three wells}
We now consider the following two dimensional potential energy
with three wells,
\begin{equation}
U(x,y) = 3\beta e^{-x^2}\left[e^{-(y-1/3)^2} - e^{-(y-5/3)^2}\right]
-5\beta
e^{-y^2}
\left[e^{-(x-1)^2} + e^{-(x+1)^2}\right]
\end{equation}
where $\beta=1/kT$ is a thermal factor. This potential has two
deep wells at $(-1,0)$ and at $(1,0)$ and a shallower well at
$(0,5/3)$, which we denote as the points $L,R,C$, respectively,
The transitions between the wells of this potential have been
analyzed in many works \cite{Schulten}. In figure
\ref{f:three_wells} we plotted on the left the results of 1400
points sub-sampled from a total of 80000 points randomly generated
from this potential confined to the region $[-2.5,2.5]^2\subset
\mathbb{R}^2$ at temperature $\beta=2$, color-coded by their local
density. On the right we plotted the first two diffusion map
coordinates $\psi_1(\mb{x}),\psi_2(\mb{x})$. Notice how in the diffusion
map space one can clearly see a triangle where each vertex
corresponds to one of the points $L,R,C$. This figure shows very
clearly that there are two possible pathways to go from $L$ to
$R$. A direct (short) way and an indirect longer way, that passes
through the shallow well centered at $C$.
\begin{figure}[t]
\mbox{
\begin{minipage}[t] {\textwidth}
\begin{center}
\begin{tabular}{c}
\psfig{figure=three_well.eps,width=8.0cm}\\
\end{tabular}
\end{center}
\end{minipage}
}
\\
\caption{Numerical results for a triple well potential in 2-D. }
\label{f:three_wells}
\end{figure}
\subsection{Iris data set}
We conclude this section with a diffusion map analysis of one of
the most popular multivariate datasets in pattern recognition, the
iris data set. This set contains 3 distinct classes of samples in
four dimensions, with 50 samples in each class. In figure
\ref{f:iris} we see on the left the result of the three
dimensional diffusion map on this dataset. This picture clearly
shows that all 50 points of class 1 (blue) are shrunk into a
single point in the diffusion map space and can thus be easily
distinguished from classes two and three (red and green). In the
right plot we see the results of re-running the diffusion map on
the 100 remaining red and green samples. The 2-D plot of the first
two diffusion maps coordinates shows that there is no perfect
separation between these two classes. However, clustering
according to the sign of $\psi_1(\mb{x})$ gives misclassifications
rates similar to those of other methods, of the order of 6-8
samples depending on the value chosen for the kernel width $\varepsilon$.
\begin{figure}[t]
\mbox{
\begin{minipage}[t] {\textwidth}
\begin{center}
\begin{tabular}{c}
\psfig{figure=iris.eps,width=8.0cm}\\
\end{tabular}
\end{center}
\end{minipage}
}
\\
\caption{Diffusion map for the iris data set. } \label{f:iris}
\end{figure}
\section{Summary and Discussion}
In this paper, we introduced a mathematical framework for the
analysis of diffusion maps, via their corresponding infinitesimal
generators. Our results show that diffusion maps are a natural
method for the analysis of the geometry and probability
distribution of empirical data sets. The identification of the
eigenvectors of the Markov chain as discrete approximations to the
corresponding differential operators provides a mathematical
justification for their use as a dimensional reduction tool and
gives an alternative explanation for their empirical success in
various data analysis applications, such as spectral clustering
and approximations of optimal normalized cuts on discrete graphs.
We generalized the standard construction of the normalized graph
Laplacian to a one-parameter family of graph Laplacians that
provides a low-dimensional description of the data combining the
geometry of the set with the probability distribution of the data
points. The choice of the diffusion map depends on the task at
hand. If, for example, data points are known to approximately lie
on a manifold, and one is solely interested in recovering the
geometry of this set, then an appropriate normalization of a
Gaussian kernel allows to approximate the Laplace-Beltrami
operator, regardless of the density of the data points. This
construction achieves a complete separation of the underlying
geometry, represented by the knowledge of the Laplace operator,
from the statistics of the points. This is important in situations
where the density is meaningless, and yet points on the manifold
are not sampled uniformly on it. In a different scenario, if the
data points are known to be sampled from the equilibrium
distribution of a Fokker-Planck equation, the long-time dynamics
of the density of points can be recovered from an appropriately
normalized random walk process. In this case, there is a subtle
interaction between the distribution of the points and the
geometry of the data set, and one must correctly account for the
density of the points.
While in this paper we analyzed only Gaussian kernels, our
asymptotic results are valid for general kernels, with the
appropriate modification that take into account the mean and
covariance matrix of the kernel. Note, however, that although
asymptotically in the limit $N\to\infty$ and $\varepsilon\to 0$, the
choice of the isotropic kernel is unimportant, for a finite data set the
choice of both $\varepsilon$ and the kernel can be crucial for the success
of the method.
Finally, in the context of dynamical systems, we showed that
diffusion maps with the appropriate normalization constitute a
powerful tool for the analysis of systems exhibiting different
time scales. In particular, as shown in the different examples,
these time scales can be separated and the long time dynamics can
be characterized by the top eigenfunctions of the diffusion
operator. Last, our analysis paves the way for fast simulations of
physical systems by allowing larger integration steps along slow
variable directions. The exact details required for the design of
fast and efficient simulations based on diffusion maps will be
described in a separate publication \cite{new}.
\noindent{\bf Acknowledgments:} The authors would like to thank
the referee for helpful suggestions and for pointing out ref.
\cite{Horn}.
|
{
"timestamp": "2005-03-22T07:46:24",
"yymm": "0503",
"arxiv_id": "math/0503445",
"language": "en",
"url": "https://arxiv.org/abs/math/0503445"
}
|
\section{Introduction}
The use and development of ion trapping techniques, which started
about 50 years ago \cite{cite1}, have led to a broad range of
discoveries and new experiments in physics and chemistry. In
particular, one can cite high precision spectroscopy, mass
measurements, particle dynamics, nuclear and atomic processes and
the measurement of fundamental constants \cite{ionTrapping}.
During the last few years, a new type of ion trap has been
developed in which ion beams, instead of ion clouds, are
trapped\cite{EIBT1,EIBT2}. This new trap stores particles using
only electrostatic fields and works on a principle similar to that
of an optical resonator. The main advantages of the trap are the
possibility to trap fast (keV) beams without need of deceleration,
the well defined beam direction, easy access to the trapped beam
by various probes, and simple requirements in terms of external
beam injection. Different types of experiments have already been
performed with these traps, such as the measurement of metastable
state lifetimes of atomic and molecular ions \cite{meta1,meta2},
the lifetimes of metastable negative ions \cite{life1,life2}, and
electron impact detachment cross sections of negative clusters
\cite{adi}. Cluster cooling has also been observed \cite{yoni}.
Interesting dynamics of the ion motion have been discovered, such
as self-bunching (due to the so-called negative mass instability
phenomenon) and the possibility of using simple phase space
manipulation to reduce the velocity spread \cite{bunch1,sarah}.
Electrostatic ion storage rings\cite{Moller:97,Moller:01} have
also been used during the last several years in a variety of
experiments\cite{He-:01,Hansen:01}.
Although the motion of the ions in the trap can be readily
simulated, no measurements of the transverse velocity
distribution (TVD) of the stored beam have hitherto been performed. The TVD is needed to
understand the trapping efficiency, as well as the beam loss
processes, especially the ones related to multiple scattering. We describe here the method
that we have developed to
characterize the TVD of the stored
ions. The results are compared to numerical trajectory simulations,
which confirm that multiple scattering is the dominant loss
process in these traps, and that the available area of the stable
transverse phase space directly influences the lifetime of the trapped ion
beam.
\section{Experimental setup}
\subsection{Ion trap}
Figure \ref{expsetup} shows a schematic view of the
electrostatic ion trap and the detection system. Two different
setups were used for creating the ions. For light ions, an
electron impact ionization source was used and the ions were mass
selected with two magnets. For heavier species, a
matrix assisted laser desorption and ionization
(MALDI)\cite{maldi} source was used to create an ion
bunch, which was mass selected using time of flight. In
both cases, the ions were accelerated to an energy of 4.2 keV.
Three different types of ions were used in this work: Au$^+$
(m=197) and singly charged angiotensin II (m=1046) (both
produced by the MALDI source), and Ar$^+$ (m=40) (produced by the
electron impact source). After focusing and collimation, the beam
is directed into the ion trap along its axis. A complete
description of the ion trap is given in Ref. \cite{EIBT2}, and only
the details relevant for the present experiment will be given
here.
The trap is made of two identical cylindrically symmetric
``electrostatic mirrors'' that both trap the beam in the
longitudinal direction and focus it in the lateral direction. Upon
injection, the entrance set of electrodes (left side in
Fig.~\ref{expsetup}) is grounded so that the ion bunch can reach
the exit mirror (right hand side in Fig.~\ref{expsetup}), where
they are reflected. Before the reflected bunch reaches the
entrance electrodes, the potentials of these electrodes are
rapidly switched on ($\sim$ 100 ns rise time) to the same values
as those of the exit electrodes. For proper choices of voltages,
the ions bounce back and forth between the two mirrors, their
lifetime being limited mainly by collisions with the residual gas
molecules. The low pressure in the trap, of the order of
5$\times$10$^{-10}$ Torr when the electron impact source was
used, and 4$\times$10$^{-11}$ Torr for the MALDI setup, is
maintained by a cryopump.
Each electrostatic mirror comprises eight electrodes. The
potentials of the electrodes labeled $V_1$ to $V_4$ and $V_z$ in Fig.
\ref{expsetup} are independently adjustable. The other
electrodes are always grounded. Thus the 228 mm long central region of the
trap between the two innermost electrodes is essentially field-free.
The diameter of the central hole is
16 mm in the outer six electrodes and 26 mm in the two innermost
electrodes. The distance between the outermost electrodes is 407
mm.
In order for the ions to be trapped, the electrode potentials have
to satisfy certain conditions. It is well known that many
principles of geometric optics can be applied to ion optics. In
fact, our trap is based on a optical resonator made of two
cylindrically symmetric mirrors \cite{Pedersen1}. For an optical
resonator with identical mirrors and a Gaussian beam, the
stability criterion (for a beam close to the symmetry axis) is
related to the focusing properties of the mirrors:
\begin{equation}
L/4 \leq f \leq \infty, \label{eq:stable}
\end{equation}
where $f$ is the focal length of each mirror and $L$ is the
distance between them. This condition is easy to fulfill with the
above design. Another obvious requirement is that the maximum
potential on the mirror axis, $V_{max}$, has to be high enough to
reflect the ions , i.e., $qV_{max} > E_k$, where $q$ is the charge
of the ions and $E_k$ is their kinetic energy.
Some important aspects of the design should be emphasized.
First, the trap is completely electrostatic, so there is no
limit on the mass that can be trapped. Second, the
trapping depends only on the ratio ${E_k}/{q}$, which means
that ions of different mass that are accelerated through the same potential difference can
be stored simultaneously. Third, the central part of the ion trap is (nearly)
field-free, so the ions travel in straight lines in this
region.
Various electrode voltage configurations are possible to achieve
trapping. We define a particular configuration by the set of
potentials $\{V_1,V_2,V_3,V_4,V_z\}$. $V_z$ is connected to the
central electrode of an Einzel lens that plays the major role in
determining the focusing properties of the mirrors. In the present
work, only symmetric configurations, i.e., where identical
potentials are applied to the two mirrors, are considered. The
potentials on the four outmost electrodes were set to
$\{V_1,V_2,V_3,V_4\}$=\{6.5, 4.875, 3.25, 1.65\} kV, while the
Einzel voltage was varied between $2700<V_z<3200$ V and
$4000<V_z<4300$ V. These two ranges correspond to the known values
where the trap is stable, i.e., they satisfy the criterion
Eq.~\ref{eq:stable}, as has been shown in Ref.~\cite{Pedersen1}.
Additional details about trapping stability and the comparison to
optical models can be found in the literature\cite{Pedersen1}.
\subsection{Detection system}
One of the ion loss processes from the trap is charge
exchange, which leads to neutralization of the stored
particles. These neutral particles pass freely through the mirrors and can be
detected by a microchannel plate
(MCP) detector located downstream of the trap (see
Fig. \ref{expsetup}). The detector is coupled to a phosphor
screen so that the spatial
distribution of the neutral particles exiting the trap can be imaged. The
location and size of the MCP was different for the two
different ion source setups used in this work: The MCP
was 25 mm in diameter, and located at a distance of 80.3 cm from
the center of the trap for the MALDI setup, while for the electron
impact ionization source the MCP was 40 mm in diameter, and
located at a distance of 90.3 cm from the center of the trap. The
imaging is performed by a charge-coupled device (CCD) camera
located outside the vacuum that is connected to a frame
grabber which digitizes the picture in real time. The first image
is taken in coincidence with the raising of the potentials on the entrance mirror, and subsequent
images are digitized at a rate of 25 Hz for the whole trapping
time ($\sim1$ s). The positions of impact (($x,y$) on the front surface of the MCP)
are determined for all hits producing an amount of
light (as measured by the CCD camera) above a preprogrammed
threshold. Images of about 50 to 150 injections are averaged to
produce statistically significant results. The radial coordinate
$r$ is calculated as
\begin{equation}
r=\sqrt{(x-x_0)^2+(y-y_0)^2} \label{eq:radial}
\end{equation}
where ($x_0, y_0$) is the point where the trap axis crosses the
detector plane. This point is determined at a later stage by
finding the center of the measured radial distribution.
\subsection{Data analysis}
In order to study the TVD inside the trap, we analyze the radial
distribution of the neutral particles hitting the MCP detector.
Fig.~\ref{expsetup} shows the relationship between the ion
position and velocity inside the trap at the instant of its
neutralization, and the point of impact of the neutralized
particle on the detector, $r$. The ion's position at the
neutralization point is given by its distance $R$ from the optical
axis of the trap and distance from the MCP, $s$. The ion's
velocity at the same point is defined in terms of its longitudinal
and transversal velocities $v_\shortparallel$ and $v_\bot$,
respectively (see Fig.~\ref{expsetup}). If we assume that the
angular scattering taking place during the charge exchange is
small compared to the angular dispersion of the beam (a very good
approximation for the heavy ions created in the MALDI
source)\cite{scattering}, then the position of impact on the MCP
can be calculated from
\begin{equation}
r=R+\frac{s v_{\bot}}{v_{\shortparallel}}.
\end{equation}
If we also use the fact that $R$ $\ll$ $r$, then we obtain for the
transverse velocity
\begin{equation}
v_{\bot}=\frac{r {v_{\shortparallel}}}{s}\approx
\frac{r}{s}\sqrt{\frac{2E_k}{m}}, \label{eq:vtrans}
\end{equation}
where $m$ is the particle mass. Two problems arise from this
simple formula: First, the velocities $v_{\shortparallel}$ and
$v_{\bot}$ are not constant in the trap, as the particles are
slowed down and focused (or defocused, see Ref.~\cite{Pedersen1})
inside the mirrors. Second, the exact distance $s$ between the
neutralization point in the trap and the MCP is unknown. The
importance of these two effects, which can smear the radial
distribution measurement, will be treated separately using
numerical simulation, as described in the next Section.
\section{Numerical simulations}
In order to verify the different approximations made in the
derivation of Eq. \ref{eq:vtrans}, and to provide a better
understanding of the trap behavior, we have performed numerical
simulations of the particle trajectories in the actual potentials of
the ion trap. The calculations were carried out using SIMION
\cite{SIMION}, which can solve the Laplace equation for a specific
potential configuration in space and propagate ions on the
computed potential grid. The program uses a fourth-order
Runge-Kutta method to solve the Newtonian equations of motion. The
density of ions in the trap is assumed to be low enough for
ion-ion interactions to be neglected, and the trajectories are
calculated for one ion at a time (the actual number of ions in the
trap was of the order of 10$^5$ ions per injection).
For different values of $V_z$, while keeping the other potentials
constant, we have traced the stable trajectories, starting from an
initial distribution that covers the whole transverse stable
(i.e., trapped) phase space of the electrostatic trap, as
described in Ref.~\cite{Pedersen1}. The stable phase space was
found by systematically varying the initial conditions of the
particles. A stable trajectory was defined as one for which a
propagated ion was trapped for more than 500 $\mu$s (about 200
oscillations for 4.2 keV Ar$^+$, or 90 for 4.2 keV Au$^+$). It was
found that ions in unstable trajectories were usually lost from
the trap after a few oscillations ($<20\;\mu$s). The calculations
were made using a constant integration time step, and the
positions and velocities of the ions were recorded in a file at
each of these time steps. Using this information, simulated
distributions for the radial distribution at the MCP were
calculated by assuming an equal probability for neutralization at
each of these integration time steps, and propagating the
(neutral) particles in straight lines, using the initial positions
and velocities as recorded. This method has the advantage of
representing faithfully the local ion density along the length of
the trap. Implicit in the assumption of equal probability of
neutralization in each time step is the assumption that the
neutralization cross section is independent of kinetic energy for
energies below 4.2 keV\cite{neutralization}, even in the mirrors
where the kinetic energies approach zero. The results can then be
directly compared to the experimental distributions.
\section{Experimental and Simulation Results}
Figure \ref{fig:r2hist} shows a comparison between the measured
(dotted line) and simulated (solid line) normalized distributions
for the distance squared ($P(r^2)$) at the MCP for 4.2 keV Au$^+$,
and $V_z$=3200 V. We choose to plot the $r^{2}$ distributions as
they display the radial density information in the most relevant
manner. The number of particles located in an interval of width
$d(r^2)=2rdr$ is proportional to the number of particles in the
ring between $r$ and $r+dr$, whose area is given by $2\pi rdr$.
Similar distributions were measured for Ar$^+$ and angiotensin
II$^+$ ions. Each of the measured distributions was characterized
by the standard deviation of the radial distribution which in the
present case is equal to the square root of the mean of r$^2$:
$\sigma_r=\sqrt{<r^2>}$. Using Eq.~\ref{eq:vtrans}, and replacing
$r$ by $\sigma_r$ and $s$ by the distance from the center of the
trap to the MCP, typical transverse velocities $v_{\bot}$ could be
obtained for the different masses and values of the Einzel lens voltage ($V_z$).
Figure ~\ref{fig:vtransexp} shows the results for the three different
ions as a function of $V_z$. Only a weak
dependence of transverse velocity $v_\bot$ on $V_z$ is observed,
except for Ar$^+$ around 3250 V. The ratio
$v_{\bot}/v_{\shortparallel}\approx$ 9$\times$10$^{-3}$ is found
to be approximately constant for all ions.
Based on the excellent agreement between the experimental data and
the simulations (see Fig.~\ref{fig:r2hist}), one can now use the
simulation to check the assumptions which led to
Eq.~\ref{eq:vtrans}, especially the assumption related to the
contribution of the neutral particles produced inside the mirrors
to $P(r^2)$ and the unknown distance between the neutralization
point and the MCP. Figure~\ref{fig:vtranssim} shows an example of
the distribution of the square of the transverse velocity
$P(v_{\bot}^2)$ from Simion simulations. The case presented is for
all stable Ar$^+$ ions in the field-free region of the trap, for
an Einzel lens voltage of $V_z$=3300 V. To compare this
distribution to the experimentally deduced typical transverse
velocity, we characterize this distribution in a similar way as
the squared radial distribution (see Fig.~\ref{fig:r2hist}), using
the square root of its mean, which is equivalent to the standard
deviation of the TVD, yielding $\sigma_{v{\bot}}$=1.16 mm/$\mu$s.
This value is slightly lower than the one derived directly from
the measured radial distributions (see Fig.~\ref{fig:vtransexp}),
as can be expected since the latter includes some contributions
from slower ions inside the mirrors that tend to have also larger
$v_\bot/v_\shortparallel$ ratio. However, the difference is
relatively small (the reduced detection efficiency of the MCP for
the slower particles also contributes to the fact that these have
a minor effect on the measured distributions), and we conclude
that the data shown in Fig.~\ref{fig:vtransexp} are an upper limit
of the transverse velocity of the trapped ions in the central
(field-free) region of the trap.
\section{Transverse phase space}
The results obtained in the previous section show that the
measured and simulated transverse velocities are in good
agreement. Since the simulated value is obtained by filling the
stable phase space of the trap, one can conclude that the ions
stored in the trap always fill the available (stable)
transverse phase space. This has an important implication as far
as the ion loss processes are concerned. As pointed out
previously \cite{Pedersen1}, two processes play an important
role in limiting the lifetime of the ions in the trap. The first
is neutralization of the ions via charge exchange
with the residual gas molecules, and the second is
multiple scattering, which increases the transverse velocity of the
ions until they reach the limit of the stable transverse phase
space. Although the importance of the neutralization process can
be observed experimentally by counting the number of neutral
particles exiting the trap, the importance of multiple
scattering is more difficult to observe experimentally. Also, it
is quite clear that the loss due to neutralization is independent of
the trap configuration, while the loss due to multiple scattering
will be strongly dependent on the available transverse phase
space, if the stable transverse phase space is always full. The
fact that the measured transverse velocity is found to be equal to
the one extracted from a simulated "full transverse phase space"
suggests that the loss due to multiple scattering is very
important, and that the lifetime of the ions is mostly limited by
this process, a conclusion which was already reached by
Pedersen et al. \cite{Pedersen1} using arguments based on known angular
scattering cross sections.
In order to demonstrate the importance of multiple scattering as a
loss process, we have calculated the area of the stable transverse
phase space for the Ar$^+$ ions as a function of the Einzel lens
voltage $V_z$. The area was calculated by recording the transverse
position and velocity for each pass of a stable ion through the
midplane of the trap. A scatter plot of these coordinates was then
made, and the area filled by the points was estimated by dividing
the phase space into a fine grid and counting the number of cells
for which at least four points were found. These cells are then
called "stable cells" (the minimum number of points required for a
cell to be defined as stable has only a small influence on the
final results). Fig.~\ref{fig:lifetime}(a) shows the results of
such a calculation as a function of V$_z$, while
Fig.~\ref{fig:lifetime}(b) shows the lifetime of the ions in the
trap, as obtained by measuring the rate of neutral Ar
hitting the MCP as a function of storage time and fitting the
decay using an exponential function. A clear correspondence
between the lifetime and the area of the transverse phase space is
observed, including the dip around V$_z$=3200 V.
\section{Conclusions}
We have measured the transverse velocity distribution of 4.2 keV
Ar$^+$, Au$^+$, and angiotensin II$^+$ stored in a linear
electrostatic ion trap. The results show that the width of the TVD
is mass dependent, and represents about 1\% of the longitudinal
beam velocity for the present trap geometry. The experimental
results are in excellent agreement with the numerical simulation.
More important, it shows that in the existing experimental setup,
the phase space of the trap is filled very soon after injection.
Thus, we can expect multiple scattering to be an important ion
loss process (the other being neutralization). This is also
demonstrated by the correlation between the area of transverse
phase space and the measured lifetimes.
A consequence of our results is that the lifetime in the trap will
also be a function of the trap length. Indeed, for a given angular
dispersion of the beam and for a given mirror geometry, a trap
with longer distance between the mirrors will be less stable, as
the particles will be further away from the central axis of the
trap when they enter the mirrors. On the other hand, although
shorter traps will probably be more stable, they can store less
ions.
The tool that we have developed to measure the transverse velocity
distribution of the stored ions can now be used in studies of
transverse cooling. Specifically, if a kicker for stochastic
cooling is installed in the field-free region of the trap, it
should be possible to shrink the $r^2$ distribution, and thereby
increase the storage lifetime. Moreover, because of Coulomb
repulsion between the ions (especially near the turning points in
the mirrors) and the radial mixing induced by the mirror, we can
expect that transverse cooling will also affect the longitudinal
velocity distribution.
This work was supported in part by the Israel Science Foundation.
Laboratoire Kastler Brossel is Unit{\'e} Mixte de Recherche du CNRS no. 8552,
of the Physics D{\'e}partement of Ecole Normale Sup{\'e}rieure and Universit{\'e} Pierre
et Marie Curie.
|
{
"timestamp": "2005-03-14T21:35:30",
"yymm": "0503",
"arxiv_id": "physics/0503117",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503117"
}
|
\section{Introduction}
\subsection{Harmonic analysis of boolean functions}
\label{sec:intro} The motivation for this paper is the study of
\emph{boolean functions} $f : \{-1,1\}^n \to \{-1,1\}$, where $\{-1,1\}^n$
is equipped with the uniform probability measure. This topic is of
significant interest in theoretical computer science; it also
arises in other diverse areas of mathematics including
combinatorics (e.g., sizes of set systems, additive
combinatorics), economics (e.g., social choice), metric spaces
(e.g., non-embeddability of metrics), geometry in Gaussian space
(e.g., isoperimetric inequalities), and
statistical physics (e.g., percolation, spin glasses).\\
Beginning with Kahn, Kalai, and Linial's landmark paper ``The
Influence Of Variables On Boolean Functions''~\cite{KaKaLi:88}
there has been much success in analyzing questions about boolean
functions using methods of harmonic analysis. Recall that KKL
essentially shows the following (see
also~\cite{Talagrand:94,FriedgutKalai:96}):
\paragraph{KKL Theorem:} If $f : \{-1,1\}^n
\to \{-1,1\}$ satisfies ${\bf E}[f] = 0$ and $\mathrm{Inf}_i(f) \leq \tau$ for all
$i$, then $\sum_{i=1}^n \mathrm{Inf}_i(f) \geq \Omega(\log(1/\tau))$.\\
\noindent We have used here the notation $\mathrm{Inf}_i(f)$ for the
\emph{influence of the $i$th coordinate on $f$},
\begin{equation} \label{eqn:influence}
\mathrm{Inf}_i(f) = \mathop{\bf E\/}_x[\mathop{\bf Var\/}_{x_i}[f(x)]] = \sum_{S \ni i}
\hat{f}(S)^2.
\end{equation}
Although an intuitive understanding of the analytic properties of
boolean functions is emerging, results in this area have used
increasingly elaborate methods, combining random restriction
arguments, applications of the Bonami-Beckner inequality, and
classical tools from probability theory. See for
example~\cite{Talagrand:94,Talagrand:96,FriedgutKalai:96,Friedgut:99,Bourgain:99,BeKaSc:99,Bourgain:02,KindlerSafra:u,DiFrKiOD:u}.\\
As in the KKL paper, some of the more refined problems studied in
recent years have involved restricting attention to functions with
low influences~\cite{BeKaSc:99,Bourgain:99,DiFrKiOD:u} (or,
relatedly, ``non-juntas''). There are several reasons for this.
The first is that large-influence functions such as ``dictators''
--- i.e., functions $f(x_1, \dots, x_n) = \pm x_i$ --- frequently trivially
maximize or minimize quantities studied in boolean analysis.
However this tends to obscure the truth about extremal behaviors
among functions that are ``genuinely'' functions of $n$ bits.
Another reason for analyzing only low-influence functions is that
this subclass is often precisely what is interesting or necessary
for applications. In particular, the analysis of low-influence
boolean functions is crucial for proving hardness of approximation
results in theoretical computer science and is also very natural
for the study of social choice. Let us describe these two
settings briefly.\\
In the economic theory of social choice, boolean functions $f :
\{-1,1\}^n \to \{-1,1\}$ often represent voting schemes, mapping $n$
votes between two candidates into a winner. In this case, it is
very natural to exclude voting schemes that give any voter an
undue amount of influence; see e.g.~\cite{Kalai:04}. In the study
of hardness of approximation and probabilistically checkable
proofs (PCPs), the sharpest results often involve the following
paradigm: One considers a problem that requires labeling the
vertices of a graph using the label set $[n]$; then one relaxes
this to the problem of labeling the vertices by functions $f :
\{-1,1\}^n \to \{-1,1\}$. In the relaxation one thinks of $f$ as
``weakly labeling'' a vertex by the \emph{set} of coordinates that
have large influence on $f$. It then becomes important to
understand the combinatorial properties of functions that weakly
label with the empty set. There are by now quite a few results in
hardness of approximation that use results on low-influence
functions or require conjectured such results; e.g.,
\cite{DinurSafra:02,Khot:02,DGKO:03,KhotRegev:03,KKMO:04}.\\
In this paper we give a new framework for studying functions on
product probability spaces with low influences. Our main tool is
a simple invariance principle for low-influence polynomials; this
theorem lets us take an optimization problem for functions on one
product space and pass freely to other product spaces, such as
Gaussian space. In these other settings the problem sometimes
becomes simpler to solve. It is interesting to note that while in
the theory of hypercontractivity and isoperimetry it is common to
prove results in the Gaussian setting by proving them first in the
$\{-1,1\}^n$ setting (see, e.g., \cite{Bakry:94}),
here the invariance principle is actually used to go the other way around.\\
As applications of our invariance principle we prove two
previously unconnected conjectures from boolean harmonic analysis;
the first was motivated by hardness of approximation in computer
science, the second by the theory of social choice from economics:
\begin{conjecture}[``Majority Is Stablest''
conjecture~\cite{KKMO:04}] \label{conj:MIST}
Let $0 \leq \rho \leq 1$ and $\epsilon > 0$ be given. Then there
exists $\tau > 0$ such that if $f : \{-1,1\}^n \to [-1,1]$ satisfies
${\bf E}[f] = 0$ and $\mathrm{Inf}_i(f) \leq \tau$ for all $i$, then
\[
\mathbb{S}_\rho(f) \leq {\textstyle \frac{2}{\pi}} \arcsin \rho + \epsilon.
\]
\end{conjecture}
Here we have used the notation $\mathbb{S}_\rho(f)$ for $\sum_S
\rho^{|S|} \hat{f}(S)^2$, the \emph{noise stability} of $f$. This
equals ${\bf E}[f(x)f(y)]$ when $(x,y) \in \{-1,1\}^n \times \{-1,1\}^n$
is chosen so that $(x_i,y_i) \in \{-1,1\}^2$ are independent random
variables with ${\bf E}[x_i] = {\bf E}[y_i] = 0$ and ${\bf E}[x_i y_i] = \rho$.
\begin{conjecture}[``It Ain't Over Till It's Over''
conjecture~\cite{Kalai:01}] \label{conj:ain't} Let $0 \leq \rho <
1$ and $\epsilon > 0$ be given. Then there exists $\delta > 0$ and
$\tau > 0$ such that if $f : \{-1,1\}^n \to \{-1,1\}$ satisfies ${\bf E}[f] =
0$ and $\mathrm{Inf}_i(f) \leq \tau$ for all $i$, then $f$ has the
following property: If $V$ is a random subset of $[n]$ in which
each $i$ is included independently with probability $\rho$, and if
the bits $(x_i)_{i \in V}$ are chosen uniformly at random, then
\[
\mathop{\bf P\/}_{V,\;(x_i)_{i \in V}} \Bigl[ \bigl|{\bf E}[f \mid (x_i)_{i \in
V}]\bigr| > 1 - \delta\Bigr] \leq \epsilon.
\]
\end{conjecture}
(In words, the conjecture states that even if a random $\rho$
fraction of voters' votes are revealed, with high probability the
election is still slightly undecided, provided $f$ has low
influences.)\\
The truth of these results gives illustration to a recurring theme
in the harmonic analysis of boolean functions: the extremal role
played the Majority function. It seems this theme becomes
especially prominent when low-influence functions are studied. To
explain the connection of Majority to our applications: In the
former case the quantity $\frac{2}{\pi} \arcsin \rho$ is precisely
$\lim_{n \to \infty} \mathbb{S}_\rho(\mathrm{Maj}_n)$; this explains the name
of the Majority Is Stablest conjecture. In the latter case, we
show that $\delta$ can be taken to be on the order of
$\epsilon^{\rho/(1-\rho)}$ (up to $o(1)$ in the exponent), which
is the same asymptotics one gets if $f$ is Majority on a large
number of inputs.
\subsection{Outline of the paper}
We begin in Section~\ref{sec:statement} with an overview of the
invariance principle, the two applications, and some of their consequences.
We prove the invariance principle in
Section~\ref{sec:invariance}. Our proofs of the two conjectures
are in Section~\ref{sec:conj}. Finally, we show in
Section~\ref{sec:counterexample} that a conjecture closely related
to Majority Is Stablest is false. Some minor proofs from
throughout the paper appear in appendices.
\subsection{Related work}
Our multilinear invariance principle has some antecedents. For
degree 1 polynomials it reduces to a version of the
Berry-Esseen Central Limit Theorems. Indeed, our proof
follows the same outlines as Lindeberg's proof of the
CLT~\cite{Lindeberg:22} (see also~\cite{Feller:71}).\\
Since presenting our proof of the invariance principle, we have
been informed by Oded Regev that related results were proved in
the past by V.~I.~Rotar$'$~\cite{Rotar:79}. As well, a
contemporary manuscript of Sourav Chatterjee~\cite{Chatterjee:u}
with an invariance principle of similar flavor has come to our
attention. What is common to our work and
to~\cite{Rotar:79,Chatterjee:u} is a generalization of Lindeberg's
argument to the non-linear case. The result of Rotar$'$ is an
invariance principle similar to ours where the condition on the
influences generalizes Lindeberg's condition. The setup is not
quite the same, however, and the proof in~\cite{Rotar:79} is of a
rather qualitative nature. It seems that even after appropriate
modification the bounds it gives would be weaker and less useful
for our type of applications. (This is quite understandable; in a
similar way Lindeberg's CLT can be less precise than the
Berry-Esseen inequality even though --- indeed, because --- it
works under weaker assumptions.) The paper~\cite{Chatterjee:u} is
by contrast very clear and explicit. However it does not seem to
be appropriate for many applications since it requires low
``worst-case'' influences, instead of the ``average-case''
influences used by this work and~\cite{Rotar:79}.\\
Finally, we would like to mention that some chaos-decomposition
limit theorems have been proved before in various settings. Among
these are limit theorems for U and V statistics and limit theorems
for random graphs; see, e.g.~\cite{Janson:97}.
\subsection{Acknowledgments}
We are grateful to Keith Ball for suggesting a collaboration among
the authors. We would also like to thank Oded Regev for referring
us to~\cite{Rotar:79} and Olivier Gu\'edon for referring us
to~\cite{CarberyWright:01}.
\section{Our results} \label{sec:statement}
\subsection{The invariance principle} \label{sec:intro-invariance}
In this subsection we present a simplified version of our invariance principle.\\
Suppose ${\boldsymbol X}$ is a random variable with ${\bf E}[{\boldsymbol X}] = 0$ and
${\bf E}[{\boldsymbol X}^2] = 1$ and ${\boldsymbol X}_1, \dots, {\boldsymbol X}_n$ are independent
copies of ${\boldsymbol X}$. Let $Q(x_1, \dots, x_n) = \sum_{i=1}^n c_i
x_i$ be a linear form and assume $\sum c_i^2 = 1$. The
Berry-Esseen CLT states that under mild conditions on the
distribution of ${\boldsymbol X}$, say ${\bf E}[|{\boldsymbol X}|^3] \leq A < \infty$, it
holds that
\[
\sup_{t} \bigl| \P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[{\boldsymbol G} \leq t]\bigr| \leq O\bigl(A \cdot {\textstyle \sum}_{i=1}^n |c_i|^3\bigr),
\]
where ${\boldsymbol G}$ denotes a standard normal random variable. Note
that a simple corollary of the above is
\begin{equation} \label{eq:berry}
\sup_{t} \bigl|\P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[Q({\boldsymbol G}_1, \dots, {\boldsymbol G}_n) \leq t] \bigr| \leq
O\bigl(A \cdot \max_i |c_i|\bigr).
\end{equation}
Here the ${\boldsymbol G}_i$'s denote independent standard normals. We
have upper-bounded the sum of $|c_i|^3$ here by a maximum, for
simplicity; more importantly though, we have suggestively replaced
${\boldsymbol G}$ by $\sum_i c_i {\boldsymbol G}_i$, which of course has the same
distribution.\\
We would like to generalize~(\ref{eq:berry}) to \emph{multilinear
polynomials} in the ${\boldsymbol X}_i$'s; i.e., functions of the form
\begin{equation} \label{eqn:Q1}
Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) = \sum_{S \subseteq [n]} c_S \prod_{i
\in S} {\boldsymbol X}_i,
\end{equation}
where the real constants $c_S$ satisfy $\sum c_S^2 = 1$. Let $d =
\max_{c_S \neq 0} |S|$ denote the degree of $Q$. Unlike in the
$d=1$ case of the CLT, there is no single random variable ${\boldsymbol G}$
which always provides a limiting distribution. However one can
still hope to prove, in light of~(\ref{eq:berry}), that the
distribution of the polynomial applied to the variables ${\boldsymbol X}_i$
is close to the distribution of the polynomial applied to
independent Gaussian random variables. This is indeed what our
invariance principle shows.\\
It turns out that the appropriate generalization of the
Berry-Esseen theorem~(\ref{eq:berry}) is to control the error
by a function of $d$ and of $\max_i \sum_{S \ni i} c_S^2$ ---
i.e., the maximum of the \emph{influences} of $Q$ (as
in~(\ref{eqn:influence})). Naturally, we also
need some conditions in addition to second moments. In our
formulation we impose the condition that the variable ${\boldsymbol X}$ is
\emph{hypercontractive}; i.e., there is some $\eta > 0$ such that
for all $a \in \mathbb R$,
\[
\|a + \eta {\boldsymbol X}\|_3 \leq \|a + {\boldsymbol X}\|_2.
\]
This condition is satisfied whenever ${\bf E}[{\boldsymbol X}]=0$ and
${\bf E}[|{\boldsymbol X}|^{3}]<\infty;$ in particular, it holds for any
mean-zero random variable ${\boldsymbol X}$ taking on only finitely many
values. Using hypercontractivity, we get a simply proved
invariance principle with explicit error bounds. The following
theorem (a simplification of Theorem~\ref{thm:supertheorem},
bound~(\ref{eq:lim_dist})) is an example of what we prove:
\begin{theorem} \label{thm:simple}
Let ${\boldsymbol X}_1, \dots, {\boldsymbol X}_n$ be independent random variables
satisfying ${\bf E}[{\boldsymbol X}_i] = 0$, ${\bf E}[{\boldsymbol X}_i^2] = 1$, and
${\bf E}[|{\boldsymbol X}_i|^{3}] \leq \beta.$ Let $Q$ be a degree $d$
multilinear polynomial as in~(\ref{eqn:Q1}) with
\[
\sum_{|S| > 0} c_S^2 = 1, \qquad \qquad \sum_{S \ni i} c_S^2 \leq
\tau \quad \text{for all $i$}.
\]
Then
\[
\sup_{t} \bigl|\P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[Q({\boldsymbol G}_1, \dots, {\boldsymbol G}_n) \leq t] \bigr|
\leq O(d\beta^{1/3}\tau^{1/8d}),
\]
where ${\boldsymbol G}_1, \dots, {\boldsymbol G}_n$ are independent standard
Gaussians.\\
If, instead of assuming ${\bf E}[|{\boldsymbol X}_i|^{3}] \leq \beta$, we assume
that each ${\boldsymbol X}_i$ takes only on finitely many values, and that
for all $i$ and all $x \in \mathbb R$ either $\Pr[{\boldsymbol X}_i = x] = 0$ or
$\Pr[{\boldsymbol X}_i = x] \geq \alpha$, then
\[
\sup_{t} \bigl|\P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[Q({\boldsymbol G}_1, \dots, {\boldsymbol G}_n) \leq t] \bigr|
\leq O(d\,\alpha^{-1/6}\, \tau^{1/8d}).
\]
\end{theorem}
Note that if $d$, $\beta$, and $\alpha$ are fixed then the above
bound tends to $0$ with $\tau$. We call this theorem an
``invariance principle'' because it shows that $Q({\boldsymbol X}_1, \dots,
{\boldsymbol X}_n)$ has essentially the same distribution no matter what
the ${\boldsymbol X}_i$'s are. Usually we will not push for the optimal
constants; instead we will try to keep our approach as simple as
possible while still giving explicit bounds useful for our
applications.\\
An unavoidable deficiency of this sort of invariance principle is
the dependence on $d$ in the error bound. In applications such as
Majority Is Stablest and It Ain't Over Till It's Over, the
functions $f$ may well have arbitrarily large degree. To overcome
this, we introduce a supplement to the invariance principle: We
show that if the polynomial $Q$ is ``smoothed'' slightly then the
dependence on $d$ in the error bound can be eliminated and
replaced with a dependence on the smoothness. For ``noise
stability''-type problems such as ours, this smoothing is
essentially harmless.\\
In fact, the techniques we use are strong enough to obtain Berry-Esseen
estimates under Lyapunov-type assumptions. In particular, we believe that the
following theorem is new even in the case of sums of independent
random variables.
\begin{theorem} \label{thm:Lyap}
Let $q \in (2,3].$
Let ${\boldsymbol X}_1, \dots, {\boldsymbol X}_n$ be independent random variables
satisfying ${\bf E}[{\boldsymbol X}_i] = 0$, ${\bf E}[{\boldsymbol X}_i^2] = 1$, and
${\bf E}[|{\boldsymbol X}_i|^{q}] \leq \beta.$ Let $Q$ be a degree $d$
multilinear polynomial as in~(\ref{eqn:Q1}) with
\[
\sum_{|S| > 0} c_S^2 = 1, \qquad \qquad \sum_{S \ni i} c_S^2 \leq
\tau \quad \text{for all $i$}.
\]
Then
\[
\sup_{t} \bigl|\P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[Q({\boldsymbol G}_1, \dots, {\boldsymbol G}_n) \leq t] \bigr|
\leq
\]
\[
O(d\beta^{\frac{d}{qd+1}}
(\sum_{i}(\sum_{S \ni i} c_{S}^{2})^{q/2})^{\frac{1}{qd+1}})
\leq
O(d\beta^{\frac{d}{qd+1}}
\tau^{\frac{q-2}{2qd+2}}),
\]
where ${\boldsymbol G}_1, \dots, {\boldsymbol G}_n$ are independent standard
Gaussians.\\
\end{theorem}
\subsection{Influences and noise stability in product
spaces} \label{sec:general}
Our proofs of the Majority Is Stablest and It Ain't Over Till It's
Over conjectures hold not just for functions on the
uniform-distribution discrete cube, but for functions on arbitrary
finite product probability spaces. Harmonic analysis results on
influences have often considered the biased product distribution
on the discrete cube (see, e.g.,
\cite{Talagrand:94,FriedgutKalai:96,Friedgut:99,Bourgain:99});
and, some recent works involving influences and noise stability
have considered functions on product sets $[q]^n$ endowed with the
uniform distribution (e.g., \cite{AlDiFrSu:04,KKMO:04}). But since
there doesn't
appear to be a unified treatment for the general case in the literature, we give the necessary definitions here.\\
Let $(\Omega_1, \mu_1), \dots, (\Omega_n, \mu_n)$ be
probability spaces and let $(\Omega, \mu)$ denote the
product probability space. Let
\[
f : \Omega_1 \times \cdots \times \Omega_n \to \mathbb R
\]
be any real-valued function on $\Omega$.
\begin{definition} \label{def:influence_general}
The \emph{influence of the $i$th coordinate on $f$} is
\[
\mathrm{Inf}_i(f) = \mathop{\bf E\/}_{\mu} [ \mathop{\bf Var\/}_{\mu_i} [f]].
\]
\end{definition}
Note that for boolean functions $f : \{-1,1\}^n \to \{-1,1\}$ this
agrees with the classical notion of influences introduced to
computer science by Ben-Or and Linial~\cite{BenorLinial:90}. When
the domain $\{-1,1\}^n$ has a $p$-biased distribution, our notion
differs from that of, say,~\cite{Friedgut:98} by a multiplicative
factor of $4p(1-p)$. We believe the above definition is more
natural, and in any case it is easy to pass between the two.\\
To define noise stability, we first define the $T_\rho$ operator
on the space of functions $f$:
\begin{definition} \label{def:T_general}
For any $0 \leq \rho \leq 1$, the operator $T_\rho$ is defined by
\begin{equation} \label{eq:T_general}
(T_\rho f)(\omega_1, \dots, \omega_n) = {\bf E}[f(\omega_1', \dots,
\omega_n')],
\end{equation}
where each $\omega_i'$ is an independent random variable defined
to equal $\omega_i$ with probability $\rho$ and to be randomly
drawn from $\mu_i$ with probability $1-\rho$.
\end{definition}
\medskip
We remark that this definition agrees with that of the
``Bonami-Beckner operator'' introduced in the context of boolean
functions by KKL~\cite{KaKaLi:88} and also with its generalization
to $[q]^n$ from~\cite{KKMO:04}. For more on this operator, see
Wolff~\cite{Wolff:u}. With this definition in place, we can
define noise stability:
\begin{definition} \label{def:Stab_general}
The \emph{noise stability of $f$ at $\rho \in [0,1]$} is
\[
\mathbb{S}_\rho(f) = \mathop{\bf E\/}_\mu[f \cdot T_\rho f].
\]
\end{definition}
\bigskip
For the It Ain't Over Till It's Over problem, we introduce a new
operator $V_\rho$:
\begin{definition} \label{def:V}
For any $\rho \in [0,1]$, the operator $V_\rho$ is defined as
follows. The operator takes a function
$f : \Omega_1 \times \cdots \times \Omega_n \to \mathbb R$ to a function
$g : \Omega_1 \times \cdots \times \Omega_n \times \{0,1\}^n \to \mathbb R$, where
$\{0,1\}^n$ is equipped with the $(1-\rho,\rho)^{\otimes n}$ measure.
It is defined as follows:
\[
(V_\rho f)(\omega_1, \dots, \omega_n,x_1,\ldots,x_n) =
\mathop{\bf E\/}_{\omega'} \left[
f \left( x_1 \omega_1 + (1-x_1) \omega'_1, \ldots,
x_n \omega_n + (1-x_n) \omega'_n \right) \right].
\]
\end{definition}
\medskip
Finally, we would like to note that our definitions are valid for
functions $f$ into the reals, although our motivation is usually
$\{-1,1\}$-valued functions. Our proofs of the Majority Is Stablest
and It Ain't Over Till It's Over conjectures will hold in the
setting of functions $f : \Omega_1 \times \cdots \times \Omega_n
\to [-1,1]$ (note that Conjecture~\ref{conj:MIST} \emph{requires}
this generalized range). For notational simplicity, though, we
will give our proofs for functions into $[0,1]$; the reader can
easily convert such results to the $[-1,1]$ case by the linear
transformation $f \mapsto 2f-1$, which interacts in a simple way
with the definitions of $\mathrm{Inf}_i$, $\mathbb{S}_\rho$ and $V_\rho$.
\subsection{Majority Is Stablest} \label{sec:misc}
\subsubsection{About the problem} \label{sec:misc-discuss}
The Majority Is Stablest conjecture, Conjecture~\ref{conj:MIST},
was first formally stated in~\cite{KKMO:04}. However the notion of
Hamming balls having the highest noise stability in various senses
has been widely spread among the community studying discrete
Fourier analysis. Indeed, already in KKL's 1998
paper~\cite{KaKaLi:88} there is the suggestion that Hamming balls
and subcubes should maximize a certain noise stability-like
quantity. In~\cite{BeKaSc:99}, it was shown that every
`asymptotically noise stable'' function is correlated with a
weighted majority function; also, in~\cite{MORSS:04} it was shown
that the majority function asymptotically maximizes a high-norm
analog of $\mathbb{S}_{\rho}$.\\
More concretely, strong motivation for getting
sharp bounds on the noise stability of low-influence functions
came from two 2002 papers, one by Kalai~\cite{Kalai:02} on social
choice and one by Khot~\cite{Khot:02} on PCPs and hardness of
approximation. We briefly discuss these two papers below.\\
\paragraph{Kalai '02 --- Arrow's Impossibility Theorem:} Suppose $n$ voters
rank three candidates, $A$, $B$, and $C$, and a \emph{social
choice} function $f : \{-1,1\}^n \to \{-1,1\}$ is used to aggregate the
rankings, as follows: $f$ is applied to the $n$ $A$-vs.-$B$
preferences to determine whether $A$ or $B$ is globally preferred;
then the same happens for $A$-vs.-$C$ and $B$-vs.-$C$. The
outcome is termed ``non-rational'' if the global ranking has $A$
preferable to $B$ preferable to $C$ preferable to $A$ (or if the
other cyclic possibility occurs). Arrow's Impossibility Theorem
from the theory of social choice states that under some mild
restrictions on $f$ (such as $f$ being odd; i.e., $f(-x) =
-f(x)$), the only functions which never admit non-rational
outcomes given rational voters are the dictator functions $f(x) =
\pm x_i$.
Kalai~\cite{Kalai:02} studied the \emph{probability} of a rational
outcome given that the $n$ voters vote independently and at random
from the 6 possible rational rankings. He showed that the
probability of a rational outcome in this case is precisely $3/4 +
(3/4) \mathbb{S}_{1/3}(f)$. Thus it is natural to ask which function
$f$ with small influences is most likely to produce a rational
outcome. Instead of considering small influences, Kalai
considered the essentially stronger assumption that $f$ is
``transitive-symmetric''; i.e., that for all $1 \leq i < j \leq n$
there exists a permutation $\sigma$ on $[n]$ with $\sigma(i) = j$
such that $f(x_1, \dots, x_n) = f(x_{\sigma(1)}, \dots,
x_{\sigma(n)})$ for all $(x_1, \dots, x_n)$. Kalai conjectured
that Majority was the transitive-symmetric function that maximized
$3/4 + (3/4) \mathbb{S}_{1/3}(f)$ (in fact, he made a stronger
conjecture, but this conjecture is false; see
Section~\ref{sec:counterexample}). He further observed that this
would imply that in any transitive-symmetric scheme the
probability of a rational outcome is at most $3/4 +
(3/2\pi)\arcsin(1/3) + o_n(1) \approx .9123$; however, Kalai could
only prove the weaker bound $.9192$.
\paragraph{Khot '02 --- Unique Games and hardness of approximating
2-CSPs:} In computer science, many combinatorial optimization
problems are NP-hard, meaning it is unlikely there are efficient
algorithms that always find the optimal solution. Hence there has
been extensive interest in understanding the complexity of
\emph{approximating} the optimal solution. Consider for example
``$k$-variable constraint satisfaction problems'' ($k$-CSPs) in
which the input is a set of variables over a finite domain,
along with some constraints on $k$-sets of the variables,
restricting what sets of values they can simultaneously take. We
say a problem has ``$(c,s)$-hardness'' if it is NP-hard, given a
$k$-CSP instance in which the optimal assignment satisfies a
$c$-fraction of the constrains, for an algorithm to find an
assignment that satisfies an $s$-fraction of the constraints. In
this case we also say that the problem is ``$s/c$-hard to
approximate''.
The PCP and Parallel Repetition theorems have led to many
impressive results showing that it is NP-hard even to give
$\alpha$-approximations for various problems, especially $k$-CSPs
for $k \geq 3$. For example, letting MAX-$k$LIN($q$) denote the
problem of satisfying $k$-variable linear equations over ${\bf
Z}_q$, it is known \cite{Hastad:01} that MAX-$k$LIN($q$) has
$(1-\epsilon, 1/q + \epsilon)$-hardness for all $k \geq 3$, and this is
sharp. However it seems that current PCP theorems are not strong
enough to give sharp hardness of approximation results for 2-CSPs
(e.g., constraint satisfaction problems on graphs). The
influential paper of Khot~\cite{Khot:02} introduced the ``Unique
Games Conjecture'' (UGC) in order to make progress on 2-CSPs; UGC
states that a certain 2-CSP over a large domain has $(1-\epsilon,
\epsilon)$-hardness.
Interestingly, it seems that using UGC to prove hardness results
for other 2-CSPs typically crucially requires strong results about
influences and noise stability of boolean functions. For example,
\cite{Khot:02}'s analysis of MAX-$2$LIN($2$) required an upper
bound on $\mathbb{S}_{1-\epsilon}(f)$ for small $\epsilon$ among balanced
functions $f : \{-1,1\}^n \to \{-1,1\}$ with small influences; to get
this, Khot used the following deep result of
Bourgain~\cite{Bourgain:02} from 2001:
\begin{theorem}[Bourgain~\cite{Bourgain:02}] \label{thm:bourgain}
If $f : \{-1,1\}^n \to \{-1,1\}$ satisfies ${\bf E}[f] = 0$ and $\mathrm{Inf}_i(f)
\leq 10^{-d}$ for all $i \in [n]$, then
\[
\sum_{|S| > d} \hat{f}(S)^2 \geq d^{-1/2 - O(\sqrt{\log \log d / \log d})} = d^{-1/2 -o(1)}.
\]
\end{theorem}
Note that Bourgain's theorem has the following easy corollary:
\begin{corollary} \label{cor:eps1/2-} If $f : \{-1,1\}^n \to \{-1,1\}$ satisfies ${\bf E}[f] = 0$ and $\mathrm{Inf}_i(f)
\leq 2^{-O(1/\epsilon)}$ for all $i \in [n]$, then
\[
\mathbb{S}_{1-\epsilon}(f) \leq 1 - \epsilon^{1/2 + o(1)}.
\]
\end{corollary}
This corollary enabled Khot to show $(1-\epsilon, 1-\epsilon^{1/2 +
o(1)})$-hardness for MAX-$2$LIN($2$), which is close to sharp (the
algorithm of Goemans-Williamson~\cite{GoemansWilliamson:95}
achieves $1-O(\sqrt{\epsilon})$). As an aside, we note that Khot and
Vishnoi~\cite{KhotVishna:u} recently used
Corollary~\ref{cor:eps1/2-} to prove that negative type metrics do
not embed into $\ell_1$ with constant distortion.
Another example of this comes from the work of~\cite{KKMO:04}.
Among other things,~\cite{KKMO:04} studied the MAX-CUT problem: Given
an undirected graph, partition the vertices into two parts so as
to maximize the number of edges with endpoints in different parts.
The paper introduced the Majority Is Stablest
Conjecture~\ref{conj:MIST} and showed that together with UGC it
implied $({\textstyle \frac12} + {\textstyle \frac12} \rho - \epsilon, {\textstyle \frac12} + {\textstyle
\frac{1}{\pi}} \arcsin \rho + \epsilon)$-hardness for MAX-CUT. In
particular, optimizing over $\rho$ (taking $\rho \approx .69$)
implies MAX-CUT is
$.878$-hard to approximate, matching the groundbreaking algorithm
of Goemans and Williamson~\cite{GoemansWilliamson:95}.
\subsubsection{Consequences of confirming the conjecture}
In Theorem~\ref{thm:MIST} we confirm a generalization of the
Majority Is Stablest conjecture.
We give a slightly simplified statement of this theorem here:
\paragraph{Theorem~\ref{thm:MIST}}\emph{Let $f : \Omega_1 \times \cdots \times \Omega_n \to [0,1]$ be a
function on a discrete product probability space and assume that
for each $i$ the minimum probability of any atom in $\Omega_i$ is
at least $\alpha \leq 1/2$. Further assume that $\mathrm{Inf}_i(f) \leq
\tau$ for all $i$. Let $\mu = {\bf E}[f]$. Then for any $0 \leq \rho <
1$,
\[
\mathbb{S}_\rho(f) \leq \lim_{n\to \infty} \mathbb{S}_\rho(\mathrm{Thr}^{(\mu)}_n) +
O\Bigl({\textstyle \frac{\log \log (1/\tau)}{\log(1/\tau)}}\Bigr),
\]
where $\mathrm{Thr}^{(\mu)}_n : \{-1,1\}^n \to \{0,1\}$ denotes the symmetric
threshold function with expectation closest to $\mu$, and the
$O(\cdot)$ hides a constant depending only on $\alpha$ and $1-\rho$.}\\
We now give some consequences of this theorem:
\begin{theorem} In the terminology of Kalai~\cite{Kalai:02}, any odd,
balanced social choice function $f$ with either
\begin{itemize}
\item $o_n(1)$ influences or
\item such that $f$ is transitive
\end{itemize}
has probability at most $3/4 + (3/2\pi)\arcsin(1/3) + o_n(1) \approx
.9123$ of producing a rational outcome. The majority function on
$n$ inputs achieves this bound, $3/4 + (3/2\pi)\arcsin(1/3) +
o_n(1)$.
\end{theorem}
By looking at the series expansion of $\frac{2}{\pi} \arcsin(1-\epsilon)$
we obtain the following strengthening of Corollary~\ref{cor:eps1/2-}.
\begin{corollary} \label{cor:eps1/2} If $f : \{-1,1\}^n \to \{-1,1\}$ satisfies ${\bf E}[f] = 0$ and $\mathrm{Inf}_i(f)
\leq \epsilon^{-O(1/\epsilon)}$ for all $i \in [n]$, then
\[
\mathbb{S}_{1-\epsilon}(f) \leq 1 - ({\textstyle \frac{\sqrt{8}}{\pi} -
o(1)})\epsilon^{1/2}.
\]
\end{corollary}
Using Corollary~\ref{cor:eps1/2} instead of
Corollary~\ref{cor:eps1/2-} in Khot~\cite{Khot:02} we obtain
\begin{corollary} MAX-$2$LIN($2$) and MAX-2SAT have $(1-\epsilon, 1 -
O(\epsilon^{1/2}))$-hardness. \rnote{Actually, for MAX-2LIN(2) we
probably exactly match (to the constant factor) the algorithm of
GW. Check?}
\end{corollary}
More generally,~\cite{KKMO:04} now implies
\begin{corollary} MAX-CUT has $({\textstyle \frac12} + {\textstyle \frac12} \rho - \epsilon, {\textstyle \frac12} + {\textstyle
\frac{1}{\pi}} \arcsin \rho + \epsilon)$-hardness for each $\rho$ and
all $\epsilon > 0$, assuming UGC only. In particular, the
Goemans-Williamson .878-approximation algorithm is best possible,
assuming UGC only.
\end{corollary}
The following two results are consequences of a generalization of
``Majority is Stablest'' as shown in~\cite{KKMO:04}:
\begin{theorem} UGC implies that for each $\epsilon > 0$ there exists
$q = q(\epsilon)$ such that MAX-$2$LIN($q$) has $(1-\epsilon,
\epsilon)$-hardness. Indeed, this statement is \emph{equivalent} to
UGC.
\end{theorem}
\begin{theorem} The MAX-$q$-CUT problem, i.e.~Approximate
$q$-Coloring, has $(1 - 1/q + q^{2+o(1)})$-hardness factor,
assuming UGC only.
This asymptotically matches the approximation factor obtained by
Frieze and Jerrum~\cite{FriezeJerrum:95}.
\end{theorem}
\subsection{It Ain't Over Till It's Over}
The It Ain't Over Till It's Over conjecture was originally made by
Kalai and Friedgut~\cite{Kalai:01} in
studying social indeterminacy~\cite{FrKaNa:02,Kalai:04}.
The setting here is similar to the
setting of Arrow's Theorem from Section~\ref{sec:misc-discuss}
except that there are an arbitrary finite number of candidates.
Let $R$ denote the (asymmetric) relation given on the candidates
when the \emph{monotone} social choice function $f$ is used. Kalai
showed that if $f$ has small influences, then the It Ain't Over
Till It's Over Conjecture implies that \emph{every} possible
relation $R$ is achieved with probability bounded away from $0$.
Since its introduction in 2001, the It Ain't Over Till It's Over
problem has circulated widely in the community studying harmonic
analysis of boolean functions. The conjecture was given as one of
the top unsolved problems in the field at a workshop at Yale in
late 2004.\\
In Theorem~\ref{thm:aint} we confirm the It Ain't Over Till It's
Over conjecture and generalize it to functions on arbitrary
finite product probability spaces with means bounded away from 0
and 1. Further, the asymptotics we give show that symmetric
threshold functions (e.g., Majority in the case of mean $1/2$) are
the ``worst'' examples. We give a slightly simplified statement
of Theorem~\ref{thm:aint} here:
\paragraph{Theorem~\ref{thm:aint}} \emph{Let $0 < \rho < 1$ and
let $f : \Omega_1 \times \cdots \times \Omega_n \to [0,1]$ be a
function on a discrete product probability space; assume that for
each $i$ the minimum probability of any atom in $\Omega_i$ is at
least $\alpha \leq 1/2$. Then there exists $\epsilon(\rho,\mu) > 0$ such that
if $\epsilon < \epsilon(\rho,\mu)$ and $\mathrm{Inf}_i(f) \leq \epsilon^{O(\sqrt{\log(1/\epsilon)})}$ for all $i$ and $\mu = {\bf E}[f]$ then
\[
\Pr[V_\rho f > 1 - \delta] \leq \epsilon
\]
and
\[
\Pr[V_\rho f < \delta] \leq \epsilon
\]
provided
\[
\delta < \epsilon^{\rho/(1-\rho) + O(1/\sqrt{\log(1/\epsilon)})},
\]
where the $O(\cdot)$ hides a constant depending only on $\alpha$,
$1-\mu$, $\rho$, and $1-\rho$.}\\
\section{The invariance principle} \label{sec:invariance}
\subsection{Setup and notation} \label{sec:setup}
In this section we will describe the setup and notation necessary
for our invariance principle. Recall that we are interested in
functions on finite product probability spaces, $f : \Omega_1
\times \cdots \times \Omega_n \to \mathbb R$. For each $i$, the space of
all functions $\Omega_i \to \mathbb R$ can be expressed as the span of a
finite set of orthonormal random variables, ${\boldsymbol X}_{i,0} = 1,
{\boldsymbol X}_{i,1}, {\boldsymbol X}_{i, 2}, {\boldsymbol X}_{i,3}, \dots$; then $f$ can be
written as a multilinear polynomial in the ${\boldsymbol X}_{i,j}$'s. In
fact, it will be convenient for us to mostly disregard the
$\Omega_i$'s and work directly with sets of orthonormal random
variables; in this case, we can even drop the restriction of
finiteness. We thus begin with the following definition:
\begin{definition}
We call a collection of finitely many orthonormal real random
variables, one of which is the constant $1$, an \emph{orthonormal
ensemble}. We will write a typical \emph{sequence} of $n$
orthonormal ensembles as ${\boldsymbol{\mathcal{X}}} = ({\boldsymbol{\mathcal{X}}}_1, \dots, {\boldsymbol{\mathcal{X}}}_n)$,
where ${\boldsymbol{\mathcal{X}}}_i = \{{\boldsymbol X}_{i,0} = 1, {\boldsymbol X}_{i,1}, \dots,
{\boldsymbol X}_{i,m_i}\}$. We call a sequence of orthonormal ensembles
${\boldsymbol{\mathcal{X}}}$ \emph{independent} if the ensembles are independent families
of random variables.
We will henceforth be concerned only with independent sequences of
orthonormal ensembles, and we will call these \emph{sequences of
ensembles}, for brevity.
\end{definition}
\begin{remark} \label{rem:simple} Given a sequence of independent random variables
${\boldsymbol X}_1, \dots, {\boldsymbol X}_n$ with ${\bf E}[{\boldsymbol X}_i] = 0$ and
${\bf E}[{\boldsymbol X}_i^2] = 1$ (as in Theorem~\ref{thm:simple}), we can view
them as a sequence of ensembles ${\boldsymbol{\mathcal{X}}}$ by renaming ${\boldsymbol X}_i =
{\boldsymbol X}_{i,1}$ and setting ${\boldsymbol X}_{i,0} = 1$ as required.
\end{remark}
\begin{definition} We denote by ${\boldsymbol{\mathcal{G}}}$ the \emph{Gaussian sequence of
ensembles}, in which ${\boldsymbol{\mathcal{G}}}_i = \{{\boldsymbol G}_{i,0} = 1, {\boldsymbol G}_{i,1},
{\boldsymbol G}_{i,2}, \dots\}$ and all ${\boldsymbol G}_{i,j}$'s with $j \geq 1$
are independent standard Gaussians.
\end{definition}
As mentioned, we will be interested in \emph{multilinear
polynomials} over sequences of ensembles. By this we mean sums of
products of the random variables, where each product is obtained
by multiplying one random variable from each ensemble.
\begin{definition}
A \emph{multi-index} ${\boldsymbol \sigma}$ is a sequence $(\sigma_1, \dots,
\sigma_n)$ in $\mathbb N^n$; the \emph{degree} of ${\boldsymbol \sigma}$, denoted
$|{\boldsymbol \sigma}|$, is $|\{i \in [n] : \sigma_i > 0\}|$. Given a
doubly-indexed set of indeterminates $\{x_{i,j}\}_{i \in [n], j
\in \mathbb N}$, we write $x_{\boldsymbol \sigma}$ for the monomial $\prod_{i =
1}^n x_{i,\sigma_i}$. We now define a \emph{multilinear
polynomial} over such a set of indeterminates to be any expression
\begin{equation} \label{eqn:Q}
Q(x) = \sum_{{\boldsymbol \sigma}} c_{\boldsymbol \sigma} x_{\boldsymbol \sigma}
\end{equation}
where the $c_{\boldsymbol \sigma}$'s are real constants, all but finitely
many of which are zero. The \emph{degree} of $Q(x)$ is
$\max\{|{\boldsymbol \sigma}| : c_{\boldsymbol \sigma} \neq 0\}$, at most $n$. We
also use the notation
\[
Q^{\leq d}(x) = \sum_{|{\boldsymbol \sigma}| \leq d} c_{\boldsymbol \sigma}
x_{\boldsymbol \sigma}
\]
and the analogous $Q^{= d}(x)$ and $Q^{> d}(x)$.
\end{definition}
Naturally, we will consider applying multilinear polynomials $Q$
to sequences of ensembles ${\boldsymbol{\mathcal{X}}}$; the distribution of these
random variables $Q({\boldsymbol{\mathcal{X}}})$ is the subject of our invariance
principle. Since $Q({\boldsymbol{\mathcal{X}}})$ can be thought of as a function on a
product space $\Omega_1 \times \cdots \times \Omega_n$ as
described at the beginning of this section, there is a consistent
way to define the notions of influences, $T_\rho$, and noise
stability from Section~\ref{sec:general}. For example, the
``influence of the $i$th ensemble on $Q$'' is
\[
\mathrm{Inf}_i(Q({\boldsymbol{\mathcal{X}}})) = {\bf E}[{\bf Var}[Q({\boldsymbol{\mathcal{X}}}) \mid {\boldsymbol{\mathcal{X}}}_1, \dots,
{\boldsymbol{\mathcal{X}}}_{i-1}, {\boldsymbol{\mathcal{X}}}_{i+1}, \dots, {\boldsymbol{\mathcal{X}}}_n]].
\]
Using independence and orthonormality, it is easy to show the
following formulas, familiar from harmonic analysis of boolean
functions:
\begin{proposition} \label{prop:infQ}
Let ${\boldsymbol{\mathcal{X}}}$ be a sequence of ensembles and $Q$ a multilinear
polynomial as in~(\ref{eqn:Q}). Then
\[
{\bf E}[Q({\boldsymbol{\mathcal{X}}})] = c_{\bf{0}}; \qquad {\bf E}[Q({\boldsymbol{\mathcal{X}}})^2] = \sum_{{\boldsymbol \sigma}}
c_{\boldsymbol \sigma}^2; \qquad
{\bf Var}[Q({\boldsymbol{\mathcal{X}}})] = \sum_{|{\boldsymbol \sigma}| > 0} c_{\boldsymbol \sigma}^2;
\]
\[
\mathrm{Inf}_i(Q({\boldsymbol{\mathcal{X}}})) = \sum_{{\boldsymbol \sigma} : \sigma_i > 0} c_{\boldsymbol \sigma}^2;
\qquad T_\rho Q({\boldsymbol{\mathcal{X}}}) = \sum_{{\boldsymbol \sigma}} \rho^{|{\boldsymbol \sigma}|} c_{\boldsymbol \sigma}
{\boldsymbol{\mathcal{X}}}_{\boldsymbol \sigma}; \qquad \mathbb{S}_\rho(Q({\boldsymbol{\mathcal{X}}})) = \sum_{{\boldsymbol \sigma}} \rho^{|{\boldsymbol \sigma}|}
c_{\boldsymbol \sigma}^2.
\]
\end{proposition}
Note that in each case above, the formula does not depend on the
sequence of ensembles ${\boldsymbol{\mathcal{X}}}$; it only depends on $Q$. Thus we
are justified in henceforth writing ${\bf E}[Q]$, ${\bf E}[Q^2]$, ${\bf Var}[Q]$,
$\mathrm{Inf}_i(Q)$, and $\mathbb{S}_\rho(Q)$, and in treating $T_\rho$ as a
formal operator on multilinear polynomials:
\begin{definition} \label{def:T_poly}
For $\rho \in [0,1]$ we define the operator $T_\rho$ as acting
formally on multilinear polynomials $Q(x)$ as in~(\ref{eqn:Q}) by
\[
(T_\eta Q)(x) = \sum_{{\boldsymbol \sigma}} \rho^{|{\boldsymbol \sigma}|} c_{\boldsymbol \sigma}
x_{\boldsymbol \sigma}.
\]
\end{definition}
Note that for every sequence of ensembles, we have that
Definition~\ref{def:T_poly} agrees with Definition~\ref{def:T_general}.
\bigskip
We end this section with a short discussion of ``low-degree
influences'', a notion that has proven crucial in the analysis of
PCPs (see, e.g., \cite{KKMO:04}).
\begin{definition} \label{def:low-degree-influence} The \emph{$d$-low-degree influence of the $i$th ensemble on $Q({\boldsymbol{\mathcal{X}}})$}
is
\[
\mathrm{Inf}^{\leq d}_i(Q({\boldsymbol{\mathcal{X}}})) = \mathrm{Inf}^{\leq d}_i(Q) = \sum_{{\boldsymbol \sigma}:
|{\boldsymbol \sigma}| \leq d, \sigma_i > 0} c_{\boldsymbol \sigma}^2.
\]
Note that this gives a way to define low-degree influences
$\mathrm{Inf}^{\leq d}_i(f)$ for functions $f : \Omega_1 \times \cdots
\Omega_n \to \mathbb R$ on finite product spaces.
\end{definition}
There isn't an especially natural interpretation of $\mathrm{Inf}_i^{\leq
d}(f)$. However, the notion is important for PCPs due to the fact
that a function with variance $1$ cannot have too many coordinates
with substantial low-degree influence; this is reflected in the
following easy proposition:
\begin{proposition} \label{prop:infD}
Suppose $Q$ is multilinear polynomial as in~(\ref{eqn:Q}). Then
\[
\sum_i \mathrm{Inf}_i^{\leq d}(Q) \leq d \cdot {\bf Var}[Q].
\]
\end{proposition}
\ignore{
\begin{proposition} $\mathbb{S}_\rho(Q) = {\bf E}[(T_{\sqrt{\rho}} Q)^2]$.
\end{proposition}
} \ignore{
\begin{remark}
Note that for a general sequence of ensembles ${\boldsymbol{\mathcal{X}}}$ it holds that
$(T_\eta Q)(x)$ is the expected value of $Q(y)$, where for each
ensemble $i$ independently with probability $\eta$ it holds that
$x_{i,j} = y_{i,j}$ for all $j$ and with probability $1-\eta$ the
vector $(y_{i,j})_j$ is drawn from ${\boldsymbol{\mathcal{X}}}_i$.
Note furthermore, that for ${\boldsymbol{\mathcal{G}}}$, the operator $T_{\eta}$ is the
usual Ornstein-Uhlenbeck operator
\[
(T_\rho f)(x) = \mathop{\bf E\/}_{y}[f(\rho x + \sqrt{1-\rho^2}\, y)],
\]
where the expected value is with respect to the Gaussian measure.
\end{remark}
}
\subsection{Hypercontractivity}
As mentioned in Section~\ref{sec:intro-invariance}, our invariance
principle requires that the ensembles involved to be
hypercontractive in a certain sense. Recall that a random
variable ${\boldsymbol Y}$ is said to be ``$(p,q,\eta)$-hypercontractive''
for $1 \leq p \leq q < \infty$ and $0 < \eta < 1$ if
\begin{equation} \label{eqn:easy-hc}
\|a + \eta {\boldsymbol Y}\|_q \leq \|a + {\boldsymbol Y}\|_p
\end{equation}
for all $a \in \mathbb R$. This type of hypercontractivity was
introduced (with slightly different notation)
in~\cite{KrakowiakSzulga:88}. Some basic facts about
hypercontractivity are explained in Appendix~\ref{app:hc}; much
more can be found in \cite{KwapienWoyczynski:92}. Here we just
note that for $q>2$ a random variable ${\boldsymbol Y}$ is
$(2,q,\eta)$-hypercontractive with some $\eta \in (0,1)$ if and
only if ${\bf E}[{\boldsymbol Y}]=0$ and ${\bf E}[|{\boldsymbol Y}|^{q}]<\infty.$ Also, if
${\boldsymbol Y}$ is $(2,q,\eta)$-hypercontractive then $\eta \leq
(q-1)^{-1/2}.$\\
We now define our extension of the notion of hypercontractivity to
sequences of ensembles:
\begin{definition}
Let ${\boldsymbol{\mathcal{X}}}$ be a sequence of ensembles. For $1 \leq p \leq q <
\infty$ and $0 < \eta < 1$ we say that ${\boldsymbol{\mathcal{X}}}$ is
\emph{$(p,q,\eta)$-hypercontractive} if
\[
\|(T_\eta Q)({\boldsymbol{\mathcal{X}}})\|_q \leq \|Q({\boldsymbol{\mathcal{X}}})\|_p
\]
for every multilinear polynomial $Q$ over ${\boldsymbol{\mathcal{X}}}$.
\end{definition}
Since $T_{\eta}$ is a contractive semi-group, we have
\begin{remark} If ${\boldsymbol{\mathcal{X}}}$ is $(p,q,\eta)$-hypercontractive then it
is $(p,q,\eta')$-hypercontractive for any $0 < \eta' \leq \eta$.
\end{remark}
There is a related notion of hypercontractivity for \emph{sets} of
random variables which considers all polynomials in the variables,
not just multilinear polynomials; see, e.g.,
Janson~\cite{Janson:97}. Several of the properties of this notion
of hypercontractivity carry over to our setting of sequences of
ensembles. In particular, the following facts can easily be
proved by repeating the analogous proofs in~\cite{Janson:97}; for
completeness, we give the proofs in Appendix~\ref{app:hc}.
\begin{proposition} \label{prop:join-hypercon}
Suppose ${\boldsymbol{\mathcal{X}}}$ is a sequence of $n_1$ ensembles and ${\boldsymbol{\mathcal{Y}}}$ is an
independent sequence of $n_2$ ensembles. Assume both are
$(p,q,\eta)$-hypercontractive. Then the sequence of ensembles
${\boldsymbol{\mathcal{X}}} \cup {\boldsymbol{\mathcal{Y}}} = ({\boldsymbol{\mathcal{X}}}_1, \dots, {\boldsymbol{\mathcal{X}}}_{n_1}, {\boldsymbol{\mathcal{Y}}}_1, \dots,
{\boldsymbol{\mathcal{Y}}}_{n_2})$ is also $(p,q,\eta)$-hypercontractive.
\end{proposition}
\begin{proposition} \label{prop:hypercon}
Let ${\boldsymbol{\mathcal{X}}}$ be a $(2,q,\eta)$-hypercontractive sequence of
ensembles and $Q$ a multilinear polynomial over ${\boldsymbol{\mathcal{X}}}$ of degree
$d$. Then
\[
\|Q({\boldsymbol{\mathcal{X}}})\|_q \leq \eta^{-d} \; \|Q({\boldsymbol{\mathcal{X}}})\|_2.
\]
\end{proposition}
In light of Proposition~\ref{prop:join-hypercon}, to check that a
sequence of ensembles is $(p,q,\eta)$-hypercontractive it is
enough to check that each ensemble individually is
$(p,q,\eta)$-hypercontractive (as a ``sequence'' of length 1); in
turn, it is easy to see that this is equivalent to checking that
for each $i$, all linear combinations of the random variables
${\boldsymbol X}_{i,1}, \dots, {\boldsymbol X}_{i, m_i}$ are hypercontractive in the
traditional sense of~(\ref{eqn:easy-hc}).\\
We end this section by recording the optimal hypercontractivity
constants for the ensembles we consider. The result for $\pm 1$
Rademacher variables is well known and due originally to
Bonami~\cite{Bonami:70} and independently
Beckner~\cite{Beckner:75}; the same result for Gaussian and uniform
random variables is also well known and in fact follows easily from the
Rademacher case. The optimal hypercontractivity constants for general
finite spaces was recently determined by Wolff~\cite{Wolff:u}
(see also~\cite{Oleszkiewicz:03}):
\begin{theorem} \label{thm:bonami} Let ${\boldsymbol X}$ denote either a uniformly random $\pm 1$
bit, a standard one-dimensional Gaussian, or a random variable
uniform on $[-\sqrt{3}, \sqrt{3}]$. Then ${\boldsymbol X}$ is $(2, q,
(q-1)^{-1/2})$-hypercontractive.
\end{theorem}
\begin{theorem} \label{thm:wolff} (Wolff)\ \ Let ${\boldsymbol X}$ be any
mean-zero random variable on a finite probability space in which
the minimum nonzero probability of any atom is $\alpha \leq 1/2$.
Then ${\boldsymbol X}$ is $(2, q, \eta_q(\alpha))$-hypercontractive, where
\[
\eta_q(\alpha) = \left(\frac{A^{1/q'} - A^{-1/q'}}{A^{1/q} -
A^{-1/q}} \right)^{-1/2}
\]
\[
\text{with } \quad A = \frac{1-\alpha}{\alpha}, \quad 1/q + 1/q' =
1.
\]
\end{theorem}
Note the following special case:
\begin{proposition} \label{prop:wolff}
\[
\eta_3(\alpha) = \left(A^{1/3} + A^{-1/3}\right)^{-1/2} \;\;
\mathop{\sim}^{\alpha \to 0} \quad \alpha^{1/6},
\]
and also
\[
{\textstyle \frac12} \alpha^{1/6} \leq \eta_3(\alpha) \leq 2^{-1/2},
\]
for all $\alpha \in [0,1/2]$.
\end{proposition}
For general random variables with bounded moments we have the
following results, proved in Appendix~\ref{app:hc}:
\begin{proposition} \label{prop:bdd}
Let ${\boldsymbol X}$ be a mean-zero random variable satisfying
${\bf E}[|{\boldsymbol X}|^{q}]< \infty$. Then ${\boldsymbol X}$ is
$(2,q,\eta_{q})$-hypercontractive with $\eta_{q}=\frac{\| {\boldsymbol X}
\|_{2}}{2\sqrt{q-1}\| {\boldsymbol X} \|_{q}}.$
\end{proposition}
In particular, when ${\bf E}[{\boldsymbol X}] = 0$, ${\bf E}[{\boldsymbol X}^{2}]=1$, and
${\bf E}[|{\boldsymbol X}|^{3}] \leq \beta$, we have that ${\boldsymbol X}$ is
$(2,3,2^{-3/2}\beta^{-1/3})$-hypercontractive.
\begin{proposition} \label{prop:add}
Let ${\boldsymbol X}$ be a mean-zero random variable satisfying
${\bf E}[|{\boldsymbol X}|^{q}]< \infty$ and let ${\boldsymbol V}$ be a random variable
independent of ${\boldsymbol X}$ with $\P[{\boldsymbol V}=0]=1-\rho$ and
$\P[{\boldsymbol V}=1]=\rho.$ Then ${\boldsymbol V}{\boldsymbol X}$ is
$(2,q,\xi_{q})$-hypercontractive with $\xi_{q}=\frac{\| {\boldsymbol X}
\|_{2}}{2\sqrt{q-1}\| {\boldsymbol X} \|_{q}} \cdot
\rho^{\frac{1}{2}-\frac{1}{q}}.$
\end{proposition}
\subsection{Hypotheses for invariance theorems --- some families of ensembles}
All of the variants of our invariance principle that we prove in
this section will have similar hypotheses. Specifically, they
will be concerned with a multilinear polynomial $Q$ over two
hypercontractive sequences of ensembles, ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$;
furthermore, ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$ will be assumed to have satisfy a
``matching moments'' condition, as described below. We will now
lay out four hypotheses --- ${\boldsymbol{H1}},\boldsymbol{H2}, \boldsymbol{H3}$, and $\boldsymbol{H4}$
that will be used in the theorems of this section. As can easily
be seen (using Theorems~\ref{thm:bonami} and~\ref{thm:wolff} and
Proposition~\ref{prop:wolff}; see also Appendix~\ref{app:hc}), the
hypothesis ${\boldsymbol{H1}}$ generalizes $\boldsymbol{H2}, \boldsymbol{H3}$, and $\boldsymbol{H4}$;
hence all proofs will be carried out only in the setting of
${\boldsymbol{H1}}$. However the amount of notation and number of parameters
under ${\boldsymbol{H1}}$ is quite cumbersome, and the reader who is
interested mainly in functions on finite product spaces
($\boldsymbol{H3}$) or just boolean functions where $\{-1,1\}^n$ has the
uniform distribution ($\boldsymbol{H4}$) may find it easier to proceed
through the
proofs and results in the restricted cases.\\
Herewith our hypotheses:
\begin{enumerate}
\item[${\boldsymbol{H1}}$]
Let $r \geq 3$ be an integer and let ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$ be
independent sequences of $n$ ensembles which are
$(2,r,\eta)$-hypercontractive; recall that $\eta \leq
(r-1)^{-1/2}$. Assume furthermore that for all $1 \leq i \leq n$
and all sets $\Sigma \subset \mathbb N$ with $|\Sigma| < r$, the
sequences ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$ satisfy the ``matching moments''
condition
\begin{equation} \label{eq:mixed_r_moments}
{\bf E} \left[\prod_{\sigma \in \Sigma} {\boldsymbol X}_{i,\sigma} \right] =
{\bf E} \left[\prod_{\sigma \in \Sigma} {\boldsymbol Y}_{i,\sigma} \right].
\end{equation}
Finally, let $Q$ be a multilinear polynomial as
in~(\ref{eqn:Q}).\\
We remark that in ${\boldsymbol{H1}}$, if $r = 3$ then the matching moment
conditions hold automatically since the sequences are orthonormal.
We also remark that we have added the condition $\eta \leq
(r-1)^{-1/2}$ so that we can take ${\boldsymbol{\mathcal{Y}}} = {\boldsymbol{\mathcal{G}}}$, the Gaussian
sequence of ensembles (see Theorem~\ref{thm:bonami}).
\item[$\boldsymbol{H2}$] Let $r = 3.$ Let ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$ be independent
sequences of ensembles in which each ensemble has only two random
variables, ${\boldsymbol X}_{i,0} = 1$ and ${\boldsymbol X}_{i,1} = {\boldsymbol X}_i$
(respectively, ${\boldsymbol Y}_{i,0} = 1$, ${\boldsymbol Y}_{i,1} = {\boldsymbol Y}_i$), as
in Remark~\ref{rem:simple}. Further assume that each ${\boldsymbol X}_i$
(respectively ${\boldsymbol Y}_i$) satisfies ${\bf E}[{\boldsymbol X}_i]=0,$
${\bf E}[{\boldsymbol X}_i^{2}]=1$ and ${\bf E}[|{\boldsymbol X}_i|^{3}] \leq \beta.$ Put
$\eta=2^{-3/2}\beta^{-1/3}$, so ${\boldsymbol{\mathcal{X}}}$ and ${\boldsymbol{\mathcal{Y}}}$ are
$(2,3,\eta)$-hypercontractive. Finally, let $Q$ be a multilinear
polynomial as in~(\ref{eqn:Q}).\\
The hypothesis $\boldsymbol{H2}$ is used to derive the multilinear version
of the Berry-Esseen inequality given in Theorem~\ref{thm:simple}.
\item[$\boldsymbol{H3}$] Let $r = 3$ and let ${\boldsymbol{\mathcal{X}}}$ be a sequence of $n$
ensembles in which the random variables in each ensemble ${\boldsymbol{\mathcal{X}}}_i$
form a basis for the real-valued functions on some finite
probability space $\Omega_i$. Further assume that the least
nonzero probability of any atom in any $\Omega_i$ is $\alpha \leq
1/2$, and let $\eta = {\textstyle \frac12} \alpha^{1/6}$. Let ${\boldsymbol{\mathcal{Y}}}$ be any
independent $(2,3,\eta)$-hypercontractive sequence of ensembles.
Finally, let $Q$ be a multilinear polynomial as
in~(\ref{eqn:Q}).\\
We remark that $Q({\boldsymbol{\mathcal{X}}})$ in $\boldsymbol{H3}$ encompasses \emph{all}
real-valued functions $f$ on finite product spaces, including
the familiar cases of the $p$-biased discrete cube (for which
$\alpha = \min\{p, 1-p\}$) and the set $[q]^n$ with uniform
measure (for which $\alpha = 1/q$). Note also that $\eta \leq
2^{-1/2}$ so we may take ${\boldsymbol{\mathcal{Y}}}$ to be the Gaussian sequence of
ensembles.
\item[$\boldsymbol{H4}$] Let $r = 4$ and $\eta = 3^{-1/2}$. Let ${\boldsymbol{\mathcal{X}}}$
and ${\boldsymbol{\mathcal{Y}}}$ be independent sequences of ensembles in which each
ensemble has only two random variables, ${\boldsymbol X}_{i,0} = 1$ and
${\boldsymbol X}_{i,1} = {\boldsymbol X}_i$ (respectively, ${\boldsymbol Y}_{i,0} = 1$,
${\boldsymbol Y}_{i,1} = {\boldsymbol Y}_i$), as in Remark~\ref{rem:simple}. Further
assume that each ${\boldsymbol X}_i$ (respectively ${\boldsymbol Y}_i$) is either a)
a uniformly random $\pm 1$ bit; b) a standard one-dimensional
Gaussian; or c) uniform on $[-3^{1/2}, 3^{1/2}]$. Hence ${\boldsymbol{\mathcal{X}}}$
and ${\boldsymbol{\mathcal{Y}}}$ are $(2,4,\eta)$-hypercontractive. Finally, let $Q$ be
a multilinear polynomial as
in~(\ref{eqn:Q}).\\
Note that this simplest of all hypotheses allows for arbitrary
real-valued functions on the uniform-measure discrete cube $f :
\{-1,1\}^n \to \mathbb R$. Also, under $\boldsymbol{H4}$, $Q$ is just a multilinear
polynomial in the usual sense over the ${\boldsymbol X}_i$'s or
${\boldsymbol Y}_i$'s; in particular, if $f : \{-1,1\}^n \to \mathbb R$ then $Q$ is
the ``Fourier expansion'' of $f$. Finally, note that the matching
moments condition~(\ref{eq:mixed_r_moments}) holds in $\boldsymbol{H4}$
since it requires ${\bf E}[X_t^3] = {\bf E}[Y_t^3]$ for each $t$, and this
is true since both equal $0$.
\end{enumerate}
\subsection{Basic invariance principle, ${\mathcal{C}}^r$ functional version}
The essence of our invariance principle is that if $Q$ is of
bounded degree and has low influences then the random variables
$Q({\boldsymbol{\mathcal{X}}})$ and $Q({\boldsymbol{\mathcal{Y}}})$ are close in distribution. The simplest
way to formulate this conclusion is to say that if $\Psi : \mathbb R
\to \mathbb R$ is a sufficiently nice ``test function'' then
$\Psi(Q({\boldsymbol{\mathcal{X}}}))$ and $\Psi(Q({\boldsymbol{\mathcal{Y}}}))$ are close in
expectation.
\begin{theorem} \label{thm:cj} Assume hypothesis ${\boldsymbol{H1}},\boldsymbol{H2},\boldsymbol{H3}$, or
$\boldsymbol{H4}.$ Further assume ${\bf Var}[Q] \leq 1$, $\deg(Q) \leq d$, and $\mathrm{Inf}_i(Q)
\leq \tau$ for all $i$. Let $\Psi : \mathbb R \to \mathbb R$ be a ${\mathcal{C}}^r$
function with $|\Psi^{(r)}| \leq B$ uniformly. Then
\[
\Bigl| {\bf E}\bigl[\Psi(Q({\boldsymbol{\mathcal{X}}}))\bigr] -
{\bf E}\bigl[\Psi(Q({\boldsymbol{\mathcal{Y}}}))\bigr] \Bigr| \leq \epsilon,
\]
where
\[
\epsilon = \left\{ \begin{array}{ll}
(2B/r!)\,d\,\eta^{-rd}\;\tau^{r/2 - 1} & \text{under } {\boldsymbol{H1}}, \\
B\,30^d\;\beta^{d}\;\tau^{1/2} & \text{under } \boldsymbol{H2}, \\
B\,(10\alpha^{-1/2})^d\;\tau^{1/2} & \text{under } \boldsymbol{H3}, \\
B\,10^d\;\tau & \text{under } \boldsymbol{H4}.
\end{array} \right.
\]
\end{theorem}
As will be the case in all of our theorems, the results under
$\boldsymbol{H2},\boldsymbol{H3}$ and $\boldsymbol{H4}$ are immediate corollaries of the
result under ${\boldsymbol{H1}}$; one only needs to substitute in $r=3$,
$\eta=2^{-3/2}\beta^{-1/3}$ or $r = 3$, $\eta = {\textstyle \frac12}
\alpha^{1/6}$ or $r = 4$, $\eta = 3^{-1/2}$ (we have also here
used that $(1/3)\,d\,2^{9d/2}$ is at most $30^d$ and that
$(1/3)\,d\,8^d$ and $(1/12)\,d\,9^d$ are at most $10^d$). Thus it
will suffice for us to carry out the proof under ${\boldsymbol{H1}}$.\\
\begin{proof}
We begin by defining intermediate sequences between ${\boldsymbol{\mathcal{X}}}$ and
${\boldsymbol{\mathcal{Y}}}$. For $i = 0, 1, \dots, n$, let ${\boldsymbol{\mathcal{Z}}}^{(i)}$ denote the
sequence of $n$ ensembles $({\boldsymbol{\mathcal{Y}}}_1, \dots, {\boldsymbol{\mathcal{Y}}}_i, {\boldsymbol{\mathcal{X}}}_{i+1},
\dots, {\boldsymbol{\mathcal{X}}}_n)$ and let ${\boldsymbol Q}^{(i)} = Q({\boldsymbol{\mathcal{Z}}}^{(i)})$. Our
goal will be to show
\begin{equation} \label{eqn:bound}
\Bigl|{\bf E}\bigl[\Psi({\boldsymbol Q}^{(i-1)})\bigr] -
{\bf E}\bigl[\Psi({\boldsymbol Q}^{(i)})\bigr]\Bigr| \leq
\left(\frac{2B}{r!}\,\eta^{-rd}\right)\cdot\mathrm{Inf}_i(Q)^{r/2}
\end{equation}
for each $i \in [n]$. Summing this over $i$ will complete the
proof since ${\boldsymbol{\mathcal{Z}}}^{(0)} = {\boldsymbol{\mathcal{X}}}$, ${\boldsymbol{\mathcal{Z}}}^{(n)} = {\boldsymbol{\mathcal{Y}}}$, and
\[
\sum_{i=1}^n \mathrm{Inf}_i(Q)^{r/2} \leq \tau^{r/2-1} \cdot \sum_{i=1}^n
\mathrm{Inf}_i(Q) = \tau^{r/2-1} \cdot \sum_{i=1}^n \mathrm{Inf}_i^{\leq d}(Q)
\leq d \tau^{r/2-1},
\]
where we used Proposition~\ref{prop:infD} and ${\bf Var}[Q] \leq 1$.\\
\newcommand{\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS}}{\tilde{{{\boldsymbol Q}}}} \newcommand{\Rt}{{{\boldsymbol R}}} \newcommand{\St}{{{\boldsymbol S}}}
Let us fix a particular $i \in [n]$ and proceed to
prove~(\ref{eqn:bound}). Given a multi-index ${\boldsymbol \sigma}$, write
${\boldsymbol \sigma} \setminus i$ for the same multi-index except with
$\sigma_i = 0$. Now write
\begin{eqnarray*}
\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} &=& \sum_{{\boldsymbol \sigma} : \sigma_i = 0} c_{\boldsymbol \sigma}
{\boldsymbol{\mathcal{Z}}}^{(i)}_{\boldsymbol \sigma}, \\
\Rt &=& \sum_{{\boldsymbol \sigma} : \sigma_i > 0} c_{\boldsymbol \sigma}
{\boldsymbol X}_{i,\sigma_i} \cdot {\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma} \setminus i}, \\
\St &=& \sum_{{\boldsymbol \sigma} : \sigma_i > 0} c_{\boldsymbol \sigma}
{\boldsymbol Y}_{i,\sigma_i} \cdot {\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma} \setminus i}.
\end{eqnarray*}
Note that $\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS}$ and the variables ${\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma}
\setminus
i}$ are independent of the variables in ${\boldsymbol{\mathcal{X}}}_i$ and ${\boldsymbol{\mathcal{Y}}}_i$ and
that ${\boldsymbol Q}^{(i-1)} = \tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \Rt$ and ${\boldsymbol Q}^{(i)} = \tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \St$.
\\
To bound the left side of~(\ref{eqn:bound}) --- i.e.,
$|{\bf E}[\Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \Rt) - \Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \St)]|$ --- we use
Taylor's theorem: for all $x, y \in \mathbb R$,
\[
\Bigl|\Psi(x+y) - \sum_{k=0}^{r-1}
\frac{\Psi^{(k}(x)\;y^k}{k!}\Bigr| \leq \frac{B}{r!}\,|y|^r.
\]
In particular,
\begin{equation} \label{eq:R_taylor}
\Bigl|{\bf E}[\Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \Rt)] - \sum_{k=0}^{r-1}
{\bf E}\Bigl[\frac{\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\;\Rt^k}{k!}\Bigr]\Bigr| \leq
\frac{B}{r!}\,{\bf E}\bigl[|\Rt|^r\bigr]
\end{equation}
and similarly,
\begin{equation} \label{eq:S_taylor}
\Bigl|{\bf E}[\Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \St)] - \sum_{k=0}^{r-1}
{\bf E}\Bigl[\frac{\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\;\St^k}{k!}\Bigr]\Bigr| \leq
\frac{B}{r!}\,{\bf E}\bigl[|\St|^r\bigr].
\end{equation}
We will see below that that $\Rt$ and $\St$ have finite $r$ moments.
Moreover,
for $0 \leq k
\leq r$ it holds that $|\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\,\Rt^k| \leq
|k!\,B\,\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS}^{r-k}\,\Rt^k|$ (and similarly for $\St$). Thus all
moments above are finite.
We now claim
that for all $0 \leq k < r$ it holds that
\begin{equation} \label{eq:S_T_moments}
{\bf E}[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\,\Rt^k] = {\bf E}[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\,\St^k].
\end{equation}
Indeed,
\begin{eqnarray} \nonumber
{\bf E}[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\,\Rt^k] &=& {\bf E} \Bigl[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})
\sum_{\substack{({\boldsymbol \sigma}^1,\dots,{\boldsymbol \sigma}^k) \\ \text{s.t. }
\forall t,\;\;\sigma^t_i > 0}} \prod_{t=1}^k c_{{\boldsymbol \sigma}^t}
\prod_{t=1}^k {\boldsymbol X}_{i,\sigma^t_i}
\prod_{t=1}^k {\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma}^t \setminus i} \Bigr] \\
&=& \label{eq:clt_ind}
\sum_{\substack{({\boldsymbol \sigma}^1,\dots,{\boldsymbol \sigma}^k) \\ \text{s.t. }
\forall t,\;\;\sigma^t_i > 0}} \prod_{t=1}^k c_{{\boldsymbol \sigma}^i}
\cdot {\bf E} \Bigl[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})
\prod_{t=1}^k {\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma}^t \setminus i} \Bigr]
\cdot {\bf E}\Bigl[\prod_{t=1}^k {\boldsymbol X}_{i,\sigma^t_i}\Bigr] \\
&=& \label{eq:clt_same_dist}
\sum_{\substack{({\boldsymbol \sigma}^1,\dots,{\boldsymbol \sigma}^k) \\ \text{s.t. }
\forall t,\;\;\sigma^t_i > 0}} \prod_{t=1}^k c_{{\boldsymbol \sigma}^i}
\cdot {\bf E} \Bigl[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})
\prod_{t=1}^k {\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma}^t \setminus i} \Bigr]
\cdot {\bf E}\Bigl[\prod_{t=1}^k {\boldsymbol Y}_{i,\sigma^t_i}\Bigr] \\
&=& \nonumber {\bf E} \left[\Psi^{(k)}(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS})\,\St^k \right].
\end{eqnarray}
\noindent The equality in (\ref{eq:clt_ind}) follows since
${\boldsymbol{\mathcal{Z}}}^{(i)}_{{\boldsymbol \sigma}^t \setminus i}$ and $\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS}$ are independent
of the variables in ${\boldsymbol{\mathcal{X}}}_i$ and ${\boldsymbol{\mathcal{Y}}}_i$. The equality in
(\ref{eq:clt_same_dist}) follows from the matching moments
condition~(\ref{eq:mixed_r_moments}).\\
From (\ref{eq:R_taylor}), (\ref{eq:S_taylor}) and
(\ref{eq:S_T_moments}) it follows that
\begin{equation} \label{eq:taylor_bd}
|{\bf E}[\Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \Rt) - \Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \St)]| \leq \frac{B}{r!}\,({\bf E}[|\Rt|^r] +
{\bf E}[|\St|^r]).
\end{equation}
We now use hypercontractivity. By
Proposition~\ref{prop:join-hypercon} each ${\boldsymbol{\mathcal{Z}}}^{(i)}$ is
$(2,r,\eta)$-hypercontractive. Thus by
Proposition~\ref{prop:hypercon},
\begin{equation} \label{eq:R_S_hyper}
{\bf E}[|\Rt|^r] \leq \eta^{-rd} {\bf E}[\Rt^2]^{r/2}, \quad
{\bf E}[|\St|^r] \leq \eta^{-rd} {\bf E}[\St^2]^{r/2}.
\end{equation}
However,
\begin{equation} \label{eq:S_R_inf}
{\bf E}[\St^2] = {\bf E}[\Rt^2] = \sum_{{\boldsymbol \sigma} : \sigma_i > 0}
c_{\boldsymbol \sigma}^2 = \mathrm{Inf}_i(Q).
\end{equation}
Combining (\ref{eq:taylor_bd}), (\ref{eq:R_S_hyper}) and
(\ref{eq:S_R_inf}) it follows that
\[
|{\bf E}[\Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \Rt) - \Psi(\tilde{{\boldQ}}} \newcommand{\Rt}{{\boldR}} \newcommand{\St}{{\boldS} + \St)]| \leq
\left(\frac{2B}{r!}\,\eta^{-rd}\right)\cdot\mathrm{Inf}_i(Q)^{r/2}
\]
confirming~(\ref{eqn:bound}) and completing the proof.
\end{proof}
\subsection{Invariance principle --- other functionals, and smoothed version}
Our basic invariance principle shows that ${\bf E}[\Psi(Q({\boldsymbol{\mathcal{X}}}))]$
and ${\bf E}[\Psi(Q({\boldsymbol{\mathcal{Y}}}))]$ are close if $\Psi$ is a
${\mathcal{C}}^r$ functional with bounded $r$th derivative. To show that
the distributions of $Q({\boldsymbol{\mathcal{X}}})$ and $Q({\boldsymbol{\mathcal{Y}}})$ are close in other
senses we need the invariance principle for less smooth
functionals. This we can obtain using straightforward
approximation arguments; we defer the proof of Theorem~\ref{thm:supertheorem}
which follows to Section~\ref{sec:proofs}.\\
Theorem~\ref{thm:supertheorem} shows closeness of distribution in
two senses. The first is closeness in \emph{L\'{e}vy's metric};
recall that the distance between two random variables ${\boldsymbol R}$ and
${\boldsymbol S}$ in L\'{e}vy's metric is
\[
d_L({\boldsymbol R}, {\boldsymbol S}) = \inf\{\lambda > 0 : \quad
\forall t \in \mathbb R,\;\; \Pr[{\boldsymbol S} \leq t - \lambda] - \lambda \leq
\Pr[{\boldsymbol R} \leq t] \leq \Pr[{\boldsymbol S} \leq t + \lambda] + \lambda\}.
\]
We also show the distributions are close in the usual sense with a
weaker bound; the proof of this goes by comparing the
distributions of $Q({\boldsymbol{\mathcal{X}}})$ and $Q({\boldsymbol{\mathcal{Y}}})$ to $Q({\boldsymbol{\mathcal{G}}})$ and
noting that bounded-degree Gaussian polynomials are known to have
low ``small ball probabilities''. Finally,
Theorem~\ref{thm:supertheorem} also shows $L^1$ closeness and, as
a technical necessity for applications, shows closeness under the
functional $\zeta : \mathbb R \to \mathbb R$ defined by
\begin{equation} \label{eq:def_trunc}
\zeta(x) = \left\{ \begin{array}{ll}
x^2 & \mbox{ if } x \leq 0,\\
0 & \mbox{ if } x \in [0,1], \\
(x-1)^2 & \mbox{ if } x \geq 1;
\end{array} \right.
\end{equation}
this functional gives the squared distance to the interval
$[0,1]$.\\
\begin{theorem} \label{thm:supertheorem} Assume Hypothesis
${\boldsymbol{H1}},\boldsymbol{H2},\boldsymbol{H3}$, or $\boldsymbol{H4}$. Further assume ${\bf Var}[Q] \leq
1$, $\deg(Q) \leq d$ and $\mathrm{Inf}_i(Q) \leq \tau$ for all $i$. Then
\begin{eqnarray}
\Bigl| \|Q({\boldsymbol{\mathcal{X}}})\|_1 - \|Q({\boldsymbol{\mathcal{Y}}})\|_1 \Bigr| &\leq& O(\epsilon^{1/r}), \label{eq:superl1} \\
d_L(Q({\boldsymbol{\mathcal{X}}}), Q({\boldsymbol{\mathcal{Y}}})) &\leq& O(\epsilon^{1/(r+1)}), \label{eq:superlevi} \\
\Bigl| {\bf E}\bigl[\zeta(Q({\boldsymbol{\mathcal{X}}}))\bigr] - {\bf E}\bigl[\zeta(Q({\boldsymbol{\mathcal{Y}}}))\bigr] \Bigl| & \leq &
O(\epsilon^{2/r}),
\label{eq:superl2}
\end{eqnarray}
where $O(\cdot)$ hides a constant depending only on $r$, and
\[
\epsilon = \left\{ \begin{array}{ll}
d\,\eta^{-rd}\;\tau^{r/2 - 1} & \text{under } {\boldsymbol{H1}},\\
30^{d}\beta^{d}\;\tau^{1/2} & \text{under } \boldsymbol{H2},\\
(10\alpha^{-1/2})^d\;\tau^{1/2} & \text{under } \boldsymbol{H3}, \\
10^d\;\tau & \text{under } \boldsymbol{H4}.\\
\end{array} \right.
\]
If in addition ${\bf Var}[Q] = 1$ then
\begin{equation} \label{eq:lim_dist}
\sup_t\;\Bigl|\P\bigl[Q({\boldsymbol{\mathcal{X}}}) \leq t\bigr] - \P\bigl[Q({\boldsymbol{\mathcal{Y}}}) \leq t\bigr]\Bigr| \leq
O\bigl(d\,\epsilon^{1/(rd+1)}\bigr).
\end{equation}
\end{theorem}
\bigskip
As discussed in Section~\ref{sec:intro-invariance},
Theorem~\ref{thm:supertheorem} has the unavoidable deficiency of
having error bounds depending on the degree $d$ of $Q$. This can
be overcome if we first ``smooth'' $Q$ by applying $T_{1 -
\gamma}$ to it, for some $0 < \gamma < 1$.
Theorem~\ref{thm:smooththeorem} which follows will be our main
tool for applications; its proof is a straightforward degree
truncation argument which we also defer to
Section~\ref{sec:proofs}. As an additional benefit of this
argument, we will show that $Q$ need only have small
\emph{low-degree influences}, $\mathrm{Inf}_i^{\leq d}(Q)$, as opposed to
small influences. As discussed at the end of
Section~\ref{sec:setup}, this feature has proven essential for
applications involving PCPs.
\begin{theorem} \label{thm:smooththeorem}
Assume hypothesis ${\boldsymbol{H1}}, \boldsymbol{H3}$,or $\boldsymbol{H4}$. Further assume
${\bf Var}[Q] \leq 1$ and \linebreak $\mathrm{Inf}_i^{\leq\,\log(1/\tau)/K}(Q)
\leq \tau \leq$ for all $i$, where
\[
K = \left\{ \begin{array}{ll}
\log(1/\eta) & \text{under } {\boldsymbol{H1}}, \\
\log(1/\alpha) & \text{under } \boldsymbol{H3}, \\
1 & \text{under } \boldsymbol{H4}.
\end{array} \right.
\]
Given $0 < \gamma < 1$, write ${\boldsymbol R} = (T_{1-\gamma} Q)({\boldsymbol{\mathcal{X}}})$
and ${\boldsymbol S} = (T_{1-\gamma} Q)({\boldsymbol{\mathcal{Y}}})$. Then
\begin{eqnarray*}
d_L({\boldsymbol R}, {\boldsymbol S}) & \leq & \tau^{\Omega(\gamma/K)}, \\
\Bigl|{\bf E}\bigl[\zeta({\boldsymbol R})\bigr] -
{\bf E}\bigl[\zeta({\boldsymbol S})\bigr]\Bigr| &\leq&
\tau^{\Omega(\gamma/K)},
\end{eqnarray*}
where the $\Omega(\cdot)$ hides a constant depending only on $r$.
More generally the statement of the theorem holds for
${\boldsymbol R} = Q({\boldsymbol{\mathcal{X}}}), {\boldsymbol S}=Q({\boldsymbol{\mathcal{Y}}})$ if
${\bf Var}[Q^{> d}] \leq (1-\gamma)^{2d}$ for all $d$.
\end{theorem}
\ignore{ Here are the painstaking bounds we can actually get:
\begin{eqnarray}
d_L({\boldsymbol R}, {\boldsymbol S}) &\leq& C_r\,\tau^{1/c_1}\cdot[\log(1/\tau)/\log(1/\rho\eta)]^{1/(r+1)}, \label{eq:levi_smooth} \\
|{\bf E}[\zeta({\boldsymbol R})] - {\bf E}[\zeta({\boldsymbol S})]| &
\leq & C_r\,\tau^{1/c_2}\cdot[\log(1/\tau)/\log(1/\rho\eta)]^{2/r}, \label{eq:l2_smooth}
\end{eqnarray}
where
\begin{equation} \label{eq:c1_c2}
c_1 = \frac{3r}{r-2} \cdot \frac{\ln(1/\eta)}{\ln(1/\rho)} +
\frac{2r+2}{r-2},
\qquad c_2 = \frac{2r}{r-2} \cdot \frac{\ln(1/\eta)}{\ln(1/\rho)} +
\frac{r}{r-2}.
\end{equation}
}
\subsection{Proofs of extensions of the invariance principle} \label{sec:proofs}
In this section we will prove Theorems~\ref{thm:supertheorem}
and~\ref{thm:smooththeorem} under hypothesis ${\boldsymbol{H1}}$. The results
under $\boldsymbol{H2},\boldsymbol{H3}$, and $\boldsymbol{H4}$ are corollaries.
\subsubsection{Invariance principle for some $C^0$ and $C^1$
functionals} \label{sec:convolve} In this section we
prove~(\ref{eq:superl1}), (\ref{eq:superlevi}), (\ref{eq:superl2})
of Theorem~\ref{thm:supertheorem}. We do it by approximating the
following functions in the sup norm by smooth functions:
\[\begin{array}{lll}
\ell_1(x) = |x|; &
\Delta_{s,t}(x) = \left\{ \begin{array}{ll}
1 & \mbox{ if } x \leq t-s, \\
\frac{t-x+s}{2s} & \mbox{ if } x \in [t-s,t+s], \\
0 & \mbox{ if } x \geq t+s;
\end{array} \right. &
\zeta(x) = \left\{ \begin{array}{ll}
x^2 & \mbox{ if } x \leq 0,\\
0 & \mbox{ if } x \in [0,1], \\
(x-1)^2 & \mbox{ if } x \geq 1.
\end{array} \right.
\end{array}
\]
\begin{lemma} \label{lem:approx-functional}
Let $r \geq 2$ be an integer. Then there exist constant $B_r$ for
which the following holds. For all $0 < \lambda \leq 1/2$ there exist
${\mathcal{C}}^\infty$ functions $\ell_1^\lambda$, $\Delta_{\lambda,t}^\lambda$
and $\zeta^\lambda$ satisfying the following:
\begin{itemize}
\item $\|\ell_1^\lambda - \ell_1\|_\infty \leq 2\lambda$; and, $\|(\ell_1^\lambda)^{(r)}\|_\infty \leq
4 B_r\,\lambda^{1-r}$.
\item $\Delta_{\lambda,t}^\lambda$ agrees with $\Delta_{\lambda,t}$ outside
the interval $(t-2\lambda, t+2\lambda)$, and is otherwise in $[0,1]$; and, $\|(\Delta_{\lambda,t}^\lambda)^{(r)}\|_\infty \leq
B_r\,\lambda^{-r}$.
\item $\|\zeta^\lambda - \zeta\|_\infty \leq 2\lambda^2$; and, $\|(\zeta^\lambda)^{(r)}\|_\infty \leq
2 B_{r-1} \,\lambda^{2-r}$.
\end{itemize}
\end{lemma}
\begin{proof} Let $f(x) = x 1_{\{x \geq 0\}}$.
We will show that for all $\lambda > 0$ there is a ${\mathcal{C}}^\infty$
function $f_\lambda$ satisfying the following:
\begin{itemize}
\item $f_\lambda$ and $f$ agree on $(-\infty, -\lambda]$ and $[\lambda,
\infty)$;
\item $0 \leq f_\lambda(x) \leq f(x) + \lambda$ on $(-\lambda, \lambda)$;
and,
\item $\|f_\lambda^{(r)}\|_\infty \leq 2 B_r\,\lambda^{1-r}$.
\end{itemize}
The construction of $f$ easily gives the construction of the other
functionals by letting $\ell_1^\lambda(x) = f_\lambda(x) +
f_\lambda(-x)$ and
\begin{equation} \label{eq:delta_lam}
\Delta_{\lambda,t}^\lambda(x) = \left\{ \begin{array}{ll}
\frac{1}{2\lambda}f_\lambda(t-x+\lambda) & \mbox{ if } x \geq t,\\
1 - \frac{1}{2\lambda}f_{\lambda}(x-t+\lambda) & \mbox{ if } x \leq
t; \end{array} \right. \qquad \qquad
\zeta^\lambda(x) = \left\{ \begin{array}{ll}
\int_{-\infty}^{x-1} f_\lambda(t)dt & \mbox{ if } x \geq 1/2,\\
\int_{-\infty}^{1-x} f_\lambda(t)dt& \mbox{ if } x \leq 0.
\end{array} \right.
\end{equation}
To construct $f$, first let $\psi$ be a nonnegative ${\mathcal{C}}^\infty$
function satisfying the following: $\psi$ is $0$ outside $(-1,1)$,
$\int_{-1}^1 \psi(x)\,dx = 1$, and $\int_{-1}^1 x \psi(x)\,dx =
0$.
It is well known that such functions $\psi$ exist. Define the
constant $B_r$ to be $\|\psi^{(r)}\|_\infty$.\\
Next, write $\psi_\lambda(x) = \psi(x/\lambda)/\lambda$, so
$\psi_\lambda$ satisfies the same three properties as $\psi$ with
respect to the interval $(-\lambda, \lambda)$ rather than
$(-1,1)$. Note that $\|\psi_\lambda^{(r)}\|_\infty =
B_r\,\lambda^{-1-r}$.\\
Finally, take $f_\lambda = f * \psi_\lambda$, which is
${\mathcal{C}}^\infty$. The first two properties demanded of $f$ follow
easily. To see the third, first note that $f_\lambda^{(r)}$ is
identically 0 outside $(-\lambda, \lambda)$ and then observe that
for $|x| < \lambda$,
\[
|f_\lambda^{(r)}(x)| = |(f * \psi_\lambda)^{(r)}(x)| = |(f
* \psi_\lambda^{(r)})(x)| \leq \|\psi_\lambda^{(r)}\|_\infty \cdot \int_{x-\lambda}^{x+\lambda}|f|
\leq 2 B_r \lambda^{1-r}.
\]
This completes the proof.
\end{proof}\\
We now prove~(\ref{eq:superl1}), (\ref{eq:superlevi}) and
(\ref{eq:superl2}).\\
\begin{proof}
Note that the properties of $\Delta^\lambda_{\lambda,t}$ imply that
\begin{equation}
\label{eqn:delta-lam-props} \Pr[{\boldsymbol R} \leq t - 2\lambda] \leq
{\bf E}[\Delta_{\lambda, t}^\lambda({\boldsymbol R})] \leq \Pr[{\boldsymbol R} \leq t +
2\lambda]
\end{equation}
holds for every random variable ${\boldsymbol R}$ and every $t$ and $0 <
\lambda \leq 1/2$.\\
Let us first prove~(\ref{eq:superl1}), with
\[
\epsilon = d\,\eta^{-rd}\;\tau^{r/2 - 1}
\]
since we assume ${\boldsymbol{H1}}$. Taking $\Psi =
\ell_1^\lambda$ in Theorem~\ref{thm:cj} we obtain
\begin{multline*}
\Bigl|{\bf E}\bigl[\ell_1(Q({\boldsymbol{\mathcal{X}}}))\bigr]-E\bigl[\ell_1(Q({\boldsymbol{\mathcal{Y}}}))\bigr]\Bigr|
\leq
\Bigl|{\bf E}\bigl[\ell_1^{\lambda}(Q({\boldsymbol{\mathcal{X}}}))\bigr]-E\bigl[\ell_1^{\lambda}(Q({\boldsymbol{\mathcal{Y}}}))\bigl]\Bigl| + 4 \lambda \\
\leq (4 B_r\,\lambda^{1-r} / r!)\,d\,\eta^{-rd}\;\tau^{r/2-1} + 4
\lambda = O(\epsilon\,\lambda^{1-r}) + 4\lambda.
\end{multline*}
Taking $\lambda = \epsilon^{1/r}$, gives the bound~(\ref{eq:superl1}).
Next, using~(\ref{eqn:delta-lam-props}) and applying
Theorem~\ref{thm:cj} with $\Psi = \Delta^\lambda_{\lambda, t}$
we obtain
\begin{multline*}
d_L(Q({\boldsymbol{\mathcal{X}}}), Q({\boldsymbol{\mathcal{Y}}})) \leq \max\left\{4\lambda, \sup_t
\Bigl|{\bf E}\bigl[\Delta^{\lambda}_{\lambda,t}(Q({\boldsymbol{\mathcal{X}}}))\bigr] -
{\bf E}\bigl[\Delta^{\lambda}_{\lambda,t}(Q({\boldsymbol{\mathcal{Y}}}))\bigr]\Bigr|\right\} \\
\leq \max\left\{(B_r\,\lambda^{-r}/r!)\,d\,\eta^{-rd}\;\tau^{r/2-1}, 4 \lambda
\right\} = \max\{O(\epsilon\,\lambda^{-r}), 4\lambda\}.
\end{multline*}
Again taking $\lambda = \epsilon^{1/(r+1)}$ we
achieve~(\ref{eq:superlevi}). Finally, using $\Psi =
\zeta^\lambda$ we get
\begin{multline*}
\Bigl| {\bf E}\bigl[\zeta(Q({\boldsymbol{\mathcal{X}}}))\bigr] -
{\bf E}\bigl[\zeta(Q({\boldsymbol{\mathcal{Y}}}))\bigr] \Bigr| \leq \Bigl|
{\bf E}\bigl[\zeta^{\lambda}(Q({\boldsymbol{\mathcal{X}}}))\bigr] - {\bf E}\bigl[\zeta^{\lambda}(Q({\boldsymbol{\mathcal{Y}}}))\bigr] \Bigr| + 4 \lambda^2\\
\leq (2B_{r-1}\,\lambda^{2-r} / r!)\,d\,\eta^{-rd}\;\tau^{r/2-1} + 4
\lambda^2 = O(\epsilon\,\lambda^{2-r}) + 4\lambda^2,
\end{multline*}
and taking $\lambda = \epsilon^{1/r}$ we get~(\ref{eq:superl2}). This
concludes the proof of the first three bounds in
Theorem~\ref{thm:supertheorem}.
\end{proof}
\subsubsection{Closeness in distribution} \label{cid} We proceed to
prove~(\ref{eq:lim_dist}) from Theorem \ref{thm:supertheorem}. By
losing constant factors it will suffice to prove the bound in the
case that ${\boldsymbol{\mathcal{Y}}} = {\boldsymbol{\mathcal{G}}}$, the sequence of independent Gaussian
ensembles. As mentioned, we will use the fact that bounded-degree
multilinear polynomials over ${\boldsymbol{\mathcal{G}}}$ have low ``small ball
probabilities''. Specifically, the following theorem is an
immediate consequence of Theorem~8 in~\cite{CarberyWright:01}
(taking $q=2d$ in their notation):
\begin{theorem} \label{thm:smallball}
There exists a universal constant $C$ such that for all
multilinear polynomials $Q$ of degree $d$ over ${\boldsymbol{\mathcal{G}}}$ and all
$\epsilon > 0$,
\[
\P[|Q({\boldsymbol{\mathcal{G}}})| \leq \epsilon] \leq C\,d\,(\epsilon/\|Q({\boldsymbol{\mathcal{G}}})\|_2)^{1/d}.
\]
\end{theorem}
Thus we have the following:
\begin{corollary} \label{cor:smallball}
For all multilinear polynomials $Q$ of degree $d$ over ${\boldsymbol{\mathcal{G}}}$
with ${\bf Var}[Q] = 1$ and for all $t \in \mathbb R$ and $\epsilon > 0$,
\[
\P[|Q({\boldsymbol{\mathcal{G}}}) - t| \leq \epsilon] \leq O(d\,\epsilon^{1/d}).
\]
\end{corollary}
We now prove~(\ref{eq:lim_dist}).\\
\begin{proof}
We will use Theorem~\ref{thm:cj} with $\Psi =
\Delta_{\lambda,t}^{\lambda}$, where $\lambda$ will be chosen later.
Writing $\Delta_t = \Delta_{\lambda,t}^{\lambda}$ for brevity and using
fact~(\ref{eqn:delta-lam-props}) twice, we have
\begin{eqnarray}
\P[Q({\boldsymbol{\mathcal{X}}}) \leq t] & \leq & {\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{X}}})] \nonumber\\
& \leq & {\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{G}}})] +
|{\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{X}}})] - {\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{G}}})]| \nonumber\\
& \leq & \P[Q({\boldsymbol{\mathcal{G}}}) \leq t + 4\lambda] +
|{\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{X}}})] - {\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{G}}})]| \nonumber\\
& = & \P[Q({\boldsymbol{\mathcal{G}}}) \leq t] + \P[t < Q({\boldsymbol{\mathcal{G}}}) \leq t+4\lambda] +
|{\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{X}}})] - {\bf E}[\Delta_{t+2\lambda}({\boldsymbol{\mathcal{G}}})]|.
\label{eqn:d1}
\end{eqnarray}
The second quantity in~(\ref{eqn:d1}) is at most
$O(d\,(4\lambda)^{1/d})$ by Corollary~\ref{cor:smallball}; the third
quantity in~(\ref{eqn:d1}) is at most $O(\epsilon\,\lambda^{-r})$ by
Lemma~\ref{lem:approx-functional} and Theorem~\ref{thm:cj}. Thus
we conclude
\[
\P[Q({\boldsymbol{\mathcal{X}}}) \leq t] \leq \P[Q({\boldsymbol{\mathcal{G}}}) \leq t] + O(d\,\lambda^{1/d}) +
O(\epsilon\,\lambda^{-r}),
\]
independently of $t$. Similarly it follows that
\[
\P[Q({\boldsymbol{\mathcal{X}}}) \leq t] \geq \P[Q({\boldsymbol{\mathcal{G}}}) \leq t] - O(d\,\lambda^{1/d}) -
O(\epsilon\,\lambda^{-r}).
\]
independently of $t$. Choosing $\lambda = \epsilon^{d/(rd+1)}$ we get
\[
\Bigl|\P\bigl[Q({\boldsymbol{\mathcal{X}}}) \leq t\bigr] - \P\bigl[Q({\boldsymbol{\mathcal{G}}}) \leq
t\bigr]\Bigr| \leq O(d\,\epsilon^{1/(rd+1)}),
\]
as required.
\end{proof}
\bigskip
The proof of Theorem \ref{thm:supertheorem} is now complete.
\subsubsection{Invariance principle for smoothed functions}
\newcommand{H(\boldR)}{H({\boldsymbol R})}
\newcommand{L(\boldR)}{L({\boldsymbol R})}
\newcommand{H(\boldS)}{H({\boldsymbol S})}
\newcommand{L(\boldS)}{L({\boldsymbol S})}
The proof of Theorem~\ref{thm:smooththeorem} is by truncating at
degree $d = c \log(1/\tau) / \log(1/\eta)$, where $c > 0$ is a
sufficiently small constant to be chosen later. Let $L(\boldR) =
(T_{1-\gamma} Q)^{\leq d}({\boldsymbol{\mathcal{X}}})$, $H(\boldR) = (T_{1-\gamma} Q)^{>
d}({\boldsymbol{\mathcal{X}}})$, and define $L(\boldS)$, and $H(\boldS)$ analogously for
${\boldsymbol{\mathcal{Y}}}$. Note that the low-degree influences of $T_{1-\gamma} Q$
are no more than those of $Q$.\\
We first prove the upper bound on $d_L({\boldsymbol R}, {\boldsymbol S})$. By
Theorem \ref{thm:supertheorem} we have
\begin{equation} \label{eqn:low}
d_L(L(\boldR), L(\boldS)) \leq
d^{\Theta(1)}\,\eta^{-\Theta(d)}\,\tau^{\Theta(1)} =
\eta^{-\Theta(d)}\,\tau^{\Theta(1)}.
\end{equation}
As for $H(\boldR)$ and $H(\boldS)$ we have ${\bf E}[H(\boldR)] = {\bf E}[H(\boldS)] = 0$
and ${\bf E}[H(\boldR)^2] = {\bf E}[H(\boldS)^2] \leq (1-\gamma)^{2d}$ (since
${\bf Var}[Q] \leq 1$). Thus by Chebyshev's inequality it follows that
for all $\lambda$,
\begin{equation} \label{eqn:high}
\P[|H(\boldR)| \geq \lambda] \leq (1-\gamma)^{2d}/\lambda^2, \qquad
\P[|H(\boldS)| \geq \lambda] \leq (1-\gamma)^{2d}/\lambda^2.
\end{equation}
Combining~(\ref{eqn:low}) and~(\ref{eqn:high}) and taking $\lambda =
(1-\gamma)^{2d/3}$ we conclude that the L\'{e}vy distance between
${\boldsymbol R}$ and ${\boldsymbol S}$ is at most
\begin{equation} \label{eqn:mainbound}
\eta^{-\Theta(d)}\,\tau^{\Theta(1)} + 4 (1-\gamma)^{2d/3} \leq
\eta^{-\Theta(d)}\,\tau^{\Theta(1)} +
\exp\bigl(-\gamma\,\Theta(d)\bigr).
\end{equation}
Our choice of $d$, with $c$ taken sufficiently small so that the
second term above dominates, completes the proof of the upper bound on $d_L({\boldsymbol R}, {\boldsymbol S})$.\\
To prove the claim about $\zeta$ we need the following simple
lemma:
\begin{lemma} For all $a, b \in \mathbb R$, $|\zeta(a+b) - \zeta(a)| \leq 2|ab| +
2b^2$.
\end{lemma}
\begin{proof}
We have
\[
|\zeta(a+b) - \zeta(a)| \leq |b| \sup_{x \in [a,a+b]}
|\zeta'(x)|.
\]
The claim follows since $\zeta'(x) = 0$ for $|x| \leq 1$ and
$|\zeta'(x)| = 2||x|-1| \leq 2|x|$ for $|x| \geq 1$.
\end{proof}
\bigskip
By~(\ref{eq:superl2}) in Theorem~\ref{thm:supertheorem} we get the
upper bound of $\eta^{-\Theta(d)}\,\tau^{\Theta(1)}$ for
$|{\bf E}[\zeta(L(\boldR)) - \zeta(L(\boldS))]|$. The Lemma above and
Cauchy-Schwartz imply
\begin{multline*}
{\bf E}\Bigl[\bigl|\zeta({\boldsymbol R})) - \zeta(L(\boldR))\bigr|\Bigr] =
{\bf E}\Bigl[\bigl|\zeta(L(\boldR) + H(\boldR)) - \zeta(L(\boldR))\bigr|\Bigr]
\leq 2 {\bf E}\bigl[|L(\boldR) H(\boldR)|\bigr] + {\bf E}\bigl[H(\boldR)^2\bigr]
\\ \leq 2\sqrt{{\bf E}[H(\boldR)^2]} + {\bf E}[H(\boldR)^2] \leq 2(1-\gamma)^d +
(1-\gamma)^{2d} \leq \exp\bigl(-\gamma\,\Theta(d)\bigr),
\end{multline*}
and similarly for ${\boldsymbol S}$. Thus
\[
|{\bf E}[\zeta({\boldsymbol R})] - {\bf E}[\zeta({\boldsymbol S})]| \leq
\eta^{-\Theta(d)}\,\tau^{\Theta(1)} +
\exp\bigl(-\gamma\,\Theta(d)\bigr)
\]
as in~(\ref{eqn:mainbound}) and we get the same upper bound.
Finally, it is easy to see that the second statement of the theorem also holds
as the only property of ${\boldsymbol R}$ we have used is that ${\bf Var}[Q^{> d}] \leq (1-\gamma)^{2d}$ for all $d$.
\subsection{Invariance principle under Lyapunov conditions}
Here we sketch a proof of Theorem~\ref{thm:Lyap}.
\begin{proof} (sketch) Let $\Delta:\mathbb R \to [0,1]$ be a nondecreasing smooth function with $\Delta(0)=0,$ $\Delta(1)=1$ and $A:=\sup_{x \in \mathbb R} |\Delta'''(x)| < \infty.$ Then $\sup_{x \in \mathbb R} |\Delta''(x)| \leq A/2$ and therefore for $x,y \in \mathbb R$ we have
\[
|\Delta''(x)-\Delta''(y)| \leq A^{3-q}|\Delta''(x)-\Delta''(y)|^{q-2} \leq
A^{3-q}(A|x-y|)^{q-2}=A|x-y|^{q-2}.
\]
For $s>0$ let $\Delta_{s}(x)=\Delta(x/s),$ so that
$|\Delta_{s}''(x)-\Delta_{s}''(y)| \leq A s^{-q}|x-y|^{q-2}$
for all $x,y \in \mathbb R.$
Let ${\boldsymbol Y}$ and ${\boldsymbol Z}$ be random variables with
${\bf E}[{\boldsymbol Y}]={\bf E}[{\boldsymbol Z}],$ ${\bf E}[{\boldsymbol Y}^{2}]={\bf E}[{\boldsymbol Z}^{2}]$
and ${\bf E}[|{\boldsymbol Y}|^{q}], {\bf E}[|{\boldsymbol Z}|^{q}]<\infty.$
Then $|{\bf E}[\Delta_{s}(x+{\boldsymbol Y})]-{\bf E}[\Delta_{s}(x+{\boldsymbol Z})]| \leq
As^{-q}({\bf E}[|{\boldsymbol Y}|^{q}]+{\bf E}[|{\boldsymbol Z}|^{q}])$ for all $x \in \mathbb R.$
Indeed, for $u \in [0,1]$ let
$\phi(u)={\bf E}[\Delta_{s}(x+u{\boldsymbol Y})]-{\bf E}[\Delta_{s}(x+u{\boldsymbol Z})].$
Then $\phi(0)=\phi'(0)=0$ and
\[
|\phi''(u)|=
|{\bf E}[{\boldsymbol Y}^{2}(\Delta_{s}''(x+u{\boldsymbol Y})-\Delta_{s}''(x))]-
{\bf E}[{\boldsymbol Z}^{2}(\Delta_{s}''(x+u{\boldsymbol Z})-\Delta_{s}''(x))]| \leq
As^{-q}u^{q-2}({\bf E}[|{\boldsymbol Y}|^{q}]+{\bf E}[|{\boldsymbol Z}|^{q}]),
\]
so that $|\phi(1)| \leq As^{-q}({\bf E}[|{\boldsymbol Y}|^{q}]+{\bf E}[|{\boldsymbol Z}|^{q}]).$
Now, using the above estimate and the fact that both ${\boldsymbol{\mathcal{X}}}$
and ${\boldsymbol{\mathcal{G}}}$ are $(2,q,\eta)$-hypercontractive with
$\eta=\frac{\beta^{-1/q}}{2\sqrt{q-1}}$ one arrives at
\[
|{\bf E}[\Delta_{s}(Q({\boldsymbol X}_{1}, \ldots, {\boldsymbol X}_{n}))]-
{\bf E}[\Delta_{s}(Q({\boldsymbol G}_{1}, \ldots, {\boldsymbol G}_{n}))]| \leq
O(s^{-q}\eta^{-qd}\sum_{i}(\sum_{S \ni i} c_{S}^{2})^{q/2}).
\]
Replacing $Q$ by $Q+t$ and using the arguments of subsection~\ref{cid}
yields
\[
\sup_{t} \bigl|\P[Q({\boldsymbol X}_1, \dots, {\boldsymbol X}_n) \leq t] -
\P[Q({\boldsymbol G}_1, \dots, {\boldsymbol G}_n) \leq t] \bigr|
\leq
\]
\[
O(ds^{1/d})+
O(s^{-q}\eta^{-qd}\sum_{i}(\sum_{S \ni i} c_{S}^{2})^{q/2}).
\]
Optimizing over $s$ ends the proof. We skip some elementary
calculations.
\end{proof}
\section{Proofs of the conjectures} \label{sec:conj}
Our applications of the invariance principle have the following
character: We wish to study certain noise stability properties of
low-influence functions on finite product probability spaces. By
using the invariance principle for slightly smoothed functions,
Theorem~\ref{thm:smooththeorem}, we can essentially analyze the
properties in the product space of our choosing. And as it
happens, the necessary result for Majority Is Stablest is already
known in Gaussian space~\cite{Borell:85} and the necessary result
for It Ain't Over Till It's Over is already known on the
uniform-measure discrete cube~\cite{MORSS:04}.\\
In the case of the Majority Is Stablest problem, one needs to find
a set of prescribed Gaussian measure which maximizes the probability
that the Ornstein-Uhlenbeck process (started at the Gaussian measure)
will belong to the set at times $0$ and time $t$ for some fixed time $t$.
This problem was solved by Borell
in~\cite{Borell:85} using symmetrization arguments. It should
also be noted that the analogous result for the sphere has been
proven in more than one place, including a paper of Feige and
Schechtman~\cite{FeigeSchechtman:02}. It fact, one can deduce Borell's result
and Majority is Stablest from the spherical result using the proximity of
spherical and Gaussian measures in high dimensions and the invariance
principle proven here.\\
In the case of the It Ain't Over Till It's Over problem, the
necessary result on the discrete cube $\{-1,1\}^n$ was essentially
proven in the recent paper~\cite{MORSS:04} using the reverse
Bonami-Beckner inequality (which is also due to
Borell~\cite{Borell:82}). This paper did not solve the conjecture
though (nor did that paper note the relevance), even when the
conjecture is set on $\{-1,1\}^n$; the reason is that reduction of
the problem to a question about $T_\rho$ already involves
transferring to a different product domain (e.g., $\{-1, 0, 1\}^n$
with biased measure) and so the invariance principle is
required.\\
Note that in both cases the necessary auxiliary result is valid
without any assumptions about low influences. This should not be
surprising in the Gaussian case, since given a multilinear
polynomial $Q$ over Gaussians it is easy to define another
multilinear polynomial $\tilde{Q}$ over Gaussians with exactly the
same distribution and arbitrarily low influences, by letting
\[
\tilde{Q}(x_{1,1},\dots,x_{1,N},\;\ldots\;,x_{n,1},\ldots,x_{n,N})
= Q\Bigl(\frac{x_{1,1} + \cdots +
x_{1,N}}{N^{1/2}},\;\ldots\;,\frac{x_{n,1} + \cdots +
x_{n,N}}{N^{1/2}}\Bigr).
\]
The fact that low influences are not required for
the the results of~\cite{MORSS:04} is perhaps more surprising.
\subsection{Noise stability in Gaussian space}
We begin by recalling some definitions and results relevant for
``Gaussian noise stability''. Throughout this section we consider
$\mathbb R^n$ to have the standard $n$-dimensional Gaussian distribution,
and our probabilities and expectations are over this
distribution.\\
Let $U_\rho$ denote the Ornstein-Uhlenbeck operator acting on
$L^2(\mathbb R^n)$ by
\[
(U_\rho f)(x) = \mathop{\bf E\/}_{y}[f(\rho x + \sqrt{1-\rho^2}\, y)],
\]
where $y$ is a random standard $n$-dimensional Gaussian. It is
easy to see that if $f(x)$ is expressible as a \emph{multilinear}
polynomial in its $n$ independent Gaussian inputs,
\[
f(x_1, \dots, x_n) = \sum_{S \subseteq [n]} c_S \prod_{i \in S}
x_i,
\]
then $U_\rho f$ is the following multilinear polynomial:
\[
(U_\rho f)(x_1, \dots, x_n) = \sum_{S \subseteq [n]} \rho^{|S|}
c_S \prod_{i \in S} x_i.
\]
Thus $U_\rho$ acts identically to $T_\rho$ for multilinear
polynomials $Q$ over ${\boldsymbol{\mathcal{G}}}$, the Gaussian sequence of
ensembles.\\
Next, given any function $f : \mathbb R^n \to \mathbb R$, recall that its
\emph{(Gaussian) nonincreasing spherical rearrangement} is defined
to be the upper semicontinuous nondecreasing function $f^* : \mathbb R
\to \mathbb R$ which is equimeasurable with $f$; i.e., for all $t \in
\mathbb R$, $f^*$ satisfies $\Pr[f > t] = \Pr[f^* > t]$ under Gaussian measure.\\
We now state a result of Borell concerning the
Ornstein-Uhlenbeck operator $U_\rho$ (see also Ledoux's
Saint-Flour lecture notes~\cite{DoGrLe:96}). Borell uses Ehrhard
symmetrization to show the following:
\begin{theorem} (Borell~\cite{Borell:85}) \label{thm:borell}
Let $f, g \in L^2(\mathbb R^n)$.
Then for all $0 \leq \rho \leq 1$ and all
$q \geq 1$,
\[
{\bf E}[(U_\rho f)^q \cdot g] \leq {\bf E}[(U_\rho f^*)^q \cdot g^*].
\]
\end{theorem}
Borell's result is more general and is stated for Lipschitz
functions, but standard density arguments immediately imply the
validity of the statement above. One immediate consequence of the
theorem is that $\mathbb{S}_\rho(f) \leq \mathbb{S}_\rho(f^*)$, where we
define
\begin{equation} \label{eqn:gauss-stab}
\mathbb{S}_\rho(f) = {\bf E}[f \cdot U_\rho f] = {\bf E}[(U_{\sqrt{\rho}} f)^2].
\end{equation}
One can think of this quantity as the ``(Gaussian) noise stability
of $f$ at $\rho$''; again, it is compatible with our earlier
definition of $\mathbb{S}_\rho$ if $f$ is a multilinear polynomial over
${\boldsymbol{\mathcal{G}}}$.\\
Note that the latter equality in~(\ref{eqn:gauss-stab}) and the
fact that $U_{\sqrt{\rho}}$ is positivity-preserving and linear
imply that $\sqrt{\mathbb{S}_\rho}$ defines an $L^2$ norm on
$L^2(\mathbb R^n),$ dominated by the usual $L^{2}$ norm, so that it is a
continuous convex functional on $L^{2}(\mathbb R^{n})$. The set of all
$[0,1]$-valued functions from $L^{2}(\mathbb R^{n})$ having the same mean
as $f$ is closed and bounded in the standard $L^{2}$ norm and one
can easily check that its extremal points are indicator functions;
hence by the Edgar-Choquet theorem (see \cite{Edgar:75}; clearly
$L^{2}(\mathbb R^{n})$ is separable and it has the Radon-Nikodym property
since it is a Hilbert space):
\[
\sqrt{\mathbb{S}_\rho}(f) \leq \sup_{\chi}\sqrt{\mathbb{S}_\rho}(\chi),
\]
where the supremum is taken over all functions $\chi:\mathbb R^{n} \to \{
0,1\}$ with ${\bf E}[\chi]={\bf E}[f].$ Since by Borell's result
$\mathbb{S}_\rho(\chi) \leq \mathbb{S}_\rho(\chi^{*})$, we have
$\mathbb{S}_\rho(f) \leq \mathbb{S}_\rho(\chi_{\mu})$ where $\chi_{\mu}:\mathbb R
\to \{ 0,1\}$ is the indicator function of a halfline with measure
$\mu={\bf E}[f]$.\\
Let us introduce some notation:
\begin{definition} Given $\mu \in [0,1]$, define $\chi_\mu : \mathbb R \to \{0,1\}$
to be the indicator function of the interval $(-\infty, t]$, where
$t$ is chosen so that ${\bf E}[\chi_\mu] = \mu$. Explicitly, $t =
\Phi^{-1}(\mu)$, where $\Phi$ denotes the distribution function of
a standard Gaussian. Furthermore, define
\[
\StabThr{\rho}{\mu} = \mathbb{S}_\rho(\chi_\mu) = \Pr[{\boldsymbol X} \leq t,
{\boldsymbol Y} \leq t],
\]
where $({\boldsymbol X}, {\boldsymbol Y})$ is a two dimensional Gaussian vector with
covariance matrix
$
\left( \begin{smallmatrix}
1 & \rho \\
\rho & 1
\end{smallmatrix} \right)
$.
\end{definition}
Summarizing the above discussion, we obtain:
\begin{corollary} \label{cor:bor}
Let $f : \mathbb R^n \to [0,1]$ be a measurable
function on Gaussian space with ${\bf E}[f] = \mu$. Then for all $0
\leq \rho \leq 1$ we have $\mathbb{S}_\rho(f) \leq
\StabThr{\rho}{\mu}$.
\end{corollary}
\noindent This is the result we will use to prove the Majority Is
Stablest conjecture. We note that in general there is no
closed form for $\StabThr{\rho}{\mu}$; however, some asymptotics
are known: For balanced functions we have
Sheppard's formula $\StabThr{\rho}{1/2} = \frac14+
\frac{1}{2\pi}\arcsin \rho$. Some other properties of
$\StabThr{\rho}{\mu}$ are given in Appendix~\ref{app:StabThr}.
\subsection{Majority Is Stablest} \label{subsec:maj_stablest}
In this section we prove a strengthened form of the Majority Is
Stablest conjecture. The implications of this result were
discussed in Section~\ref{sec:misc}.
\begin{theorem} \label{thm:MIST}
Let $f : \Omega_1 \times \cdots \times \Omega_n \to [0,1]$ be a
function on a finite product probability space and assume that
for each $i$ the minimum probability of any atom in $\Omega_i$ is
at least $\alpha \leq 1/2$. Write $K = \log(1/\alpha)$. Further
assume that there is a $0 < \tau < 1/2$ such that $\mathrm{Inf}_i^{\leq
\log(1/\tau)/K}(f) \leq \tau$ for all $i$. (See
Definition~\ref{def:low-degree-influence} for the definition of
low-degree influence.) Let $\mu = {\bf E}[f]$. Then for any $0 \leq
\rho < 1$,
\[
\mathbb{S}_\rho(f) \leq \StabThr{\rho}{\mu} + \epsilon,
\]
where
\[
\epsilon = O\Bigl(\frac{K}{1-\rho}\Bigr) \cdot \frac{ \log \log
(1/\tau)}{\log(1/\tau)}.
\]
\end{theorem}
\rnote{In the most basic case, with rademachers and $\mu = 1/2$,
you can probably slightly improve the error term. This may well
be important for hardness of approximation / metric space
problems, and we should check the details. Specifically, can one
show that for any function that is not a $(1/n^{.001},
n/2)$-junta, the noise stability at $1-O(1/\log)$ is at most $1 -
\Omega(1/\sqrt{\log n})$?}
\noindent For the reader's convenience we record here two facts
from Appendix~\ref{app:StabThr}:
\begin{eqnarray*}
\StabThr{\rho}{{\textstyle \frac12}} & = & \frac14+ \frac{1}{2\pi}\arcsin \rho \\
\StabThr{\rho}{\mu} & \sim & \mu^{2/(1+\rho)}\,(4\pi\ln
(1/\mu))^{-\rho/(1+\rho)}\,\frac{(1+\rho)^{3/2}}{(1-\rho)^{1/2}}
\qquad \text{as $\mu \to 0$.}
\end{eqnarray*}
\begin{proof}
As discussed in Section~\ref{sec:setup}, let ${\boldsymbol{\mathcal{X}}}$ be the
sequence of ensembles such that ${\boldsymbol{\mathcal{X}}}_i$ spans the functions on
$\Omega_i$, and express $f$ as the multilinear polynomial $Q$.
We use the invariance principle under hypothesis $\boldsymbol{H2}$.
Express
$\rho = \rho' \cdot (1-\gamma)^2$, where $0 < \gamma \ll 1-\rho$
will be chosen later. Writing $Q(x) = \sum c_{{\boldsymbol \sigma}}
x_{{\boldsymbol \sigma}}$ (with $c_{\mathbf{0}} = \mu$) we see that
\[
\mathbb{S}_{\rho}(Q({\boldsymbol{\mathcal{X}}})) = \sum (\rho' \cdot
(1-\gamma)^2)^{|{\boldsymbol \sigma}|} c_{{\boldsymbol \sigma}}^2 =
\mathbb{S}_{\rho'}((T_{1-\gamma} Q)({\boldsymbol{\mathcal{G}}})),
\]
where ${\boldsymbol{\mathcal{G}}}$ is the sequence of independent Gaussian ensembles.\\
Since $Q({\boldsymbol{\mathcal{X}}})$ is bounded in $[0,1]$ the same is true of ${\boldsymbol R}
= (T_{1-\gamma} Q)({\boldsymbol{\mathcal{X}}})$. In other words, ${\bf E}[\zeta({\boldsymbol R})] =
0$, where $\zeta$ is the function from~(\ref{eq:def_trunc}).
Writing ${\boldsymbol S} = (T_{1-\gamma} Q)({\boldsymbol{\mathcal{G}}})$, we conclude from
Theorem~\ref{thm:smooththeorem} that ${\bf E}[\zeta({\boldsymbol S})] \leq
\tau^{\Omega(\gamma/K)}$. That is, $\|{\boldsymbol S} - {\boldsymbol S}'\|_2^2 \leq
\tau^{\Omega(\gamma/K)}$, where ${\boldsymbol S}'$ is the random variable
depending on ${\boldsymbol S}$ defined by
\[
{\boldsymbol S}' = \left\{ \begin{array}{rl}
0 & \text{if ${\boldsymbol S} \leq 0$,} \\
{\boldsymbol S} & \text{if ${\boldsymbol S} \in [0,1]$,} \\
1 & \text{if ${\boldsymbol S} \geq 1$.}
\end{array}
\right.
\]
Then
\begin{eqnarray*}
|\mathbb{S}_{\rho'}({\boldsymbol S}) - \mathbb{S}_{\rho'}({\boldsymbol S}')| &=&
|{\bf E}[{\boldsymbol S} \cdot U_{\rho'} {\boldsymbol S}] - {\bf E}[{\boldsymbol S}' \cdot U_{\rho'} {\boldsymbol S}']| \\
&\leq&
|{\bf E}[{\boldsymbol S} \cdot U_{\rho'} {\boldsymbol S}] - {\bf E}[{\boldsymbol S}' \cdot U_{\rho'} {\boldsymbol S}]| +
|{\bf E}[{\boldsymbol S}' \cdot U_{\rho'} {\boldsymbol S}] - {\bf E}[{\boldsymbol S}' \cdot U_{\rho'} {\boldsymbol S}']| \\
&\leq&
(\|{\boldsymbol S}\|_2 + \|{\boldsymbol S}'\|_2)\|{\boldsymbol S}-{\boldsymbol S}'\|_2 \leq \tau^{\Omega(\gamma/K)},
\end{eqnarray*}
where we have used the fact that $U_{\rho'}$ is a contraction on $L^2$.
Writing $\mu' = {\bf E}[{\boldsymbol S}']$ it follows from Cauchy-Schwartz that
$|\mu-\mu'| \leq \tau^{\Omega(\gamma/K)}$.
Since ${\boldsymbol S}'$ takes values in
$[0,1]$ it follows from Corollary~\ref{cor:bor} that
$\mathbb{S}_{\rho'}({\boldsymbol S}') \leq \StabThr{\rho'}{\mu'}$. We thus
conclude
\[
\mathbb{S}_\rho(Q({\boldsymbol{\mathcal{X}}})) = \mathbb{S}_{\rho'}({\boldsymbol S}) \leq
\mathbb{S}_{\rho'}({\boldsymbol S}') + \tau^{\Omega(\gamma/K)} \leq
\StabThr{\rho'}{\mu'} + \tau^{\Omega(\gamma/K)}.
\]
We can now bound the difference $|\StabThr{\rho}{\mu} -
\StabThr{\rho'}{\mu'}|$ using Lemmas~\ref{lem:I1} and
Corollary~\ref{cor:I2} in Appendix~\ref{app:StabThr}. We get a
contribution of $2|\mu - \mu'| \leq \tau^{\Omega(\gamma/K)}$ from
the difference in the $\mu$'s and a contribution of at most
$O(\gamma/(1-\rho))$ from the difference in the $\rho$'s. Thus we
have
\[
\mathbb{S}_\rho(Q({\boldsymbol{\mathcal{X}}})) \leq \StabThr{\rho}{\mu} +
\tau^{\Omega(\gamma/K)} + O(\gamma/(1-\rho)).
\]
Taking
\[
\gamma = C \cdot K \cdot \frac{\log \log (1/\tau)}{\log(1/\tau)}
\]
for some large enough constant $C$ and this gives the claimed
bound.
\end{proof}
\subsection{It Ain't Over Till It's Over}
As mentioned, our proof of the It Ain't Over Till It's Over
conjecture will use a result due essentially to~\cite{MORSS:04}:
\begin{theorem} \label{thm:coins} Let $f : \{-1,1\}^n \to [0,1]$ have ${\bf E}[f] = \mu$
(with respect to uniform measure on $\{-1,1\}^n$). Then for any $0 <
\rho < 1$ and any $0 < \epsilon \leq 1 - \mu$ we have
\[
\Pr[T_\rho f > 1-\delta] < \epsilon
\]
provided
\[
\delta < \epsilon^{\rho^2/(1-\rho^2) + O(\kappa)},
\]
where
\[
\kappa = \frac{\sqrt{c(\mu)}}{1-\rho} \cdot
\frac{1}{\sqrt{\log(1/\epsilon)}}, \qquad c(\mu) = \mu
\log(e/(1-\mu)).
\]
\end{theorem}
This theorem follows from the proof of Theorem 4.1
in~\cite{MORSS:04}; for completeness we give an explicit
derivation in Appendix~\ref{app:coins}.
\begin{remark} Since the only fact about $\{-1,1\}^n$ used in the proof of
Theorem~\ref{thm:coins} is the reverse Bonami-Beckner inequality,
and since this inequality also holds in Gaussian space, we
conclude that Theorem~\ref{thm:coins} also holds for
measurable functions on Gaussian space $f : \mathbb R^n \to
[0,1]$. In this setting the result can be proved using Borell's
Corollary~\ref{cor:bor} instead of using the reverse Bonami-Beckner
inequality.
\end{remark}
\bigskip
The first step of the proof of It Ain't Over Till It's Over is to
extend Theorem~\ref{thm:coins} to functions on arbitrary product
probability spaces. Note that if we only want to solve the
problem for functions on $\{-1,1\}^n$ with the uniform measure, this
step is unnecessary. The proof of the extension is very similar
to the proof of Theorem~\ref{thm:MIST}. In order to state the theorem it would
be helpful to let $u > 0$ be a constant such that
Theorem~\ref{thm:smooththeorem} holds with the bound $\tau^{u \gamma/K}$.
\begin{theorem} \label{thm:general-coins}
Let $f : \Omega_1 \times \cdots \times \Omega_n \to [0,1]$ be a
function on a finite product probability space and assume that
for each $i$ the minimum probability of any atom in $\Omega_i$ is
at least $\alpha \leq 1/2$. Let $K \geq \log(1/\alpha)$. Further
assume that there is a $\tau > 0$ such that $\mathrm{Inf}_i^{\leq
\log(1/\tau)/K}(f) \leq \tau$ for all $i$ (recall
Definition~\ref{def:low-degree-influence}).
Let $\mu = {\bf E}[f]$. Then for any $0 < \rho
< 1$ there exists $\epsilon(\mu,\rho)$ such that if $0 < \epsilon < \epsilon(\mu,\rho)$
we have
\[
\Pr[T_\rho f > 1 - \delta] \leq \epsilon
\]
provided
\[
\delta < \epsilon^{\rho^2/(1-\rho^2) + C \kappa}, \qquad
\tau \leq \epsilon^{(100K/u(1-\rho))(1 /(1-\rho)^3 + C \kappa)}
\]
where
\[
\kappa = \frac{\sqrt{c(\mu)}}{1-\rho} \cdot
\frac{1}{\sqrt{\log(1/\epsilon)}}, \qquad c(\mu) = \mu
\log(e/(1-\mu)) + \epsilon
\]
and $C > 0$ is some constant.
\end{theorem}
\begin{proof}
Without loss of generality we assume that
$\delta = \epsilon^{\rho^2/(1-\rho^2) + C \kappa}$
as taking a smaller $\delta$ yields a smaller tail probability.
We can also assume $\epsilon(\mu,\rho)<1/10.$
Let ${\boldsymbol{\mathcal{X}}}$ and $Q$ be as in the proof of Theorem~\ref{thm:MIST}
and this time decompose $\rho = \rho' \cdot (1-\gamma)$ where we
take $\gamma = \kappa \cdot (1-\rho)^2$. Note that taking
$\epsilon(\mu,\rho)$ sufficiently small we have $\kappa <1, \gamma < 0.1$ and
$(1-\rho)/(1-\rho') \leq 2$.
Let ${\boldsymbol R} =
(T_{1-\gamma} Q)({\boldsymbol{\mathcal{X}}})$ as before, and let ${\boldsymbol S} =
(T_{1-\gamma} Q)({\boldsymbol{\mathcal{Y}}})$, where ${\boldsymbol{\mathcal{Y}}}$ denotes the Rademacher
sequence of ensembles (${\boldsymbol Y}_{i,0} = 1$, ${\boldsymbol Y}_{i,1} = \pm 1$
independently and uniformly random). Since ${\bf E}[\zeta({\boldsymbol R})] =
0$ as before, we conclude from Theorem~\ref{thm:smooththeorem}
that we have ${\bf E}[\zeta({\boldsymbol S})] \leq
\tau^{u \gamma/K} \leq \epsilon^{10/(1-\rho) + 2C \kappa}$; i.e.,
\begin{equation} \label{eq:most_ugly}
\|{\boldsymbol S} - {\boldsymbol S}'\|_2^2 \leq \epsilon^{10/(1-\rho) + 2C \kappa}
\end{equation}
where ${\boldsymbol S}'$ is the truncated version of ${\boldsymbol S}$ as in the
proof of Theorem~\ref{thm:MIST}. Now ${\boldsymbol S}'$ is a function
$\{-1,1\}^n \to [0,1]$ with mean $\mu'$ differing from $\mu$ by at
most $\epsilon^{5}$ (using Cauchy-Schwartz, as before).
This implies that $c(\mu') \leq O(c(\mu))$.
Furthermore, our assumed upper
bound on $\delta$ also holds with $\rho'$ in place of $\rho$.
This is because
\[
\frac{{\rho'}^2}{1-{\rho'}^2} - \frac{{\rho}^2}{1-{\rho}^2} =
\frac{1}{1-\rho'^2}-\frac{1}{1-\rho^2} \leq
(\rho'^2-\rho^2)\frac{1}{(1-\rho'^2)^2} \leq \frac{2 \gamma}{(1-\rho')^2}
\leq \frac{8 \gamma}{(1-\rho)^2} = 8 \kappa.
\]
Thus Theorem~\ref{thm:coins} implies that if $C$ is sufficiently large then
\[
\Pr[T_{\rho'} {\boldsymbol S}' > 1 - 4 \delta] < \epsilon/2.
\]
This, in turn implies that
\[
\Pr[T_{\rho'} {\boldsymbol S} > 1 - 2 \delta] < 3\epsilon/4.
\]
This follows by (\ref{eq:most_ugly}) since,
\[
\Pr[T_{\rho'} {\boldsymbol S} > 1 - 4 \delta] - \Pr[T_{\rho'} {\boldsymbol S}' > 1 - 2 \delta]
\leq \delta^{-2} \|T_{\rho'} {\boldsymbol S} - T_{\rho'} {\boldsymbol S}'\|_2^2 \leq
\delta^{-2} \|{\boldsymbol S} - {\boldsymbol S}'\|_2^2.
\]
We now use Theorem~\ref{thm:smooththeorem} again, bounding the
L\'{e}vy distance of $(T_\rho Q)({\boldsymbol{\mathcal{Y}}})$ and $(T_\rho Q)({\boldsymbol{\mathcal{X}}})$
by $\tau^{u(1-\rho)/K}$, which is
smaller than $\delta$ and $\epsilon/8$.
Thus
\[
\Pr[(T_\rho Q)({\boldsymbol{\mathcal{X}}}) > 1 - \delta] \leq
\Pr[T_\rho f > 1- 2 \delta] + \epsilon/8 < \epsilon,
\]
as needed.
\end{proof}
\bigskip
The second step of the proof of It Ain't Over Till It's Over is to
use the invariance principle to show that the random variable
$V_\rho f$ (recall Definition~\ref{def:V}) has essentially the
same distribution as $T_{\sqrt{\rho}} f$.
\begin{theorem} \label{thm:vote-dist}
Let $0 < \rho < 1$ and let $f : \Omega_1 \times \cdots \times
\Omega_n \to [0,1]$ be a function on a finite product
probability space; assume that for each $i$ the minimum
probability of any atom in $\Omega_i$ is at least $\alpha \leq
1/2$. Further assume that there is a $0 < \tau < 1/2$ such that
$\mathrm{Inf}_i^{\leq\,\log(1/\tau)/K'} \leq \tau$ for all $i$, where $K'
= \log(1/(\alpha \rho (1-\rho)))$. Then
\[
d_L(V_\rho f, T_{\sqrt{\rho}} f) \leq \tau^{\Omega((1-\rho)/K')}.
\]
\end{theorem}
\begin{proof}
Introduce ${\boldsymbol{\mathcal{X}}}$ and $Q$ as in the proof of
Theorems~\ref{thm:MIST} and~\ref{thm:general-coins}. We now
define a new independent sequence of orthonormal ensembles
${\boldsymbol{\mathcal{X}}}^{(\rho)}$ as follows. Let ${\boldsymbol V}_1, \dots, {\boldsymbol V}_n$ be
independent random variables, each of which is $1$ with
probability $\rho$ and $0$ with probability $1-\rho$. Now define
${\boldsymbol{\mathcal{X}}}^{(\rho)} = ({\boldsymbol{\mathcal{X}}}_1^{(\rho)}, \dots, {\boldsymbol{\mathcal{X}}}_n^{(\rho)})$ by
${\boldsymbol X}^{(\rho)}_{i, 0} = 1$ for each $i$, and
${\boldsymbol X}^{(\rho)}_{i,j} = \rho^{-1/2} {\boldsymbol V}_i {\boldsymbol X}_{i,j}$ for
each $i$ and $j > 0$. It is easy to verify that ${\boldsymbol{\mathcal{X}}}^{(\rho)}$
is indeed an independent sequence of orthonormal ensembles. We
will also use the fact that each atom in the ensemble
${\boldsymbol{\mathcal{X}}}_i^{(\rho)}$ has weight at least $\alpha' = \alpha \cdot
\min\{\rho, 1-\rho\} \geq \alpha \rho(1-\rho)$.
(one can also use Proposition~\ref{prop:add} to get a bit better
estimate on $K'$).\\
The crucial observation is now simply that the random variable
$V_\rho f$ has precisely the same distribution as the random
variable $(T_{\sqrt{\rho}} Q)({\boldsymbol{\mathcal{X}}}^{(\rho)})$. The reason is that
when the randomness in the ${\boldsymbol V}_i = 1$ ensembles is fixed, the
expectation of the restricted function is given by substituting
$0$ for all other random variables ${\boldsymbol X}_{i,j}$. The
$T_{\sqrt{\rho}}$ serves to cancel the factors $\rho^{-1/2}$
introduced in the definition of ${\boldsymbol X}^{(\rho)}_{i,j}$ to ensure orthonormality.\\
It now simply remains to use Theorem~\ref{thm:smooththeorem} to
bound the L\'{e}vy distance of $(T_{\sqrt{\rho}}
Q)({\boldsymbol{\mathcal{X}}}^{(\rho)})$ and $(T_{\sqrt{\rho}} Q)({\boldsymbol{\mathcal{X}}})$, where here
${\boldsymbol{\mathcal{X}}}$ denotes a copy of this sequence of ensembles.
We use hypothesis $\boldsymbol{H3}$ and
get a bound of $\tau^{\Omega((1-\sqrt{\rho})/K')} =
\tau^{\Omega((1-\rho)/K')}$, as required.
\end{proof}
\bigskip
Our generalization of It Ain't Over Till It's Over is now simply a
corollary of Theorems~\ref{thm:general-coins}
and~\ref{thm:vote-dist}; by taking $K'$ instead of $K$ in the
upper bound on $\tau$ and taking $\delta$ to have its maximum possible value,
we make the error of
\[
\tau^{u((1-\rho)/K')} \leq
\epsilon^{(100/(1-\rho))(1 /(1-\rho)^3 + C \kappa)}
\]
from Theorem~\ref{thm:vote-dist} which is
negligible compared to both $\epsilon$ and $\delta$ below.
\begin{theorem} \label{thm:aint}
Let $0 < \rho < 1$ and let $f : \Omega_1 \times \cdots \times
\Omega_n \to [0,1]$ be a function on a finite product
probability space; assume that for each $i$ the minimum
probability of any atom in $\Omega_i$ is at least $\alpha \leq
1/2$. Further assume that there is a $0 < \tau < 1/2$ such that
$\mathrm{Inf}_i^{\leq\,\log(1/\tau)/K'} \leq \tau$ for all $i$, where $K'
= \log(1/(\alpha \rho (1-\rho)))$. Let $\mu = {\bf E}[f]$.
Then there exists an $\epsilon(\rho,\mu) > 0$ such that if
$\epsilon \leq \epsilon(\rho,\mu)$ then
\[
\Pr[V_\rho f > 1 - \delta] \leq \epsilon
\]
provided
\[
\delta < \epsilon^{\rho^2/(1-\rho^2) + C \kappa}, \qquad
\tau \leq \epsilon^{(100K'/u(1-\rho))(1 /(1-\rho)^3 + C \kappa)}
\]
where
\[
\kappa = \frac{\sqrt{c(\mu)}}{1-\rho} \cdot
\frac{1}{\sqrt{\log(1/\epsilon)}}, \qquad c(\mu) = \mu
\log(e/(1-\mu)) + \epsilon,
\]
where $C > 0$ is some finite constant.
\end{theorem}
\begin{remark}
To get $V_\rho f$ bounded away from both $0$ and $1$ as desired in
Conjecture~\ref{conj:ain't}, simply use Theorem~\ref{thm:aint}
twice, once with $f$, once with $1-f$.
\end{remark}
\section{Weight at low levels --- a counterexample}
\label{sec:counterexample}
The simplest version of the Majority Is
Stablest result states roughly that among all balanced functions
$f : \{-1,1\}^n \to \{-1,1\}$ with small influences, the Majority
function maximizes $\sum_S \rho^{S} \hat{f}(S)^2$ for each $\rho$.
One might conjecture that more is true; specifically, that
Majority maximizes $\sum_{|S| \leq d} \hat{f}(S)^2$ for each $d =
1, 2, 3, \dots$. This is known to be the case for $d = 1$
(\cite{KKMO:04}) and is somewhat suggested by the theorem of
Bourgain~\cite{Bourgain:02} which says that $\sum_{|S| \leq d}
\hat{f}(S)^2 \leq 1 - d^{-1/2 - o(1)}$ for functions with low
influences. An essentially weaker conjecture was made
Kalai~\cite{Kalai:02}:
\begin{conjecture} \label{conj:kalai}
Let $d \geq 1$ and let $C_n$ denote the collection of all
functions $f : \{-1,1\}^n \to \{-1,1\}$ which are odd and
transitive-symmetric (see Section~\ref{sec:misc-discuss}'s
discussion of~\cite{Kalai:02}). Then
\[
\limsup_{n \to \infty} \sup_{f \in C_n} \sum_{|S| \leq d}
\hat{f}(S)^2 = \lim_{\text{$n$ odd } \to \infty} \sum_{|S| \leq d}
\widehat{\mathrm{Maj}_n}(S)^2.
\]
\end{conjecture}
We now show that these conjectures are false: We construct a
sequence $(f_n)$ of completely symmetric odd functions with small
influences that have more weight on levels $1$, $2$, and $3$ than
Majority has. \ignore{By odd we mean that $f(-x) = -f(x)$;} By
``completely symmetric'' we mean that $f_n(x)$ depends only on
$\sum_{i=1}^n x_i$; because of this symmetry our counterexample is
more naturally viewed in terms of the Hermite expansions of
functions $f : \mathbb R \to \{-1,1\}$ on one-dimensional Gaussian
space.\\
There are several normalizations of the Hermite polynomials in the
literature. We will follow~\cite{LedouxTalagrand:91} and define
them to be the orthonormal polynomials with respect to the
one-dimensional Gaussian density function $\phi(x) =
e^{-x^2/2}/\sqrt{2\pi}$. Specifically, we define the Hermite
polynomials $h_d(x)$ for $d \in \mathbb N$ by
\[
\exp(\lambda x - \lambda^2/2) = \sum_{d=0}^\infty
\frac{\lambda^d}{\sqrt{d!}}\;h_d(x).
\]
The first few such polynomials are $h_0(x) = 1$, $h_1(x) = x$,
$h_2(x) = (x^2 - 1)/\sqrt{2}$, and $h_3(x) = (x^3 - 3x)/\sqrt{6}$.
The orthonormality condition these polynomials satisfy is
\[
\int_\mathbb R h_d(x) h_{d'}(x) \phi(x)\,dx = \left\{\begin{array}{cl} 1
& \text{if $d = d'$,} \\ 0 & \text{else.} \end{array} \right.
\]
\ignore{
More concretely, we will use the following easy fact.
\begin{lemma} \label{lem:hermite}
Let $f : \mathbb R \to \mathbb R$ be a bounded Riemann measurable function.
Define $f_n : \{-1,1\}^n \to \mathbb R$ by letting
\[
f_n(x_1,\ldots,x_n) = f \left(\frac{1}{\sqrt{n}} \sum_{i=1}^n x_i \right).
\]
Then for all $d$ it holds that
\[
\lim_{n \to \infty} \sum_{|S|=d} \hat{f_n}^2(S) =
\int_{\mathbb R} h_d(x) f(x) \phi(x) dx.
\]
\end{lemma}
}
We will actually henceforth consider functions whose domain is
$\mathbb R^* = \mathbb R \setminus \{0\}$, for simplicity; the value of a
function at a single point makes no difference to its Hermite
expansion. Given a function $f : \mathbb R^* \to \mathbb R$ we write
$\hat{f}(d)$ for $\int h_d(x) f(x) \phi(x)\,dx$. Let us also use
the notation $\mathrm{Maj}$ for the function which is $1$ on $(0, \infty)$
and $-1$ on $(-\infty, 0)$.
\begin{theorem} \label{thm:counter} There is an odd function $f : \mathbb R^* \to \{-1,1\}$ with
\[
\sum_{d \leq 3} \hat{f}(d)^2 \geq .75913 > \frac{2}{\pi} +
\frac{1}{3\pi} = \sum_{d \leq 3} \widehat{\mathrm{Maj}}(d)^2.
\]
\end{theorem}
\begin{proof}
Let $t > 0$ be a parameter to be chosen later, and let $f$ be the
function which is $1$ on $(-\infty, -t]$ and $(0, t)$, and $-1$ on
$(-t, 0)$ and $[t, \infty)$. Since $f$ is odd, $\hat{f}(0) =
\hat{f}(2) = 0$. Elementary integration gives
\[
F_1(t) = \int h_1(x) \phi(x)\,dx = -e^{-t^2/2}/\sqrt{2\pi}, \qquad
F_3(t) = \int h_3(x) \phi(x)\,dx = (1-t^2)e^{-t^2/2}/\sqrt{12\pi};
\]
thus
\begin{eqnarray*}
\hat{f}(1) & = & 2(F_1(t) + F_1(-t) - F_1(0)) - F_1(\infty) -
F_1(-\infty) = \sqrt{2/\pi}\,(1-2e^{-t^2/2}), \\
\hat{f}(3) & = & 2(F_1(t) + F_1(-t) - F_1(0)) - F_1(\infty) -
F_1(-\infty) = -\sqrt{1/3\pi}\,(1 - 2(1-t^2)e^{-t^2/2}).
\end{eqnarray*}
We conclude
\begin{equation} \label{eqn:formula}
\sum_{d \leq 3} \hat{f}(d)^2 = \frac{2}{\pi} \Bigl(1 -
2e^{-t^2/2}\Bigr)^2 + \frac{1}{3\pi} \Bigl(1 -
2(1-t^2)e^{-t^2/2}\Bigr)^2.
\end{equation}
As $t \to 0$ or $\infty$ we recover the fact, well known in the
boolean regime (see, e.g., \cite{Bernasconi:98}), that $\sum_{d
\leq 3} \widehat{\mathrm{Maj}}(d)^2 = 2/\pi + 1/3\pi$. But the above
expression is not maximized for these $t$; rather, it is maximized
at $t = 2.69647$, where the expression becomes roughly $.75913$.
Fixing this particular $t$ completes the proof.
\end{proof}
\bigskip
It is now clear how to construct the sequence of completely
symmetric odd functions $f_n : \{-1,1\}^n \to \{-1,1\}$ with the same
property --- take $f_n(x) = f((x_1 + \cdots + x_n)/\sqrt{n})$. The
proof that the property holds follows essentially from the fact
that the limits of Kravchuk polynomials are Hermite polynomials.
For completeness, give a direct proof of
Corollary~\ref{cor:counter} in Appendix~\ref{app:counter}.
\begin{corollary} \label{cor:counter}
For $n$ odd there is a sequence of completely symmetric odd
functions $f_n : \{-1,1\}^n \to \{-1,1\}$ satisfying $\mathrm{Inf}_i(f_n)
\leq O(1/\sqrt{n})$ for each $i$, and
\[
\lim_{n \mbox{ odd }\to \infty} \sum_{|S| \leq 3}
\widehat{f_n}(S)^2 \geq 0.75913 > \frac{2}{\pi} + \frac{1}{3\pi} =
\lim_{n \mbox{ odd }\to \infty} \sum_{|S| \leq 3}
\widehat{\mathrm{Maj}_n}(S)^2.
\]
\end{corollary}
\bigskip
In light of this counterexample, it seems we can only hope to
sharpen Bourgain's Theorem~\ref{thm:bourgain} in the asymptotic
setting; one might ask whether its upper bound can be improved to
\[
1 - (1-o(1)) (2/\pi)^{3/2}\;d^{-1/2},
\]
the asymptotics for Majority.
\bibliographystyle{abbrv}
|
{
"timestamp": "2005-05-24T01:54:02",
"yymm": "0503",
"arxiv_id": "math/0503503",
"language": "en",
"url": "https://arxiv.org/abs/math/0503503"
}
|
\section{\label{Intro}Introduction}
Entanglement is a property of correlated quantum systems that cannot
be accounted for classically. Entangled states of distinct (possibly
interacting) quantum systems, which are those that cannot be
factorized into product states of the subsystems, are of fundamental
interest in quantum mechanics. The production of pairwise entangled
states is an essential requirement for the operation of the quantum
gates that make quantum information and quantum computation possible
\cite{Bouwmeester}. Considerable attention has been devoted to
interacting Heisenberg spin systems
\cite{Arnesen,Wang,gerard,Korepin}, which serve as a model for
various solid state \cite{Loss1, Loss2, Imamoglu} or NMR
\cite{Ernst, Nielson} quantum computation schemes and for simulating
magnetic phenomena in condensed matter systems using atoms in
optical lattices~\cite{SorensenMolmer, Duan}. Indeed, general
Hamiltonians that include Heisenberg spin-spin interactions have
been proposed as ``generic''~\cite{Shepelyansky} or
``ideal''~\cite{Makhlin} model Hamiltonians for quantum computation
systems. A key question for entangled quantum states is the effect
of decoherence due to the environment (see, e.g.,~\cite{Bloch,
Zurek, Albrecht,Das Sarma, Dodd} and references therein), which is
not only a fundamental issue for quantum computation
devices~\cite{Lloyd, Knight} but also for the relation between
quantum and classical physics~\cite{Zurek, Braun}. Although there
have been many investigations of decoherence in recent years,
careful investigation of well-understood model systems continue to
produce surprises that add to fundamental understanding. For
example, Yu and Eberly~\cite{Eberly} have recently shown that the
entanglement of a pair of non-interacting qubits in the presence of
spontaneous decay of the upper states may decohere in a finite time
instead of exponentially.
In this paper we examine decoherence due to both population
relaxation and thermal effects for an entangled (and interacting)
two qubit system. The Hamiltonian for our two-qubit system has the
form of the well-known Heisenberg XY model for two interacting spins
in the presence of an external magnetic field, where the effective
magnetic field is defined by the energy separation of the two-level
system that we associate with each (spin 1/2) qubit. As noted
above, this form of Hamiltonian is very common in models for quantum
computing~\cite{SorensenMolmer, Duan, Shepelyansky,Makhlin}. Our
analysis of decoherence complements that of Ref.~\cite{Eberly} by
examining a system in which the qubits interact. For our two-qubit
model system at zero temperature, we find that for any initial
state, including the common one in which the two qubits are
initially unentangled, the system reaches a steady state of pairwise
entanglement in spite of population relaxation. The extent of
steady-state entanglement is sensitive to both the spatial
anisotropy of the interaction between qubits and to the energy level
separation of the two levels associated with each qubit. To the
extent that these two parameters can be varied in some particular
physical realization of our model system, the magnitude of steady
state entanglement may thus be controlled. We analyze both
analytically and numerically the time-dependent evolution of the
entanglement (as measured quantitatively by the
concurrence~\cite{Bennett, Wootters}) of our model two-qubit system
for some typical initial states: a pure, separable initial state; a
pure, entangled initial state; and a mixed initial state. We also
obtain an analytic formula for the steady state concurrence that
shows its dependence on both the system parameters and the
decoherence rate and that enables us to specify optimal values for
these parameters to achieve the maximum possible concurrence. In a
separate section, we consider the case of finite temperature and
present an analytic formula for the concurrence, which remains
non-zero over a finite range of low temperatures. In our concluding
section, we discuss some implications of these results.
\section{\label{Hamiltonian}Two-Qubit Hamiltonian}
We note that the Hamiltonian of a Heisenberg chain of $N$ spin
$\frac{1}{2}$ particles with nearest-neighbor interactions
is~\cite{Korepin}:
\begin{equation}
H=\sum_{n=1}^N(J_xS_{n}^{x}S_{n+1}^{x}+J_yS_{n}^{y}S_{n+1}^{y}+J_zS_{n}^{z}S_{n+1}^{z})
\end{equation} where $S_{n}^{\alpha}=\frac{1}{2}\sigma_{\alpha}^{n}
(\alpha=x,y,z)$ are the local spin $\frac{1}{2}$ operators at site
$n$, the $\sigma_{\alpha}^{n}$ operators are the Pauli matrices at
site $n$, the periodic boundary condition $S_{N+1}=S_1$ applies,
and $\hbar=1$. For arbitrary $J_{\alpha}$'s, the Heisenberg chain is
often called the $XYZ$ model. The chain is said to be
antiferromagnetic for $J_{\alpha}>0$ and ferromagnetic for
$J_{\alpha}<0$. The $XY (J_z=0)$ and the Heisenberg-Ising
$(J_y=J_z=0)$ interactions have been analyzed for nuclear spin
systems \cite{Ernst}, in particular for nuclear magnetic resonance
approaches to quantum computation (see, e.g., Section 7.7 of
Ref.~\cite{Nielson}).
The Hamiltonian $H$ for an anisotropic two-qubit Heisenberg $XY$
system in an (effective) external magnetic field $\omega$ along the
z-axis is: \begin{equation}\label{Hamilton}
H=\omega(S_{1}^{z}+S_{2}^{z})+J(S_{1}^{+}S_{2}^{-}+S_{1}^{-}S_{2}^{+})+\Delta(S_{1}^{+}S_{2}^{+}+S_{1}^{-}S_{2}^{-})
\end{equation} where $J=(J_x+J_y)/2$, $\Delta=(J_x-J_y)/2$, and
$S^{\pm}=S^{x}{\pm}{i}S^{y}$ are the spin raising and lowering
operators. The first term in the Hamiltonian describes the energy of
the spins in the effective external magnetic field. This effective
field is defined by the energy levels of our qubits: we assume that
each of our two qubits represents an identical two level system
whose two energies are defined by $\pm \omega/2$. The spin
interaction Hamiltonian, described by the second and the third terms
in Eq.~({\ref{Hamilton}}), produces the coherence of the two qubits
that is necessary for their entanglement in the presence of
decoherence. As shown below, the third term, whose magnitude is
proportional to the parameter $\Delta$, which describes the spatial
anisotropy of the spin-spin interaction, is essential for the
production of steady state entanglement.
\section{\label{Zero}Time Evolution of the Concurrence at Zero Temperature}
The time evolution of the system for zero temperature, $T=0$, is
given by the following master equation (see, e.g., \cite{Carmichael,
Gardiner} and Section 8.4.1 of \cite{ Nielson}):
\begin{equation} \label{Lindblad}\dot{\rho}=-i\left[H,\rho\right] + \gamma{\cal
D}\left[S_{1}^{-}\right]\rho+\gamma{\cal
D}\left[S_{2}^{-}\right]\rho.\end{equation} Here $\rho$ is the density matrix,
which in the presence of population relaxation represents the mixed
state of the system. The Lindblad super operator $\cal D$
{~\cite{lindblad}} is defined by ${\cal D}[A]B \equiv ABA^\dagger -
\{A^\dagger A,B\}/2$,
which describes the population relaxation of the upper state of each qubit due to the environment;
$\gamma$ is the phenomenological rate of population relaxation,
which for simplicity is assumed to be the same for each of the two
qubits (i.e., we assume each qubit has the same interaction with the
environment). As discussed below, the assumption of a single decay
rate, $\gamma$, requires us to place restrictions on the magnitude
of the coupling between qubits.
Entanglement is increasingly regarded as a physical resource of a
quantum information system (see, e.g., Section 12.5 of
Ref.~\cite{Nielson}) and many measures for quantifying entanglement
have been developed (see, e.g., \cite{Bennett, Wootters, Vedral,
Horodecki, Rains, Preskill}). Since decoherence processes cause the
system state to become mixed, we use the measure of entanglement
termed concurrence \cite{Bennett, Wootters}. For a system
described by the density matrix $\rho$, the concurrence $C$ is
\begin{equation} \label{C} C={\rm
max}\left(\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4},0\right),
\end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_4$
are the eigenvalues (with $\lambda_1$ the
largest one) of the ``spin-flipped'' density operator $R$, which is
defined by \begin{equation} R =\rho
\left(\sigma_{y}\otimes\sigma_{y}\right)\rho^{*}
\left(\sigma_{y}\otimes\sigma_{y}\right), \end{equation} where $\rho^{*}$
denotes the complex conjugate of ${\rho}$ and ${\sigma}_{y}$ is the
usual Pauli matrix. $C$ ranges in magnitude from $0$ to $1$;
nonzero $C$ denotes an entangled state.
The basis states $\ket{\psi_{i}}$ for our two-qubit system are the
separable product states of the individual qubits: \begin{eqnarray}
\label{basis4}
\ket{\psi_{1}}&=&\ket{e}_{1}\otimes\ket{e}_{2},\nonumber \\
\ket{\psi_{2}}&=&\ket{e}_{1}\otimes\ket{g}_{2},\nonumber \\
\ket{\psi_{3}}&=&\ket{g}_{1}\otimes\ket{e}_{2},\nonumber \\
\ket{\psi_{4}}&=&\ket{g}_{1}\otimes\ket{g}_{2}. \end{eqnarray} In general, a
two-qubit system is represented by a density matrix having sixteen
non-zero elements. For our Hamiltonian, however, the density matrix
can be represented as the sum of two submatrices that evolve
independently of one another, \begin{eqnarray} \label{densitymatrix}
\rho=\left(\begin{array}{cccc}
\rho_{11}& 0 & 0 &\rho_{14}\\
0& \rho_{22} & \rho_{23} &0 \\
0& \rho_{32} & \rho_{33} &0 \\
\rho_{41}& 0 & 0 &\rho_{44} \\
\end{array}\right)+
\left(\begin{array}{cccc}
0 & \rho_{12} & \rho_{13} & 0 \\
\rho_{21}& 0 & 0 & \rho_{24} \\
\rho_{31}& 0 & 0 & \rho_{34} \\
0 & \rho_{42} & \rho_{43} & 0 \\
\end{array}\right),
\end{eqnarray} i.e., in solving Eq.~({\ref{Lindblad}}) for $\rho(t)$ the
forms of each of the two submatrices in Eq.~(\ref{densitymatrix})
are preserved. (Note that the second matrix on the right hand side of Eq.~(\ref{densitymatrix}) does not have the form of a density matrix.) Each of the examples in this paper has an initial density matrix whose form is that of the first matrix on the right of Eq.~(\ref{densitymatrix}). This limitation is not very restrictive, as unentangled, entangled, and mixed states can all be described.
Furthermore, for a state having a density matrix of the
form of the first matrix on the right of Eq.~(\ref{densitymatrix}),
the concurrence has the following analytic form: \begin{eqnarray}\label{CR} C=\max{\{0, C_1,
C_2\}},\end{eqnarray} where
\begin{eqnarray}\label{C1C2}
C_1&=&2(|\rho_{41}|-\sqrt{\rho_{33}\rho_{22}})\nonumber\\
C_2&=&2(|\rho_{32}|-\sqrt{\rho_{44}\rho_{11}}).
\end{eqnarray}
The solutions of the master equation in Eq.~({\ref{Lindblad}})
simplify by transforming from the product state basis
$\ket{\psi_{i}}$ in Eq.~({\ref{basis4}}) to the basis of eigenstates
$\ket{\Phi_{\alpha}}$ of the Hamiltonian in Eq.~({\ref{Hamilton}}),
\begin{eqnarray} \label{basis5}
\ket{\Phi_{1}}&=&N^+(\ket{gg} + \frac{\Delta}{\Omega-\omega}\ket{ee}),\nonumber \\
\ket{\Phi_{2}}&=&\frac{1}{\sqrt{2}}(\ket{eg}+\ket{ge}),\nonumber \\
\ket{\Phi_{3}}&=&\frac{1}{\sqrt{2}}(\ket{ge}-\ket{eg}),\nonumber \\
\ket{\Phi_{4}}&=&N^-(\ket{gg}-\frac{\Delta}{\Omega+\omega}\ket{ee}),\\
\label{omega}
\Omega &=& \sqrt{\omega^2+\Delta^2},\\
N^{\pm}&=&(\Omega \mp \omega)/\sqrt{\Delta^2 + (\Omega \mp
\omega)^2}. \end{eqnarray}
After transformation, the solutions for each element
$\bar{\rho}_{{\alpha}{\alpha'}}$ of the density matrix (where
$\bar{\rho}$ denotes $\rho$ in the eigenstate basis) can be found
analytically. For the interesting special case that both qubits are
initially in their ground states (i.e., the system is initially in
state $\ket{\psi_{4}}$ in Eq.~(\ref{basis4})), the analytic
solutions for $\bar{\rho}_{{\alpha}{\alpha'}}(t)$ are:
\begin{eqnarray}\label{element}
\label{p11}
\bar{\rho}_{11}(t)&=&\frac{1}{2\Omega{\alpha}}\Big[-\omega\alpha+2\Omega{\Delta^2}e^{-2\gamma{t}}\nonumber\\
&&+\Omega(\alpha-2\Delta^2)+2e^{-\gamma{t}}\Delta^2\gamma{\sin{[2\Omega{t}]}}\Big]\\
\label{p22}
\bar{\rho}_{22}(t)&=&\frac{\Delta^2}{\Omega{\alpha}}\Big[\Omega-{\Omega}e^{-2\gamma{t}}-e^{-\gamma{t}}\gamma{\sin{[2\Omega{t}]}}\Big]\\
\label{p33}
\bar{\rho}_{33}(t)&=&\bar{\rho}_{22}(t)\\
\label{p44}
\bar{\rho}_{44}(t) &=&1-\bar{\rho}_{11}(t)-\bar{\rho}_{22}(t)-\bar{\rho}_{33}(t)\\
\label{p14}
\bar{\rho}_{14}(t)&=&\frac{\Delta}{4i\Omega^2+2\Omega\gamma}\Big[2i\Omega
e^{-\gamma t}\cos{[2\Omega t]}\nonumber\\
&&+2\Omega e^{-\gamma t}\sin{[2\Omega t]}+\gamma\Big]\\
\label{p41} \bar{\rho}_{41}(t)&=&\bar{\rho}_{14}^*(t)
\end{eqnarray}
where all other matrix elements are zero and where
\begin{eqnarray}
\label{alpha} \alpha&=&4\Omega^2+\gamma^2.
\end{eqnarray}
From Eqs.~(\ref{p11}-\ref{p41}) it is evident that both the
off-diagonal (coherence) matrix elements $\bar{\rho}_{14}$ and
$\bar{\rho}_{41}$ in Eqs.~(\ref{p14}-\ref{p41}) and the diagonal
(population) matrix elements $\bar{\rho}_{{\alpha}{\alpha}}$ in
Eqs.~(\ref{p11}-\ref{p44}) have terms that oscillate with frequency
$2\Omega$. Note that the coherence matrix elements $\bar{\rho}_{14}$
and $\bar{\rho}_{41}$ are non-zero only when the spin-spin
interactions are anisotropic (i.e., $\Delta \neq 0$); also, the
value of $\Omega$ is sensitive to this anisotropy (cf.
Eq.~(\ref{omega})). From Eqs.~(\ref{p11}-\ref{p41}) it can be seen
that the coherence matrix elements have terms that decay at the rate
$\gamma$ while the population matrix elements also have terms that
decay at the rate $2\gamma$. Analytic solutions similar to
Eqs.~(\ref{p11}-\ref{p41}) can be given for some other initial
states.
The assumption of a single decay rate, $\gamma$, in the master
equation ({\ref{Lindblad}}) requires some discussion. Owing to the
interaction between qubits described by the Hamiltonian
({\ref{Hamilton}}), the two-qubit energy level structure is altered
from that describing non-interacting, identical qubits.
Nevertheless, the assumption of a single decay rate, $\gamma$, is
reasonable provided the interaction does not significantly alter the
effective energy level separations, or, more precisely, provided the
rotating wave approximation remains valid~\cite{ZollerPC} (see,
e.g., pp. 160-161 of Ref.~\cite{Gardiner}). The eigenstates in
Eq.~(\ref{basis5}) have the following eigenenergies~\cite{gerard}:
the Bell states have eigenvalues $\pm J$ while the other eigenstates
have eigenvalues $\pm \Omega$. Thus if we restrict the magnitudes
of the coupling parameter $J$ and the anisotropy parameter $\Delta$
to values such that,
\begin{eqnarray}
&&|J|/\omega \le{0.1}\label{restriction1}\\
&&(\Omega - \omega)/\omega \le{0.1} \label{restriction2},
\end{eqnarray}
we shall ensure that the energy level separations of the interacting
qubit system do not change by more than~10\% from that of the
non-interacting qubit system. Except where it is explicitly
mentioned otherwise, all examples given below have parameter values
for which the above inequalities are satisfied.
Perhaps surprisingly, the decoherence due to population relaxation
does not prevent the creation of a steady state level of
entanglement, regardless of the initial state of the system. This is
demonstrated in Figs.~{\ref{fig1}} and~\ref{fig2}, which show the
time evolution of the concurrence (cf. Eq.~(\ref{C})) for three
different initial states: (1) An unentangled, separable state,
$\ket{\psi_{4}}$ (cf. Eq.~(\ref{basis4})); (2) a completely
entangled state, the Bell state $\ket{\Psi}=
\frac{1}{\sqrt{2}}(\ket{gg}-\ket{ee})$; and (3) a mixed state,
defined as an equal mixture of $\ket{\psi_{4}}$ and the Bell state
$\ket{\Phi_{2}}$. In Fig.~\ref{fig1} we consider the case that $J
= \Delta = \omega/10$, which implies that $J_y = 0$ and which thus
corresponds to the ``generic'' quantum computation model Hamiltonian
of Ref.~\cite{Shepelyansky}. In Fig.~\ref{fig2} we consider the
case that $J = \omega/10$ and that $\Delta = 0.458 \omega$, which
corresponds to a general case in which $J_x$ and $J_y$ have opposite
signs, which may possibly be achieved for an optical lattice
system~\cite{SorensenMolmer, Duan}. For each of the three initial
states considered, the corresponding curves in Figs.~{\ref{fig1}}
and~\ref{fig2} give the numerical results for the concurrence
defined by Eq.~(\ref{C}), after solving Eq.~(\ref{Lindblad})
numerically for the density matrix in the separable representation
(cf. Eq.~(\ref{basis4})). Since each initial state has a density
matrix of the form of the first matrix on the right of
Eq.~(\ref{densitymatrix}), the concurrence for each of these states
is given also by Eqs.~(\ref{CR}-\ref{C1C2}). (Note that
discontinuities in the time derivatives of $C(t)$ for the dashed
curve in Fig.~\ref{fig1} in the range $2.0~{\leq}~t ~{\leq} ~2.5$
stem from the definition in Eq.~(\ref{C}); all density matrix
elements are smooth functions of $t$.) The solid circles on the
curves for the initial state $\ket{gg}$ in Figs.~{\ref{fig1}}
and~\ref{fig2} give the concurrence obtained from the analytic
Eqs.~(\ref{CR}-\ref{C1C2}) (after transforming the analytic
expressions in Eqs.~(\ref{p11}) - (\ref{p41}) for this state's
density matrix $\bar{\rho}_{{\alpha}{\alpha'}}$ to $\rho_{ij}$).
These analytic results coincide with those obtained by direct
numerical solution of Eq.~(\ref{Lindblad}).
Despite the presence of decoherence, the results in Figs.~\ref{fig1}
and~\ref{fig2} show that the concurrence reaches the same steady
state value (after some oscillatory behavior) for a given set of
system parameters, regardless of the initial state of the system.
(This is true even for initial states having non-zero matrix
elements belonging to the second matrix on the right of
Eq.~(\ref{densitymatrix}); for our system, such matrix elements
vanish in the steady state.) Clearly the Heisenberg spin-spin
interaction in Eq.~(\ref{Hamilton}) serves to maintain an entangled
state despite the presence of decoherence in Eq.~(\ref{Lindblad}).
We find a steady state concurrence of 0.09309 for the system
parameter values chosen in Fig.~\ref{fig1} and a steady state
concurrence of 0.28916 for the system parameter values chosen in
Fig.~\ref{fig2}.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig1.eps}
\end{center}
\caption{\label{fig1}Plot of $T=0$ concurrence vs. scaled time,
$\gamma t$, for three different initial states: (1) An initially
unentangled state, $\ket{\Psi}=\ket{gg}$ (solid line); an initially
entangled state, the Bell state $\ket{\Psi}= \frac{1}{\sqrt{2}}(
\ket{gg}-\ket{ee})$ (dashed line); and (3) an initially mixed state,
defined as an equal mixture of $\ket{gg}$ and $\frac{1}{\sqrt{2}}(
\ket{eg}+\ket{ge})$ (solid squares). The system parameters are:
$\gamma=0.3$, $\omega=1.0$, $J=0.1$, and $\Delta=0.1$.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig2.eps}
\end{center}
\caption{\label{fig2}Plot of $T=0$ concurrence vs. scaled time,
$\gamma t$, for the same three initial states as in Fig.~\ref{fig1},
but for the following different system parameters: $\gamma=0.458$,
$\omega=1.0$, $J=0.1$, and $\Delta=0.458$.}
\end{figure}
The analytic expressions for the $T=0$ steady state values of
$\rho_{ij}(t)$ are as follows:
\begin{eqnarray}
\label{p11a}
\rho_{11}&=& \rho_{22}=\rho_{33}=\frac{\Delta^2}{\alpha}\\
\label{p44a}
\rho_{44}&=& 1 - \frac{3\Delta^2}{\alpha}\\
\label{p14a}
\rho_{14}&=&\frac{-2\omega\Delta-i\Delta\gamma}{\alpha}\\
\label{p41a} \rho_{41}&=&\rho_{14}^*
\end{eqnarray}
The corresponding $T=0$ steady state concurrence is found to be:
\begin{equation} \label{steadycon}
C_{steady}=\frac{2\sqrt{\Delta^2(4\omega^2+\gamma^2)}-2\Delta^2}{\alpha}
\end{equation} This result stems from $C_1$ in Eq.~({\ref{C1C2}}) calculated
for the separable basis density matrix $\rho_{ij}$ in
Eqs.~(\ref{p11a}-\ref{p41a}). The steady-state concurrence is seen
to depend on the system parameters $\omega$, $\Delta$, and $\gamma$
(but not on $J$). Also, $\gamma$ serves as a scale factor, i.e.,
$C_{steady}$ depends only on the scaled variables,
$\bar{\omega}=\omega/\gamma$ and $\bar{\Delta}=\Delta/\gamma$. These
parameters may be varied in order to maximize $C_{steady}$. The
function $C_{steady}(\bar{\omega},\bar{\Delta})$ is shown in
Fig.~{\ref{fig3}; one sees that the surface has a ridge along which
it takes its maximum value. The coordinates of the ridge and the
value of $C_{steady}$ on the ridge may be determined analytically.
For fixed $\omega$, $C_{steady}$ (cf. Eq.~(\ref{steadycon})) has its
maximum at the following value of $\Delta$:
\begin{eqnarray}
\label{Deltamax} \Delta_{max}&=& \frac{\sqrt{4 \omega^2 +
\gamma^2}}{(1+\sqrt{5})}.
\end{eqnarray}
The solid line in the $\bar{\omega}$-$\bar{\Delta}$ plane of
Fig.~{\ref{fig3}} represents the locus of points
$\bar{\Delta}_{max}(\bar{\omega})$ given by Eq.~(\ref{Deltamax})
(upon division by $\gamma$). Substitution of $\Delta_{max}$ into
Eq.~(\ref{steadycon}) gives the parameter-independent maximum value
of the concurrence, represented by the solid line in
Fig.~{\ref{fig3}} along the ridge of $C_{steady}$:
\begin{eqnarray}
\label{Csteadybmax} C_{steady}(\Delta_{max})&=&
(1+\sqrt{5})^{-1}=0.309.
\end{eqnarray}
Eq.~(\ref{steadycon}) shows that in order to have a positive value
of $C_{steady}$, one must have $4\omega^2+\gamma^2 \geq\Delta^2$.
Note finally that Eq.~(\ref{Csteadybmax}) was derived from
Eq.~(\ref{steadycon}) without taking into account the restrictions
on the parameter values imposed by the conditions in
Eqs.~(\ref{restriction1}) and (\ref{restriction2}) that are
necessitated by our assumption of a single decay rate, $\gamma$.
Nevertheless, one sees for the example plotted in Fig.~{\ref{fig2}}
that there do exist values of the parameters that satisfy
Eqs.~(\ref{restriction1}) and (\ref{restriction2}) for which one
obtains a steady state level of concurrence that is close to the
global maximum value given by Eq.~(\ref{Csteadybmax}) (and shown by
the solid line in Fig.~{\ref{fig3}}).
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig3.eps}
\end{center}
\caption{\label{fig3}Plot of the $T=0$ steady state concurrence
(cf. Eq.~(\ref{steadycon})) as a function of the scaled energy
$\bar{\omega}$ and the scaled anisotropy parameter $\bar{\Delta}$
(ranging from $0.309$ to $1$), where $\bar{\omega}=\omega/\gamma$
and $\bar{\Delta}=\Delta/\gamma$. The solid lines locate the maximum
value of concurrence (cf. Eq.~(\ref{Csteadybmax})); see text for
discussion.}
\end{figure}
\section{\label{Finite}Temperature Dependence of the Steady State Concurrence}
It is of interest to examine how the steady state entanglement
obtained for zero temperature in the prior section changes when the
temperature is finite. For simplicity, we assume that each qubit
interacts with the same thermal bath. It is known that the
equilibrium entanglement must vanish at some finite temperature
{\cite{Fine}}. In order to examine the effect of thermal
decoherence on the entanglement for our system we consider the
following master equation {\cite{Carmichael2, Mintert}:
\begin{eqnarray} \label{Lindblad2}\dot{\rho}&=&-i\left[H,\rho\right] +
\gamma{(\bar{n}+1)}{\cal
D}\left[S_{1}^{-}\right]\rho+\gamma{(\bar{n}+1)}{\cal
D}\left[S_{2}^{-}\right]\rho \nonumber\\&&+\gamma\bar{n}{\cal
D}\left[S_{1}^{+}\right]\rho+\gamma\bar{n}{\cal
D}\left[S_{2}^{+}\right]\rho, \end{eqnarray} where $\bar{n}$, the average
excitation of the thermal bath, parametrizes the temperature. Note
that $\bar{n}$ is zero at zero temperature, whereupon one observes
that Eq.~(\ref{Lindblad2}) reduces to Eq.~(\ref{Lindblad}); also,
$\bar{n}$ becomes infinite as the temperature becomes infinite. The
master equation (\ref{Lindblad2}) may be solved to obtain the
following analytic expressions for the steady state density matrix
of our system:
\begin{eqnarray}\label{densitymatrixT}
\rho_{11}&=&\frac{\bar{n}^2(4\bar{\omega}^2+(1+2\bar{n})^2)+\bar{\Delta}^2(1+2\bar{n})^2}{(1+2\bar{n})^2(4\bar{\omega}^2+(1+2\bar{n})^2+4\bar{\Delta}^2)}\nonumber\\
\rho_{22}&=&\rho_{33}=\frac{1}{4}[1-\frac{4\bar{\omega}^2+(1+2\bar{n})^2}{(1+2\bar{n})^2(4\bar{\omega}^2+(1+2\bar{n})^2+4\bar{\Delta}^2)}]\nonumber\\
\rho_{44}&=&\frac{4\bar{\omega}^2(1+\bar{n})^2+(1+2\bar{n})^2((1+\bar{n})^2+\bar{\Delta}^2)}{(1+2\bar{n})^2(4\bar{\omega}^2+(1+2\bar{n})^2+4\bar{\Delta}^2)}\nonumber\\
\rho_{14}&=&-\frac{\bar{\Delta}(2\bar{\omega}+i(2\bar{n}+1))}{(1+2\bar{n})(4\bar{\omega}^2+(1+2\bar{n})^2+4\bar{\Delta}^2)}
\end{eqnarray} In the limit of zero temperature (i.e., $\bar{n}\rightarrow
0$), the density matrix elements in Eq.~(\ref{densitymatrixT})
reduce to the results in Eqs.~(\ref{p11a}) - (\ref{p41a}). In the
limit of infinite temperature (i.e., $\bar{n}\rightarrow \infty$),
the density matrix becomes diagonal, with each diagonal element
equal to $1/4$, indicating, as expected {\cite{Fine}}, that all
entanglement vanishes.
The concurrence may be calculated for the finite temperature,
steady-state density matrix in Eq.~(\ref{densitymatrixT}) to obtain:
\begin{eqnarray}\label{CT}
C(\bar{\omega},\bar{\Delta},\bar{n})&=&2\frac{\sqrt{\bar{\Delta}^2(4\bar{\omega}^2+(1+2\bar{n})^2)}}{(1+2\bar{n})(4\bar{\Omega}^2+(1+2\bar{n})^2)}-\frac{1}{2}\nonumber\\&&+\frac{(4\bar{\omega}^2+(1+2\bar{n})^2)}{2[(1+2\bar{n})^2(4\bar{\Omega}^2+(1+2\bar{n})^2)]},
\end{eqnarray} where all system parameters have been normalized by the
relaxation rate $\gamma$: $\bar{\Delta}=\Delta/\gamma$,
$\bar{\omega}=\omega/\gamma$, and $\bar{\Omega} =
\Omega/\gamma=\sqrt{\bar{\omega}^2+\bar{\Delta}^2}$ (cf.
Eq.~(\ref{omega})). In the limit of zero temperature (i.e.,
$\bar{n}\rightarrow 0$), the concurrence in Eq.~(\ref{CT}) reduces
to that in Eq.~(\ref{steadycon}). The behavior of this finite
temperature, steady state concurrence is shown in Fig.~{\ref{fig4}}
for the same two sets of system parameters considered in
Figs.~\ref{fig1} and~\ref{fig2} respectively. One sees that both
curves decrease with increasing $\bar{n}$ until eventually the
concurrence vanishes at a finite value of $\bar{n}$, as expected
\cite{Fine}. One sees also that the larger the value of the
interaction asymmetry parameter $\bar{\Delta}$, the larger the value
of the concurrence at any finite value of $\bar{n}$. For any fixed
temperature (i.e., $\bar{n}$), as the effective magnetic field,
$\bar{\omega}$, increases, the concurrence takes a finite, non-zero
value. In the limit $\bar{\omega}\rightarrow \infty$, one has that
$\bar{n}\rightarrow 0$ and $C\rightarrow
|\bar{\Delta}|/\bar{\omega}$. This decrease with
$\bar{\omega}^{-1}$ as well as the fact that $C \ge 0$ only if
$\bar{\Delta} \not= 0$ is consistent with the results of
Ref.~\cite{gerard}.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig4.eps}
\end{center}
\caption{Plots of the finite temperature, steady state concurrence
(cf. Eq.~(\ref{CT})) as a function of the average thermal excitation
function, $\bar{n}$, for the same two sets of system parameters as
in Figs.~\ref{fig1} and~\ref{fig2}. The curve in the lower left
corner of the figure corresponds to the same system parameters as in
Fig.~\ref{fig1}; the curve close to the diagonal corresponds to the
same system parameters as in Fig.~\ref{fig2}. Note that at $\bar{n}
= 0$ both curves begin at the steady state values of the concurrence
shown in Figs.~\ref{fig1} and~\ref{fig2}.}\label{fig4}
\end{figure}
\section{\label{Discussion} Discussion and Conclusions}
Quantum coherence is a necessary requirement for the existence of
entanglement. One may define coherence existing in a single qubit as
``local coherence'' while coherence between two qubits $A$ and $B$
may be defined as ``global coherence'' \cite{Eberly}. How do local
and global coherence relate to entanglement? The answer for our
model system may be understood by considering the relation between a
general two-qubit density matrix, $\rho_{AB}$, and the reduced
density matrices, $\rho^A$ and $\rho^B$, for each of the two qubits,
where $\rho^A=tr_B(\rho^{AB})$ is obtained by tracing over the
degrees of freedom of qubit $B$, and similarly for $\rho^B$. In our
model, after doing partial traces, we find that in the steady
state the local coherence of each qubit is zero, i.e., $\rho^A$ and
$\rho^B$ are diagonal matrices. However, there exist global
coherence terms for our system (i.e., $\rho_{14}$ and $\rho_{41}$)
that are non-zero, indicating that global, not local, coherence is
responsible for this system's entanglement. For time $t>0$,
$\rho_{14}$ and $\rho_{41}$ for our system are always non-zero;
$\rho_{23}$ and $\rho_{32}$ may be non-zero for finite times, but
vanish in the steady state. Ref.~{\cite{Eberly}} considered a
decohering system of two entangled (but non-interacting) qubits. In
that system, both local and global coherence vanished in the
asymptotic time limit; however, in some cases, the latter vanished
for finite times \cite{Eberly}.
An interesting question regarding the steady state of our model
system is whether or not it is a decoherence-free subspace
\cite{DFS}. Typically, a decoherence-free subspace is defined to be
one for which the decoherence terms in the system's master equation
vanish \cite{Whaley}. For our system, this would mean that the
second and third terms on the right of Eq.~(\ref{Lindblad}) vanish
for the case of zero temperature or, for the case of finite
temperature, that all terms except for the first one on the right of
Eq.~(\ref{Lindblad2}) vanish. However, in our system, the
decoherence terms in Eqs.~(\ref{Lindblad}) and~(\ref{Lindblad2}) do
not vanish; rather the sum of all terms on the right hand sides of
these two master equations vanish. This implies that population
relaxation and thermal decoherence (in the case of finite
temperature) are competing with the spin-spin interaction terms to
create a steady state level of entanglement, as measured by the
concurrence.
We note, finally, that a somewhat different model system studied by
S. Montangero, G. Benenti, and R. Fazio \cite{Montangero} has found
results for the pairwise concurrence that are somewhat similar to
those we find for our system. Specifically, they have considered
the entanglement of a pair of spins within a qubit lattice in which
there is disorder in the spin-spin couplings. They have identified
a regime in which the pairwise concurrence is stable against such
disorder in the couplings and has a value in the range of
0.2-0.3~\cite{Montangero}. We note that the numerical maximum for
their ``saturation value'' of the concurrence is quite close to the
analytical maximum we have derived in this paper (cf.
Eq.~(\ref{Csteadybmax})). It is interesting to observe that our
analytic result for the maximum value of the steady state
concurrence is constant for a range of system parameters (cf.
Eqs.~(\ref{Deltamax})-(\ref{Csteadybmax})). Whether or not this
analytical maximum holds also for other systems, such as the
different one considered in~\cite{Montangero}, is an open question.
In summary, we have provided a detailed analytical and numerical
analysis of decoherence for an interacting two-qubit model system
having a Hamiltonian identical in form to that for the well-known
two-qubit Heisenberg XY spin 1/2 system in the presence of an
(effective) external uniform magnetic field. For $T=0$, we have
presented an analytic solution for the evolution of entanglement,
measured by concurrence, for the case that both qubits are initially
in their ground states; we have presented also numerical solutions
for two other typical initial states. We find that our system is
robust against decoherence: a steady state level of entanglement,
controllable by the values of the system parameters, is always
reached for zero or finite, low temperatures. For the $T=0$ case,
we have defined this steady state analytically and obtained the
parameter values that maximize its entanglement. For $T>0$, the
steady state level of entanglement is found to vanish at a finite
temperature. Since our model interaction Hamiltonian describes also
mesoscopic objects that interact via their spins (e.g., cf.
Ref.~\cite{Skomski}), it may be that a certain level of entanglement
is robust against decohering interactions with an environment even
for such objects. As noted by Ghosh {\it et al.}~\cite{Ghosh},
even ``the slightest degree of entanglement can have profound
effects'' on the properties of mesoscopic spin systems.
We acknowledge stimulating discussions with Joseph H. Eberly, Hong
Gao, Andrei Y. Istomin, Murray Holland, Ting Yu, and Peter Zoller.
This work is supported in part by grants from the Nebraska Research
Initiative and the W. M. Keck Foundation.
\section*{References}
|
{
"timestamp": "2006-09-04T17:15:28",
"yymm": "0503",
"arxiv_id": "quant-ph/0503116",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503116"
}
|
\section{Introduction}
\indent Ebola hemorrhagic fever is a highly infectious and lethal disease named after a
river in the Democratic Republic of the Congo (formerly Zaire) where it
was first identified in 1976 \cite{CDC1}. Twelve outbreaks of Ebola
have been reported in Congo, Sudan, Gabon, and Uganda as of September 14, 2003
\cite{CDC2,WHO0}. Two different strains of the Ebola virus (Ebola-Zaire and the Ebola-Sudan)
have been reported in those regions. Despite extensive search,
the reservoir of the Ebola virus has not yet been identified
\cite{Breman2, Leirs1}. Ebola is transmitted by physical contact with
body fluids, secretions, tissues or semen from infected persons \cite{CDC1, WHO1}. Nosocomial
transmission (transmission from patients within hospital settings) has
been typical as patients are often treated by
unprepared hospital personnel (barrier nursing techniques need to be observed).
Individuals exposed to the virus who become infectious do so after a mean incubation
period of $6.3$ days ($1-21$ days) \cite{Breman1}. Ebola is
characterized by initial flu-like symptoms
which rapidly progress to vomiting, diarrhea, rash, and internal and external bleeding.
Infected individuals receive limited care as no specific treatment
or vaccine exists. Most infected persons die within $10$ days of
their initial infection \cite{Nature1} ($50\%-90\%$ mortality \cite{WHO1}).\\
\noindent Using a simple SEIR (susceptible-exposed-infectious-removed)
epidemic model (Figure \ref{myfig0}) and data from two
well-documented Ebola outbreaks (Congo $1995$ and Uganda $2000$), we
estimate the number of secondary cases generated by an index case
in the absence of control interventions ($R_0$). Our estimates of
$R_0$ are $1.83$ (SD $0.06$) for Congo (1995) and
$1.34$ (SD $0.03$) for Uganda (2000). We model the course of the outbreaks via
an SEIR epidemic model that includes a smooth transition in the transmission rate after
control interventions are put in place. We also perform an uncertainty
analysis on the basic reproductive number $R_0$ to
account for its sensitivity to disease-related parameters and analyze the
model sensitivity of the final epidemic size to the time at which
interventions begin. We provide a distribution for the final epidemic size. A
two-week delay in implementing public health measures results in an
approximated doubling of the final epidemic size.
\section{Methods}
We fit data from Ebola hemorrhagic fever outbreaks in Congo (1995) and Uganda (2000)
to a simple deterministic (continuous
time) SEIR epidemic model (Figure \ref{myfig0}). The least-squares fit
of the model provides estimates for the epidemic parameters. The fitted model can then be used to estimate
the basic reproductive number $R_0$ and quantify the impact of intervention measures on the
transmission rate of the disease. Interpreting the fitted model as an
expected value of a Markov process, we use multiple stochastic
realizations of the epidemic to estimate a distribution for the final
epidemic size. We also study the sensitivity
of the final epidemic size to the timing of interventions
and perform an uncertainty analysis on $R_0$ to account for
the high variability in disease-related parameters in our model.
\subsection{Epidemic Models}
Individuals are assumed to be in one of the following epidemiological
states (Figure \ref{myfig0}): susceptibles (at risk of contracting the disease),
exposed (infected but not yet infectious), infectives (capable of
transmitting the disease), and removed (those who recover or die from the
disease).
\subsubsection{Differential Equation Model}
Susceptible individuals in class $S$ in contact with the
virus enter the exposed class $E$ at the per-capita rate $\beta
I/N$, where $\beta$ is transmission rate per
person per day, $N$ is the total effective population size,
and $I/N$ is the probability that a contact is made with a infectious
individual (i.e. uniform mixing is assumed).
Exposed individuals undergo an average incubation period (assumed
asymptomatic and uninfectious) of $1/k$ days before progressing to the
infectious class $I$. Infectious individuals move to the $R$-class
(death or recovered) at the per-capita rate $\gamma$ (see Figure
\ref{myfig0}). The above transmission process is modeled by the following system of
nonlinear ordinary differential equations \cite{AM, BC}:\\
\begin{equation}
\label{eqn1}
\begin{array}{rcl}
{\displaystyle \dot{S}(t)}&=& -\beta S(t) I(t)/N\\
{\displaystyle \dot{E}(t)}&=& \beta S(t) I(t)/N -k E(t)\\
{\displaystyle \dot{I}(t)}&=& k E(t) - \gamma I(t)\\
{\displaystyle \dot{R}(t)}&=& \gamma I(t)\\
{\displaystyle \dot{C}(t)}&=& k E(t),\\
\end{array}
\end{equation}
\noindent where $S(t)$, $E(t)$, $I(t)$, and $R(t)$ denote the number of
susceptible, exposed, infectious, and removed individuals at
time $t$ (the dot denotes time derivatives). $C(t)$ is not an epidemiological
state but serves to keep track of
the cumulative number of Ebola cases from the time of onset of symptoms.\\
\subsubsection{Markov Chain Model}
\noindent The analogous stochastic model (continuous time Markov
chain) is constructed by considering three events: \textit{exposure}, \textit{infection} and
\textit{removal}. The transition rates are defined as:\\
\begin {tabular}{l c l}
\hline
Event & Effect & Transition rate\\
\hline
Exposure & (S, E, I, R) \ $\rightarrow$ \ (S-1, E+1, I, R) & $\beta(t) S I/N$ \\
Infection & (S, E, I, R) \ $\rightarrow$ \ (S, E-1, I+1, R) & $k E$ \\
Removal & (S, E, I, R) \ $\rightarrow$ \ (S, E, I-1, R+1) & $\gamma I$ \\
\hline
\\
\\
\end{tabular}
\noindent The event times $0 < T_1 < T_2 < ...$ at which an
individual moves from one state to another are modeled as a renewal
process with increments distributed exponentially, \\
$$P(T_k - T_{k-1} > t | T_j, j \le k-1) = e^{-t \mu(T_{k-1})}$$
\noindent where $\mu(T_{k-1}) = (\beta(T_{k-1}) S(T_{k-1})I(T_{k-1})/N
+ kE(T_{k-1}) +\gamma I(T_{k-1}))^{-1}$.\\
\noindent The final epidemic size is $Z=C(T)$ where
$T= min\{t>0, E(t)+I(t)=0 \}$, and its empirical distribution can be
computed via Monte Carlo simulations \cite{Renshaw1}.\\
\subsection{The Transmission Rate and the Impact of Interventions}
\noindent The intervention strategies to control the spread of Ebola
include surveillance, placement of suspected cases in quarantine
for three weeks (the maximum estimated length of the incubation
period), education of hospital personnel and community members on the
use of strict barrier nursing techniques (i.e protective clothing and
equipment, patient management), and the rapid burial or cremation of patients
who die from the disease \cite{WHO1}. Their net effect, in our model, is
to reduce the transmission rate $\beta$ from
$\beta_0$ to $\beta_1 < \beta_0$. In practice, the impact of the
intervention is not instantaneous. Between the time of the onset of
the intervention to the time of full compliance, the transmission rate
is assumed to decrease gradually from $\beta_0$ to $\beta_1$ according
to\\
$$
\beta(t) = \left \{ \begin{array}{ll}
\beta_0 & t<\tau \\
\beta_1+ (\beta_0 - \beta_1)e^{-q (t-\tau)} & t \ge \tau
\end{array} \right.
$$
\noindent where $\tau$ is the time at which interventions start and
$q$ controls the rate of the transition from $\beta_0$ to
$\beta_1$. Another interpretation of
the parameter $q$ can be given in terms of $t_h = \frac{ln(2)}{q}$,
the time to achieve $\beta(t) = \frac{\beta_0 + \beta_1}{2}$.
\subsection{Epidemiological data}
The data for the Congo (1995) and Uganda (2000) Ebola hemorrhagic
fever outbreaks include the identification dates of
the causative agent and data sources. The reported data are ($t_i$,
$y_i$), $i=1,...,n$ where $t_i$ denotes the $i^{th}$ reporting time
and $y_i$ the cumulative number of infectious cases from the beginning of the
outbreak to time $t_i$.\\
\noindent \textbf{Congo 1995.} This outbreak began in the Bandundu region, primarily
in Kikwit, located on the banks of the Kwilu River. The first case
(January 6) involved a 42-year old male charcoal worker and farmer
who died on January 13. The Ebola virus was not
identified as the causative agent until May $9$. At that time, an international team
implemented a control plan that involved active
surveillance (identification of cases) and education
programs for infected people and their family members. Family members
were visited for up to three weeks (maximum incubation period) after
their last identified contact
with a probable case. Nosocomial transmission occurred in Kikwit General Hospital but it
was halted through the institution of strict barrier nursing
techniques that included the use of protective equipment and
special isolation wards. A total of $315$ cases of Ebola
were identified ($81\%$ case fatality). Daily Ebola cases by date of
symptom onset from March $1$ through
July $12$ are available (Figure \ref{figDailycases}) \cite{Khan1}.\\
\noindent \textbf{Uganda 2000.} A total of $425$ cases ($53\%$ case
fatality) of Ebola were identified in three
districts of Uganda: Gulu, Masindi and Mbara. The onset of symptoms for
the first reported case was on August $30$, but the cause was
not identified as Ebola until October $15$ by the National Institute of
Virology in Johannesburg (South Africa). Active
surveillance started during the third week of October. A plan that
included the voluntary hospitalization of probable cases
was then put in place. Suspected cases were closely followed
for up to three weeks. Other control measures included community
education (avoiding crowd gatherings
during burials) and the systematic implementation of
protective measures by health care personnel and
the use of special isolation wards in hospitals. Weekly Ebola cases
by date of symptom onset are available from the WHO (World Health
Organization) \cite{WHO2} (from August $20$, $2000$ through January $7$, $2001$)
(Figure \ref{figDailycases}).\\
\subsection{Parameter Estimation}
Empirical studies in Congo suggest that the
incubation period is less than $21$ days with a mean of $6.3$ days
\cite{Breman1} and the infectious period is between $3.5$ and $10.7$
days. The model parameters $\Theta =(\beta_0$, $\beta_1$, $k$, $q$, $\gamma$) are
fitted to the Congo (1995) and Uganda (2000) Ebola outbreak data by
\textit{least squares} fit to the cumulative number of cases $C(t,\Theta)$ in
eqn. (\ref{eqn1}). We used a computer program
(Berkeley Madonna, Berkeley, CA) and appropriate initial
conditions for the parameters ($0<\beta<1$, $0<q<100$, $1<1/k<21$
\cite{Breman1}, $3.5<1/\gamma<10.7$ \cite{Piot1}). The optimization
process was repeated $10$ times (each time the program is fed with two
different initial conditions for each parameter) before the ``best fit'' was
chosen. The asymptotic variance-covariance $AV(\hat{\theta})$ of
the least-squares estimate is\\
$$AV(\hat{\theta}) = \sigma^2 (\sum^n_{i=1} \nabla C(t_i, \Theta_0) \nabla C(t_i,\Theta_0)^{T})^{-1}$$ \\
\noindent which we estimate by \\
$$\hat{\sigma}^2 (\sum_{i=1}^n \hat{\nabla C}(t_i,\hat{\Theta}) \hat{\nabla
C}(t_i,\hat{\Theta})^{T})^{-1}$$\\
\noindent where $n$ is the total number of observations,
$\hat{\sigma}^2 = \frac{1}{n-5} \sum(y_i - C(t_i, \hat{\Theta}))^2$
and $\hat{\nabla C}$ are numerical derivatives of $C$.\\
\noindent For small samples, the confidence intervals based on these
variance estimates may not have the nominal coverage probability. For
example, for the case of Zaire $1995$, the $95 \%$ confidence interval for $q$ based on
asymptomatic normality is ($-0.26, 2.22$). It should be obvious that this interval
is not ``sharp'' as it covers negative values whereas we know $q \ge
0$. The likelihood ratio provides an attractive alternative to build
confidence sets (Figure \ref{figq}). Formally, these sets are of the form\\
$$\left \{ \Theta: \frac{\sum(y_i - C(t_i,\Theta))^2}{\sum(y_i - C(t_i,\hat{\Theta}))^2} \le A_{\alpha} \right \}$$ \\
\noindent where $A_{\alpha}$ is the $1-\alpha$ quantile of an $F$ distribution with
appropriate degrees of freedom. Parameter estimates are given in
Table \ref{TableParameters}.
\subsection{The Reproductive Number}
\noindent The basic reproductive number $R_0$ measures the average number
of secondary cases generated by a primary case in a pool of mostly
susceptible individuals \cite{AM,BC} and is an estimate of the
epidemic growth at the start of an outbreak if everyone is susceptible. That is, a primary case
generates $R_0 = \frac{\beta_0}{\gamma}$ new cases on the average where $\beta_0$ is the
pre-interventions transmission rate and $1/\gamma$ is the mean
infectious period. The effective reproductive number
at time $t$, $R_{eff}(t) = \frac{\beta(t)}{\gamma} x(t)$, measures the average number of
secondary cases per infectious
case $t$ time units after the introduction of the initial
infections and $x(t) = \frac{S(t)}{N} \approx 1$ as the population size is much larger than the resulting size of the outbreak (Table \ref{TableOutbreaks}). Hence, $R_{eff}(0)=R_0$. In a closed population,
the effective reproductive number $R_{eff}(t)$ is
non-increasing as the size of the susceptible population
decreases. The case $R_{eff}(t) \le 1$ is of special interest as it highlights
the crossing of the threshold to eventual control of the outbreak.
An intervention is judged successful if it reduces the effective
reproductive number to a value less than one. In our model, the
post-intevention reproductive number $R_p = \frac{\beta_1}{\gamma}$ where $\beta_1$ denotes
the post-intervention transmission rate. In general, the smaller $\beta_1$, the faster
an outbreak is extinguished. By the delta method \cite{Bickel1}, the variance of the estimated
basic reproductive number $\hat{R_0}$ is approximately\\
$$ V(\hat{R_0}) \approx \hat{R_0}^2 \ \{ \frac{V(\hat{\beta_0})}{\hat{\beta_0}^2} +
\frac{V(\hat{\gamma})}{\hat{\gamma}^2} - \frac{2
Cov(\hat{\beta_0}, \hat{\gamma})}{\hat{\beta_0}
\hat{\gamma}} \}. $$
\subsection {The Effective Population Size}
\noindent A rough estimate of the population size in the Bandundu
region of Congo (where the epidemic developed) in 1995 is computed
from the population size of the Bandundu region in $1984$
\cite{WGazzetter1} and annual population growth
rates \cite{unhabitat1} (Table \ref{TableOutbreaks}). For the case
of Uganda (2000), we adjusted the population sizes of the
districts of Gulu, Masindi and Mbara in $1991$ and
annual population growth rates \cite{UBOS1} (Table
\ref{TableOutbreaks}). These estimates are an upper bound of the
effective population size (those at risk of becoming infected) for
each region. Estimates of the effective population
size are essential when the incidence is modeled with the pseudo mass-action
assumption ($\beta(t) S I$) which implies that transmission grows linearly with the
population size and hence the basic reproductive number $R_0 (N) = \beta_0 N
/\gamma$. In our model, we use the true mass-action assumption
($\beta(t) S I/N$) which makes the model parameters (homogeneous
system of order $1$) independent of $N$ and hence the basic reproductive number can be
estimated by $R_0 = \beta_0 / \gamma$ \cite{CVF}. In fact, comparisons between
the pseudo mass-action and the true mass-action assumptions with
experimental data have concluded in favor of the later
\cite{JDH}. The model assumption that $N$ is constant is not critical as the
outbreaks resulted in a small number of cases compared to the size of the
population.
\subsection{Uncertainty Analysis on $R_0$}
Log-normal distributions seem to model well the incubation period distributions for
a large number of diseases \cite{Sartwell1}. Here, a log-normal
distribution is assumed for the incubation period of Ebola in our
uncertainty analysis. Log-normal distribution parameters are set from empirical
observations (mean incubation period is $6.3$ and the $95\%$ quantile
is $21$ days \cite{Breman1}). The infectious period is assumed to be uniformly
distributed in the range ($3.5-10.7$) days \cite{Piot1}. \\
\noindent A formula for the basic reproductive number $R_0$ that depends on the initial per-capita rate of
growth $r$ in the number of cases (Figure \ref{figR0uncertainty}), the incubation period
($1/k$) and the infectious period ($1/\gamma$) can be obtained by linearizing
equations $\dot{E}$ and $\dot{I}$ of system (\ref{eqn1}) around the disease-free equilibrium
with $S=N$. The corresponding Jacobian matrix is given by:\\
\[
J=\left(\begin{array}{cc}
-k & \beta \\
k & -\gamma \\
\end{array}\right),
\]
\noindent and the characteristic equation is given by:
\[
r^2 + (k+\gamma) r + (\gamma - \beta) k = 0
\]
\noindent where the early-time and per-capita free growth $r$ is essentially the dominant eigenvalue. By solving for $\beta$ in terms of $r$, $k$ and $\gamma$, one can
obtain the following expression for $R_0$ using the fact that $R_0 = \beta/\gamma$:\\
$$ {R_0} = 1 + \frac{r^2 + (k+\gamma) r}{k \gamma}.$$ \\
\noindent Our estimate of the initial rate of growth $r$ for the Congo 1995 epidemic
is $r=0.07$ day$^{-1}$, obtained from the time series $y(t)$, $t<\tau$ of the
cumulative number of cases and assuming exponential growth ($y(t)
\propto e^{rt}$). The distribution of $R_0$ (Figure \ref{figR0uncertainty})
lies in the interquartile range (IQR) ($1.66-2.28$) with a median of $1.89$, generated from
Monte Carlo sampling of size $10^5$ from the distributed epidemic
parameters ($1/k$ and $1/\gamma$) for fixed $r$ \cite{Blower1}. We give the median of $R_0$ (not the mean) as the resulting distribution of $R_0$ from our uncertainty analysis is skewed to the right.
\section{Results}
\indent Using our parameter estimates (Table \ref{TableParameters}), we
estimate an $R_0$ of $1.83$ (SD $0.06$) for Congo (1995) and
$1.34$ (SD $0.03$) for Uganda (2000).
\noindent The effectiveness of interventions is often quantified in terms of the
reproductive number $R_p$ after interventions are put in place. For
the case of Congo $R_p = 0.51$ (SD $0.04$) and $R_p = 0.66$ (SD
$0.02$) for Uganda allowing us to conclude that in both cases, the intervention
was successful in controlling the epidemic. Furthermore, the time to
achieve a transmission rate of $\frac{\beta_0 + \beta_1}{2}$ ($t_h$)
is $0.71$ ($95\%$ CI ($0.02, 1.39$)) days and $0.11$ ($95 \%$ CI ($0, 0.87$)) days
for the cases of Congo and Uganda respectively after the time at which interventions begin.\\
\noindent We use the estimated parameters to simulate the Ebola outbreaks in Congo (1995)
and Uganda (2000) via Monte Carlo simulations of the stochastic model of Section $2.1$ \cite{Renshaw1}.
There is very good agreement between the mean of the stochastic
simulations and the reported cases despite the
``wiggle'' captured in the residuals around the time $\tau$ of the
start of interventions (Figure \ref{figmodel2}). The
empirical distribution of the final epidemic sizes for the cases of Congo
$1995$ and Uganda $2000$ are given in Figure \ref{figOutbreaksizedistr}.\\
\noindent The final epidemic size is sensitive to
the start time of interventions $\tau$. Numerical solutions
(deterministic model) show that the final epidemic
size grows exponentially fast with the initial time of
interventions (not surprising as the intial epidemic growth is driven by
exponential dynamics). For instance, for the case of
Congo, our model predicts that there would have been $20$ more cases
if interventions had started one day later (Figure \ref{figSensInterv}).
\section{Discussion}
\indent Using epidemic-curve data from two major Ebola hemorrhagic fever
outbreaks \cite{Khan1, WHO2}, we have estimated the basic reproductive
number ($R_0$) (Table \ref{TableOutbreaks}). Our estimate of $R_0$ (median is $1.89$)
obtained from an uncertainty analysis \cite{Blower1} by simple random sampling (Figure
\ref{figR0uncertainty}) of the parameters $k$ and $\gamma$ distributed
according to empirical data from the Zaire (now the Democratic Republic of Congo) $1976$ Ebola outbreak \cite{Breman1, Piot1} is in agreement with our estimate of $R_0 = 1.83$ from the outbreak in Congo $1995$ (obtained from least squares fitting
of our model (\ref{eqn1}) to epidemic curve data). \\
The difference in the basic reproductive
numbers $R_0$ between Congo and Uganda is due to our different estimates for the
infectious period ($1/\gamma$) observed in these two places. Their
transmission rates $\beta_0$ are quite similar (Table \ref{TableParameters}). Our
estimate for the infectious period for the case of Congo ($5.61$ days)
is slightly larger than that of Uganda ($3.50$ days). Clearly, a larger infectious
period increases the likelihood of infecting a susceptible
individual and hence increases the basic reproductive number.
The difference in the infectious periods might be due to differences
in virus subtypes \cite{Niikura1}. The Congo outbreak was caused by the Ebola-Zaire
virus subtype \cite{Khan1} while the Uganda outbreak was caused by
the Ebola-Sudan virus subtype \cite{WHO2}.\\
\noindent The significant reduction from the basic reproductive number ($R_0$) to the post-intervention reproductive number ($R_p$) in our estimates for Congo and Uganda shows that the implementation of control measures such as education, contact tracing and quarantine will have a significant effect on lowering the effective reproductive rate of Ebola.
Furthermore, estimates for the time to achieve
$\frac{\beta_0 + \beta_1}{2}$ have been provided (Table \ref{TableParameters}).\\
\noindent We have explored the sensitivity of the final epidemic size to the
starting time of interventions. The exponential increase of the final
epidemic size with the time of start of interventions (Figure
\ref{figSensInterv}) supports the idea that the rapid
implementation of control measures should be considered as a critical
component in any contingency plan against disease outbreaks specially
for those like Ebola and SARS for which no specific treatment or
vaccine exists. A two-week delay in implementing public health
measures results in an approximated doubling of the final outbreak
size. Because the existing control measures cut the transmission rate to
less than half, we should seek and support further improvement in the effectiveness of
interventions for Ebola. A mathematical model that considers basic public health interventions for SARS control in Toronto supports this conclusion \cite{Chowell1, Chowell2}. Moreover, computer simulations show that small perturbations to the rate $q$ at which interventions are put fully in place do not have a significant effect on the final epidemic size. The rapid identification of an outbreak, of course, remains the strongest determinant of the final outbreak size.\\
\noindent Field studies of Ebola virus are difficult to conduct due to
the high risk imposed on the scientific and medical personnel
\cite{Nature2}. Recently, a new vaccine that makes use of an
\textit{adenovirus technology} has been shown to give cynomolgus macaques
protection within $4$ weeks of a single jab \cite{Nature3,
Nature4}. If the vaccine turns out to be effective in humans, then
its value should be tested. A key question would be ``What are the
conditions for a successful target vaccination campaign during an Ebola outbreak?''
To address questions of this type elaborate models need to be developed.
|
{
"timestamp": "2005-03-02T00:52:48",
"yymm": "0503",
"arxiv_id": "q-bio/0503006",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503006"
}
|
\section{introduction}
One of the major problems in mathematical physics is concerned
with the geometrical information stored in the spectrum of the Laplace
Beltrami operator
\begin{equation}
-\triangle\psi_j({\bf r}) = E_j\psi_j({\bf r}); \;\; {\bf r}
\in \Omega(\alpha) \ .
\end{equation}
\noindent The spectrum is ordered such that $E_{j-1}\ \le E_j\le E_{j+1}$ and
$\Omega(\alpha)$ is a connected compact region, parameterized by $\alpha$,
on a 2D Riemannian manifold. If $\Omega(\alpha)$ has a boundary, Dirichlet
boundary conditions are assumed. The corresponding physical system could be
a vibrating drum. In 1911 H. Weyl showed that the number of eigenvalues up
to energy E is
\begin{equation}
N(E) \sim {AE\over 4\pi}, \;\;\;\; \hbox{as} \;\;
E\rightarrow \infty
\end{equation}
\noindent where $A$ is the area of $\Omega$. Subsequent research have shown
(see e.g., \cite{clark67}) that each of the terms in the asymptotic series
of $N(E)$ provides further geometrical information on the boundary. This
prompted M. Kac to ask, `can one hear the shape of a drum ?' \cite{kac66}.
That is, `is it possible to uniquely define the shape of the drum from
the spectrum ?' It is known by now that for certain classes of domains
the answer to Kac's question is positive, whereas there exists a large
set of {\it isospectral} domains which are not {\it isometric}.
(Ref. \cite{zeldich} gives an updated review of this subject.)
In the present note we would like to investigate the geometrical information
stored in yet another sequence of numbers which are derived from the
eigenfunctions $\psi_j$. Considering real eigenfunctions $\psi_j$, we count
the number $\nu_j$ of {\it nodal domains} which are the connected domain
where $\psi_j$ has a constant sign. The nodal domains are separated by
the {\it nodal lines} where $\psi_j=0$. The sequence
$\left \{\nu_j\right\}_{j=1}^{\infty}$ is the sequence of nodal counts.
According to Courant's Nodal theorem $\nu_j \le j$. This fundamental
theorem reveals the deep connection between the spectrum and the nodal
count. It is convenient to define the {\it normalized} nodal domain
numbers $\xi_j = \nu_j/j$. Because of Courant's theorem
$0 \le \xi_j \le 1$. This estimate has been further refined (for domains
in $\mathbf{R}^2$ ) \cite{pleijel56}
\begin{equation}
\limsup_{j\rightarrow\infty}\; \xi_j = 0.691 \ldots
\end{equation}
\noindent Following \cite{uzy02}, we study the distribution of the normalized nodal
numbers in the spectral interval $I=[E^0,E^1]$
\begin{equation}
P(\xi,I) = {1\over N_I}\sum_{E_j\in I} \delta(\xi-\xi_j)
\label{dbn}
\end{equation}
\noindent where $N_I$ is the number of levels in the interval $I$.
In Ref. \cite{uzy02} the above distribution has been introduced as a tool
to distinguish between systems which are integrable (separable) or classically
chaotic. For the class of separable domains, it was shown that the limit
distribution
\begin{equation}
P(\xi) = \lim_{E\rightarrow\infty} P(\xi,I)
\end{equation}
\noindent exists. This has universal features: ({\it a}) there exists a system
dependent parameter $\xi'$, maximum value of the nodal domain number, such
that $P(\xi)=0$ for $\xi > \xi'$ and ({\it b}) for $\xi\approx\xi'$,
\begin{equation}
P(\xi) = {C\over \sqrt{1 -\xi/\xi'}} \, .
\label{uexp}
\end{equation}
\noindent The constant $C$ is system dependent, but the order of the singularity
is universal and depends only on the dimensionality. (It was recently shown
that the exponent for domains in $d$ dimensions is $(d-3)/2$.)
The dependence on the geometry of the domain can come only through the
parameters $\xi',C $ or the details of the function $P(\xi)$ away from
the universal domain. Indeed, the limiting distributions for the rectangular
and circular boundaries were computed in \cite{uzy02} and found to be
different as expected. However, as will be shown below, the function
$P(\xi)$ does not distinguish between different rectangles. That is
$\xi' = 2/\pi$ and
\begin{equation}
P(\xi) = {\left[1 - {(\pi\xi/2)}^2 \right]}^{-1/2}
\end{equation}
\noindent for all rectangles! Note that for $\xi\approx 2/\pi$, this result
coincides with the universal expression (\ref{uexp}) with $C=1/\sqrt 2$.
The new result of the present note is that the dependence of $P(\xi,I)$
on the {\it finite} spectral interval $I$ contains sufficient information
to resolve between different rectangles. Thus, by counting nodal domains
one can deduce the shape of the (rectangular) drum. It should be emphasized
at the outset that the nodal count sequence involves dimensionless integers,
and therefore it cannot provide any scale information. Hence, when we say
``resolve'' we mean ``resolve up to a scale''.
\section{rectangles}
We consider the Dirichlet spectrum of a domain bounded in a rectangle with
sides $L_x$ and $L_y$. Denoting $\alpha = L_x/L_y$ and choosing $L_x=\pi$,
the spectrum is given by
\begin{equation}
E = n^2 + \alpha^2m^2\ ,
\end{equation}
\noindent where $n,m=1,2,3\ldots$ and $0 < \alpha < 1$. Since the system is separable
in rectangular co-ordinates the nodal domain number is simply $\nu_j=n m$,
and $j=N(n^2 + \alpha^2m^2)$ where $N(E)$ is the spectral counting function.
The leading terms in the asymptotic expansion of $N(E)$ are
\begin{equation}
N(E) \simeq {1\over 4\pi} \Big[AE - L\sqrt{E}\Big] \label{weyl}
\end{equation}
\noindent where $A,L$ are the area and perimeter of the boundary
respectively \cite{morse}. In terms of $\alpha$,
\begin{equation}
N(E) \simeq {\pi E\over 4\alpha}
\left(1-{2\over\pi}{1+\alpha\over\sqrt{E}}\right)\, .
\end{equation}
\noindent Introducing the transformation
\begin{equation}
n(E,\theta) = \sqrt{E}\cos\theta \;\; ; \;\;
m(E,\theta) = \sqrt{E}\sin\theta/\alpha \, ,
\end{equation}
\noindent the normalized nodal-domain number can be approximated by
\begin{equation}
\xi_j(E,\theta) = {2\over\pi} \sin 2\theta \left[1 - {2\over\pi}
{(1+\alpha)\over\sqrt{E}}\right]^{-1} \, .
\end{equation}
\noindent Converting the summation in eq.(\ref{dbn}) into an integral, we obtain
the leading terms in the asymptotic expansion of $P(\xi,I)$ in the large
$E$ limit
\begin{equation}
P(\xi,I) \simeq {1\over 2\alpha N_I}
\int_{E^0}^{E^1}\int_0^{\pi/2} \delta
\Big[\xi - \xi_j(E,\theta)\Big]\;dE\;d\theta
\end{equation}
\noindent where
\begin{equation}
N_I \simeq {\pi\over 4\alpha}\left\{(E^1-E^0)-{2\over\pi}
(1+\alpha)\left(\sqrt{E^1}-\sqrt{E^0}\right)\right\} \, .
\end{equation}
\noindent Introducing the variable $x=\sqrt{E/E^0}$
\begin{equation}
P(\xi,I) = {E^0\over\alpha N_I} \int_1^g\int_0^{\pi/2} x\;\delta
\left[\xi - {2\over\pi}{\sin 2\theta\over(1-\epsilon/x)}\right] \;
dx\; d\theta
\end{equation}
\noindent where
\begin{equation}
g=\sqrt{E^1\over E^0}, \;\; \epsilon(\alpha) =
{2\over\pi}{(1+\alpha)\over\sqrt{E^0}}.
\end{equation}
\noindent The integral reduces to
\begin{equation}
P(\xi,I) = {E^0\over\alpha N_I} \int_1^l x {\left[{2\over\pi}
{\cos 2\theta_0\over(1-\epsilon/x)}\right]}^{-1} \; dx
\end{equation}
\noindent where $\sin 2\theta_0 = {\pi\xi\over 2} \left(1-{\epsilon\over x}\right)$
and
\begin{equation}
l = \left\{
\begin{array}{ll}
g,&\hbox{if} \;\; \xi < {2\over\pi} \\
\hbox{min}\left[g,\epsilon{\pi\xi\over 2}{\left({\pi\xi\over 2}-1\right)}^{-1}
\right], &\hbox{if} \;\; {2\over\pi} < \xi \le {2\over\pi}{1\over 1-\epsilon}
\end{array} \right . \, .
\end{equation}
\noindent Note that $P(\xi,I)=0$ for $\xi > {2\over\pi}{1\over 1-\epsilon}$. The
above integral can be rewritten as
\begin{equation}
P(\xi,I) = {\pi E^0\over 2\alpha N_I}\int_1^l
{x(x-\epsilon)\over\sqrt{a+bx+cx^2}} \; dx
\label{dbn1}
\end{equation}
\noindent where
\begin{equation}
a = - \epsilon^2 {\left({\pi\xi\over 2}\right)}^2, \;\;
b = 2\epsilon {\left({\pi\xi\over 2}\right)}^2, \;\;
c = 1 - {\left({\pi\xi\over 2}\right)}^2 \, .
\end{equation}
\noindent This integral can be computed for any given value of the
parameters \cite{grad}.
\begin{figure}[h]
\centerline{\psfig{figure=rec_deriv1.ps,height=9cm,width=6.5cm,angle=-90}}
\caption{Typical behavior of the derivative $P'$ for $\xi<2/\pi$.
For $\xi>2/\pi$, $P'$ is not defined as the function $P$ is not smooth.}
\label{fig1}
\end{figure}
\section{results}
Using the above expression, it is possible to show that the derivative
\begin{equation}
P' = \left.{\partial
P\over\partial\alpha}\right|_{\alpha=\alpha_0}
\end{equation}
\noindent is positive for all values of $\xi<2/\pi$. Moreover, $P'$, and hence the
sensitivity to $\alpha$, is maximal in the vicinity of the critical value
$\xi'=2/\pi$, as can be seen in Figure \ref{fig1}.
\begin{figure}[ht]
\centerline{\psfig{figure=rec_dbn.ps,height=9cm,width=8cm,angle=-90}}
\caption{Nodal domain distribution for the rectangular boundary. Solid, dashed
and dotted curves are the approximate distribution (\ref{dbn1}) with $\alpha =
0.13,0.44,0.92$ respectively for the energy range $I=[10^2,10^4]$. This may be
compared with the limiting distribution (\ref{limit}) shown as a
thick curve.}
\label{fig2}
\end{figure}
In Figure \ref{fig2}, the nodal domain distribution given by the
eq. (\ref{dbn1}) is shown for different $\alpha$, along with the
corresponding numerical data. The limiting distribution is obtained by
taking the spectral interval to infinity. In this limit, $\epsilon=0$ and
\begin{equation}
P(\xi) = \left\{ \begin{array}{ll}
{\left[ 1 - {(\pi\xi/2)}^2 \right]}^{-1/2}, & \xi < 2/\pi \\[10pt]
0, & \xi > 2/\pi \end{array} \right .
\label{limit}
\end{equation}
\noindent which is independent of $\alpha$. Thus the parameter dependence is
arising from the leading finite energy correction to $P(\xi,I)$.
The problem studied above shows clearly that the nodal sequence stores
geometrical information, which, in the present case suffices to determine
unambiguously the rectangle for which the nodal sequence is given. Attempts
to generalize these ideas to other separable systems such as e.g., smooth
surfaces of revolutions or flat tori are under way. \\
\noindent {\bf Acknowledgments}
This research was supported in part by the Minerva center for complex
systems and the Einstein (Minerva) center at the Weizmann Institute.
Grants from the German-Israeli Foundation and the Israel Science Foundation
are acknowledged.
|
{
"timestamp": "2005-03-03T12:54:32",
"yymm": "0503",
"arxiv_id": "nlin/0503002",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503002"
}
|
\section{Introduction}
Let $\left( \Omega ,\mathcal{F},\left( \mathcal{F}_{t}\right) _{t\geq 0},%
\mathbf{P}\right) $ be a filtered probability space satisfying the
usual hypotheses (right continuous and complete). Given the end $L$\
of an $\left(
\mathcal{F}_{t}\right) $\ predictable set $\Gamma $, i.e\textbf{\ }%
\begin{equation*}
L=\sup \left\{ t:\left( t,\omega \right) \in \Gamma \right\} ,
\end{equation*}%
(these times are also refered to as honest times), M. Barlow
(\cite{barlow})
and Jeulin and Yor (\cite{yorjeulin}) have shown that the supermartingale:%
\begin{equation*}
Z_{t}^{L}=\mathbf{P}\left( L>t\mid \mathcal{F}_{t}\right) ,
\end{equation*}%
chosen to be c\`{a}dl\`{a}g, plays an essential role in the enlargement
formulae with respect to $L$, i.e: in expressing a general $\left( \mathcal{F%
}_{t}\right) $ martingale $\left( M_{t}\right) $ as a semimartingale in $%
\left( \mathcal{F}_{t}^{L}\right) _{t\geq 0}$, the smallest filtration which
contains $\left( \mathcal{F}_{t}\right) $, and makes $L$\ a stopping time.
This enlargement formula is:%
\begin{equation}
M_{t}=\widetilde{M}_{t}+\int_{0}^{t\wedge L}\frac{d<M,Z>_{s}}{Z_{s_{-}}}%
+\int_{L}^{t}\frac{d<M,1-Z>_{s}}{1-Z_{s_{-}}%
}, \label{grossform}
\end{equation}
where $\left( \widetilde{M}_{t}\right) _{t\geq 0}$ denotes an
$\left( \left( \mathcal{F}_{t}^{L}\right) ,\mathbf{P}\right) $ local
martingale. Hence it is important to dispose of an explicit formula
for $\left( Z_{t}^{L}\right) _{t\geq 0}$. In the literature about
progressive enlargements of filtrations, not so many examples are
fully developed (see e.g. for example \cite{zurich},
\cite{jeulinyor} or \cite{jeulin}); indeed, the computation of
$\left( Z_{t}^{L}\right) $\ is sometimes difficult. Moreover, the
examples are developed essentially in the Brownian setting, where as
we shall see, $\left( Z_{t}^{L}\right) $\ is continuous, and no
examples of discontinuous $\left( Z_{t}^{L}\right)'s$ are
given.\bigskip
In this paper, we first consider a special family of honest times
$g$, and then we later prove that this family is generic in the
sense that every honest time is in fact of this form (under some
reasonable assumptions).
More precisely, we consider the following class of local
martingales.
\begin{defn}
We say that an $\left( \mathcal{F}_{t}\right) $ local martingale
$\left( N_{t}\right) $ belongs to the class
$\left(\mathcal{C}_{0}\right)$, if it is strictly positive, with no
positive jumps, and $\lim_{t\rightarrow\infty}N_{t}=0$.
\end{defn}
\begin{rem}
Let $\left( N_{t}\right) $ be a local martingale of class
$\left(\mathcal{C}_{0}\right)$. Then: $$S_{t}\equiv \sup_{s\leq
t}N_{s},$$its supremum process, is continuous. This property is
essential in our paper. Hence, most of the results we shall state
remain valid for positive local martingales, which go to zero at
infinity, and whose suprema are continuous.
\end{rem}
We associate with a local martingale of class $\left(\mathcal{C}_{0}\right)$, the supermartingale $\left( \frac{N_{t}}{S_{t}}\right) _{t\geq 0}$%
, and the random time $g$ defined as:
\begin{eqnarray*}
g &\equiv &\sup \left\{ t\geq 0:\quad N_{t}=S_{\infty }\right\} \\
&=&\sup \left\{ t\geq 0:\quad S_{t}-N_{t}=0\right\}.
\end{eqnarray*}
In Section 2, we prove that the associated supermartingale $Z$
satisfies:
\begin{equation}
Z_{t}\equiv \mathbf{P}\left( g>t\mid \mathcal{F}_{t}\right) =\frac{N_{t}}{%
S_{t}}, \label{decomult}
\end{equation}%
and then give the decomposition formula (\ref{grossform}) in terms
of the local martingale $\left( N_{t}\right) $. This will provide us
with some new, and explicit examples of such supermartingales
$\left(Z_{t}\right)$ which are discontinuous. We also establish some
relationship between the multiplicative representation
(\ref{decomult}) and the Doob-Meyer (additive) decomposition of
$\left( Z_{t}\right) $.
In Section 3, we study the problem of the initial enlargement of $\left(\mathcal{F}%
_{t}\right)$ with the variable $S_{\infty}$, and then give a new proof of (\ref%
{grossform}).
In Section 4, we show that the formula (\ref{decomult}) is in fact very
general. More precisely, for any end of a predictable set $L$, under the
assumptions \textbf{(CA)}:
\begin{itemize}
\item all $\left( \mathcal{F}_{t}\right) $-martingales are \textbf{\underline{c}}%
ontinuous (e.g: the Brownian filtration);
\item $L$ \textbf{\underline{a}}voids every $\left( \mathcal{F}_{t}\right)$ -stopping
time $T$, i.e. $P\left[ L=T\right] =0$,
\end{itemize}
the supermartingale $Z_{t}^{L}=\mathbf{P}\left( L>t\mid \mathcal{F}%
_{t}\right) $ may be represented as (\ref{decomult}).
In Section 5, we give some new examples of enlargements of
filtrations. Moreover, as an illustration of our approach and the
method of enlargements of filtrations, we recover and complete some
known results of D. Williams (\cite{williams2}) about path
decompositions of some diffusion processes, given their minima. We
add a new fragment in these path decompositions, by introducing a
new family of random times, as defined in \cite{ANMY} and called
pseudo-stopping times, which generalize the fundamental notion of
stopping times, introduced by J.L. Doob. We take this opportunity to
quote two passages, resp. in the appendix of Meyer's book (1966):
\begin{quote} Les temps d'arr\^{e}t ont \'{e}t\'{e}
utilis\'{e}s, sans d\'{e}finition formelle, depuis le d\'{e}but de
la th\'{e}orie des processus. La notion appara\^{\i}t tout \`{a}
fait clairement pour la premi\`{e}re fois chez Doob en 1936.
\end{quote}and in Dellacherie-Meyer's book, volume I
(\cite{dellachmeyer}), p.184: 0194 \begin{quote} Il a sans doute
fallu autant de g\'{e}nie aux cr\'{e}ateurs du calcul
diff\'{e}rentiel pour expliciter la notion si simple de
d\'{e}riv\'{e}e, qu'\`{a} leurs successeurs pour faire tout le
reste. L'invention des temps d'arr\^{e}t par Doob est tout \`{a}
fait comparable. \end{quote}
\section{A multiplicative representation formula}
\subsection{Doob's maximal identity}
Let $\left( N_{t}\right) _{t\geq 0}$ be a local martingale which
belongs to the class $(\mathcal{C}_{0})$, with $N_{0}=x$.
Let $S_{t}=\sup_{s\leq t}N_{s}$. We consider:%
\begin{eqnarray}
g &=&\sup \left\{ t\geq 0:\quad N_{t}=S_{\infty }\right\} \notag \\
&=&\sup \left\{ t\geq 0:\quad S_{t}-N_{t}=0\right\} . \label{defdeg}
\end{eqnarray}
To establish our main proposition, we shall need the following
variant of Doob's maximal inequality, which we call Doob's maximal
identity:
\begin{lem}[Doob's maximal
identity]
\label{maxeq} For any $a>0$, we have:%
\begin{enumerate}
\item
\begin{equation} \mathbf{P}\left( S_{\infty }>a\right) =\left(
\frac{x}{a}\right) \wedge 1. \label{loimax}
\end{equation}Hence, $\dfrac{x}{S_{\infty }}$ is a uniform random variable on $%
\left(0,1\right)$.
\item For any stopping time $T$:%
\begin{equation}
\mathbf{P}\left( S^{T}>a\mid \mathcal{F}_{T}\right) =\left( \frac{N_{T}}{a}%
\right) \wedge 1 , \label{loimaxcond}
\end{equation}%
where
\begin{equation*}
S^{T}=\sup_{u\geq T}N_{u}.
\end{equation*}%
Hence $\dfrac{N_{T}}{S^{T}}$ is also a uniform random variable on
$\left(0,1\right)$, independent of $\mathcal{F}_{T}$.
\end{enumerate}
\end{lem}
\begin{proof}
Formula (\ref{loimaxcond}) is a consequence of (\ref{loimax}) when applied
to the martingale $\left( N_{T+u}\right) _{u\geq 0}$ and the filtration $%
\left( \mathcal{F}_{T+u}\right) _{u\geq 0}$. Formula (\ref{loimax})
itself is obvious when $a\leq x$, and for $a>x$, it is obtained by
applying Doob's optional stopping theorem to the local martingale
$\left( N_{t\wedge T_{a}}\right) $, where $T_{a}=\inf \left\{ u\geq
0:\text{ }N_{u}>a\right\} $.
\end{proof}
The next proposition gives an explicit formula for $Z_{t}\equiv \mathbf{P}%
\left( g>t\mid \mathcal{F}_{t}\right) $, in terms of the local martingale $%
\left( N_{t}\right) $. Without loss of generality, \textbf{we assume
from now on that $\mathbf{x=1}$}. Indeed, if $N_{0}=x$, we consider
the local martingale $\left( \frac{N_{t}}{x}\right) $ which starts
at $1$.
\begin{prop}\label{applicationmax}
\begin{enumerate}
\item In our setting, the formula:%
\begin{equation*}
Z_{t}=\frac{N_{t}}{S_{t}},\text{ }t\geq 0
\end{equation*}%
holds.
\item The Doob-Meyer additive decomposition of $\left( Z_{t}\right) $\ is:%
\begin{equation}
Z_{t}=\mathbf{E}\left[ \log S_{\infty }\mid \mathcal{F}_{t}\right] -\log
\left( S_{t}\right) . \label{DB}
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
We first note that:%
\begin{eqnarray*}
\left\{ g>t\right\} &=&\left\{ \exists \text{ }u>t:\text{ }%
S_{u}=N_{u}\right\} \\
&=&\left\{ \exists \text{ }u>t:\text{ }S_{t}\leq N_{u}\right\} \\
&=&\left\{ \sup_{u\geq t}N_{u}\geq S_{t}\right\} .
\end{eqnarray*}%
Hence, from (\ref{loimaxcond}), we get: $\mathbf{P}\left( g>t\mid \mathcal{F}%
_{t}\right) =\frac{N_{t}}{S_{t}}$.
To establish (\ref{DB}), we develop $\left( \frac{N_{t}}{S_{t}}\right) $\
thanks to Ito's formula, to obtain:%
\begin{equation*}
Z_{t}=1+\int_{0}^{t}\frac{1}{S_{s}}dN_{s}-\int_{0}^{t}\frac{N_{s}}{\left(
S_{s}\right) ^{2}}dS_{s}.
\end{equation*}%
Now, we remark that the measure $dS_{s}$\ is carried by the set $\left\{ s:%
\text{ }Z_{s}=1\right\} $; hence:%
\begin{eqnarray*}
Z_{t} &=&1+\int_{0}^{t}\frac{1}{S_{s}}dN_{s}-\int_{0}^{t}\frac{1}{S_{s}}%
dS_{s} \\
\dfrac{N_{t}}{S_{t}}&=&1+\int_{0}^{t}\frac{1}{S_{s}}dN_{s}-\log
\left( S_{t}\right) .
\end{eqnarray*}%
From the unicity of the Doob-Meyer decomposition, $\log \left(
S_{t}\right) $ is the predictable increasing part of $\left(
Z_{t}\right) $\ whilst $\left(
\int_{0}^{t}\frac{1}{S_{s}}dN_{s}\right) $\ is its martingale part. As $%
\left( Z_{t}\right) $\ is of class $\left( D\right) $, $\left( \int_{0}^{t}%
\frac{1}{S_{s}}dN_{s}\right) $\ is a uniformly integrable martingale. Now,
let $t\rightarrow \infty $: as $Z_{\infty }=0$, $\log S_{\infty
}=1+\int_{0}^{\infty }\frac{1}{S_{s}}dN_{s}$ and thus:
\begin{equation}\label{qqrrr}
1+\int_{0}^{t}\frac{1}{S_{s}}dN_{s}=\mathbf{E}\left[ \log S_{\infty
}\mid \mathcal{F}_{t}\right] ,
\end{equation}%
which proves (2).
\end{proof}
\begin{rem}
It is well known, and it follows from (\ref{DB}), that the
martingale in (\ref{qqrrr}) is in fact in BMO.
\end{rem}
\begin{cor}
Assuming that all $\left(\mathcal{F}_{t}\right)$ martingales are
continuous, the following hold:
\begin{enumerate}
\item $\log \left( S_{t}\right) $ is the dual predictable projection
of $\mathbf{1}_{\left\{ g\leq t\right\} }$: for any positive
predictable process $\left(k_{s}\right)$,
$$\mathbf{E}\left(k_{g}\right)=\mathbf{E}\left(\int_{0}^{\infty}k_{s}\dfrac{dS_{s}}{S_{s}}\right);$$
\item The random time $g$ is honest and avoids any $\left( \mathcal{F}_{t}\right) $%
stopping time $T$\textit{, i.e. }$P\left[ g=T\right] =0$.
\end{enumerate}
\end{cor}
\begin{proof}
Under our assumptions, the predictable and optional sigma algebras
are equal. Thus, it suffices to prove that $g$ avoids stopping
times, the other assertions being obvious. Since $\log \left(
S_{t}\right) $ is the dual predictable projection of
$\mathbf{1}_{\left\{ g\leq t\right\} }$ and is continuous, then for
any $\left( \mathcal{F}_{t}\right) $ stopping time $T$,
\begin{equation*}
\mathbf{E}\left[ \mathbf{1}_{\left\{ g=T\right\}
}\right]=\mathbf{E}\left[\left(\Delta \log \left(
S_{\bullet}\right)\right)_{T}\right]=0.
\end{equation*}%
Thus we get $P\left( g=T\right) =0$.
\end{proof}
\bigskip
We can now write the formula (\ref{grossform}) in terms of the martingale $%
\left( N_{t}\right) $.
\begin{prop}
Let $\left( X_{t}\right) _{t\geq 0}$ be a local $\left( \mathcal{F}%
_{t}\right) $ martingale. Then, $X$\ has the following decomposition as a
semimartingale in $\left( \mathcal{F}_{t}^{g}\right) $:%
\begin{equation*}
X_{t}=\widetilde{X}_{t}+\int_{0}^{t\wedge g}\frac{d<X,N>_{s}}{N_{s-}}%
-\int_{g}^{t}\frac{d<X,N>_{s}}{S _{\infty}-N_{s-}}
\end{equation*}%
where $\left( \widetilde{X}_{t}\right) $ is an $\left( \mathcal{F}%
_{t}^{g}\right) $\ local martingale.
\end{prop}
\begin{proof}
This is a consequence of formula (\ref{grossform}) and Proposition
\ref{applicationmax}.
\end{proof}
We shall now give a relationship between $\left( S_{t}\right) $\ and $%
\mathbf{E}\left[ \log S_{\infty }\mid \mathcal{F}_{t}\right] $. For
this, we shall need the following easy extension of Skorokhod's
reflection lemma (see \cite{Mckean}, p.72):
\begin{lem}\label{lemmreflection}
Let $y$ be a real-valued c\`{a}dl\`{a}g function on
$\left[0,\infty\right)$, such that $y$ has no negative jumps, and
$y(0)=0$. Then, there exists a unique pair $\left(z,a\right)$ of
functions on $\left[0,\infty\right)$ such that:
\begin{enumerate}
\item z=y+a
\item z is positive, c\`{a}dl\`{a}g and has no negative jumps,
\item a is increasing, continuous, vanishing at zero and the
corresponding measure $da_{s}$ is carried by
$\left\{s:\;z(s)=0\right\}$.
\end{enumerate} The function $a$ is moreover given by
$$a(t)=\sup_{s\leq t}\left(-y(s)\right).$$
\end{lem}
\begin{prop}
\label{sko}With
\begin{equation*}
\mu _{t}=\mathbf{E}\left[ \log S_{\infty }\mid
\mathcal{F}_{t}\right] ,
\end{equation*}%
we have:%
\begin{equation*}
\log \left( S_{t}\right) =\sup_{s\leq t}\mu _{s}-1\equiv \overline{\mu }%
_{t}-1,
\end{equation*}%
or equivalently:%
\begin{equation*}
S_{t}=\exp \left( \overline{\mu }_{t}-1\right)
\end{equation*}
\end{prop}
\begin{proof}
From (\ref{DB}), we can write:%
\begin{equation*}
1-Z_{t}=\left( 1-\mu _{t}\right) +\log \left( S_{t}\right) .
\end{equation*}%
From Lemma \ref{lemmreflection}, we deduce that
\begin{equation*}
\log \left( S_{t}\right) =\sup_{s\leq t}\mu _{s}-1.
\end{equation*}
\end{proof}
\subsection{Some hidden Az\'{e}ma-Yor martingales}
We shall now associate with the two dimensional process
\begin{equation*}
\left( \log \left( S_{t}\right) ,\text{ }Z_{t}\right) _{t\geq 0}
\end{equation*}%
a family of martingales reminiscent of Az\'{e}ma-Yor martingales
(see, e.g., \cite{AY}) which we shall now discuss. In fact, once
again, we have to introduce a slightly generalized version of what
are usually called Az\'{e}ma-Yor martingales. Indeed, these
martingales were originally defined for continuous local martingales
(see \cite{revuzyor}, Chapter VI), while we would like to define
them for local martingales without positive jumps. This extension
can be dealt with the following balayage argument:
\begin{lem}
Let $Y=M+A$ be a special semimartingale, where $M$ is a
c\`{a}dl\`{a}g local martingale, and $A$ a continuous increasing
process. Set $H=\left\{t:\;Y_{t}=0\right\}$, and define $g_{t}\equiv
\sup\left\{s<t:\;Y_{s}=0\right\}$. Then, for any locally bounded
predictable process $\left(k_{t}\right)$, $\left(k_{g_{t}}\right)$
is predictable and
\begin{equation}\label{balay}
k_{g_{t}}Y_{t}=k_{0}Y_{0}+\int_{0}^{t}k_{g_{s}}dY_{s}.
\end{equation}
\end{lem}
\begin{proof}
The proof is the same as the proof for continuous semimartingales.
The reader can refer to \cite{delmaismey}, p.144, for even more
general versions of the balayage formula.
\end{proof}Now, we can state the following generalization of the
classical Az\'{e}ma-Yor martingales:
\begin{prop}\label{azemayorgeneralisee}
Let $\left(N_{t}\right)_{t\geq 0}$ be a local martingale such that
its supremum process $\left(S_{t}\right)$ is continuous (this is the
case if $N_{t}$ is in the class $\mathcal{C}_{0}$). Let $f$ be a
locally bounded Borel function and define
$F\left(x\right)=\int_{0}^{x}dyf\left(y\right)$. Then, $X_{t}\equiv
F\left(S_{t}\right)-f\left(S_{t}\right)\left(S_{t}-N_{t}\right)$ is
a local martingale and:
\begin{equation} \label{ayor}
F\left(S_{t}\right)-f\left(S_{t}\right)\left(S_{t}-N_{t}\right)=%
\int_{0}^{t}f\left(S_{s}\right)dN_{s}+F\left(S_{0}\right),
\end{equation}
\end{prop}
\begin{proof}
In (\ref{balay}), take $k_{t}\equiv f\left(S_{t}\right)$, and
$Y_{t}\equiv S_{t}-N_{t}$. Then, we have:
\begin{equation*}
f\left(S_{g_{t}}\right)\left(S_{t}-N_{t}\right)=\int_{0}^{t}f\left(S_{g_{s}}%
\right)d\left(S_{s}-N_{s}\right).
\end{equation*}
But $S_{g_{t}}=S_{t}$, hence:
\begin{equation*}
F\left(S_{t}\right)-f\left(S_{t}\right)\left(S_{t}-N_{t}\right)=%
\int_{0}^{t}f\left(S_{s}\right)dN_{s}+F\left(S_{0}\right).
\end{equation*}%
In conclusion, for any locally bounded function $f$,
\begin{equation*}
F\left(S_{t}\right)-f\left(S_{t}\right)\left(S_{t}-N_{t}\right)=%
\int_{0}^{t}f\left(S_{s}\right)dN_{s}+F\left(S_{0}\right),
\end{equation*}
is a local martingale. \end{proof}
\begin{rem}
Although very simple, these martingales played an essential role in
the resolution by Az\'{e}ma and Yor of Skorokhod's embedding problem
(see \cite{revuzyor}, chapter VI for more details and references).
\end{rem}
\begin{rem}
In \cite{laurentyor}, a special case of Proposition
\ref{azemayorgeneralisee}, for spectrally negative L\'{e}vy
martingales is obtained by different means.
\end{rem}
Now, we associate with the two dimensional process $\left( \log
\left( S_{t}\right) ,\text{ }Z_{t}\right) _{t\geq 0}$, a canonical
family of local martingales which are in fact of the form
(\ref{ayor}).
\begin{prop}
Let $f$ be a locally bounded and Borel function, and let $%
F\left(x\right)=\int_{0}^{x}dyf\left(y\right)$.
\begin{enumerate}
\item The following processes are local martingales:%
\begin{equation}
F\left( \log \left( S_{t}\right) \right) -f\left( \log \left( S_{t}\right)
\right) \left( 1-Z_{t}\right) ,\text{ }t\geq 0. \label{azemayordeg}
\end{equation}
\item Denoting $K\left( x\right) =F\left( x-1\right) $ and $k\left( x\right)
=f\left( x-1\right) $, then the local martingales in (\ref{azemayordeg}) are
seen to be equal to:%
\begin{equation}
K\left( \overline{\mu }_{t}\right) -k\left( \overline{\mu
}_{t}\right) \left( \overline{\mu }_{t}-\mu _{t}\right) ,\text{
}t\geq 0. \label{f}
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
(1). The fact that (\ref{azemayordeg}) defines a local martingale
may be seen as an application of Ito's lemma (when $f$ is regular),
followed by a monotone class argument.
(2). Formula (\ref{f}) is obtained by a trivial change of variables,
and the fact that: $1-Z_{t}=\overline{\mu }_{t}-\mu _{t}$, which was
derived in Proposition \ref{sko}.
\end{proof}
\begin{rem}
Similar formulas are derived in \cite{ANMYII} from different
considerations.
\end{rem}
\section{Initial expansion with $S_{\infty}$ and enlargement formulae}
In this Section, we shall deal with the question of initial
enlargement of
the filtration $\left(\mathcal{F}_{t}\right)$ with the variable $S_{\infty}$%
. This problem cannot be dealt with the powerful enlargement theorem
of Jacod (see \cite{jeulinyor}), but can be treated by a careful
combination of different propositions in \cite{jeulin}. However, we
shall give a simple proof which can also be adapted to deal with
some other
situations. Eventually, we will use our result about the initial expansion of $%
\left(\mathcal{F}_{t}\right)$ with the variable $S_{\infty}$ to
recover formula (\ref{grossform}).
Let us define the new filtration
\begin{equation*}
\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\equiv
\bigcap_{\varepsilon>0}\left(\mathcal{F}_{t+\varepsilon}\vee
\sigma\left(S_{\infty}\right)\right),
\end{equation*}%
which satisfies the usual assumptions. The new information $%
\sigma\left(S_{\infty}\right)$ is brought in at the origin of time
and $g$ is a stopping time for this larger filtration. More
precisely:
\begin{lem}\label{lemminclusion}
The following hold:
\begin{enumerate}
\item \begin{equation*}
g=\inf\left\{t:\;N_{t}=S_{\infty}\right\};
\end{equation*} and hence $g$ is an
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$
stopping time.
\item Consequently: \begin{equation*} \mathcal{F}_{t}^{g}\subset
\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}.
\end{equation*}
\end{enumerate}
\end{lem}
\begin{proof}
$(1)$ The measure $dS_{t}$ is carried by the set
$\left\{t:\;N_{t}=S_{t}\right\}$. As
$g=\sup\left\{t:\;N_{t}=S_{t}\right\}$, the process
$\left(S_{t}\right)$ does not grow after $g$, which also satisfies:
$$g=\inf\left\{t:\;S_{t}=S_{\infty}\right\};$$hence $g$ is an
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$
stopping time.
$(2)$ It is obvious.
\end{proof}
Now we introduce some standard terminology.
\begin{defn}
We shall say that the pair of filtrations $\left(\mathcal{F}_{t}, \mathcal{F}%
_{t}^{\sigma\left(S_{\infty}\right)}\right)$ satisfies the $%
\left(H^{\prime}\right)$ hypothesis if every $\left(\mathcal{F}_{t}\right)$
(semi)martingale is a $\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}%
\right)$ semimartingale.
\end{defn}
We shall now show that the pair of filtrations $\left(\mathcal{F}_{t},
\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$ satisfies the $%
\left(H^{\prime}\right)$ hypothesis and give the decomposition of a $\left(%
\mathcal{F}_{t}\right)$ local martingale in $\left(\mathcal{F}%
_{t}^{\sigma\left(S_{\infty}\right)}\right)$. For this, we need to know the
conditional law of $S_{\infty}$ given $\mathcal{F}_{t}$.
\begin{prop}
For any Borel bounded or positive function $f$, we have:
\begin{eqnarray}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right) &=&
f\left(S_{t}\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+%
\int_{0}^{N_{t}/S_{t}}dxf\left(\dfrac{N_{t}}{x}\right) \label{grosavecs} \\
&=& f\left(S_{t}\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+N_{t}%
\int_{S_{t}}^{\infty}dx\frac{f\left(x\right)}{x^{2}}. \notag
\end{eqnarray}
\end{prop}
\begin{proof}
The proof is based on Lemma \ref{maxeq}; in the following, $U$ is a random
variable, which follows the standard uniform law and which is independent of
$\mathcal{F}_{t}$.
\begin{eqnarray*}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right) &=& \mathbf{E%
}\left(f\left(S_{t}\vee S^{t}\right)|\mathcal{F}_{t}\right) \\
&=& \mathbf{E}\left(f\left(S_{t}\right)\mathbf{1}_{\left\{S_{t}\geq
S^{t}\right\}}|\mathcal{F}_{t}\right)+\mathbf{E}\left(f\left(S^{t}\right)%
\mathbf{1}_{\left\{S_{t}< S^{t}\right\}}|\mathcal{F}_{t}\right) \\
&=& f\left(S_{t}\right)\mathbf{P}\left(S_{t}\geq S^{t}|\mathcal{F}%
_{t}\right)+ \mathbf{E}\left(f\left(S^{t}\right)\mathbf{1}_{\left\{S_{t}<
S^{t}\right\}}|\mathcal{F}_{t}\right) \\
&=& f\left(S_{t}\right)\mathbf{P}\left(U\leq \dfrac{N_{t}}{S_{t}}|\mathcal{F}_{t}\right)+%
\mathbf{E}\left(f\left(\dfrac{N_{t}}{U}\right)\mathbf{1}_{\left\{U<\frac{%
N_{t}}{S_{t}}\right\}}|\mathcal{F}_{t}\right) \\
&=& f\left(S_{t}\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+%
\int_{0}^{N_{t}/S_{t}}dxf\left(\dfrac{N_{t}}{x}\right).
\end{eqnarray*}%
A straightforward change of variable in the last integral also gives:
\begin{equation*}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right)=f\left(S_{t}%
\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+N_{t}\int_{S_{t}}^{\infty}dy\frac{%
f\left(y\right)}{y^{2}}.
\end{equation*}
\end{proof}
One may now ask if $\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}%
_{t}\right)$ is of the form (\ref{ayor}). The answer to this question is
positive. Indeed:
\begin{eqnarray*}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right)&=&
f\left(S_{t}\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+N_{t}\int_{S_{t}}^{%
\infty}dy\frac{f\left(y\right)}{y^{2}} \\
&=& S_{t}\int_{S_{t}}^{\infty}dy\frac{f\left(y\right)}{y^{2}}%
-\left(S_{t}-N_{t}\right)\left(\int_{S_{t}}^{\infty}dy\frac{f\left(y\right)}{%
y^{2}}-\dfrac{f\left(S_{t}\right)}{S_{t}}\right).
\end{eqnarray*}%
Hence,
\begin{equation*}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right)=H\left(1%
\right)+H\left(S_{t}\right)-h\left(S_{t}\right)\left(S_{t}-N_{t}\right),
\end{equation*}
with
\begin{equation*}
H\left(x\right)=x\int_{x}^{\infty}dy\frac{f\left(y\right)}{y^{2}},
\end{equation*}
and
\begin{equation*}
h\left(x\right)=h_{f}\left(x\right)\equiv\int_{x}^{\infty}dy\frac{f\left(y\right)}{y^{2}}-\dfrac{%
f\left(x\right)}{x}=\int_{x}^{\infty}\frac{dy}{y^{2}}\left(f\left(y\right)-f\left(x\right)\right).
\end{equation*}
Moreover, again from formula (\ref{ayor}), we have the following
representation of $\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}%
_{t}\right)$ as a stochastic integral:
\begin{equation} \label{represstoc}
\mathbf{E}\left(f\left(S_{\infty}\right)|\mathcal{F}_{t}\right)=\mathbf{E}%
\left(f\left(S_{\infty}\right)\right)+\int_{0}^{t}h\left(S_{s}\right)dN_{s}.
\end{equation}%
Let us sum up these results, introducing some notations:
\begin{eqnarray}
\lambda_{t}\left(f\right) &\equiv& \mathbf{E}\left(f\left(S_{\infty}\right)|%
\mathcal{F}_{t}\right) \\
&=& f\left(S_{t}\right)\left(1-\dfrac{N_{t}}{S_{t}}\right)+N_{t}%
\int_{S_{t}}^{\infty}dx\frac{f\left(x\right)}{x^{2}};
\end{eqnarray}%
and
\begin{equation}
\lambda_{t}\left(f\right)=\mathbf{E}\left(f\left(S_{\infty}\right)\right)+%
\int_{0}^{t}\dot{\lambda}_{s}\left(f\right)dN_{s},
\end{equation}%
where:
\begin{equation}
\dot{\lambda}_{s}\left(f\right)=h_{f}\left(S_{s}\right).
\end{equation}%
Moreover, there exist two families of random measures $%
\left(\lambda_{t}\left(dx\right)\right)_{t\geq 0}$ and $\left(\dot{\lambda}%
_{t}\left(dx\right)\right)_{t\geq 0}$, with
\begin{eqnarray}
\lambda_{t}\left(dx\right) &=& \left(1-\dfrac{N_{t}}{S_{t}}%
\right)\delta_{S_{t}}\left(dx\right)+N_{t}\mathbf{1}_{\left\{x>S_{t}\right\}}%
\dfrac{dx}{x^{2}} \\
\dot{\lambda}_{t}\left(dx\right) &=& -\dfrac{1}{S_{t}}\delta_{S_{t}}\left(dx%
\right)+\mathbf{1}_{\left\{x>S_{t}\right\}}\dfrac{dx}{x^{2}},
\end{eqnarray}
such that
\begin{eqnarray}
\lambda_{t}\left(f\right) &=& \int\lambda_{t}\left(dx\right)f\left(x\right)
\\
\dot{\lambda}_{t}\left(f\right) &=& \int\dot{\lambda}_{t}\left(dx\right)f%
\left(x\right).
\end{eqnarray}%
Eventually, we notice that there is an absolute continuity relationship
between $\lambda_{t}\left(dx\right)$ and $\dot{\lambda}_{t}\left(dx\right)$;
more precisely,
\begin{equation}
\dot{\lambda}_{t}\left(dx\right)=\lambda_{t}\left(dx\right)\rho\left(x,t%
\right),
\end{equation}%
with
\begin{equation} \label{absolucontrel}
\rho\left(x,t\right)=\dfrac{-1}{S_{t}-N_{t}}\mathbf{1}_{\left\{S_{t}=x\right%
\}}+\dfrac{1}{N_{t}}\mathbf{1}_{\left\{S_{t}<x\right\}}.
\end{equation}%
Now, we can state the main theorem of this section.
\begin{thm}
\label{decoinitial} Let $\left(N_{t}\right)_{t\geq 0}$ be a local
martingale in the class $\mathcal{C}_{0}$ (recall $N_{0}=1$).
Then, the pair of filtrations $\left(\mathcal{F}_{t}, \mathcal{F}%
_{t}^{\sigma\left(S_{\infty}\right)}\right)$ satisfies the $%
\left(H^{\prime}\right)$ hypothesis and every $\left(\mathcal{F}_{t}\right)$
local martingale $\left(X_{t}\right)$ is an $\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}%
\right)}\right)$ semimartingale with canonical decomposition:
\begin{equation*}
X_{t}=\widetilde{X}_{t}+\int_{0}^{t}\mathbf{1}_{\left\{ g>s\right\} }\frac{%
d<X,N>_{s}}{N_{s-}}-\int_{0}^{t}\mathbf{1}_{\left\{ g\leq s\right\} }\frac{%
d<X,N>_{s}}{S_{\infty}-N_{s-}},
\end{equation*}%
where $\left( \widetilde{X}_{t}\right) $ is a $\left(\mathcal{F}%
_{t}^{\sigma\left(S_{\infty}\right)}\right)$\ local martingale.
\end{thm}
\begin{rem}
The following proof is tailored on the arguments found in
\cite{zurich}, although our framework is more general: we do not
assume that our filtration has the predictable representation
property with respect to some martingale nor that all martingales
are continuous.
\end{rem}
\begin{proof}
We can first assume that $X$ is in $\mathcal{H}^{1}$; the general
case follows by localization. Let $\Lambda_{s}$ be an
$\mathcal{F}_{s}$ measurable set, and take $t>s$. Then, for any
bounded test function $f$, $\lambda_{t}\left(f\right)$ is a bounded
martingale, hence in $BMO$, and we
have:%
\begin{eqnarray*}
\mathbf{E}\left(\mathbf{1}_{\Lambda_{s}}f\left( A_{\infty
}\right)\left(X_{t}-X_{s}\right)\right) &=& \mathbf{E}\left(\mathbf{1}%
_{\Lambda_{s}}\left(\lambda_{t}\left(f\right)X_{t}-\lambda_{s}\left(f%
\right)X_{s}\right)\right) \\
&=& \mathbf{E}\left(\mathbf{1}_{\Lambda_{s}}\left(<\lambda\left(f%
\right),X>_{t}-<\lambda\left(f\right),X>_{s}\right)\right) \\
&=& \mathbf{E}\left(\mathbf{1}_{\Lambda_{s}}\left(\int_{s}^{t}\dot{\lambda}%
_{u}\left(f\right)d<X,N>_{u}\right)\right) \\
&=& \mathbf{E}\left(\mathbf{1}_{\Lambda_{s}}\left(\int_{s}^{t}\int%
\lambda_{u}\left(dx\right)\rho\left(x,u\right)f\left(x\right)d<X,N>_{u}%
\right)\right) \\
&=& \mathbf{E}\left(\mathbf{1}_{\Lambda_{s}}\left(\int_{s}^{t}d<X,N>_{u}\rho%
\left(S_{\infty },u\right)\right)\right).
\end{eqnarray*}%
But from (\ref{absolucontrel}), we have:%
\begin{equation*}
\rho\left(S_{\infty },t\right)=\dfrac{-1}{S_{t}-N_{t}}\mathbf{1}%
_{\left\{S_{t}=S_{\infty}\right\}}+\dfrac{1}{N_{t}}\mathbf{1}_{\left\{S_{t}<S_{\infty}\right%
\}}.
\end{equation*}
It now suffices to note (from Lemma \ref{lemminclusion}) that
$\left(S_{t}\right)$ is constant after $g$ and $g$ is the first
time when $S_{\infty}=S_{t}$, or in other words:
\begin{equation*}
\mathbf{1}_{\left\{S_{\infty}>S_{t}\right\}}=\mathbf{1}_{\left\{g>t\right\}},%
\text{ and }\mathbf{1}_{\left\{S_{\infty}=S_{t}\right\}}=\mathbf{1}%
_{\left\{g\leq t\right\}}.
\end{equation*}%
This completes the proof.
\end{proof}
Theorem \ref%
{decoinitial} yields a new proof of the decomposition formula in the
progressive enlargement case. More precisely, we have:
\begin{cor}
\label{hyphprimepourN} The pair of filtrations $\left(\mathcal{F}_{t},%
\mathcal{F}_{t}^{g}\right)$ satisfies the $\left(H^{\prime}\right)$
hypothesis. Moreover, every $\left(\mathcal{F}_{t}\right)$ local martingale $%
X$ decomposes as:
\begin{equation*}
X_{t}=\widetilde{X}_{t}+\int_{0}^{t}\mathbf{1}_{\left\{ g>s\right\} }\frac{%
d<X,N>_{s}}{N_{s}}-\int_{0}^{t}\mathbf{1}_{\left\{ g\leq s\right\} }\frac{%
d<X,N>_{s}}{S_{\infty}-N_{s}},
\end{equation*}%
where $\left( \widetilde{X}_{t}\right) $ is a $\left(\mathcal{F}%
_{t}^{g}\right)$\ local martingale.
\end{cor}
\begin{proof}
Let $X$ be an $\left(\mathcal{F}_{t}\right)$ martingale which is in
$\mathcal{H}^{1}$; the general case follows by localization. From
Theorem \ref{decoinitial}
\begin{equation*}
X_{t}=\widetilde{X}_{t}+\int_{0}^{t}\mathbf{1}_{\left\{ g>s\right\} }\frac{%
d<X,N>_{s}}{N_{s}}-\int_{0}^{t}\mathbf{1}_{\left\{ g\leq s\right\} }\frac{%
d<X,N>_{s}}{S_{\infty}-N_{s}},
\end{equation*}%
where $\left(\widetilde{X}_{t}\right) _{t\geq 0}$ denotes an $\left(%
\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$ martingale.
Thus, $\left(\widetilde{X}_{t}\right) $, which is equal to:
\begin{equation*}
X_{t}-\left(\int_{0}^{t}\mathbf{1}_{\left\{ g>s\right\} }\frac{d<X,N>_{s}}{%
N_{s}}-\int_{0}^{t}\mathbf{1}_{\left\{ g\leq s\right\} }\frac{d<X,N>_{s}}{%
S_{\infty}-N_{s}},\right),
\end{equation*}
is $\left(\mathcal{F}_{t}^{g}\right)$ adapted (recall that $\mathcal{F}%
_{t}^{g}\subset \mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}$),
and hence it is an $\left(\mathcal{F}_{t}^{g}\right)$ martingale.
\end{proof}
\section{A multiplicative characterization of $Z_{t}$}
Usually, in the literature about progressive enlargements of
filtrations, it is assumed that the conditions \textbf{(CA)} are
satisfied. Now, we shall prove that under this assumption the supermartingale $%
Z_{t}^{L}=\mathbf{P}\left( L>t\mid \mathcal{F}_{t}\right) $, associated with
an honest time, can be represented as $\left( \dfrac{N_{t}}{S_{t}}\right)
_{t\geq 0}$, where $N_{t}$ is a positive local martingale. More precisely,
we have the following:
\begin{thm}
\label{multiplicatcarac} Let $L$\ be an honest time. Then, under the
conditions \textbf{(CA)}, there exists a continuous and nonnegative
local martingale $\left( N_{t}\right) _{t\geq 0}$,
with $N_{0}=1$ and $\lim_{t\rightarrow \infty }N_{t}=0$, such that:%
\begin{equation*}
Z_{t}=\mathbf{P}\left( L>t\mid \mathcal{F}_{t}\right) =\dfrac{N_{t}}{S_{t}}
\end{equation*}
\end{thm}
\begin{proof}
Under the conditions \textbf{(CA)}, $\left( Z_{t}\right) _{t\geq 0}$ is
continuous and can be written as (see \cite{azema} or \cite{delmaismey} for
details):%
\begin{equation*}
Z_{t}=M_{t}-A_{t},
\end{equation*}%
where $\left( M_{t}\right) $ and $\left( A_{t}\right) $ are continuous, $%
Z_{0}=1$ and $dA_{t}$ is carried by $\left\{ t:\text{ }Z_{t}=1\right\} $.
Then, for $t<T_{0}\equiv \inf\left\{t:\;Z_{t}=0\right\}$, we have:%
\begin{equation*}
\log \left( Z_{t}\right) =\int_{0}^{t}\frac{dM_{s}}{Z_{s}}-\frac{1}{2}%
\int_{0}^{t}\frac{d<M>_{s}}{Z_{s} ^{2}}-A_{t},
\end{equation*}%
hence:%
\begin{equation}
-\log \left( Z_{t}\right) =-\left( \int_{0}^{t}\frac{dM_{s}}{Z_{s}}-\frac{1}{%
2}\int_{0}^{t}\frac{d<M>_{s}}{Z_{s} ^{2}}\right) +A_{t}; \label{a}
\end{equation}%
and, from Skorokhod's reflection lemma, we have:
\begin{equation} \label{logito}
A_{t}=\sup_{u\leq t}\left( \int_{0}^{u}\frac{dM_{s}}{Z_{s}}-\frac{1}{2}%
\int_{0}^{u}\frac{d<M>_{s}}{Z_{s} ^{2}}\right) .
\end{equation}%
Now, combining (\ref{a}) and (\ref{logito}), we obtain:
\begin{equation*}
Z_{t}=\frac{N_{t}}{S_{t}},
\end{equation*}%
where
\begin{equation*}
N_{t}=\exp \left( \int_{0}^{t}\frac{dM_{s}}{Z_{s}}-\frac{1}{2}\int_{0}^{t}%
\frac{d<M>_{s}}{Z_{s} ^{2}}\right)
\end{equation*}%
is a local martingale starting from $1$, and
\begin{eqnarray*}
S_{t} &=&\sup_{u\leq t}\left(\exp \left( \int_{0}^{u}\frac{dM_{s}}{Z_{s}}-\frac{1}{%
2}\int_{0}^{u}\frac{d<M>_{s}}{Z_{s} ^{2}}\right)\right) \\
&=&\exp \left( \sup_{u\leq t}\left( \int_{0}^{u}\frac{dM_{s}}{Z_{s}}-\frac{1%
}{2}\int_{0}^{u}\frac{d<M>_{s}}{Z_{s} ^{2}}\right) \right) \\
&=&\exp \left( A_{t}\right) .
\end{eqnarray*}We finally note that, since $Z_{T_{0}}=0$, $\lim_{t\uparrow
T_{0}}N_{t}=0$, which allows to define $N_{t}$ for all $t\geq 0$.
\end{proof}
\begin{cor}
The supermartingale $Z_{t}=\mathbf{P}\left( L>t\mid
\mathcal{F}_{t}\right)$ admits the following additive and
multiplicative representations:
\begin{eqnarray*}
Z_{t} &=& \dfrac{N_{t}}{S_{t}} \\
Z_{t} &=& M_{t}-A_{t}.
\end{eqnarray*} Moreover, these two representations are related as follows:
\begin{eqnarray*}
N_{t} &=& \exp \left( \int_{0}^{t}\frac{dM_{s}}{Z_{s}}-\frac{1}{2}\int_{0}^{t}%
\frac{d<M>_{s}}{Z_{s} ^{2}}\right) \\
S_{t} &=& \exp\left(A_{t}\right);
\end{eqnarray*}and
\begin{eqnarray*}
M_{t} &=& 1+\int_{0}^{t}\dfrac{dN_{s}}{S_{s}}=\mathbf{E}\left(\log S_{\infty}\mid \mathcal{F}_{t}\right), \\
A_{t} &=& \log S_{t}.
\end{eqnarray*}
\end{cor}
\begin{proof}
It is a consequence of Proposition \ref{applicationmax} and Theorem
\ref{multiplicatcarac}.
\end{proof}
\bigskip
Now, as a consequence of Theorem \ref{multiplicatcarac}, we can recover the
enlargement formulae and the fact that the pair of filtrations $\left(%
\mathcal{F}_{t}, \mathcal{F}_{t}^{L}\right)$ satisfies the $(H^{\prime})$
hypothesis:
\begin{cor}
Let $L$\ be an honest time. Then under the conditions \textbf{(CA)},
the pair of filtrations $\left(\mathcal{F}_{t},
\mathcal{F}_{t}^{L}\right)$ satisfies the $(H^{\prime})$ hypothesis
and
every $\left(\mathcal{F}_{t}\right)$ local martingale $X$ is an $\left(%
\mathcal{F}_{t}^{L}\right)$ semimartingale with canonical decomposition:
\begin{equation*}
X_{t}=\widetilde{X}_{t}+\int_{0}^{t\wedge L}\dfrac{d<X,Z>_{s}}{Z_{s}}%
+\int_{L}^{t}\dfrac{d<X,1-Z>_{s}}{1-Z_{s}%
},
\end{equation*}
where $\left( \widetilde{X}_{t}\right) _{t\geq 0}$ denotes an
$\left( \left( \mathcal{F}_{t}^{L}\right)\right) $ local martingale.
\end{cor}
\begin{proof}
It is a combination of Theorem \ref{multiplicatcarac} and Corollary \ref%
{hyphprimepourN}.
\end{proof}
\begin{rem}
We then see that under the assumptions \textbf{(CA)}, the initial
enlargement of filtrations with $A_{\infty}$ amounts to enlarging
initially the filtration with $S_{\infty}$, the terminal value of
the supremum process of a continuous local martingale in
$\mathcal{C}_{0}$.
\end{rem}
We shall now outline another nontrivial consequence of Theorem
\ref{multiplicatcarac} here. In \cite{azemjeulknightyor}, the
authors are interested in giving explicit examples of dual
predictable projections of processes of the form $\mathbf{1}_{g\leq
t}$, where $g$ is an honest time. Indeed, these dual projections are
natural examples of increasing injective processes (see
\cite{azemjeulknightyor} for more details and references). With
Theorem \ref{multiplicatcarac}, we have a complete characterization
of such projections:
\begin{cor}
Assume the assumption \textbf{(C)} holds, and let
$\left(C_{t}\right)$ be an increasing process. Then $C$ is the dual
predictable projection of $\mathbf{1}_{g\leq t}$, for some honest
time $g$ that avoids stopping times, if and only if there exists a
continuous local martingale $N_{t}$ in the class $\mathcal{C}_{0}$
such that
$$C_{t}=\log S_{t}.$$
\end{cor}
\bigskip
The previous results can be naturally extended to the case where the
supermartingale $Z_{t}$ has only negative jumps; we gave a special
treatment under the hypothesis \textbf{(CA)} because of its
practical importance. We just give here the extension of Theorem
\ref{multiplicatcarac}; the corollaries are easily deduced.
\begin{prop}
Let $L$\ be an honest time that avoids stopping times. Assume that
$Z_{t}^{L}$ has no positive jumps. Then, there exists a local
martingale $\left( N_{t}\right) _{t\geq 0}$, in the class
$\mathcal{C}_{0}$,
with $N_{0}=1$, such that:%
\begin{equation*}
\left(Z_{t}^{L}=\right)Z_{t}=\mathbf{P}\left( L>t\mid
\mathcal{F}_{t}\right) =\dfrac{N_{t}}{S_{t}}
\end{equation*}
\end{prop}
\begin{proof}
We use the same notations as in the proof of Theorem
\ref{multiplicatcarac}. For $t<T_{0}\equiv \inf\left\{t:\;Z_{t}=0\right\}$, we have:%
\begin{equation*}
-\log \left( Z_{t}\right) =-\left(\int_{0}^{t}\left(\frac{dM_{s}}{Z_{s-}}-\frac{1}{2}%
\frac{d<M^{c}>_{s}}{Z_{s-} ^{2}}\right)+\sum_{0<s\leq t}\left(\log
\left(1+\dfrac{\Delta Z_{s}}{Z_{s-}}\right)-\dfrac{\Delta
Z_{s}}{Z_{s-}}\right)\right)+A_{t}.
\end{equation*}Now, from Lemma \ref{lemmreflection},
$$A_{t}=\sup_{s\leq t}\left(\int_{0}^{t}\left(\frac{dM_{s}}{Z_{s-}}-\frac{1}{2}%
\frac{d<M^{c}>_{s}}{Z_{s-} ^{2}}\right)+\sum_{0<s\leq t}\left(\log
\left(1+\dfrac{\Delta Z_{s}}{Z_{s-}}\right)-\dfrac{\Delta
Z_{s}}{Z_{s-}}\right)\right).$$ Now, combining the last two
equalities, we obtain:
$$Z_{t}=\dfrac{N_{t}}{S_{t}},$$where $$N_{t}=\exp\left(\int_{0}^{t}\left(\frac{dM_{s}}{Z_{s-}}-\frac{1}{2}%
\frac{d<M^{c}>_{s}}{Z_{s-} ^{2}}\right)\right)\prod_{0<s\leq
t}\left(1+\dfrac{\Delta
Z_{s}}{Z_{s-}}\right)\exp\left(-\dfrac{\Delta
Z_{s}}{Z_{s-}}\right).$$
\end{proof}
\section{Examples and applications}
In this section, we look at some specific local martingales $N_{t}$,
and use the initial enlargement formula with $S_{\infty}$, to get
some path decompositions, given the maximum or the minimum of some
stochastic processes. Our aim here is to illustrate how techniques
from enlargement of filtrations can be applied. To have a complete
description for the path decompositions, we associate with $g$ a
random time, called pseudo-stopping time, which occurs before $g$.
Eventually, we give some explicit examples of supermartingales
$Z_{t}$ with jumps.
\subsection{Pseudo-stopping times}\label{secpta}
In \cite{ANMY}, we have proposed the following generalization of
stopping times:
\begin{defn}
Let $\rho:\;(\Omega,\mathcal{F})\rightarrow\mathbf{R}_{+}$ be a
random time; $\rho$ is called a pseudo-stopping time if for every
bounded $\left(\mathcal{F}_{t}\right)$ martingale we have:
$$\mathbf{E}\left(M_{\rho}\right)=\mathbf{E}\left(M_{0}\right).$$
\end{defn}David Williams (\cite{williams}) gave the first example of such a random
time and the following systematic construction is established in
\cite{ANMY}:
\begin{prop}\label{ptaconstruction}
Let $L$ be an honest time. Then, under the conditions \textbf{(CA)},
$$\rho\equiv \sup\left\{t<L:\;Z_{t}^{L}=\inf_{u\leq
L}Z_{u}^{L}\right\},$$is a pseudo-stopping time, with
$$Z_{t}^{\rho}\equiv
\mathbf{P}\left(\rho>t\mid\mathcal{F}_{t}\right)=\inf_{u\leq
t}Z_{u}^{L},$$and $Z_{\rho}^{\rho}$ follows the uniform distribution
on $(0,1)$.
\end{prop}The following property, also proved in \cite{ANMY}, is
essential in studying path decompositions:
\begin{prop}\label{regenrative}
Let $\rho$ be a pseudo-stopping time and let $M_{t}$ be an
$\left(\mathcal{F}_{t}\right)$ local martingale. Then
$\left(M_{t\wedge \rho}\right)$ is an
$\left(\mathcal{F}_{t}^{\rho}\right)$ local martingale.
\end{prop}In our setting, Proposition \ref{ptaconstruction} gives:
\begin{prop}\label{ptamult}
Define the nonincreasing process $\left(r_{t}\right)$ by:
$$r_{t}\equiv \inf_{u\leq t}\dfrac{N_{u}}{S_{u}}.$$Then,
$$\rho\equiv \sup\left\{t<g:\;\dfrac{N_{t}}{S_{t}}=\inf_{u\leq
g}\dfrac{N_{u}}{S_{u}}\right\},$$is a pseudo-stopping time and
$r_{\rho}$ follows the uniform distribution on $(0,1)$.
\end{prop}
\subsection{Path decompositions given the maxima or the minima of a diffusion}
Now, we shall apply the techniques of enlargements of filtrations to
establish some path decompositions results. Some of the following
results have been proved by David Williams in \cite{williams2},
using different methods. Jeulin has also given a proof based on
enlargements techniques in the case of transient diffusions (see
\cite{jeulin}). Here, we complete the results of David Williams by
introducing the pseudo-stopping times $\rho$ defined in Proposition
\ref{ptamult}, and we detail some interesting examples.
\subsubsection{The killed Brownian Motion}
Let $$N_{t}\equiv B_{t},$$where $\left(B_{t}\right)_{t\geq 0}$ is a
Brownian Motion starting at $1$, and stopped at
$T_{0}=\inf\left\{t:\;B_{t}=0\right\}$. Let
$$S_{t}\equiv \sup_{s\leq t}B_{s}.$$ Let $$g=\sup\left\{t:B_{t}=S_{t}\right\}$$ and $$\rho=\sup\left\{t<g:\;\dfrac{B_{t}}{S_{t}}=\inf_{u\leq
g}\dfrac{B_{u}}{S_{u}}\right\}.$$ From Doob's maximal identity,
$S_{T_{0}}=S_{g}$ is distributed as the reciprocal of a uniform
distribution $\left(0,1\right)$, i.e. it has the density:
$\mathbf{1}_{\left[1,\infty\right)}\left(x\right)\dfrac{1}{x^{2}}$.
\begin{prop}\label{madeco}
Let $\left(B_{t}\right)_{t\geq 0}$ be a Brownian Motion starting at
$1$ and stopped when it first hits $0$. Then:
\begin{itemize}
\item $\dfrac{B_{\rho}}{S_{\rho}}$ follows the uniform law on $(0,1)$, and conditionally on $\dfrac{B_{\rho}}{S_{\rho}}=r$,
$\left(B_{t}\right)$ is a Brownian Motion up to the first time when
$B_{t}=rS_{t}$.
\item $\left(B_{t}\right)$
is an $\left(\mathcal{F}_{t}^{g}\right)$ and
$\left(\mathcal{F}_{t}^{\sigma\left(S_{T_{0}}\right)}\right)$
semimartingale with canonical decomposition:
\begin{equation}\label{decobessel}
B_{t}=\widetilde{B}_{t}+\int_{0}^{t\wedge g}\dfrac{ds}{B_{s}}-\int_{g}^{t\wedge
T_{0}}\dfrac{ds}{S_{T_{0}}-B_{s}},
\end{equation}where $\left(\widetilde{B}_{t}\right)$ is an
$\mathcal{F}_{t}^{\sigma\left(S_{T_{0}}\right)}$ Brownian Motion,
stopped at $T_{0}$ and independent of $S_{T_{0}}$. Consequently, we
have the following path decomposition: conditionally on
$S_{T_{0}}=m$:
\begin{enumerate}
\item the process $\left(B_{t};\;t\leq g\right)$ is a Bessel process of
dimension $3$, started from $1$, considered up to $T_{m}$, the first
time when it hits $m$;
\item the process $\left(S_{g}-B_{g+t};\;t\leq
T_{0}-g\right)$ is a $\left(\mathcal{F}_{g+t}\right)$ three
dimensional Bessel process, started from $0$, considered up to
$T_{m}$, the first time when it hits $m$, and is independent of
$\left(B_{t};\;t\leq g\right)$.
\end{enumerate}
\end{itemize}
\end{prop}
\begin{proof}
The results concerning the decomposition until $\rho$ are
consequences of the results of Subsection \ref{secpta}. The
decomposition formula is a consequence of Theorem \ref{decoinitial}.
Since $\left(\widetilde{B}_{t}\right)$ is an
$\mathcal{F}_{t}^{\sigma\left(S_{T_{0}}\right)}$ local martingale,
with $t\wedge T_{0}$ as its bracket, it follows from L\'{e}vy's
theorem that it is an
$\mathcal{F}_{t}^{\sigma\left(S_{T_{0}}\right)}$ Brownian Motion.
Moreover, it is independent of
$\mathcal{F}_{0}^{\sigma\left(S_{T_{0}}\right)}=\sigma\left(S_{T_{0}}\right)$.
Now, conditionally on $S_{T_{0}}=m$, with
$T_{m}=\inf\left\{t:\;B_{t}=m\right\}$, $\left(B_{t}\right)$
satisfies the following stochastic differential equation:
$$B_{t}=\widetilde{B}_{t}+\int_{0}^{t\wedge
T_{m}}\frac{ds}{B_{s}}.$$Hence it is a three dimensional Bessel
process up to $T_{m}$.
It also follows from the decomposition formula that:
$$B_{g+t}=\widetilde{B}_{g+t}+\int_{0}^{g}\dfrac{ds}{B_{s}}-\int_{0}^{t\wedge
(T_{0}-g)}\dfrac{ds}{S_{g}-B_{g+s}}.$$ This equation can also be written
as:$$S_{g}-B_{g+t}=-\left(\widetilde{B}_{g+t}-\widetilde{B}_{g}\right)+\int_{0}^{t\wedge
(T_{0}-g)}\dfrac{ds}{S_{g}-B_{g+s}}.$$ Now,
$\left(\widetilde{B}_{g+t}-\widetilde{B}_{g}\right)$ is an
$\left(\mathcal{F}_{g+t}\right)$ Brownian Motion, starting from $0$, and is independent of $\mathcal{F}_{g}$. Taking
$\widetilde{\beta}_{t}\equiv
-\left(\widetilde{B}_{g+t}-\widetilde{B}_{g}\right)$, which is
also an
$\left(\mathcal{F}_{g+t}\right)$ Brownian Motion, starting from $0$, independent of
$\mathcal{F}_{g}$, the process $\xi_{t}\equiv S_{g}-B_{t}$
satisfies the stochastic differential equation:
$$\xi_{t}=\widetilde{\beta}_{t}+\int_{0}^{t\wedge
(T_{0}-g)}\frac{ds}{\xi_{s}};$$hence it is a three dimensional
Bessel process, started at $0$, and considered up to $T_{m}$, and
conditionally on $S_{g}$, is independent of $\mathcal{F}_{g}$.
\end{proof}
\subsubsection{Some recurrent diffusions}
The previous example can be generalized to a wider class of
recurrent diffusions $\left(X_{t}\right)$, satisfying the stochastic
differential equation:
\begin{equation}\label{equationrecurrence}
X_{t}=x+B_{t}+\int_{0}^{t}b\left(X_{s}\right)ds,\;x>0
\end{equation}where $\left(B_{t}\right)$ is the standard Brownian
Motion, and $b$ is a Borel integrable function which allows
existence and uniqueness for equation (\ref{equationrecurrence})
(for example $b$ bounded or Lipschitz continuous). The infinitesimal
generator $L$ of this diffusion is:
$$L=\frac{1}{2}\dfrac{d^{2}}{dx^{2}}+b\left(x\right)\dfrac{d}{dx}.$$Let $T_{0}\equiv
\inf\left\{t:\;X_{t}=0\right)$, and denote by $s$ the scale function
of $X$, which is strictly increasing and which vanishes at zero,
i.e:
$$s\left(z\right)=\int_{0}^{z}\exp\left(-2\widehat{b}\left(y\right)\right)dy,$$where$$\widehat{b}\left(y\right)=\int_{0}^{y}b\left(u\right)du.$$ Hence,
$$N_{t}\equiv\dfrac{s\left(X_{t\wedge T_{0}}\right)}{s\left(x\right)}$$ is a continuous local
martingale belonging to the class $\mathcal{C}_{0}$. If $S_{t}$
denotes the supremum process of $N_{t}$ and $\overline{X}_{t}$ the
supremum process of $X_{t}$, we
have:$$S_{t}=\dfrac{s\left(\overline{X}_{t\wedge
T_{0}}\right)}{s\left(x\right)}.$$Now,
let$$g=\sup\left\{t<T_{0}:\;X_{t}=\overline{X}_{t}\right\},$$and$$\rho=\sup\left\{t<g:\;\dfrac{X_{t}}{\overline{X}_{t}}=\inf_{u\leq
g}\dfrac{X_{u}}{\overline{X}_{u}}\right\}.$$
\begin{prop}
Let $\left(X_{t}\right)$ be a diffusion process satisfying equation
(\ref{equationrecurrence}). Then:
\begin{itemize}
\item $\dfrac{X_{\rho}}{\overline{X}_{\rho}}$ follows the uniform law on $(0,1)$, and conditionally on $\dfrac{X_{\rho}}{\overline{X}_{\rho}}=r$,
$\left(X_{t},\;t\leq\rho\right)$ is a diffusion process, up to the
first time when $X_{t}=r\overline{X}_{t}$, with the same
infinitesimal generator as $X$.
\item $\left(X_{t}\right)$
is an $\left(\mathcal{F}_{t}^{g}\right)$ and an
$\left(\mathcal{F}_{t}^{\sigma\left(\overline{X}_{T_{0}}\right)}\right)$
semimartingale with canonical decomposition:
\begin{equation}\label{decobessel2}
X_{t}=\widetilde{B}_{t}+\int_{0}^{t}b\left(X_{u}\right)du+\int_{0}^{t\wedge g}\dfrac{s'\left(X_{u}\right)}{s\left(X_{u}\right)}du-\int_{g}^{t\wedge
T_{0}}\dfrac{s'\left(X_{u}\right)}{s\left(\overline{X}_{T_{0}}\right)-s\left(X_{u}\right)}du,
\end{equation}where $\left(\widetilde{B}_{t}\right)$ is an
$\mathcal{F}_{t}^{\sigma\left(\overline{X}_{T_{0}}\right)}$ Brownian
Motion, stopped at $T_{0}$ and independent of
$\overline{X}_{T_{0}}$. Consequently, we have the following path
decomposition: conditionally on $\overline{X}_{T_{0}}=m$:
\begin{enumerate}
\item the process $\left(X_{t};\;t\leq g\right)$ is a diffusion process started from $x>0$, considered up to $T_{m}$, the first time when it
hits $m$, with infinitesimal generator
$$\frac{1}{2}\dfrac{d^{2}}{dx^{2}}+\left(b\left(x\right)+\dfrac{s'\left(x\right)}{s\left(x\right)}\right)\dfrac{d}{dx}.$$
\item the process $\left(X_{g+t};\;t\leq
T_{0}-g\right)$ is a $\left(\mathcal{F}_{g+t}\right)$ diffusion
process, started from $m$, considered up to $T_{0}$, the first time
when it hits $0$, and is independent of $\left(X_{t};\;t\leq
g\right)$; its infinitesimal generator is given by:
$$\frac{1}{2}\dfrac{d^{2}}{dx^{2}}+\left(b\left(x\right)+\dfrac{s'\left(x\right)}{s\left(x\right)-s\left(m\right)}\right)\dfrac{d}{dx}.$$
\item $\overline{X}_{T_{0}}$ follows the same law as
$s^{-1}\left(\dfrac{1}{U}\right)$, where $U$ follows the uniform law
on $(0,1)$.
\end{enumerate}
\end{itemize}
\end{prop}
\begin{proof}
The proof is exactly the same as the proof of Proposition
\ref{madeco}, so we will not reproduce it here.
\end{proof}
\subsubsection{Geometric Brownian Motion with negative drift} Let $$N_{t}\equiv
\exp\left(2\nu B_{t}-2\nu^{2}t\right),$$where $\left(B_{t}\right)$
is a standard Brownian Motion, and $\nu>0$. With the notation of
Theorem \ref{decoinitial}, we have:
$$S_{t}=\exp\left(\sup_{s\leq t}2\nu\left(
B_{s}-\nu s\right)\right),$$and
$$g=\sup\left\{t:\;\left(
B_{t}-\nu t\right)=\sup_{s\geq 0}\left( B_{s}-\nu
s\right)\right\}.$$ Before stating our proposition, let us mention
that we could have worked with more general continuous exponential
local martingales, but we preferred to keep the discussion as simple
as possible (the proof for more general cases is exactly the same).
\begin{prop}
With the assumptions and notations used above, we have:
\begin{enumerate}
\item The variable $\sup_{s\geq 0}\left(
B_{s}-\nu s\right)$ follows the exponential law of parameter $2\nu$.
\item Every local martingale $X$ is an
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$
semimartingale and decomposes as:$$X_{t}=\widetilde{X}_{t}+2\nu
<X,B>_{t\wedge
g}-2\nu\int_{g}^{t}\dfrac{N_{s}}{S_{\infty}-N_{s}}d<X,B>_{s},$$where
$\widetilde{X}_{t}$ is an
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$ local
martingale.
\item Conditionally on $S_{\infty}=m$, the process $\left(B_{t}-\nu t;\;t\leq
g\right)$ is a Brownian Motion with drift $+\nu$ up to the first
hitting time of its maximum $m/2\nu$.
\end{enumerate}
\end{prop}
\begin{proof}
From Doob's maximal equality, $\left(\exp\left(\sup_{s\leq
g}\left(2\nu B_{s}-2\nu^{2}s\right)\right)\right)^{-1}$ follows the
uniform law and hence $\sup_{s\geq 0}\left(B_{s}-\nu s\right)$
follows the exponential law of parameter $2\nu$.
The decomposition formula is a consequence of Theorem
\ref{decoinitial} and the fact that: $dN_{t}=2\nu N_{t}dB_{t}$.
To show $(3)$, it suffices to notice that $B_{t}-\nu t$ is equal to
$\widetilde{B}_{t}+\nu t$ in the filtration
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$, with
$\left(\widetilde{B}_{t}\right)$ an
$\left(\mathcal{F}_{t}^{\sigma\left(S_{\infty}\right)}\right)$
Brownian Motion which is independent of $S_{\infty}$.
\end{proof}
\subsubsection{General transient diffusions} Now, we consider
$\left(R_{t}\right)$, a transient diffusion with values in
$\left[0,\infty\right)$, which has $\left\{0\right\}$ as entrance
boundary. Let $s$ be a scale function for $R$, which we can choose
such that: $$s\left(0\right)=-\infty, \text{ and }
s\left(\infty\right)=0.$$ Then, under the law $\mathbf{P}_{x}$, for
any $x>0$, the local martingale
$\left(N_{t}=\dfrac{s\left(R_{t}\right)}{s\left(x\right)},\;t\geq
0\right)$ satisfies the conditions of Theorem \ref{decoinitial}, and
we have:$$\mathbf{P}_{x}\left(g>
t|\mathcal{F}_{t}\right)=\dfrac{s\left(R_{t}\right)}{s\left(I_{t}\right)}\,$$where
$$g=\sup\left\{t:\; R_{t}=I_{t}\right\},$$and
$$I_{t}=\inf_{s\leq t}R_{s}.$$We thus recover results of Jeulin
(\cite{jeulin}, Proposition 6.29, p.112) by other means. Jeulin used
this formula and gave a quick proof of a theorem of David Williams
(\cite{williams2}), using initial enlargement of filtrations
arguments. Our proof would follow the same lines and so we refer to
the book of Jeulin. We would rather detail an interesting example:
the three dimensional Bessel process.
\begin{prop}
Let $\left(R_{t}\right)$ be a three dimensional Bessel process
starting from $1$, and set, as above, $I_{t}=\inf_{s\leq t}R_{s}$,
and $g=\sup\left\{t:\; R_{t}=I_{t}\right\}$. Define $\rho$ by:
$\rho= \sup\left\{t<g:\;\dfrac{I_{t}}{R_{t}}=\inf_{u\leq
g}\dfrac{I_{u}}{R_{u}}\right\}.$Then:
\begin{enumerate}
\item The variable $\dfrac{I_{\rho}}{R_{\rho}}$ follows the uniform law on $(0,1)$ and, conditionally on $I_{\rho}=rR_{\rho}$, $\left(R_{t},t\leq T_{r}\right)$ is a three
dimensional Bessel process starting from $1$, up to the first time
$T_{r}$ when $I_{t}=rR_{t}$.
\item $I_{\infty}\equiv I_{g}$ follows the uniform law on $(0,1)$;
\item Conditionally on $I_{\infty}=r$, the process $\left(R_{t},\;t\leq
g\right)$ is a Brownian Motion starting from $1$ and stopped when it
first hits $r$.
\end{enumerate}
\end{prop}
\begin{proof}
There exists $\left(\beta\right)_{t\geq 0}$, a Brownian Motion, such
that
$$R_{t}=1+\beta_{t}+\int_{0}^{t}\dfrac{ds}{R_{s}}.$$
$(1)$ follows easily from the results of Subsection
\ref{secpta}.
Now, from Ito's formula, it follows that
$$\dfrac{1}{R_{t}}=1-\int_{0}^{t}\dfrac{d\beta_{s}}{R_{s}^{2}};$$hance, it is a local
martingale. In
$\left(\mathcal{F}_{t}^{\sigma\left(I_{\infty}\right)}\right)$,
$$\beta_{t\wedge g}=\widetilde{\beta}_{t}-\int_{0}^{t\wedge
g}\dfrac{ds}{R_{s}},$$where $\left(\widetilde{\beta}_{t}\right)$ is
an $\left(\mathcal{F}_{t}^{\sigma\left(I_{\infty}\right)}\right)$
Brownian Motion independent of $I_{\infty}$. Hence, $R_{t\wedge g}$
decomposes as
$$R_{t\wedge g}=\widetilde{\beta}_{t}$$ in
$\left(\mathcal{F}_{t}^{\sigma\left(I_{\infty}\right)}\right)$, and
this completes the proof for $(3)$, and $(2)$ is an immediate
consequence of Doob's maximal identity.
\end{proof}
\begin{rem}
The previous method applies to any transient diffusion
$\left(R_{t}\right)_{t\geq 0}$, with values in
$\left(0,\infty\right)$, and which satisfies:
$$R_{t}=x+B_{t}+\int_{0}^{t}duc\left(R_{u}\right),$$where
$c:\mathbb{R}_{+}\rightarrow\mathbb{R}$ allows uniqueness in law for
this equation. These diffusions were studied in
\cite{saichotanemura} to obtain some extension of Pitman's theorem
(see also \cite{zurich}).
\end{rem}
\subsection{Some examples of $Z_{t}$ with jumps}
We shall conclude this paper by giving some explicit examples of
discontinuous $Z's$. Let $X$ be a Poisson process with parameter $c$
and let $N_{t}=X_{t}-ct$. $N$ is a martingale in the natural
filtration $\left(\mathcal{F}_{t}\right)$ of $X$. Every local
martingale $Y$ in this filtration may be written as:
$$Y_{t}=Y_{0}+\int_{0}^{t}k_{s}dN_{s},$$ where $k$ is an
$\left(\mathcal{F}_{t}\right)$ predictable process. Now, for
$f:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ a locally bounded and
Borel function, let
$$\mathcal{E}_{t}^{f}=\exp\left(-\int_{0}^{t}f\left(s\right)dX_{s}+c\int_{0}^{t}\left(1-\exp\left(-f\left(s\right)\right)\right)ds\right)$$
$\mathcal{E}_{t}^{f}$ is an $\mathcal{F}_{t}$ local martingale which
can be represented as:
$$\mathcal{E}_{t}^{f}=1+\int_{0}^{t}\mathcal{E}_{s-}^{f}\left(\exp\left(-f\left(s\right)\right)-1\right)dN_{s}.$$
If $\int_{0}^{\infty}f\left(s\right)ds=\infty$, then
$\lim_{t\rightarrow\infty}\mathcal{E}_{t}^{f}=0$.
\begin{prop}
Let $f$ be a nonnegative locally bounded and Borel function on
$\mathbf{R}_{+}$, such that
$\lim_{t\rightarrow\infty}\mathcal{E}_{t}^{f}=0$.
Define:$$g=\sup\left\{t:\;\mathcal{E}_{t}^{f}=\overline{\mathcal{E}}_{t}^{f}\right\},$$where$$\overline{\mathcal{E}}_{t}^{f}=
\sup_{s\leq t}\mathcal{E}_{s}^{f}.$$ Then:
\begin{enumerate}
\item $\sup_{s\geq
0}\left(-\int_{0}^{t}f\left(s\right)dX_{s}+c\int_{0}^{t}\left(1-\exp\left(-f\left(s\right)\right)\right)ds\right)$
is distributed as a random variable with the exponential law with
parameter $1$;
\item The supermartingale $Z_{t}^{g}$ associated with $g$ is given
by:
$$ \mathbf{P}\left(g>t\mid\mathcal{F}_{t}\right)=\dfrac{\mathcal{E}_{t}^{f}}{\overline{\mathcal{E}}_{t}^{f}};$$
\item Every $\mathcal{F}_{t}$ local martingale $Y_{t}\left(=\int_{0}^{t}k_{s}dN_{s}\right)$ is a
semimartingale in the filtration
$\mathcal{F}_{t}^{\sigma\left(\overline{\mathcal{E}}_{\infty}^{f}\right)}$,
with canonical
decomposition:$$Y_{t}=\widetilde{Y}_{t}+c\int_{0}^{t\wedge
g}k_{s}\left(\exp\left(-f\left(s\right)\right)-1\right)ds-c\int_{g}^{t}k_{s}\left(\exp\left(-f\left(s\right)\right)-1\right)\dfrac{\mathcal{E}_{s}^{f}}{\overline{\mathcal{E}}_{\infty}^{f}-\mathcal{E}_{s}^{f}}ds,$$
where $\widetilde{Y}_{t}$ is an
$\mathcal{F}_{t}^{\sigma\left(\overline{\mathcal{E}}_{\infty}^{f}\right)}$
local martingale.
\end{enumerate}
\end{prop}
\newpage
|
{
"timestamp": "2007-08-02T23:45:56",
"yymm": "0503",
"arxiv_id": "math/0503386",
"language": "en",
"url": "https://arxiv.org/abs/math/0503386"
}
|
\section{\label{intro}Introduction}
Let $X$ be a finite set. A (symmetric) {\it association scheme} with $d$ classes on $X$ is a partition of
$X\times X$ into sets $R_0$, $R_1, \ldots , R_d$ (called {\it associate classes} or {\it relations}) such that
\begin{enumerate}
\item $R_0=\{(x,x) \mid x\in X\}$ (the diagonal relation),
\item $R_i$ is symmetric for $i=1,2,\ldots ,d$,
\item for all $i,j,k$ in $\{0,1,2,\ldots ,d\}$ there is an integer $p_{ij}^k$ such that, for all $(x,y)\in R_k$,
$$|\{z\in X \mid (x,z)\in R_i\; {\rm and}\; (z,y)\in R_j\}|=p_{ij}^k.$$
\end{enumerate}
We denote such an association scheme by $(X, \{R_i\}_{0\leq i\leq d})$. Elements $x$ and $y$ of $X$ are called {\it $i$-th associates} if $(x,y)\in R_i$. The numbers $p_{ij}^k$, $0\leq k,i,j\leq d$, are called the {\it intersection parameters} of the scheme. That $p_{ii}^0$ exists means that there is a constant number of $i$-th associates of any element of $X$, which is usually denoted by $n_i$. The numbers $n_0,n_1,\ldots ,n_d$ are called the {\it valencies} (or {\it degrees}) of the scheme. We have
\begin{enumerate}
\item $n_0=1$, $n_0+n_1+\cdots +n_d=|X|,$
\item $p_{0j}^k=\delta_{j,k}$ (Kronecker delta), $p_{ij}^0=\delta_{i,j}n_j$,
\item $p_{ij}^k=p_{ji}^k$, $p_{ij}^kn_k=p_{ik}^jn_j$.
\end{enumerate}
For $i\in \{0,1,\ldots ,d\}$, let $A_i$ be the adjacency matrix of the relation $R_i$, that is, the rows and columns of $A_i$ are both indexed by $X$ and
$$(A_i)_{xy}:=\biggm\{
\begin{array}{c} 1 \quad \mbox{ if }\quad (x,y)\in R_i, \\
0 \quad \mbox{ if }\quad (x,y)\notin R_i. \\
\end{array} $$
The matrices $A_i$ are symmetric $(0,1)$-matrices and
$$A_0=I, \; A_0+A_1+\cdots +A_d=J,$$
where $J$ is the all one matrix of size $|X|$ by $|X|$.
By the definition of an association scheme, we have
$$A_iA_j=\sum_{k=0}^d p_{ij}^kA_k $$
for any $i,j\in \{0,1,\ldots ,d\}$. So $A_0,A_1,\cdots , A_d$ form a basis of the commutative algebra generated
by $A_0,A_1,\cdots , A_d$ over the reals (which is called the {\it Bose-Mesner algebra} of the association
scheme). Moreover this algebra has a unique basis $E_0,E_1,\cdots , E_d$ of primitive idempotents; one of the
primitive idempotents is $\frac {1}{|X|}J$. So we may assume that $E_0=\frac {1} {|X|}J$. Let $m_i={\rm
rank}\;E_i$. Then
$$m_0=1,\; m_0+m_1+\cdots +m_d=|X|.$$
The numbers $m_0,m_1,\ldots ,m_d$ are called the {\it multiplicities} of the scheme. Since we have two bases of
the Bose-Mesner algebra, we may consider the transition matrices between them. Define
$P=\left(p_j(i)\right)_{0\le i,j\le d}$ (the {\it first eigenmatrix}) and $Q=\left(q_j(i)\right)_{0\le i,j\le d}$
(the {\it second eigenmatrix}) as the $(d+1)\times (d+1)$ matrices with rows and columns indexed by $0,1,2,\ldots
,d$ such that
$$(A_0,A_1, \ldots ,A_d)=(E_0,E_1, \ldots ,E_d)P,$$
and
$$|X|(E_0,E_1, \ldots ,E_d)=(A_0,A_1, \ldots ,A_d)Q.$$
Of course, we have
$$P=|X|Q^{-1}, \;\; Q=|X|P^{-1}.$$
Note that $\{p_j(i)\ |\ 0\le i\le d\}$ is the set of eigenvalues of $A_j$ and the zeroth row and column of $P$ and $Q$ are as indicated below.
$$P=\left(\matrix{
1&n_1&\cdots&n_d\cr
1\cr
\vdots & & \cr
1
}\right),\;\; Q=\left(\matrix{
1&m_1&\cdots&m_d\cr
1\cr
\vdots & & \cr
1
}\right)$$
Before we proceed further, we give some examples of association schemes.
\begin{examp}\label{tranexamp}
Let $X$ be a finite set and let $G$ be a group acting transitively on $X$. We say that $G$ acts {\em generously transitively} on $X$ if the orbits of the induced action of $G$ on $X\times X$ are all symmetric. (The orbits of $G$ on $X\times X$ are usually called the {\em orbitals} of the action of $G$ on $X$.) It is clear that if $G$ acts generously transitively on $X$, then the orbitals of $G$ on $X$ can be taken as the relations of an association scheme, which will be called the {\em orbital scheme} of $G$ on $X$. The next example arises in this way.
\end{examp}
\begin{examp}\label{cycexamp}
We consider {\em cyclotomic schemes} defined as follows. Let $q$ be a prime power and let $q-1=ef$ with $e>1$. Let $C_0$ be the subgroup of the multiplicative group of ${\bf F}_q$ of index $e$, and let $C_0,C_1,\ldots ,C_{e-1}$ be the cosets of $C_0$. We require $-1\in C_0$. Define $R_0=\{(x,x) : x\in {\bf F}_q\}$, and for $i\in \{1,2,\ldots ,e\}$, define $R_i=\{(x,y)\mid x,y\in {\bf F}_q, x-y\in C_{i-1}\}$. Then $({\bf F}_q, \{R_i\}_{0\leq i\leq e})$ is an $e$-class symmetric association scheme (the $R_i$ are the orbitals of the action of $G$ on ${\bf F}_q$, where $G=\{x\mapsto ax+b\mid a\in C_0, b\in {\bf F}_q\}$). The intersection parameters of the cyclotomic scheme are related to the cyclotomic numbers (\cite[p.~25]{st}). Namely, for $i,j,k\in \{1,2,\ldots ,e\}$, given $(x,y)\in R_k$,
\begin{equation}\label{cycparam}
p_{ij}^k=|\{z\in {\bf F}_q\mid x-z\in C_{i-1}, y-z\in C_{j-1}\}|=|\{z\in C_{i-k}\mid 1+z\in C_{j-k}\}|.
\end{equation}
The first eigenmatrix $P$ of this scheme is the following $(e+1)$ by $(e+1)$ matrix (with the rows of $P$ arranged in a certain way)
$$P=\left(\matrix{
1&f&\cdots&f\cr
1\cr
\vdots & & P_0\cr
1
}\right)$$
with $P_0=\sum_{i=1}^{e}\eta_iC^i$, where $C$ is the $e$ by $e$ matrix:
$$C=\left(\matrix{
& 1\cr & & 1\cr & & & \ddots\cr & & & & 1\cr 1}\right)$$ and $\eta_i=\sum_{\beta\in C_i}\psi(\beta)$, $1\leq
i\leq e$, for a fixed nontrivial additive character $\psi$ of ${\bf F}_q$. See \cite{bandm} for more details.
\end{examp}
Next we introduce the notion of a pseudocyclic association scheme.
\begin{defi}
Let $(X, \{R_i\}_{0\leq i\leq d})$ be an association scheme. We say that $(X, \{R_i\}_{0\leq i\leq d})$ is {\it pseudocyclic} if there exists an integer $t$ such that $m_i=t$ for all $i\in \{1,\cdots , d\}$.
\end{defi}
The following theorem gives combinatorial characterizations for an association scheme to be pseudocyclic.
\begin{teor}\label{pseudocyc}
Let $(X, \{R_i\}_{0\leq i\leq d})$ be an association scheme, and for $x\in X$ and $1\leq i\leq d$, let $R_i(x)=\{y\mid (x,y)\in R_i\}$. Then the following are equivalent.\\
(1). $(X, \{R_i\}_{0\leq i\leq d})$ is pseudocyclic.\\
(2). For some constant $t$, we have $n_j=t$ and $\sum_{k=1}^{d}p_{kj}^k=t-1$, for $1\leq j\leq d$.\\
(3). $(X, {\mathcal B})$ is a $2-(v,t,t-1)$ design, where ${\mathcal B}=\{R_i(x)\mid x\in X, 1\leq i\leq d\}$.
\end{teor}
For a proof of this theorem, we refer the reader to \cite[p.~48]{bcn} and \cite[p.~84]{henkthesis}. Part (2) in the above theorem is very useful. For example, we may use it to prove that the cyclotomic scheme in Example~\ref{cycexamp} is pseudocyclic. The proof goes as follows. First, the nontrivial valencies of the cyclotomic scheme in Example~\ref{cycexamp} are all equal to $f$. Second, by (\ref{cycparam}) and noting that $-1\in C_0$, we have
\begin{eqnarray*}
\sum_{k=1}^ep_{kj}^k&=&\sum_{k=1}^e|\{z\in C_{0}\mid 1+z\in C_{j-k}\}|\\
&=&|C_0|-1=f-1\\
\end{eqnarray*}
Pseudocyclic schemes can be used to construct strongly regular graphs and distance regular graphs of diameter 3
(\cite{bm}, \cite[p.~388]{bcn}). In view of this, it is of interest to construct pseudocyclic association
schemes, as remarked by the authors of \cite{bcn} (see \cite[p.~389]{bcn}). The cyclotomic schemes are examples of
pseudocyclic association schemes on prime-power number of points. Very few examples of pseudocyclic association
schemes on nonprime-power number of points are currently known (see \cite{mathon}, \cite[p.~390]{bcn} and
\cite{henkthesis}). One class of such examples comes from the action of ${\rm PGL}(2,2^m)$ on the set of exterior
lines to a nonsingular conic in ${\rm PG}(2,2^m)$. We will give a quick review of this class of association schemes in
Section 2, and also include a proof of the pseudocyclicity of these association schemes. In \cite{henkthesis}, it
was further conjectured that the orbital scheme of ${\rm P\Gamma L}(2,2^m)$ on the set of exterior lines to a
nonsingular conic in ${\rm PG}(2,2^m)$ is also pseudocyclic if $m$ is an odd prime. We will confirm this conjecture in
Section 3. As a by-product, we obtain a class of Latin square type strong regular graphs on nonprime-power number
of points.
\section{The Elliptic Schemes}
In the rest of this paper, we always assume that $q=2^m$, where $m$ is a positive integer. Let
$${\cal O}=\{(\xi, \xi^2, 1)\mid \xi\in {\bf F}_q\}\cup\{(0,1,0)\}.$$
Then ${\cal O}$ is a nonsingular conic in ${\rm PG}(2,q)$. A line of ${\rm PG}(2,q)$ is called {\it exterior} (resp. {\it
secant}) if it meets ${\cal O}$ in 0 (resp. 2) points. Let ${\cal E}$ (resp. ${\cal H}$) be the set of exterior (resp. secant)
lines to ${\cal O}$. Then $$|{\cal E}|=\frac {q(q-1)} {2},\; \mbox{and}\; |{\cal H}|=\frac {(q+1)q} {2}.$$ The subgroup of
${\rm PGL}(3,q)$ fixing ${\cal O}$ setwise is isomorphic to ${\rm PGL}(2,q)$ (cf. \cite[p.~158]{hirsch}). Hence ${\rm PGL}(2,q)$ acts
on ${\cal E}$ and ${\cal H}$, respectively. Moreover, it is shown in \cite{hxmay2004} that ${\rm PGL}(2,q)$ acts generously
transitively on both ${\cal E}$ and ${\cal H}$. Therefore we obtain two association schemes, one on ${\cal E}$ and the other on
${\cal H}$. The association scheme on ${\cal E}$ will be called the {\it elliptic} scheme, and the association scheme on
${\cal H}$ is called the {\it hyperbolic} scheme.
Since the point $(1,0,0)$ is the nucleus of ${\cal O}$ (i.e., the point at which all tangent lines to ${\cal O}$ meet), we
see that each line in ${\cal E}\cup{\cal H}$ can be written as $(1,x,y)^{\perp}=\{(a_0,a_1,a_2)\in {\bf F}_q^3\mid
a_0+a_1x+a_2y=0\}$ for some $x,y\in{\bf F}_q$. Let ${\rm Tr}: {\bf F}_q\rightarrow {\bf F}_2$ be the trace map. Also for $e\in{\bf F}_2$
we define
$${\bf T}_e=\{x\in{\bf F}_q\mid {\rm Tr}(x)=e\},$$
and ${\bf T}_e^*={\bf T}_e\setminus\{0\}$. Then $(1,x,y)^{\perp}$ is in ${\cal E}$ (resp. ${\cal H}$) if and only if ${\rm Tr}(xy)=1$ (resp. ${\rm Tr}(xy)=0$). Given two lines $\ell=(1,x,y)^{\perp}$ and $m=(1,z,u)^{\perp}$, we define
$${\hat \rho}(\ell,m)=x^2u^2+y^2z^2 +(x+z)(y+u).$$
We remark that the function ${\hat \rho}$ comes from the cross-ratio of four points on a projective line (see
\cite{hxmay2004} for details). The following theorem in \cite{hxmay2004} gives a simple description of the
orbitals of the action of ${\rm PGL}(2,q)$ on ${\cal E}$ by using the function ${\hat \rho}$.
\begin{teor}\label{description}
The orbitals of the action of ${\rm PGL}(2,q)$ on ${\cal E}$ are $\Gamma_0$ (the diagonal class), and $\Gamma_{a}=\{(\ell, m)\mid {\hat \rho}(\ell, m)=a\}$ for all $a\in {\bf T}_0^*$.\end{teor}
There is a similar description of the orbitals of ${\rm PGL}(2,q)$ on ${\cal H}$ in \cite{hxmay2004}. Since we are only concerned with the elliptic scheme in this paper, we omit that description.
The pair $({\cal E}, \{\Gamma_a\})$ is an association scheme on ${\cal E}$ with $\frac {(q-2)}{2}$ classes. The intersection parameters of this scheme are computed in \cite{hxmay2004}. For $a,b,c\in {\bf T}_0^*$, given $(\ell, m)\in\Gamma_c$, we use $p_{a,b}^c$ to denote $|\{k\in {\cal E} \mid (\ell,k)\in \Gamma_a\; {\rm and}\; (k,m)\in \Gamma_c\}|$. We have:
\begin{teor}\label{parameters}
Let $a,b,c\in {\bf T}_0^*$. Then for any $v\in {\bf T}_1$,
\beql{pexp} p^c_{a,b}=\left\{ \begin{array}{ll}
1+2\delta_{{\rm Tr}(ac),1}, \; & \mbox{if $a+b+c=0$;} \\
\sum_\tau |\{z\in{\bf F}_q \mid z^2+z=v+ac/\tau^2\}|, & \mbox{otherwise,}
\end{array}
\right.
\end{equation}
where the last sum is over the two elements $\tau\in {\bf F}_q$ satisfying $\tau^2+\tau=a+b+c$. Also for all $a\in {\bf T}_0^*$, the valency $n_a=q+1$.
\end{teor}
The association scheme $({\cal E}, \{\Gamma_a\})$ is pseudocyclic. This is already known in \cite{henkthesis}. For convenience of the reader, we include a proof here.
\begin{teor}\label{ellipseudo}
The association scheme $({\cal E}, \{\Gamma_a\})$ is pseudocyclic.
\end{teor}
\begin{proof}
By Theorem~\ref{parameters}, the nontrivial valencies of the association scheme $({\cal E}, \{\Gamma_a\})$ are all
equal to $q+1$. By Part (2) of Theorem~\ref{pseudocyc}, it suffices to prove that $\sum_{a\in
{\bf T}_0^*}p_{a,b}^a=q$ for all $b\in {\bf T}_0^*$.
By Theorem~\ref{parameters}, for $a,b\in {\bf T}_0^*$, we have
$$p_{a,b}^a=\sum_{\tau^2+\tau=b}(1-(-1)^{{\rm Tr}(a/\tau)}).$$
Fixing $\tau\in{\bf F}_q\setminus\{0,1\}$ with $\tau^2+\tau=b$, we have
\begin{eqnarray*} \sum_{a\in {\bf T}_0^*} p^a_{a,b} &=& \sum_{a\in {\bf T}_0^*}(1-(-1)^{{\rm Tr}(a/\tau)}+1-(-1)^{{\rm Tr}(a/(\tau+1))}) \\
&=& 2(q/2 -1)- \sum_{a\in {\bf T}_0^*}((-1)^{{\rm Tr}(a/\tau)}+(-1)^{{\rm Tr}(a/(\tau+1))})\\
&=& 2(q/2 -1)-(-1-1)\\
&=& q
\end{eqnarray*}
This completes the proof.
\end{proof}
\section{Pseudocyclic fusion schemes of the elliptic schemes}
As we have seen in the last section, the elliptic scheme $({\cal E},\{\Gamma_a\})$ is pseudocyclic. In this section, we will consider the fusion scheme of $({\cal E},\{\Gamma_a\})$ obtained by merging the classes $\Gamma_a$ via the Frobenius automorphism $x\mapsto x^2$ of ${\bf F}_q$. Specifically, for $a\in {\bf T}_0^*$, define
$$\Delta_a=\cup_{i\in C_a} \Gamma_i,$$
where $C_a:=\{a, a^2, a^4, \ldots, a^{2^{m-1}}\}$. Let ${\cal R}$ be a set of orbit representatives of ${\bf T}_0^*$ under
the action of the Frobenius automorphism. Then $\Delta_0:=\Gamma_0$, and $\Delta_a$, $a\in{\cal R}$ are the orbitals
of ${\rm P\Gamma L}(2,q)$ on ${\cal E}$. Therefore $({\cal E}, \{\Delta_a\})$ is also an association scheme. The (nontrivial)
intersection parameters of this fusion scheme will be denoted by $P^c_{a,b}$, where $a,b,c\in {\cal R}$. We have for
$a,b,c\in{\cal R}$,
\[ P^c_{a,b}=\sum_{e\in C_a} \sum_{f\in C_b} p^g_{e,f},\]
where $g\in C_c$. (This is independent of the choice of $g\in C_c$.)
Now, if $m$ is prime, then each $C_a$, $a\in {\cal R}$, has size $m$, so the nontrivial valencies of the fusion scheme $({\cal E}, \{\Delta_a\})$ are all equal to $m(q+1)$. Hollmann \cite[p.~133]{henkthesis} made the following conjecture.
\begin{con}\label{pseudoconj}
If $m$ is an odd prime, then $({\cal E}, \{\Delta_a\})$ is pseudocyclic.
\end{con}
As far as we know, there is no published proof of this conjecture. There is one sentence on Page 390 of \cite{bcn} stating the above conjecture as a fact. But this was not backed up by a proof.
Note that the nontrivial valencies of $({\cal E}, \{\Delta_a\})$ are all equal to $m(q+1)$ when $m$ is prime. So in
order to prove Conjecture~\ref{pseudoconj}, by Part (2) of Theorem~\ref{pseudocyc}, we need to show that
\beql{original} \sum_{c\in {\cal R}}P_{c,c}^b=m(q+1)-1, \end{equation} for all $b\in{\cal R}$. (Here we implicitly used the fact
that $P_{c,c}^b=P_{c,b}^c$ since all nontrivial valencies are all equal when $m$ is prime.) Simplifying the left
hand side of (\ref{original}), we see that (\ref{original}) is equivalent to \beql{pc1}
\sum_{k=0}^{m-1}\sum_{c\in {\bf T}_0^*} p^b_{c,c^{2^k}}=m(q+1)-1. \end{equation} Now, the $k=0$ term of the left hand side of
(\ref{pc1}) is equal to $q$ as seen in the proof of Theorem~\ref{ellipseudo}. So in order to prove (\ref{pc1}),
we have to show that \beql{pc2}\sum_{k=1}^{m-1} \sum_{c\in {\bf T}_0^*} p^b_{c,c^{2^k}}=(m-1)(q+1), \end{equation} for all
$b\in{\bf T}_0^*$.
We will prove a stronger result:
\begin{teor}\label{strong}
Let $m$ be an odd integer, and let $k$ be any integer in $\{1,2,\ldots ,m-1\}$ satisfying $\gcd(k,m)=1$. Write $\sigma=2^k$. Then
\beql{pc0} \sum_{c\in {\bf T}_0^*} p^b_{c,c^\sigma} =q+1, \end{equation}
for all $b\in{\bf T}_0^*$.
\end{teor}
The most important ingredient in our proof of Theorem~\ref{strong} is a family of polynomials $H_{\alpha, \gamma}(X)$ introduced in \cite{hxpermpoly}. In fact we discovered these polynomials while working on a proof of Theorem~\ref{strong}. We now define the polynomials $H_{\alpha, \gamma}(X)$ and quote the main theorem from \cite{hxpermpoly}.
Let $m\geq1$ be an integer, let $k$ be any integer in $\{1,\ldots,
m-1\}$ with $\gcd(k,m)=1$, and let $r\in\{1,\ldots, m-1\}$ be such
that $kr\equiv 1$ (mod $m$). Write $\sigma=2^k$ and use ${\rm Tr}(X)$ to denote the following polynomial in ${\bf F}_2[X]$.
$${\rm Tr}(X):=X+X^2+\cdots+X^{2^{m-1}}.$$
For $\alpha, \gamma$ in $\{0,1\}$, we define the polynomial
\[H_{\alpha,\gamma}(X):= \gamma {\rm Tr}(X) + \frac{\left(\alpha {\rm Tr}(X) + \sum_{i=0}^{r-1} X^{\sigma^i}\right)^{\sigma+1}} {X^2}.\]
(Note that $H_{\alpha,\gamma}(X)$ is indeed a polynomial in $X$ with coefficients in ${\bf F}_2$ and $H_{\alpha,\gamma}(0)=0$. Also see \cite{hxpermpoly} for connections between $H_{\alpha,\gamma}(X)$ and the Dickson polynomials.)
The following is the main theorem from \cite{hxpermpoly}.
\begin{teor}\label{mainthm} Let $m, k$ be positive integers with $\gcd(k,m)=1$,
let $r\in\{1,\ldots, m-1\}$ be such that $kr\equiv 1 \;(\bmod
\;m)$, and let $\alpha, \gamma\in \{0,1\}$. Then the mapping $H_{\alpha, \gamma}: x\mapsto H_{\alpha,\gamma}(x)$, $x\in{\bf F}_q$, maps ${\bf T}_0$ bijectively to ${\bf T}_0$, and maps ${\bf T}_1$ bijectively to ${\bf T}_{r+(\alpha+\gamma)m}$. In particular, the polynomial $H_{\alpha,\gamma}(X)$ is a permutation polynomial of ${\bf F}_q$ if and only if $r+(\alpha+\gamma)m \equiv 1$ {\em (mod 2)}.
\end{teor}
We are now ready to give the proof of Theorem~\ref{strong}.
\vspace{0.1in}
\noindent{\bf Proof of Theorem~\ref{strong}:} Recall that from Theorem~\ref{parameters}, for $b,c\in {\bf T}_0^*$,
\begin{eqnarray*}
p^b_{c,c^\sigma}=\left\{ \begin{array}{ll}
1+2\delta_{{\rm Tr}(bc),1}, \; & \mbox{if $c^\sigma+c+b=0$;} \\
\sum_{\tau^2+\tau=c^\sigma+c+b} |\{z\in{\bf F}_q \mid z^2+z=v+bc/\tau^2\}|, & \mbox{if $c^\sigma+c+b\neq 0$,}
\end{array}
\right.
\end{eqnarray*}
where $v$ is any element with ${\rm Tr}(v)=1$. Since $b\in{\bf T}_0^*$ and $m$ is odd, we can find a unique $c_0\in {\bf T}_0^*$ such that $c_0^\sigma +c_0=b$. So
\begin{eqnarray*}
\sum_{c\in {\bf T}_0^*} p^b_{c,c^\sigma} &=& 1+2\delta_{{\rm Tr}(bc_0),1}+2\sum_{c\in {\bf T}_0^*,\; c^\sigma +c+b\neq 0}\sum_{\tau^2+\tau=c^\sigma+c+b}\delta_{{\rm Tr}(bc/\tau^2), 1}\\
&=&1+2|\{(c,\tau)\in {\bf F}_q^*\times {\bf F}_q^*\mid \tau^2+\tau=c^\sigma+c+b, {\rm Tr}(c)=0, {\rm Tr}(bc/\tau^2)=1\}|.
\end{eqnarray*}
For convenience, we define
$$N_{k}(b):=|\{(c,\tau)\in {\bf F}_q^*\times {\bf F}_q^*\mid \tau^2+\tau=c^\sigma+c+b, {\rm Tr}(c)=0, {\rm Tr}(bc/\tau^2)=1\}|.$$
Our goal is to prove that $N_{k}(b)=q/2$ for all $b\in{\bf T}^*_0$.
For later use, we define the polynomial
\[ f(X):= \sum_{i=0}^{r-1} X^{\sigma^i}\in {\bf F}_2[X], \]
where $r$ is an integer satisfying $kr\equiv 1$ (mod $m$).
Since $b\in {\bf T}_0^*$ and $m$ is odd, we can write $b=\beta +\beta^2$ with $\beta\in{\bf T}_0^*$. Then the equation
$\tau^2+\tau=c^\sigma +c+b$ involved in the definition of $N_{k}(b)$ becomes \beql{newequ} c^\sigma +c=(\beta
+\tau)+(\beta +\tau)^2. \end{equation} Noting that $m$ is odd, we see that for any $\tau\in{\bf F}_q$, there is a unique
solution $c\in{\bf T}_0$ of (\ref{newequ}), namely
\[c=f(\tau +\beta)+r{\rm Tr}(\tau +\beta)=f(\tau +\beta)+r{\rm Tr}(\tau),\]
where in the last equality we used the fact that $\beta\in {\bf T}_0$. Therefore we have
\begin{eqnarray*}
N_{k}(b)=\left\{ \begin{array}{ll}
|\{\tau\in{\bf F}_q^* : \frac{b (f(\tau +\beta)+{\rm Tr}(\tau))}{\tau^2}\in {\bf T}_1\}|, & \mbox{if $r$ is odd;} \\
|\{\tau\in{\bf F}_q^* : \frac{b f(\tau +\beta)}{\tau^2}\in {\bf T}_1\}| , & \mbox{if $r$ is even.}
\end{array}
\right.
\end{eqnarray*}
We will consider the $r$ odd case and the $r$ even case separately.
\noindent{\bf Case 1}. $r$ is odd. Let $x=b/\tau^2$, where $b=\beta +\beta^2\in {\bf T}_0^*$ and $\tau\in{\bf F}_q^*$. Then
\begin{eqnarray*}
{\rm Tr}\left(\frac{b (f(\tau +\beta)+{\rm Tr}(\tau))}{\tau^2}\right)&=&{\rm Tr}\left(x\sum_{i=0}^{r-1}(\beta+\sqrt{b/x})^{\sigma^i}+x{\rm Tr}(b/x)\right) \\
&=&{\rm Tr}\left(\sum_{i=0}^{r-1} x^2(\beta^2+b/x)^{\sigma^i}\right)+{\rm Tr}(x){\rm Tr}(b/x) \\
&=&{\rm Tr}\left(\sum_{i=0}^{r-1} x^{\sigma^{r-i}}(\beta^2+b/x)\right)+{\rm Tr}(x){\rm Tr}(b/x)\\
&=&{\rm Tr}\left((\beta^2+b/x)(f(x)+x^2+x)\right)+{\rm Tr}(x){\rm Tr}(b/x) \\
&=&{\rm Tr}\left(\beta^2(f(x)+\frac {f(x)}{x}+\frac{f(x)^2}{x^2})\right)+{\rm Tr}(x){\rm Tr}\left(\frac{b}{x}\right),
\end{eqnarray*}
where in the last step, we used $b=\beta+\beta^2$. Now noting that for $x\in {\bf F}_q^*$,
$$H_{0,0}(x)=f(x)+\frac {f(x)}{x}+\frac{f(x)^2}{x^2}.$$
(One can prove this directly, or see Lemma 3.1 in \cite{hxpermpoly}.) Therefore, in this case, we have
\beql{twoterms}N_{k}(b)=|\{x\in{\bf T}_0^*\mid \beta^2 H_{0,0}(x)\in {\bf T}_1\}|+|\{x\in{\bf T}_1\mid \beta^2 H_{0,0}(x)+b/x\in {\bf T}_1\}|.\end{equation}
For the first summand in (\ref{twoterms}), noting that $H_{0,0}(0)=0$ and $H_{0,0}$ maps ${\bf T}_0$ to ${\bf T}_0$ bijectively (Theorem~\ref{mainthm}), we have
\begin{eqnarray*}
|\{x\in{\bf T}_0^*\mid \beta^2 H_{0,0}(x)\in {\bf T}_1\}| &=& |\beta^2 {\bf T}_0^*\cap{\bf T}_1|\\
&=& (q/2 -1)-|\beta^2{\bf T}_0^*\cap {\bf T}_0^*|.
\end{eqnarray*}
Since ${\bf T}_0^*$ is a $(q-1,q/2 -1,q/4 -1)$ (Singer) difference set in the cyclic group ${\bf F}_q^*$, and $\beta\neq 0, 1$, we see that $|\beta^2{\bf T}_0^*\cap {\bf T}_0^*|=q/4 -1$. Hence
\[|\{x\in{\bf T}_0^*\mid \beta^2 H_{0,0}(x)\in {\bf T}_1\}|=(q/2-1)-(q/4 -1)=q/4.\]
For the second summand in (\ref{twoterms}), using $b=\beta+\beta^2$, we see that
\[{\rm Tr}(\beta^2 H_{0,0}(x)+ b/x)={\rm Tr}(\beta^2(H_{0,0}(x)+1/x +1/x^2)).\]
For any $x\in{\bf T}_1$, we have
\begin{eqnarray*}
H_{1,0}(x) &=& \frac {(1+ f(x))^{\sigma +1}} {x^2} \nonumber\\
&=& 1+f(x)+ (1+f(x))/x +(1 +f(x))^2/x^2 \nonumber \\
&=& 1+ 1/x +1/x^2 + H_{0,0}(x).
\end{eqnarray*}
Also by Theorem~\ref{mainthm}, $H_{1,0}$ maps ${\bf T}_1$ bijectively to ${\bf T}_{r+m}={\bf T}_0$. Hence
\begin{eqnarray*}
|\{x\in{\bf T}_1\mid \beta^2 H_{0,0}(x)+b/x\in {\bf T}_1\}|&=&|\{x\in{\bf T}_1\mid \beta^2 (H_{0,0}(x)+1/x+1/x^2)\in {\bf T}_1\}|\nonumber\\
&=&|\beta^2 {\bf T}_1\cap {\bf T}_1|\nonumber\\
&=&q/4.
\end{eqnarray*}
Therefore we have $N_{k}(b)=q/4 +q/4=q/2.$
\noindent{\bf Case 2.} $r$ is even. This case is similar to Case 1 and actually easier. Let $x=b/\tau^2$. By the
same computations as those in the $r$ odd case, we find that
\[{\rm Tr}(\frac{b f(\tau +\beta)}{\tau^2})={\rm Tr}\left(\beta^2H_{0,0}(x)\right).\]
By Theorem~\ref{mainthm}, $H_{0,0}$ maps ${\bf T}_0^*$ bijectively to ${\bf T}_0^*$, and maps ${\bf T}_1$ bijectively to ${\bf T}_r={\bf T}_0$. Therefore,
\begin{eqnarray*}
|\{\tau\in{\bf F}_q^*\mid \frac {b f(\tau +\beta)}{\tau^2}\in {\bf T}_1\}|&=&|\{x\in {\bf F}_q^*\mid \beta^2 H_{0,0}(x)\in{\bf T}_1\}|\nonumber\\
&=&|\{x\in {\bf T}_0^*\mid \beta^2 H_{0,0}(x)\in{\bf T}_1\}|+|\{x\in {\bf T}_1\mid \beta^2 H_{0,0}(x)\in{\bf T}_1\}| \nonumber\\
&=&|\beta^2{\bf T}_0^*\cap {\bf T}_1|+|\beta^2{\bf T}_0\cap {\bf T}_1|\nonumber\\
&=&2|\beta^2{\bf T}_0^*\cap {\bf T}_1|\nonumber\\
&=&q/2.
\end{eqnarray*}
In summary, in both cases, we have shown that $N_{k}(b)=q/2$ for all $b\in{\bf T}_0^*$. The proof is complete.
\begin{remark} More general results can be proved in the same fashion as above. Let $e,f\in{\bf F}_2$. Define
$$N_{k,e,f}(b):=|\{(c,\tau)\in {\bf F}_q^*\times {\bf F}_q^*\mid \tau^2+\tau=c^\sigma+c+b, {\rm Tr}(c)=e, {\rm Tr}(bc/\tau^2)=f\}|.$$
Then using the same arguments as those in the proof of Theorem~\ref{strong}, we find that $N_{k,0,0}(b)=q/2 -3$, $N_{k,1,0}(b)=q/2 -1$, and $N_{k,1,1}(b)=q/2$, for all $b\in {\bf T}_0^*$.
\end{remark}
Now we can finish the proof of Conjecture~\ref{pseudoconj}.
\begin{teor}
If $m$ is an odd prime, then $({\cal E}, \{\Delta_a\})$ is pseudocyclic.
\end{teor}
\begin{proof} Since $m$ is prime, the nontrivial valencies of the scheme are all equal to $m(q+1)$. To finish the proof, we need to prove (\ref{original}) for all $b\in {\cal R}$. As we have seen in the analysis before the statement of Theorem~\ref{strong}, (\ref{original}) is equivalent to (\ref{pc2}). Since $m$ is an odd prime, any integer $k\in\{1,2,\ldots ,m-1\}$ is relatively prime to $m$. So we can apply Theorem~\ref{strong} to obtain
\[\sum_{c\in {\bf T}_0^*} p^b_{c,c^\sigma} =q+1,\]
for all $b\in{\bf T}_0^*$. Now (\ref{pc2}) follows. This completes the proof.
\end{proof}
\section{Latin square type strongly regular graphs}
A {\em strongly regular graph srg} $(v,k,\lambda,\mu)$ is a graph with $v$ vertices that is regular of valency $k$ and that has the following properties:
\begin{enumerate}
\item For any two adjacent vertices $x,y$, there are exactly $\lambda$ vertices adjacent to both $x$ and $y$.
\item For any two nonadjacent vertices $x,y$, there are exactly $\mu$ vertices adjacent to both $x$ and $y$.
\end{enumerate}
It is well known \cite[p.~407]{vanlint} that strongly regular graphs are equivalent to two-class association schemes. An srg $(v,k,\lambda,\mu)$ is said to be of {\it Latin square type} if
$$(v,k,\lambda, \mu)=(n^2, t(n-1), n+t^2-3t, t^2-t),$$
where $1\leq t\leq n+1$. Any Latin square of order $n$ gives rise to a Latin square type srg (actually called Latin square graph in this case) with parameters $(n^2, 3(n-1), n-2, 6)$ (see \cite[p.~273]{vanlint}). Many examples of Latin square type srg on prime-power number of points are known \cite{ma}. In contrast, not too many examples of Latin square type srg on nonprime-power number of points are known.
In \cite{bm}, it was shown that pseudocyclic association schemes can give rise to Latin square type srg. We quote the following theorem from \cite{bm}. A proof can be found in \cite{tf}.
\begin{teor}\label{srg}
Let $(X, \{R_i\}_{0\leq i\leq d})$ be a pseudocyclic association scheme on $dt+1$ points. Then the graph $G$ whose vertex set is $X\times X$, and where two distinct vertices $(x,y)$ and $(x',y')$ are adjacent if and only if $(x,x')\in R_i$ and $(y,y')\in R_i$ for some $i\neq 0$, is a Latin square type srg with parameters
$$(|X|^2, t(|X|-1), |X|+t^2-3t, t^2-t).$$
\end{teor}
Using Theorem~\ref{srg}, one can obtain Latin square type srg from the pseudocyclic association scheme $({\cal E}, \{\Gamma_a\})$ (the elliptic scheme). These srg have parameters
$$(\frac{1}{2}q^2(q-1)^2, \frac{1}{2}(q-2)(q+1)^2, \frac{1}{2}(3q^2-3q-4), q(q+1)).$$
We note that the Latin square type srg arising from $({\cal E}, \{\Gamma_a\})$ were mentioned in \cite{tf}, in which another construction of these srg was given.
Now since we have shown that the fusion scheme $({\cal E}, \{\Delta_a\})$ of the elliptic scheme $({\cal E}, \{\Gamma_a\})$ is also pseudocyclic when $m$ is an odd prime. We obtain more Latin square type srg via Theorem~\ref{srg}.
\begin{teor}
Let $q=2^m$, where $m$ is an odd prime. Then there exists a Latin square type srg with parameters
$$(\frac{1}{2}q^2(q-1)^2, m(q+1)(\frac{q(q-1)}{2}-1), \lambda, \mu),$$
where $\lambda=\frac{q(q-1)}{2} + m^2(q+1)^2 -3m(q+1)$ and $\mu=m^2(q+1)^2-m(q+1)$.
\end{teor}
\begin{proof} Straightforward.\end{proof}
\noindent{\bf Acknowledgements:} The second author thanks Philips Research Eindhoven, the Netherlands, where part
of this work was carried out. The research of the second author is supported in part by NSF grant DMS 0400411.
|
{
"timestamp": "2005-03-24T23:34:25",
"yymm": "0503",
"arxiv_id": "math/0503570",
"language": "en",
"url": "https://arxiv.org/abs/math/0503570"
}
|
\section{Introduction}
\label{intro}
Donald Coxeter's work on regular polytopes and groups of reflexions is often viewed as his most important contribution. At its heart lies a dialogue between geometry and algebra which was so characteristic for his mathematics (see, for example, \cite{c_rp,c_rcp,cm}). This paper is yet more evidence for his lasting influence on generations of geometers.
In \cite{ms_rpo} (see also \cite[Sections~7E, 7F]{arp}), we classified completely all the faithfully realized regular polytopes and discrete regular apeirotopes in dimensions up to three. Further, in \cite{m_rpfr}, the first author classified the regular polytopes and apeirotopes of maximal rank in each higher dimension, and showed that chiral polytopes could not have full rank. Last, in \cite{s_cp1,s_cp2}, the second author has found all the chiral apeirohedra in three dimensions.
The present paper surveys the developments on realizations of regular or chiral polytopes, which have occurred since the publication of our book~\cite{arp}. There are two quite different ways to approach realizations. The first, for which a fairly complete theory exists (at least, in the finite case), asks for a description of the space of all realizations (a kind of ``moduli space") of a given abstract regular polytope or apeirotope, with rank playing only a minor r\^ole (see \cite[Sections~5B, 5C]{arp} for further details). The second, about which much less is known in general terms, asks for a classification of the realizations of all these polytopes and apeirotopes in a euclidean space of given dimension (in this case, it is usual to impose conditions such as faithfulness and discreteness). This problem is solved in three dimensions. The finite regular polyhedra have long been known; adding to the Petrie-Coxeter apeirohedra of \cite{c_rsp}, Gr\"unbaum \cite{g_on} found all but one of the remaining regular apeirohedra, while Dress \cite{d1,d2} found the missing example, and proved that the classification was then complete. We refer the reader to \cite{ms_rpo} for a quick method of arriving at the full characterization, including a discussion of the geometry of the regular apeirohedra and presentations of their symmetry groups, as well as for the enumeration of the regular $4$-apeirotopes in three dimensions.
In four dimensions, the currently open problems are those of classifying the finite regular polyhedra, and the regular apeirohedra and $4$-apeirotopes; \cite{m_rpfr} solves the problems of the regular $4$-polytopes and $5$-apeirotopes. The paper \cite{m_fdrp} in preparation actually settles the first of these problems (the polytopes with planar faces were classified in \cite{abm,bracho}); however, the other two, together with the corresponding classification problems for chiral polytopes, are still open, although some progress has been made on them.
\section{Regular and chiral polytopes}
\label{regchirpol}
For the general background on abstract regular polytopes, we refer the reader to the recently published monograph \cite{arp}; for the most part, we shall not cite original papers directly. In this paper, we largely concentrate on the geometric aspects of the theory, that is, on realizations of regular polytopes.
However, we begin with the more combinatorial picture. An \emph{abstract polytope} of \emph{rank} $n$, or simply an (\emph{abstract}) \emph{$n$-polytope}, is a partially ordered set $\mathcal{P}$ with a strictly monotone rank function, taking values in $\{-1,0,\ldots,n\}$. The elements of rank $j$ are the \emph{$j$-faces} of $\mathcal{P}$, or \emph{vertices}, \emph{edges} and \emph{facets} of $\mathcal{P}$ if $j = 0$, $1$ or $n-1$, respectively. The maximal chains are the \emph{flags} of $\mathcal{P}$ and contain exactly $n + 2$ faces, including a unique minimal face and a unique maximal face (usually omitted from the notation). Two flags are called \emph{adjacent} if they differ by one element; then $\mathcal{P}$ is \emph{strongly flag-connected}, meaning that, if $\mathnormal{\Phi}$ and $\mathnormal{\Psi}$ are two flags, then they can be joined by a sequence of successively adjacent flags $\mathnormal{\Phi} = \mathnormal{\Phi}_0,\mathnormal{\Phi}_1,\ldots,\mathnormal{\Phi}_k = \mathnormal{\Psi}$, each of which contains $\mathnormal{\Phi} \cap \mathnormal{\Psi}$. Finally, if $F$ and $G$ are a $(j-1)$-face and a $(j+1)$-face with $F < G$, then there are exactly \emph{two} $j$-faces $H$ such that $F < H < G$. An $n$-polytope $\mathcal{P}$ is then called \emph{regular} if its combinatorial automorphism group $\mathnormal{\Gamma}(\mathcal{P})$ (preserving the partial ordering) is (simply) transitive on its flags; in this case, if $\mathnormal{\Phi}$ is a (fixed) \emph{base flag} and, for $j = 0,\ldots,n-1$, $\rho_j$ is the automorphism which maps $\mathnormal{\Phi}$ to the adjacent flag $\mathnormal{\Phi}^{j}$ with a different $j$-face, then $\mathnormal{\Gamma}(\mathcal{P})$ is generated by $\rho_{0},\ldots,\rho_{n-1}$.
We can adopt (see \cite[Theorem~2E11]{arp}) the viewpoint that an abstract regular polytope is to be identified with its group. The latter is precisely what is called a \emph{string C-group}; here, the ``C" stands for ``Coxeter", though not every C-group is a Coxeter group. A string C-group $\mathnormal{\Gamma}$ is a group generated by $n$ involutions $\rho_j$ (the \emph{distinguished generators}) with $j \in \mathsf{N} := \{0,\ldots,n-1\}$, such that $\rho_j$ and $\rho_k$ commute if $0 \leq j \leq k - 2 \leq n-3$, and
\beql{intprop}
\scl{\rho_i \mid i \in \mathsf{J}} \cap \scl{\rho_i \mid i \in \mathsf{K}} = \scl{\rho_i \mid i \in \mathsf{J} \cap \mathsf{K}}
\end{equation}
for each $\mathsf{J},\mathsf{K} \subseteq \mathsf{N}$; the last is the \emph{intersection property}. Each string C-group $\mathnormal{\Gamma}$ then determines (uniquely) a regular $n$-polytope $\mathcal{P}$ with $\mathnormal{\Gamma} = \mathnormal{\Gamma}(\mathcal{P})$. The \emph{$j$-faces} of $\mathcal{P}$ are the right cosets $\mathnormal{\Gamma}_j\sigma$ of the \emph{distinguished subgroup}
\[ \mathnormal{\Gamma}_j := \scl{\rho_i \mid i \neq j} \]
for each $j \in \mathsf{N}$, and two faces are incident just when they intersect (as cosets). In fact, incidence actually induces an order relation:
\[ \mathnormal{\Gamma}_j\sigma \leq \mathnormal{\Gamma}_k\tau \iff \mathnormal{\Gamma}_j\sigma \cap \mathnormal{\Gamma}_k\tau \neq \emptyset \mbox{ and } j \leq k. \]
Formally, we also adjoin two copies of $\mathnormal{\Gamma}$ itself, as the (unique) $(-1)$- and $n$-faces of $\mathcal{P}$. The maximal chains (with respect to this ordering) are the \emph{flags} of $\mathcal{P}$; the group $\mathnormal{\Gamma}$ is then simply transitive on the flags of $\mathcal{P}$. In particular, for $j = 0,\ldots,n-1$ the distinguished generator $\rho_j$ of $\mathnormal{\Gamma}$ takes the \emph{base flag} $\mathnormal{\Phi} := \{\mathnormal{\Gamma}_{-1},\mathnormal{\Gamma}_0,\mathnormal{\Gamma}_1,\ldots,\mathnormal{\Gamma}_{n-1},\mathnormal{\Gamma}_n\}$ into the adjacent flag $\mathnormal{\Phi}^j$ which differs from it in $\mathnormal{\Gamma}_j$. Note that the distinguished subgroups $\mathnormal{\Gamma}_{n-1} = \scl{\rho_0,\ldots,\rho_{n-2}}$ and $\mathnormal{\Gamma}_0 = \scl{\rho_1,\ldots,\rho_{n-1}}$ are themselves string C-groups; the corresponding polytopes are the \emph{facet} and \emph{vertex-figure} of $\mathcal{P}$, respectively (the latter consists of the faces of $\mathcal{P}$ with vertex $\mathnormal{\Gamma}_{0}$). As we said earlier, \cite[Theorem~2E11]{arp} shows that this description of a regular polytope $\mathcal{P}$ in terms of (its C-group) $\mathnormal{\Gamma}(\mathcal{P})$ and the previous one in terms of the face poset are equivalent.
The distinguished generators of $\mathnormal{\Gamma} = \mathnormal{\Gamma}(\mathcal{P})$ satisfy relations
\begin{equation}
\label{cgp}
(\rho_{i}\rho_{j})^{p_{ij}} = \varepsilon \quad (i,j = 0,\ldots,n-1),
\end{equation}
with $p_{ii} = 1$, $p_{ij} = p_{ji} \geq 2$ if $i \neq j$, and $p_{ij} = 2$ if $|i-j| \geq 2$ (hence the term ``string" C-group). The numbers $p_{j} := p_{j-1,j}$ ($j = 1,\ldots,n-1$) determine the \emph{Schl\"afli type} $\{p_{1},\ldots,p_{n-1}\}$ of $\mathcal{P}$. To avoid cases which, in our context, turn out to be trivial, we always assume that adjacent generators $\rho_{j-1}$ and $\rho_j$ of $\mathnormal{\Gamma}$ do not commute (this is justified in \sectref{real}); in other words, $p_j > 2$ (possibly, $p_j = \infty$). If the polytope is determined just by the $p_j$, then we have the \emph{universal} regular polytope (of that Schl\"afli type), for which we use the same symbol $\{p_1,\ldots,p_{n-1}\}$ (but without qualification); we write $[p_1,\ldots,p_{n-1}]$ for the corresponding \emph{Coxeter} group. Generally, however, the group $\mathnormal{\Gamma}$ will satisfy additional relations as well, for some of which we introduce special notation later.
The underlying face-set of a polytope $\mathcal{P}$ can be finite or infinite. An infinite $n$-polytope is also called an (\emph{abstract}) \emph{$n$-apeirotope}; when $n = 2$, we also refer to it as an \emph{apeirogon}, and when $n = 3$ as an \emph{apeirohedron}.
A central question in the abstract theory is that of the amalgamation of polytopes of lower rank. If a regular $(n+1)$-polytope has facets (of type) the $n$-polytope $\mathcal{P}$ and vertex-figures the $n$-polytope $\mathcal{Q}$, then the facets of $\mathcal{Q}$ must be isomorphic to the vertex-figures of $\mathcal{P}$. Conversely, if $\mathcal{P}$ and $\mathcal{Q}$ satisfy this latter criterion, then we write $\scl{\mathcal{P},\mathcal{Q}}$ for the class of all regular $(n+1)$-polytopes with facet $\mathcal{P}$ and vertex-figure $\mathcal{Q}$. The question has two parts. First, is $\scl{\mathcal{P},\mathcal{Q}} \neq \emptyset$; in other words, does there exist any such regular $(n+1)$-polytope at all? If so, then there is a \emph{universal} member $\{\mathcal{P},\mathcal{Q}\}$ in the family $\scl{\mathcal{P},\mathcal{Q}}$, of which every other one is a quotient (in the sense that its group is an appropriate quotient). Second, given that it exists, we ask what $\{\mathcal{P},\mathcal{Q}\}$ is. (See \cite[Section~4B]{arp} for further details.) In the present context, we often pose this question in the form: is a given regular polytope, whose facet and vertex-figure are known, actually universal of its kind?
There are several general techniques for constructing new regular polytopes from old ones. In particular, two different regular polytopes may be related by what is called a \emph{mixing operation}; the distinguished generators of the second group are certain products of those of the first (see \cite[Chapter~7]{arp}). Apart from the \emph{duality} operation $\delta$, which just reverses the order of the distinguished generators (and the order relation on the faces), there are two others we mention here; one further operation (for chiral polyhedra) will occur in Section~\ref{chirpol3}. Let $\mathnormal{\Gamma} = \scl{\rho_i \mid i \in \mathsf{N}}$ be a string C-group, let $j \neq k$, and consider the operation
\[ (\rho_0,\ldots,\rho_{n-1}) \mapsto (\rho_0,\ldots,\rho_{j-1},\rho_j\rho_k,\rho_{j+1},\ldots,\rho_{n-1}) =: (\sigma_0,\ldots,\sigma_{n-1}). \]
Since adjacent generators of $\mathnormal{\Gamma}$ do not commute, we easily see that the group $\mathnormal{\Delta} := \scl{\sigma_0,\ldots,\sigma_{n-1}}$ cannot possibly be a string C-group unless $(j,k) = (2,0)$ or $(n-3,n-1)$. The former will rule itself out later for geometric reasons (see Section~\ref{real}); the latter, namely,
\beql{petrie}
\pi\colon\ (\rho_0,\ldots,\rho_{n-1}) \mapsto (\rho_0,\ldots,\rho_{n-4},\rho_{n-3}\rho_{n-1},\rho_{n-2},\rho_{n-1}) =: (\sigma_0,\ldots,\sigma_{n-1}),
\end{equation}
which we denote by $\mathnormal{\Gamma} \mapsto \mathnormal{\Gamma}^\pi$, is called the \emph{Petrie operation}, since it generalizes the operation with the same name when $n = 3$. Even when $n = 3$, the Petrie operation $\pi$ does not always yield a C-group (though such cases are rather exceptional), but, for higher rank, each application has to be checked directly. However, if in fact $\mathnormal{\Gamma}^\pi$ is a C-group, then we write $\mathcal{P} \mapsto \mathcal{P}^\pi$ to indicate the effect of the operation on the corresponding polytope $\mathcal{P}$; the new polytope $\mathcal{P}^\pi$ is called the \emph{Petrial} of $\mathcal{P}$. One general case (see \cite{m_rpfr}) can be settled easily.
\bpropl{nonpetrie}
If $\mathnormal{\Gamma} = \scl{\rho_0,\ldots,\rho_{n-1}}$ is a string C-group with $n \geq 4$ for which $p_{n-3}$ is odd, then the Petrial $\mathnormal{\Gamma}^{\pi}$ is not a C-group.
\end{proposition}
Mixing operations are particularly powerful when applied to regular polyhedra or apeirohedra $\mathcal{P}$. For example, the Petrial $\mathcal{P}^{\pi}$ can be obtained from $\mathcal{P}$ by replacing the $2$-faces by the \emph{Petrie polygons} of $\mathcal{P}$ (while keeping the vertices and edges); the geometric picture of a Petrie polygon here is one which shares two successive edges of each $2$-face which it meets, but not a third. An important class of regular polyhedra or apeirohedra consists of those which are completely determined by their Schl\"afli type and the length of their Petrie polygons. We write $\{p,q\}_r$ for the polyhedron (possibly infinite) of Schl\"afli type $\{p,q\}$, whose Petrie polygons of length $r$ determine it. Its group is the Coxeter group $\scl{\rho_0,\rho_1,\rho_2} = [p,q]$, with the imposition of the single extra relation
\beql{petriepol}
(\rho_0\rho_1\rho_2)^r = \varepsilon.
\end{equation}
We note that, if it is a genuine polyhedron, then the Petrial of $\{p,q\}_r$ is $\{r,q\}_p$.
In the context of polyhedra, another operation is also of great importance. The (\emph{second})
\emph{facetting operation} $\varphi_{2}$ is given by
\beql{facett}
\varphi_{2}\colon\ (\rho_{0},\rho_{1},\rho_{2}) \mapsto (\rho_{0},\rho_{1}\rho_{2}\rho_{1},\rho_{2}),
\end{equation}
and replaces the $2$-faces of a polyhedron $\mathcal{P}$ by the \emph{holes} (while keeping the vertices and edges); a hole of $\mathcal{P}$ is an edge-circuit which exits from the \emph{second} edge (in some local orientation) emanating from a vertex from the edge by which it entered. The designation of a (possibly infinite) regular polyhedron of Schl\"afli type $\{p,q\}$, which is determined by its holes of length $h$, is
$\{p,q {\mkern2mu|\mkern2mu} h\}$. The corresponding relation to be imposed on the Coxeter group $\scl{\rho_0,\rho_1,\rho_2} = [p,q]$ is
\beql{holepol}
(\rho_0\rho_1\rho_2\rho_1)^h = \varepsilon.
\end{equation}
Various examples of such polyhedra occur later; for now, let us observe that the three Petrie-Coxeter apeirohedra are, as abstract regular polyhedra, $\{4,6 {\mkern2mu|\mkern2mu} 4\}$, $\{6,4 {\mkern2mu|\mkern2mu} 4\}$ and $\{6,6 {\mkern2mu|\mkern2mu} 3\}$.
In \cite[Section 7A]{arp} we also introduced the notion of a mix of two regular polytopes (or corresponding C-groups). The following abstract construction is a special case of this mix and occurs when one polytope is $1$-dimensional, that is, a segment. Again, suppose that $\mathnormal{\Gamma} = \scl{\rho_i \mid i \in \mathsf{N}}$ is a string C-group. Let $\tau$ be an involution which commutes with all $\rho_j$, and consider the operation
\beql{mixseg}
(\rho_0,\ldots,\rho_{n-1},\tau) \mapsto (\rho_0\tau,\rho_1\ldots,\rho_{n-1}) =: (\sigma_0,\ldots,\sigma_{n-1}).
\end{equation}
This is called \emph{mixing with a segment}, because $\tau$ can be regarded as the generating involution of the group of the segment $\{\mkern4mu\}$ (see \sectref{real} for the notation). We have (see \cite[Theorem~7A8]{arp})
\bthml{mixsegpol}
Mixing a string C-group $\mathnormal{\Gamma}$ with the group of a segment always yields another C-group. This is isomorphic to $\mathnormal{\Gamma}$ if all edge-circuits in the associated regular polytope $\mathcal{P}$ have even length; otherwise, it is isomorphic to the direct product $\mathnormal{\Gamma} \times \mathcal{C}_2$ of $\mathnormal{\Gamma}$ with a cyclic group $\mathcal{C}_2$ of order $2$.
\end{theorem}
The resulting regular polytope (which we again say is obtained from $\mathcal{P}$ by mixing with a segment) is denoted by $\mathcal{P} \mathbin{\Diamond} \{\mkern4mu\}$. This has twice as many vertices as $\mathcal{P}$ precisely when some edge-circuit of $\mathcal{P}$ has odd length.
We also require another basic technique for constructing regular polytopes from certain groups by what are called \emph{twisting operations} (see \cite[Chapter~8]{arp}). In this, a given group (usually itself a C-group) is augmented by means of one or more group automorphisms. This technique has been extremely successful in various classification problems for regular polytopes. In the present context, it assumes great importance in the enumeration of the regular polyhedra in $\mathbb{E}^4$; see \sectref{regpol4} below.
Roughly speaking, chiral polytopes have half as many possible automorphisms as have regular polytopes. More technically, the $n$-polytope $\mathcal{P}$ is \emph{chiral} if it has two orbits of flags under its group $\mathnormal{\Gamma}(\mathcal{P})$, with adjacent flags in different orbits. A chiral $n$-polytope $\mathcal{P}$ is then identified with a group of the form $\mathnormal{\Gamma} = \scl{\sigma_1,\ldots,\sigma_{n-1}}$, on which there are relations
\beql{chirpolrel}
\begin{cases}
\sigma_j^{p_j} = \varepsilon, & \text{$j = 1,\ldots,n-1$,} \cr
(\sigma_j \sigma_{j+1} \cdots \sigma_k)^2 = \varepsilon, & \text{$1 \leq j < k \leq n-1$.}
\end{cases}
\end{equation}
We again refer to $\{p_1,\ldots,p_{n-1}\}$ as the \emph{Schl\"afli type} of $\mathcal{P}$.
The relationship between the group and the corresponding (abstract) polytope is a little less obvious than is the case for regular polytopes (see \cite{sw_c} for more details). The distinguished generator $\sigma_j$ permutes the $(j-1)$- and $j$-faces cyclically in the appropriate section of the base flag $\mathnormal{\Phi} = \{F_0,F_1,\ldots,F_{n-1}\}$; if $F_j'$ replaces $F_j$ in the adjacent flag $\mathnormal{\Phi}^j$, then $F_{j-1}'\sigma_j = F_{j-1}$ and $F_j\sigma_j = F_j'$. The vertices of $\mathcal{P}$ are (identified with) the right cosets of the subgroup $\mathnormal{\Gamma}_0 := \scl{\sigma_2,\ldots,\sigma_{n-1}}$, with $F_0 = \mathnormal{\Gamma}_0$ itself the base vertex. The involutory element $\tau := \sigma_1\sigma_2$ interchanges the two vertices of the base edge, taking $\mathnormal{\Phi}$ into $(\mathnormal{\Phi}^{0})^{2} = (\mathnormal{\Phi}^{2})^{0}$; it is often useful to replace $\sigma_1$ as a generator by $\tau$ (compare \cite{s_cp1}).
In a chiral polytope, adjacent flags are not equivalent under the group. If $\mathnormal{\Phi}$ is replaced by an adjacent flag, $\mathnormal{\Phi}^{0}$ (say), then the respective generators are $\sigma_{1}^{-1}, \sigma_{1}^{2}\sigma_{2}, \sigma_{3}, \ldots,\sigma_{n-1}$. Thus a chiral polytope occurs in two ({\em combinatorially}) {\em enantiomorphic forms\/}, each specified by the choice of an orbit of base flags ($\mathnormal{\Phi}$ or $\mathnormal{\Phi}^{0}$), or, equivalently, a conjugacy class of sets of generators (represented by $\sigma_{1},\ldots,\sigma_{n-1}$ or $\sigma_{1}^{-1}, \sigma_{1}^{2}\sigma_{2}, \sigma_{3}, \ldots, \sigma_{n-1}$, respectively). For a regular polytope, these two enantiomorphic forms can be identified (under the generator $\rho_{0}$ of $\mathnormal{\Gamma}$).
\section{Realizations}
\label{real}
There are many candidates for spaces in which regular polytopes $\mathcal{P}$ might be realized geometrically. The usual (and generally most useful) context of realizations is of those in euclidean spaces, because it is in these that we obtain the richest structure. However, initially at least, it is appropriate for us to broaden the definition. Thus, for the time being, $E$ is a $k$-dimensional spherical space $\mathbb{S}^k$, euclidean space $\mathbb{E}^k$ or hyperbolic space $\mathbb{H}^k$, for some $k$. If $\mathcal{P}$ is a finite polytope, then $E$ will be spherical; if $\mathcal{P}$ is an apeirotope, then, since we are generally interested only in discrete realizations, $E$ will be euclidean or hyperbolic.
We begin with a brief review of some definitions (see \cite[Chapter~5]{arp} for the general background here). Let $\mathcal{P}$ be an abstract regular polytope (or apeirotope -- for the moment, we use the generic term, not distinguishing between the finite and infinite cases), and let $\mathnormal{\Gamma}:=\mathnormal{\Gamma}(\mathcal{P})$. For a \emph{faithful realization} of $\mathcal{P}$ we have two ingredients. First, we need a suitable space $E$ which admits a group $\mathcal{G}$ of isometries isomorphic to $\mathnormal{\Gamma}$; this is the \emph{symmetry group} of the realization of $\mathcal{P}$. It is convenient to identify the \emph{reflexion} $R_j$ in $\mathcal{G}$ corresponding to the involution $\rho_j$ in $\mathnormal{\Gamma}$ with its \emph{mirror}
\[ \{x \in E \mid xR_j = x\} \]
of fixed points; we thus use the same symbol $E$ for the ambient space to denote the identity mapping. The intersection
\[ W := R_1 \cap \cdots \cap R_{n-1} \]
is called the \emph{Wythoff space} of the realization. The realization of $\mathcal{P}$ associated with
$\mathcal{G}$ and its generators $R_{j}$ then arises from some choice of \emph{initial vertex} $v \in W$. The vertex-set of the realization is $V := v\mathcal{G}$, the orbit of $v$ under $\mathcal{G}$, and we always assume that $E$ is spanned by $V$ (as a subspace of the appropriate kind), so that $E$ is thought of as the \emph{ambient space} of the realization, namely, the space (of one of the three kinds) of smallest dimension which contains it.
Note that, if $\mathcal{G}$ were to be such that $R_j = E$, the identity mapping, then $R_k = E$ for all $k > j$ as well and the realization would not be faithful. In particular, this will happen if $p_j = 2$, which is why we excluded this possibility in \sectref{regchirpol}.
The induced geometric structure, the actual \emph{realization} $P$ of $\mathcal{P}$, is defined as follows. Write $F_0 := v$, and, for $j\geq 1$, let
\[ F_j := F_{j-1}\scl{R_0,\ldots,R_{j-1}}; \]
these are the basic faces. Then the $j$-faces of the realization are the $F_jG$ with $G\in\mathcal{G}$, with the order relation given by iterated membership. Thus \emph{edges} are composed of the two vertices which belong to them (we also think of an edge as the line-segment between its vertices -- there will be no ambiguity, even in the spherical case, because antipodal points of the sphere will never determine an edge), $2$-faces of the edges which belong to them, and so on up to the \emph{ridges} or $(n-2)$-faces and \emph{facets} or $(n-1)$-faces. We sometimes refer to the realization $P$ as a \emph{geometric polytope}. Its \emph{dimension} is defined by $\dim P := \dim E$, and its vertex-set is denoted by $V(P) := V$. Finally, for the realization to be \emph{faithful\/}, we demand that, for each $j = 1,\ldots,n-1$, a $j$-face be uniquely determined by the $(j-1)$-faces which belong to it. Recall here our initial assumption that $\mathcal{G}$ and $\mathnormal{\Gamma}$ be isomorphic, so for a faithful realization we then have natural bijections between the sets of $j$-faces of $\mathcal{P}$ and $P$ for each $j$. Some regular polytopes do not admit faithful realizations, because this latter condition implies a corresponding purely combinatorial condition on $\mathcal{P}$.
A realization of an abstract regular $n$-polytope $\mathcal{P}$ determines a realization of each of its faces or co-faces (iterated vertex-figures). In particular, $F_{n-1}$ (and its induced structure, with the same initial vertex $v$) gives a realization of the facet of $\mathcal{P}$; its symmetry group is the image $\mathcal{G}_{n-1}$ of $\mathnormal{\Gamma}_{n-1}$. If we write $w$ for the mid-point of the edge between $v$ and $vR_0$, then $w$ is the initial vertex of a realization of the vertex-figure of $\mathcal{P}$, with symmetry group the image $\mathcal{G}_0$ of $\mathnormal{\Gamma}_0$. (This suffices for our purposes. However, in the hyperbolic case of a polytope with vertices on the absolute, then the initial vertex $w$ is well-defined as the intersection of the mirror $R_0$ with the line between $v$ and $vR_0$ -- in any event, $w$ will always lie in this intersection.) Faithfulness is hereditary; that is, if the original realization of $\mathcal{P}$ is faithful, then the realizations of the facet and vertex-figure of $\mathcal{P}$ are also faithful. In a similar way, $\scl{R_0,\ldots,R_{j-1}}$ is the symmetry group of the basic $j$-face $F_j$ of $P$, while $\scl{R_{j+1},\ldots,R_{n-1}}$ is that of the basic \emph{co-$j$-face} $P/F_j$, which is the $(j+1)$-fold iterated vertex-figure. Thus the vertex-figure itself is $P/F_0$. Even more generally, $\scl{R_{j+1},\ldots,R_{k-1}}$ is the symmetry group of the \emph{section} $F_k/F_j$ (for $j \leq k-2$), the $(j+1)$-fold iterated vertex-figure of the basic $k$-face $F_{k}$.
We often find it more convenient to use $vR_0$ rather than $w$ as the initial vertex of the
vertex-figure; for most purposes, this makes little difference, since the combinatorics are not altered.
For regular polytopes of rank at most $2$ we have the following spherical or euclidean realizations. In $\mathbb{E}^0$ we just have the point (realizing the $0$-polytope), the finite regular $1$-polytopes are segments $\{\mkern4mu\}$, which are naturally realized in the $0$-sphere $\mathbb{S}^0$, while the regular apeirogon $\{\infty\}$ is naturally realized discretely in $\mathbb{E}^1 = \mathbb{R}$. In the unit circle $\mathbb{S}^1$, there is an infinite family of (finite) regular polygons. Their mirrors $R_0$ and $R_1$ are lines through its centre at a \emph{rational} angle $\pi/p$, meaning that $p > 2$ is a rational number (always in its lowest terms); the resulting regular polygon is denoted $\{p\}$. In addition, $\{\infty\}$ has non-discrete faithful realizations in $\mathbb{S}^1$. As we mentioned before, we shall not address here the question of finding all possible realizations of a given abstract regular polytope; a fairly complete theory has been described in \cite[Sections~5B, 5C]{arp}. Suffice it to remark that the realization space has been determined for several interesting classes of polytopes; see, for example, \cite{mowe}.
There are important restrictions on faithful realizations; we refer to \cite[Sections~5B, 5C]{arp} for proofs.
\bthml{rankdimpol}
Let $P$ be a faithful realization of an abstract regular polytope $\mathcal{P}$, whose ambient space $E$ is a spherical, euclidean or hyperbolic space. Then $\dim P \geq \mathop{\rm rank}\mathcal{P} - 1$.
\end{theorem}
\bthml{dimmirror}
Let $P$ be a faithful realization of an abstract regular $n$-polytope in $E$, with group $\mathcal{G} = \scl{R_0,\ldots,R_{n-1}}$. Then $\dim R_j \geq j$ for $j = 0,\ldots,n-2$, and $\dim R_{n-1} \geq n-2$.
\end{theorem}
In both theorems, if the polytope is finite, so that the ambient space is spherical, then, regarded as euclidean realizations, each of the dimensions must be increased by $1$.
If we have (not necessarily faithful) realizations of the abstract regular polytope (or apeirotope) $\mathcal{P}$ in two euclidean spaces, say $P$ with mirrors $S_0,\ldots,S_{n-1}$ in $L$ and $Q$ with mirrors $T_0,\ldots,T_{n-1}$ in $M$ (possibly some $S_j = L$ or $T_j = M$), then their \emph{blend} has mirrors $S_j \times T_j$ in $L \times M$ for $j = 0,\ldots,n-1$. Indeed, if $v \in S_1 \cap \cdots \cap S_{n-1}$ and $w \in T_1 \cap \cdots \cap T_{n-1}$ are the initial vertices of the two realizations, then $(v,w)$ can be chosen as the initial vertex of the blend, which we then write $P \# Q$. A realization which cannot be expressed as a blend in a non-trivial way is called \emph{pure}.
One main tool for classifying regular polytopes of a fixed rank $n$ in a fixed dimension is the \emph{dimension vector} $(\dim R_0,\dim R_1,\ldots,\dim R_{n-1})$ of the possible realizations; the first step in any enumeration is to determine which dimension vectors can occur.
It is worth noting that, in general, duals of faithfully realizable regular polytopes are not necessarily faithfully realizable at all (Petrials are particular examples), let alone in the same space.
There is a similar realization theory for chiral polytopes. Indeed, let us call a realization $P$ of an abstract polytope $\mathcal{P}$ \emph{chiral} if $P$ has two orbits of flags under its symmetry group $\mathcal{G}(P)$, with adjacent flags lying in different orbits. It is clear that the original polytope $\mathcal{P}$ must be regular or chiral. Note that there exist (already in $\mathbb{E}^3$) faithful realizations of polytopes with two flag orbits under $\mathcal{G}(P)$ which are not chiral (see \cite{w_i2} for examples).
It is helpful to remark that, if $\mathcal{P}$ is a regular $n$-polytope with group $\mathnormal{\Gamma} = \scl{\rho_0,\ldots,\rho_{n-1}}$, then its combinatorial rotation subgroup $\mathnormal{\Gamma}^+(\mathcal{P})$ has generators
\[ \sigma_j := \rho_{j-1}\rho_j, \qquad j = 1,\ldots,n-1. \]
Thus a chiral realization of a polytope may be thought of as having only rotational symmetries. Moreover, if the abstract polytope $\mathcal{P}$ is at least chiral, in that its group $\mathnormal{\Gamma}$ contains the automorphisms $\sigma_1,\ldots,\sigma_{n-1}$ in the definition of chirality, then $\mathcal{P}$ is actually regular if we can adjoin any one of the involutions $\rho_j$ for $j = 0,\ldots,n-1$. (We then have $\rho_i = \sigma_{i+1}\rho_{i+1}$ for $i = 0,\ldots,j-1$, or $\rho_i = \rho_{i-1}\sigma_i$ for $i = j+1,\ldots,n-1$.)
Chiral realizations are derived by a variant of Wythoff's construction, applied to a suitable representation $\mathcal{G} = \scl{S_1,\ldots,S_{n-1}}$ of the underlying combinatorial group $\mathnormal{\Gamma} := \scl{\sigma_{1},\ldots,\sigma_{n-1}}$; the latter is $\mathnormal{\Gamma}(\mathcal{P})$ or $\mathnormal{\Gamma}^{+}(\mathcal{P})$ according as the abstract polytope $\mathcal{P}$ is chiral or regular. The Wythoff space now is the fixed set of the subgroup $\mathcal{G}_0 := \scl{S_2,\ldots,S_{n-1}}$. We describe the $3$-dimensional case in more detail in Section~\ref{chirpol3}.
It is clear that an abstract regular polytope may have chiral realizations, though not necessarily faithful ones; it is an interesting open question whether it could actually have faithful chiral realizations. It is an elementary observation that a realized polygon with full rotational symmetry group is actually regular. Similar arguments to those used in the proof of Theorem~\ref{rankdimpol} then yield
\bpropl{rankchirreal}
If $P$ is a faithful chiral realization of an abstract polytope, whose ambient space is a spherical, euclidean or hyperbolic space $E$, then $\dim P \geq \mathop{\rm rank} \mathcal{P} - 1$.
\end{proposition}
When the abstract polytope $\mathcal{P}$ is finite, we usually assume that the centroid of the vertex-set $V$ of its (chiral or regular) realization $P$ is the origin $o$ of $E$, so that $\mathcal{G}$ is an orthogonal group. If $\mathcal{P}$ is infinite, in which case we again call $P$ a (\emph{geometric}) \emph{apeirotope}, we will additionally demand of $P$ that it be discrete, so that the group $\mathcal{G}$ acts discretely on the ambient space $E$. Moreover, in order to avoid constant repetition of various fixed phrases subsequently, we adopt the conventions that, in the geometric context of realizations, \emph{regular polytope} will mean ``faithfully realized finite abstract regular polytope'', while \emph{regular apeirotope} will mean ``discrete faithfully realized abstract regular apeirotope''; we also adopt the corresponding terminology for chiral polytopes and chiral apeirotopes.
We end the section with two general remarks. Let $S$ and $T$ be linear reflexions. First, since $ST = (-S)(-T) = S^\perp T^\perp$ (thus identifying $-S$ with its mirror $S^\perp$, and so on), then $S \cap T$ and $S^\perp \cap T^\perp$ are both pointwise fixed by the product. That is, the axis (fixed set) of $ST$ is
\beql{prodaxis}
(S \cap T) + (S^\perp \cap T^\perp)\;\, (\,= (S \cap T) + (S+T)^\perp\,).
\end{equation}
In particular, if $S$ and $T$ commute, then \eqref{prodaxis} is the mirror of their product $ST = TS$, which is again a reflexion.
Second, we have a general construction from \cite{m_rpfr}, of which special cases already occur in \cite{ms_rpo}. Let $X$ be a point-set in a euclidean space $E$. We call $X$ \emph{rational} if the points of $X$ can be chosen to have rational coordinates with respect to some (linear or affine) coordinate system in $E$. The following remark is obvious.
\bleml{apeirset}
Let $E$ be a euclidean space, and let $X$ be a finite point-set in $E$. Let $\mathcal{R}(X)$ be the group generated by the point-reflexions (inversions) in the points of $X$. Then $\mathcal{R}(X)$ is discrete if and only if $X$ is rational.
\end{lemma}
If $P$ is a regular polytope with ambient space $E$, then we similarly call $P$ \emph{rational} if its vertex-set is rational. We have the following.
\bthml{apeirpol}
Let $P$ be a rational regular $n$-polytope in the euclidean space $E$, with symmetry group $\mathcal{G}_0 = \scl{R_1,\ldots,R_n}$ and initial vertex $w$, and suppose that $v \in R_1 \cap \cdots \cap R_n$. Let $R_0 = \{w\}$ be the point-reflexion in the point $w$. Then $\mathcal{G} := \scl{R_0,\ldots,R_n}$ is the group of a discrete regular $(n+1)$-apeirotope $\mathop{\rm apeir}\nolimits P$, with $2$-faces apeirogons, and vertex-figure $P$ at the initial vertex $v$.
\end{theorem}
We call $\mathop{\rm apeir}\nolimits P$ the \emph{free abelian apeirotope on} $P$, or \emph{with vertex-figure} $P$, and \emph{base vertex} $v$. When we apply this construction, it will usually be the case that $P$ itself is finite and full-dimensional in $E$, so that $v$ is the centre of $P$.
\section{Regular polytopes of full rank}
\label{fullrank}
If $P$ is a realization of a regular polytope $\mathcal{P}$ for which equality holds in \thmref{rankdimpol}, then we say that $P$ is \emph{of full rank}. The emphasis is placed this way round, because our aim (as explained in \sectref{intro}) is to classify regular (and chiral) polytopes by dimension. In this case, we can go further than \thmref{dimmirror}, and place further restrictions on the dimensions of the mirrors of the generating reflexions of the realizations. We refer to \cite{m_rpfr} for a proof.
\bthml{fullrankmir}
Let $P$ be a faithful realization of full rank of a regular $n$-polytope $\mathcal{P}$ in the ambient space $E$, with symmetry group $\mathcal{G} = \scl{R_0,\ldots,R_{n-1}}$. Then $\dim R_j = j$ or $n-2$ for $j = 0,\ldots,n-3$, and $\dim R_{n-2} = \dim R_{n-1} = n-2$.
\end{theorem}
For finite polytopes, we now find it convenient to revert to the former definition of realization in euclidean spaces. In other words, henceforth we regard a sphere which carries the vertices of a realization $P$ of a finite regular polytope as sitting in the euclidean space of one larger dimension with centre the origin $o$. The mirrors $R_j$ of its euclidean group $\mathcal{G}$ are then thought of as linear subspaces, also of one larger dimension than before; in particular, in the minimal case, $R_0$ is either a line or a hyperplane. Finally, we shall use the more familiar $I$ for the identity (in a sense, $E$ is no longer quite appropriate).
\breml{reflgpquery}
If $R$ is a linear reflexion in a euclidean space $E$, then $-R = (-I)R$, the product of $R$ with the central inversion $-I$, is the reflexion in the orthogonal complement $R^\perp$ of $R$. Replacing a mirror by its orthogonal complement is often a useful tool in studying realizations. In particular, in the case of a faithful realization of full rank of a finite regular $n$-polytope with centre $o$, if the mirror $R_0$ is a line, then $-R_0$ is a hyperplane reflexion. If we replace $R_0$ by $-R_0$, then at worst we have replaced the symmetry group $\mathcal{G}$ by $\mathcal{G} \times \mathcal{C}_2$, with $\mathcal{C}_2 = \{\pm I\}$; in any event, we always have another finite group. Thus the mirror replacement often produces groups closely related to finite groups generated by hyperplane reflexions.
\end{remark}
\remref{reflgpquery} enables us to introduce some important geometric operations on finite polytopes of full rank, which are the key to their enumeration. For such polytopes $P$, since $o$ is the sole fixed point of the ambient space $E$ under the group $\mathcal{G}$, it follows that
\[ K_0 := R_0 \cap \cdots \cap R_{n-1} = \{o\}. \]
Thus the central reflexion $-I$, identified with its mirror $\{o\}$, is $K_0$, so the mirror replacement of \remref{reflgpquery} is $R_0 \mapsto R_0K_0$. Moreover, it is extremely useful to have variant operations, which act on the co-$(j-1)$-face $P/F_{j-1}$ for some $j$ and also apply to apeirotopes when their co-$(j-1)$-faces are finite. With
\[ K_k := R_k \cap \cdots \cap R_{n-1} \quad (0 \leq k \leq n-1), \]
we see that (the reflexion in) $K_k$ induces the central inversion on the affine hull of $P/F_{k-1}$; recall here our assumption of full rank. For $0 \leq j \leq k \leq n-1$, we then define the operation $\kappa_{jk}$ on $\mathcal{G}$ by
\beql{kappadef}
\kappa_{jk}\colon\ (R_0,\ldots,R_{n-1}) \mapsto (R_0,\ldots,R_{j-1},R_jK_k,R_{j+1},\ldots,R_{n-1}) =: (S_0,\ldots,S_{n-1}).
\end{equation}
This produces a new group with generators $S_0,\ldots,S_{n-1}$. We abbreviate $\kappa_{jj}$ to $\kappa_j$, because this is the most important case (and here usually only with $j = 0,1$), but $\kappa_{02}$ is also useful. Thus $\kappa_j$ interchanges the two possibilities for $R_j$ which can occur in \thmref{fullrankmir}. Just as with the Petrie operation, though, it must be emphasized that it is by no means generally the case that $\kappa_{jk}$ will yield a C-group when it is applied to another; for example, for $S_j$ to be an involution, we need $j = k$ or $j \leq k - 2$. Observe also that $K_{n-1} = R_{n-1}$, so that the Petrie operation of \eqref{petrie} can be written as $\pi = \kappa_{n-3,n-1}$.
One result, for which we only have a case-by-case (but not a general) proof, is the following.
\bthml{kappa0}
If $P$ is a finite regular polytope of full rank, then $P^{\kappa_0}$ is also a finite regular polytope of full rank.
\end{theorem}
It is instructive to see how the operation $\kappa_0$ acts geometrically on simple examples. In fact, $\kappa_0$ may do one of three things, even when the original group $\mathcal{G}$ is a hyperplane reflexion group: it may double the order, leave it the same, or even halve it. To illustrate this, in $\mathbb{E}^3$ take, respectively, the (group of the) tetrahedron, octahedron and cube; note that, in each case, whereas the old facets were of full rank, the new ones (of the polyhedron associated with the new group) are skew polygons, and so are not. In the planar case, we have $\{p\}^{\kappa_0} = \{q\}$, where $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$.
\breml{kappamix}
If $K_k \in \scl{R_j,\ldots,R_{n-1}}$, then $\kappa_{jk}$ results in a mixing operation.
\end{remark}
It would be inappropriate to reproduce all the details of \cite{m_rpfr} here, even in outline form. However, let us note a few of the salient facts. We shall say more about three and four dimensions in later sections; from five dimensions on, things settle in a common pattern. Recall our conventions that ``regular (or chiral) polytope" will mean ``faithfully realized finite abstract regular (or chiral) polytope'', while ``regular (or chiral) apeirotope" will mean ``discrete faithfully realized abstract regular (or chiral) apeirotope''.
For the regular $n$-polytopes in $\mathbb{E}^n$, we add to the simplex, cross-polytope and cube the results of applying $\kappa_0$ to each. From the $n$-simplex $\{3^{n-1}\}$ we obtain a polytope $\{3^{n-1}\}^{\kappa_0}$ with $2(n+1)$ vertices, those of the simplex and its dual; its group is $S_{n+1} \times C_2$. For the $n$-cross-polytope, $\{3^{n-2},4\}^{\kappa_0}$ has the same vertices and symmetry group as $\{3^{n-2},4\}$. With the $n$-cube $\{4,3^{n-2}\}$, there is a distinction between even and odd dimensions $n$. When $n$ is even, $\{4,3^{n-2}\}^{\kappa_0}$ has the same vertices and symmetry group; however, when $n$ is odd, $\{4,3^{n-2}\}^{\kappa_0}$ is isomorphic to the \emph{half-cube} $\{4,3^{n-2}\}/2 \cong \{4,3^{n-2}\}_n$, obtained by identifying opposite vertices of the cube.
For the regular $(n+1)$-apeirotopes in $\mathbb{E}^n$, we can apply the ``apeir'' construction to each of the six $n$-polytopes of the last paragraph. We also have $\{4,3^{n-2},4\}$, the tiling of space by $n$-cubes, and, finally, $\{4,3^{n-2},4\}^{\kappa_1}$, which is obtained from it by replacing its vertex-figure $\{3^{n-2},4\}$ with $\{3^{n-2},4\}^{\kappa_0}$. This last is very interesting; its $3$-face is the Petrie-Coxeter apeirohedron $\{4,6 {\mkern2mu|\mkern2mu} 4\}$, and, more generally, its facet is the $n$-face of $\{4,3^{m-2},4\}^{\kappa_1}$ for each $m \geq n$.
The following table lists the numbers of regular polytopes and apeirotopes of full rank, according to dimension.
\begin{center}
\begin{tabular}{||c|c|c||}
\hline
dimension & polytopes & apeirotopes \\
\hline \hline
0 & 1 & - \\
1 & 1 & 1 \\
2 & $\infty$ & 6 \\
3 & 18 & 8 \\
4 & 34 & 18 \\
$\geq 5$ & 6 & 8 \\
\hline
\end{tabular}
\end{center}
We end the section by quoting another result from \cite{m_rpfr}. If equality occurs in \propref{rankchirreal}, then (as before) we say that $P$ is \emph{of full rank}. This result shows that including chiral polytopes does not add any new examples to the previous classification.
\bthml{fullrankchir}
There are no chiral realizations of polytopes of full rank.
\end{theorem}
\section{Regular polytopes in three dimensions}
\label{regpol3}
The paper \cite{ms_rpo} was devoted to the complete classification of the regular polytopes and apeirotopes in $\mathbb{E}^3$, and so we confine ourselves here to the briefest mention of the techniques employed.
With rank at most $2$, we have the segment in rank $1$, and the polygons (planar and zigzag) and apeirogons (linear, zigzag and helical) in rank $2$. We say no more about them.
With rank $3$, we first note that the three regular planar tessellations and their Petrials are planar. There are nine ``classical'' regular polyhedra (the so-called Platonic solids and the Kepler-Poinsot polyhedra -- see \cite[Section~1A]{arp} for discussion of truer attributions), and nine others, which (as a family) can be regarded either as their Petrials, or as the result of applying $\kappa_0$ to them. There are twelve apeirohedra which are blends of the six planar ones with a segment or apeirogon, and twelve others which are pure (unblended); of these, except for the Petrie-Coxeter apeirohedra of \cite{c_rsp}, all but one were found by Gr\"unbaum \cite{g_on}, and the last was discovered by Dress \cite{d1,d2}.
The last case of the twelve pure apeirohedra is possibly the most interesting, at least for the methods employed. A geometric discussion shows that the possible dimension vectors (of the mirrors of the generating reflexions) are given by $(2,1,2),\ (1,1,2),\ (1,2,1)$ and $(1,1,1)$. If these mirrors are $R_0,R_1,R_2$ (we assume that the initial vertex is $o$, so that $R_1,R_2$ are linear mirrors), define $S_0'$ to be the translate of $R_0$ through $o$, $S_j' := R_j$ for $j = 1,2$, and finally $S_j := S_j'$ or $-S_j'$, according as $R_j$ is a plane or line. This relates the original symmetry group to one of the crystallographic Coxeter groups $[3,3], [3,4]$ or $[4,3]$ (we need both the latter forms) or the corresponding regular polyhedra; then the three groups, each with four dimension vectors, result in the twelve apeirohedra.
These apeirohedra are listed in the following table; for any notation not introduced hitherto, we refer to \cite{ms_rpo} or \cite[Section~7E]{arp}.
\begin{center}
\begin{tabular}{|c||ccc|}
\hline
& $\{3,3\}$ & $\{3,4\}$ & $\{4,3\}$ \\
\hline \hline
(2,1,2) & $\{6,6 {\mkern2mu|\mkern2mu} 3\}$ & $\{6,4 {\mkern2mu|\mkern2mu} 4\}$ & $\{4,6 {\mkern2mu|\mkern2mu} 4\}$ \\
(1,1,2) & $\{\infty,6\}_{4,4}$ & $\{\infty,4\}_{6,4}$ & $\{\infty,6\}_{6,3}$
\\
(1,2,1) & $\{6,6\}_{4}$ & $\{6,4\}_{6}$ & $\{4,6\}_{6}$ \\
(1,1,1) & $\{\infty,3\}^{(a)}$ & $\{\infty,4\}_{\cdot,*3}$ &
$\{\infty,3\}^{(b)}$ \\
\hline
\end{tabular}
\end{center}
The entries in the left column are the dimension vectors $(\dim R_{0},\dim R_{1},\dim R_{2})$, and the remaining columns are indexed by the corresponding finite regular polyhedra. Of these twelve apeirohedra, nine occur naturally as distinguished members in large families of polyhedra (generally apeirohedra), in which all but two polyhedra are chiral (the two exceptional polyhedra are regular); we elaborate on this in Section~\ref{chirpol3}.
Finally, there are eight regular $4$-apeirotopes in $\mathbb{E}^3$ (see \cite[Section~7F]{arp}). There is the
regular tiling $\{4,3,4\}$ of space by cubes, the result $\{\{4,6 {\mkern2mu|\mkern2mu} 4\},\{6,4\}_3\}$ of applying
$\kappa_1$ (or $\pi$) to it, and six more obtained by applying the ``apeir'' operation to the six rational
regular polyhedra, namely, the tetrahedron, octahedron and cube and their Petrials.
\smallskip
\section{Chiral polytopes in three dimensions}
\label{chirpol3}
We now proceed with the enumeration of the (discrete and faithful) chiral polyhedra in $\mathbb{E}^3$, following \cite{s_cp1,s_cp2}. Again, we shall not go into details and therefore only briefly summarize the results.
The symmetry group $\mathcal{G} := \mathcal{G}(P)$ of a chiral polyhedron $P$ has two orbits on the flags, such that adjacent flags are in distinct orbits. If $\mathcal{P}$ is the underlying abstract polyhedron, then $\mathcal{G}$ is isomorphic to $\mathnormal{\Gamma}(\mathcal{P})$ or $\mathnormal{\Gamma}^+(\mathcal{P})$ according as $\mathcal{P}$ is chiral or regular. In either case, $\mathcal{G} = \scl{S_1,S_2}$, where $S_1,S_2$ are the distinguished generators of $\mathcal{G}$ associated with a base flag $\mathnormal{\Phi}$ of $P$ and corresponding to the generators $\sigma_1,\sigma_2$ of $\mathnormal{\Gamma}(\mathcal{P})$ or $\mathnormal{\Gamma}^+(\mathcal{P})$, respectively. If $P$ is of type $\{p,q\}$, then
\[ S_1^p = S_2^q = (S_1S_2)^2 = I , \]
but in general there are also other independent relations. If $\mathnormal{\Phi}$ is replaced by $\mathnormal{\Phi}^2$ (the adjacent flag with a different $2$-face), then the new pair of generators of $\mathcal{G}$ are $S_1S_2^2,S_2^{-1}$. Thus $S_1,S_2$ and $S_1S_2^2,S_2^{-1}$ are the pairs of generators representing the two enantiomorphic forms of $P$.
As we remarked in Section~\ref{real}, a chiral polyhedron $P$ can be obtained from a variant of Wythoff's construction, applied to a group $\mathcal{G}=\scl{S_1,S_2}$ with initial vertex a point $v$ fixed by $S_2$ (but not $S_1$). If we set $T := S_1S_2$, which must be a reflexion in a line or plane, then the base vertex, edge and facet of $P$ are $v$, $v\scl{T}$ and $(v\scl{T})\scl{S_1}$, respectively; as usual, the other vertices, edges and facets are their images under $\mathcal{G}$.
The first step is to determine the possible special groups and their generators. Recall that, if $R\colon x\mapsto xR' + t$ is a general element of $\mathcal{G}$, with $R' \in \mathrm{O}_3$, the orthogonal group, and $t \in \mathbb{E}^{3}$ a translation vector, then the linear mappings $R'$ form the \emph{special group} $\mathcal{G}_0$ of $\mathcal{G}$. In the present context, $\mathcal{G}$ must be a crystallographic group in $\mathbb{E}^3$ and $\mathcal{G}_0 = \scl{S_1',S_2'}$ a finite subgroup of $\mathrm{O}_3$. If $T(\mathcal{G})$ denotes the subgroup of all translations in $\mathcal{G}$, then $\mathcal{G}_0 \cong \mathcal{G}/T(\mathcal{G})$. It turns out that the only possible special groups are $[3,3]$ and $[3,4]$ (possibly as $[4,3]$), the full tetrahedral and octahedral group, respectively, and their rotation subgroups $[3,3]^+$ and $[3,4]^+$ (possibly as $[4,3]^+$), as well as the group $[3,3]^*$ obtained from $[3,3]^+$ by adjoining the central inversion in the invariant point of $[3,3]^+$. In particular, this limits the possible Schl\"afli types to $\{4,6\}$, $\{6,4\}$, $\{6,6\}$, $\{\infty,3\}$ and $\{\infty,4\}$.
A chiral polyhedron in $\mathbb{E}^3$ cannot be finite (by Theorem~\ref{fullrankchir}) or be a blend (its group must be affinely irreducible). Thus each chiral polyhedron is infinite and pure.
The possible apeirohedra fall into six infinite $2$-parameter families (up to congruence). In each family, all but two polyhedra are chiral; the two exceptional polyhedra are regular. The following table lists the families of polyhedra by the structure of their special group, along with the two regular polyhedra occurring in each family; in three families, one exceptional polyhedron is finite.
\medskip
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$[3,3]^*$ & $[4,3]$ & $[3,4]$ & $[3,3]^+$ & $[4,3]^+$ & $[3,4]^+$ \\
\hline\hline
$P(a,b)$ & $Q(c,d)$ & $Q(c,d)^*$ & $P_1(a,b)$ & $P_2(c,d)$ & $P_3(c,d)$ \\
\hline
$\{6,6\}_{4}$ & $\{4,6\}_{6}$ & $\{6,4\}_{6}$ & $\{\infty,3\}^{(a)}$ &
$\{\infty,3\}^{(b)}$ & $\{\infty,4\}_{\cdot,*3}$ \\
$\{6,6 {\mkern2mu|\mkern2mu} 3\}$ & $\{4,6 {\mkern2mu|\mkern2mu} 4\}$ & $\{6,4 {\mkern2mu|\mkern2mu} 4\}$ & $\{3,3\}$ & $\{4,3\}$ & $\{3,4\}$ \\
\hline
\end{tabular}
\end{center}
\medskip
The columns are indexed by the special groups to which the respective polyhedra correspond; some groups occur twice but with different pairs of generators. The second row contains the six families; as we said before, possibly with one exception, all members of a family are apeirohedra. For the first three families, discreteness forces the parameter pairs $a,b$ and $c,d$, respectively, to be relatively prime integers; however, for the last three families, the parameters can be reals. (Thus, when the polyhedra are considered up to similarity, there is a single rational or real parameter, as appropriate.)
In particular, the chiral polyhedra $P(a,b)$, $Q(c,d)$ and $Q(c,d)^*$ (the dual of $Q(c,d)$) have finite skew faces and skew vertex-figures, and are of types $\{6,6\}$, $\{4,6\}$ or $\{6,4\}$, respectively; remarkably, in each family, the two regular polyhedra have planar faces or vertex-figures. Recall that no regular polyhedron has finite skew faces and skew vertex-figures (see \cite[Section~7E]{arp}). On the other hand, the polyhedra $P_1(a,b)$, $P_2(c,d)$ and
$P_3(c,d)$ have infinite faces consisting of helices over triangles, squares or triangles, respectively, and are of types $\{\infty,3\}$, $\{\infty,3\}$ or $\{\infty,4\}$.
The last two rows of the table comprise nine of the twelve pure regular apeirohedra in $\mathbb{E}^{3}$, namely those listed in the table of Section~\ref{regpol3} with dimension vectors $(1,2,1)$, $(1,1,1)$ or $(2,1,2)$, as well as the three (finite) ``crystallographic'' Platonic polyhedra. The three remaining pure regular apeirohedra $\{\infty,6\}_{4,4}$, $\{\infty,4\}_{6,4}$ and
$\{\infty,6\}_{6,3}$ all have dimension vector $(1,1,2)$ and do not occur in families alongside chiral polyhedra.
We now display the families of polyhedra, with the various known relationships among them. These complement the known relationships between regular polyhedra (see \cite[Section~7E]{arp}). Three operations on (chiral or regular) polyhedra and their groups $\mathcal{G}$ are involved:\ the duality operation $\delta$, the second facetting operation $\varphi_2$ and the halving operation $\eta$ (see Section~\ref{regchirpol} or \cite{s_cp1}). In terms of the generators of $\mathcal{G}$ they are defined as follows:
\[\begin{array}{rlll}
\delta\colon & (S_1,S_2) &\mapsto& (S_2^{-1},S_1^{-1}) ,\\
\varphi_2\colon & (S_1,S_2) &\mapsto& (S_1S_2^{-1},S_2^2), \\
\eta\colon & (S_1,S_2) &\mapsto& (S_1^2S_2,S_2^{-1}).
\end{array} \]
In each case, the pair of elements on the right are the generators for the group of a new polyhedron, namely the image of the given polyhedron under $\delta$, $\varphi_2$ or $\eta$, respectively.
The following diagram emphasizes operations relating families rather than individual polyhedra. In particular, we drop the parameters from the notation; for example, $P_1$ denotes the family of polyhedra $P_1(a,b)$.
\begin{equation}
\label{displayone}
\begin{matrix}
Q^* & \stackrel{\delta}{\longleftrightarrow} \mkern-30mu & Q & \mkern-30mu
\stackrel{\varphi_2}{\longrightarrow}& P_2 \cr \cr
& &\;\;\downarrow\! {\scriptstyle\eta} & & & \cr \cr
P_3 & & P & \mkern-30mu \stackrel{\varphi_2}{\longrightarrow}& P_1 \cr
& &
\begin{picture}(60,60)
\put(9,46){\oval(42,42)[b]}
\put(9,46){\oval(42,42)[tl]}
\put(9,67){\vector(1,0){2}}
\put(-19,34){$\scriptstyle\delta$}
\end{picture}
&&
\end{matrix}
\end{equation}
Observe that, in the diagram, $P_3$ is not connected to any other family; it is an interesting open question if there exists a relationship between $P_3$ and any other family. The circular arrow in the diagram indicates the self-duality of the family (in fact, of each of its polyhedra). The operations $\delta$ and $\varphi_2$ map a polyhedron to one with the same parameter pair, either $a,b$ or $c,d$. However, $\eta$ replaces $c,d$ by the new pair $c-d,c+d$. Moreover, note that $\varphi_2$, when applicable, maps a polyhedron to one in the same row of the table.
For a discussion of other classes of highly symmetric polyhedra in $\mathbb{E}^3$ we refer the reader to, for example, \cite{g_ap}.
\section{Regular polytopes in four dimensions}
\label{regpol4}
Just as is the case with the classical regular polytopes and apeirotopes, the richest family of full rank occurs in $\mathbb{E}^4$. Again, we do not wish to go into the results of \cite{m_rpfr} in great detail; instead, we shall concentrate on a few plums.
We have already accounted for the effects of $\kappa_0$; we merely note that the sixteen classical regular (convex and star-) $4$-polytopes give rise to another sixteen in this way. However, we can also apply $\pi$ to the $4$-cube $\{4,3,3\}$, to obtain
\[ \{4,3,3\}^\pi = \{\{4,4 {\mkern2mu|\mkern2mu} 4\},\{4,3\}_3\}. \]
That is, the facets are toroids, and the vertex-figure is the half-$3$-cube; moreover, the polytope is universal of this kind. The final instance (of the $34$ in the table of \sectref{fullrank}) is obtained by applying $\kappa_0$ to this last.
These two finite polytopes just mentioned contribute two regular $5$-apeirotopes via the ``apeir" construction. Leaving aside the examples already discussed in \sectref{fullrank}, then for the $5$-apeirotopes there remain those obtained from $\{3,3,4,3\}$ and its dual $\{3,4,3,3\}$. For the first, we can apply $\kappa_1$ (that is, apply $\kappa_0$ to its vertex-figure $\{3,4,3\}$); we get an apeirotope whose $3$-faces are Petrie-Coxeter apeirohedra $\{6,6 \hole3\}$. For the other, we can first apply $\kappa_1$; the resulting apeirotope has $3$-faces the last Petrie-Coxeter apeirohedron $\{6,4 {\mkern2mu|\mkern2mu} 4\}$. To both of these (that is, $\{3,4,3,3\}$ and $\{3,4,3,3\}^{\kappa_{1}}$), we can now apply $\pi$ as well; the $3$-faces remain as they were (that is, octahedra $\{3,4\}$ or $\{6,4 {\mkern2mu|\mkern2mu} 4\}$, respectively); the facet of the first is the universal apeirotope $\{\{3,4\},\{4,4 {\mkern2mu|\mkern2mu} 4\}\}$ (we comment on this further in \sectref{opprob}).
It is a striking fact that all three Petrie-Coxeter apeirohedra in $\mathbb{E}^3$ occur as $3$-faces of regular $5$-apeirotopes in $\mathbb{E}^4$ (one of them twice).
We next discuss the recent (as yet unpublished) classification of the four-dimensional (finite) regular polyhedra. Those polyhedra with planar faces were all found in \cite{abm,bracho}; the methods we employ in \cite{m_fdrp} are akin to those used in \cite{m_rpfr,ms_rpo}, and are, we feel, much simpler.
As we have already pointed out in \sectref{real}, our strategy is to determine what possible dimension vectors can occur, and then to enumerate every polytope in the corresponding subclasses. \thmref{dimmirror} provides a starting point; in the current case, the dimension vector must satisfy
\[ \dim R_0 \geq 1, \quad \dim R_1 \geq 2, \quad \dim R_2 \geq 2. \]
We now proceed as follows. As essentially the same trick we perform in $\mathbb{E}^3$, if the mirror $R_0$ satisfies $\dim R_0 = 1$, then we can replace it by
\[ -R_0 = R_0^\perp, \]
its orthogonal complement, which (as an isometry) is its product with the central inversion $-I$; we refer to this more general operation as $\kappa_0$ as well. We always obtain another finite group $\mathcal{G}'$; in fact,
\[ |\mathcal{G}'| = {\textstyle \frac{1}{2}}|\mathcal{G}|,\ |\mathcal{G}|,\ \mbox{or } 2|\mathcal{G}|. \]
Next, if $\dim R_0 = 2$ and $\dim R_2 = 3$ (or vice versa, but this case will have to be excluded), then we can replace $R_0$ by $R_0R_2$, that is, apply (or reverse) the Petrie operation $\pi$; bearing in mind \eqref{prodaxis}, the new $R_0$ has $\dim R_0 = 1$ or $3$, and in the former case we can proceed as previously.
Finally, as long as our (possibly new) group contains a hyperplane reflexion (that is, $\dim R_j = 3$ for some $j$), we can regard $\mathcal{G}$ as a reflexion (Coxeter) group, on which certain involutions with $2$-dimensional mirrors act as automorphisms (more precisely, $\mathcal{G}$ is the corresponding semi-direct product). When we have carried out the foregoing procedures, only the dimension vectors $(3,2,3)$ and $(2,3,2)$ need to be considered. For classification purposes, we then reverse the procedure: the starting point is a Coxeter group, not necessarily with standard
generators, which can be represented by a diagram that permits permutation of its nodes.
We give a couple of simple examples of what happens in the cases $(3,2,3)$ and $(2,3,2)$ in a little detail, and then comment on the remaining cases (with the exception of $(2,2,2)$) more briefly. We list them according to their dimension vectors.
\begin{itemize}} %\itemsep-1.5mm
\item $(3,2,3)$: from the group $[3,4,3]$ of the regular $24$-cell, we derive the diagrams
\begin{center}
\begin{picture}(250,70)
\Horone{0,11}
\Horone{0,59}
\Verone{48,11}
\put(52,33){$4$}
\Resq{107,11}
\put(96,33){$\frac{4}{3}$}
\put(159,33){$4$}
\Horone{204,11}
\Horone{204,59}
\Verone{252,11}
\put(256,33){$\frac{4}{3}$}
\end{picture}
\end{center}
each of which permits a top-to-bottom flip, and thereby gives two dual regular polyhedra with dimension vectors $(3,2,3)$. (From the first diagram, we obtain the polyhedra $\{4,8 {\mkern2mu|\mkern2mu} 3\}$ and $\{8,4 {\mkern2mu|\mkern2mu} 3\}$ of \cite{c_rsp}.) Similar examples derive from the diagram
\begin{center}
\begin{picture}(48,70)
\Horone{0,11}
\Horone{0,59}
\Verone{48,11}
\end{picture}
\end{center}
\item $(2,3,2)$: the general case is derived from a diagram
\begin{center}
\begin{picture}(110,100)
\REsq{19,14}
\thicklines
\put(19,14){\line(1,1){72}}
\put(19,86){\line(1,-1){72}}
\multiput(8,48)(88,0){2}{$p$}
\multiput(32,37)(42,0){2}{$r$}
\multiput(53,3)(0,89){2}{$q$}
\end{picture}
\end{center}
with horizontal and vertical flips. This gives rise to a polyhedron of type $\{2p,2q\}_{2r}$, from which are obtained up to five others by duality and Petriality. (There is a restriction on $q$: it must not be a fraction with even denominator.) As a specific instance, the full family of six is obtained when $\{p,q,r\} = \{3,4,\frac{4}{3}\}$.
\item $(3,3,3)$: this corresponds to three-dimensional polyhedra, and so is excluded (but only on these grounds).
\item $(1,3,3)$: this is allowed; $\kappa_0$ can be applied to the case $(3,3,3)$.
\item $(2,3,3)$: this is obtained from $(3,3,3)$ or $(1,3,3)$ by Petriality; therefore, the first possibility must be excluded.
\item $(3,3,2)$: this would be obtained from $(2,3,3)$ by duality; however, in the allowed case, the faces of the original are centred at $o$, and so the dual must be excluded.
\item $(1,3,2)$: this would be obtained from $(3,3,2)$ by applying $\kappa_0$, and so is also disallowed.
\item $(1,2,3)$: this arises from $(3,2,3)$ by applying $\kappa_0$.
\item $(2,2,3)$: this is obtained from $(3,2,3)$ or $(1,2,3)$ by Petriality.
\item $(3,2,2)$: this would arise from $(2,2,3)$ by duality. However, it may be seen that (with either possibility) the product $R_0R_1$ of the corresponding reflexions $R_0$ and $R_1$ in the original is a double rotation (in two orthogonal planes), since $R_0 \cap R_1 = \{o\}$; it follows that the class cannot occur.
\item $(1,2,2)$: this would be obtained from $(3,2,2)$ by applying $\kappa_0$, and so it too must be excluded.
\end{itemize}
It is notable that only the groups $[3,3,3]$ and $[3,4,3]$ give rise to polyhedra in the classes $(3,2,3)$ and $(2,3,2)$ and those derived from them. Even though other finite reflexion groups in $\mathbb{E}^4$ permit diagram automorphisms (for suitably chosen generators), these are inner, and then the corresponding ``polyhedra'' degenerate.
The anomalous case is dimension vector $(2,2,2)$, to which the notion of a Coxeter group with outer automorphisms is inapplicable. Indeed, some examples of this kind cannot be related to Coxeter groups in any meaningful way. The approach here is through quaternions. Each isometry which occurs in such a group is a rotation (that is, lies in $\mathrm{SO}_4$), and so can be represented by a quaternionic transformation of the form
\beql{quatmap}
x \mapsto \ol{a}xb,
\end{equation}
where $a,\ b$ are unit quaternions (recall that $a^{-1} = \ol{a}$). In keeping with our usual conventions, mappings are thought of as acting on the right; thus it must be the inverse of a quaternion which provides an appropriate mapping when acting on the left. For the mapping \eqref{quatmap} to be a reflexion, both $a$ and $b$ must be pure imaginary. Our symmetry group $\mathcal{G}$ gives rise to two groups $\mathcal{G}_L$ and $\mathcal{G}_R$ of the left-acting quaternions $a$ and right-acting quaternions $b$; then $\mathcal{G}$ is a certain quotient of $\mathcal{G}_L \times \mathcal{G}_R$ (for further details at this stage, we refer the reader to \cite{dv}). Further, there are then quotients $G_L,\ G_R$ of $\mathcal{G}_L,\ \mathcal{G}_R$ in $\mathrm{SO}_3$, each by normal subgroups of index $2$, and these
are generated by half-turns about lines in $\mathbb{E}^3$. If $a = \cos\vartheta + u\sin\vartheta$, with $u$ pure imaginary, then the image of $a$ under the homomorphism from $\mathcal{G}_L$ to $G_L$ is a rotation through $2\vartheta$ about the axis in $\mathbb{E}^3$ through $u$, when the latter is regarded as a unit vector in $\mathbb{E}^3$. Thus, for example, if $a$ is pure imaginary, then its image is the half-turn about the axis in $\mathbb{E}^3$ through $a$; it is important to note that this half-turn lifts to two pure imaginary quaternions $\pm a$. The only groups which can occur as such groups $G_L$ or $G_R$ are dihedral, octahedral or icosahedral; the cyclic and tetrahedral groups do not contain enough half-turns. Finally, if the generating reflexions are
\[ xR_j := \ol{a}_jxb_j = -a_jxb_j \quad (j = 0,1,2), \]
then (as scalar products of vectors in $\mathbb{E}^3$),
\[ \scl{a_1,a_2} = \pm\scl{b_1,b_2}, \]
because the product $R_1R_2$ must have a $2$-dimensional axis. However, the opposite must be true for the product $R_0R_1$, because this has to be a double rotation.
In summary, the following ingredients go into the enumeration. First, two groups in $\mathbb{E}^3$ generated by half-turns: these are a dihedral group $D_{2k}$ ($k$ can only take the values $2$, $3$ or $5$), the octahedral group $S_4 = [3,3] = [3,4]_3$ or the isosahedral group $A_5 = [3,5]_5$. Second, for $j = 1,2$, two regular polyhedra of type $\{r_j,q\}$ (with the same $q$); here, we must allow $r_j > 1$, rather than the usual $r_j \geq 2$, to account for two possible liftings of the half-turns contributing to $R_0$. We then obtain a polyhedron of type $\{p,q\}$, where the face $\{p\}$ is of the form $\{p_1\} \# \{p_2\}$, with
\[ \frac{1}{p_j} = \frac{1}{2}\left( \pm \frac{1}{r_1} \pm \frac{1}{r_2} \right), \]
where the signs are chosen so that $p_j > 2$ for $j = 1,2$. It is convenient to write the face, instead, as
\[ \left\{\frac{p}{d_1,d_2}\right\}, \qquad \mbox{with} \quad p_j = \frac{p}{d_j} \]
(in lowest terms) for $j = 1,2$.
As a specific example, if $r_1 = 3,\ r_2 = {\textstyle \frac{5}{2}}$ and $q = 5$, then we obtain a polyhedron of type
\[ \{\tfrac{30}{1,11},5\}. \]
However, if we replace ${\textstyle \frac{5}{2}}$ by $\frac{5}{3}$ (or $3$ by $\frac{3}{2}$), indicating a different choice of lifting for $R_0$, then we obtain type
\[ \{\tfrac{15}{2,7},5\}. \]
\begin{remark}
A further comment is in order here. An opposite orthogonal transformation of $\mathbb{E}^4$ is of the form
\[ x \mapsto\ol{a}\,\ol{x}b, \]
with $a,\ b$ as before. In a group $\mathcal{G}$ containing such transformations, the corresponding left and right groups $\mathcal{G}_L$ and $\mathcal{G}_R$ must be conjugate in the whole group of unit quaternions. Thus one could also use quaternions to investigate the classes other than $(2,2,2)$; however, the methods which we have already described are more efficacious.
\end{remark}
\section{Open problems}
\label{opprob}
As the dimension increases, so there are more possibilities for the ranks of faithfully realized regular or chiral polytopes or apeirotopes. In full rank, the regular cases are classified, and chirality does not occur. In $\mathbb{E}^4$, therefore, the open cases are the (finite) chiral polytopes of rank $3$, and the regular or chiral apeirotopes of ranks $3$ and $4$.
We look at the regular cases first; we begin with rank $4$. Each of the eight regular $4$-apeirotopes in $\mathbb{E}^3$ can be blended (mixed) with a segment or an apeirogon; this gives $16$ blended examples. Next, the ``apeir'' construction described at the end of \sectref{real} can be applied to any of the four-dimensional rational regular polyhedra. Finally, certain of the facets of the regular apeirotopes of full rank in $\mathbb{E}^4$ are $4$-apeirotopes. It is possible that there are not too many more examples which do not fall under one of these three categories,
and maybe even none at all.
Incidentally, there is only one four-dimensional $4$-apeirotope whose facets are finite regular polyhedra. This is the universal $\{\{3,4\},\{4,4 {\mkern2mu|\mkern2mu} 4\}\}$, with facet the octahedron $\{3,4\}$ and vertex-figure the toroid $\{4,4 {\mkern2mu|\mkern2mu} 4\}$, which, as noted in \sectref{regpol4}, is the facet of the $5$-apeirotope $\{3,4,3,3\}^\pi$. (Compare \cite[Theorem~10B3]{arp} with $s = 4$ in the dual form, and the preceding discussion.) To see that this is the only example, observe that there are no four-dimensional (finite) regular polyhedra with triangular faces (nor
with pentagons or pentagrams either, but these must be excluded on crystallographic grounds). Hence, the only possible vertex-figure has square faces, which means that the facet must be an octahedron or its Petrial $\{6,4\}_3$. In turn, the vertex-figure must be a regular polyhedron with square faces, and circumradius equal to its edge-length; this forces it to be $\{4,4 {\mkern2mu|\mkern2mu} 4\}$. Finally, direct calculation shows that, in fact, $\{6,4\}_3$ cannot actually be a facet in such a way.
As for the four-dimensional regular apeirohedra, a mere glance at some of the possibilities shows that the enumeration problem is likely to be rather hard. For example, in $\mathbb{E}^2$ the apeirohedron $\{{\textstyle \frac{5}{2}},10\}$ is non-discrete; however, when it is blended with its isomorphic copy $\{5,{\textstyle \frac{10}{3}}\}$ in $\mathbb{E}^4$, a discrete regular apeirohedron of type $\{5,10\}$ is obtained. Several similar examples also occur.
There are also examples derived from complex reflexion groups in $\mathbb{C}^2$, which we regard as real groups in $\mathbb{E}^4$ generated by reflexions with $2$-dimensional mirrors. A curiosity is the following. We can twist the first of the two diagrams below by the dihedral group $D_3$ (or symmetric group $S_3$), and the second by $C_2$. We then actually obtain the same geometric group; however, the outer automorphisms of one correspond to the generating reflexions of the other (and vice versa). We refer to \cite[Section~9D]{arp} for the background here.
\begin{center}
\begin{picture}(150,68)
\Rtri{10,10} \Rtri{110,10}
\multiput(30,11)(0,38){2}{$4$}
\put(21,31){$4$}
\put(120,31){$6$}
\multiput(0,31)(100,0){2}{$4$}
\end{picture}
\end{center}
We now turn to chiral polytopes and apeirotopes. For the latter, various infinite families of chiral apeirohedra were described in \cite{s_cp1,s_cp2} (see Section~\ref{chirpol3}); each such apeirohedron can be blended with a segment or an apeirogon to give a four-dimensional chiral apeirohedron. Finally, there are plenty of finite chiral polyhedra in $\mathbb{E}^4$; for example, each chiral toroid $\{4,4\}_{(s,t)}$ is realizable. Whether there exist non-toroidal finite chiral polyhedra in $\mathbb{E}^4$ is a nice open question.
Finally, presentations for the symmetry groups have only been fully worked out for the $3$-dimensional regular polyhedra and apeirotopes (see \cite[Sections~7E, 7F]{arp}). For higher dimensions, presentations are known for certain classes of polytopes, for example, the regular star-polytopes (see \cite[Section 7D]{arp} or \cite{m_grsp}). In this context, the main tool is the so-called ``circuit criterion", which states that the automorphism group of an abstract polytope $\mathcal{P}$ (and thus the symmetry group of a faithful realization) is determined by the group of its vertex-figure and the circuit structure of the edge-graph of $\mathcal{P}$ (see \cite[Section 2F]{arp} for more details). A variant of this method should also succeed in the chiral case. In particular, there is an interest in presentations for the symmetry groups of the $3$-dimensional chiral apeirohedra. Here we do not know if the corresponding abstract apeirohedra are also chiral or if they are regular. Settling this question may have to be the first step in arriving at presentations for their symmetry groups.
|
{
"timestamp": "2005-03-18T20:24:09",
"yymm": "0503",
"arxiv_id": "math/0503389",
"language": "en",
"url": "https://arxiv.org/abs/math/0503389"
}
|
\section{Introduction}
\lbl{sec.intro}
\subsection{The volume conjecture for small angles}
\lbl{sub.volume}
In an earlier publication, the authors stated and proved the {\em Volume
Conjecture}
for small purely imaginary angles; see \cite{GL2}. More precisely, the authors
proved that for every knot $K$ in $S^3$ there exists a positive angle
$\alpha(K) >0$ such that
\begin{equation}
\lbl{eq.VC}
\lim_{n \to \infty} \frac{\log|J_{K,n}(e^{\alpha/n})|}{n}=0
\end{equation}
for all $\alpha \in i [0, \alpha(K))$, where
\begin{itemize}
\item
$f(e^{\alpha/n})$ denotes the evaluation of a rational function $f(q)$ at
$q=e^{\alpha/n}$,
\item
$J_{K,n}(q) \in \mathbb Z[q^{\pm}]$ is the {\em Jones polynomial} of a knot
{\em colored} with the
$n$-dimensional irreducible representation of $\mathfrak{sl}_2$,
normalized so that it equals to $1$ for the unknot (see \cite{J, Tu}).
\end{itemize}
In the following, we will refer to the complex parameter $\alpha$ as
{\em the angle}, making contact with standard terminology from
hyperbolic geometry.
As was explained in \cite{GL2}, the above result agrees with the fact that
$$
\mathrm{vol}(\rho_{\alpha})=0
$$
where
\begin{equation}
\lbl{eq.rho}
\rho_{\alpha}: \pi_1(S^3-K) \longrightarrow \mathrm{SL}_2(\mathbb C),
\qquad \rho_{\alpha}(\mathfrak{m})= \mat {e^{\alpha/n}} 0 0
{e^{-\alpha/n}}.
\end{equation}
is a {\em reducible} representation of the knot group in $\mathrm{SL}_2(\mathbb C)$
with prescribed behavior on a meridian $\mathfrak{m}$ of the knot $K$.
For further reading concerning the history of the volume conjecture, we refer
the reader to \cite{Gu,K,MM}, as well as \cite{GL2}.
Notice that $\rho_{\alpha}$ is a 1-parameter deformation of the {\em trivial
representation} $\rho_0=I$.
Moreover, Equation \eqref{eq.VC} implies that the sequence
$J_{K,n}(e^{\alpha/n})$ grows at a subexponential rate,
as $n$ approaches infinity, and $\alpha$ is small and purely imaginary.
The purpose of the present paper is to identify the polynomial
growth rate of $J_{K,n}(e^{\alpha/n})$ in terms of the
inverse Alexander polynomial $\Delta_K$ of $K$, symmetrized by $\Delta_K(t^{-1})=
\Delta_K(t)$, and normalized by $\Delta_K(1)=1$, and $\Delta_{\text{unknot}}(t)=1$.
More precisely, we have the following theorem.
\begin{theorem}
\lbl{thm.11}
For every knot $K$ there exists an open neighborhood $U_K$ of $0 \in \mathbb C$
such that for all complex angles $\alpha \in U_K$, we have:
\begin{equation}
\lbl{eq.thm11}
\lim_{n\to\infty}
J_{K,n}(e^{\alpha/n}) =
\frac{1}{\Delta_K(e^{\alpha})} \in \mathbb C.
\end{equation}
Moreover, the convergence with respect to $\alpha$ is uniform on compact subsets
of $U_K$.
\newline
In particular since $\Delta_K(1)=1$, \eqref{eq.thm11} implies \eqref{eq.VC}.
\end{theorem}
The reader may compare the above theorem with the famous
Melvin-Morton-Rozansky (MMR, in short) Conjecture, which was settled
by Bar-Natan and the first author in \cite{B-NG}. Let $\mathbb Q[[h]]$ denote
the ring of {\em formal power series} in a variable $h$ with rational
coefficients.
\begin{theorem}
\lbl{thm.MMR}\cite{B-NG}
For every knot $K$ we have the following equality in the
ring $\mathbb Q[[h]]$:
\begin{equation}
\lbl{eq.MMR}
\lim_{n \to \infty} \,\, J_{K,n}(e^{h/n}) = \frac{1}{\Delta_K(e^{h})}
\in \mathbb Q[[h]],
\end{equation}
\end{theorem}
To avoid confusion, let us point out that Equation
\eqref{eq.MMR} is a statement about coefficients of formal power series.
In other words, \eqref{eq.MMR} can be phrased as follows:
for every $m \geq 0$, we have:
\begin{equation}
\lbl{eq.MMRalt}
\lim_{n \to \infty}
\mathrm{coeff} \left( J_{K,n}(e^{h/n}) , h^m \right)=
\mathrm{coeff} \left( \frac{1}{\Delta_K(e^{h})} , h^m\right),
\end{equation}
where for an analytic function $f(x)$ we define:
$$
\mathrm{coeff}( f(h), h^m)=\frac{1}{m!} \frac{d^m}{dh^m}|_{h=0} f(h).
$$
Actually, for every $m \geq 0$, $
\mathrm{coeff} \left( J_{K,n}(e^{h/n}) , h^m \right) $ is a polynomial in $1/n$
of degree $m$ (see also Section \ref{sub.fti} below). Thus, the limit
with respect to $n \to\infty$ in \eqref{eq.MMRalt} exists and is simply
the constant term of the above-mentioned polynomial. Identifying that constant
term with the right hand side of \eqref{eq.MMRalt} is the non-trivial
part of the MMR Conjecture.
Let us compare Theorems \ref{thm.11} and \ref{thm.MMR}.
Since convergence with respect to $\alpha$ is uniform on compact subsets,
it is easy to see that Theorem \ref{thm.11} implies Theorem \ref{thm.MMR}.
In that sense, we may say
that Theorem \ref{thm.11} is an analytic form of the MMR Conjecture.
Thus, Theorem \ref{thm.11} can be viewed as a statement about the volume
conjecture for small angles, as well as an analytic form of the MMR Conjecture.
Armed with Theorem \ref{thm.11} one may ask for a full asymptotic expansion
of the left hand side of \eqref{eq.thm11} in terms of powers of $1/n$.
Before we answer this question, let us recall what is known on the level
of formal power series, that is, about the $1/n$ terms of
\eqref{eq.MMRalt}.
Rozansky discovered that after resummation, for every fixed $m \geq 0$,
the $1/n^m$ terms of \eqref{eq.MMRalt} are rational
functions in a variable $e^h$. Let us state Rozansky's discovery concretely.
\begin{theorem}
\lbl{thm.Zrat}\cite{Ro}
For every knot $K$ there exists a sequence $P_{K,k}(q) \in \mathbb Q[q^{\pm}]$
of Laurent polynomials with $P_{K,0}(q)=1$ such that
\begin{equation}
\lbl{eq.Zrat}
J_{K,n}(e^{h/n}) \sim_{n \to \infty}
\sum_{k=0}^\infty
\frac{P_{K,k}(e^{h})}{\Delta_K(e^{h})^{2k+1}} \left(\frac{h}{n}\right)^k
\in \mathbb Q[[h]]
\end{equation}
in the ring $\mathbb Q[[h]]$ of formal power series in $h$.
\end{theorem}
A different proof, valid for all simple Lie groups, was given in \cite{Ga1},
using work of \cite{GK}.
Let us point out that \eqref{eq.Zrat} means the following:
for every $N \geq 0$ we have:
\begin{equation}
\lbl{eq.Zratalt}
\lim_{n\to\infty}
\left( \frac{n}{h} \right)^N \left(
J_{K,n}(e^{h/n})-\sum_{k=0}^{N-1}
\frac{P_{K,k}(e^{h})}{\Delta_K(e^{h})^{2k+1}} \left(\frac{h}{n}\right)^k
\right)=
\frac{P_{K,N}(e^h)}{\Delta_K^{2N+1}(e^h)} \in \mathbb Q[[h]].
\end{equation}
\subsection{Asymptotics to all orders}
\lbl{sub.results}
Our results are the following:
\begin{theorem}
\lbl{thm.1}
For every knot $K$ there exists an open neighborhood $U_K$ of $0 \in \mathbb C$
such that for all complex angles $\alpha \in U_K$, we have
an asymptotic expansion (uniform on compact subsets of $U_K$
with respect to $\alpha$):
\begin{equation}
\lbl{eq.thm1}
J_{K,n}(e^{\alpha/n}) \sim_{n \to \infty} \sum_{k=0}^\infty
\frac{P_{K,k}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2k+1}} \left(\frac{\alpha}{n}\right)^k
.
\end{equation}
\end{theorem}
In other words, for $\alpha \in U_K$ and every $N \geq 0$,
\begin{equation}
\lbl{eq.thm1alt}
\lim_{n\to\infty}
\left( \frac{n}{\alpha} \right)^N \left(
J_{K,n}(e^{\alpha/n})-\sum_{k=0}^{N-1}
\frac{P_{K,k}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2k+1}} \left(\frac{\alpha}{n}\right)^k
\right)=
\frac{P_{K,N}(e^{\alpha})}{\Delta_K^{2N+1}(e^{\alpha})} \in \mathbb C.
\end{equation}
Moreover, convergence with respect to $\alpha$ is uniform on compact subsets
of $U_K$.
Thus, the above theorem determines to all orders the asymptotic expansion
of the volume conjecture for small angles.
\subsection{A small dose of physics}
\lbl{sub.physics}
One does not need to know the relation of the colored Jones
function and quantum field theory in order to understand the statement
and proof of Theorem \ref{thm.1}. Nevertheless, we want to add some
philosophical comments, for the benefit of the willing reader.
According to Witten (see \cite{Wi}), the Jones polynomial
$J_{K,n}$ can be expressed by a partition function of a topological quantum
field theory in $3$ dimensions---a gauge theory with Chern-Simons Lagrangian.
The stationary points of the Lagrangian
correspond to $SU(2)$-flat connections on an ambient manifold, and the
observables are knots, colored by the $n$-dimensional irreducible
representation of $SU(2)$. In case of a knot in $S^3$, there is only one
ambient flat connection, and the corresponding perturbation theory
is a formal power series in $h=\log q$.
Rozansky exploited a cut-and-paste property of the Chern-Simons path integral
and considered perturbation theory of the knot complement, along an abelian
flat connection with monodromy given by \eqref{eq.rho}. In fact, Rozansky
calls such an expansion the $U(1)$-{\em RCC connection contribution} to the
Chern-Simons path integral, where RCC stands for {\em reducible connection
contribution}, and $U(1)$ stands for the fact that the flat $SU(2)$
connections are actually $U(1)$-valued abelian connections.
Formal properties of such a perturbative expansion, enabled Rozansky to
deduce (in physics terms) the loop expansion of the colored Jones function.
In a later publication, Rozansky proved the existence of the loop expansion
using an explicit state-sum description of the colored Jones function.
Of course, perturbation theory means studying formal power series that rarely
converge. Perturbation theory at the trivial flat connection in a knot
complement converges, as it resums to a Laurent polynomial in $e^h$; namely
the colored Jones polynomial.
The volume conjecture for small complex angles is precisely the
statement that perturbation theory for abelian flat connections (near the
trivial one) does converge.
At the moment, there is no physics (or otherwise) formulation of perturbation
theory of the Chern-Simons path integral along a discrete and faithful
$\mathrm{SL}_2(\mathbb C)$ representation. Nor is there an adequate explanation of the
relation between $SU(2)$ gauge theory (valid near $\alpha=0$) and
a complexified $\mathrm{SL}_2(\mathbb C)$ gauge theory, valid near $\alpha=2\pi i$.
These are important and tantalizing questions, with no answers at present.
\subsection{WKB}
\lbl{sec.WKB}
Since we are discussing physics interpretations of Theorem \ref{thm.1}
let us make some more comments. Obviously, when the angle
$\alpha$ is sufficiently big, the asymptotic expansion of Equation \eqref{eq.thm1}
may break down. For example, when $e^{\alpha}$ is a complex root of the Alexander
polynomial, then the right hand side of \eqref{eq.thm1} does not make sense,
even to leading order. In fact, when $\alpha$ is near $2 \pi i$, then the solutions
are expected to grow exponentially, and not polynomially, according to the
Volume Conjecture.
The breakdown and change of rate of asymptotics is
a well-documented phenomenon well-known in physics, associated with WKB
analysis, after Wentzel-Krammer-Brillouin; see for example \cite{O}.
In fact, one may obtain an independent proof of Theorem \ref{thm.1}
using {\em WKB analysis}, that is, the study of asymptotics of solutions of
difference equations with a small parameter.
The key idea is that the sequence of colored Jones
functions is a solution of a linear $q$-difference equation, as was
established in \cite{GL1}. A discussion on WKB analysis of $q$-difference
equations was given by Geronimo and the first author in \cite{GG}.
The WKB analysis can, in particular, determine {\em small exponential
corrections} of the form $e^{-c_{\alpha} n}$ to the asymptotic expansion of
Theorem \ref{thm.1}, where $c_{\alpha}$ depends on $\alpha$, with $\text{Re}(c_{\alpha})
<0$ for $\alpha$ sufficiently small.
These exciting small exponential corrections
cannot be captured by classical asymptotic analysis (since they vanish
to all orders in $n$), but they are important and dominant (i.e.,
$\text{Re}(c_{\alpha})>0)$ when $\alpha$ is
near $2 \pi i$, according to the volume conjecture. Understanding the change of
sign of $\text{Re}(c_{\alpha})$ past certain so-called Stokes directions
is an important question that WKB addresses.
We will not elaborate or use the WKB analysis in the present paper.
Let us only mention that the loop expansion of the colored Jones function
can be interpreted as WKB asymptotics on a $q$-difference equation satisfied
by the colored Jones function.
\subsection{The main ideas}
\lbl{sub.main}
The main ideas of Theorem \ref{thm.1} is to compare three different
views of the Jones polynomial: one coming from perturbative quantum field
theory, one from a resummation of quantum field theory (known as the loop
expansion), and a third non-perturbative view, in terms of the cyclotomic
function.
The main advantage of the cyclotomic function of a knot is a key integrality
property, due to Habiro, and a priori exponential estimates for the $l^1$-norm
and quadratic bounds for the degrees of the revelant polynomials.
The latter were established in \cite{GL2}. Using these bounds, we can prove
that for small enough complex angles, a sequence of holomorphic functions
is uniformly bounded, and the limit of derivatives of any order (at zero)
exists; see Theorem \ref{thm.boundC3}.
A key lemma from complex analysis on normal families
guarantees under the above hypothesis that the sequence of holomorphic
functions converges, uniformly on compact sets, to a holomorphic function
whose derivatives (at zero) are the limits of the derivatives of the original
sequence of holomorphic functions.
\subsection{Acknowledgement}
Soon after the completion of the authors' work \cite{GL2}, H. Murakami
posted an interesting paper, in which he identified the polynomial growth
of the volume conjecture for small angles, for the case of the $4_1$ knot;
see \cite{M}.
Upon reading Murakami's paper, it became clear that the methods of \cite{GL2}
can be adapted to all knots, and to all orders, for small complex angles.
We wish to thank Murakami who motivated our present work.
\section{Three expansions of the Jones polynomial}
\lbl{sec.fti}
\subsection{Finite type invariants and the Jones polynomial}
\lbl{sub.fti}
The colored Jones function of a knot is a 2-parameter invariant, that
depends on the color $n$ and the formal parameter
$$
h=\log q.
$$
{\em Perturbative quantum field theory} (formalized mathematically by the
{\em Kontsevich integral} of a knot, and its image under the $\mathfrak{sl}_2$
{\em weight system}, described for example in \cite{B-N})
gives the following expansion of the colored Jones function:
\begin{eqnarray}
\lbl{eq.pert}
J_{K,n}(e^h) &=& \sum_{0 \leq i, 0 \leq j \leq i} c_{K,i,j} n^j h^i \\
\notag
&=& \sum_{0 \leq i, 0 \leq j \leq i} c_{K,i,j} (nh)^j h^{i-j} \\
\notag
&=& \sum_{0 \leq j, k} c_{K,j+k,j} (nh)^j h^k.
\end{eqnarray}
Here, $K\to c_{K,i,j}$ are {\em finite type knot invariants} of type $i$;
see \cite{B-N}.
The important property is that $c_{K,i,j}=0$ in the $(i,j)$ plane and
above the diagonal $i=j$. Thus, one can resum the formal power series
as follows:
\begin{eqnarray}
\lbl{eq.R}
J_{K,n}(e^h) &=& \sum_{k=0}^\infty R_{K,k}(nh) h^k,
\end{eqnarray}
where
$$
R_{K,k}(x)=\sum_{0 \leq j} c_{K,j+k,j} x^j \in \mathbb Q[[x]].
$$
\subsection{The loop expansion of the Jones polynomial}
\lbl{sub.loop}
The Melvin-Morton-Rozansky Conjecture states that
$$
R_{K,0}(x)=\frac{1}{\Delta_K(e^x)}.
$$
More generally, in \cite{Ro}, Rozansky proves that
$$
R_{K,k}(x)=\frac{P_{K,k}(e^x)}{\Delta_K(e^x)^{2k+1}}
$$
for Laurent polynomials $P_{K,k}(q) \in \mathbb Q[q^{\pm}]$.
Although the polynomials $P_{K,k}(q)$ are not finite type invariants
(with respect to the usual crossing change of knots), they are indeed
finite type invariants with respect to a loop move described in \cite{GR}.
We will not use this fact in our paper.
Rozansky conjectured that the resummation given by the above equations
could be preformed on the level of a universal perturbative invariant
(the Kontsevich integral of a knot; see \cite{B-N}), and this was proven
to be the case in \cite{GK}. As a result, one obtains a proof of this
resummation property valid for all simple Lie algebras, see \cite{Ga1}.
\subsection{The cyclotomic expansion of the Jones polynomial}
\lbl{sub.cyclotomic}
In \cite{Ha}, Habiro introduced an alternative packaging of the colored Jones
function $J_{K,n}$; using the so-called {\em cyclotomic function} $C_{K,n}$.
The latter is related to the former by the following
\begin{equation}
\lbl{eq.J2C}
J_{K,n}(q)=\sum_{k=0}^n C_{n,k}(q) C_{K,k}(q),
\end{equation}
where
\begin{eqnarray*}
\lbl{eq.cyclokernel}
C_{n,k}(q) &:=& \frac{1}{q^{n/2}-q^{-n/2}}
\prod_{j=n-k}^{n+k} (q^{j/2}-q^{-j/2}) \\
& = & \prod_{j=1}^k (( q^{n/2}-q^{-n/2})^2 - (q^{j/2}-q^{-j/2})^2) \\
& = & \prod_{j=1}^k (( q^{n/2}+q^{-n/2})^2 - (q^{j/2}+q^{-j/2})^2).
\end{eqnarray*}
Thus, in a sense $J_{K,n}$ and $C_{K,n}$ are related by a lower-diagonal
invertible matrix. For an explicit inversion of the above equation (which we
will not use in the present paper), we refer the reader to \cite[Sec.4]{GL1}.
\subsection{Comparing the cyclotomic and the loop expansion}
\lbl{sub.comparing}
So far, we have three expansions: the finite type expansion, the loop
expansion and the cyclotomic expansion. Now, we'll compare the last two.
In other words, we'll compare Equations \eqref{eq.R} and \eqref{eq.J2C}.
Let
$$
q=e^h, \qquad x=nh.
$$
For a function $f(q)$, let us denote by $\langle f \rangle_k$ the $k$-th coefficient
in the Taylor expansion of $f(e^h)$ around $h=0$.
Of course,
$$
\langle f \rangle_k=\frac{1}{k!} \frac{d^k}{dh^k}|_{h=0} f(e^h).
$$
In other words, we have:
$$
f(e^h)=\sum_{k=0}^\infty \langle f \rangle_k h^k \in \mathbb Q[[h]].
$$
\begin{lemma}
\lbl{lem.compare2}
\rm{(a)} For every knot $K$, we have the following equality
in $\mathbb Q[[x,h]]$:
$$
\sum_{k=0}^\infty R_{K,k}(x) h^k = \sum_{k=0}^\infty C_{K,k}(e^h)
\prod_{j=1}^k (e^{x/2}-e^{-x/2})^2-(e^{jh/2}-e^{-jh/2})^2) \in \mathbb Q[[x,h]].
$$
\rm{(b)} It follows that for every $k$,
$$
R_{K,k}(x)=\sum_{l=0}^\infty
\sum_{j=0}^k \langle C_{K,l} \rangle_j z^{2l-[j/2]} p_{l,j,k}(z)
$$
where
$$
z=e^{x/2}-e^{-x/2},
$$
and $p_{l,j,k}(z)$ is an even polynomial of $z$ of degree $[j/2]$, with
coefficients polynomials of $l$ of degree $k+1$.
\newline
\rm{(c)}
In particular, we have:
\begin{eqnarray*}
R_{K,0}(x) &=& \sum_{l=0}^\infty \langle C_{K,l} \rangle_0 z^{2l} \\
R_{K,1}(x) &=& \sum_{l=0}^\infty \langle C_{K,l} \rangle_1 z^{2l} \\
R_{K,2}(x) &=& \sum_{l=0}^\infty \langle C_{K,l} \rangle_2 z^{2l} -
\sum_{l=0}^\infty \langle C_{K,l} \rangle_0 \frac{l(l+1)(2l+1)}{6} z^{2l-2} \\
R_{K,3}(x) &=& \sum_{l=0}^\infty \langle C_{K,l} \rangle_3 z^{2l} -
\sum_{l=0}^\infty \langle C_{K,l} \rangle_1 \frac{l(l+1)(2l+1)}{6} z^{2l-2}
\end{eqnarray*}
in $\mathbb Q[[x]]$.
\end{lemma}
\begin{proof}
It follows easily, working in the ring $\mathbb Q[[x,h]]$, and using the fact
that the map:
$$
\mathbb Q(e^x)[[h]] \longrightarrow \mathbb Q[[x,h]]
$$
given by $e^x=\sum_{k=0}^\infty x^k/k!$ is 1-1.
\end{proof}
\section{Proof of Theorem \ref{thm.11}}
\lbl{sec.proofs}
Let us assume for the moment the following theorem, whose proof
will be given in the next section.
\begin{theorem}
\lbl{thm.boundC3}
\rm{(a)}
For every knot $K$ there exist an open neighborhood $U_K$ of $0 \in \mathbb C$
and a positive number $M$ such
that for $\alpha \in U_K$, and all $n \geq 0$, we have:
$$
|J_{K,n}(e^{\alpha/n})| < M.
$$
\rm{(b)}
Moreover, for every $m \geq 0$, the following limit exists and given by:
$$
\lim_{n\to\infty} \frac{d^m}{d \alpha^m}|_{\alpha=0} J_{K,n}(e^{\alpha/n})
=m! \,\, \mathrm{coeff}\left( \frac{1}{\Delta_K(e^{\alpha})}, \alpha^m \right).
$$
\end{theorem}
\subsection{A lemma from complex analysis}
\lbl{sub.complex}
The proof of Theorem \ref{thm.1} will use the following lemma on normal
families that is sometimes refered to by the name of Vitali and
Montel's theorem. For a reference, see \cite{Hi,Sch}. The lemma exhibits
the power of holomorphy, coupled with uniform boundedness.
Let $\Delta_r=\{z \in \mathbb C \, : \, |z| < r \}$
denote the open complex disk around $0$ of radius $r >0$.
\begin{lemma}
\lbl{lem.complex}
If $f_n: \Delta_r \to \bar\Delta_M$ is a sequence of holomorphic functions such
that for every $m \geq 0$, we have:
$$
\lim_{n \to \infty} f^{(m)}_n(0) =a_m.
$$
Then,
\begin{itemize}
\item
The limit $f(z)=\lim_n f_n(z)$ exists pointwise for $z \in D_r$.
\item
$f: D_r \to \bar\Delta_M$ is holomorphic,
\item
The convergence is uniform on compact subsets, and
\item
For every $m$, $f^{(m)}(0)=a_m$.
\end{itemize}
\end{lemma}
\begin{proof}
$\{f_n\}_n$ is uniformly bounded, so it is a normal family, and contains
a convergent subsequence $f_j\to f$. Convergence is uniform on compact sets,
and $f$ is holomorphic, and for every $m \geq 0$,
$\lim_j f_j^{(m)}(0)=f^{(m)}(0)=a_m$.
If $\{f_n\}_n$ is not convergent, since it is a normal family,
then there exist two subsequences that converge to $f$ and $g$ respectively,
with $f \neq g$.
Applying the above discussion, it follows that $f$ and $g$ are holomorphic
functions with equal derivatives of all orders at $0$. Thus, $f=g$, giving
a contradiction. Thus, $\{f_n\}_n$ is convergent and the result follows
from the above discussion.
\end{proof}
\begin{remark}
\lbl{rem.necessarynormal}
We have seen that the hypotheses in Lemma \ref{lem.complex} are sufficient
to ensure existence of the limit and uniform convergence on compact sets.
It is easy to see that these hypotheses are also necessary.
\end{remark}
\subsection{Proof of Theorem \ref{thm.11}}
\lbl{sub.proofthm11}
Fix a knot $K$ and an open neighborhood $U_K$ of $0 \in \mathbb C$ as in
Theorem \ref{thm.boundC3}.
Theorem \ref{thm.boundC3} and Lemma \ref{lem.complex} imply that
for $\alpha \in U_K$,
$$
\lim_{n\to\infty}J_{K,n}(e^{\alpha/n})
= \frac{1}{\Delta_K(e^{\alpha})}.
$$
Moreover, convergence with respect to $\alpha$ is uniform on compact subsets
of $U_K$. This proves Theorem \ref{thm.11}.
\qed
\section{Estimates of the cyclotomic function}
\lbl{sec.estimates}
This section is devoted to the proof of Theorem \ref{thm.boundC3}.
Our main tool will be estimates in the cyclotomic expansion of a knot,
similar to the ones used in \cite{GL2}.
A key result of Habiro is an {\em integrality property}
of the cyclotomic function $n\to C_{K,n}$ of a knot. Namely,
$$
C_{K,n}(q) \in \mathbb Z[q^{\pm}]
$$
for all knots $K$ and all $n$; see \cite{Ha}.
We will use two further results from \cite{GL2}: an exponential
bound on the size of the coefficients of $C_{K,n}$,
and a quadratic bound on the min and max degrees of $C_{K,n}$.
Recall that for a Laurent polynomial
$f(q)=\sum_k a_k q^k$, we define its $l^1$ norm by
$$
||f||_1=\sum_k |a_k|.
$$
\begin{theorem}
\lbl{thm.boundC}
\rm{(a)} For every knot $K$ we have:
\begin{equation}
\lbl{eq.LC}
||C_{K,n}||_1 \leq e^{C n + C' \log n}
\end{equation}
\rm{(b)} Moreover,
$$
\mathrm{maxdeg}_q (C_{K,n}) =O(n^2), \qquad \mathrm{mindeg}_q (C_{K,n})
=O(n^2).
$$
\end{theorem}
Here, and below, the $O(f(n))$ notation means that a quantity bounded
by a constant times $f(n)$.
\begin{theorem}
\lbl{thm.boundC2}
For every knot $K$, there exist constants $C, C', C''$
and $C'''$ (that depend on $K$) such that for all $n \geq 0$ and $k \geq 0$
we have:
\begin{equation}
\lbl{eq.boundC2}
|C_{K,n}^{(k)}(e^{\alpha})| \leq e^{C n + C' (k+1)
\log n + \Re(\alpha) C'' n^2 + C'''},
\end{equation}
where $C_{K,n}^{(k)}$ denotes the $k$-th derivative of $C_{K,n}(e^h)$
with respect to $h$.
\end{theorem}
\begin{proof}
Let us write
$$
C_{K,n}(q)=\sum_{j=-C_1 n^2}^{C_1 n^2} a_{j,n} q^j.
$$
Then,
$$
C_K^{(k)}(e^h)=\sum_{j=-C_1 n^2}^{C_1 n^2} a_{j,n} j^k e^{jh}.
$$
We will estimate each coefficient and each monomial by:
\begin{eqnarray*}
|a_{j,n}| & \leq & ||C_{K,n}||_1 \leq e^{C n + C' \log n} \\
|j|^k & \leq & (C_1 n^2)^k \\
|e^{\alpha j}| & \leq & e^{|\Re(\alpha)| C_1 n^2}.
\end{eqnarray*}
The result follows.
\end{proof}
\begin{corollary}
\lbl{cor.boundC2}
With the notation of Theorem \ref{thm.boundC2}, for every $n \geq 0$
and $0 \leq k \leq n$, and $0 \leq l \leq n$, we have:
$$
|C_{K,k}^{(l)}(e^{\alpha/n})| \leq e^{C k + C'(l+1) \log k
+ |\Re(\alpha)| C'' k + C'''}
$$
\end{corollary}
Let us recall an elementary estimate from \cite[Sec.3]{GL2}.
\begin{lemma}
\lbl{lem.estimate}
There exist positive constants $C_1, C_2$ and $C_3$, so that for all complex
numbers $\alpha$ with $0 < \Re(\alpha) < 1/6$, and for every $0 \leq k < n$
we have:
$$
|C_{n,k}(e^{\alpha/n})| \leq e^{C_1 k \log|\alpha| + C_2 \log k + C_3}.
$$
\end{lemma}
\begin{proof}(of Theorem \ref{thm.boundC3})
Combining Corollary \ref{cor.boundC2}
and Lemma \ref{lem.estimate}, it follows that for all $0 \leq k \leq n$,
we have:
$$
|C_{n,k}(e^{\alpha/n}) C_{K,k}(e^{\alpha/n})| \leq
e^{C k + C' \log k + |\Re(\alpha)| C'' k + C''' + C_1 k \log|\alpha| +
C_2 \log k + C_3}.
$$
Let us choose $\alpha \in U_K$, where
\begin{equation}
\lbl{eq.UK}
U_K = \{\alpha \in \mathbb C \, | \, C+C'' |\Re(\alpha)| + C_1 \log|\alpha| <0 \}.
\end{equation}
Then, equation \eqref{eq.J2C} and the
above estimate conclude the first part of Theorem \ref{thm.boundC3}.
The second part follows from Equation \eqref{eq.R} and the MMR Conjecture.
Indeed, consider the sequence
$$
f_n: U_K \to \{z: \, |z| < N\}, \qquad
\alpha \to f_n(\alpha)=J_{K,n}(e^{\alpha/n}).
$$
Since $J_{K,n}(q)$ is a Laurent polynomial in $q$, it follows that
$f_n$ is an entire function. Equation \eqref{eq.R} implies that
$$
f_n(\alpha)=\sum_{k=0}^\infty R_{K,k}(\alpha) \left( \frac{\alpha}{n} \right)^k.
$$
Thus, for every $m \geq 0$,
$$
f_n^{(m)}(0)=m! \left( \mathrm{coeff}(R_{K,0}(\alpha),\alpha^m) +
\frac{1}{n} \mathrm{coeff}(R_{K,1}(\alpha),\alpha^{m-1}) + \dots
\frac{1}{n^m} \mathrm{coeff}(R_{K,m}(\alpha),\alpha^0) \right).
$$
Thus, using the MMR Conjecture, we obtain:
\begin{eqnarray*}
\lim_{m\to\infty} f_n^{(m)}(0) &=& m! \,\, \mathrm{coeff}(R_{K,0}(\alpha),\alpha^m) \\
&=& m! \,\, \mathrm{coeff} \left( \frac{1}{\Delta_K(e^{\alpha})},\alpha^m \right).
\end{eqnarray*}
The result follows.
\end{proof}
\section{Proof of Theorem \ref{thm.1}}
\lbl{sec.allorders}
To leading order (i.e., $N=0$ in \eqref{eq.Zratalt}) Theorem \ref{thm.1}
is Theorem \ref{thm.11}.
By now, it should be clear the strategy for proving Theorem \ref{thm.1}
to all orders. To simplify notation, let us define:
\begin{equation}
\lbl{eq.JN}
J_{K,n}^{(N)}(e^{\alpha/n})=J_{K,n}(e^{\alpha/n})-\sum_{k=0}^{N-1}
\frac{P_{K,k}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2k+1}} \left(\frac{\alpha}{n}\right)^k.
\end{equation}
Theorem \ref{thm.1} follows from the following result
and the argument of Section \ref{sub.proofthm11}.
\begin{theorem}
\lbl{thm.boundC4}
\rm{(a)}
For every knot $K$
there exists an open neighborhood $U_K$ of $0 \in \mathbb C$ such that
for every $N \geq 0$ there exists a positive number $M_N$ such
that for $\alpha \in U_K$, and all $n \geq 0$, we have:
$$
\left|\left( \frac{n}{\alpha} \right)^N
J_{K,n}^{(N)}(e^{\alpha/n}) \right| < M_N.
$$
\rm{(b)}
Moreover, for every $m \geq 0$, the following limit exists and given by:
$$
\lim_{n\to\infty} \frac{d^m}{d \alpha^m}|_{\alpha=0}
\left( \left( \frac{n}{\alpha} \right)^N J_{K,n}^{(N)}(e^{\alpha/n}) \right)
=m! \,\, \mathrm{coeff}\left(
\frac{P_{K,N}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2N+1}}, \alpha^m \right).
$$
\end{theorem}
\begin{proof}
We will prove the theorem by induction on $N$. For $N=0$, this
is Theorem \ref{thm.11} proven in Section \ref{sec.proofs}.
Let us assume that it is true for $N-1$.
Let us define for every $k \geq 0$,
two auxiliary biholomorphic functions
\begin{eqnarray*}
c_k(x,\epsilon) &=&
\prod_{j=1}^k (e^{x/2}-e^{-x/2})^2-(e^{jh/2}-e^{-jh/2})^2), \\
g_{K,k}(x,\epsilon) &=& c_k(x,\epsilon) C_{K,k}(e^{\epsilon}).
\end{eqnarray*}
Thus, using the definition of $C_{n,k}$ and Equation \eqref{eq.J2C},
it follows that:
\begin{equation}
\lbl{eq.now1}
C_{n,k}(e^{\alpha/n})=c_k(\alpha,\alpha/n), \qquad
J_{K,n}(e^{\alpha/n})=\sum_{k=0}^n g_{K,k}(\alpha,\alpha/n).
\end{equation}
For a function $h=h(x)$, let us define the $N$-th Taylor approximation by:
$$
\mathrm{Taylor}^N(h,x)=\sum_{j=0}^N \frac{h^{(j)}(0)}{j!} x^j.
$$
Applying Lemma \ref{lem.compare2} to the function $\epsilon\to g_{K,k}(\alpha,\epsilon)$,
and evaluating at $\epsilon=\alpha/n$, it follows that:
\begin{eqnarray}
\sum_{k=0}^{N-1}
\frac{P_{K,k}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2k+1}} \left(\frac{\alpha}{n}\right)^k &=&
\sum_{k=0}^\infty \mathrm{Taylor}^{N-1}(g_{K,k}(\alpha,\cdot), \frac{\alpha}{n}) \\
&=&
\lbl{eq.now2}
\sum_{k=0}^n \mathrm{Taylor}^{N-1}(g_{K,k}(\alpha,\cdot), \frac{\alpha}{n})
+ \text{err}_n(\alpha).
\end{eqnarray}
Equations \eqref{eq.JN}, \eqref{eq.now1} and \eqref{eq.now2} and Taylor's
theorem imply that:
\begin{eqnarray*}
J_{K,n}^{(N)}(e^{\alpha/n})&=&
J_{K,n}(e^{\alpha/n})-\sum_{k=0}^{N-1}
\frac{P_{K,k}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2k+1}} \left(\frac{\alpha}{n}\right)^k \\
&=& \sum_{k=0}^n g_{K,k}(\alpha,\alpha/n)-
\sum_{k=0}^n \mathrm{Taylor}^{N-1}(g_{K,k}(\alpha,\cdot), \frac{\alpha}{n})
- \text{err}_n(\alpha) \\
&\approx& \left(\frac{\alpha}{n}\right)^N \sum_{k=0}^n
\,
\frac{1}{N!} \frac{\partial^N}{\partial \epsilon^N}|_{\epsilon \approx \alpha/n}
g_{K,k}(\alpha,\epsilon)
-\text{err}_n.
\end{eqnarray*}
The analytiticy of $g_{K,k}$ and Theorem \ref{thm.boundC2} implies that
there exists a positive $M'_N$ such that for all $\alpha \in U_K$
(defined in \ref{eq.UK}), we have:
$$
|\text{err}_n(\alpha)| < M'_N.
$$
Corollary \ref{cor.boundC2} and Equation \eqref{eq.now1} imply that there
exists a positive $M_N$ such that
$$
|\left(\frac{n}{\alpha}\right)^N J_{K,n}^{(N)}(e^{\alpha/n})| < M_N
$$
for all $n \geq 0$ and for all $\alpha \in U_K$.
This proves part (a) of Theorem \ref{thm.boundC4}.
For part (b), we will use Equation \eqref{eq.R}, which implies that:
$$
J_{K,n}^{(N)}(e^{\alpha/n})=\sum_{k=N}^\infty R_{K,k}(\alpha)
\left(\frac{\alpha}{n}\right)^{k}.
$$
Thus, for every $m \geq 0$,
\begin{eqnarray*}
\frac{d^m}{d \alpha^m}|_{\alpha=0}
\left( \left( \frac{n}{\alpha} \right)^N J_{K,n}^{(N)}(e^{\alpha/n}) \right)
&=& m! \left( \mathrm{coeff}(R_{K,N}(\alpha),\alpha^m) +
\frac{1}{n} \mathrm{coeff}(R_{K,N+1}(\alpha),\alpha^{m-1}) + \right. \\ & & \dots +
\left. \frac{1}{n^m} \mathrm{coeff}(R_{K,N+m}(\alpha),\alpha^0) \right).
\end{eqnarray*}
Using Rozansky's theorem \ref{thm.Zrat} and Equation \eqref{eq.Zratalt},
we obtain:
\begin{eqnarray*}
\lim_{m\to\infty} \frac{d^m}{d \alpha^m}|_{\alpha=0}
\left( \left( \frac{n}{\alpha} \right)^N J_{K,n}^{(N)}(e^{\alpha/n}) \right)
&=& m! \,\, \mathrm{coeff} (R_{K,N}(\alpha),\alpha^m) \\
&=&
m! \,\, \mathrm{coeff} \left(\frac{P_{K,N}(e^{\alpha})}{\Delta_K(e^{\alpha})^{2N+1}},
\alpha^m \right).
\end{eqnarray*}
The result follows.
\end{proof}
\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\,}
\fi
|
{
"timestamp": "2005-04-04T16:14:11",
"yymm": "0503",
"arxiv_id": "math/0503641",
"language": "en",
"url": "https://arxiv.org/abs/math/0503641"
}
|
\section{Introduction}
Alternative models for quantum computation based on projective measurements \cite{Raussen01, Nielsen01, Leung03} have recently attracted much attention. A common concept of these models is the simulation of individual quantum circuit operations and how simulations can be composed together \cite{Childs04}. More specifically, in these models a sequence of single- or two-qubit measurements is applied to a collection of fixed initial quantum states thereby in effect simulating unitary transformations on a smaller subspace of states.
Although our results are applicable to a much larger class of measurement-based models, our analysis will focus on a variation of Raussendorf and Briegel's one-way quantum computer model \cite{Raussen01} where computation is performed by single-qubit projective measurements on
some initial \emph{graph state} \cite{Raussen03}. Henceforth, we will refer to this model as the {\em graph-state model} and the computation realized in it as a {\em graph-state simulation}.
The graph-state model offers a decomposition of a quantum algorithm in terms of alternative elementary primitives, as well as potential advantages in certain physical implementations. For example, suppose entangling gates can only be realized nondeterministically with flagged faults as, e.g., in optical quantum computation \cite{Knill00}. Then, graph-state simulation offers much advantage since entangling gates are only used for the preparation of the graph state, which can be done independently from the main computation \cite{Nielsen04}.
In any physical realization of quantum computation, unknown errors will always be present and they will have to be corrected using quantum error-correcting codes in a fault-tolerant manner.
An important question is therefore under what conditions computation can be executed reliably in the graph-state model in the presence of physical noise. In the circuit model, fault-tolerant methods \cite{Shor96} are available for the reliable execution of any desired computation, if the noise is sufficiently weak. Now, if such a fault-tolerant circuit is {\em simulated} in the graph-state model with sufficiently weak noise, will the same desired computation be reliably executed? More specifically, this is answered by first analyzing noisy simulations of individual operations and then how the noisy simulations compose together.
The first results on this problem were reported in the Ph.D.\ thesis of Raussendorf \cite{Raussen03b}. This work proved the existence of an accuracy threshold for cluster-state computation for various independent stochastic error models. More recently, Nielsen and Dawson \cite{Nielsen04b} obtained proofs for the existence of an accuracy threshold in the graph-state model that apply to more general error models (including errors due to nondeterministic gates) by reduction to a threshold theorem for local non-Markovian noise \cite{Terhal04}. In addition, they established a conceptual framework and two technical theorems that are of independent interest.
In this paper, we use the concept of composable simulations \cite{Childs04} and the threshold theorem derived in the circuit model \cite{Aharonov99comb, Knill96b, Kitaev97b, Terhal04, Aliferis05b} to obtain a simple proof for the existence of an accuracy threshold in the graph-state model.
Furthermore, for any specific form of noise, our proof allows known lower bounds on the threshold in the circuit model to be translated to equivalent bounds in the graph-state model. We discuss in particular how the two lower bounds are related for commonly used fault-tolerant architectures based on self-dual CSS codes.
\section{Review of the graph-state model}
We begin by briefly introducing the graph-state model
in the ideal noiseless case. Various recent interpretations of this model have been reported and reviewed recently \cite{Verstraete03}, \cite{Aliferis04}, \cite{Childs04}, \cite{Perdrix04b}, \cite{Nielsen05b}, \cite{Jozsa05}.
We follow the language in Ref.$\,$\cite{Childs04}, which explicitly uses the notion of composable simulations that forms the core of our subsequent analysis. Our discussion in this section is intended to introduce the basic notions and terminology that we will use later in our proof.
In the circuit model, an arbitrary quantum computation can be decomposed into state preparation, measurements, and a universal set of gates. To show the universality of the graph-state model of quantum computation, it suffices to show that (i) each element for universality in the standard model can be simulated and (ii) the simulation can be composed to simulate the entire computation. The approach is to first define an appropriate notion of simulation that is composable, followed by a complete recipe to composably simulate each element needed for universality.
We first describe the notion of composable simulations. Let $\mathcal{F}$ be an operation (a superoperator, or completely positive trace-preserving map) to be simulated and $\mathcal{S}_\mathcal{F}$ be the associated operation that simulates $\mathcal{F}$. For simplicity, let $\mathcal{F}$ act on $n$ qubits. In the general case, $\mathcal{F}$ can have quantum and classical input and output of arbitrary dimensions, but this only requires extra notations and therefore will not be written out explicitly here. For a $2n$-bit string $x$, let $\mathcal{P}_x$ be the superoperator corresponding to conjugation by
the Pauli operator indexed by $x$.
Our composable simulation $\mathcal{S}_\mathcal{F}$ takes two inputs, a classical $2n$-bit string $e_{\rm in}$ and an $n$-qubit quantum state $\mathcal{P}_{e_{\rm in}} (\rho_{\rm in})$, so that $\forall \rho_{\rm in}$, $\forall e_{\rm in}$, it acts as
\begin{equation}
\label{eq:compdef}
\mathcal{S}_\mathcal{F} ( e_{\rm in} \otimes \mathcal{P}_{e_{\rm in}} (\rho_{\rm in}) )
= \sum\limits_{e_{\rm out}} p_{e_{\rm out}} e_{\rm out} \otimes (\mathcal{P}_{e_{\rm out}} \! \circ \mathcal{F}) (\rho_{\rm in}) \, ,
\end{equation}
\noindent where $e_{\rm out}$ is some new $2n$-bit string that appears with probability $p_{e_{\rm out}}$. (Throughout the paper, the symbol for a bit string such as $e_{\rm in}$ also labels the corresponding density matrix.) To rephrase the above definition, for each specific classical output $e_{\rm out}$, $\mathcal{S}_\mathcal{F}$ evolves the arbitrary state $\rho_{\rm in}$ according to the intended operation $\mathcal{F}$ up to a new known succeeding Pauli operation $\mathcal{P}_{e_{\rm out}}$, despite the $\mathcal{P}_{e_{\rm in}}$ occurring to $\rho_{\rm in}$ prior to the simulation.
Note that $e_{\rm out}$ is a function of $e_{\rm in}$ and the measurement outcomes obtained in $\mathcal{S}_\mathcal{F}$, and this function depends on $\mathcal{S}_\mathcal{F}$.
However, the statistics of $e_{\rm out}$ has no consequence, because composable simulations work for \emph{all} measurement outcomes and for all $e_{\rm in}$---all outcome histories lead to valid simulations, where an ``outcome history'' denotes the set of all measurement outcomes collected in a specific run of the simulation.
As we will see next, this is important as it will allow us to compose simulations of individual operations to obtain a simulation of the combined operation.
Now, consider simulating a sequence of $l$ operations $\{\mathcal{F}_j\}$, and we will see that it can be done by composing the sequence of simulations $\{\mathcal{S}_{\mathcal{F}_j}\}$. By repeated applications of \eq{compdef}, $\forall \rho_{\rm in}$, $\forall e_{\rm in}$,
\begin{equation}
\begin{array}{c}
\label{sequence}
\mathcal{S}_{\mathcal{F}_l} \circ \cdots \circ \mathcal{S}_{\mathcal{F}_1} (e_{\rm in} \otimes \mathcal{P}_{e_{\rm in}} (\rho_{\rm in}) ) \hspace*{15ex} \\[1.2ex]
= \sum\limits_{e_{\rm out}} p_{e_{\rm out} } e_{\rm out} \otimes (\mathcal{P}_{e_{\rm out}} \circ \mathcal{F}_l \circ \cdots \circ \mathcal{F}_1) (\rho_{\rm in}) \, ,
\end{array}
\end{equation}
\noindent which states that, for all outcome histories, the entire sequence of operations $\{\mathcal{F}_j\}$ is simulated properly, up to a final overall $\mathcal{P}_{e_{\rm out}}$ (which just redefines the final classical outcome of the computation).
We will now describe how composable simulations are realized in the graph-state model. Let $\Gamma$ denote a graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$.
One way to specify and to create the graph state corresponding to $\Gamma$ is to start with the initial state $\bigotimes_{i\in V(\Gamma)} |+\>$ and then apply a controlled-phase ({\sc cphase}) gate to each pair of qubits in $E(\Gamma)$ (where {\sc cphase}$\,|ab\> = (-1)^{ab}|ab\>$ in the computation basis). In other words, each vertex corresponds to a qubit initially in the state $|+\>$, and each edge corresponds to a subsequent {\sc cphase}.
As precursor to a graph-state simulation, our next step is to composably simulate a universal set of circuit elements (state preparation, measurements, and a universal set of gates), using single-qubit measurements and {\sc cphase}.
In the circuit model, it suffices to prepare any Pauli eigenvector and measure along any Pauli basis. Both of these can be trivially simulated in the graph-state model using single-qubit measurements. For the universal set of gates, we choice the Clifford group generators $\{H, S\equiv e^{-i \sigma_{\rm z} \pi /4},$ {\sc cphase}$\}$ and the additional non-Clifford $T\equiv e^{-i \sigma_{\rm z} \pi /8}$. Here $\{\sigma_{\rm x}, \sigma_{\rm z} \}$ denote the standard Pauli operators.
Figure \ref{meas-patt} shows how to composably simulate these gates,
with the classical registers omitted for simplicity.
In Fig.\ \ref{meas-patt}, qubits are represented as circles. The boxed circles contain the quantum inputs, unboxed ones are prepared in $|+\>$, and open circles (unmeasured qubits) contain the quantum outputs.
Edges denote {\sc cphase} gates acting on the adjoined qubits. The measurement bases for each qubit are given in the circle.
The quantum state at the input of each pattern has known Pauli corrections labeled by the classical register $e_{\rm in}$ (not shown), which depends on past measurement outcomes. In the simulation of $T$, $e_{\rm in}$ is used to control one of the quantum measurements. The output quantum state also has Pauli corrections labeled by an updated string $e_{\rm out}$. Each simulation pattern defines an update rule, mapping $e_{\rm in}$ and measurement outcomes obtained in the pattern to $e_{\rm out}$.
\begin{figure}[h]
\begin{center}
\epsfig{file=1.eps}
\vspace{0.1cm}
\caption{\label{meas-patt}
\footnotesize{
Composable simulations for (a) the Hadamard gate ($H$), (b) the rotation around the $z$-axis by $\pi/2$ ($S$), (c) the {\sc cphase} and (d) the rotation around the $z$-axis by $\pi/4$ ($T$). Note that we use the {\sc cphase} to simulate itself, since it can be built in as a vertical edge of the graph. We have omitted the input classical registers and their updates for simplicity. The symbols $M_{\!X}$ and $M_{\!Y}$ indicate measurements of $\sigma_{\rm x}$ or $\sigma_{\rm y}$ on the corresponding qubits, and $M_{T}$ indicates a measurement of the observable $(\sigma_{\rm x} {\pm} \sigma_{\rm y}) / \sqrt{2}$ depending on whether there is a $\sigma_{\rm x}$ correction in the input qubit.
}
}
\end{center}
\end{figure}
Any circuit (sequence of gates and measurements on standard initial states) can then be simulated by composing a sequence of simulations by identifying the quantum output of one simulation (the open circle) with the input to the next (the boxed circles) and similarly for the classical registers. The combined simulation thus consists of single-qubit measurements on qubits prepared in a graph state (with the {\sc cphase} being part of the graph state preparation), giving a complete recipe for the entire graph-state simulation. Note that evolutions of single qubits and their interactions ({\sc cphase}) in the simulated circuit are represented in the graph as linear paths and the links between them, respectively. As an example, Fig.$\,$\ref{meas-patt:cnot} shows how a composition of the measurement patterns for the simulation of $H$ and {\sc cphase} leads to a new pattern for the simulation of the operation {\sc cnot}$\,=(I\otimes H)${\sc cphase}$(I\otimes H)$.
\begin{figure}[h]
\begin{center}
\epsfig{file=2.eps}
\vspace{0.2cm}
\caption{\label{meas-patt:cnot}
\footnotesize{
A schematic diagram of the composition of the patterns in Fig.$\,$\ref{meas-patt}(a), Fig.$\,$\ref{meas-patt}(c), and Fig.$\,$\ref{meas-patt}(a) that simulates {\sc cnot}$\,=(I\otimes H)\,${\sc cphase}$\,(I\otimes H)$. In the dashed ellipse on the left, the output of the measurement pattern simulating the first $H$ is identified with the input for the lower qubit of the {\sc cphase} simulation, whose output for the same qubit is identified with the input qubit of the simulation of the second $H$ (right ellipse). On the right is the result of the composition.
}
}
\end{center}
\end{figure}
\section{Noisy graph-state computation}
We now investigate how noise at the level of the graph-state simulation maps to noise in the simulated operations and, most importantly, whether such ``simulated noise'' can be tolerated by simulating a fault-tolerant circuit described in the circuit model. We begin by mentioning a modification to the graph-state model that is necessary for fault tolerance. Since the ability to prepare fresh qubits and interact them with the existing ones is essential for all fault-tolerant constructions \cite{Aharonov96b}, instead of creating the entire graph state before computation, a minimal modification to the model is to build the required graph state dynamically as the computation proceeds \cite{Raussen01,Raussen03b,Nielsen04,Nielsen04b}.
The simulated circuit defines a partial time ordering of the simulations and the measurements used therein, inducing a partial ordering of the qubits in the graph state. The qubits can be added slightly before their preceding neighbors are measured, as long as the {\sc cphase} gates are applied according to the time ordering of the simulations.
This change in the model still preserves the appealing feature of the graph-state model in that all unitary interactions are applied prior to and independent of the measurements that realize the computation.
Coming to the main part of this paper, we must analyze how physical noise affects the elementwise simulations and how the noisy simulations compose together. The elementary steps in the simulation are the preparation of $|+\>$, the {\sc cphase}, the single-qubit measurements, and the storage of qubits. Moreover, each operation belongs to a unique simulation. Thus, noise afflicting a given operation only acts within one simulation. In particular, an erroneous {\sc cphase} cannot affect two successive simulations.
In any noise model and without loss of generality, each noisy state preparation or noisy gate can be expressed as the ideal operation followed by a \emph{noise operation}. Hence, noise operations intersperse pairs of successive ideal operations. A noise operation is a system-environment coupling,
and it can always be
described by some unitary joint evolution
\begin{equation}
\label{eq:expansion}
U_{\rm fault} = I \otimes A_{0} + \sum_i P_i \otimes A_{i} \, ,
\vspace*{-1ex}
\end{equation}
\noindent where $P_i$ ranges over all nontrivial Pauli operators indexed by $i$ acting on the output system of the preceding ideal operation and each $A_{i}$ is an arbitrary operator acting on the environment, subject to the condition that $U_{\rm fault}$ is unitary. A noisy measurement is modeled as the ideal measurement \emph{preceded} by a noise operation given by Eq.$\,$(\ref{eq:expansion}), with $P_i$ acting on the qubits to be measured.
We first consider independent stochastic noise processes. In this case, each noise operation is {\em by assumption} acting on a separate environmental register, which is mapped to orthogonal states by the two terms in Eq.$\,$(\ref{eq:expansion}). Physically, this assumption corresponds to the requirement that a {\em record} be kept in the environment whenever faults occur, which can in principle be read to indicate the location of faults.
In more detail, the two terms result in perfectly distinguishable environmental states, so that the corresponding states in the system {\em do not interfere} with one another, and their normalization can be interpreted as the probabilities of the first or second term in Eq.$\,$(\ref{eq:expansion}) occurring. These two terms thus correspond to the two events of not having or having a fault. We call the second term in Eq.$\,$(\ref{eq:expansion}) the \emph{fault operator} or simply the fault. A \emph{fault path} for the entire computation is an event occurring with some definite probability describing whether each noise operation results in a fault or not.
Our first goal is to show that faults within one simulation only affect that simulated operation, even though classical registers that carry the Pauli corrections and control the simulation are shared by many simulations. Consider a sequence of simulations $\{ \mathcal{S}_{\mathcal{F}_j} \}$ applied to an input $\sum_{e_{\rm in}} p_{e_{\rm in}} e_{\rm in} \otimes \mathcal{P}_{e_{\rm in}}(\rho_{\rm in})$.
Suppose some number of faults occur within $\mathcal{S}_{\mathcal{F}_1}$.
Each term in the expansion of Eq.$\,$(\ref{eq:expansion}) of all these fault operators consists of Pauli operators acting on the simulation qubits which can be commuted to the end of the simulation (since, as shown in Fig.$\,$\ref{meas-patt}, each simulation is realized by a sequence of unitary {\sc cphase}(s) and single-qubit measurements). This results in a combined fault operator, each term in the Pauli expansion of which contains some Pauli operator acting on either the output classical registers of $\mathcal{S}_{\mathcal{F}_1}$ or its quantum output, or both.
The most general erroneous output is thus given by $\sum_{e_{\rm out}} p_{e_{\rm out}}^{(1)} e_{\rm out} \otimes \rho_{\rm out}$ for some distribution $\{p_{e_{\rm out}}^{(1)}\}$, where $e_{\rm out}$ is some possibly erroneous classical output and $\rho_{\rm out} = \mathcal{E}_{e_{\rm out}} (\mathcal{P}_{e^{\rm ideal}_{\rm out} } \circ \mathcal{F}_1 (\rho_{\rm in}))$, $\mathcal{E}_{e_{\rm out}}$ is the completely positive trace non-increasing map induced by the combined fault operator on the quantum output and is conditioned on $e_{\rm out}$, and $e^{\rm ideal}_{\rm out}$ labels the ideal corrections at the output in the absence of faults inside $\mathcal{S}_{\mathcal{F}_1}$ and depends on $e_{\rm in}$.
Let $\tilde{\mathcal{E}}_{e_{\rm out}} = \mathcal{P}_{e_{\rm out}}^\dagger \circ \mathcal{E}_{e_{\rm out}} \circ \mathcal{P}_{e^{\rm ideal}_{\rm out}}$. Then, the output of $\mathcal{S}_{\mathcal{F}_1}$ can be rewritten as $\sum_{e_{\rm out}} p_{e_{\rm out}}^{(1)} e_{\rm out} \otimes \mathcal{P}_{e_{\rm out}}(\tilde{\mathcal{E}}_{e_{\rm out}} \circ \mathcal{F}_1(\rho_{\rm in}))$.
Hence, besides the extra $\tilde{\mathcal{E}}_{e_{\rm out}}$, the noisy output state is of the same form as some ideal noiseless output, with the classical register reflecting the Pauli correction on the quantum state.
In particular, this means that we can include errors in both the quantum and classical registers in $\tilde{\mathcal{E}}_{e_{\rm out}}$ and interpret it as a simulated fault operation following the simulated $\mathcal{F}_1$.
The above analysis can now be repeated to subsequent simulations, so that a simulated fault appears after each erroneous simulation. In each term labeled by $e_{\rm out}$, the simulated evolution on the system and the environment is the intended computation (the sequence $\{\mathcal{F}_j\}$) interspersed by the action of simulated fault operators (whose particular type may depend on $e_{\rm in}$ or $e_{\rm out}$ at the corresponding erroneous simulation).
We pause to discuss the above argument again. The composable simulation has been described in many different ways in the literature, such as feed-forward of measurement outcomes and propagation of by-product Pauli operations. Since the classical knowledge (correct or not) of these by-product Pauli operations from one simulation step is input to the next, it is worrisome that an error in them will feed forward, inducing highly \emph{correlated} simulated faults in the simulated circuit, even if initial faults in the simulation are uncorrelated. It only takes a shift in one's perspective and inspection of the composability requirement to recognize a simpler interpretation of the error action. In particular, errors in the classical information of the by-product Pauli operations $e_{\rm out}$ are {\em equivalent to} unknown Pauli errors in the quantum output $\rho_{\rm out}$ of the erroneous simulation. The above argument takes full advantage of the equivalence and mathematically redefines $e_{\rm out}$ to indicate {\em the} by-product Pauli operation, attributing any ``mismatch'' with an ideal simulation to noise acting on the quantum output of the simulation of $\mathcal{F}_1$ {\em alone}. {From} this point of view, the errors in classical information are localized and do not propagate. Being able to localize errors to individual simulated operations achieves a simple and direct mapping from the noise in the simulation to noise in the simulated circuit.
We can now finish the proof of the existence of an accuracy threshold for independent stochastic noise in the graph-state model, using the threshold theorem for standard quantum computation \cite{Aharonov99comb, Knill96b,Kitaev97b,Aliferis05b}:
In the circuit-model proof, certain fault paths are ``good'' and can be proved to give the ideal computation results. ``Bad'' fault paths form the complement of the ``good'' ones and have suppressed probability if the physical fault probability is below a certain critical value, the accuracy threshold.
Consider the noisy graph-state simulation of a \emph{fault-tolerant circuit}.
In the final output of the fault-tolerant circuit simulation, consider each $e_{\rm out}$ term.
Our arguments based on composability ensures that the evolution of the quantum state is simply the intended simulated operations, interspersed by the action of faults.
Since each fault path in the simulation is mapped to a unique fault path in the simulated circuit due to error localization, good fault paths in a graph-state simulation can be defined as those resulting in good fault paths in the simulated circuit \cite{note3}.
All other fault paths in the simulation are bad, and their probability will be suppressed below a certain accuracy threshold just as in the circuit model, because a simulated fault appears after some simulated operation only if there is at least one fault in its simulation.
Furthermore, the probability of this happening is at most the sum of the fault probabilities of all the elementary steps in the simulation.
Then, with reference to Fig.\ \ref{meas-patt}, we note that the simulation of each gate in our universal set involves the use of one to two {\sc cphase} gates and zero to two measurements. Therefore the probability of any simulation containing faults is bounded by $p_{\rm sim} \leq 4 p$.
More specifically, if $p_0$ is the threshold value of the fault-tolerant architecture used in the circuit model and if $p \leq p_0 \, / 4$ in the simulation, then $p_{\rm sim} \leq p_0$ and
the final measurement outcome will provide the correct computation results with the desired accuracy.
This holds for each $e_{\rm out}$ term in the final state of the simulation, thereby establishing a threshold lower bound of $p_0 \, / 4$ for the graph-state model.
In the above, we have related the accuracy threshold
in the graph-state model to that in the circuit model by the direct simulation of fault-tolerant architectures designed in the latter. However, we note that, in order to obtain the above threshold bound, we assumed that the fault-tolerant simulated circuit makes use of the same universal set as ours. In general, the same gate sets need not be used in both models, and elementary measurement patterns need to be composed to simulate a \emph{single} operation in the simulated circuit.
In particular, in most studies, {\sc cnot} rather than {\sc cphase} is used as the elementary interaction.
In this case, the measurement pattern in Fig.$\,$\ref{meas-patt:cnot} for the simulation of {\sc cnot} implies the threshold condition $p \leq p_0/5$. However, in many cases of interest this lower bound is pessimistic.
For example, in fault-tolerant designs based on self-dual CSS codes (e.g., \cite{Steane02,
Knill04, Aliferis05b}), {\sc cphase} can replace {\sc cnot} as an
alternative bitwise encoded operation and can also be used in
error correction with a small number of additional $H$
gates. Since there is no overhead for simulating single-qubit state preparation, measurement, or the {\sc cphase} in the graph-state model, the thresholds for circuits based on these codes in the circuit and graph-state models will be essentially the same.
We now proceed to prove the existence of an accuracy threshold for the
graph-state simulation for local non-Markovian noise. We will make use
of our observation of the localization of errors
and the threshold results in the circuit model \cite{Terhal04,
Aliferis05b}.
In the local non-Markovian error model, the noise operations still
have the form given by \eq{expansion} and they act on the system in the
same way as in the local Markovian model. However, different noise
operations may now act on the same environmental register, and the
term acting trivially and nontrivially on the system may not map the
environmental register to orthogonal states. Altogether, faults can
combine coherently.
Furthermore, a fault no longer corresponds to an ``event,''
in the sense that probabilities cannot be assigned. Instead, one imposes that the
strength of the fault operator at each location is bounded below a
certain value $\eta$---i.e. $|| \sum_i P_i \otimes A_i ||_{\rm
sup} \leq \eta$.
To simplify the analysis,
we consider the \emph{purification} of the graph-state simulation, where measurements are replaced by coherent operations by attaching extra ancillary qubits.
In our noise model, noisy measurements are modeled as being ideal with noise factored into the preceding noise operations, so that changing our description of the measurements does not affect the analysis.
Likewise, the classical $2n$-bit string carrying $e_{\rm in}$ can be
replaced by a $2n$-qubit register in the state $|e_{\rm in}\rangle$ and any
adaptive operations inside these equivalent simulations will be controlled
by this quantum register. The update of this register to obtain
$|e_{\rm out}\rangle$ can also be done coherently by controlling gates from the
extra ancillary qubits and also by doing the classical processing reversibly.
We emphasize that this alternative coherent description is purely mathematical and is also employed in the circuit-model proofs in Refs.~\cite{Terhal04, Aliferis05b}.
The composable simulation $\mathcal{S}_\mathcal{F}$ is now a conjugation by a unitary
operator $S_F$ taking two inputs $|e_{\rm in}\rangle$ and $P_{e_{\rm in}} |\psi_{\rm in}\rangle$ and some ancillary qubits starting in the fixed state $|+\rangle^{\otimes k}$ \cite{note2}, so that $\forall |\psi_{\rm in}\rangle$, $\forall |e_{\rm in}\rangle$ it acts as
\begin{equation}
\begin{array}{c}
\label{eq:compdef-coh}
S_F (|e_{\rm in}\rangle \otimes P_{e_{\rm in}} |\psi_{\rm in}\rangle \otimes |+\rangle^{\otimes k} )
\hspace*{15ex}\\[1.2ex]
= \sum\limits_{i} c_{i} \, |e_{\rm out}\rangle \otimes |i\> \otimes |\phi_i\> \otimes
P_{e_{\rm out}} F |\psi_{\rm in}\rangle \, ,
\vspace*{-1ex}
\end{array}
\end{equation}
\noindent where $\{ |\phi_i \> \}$ is the orthonormal basis on which measurements are to be performed, $\{ |i \rangle \}$ is the computation basis with $i$ labeling the possible measurement outcomes carried by the extra ancillas we have introduced, $c_{i}$ is the \emph{amplitude} of the $i$th term, $|e_{\rm out}\rangle$ is a $2n$-qubit state that depends on $e_{\rm in}$ and $i$, and $F$ is the simulated unitary operator.
Having expressed the fault-tolerant circuit to be simulated as well as
the graph-state simulation itself
unitarily, a unitary noise operation of
the form of \eq{expansion} is inserted \emph{at every} location in the
simulation (where locations are specified by the original graph-state
simulation before the unitary idealization). The output state is a
{\em linear superposition} of states, each evolved according to a
specific set of fault operators and expanded in the eigenbasis of all measured operators (including both measurements part of the graph-state simulation and also measurements originally in the simulated circuit).
Fault paths can again be ``good'' or ``bad,'' defined as in our discussion for independent stochastic noise.
For each term evolved by a good fault path, a final quantum state that
will provide the correct statistics will be generated, independent of
the state of the register $|i\>$ coherently carrying the measurement information
due to the localization of errors.
This is because, for each term in the Pauli expansion of faults acting
on $|e_{\rm out}\rangle$, the register $|e_{\rm out}\rangle$ can always be taken to carry the
correct Pauli correction by redefining the error acting on $P_{e_{\rm out}}
F |\psi_{\rm in}\rangle$ exactly as in our previous discussion. Therefore, for each
term in this Pauli expansion, good fault paths in the
simulation are mapped to good fault paths in the simulated circuit that produce the ideal computation results, using the threshold theorem in the circuit model.
Hence, by linearity, the whole coherent sum of these terms will also produce the ideal computation results.
It remains to bound the {\em sup norm} of the bad fault paths of the
graph-state simulation, which can combine coherently.
Following the threshold theorem in the circuit model for local
non-Markovian noise \cite{Terhal04, Aliferis05b}, it suffices to bound
the sup norm of the ``bad'' part of a given simulation (i.e., the sum
over terms of the form $\sum_i P_i \otimes A_{i}$ in at least one location
within this simulation).
But this sup norm is simply bounded by $\eta_{\, \rm sim} \leq 4 \eta$,
where $\eta$ is a bound on the sup norm of the fault operator acting
on each location in the simulation (by the triangular inequality of the sum norm). Thus $\eta \leq \eta_0 /4 $ is the
threshold condition for the graph-state model if $\eta_0$ is the
established threshold strength for the circuit model.
\section{Conclusion}
To conclude, we have invoked the composability property of simulations in the graph-state model to show that faults in the graph-state simulation of any quantum circuit (and of a fault-tolerant circuit, in particular) can be viewed as affecting the simulated operations alone. Thus, the existence of an accuracy threshold for the graph-state model follows from the threshold theorem in the circuit model for the same noise process. As an aside, the same insight can be applied to other measurement-based models of quantum computation and the teleportation of gates.
Although other proofs for the existence of an accuracy threshold in the graph-state model have already been reported for a variety of error models \cite{Raussen03b, Nielsen04b}, we believe our analysis provides an alternative, conceptually different and in many respects simpler way of thinking about fault-tolerant circuit simulations.
We note that in optical implementations of graph-state computation \cite{Nielsen04}, gate nondeterminism and photon losses give additional sources of faults not treated in this work.
The works in Refs.$\,$\cite{Nielsen04,Nielsen04b,Browne04} show how to control these faults by preparing
microclusters.
A precise threshold analysis in this setting is pursued elsewhere \cite{Nielsen05}.
\begin{acknowledgments}
We thank Michael Nielsen and Robert Raussendorf for helpful discussions on their work in this problem. P.A. and D.L. are supported by the US NSF under grant no.$\,$EIA-0086038. D.L. is also supported by the Richard Tolman Foundation and the Croucher Foundation.
\end{acknowledgments}
|
{
"timestamp": "2006-03-27T21:52:23",
"yymm": "0503",
"arxiv_id": "quant-ph/0503130",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503130"
}
|
\section{Introduction}
Consider a region ${\mathcal D}$ in ${\mathbb R}^2$ with piecewise smooth boundary and finite area. The {\em billiard flow} on the unit cotangent bundle of ${\mathcal D}$ is defined as the motion along straight lines with specular reflections at its boundary $\partial{\mathcal D}$. The quantum states and energy levels of the flow are determined by the eigenvalue problem for the Dirichlet Laplacian,\footnote{Our results can easily be adapted to the case of Neumann boundary conditions provided the spectrum of the Laplacian is discrete (which, in contrast to Dirichlet conditions, is not generally the case for non-compact regions with finite area).}
\begin{equation}
\begin{cases}
(\Delta+\lambda)\varphi =0 \\
\varphi\big|_{\partial{\mathcal D}} =0 ,
\end{cases}
\end{equation}
where $\Delta=\partial_x^2+\partial_y^2$. It is well known that the spectrum is discrete. The asymptotic distribution of the eigenvalues
\begin{equation}
0<\lambda_1\leq\lambda_2\leq\ldots\to\infty
\end{equation}
is governed by Weyl's law (cf. \cite{Simon79,Berg92a,Berg92b,Ivrii98,Berg01} and references therein)
\begin{equation}\label{weyl}
\lim_{\lambda\to\infty}\frac{\#\{ j : \lambda_j < \lambda \}}{\lambda} = \frac{\operatorname{Area}({\mathcal D})}{4\pi}.
\end{equation}
The mean spacing between consecutive eigenvalues is therefore asymptotically constant. We denote by $\{\varphi_j\}_j$ an orthonormal basis of eigenfunctions, and consider the probability measure
\begin{equation}
d\nu_j = |\varphi_j(x,y)|^2 dx\,dy
\end{equation}
associated with the $j$th eigenstate. One of the central problems in quantum chaos is to classify all weak limits of $d\nu_j$ as $j\to\infty$. The {\em quantum ergodicity theorem}, due to Schnirelman, Zelditch and Colin de Verdi\`ere \cite{Schnirelman74,Zelditch87,Colin85} (adapted for billiard flows on domains of the above type in \cite{Zelditch96}), asserts that, if the underlying dynamics is ergodic, there is a subsequence $\lambda_{j_1},\lambda_{j_2},\ldots$ of full density\footnote{A subsequence $\{\lambda_{j_i}\}_i$ is of full density if $\lim_{\lambda\to\infty} \#\{ i : \lambda_{j_i} < \lambda \}/\#\{ j : \lambda_{j} < \lambda \} = 1$.}
such that the corresponding eigenfunctions $\varphi_{j_i}$ $(i\to\infty)$ become uniformly distributed on the unit cotangent bundle of ${\mathcal D}$. This implies for instance that for any set ${\mathcal A}\subset{\mathcal D}$ with smooth boundary,
\begin{equation}
\lim_{i\to\infty} \int_{{\mathcal A}} d\nu_{j_i} = \frac{\operatorname{Area}({\mathcal A})}{\operatorname{Area}({\mathcal D})}.
\end{equation}
The proof of this theorem does not indicate whether in fact {\em all} eigenfunctions become uniformly distributed (a phenomenon called {\em quantum unique ergodicity} since there is only one possible quantum limit \cite{Rudnick94,Sarnak03}), or if there may exist sparse subsequences that have a singular limit, e.g., measures concentrated on periodic orbits of the billiard flow. Such exceptional subsequences have been observed in numerical experiments and are referred to as {\em scars} or {\em bouncing ball modes}. Following earlier results for quantum maps \cite{Degli95,Marklof00,Kurlberg00,Kurlberg01}, recent seminal contributions on the question of quantum unique ergodicity include the work of Faure, Nonnenmacher and De Bi\`evre \cite{Faure03,Faure04} who prove the existence of localized eigenstates for quantum cat maps, and Lindenstrauss' proof \cite{Lindenstrauss03} of quantum unique ergodicity in the case of Hecke eigenstates\footnote{Hecke eigenstates are simultaneous eigenfunctions of the Laplacian and all Hecke operators. If the spectrum of the Laplacian is simple, as conjectured e.g. for the modular surface, any eigenfunction of the Laplacian is a Hecke eigenstate.} of the Laplacian on compact arithmetic hyperbolic surfaces of congruence type.
\setlength{\unitlength}{0.00008\textwidth}
\begin{figure}
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\begin{picture}(8774,1227)(0,-10)
\drawline(1175,1200)(2137,1200)(2137,612)
(3500,612)(3500,187)(5687,187)
(5687,62)(8762,62)
\drawline(8762,12)(12,12)
\drawline(12.000,25.000)(85.790,43.131)(158.583,64.924)
(230.196,90.326)(300.451,119.272)(369.173,151.691)
(436.190,187.502)(501.335,226.615)(564.445,268.933)
(625.363,314.350)(683.937,362.753)(740.020,414.020)
(793.473,468.025)(844.162,524.631)(891.961,583.699)
(936.750,645.080)(978.418,708.621)(1016.860,774.164)
(1051.981,841.545)(1083.693,910.597)(1111.917,981.145)
(1136.582,1053.015)(1157.627,1126.028)(1174.999,1200.000)
\end{picture}
}
\caption{Leaky Sinai billiard} \label{fig1}
\end{figure}
\begin{figure}
{\renewcommand{\dashlinestretch}{30}
\begin{picture}(8784,1235)(0,-10)
\drawline(1185,1208)(2147,1208)(2147,620)
(3510,620)(3510,195)(5697,195)
(5697,70)(8772,70)
\drawline(8772,20)(22,20)
\drawline(1197.000,1208.000)(1121.977,1205.533)(1047.258,1198.336)
(973.144,1186.436)(899.928,1169.882)(827.903,1148.739)
(757.356,1123.092)(688.569,1093.043)(621.816,1058.712)
(557.363,1020.236)(495.467,977.768)(436.376,931.478)
(380.324,881.550)(327.536,828.183)(278.221,771.591)
(232.577,711.999)(190.786,649.644)(153.014,584.776)
(119.412,517.653)(90.114,448.542)(65.237,377.721)
(44.880,305.470)(29.124,232.078)(18.033,157.838)
(11.649,83.046)(10.000,8.000)
\end{picture}
}
\caption{Leaky Bunimovich billiard} \label{fig2}
\end{figure}
\begin{figure}
{\renewcommand{\dashlinestretch}{30}
\begin{picture}(8774,1227)(0,-10)
\drawline(8762,12)(12,12)
\drawline(1200,1200)(12,1200)(12,25)
\drawline(1175,1200)(2137,1200)(2137,612)
(3500,612)(3500,187)(5687,187)
(5687,62)(8762,62
\end{picture}
}
\caption{Leaky polygonal billiard} \label{fig3}
\end{figure}
In the present paper we show that for certain non-compact domains ${\mathcal D}\subset{\mathbb R}^2$ with finite area the sequence of measures $d\nu_j$ is not tight,\footnote{A sequence of probability measures $d\nu_j$ is {\em tight} if for any $\epsilon>0$ there is a compact domain ${\mathcal K}\subset{\mathcal D}$ such that $\limsup_{j\to\infty} \int_{{\mathcal D}-{\mathcal K}} d\nu_{j} < \epsilon$.} provided there is no extreme clustering of eigenvalues. Hence there exist subsequences of eigenstates $\varphi_{j_i}$ that leak to infinity, and quantum unique ergodicity is not satisfied for such a system.
Let ${\mathcal D}$ be given by
\begin{equation}
{\mathcal D}=\{ (x,y)\in{\mathbb R}^2 : x> 0,\; 0< y < f(x)\}
\end{equation}
where $f:(0,\infty)\to (0,\infty)$ is right-continuous and decreasing to $0$ as $x\to\infty$. More specifically, we assume that $f$ is constant on the intervals $[a_i,a_{i+1})$, $i=1,2,3,\ldots$. Examples of such domains are displayed in figs. \ref{fig1}--\ref{fig3}. The condition
\begin{equation}
\sum_{i=1}^\infty \ell_i\delta_i < \infty, \qquad \text{with $\delta_i:=f(a_i)$ and $\ell_i:=a_{i+1}-a_i$,}
\end{equation}
ensures ${\mathcal D}$ has finite area.
To illustrate our main result, let us for example choose $\delta_i=i^{-(1+\sigma)}$ and $\ell_i=i^\rho$ where $\sigma>\rho>0$ are abribrary fixed constants. Theorem \ref{thm1} in Section \ref{secLeaky} implies that there is a constant $C>0$ such that (at least) one of the following two statements is true:
\begin{itemize}
\item[$\Box$] There is a subsequence of eigenfunctions $\varphi_{j_i}$ ($i=1,2,\ldots$) with eigenvalues $\lambda_{j_i}\in\pi^2 i^{2(1+\sigma)}+[-C i^{-2\rho},C i^{-2\rho}]$ and some $c>0$ such that for any compact ${\mathcal K}\subset{\mathcal D}$ we have
\begin{equation}
\liminf_{i\to\infty} \int_{{\mathcal D}-{\mathcal K}} d\nu_{j_i} >c.
\end{equation}
\item[$\Box$] The number of eigenvalues $\lambda_j$ in the interval $\pi^2 i^{2(1+\sigma)}+[-C i^{-2\rho},C i^{-2\rho}]$ is unbounded as $i\to\infty$.
\end{itemize}
The first statement implies that eigenfunctions loose a positive proportion of mass. The second alternative implies extreme level clustering; this seems unlikely for a generic billiard of the above type, but cannot a priori be ruled out. To get a rough idea on whether to expect more level clustering than in the case of compact domains ${\mathcal D}$, we show in Section \ref{secThm2} that the spectral counting function has the asymptotics (Theorem \ref{thm2})
\begin{equation}
\#\{ j : \lambda_j < \lambda \} = \frac{\operatorname{Area}({\mathcal D})}{4\pi}\,\lambda - \frac{L(\lambda)}{4\pi} \sqrt\lambda + \frac{1}{2\pi} \sqrt\lambda
\sum_{\substack{i=1\\ \delta_i\sqrt\lambda>\pi}}^\infty \ell_i \sum_{r=1}^\infty \frac1r J_1\bigg(2 r \delta_i \sqrt{\lambda}\bigg) +O(\sqrt\lambda),
\end{equation}
where
\begin{equation}
L(\lambda) = 2 \sum_{\substack{i=1\\ \delta_i\sqrt\lambda>\pi}}^\infty \ell_i
\end{equation}
is an `effective length' of the boundary $\partial{\mathcal D}$ and $J_1$ is the $J$-Bessel function. The fluctuations are therefore larger than in the compact case, where the error term is of order $O(\sqrt\lambda)$; cf. Section \ref{secThm2} for a more detailed discussion.
The proof of Theorem \ref{thm1} is elementary and based on the construction of `bouncing ball' quasimodes \cite{Heller88,Backer97,Tanner97,Donnelly03,Burq04,Burq03a,Burq03b,Zelditch04,Hillairet05} (see also Bogomolny and Schmit's recent work on eigenfunctions in pseudo-integrable billiards \cite{Bogomolny04}). The non-compactness of the domain allows for quasimodes with discrepancy almost as small as $O(\mu^{-1})$, where $\mu$ is the quasi-eigenvalue. The best rigorous bound for the discrepancy in the compact case is $O(1)$, cf. \cite{Donnelly03}.
Our construction is completely independent on the choice of $f$ on the interval $(0,a_1)$, and one may use this additional freedom to tune $f$ on $(0,a_1)$ in such a way that the billiard flow on ${\mathcal D}$ is ergodic. It seems plausible that this is the case if the billiard flow on the restricted compact region ${\mathcal D}_0=\{ (x,y)\in{\mathbb R}^2 : 0<x<a_1,\; 0< y < f(x)\}$ is ergodic (as in the examples displayed in figs. \ref{fig1} and \ref{fig2}), but to the best of my knowledge there are no rigorous results in this direction (see however \cite{Lenci02,Lenci03,Graffi04} for proofs of ergodicity for different classes of non-compact domains). A further interesting class of examples are infinite pseudo-integrable billiards (fig. \ref{fig3}) that are known to be ergodic\footnote{Since the modulus of the momentum components in both $x$- and $y$-directions are constants of motion, ergodicity is here understood with respect to a two-dimensional submanifold of the unit cotangent bundle.} for almost all initial directions \cite{Degli00}.
\section{Quasimodes}
A function $\psi\in H_0^1({\mathcal D})$ is called a {\em quasimode} for $-\Delta$ with {\em quasi-eigenvalue $\mu$} and {\em discrepancy $\epsilon$}, if
\begin{equation}\label{qmode}
\begin{cases}
\| (\Delta+\mu)\psi \| \leq \epsilon \|\psi\| , \\
\psi\big|_{\partial{\mathcal D}} = 0,
\end{cases}
\end{equation}
where $\|\,\cdot\,\|$ denotes the $L^2$ norm.
A sequence of quasimodes $\{\psi_i\}_i$ with quasi-eigenvalues $\mu_i$ {\em is of order $s$}, if
\begin{equation}\label{qmode2}
\| (\Delta+\mu_i)\psi_i \| = O(\mu_i^{-s/2}) \|\psi_i\| .
\end{equation}
We summarize a few important properties of quasimodes; more details can be found in \cite{Colin77,Lazutkin93,Donnelly03,Zelditch04}.
By expanding $\psi$ in an orthonormal basis of eigenfunctions, $\psi=\sum_j \langle \psi,\varphi_j \rangle \varphi_j$, it is easy to see that \eqref{qmode} implies
\begin{equation}
\sum_j |\langle \psi,\varphi_j \rangle|^2 (\lambda_j-\mu)^2 \leq \epsilon^2 \|\psi\|^2
=
\epsilon^2 \sum_j |\langle \psi,\varphi_j \rangle|^2.
\end{equation}
Hence $|\lambda_j-\mu|\leq \epsilon$ for at least one $j$, i.e., there is at least one eigenvalue $\lambda_j$ in the interval $[\mu-\epsilon,\mu+\epsilon]$. Consider the larger interval $J=[\mu-b\epsilon,\mu+b\epsilon]$, $b>1$. We have
\begin{equation}\label{ring}
\sum_{\lambda_j \notin J} |\langle \psi,\varphi_j \rangle|^2
\leq (b\epsilon)^{-2} \sum_{\lambda_j \notin J} |\langle \psi,\varphi_j \rangle|^2 (\lambda_j-\mu)^2
\leq b^{-2} \|\psi\|^2 .
\end{equation}
For a domain ${\mathcal A}\subset{\mathcal D}$ define
\begin{equation}
\| \psi \|_{\mathcal A}= \sqrt{\int_{{\mathcal A}} |\psi(x,y)|^2 dx\,dy} .
\end{equation}
Triangle and Cauchy-Schwarz inequality imply
\begin{equation}
\begin{split}
\| \psi \|_{\mathcal A}
& \leq \bigg\|\sum_{\lambda_j\in J} \langle\psi,\varphi_j\rangle \varphi_j\bigg\|_{\mathcal A} +
\bigg\|\sum_{\lambda_j\notin J} \langle\psi,\varphi_j\rangle \varphi_j\bigg\|_{\mathcal A} \\
& \leq \sqrt{\sum_{\lambda_j\in J} |\langle\psi,\varphi_j\rangle|^2} \sqrt{\sum_{\lambda_j\in J} \|\varphi_j\|_{\mathcal A}^2} +
\bigg\|\sum_{\lambda_j\notin J} \langle\psi,\varphi_j\rangle \varphi_j\bigg\| \\
& \leq \|\psi\| \sqrt{\sum_{\lambda_j\in J} \|\varphi_j\|_{\mathcal A}^2} + \sqrt{\sum_{\lambda_j\notin J} |\langle\psi,\varphi_j\rangle|^2 }
\end{split}
\end{equation}
and hence, together with \eqref{ring},
\begin{equation}
\sqrt{\sum_{\lambda_j\in J} \|\varphi_j\|_{\mathcal A}^2} \geq
\frac{\| \psi \|_{\mathcal A}}{\|\psi\|} - b^{-1} .
\end{equation}
Now suppose that
\begin{equation}\label{assu}
\text{\begin{minipage}{0.8\columnwidth}{\em for a sequence of quasimodes $\psi_i$ with quasi-eigenvalue $\mu_i$ and discrepancy $\epsilon_i$ the intervals $J_i=[\mu_i-b\epsilon_i,\mu_i+b\epsilon_i]$ each contain at most $k$ eigenvalues $\lambda_j$.}\end{minipage}}
\end{equation}
Then, in each interval $J_i$ there is a $\lambda_{j_i}$ such that
\begin{equation}\label{tru}
\|\varphi_{j_i}\|_{\mathcal A} \geq \frac{1}{\sqrt k}\bigg(
\frac{\| \psi_i \|_{\mathcal A}}{\|\psi_i\|} - b^{-1} \bigg).
\end{equation}
\section{Leaky domains\label{secLeaky}}
Let $f:(0,\infty)\to (0,\infty)$ be a right-continuous function, monotonically decreasing to 0 on the half-line $[a_1,\infty)$ (for some $a_1>0$), and $\int f(x)dx < \infty$. We are interested in the domain ${\mathcal D}=\{ (x,y)\in{\mathbb R}^2 : x> 0,\; 0< y < f(x)\}$. In the following we will assume that $f$ is chosen so that
\begin{equation}\label{fina}
\int_{a_1}^\infty f(x) h(\pi^2 f(x)^{-2}) dx < \infty,
\end{equation}
where $h:[0,\infty)\to[0,\infty)$ is a fixed increasing function bounded by $h(x)\leq\sqrt x$.
The central result is the following.\footnote{The notation $A\ll B$ for two positive quantities $A,B$ means {\em there is a constant $C>0$ such that $A\leq C B$}. We write $A\asymp B$ if $A\ll B \ll A$.}
\begin{thm}\label{thm1}
For any given decreasing function $\tau:[0,\infty)\to(0,\infty)$, and any infinite sequence of real numbers
\begin{equation}\label{seq}
0<\mu_1 \leq \mu_2 \leq \ldots \to \infty
\end{equation}
satisfying
\begin{equation}\label{one}
\sum_{i=1}^\infty \tau(\mu_i) < \infty,
\end{equation}
there is a domain ${\mathcal D}$ of the above type whose Dirichlet Laplacian has an infinite sequence of quasimodes $\psi_{i,m,n}$ with quasi-eigenvalues
\begin{equation}\label{quas}
\mu_{i,m,n}= n^2 \mu_i + m^2 \xi_i, \qquad i,m,n\in{\mathbb N},
\end{equation}
and
\begin{equation} \label{eps}
\xi_i \asymp \frac{h(\mu_i)^2}{\mu_i\, \tau(\mu_i)^2},
\end{equation}
so that
\begin{itemize}
\item[(i)]
$\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \| = O(m\xi_i)
\| \psi_{i,m,n} \|$,
\item[(ii)]
$\langle \psi_{i,m,n}, \psi_{i',m',n'} \rangle = 0$ for $i\neq i'$ or $n\neq n'$,
\item[(iii)]
$|\langle \psi_{i,m,n}, \psi_{i,m',n} \rangle |\ll \min\{0.001, |m-m'|^{-1}\} \|\psi_{i,m,n}\|\,\|\psi_{i,m',n}\|$ for $m\neq m'$,
\item[(iv)]
for any compact set ${\mathcal K}\subset{\mathcal D}$,
\begin{equation}\label{three}
\frac{\|\psi_{i,m,n}\|_{{\mathcal D}-{\mathcal K}}}{\|\psi_{i,m,n}\|} \to 1
\end{equation}
uniformly for all $m,n\in{\mathbb N}$ as $i\to\infty$.
\end{itemize}
\end{thm}
\begin{remark}
Note that the set $\{\mu_{i,m,n}: i,m,n\in{\mathbb N}\}$ is a discrete subset of ${\mathbb R}_+$, with mean density
\begin{equation}\label{weyl2}
\lim_{\lambda\to\infty}\frac{\#\{ (i,m,n) : \mu_{i,m,n} < \lambda \}}{\lambda} = \frac{C}{4\pi},
\end{equation}
where
\begin{equation}\label{see}
C= \pi^2 \sum_i \frac{1}{\sqrt{\mu_i\xi_i}} \leq \operatorname{Area}({\mathcal D}).
\end{equation}
This may either be verified directly, or concluded from the observation (cf. Sections \ref{secProof} and \ref{secProof2}) that $\{\mu_{i,m,n}\}$ can be identified with the spectrum of the Dirichlet Laplacian on an infinite union of rectangles ${\mathcal D}_i$ with sides $\ell_i=\pi\xi_i^{-1/2}$, $\delta_i=\pi\mu_i^{-1/2}$, and thus total area $C=\sum_i\operatorname{Area}({\mathcal D}_i)$. In this interpretation, \eqref{weyl2} represents Weyl's law \eqref{weyl}.
\end{remark}
\begin{remark}
If assumption \eqref{assu} holds e.g. for the quasimodes $\psi_{i,1,1}$, eqs. \eqref{tru} and \eqref{three} imply there is an infinite sequence of eigenfunctions $\varphi_{j_i}$, such that for any compact ${\mathcal K}\subset{\mathcal D}$
\begin{equation}
\liminf_{i\to\infty} \|\varphi_{j_i}\|_{{\mathcal D}-{\mathcal K}} \geq \frac{1-b^{-1}}{\sqrt{k}} .
\end{equation}
That is, the eigenstates $\varphi_{j_i}$ loose a positive proportion of mass. It should be stressed that we have not ruled out the probably very remote possibility that assumption \eqref{assu} with $\epsilon_i=O(m\xi_i)$ can never be satisfied for the domains ${\mathcal D}$ considered in the theorem (an explicit construction of ${\mathcal D}$ is given in Section \ref{secProof}). It would be interesting to see whether \eqref{assu} can be established at least for generic choices of such ${\mathcal D}$, i.e., generic choices of $\delta_i$.
In Section \ref{secThm2} we will prove an upper bound for the error term in Weyl's law, which in turn yields a rough estimate on possible level clustering.
\end{remark}
\begin{remark}
For $m,n$ bounded as $i\to\infty$ the theorem establishes quasimodes with very small discrepancy,
\begin{equation} \label{two5}
\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \| = O\bigg(\frac{h(\mu_{i,m,n})^2}{\mu_{i,m,n}\, \tau(\mu_{i,m,n})^2}\bigg) \| \psi_{i,m,n} \| .
\end{equation}
Since $h$ and $\tau$ can be arbitrarily slowly increasing/decreasing functions (respectively), this yields quasimodes of order arbitrarily close to 2; cf. example \ref{exalg} below. The number of such quasimodes with $\mu_{i,m,n}<\lambda$,
\begin{equation}
\begin{split}
N_{\text{bb}}(\lambda) & = \#\{ (i,m,n) :\, m,n=O(1),\; \mu_{i,m,n}<\lambda \}\\
& \asymp \#\{ i:\; \mu_i<\lambda \},
\end{split}
\end{equation}
is determined by the restriction that
\begin{equation}
\int \tau(\lambda) dN_{\text{bb}}(\lambda) <\infty.
\end{equation}
Hence the higher the desired accuracy of quasimodes (achieved by choosing a sufficiently slowly decreasing $\tau$), the thinner the corresponding sequence of quasimodes becomes.
\end{remark}
\begin{remark}
The theorem also implies that there can be sequences of quasimodes of order zero that have almost full density. `Order zero' means that
\begin{equation} \label{two6}
\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \| = O(1)
\| \psi_{i,m,n} \|,
\end{equation}
i.e., $m\xi_i \leq C_1$ for some constant $C_1>0$. Since in view of \eqref{eps} there is a constant $C_2>0$ such that $\xi_i\mu_i \geq C_2$, we have
\begin{equation}\label{low}
\begin{split}
N_{\text{BB}}(\lambda) & = \#\{ (i,m,n) :\, \mu_{i,m,n}= n^2 \mu_i + m^2 \xi_i <\lambda, \; m\xi_i \leq C_1 \} \\
& \geq \#\left\{ (i,m,n) :\, n^2<\frac{\lambda}{\mu_i} - \frac{C_1^2}{C_2} , \; m \leq \frac{C_1}{\xi_i} \right\} \\
& \asymp \sqrt\lambda \sum_{\mu_i<\lambda} \frac{\sqrt{\mu_i}\,\tau(\mu_i)^2}{h(\mu_i)^2}.
\end{split}
\end{equation}
For suitable choices of $h$ and $\tau$ this quantity can be arbitrarily close to a function $\asymp\lambda$, cf. \eqref{low2}. On the other hand, it is bounded from below by $\gg\sqrt\lambda$. This bound is attained in the case when
\begin{equation}
\sum_{i=1}^\infty \frac{\sqrt{\mu_i}\,\tau(\mu_i)^2}{h(\mu_i)^2} < \infty,
\end{equation}
and coincides with the bound for compact domains, cf. \cite{Donnelly03}. Note that the heuristic approaches in \cite{Backer97,Tanner97} predict a greater number of bouncing ball modes.
\end{remark}
\begin{ex}\label{exalg}
Take $h(x)=x^{\beta}$ with $0\leq\beta<1/2$.
For any given infinite sequence of real numbers $\mu_i$ with
\begin{equation}\label{one2}
\#\{ j : \mu_j \leq \lambda \} \asymp \lambda^{\alpha} ,
\end{equation}
there is a domain ${\mathcal D}$ with $\int f(x)^{1-2\beta} dx < \infty$, so that the corresponding quasimodes $\psi_j$ have order $2-2\sigma$, for any fixed $\sigma>2(\alpha+\beta)$. That is,
\begin{equation} \label{two1}
\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \| =O(m \mu_{i,m,n}^{-1+\sigma})
\| \psi_{i,m,n} \|,
\end{equation}
The fact that \eqref{one2} implies \eqref{one} with $\tau(x)=x^{-\alpha'}$ ($\alpha'>\alpha$) is seen by summation by parts. In view of Weyl's law \eqref{weyl} and the small discrepancy $O(\mu_{i,m,n}^{-1+\sigma})$ for bounded $m$, a failure of assumption \eqref{assu} would imply an extreme clustering of eigenvalues. As we shall see in Section \ref{secThm2}, the bounds on the error term in Weyl'a law worsen as $\sigma\to 0$, and hence clustering cannot be ruled out.
An evaluation of the lower bound for the number of order-zero quasimodes in \eqref{low} yields
\begin{equation}\label{low2}
N_{\text{BB}}(\lambda) \gg \lambda^\theta,
\end{equation}
with $\theta=\max\{1+\alpha-2\alpha'-2\beta,1/2\}$.
Note that $\theta$ can be arbitrarily close to 1 for suitable parameter choices.
\end{ex}
\begin{ex}
A second interesting choice that yields a domain ${\mathcal D}$ with exponentially narrow cusps, is $h(x)=\sqrt{x}/\log^\gamma(1+x)$ with $\gamma>0$.
For any given infinite sequence of real numbers $\mu_i$ with
\begin{equation}\label{one3}
\#\{ j : \mu_j \leq \lambda \} \asymp \log^\alpha\lambda ,
\end{equation}
there is a domain ${\mathcal D}$ with $\int |\log f(x)|^{-\gamma}dx < \infty$, so that
\begin{equation} \label{two2}
\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \| =O(m\log^{-\sigma}\mu_{i,m,n})
\| \psi_i \|,
\end{equation}
for any fixed $\sigma<2(\gamma-\alpha)$. Choose here $\tau(x)=\log^{-\alpha'}x$ with $\alpha'>\alpha$, and \eqref{one} can again be checked using summation by parts.
In this case the number of order-zero quasimodes is bounded from below by
\begin{equation}\label{low3}
N_{\text{BB}}(\lambda) \gg \sqrt\lambda.
\end{equation}
\end{ex}
\section{Proof of Theorem \ref{thm1}\label{secProof}}
We begin by constructing accurate quasimodes on the rectangle
$[a,a+\ell] \times [0,\delta]$ with Dirichlet boundary conditions at $y=0,\delta$.
Let $\chi\in C_0^\infty({\mathbb R})$ be a mollified characteristic function of the interval $[0,1]$. That is, $0\leq\chi(x)\leq 1$, $\chi(x)=0$ for $x\notin[0,1]$ and $\chi(x)=1$ for $x\in[\epsilon,1-\epsilon]$ for some fixed, small $\epsilon>0$. We assume also that $\chi'(x)=O(\epsilon^{-1})$ (such a choice is always possible).
For $m,n\in{\mathbb N}$, $a\in{\mathbb R}$ and $\ell,\delta>0$ put
\begin{equation}\label{qm}
\psi_{m,n}(x,y)= \chi\bigg(\frac{x-a}{\ell}\bigg) \sin\bigg(\frac{\pi m (x-a)}{\ell}\bigg) \sin\bigg(\frac{\pi n y}{\delta}\bigg)
\end{equation}
and
\begin{equation}\label{qmu}
\mu_{m,n} = \pi^2 \bigg[ \bigg(\frac{m}{\ell}\bigg)^2 + \bigg(\frac{n}{\delta}\bigg)^2 \bigg].
\end{equation}
Straightforward differentiation yields
\begin{multline}
(\Delta+\mu_{m,n}) \psi_{m,n}(x,y)
= \frac{1}{\ell^2} \bigg[ 2\pi m \chi'\bigg(\frac{x-a}{\ell}\bigg) \cos\bigg(\frac{\pi m (x-a)}{\ell}\bigg) \\ + \chi''\bigg(\frac{x-a}{\ell}\bigg) \sin\bigg(\frac{\pi m (x-a)}{\ell}\bigg) \bigg] \sin\bigg(\frac{\pi n y}{\delta}\bigg),
\end{multline}
and hence
\begin{equation}
\| (\Delta+\mu_{m,n}) \psi_{m,n} \|^2 = O_\chi\bigg(\frac{m^2\delta}{\ell^3}\bigg).
\end{equation}
where the implied constant only depends on the choice of $\chi$.
Because of this and
\begin{equation}
\| \psi_{m,n} \|^2 = \frac{\ell\delta}{4} (1+O(\epsilon)),
\end{equation}
we obtain
\begin{equation}\label{err}
\| (\Delta+\mu_{m,n}) \psi_{m,n} \| = O_\chi\bigg(\frac{m}{\ell^2}\bigg)
\| \psi_{m,n} \| .
\end{equation}
Furthermore, for $n\neq n'$ we have $\langle \psi_{m,n},\psi_{m',n'} \rangle = 0$, and for $n=n'$, $m\neq m'$,
\begin{equation}
\begin{split}
\langle \psi_{m,n},\psi_{m',n} \rangle
& = \frac{\delta}{2} \int_0^{\ell} \chi\bigg(\frac{x}{\ell}\bigg)^2 \sin\bigg(\frac{\pi m x}{\ell}\bigg)
\sin\bigg(\frac{\pi m' x}{\ell}\bigg) dx \\
& = \frac{\delta}{2} \bigg\{ \int_0^{\epsilon\ell} + \int_{(1-\epsilon)\ell}^\ell \bigg\} \bigg[\chi\bigg(\frac{x}{\ell}\bigg)^2-1\bigg] \sin\bigg(\frac{\pi m x}{\ell}\bigg)
\sin\bigg(\frac{\pi m' x}{\ell}\bigg)dx \\
& = \frac{\ell\delta}{4} \bigg\{ \int_0^{\epsilon} + \int_{1-\epsilon}^1 \bigg\} [\chi(x)^2-1] [\cos(\pi (m-m') x) -\cos(\pi (m+m') x)] dx\\
& = \frac{\ell\delta}{4} O(\epsilon).
\end{split}
\end{equation}
On the other hand, using integration by parts, we have
\begin{multline}
\int_0^{\epsilon} [\chi(x)^2-1] \cos(\pi (m-m') x) dx \\
=
\frac{1}{\pi(m-m')} \bigg\{ \bigg[ [\chi(x)^2-1] \sin(\pi (m-m') x) \bigg]_0^\epsilon
\\ - \int_0^{\epsilon} 2\chi(x)\chi'(x) \sin(\pi (m-m') x) dx \bigg\}.
\end{multline}
Since $\chi(\epsilon)^2=1$, $\sin(0)=0$ the first term vanishes, and since $\chi'(x)=O(\epsilon^{-1})$ the integral is of $O(1)$. The analogous argument works for the remaining integrals. Hence
\begin{equation}\label{2b}
|\langle \psi_{m,n},\psi_{m',n} \rangle| \ll \min\bigg\{\epsilon,\frac{1}{|m-m'|}\bigg\}
\|\psi_{m,n}\|\,\|\psi_{m',n}\|.
\end{equation}
We will now give an explicit construction of ${\mathcal D}$. The function $f$ is chosen constant on the intervals $[a_i,a_{i+1})$, $i=1,2,3,\ldots$; set $\delta_i=f(a_i)$ and $\ell_i=a_{i+1}-a_i$. As quasimodes we take
\begin{equation}
\psi_{i,m,n}(x,y)= \chi\bigg(\frac{x-a_i}{\ell_i}\bigg) \sin\bigg(\frac{\pi m(x-a_i)}{\ell_i}\bigg) \sin\bigg(\frac{\pi n y}{\delta_i}\bigg) ,
\end{equation}
with quasi-eigenvalues
\begin{equation}\label{qmu2}
\mu_{i,m,n} = \pi^2 \bigg[ \bigg(\frac{m}{\ell_i}\bigg)^2 + \bigg(\frac{n}{\delta_i}\bigg)^2 \bigg].
\end{equation}
By construction, these are completely localized in the rectangle $[a_i,a_{i+1}]\times[0,\delta_i]$ and hence satisfy requirement (iv) of the theorem. Setting $\mu_i = \pi^2 \delta_i^{-2}$, every given sequence of $\mu_i$ having property \eqref{one} determines a sequence of $\delta_i$. Because of \eqref{err},
\begin{equation}
\frac{\| (\Delta+\mu_{i,m,n}) \psi_{i,m,n} \|}{\| \psi_{i,m,n} \|} =
O_\chi(m \ell_i^{-2})=O_\chi(m \delta_i^2 A_i^{-2})=O_\chi(m \mu_i^{-1} A_i^{-2}).
\end{equation}
To minimize the discrepancy, we would like to choose $A_i$ as large as possible. The choice $A_i=\tau(\mu_i) h(\mu_i)^{-1}$ yields condition (i) and determines $f$. Since
\begin{equation}\label{fina1}
\begin{split}
\int_{a_1}^\infty f(x) h(\pi^2 f(x)^{-2}) dx & = \sum_i \ell_i f(a_i) h(\pi^2 f(a_i)^{-2})\\
& = \sum_i A_i h(\pi^2 \delta_i^{-2}) \\
& = \sum_i \tau(\mu_i) < \infty ,
\end{split}
\end{equation}
the function $f$ is in the required class satisfying \eqref{fina}.
Condition (ii) is evident from \eqref{qm}, and (iii) from \eqref{2b}.
\section{Asymptotic distribution of eigenvalues\label{secThm2}}
In view of condition \eqref{assu} we would like to control the number of eigenvalues in small intervals. The following theorem illustrates that extreme level clustering cannot a priori be ruled out.
\begin{thm}\label{thm2}
The spectral counting function $N(\lambda)=\#\{j:\lambda_j < \lambda\}$ of the Dirichlet Laplacian for the domain ${\mathcal D}$ (as in Section \ref{secProof}) satisfies
\begin{equation}
N(\lambda) = \frac{\operatorname{Area}({\mathcal D})}{4\pi}\,\lambda - \frac{L(\lambda)}{4\pi} \sqrt\lambda + \frac{1}{2\pi} \sqrt\lambda
\sum_{\substack{i=1\\ \delta_i\sqrt\lambda>\pi}}^\infty \ell_i \sum_{r=1}^\infty \frac1r J_1\bigg(2 r \delta_i \sqrt{\lambda}\bigg) +O(\sqrt\lambda),
\end{equation}
where
\begin{equation}
L(\lambda) = 2 \sum_{\substack{i=1\\ \delta_i\sqrt\lambda>\pi}}^\infty \ell_i
\end{equation}
and $J_1$ is the $J$-Bessel function.
\end{thm}
\begin{remark}
The standard bound
\begin{equation}\label{Jbound}
|J_1(x)|\ll x^{-1/2}\quad \text{for $x$ large}
\end{equation}
implies that
\begin{equation}\label{Neee}
N(\lambda) = \frac{\operatorname{Area}({\mathcal D})}{4\pi}\,\lambda + O(L(\lambda)\sqrt\lambda),
\end{equation}
where
\begin{equation}
L(\lambda) = 2\pi \sum_{\substack{i=1\\ \mu_i<\lambda}}^\infty \frac{1}{\sqrt{\xi_i}} \ll
\sum_{\substack{i=1\\ \mu_i<\lambda}}^\infty \frac{\sqrt{\mu_i}\,\tau(\mu_i)}{h(\mu_i)} ;
\end{equation}
recall that $\mu_i=\pi^2/\delta_i^2$ and $\xi_i=\pi^2/\ell_i^2$.
As the examples following Theorem \ref{thm1} illustrate, a good quasimode discrepancy ($\xi_i$ small) is thus traded with an error bound in \eqref{Neee} approaching $o(\lambda)$. But as we shall see in the following section, cf. eq. \eqref{DiDi}, the number of eigenvalues in the interval $[\lambda,\lambda+\sigma]$ with $\sigma<\sqrt\lambda$ is
\begin{equation}
N(\lambda+\sigma)-N(\lambda)= \#\{ (i,m,n)\in{\mathbb N}^3 : \lambda\leq \mu_{i,m,n} < \lambda+\sigma \} +O(\sqrt\lambda),
\end{equation}
with quasi-eigenvalues $\mu_{i,m,n}$ as in \eqref{quas}.
That is, all extreme fluctuations beyond $O(\sqrt\lambda)$ are due to the presence of bouncing ball quasimodes.
\end{remark}
\section{Proof of Theorem \ref{thm2}\label{secProof2}}
Consider the domains ${\mathcal D}_i=\{ (x,y)\in{\mathbb R}^2 : a_i<x<a_{i+1},\; 0< y < f(x)\}$ where $i=0,1,2,\ldots$ and $a_0=0$. Let $N_{\operatorname{D}}^{(i)}(\lambda)$ be the spectral counting function for the Dirichlet Laplacian for ${\mathcal D}_i$, and $N_{\operatorname{N}}^{(i)}(\lambda)$ the counting function with Neumann conditions on the boundary lines $x=a_i$ and $x=a_{i+1}$ and Dirichlet conditions on the remaining boundary. Set
\begin{equation}
N_{\operatorname{D}}(\lambda)=\sum_{i=0}^\infty N_{\operatorname{D}}^{(i)}(\lambda), \qquad
N_{\operatorname{N}}(\lambda)=\sum_{i=0}^\infty N_{\operatorname{N}}^{(i)}(\lambda).
\end{equation}
It is well known (`Dirichlet-Neumann bracketing' \cite{Berg92a,Berg01}) that
\begin{equation}
N_{\operatorname{D}}(\lambda) \leq N(\lambda) \leq N_{\operatorname{N}}(\lambda).
\end{equation}
For $i=0$ the general error estimate in Weyl's law for compact domains yields
\begin{equation}
N_{\operatorname{D}}^{(0)}(\lambda) = \frac{\operatorname{Area}({\mathcal D}_0)}{4\pi} \lambda + O(\sqrt\lambda),
\qquad
N_{\operatorname{N}}^{(0)}(\lambda) = \frac{\operatorname{Area}({\mathcal D}_0)}{4\pi} \lambda + O(\sqrt\lambda).
\end{equation}
For the remaining domains we have
\begin{equation}
N_{\operatorname{D}}^\Box(\lambda):=\sum_{i=1}^\infty N_{\operatorname{D}}^{(i)}(\lambda) =\#\{ (m,n,i)\in{\mathbb N}^3 : n^2 \mu_i + m^2 \xi_i <\lambda \}
\end{equation}
and
\begin{equation}
N_{\operatorname{N}}^\Box(\lambda):=\sum_{i=1}^\infty N_{\operatorname{N}}^{(i)}(\lambda) =
N_{\operatorname{D}}^\Box(\lambda) + \#\{ (n,i)\in{\mathbb N}^2 : n^2 \mu_i <\lambda \} .
\end{equation}
Note that
\begin{equation}\label{sixsix}
N_{\operatorname{N}}^\Box(\lambda)-N_{\operatorname{D}}^\Box(\lambda) \leq
\sum_{\mu_i<\lambda} \sqrt{\frac{\lambda}{\mu_i}} = O(\sqrt\lambda)
\end{equation}
since $\sum_i \mu_i^{-1/2} <\infty$, cf. \eqref{see}. Therefore
\begin{equation}\label{DiDi}
N(\lambda) = \frac{\operatorname{Area}({\mathcal D}_0)}{4\pi} \lambda + N_{\operatorname{D}}^\Box+O(\sqrt{\lambda}) .
\end{equation}
Now
\begin{equation}
N_{\operatorname{D}}^\Box(\lambda) = \sum_{\substack{i,n=1\\ n^2\mu_i<\lambda}}^\infty \left[\sqrt{\frac{\lambda-n^2\mu_i}{\xi_i}} +O(1) \right]
= \sum_{\substack{i,n=1\\ n^2\mu_i<\lambda}}^\infty \sqrt{\frac{\lambda-n^2\mu_i}{\xi_i}} +O(\sqrt\lambda),
\end{equation}
recall the argument in \eqref{sixsix}.
The main term is
\begin{equation}\label{maint}
\begin{split}
\sum_{\substack{i,n=1\\ n^2\mu_i<\lambda}}^\infty \sqrt{\frac{\lambda-n^2\mu_i}{\xi_i}}
& =\sqrt\lambda \sum_{\substack{i=1\\ \mu_i<\lambda}}^\infty \sum_{n=1}^\infty\frac{1}{\sqrt{\xi_i}} F\bigg(n\sqrt{\frac{\mu_i}{\lambda}}\bigg) \\
& =\frac12 \sqrt\lambda \sum_{\substack{i=1\\ \mu_i<\lambda}}^\infty \sum_{n=-\infty}^\infty\frac{1}{\sqrt{\xi_i}} F\bigg(n\sqrt{\frac{\mu_i}{\lambda}}\bigg) -\frac12 \sqrt{\lambda} \sum_{\mu_i<\lambda} \frac{1}{\sqrt{\xi_i}}
\end{split}
\end{equation}
where $F(x)=\sqrt{\max\{ 1-x^2,0 \}}$. The Poisson summation formula yields for the sum over $n$
\begin{equation}\label{series}
\sum_{n=-\infty}^\infty F\bigg(n\sqrt{\frac{\mu_i}{\lambda}}\bigg)
=
\sqrt{\frac{\lambda}{\mu_i}} \sum_{r=-\infty}^\infty\widehat F\bigg(r\sqrt{\frac{\lambda}{\mu_i}}\bigg)
\end{equation}
where $\widehat F(0)=\pi/2$ and for $y\neq 0$
\begin{equation}
\begin{split}
\widehat F(y) & = \int_{-1}^1 \sqrt{1-x^2}\, \cos(2\pi xy) dx \\
& = \frac{1}{2y}\, J_1(2\pi y) .
\end{split}
\end{equation}
So
\begin{equation}\label{series2}
\sum_{n=-\infty}^\infty F\bigg(n\sqrt{\frac{\mu_i}{\lambda}}\bigg)
=
\frac{\pi}{2} \sqrt{\frac{\lambda}{\mu_i}}+ \sum_{r=1}^\infty \frac1r J_1\bigg(2\pi r\sqrt{\frac{\lambda}{\mu_i}}\bigg) .
\end{equation}
The bound \eqref{Jbound} proves the convergence of the series on the right hand side of \eqref{series2}. This concludes the proof of Theorem \ref{thm2}.
\section*{Acknowledgments}
I thank M. van den Berg, M. Degli Esposti, J. Keating, M. Lenci, Z. Rudnick and R. Schubert for stimulating discussions.
|
{
"timestamp": "2005-03-28T16:58:14",
"yymm": "0503",
"arxiv_id": "math-ph/0503066",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503066"
}
|
\section{Introduction}
Systems made up of entities that interact pairwise can be modeled as
networks. To comprehend the emergent properties of such systems---the
objective of the study of complex systems and systems biology---one
approach is to investigate the global properties of the corresponding
networks \cite{mejn:rev,ba:rev,harary,wf}. In many cases the
individual entities (or vertices) have distinct functions in the
system. In such cases, provided the wiring of the edges relates to the
function of vertices, one can predict these functions from the
vertices' position in the network. For example, a corporate hierarchy
may be topped by a CEO, followed by a CFO and COO, so a chart of
who reports to whom is enough to identify these positions. Another
problem in this category of much recent interest is to predict protein
functions \cite{hodg:pfp} from the networks of protein interactions
\cite{yook:protein,deng:pfp,hish:pfp,leto:pfp,sama:pfp,vaz:pfp}.
These methods, like other methods based on e.g. protein sequences,
are important because to confirm a protein function one needs
function-specific and possibly hard-to-design \textit{in vivo},
genetic or biochemical tests, while interaction and sequence data can
be obtained fairly easily.
In this paper we propose a general method of predicting the functions
of vertices in networked systems where the functions are partly mapped
out. The rationale of our algorithm is to match unknown vertices with
the most similar (judging from the network structure) categorized
vertex and take the functions of the latter vertex as our
forecast. The network similarity concept we ground our method on is
related to the notion of regular equivalence \cite{eve:sim,wf} or role
similarity \cite{regeeco1} of social network theory. Roughly speaking,
two vertices are similar, in this sense, if the network looks alike from
their respective perspectives. We evaluate our method on model
networks where the categories of vertices reflect their placement in
the network. We also apply the method to \textit{S.\ cerevisiae}
protein data obtained from the MIPS data base \cite{pagel:mips} (data
extracted January 23, 2005).
\section{Role similarity and definition of the prediction scheme}
\begin{figure}
\resizebox*{0.95\linewidth}{!}{\includegraphics{equ.eps}}
\caption{
Illustration of structural and regular equivalence. $i$ and $j$
are structurally equivalent in (a) since they have the same
neighborhoods, and regularly equivalent in (b) since there is a
matching of regularly equivalent vertices between the
neighborhoods. In (b) vertices of the same color are regularly
equivalent.
}
\label{fig:equ}
\end{figure}
Role similarity refers to rather broad set of concepts and related
measures. Basically, the \textit{role} of a vertex is determined by
the characteristics of the vertices it is connected to
\cite{wf}.\footnote{Note that the nomenclature is somewhat ambiguous. Another
use of ``role'' is to say that vertices with the similar values of
vertex-specific structural measures have the same role
\cite{gui:meta,luss:dolphin}.} Consider
two vertices $i$ and $j$. If their neighborhoods
are similar, we say $i$ and $j$ have high role similarity. The
question how to define the similarity of the neighborhoods $\Gamma_i$
and $\Gamma_j$ leads to two different concepts. One choice matches the
identity of vertices in the neighborhood. This leads to the
\textit{structural equivalence} relation which is true if
$\Gamma_i=\Gamma_j$. Another way to compare neighborhoods is to match
the similarity of vertices in the neighborhood which gives the concept
of \textit{regular equivalence}---if one can pair the vertices of
$\Gamma_i$ with vertices in $\Gamma_j$ such that each pair is
regularly equivalent, then $i$ and $j$ are also regularly
equivalent. Since vertices with the same functions need not, in
general, be close, we will need a similarity score measuring how close
to regular equivalence two vertices are. Following
Refs.\ \cite{simrank,blondel:sim} we define a similarity score based on
iterating the regular equivalence principle ``two vertices are similar
if they are pointed to, or point to, vertices that themselves
similar.'' In the general case of a directed network with $R$
different types of edges, one implementation of this argument is just
to sum the similarities between vertices of the neighborhoods:
\begin{equation}\label{eq:simdef_i}
\sigma^\mathrm{I}_{n+1}(i,j) = \sum_{r=1}^R\left[
\sum_{i'\in\Gamma_{i,r}^{\mathrm{in}}}
\sum_{j'\in\Gamma_{j,r}^{\mathrm{in}}} \sigma^\mathrm{I}_n (i',j') +
\sum_{i'\in\Gamma_{i,r}^{\mathrm{out}}}
\sum_{j'\in\Gamma_{j,r}^{\mathrm{out}}} \sigma^\mathrm{I}_n
(i',j')\right],
\end{equation}
where $\sigma^\mathrm{I}_n(i,j)$ is the similarity between $i$ and $j$
after the $n$'th iteration and $\Gamma_{i,r}^{\mathrm{in}}$ is the
in-neighborhood of $i$ with respect to $r$-edges. To avoid
overflow problems we rescale all similarities so that
$\max_{ij}|\sigma^\mathrm{I}_n(i,j)|=S$ after each iteration. We
break the iteration when the sum, before the normalization, has not
changed by more than a $10^{-8}$th of its previous value.
By the Eq.~\ref{eq:simdef_i} definition, high degree vertices will
appear more similar to the average other vertex than low-degree
vertices. To compensate for this effect one may divide by the
appropriate degrees (numbers of neighbors) to obtain:
\begin{widetext}
\begin{equation}\label{eq:simdef_ii}
\sigma^\mathrm{II}_{n+1}(i,j) = \sum_{r=1}^R\left[
\frac{1}{k_{i,r}^{\mathrm{in}}\:k_{j,r}^{\mathrm{in}}}
\sum_{i'\in\Gamma_{i,r}^{\mathrm{in}}}
\sum_{j'\in\Gamma_{j,r}^{\mathrm{in}}} \sigma^\mathrm{II}_n (i',j') +
\frac{1}{k_{i,r}^{\mathrm{out}}\:k_{j,r}^{\mathrm{out}}}
\sum_{i'\in\Gamma_{i,r}^{\mathrm{out}}}
\sum_{j'\in\Gamma_{j,r}^{\mathrm{out}}} \sigma^\mathrm{II}_n
(i',j')\right],
\end{equation}
\end{widetext}
where $k_{i,r}^{\mathrm{in}}$ is the in-degree of $i$ with respect to
$r$-edges. From now on we call $\sigma^\mathrm{I}(i,j)=
\sigma^\mathrm{I}_\infty(i,j)$ of Eq.~\ref{eq:simdef_i} and
$\sigma^\mathrm{II}(i,j)$ of Eq.~\ref{eq:simdef_ii} the I- and
II-similarity between $i$ and $j$ respectively.
As mentioned, we suppose some of the vertices are functionally
categorized. In general we assume one vertex can have many
functions. For pairs of such functionally determined vertices the
above similarities will add no information. Instead we define
a functional similarity
\begin{equation}\label{eq:simdef_f}
\sigma_f(i,j) = J(F_i,F_j) - \langle J \rangle ,
\end{equation}
for such pairs, where $F_i$ is $i$'s function set (we assume a finite
number of functions) and $J(\:\cdot\:)$ denotes the Jackard index
$J(A,B) = |A\cap B|\:/\:|A\cup B|$ and the average is over all pairs of
categorized vertices. We will later need $\sigma(i,j)=0$ to represent
neutrality which is why we subtract the mean. Whenever a pair of
classified vertices $(i,j)$ appears in the sums of
Eqs.~\ref{eq:simdef_i} or \ref{eq:simdef_ii} we use the
$\sigma_f(i,j)$ value of Eq.~\ref{eq:simdef_f} instead of
$\sigma^\mathrm{I}(i,j)$ or $\sigma^\mathrm{II}(i,j)$. I.e., we assume
the functional classification is more accurate than the
role-similarities and hence do not update the former.
In general we can now define our prediction scheme as follows:
\begin{enumerate}
\item \label{enu:init} For vertex pairs with at least one unclassified
vertex initialize $\sigma_0(i,j)$ to $0$ if $i\neq j$ and
to $1 - \langle J \rangle$ otherwise.
\item \label{enu:sim} Calculate the similarity scores for all pairs of
unique vertices such that at least one is unclassified.
\item \label{enu:choose} For an unclassified vertex $i$, predict the
function set $F_{\hat{i}}$, where $\hat{i}$ is the classified
vertex with highest similarity to $i$. If $\hat{i}$ is not unique,
but a set $\hat{I} = \{\hat{i}_1,\cdots,\hat{i}_m\}$ has the highest
similarity to $i$, then let the set $G$ of functions present in more
than half of the set of $j$'s be your guess. If $G$ is empty, let
$F_j$ for a random $j\in\hat{I}$ be the guess.
\end{enumerate}
The diagonal elements will have maximal functional similarity (which
is why we set them to $1-\langle J \rangle$ in step~\ref{enu:init}),
otherwise we assume neutrality. The backup selection rules in
step~\ref{enu:choose} will typically be needed when unclassified
vertices are structurally equivalent to classified vertices, the use
of the majority rule instead of only a random guess will compensate
for occasional errors in the assignment of functions to classified
proteins. Our parameter $S$ sets the relative importance of the
functional similarities to the subsequent assessments of
$\sigma$. As mentioned above, the functional classification is assumed
to be more accurate than the role-similarities, and it is thus sensible to
choose a $\sigma\in [0,1-\langle J\rangle]$. The appropriate $S$ value
is problem dependent. We will use $S=0{.}8$ which is in this interval
for both our two test cases. To summarize, we have proposed two
versions of our prediction scheme, scheme I and II, corresponding to
I- and II-similarity.
\section{Application to model networks}
To test our prediction algorithm we construct model networks where the
assigned functions of the vertices correspond to their position in the
network. We test the algorithm's size scaling and performance in
sub-ideal conditions by randomly perturbing the network.
\subsection{Definition of the model networks}
\begin{figure*}
\includegraphics{ill.eps}
\caption{
Model networks where vertex function and position are related. (a)
shows the initial network. (b) shows a realization with 30
vertices and rewiring probability $r=0{.}1$. ``\textbf{*}''
indicates a rewired edge.
}
\label{fig:ill}
\end{figure*}
In defining our model, we will metaphorically use the flow of raw
material, products and information in a manufacturing system. For our
purpose we only need networks where the functions of vertices correspond to
their position in their network surroundings---we will not further
motivate its relevance as a model for manufacturing networks. We
assign five distinct functional classes of the vertices: The
\textit{supply} vertices are the source of the raw material which
flows along \textit{A-edges} to \textit{assembler} vertices. The
assembled products are transported via \textit{B-edges} to
\textit{delivery} vertices that dispatch the products. From the
delivery vertices informational feedback is sent to the supply
vertices through \textit{C-edges}. Furthermore, the A and B-edges can
fork at \textit{A-} and \textit{B-distributor} vertices.
The precise definition of the model is as follows: Start with the
kernel shown in Fig.~\ref{fig:ill}(a), then grow the network vertex by
vertex. At each iteration, assign, with equal probability, one of the
above functions to the new vertex. Then, depending on the assigned
function, form edges including the new vertex as follows.
\begin{description}
\item[Supply.] Add an A-edge to an assembler or A-distributor, and a
C-edge from a delivery vertex.
\item[Assembly.] Add an A-edge from an assembler or A-distributor
vertex, and a B-edge to an assembler or A-distributor.
\item[Delivery.] Add a B-edge from an assembler or B-distributor, and
a C-edge to a supplier.
\item[A(B)-distribution.] Add an A(B)-edge from an assembler or
A(B)-distributor vertex, and an A(B)-edge to an assembler or
A(B)-distributor.
\end{description}
The choice of vertex to attach the new vertex to, given its functional
category, is done with uniform randomness. Note that the number of
edges will on average be twice the number of vertices (two edges are
added per vertex).
From the definition so far, any vertex is identifiable from its
neighborhood---a vertex with incoming C-edges and out-going A-edges is
a supplier, and so on. Real data-sets are seldom perfect---neither in
the wiring of the edges, nor in the functional classification. To test
the prediction scheme under more realistic circumstances we randomize
the network as follows: After generating a network according to the
above scheme, we go through all edges sequentially. With a probability
$r$ detach the from-side of an edge and re-attach it to a randomly
chosen vertex such that no self-edge or multiple edge (of the same
type---A, B or C) is formed. Rewire the to-side likewise with the same
probability. A realization of the algorithm is displayed in
Fig.~\ref{fig:ill}(b). After the rewiring there is not necessarily
enough information to classify a vertex---$i$ in Fig.~\ref{fig:ill}(b)
is an assembler but could just as well have been a B-distributor.
\subsection{Prediction performance}
\begin{figure}
\resizebox*{\linewidth}{!}{\includegraphics{mod.eps}}
\caption{
The fraction of correctly predicted functions $s$ for our model
networks as a function of the rewiring probability $r$. (a) show
the results based on I-similarities, (b) is the corresponding plot
for II-similarities. The points are averaged over $\sim 1000$ runs
of the network construction and prediction scheme with
$a=1/50$. Errorbars are smaller than the symbol size. The
horizontal line marks the limit of random guessing $0{.}2$.
}
\label{fig:mod}
\end{figure}
To test the our prediction scheme we mark a random set of $aN$,
$a\in(0,N)$, vertices unclassified. Then we predict the function of these
vertices and let the average fraction of correctly predicted vertices
$s$ be our performance measure. Fig.~\ref{fig:mod} shows $s$ for
$a=1/50$ and different network sizes, as a function of the the
rewiring probability $r$. In the small-$r$ limit the I-similarity
prediction scheme makes an almost flawless job with $s>99{.}9\%$ for
$N\geqslant 500$. Note, since we have five distinct functions, random
guessing could not do better than $s=1/5$. This value, $s=1/5$, is by
necessity attained in the random limit $r=1$. For small $r$-values the
scheme II performs best, but if $r\lesssim 0{.}2$ scheme I performs
slightly better. The size convergence for scheme I is faster, so in
the large network limit II may outperform I. To understand the
performance of the different schemes we note that scheme I has a
tendency to match an unknown vertex to a known vertex of high
degree. When $r=0$ this effect leads to some mispredictions for scheme
I. But the redundant information about high degree vertices makes the
more robust to minor perturbations, thus the slower decay of the
$s(r)$-curves compared with scheme II.
We observe that the performance increases with the systems size for
both schemes. This is important effect since databases in general grow
in size--our prediction scheme will thus be more accurate with time.
We surmise the explanation lies in, roughly speaking, that the bigger
the network gets, the more likely it is that there is a very good
matching. This is an effect local methods (taking only the surrounding
of a vertex into account) could not utilize. A full explanation of
this effect lies beyond the scope of this paper.
\section{Predicting protein function in yeast}
\begin{figure}
\resizebox*{0.85\linewidth}{!}{\includegraphics{pex.eps}}
\caption{
Example from the yeast protein prediction by scheme II on the
first level functional data. When YJL191w is marked
unknown it gets matched with YOR133w because their surroundings
looks similar. The arrowed lines mark genetic regulation edges,
other lines represent physical interaction.
}
\label{fig:pex}
\end{figure}
\subsection{Functional prediction of proteins}
Specifying protein functions experimentally requires demanding and
potentially expensive tests. If one can obtain good guesses of the
functions of an unknown protein, much is gained. During last decade,
there has been a great number of methods suggested for protein
functional prediction, including methods based on based on sequence
or structure alignments \cite{paw:seq,irving:struct}, attributes
derived from collections of sequences or
structures \cite{jensen:seq,dobson:struct}, phylogenetic profiles
\cite{pelle:pfp}, or analysis of protein complexes
\cite{gavin:complexes}. Much of recent work has concentrated on
functional prediction based on protein-protein interaction data. Many
of these are specialized methods that exploit specific features of
protein-protein interaction data \cite{vaz:pfp,schw:pfp,marc:pfp1,%
marc:pfp2,hodg:pfp,leto:pfp,sama:pfp} (such as that vertices that
interact physically are likely to share some functionality). The more
general approaches \cite{deng:pfp,hish:pfp} are local in the sense
that they are only based on pairwise statistics. For this reason they
may not share the advantageous size scaling properties of our method.
\subsection{Applying the method to protein data}
There are two types of large scale network data available for
\textit{S.\ cerevisiae}: ``physical'' and ``genetic'' protein-protein
interactions. The terms ``physical'' and ``genetic'' refer to the type of
experiment used to deduce the interaction. The genetic experiments
are based on mutation studies, and the evidence from them is of
a more indirect nature. We therefore distinguish
between physical and genetic edges. All edges are undirected. Our data
set, derived from the MIPS data base, has $N=4580$ linked together by
$5129$ genetic regulation edges and $7434$ physical interaction
edges. We removed duplicates, self-edges and interactions where one or
both of the interacting substances were not proteins. The assigned
functions are arranged in a hierarchical fashion, according to the
FunCat categorization scheme \cite{ruepp:funcat} used by the MIPS
database. The first level contains the coarsest description of a
protein's function, such as ``metabolism,'' the second level is more
specified e.g.\ ``amino acid metabolism,'' and so on. We will test our
algorithm of the first and second level of this hierarchy and thus
treat functions that differ in a finer classification as equal. There
are three categories with no substantial functional
information---``ubiquitous expression,'' ``classification not yet
clear-cut'' and ``unclassified proteins.'' We considered vertices with
no other assigned categories than these three uncategorized.
In Fig.~\ref{fig:pex} we show a small example of scheme II in action
on the yeast data. Suppose YJL191w is to be classified (we know it has
the level-1 functions ``protein with binding function \ldots'' and
``protein synthesis''). The classified protein with highest similarity
is YOR133w. This is because YNL041c, which interacts physically with
YJL191w, is functionally identical (at level one of the hierarchy) to
YBR068c that is physically linked to YOR133w. Similarly, YJL191w is
genetically linked with YCR031c, which shares one functional category
with YDR385w, which is genetically linked with YOR133w. These two
features give a high similarity score to the pair YJL191w and YOR133w,
so scheme II guesses that YJL191w has the functional category
``protein synthesis'' but misses the ``protein with binding function
\ldots'' category.
\subsection{Performance of the scheme}
\begin{table}
\caption{\label{tab:perf} The performance of our methods compared to
the neighborhood counting method of Ref.\ \cite{schw:pfp}. $s_+$ is
the average fraction of correct predictions among the predicted
functions averaged over all the classified proteins. $s_-$ is the
average fraction of correct predictions among the actual
functions.}
\begin{ruledtabular}
\begin{tabular}{r|cccccc}
& \multicolumn{3}{c}{level 1} & \multicolumn{3}{c}{level 2}\\
& NCM & Scheme I & Scheme II & NCM & Scheme I & Scheme II\\\hline
$s_+$ & 0{.}269(6) & 0{.}392(6) & 0{.}337(6) &
0{.}199(5) & 0{.}238(6) & 0{.}220(6) \\
$s_-$ & 0{.}354(6) & 0{.}291(5) & 0{.}346(7) &
0{.}252(6) & 0{.}199(5) & 0{.}231(6) \\
\end{tabular}
\end{ruledtabular}
\end{table}
For the previously described test networks we know \textit{a priori}
that the number of functions to be predicted is one. The same may be
true for a variety of systems, but not for proteins. With the number
of functions as one variable in the prediction problem we proceed to
replace the success rate $s$ by the two measures \textit{precision}
$s_+$ and \textit{recall} $s_-$ (the names borrowed from corresponding
quantities in the text-mining literature, see e.g.\ Ref.~\cite{rag:tm}
and references therein):
\begin{equation}\label{eq:spm}
s_+ = \left\langle\frac{n_c}{f_*}\right\rangle \mbox{~and~}
s_- = \left\langle\frac{n_c}{f}\right\rangle ,
\end{equation}
where $n_c$ is the number of correctly predicted functions, $f$ is the
real number of functions and $f_*$ is the number of predicted
functions. $1-s_+$ is thus the expected fraction of false positive
predictions (and similarly for $s_-$). Both these measures take values
in the interval $[0,1]$ with $0$ meaning that no function is predicted
correctly and $1$ represents perfect prediction. The averages are over
the set of predicted functions in the same kind of leave-one-out
estimates as performed for the test networks.
We follow Refs.\ \cite{vaz:pfp,deng:pfp} and use the neighborhood
counting method (NCM) of Ref.\ \cite{schw:pfp} for reference
values. This method assigns the $f_*$ most frequent functions among
the neighbors of the physical interaction network to the unknown
protein. Considering its simplicity, compared with the more elaborate
procedures listed above, this is a remarkably efficient method. (I.e.,
$f_*$ is a parameter of this model.) In our implementation, if the
$f_*$'th function is not unique we select that randomly. Thus proteins
with no neighbors are assigned $f_*$ functions randomly. Precision and
recall values are displayed in Tab.~\ref{tab:perf}. We use $f_*=2$ for
the NCM which is the closest value to the average number of functions
per protein for both levels one and two in our data set. The values
may look low compared to similar tables in other papers on protein
prediction, but these often do not include low-degree vertices, or use
other performance measures (such as counting the fraction of proteins
with at least one correctly predicted function, and so on). We note
that, like the more disordered test networks, scheme II gives better
performance in general (typically having better recall- but slightly
worse precision-values).
\section{Summary and discussion}
We have proposed methods for predicting the function of vertices in
networked systems where the function of a vertex relates to its
position. The principle behind our scheme is role equivalence as
related to the regular equivalence concept of social network
analysis. I.e., vertices are similar if the network, as seen from the
respective vertices, look similar. We make two extensions to the method
proposed in Refs.\ \cite{simrank,blondel:sim} to networks where some of
the vertices are functionally categorized. The prediction of an
uncategorized protein is then done by copying the functions of the
other vertex with highest role similarity. Our schemes, corresponding
to our two role similarities, are tested on model networks. These are
designed to have a correspondence between the function of the vertex
and their network surrounding. This correspondence can be tuned by a
randomization parameter. We find that the performance of both schemes
increases with the system size (the fraction of unknown vertices and
rewired edges is fixed), which makes the applicability of our methods
increasing with time (as data bases, in general, tend to grow). The
differences between scheme I and II can be described by the fact that,
scheme I gives (compared with scheme II) a higher similarity to
vertex-pairs containing a high-degree vertex. Furthermore, we apply
our method to the \textit{S.\ cerevisiae} proteome. We use the
networks of protein-protein interactions and obtain results that
compare well with standard methods designed solely with protein
functional prediction in mind. We do not claim that our method
outperform the best specialized protein prediction methods---our aim
is to construct a global method for general functional prediction, and
most protein functional prediction schemes would perform poorly on our
test networks. The ideas of this paper might however contribute to
future, more elaborate, methods for prediction of protein functions.
The basic advantage of our method, as we see it, is that is a very
general method that should apply to functional prediction in many
systems. Moreover, it makes use of global network information,
giving performance that does not decrease as the systems gets
larger. The fact that it is a truly global algorithm---the prediction
of every vertex' functions takes wiring of the whole network into
account---makes it rather slow (compared to e.g.\ specialized protein
functional prediction methods, such as the one proposed in
Ref.\ \cite{schw:pfp}). The execution time scales as $O(M^2)$ (where
$M$ is the total number of edges). But data sets of $10^4$-$10^5$,
which cover e.g.\ the size of proteomes of known organisms, should be
manageable to present day computers. We believe the problem of
functional prediction in different types of networked systems is far
from concluded---both in its full generality and the question how to
utilize the characteristics of more specific systems.
\subsection*{Acknowledgments}
The authors thank Micha Enevoldsen, Elizabeth Leicht and Mark Newman
for comments.
|
{
"timestamp": "2005-03-06T19:52:49",
"yymm": "0503",
"arxiv_id": "q-bio/0503010",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503010"
}
|
\section{Introduction}
The general structures ruling the evolution of classical and quantum
systems are not essentially different. For instance both systems are
Hamiltonian vector fields and both are derivations on the Lie
algebra of observables with respect to the Poisson bracket and the
commutator bracket respectively. Besides, in some appropriate limit,
quantum mechanics should reproduce classical mechanics.\cite{dirac}
So the question arises of which alternative quantum descriptions for
a given quantum system would reproduce the alternative classical
descriptions of Hamiltonian systems.These systems are usually known
as bi-Hamiltonian systems. Completely integrable systems are often
associated with alternative compatible Poisson structures. We recall
that by compatibility is usually understood that any combination,
with real coefficients, of the two Poisson brackets satisfies the
Jacobi identity. In this respect, we should remark that while on a
vector space the imaginary part of the hermitian structures, i.e.
constant symplectic structures, are always mutually compatible, this
is not true for the full hermitian structures. In this case the
compatibility of the complex structures gives non trivial conditions
even in the vector space situation. As a matter of fact the complex
structure, related to the indetermination relation, plays no role in
the classical limit of quantum mechanics.\cite{bedlevo}
In the study of bi-Hamiltonian systems one usually starts with a
given dynamics and looks for alternative Hamiltonian descriptions
(see a partial list of references for classical \cite{ma} and
for quantum \cite{blo} systems).
In this paper we deal with a kind of converse problem \cite{msv},
i.e. we start with two Hermitian structures on a complex Hilbert
space and look for all dynamical quantum evolutions which turn out
to be bi-unitary with respect to them. This study generalizes our
previous results on finite-dimensional bi-Hamiltonian systems in
reference \cite{mor} to the infinite-dimensional case.
This paper is organized as follows. In section 2, we consider two
Hermitian structures on a finite-dimensional Hilbert space and show
the equivalence of the following three properties for the Hermitian
positive operator $G$ which connects them: the non-degeneracy, the
cyclicity and the genericity. A short description of a bi-unitary
group is also given. In section 3, we introduce the
infinite-dimensional case recalling the direct integral
decomposition of a Hilbert space with respect to a commutative ring
of operators, which is a suitable mathematical tool to deal with
such a situation \cite{nai}. In section 4, we extend to the
infinite-dimensional Hilbert spaces the analysis drawn in section 2.
In particular, we prove that the component spaces in the
decomposition are one-dimensional if and only if the Hermitian
structures are in relative generic position. Also, we show that this
happens if and only if the operator $G$ connecting the two Hermitian
structures is cyclic. This allows to conclude that all the quantum
systems, which are bi-unitary with respect to two Hermitian
structures in generic relative position, commute among themselves.
Moreover, the bi-unitary group is explicitly exhibited both in the
generic and non generic case. In section 5, the analysis starts from
different complexifications of a real Hilbert space to discuss the
previous results in the light of the notion of compatible
triples.\cite{mor, dasilva} In section 6 we discuss a simple example
of some physical interest and finally, in the last section, we draw
a few conclusions.
\section{Bi-unitary group on a finite-dimensional space}
In quantum mechanics the Hilbert space $\mathcal{H}$ is given as a \emph{%
complex} vector space, because the complex structure enters directly
the Schroedinger equation of motion.
Denoting with $h_{1}(.,.)$ and $h_{2}(.,.)$ two Hermitian structures
given on $\mathcal{H}$\ (both linear, for instance, in the second
factor), we search for the group of transformations which leave both
$h_{1}$ and $h_{2}$ invariant, that is the bi-unitary transformation
group.
By using the Riesz's theorem a bounded, positive operator $G$ may be
defined, which is self-adjoint both with respect to $h_{1}$ and
$h_{2}$, as:
\begin{equation}
h_{2}(x,y)=h_{1}(Gx,y),\ \ \ \ \forall x,y\in \mathcal{H}.
\end{equation}
Moreover, any bi-unitary transformation $U$ must commute with $G$.
Indeed:
\begin{equation*}
\fl h_{1}(x,U^{\dagger
}GUy)=h_{1}(Ux,GUy)=h_{2}(Ux,Uy)=h_{2}(x,y)=h_{1}(Gx,y)=h_{1}(x,Gy)
\end{equation*}
and from this
\begin{equation}
U^{\dagger }GU=G \Leftrightarrow [G,U]=0.
\end{equation}
Therefore the group of bi-unitary transformations is contained in the commutant $%
G^{\prime }$ of the operator $G$.
To visualize these transformations, let us consider the bi-unitary
group of transformations when $\mathcal{H}$ is finite-dimensional.
In this case $G$ is diagonalizable and the two Hermitian structures
result proportional in each eigenspace of $G$ \emph{via} the
eigenvalue. Then the group of bi-unitary transformations is given by
\begin{equation}
U(n_{1})\times U(n_{2})\times ...\times U(n_{m}), \ \ \ %
n_{1}+n_{2}+...+n_{m}=n=\dim \mathcal{H},
\end{equation}
where $n_{k}$ denotes the degeneracy of the $k$-th eigenvalue of
$G$.
The picture should be clear now. Each Hermitian structure on
$\mathcal{H}$ defines a different realization of the unitary group
as a group of transformations. The intersection of these two groups
identifies the group of bi-unitary transformations.
In finite-dimensional complex Hilbert spaces the following
definition can be introduced \cite{mor}:
\noindent \textbf{Definition 1 }\textit{Two Hermitian forms are said
to be in generic relative position when the eigenvalues of
}$G$\textit{\ are non-degenerate.}
Then, if \ $h_{1}$ and $h_{2}$ are in generic position, the group of
bi-unitary transformations becomes
\begin{eqnarray*}
&&\underbrace{U(1)\times U(1)\times ...\times U(1)}. \nonumber\\
&&\ \ \ \ \ \ \ \ \ \ n\ \ factors \nonumber
\end{eqnarray*}
In other words, this means that $G$ generates a complete set of
commuting observables.
Now, recalling that an operator is cyclic when a vector $x_{0}$
exists such
that the set $\{x_{0},$ $Gx_{0},...,$ $G^{n-1}x_{0}\}$ spans the whole $n-$%
dimensional Hilbert space, we show that:
\noindent \textbf{Proposition 1}
\textit{Two Hermitian forms are in generic relative position if and only if
their connecting operator }$G$\textit{\ is cyclic}.
\textbf{Proof }The non singular operator $G$ has a discrete spectrum
and is diagonalizable so, when $h_{1}$ and $h_{2}$ are in generic
position, $G$ admits $n$ distinct eigenvalues $\lambda _{k}$. Let
now $\{e_{k}\}$ be the eigenvector basis of $G$ and $\{\mu ^{k}\}$
an $n$-tuple of nonzero complex numbers. The vector
\begin{equation}
x_{0}=\sum\nolimits_k\mu ^{k}e_{k}
\end{equation}
is a cyclic vector for $G$. In fact one obtains
\begin{equation}
G^{m}x_{0}=\sum\nolimits_k\mu ^{k}\lambda _{k}^{m}e_{k}\ ,\ \ \
m=0,1,...,n-1.
\end{equation}
The vectors $\{G^{m}x_{0}\}$ are linearly independent because the
determinant of their components is given by
\begin{equation}
(\prod\limits_{k}\mu ^{k})V(\lambda _{1},...,\lambda _{n}),
\end{equation}
where $V$ denotes the Vandermonde determinant which is different
from zero when all the eigenvalues $\lambda _{k}$ are distinct. The
converse is also true.$\ \ \Box$
This shows that definition $(1)$ may be equivalently formulated as:
\noindent\textbf{Definition 2} \textit{Two Hermitian forms are said
to be in generic relative position when their connecting operator
}$G$\textit{\ is cyclic.}
The genericity condition can also be restated in a purely algebraic
form as follows:
\noindent \textbf{Definition 3} \textit{Two Hermitian forms are said
to be in generic relative position when }$G^{\prime \prime
}=G^{\prime }$\textit{, i.e. when the bi-commutant of }$G$
\textit{coincides with the commutant of} $G$.
Equivalence of definitions $(3)$ and $(1)$ is apparent.
The last two equivalent properties of $G$ are readily suitable for
an extension of the genericity condition to the infinite-dimensional
case while, at a first glance, the definition based on
non-degeneracy of the spectrum of $G$ looks hardly generalizable.
\section{Decomposing an infinite-dimensional Hilbert space}
Now we deal with the infinite-dimensional case, when the connecting
operator $G$ may have a point part and a continuum part in its
spectrum.
As regards to the point part, the bi-unitary group is
$U(n_{1})\times ...\times U(n_{k})\times ...,$ where now $n_{k}$ may
also be $\infty .$ When $G$ admits a continuum spectrum, the
characterization of the bi-unitary group is more involved and
suitable mathematical tools are needed from the spectral theory of
operators and the theory of rings of operators on Hilbert spaces.
We recall that each commutative (weakly closed) ring of operators
$C$ in a Hilbert space, containing the identity, corresponds to a
direct integral of Hilbert spaces.
The following theorems \cite{nai} are useful:
\noindent \textbf{Theorem 1 }\textit{To each direct integral of
Hilbert spaces with respect to a measure }$\sigma $\textit{\ on a
real interval }$\Delta :$
\begin{equation*}
\mathcal{H}=\int_{\Delta }H_{\lambda }\textrm{d}\sigma (\lambda ),
\end{equation*}
\textit{there corresponds a commutative weakly closed ring
}$C=L_{\sigma }^{\infty }(\Delta ),$\textit{\ where to each
}$\varphi \in L_{\sigma }^{\infty }(\Delta )$\textit{\ there
corresponds the operator }$L_{\varphi }:(L_{\varphi }\xi )=\varphi
(\lambda )\xi _{\lambda }$ \textit{with} $\xi
\in \mathcal{H},$ $\xi _{\lambda }\in H_{\lambda }$\textit{\ and }$%
||L_{\varphi }||=||\varphi ||_{\infty }.$
\bigskip
\emph{Vice versa}:
\bigskip
\noindent \textbf{Theorem 2 }\textit{To each commutative weakly closed ring }$C$%
\textit{\ of operators in a Hilbert space }$\mathcal{H}$\textit{\
there corresponds a decomposition of }$\mathcal{H}$\textit{\ into a
direct
integral, for which }$C$\textit{\ is the set of operators of the form }$%
L_{\varphi },$ $\varphi \in L^{\infty }$\textit{.}
To apply the previous theorems to the ring $R(G)$ generated by the
connecting operator $G$, we preliminarily remark that:
\noindent \textbf{Proposition 2} \textit{The weakly closed commutative ring }$R(G)$%
\textit{\ generated by the connecting operator }$G$ \textit{contains
the identity.}
\textbf{Proof} Let $E_{0}$ be the principal identity of $G$ in the ring of all bounded operators $\mathcal{B}(\mathcal{H}%
) :$ by definition $E_{0}$ is the projection operator on the
orthogonal complement of the set $\ker G.$
We recall \cite{nai} that the minimal weakly closed ring $R(G)$ containing $%
G $ contains only those elements $A\in G^{\prime \prime }$ which
satisfy, like $G,$ the following condition:
\begin{equation}
E_{0}A=AE_{0}=A.
\end{equation}
Now the positiveness of the operator $G$ ensures that $\ker G=0.$ This implies that $E_{0}=%
\mathbf{1}\in R(G).\ \ \ \square $
Then, by theorem (2), the ring $R(G)$ induces a decomposition of the Hilbert space $%
\mathcal{H}$ into the direct integral
\begin{equation}
\mathcal{H}=\int_{\Delta }H_{\lambda }\textrm{d}\sigma (\lambda ),
\label{Hilbert decomposition}
\end{equation}
where $\Delta =[a,b]$ contains the entire spectrum of the positive
self-adjoint operator $G.$ The measure $\sigma (\lambda )$ in
equation (\ref{Hilbert decomposition}) is obtained by the spectral
family $\{P_{G}(\lambda )\}$ of $G$ and cyclic vectors in the usual
way.\cite{nai}
We remark that it results $R(G)\equiv G^{\prime \prime }$. Therefore $%
G^{\prime \prime }$ is commutative.
Now every operator $A$ from the commutant $G^{\prime }$ is
representable in the form of a direct integral of operators
\begin{equation}
A\ \cdot=\int_{\Delta }A(\lambda )\ \cdot\ \textrm{d}\sigma (\lambda
),
\end{equation}
where $A(\lambda )$ is a bounded operator in $H_{\lambda }$, for
almost every $\lambda \in \Delta $.
Thus the bi-unitary transformations, as they belong to $G^{\prime}
,$ are in general a direct integral of unitary operators $U(\lambda
)$ acting on $H_{\lambda }$.
In particular, every operator
$B$ of the bi-commutant $G^{\prime \prime }=R(G)$ is a
multiplication by a number $b(\lambda )$ on $H_{\lambda },$ for
almost every $\lambda :$
\begin{equation}
B(\lambda )= b(\lambda )\ 1_{\lambda} .
\end{equation}
\section{Bi-unitary group on an infinite-dimensional Hilbert space}
More insight can be gained from a more specific analysis of the
direct integral decomposition of $\mathcal{H}\mathbb{\ },$ which can
be written as
\begin{equation}
\mathcal{H}=\int_{\Delta }H_{\lambda }\textrm{d}\sigma (\lambda
)=\bigoplus\limits_{k}\int_{\Delta _{k}}H_{\lambda }\textrm{d}\sigma
(\lambda )=\bigoplus\limits_{k}\mathcal{H}_{k},
\label{hilbertdecompfine}
\end{equation}
where now the spectrum $\Delta $ of $G$ is the union of a countable
number of measurable sets $\Delta _{k}$, such that for $\lambda \in
\Delta _{k}$ the spaces $H_{\lambda }$ have the same dimension
$n_{k}$ ($n_{k}$ may be $\infty $).
The measure $\sigma (\lambda )$ is obtained by the measures $\sigma
_{k}(\lambda )$'s \textit{via } the spectral family $\{P_{G}(\lambda
)\}$ of $G$\ and cyclic vectors $u_{k}$ , with $\sigma _{k}(\lambda
)=(P_{G}(\lambda )u_{k},u_{k})$.
The dimension $n_{k}$ of the spaces $H_{\lambda }$ is the analog of
the degeneracy of the eigenvalues $\lambda $ of the point part of
the spectrum of $G$ .
According to the decomposition of equation
(\ref{hilbertdecompfine}), any operator $A$ in the commutant
$G^{\prime }$ is representable as:
\begin{equation}
A\ \cdot=\bigoplus\limits_{k}\int_{\Delta _{k}}A(\lambda ) \ \cdot\
\textrm{d}\sigma (\lambda ). \label{operatordecomp}
\end{equation}
In particular, the connecting operator $G$ is a multiplication by
$\lambda $ on each $H_{\lambda }$, so we get the following result at
once:
\noindent\textbf{Proposition 3 }\textit{Let two Hermitian structures} $h_{1}$ \textit{%
and} $h_{2}$ \textit{be given on the Hilbert space
}$\mathcal{H}$\textit{. Then there exists a decomposition of
}$\mathcal{H}$ \textit{into a direct
integral of Hilbert spaces }$H_{\lambda }$\textit{\ such that in each space }%
$H_{\lambda }$\textit{\ the structures\ }$h_{1}|_{H_{\lambda }}$
\textit{and} $h_{2}|_{H_{\lambda }}$\textit{\ are proportional:
}$h_{2}|_{H_{\lambda }}=\lambda \ h_{1}|_{H_{\lambda }}$\textit{.}
Moreover, as $G$ acts like a multiplicative operator on each
component space $H_{\lambda },$ the expressions of $h_{1}$ and
$h_{2}$ on $\mathcal{H}$ are:
\begin{equation*}
h_{1}(x,y)=\sum_{k}\int\nolimits_{\Delta
_{k}}<x_{\lambda},y_{\lambda}>_{\lambda}\textrm{d}\sigma (\lambda )\
\ ,
\end{equation*}
\begin{equation}
h_{2}(x,y)=\sum_{k}\int\nolimits_{\Delta _{k}}\lambda
<x_{\lambda},y_{\lambda}>_{\lambda }\textrm{d}\sigma (\lambda )
\label{inner}
\end{equation}
where $<x_{\lambda},y_{\lambda}>_{\lambda }$ is the inner product on
the component $H_{\lambda }$.
As a consequence of proposition (3) and equation
(\ref{operatordecomp}), the elements $U$ of the bi-unitary group
acting on $\mathcal{H}$ have the form:
\begin{equation}
U\ \cdot=\bigoplus\limits_{k}\int_{\Delta _{k}}U_{n_{k}}(\lambda )\
\cdot \ \textrm{d}\sigma (\lambda ), \label{unitdecomp}
\end{equation}
where $U_{n_{k}}(\lambda )$ is an element of the unitary group
$U(n_k) $ for each $\lambda \in \Delta _{k}.$
As regards to the notion of two Hermitian forms in generic position,
the following statement \cite{lecce} holds:
\noindent\textbf{Proposition 4}\textit{\ Two Hermitian structures }$h_{1}$ \textit{and%
} $h_{2}$\ \textit{are in generic relative position if and only if
the component
spaces }$H_{\lambda }$\textit{\ of the decomposition of }$\mathcal{H}$%
\textit{\ into a direct integral\ with respect to }$R(G)$
\textit{are one-dimensional. }
\textbf{Proof} Let us suppose that two Hermitian forms are given in
generic relative position. Then, by definition (3), $R(G)=G^{\prime \prime }=G^{\prime }$, so $%
G^{\prime }$ is commutative and any component operator $A(\lambda )$
in
equation (\ref{operatordecomp}) acts on an one-dimensional component space $%
H_{\lambda }$, for almost every $\lambda \in \Delta $.
In order to prove the converse, observe that if $R(G)=G^{\prime
\prime }\neq G^{\prime }$, then $G^{\prime }$ is not commutative. So
a subset $\Delta _{0}$ of $\Delta $ exists such that $\dim
H_{\lambda }>1$ for $\lambda \in \Delta _{0}.\ \ \ \square $
This shows the equivalence of definitions (1) and (3) also in the
infinite-dimensional case.
Propositions (3) and (4) extend to infinite-dimensional complex
Hilbert spaces some results of our previous work \cite{mor}, so that
we can say that all quantum dynamical bi-Hamiltonian systems are
pairwise commuting if (and only if) the two Hermitian structures are
in generic relative position.
In the generic case, the unitary component operators
$U_{n_k}(\lambda )$ in
equation (\ref{unitdecomp}) reduce to a multiplication by a phase factor $%
\textrm{exp}(\textrm{i} \vartheta (\lambda ))$ on $H_{\lambda }$ for
almost every $\lambda $, so that the elements of the bi-unitary
group read
\begin{equation}
U\ \cdot=\int_{\Delta }\rm{e}^{\rm{i}\vartheta (\lambda )}\ \cdot \
\textrm{d}\sigma (\lambda ).
\end{equation}
Therefore in the generic case the group of bi-unitary
transformations is parameterized by the $\sigma -$measurable real
functions $\vartheta $ on $\Delta .$ This shows that the bi-unitary
group may be written as
\begin{equation}
U_{\vartheta }=\textrm{exp}(\textrm{i}\vartheta (G))\; .
\end{equation}
Finally, like in the finite-dimensional case, an equivalence may be
stated between the genericity condition and the cyclicity of the
operator $G$. In fact, we have:
\noindent\textbf{Proposition 5\ }\textit{Let }$G$\textit{\ be a
bounded positive self-adjoint operator in }$\mathcal{H}.$\textit{
Then }$G$\ \textit{is cyclic if and only if }$G^{\prime \prime
}=G^{\prime }.$
\textbf{Proof} Let us suppose $G^{\prime \prime }=G^{\prime }$. Then
$R(G)=G^{\prime \prime }=G^{\prime }$ and $G^{\prime }$ is
commutative. Hence the decomposition of the Hilbert space yields
one-dimensional component
spaces $H_{\lambda }$ where $G$ acts as a multiplication by $\lambda $ in $%
L_{2}(\Delta ,\sigma ).$ Then the vector $x_{0}=1/\lambda $ is a
cyclic vector in $L_{2}(\Delta ,\sigma )$, so $G$ is cyclic.
Conversely, let $G$ be cyclic. Then each space $H_{\lambda }$ is
one-dimensional and any operator from $G^{\prime }$ acts as a
multiplication by a number in $H_{\lambda }$. Hence $G^{\prime
}=R(G)=G^{\prime \prime }.\ \ \ \square $
Summarizing, we have shown the equivalence of definitions (1), (2)
and (3) in the infinite-dimensional case.
\section{Compatible structures on a real infinite-dimensional Hilbert space }
In the previous section we have analyzed the setting of a complex
Hilbert
space $\mathcal{H}$ with two Hermitian structures\ $h_{1}(.,.)$ and $%
h_{2}(.,.)$ and now, to make contact with real linear Hamiltonian
mechanics \cite{mor} on infinite dimensional spaces, we analyze the
consequences of this on real Hilbert spaces. Besides, the real
context\ displays richer contents and is a more general setting for
the analysis of our geometric structures.
We start therefore with a real vector space
$\mathcal{H}^{\mathcal{R}}$ (isomorphic to the realification of
$\mathcal{H}$).
From the two Hermitian structures on the previous complex Hilbert space, $%
h_{1}(.,.)$ and $h_{2}(.,.),$\ we get on $\mathcal{H}^{\mathcal{R}}$
two metric tensors $g_{a}$ and two symplectic forms $\omega _{a}$
\textit{via }:
\begin{equation*}
g_{a}(x,y)=\Re \ h_{a}(x,y);\ \ \omega _{a}(x,y)=\Im \ h_{a}(x,y)\ ,
\ a=1,2.
\end{equation*}
On $\mathcal{H}^{\mathcal{R}}$
the multiplication by
the imaginary unit appears as the action of a linear operator $J$ , $%
J^{2}=-1,$ which is skew-adjoint with respect to both $g$'s.
The structures are related by the equation $\omega
_{a}(x,y)=g_{a}(Jx,y)$ which defines the \emph{admissible} triples
$(g_{a},\omega _{a},J)$.
Then the three linear operators $G^{\mathcal{R}}=g_{1}^{-1}\circ
g_{2},T=\omega _{1}^{-1}\circ \omega _{2}=-J\circ
G^{\mathcal{R}}\circ J$ and $J$ are a set of mutually commuting
linear operators, $G^{\mathcal{R}}$ and $T$ being self-adjoint with
respect to both metric tensors. We remark, by the way, that $T$ is
the recursion operator for symplectic structures.
For instance, to check that $[G^{\mathcal{R}},J]=0,$ consider the
equation $h_{2}(x,y)= h_{1}(G x,y)$ which defines the connecting
operator $G.$ Then:
\begin{eqnarray}
\fl h_{1}(G x,y) =g_{1}(G x,y)+\textrm{i}g_{1}(J G x,y)=h_{2}(x,y) \nonumber\\
=g_{2}(x,y)+\textrm{i}g_{2}(J
x,y)=g_{1}(G^{\mathcal{R}}x,y)+\textrm{i}g_{1}(G^{\mathcal{R}}J
x,y). \nonumber
\end{eqnarray}
This shows, by equating real and imaginary parts, that $G^{\mathcal{R}}=G$ and $%
[G,J]=0 .$ It is trivial now that $[T,G]=[T,J]=0$ as well. By
definition this means that these two triples are
\emph{compatible}.\cite{mor}
Quantum theory in the usual complex context leads quite naturally to
consider identical complex structures in the two triples. On the
contrary, in the real context it is possible to consider the case of
two distinct complex
structures $J_{1},J_{2}$. In other words, on a real Hilbert space $\mathcal{H}%
^{\mathcal{R}}$ let two admissible triples $(g_{1},J_{1},\omega _{1})$ and $%
(g_{2},J_{2},\omega _{2})$ be given which are compatible, that is
the commuting operators $\left\{ G,T,J_{1},J_{2}\right\} $ have the
correct bi-Hermiticity properties.\cite{dasilva}
Now it is possible to complexify $\mathcal{H}^{\mathcal{R}}$ and to
get a complex Hilbert space $\mathcal{H}_{1}$ with a Hermitian
scalar
product $<.,.>_{1}\ $ \textit{ via }$\ (g_{1},J_{1},\omega _{1}).$ Since by hypothesis the operators $%
\left\{ G,T,J_{2}\right\} $ commute with $J_{1}$, they become
complex-linear operators on $\mathcal{H}_{1}$. In particular $G$
becomes a complex-linear bounded positive self-adjoint operator,
therefore $G$ acts as a multiplication by $\lambda $ on the
component spaces in the associated direct integral decomposition
\begin{equation}
\mathcal{H}=\int_{\Delta }H_{\lambda }\textrm{d}\sigma (\lambda ).
\end{equation}
Now $J_{2}$ commutes with $G$, i.e. $J_{2}\in G^{^{\prime }}$ , so
$J_{2}$
is block-diagonal on $\mathcal{H}$. In each $H_{\lambda },$ we have $%
J_{2}^{2}(\lambda )=-1_{\lambda }$ and $J_{2}^{\dagger }(\lambda
)=-J_{2}(\lambda )$ . Then $H_{\lambda }$ splits in two parts
corresponding to the eigenvalues $\pm $i of $J_{2}(\lambda ):
H_{\lambda }=H_{\lambda }^{+}\oplus H_{\lambda }^{-},$ where on
$H_{\lambda }^{+}:J_{2}=J_{1}=$i, while on $H_{\lambda
}^{-}:J_{2}=-J_{1}=-$i. The direct integral decomposition becomes:
\begin{equation}
\fl \mathcal{H}=\int_{\Delta }H_{\lambda }^{+}\oplus H_{\lambda
}^{-}\;\textrm{d}\sigma (\lambda )=\mathcal{H}^{+}\oplus
\mathcal{H}^{-}=\int_{\Delta ^{+}}H_{\lambda }^{+}\;
\textrm{d}\sigma (\lambda )\oplus \int_{\Delta ^{-}}H_{\lambda
}^{-}\;\textrm{d}\sigma (\lambda ), \label{pmdecomp}
\end{equation}
where $\Delta ^{+}$ and $\Delta ^{-}$, subsets of $\Delta $ not
necessarily disjoint, are support of $H_{\lambda }^{+}$ and
$H_{\lambda }^{-}$ respectively. This completely extends the
finite-dimensional result in \cite{mor}.
At this point we can draw a complete picture: starting from two
admissible triples $(g_{a},J_{a},\omega _{a}),\ a=1,2,$ on
$\mathcal{H}^{\mathcal{R}}$ we may construct the corresponding
Hermitian structures $h_{a}=g_{a}+\textrm{i}\omega _{a}$. We stress
that $h_{a}$ is a Hermitian structure on $\mathcal{H}_{a},$ which is
the
complexification of $\mathcal{H}^{\mathcal{R}}\ $ \emph{via }$\ %
J_{a}$ , so that in general $h_{1}$ and $h_{2}$ are not Hermitian
structures on the $\emph{same}$ complex vector space.
When the triples are compatible
the decomposition of the space in equation (\ref{pmdecomp}) holds, so that $%
\mathcal{H}^{\mathcal{R}}$ can be decomposed into the direct sum of
the spaces $ \mathcal{H}_{\mathcal{R}}^{+}$ and
$\mathcal{H}_{\mathcal{R}}^{-}$
on which $J_{2}=\pm J_{1},$ respectively. The comparison of $h_{1}$ and $%
h_{2} $ requires a fixed complexification of
$\mathcal{H}^{\mathcal{R}}$, for instance
$\mathcal{H}_{1}=\mathcal{H}_{1}^{+}\oplus \mathcal{H}_{1}^{-}.$
Then, using equations (\ref{inner}) and (\ref{pmdecomp}), we can
write
\begin{equation}
h_{1}(x,y)=\int\nolimits_{\Delta
^{+}}<x_{\lambda},y_{\lambda}>_{\lambda}\textrm{d}\sigma (\lambda
)+\int\nolimits_{\Delta ^{-}}<x_{\lambda},y_{\lambda}>_{\lambda
}\textrm{d}\sigma (\lambda )\ \ ,
\end{equation}
while
\begin{equation}
h_{2}(x,y)=\int\nolimits_{\Delta ^{+}}\lambda
<x_{\lambda},y_{\lambda}>_{\lambda }\textrm{d}\sigma (\lambda
)+\int\nolimits_{\Delta ^{-}}\lambda
<y_{\lambda},x_{\lambda}>_{\lambda }\textrm{d}\sigma (\lambda )\ .
\label{scalarprod}
\end{equation}
It is apparent that $h_{2}$ is not a Hermitian structure as it is
neither linear nor anti-linear on the whole space $\mathcal{H}_{1}.$
\section{Example: Particle in a box, a double case}
Consider the operator $G=1+X^{2}$ , with $X$ position operator, on $%
L_{2}([-\alpha ,\alpha ],dx)$. It is Hermitian with spectrum $\Delta
=[1,1+\alpha ^{2}]$. From the spectral family of $X:$%
\begin{equation}
P(\lambda )f=\chi _{[-\alpha ,\lambda ]}f \ ,
\end{equation}
where $\chi _{[-\alpha ,\lambda ]}$ is the characteristic function
of
the interval $[-\alpha ,\lambda ]$, we get the spectral family $%
P_{G}(\lambda )$ of $G$:
\begin{equation}
P_{G}(\lambda )=P(\sqrt{\lambda -1})-P(-\sqrt{\lambda -1})\ .
\label{spectralfam}
\end{equation}
In fact, by a simple computation it is immediate to check that
$P_{G}$ is a projection operator:
\begin{equation}
P_{G}^{2}=P_{G},\ \ P_{G}(0)=0,\ \ P_{G}(\alpha ^{2})=1.
\end{equation}
Furthermore, write $G$ as
\begin{equation}
\fl G\ \cdot=\int\limits_{[-\alpha ,\alpha ]}(1+\lambda
^{2})\cdot\textrm{d} P(\lambda )=\int\limits_{[-\alpha
,0]}(1+\lambda ^{2})\cdot\textrm{d} P(\lambda
)+\int\limits_{[0,\alpha ]}(1+\lambda ^{2})\cdot\textrm{d} P(\lambda
)\ ,
\end{equation}
and change variable putting $\lambda =-\sqrt{\mu -1}$ in the first
integral and $\lambda =\sqrt{\mu -1}$ \ in the second one.
Eventually, the spectral decomposition of $G$ reads
\begin{equation}
G\ \cdot=\int\limits_{[1,1+\alpha ^{2}]}\lambda \ \cdot \ \textrm{d}
P_{G}(\lambda )\ ,
\end{equation}
where $P_{G}(\lambda )$ is given by equation (\ref{spectralfam}).
Now $G$ does not have cyclic vectors on the whole $L_{2}([-\alpha
,\alpha ],\textrm{d}x)$, because if $f$ is any vector, $xf(-x)$ is
non-zero and orthogonal to all powers $G^{n}f$. In other words
$G^{\prime}$, which contains both $X$ and the parity operator, is
not commutative.
This argument fails on $L_{2}([0,\alpha ],\textrm{d}x)$, where $\chi
_{[0,\alpha ]}$ is cyclic. Analogously, $\chi _{[-\alpha ,0]}$ is
cyclic on $L_{2}([-\alpha ,0],\textrm{d}x),$ so we get the splitting
in two $G$-cyclic spaces
\begin{equation}
L_{2}[-\alpha ,\alpha ]=L_{2}[-\alpha ,0]\oplus L_{2}[0,\alpha ]\ .
\end{equation}
From $P_{G}$ and those cyclic vectors we obtain the measure
\begin{equation}
\sigma (\lambda )=(P_{G}(\lambda )\chi _{\lbrack 0,\alpha ]},\chi
_{\lbrack 0,\alpha ]})=\sqrt{\lambda -1}
\end{equation}
for the decomposition of the Hilbert space
\begin{equation}
\mathcal{H}=\int\limits_{[1,1+\alpha ^{2}]}H_{\lambda }\ \textrm{d}\sigma (\lambda )%
\ ,
\end{equation}
where $H_{\lambda }$ is one-dimensional for the particle in the
$[0,\alpha ]$ box while is bi-dimensional for the $[-\alpha ,\alpha
]$ box.
The general case of an asymmetric box $[-\alpha ,\beta ]$ is a
direct superposition of the two previous cases, as we have shown in
section 4: in fact, assuming $ \beta
>\alpha$ for instance, the decomposition becomes the direct sum of
bi-dimensional spaces for the $[-\alpha ,\alpha ]$ box plus
one-dimensional spaces for the $[\alpha ,\beta ]$ box.
The bi-unitary transformations $U$ read
\begin{equation}
U\ \cdot=\int\limits_{[1,1+\alpha
^{2}]}\textrm{e}^{\textrm{i}\varphi (\lambda )}\ \cdot \
\textrm{d}\sqrt{\lambda -1}
\end{equation}
in the $[0,\alpha ]$ box, and
\begin{equation}
U\ \cdot=\int\limits_{[1,1+\alpha ^{2}]}U_{2}(\lambda )\ \cdot \
\textrm{d}\sqrt{\lambda -1}
\end{equation}
in the $[-\alpha ,\alpha ]$ box. Finally, in the $[-\alpha ,\beta ]$
box:
\begin{equation}
U\ \cdot=\int\limits_{[1,1+\alpha ^{2}]}U_{2}(\lambda )\ \cdot \
\textrm{d}\sqrt{\lambda -1}\ \oplus\ \int\limits_{[1+\alpha
^{2},1+\beta ^{2}]}\textrm{e}^{\textrm{i}\varphi (\lambda )}\ \cdot
\ \textrm{d}\sqrt{\lambda -1}\ .
\end{equation}
\section{Concluding remarks}
In this paper we have shown how to extend to the more realistic case
of infinite dimensions the results of our previous paper dealing
mainly with finite level quantum systems. Our approach shows, in the
framework of quantum systems, how to deal with ``pencils of
compatible Hermitian structures'' in the same spirit of ``pencils
of compatible Poisson structures'' \cite{ge, imm}. We hope to be
able to extend these results to the evolutionary equations for
classical and quantum field theories.
\bigskip
\textbf{References}
\bigskip
|
{
"timestamp": "2005-03-15T17:04:05",
"yymm": "0503",
"arxiv_id": "math-ph/0503040",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503040"
}
|
\section{Introduction}
The Galton board is an upright board with evenly
spaced nails driven into its upper half. The
nails are arranged in staggered order. The lower half of the board is
divided with vertical slats into a number of narrow rectangular
slots. From the front, the whole installation is covered with a
glass cover. In the middle of the upper edge, there is a funnel in
which balls can be poured, the diameter of the balls
being much smaller than the distance between the nails. The funnel is
located precisely above the central nail of the second row, i.\,e. the
ball, if perfectly centered, would fall vertically and directly
onto the uppermost point of this nail's surface (Fig. 1).
\wfig<bb=0 0 50.8mm 47.9mm>{ris1.eps}
Theoretically, the ball would repeatedly bounce off this nail's uppermost
point. Obviously, such a motion of the ball is unstable. In fact, due to
unavoidable inaccuracy in the board's positioning and impossibility to
completely exclude the lateral component, no matter how small, of the
ball's velocity, each ball, generally speaking, would meet the nail
somewhat obliquely. The ball would then deviate from the vertical line
and, after having collided with many other nails, fall into one of the
slots. If the experiment is run with a large number of balls,
dropped one after the other, then the following results are obtained: the
balls are distributed evenly to the left and to the right of the central
compartment (left and right deviations are equiprobable). Besides, the
balls would more rarely fall into the leftmost and rightmost compartments,
for large deviations are more rare to appear than small ones. However, despite the presence of nails and all the imperfections in the construction, the majority of the balls
will agglomerate in the central compartment as this provides
the smallest deviation. The number of balls in the
compartments would approximately correspond to the Gaussian law of errors.
In the earlier experiments with the Galton board the funnel was
filled with pellets or millet grains.
\section{Statement of problem}
In Galton board experiments
ball-to-nail impacts have always been inelastic. In this paper, we present results of simulations of a model of the Galton board for
various degrees of elasticity of the ball-to-nail collision.
We model the ball as a mass point. Hence, the ball's motion can be
regarded as the motion of a mass point in a vertical plane under the
action of gravity accompanied with multiple collisions with the nails. These
collisions are characterized by the coefficient of restitution~$e$, which
affects only the normal component of the ball's velocity after the impact.
The coefficient of restitution is the first parameter of the problem. It
can vary from 0 to 1. A value of~$e=1$ corresponds to
absolutely elastic impact for which the ball's energy does not change. The
other extreme case,~$e=0$, corresponds to absolutely inelastic impact: the
ball ``sticks'' to the nail. The nail's radius~$R$ is the second parameter
of the problem. Since the ball leaves the funnel and falls onto the nail
centrally, but possibly with some small departure to the left or to the right, we adopt that the first drop of the ball obeys the Gaussian law. On the other
hand, if the balls distribute uniformly over the funnel's
opening, then their distribution over the rectangular compartments will be
far from normal (Fig.~2). This distribution resembles \emph{the arcsine
law\/}. Incidentally, according to Paul Levy, the distribution of time
intervals over which a Brownian particle is located on the positive semi-axis, has a similar form. This observation is, probably, not just a
coincidence. The point is that a particle in Brownian motion experiences a
large number of random collisions with molecules of the surrounding fluid
(in our case, the ``molecules'' are regularly placed and fixed).
\fig{ris2.eps}
Accordingly, the problem's third parameter is the variance of the
distribution of the balls over the funnel's opening. Thus, we
introduce three parameters for the problem: 1) the coefficients of
restitution~$e$, 2) the nail's radius~$R$, 3) the variance~$\sigma_{0}$
of the normal distribution of the first ball-to-nail impact.
It is also necessary to choose the dimensions of the model board. The geometry of the board should meet the two requirements:
\begin{itemize}
\item the balls should not reach the vertical boundaries of the Galton
board;
\item each ball should experience at least several collisions with the
nails before it gets into one of the rectangular compartments.
\end{itemize}
\begin{figure}[!ht]
\begin{center}\it
\begin{tabular}{c c}
\includegraphics[width=2.2in,height=1.1in]{ris3a.eps} & \includegraphics[width=2.2in,height=1.1in]{ris3b.eps} \\
a & b \\
\includegraphics[width=2.2in,height=1.1in]{ris3c.eps} & \includegraphics[width=2.2in,height=1.1in]{ris3d.eps} \\
c & d \\
\end{tabular}
\end{center}\vspace{-3mm}
\caption{}
\end{figure}
Figure 3 (a--d) shows the balls' distribution over the
rectangular compartments for different dimensions of the model board,
namely,~$50 \times 50$ (a), $100 \times 100$ (b), $200 \times 200$ (c),
and $400 \times 300$ (d). In these cases, the three parameters of the
problem are:~$e=0.8$, $R=0.7$, and~$\sigma_{0}=0.05$. As we can see, the
distribution of the balls looks similarly for all the specified dimensions
of the board, but in the case of~$50 \times 50$ (Fig.~3a) the balls do reach the
vertical boundaries of the model Galton board. For the~$100 \times 100$
board, the balls no longer reach the boundaries (Fig.~3b). Further
enlargement of the model board's dimensions (its length and its
height) does not affect the pattern of the balls' distribution over the
compartments (Figs.~3c, d), but greatly increases the computation time.
Thus, the dimensions of the model Galton
board can be set to~$100 \times 100$ without any loss of quality.
So, we are going to investigate the properties of the balls' distribution
over the compartments of the Galton board and the dependency of the
variance of this distribution on the three specified parameters.
\section{Mathematical model}
The method of investigation consists in
simulating the motion of the balls (mass points) and taking into account
their collisions with obstacles (the nails) for different values of the
three specified parameters of the problem.
On the Galton board, we introduce an orthogonal coordinate system~$Oxy$ in
the following way: the axis~$Ox$ is directed horizontally and passes through
the upper boundaries of the rectangular compartments, in which the falling
balls are to be collected (for brevity, from this point on, we
will say {\it compartments} instead of {\it rectangular compartments}).
The axis~$Oy$ is directed vertically and goes
through the center of the nail that a ball is to hit first. The length of
the board is taken large enough for the balls not to reach its vertical
boundaries (as was specified earlier).
Thus, a pair~$x,\,y$ represents the position of a ball in the plane of the
Galton board. The fall of the ball is described with a set of two ordinary
differential equations:
\begin{equation}\label{1}
\ddot{x} = 0, \qquad \ddot{y} = - g,
\end{equation}
where $g$ is the gravitational acceleration.
Since the ball falls from the funnel and onto the first nail under
gravity, the velocity of the ball at the point of the first impact is
~$v_0=\sqrt{2g(h_0-R\sin\alpha_0)}$, where~$h_0$ is the distance
between the funnel's opening and the center of the first nail,~$R$ is the
nail's radius, while~$\alpha_0$ is the angle between the axis~$Ox$ and the
radius drawn to the point where the ball hits the nail (Fig.~4).
\fig<bb=0 0 77.1mm 63.2mm>{ris4.eps}
We will investigate the further motion of the ball according to the
following plan:
\begin{enumerate}
\item Introduce a coordinate system fixed to the nail: its origin is at
the ball-to-nail impact point, and its axes are the tangent and the
normal to the nail's surface at this point. Thus, with respect to this coordinate system, the velocity of the
particle at the first impact point has the
following components:~$v_0^{\tau}=v_0\cos\alpha_0$,
$v_0^n=-v_0\sin\alpha_0$.
\item After the impact with the nail, the velocity components will
change and take the form: $v_1^{\tau}=v_0\cos\alpha_0$,
$v_1^n=-ev_0^n=ev_0\sin\alpha_0$, where~$e$ is the coefficient of
restitution.
\item Then the ball will move in a parabola. To find its path, we solve
the equations~(1) with the following initial values: $x\,(0)=x_0$,
$y\,(0)=y_0$, $\dot{x}\,(0)=v_1\cos\gamma_0$,
$\dot{y}\,(0)=v_1\sin\gamma_0$, where~$(x_0,\,y_0)$ are the coordinates
of the ball at the time it hits the
nail,~$v_1=\sqrt{(v_1^{\tau})^2+(v_1^n)^2}$, while~$\gamma_0$ is the
angle between the axis~$Ox$ and the velocity vector~$v_1$. Thus, the
ball's path is the following parabola:
\begin{equation}\label{2}
y = - \frac{g}{2v_{1}^{2}\cos^2\gamma_{0}}(x - x_{0})^{2} + (x -
x_{0})\tan\gamma_{0} + y_{0}.
\end{equation}
The portion of the parabola the ball will take is determined by the direction
of the ball's velocity vector after its impact with the nail.
\item From (1) we find the velocity with which the ball will approach
the next nail. Let~$(x_1,\,y_1)$ be the coordinates of the point of the
next ball-to-nail impact. Then the velocity of the ball on the surface of
this nail has the following components:
$$
v_{2}^{x} = v_{1}\cos\gamma_{0}, \qquad v_{2}^{y} = - \frac{g(x_{1} -
x_{0})}{v_{1}\cos\gamma_{0}}+ v_{1}\sin\gamma_{0}.
$$
\end{enumerate}
Then another collision occurs, and again the ball's motion is calculated
according to the procedure described above. The whole operation is
repeated until the ball crosses the axis~$Ox$. As soon as the ball's path
crosses the axis~$Ox$, we find the intersection point and thus determine the
compartment the ball falls into.
One of the most important things about this model is to find the nail that
the ball is going to hit next. To that end, consider
the perpendiculars to the ball's path which go through the nails'
centers. Such perpendiculars are described with linear equations:
\begin{equation}\label{3}
x - x_{n} + \left( - \frac{g}{v^{2}\cos^{2}\gamma}(x_{n}-x_{\text{imp}})+
\tan\gamma\right)(y - y_{n}) = 0,
\end{equation}
where $(x_{\text{imp}},\,y_{\text{imp}})$ are the coordinates of the
previous ball-to-nail impact, $(x_n,\,y_n)$ are the coordinates of the
path's point through which a perpendicular is drawn, $v$ is the magnitude of the
ball's velocity after the previous impact, and $\gamma$ is the angle
between the axis~$Ox$ and the velocity vector~$v$.
Since our goal is to find perpendiculars through the nails'
centers, we insert the coordinates of the center of one of the
nails~$(x_{c},\,y_{c})$ into~(3). From this equation, we find a
pair~$(x_{n},\,y_{n})$ which meets the following requirements:
\begin{itemize}
\item the distance between the nail's center and the
point~$(x_{n},\,y_{n})$ is smaller than the nail's radius;
\item the absolute value of the difference between the abscissa of the
previous impact point and the abscissa of the path's point, through which the
perpendicular is drawn, is as small as possible.
\end{itemize}
The first requirement is to ensure that the ball's path meets the nail,
i.\,e. the coordinates of the next impact
point~$(x,\,y)$ can be found. These coordinates satisfy the following set of
equations:
$$
\label{5} \left\{
\begin{array}{cc}
(x - x_{c})^{2} + (y - y_{c})^{2} = R^{2}, \\
y = - \frac{g}{2v^{2}\cos^{2}\gamma}(x - x_{\text{imp}})^{2} + (x - x_{\text{imp}})\tan\gamma + y_{\text{imp}}, \\
\end{array}
\right.
$$
where $(x_{c},\,y_{c})$ are the coordinates of the nail's center, $R$ is
the nail's radius, and~$(x_{\text{imp}},\,y_{\text{imp}})$ are the
coordinates of the previous impact point.
The second requirement is to take the impacts in their sequence. This
follows from the parametric form of the ball's path. We solve the
equations~(1) with the following initial values:~$x\,(0)=x_{\text{imp}}$,
$y\,(0)=y_{\text{imp}}$, $\dot{x}\,(0)=v\cos\gamma$,
$\dot{y}\,(0)=v\sin\gamma$, where~$(x_{\text{imp}},\,y_{\text{imp}})$ are
the coordinates of the previous impact point,~$v$ is the magnitude of the
ball's velocity after the previous impact, while~$\gamma$ is the angle
between the axis~$Ox$ and the velocity vector~$v$. The result is the
parametric form of the ball's path after it has hit the nail:
$$
x = v t \cos\gamma + x_{\text{imp}}, \qquad y = - \frac{g}{2} t^2 + v
t\sin\gamma + y_{\text{imp}}.
$$
\textbf{5. Simulation results.} The model Galton board has been
implemented as an interactive Microsoft Visual
C++ application. The application offers the opportunity to vary every parameter of the
model: the nail's radius, the coefficient of restitution, and the
variance of the initial distribution when the ball hits the first nail;
it also allows varying the number of dropped balls. The application
outputs a histogram of the balls in the compartments and the variance of the final
distribution of the balls over the compartments. Besides, the experiment's
results can be visualized.
First, we get a histogram of the distribution of the balls over the
compartments (Fig.~5). Form this histogram, the variance is calculated
using the following well-known formulas:
$$
\bar{x} = \sum\limits_{i=1}^n x_{i} \frac{N_{i}}{N}, \qquad
\sigma^{2} = \sum\limits_{i=1}^{n} (x_{i} - \bar{x})^{2}
\frac{N_{i}}{N},
$$
where $\bar{x}$ is the mathematical expectation, $x_{i}$ is the coordinate
of the center of the~\textit{i}th compartment,~$n$ is the number of
compartments,~$N$ is the number of dropped balls,~$N_{i}$ is the final number of
balls accumulated in the~\textit{i}th compartment, and~$\sigma$ is the
variance of the distribution.
Second, using the variance obtained, we plot the normal distribution
(Gaussian) curve using the Gauss
formula~$f\,(x)=\frac1{\sigma\sqrt{2\pi}}e^{-(x-\bar{x})^2/(2\sigma^2)}$.
Next, we compare the theoretical curve with the model curve (Fig. 6). The
model curve is plotted with squares, while the theoretical curve is
plotted as a solid line.
\ffig<width=0.48\textwidth>{ris5.eps}<width=0.48\textwidth>{ris6.eps}
The results given in this paper were obtained for~100\,000 dropped balls.
This number is optimal from the viewpoint of the result's accuracy and the
processing time needed for the experiment. Figures~7 (a--d) show the
histograms of the balls' distribution over the compartments for~1\,000~(a),
10\,000~(b), 100\,000~(c), and~1\,000\,000 (d) of dropped balls.
These results were obtained with the following value of the parameters:~$e=1$, $R=0.1$, and~$\sigma_0=0.04$. We can see that the
histograms shown in Figs.~7c and d are, for all practical purposes,
identical. The accuracy of the results can also be judged
by the figures given in Table~1. These figures are the values of the
variance of the final distribution for~10\,000, 100\,000 and~1\,000\,000
dropped balls in eight series of simulations.
\begin{figure}[!ht]
\begin{center}\it
\begin{tabular}{c c}
\includegraphics[width=2.2in,height=1.1in]{ris7a.eps} & \includegraphics[width=2.2in,height=1.1in]{ris7b.eps} \\
a & b \\
\includegraphics[width=2.2in,height=1.1in]{ris7c.eps} &
\includegraphics[width=2.2in,height=1.1in]{ris7d.eps} \\
c & d \\
\end{tabular}
\end{center}\vspace{-3mm}
\caption{}\vspace{-6mm}
\end{figure}
\begin{table}[!ht]
\caption{$e = 1$, $R = 0.1$, and $\sigma_{0} = 0.04$}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
N & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
10000 & 7.3820 & 7.3381 & 7.3367 & 7.4097 & 7.4266 & 7.4201 & 7.3159 & 7.3559 \\ \hline
100000 & 7.4066 & 7.3817 & 7.3841 & 7.3938 & 7.3779 & 7.4387 & 7.4063 & 7.3713 \\ \hline
1000000 & 7.4028 & 7.3988 & 7.3901 & 7.3917 & 7.4066 & 7.3958 & 7.3902 & 7.4104 \\ \hline
\end{tabular}
\vspace{-3mm}
\end{center}
\end{table}
Let us first consider the case where the balls are distributed uniformly on
the width of the funnel's opening. Instead of the normal
distribution of the balls over the compartments (as it might be expected), we get a somewhat unusual distribution with peripheral peaks and
two distinctive gaps near the center (Fig.~2). These gaps are located symmetrically with respect to the vertical axis through the funnel's center. For this
case, the~$200 \times 100$ model Galton board was taken, otherwise the
balls reach its vertical boundaries.
More precisely, Fig.~2 corresponds to the case of absolutely elastic
impact ($e=1$) and~$R=0.1$.
\begin{figure}[!ht]
\begin{center}\it
\begin{tabular}{c c}
\includegraphics[width=2.2in,height=1.1in]{ris8a.eps} & \includegraphics[width=2.2in,height=1.1in]{ris8b.eps} \\
a & b \\
\includegraphics[width=2.2in,height=1.1in]{ris8c.eps} & \includegraphics[width=2.2in,height=1.1in]{ris8d.eps} \\
c & d \\
\end{tabular}
\end{center}\vspace{-3mm}
\caption{}
\end{figure}
For a smaller value of the coefficient of restitution ($e=0.8$) and an increased value of the
nail's radius to~$R=0.3$, the balls' distribution over the
compartments changes not very significantly:
distinctive peripheral peaks are still present, while instead of two pronounced gaps we
have several symmetrically located small pits (Fig.~8a). With a further
decrease of the coefficient of restitution the depth and structure of
these pits changes. In Figs.~8b, c, and d, the histograms are shown
for~$e=0.6$, $e=0.4$, and~$e=0.1$, respectively (the nail's radius is the same,~$R=0.3$).
If the balls are fed into the funnel according to a Gaussian law with large
dispersion~$\sigma_0$, then the form of the histograms will not change
qualitatively. Therefore, the case of small dispersion~$\sigma_0$ becomes
especially interesting.
Let~$\sigma_0=0.05$ and~$R=0.4$. We are going to investigate the
form of the histogram, as the coefficient of restitution~$e$ decreases
from 1 to 0. Figures~9 (a--d) show the balls' distributions over the
compartments for~$e=1$~1 (a), 0.9 (b), 0.8 (c), and~0.7 (d).
\begin{figure}[!ht]
\begin{center}\it
\begin{tabular}{c c}
\includegraphics[width=2.2in,height=1.1in]{ris9a.eps} & \includegraphics[width=2.2in,height=1.1in]{ris9b.eps} \\
a & b \\
\includegraphics[width=2.2in,height=1.1in]{ris9c.eps} & \includegraphics[width=2.2in,height=1.1in]{ris9d.eps} \\
c & d \\
\multicolumn{2}{c}{\includegraphics[width=2.2in,height=1.2in]{ris9e.eps}} \\
\multicolumn{2}{c}{e}
\end{tabular}\vspace{-4mm}
\end{center}
\caption{}\vspace{4mm}
\end{figure}
We can see that the distribution in the case of absolutely elastic impact
is almost Gaussian with two noticeable pits. As the coefficient of
restitution decreases, the ``normal'' distribution gets ``corrupted'';
instead of the pits, distinctive gaps appear, which become deeper with a
decrease of~$e$. However, this picture holds only for $e$ below a value
of~$e\approx 0.7$. A further decrease of the coefficient of restitution
makes the gaps disappear, and the distribution becomes practically
indistinguishable from the Gaussian distribution (Fig.~9e).
Figures~10 (a--e) show a similar series of histograms for~$R=1$, while the coefficient of restitution takes successively
the values~0.7, 0.6, 0.3, 0.2, and~0.1. We can see that for~$e=0.2$ there
are two gaps, while for larger and smaller values
of~$e$ the distribution is, practically, Gaussian. A further increase in~$e$
results in a distribution which is very different from Gaussian.
\begin{figure}[!ht]
\begin{center}\it
\begin{tabular}{c c}
\includegraphics[width=2.2in,height=1.1in]{ris10a.eps} & \includegraphics[width=2.2in,height=1.1in]{ris10b.eps} \\
a & b \\
\includegraphics[width=2.2in,height=1.1in]{ris10c.eps} & \includegraphics[width=2.2in,height=1.1in]{ris10d.eps} \\
c & d \\
\multicolumn{2}{c}{\includegraphics[width=2.2in,height=1.2in]{ris10e.eps}} \\
\multicolumn{2}{c}{e}
\end{tabular}
\end{center}\vspace{-3mm}
\caption{}
\end{figure}
The mentioned ``occasions'' of deviation of the final distribution of the
balls from a Gaussian distribution are intriguingly associated with non-monotonic behavior
of the variance~$\sigma$ as a function of
two variables~$R$ and~$e$ (with~$\sigma_0$ being fixed). Table~2 gives the
values of~$\sigma$ for~$\sigma_0=0.05$. We can see that for a fixed~$R$ ,
the variance ~$\sigma$ a local maximum. It is exactly that value of the coefficient of restitution in the vicinity of which a
substantial deviation from the Gaussian distribution occurs. For example,
for~$R=0.4$, the variance~$\sigma$ has a maximum at~$e\simeq 0.7$; this
value has already been mentioned in connection with the analysis of the
series of histograms in Fig.~9. Similarly, for~$R=1$, the local maximum is
reached at~$e\simeq 0.2$ (as it should be, according to
Fig.~10).
\begin{table}[!ht]
\caption{$\sigma_0=0.05$}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$e \setminus R$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 1 & 1.2 & 1.5 \\
\hline
1 & 9.24 & 9.19 & 9.26 & 9.44 & 9.61 & 9.94 & 10.29 & 10.38 & 11.27 & 12.80 \\ \hline
0.9 & 9.06 & 8.97 & 9.05 & 9.18 & 9.22 & 9.32 & 9.57 & 10.63 & 12.27 & 11.36 \\ \hline
0.8 & 8.79 & 8.47 & 8.32 & 8.07 & 7.83 & 7.52 & 7.78 & 8.22 & 8.05 & 8.47 \\ \hline
0.7 & 8.57 & 8.56 & 8.46 & 8.41 & 8.34 & 8.26 & 8.22 & 7.14 & 7.69 & 8.82 \\ \hline
0.6 & 8.33 & 8.26 & 8.30 & 8.21 & 8.18 & 8.15 & 8.10 & 7.96 & 7.89 & 7.72 \\ \hline
0.5 & 8.06 & 8.00 & 7.98 & 7.94 & 7.91 & 7.84 & 7.79 & 7.65 & 7.60 & 7.75 \\ \hline
0.4 & 7.77 & 7.72 & 7.68 & 7.61 & 7.64 & 7.57 & 7.52 & 7.36 & 7.22 & 7.06 \\ \hline
0.3 & 7.53 & 7.42 & 7.34 & 7.30 & 7.22 & 7.23 & 7.17 & 7.02 & 6.86 & 6.76 \\ \hline
0.2 & 7.03 & 7.08 & 7.05 & 7.07 & 7.08 & 7.08 & 7.18 & 7.23 & 7.20 & 6.64 \\ \hline
0.1 & 6.67 & 6.72 & 6.88 & 7.03 & 6.94 & 6.96 & 6.83 & 6.68 & 6.68 & 6.72 \\ \hline
\end{tabular}
\end{center}
\end{table}
The specified features of the histograms require theoretical
treatment and interpretation. The problem of the gaps should be especially
emphasized because this problem is likely to be most directly relevant to
the famous Kirkwood gaps in the distribution of asteroids in the main
asteroid belt between Mars and Jupiter. It is well known that these gaps
cannot be satisfactorily explained with the resonance ratios of the
orbital periods of the major planets. Meanwhile it would be useful to
investigate a simple model, where small planets (large asteroids) move
along circular orbits, and there also is a flow of small asteroids
colliding with the large ones without perturbing their paths. In this case, the
impact is not absolutely elastic ($0<e<1$). After a large
number of collisions, one would obtain a distribution of the asteroids'
flow over the semi-major axes. This distribution may contain a series of
gaps, as that in the case of Galton board.
This work was partially supported by the grant ``Principal Scientific
Schools'' (00-15-96146).
\end{document}
|
{
"timestamp": "2005-03-10T14:51:54",
"yymm": "0503",
"arxiv_id": "nlin/0503024",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503024"
}
|
\section{Introduction}
This work is motivated by the problem of multi-vehicle formation
(or swarm) control, e.g., for meter-scale UAVs (unmanned aerial
vehicles), and builds on our earlier work on planar formation
control laws \cite{scltechrep,scl02,cdc03} by extending the key
results to the three-dimensional setting. Some objectives of our
formation control laws are to avoid collisions between vehicles,
maintain cohesiveness of the formation, be robust to loss of
individuals, and scale favorably to large swarms.
In considering the problem of multi-vehicle formation control,
there is special
significance, both practically and theoretically, to modeling the
vehicles as point particles moving at a common (constant) speed.
In the language of mechanics, the individual particles are subject
to {\it gyroscopic} forces; i.e., forces which alter the direction of
motion of the particles, but which leave their speed (and hence their
kinetic energy) unchanged. A formation control law is then a
feedback law which specifies these gyroscopic forces based on the
positions and directions of motion of the particles.
In the planar setting, gyroscopic forces serve as steering controls
\cite{scl02}.
For particles moving in three dimensional space, we need to introduce
the notion of {\it framing of curves} to describe the effects of
gyroscopic forces on particle motion \cite{bishop,calini}.
Recently, a growing literature has emerged on planar
formation control for unit-speed vehicles, using tools from dynamical
systems theory (including pursuit models \cite{francis} and phase-coupled
oscillator models \cite{sepulchre}), as well as graph-theoretic methods
\cite{jadbabaie}. An early (discrete-time) unit-speed model for
biological flocking behavior is the Vicsek model \cite{vicsek}.
Interacting particle models similar to those described in this paper
have also found application in obstacle
avoidance and boundary following \cite{zhang}.
\section{Curves and moving frames}
A single particle moving in three dimensional space
traces out a trajectory
$ \mbox{\boldmath$\gamma$\unboldmath}: [0,\infty)
\rightarrow \mathbb{R}^3 $,
which we assume to be at least twice continuously differentiable,
satisfying
$ |\mbox{\boldmath$\gamma$\unboldmath}'(s)| = 1, $ $ \forall s; $
i.e., $ s $ is the arc-length parameter of the curve (and the prime
denotes differentiation with respect to $ s $).
The direction of motion of the particle at $ s $ is the
unit tangent vector to the trajectory,
$ {\bf T}(s) = \mbox{\boldmath$\gamma$\unboldmath}'(s) $.
If we further restrict the speed of particle motion to be unit speed,
then the arclength parameter $ s $ is equivalent to time $ t $,
and $ {\bf T}(t) = \dot{\mbox{\boldmath$\gamma$\unboldmath}}(t) $.
The gyroscopic
force vector always lies in the plane perpendicular to $ {\bf T} $, so to
describe the effects of this force, we are compelled to introduce orthonormal
unit vectors which span this {\it normal plane}. Taken together with
$ {\bf T} $, these unit vectors constitute a {\it framing} of the curve
\boldmath$\gamma$ \unboldmath representing the particle trajectory.
There are different framings one can choose, as is best illustrated by
examples (see figure \ref{frame_3d_fig}).
For a curve \boldmath$\gamma$\unboldmath $(s) $ which is three times
continuously differentiable,
and for which \boldmath$\gamma$\unboldmath ${}''(s) \ne 0 $ for all
$ s $, the Frenet-Serret frame
$({\bf T},{\bf N},{\bf B})$ is uniquely defined, and satisfies
\begin{eqnarray}
\label{frenetserret}
\mbox{\boldmath$\gamma$\unboldmath}'(s)
\hspace{-.2cm} & = & \hspace{-.2cm} {\bf T}(s), \nonumber \\
{\bf T}'(s) \hspace{-.2cm} & = & \hspace{-.2cm} \kappa(s) {\bf N}(s),
\nonumber \\
{\bf N}'(s) \hspace{-.2cm} & = & \hspace{-.2cm}
- \kappa(s) {\bf T}(s) + \tau(s) {\bf B}(s),
\nonumber \\
{\bf B}'(s) \hspace{-.2cm} & = & \hspace{-.2cm} -\tau(s) {\bf N}(s).
\end{eqnarray}
In (\ref{frenetserret}),
$ {\bf N}(s) $
is the unit normal vector to the curve \boldmath$\gamma$ \unboldmath
at $ s $, and $ {\bf B}(s) $ is the unit
binormal vector
(which completes the right-handed orthonormal frame).
The curvature function $ \kappa $ and the torsion function $ \tau $
are given by expressions involving the derivatives of
\boldmath$\gamma$\unboldmath, and
\boldmath$\gamma$\unboldmath ${}''(s) \ne 0 $ is required for
$ \tau(s) $ to be well-defined.
Although the Frenet-Serret frame for a curve (when it exists) has a
special status (because it is uniquely defined by the derivatives of
the curve), it is not the only choice of frame, nor is it
necessarily the best choice. In particular,
the requirement that $ \mbox{\boldmath$\gamma$\unboldmath}''(s) \ne 0 $
presents serious difficulties for the interaction laws we consider
in this paper.
We therefore use an alternative framing of the curve
\boldmath$\gamma$\unboldmath, the natural Frenet frame, which is also
referred to as the Fermi-Walker frame or Relatively Parallel Adapted Frame
(RPAF):
\begin{eqnarray}
\label{naturalfrenet}
\mbox{\boldmath$\gamma$\unboldmath}'(s)
\hspace{-.2cm} & = & \hspace{-.2cm} {\bf T}(s), \nonumber \\
{\bf T}'(s) \hspace{-.2cm} & = & \hspace{-.2cm}
k_1(s){\bf M}_1 + k_2(s) {\bf M}_2, \nonumber \\
{\bf M}_1'(s) \hspace{-.2cm} & = & \hspace{-.2cm} - k_1(s){\bf T}(s),
\nonumber \\
{\bf M}_2'(s) \hspace{-.2cm} & = & \hspace{-.2cm} - k_2(s){\bf T}(s).
\end{eqnarray}
In (\ref{naturalfrenet}),
$ {\bf M}_1(s) $ and $ {\bf M}_2(s) $ are unit normal vectors which
(along with $ {\bf T}(s) $) complete a right-handed orthonormal frame.
However, there is freedom in the choice of initial conditions
$ {\bf M}_1(0) $ and $ {\bf M}_2(0) $; once these are
specified, the corresponding natural Frenet frame for a
twice-continuously-differentiable curve
\boldmath$\gamma$ \unboldmath
is unique.
\begin{figure}
\epsfxsize=9cm
\epsfbox{cdc05fig1.eps}
\caption{\label{frame_3d_fig} The Frenet-Serret frame (left), and
natural Frenet frame (right), illustrated for a three-dimensional curve.}
\end{figure}
Both (\ref{frenetserret}) and (\ref{naturalfrenet}) can be packaged
as control systems on the Lie group $ SE(3) $, the group of rigid motions in
three-dimensional space.
(A modern reference for control systems
on Lie groups is Jurdjevic \cite{jurdjevic}.)
Here we think of $ (\kappa, \tau) $ or the {\it natural curvatures}
$ (k_1,k_2) $ as controls,
which drive the evolution of the frame and the particle position
\boldmath$\gamma$\unboldmath.
\section{Formation model}
Figure \ref{motion3dfig} illustrates the trajectories of two
vehicles moving at unit speed, and their respective natural Frenet frames.
The particle (i.e., vehicle) positions are denoted by $ {\bf r}_1 $
and $ {\bf r}_2 $, and the frames by $ ({\bf x}_1,{\bf y}_1,{\bf z}_1) $
and $ ({\bf x}_2,{\bf y}_2,{\bf z}_2) $, so that
\begin{eqnarray}
\label{twouavsystem3d}
\dot{\bf r}_1 = {\bf x}_1, \hspace{1.8cm}
&& \dot{\bf r}_2 = {\bf x}_2,\nonumber \\
\dot{\bf x}_1 = {\bf y}_1 u_1 + {\bf z}_1 v_1, \hspace{.35cm}
&& \dot{\bf x}_2 = {\bf y}_2 u_2 + {\bf z}_2 v_2, \nonumber \\
\dot{\bf y}_1 = -{\bf x}_1 u_1, \hspace{1.15cm}
&& \dot{\bf y}_2 = -{\bf x}_2 u_2, \nonumber \\
\dot{\bf z}_1 = -{\bf x}_1 v_1, \hspace{1.2cm}
&& \dot{\bf z}_2 = -{\bf x}_2 v_2.
\end{eqnarray}
where the controls $ (u_1,v_1) $ and $ (u_2,v_2) $
may be feedback functions of the position and frame variables.
\begin{figure}
\hspace{.5cm}
\epsfxsize=7cm
\epsfbox{cdc05fig2.eps}
\caption{\label{motion3dfig} Three-dimensional trajectories for two
vehicles, and their respective frames.}
\end{figure}
We consider control laws which depend only on relative vehicle
positions and orientations; i.e., which depend only on the {\it shape}
of the formation. Furthermore, the {\it effect} of the controls
on each trajectory is assumed to depend only on
$ {\bf r}_1 $, $ {\bf r}_2 $, $ {\bf x}_1 $, and $ {\bf x}_2 $,
and not on the orientation of the normal vectors within their
respective normal planes.
The controls for the first vehicle can then be functions
of the relative vehicle position, $ {\bf r} = {\bf r}_2 - {\bf r}_1 $,
the heading direction
of the second vehicle, $ {\bf x}_2 $, and the frame variables for
the first vehicle, $ ({\bf x}_1,{\bf y}_1, {\bf z}_1) $. Thus,
\begin{eqnarray}
\label{vehctrlrestrict1}
u_1 \hspace{-.2cm} & = & \hspace{-.2cm}
u_1({\bf r},{\bf x}_1,{\bf y}_1,{\bf z}_1,{\bf x}_2),
\nonumber \\
v_1 \hspace{-.2cm} & = & \hspace{-.2cm}
v_1({\bf r},{\bf x}_1,{\bf y}_1,{\bf z}_1,{\bf x}_2),
\end{eqnarray}
and similarly,
\begin{eqnarray}
u_2 \hspace{-.2cm} & = & \hspace{-.2cm}
u_2({\bf r},{\bf x}_2,{\bf y}_2,{\bf z}_2,{\bf x}_1),
\nonumber \\
v_2 \hspace{-.2cm} & = & \hspace{-.2cm}
v_2({\bf r},{\bf x}_2,{\bf y}_2,{\bf z}_2,{\bf x}_1).
\end{eqnarray}
Furthermore, because the overall motion of the first vehicle
should be independent of $ {\bf y}_1 $ and $ {\bf z}_1 $, we require
\begin{equation}
v_1({\bf r},{\bf x}_1,{\bf y}_1,{\bf z}_1,{\bf x}_2) =
u_1({\bf r},{\bf x}_1,{\bf z}_1,-{\bf y}_1,{\bf x}_2),
\end{equation}
and similarly,
\begin{equation}
\label{vehctrlrestrict2}
v_2({\bf r},{\bf x}_2,{\bf y}_2,{\bf z}_2,{\bf x}_1) =
u_2({\bf r},{\bf x}_2,{\bf z}_2,-{\bf y}_2,{\bf x}_1).
\end{equation}
Finally, we require that our control laws have a discrete
(relabling) symmetry, which corresponds to the intuitive notion
that both vehicles ``run the same algorithm.'' This implies
\begin{eqnarray}
\label{vehctrlrestrict3}
u_1(-{\bf r},{\bf x}_1,{\bf y}_1,{\bf z}_1,{\bf x}_2)
= u_2({\bf r},{\bf x}_2,{\bf y}_2,{\bf z}_2,{\bf x}_1), \nonumber \\
v_1(-{\bf r},{\bf x}_1,{\bf y}_1,{\bf z}_1,{\bf x}_2)
= v_2({\bf r},{\bf x}_2,{\bf y}_2,{\bf z}_2,{\bf x}_1).
\end{eqnarray}
In this paper, the specific control laws we consider
have the form
\begin{eqnarray}
\label{twovehiclelaw3d}
u_1 \hspace{-.2cm} & = & \hspace{-.2cm}
F(-{\bf r},{\bf x}_1,{\bf y}_1,{\bf x}_2)
- f(|{\bf r}|)\left(-\frac{\bf r}{|{\bf r}|}\cdot{\bf y}_1\right),
\nonumber \\
u_2 \hspace{-.2cm} & = & \hspace{-.2cm}
F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1)
- f(|{\bf r}|)\left(\frac{\bf r}{|{\bf r}|}\cdot{\bf y}_2\right),
\nonumber \\
v_1 \hspace{-.2cm} & = & \hspace{-.2cm}
F(-{\bf r},{\bf x}_1,{\bf z}_1,{\bf x}_2)
- f(|{\bf r}|)\left(-\frac{\bf r}{|{\bf r}|}\cdot{\bf z}_1\right),
\nonumber \\
v_2 \hspace{-.2cm} & = & \hspace{-.2cm}
F({\bf r},{\bf x}_2,{\bf z}_2,{\bf x}_1)
- f(|{\bf r}|)\left(\frac{\bf r}{|{\bf r}|}\cdot{\bf z}_2\right),
\end{eqnarray}
which is a further restricted class of laws consistent with
(\ref{vehctrlrestrict1}) - (\ref{vehctrlrestrict3}). (We discuss
later how $ F $ and $ f $ are chosen.)
\section{Shape variables and equilibria}
The geometry of the problem of interacting particles moving at unit
speed in the plane has been considered in earlier work
\cite{scltechrep,scl02,cdc03}.
The unit speed constraint leads to the study of gyroscopic
interaction forces, and the identification of the constant
kinetic energy hyper-surface with the group $ SE(2) $ of rigid motions
in the plane. Formations or steady patterns of motion in the plane
thus become relative equilibria for particle dynamics on $ SE(2) $
\cite{scltechrep,scl02,cdc03}.
A key difficulty in extending the above geometric perspective to
three dimensions arises from the fact that the corresponding
constant kinetic energy hyper-surface cannot be identified with
$ SE(3) $, the rigid motion group in three dimensions. It is a
homogeneous space $ SE(3)/SO(2) $. However, there is considerable
advantage, particularly in the multi-particle context, to
formulating the dynamics in terms of interacting particles in
$ SE(3) $.
The dynamics (\ref{twouavsystem3d})
can be expressed in terms of the group variables
$ g_1, g_2 \in G = SE(3) $ as a pair of left-invariant systems
\begin{equation}
\label{se3system}
\dot{g}_1 = g_1 \xi_1, \;\; \dot{g}_2 = g_2 \xi_2,
\end{equation}
where $ \xi_1, \xi_2 \in {\mathfrak g}= $ the Lie algebra of $ G $.
The dynamics for $ g = g_1^{-1} g_2 $ are given by
\begin{eqnarray}
\label{gdot}
\dot{g} && \hspace{-.6cm}
= -g_1^{-1}\dot{g}_1 g_1^{-1} g_2 + g_1^{-1} \dot{g}_2 \nonumber \\
&& \hspace{-.6cm} = -g_1^{-1}g_1 \xi_1 g + g_1^{-1} g_2 \xi_2 \nonumber \\
&& \hspace{-.6cm} = - \xi_1 g + g \xi_2 \nonumber \\
&& \hspace{-.6cm} = g \xi,
\end{eqnarray}
where $ \xi = \xi_2 - \mbox{Ad}_{g^{-1}} \xi_1 \in {\mathfrak g} $.
Equation (\ref{gdot}), where $ \xi $ incorporates the control inputs
$ (u_1,v_1) $ and $ (u_2,v_2) $, describes the evolution of the
{\it relative} position and {\it relative} natural Frenet frame orientation
of the pair of vehicles. It is thus natural to consider what
equilibria of (\ref{gdot}) exist, and then to design control laws
which stabilize those equilibria. Equilibria of the shape dynamics
(\ref{gdot}) correspond to {\it relative equilibria} of the system
(\ref{se3system}) on $ G \times G $.
\subsection{Shape equilibria for a two-particle system on SE(3)}
At an equilibrium shape $ g_e $ of the shape dynamics
(\ref{gdot}), we have
\begin{equation}
\label{gexi2xi1}
g_e \xi_2(g_e) = \xi_1(g_e) g_e.
\end{equation}
To facilitate calculation, we define
\begin{eqnarray}
g_e \hspace{-.2cm} & = & \hspace{-.2cm}
\left[ \begin{array} {c c} Q & {\bf b} \\ {\bf 0} & 1 \end{array} \right],
\mbox{ where $ Q \in SO(3) $ and $ {\bf b} \in \mathbb{R}^3 $}, \nonumber \\
\xi_1(g_e) \hspace{-.2cm} & = & \hspace{-.2cm}
\left[ \begin{array} {c c} \hat{\Omega}_1 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right], \;\;
\xi_2(g_e) =
\left[ \begin{array} {c c} \hat{\Omega}_2 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right].
\end{eqnarray}
Then (\ref{gexi2xi1}) becomes
\begin{equation}
\label{equilblockmatrix}
\left[ \begin{array} {c c} Q & {\bf b} \\ {\bf 0} & 1 \end{array} \right]
\left[ \begin{array} {c c} \hat{\Omega}_2 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right] =
\left[ \begin{array} {c c} \hat{\Omega}_1 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right]
\left[ \begin{array} {c c} Q & {\bf b} \\ {\bf 0} & 1 \end{array} \right],
\end{equation}
where
$ {\bf e}_1 = \left[ \begin{array} {c c c} 1 & 0 & 0 \end{array} \right]^T $,
\begin{equation}
\label{equilblockmatrixdefn}
\Omega_1 =
\left[ \begin{array} {c} w_1 \\ -v_1 \\ u_1 \end{array} \right], \;\;
\Omega_2 =
\left[ \begin{array} {c} w_2 \\ -v_2 \\ u_2 \end{array} \right],
\end{equation}
and for any 3-vector $ \Gamma = (\Gamma_1,\Gamma_2,\Gamma_3) $,
$\hat{\Gamma} $ is the skew-symmetric matrix defined by
\begin{equation}
\hat{\Gamma} = \left[ \begin{array} {r r r} 0 \;\; & -\Gamma_3 & \Gamma_2 \\
\Gamma_3 & 0 \;\; & -\Gamma_1 \\ -\Gamma_2 & \Gamma_1 & 0 \;\;
\end{array} \right].
\end{equation}
Note that here we allow $ \Omega_1 $ and $ \Omega_2 $ to each have the
full three degrees of
freedom - not just the two corresponding to the natural curvatures.
The reason
for proceeding in this manner is that ultimately we recover not only the
relative equilibria of (\ref{se3system}) and (\ref{twouavsystem3d}), but
also an interesting class
of relative {\it periodic} solutions for (\ref{twouavsystem3d}).
From (\ref{equilblockmatrix}) we see that
$ Q \hat{\Omega}_2 = \hat{\Omega}_1 Q $, from which it follows that
\begin{equation}
\label{rotmatrixident}
\Omega_1 = Q \Omega_2.
\end{equation}
From (\ref{equilblockmatrix}) we also obtain
$ Q {\bf e}_1 = \hat{\Omega}_1 {\bf b} + {\bf e}_1 $. It can then be
shown that $ w_1 = w_2 $, and $ u_1^2 + v_1^2 = u_2^2 + v_2^2 $.
Introducing new variables $ w $, $ a $, $ \psi_1 $, and
$ \psi_2 $, we can express $ \Omega_1 $ and $ \Omega_2 $ as
\begin{equation}
\label{ctrlvectors}
\Omega_1 = \left[ \begin{array} {c} w \\ a\sin\psi_1 \\ a\cos\psi_1
\end{array} \right], \;\;\;\;
\Omega_2 = \left[ \begin{array} {c} w \\ a\sin\psi_2 \\ a\cos\psi_2
\end{array} \right].
\end{equation}
If (for $ a^2+w^2 \ne 0 $) we further define
\begin{equation}
\label{varphidefn}
\cos\varphi = \frac{a}{\sqrt{a^2+w^2}}, \;\;
\sin\varphi = \frac{w}{\sqrt{a^2+w^2}},
\end{equation}
along with
\begin{eqnarray}
\label{rotmatrixdefn}
R_{\psi_j} \hspace{-.3cm} & = & \hspace{-.3cm}
\left[ \hspace{-.15cm} \begin{array} {c c c} 1 & 0 & 0 \\
0 & \cos\psi_j & -\sin\psi_j \\
0 & \sin\psi_j & \cos\psi_j \end{array} \hspace{-.15cm} \right]
\hspace{-.1cm}, \;
R_{\varphi} \hspace{-.1cm} = \hspace{-.1cm}
\left[ \hspace{-.15cm}
\begin{array} {c c c} \cos\varphi & 0 & -\sin\varphi \\ 0 & 1 & 0 \\
\sin\varphi & 0 & \cos\varphi \end{array} \hspace{-.15cm} \right]
\hspace{-.1cm},
\nonumber \\
R_{\vartheta} \hspace{-.3cm} & = & \hspace{-.3cm}
\left[ \hspace{-.15cm} \begin{array} {c c c}
\cos\vartheta & -\sin\vartheta & 0 \\
\sin\vartheta & \cos\vartheta & 0 \\
0 & 0 & 1 \end{array} \hspace{-.15cm} \right],
\end{eqnarray}
where $ \vartheta \in [0,2\pi) $ is arbitrary, we see that
(\ref{ctrlvectors}) becomes
\begin{equation}
\Omega_j = \sqrt{a^2+w^2}\; R_{\psi_j}^T R_{\varphi}^T {\bf e}_3,
\;\; j=1,2,
\end{equation}
and from (\ref{rotmatrixident}) we obtain
\begin{eqnarray}
Q R_{\psi_2}^T R_{\varphi}^T {\bf e}_3 \hspace{-.2cm} & = & \hspace{-.2cm}
R_{\psi_1}^T R_{\varphi}^T {\bf e}_3 \nonumber \\
R_{\varphi} R_{\psi_1} Q R_{\psi_2}^T R_{\varphi}^T {\bf e}_3
\hspace{-.2cm} & = & \hspace{-.2cm}
{\bf e}_3 \nonumber \\
R_{\varphi} R_{\psi_1} Q R_{\psi_2}^T R_{\varphi}^T
\hspace{-.2cm} & = & \hspace{-.2cm}
R_{\vartheta} \nonumber \\
Q \hspace{-.2cm} & = & \hspace{-.2cm}
R_{\psi_1}^T R_{\varphi}^T R_{\vartheta} R_{\varphi} R_{\psi_2}.
\end{eqnarray}
Note that $ R_{\vartheta} $, for arbitrary $ \vartheta $, is a
rotation matrix that fixes the basis vector $ {\bf e}_3 $.
Defining $ \tilde{\bf b} $ by
$ {\bf b} = R_{\psi_1}^T R_{\varphi}^T \tilde{\bf b} $, after
some calculation, one can show that
\begin{equation}
\label{gedecomp}
\left[ \hspace{-.15cm}
\begin{array} {c c} Q & {\bf b} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right]
\hspace{-.1cm} = \hspace{-.1cm} \left[ \hspace{-.15cm}
\begin{array} {c c} R_{\psi_1}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right] \hspace{-.15cm}
\left[ \hspace{-.15cm}
\begin{array} {c c} R_{\varphi}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right] \hspace{-.15cm}
\left[ \hspace{-.15cm}
\begin{array} {c c} R_{\vartheta} & \tilde{\bf b} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right] \hspace{-.15cm}
\left[ \hspace{-.15cm} \begin{array} {c c} R_{\varphi} & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right] \hspace{-.15cm}
\left[ \hspace{-.15cm} \begin{array} {c c} R_{\psi_2} & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.15cm} \right] \hspace{-.1cm},
\end{equation}
\begin{equation}
\label{tildebfinal}
\tilde{\bf b} = \left[ \hspace{-.1cm}
\begin{array} {c} \frac{a}{a^2+w^2}\sin\vartheta \\
\frac{a}{a^2+w^2}(1-\cos\vartheta) \\ \tilde{b}_3 \end{array}
\hspace{-.1cm} \right].
\end{equation}
Thus, $ g_e $ can be decomposed as a product of five rigid motions
(four of which represent pure rotations), and contains two free
parameters - $ \vartheta $ and $ \tilde{b}_3 $ - once the control vectors
$ \Omega_1 $ and $ \Omega_2 $ are specified.
\vspace{.25cm}
\noindent
{\bf Remark}: For purposes of interpretation of (\ref{gedecomp})
in the context of particle trajectories, we may take
$ R_{\psi_1} = R_{\psi_2} = I $, so that (\ref{gedecomp}) reduces
to
\begin{equation}
\label{gedecompinterp}
\left[ \hspace{-.1cm}
\begin{array} {c c} Q & {\bf b} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right]
= \left[ \hspace{-.1cm}
\begin{array} {c c} R_{\varphi}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right]
\left[ \hspace{-.1cm}
\begin{array} {c c} R_{\vartheta} & \tilde{\bf b} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right]
\left[ \hspace{-.1cm} \begin{array} {c c} R_{\varphi} & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right].
\end{equation}
To see this, recall that by definition $ g = g_1^{-1} g_2 $.
Let $ \tilde{g}_e $ be defined by
\begin{equation}
\label{getildedefn}
g_e = \left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_1}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right]
\tilde{g}_e
\left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_2} & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right].
\end{equation}
Then
\begin{eqnarray}
\label{rightmulrot}
\tilde{g}_e \hspace{-.2cm} & = & \hspace{-.2cm} \left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_1} & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right]
g_1^{-1} g_2
\left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_2}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right] \nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm} \left(g_1 \left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_1}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right] \right)^{-1}
\left(g_2 \left[ \hspace{-.1cm}
\begin{array} {c c} R_{\psi_2}^T & {\bf 0} \\
{\bf 0} & 1 \end{array} \hspace{-.1cm} \right] \right).
\end{eqnarray}
Thus, if we exhibit a shape equilibrium
$ \tilde{g}_e $ of the form (\ref{gedecompinterp}), we can
always write down a family of shape equilibria (\ref{getildedefn})
parameterized by $ \psi_1 $ and $ \psi_2 $, which differ only
in the orientation of the unit normal vectors of the two frames
(and are therefore indistinguishable if only the particle
trajectories in $ \mathbb{R}^3 $ are observed).
$ \Box $
\vspace{.25cm}
\noindent
{\bf Proposition 1}:
Consider the two-particle system on $ G \times G $ given by
\begin{equation}
\label{se3xse3system}
\dot{g}_1 = g_1
\left[ \begin{array} {c c} \hat{\Omega}_1 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right], \;\;
\dot{g}_2 = g_2
\left[ \begin{array} {c c} \hat{\Omega}_2 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right],
\end{equation}
where $ \Omega_1 = \Omega_1(g) $, $ \Omega_2 = \Omega_2(g) $, and
$ g = g_1^{-1} g_2 $ (i.e., the controls $ \Omega_1 $ and
$ \Omega_2 $ are arbitrary, but are $ G $-invariant).
Then there is a corresponding reduced system on $ G $
(the ``shape space'') given by
\begin{equation}
\dot{g} = - \left[ \begin{array} {c c} \hat{\Omega}_1 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right] g + g
\left[ \begin{array} {c c} \hat{\Omega}_2 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right],
\end{equation}
(c.f. (\ref{gdot})) whose equilibria are given by (\ref{equilblockmatrix}).
Solutions of (\ref{equilblockmatrix}), with (\ref{equilblockmatrixdefn}),
require that (\ref{ctrlvectors}) hold.
\begin{itemize}
\item[(1)] If $ w = a = 0 $, then
$ Q $ satisfies $ Q {\bf e}_1 = {\bf e}_1 $, and $ {\bf b} $ is arbitrary.
Then $ Q $ yields one free parameter, and
$ {\bf b} $ yields three free parameters.
\item[(2)] If $ w^2 + a^2 \ne 0 $, then $ (Q,{\bf b}) $ satisfies
(\ref{gedecomp}), with $ R_{\psi_1} $, $ R_{\psi_2} $,
$ R_{\varphi} $, and $ R_{\vartheta} $ given by (\ref{rotmatrixdefn})
and with $ \tilde{\bf b} $ given by (\ref{tildebfinal}).
The angle $ \varphi $ is related to $ w $ and $ a $
through (\ref{varphidefn}), and $ \vartheta $ and $ \tilde{b}_3 $
are free parameters.
\end{itemize}
The resulting $ (Q,{\bf b}) $ then describe the shape equilibria
(i.e., the relative equilibria) for (\ref{se3xse3system}).
\vspace{.25cm}
\noindent
{\bf Proof}: Follows from the calculations outlined above. $ \Box $
\vspace{.25cm}
\noindent
{\bf Proposition 2}:
Consider (\ref{se3xse3system}) as the underlying dynamics for
the evolution of two particle trajectories in $ \mathbb{R}^3 $
and their corresponding natural Frenet frames.
Then relative equilibria $ (Q,{\bf b}) $ for (\ref{se3xse3system})
correspond to the following steady-state formations of
the two particles in $ \mathbb{R}^3 $:
\begin{itemize}
\item[(1)] If $ w = a = 0 $, then the two particles move in
the same direction with arbitrary relative positions.
\item[(2)] If $ w = 0 $ but $ a \ne 0 $, then the particles
move on circular orbits with a common radius, in planes
perpendicular to a common axis.
\item[(3)] If $ w \ne 0 $ but $ a = 0 $, then the particles
move in the same direction on collinear trajectories.
\item[(4)] If $ w \ne 0 $ and $ a \ne 0 $, then the particles
follow circular helices with the same radius, pitch, axis,
and axial direction of motion.
\end{itemize}
\vspace{.25cm}
\noindent
{\bf Proof}: Omitted due to space constraints, but follows from
{\bf Proposition 1}, along with the {\bf Remark}
and calculations outlined above. $ \Box $
\subsection{Shape equilibria for an n-particle system on SE(3)}
Our definition of the shape variable $ g $ for the two-particle problem
extends naturally to the
$ n $-particle problem (under the assumption that the $ n $-particle
interaction law has $ G $ as a symmetry group). We define
\begin{equation}
\tilde{g}_j = g_1^{-1} g_j, \;\; j=2,...,n,
\end{equation}
where $ g_1,g_2,...,g_n $ are the
group variables (each representing one of the particles), and
$ \tilde{g}_2,\tilde{g}_3,...,\tilde{g}_n $ are shape variables.
(This is analogous to the approach taken in the planar problem,
where the corresponding group is SE(2) \cite{scltechrep,scl02,cdc03}.)
\vspace{.25cm}
\noindent
{\bf Proposition 3}:
Consider
\begin{equation}
\label{se3xse3nsystem}
\dot{g}_1 =
g_1 \left[ \begin{array} {c c} \hat{\Omega}_1 & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right], \;\; ..., \;\;
\dot{g}_n =
g_n \left[ \begin{array} {c c} \hat{\Omega}_n & {\bf e}_1 \\ {\bf 0} & 0
\end{array} \right],
\end{equation}
where $ \Omega_1,...,\Omega_n $ are $ G $-invariant controls,
as the underlying dynamics for
the evolution of $ n $ particle trajectories in $ \mathbb{R}^3. $
Then relative equilibria $ (Q_2,{\bf b}_2),...,(Q_n,{\bf b}_n) $
for (\ref{se3xse3nsystem})
correspond to the following steady-state formations of
the $ n $ particles in $ \mathbb{R}^3 $ (see figure \ref{rel_eq_3d_fig}):
\begin{itemize}
\item[(1)] If $ w = a = 0 $, then the $ n $ particles all move in
the same direction with arbitrary relative positions.
\item[(2)] If $ w = 0 $ but $ a \ne 0 $, then the particles
move on circular orbits with a common radius, in planes
perpendicular to a common axis.
\item[(3)] If $ w \ne 0 $ but $ a = 0 $, then the particles
move in the same direction on collinear trajectories.
\item[(4)] If $ w \ne 0 $ and $ a \ne 0 $, then the particles
follow circular helices with the same radius, pitch, axis,
and axial direction of motion.
\end{itemize}
\vspace{.25cm}
\noindent
{\bf Proof}: Omitted due to space constraints, but analogous to the
proof of {\bf Proposition 2}. $ \Box $
\vspace{.25cm}
\begin{figure}
\epsfxsize=8.5cm
\epsfbox{cdc05fig3.eps}
\caption{\label{rel_eq_3d_fig} Rectilinear, circling, and helical
formations, illustrated for five particles. The arrows represent
the unit tangent vectors to the particle trajectories.}
\end{figure}
\noindent
{\bf Remark}:
When $ w \ne 0 $ at a relative equilibrium for our model
(\ref{se3xse3system}) of particles
evolving in $ G \times G $, the corresponding natural curvatures
in (\ref{twouavsystem3d}) are then
in fact periodic functions of time (or arc-length parameter). $ \Box $
\section{Rectilinear formation law}
The two types of equilibrium formations for which we consider
specific stabilizing control laws (for a pair of vehicles)
are rectilinear formations (in which both vehicles head in
the same direction) and circling
formations (in which both vehicles follow the same circular orbit).
Figure \ref{rectcirc} shows simulations which converge to these
two types of equilibrium formations.
For concreteness, we use the variables $ ({\bf r}_1,{\bf x}_1,{\bf y}_1) $
and $ ({\bf r}_2,{\bf x}_2,{\bf y}_2) $,
rather than the group variables $ g_1 $ and $ g_2 $.
\begin{figure}
\epsfxsize=5.5cm
\epsfbox{cdc05fig4a.eps}
\epsfxsize=2.8cm
\epsfbox{cdc05fig4b.eps}
\caption{\label{rectcirc} Convergence to a rectilinear formation (left),
and to a circling formation (right). The trajectories, which are
three-dimensional, are viewed perpendicular to
the plane of the equilibrium formation. }
\end{figure}
Consider the Lyapunov function candidate
\begin{equation}
\label{vrect}
V_{\mathit rect} = -\ln(1+{\bf x}_2\cdot {\bf x}_1) + h(|{\bf r}|),
\end{equation}
where we assume that
\begin{itemize}
\item[$ ( \hspace{-.05cm} \mbox{A}1 \hspace{-.05cm} ) $]
$ dh/d\rho = f(\rho) $, where $ f(\rho) $ is a Lipschitz
continuous function on $ (0,\infty) $, so that
$ h(\rho) $ is continuously differentiable on $ (0,\infty) $;
\item[$ ( \hspace{-.05cm} \mbox{A}2 \hspace{-.05cm} ) $]
$ \lim_{\rho \rightarrow 0} h(\rho) = \infty $,
$ \lim_{\rho \rightarrow \infty} h(\rho) = \infty $, and
$ \exists \tilde{\rho} \mbox{ such that } h(\tilde{\rho}) = 0 $.
\end{itemize}
Figure \ref{fhfig} shows an example of functions $ f(\cdot) $ and
$ h(\cdot) $ satisfying conditions (A1) and (A2).
An example of a suitable function $ f(\cdot) $ is
\begin{equation}
\label{fofr}
f(|{\bf r}|)=\alpha \left[1-\left({r_o}/{|{\bf r}|}\right)^2\right],
\end{equation}
where $ \alpha $ and $ r_o $ are positive constants.
Observe that the term
$ -\ln(1+{\bf x}_2 \cdot {\bf x}_1) $ in (\ref{vrect})
penalizes heading-direction misalignment
between the two vehicles, and the term $ h(|{\bf r}|) $
penalizes vehicle separations which are too large or too small.
\begin{figure}
\hspace{1.5cm}
\epsfxsize=5cm
\epsfbox{cdc05fig5.eps}
\caption{\label{fhfig} An example of suitable functions $ f(\cdot) $
and $ h(\cdot) $ satisfying conditions (A1) and (A2) \cite{scl02}.}
\end{figure}
Differentiating $ V_{\mathit rect} $
with respect to time along trajectories of (\ref{twouavsystem3d}) gives
\begin{eqnarray}
\label{dotvrect}
\dot{V}_{\mathit rect} \hspace{-.2cm} & = & \hspace{-.2cm}
-\frac{\dot{\bf x}_2 \cdot {\bf x}_1 + {\bf x}_2 \cdot
\dot{\bf x}_1}{1+{\bf x}_2\cdot{\bf x_1}}+f(|{\bf r}|)
\frac{d}{dt}|{\bf r}| \nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm}
- \frac{({\bf y}_2 u_2 + {\bf z}_2 v_2)\cdot{\bf x}_1
+ {\bf x}_2 \cdot({\bf y}_1 u_1 + {\bf z}_1 v_1)}
{1+{\bf x}_2\cdot{\bf x_1}} \nonumber \\ & & \hspace{.5cm}
+f(|{\bf r}|) \left[
\frac{\bf r}{|{\bf r}|}\cdot ({\bf x}_2 - {\bf x}_1)\right]
\nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm}
-\frac{1}{1+{\bf x}_2\cdot{\bf x_1}} \bigg\{
({\bf x}_1 \cdot {\bf y}_2)u_2 +
({\bf x}_2 \cdot {\bf y}_1) u_1 \nonumber \\ & & \hspace{2cm} +
({\bf x}_1 \cdot {\bf z}_2) v_2 + ({\bf x}_2\cdot{\bf z}_1) v_1
\nonumber \\ & & \hspace{-.2cm}
-f(|{\bf r}|)\left(1+{\bf x}_2\cdot{\bf x_1}\right)\left[
\frac{\bf r}{|{\bf r}|}\cdot ({\bf x}_2 - {\bf x}_1) \right]
\hspace{-.1cm} \bigg\}.
\end{eqnarray}
If we consider control laws of the form (\ref{twovehiclelaw3d}),
then (\ref{dotvrect}) becomes (after some calculation)
\begin{eqnarray}
\label{dotvrect2}
\dot{V}_{\mathit rect} \hspace{-.2cm} & = & \hspace{-.2cm}
-\frac{1}{1+{\bf x}_2\cdot{\bf x_1}} \nonumber \\
& & \hspace{-1.3cm} \times \hspace{-.05cm} \bigg[ \hspace{-.05cm}
({\bf x}_1 \cdot {\bf y}_2) F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1)
+ ({\bf x}_2 \cdot {\bf y}_1) F(-{\bf r},{\bf x}_1,{\bf y}_1,{\bf x}_2)
\nonumber \\ & & \hspace{-1.3cm} +
({\bf x}_1 \cdot {\bf z}_2) F({\bf r},{\bf x}_2,{\bf z}_2,{\bf x}_1)
+ ({\bf x}_2\cdot{\bf z}_1) F(-{\bf r},{\bf x}_2,{\bf z}_2,{\bf x}_1)
\hspace{-.05cm} \bigg]. \nonumber \\
\end{eqnarray}
It is clear from (\ref{dotvrect2}) that one choice of $ F $ which
makes $ \dot{V}_{\mathit rect} \le 0 $ is
$ F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1) = \mu {\bf x}_1 \cdot {\bf y}_2, $
where $ \mu = \mu(|{\bf r}|) > 0 $. But more generally, we consider
\begin{equation}
\label{formoff}
F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1) =
\mp \eta \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_2 \right) +
\mu {\bf x}_1 \cdot {\bf y}_2,
\end{equation}
where $ \mu $ and $ \eta $ satisfy
\begin{itemize}
\item[$ ( \hspace{-.05cm} \mbox{A}3 \hspace{-.05cm} ) $]
$ \mu(\rho) $ and $ \eta(\rho) $ are Lipschitz continuous
on $ (0, \infty) $;
\item[$ ( \hspace{-.05cm} \mbox{A}4 \hspace{-.05cm} ) $]
$ \mu(|{\bf r}|) > \frac{1}{2}\eta(|{\bf r}|) > 0 $,
$ \forall |{\bf r}| \ge 0. $
\end{itemize}
(For simplicity, $ \mu $ and $ \eta $ can be taken to be constants,
rather than functions of $ |{\bf r}| $.)
The control law given by (\ref{twovehiclelaw3d}) with (\ref{formoff})
is the natural generalization to three dimensions of the planar
two-vehicle rectilinear law analyzed in \cite{scltechrep,scl02,cdc03}.
As in the planar setting, we can interpret the terms involving $ f $
as steering the vehicles apart to avoid collisions (or steering them
together into formation if they are too far apart).
The terms involving $ \mu $ serve to align the vehicle headings,
and the terms involving $ \eta $ serve to align the vehicle headings
perpendicular to (or parallel to) the baseline between the vehicles.
The key to proving $ \dot{V}_{\mathit rect} \le 0 $ rests with the inequality
\begin{eqnarray}
\label{baseineq}
({\bf x}_1\cdot {\bf y}_2)\left[\frac{1}{2}({\bf x}_1\cdot {\bf y}_2)
\mp \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_2\right) \right]
\hspace{-7cm} & & \nonumber \\
& & + ({\bf x}_2\cdot {\bf y}_1)\left[\frac{1}{2}({\bf x}_2\cdot {\bf y}_1)
\mp \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_1\right) \right]
\nonumber \\
& & + ({\bf x}_1\cdot {\bf z}_2)\left[\frac{1}{2}({\bf x}_1\cdot {\bf z}_2)
\mp \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf z}_2\right) \right]
\nonumber \\
& & + ({\bf x}_2\cdot {\bf z}_1)\left[\frac{1}{2}({\bf x}_2\cdot {\bf z}_1)
\mp \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf z}_1\right) \right]
\ge 0, \nonumber \\
\end{eqnarray}
which after some algebra can be shown to be equivalent to
\begin{eqnarray}
\label{baseineq2}
\left[1-({\bf x}_1 \cdot {\bf x}_2)^2\right]
\hspace{-.3cm} & \pm & \hspace{-.3cm}
\bigg\{ \hspace{-.05cm} ({\bf x}_1 \cdot {\bf x}_2) \bigg[ \hspace{-.1cm}
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)^2
+ \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)^2
\bigg] \nonumber \\ & & \hspace{-.2cm}
- 2 \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right) \bigg\} \ge 0.
\end{eqnarray}
If $ {\bf x}_1 = \pm {\bf x}_2 $, then (\ref{baseineq2}) holds with
equality for any choice of $ {\bf r} $. So suppose
$ {\bf x}_1 \ne \pm {\bf x}_2 $, and consider minimizing the expression
in (\ref{baseineq2}) over all unit vectors $ {\bf r}/|{\bf r}| $.
It is not difficult to see that (\ref{baseineq2}) achieves its minimum
for some $ {\bf r}/|{\bf r}| $ lying in the unique plane $ P $
containing $ {\bf x}_1 $ and $ {\bf x}_2 $ (indeed, any component of
$ {\bf r}/|{\bf r}| $ which is perpendicular to $ P $ will not contribute
to expression (\ref{baseineq2}).) Thus, (\ref{baseineq2}) may be viewed
as a planar inequality, and we can define angle variables $ \phi_1 $
and $ \phi_2 $ such that
\begin{eqnarray}
\label{anglevars}
& & \hspace{-1cm} {\bf x}_1 \cdot {\bf x}_2
= \cos(\phi_2 - \phi_1), \nonumber \\
& & \hspace{-1cm} \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)
= \sin\phi_1, \;\;
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)
= \sin\phi_2.
\end{eqnarray}
After substituting (\ref{anglevars}) and applying some trigonometric
identities, inequality (\ref{baseineq2}) becomes
\begin{equation}
\label{baseineq3}
\sin(\phi_2 - \phi_1)\left[\sin(\phi_2 - \phi_1) \pm \frac{1}{2}
(\sin2\phi_2-\sin2\phi_1)\right] \ge 0.
\end{equation}
It can be shown that inequality (\ref{baseineq3}) does indeed hold
\cite{scltechrep}.
In the previous section, we defined shape variables
in terms of group variables in $ SE(3) $.
However, for the two-vehicle problem at hand, we can
use the variables $ ({\bf r},{\bf x}_1,{\bf x}_2) $
instead, because equilibria of the
$ ({\bf r},{\bf x}_1,{\bf x}_2) $ dynamics will include all
possible rectilinear formations.
Note that $ V_{\mathit rect} $
depends only on $ ({\bf r},{\bf x}_1,{\bf x}_2) $, as does
$ \dot{V}_{\mathit rect} $ (due to the restrictions on the control
laws we consider). Furthermore, the
$ ({\bf r},{\bf x}_1,{\bf x}_2) $ dynamics are self-contained
as a result of (\ref{vehctrlrestrict1})-(\ref{vehctrlrestrict2}).
\vspace{.25cm}
\noindent
{\bf Proposition 4}:
Consider the system $ ({\bf r},{\bf x}_1,{\bf x}_2) $ evolving on
$ \mathbb{R}^3 \times S^2 \times S^2 $, where $ S^2 $ is the two-sphere,
according to (\ref{twouavsystem3d}), (\ref{twovehiclelaw3d}),
and (\ref{formoff}). In addition, assume (A1), (A2), (A3), and (A4).
Define the set
\begin{equation}
\Lambda = \bigg\{ ({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
{\bf x}_2 \cdot {\bf x}_1 \ne -1 \mbox{ and } |{\bf r}|>0 \bigg\}.
\end{equation}
Then any trajectory starting in $ \Lambda $ converges to the set of
equilibrium points for the $ ({\bf r},{\bf x}_1,{\bf x}_2) $-dynamics.
\vspace{.25cm}
\noindent
{\bf Proof}:
Observe that $ V_{\mathit rect} $ given by (\ref{vrect}) is continuously
differentiable on $ \Lambda $. By assumption (A2) and the form of
$ V_{\mathit rect} $, we conclude that $ V_{\mathit rect} $ is radially
unbounded (i.e., $ V_{\mathit rect} \rightarrow \infty $ as
$ {\bf x}_1 \cdot {\bf x}_2 \rightarrow -1 $, as $ |{\bf r}| \rightarrow 0 $,
or as $ |{\bf r}| \rightarrow \infty $). Therefore, for each trajectory
starting in $ \Lambda $ there exists a compact sublevel set $ \Omega $
of $ V_{\mathit rect} $ such that the trajectory remains in $ \Omega $
for all future time. Then by LaSalle's Invariance Principle \cite{khalil},
the trajectory converges to the largest invariant set $ M $ of the
set $ E $ of all points in $ \Omega $ where $ \dot{V}_{\mathit rect} = 0 $.
The set $ E $ in this case is the set of all points
$ ({\bf r},{\bf x}_1,{\bf x}_2) \in \Omega $ such that
$ {\bf x}_2 = {\bf x}_1 $. Certainly if $ {\bf x}_1 = {\bf x}_2
= \pm {\bf r}/|{\bf r}| $, then $ u_1 = u_2 = v_1 = v_2 = 0 $ and
the trajectory remains in $ E $ for all future time.
Similarly, if $ {\bf r} \cdot {\bf x}_1 = {\bf r} \cdot {\bf x}_2 = 0 $
and $ f(|{\bf r}|) = 0 $, then $ u_1 = u_2 = v_1 = v_2 = 0 $ and
the trajectory remains in $ E $ for all future time.
Otherwise, we have the following expressions for the time-evolution of
the quantities $ {\bf r}\cdot{\bf x}_1 $ and $ {\bf r}\cdot {\bf x}_2 $
at points in $ E $:
\begin{eqnarray}
\label{ddtrdotx1}
\frac{d}{dt} ({\bf r}\cdot {\bf x}_1) \hspace{-1.6cm} & & \hspace{.7cm} =
\dot{\bf r}\cdot {\bf x}_1 + {\bf r}\cdot \dot{\bf x}_1 \nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm}
({\bf x}_2 - {\bf x}_1)\cdot {\bf x}_1 + {\bf r}\cdot({\bf y}_1 u_1
+ {\bf z}_1 v_1) \nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm}
({\bf r}\cdot {\bf y}_1) \hspace{-.05cm} \left[ \hspace{-.05cm} \mp \eta
\hspace{-.05cm}
\left(\hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1
\hspace{-.05cm}\right)
\left(\hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf y}_1
\hspace{-.05cm}\right) \hspace{-.05cm} - \hspace{-.05cm}
f(|{\bf r}|)\left( \hspace{-.05cm} -\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_1
\hspace{-.05cm}\right) \hspace{-.05cm} \right]
\nonumber \\
\hspace{-.2cm} & + & \hspace{-.2cm}
({\bf r}\cdot {\bf z}_1) \hspace{-.05cm} \left[ \hspace{-.05cm} \mp \eta
\hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1
\hspace{-.05cm} \right)
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf z}_1
\hspace{-.05cm} \right) \hspace{-.05cm} - \hspace{-.05cm}
f(|{\bf r}|)\left(\hspace{-.05cm} -\frac{\bf r}{|{\bf r}|}\cdot {\bf z}_1
\hspace{-.05cm}\right) \hspace{-.05cm} \right]
\nonumber \\
\hspace{-.2cm} & = & \hspace{-.2cm}
|{\bf r}|\left[1 - \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)^2
\right] \left[\mp \eta\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)
+ f(|{\bf r}|)\right],
\end{eqnarray}
and similarly,
\begin{equation}
\label{ddtrdotx2}
\frac{d}{dt} ({\bf r}\cdot {\bf x}_2) =
|{\bf r}| \hspace{-.1cm}
\left[1 - \left( \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)^2
\right] \hspace{-.1cm}
\left[\mp \eta\left( \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)
- f(|{\bf r}|)\right] \hspace{-.05cm} .
\end{equation}
If $ {\bf x}_1 \ne \pm {\bf r}/|{\bf r}| $ and $ f(|{\bf r}|) \ne 0 $, then
$ \frac{d}{dt} ({\bf r}\cdot {\bf x}_1) \ne
\frac{d}{dt} ({\bf r}\cdot {\bf x}_2), $
and it follows that the trajectory leaves $ E $.
If $ f(|{\bf r}|) = 0 $, then the trajectory remains in $ E $, but
the only invariant subset of $ E $ with $ f(|{\bf r}|) = 0 $
also has $ {\bf r} \cdot {\bf x}_1 = 0 $ (or
$ {\bf x}_1 = \pm {\bf r}/|{\bf r}| $).
Therefore, the largest invariant set contained in $ E $ may be expressed as
\begin{eqnarray}
M \hspace{-.2cm} & = & \hspace{-.2cm}
\bigg(\bigg\{({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
{\bf x}_1 = {\bf x}_2, \; \; {\bf r}\cdot {\bf x}_1 = 0, \;\;
f(|{\bf r}|) = 0 \bigg\}
\nonumber \\ & & \cup
\bigg\{({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
{\bf x}_1 = {\bf x}_2 = \pm \frac{\bf r}{|{\bf r}|} \bigg\} \bigg)\cap
\Omega.
\end{eqnarray}
Clearly $ M $ is contained in the set of equilibria
of the $ ({\bf r},{\bf x}_1,{\bf x}_2) $-dynamics. To see that
there are no other equilibria in $ \Omega $, we observe
that at equilibrium, $ \dot{\bf r} = {\bf x}_2 - {\bf x}_1 = 0 $,
and hence $ {\bf x}_2 = {\bf x}_1 $. Since at equilibrium, we
must also have
$ \frac{d}{dt} ({\bf r}\cdot {\bf x}_1)
= \frac{d}{dt} ({\bf r}\cdot {\bf x}_2) = 0, $
we see from equations (\ref{ddtrdotx1}) and (\ref{ddtrdotx2})
that there are no equilibria in $ \Omega $ apart from those
contained in $ M $. $ \Box $
\vspace{.25cm}
\noindent
{\bf Remark}:
If $ f $ is given by (\ref{fofr}), then $ f(|{\bf r}|) = 0 $
is equivalent to $ |{\bf r}| = r_o $. Thus, the set of equilibria consists
of formations with both vehicles heading in the same direction, and
for one type of formation, the motion of the vehicles is
perpendicular to the baseline between them with an intervehicle
distance equal to $ r_o $. For the other type of formation,
both vehicles follow the same straight-line trajectory, with
one leading the other by an arbitrary distance.
The stability of these equilibria depend on the choice of parameters,
and can be further analyzed using linearization.
\vspace{.25cm}
\noindent
{\bf Remark}:
We can express $ V_{\mathit rect} $ in terms of the group variable
$ g = g_1^{-1} g_2 $ as
\begin{equation}
V_{\mathit rect} = -\ln(1+g_{11})+h(r),
\end{equation}
and the control law as
\begin{eqnarray}
\label{rectctrlgroup}
u_1 \hspace{-.3cm} & = & \hspace{-.3cm}
\mp \eta(r) \left(\frac{g_{14}g_{24}}{r^2}\right)
\hspace{-.025cm} + \hspace{-.025cm} \mu(r) g_{21}
\hspace{-.025cm} + \hspace{-.025cm} f(r)\left(\frac{g_{24}}{r}\right),
\nonumber \\
u_2 \hspace{-.3cm} & = & \hspace{-.3cm}
\mp \eta(r) \left(\frac{g^{14}g^{24}}{r^2}\right)
\hspace{-.025cm} + \hspace{-.025cm} \mu(r) g^{21}
\hspace{-.025cm} + \hspace{-.025cm} f(r)\left(\frac{g^{24}}{r}\right),
\nonumber \\
v_1 \hspace{-.3cm} & = & \hspace{-.3cm}
\mp \eta(r) \left(\frac{g_{14}g_{34}}{r^2}\right)
\hspace{-.025cm} + \hspace{-.025cm} \mu(r) g_{31}
\hspace{-.025cm} + \hspace{-.025cm} f(r)\left(\frac{g_{34}}{r}\right),
\nonumber \\
v_2 \hspace{-.3cm} & = & \hspace{-.3cm}
\mp \eta(r) \left(\frac{g^{14}g^{34}}{r^2}\right)
\hspace{-.025cm} + \hspace{-.025cm} \mu(r) g^{31}
\hspace{-.025cm} + \hspace{-.025cm} f(r)\left(\frac{g^{34}}{r}\right),
\end{eqnarray}
where $ g = \{g_{ij} \} $, $ g^{-1} = \{ g^{ij} \} $, and
$ r = \sqrt{g_{14}^2+g_{24}^2+g_{34}^2} $. $ \Box $
\section{Circling formation law}
Consider the Lyapunov function candidate
\begin{equation}
\label{vcircdefn}
V_{\mathit circ} = -\ln\left[1 \hspace{-.05cm} - \hspace{-.05cm}
{\bf x}_2\cdot {\bf x}_1
\hspace{-.05cm} + \hspace{-.05cm} 2
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2
\hspace{-.05cm} \right) \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \hspace{-.05cm}
\right) \right] + h(|{\bf r}|),
\end{equation}
where we assume
\begin{itemize}
\item[$( \hspace{-.05cm} \mbox{A} \hspace{-.05cm} 1 \hspace{-.05cm} \mbox{'}
\hspace{-.05cm})$]
$ dh/d\rho = f(\rho)-2/\rho $, where $ f(\rho) $ is a Lipschitz
continuous function on $ (0,\infty) $, so that
$ h(\rho) $ is continuously differentiable on $ (0,\infty) $;
\end{itemize}
and (A2). It can be shown that
\begin{equation}
\label{vcirclnterm}
1 - {\bf x}_2\cdot {\bf x}_1 + 2
\left( \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)
\left( \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right) \ge 0,
\end{equation}
and the function $ f $ given by (\ref{fofr}) can be used here, as well.
The term $ h(|{\bf r}|) $ in (\ref{vcircdefn}) penalizes vehicle
separations which are two large or too small. The natural-log term
in (\ref{vcircdefn}) involves the relative headings of the vehicles,
as well as the relative orientations of the headings with respect to
the baseline between the vehicles.
Differentiating $ V_{\mathit circ} $ along trajectories of
(\ref{twouavsystem3d})
and plugging in (\ref{twovehiclelaw3d}) gives
\begin{eqnarray}
\dot{V}_{\mathit circ} \hspace{-.25cm} & = & \hspace{-.25cm}
-\frac{1}{1-{\bf x}_2 \cdot {\bf x}_1 +
2\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)} \nonumber \\
& & \hspace{-1.3cm} \times \hspace{-.05cm}
\Bigg\{ \hspace{-.1cm} \left[ \hspace{-.05cm} -{\bf x}_1 \cdot {\bf y}_2
\hspace{-.05cm} + \hspace{-.05cm} 2 \hspace{-.05cm} \left( \hspace{-.05cm}
\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \hspace{-.05cm} \right) \hspace{-.05cm}
\left(\hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf y}_2 \hspace{-.05cm}
\right) \hspace{-.05cm} \right] \hspace{-.05cm}
F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1)
\nonumber \\ & & \hspace{-1.1cm}
+ \hspace{-.05cm} \left[ \hspace{-.05cm} -{\bf x}_2 \cdot {\bf y}_1
\hspace{-.05cm} + \hspace{-.05cm} 2 \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \hspace{-.05cm}
\right) \hspace{-.05cm}
\left(\hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf y}_1 \hspace{-.05cm}
\right) \hspace{-.05cm} \right] \hspace{-.05cm}
F(-{\bf r},{\bf x}_1,{\bf y}_1,{\bf x}_2)
\nonumber \\ & & \hspace{-1.1cm}
+ \hspace{-.05cm} \left[ \hspace{-.05cm} -{\bf x}_1 \cdot {\bf z}_2
\hspace{-.05cm} + \hspace{-.05cm} 2 \hspace{-.05cm}
\left(\hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \hspace{-.05cm}
\right) \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf z}_2 \hspace{-.05cm}
\right) \hspace{-.05cm} \right] \hspace{-.05cm}
F({\bf r},{\bf x}_2,{\bf z}_2,{\bf x}_1)
\nonumber \\ & & \hspace{-1.1cm}
+ \hspace{-.05cm} \left[ \hspace{-.05cm} -{\bf x}_2 \cdot {\bf z}_1
\hspace{-.05cm} + \hspace{-.05cm} 2 \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \hspace{-.05cm}
\right) \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf z}_1 \hspace{-.05cm}
\right) \hspace{-.05cm} \right] \hspace{-.05cm}
F(-{\bf r},{\bf x}_1,{\bf z}_1,{\bf x}_2)
\hspace{-.05cm} \Bigg\} \hspace{-.02cm}. \nonumber \\
\end{eqnarray}
In place of (\ref{formoff}), we use
\begin{eqnarray}
\label{formoffcirc}
F({\bf r},{\bf x}_2,{\bf y}_2,{\bf x}_1)
\hspace{-.2cm} & = & \hspace{-.2cm}
\pm \eta \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_2\right)
\nonumber \\ & & \hspace{-2cm} + \mu \hspace{-.05cm} \left[ \hspace{-.05cm}
-{\bf x}_1 \cdot {\bf y}_2
+ 2\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf y}_2 \right) \hspace{-.05cm} \right]
\hspace{-.1cm},
\end{eqnarray}
where we assume (A3) and (A4).
The key to proving $ \dot{V}_{\mathit circ} \le 0 $ can then be shown to
rest with the inequality
\begin{eqnarray}
\label{circineqxr}
1 - \left[-{\bf x}_2 \cdot {\bf x}_1
+2\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \right)\right]^2
\hspace{-6.2cm} & & \nonumber \\
\hspace{-.2cm} & \pm & \hspace{-.2cm}
\Bigg\{{\bf x}_2 \cdot {\bf x}_1 + \left[-{\bf x}_2 \cdot {\bf x}_1
+ 2\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)
\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right) \right]
\nonumber \\ & & \hspace{1.5cm} \times
\left[1-\left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1\right)^2
- \left(\frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2 \right)^2 \right] \Bigg\}
\nonumber \\
\hspace{-.2cm} & \ge & \hspace{-.2cm}
0.
\end{eqnarray}
Using a similar technique as was used above to pass from inequality
(\ref{baseineq2}) to inequality (\ref{baseineq3}), we can show that
(\ref{circineqxr}) also becomes (essentially) inequality (\ref{baseineq3}).
\vspace{.25cm}
\noindent
{\bf Proposition 5}:
Consider the system $ ({\bf r},{\bf x}_1,{\bf x}_2) $ evolving on
$ \mathbb{R}^3 \times S^2 \times S^2 $,
according to (\ref{twouavsystem3d}), (\ref{twovehiclelaw3d}),
and (\ref{formoffcirc}). In addition, assume (A1'), (A2), (A3), and (A4).
Define the set
\begin{eqnarray}
\Lambda' \hspace{-.25cm} & = & \hspace{-.25cm}
\bigg\{ ({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
1 \hspace{-.05cm} - \hspace{-.05cm}
{\bf x}_2\cdot {\bf x}_1
\hspace{-.05cm} + \hspace{-.05cm} 2
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_2
\hspace{-.05cm} \right) \hspace{-.05cm}
\left( \hspace{-.05cm} \frac{\bf r}{|{\bf r}|}\cdot {\bf x}_1 \hspace{-.05cm}
\right) \hspace{-.05cm} \ne \hspace{-.05cm} 0
\nonumber \\ & & \hspace{1.5cm}
\mbox{ and } |{\bf r}|>0 \bigg\}.
\end{eqnarray}
Then any trajectory starting in $ \Lambda' $ converges to the set
\begin{eqnarray}
\tilde{M}' \hspace{-.3cm} & = & \hspace{-.3cm}
\bigg( \hspace{-.05cm} \bigg\{ \hspace{-.05cm}
({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
{\bf x}_1 \hspace{-.025cm} = \hspace{-.025cm} -{\bf x}_2, \;
{\bf r}\cdot {\bf x}_1 \hspace{-.025cm} = \hspace{-.025cm} 0, \;
f(|{\bf r}|) \hspace{-.025cm} = \hspace{-.025cm} \frac{2}{|{\bf r}|} \bigg\}
\nonumber \\ & & \cup
\bigg\{({\bf r},{\bf x}_1,{\bf x}_2) \bigg|
{\bf x}_1 = {\bf x}_2 = \pm \frac{\bf r}{|{\bf r}|} \bigg\} \bigg)
\cap \Lambda'.
\end{eqnarray}
Note that elements of $ \tilde{M}' $ with $ {\bf x}_1 = -{\bf x}_2 $
correspond to the two vehicles following the same circular orbit,
separated by the diameter of the orbit, which is prescribed by the
function $ f $. Elements of $ \tilde{M}' $ with $ {\bf x}_1 = {\bf x}_2 $
correspond to rectilinear formations in which one vehicle
leads the other by an arbitrary distance.
\vspace{.25cm}
\noindent
{\bf Proof}:
Omitted due to space constraints, but a similar approach is used as
in the proof of {\bf Proposition 4}. $ \Box $
\vspace{.25cm}
\noindent
{\bf Remark}:
We can express $ V_{\mathit circ} $ in terms of the group variables
as
\begin{equation}
V_{\mathit circ} = -\ln\left(1-g_{11}-2\frac{g_{14}g^{14}}{r^2}\right)+h(r),
\end{equation}
and the control law for circling
can also be expressed in terms of the group variables,
analogously to (\ref{rectctrlgroup}). $ \Box $
\section{Multi-vehicle formations}
One way to generalize the two-vehicle laws discussed above to $ n $ vehicles
is to use an average of the pairwise interaction terms
used for the two-vehicle problem \cite{scltechrep,scl02,cdc03}, i.e.,
\begin{eqnarray}
\label{multivehctrl}
u_j \hspace{-.2cm} & = & \hspace{-.2cm}
\frac{1}{n} \sum_{k\ne j} \bigg[
F({\bf r}_j-{\bf r}_k,{\bf x}_j,{\bf y}_j,{\bf x}_k) \nonumber \\ & &
\hspace{1cm} - f(|{\bf r}_j-{\bf r}_k|)\left(\frac{ {\bf r}_j-{\bf r}_k }
{ |{\bf r}_j-{\bf r}_k| } \cdot {\bf y}_j \right) \bigg], \nonumber \\
v_j \hspace{-.2cm} & = & \hspace{-.2cm}
\frac{1}{n} \sum_{k\ne j} \bigg[
F({\bf r}_j-{\bf r}_k,{\bf x}_j,{\bf z}_j,{\bf x}_k) \nonumber \\ & &
\hspace{1cm} - f(|{\bf r}_j-{\bf r}_k|)\left(\frac{ {\bf r}_j-{\bf r}_k }
{ |{\bf r}_j-{\bf r}_k| } \cdot {\bf z}_j \right) \bigg],
\end{eqnarray}
$ j=1,...,n $. In (\ref{multivehctrl}), $ {\bf r}_j $ is the position of
the $ j^{\mbox{th}} $ vehicle, $ ({\bf x}_j,{\bf y}_j,{\bf z}_j) $ is
the corresponding natural Frenet frame, and $ (u_j,v_j) $ are
the associated natural curvatures.
Figures \ref{multiveh} and \ref{multivehcir} show simulation results
for multi-vehicle interactions of this type. Their analysis is a
topic of ongoing research.
\begin{figure}
\hspace{.5cm}
\epsfxsize=7.5cm
\epsfbox{cdc05fig6.eps}
\caption{\label{multiveh} Simulation results for ten vehicles using
generalization (\ref{multivehctrl}) of the two-vehicle rectilinear
formation control law (\ref{twovehiclelaw3d}) with (\ref{formoff})
and (\ref{fofr}).}
\end{figure}
\begin{figure}
\epsfxsize=4cm
\epsfbox{cdc05fig7a.eps}
\epsfxsize=4cm
\epsfbox{cdc05fig7b.eps}
\caption{\label{multivehcir} Simulation results for ten vehicles using
generalization (\ref{multivehctrl}) of the two-vehicle circling
formation control law (\ref{twovehiclelaw3d}) with (\ref{formoffcirc})
and (\ref{fofr}). (The same simulation results are viewed from two
different angles.)}
\end{figure}
\section{Acknowledgements}
This research was supported in part by the Naval Research Laboratory under
Grants No.~N00173-02-1G002, N00173-03-1G001, N00173-03-1G019, and
N00173-04-1G014; by the
Air Force Office of Scientific Research under AFOSR Grants
No.~F49620-01-0415 and FA95500410130; by the Army Research Office
under ODDR\&E MURI01 Program Grant No.~DAAD19-01-1-0465 to the Center for
Communicating Networked Control Systems (through Boston University);
and by NIH-NIBIB grant
1 R01 EB004750-01, as part of the NSF/NIH Collaborative
Research in Computational Neuroscience Program.
|
{
"timestamp": "2005-03-18T22:37:41",
"yymm": "0503",
"arxiv_id": "math/0503390",
"language": "en",
"url": "https://arxiv.org/abs/math/0503390"
}
|
\section{Introduction}
In a preceding article \cite{FHPR02} {\it
coherence in a relative sense}, i. e., understood
as a relation between a given observable and a
given quantum state, was postulated to be {\it
identical with incompatibility} between
observable and state as far as its {\it quantity}
$I_C$ is concerned. (For notation see the passage
immediately following the proof of Proposition 5
below.) Then it was shown that bipartite pure
state entanglement is expressible as $I_C$ (with
a suitable observable).
Pure states cannot be obtained as mixtures.
Therefore, the question if $I_C$ is concave, i.
e., a genuine entropy quantity, or convex, i. e.,
a genuine information one, or something third,
could not be put in this context. The first aim
of this study is to clarify this point. (This is
done in Proposition 5.) To enable this, the
mixing property of relative entropy (paralleling
the mixing property of entropy and Donald's
identity for relative entropy, see the Remark) is
derived.
In a follow up of the mentioned article
\cite{ent-meas} the special case of the final
bipartite pure state $\ket{\psi}_{12}$ in
repeatable measurement, when the initial state is
pure, was studied. It was shown that the initial
quantity of incompatibility between the measured
observable and the initial state reappears as the
amount of entanglement in $\ket{\psi}_{12}$, and
is further preserved when it is shifted in
reading the measurement result. This completes
Vedral's result \cite{Vedral} that the
information transfer from object (subsystem $1$)
to measuring apparatus (subsystem $2$) does not
exhaust the mutual information $I_{12}$ in the
final state.
I think it is of interest to find out if the
mentioned preservation of the quantity of
incompatibility between the measured observable
and the initial pure state is restricted to pure
state, or it can be generalized to mixed initial
state. This is not a straightforward
generalization. It requires more knowledge on
$I_C$. The second aim of this study is to provide
such knowledge, which will be possible due to the
mentioned auxiliary relative-entropy relations
(see section 3).
In a further preceding article \cite{Roleof} an
arbitrary discrete incomplete observable $A$ and
its completion $A^c$ to a complete observable
were investigated and it was shown that
$I_C(A,\rho ) \leq I_C(A^c,\rho )$ for any state
$\rho$. This inequality is expected if the
assumption on the identity of the amount of
coherence and that of incompatibility is correct.
But it is desirable to evaluate $I_C(A^c,\rho
)-I_C(A,\rho )$ and thus to try to acquire more
insight into the nature of $I_C$. This is the
third aim of this article. (See the discussion
after the proof of the theorem below.)
The fourth aim of this paper is to present an
argument that starts with the mentioned identity
assumption and leads to an expression for the
quantity of coherence in a natural way. Will this
expression be the same as the {\it ad hoc}
introduced one? This is done in section 2 and an
affirmative answer is obtained. It is summed up
in the conclusion (subsection 5.2.).
The fifth and last aim of this investigation is
perhaps the most important one. Namely, in
\cite{Roleof} it was established that $I_C$ plays
an important role also in some mixed bipartite
states. This line of research should be continued
in a follow up because it may contribute to our
understanding how mutual information in general
bipartite states breaks up into a quasi-classical
part and entanglement, which is the object of
study of a wide circle of researchers, e. g.
\cite{VedralHend}, \cite{ZurekOliv}. To this
purpose, one may need more detailed knowledge of
the properties of $I_C$. To acquire such
knowledge is the fifth aim of this article (see
section 4).
\subsection{Background in classical
statistical physics}
To obtain a background for our quantum
study of coherence, we assume that a
classical discrete variable $\quad
A(q)=\sum_la_l\chi_l(q)\quad$ is given
(all $a_l\in {\bf R}$ being distinct).
The symbol $q$ denotes the continuous
state variables (as a rule, it consists
of twice as many variables as there are
degrees of freedom in the system);
$\chi_l$ are the characteristic functions
$\quad\forall l:\quad \chi_l(q)\equiv
1\quad$ if $\quad q\in {\cal A}_l\quad$,
and zero otherwise. Naturally, ${\cal
A}_l$ are (Lebesgue measurable) sets such
that $A(q)=a_l$ if and only if $q\in
{\cal A}_l$, and $\quad \sum_l{\cal
A}_l={\cal Q},\quad$ where ${\cal Q}$ is
the entire state space (or phase space)
and the sum is the union of disjoint
sets.
Let $\rho(q)$ be a continuous probability
distribution in $\cal Q$ with the
physical meaning of a statistical 'state'
of the system. One can think of $\rho(q)$
as of a mixture
$$\rho(q)=\sum_lp_l\rho_l(q),\eqno{(1)}$$
where $\quad \forall l:\quad p_l\equiv
\int_{\cal Q}\rho(q)\chi_l(q)dq\quad$ are
the statistical weights (probabilities of
the results $a_l$ if $A(q)$ is measured
in $\rho(q)$), and $\quad \forall
l,\enskip p_l>0:\quad \rho_l(q)\equiv
\rho(q)\chi_l(q)/p_l\quad$ are the
'states' with definite (or sharp) values
of $A(q)$.
Let $B(q)$ be any other continuous or
discrete variable. Then, utilizing (1),
its average can be written
$$\average{B}_{\rho}\equiv \int_{\cal Q}\rho(q)
B(q)dq=\sum_lp_l
\average{B}_{\rho_l}.\eqno{(2)}$$ {\it
One distinguishes the contributions of
the individual eigenvalues $a_l$ of
$A(q)$ through the terms on the RHS.}
They contribute to $\average{B}_{\rho }$
each separately.
All this serves only as a classical
background to help us to understand the
non-classical, i. e., purely quantum
relations between the analogous quantum
entities.
\subsection{Transition to the quantum mechanical case}
The quantum mechanical analogues of the
mentioned classical entities are the
following.
Discrete observables
(Hermitian operators) $A=\sum_la_lP_l$
(spectral form in terms of distinct
eigenvalues), $\rho$ quantum state
(density operator), and $B$ an arbitrary
observable (Hermitian operator). The
quantum average is $\quad
\average{B}_{\rho}\equiv {\rm tr} (\rho B)$.
In the transition from classical to
quantum one runs into a surprise, that is
known but, perhaps, not sufficiently well
known. Before we formulate it in the form
of a lemma, let us introduce the
L\"{u}ders state $\rho_L$ \cite{Lud} in
order to obtain the quantum analogues of
relations (1) and (2). It is that mixture
of states, each with a definite value of
$A$, which has a {\it minimal}
Hilbert-Schmidt distance from the given
state $\rho$ \cite{rhorho'}. It is
defined as
$$\rho_L\equiv \sum_lp_l\rho_L^l,
\eqno{(3a)}$$ where
$$\forall l:\quad p_l\equiv {\rm tr} (\rho
P_l)\eqno{(3b)}$$ are again the
statistical weights in (3a) (or the
probabilities of the results $a_l$ when
$A$ is measured in $\rho$), and
$$\forall l,\enskip p_l>0:\quad
\rho^l_L\equiv P_l\rho
P_l/p_l\eqno{(3c)}$$ are the states with
definite values $a_l$ of $A$. Finally,
$$\average{B}
_{\rho_L}=\sum_lp_l\average{B}_{\rho^l_L}
. \eqno{(3d)}$$
Decomposition (3a) is the analogue of
(1), and (3d) is that of (2).
{\bf Lemma 1.} {\it The following four
statements are equivalent:
(i) The state $\rho$ cannot be written as
a mixture of states in each of which the
observable $A$ has a definite value.
(ii) The observable $A$ and the state
$\rho$ are incompatible, i. e., the
operators do not commute $[A,\rho]\not=
0$.
(iii) The L\"{u}ders state $\rho_L$ given
by (3a)-(3c) is distinct from the
original state $\rho$.
(iv) There exists an observable $B$ such
that
$$\average{B}_{\rho }\not=
\average{B}_{\rho_L},\eqno{(4)}$$ where
the RHS is given by (3d).}
Proof is given in Appendix 1.
The physical meaning of lemma 1 is that it
defines a kind of {\it quantum coherence} as a
special relation between observable and state.
Experimentally it is exhibited in {\it
interference}. In this relative sense (relation
between variable and state) it is lacking in
classical physics because there a state can
always be written as a mixture of states in each
of which the variable in question has a definite
value (negation of (i), cf (1)). Though classical
waves do exhibit a kind of coherence and show
interference, but this is in a different sense
(cf section 5).
One should note that the L\"{u}ders state needs
no other characterization than its role in lemma
1 (in particular (iii)). The fact that it is
"closest" to $\rho$ in Hilbert-Schmidt metrics,
though actually not important for this study,
raises the thought-provoking questions if
"closest" is true also in other metrics; if not,
why is the Hilbert-Schmidt metrics more suitable.
We take {\it two-slit interference} \cite{Young}
to serve as an illustration for lemma 1.
Let $A$ be a dichotomic position observable with
two eigenvalues: localization at the left slit,
and localization at the right slit on the first
screen. Let $\rho$ be a wave packet that has just
arrived at this two-slit screen. Next, one has to
find a suitable observable $B$ such that
inequality (4) be satisfied at the mentioned
moment. Moreover, one wants to observe
experimentally the LHS of (4), or rather the
individual probabilities of the eigenvalues of
$B$ (that go into the LHS).
To this purpose, one actually replaces $B$ by
another localization observable $A'$ on a second
screen, to which the photon will arrive some time
later. This observable is suitable for
observation (of its localization probabilities).
Hence, one can define $\quad B\equiv
U^{-1}A'U,\quad$ $U$ being the evolution operator
expressing the movement of the particle from the
two-slit screen to the second one. One should
note that $B$ is not a position observable though
$A'$ is because the hamiltonian that generates
$U$ contains the kinetic energy (square of linear
momentum).
Claim (i) of lemma 1 says that the
particle is not moving through either the
left or the right slit. Claim (ii)
expresses the same fact algebraicly.
Namely, $\rho$, being a pure state
$\ket{\psi }\bra{\psi }$, would commute
with $A$ only if $\ket{\psi }$ lay in an
eigensubspace of $A$. In our case this
would mean that the particle traverses
one of the slits.
The L\"{u}ders state $\rho_L$ is, in some
sense, the best approximation to $\rho$
of a state traversing one or the other of
the slits. Naturally, $\quad \rho \not=
\rho_L\quad$ as claimed by (iii). Claim
(iv), i. e., relation (4), amounts to the
same as the fact that the interference
pattern on the second screen is not equal
to the sum of those that would be
obtained when only one of the slits were
open (for some time) and then the other
(for another, disjoint, equally long
time).
In the two-slit experiment one actually
observes the time-delayed equivalent of
(4):
$$\average{A'}_{U\rho U^{-1}}\not=
\average{A'}_{U\rho_LU^{-1}}.\eqno{(5)}$$ Since
the LHS of (5) is {\it distinct} from the RHS,
one speaks of the former as {\it interference}.
In the described two-slit case the LHS of (5)
gives fringes, whereas the RHS does not.
Nevertheless, it is not always true that the LHS
of (5) itself means interference. This is the
case only with a suitable pair of $A$ and $\rho$
(cf (ii) in lemma 1). Let me give a
counterexample.
Let us take another two-slit experiment in which
the slits have polarizers that give opposite
linear polarization to the light passing the
slits \cite{HZ}. The state $\rho$ in the slits is
then such that we have equality in (5) (though
$A'$ is the same), and there is no interference
because $[A,\rho ]=0$. (The state $\rho
=\ket{\psi }\bra{\psi }$ is now in the composite
spatial-polarization state space, and the spatial
subsystem state - the reduced statistical
operator - is a L\"{u}ders state.)
One should note that when interference is
displayed, one has three ingredients: the state
$\rho$, the observable $A$ the two eigenvalues of
which {\it play a cooperative role}, and the
second observable $A'$ the probabilities of
eigenvalues of which are observed. Since in
theory there can be many observables like $A'$,
or $B$ in (4), one likes to omit them. Then one
speaks of {\it coherence} of the observable $A$
in the state $\rho$. We make use of the same
concepts in the general theory.
{\bf Definition 1.} {\it The LHS of relation (4),
in case inequality (4) is valid, is called
interference. If an observable $A$ and a state
$\rho$ stand in such a mutual relation that any
of the four claims of lemma 1 is known to be
valid, then one speaks of coherence.}
One should note that the concepts of interference
and of coherence stand in a peculiar relation to
each other: There is no coherence (between $A$
and $\rho$) unless an observable $B$ that
exhibits interference can be, in principle,
found; if the latter is the case, and only then,
one may forget about $B$, and concentrate on the
relation between $A$ and $\rho$, i. e., on
coherence. The kind of quantum coherence
investigated in this paper can be more fully
called "eigenvalue coherence of an observable in
relation to a state" in view of the cooperative
role of some eigenvalues (or, more precisely,
their quantum numbers, because the values of the
eigenvalues play no role) as seen in (4).
Thus, any of the four (equivalent) claims in
lemma 1 defines coherence. But for the
investigation in this article the important claim
is (ii): coherence exists if and only if $A$ and
$\rho$ do not commute. This remark is the corner
stone of the expounded approach to investigating
coherence (as in the preceding studies
\cite{FHPR02}, \cite{Roleof}).
\section{How to obtain a quantum measure
of coherence?}
We start with the assumption that
coherence of an observable $A$ with
respect to a state $\rho$ is {\it
essentially the same thing} as
incompatibility of $A$ and $\rho$:
$[A,\rho ]\not= 0$. The quantum measure
will be called {\it coherence} or
incompatibility {\it information}, and it
will be denoted by $I_C(A,\rho )$ or
shortly $I_C$ (cf (10) below).
One wonders what the meaning of a larger value of
$I_C$ for coherence is. It is more of what? The
only answer I can think of is in accordance with
the above assumption: More of incompatibility of
$A$ and $\rho$.
The next question is: Do we know what is
a "larger amount of incompatibility"?
The seminal review on entropy of Wehrl
\cite{Wehrl} (section III.C there)
explains that each member of the
Wigner-Yanase-Dyson family of skew
informations $$I_p(\rho ,A)\equiv
-S_p(\rho ,A)\equiv (1/2){\rm tr} ([\rho^p,A]
[\rho^{1-p},A]),\qquad 0<p<1,\eqno{(6)}
$$ is a good measure of incompatibility
of $\rho$ and $A$. Namely, $I_p(\rho ,A)$
is positive unless $\rho$ and $A$
commute, when it is zero. It is also
convex as an information quantity should
be.
Substituting the spectral form of $A$ in
(6), one obtains $$I_p=(1/2){\rm tr} (\sum_l
\sum_{l'}a_l[\rho^p,P_l]a_{l'}[\rho^{1-p},P_{l'}]).
$$ One can see that $I_p$ depends on the
eigenvalues of $A$.
As well known, $A$ and $\rho$ are
compatible if and only if all
eigenprojectors $P_l$ of the former are
compatible with the latter. The
eigenvalues of $A$ do not enter this
relation. Hence, $I_p(\rho ,A)$ given by
(6) is not the kind of incompatibility
measure that we are looking for. One
wonders if there is any other kind.
To obtain an answer, we turn to a
neighboring quantity: the quantum amount
of {\it uncertainty} of $A$ in $\rho$. It
is the entropy $S(A,\rho )$: $$S(A,\rho
)\equiv H(p_l),\eqno{(7a)}$$ where
$H(p_l)$ is the Shannon entropy
$$H(p_l)\equiv
-\sum_lp_llogp_l,\eqno{(7b)}$$ and
$$\forall l:\quad p_l\equiv {\rm tr} (P_l\rho
).\eqno{(7c)}$$.
It is known that whenever $A$ and $\rho$
are incompatible, and $A$ is a complete
observable, i. e., if all its eigenvalues
are nondegenerate (we'll write it as
$A^c$), then always $S(A^c,\rho )>S(\rho
)$. When $A^c$ is compatible with $\rho$,
the two quantities are equal. The
interpretation that the larger the
difference $S(A^c,\rho )-S(\rho )$, the
more incompatible $A^c$ and $\rho$ are
seems plausible. Hence, we require for
complete observables $A^c$, that
$I_C(A^c,\rho )$ should equal this
quantity: $I_C(A^c,\rho )\equiv
S(A^c,\rho )-S(\rho )$. Equivalently, one
can require that the following peculiar
decomposition of the entropy in case of a
complete observable should hold:
$$S(\rho )=S(A^c,\rho )-I_C(A^c,\rho
).\eqno{(8)}$$
On the other hand, if $A$ is a discrete
observable that is complete or incomplete
but {\it compatible} with $\rho$, then
the following decomposition parallels
(8):
$$S(\rho )=S(A,\rho )+ \sum_lp_lS(P_l\rho
P_l/p_l)\eqno{(9)}$$ (cf (7a), (7b) and
(7c)). If $p_l=0$, the corresponding
term in the sum is by definition zero.
Decomposition (9) is obtained by application of
{\it the mixing property of entropy} \cite{Wehrl}
(see Sections II.F. and II.B. there). It applies
to {\it orthogonal state decomposition}, in this
case to $\quad \rho =\sum_lp_l(P_l\rho
P_l/p_l),\quad$ and it reads $\quad S(\rho
)=H(p_l)+\sum_lp_lS(P_l\rho P_l/p_l)\quad$ (cf
(7b)).
The coherence information $I_C$ does not
appear in (9). This is as it should be
because it is zero due to the assumed
compatibility of $A$ and $\rho$.
In case of a general discrete $A$, which
is complete or incomplete, compatible
with $\rho$ or not, we must interpolate
between (8) and (9). This can be done by
observing that both decompositions can be
rewritten in a unified way as
$$I_C(A,\rho )=S\Big(\sum_lP_l\rho
P_l\Big)-S(\rho )\eqno{(10)}$$ (valid for either
$A=A^c$ or for $[A,\rho ]=0$). The searched for
interpolated formula should thus be the same
relation (10), but valid this time for all
discrete $A$. Thus, $I_C(A,\rho )$ is obtained by
the presented argument.
Making use of the mixing property of
entropy, we can rewrite (10) equivalently
as the following general decomposition of
entropy:
$$S(\rho )=S(A,\rho )+ \sum_lp_lS(P_l\rho
P_l/p_l)-I_C(A,\rho ). \eqno{(11)}$$
(Note that $A$ is any discrete observable
in (11).)
In order to derive a number of properties
of coherence information, we make a
deviation into relative entropy theory.
\section{Useful relative-entropy
relations}
The {\it relative entropy}
$S(\rho||\sigma)$ of a state (density
operator) $\rho$ with respect to a state
$\sigma$ is by definition
$$S(\rho||\sigma)\equiv {\rm tr} [\rho log(\rho )]-{\rm tr}
[\rho log(\sigma)]\eqno{(12a)}$$
$$\mbox{if}\quad \mbox{supp}(\rho ) \subseteq
\mbox{supp}(\sigma );\eqno{(12b)}$$ or
else $\quad S(\rho||\sigma)=+\infty
\quad$ (see p. 16 in \cite{O-P}). By
'support', denoted by 'supp', is meant
the subspace that is the topological
closure of the range.
If $\sigma$ is singular and condition
(12b) is valid, then the orthocomplement
of the support (i. e., the null space) of
$\rho$, contains the null space of
$\sigma$, and both operators reduce in
supp$(\sigma )$. Relation (12b) is valid
in this subspace. Both density operators
reduce also in the null space of
$\sigma$. Here the $log$ is not defined,
but it comes after zero, and it is
generally understood that zero times an
undefined quantity is zero. We'll refer
to this as {\it the zero convention}.
The more familiar concept of (von
Neumann) quantum entropy, $S(\rho )\equiv
-{\rm tr} [\rho log(\rho )]$, also requires
the zero convention. If the state space
is infinite dimensional, then, in a
sense, entropy is almost always infinite
(cf p.241 in \cite{Wehrl}). In
finite-dimensional spaces, entropy is
always finite.
There is an {\it equality for entropy}
that is much used, and we have utilized
it, {\it the mixing property} concerning
{\it orthogonal state decomposition} (cf
p. 242 in \cite{Wehrl}):
$$\sigma =\sum_k w_k\sigma_k,\eqno{(13)}$$
$\forall k:\enskip w_k\geq 0$; for $w_k>0$,
$\sigma_k>0,\enskip {\rm tr} \sigma_k=1$; $\forall
k\not= k': \sigma_k\sigma_{k'}=0$; $\sum_kw_k=1$.
Then $\quad S(\sigma )=H(w_k)+
\sum_kw_kS(\sigma_k),\quad$ $H(w_k)\equiv
-\sum_k[w_klog(w_k)]\quad$ being the Shannon
entropy of the probability distribution
$\{w_k:\forall k\}$.
The first aim of this section is to
derive an analogue of the mixing property
of entropy. The second aim is to derive
two corollaries that we shall need in
this paper.
We will find it convenient to make use of
an {\it extension} $log^e$ of the
logarithmic function to the entire real
axis: $\quad \mbox{if}\quad 0<x:\qquad
log^e(x)\equiv log(x)\quad$, $\quad
\mbox{if}\quad x\leq 0:\enskip
log^e(x)\equiv 0\quad$.
The following elementary property of the
extended logarithm will be utilized.
{\bf Lemma 2.} {\it If an orthogonal
state decomposition (13) is given, then
$$log^e(\sigma )
=\sum'_k [log(w_k)]Q_k+\sum'_k log^e
(\sigma_k),\eqno{(14)}$$ where $Q_k$ is
the projector onto the support of
$\sigma_k$, and the prim on the sum means
that the terms corresponding to $w_k=0$
are omitted.}
{\bf Proof.} Spectral forms $\forall k,
\enskip w_k>0:\enskip
\sigma_k=\sum_{l_k}s_{l_k}\ket{l_k}
\bra{l_k}\quad$ (all $s_{l_k}$ positive)
give a spectral form $\sigma =
\sum_k\sum_{l_k}w_ks_{l_k}\ket{l_k}\bra{l_k}$
of $\sigma$ on account of the
orthogonality assumed in (13) and the
zero convention. Since numerical
functions define the corresponding
operator functions via spectral forms,
one obtains further
$$log^e(\sigma
)\equiv
\sum_k\sum_{l_k}[log^e(w_ks_{l_k})]\ket{l_k}
\bra{l_k}=
\sum_k'\sum_{l_k}[log(w_k)+log(s_{l_k})]
\ket{l_k} \bra{l_k}=$$ $$
\sum_k'[log(w_k)]Q_k+\sum_k'
\sum_{l_k}[log(s_{l_k})]\ket{l_k}
\bra{l_k}.$$ (In the last step
$Q_k=\sum_{l_k}\ket{l_k}\bra{l_k}$ for
$w_k>0$ was made use of.) The same is
obtained from the RHS when the spectral
forms of $\sigma_k$ are substituted in
it. \hfill $\Box$
{\bf Proposition 1.} {\it Let condition
(12b) be valid for the states $\rho$ and
$\sigma$, and let an orthogonal state
decomposition (13) be given. Then one has
$$S(\rho||\sigma)=S\Big(\sum_kQ_k\rho
Q_k\Big)-S(\rho )+H(p_k||w_k)+\sum_kp_k
S(Q_k\rho
Q_k/p_k||\sigma_k),\eqno{(15)}$$ where,
for $w_k>0$, $Q_k$ projects onto the
support of $\sigma_k$, and $Q_k\equiv 0$
if $w_k=0$, $p_k\equiv {\rm tr} (\rho Q_k)$,
and
$$H(p_k||w_k)\equiv
\sum_k[p_klog(p_k)]-\sum_k[p_klog(w_k)]
\eqno{(16)}$$ is the classical discrete
counterpart of the quantum relative
entropy, valid because $(p_k>0)\enskip
\Rightarrow (w_k>0)$.}
One should note that the claimed validity
of the classical analogue of (12b) is due
to the definitions of $p_k$ and $Q_k$.
Besides, (13) implies that $(\sum_kQ_k)$
projects onto supp$(\sigma )$. Further,
as a consequence of (12b),
$(\sum_kQ_k)\rho =\rho$. Hence, ${\rm tr}
\Big(\sum_kQ_k\rho Q_k\Big)={\rm tr}
(\sum_kQ_k\rho )=1$.
We call decomposition (15) {\it the
mixing property of relative entropy}.
{\bf Proof} of proposition 1: We define
$$\forall k,\enskip p_k>0:\quad
\rho_k\equiv Q_k\rho
Q_k/p_k.\eqno{(17)}$$ First we prove that
(12b) implies
$$\forall k,\enskip p_k>0:\quad
\mbox{supp}(\rho_k)\subseteq \mbox{supp}
(\sigma_k).\eqno{(18)}$$
Let $k$, $p_k>0$, be an arbitrary fixed
value. We take a pure-state decomposition
$$\rho
=\sum_n\lambda_n\ket{\psi_n}\bra{\psi_n}
\eqno{(19a)},$$ $\forall n:\enskip
\lambda_n>0$. Applying $Q_k...Q_k$ to
(19a), one obtains another pure-state
decomposition
$$Q_k\rho Q_k=p_k\rho_k
=\sum_n\lambda_nQ_k\ket{\psi_n}\bra{\psi_n}
Q_k\eqno{(19b)}$$ (cf (17)). Let
$Q_k\ket{\psi_n}$ be a nonzero vector
appearing in (19b). Since (19a) implies
that $\ket{\psi_n}\in \mbox{supp}(\rho )$
(cf Appendix 2(ii)), condition (12b)
further implies $\ket{\psi_n}\in
\mbox{supp}(\sigma )$. Let us write down
a pure-state decomposition
$$\sigma =\sum_m
\lambda'_m\ket{\phi_m}\bra{\phi_m}
\eqno{(20)}$$ with $\ket{\phi_1}\equiv
\ket{\psi_n}$. (This can be done with
$\lambda'_1>0$ cf \cite{Hadji}.) Then,
applying $Q_k...Q_k$ to (20) and taking
into account (13), we obtain the
pure-state decomposition
$$Q_k\sigma Q_k=w_k\sigma_k=\sum_m
\lambda'_mQ_k\ket{\phi_m}\bra{\phi_m}
Q_k.$$ (Note that $w_k>0$ because $p_k>0$
by assumption.) Thus,
$Q_k\ket{\psi_n}=Q_k\ket{\phi_1}\in
\mbox{supp}(\sigma_k)$. This is valid for
any nonzero vector appearing in (19b),
and these span supp$(\rho_k)$ (cf
Appendix 2(ii)). Therefore, (18) is
valid.
On account of (12b), the standard
logarithm can be replaced by the extended
one in definition (12a) of relative
entropy: $\quad S(\rho ||\sigma
)=-S(\rho)-{\rm tr} [\rho log^e(\sigma
)]\quad$. Substituting (13) on the RHS,
and utilizing (14), the relative entropy
$S(\rho ||\sigma )$ becomes
$$-S(\rho )-{\rm tr} \Big\{\rho
\Big[\sum_k'[log(w_k)]Q_k+\sum_k'[
log^e(\sigma_k)]\Big]\Big\}=-S(\rho
)-\sum_k'[p_klog(w_k)]-\sum_k'{\rm tr} [\rho
log^e(\sigma_k)].$$ Adding and
subtracting $H(p_k)$, replacing
$log^e(\sigma_k)$ by
$Q_k[log^e(\sigma_k)]Q_k$, and taking
into account (16) and (17), one further
obtains
$$S(\rho ||\sigma
)=-S(\rho )+H(p_k)+H(p_k||w_k)
-\sum_k'p_k{\rm tr} [\rho_klog^e(\sigma_k)].$$
(The zero convention is valid for the
last term because the density operator
$Q_k\rho Q_k/p_k$ may not be defined.
Note that replacing $\sum_k$ by $\sum_k'$
in (16) does not change the LHS because
only $p_k=0$ terms are omitted.)
Adding and subtracting the entropies
$S(\rho_k)$ in the sum, one further has
$$S(\rho ||\sigma
)=-S(\rho )+H(p_k)+H(p_k||w_k)+
\sum_k'p_kS(\rho_k)+\sum_k'p_k\{-S(\rho_k)
-{\rm tr} [\rho_klog^e(\sigma_k)]\}.$$
Utilizing the mixing property of entropy,
one can put $S\Big(\sum_kp_k\rho_k\Big)$
instead of
$[H(p_k)+\sum_k'p_kS(\rho_k)]$. Owing to
(18), we can replace $log^e$ by the
standard logarithm and thus obtain the
RHS(15). \hfill $\Box$
{\bf Remark.} {\it In a sense, (15) runs parallel
to Donald's identity
$$S(\rho||\sigma)=
\sum_kp_kS(\rho_k||\sigma )-H(p_k),$$ when an
orthogonal decomposition $\rho =\sum_kp_k\rho_k$
of the first state $\rho$ in relative entropy is
given.}
For a general decomposition $\rho
=\sum_kp_k\rho_k$ of the first state Donald's
identity reads
$$S(\rho ||\sigma
)=\sum_kp_kS(\rho_k||\sigma
)-\sum_kp_kS(\rho_k||\rho )$$ \cite{Donald},
\cite{Schum} (relation (5) in the latter). The
more special relation in the remark follows from
this on account of the relation that generalizes
the mixing property of entropy: If $\rho
=\sum_kp_k\rho_k$ is any state decomposition,
then $$S(\rho )= \sum_kp_k S(\rho_k||\rho
)+\sum_kp_kS(\rho_k)$$ is valid (cf Lemma 4 and
Remark 1 in \cite{Mutual}).
Now we turn to the derivation of some
consequences of proposition 1.
Let $\rho$ be a state and
$A=\sum_ia_iP_i+\sum_ja_jP_j$ a spectral
form of a discrete observable (Hermitian
operator) $A$, where the eigenvalues
$a_i$ and $a_j$ are all distinct. The
index $i$ enumerates all the detectable
eigenvalues, i. e., $\forall i:\enskip
{\rm tr} (\rho P_i)>0$, and ${\rm tr} [\rho
(\sum_iP_i)]=1$.
The simplest quantum measurement of $A$ in $\rho$
changes this state into the L\"{u}ders state:
$$\rho_L(A)\equiv \sum_iP_i\rho
P_i\eqno{(21)}$$ (cf (3a) and (3c)). Such a
measurement is often called "ideal".
{\bf Corollary 1.} {\it The relative-entropic
"distance" from any quantum state to its
L\"{u}ders state is the difference between the
corresponding quantum entropies:}
$$S\Big(\rho ||\sum_iP_i\rho
P_i\Big)=S\Big(\sum_iP_i\rho
P_i\Big)-S(\rho ).$$
{\bf Proof.} First we prove that
$$\mbox{supp}(\rho )\subseteq
\mbox{supp}\Big(\sum_iP_i\rho
P_i\Big).\eqno{(22)}$$ To this purpose,
we write down a decomposition (19a) of
$\rho$ into pure states. One has
$\mbox{supp}(\sum_iP_i)\supseteq
\mbox{supp}(\rho )$ (equivalent to the
certainty of $(\sum_iP_i)$ in $\rho$, cf
\cite{Roleof}), and the decomposition
(19a) implies that each $\ket{\psi_n}$
belongs to $\mbox{supp}(\rho )$ (cf
Appendix 2(ii)). Hence, $\ket{\psi_n}\in
\mbox{supp}(\sum_iP_i)$; equivalently,
$\ket{\psi_n}=(\sum_iP_i)\ket{\psi_n}$.
Therefore, one can write
$$\forall n:\quad \ket{\psi_n}=\sum_i(P_i
\ket{\psi_n}).\eqno{(23a)}$$ On the other
hand, (19a) implies
$$\sum_iP_i\rho
P_i=\sum_i\sum_n\lambda_nP_i\ket{\psi_n}
\bra{\psi_n}P_i.\eqno{(23b)}$$ As seen
from (23b), all vectors
$(P_i\ket{\psi_n})$ belong to
supp$(\sum_iP_i\rho P_i)$. Hence, so do
all $\ket{\psi_n}$ (due to (23a)). Since
$\rho$ is the mixture (19a) of the
$\ket{\psi_n}$, the latter span
$\mbox{supp}(\rho )$ (cf Appendix 2(ii)).
Thus, finally, also (22) follows.
In our case $\sigma \equiv \sum_iP_i\rho
P_i$ in (15). We replace $k$ by $i$.
Next, we establish
$$\forall i:\quad Q_i\rho Q_i=P_i\rho
P_i.\eqno{(24)}$$ Since $Q_i$ is, by
definition, the support projector of
$(P_i\rho P_i)$, and $P_i(P_i\rho
P_i)=(P_i\rho P_i)$, one has $P_iQ_i=Q_i$
(see Appendix 2(i)). One can write
$P_i\rho P_i=Q_i( P_i\rho P_i)Q_i$, from
which then (24) follows.
Realizing that $w_i\equiv {\rm tr} (Q_i\rho
Q_i)={\rm tr} (P_i\rho P_i)\equiv p_i$ due to
(24), one obtains $H(p_i||w_i)=0$ and
$\quad \forall i:\quad S(Q_i\rho Q_i/p_i
||P_i\rho P_i/w_i)=0\quad$ in (15) for
the case at issue. This completes the
proof.\hfill $\Box$
Now we turn to a peculiar further
implication of Corollary 1.
Let $B=\sum_k\sum_{l_k}b_{kl_k}P_{kl_k}$
be a spectral form of a discrete
observable (Hermitian operator) $B$ such
that all eigenvalues $b_{kl_k}$ are
distinct. Besides, let $B$ be more
complete than $A$ or, synonymously, a
refinement of the latter. This, by
definition means that
$$\forall k:\quad
P_k=\sum_{l_k}P_{kl_k}\eqno{(25)}$$ is
valid. Here $k$ enumerates both the $i$
and the $j$ index values in the spectral
form of $A$.
Let $\rho_L(A)$ and $\rho_L(B)$ be the
L\"{u}ders states (21) of $\rho$ with
respect to $A$ and $B$ respectively.
{\bf Corollary 2.} {\it The states
$\rho$, $\rho_L(A)$, and $\rho_L(B)$ lie
on a straight line with respect to
relative entropy, i. e., $\quad
S\Big(\rho || \rho_L(B)\Big)=S\Big(\rho
||\rho_L(A)\Big)+S\Big(\rho_L(A))||
\rho_L(B)\Big)\quad$, or explicitly:}
$$S\Big(\rho
||\sum_i\sum_{l_i}(P_{il_i}\rho
P_{il_i})\Big)=S\Big(\rho
||\sum_i(P_i\rho P_i)\Big)+
S\Big(\sum_i(P_i\rho P_i)||
\sum_i\sum_{l_i}(P_{il_i} \rho
P_{il_i})\Big).$$
Note that all eigenvalues $b_{kl_k}$ of
$B$ with indices others than $il_i$ are
undetectable in $\rho$.
{\bf Proof.} Corollary 1 immediately
implies
$$S\Big(\rho ||\rho_L(B)\Big)
=\Big[S\Big(\rho_L(B)\Big)-
S\Big(\rho_L(A)\Big)\Big]+
\Big[S\Big(\rho_L(A)\Big)-S(\rho
)\Big],$$ and, as easily seen from (21),
$\rho_L(B)= \Big(\rho_L(A)\Big)_L(B)$ due
to $P_{il_i}P_{i'}=\delta_{i,i'}P_{il_i}$
(cf (25)).
\hfill $\Box$
\section{Properties of coherence
information}
To begin with, we notice in (10) that
$I_C$ depends on $\rho$ and $A$, actually
only on the eigenprojectors of the
latter.
As a consequence of (10), one can also
write the definition of $I_C$ in the form
of a relative entropy:
$$I_C=S\Big(\rho ||\sum_lP_l\rho
P_l\Big)\eqno{(26)}$$ as follows from
corollary 1.
It was proved long ago \cite{Lind} that
$S\Big(\sum_l P_l\rho P_l\Big)>S(\rho )$
if and only if $A$ and $\rho$ are
incompatible, and the two entropies are
equal otherwise. Thus, in case of
compatibility $[A,\rho ]=0$, $I_C$ is
zero, otherwise it is positive. This is
what we would intuitively expect.
It was proved in \cite{Roleof} (theorem 2
there) that
$$I_C=w_{inc}I_C\Big(\sum_l^{inc}a_lP_l,
(\sum_l^{inc}P_l)\rho (\sum_l^{inc}P_l)
/w_{inc}\Big),\eqno{(27)}$$ where "inc"
on the sum denotes summing only over all
those values of $l$ the corresponding
$P_l$ of which are incompatible with
$\rho$, and $\quad w_{inc}\equiv {\rm tr}
(\rho \sum_l^{inc}P_l)$.
This corresponds to an intuitive
expectation that the quantity $I_C$
should depend only on those
eigenprojectors $P_l$ of $A$ that do not
commute with $\rho$, and not at all on
those that do.
We obtain (27) as a special case of a much more
general result below (cf the theorem and
propositions 2 and 3).
We shall need another known concept. For the sake
of precision and clarity, we define it.
{\bf Definition 2.} {\it One says that a
discrete observable $\bar A=\sum_m\bar
a_m\bar P_m$ (spectral form in terms of
distinct eigenvalues $\bar a_m$) is
coarser than or a coarsening of
$A=\sum_la_lP_l$ if there is a
partitioning $\Pi$ in the set
$\{l:\forall l\}$ of all index values of
the latter
$$\Pi:\qquad
\{l:\forall l\}=\sum_mC_m,$$ such that
$$\forall m:\quad \bar P_m=\sum_{l\in
C_m}P_l$$ ($C_m$ are classes of values of
the index $l$, and the sum is the union
of the disjoint classes). One also says
that $A$ is finer than or a refinement of
$\bar A$.}
{\bf Theorem.} {\it Let $\bar A$ be any
coarsening of $A$ (cf definition 2). Then
$$I_C(A,\rho )=I_C(\bar A,\rho )+
\sum_m\Big[p_mI_C\Big(\bar P_mA,\bar P_m\rho \bar
P_m/p_m\Big)\Big],\eqno{(28)}$$ and $\forall
m:\enskip p_m\equiv {\rm tr} (\rho \bar P_m)$. (If
$p_m=0$, then, by the zero convention, the
corresponding $I_C$ in (28) need not be defined.
The product is by definition zero.)}
Before we prove the theorem, we apply
corollary 2 to our case.
Under the assumptions of the theorem, one
has $$S\Big(\rho ||\sum_l (P_l\rho
P_l)\Big)=S\Big(\rho ||\sum_m(\bar P_m
\rho \bar P_m)\Big)+ S\Big(\sum_m(\bar
P_m\rho \bar P_m)||\sum_l (P_l\rho
P_l)\Big).\eqno{(29)}$$
{\bf Proof} of the Theorem. On account of
(26), (29) takes the form
$$I_C(A,\rho )=I_C(\bar A,\rho )+I_C\Big(A,
\sum_m(\bar P_m\rho \bar P_m)\Big).\eqno{(30)}$$
Utilizing (10) for the second term on the RHS,
the latter becomes $S\Big(\sum_l(P_l\rho
P_l)\Big)-S\Big(\sum_m(\bar P_m\rho \bar
P_m)\Big)$. Making use of the mixing property of
entropy in both these terms, and cancelling out
$H(p_m)$ (cf (7b) {\it mutatis mutandis}), this
difference, further, becomes
$\sum_mp_mS\Big((\sum_{l\in C_m}P_l\rho
P_l)/p_m\Big)-\sum_mp_mS\Big(\bar P_m\rho \bar
P_m/p_m)\Big)$. Its substitution in (30) with the
help of (10) (and definition 2) then gives the
claimed relation (28). (Naturally, one must be
aware of the fact that $\bar A$ is a coarsening
of $A$, hence $\enskip \forall m:\enskip [\bar
P_m,A]=0,\enskip$ implying $\enskip A\equiv
\sum_m\sum_{m'}\bar P_mA\bar P_{m'}=\sum_m\bar
P_mA$.) \hfill $\Box$
If $\bar A$ is any coarsening of $A$, then the
index values $m$ of the former replace classes
$C_m$ of index values $l$ of the latter. Hence,
coherence in $\bar A$ - as a cooperative role of
index values - must be poorer than in $A$.
Therefore, one would intuitively expect that
$I_C(\bar A,\rho )$ must not be larger than
$I_C(A,\rho )$. The theorem confirms this, and
tells more: it gives the expression by which
$I_C(A,\rho )$ exceed $I_C(\bar A,\rho )$. One
wonders what the intuitive meaning of this is.
{\it Discussion of the theorem.} Let us think of
$\rho$ as describing a laboratory ensemble, and
let us imagine that an ideal measurement of $\bar
A$ is performed on each quantum system in the
ensemble. The ensemble $\rho$ is then replaced by
the mixture $\quad \sum_mp_m(\bar P_m\rho \bar
P_m/p_m)\quad$ of subensembles $\quad (\bar
P_m\rho \bar P_m/p_m)$. One can think of the
measurement of the more refined observable $A$ as
taking place in two steps: the first is the
mentioned measurement of the coarser observable
$\bar A$, and the second is a continuation of
measurement of $A$ in each subensemble $\quad
(\bar P_m\rho \bar P_m/p_m)$. Let us assume {\it
additivity} of $I_C$ in two-step measurement.
Further, let us bear in mind that, though $I_C$
is meant to be a property of each individual
member of the ensemble $\rho$, it is {\it
statistical}, i. e., it is given in terms of the
ensemble. Finally, in the second step we have an
ensemble of subensembles (a superensemble). Since
our system is anywhere in the entire ensemble
$\quad \sum_mp_m(\bar P_m\rho \bar P_m/p_m)\quad$
of the second step, one must average over the
superensemble with the statistical weights $p_m$
of its subensemble-members $\quad (\bar P_m\rho
\bar P_m/p_m)$.
If $m'\not= m$, then the part $\quad \bar
P_{m'}A\quad$ of $\quad A=\sum_{m''}\bar
P_{m''}A\quad$ is evidently undetectable in the
subensemble $\rho_m$. Hence, only $\bar P_mA$ is
relevant from the entire $A$, i. e., $I_C(A,\rho
)$ reduces to $I_C(\bar P_mA,\rho_m)$ there.
In this way one can understand relation (28).
What have we learnt from this? It is that $I_C$
is additive and statistical. This conclusion is
in keeping with the neighboring quantity
$S(A,\rho )$. Namely, one can easily derive a
relation similar to (28) for it:
$$S(A,\rho )=S(\bar A,\rho )+\sum_mp_mS(\bar
P_mA,\bar P_m\rho \bar P_m/p_m).$$ That $I_C$ and
$S(A,\rho )$ behave equally in an additive and
statistical way is no surprise since they are
terms in the same general decomposition (11) of
the entropy $S(\rho )$ of the state $\rho$.
The theorem is a substantially stronger form of a
previous result (theorem 3 in \cite{Roleof}), in
which $I_C(A,\rho )\geq I_C(\bar A,\rho )$ was
established with necessary and sufficient
conditions for equality, which are obvious in the
theorem. ($I_C$ was denoted by $E_C$ in previous
work, cf my comment following proposition 5
below.)
The theorem has the following immediate
consequences.
{\bf Proposition 2.} {\it If the coarsening $\bar
A$ defined in definition 2 is {\it compatible}
with $\rho$, then (28) reduces to}
$$I_C(A,\rho )= \sum_m\Big[p_mI_C\Big(\bar P_mA,\bar
P_m\rho \bar
P_m/p_m\Big)\Big].\eqno{(31)}$$
{\bf Proposition 3.} {\it Let us define a
coarsening $\Pi$ (cf definition 2) that
partitions $\{l:\forall l\}$ into at most
three classes: $C_{inc}$ comprising all
index values $l$ for which $a_l$ is
detectable (i. e., of positive
probability) and $P_l$ is incompatible
with $\rho$, $C_{comp}$ consisting of all
$l$ for which $a_l$ is detectable and
$P_l$ is compatible with $\rho$, and,
finally, $C_{und}$ which is made up of
all $l$ for which $a_l$ is undetectable.
The coarsening thus defined is compatible
with $\rho$, and (31) reduces to (27).}
{\bf Proof.} In the coarsening $\Pi$ of
proposition 3 the index $m$ takes on
three 'values': 'inc', 'comp', and 'und'.
It is easily seen that the coarser
observable $\bar A$ thus defined is
compatible with $\rho$. Hence, (31)
applies. Further, the second and third
terms are zero. In this way, (27) ensues.
\hfill $\Box$
{\bf Proposition 4.} {\it Coherence information
$I_C$ is unitary invariant, i. e., $\quad
I_C(A,\rho )=I_C(UAU^{\dagger},U\rho
U^{\dagger}),\quad$ where $U$ is an arbitrary
unitary operator.}
{\bf Proof.} Relative entropy is known to
be unitary invariant. On account of (26),
so is $I_C$.\hfill $\Box$
This is as it should be because $I_C$
should not depend on the basis in the
state space: $UAU^{-1}$ and $U\rho
U^{-1}$ can be understood as $A$ and
$\rho$ respectively viewed in another
basis.
{\bf Proposition 5.} {\it Coherence
information $I_C$ is convex.}
{\bf Proof.} This is an immediate
consequence of the known convexity of
relative entropy (cf (26)) under joint
mixing of the two states in it.
On account of convexity we know that $I_C$ is an
{\it information entity}, and not an entropy one
(or else it would be concave). In previous work
\cite{FHPR02}, \cite{Roleof}, \cite{ent-meas} the
same quantity (the RHS of (10)) was erroneously
denoted by $E_C(A,\rho )$ and treated as an
entropy quantity. But this does not imply that
any of the applications of $E_C(A,\rho )$ was
erroneous. All one has to do is to replace this
symbol by $I_C(A,\rho )$ and keep in mind that
one is dealing with an information quantity.
\section{Conclusion}
Perhaps it is of interest to comment upon the
more standard uses of the term "coherence" in the
literature.
One encounters the basic use of the word
"coherence" in the properties of light waves. One
distinguishes two types of coherence there: (i)
Temporal coherence, which is a measure of the
correlation between the phases of a light wave at
different points along the direction of
propagation, and (ii) spatial coherence, which is
a measure of the correlation between the phases
of a light wave at different points transverse to
the direction of propagation. (The fascinating
phenomenon of holography requires a large measure
of both temporal and spatial coherence of light.)
Quantum "coherence" refers also to large numbers
of particles that cooperate collectively in a
single quantum state. The best known examples are
superfluidity, superconductivity, and laser
light, all macroscopic phenomena. In the last
example different parts of the laser beam are
related to each other in phase, which can lead to
interference effects. "Coherence" is often
related to different kinds of correlations, see,
e. g., \cite{JS}.
In all mentioned examples "coherence" refers to
an {\it absolute} property of the quantum state
of the system; in contrast with the use of the
term in this article, which expresses a {\it
relative} property: relation between observable
and state. As it was mentioned, the kind of
quantum coherence studied in this article can be
more fully called "eigenvalue coherence of an
observable in relation to a state" in view of the
cooperative role of the eigenvalues (or rather
their quantum numbers, because the values of the
eigenvalues play no role) as seen in (4).
In the literature one often finds the claim that
quantum pure states are coherent. From the
analytical point of view of this article one can
say that a pure state $\ket{\psi }$ is {\it not
coherent} with respect to any observable for
which $\ket{\psi }\bra{\psi }$ is an
eigenprojector. But it is coherent with respect
to all other observables.
\subsection{On generality of the
results}
A question may linger on to the
end of this study: What if the observable
is not a discrete one? Can one still
speak of eigenvalue coherence in relation
to a given state $\rho$?
It seems to me that the answer is that
one should write down the following
partial spectral form of a general
observable $A'$: $$A'=\sum_la_lP_l+
P^{\perp}A'P^{\perp},$$ where the
summation goes over {\it all eigenvalues}
of $A'$, and $P\equiv \sum_lP_l$. One
should take the discrete coarsening $A$
of $A'$: $$A\equiv
\sum_la_lP_l+aP^{\perp},$$ where the
eigenvalue $a$ is arbitrary but distinct
from all $\{a_l:\forall l\}$. Then the
expounded eigenvalue coherence theory
should by applied to $A$, and it should
be valid for $A'$ (as the best we can do
for the latter). In a preceding article
\cite{Roleof} the case when
$P^{\perp}\not= 0$ with the eigenvalue
$a$ undetectable was studied.
One has eigenvalue coherence of a general
observable $A'$ in relation to a state
$\rho$ if either $A'$ has at least two
eigenvalues or if $A'$ has at least one
eigenvalue and $P^{\perp}\not= 0$.
Another question that may linger on is
whether the state $\rho$ that was used in
this paper is really general. If $\rho$
has an infinite-dimensional range and $A$
has infinitely-many eigenvalues, it may
happen that there are infinitely-many
detectable ones. The expounded theory
covers also this case.
\subsection{Summing up}
In an attempt to understand the essential
features of two-slit interference (see lemma 1
followed by its application to two-slit
interference in subsection 1.2), a general
coherence theory was developed based on the
assumption that 'coherence' equals
'incompatibility' $[A,\rho ]\not= 0$ between
observable and state. Since this relation means
that $\rho$ is incompatible with at least one
eigenevent (eigenprojector) $P_l$ of $A$, and
this property is independent of the eigenvalues,
it was argued that the entire family of
observables with one and the same decomposition
of the identity $\sum_lP_l=I$ (the latter is
called "closure relation" if $A$ is complete)
should have the same amount of incompatibility.
This discarded the Wigner-Yanase-Dyson family of
skew informations (6). Further, it was argued
that the necessarily nonnegative quantity
$\enskip S(A^c,\rho )-S(\rho )\enskip$ was a
natural measure of incompatibility between a
complete observable $A^c$ and the state $\rho$
satisfying the stated claim. Finally,
interpolating between the case of a complete and
that of a compatible observable (see (8), (9) and
(10)), the general expression (10) was obtained.
Thus, a natural quantum measure of how much of
coherence, and, equivalently, incompatibility,
there is if a discrete observable
$A=\sum_la_lP_l$ and a state $\rho$ are given was
derived along the expounded argument. It was
called coherence or incompatibility information
(denoted by $I_C(A,\rho )$ or shortly $I_C$) in
section 2.
A deviation into a general
relative-entropy investigation was made
in section 3. What was called 'the mixing
property of relative entropy'
(parallelling that of entropy) was
derived, and so were two corollaries.
The relative-entropy results were utilized to
express coherence information $I_C(A,\rho )$ in
the form of a relative entropy (cf (26)) in
section 4. Connection between the coherence
information $I_C(\bar A,\rho )$ of any coarsening
$\bar A$ (cf definition 2) of an observable $A$
and $I_C(A,\rho )$ was obtained in the theorem.
Its intuitive meaning was discussed. It was
concluded that $I_C$ is additive in two-step
measurement and statistical.
The corresponding relation took a much simpler
form in case $\bar A$ was compatible with $\rho$
(cf proposition 2). In a special case of this a
result from previous work was recognized (cf
proposition 3 and (27)). Coherence information
was shown to be unitary invariant (proposition 4)
and convex (proposition 5).
In previous work \cite{FHPR02}, \cite{Roleof},
\cite{ent-meas} the coherence information $I_C$
was successfully utilized in analyzing bipartite
quantum correlations. The last one of them filled
in an information-theoretical gap noted in
preceding investigation of the measurement
process \cite{Vedral}.
Since a number of new properties of $I_C$ have
now been obtained, even more fruitful
applications can be expected.\\
\noindent {\bf Appendix 1.}\\
\indent We prove the equivalence of the
negations of the four claims in lemma 1.
("$\neg$ (i)" is the negation of (i)
etc., and "$(\Leftrightarrow )$" is the
claim of "$\Leftrightarrow$") The logical
scheme of the proof is: $\neg$ (ii)
$\Leftrightarrow$ $\neg$ (iii)
$\Leftrightarrow$ $\neg$ (iv); $\neg$
(ii) $\Rightarrow$ $\neg$ (i)
$\Rightarrow$ $\neg$ (iii).
$\neg$ (ii) $(\Leftrightarrow )$ $\neg$
(iii): One can always write $\rho
=\sum_l\sum_{l'}P_l\rho P_{l'}$. Since
$A$ and $\rho$ commute if and only if
each eigenprojector $P_l$ of $A$ commutes
with $\rho$, the claimed equivalence is
obvious.
$\Big(\neg$ (iii) $\Rightarrow$ $\neg$
(iv)\Big) is obvious. To prove
\Big($\neg$ (iv) $\Rightarrow$ $\neg$
(iii)\Big), we restrict the operators $B$
to ray projectors $\ket{a}\bra{a}$. Then
$\neg$ (iv) implies ${\rm tr} (\rho
\ket{a}\bra{a})=\bra{a}\rho
\ket{a}=\bra{a}\rho_L\ket{a}$ for every
state vector $\ket{a}$. But then, as well
known, one must have $\rho =\rho_L$,
which is $\neg$ (iii).
$\neg$ (ii) $(\Rightarrow )$ $\neg$ (i):
In view of $\rho =\sum_l\sum_{l'}P_l\rho
P_{l'}$, commutation of $\rho$ with each
$P_l$ implies $\neg$ (i).
$\neg$ (i) $(\Rightarrow )$ $\neg$ (iii):
Let us assume that $\rho
=\sum_lp_l\rho_l$, and that each state
$\rho_l$ has the sharp value of the
corresponding eigenvalue $a_l$ of $A$.
Then $\rho_l =P_l\rho_lP_l$ (cf lemma
A.4. in \cite{FHFoundPL}). Substituting
this in the state decomposition, and
subsequently evaluating $\rho_L$
according to (3a)-(3c), one can see that
$\neg$ (iii) follows.\hfill
$\Box$\\
\noindent {\bf Appendix 2.}\\
Let $\rho =\sum_n\lambda_n\ket{n}\bra{n}$
be an arbitrary decomposition of a
density operator into ray projectors, and
let $E$ be any projector. Then $$E\rho
=\rho \quad \Leftrightarrow \quad \forall
n:\enskip E\ket{n}=\ket{n}\eqno{(A.1)}$$
(cf Lemma A.1. and A.2. in
\cite{FHJP94}).
(i) If the above decomposition is an
eigendecomposition with positive weights,
then $\sum_n\ket{n}\bra{n}=Q$, $Q$ being
now the support projector of $\rho$, and,
on account of (A.1),
$$E\rho =\rho \quad \Rightarrow \quad
EQ=Q.\eqno{(A.2)}$$.
(ii) Since one can always write $Q\rho
=\rho$, (A.1) implies that all $\ket{n}$
in the arbitrary decomposition belong to
supp$(\rho )$. Further, defining a
projector $F$ so that supp$(F)\equiv$
span$(\{\ket{n}:\forall n\})$, one has
$FQ=F$. Equivalence (A.1) implies $F\rho
=\rho$. Hence, (A.2) gives $QF=Q$.
Altogether, $F=Q$, i. e., the unit
vectors $\{\ket{n}:\forall n\}$
span supp$(\rho)$.\\
\noindent {\bf References}\\
|
{
"timestamp": "2005-03-08T10:22:35",
"yymm": "0503",
"arxiv_id": "quant-ph/0503077",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503077"
}
|
\section{Introduction}
Generalizations $\Gamma_n$ of Hermite polynomials $H_n$ were recently
\cite{GiMeWe} proposed to describe, for instance, density perturbations
constrained by a condition of matter conservation. Because of the constraint,
such polynomials cannot form a complete set, but span a subspace well suited
to specific applications. In particular, the polynomials $\Gamma_n$ used in
\cite{GiMeWe} were motivated by the consideration in nuclear physics of the
Hohenberg-Kohn functional \cite{HK} and similar functionals along the
Thomas-Fermi method \cite{Mer,WKS}. Indeed, in such approaches, the ground
state of a quantum system is shown to be a functional of its density
$\rho(r),$ and there is a special connection between $\rho(r)$ and the mean
field $u(r)$ driving the system. It was thus convenient to expand variations
of $\rho$ in a basis $\{w_m(r)\}$ of particle number conserving
components, $\delta \rho(r)=\sum_m\, \delta \rho_m\, w_m(r),$ with the
term-by-term constraint, $\forall m,\ \int dr\, w_m(r)=0.$ This spares, in the
formalism, the often cumbersome use of a Lagrange multiplier. Simultaneously,
it was convenient to expand variations of $u$ in a basis orthogonal to the
flat potential, because, trivially, a flat $\delta u,$ as just a change in
energy reference, cannot influence the density. The same basis can thus be
used for $\delta u(r)=\sum_n \delta u_n\, w_n(r),$ since the very same
condition, $\int dr\, w_n(r)=0,$ induces orthogonality to a constant
$\delta u.$ Because of the nuclear physics context of \cite{GiMeWe},
harmonic oscillators shell models were considered and the basis contained
a Gaussian factor, $e^{-\frac{1}{2}r^2}.$
The same functional approaches \cite{HK,Mer,WKS} are also of a general use
in atomic and molecular physics, where Gaussian weights would be clumsy and
radial properties are best fitted with simple exponential weights
\cite{Messiah}. Furthermore, in \cite{GiMeWe}, the discussion was restricted
to one dimensional problems. In the present note, we want to include two and
three dimensional situations. We shall thus use weights of the form
$e^{-\frac{1}{2}r},$ with $0 \le r < \infty,$ but integrals will carry a
factor $r^{\nu},$ with $\nu$ a positive exponent, suitable for dimension $d.$
This will lead to generalizations of Laguerre polynomials.
This note is also concerned with compact domains, of the form $0 \le r \le 1$
for instance. This might correspond for instance to expansions of density
fluctuations in cylindrical vessels used for chemical processes, where mass
conservation is also in order, or maybe in centrifuges. Radial integrals
with factors $r$ and $r^2$ in both the constraint and orthogonalization
conditions will lead to generalizations of Legendre polynomials.
For any positive weight $\mu(r),$ and any dimension $d,$ a constraint of
vanishing average, $\int dr\, r^{\nu}\, \mu(r)\, \Gamma_n(r) =0,$ is
incompatible with a polynomial $\Gamma$ of order $n=0.$ Therefore, in the
following, the order hierarchy for the constrained polynomials runs
from $n=1$ to $\infty,$ while that for the traditional polynomials runs from
$0$ to $\infty.$ We study in some generality the ``Laguerre'' case in
Section II. In turn, the ``Legendre'' case is the subject of Section III.
A brief Section IV discusses possible applications to the study of density
fluctuations in centrifuges. Section V answers a question which was omitted
in \cite{GiMeWe}, that of the nature of the projector onto the subspace
spanned by the constrained states and the nature of the codimension of this
subspace. A numerical application is provided in Section VI. A discussion and
conclusion make Section VII.
\section{Modification of Laguerre polynomials by a constraint of zero average}
In this Section we consider basis states carrying a weight
$e^{-\frac{1}{2}r},$ in the form $w_n(r)=e^{-\frac{1}{2}r}\, G_n^d(r),$
where $G_n^d$ is a polynomial. It is clear that $G_0^d$ cannot be a finite,
non vanishing constant if the constraint,
$\int_0^{\infty}dr\, r^{d-1}\, e^{-\frac{1}{2}r}\, G_0^d(r)=0,$ must be
implemented. Hence set integer labels $m \ge 1$ and $n \ge 1$ and define
polynomials $G_n^d$ by the conditions,
\begin{equation}
\int_0^\infty dr\ r^{d-1}\, e^{-r}\, G_m^d(r)\, G_n^d(r) = g_n^d\,
\delta_{mn}\, ,
\ \ \ \ \
\int_0^\infty dr\ r^{d-1}\, e^{-\frac{1}{2}r}\, G_n^d(r) = 0\, ,
\label{definL}
\end{equation}
where $\delta_{mn}$ is the usual Kronecker symbol and the positive numbers
$g_n^d$ are normalizations, to be defined later.
It is elementary to generate such polynomials numerically, in two steps by
brute force, namely i) first create ``trivial seeds'' of the form,
$s_n^d(r)=r^n-\langle r^n \rangle_d,$ where the subtraction of the average,
$\langle r^n \rangle_d=2^n\, (d-1+n)!/(d-1)!,$ ensures that each trivial
seed fulfills the constraint, then ii) orthogonalize such seeds by a
Gram-Schmidt algorithm. The first polynomials read,
\begin{mathletters} \begin{eqnarray}
G_1^1 = r-2,\ \ \ \, G_2^1 = r^2-5r+2,\ \ \ \ \ G_3^1 = r^3-10r^2+20r-8,\ \
\ \ \ \ \ G_4^1 = r^4-17r^3+78r^2-108r+24, \label{G1} \\
G_1^2 = r-4,\ \ \ \ G_2^2 = r^2-8r+8,\ \ \ \ G_3^2 = r^3-14r^2+44r-32,\ \ \
G_4^2 = r^4-22r^3+138r^2-288r+144, \label{G2} \\
G_1^3 = r-6,\ \ \ G_2^3 = r^2-11r+18,\ \ \ G_3^3 = r^3-18r^2+78r-84,\ \ \
G_4^3 = r^4-27r^3+216r^2-606r+468. \label{G3}
\end{eqnarray} \end{mathletters}
All these are defined to be ``monic'', namely the coefficient of $r^n$ is
always $1.$ For an illustration we show in Figure 1 the new
polynomials $G_1^1$ and $G_1^2,$ together with Laguerre polynomial $ L_1.$
The same Fig. 1 also shows $G_2^1,$ $G_2^2$ and $L_2.$
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol1.eps}
}
\caption{Comparison of Laguerre polynomials $L_1,$ $L_2$ (full lines)
with new polynomials $G_1^1,$ $G_2^1$ (long dashes), $G_1^2,$ $G_2^2$
(dashes).}
\end{figure}
Rather using the Gram-Schmidt method, we find it easier, and more elegant, to
generate the polynomials $G_n^d,$ starting from the initial table,
Eqs. (\ref{G1},\ref{G2},\ref{G3}), by means of the following recursion formula,
\begin{equation}
G_n^d(r) = (r-d)\, G_{n-1}^d(r)-2\, r\, G_{n-1}^{d\, \prime}(r)+(n+d-1)\, (n-2)
\, G_{n-2}^d(r),
\label{recurG}
\end{equation}
where the prime denotes the derivative with respect to $r.$ Its simple
structure can be proven analytically as follows:
i) Let us first create some kind of a ``less trivial seed'' at order $n,$
assuming the polynomial $G_{n-1}^d$ is known. For this, try $r\, G_{n-1}^d.$
By partial integration, we see that,
\begin{equation}
\int_0^\infty dr\ r^{d-1}\, e^{-\frac{1}{2}r} \left[r\, G_{n-1}^d(r)\right] =
2 \int_0^\infty dr\ e^{-\frac{1}{2}r} \left[r^d\, G_{n-1}^d(r)\right]' \, ,
\label{seedG}
\end{equation}
where again a prime means derivation with respect to $r.$ Thus
$\sigma_n^d \equiv \left(r\, G_{n-1}^d - 2\, r\, G_{n-1}^{d\, \prime} - 2\,
d\, G_{n-1}^d \right)$ makes indeed a less trivial seed, compatible with
the constraint. Notice that the order $n$ of this seed polynomial $\sigma_n^d$
comes from the term $r\, G_{n-1}^d$ only, the other two terms having
order $n-1.$ Notice again that, in the table, Eqs. (2), all polynomials
$G_n^d$ are monic. We can define $G_n^d$ as monic, systematically. Since
the product $r\, G_{n-1}^d$ respect this ``monicity'', and since $\sigma_n^d$
fulfills the constraint, we conclude that $\sigma_n^d$ is a linear
combination of $G_n^d,$ with coefficient 1, and of all the lower order
polynomials $G_m^d,$ with $1\le m<n,$ but with yet unknown coefficients.
ii) It turns out that such coefficients vanish if $m < n-2.$ Indeed, an
integration of $\sigma_n^d$ against $G_m^d,$ weighted by $r^{d-1} e^{-r},$
gives, by partial integration of the $G_{n-1}^{d\, \prime}$ term,
\begin{eqnarray}
&\int_0^{\infty} dr\ e^{-r}\, r^{d-1}\, \sigma_n^d(r)\, G_m^d(r) \equiv
\int_0^{\infty} dr\ r^{d-1} e^{-r}
\left[ (r-2d)\, G_{n-1}^d(r) - 2r\, G_{n-1}^{d\, \prime}(r) \right]
\, G_m^d(r) = \nonumber \\
&\int_0^{\infty} dr\ e^{-r}\, r^{d-1}\, G_{n-1}^d(r)\, (r-2d)\, G_m^d(r)
+ 2\int_0^{\infty} dr\ G_{n-1}^d(r)\, \left[ e^{-r}\, r^d\, G_m^d(r)\right]' =
\nonumber \\
&\int_0^{\infty} dr\ e^{-r}\, r^{d-1}\, G_{n-1}^d(r)
\left[-\sigma_{m+1}^d(r)-2d\, G_m^d(r)\right] .
\label{simplifG}
\end{eqnarray}
In the bracket $[\ ]$ in the last right-hand side of Eq. (\ref{simplifG}) the
seed $\sigma_{m+1}^d$ has order $m+1$ and, by definition, $G_m^d$ is of order
$m.$ By definition also, $G_{n-1}^d,$ of order $n-1,$ is orthogonal to all
those polynomials of lower order, that are compatible with the constraint.
This integral, Eq. (\ref{simplifG}), thus vanishes as long as $m+1 < n-1.$ It
can be concluded that the difference, $\sigma_n^d-G_n^d,$ contains only two
contributions, namely those from $G_{n-2}^d$ and $G_{n-1}^d.$ Explicit forms
for their coefficients are obtained by elementary manipulations, leading to
Eq. (\ref{recurG}).
Elementary manipulations also give,
\begin{equation}
2\, r\, G_n^{d\, \prime \prime} - (r-2d)\, G_n^{d\, \prime} + n\, G_n^d =
(n-1)\, (n+d)\, G_{n-1}^d\, .
\end{equation}
Here, in the same way as a prime means first derivative with respect to $r,$
we used double primes for second derivatives.
Finally the normalization of the polynomials is obtained easily as,
\begin{equation}
g_n^d \equiv \int_0^{\infty} dr\ e^{-r}\, r^{d-1}\, [G_n^d(r)]^2 =
(n-1)!\, (n+d)!\, .
\end{equation}
\section{Modification of Legendre polynomials by a constraint of zero average}
Legendre polynomials, and their associates and generalizations
(Gegenbauer, Chebyshev, Jacobi) are defined with respect to the $[-1,1]$
segment. Exceptionally in the literature, one finds shifted Legendre
polynomials, adjusted to the $[0,1]$ segment. We are here interested in
applications to radial densities in cylinders, or the small circles of
toruses, or spheres. Hence we shall use $0 \le r \le 1.$ It is clear that the
case, $d=1,$ does not make an original problem, since Legendre polynomials,
whether translated and/or scaled or not, already average to $0$ as soon as
their order $n$ is $\ge 1.$ We keep the case, $d=1,$ for the sake only of
completeness and in this Section we consider $d=1,2,3,$ with a geometry
factor $r^{d-1}.$ The weight is $\mu(r)=1,$ hence our states are described
by just a polynomial ${\cal G}_n^d$ of order $n.$ It is again obvious that
${\cal G}_0^d$ cannot be a non vanishing constant if the constraint,
$\int_0^1 dr\, r^{d-1}\, {\cal G}_0^d(r)=0,$ is implemented. Hence set
$m \ge 1,$ $n \ge 1,$ and define polynomials ${\cal G}_n^d$ from conditions,
\begin{equation}
\int_0^1 dr\ r^{d-1}\, {\cal G}_m^d(r)\, {\cal G}_n^d(r) = \gamma_n^d\,
\delta_{mn}\, ,
\ \ \ \ \
\int_0^1 dr\ r^{d-1}\, {\cal G}_n^d(r) = 0
\, ,
\label{definl}
\end{equation}
where the normalizations $\gamma_n^d$ are again to be defined later.
It is obvious that the shifted (and shrunk) Legendre polynomials
${\cal L}_n(2r-1),\ n \ge 1,$ satisfy both constraint and orthogonality
relations for $d=1,$ because they are orthogonal to any constant polynomial,
of order $0.$ The polynomials ${\cal G}_n^1={\cal L}_n(2r-1)$ thus make
nothing new. We turn therefore to $d=2$ and $d=3,$ with a brute force
construction as in the previous Section. But the defining conditions,
Eqs. (\ref{definl}), show a difference with Eqs. (\ref{definL}): both
orthogonality and constraint conditions now carry the same weight,
namely $\mu^2=\mu,$ while in the previous case, Eqs. (\ref{definL}), there
were different weights, because of the exponentials $e^{-r}$ and
$e^{-\frac{1}{2}r}.$ A similar difference between $\mu^2$ and $\mu$ happened
in the ``Hermite'' case, naturally.
Hence now, in this Legendre case, we can Gram-Schmidt orthogonalize
even more trivial seeds $r^n,$ without subtractions, and accept those
orthogonal polynomials with order $m \ge 1.$ The table of first results reads,
\begin{mathletters} \begin{eqnarray}
{\cal G}_1^1 = 2r-1,\ \ \ \ \, {\cal G}_2^1 = 6r^2-6r+1,\ \ \ \
{\cal G}_3^1 = (2r-1) (10 r^2-10r+1),\ \ \ \
{\cal G}_4^1 = 70r^4-140r^3+90r^2-20r+1, \label{Gl1}
\\
{\cal G}_1^2 = 3r-2,\ \, {\cal G}_2^2 = 10r^2-12r+3,\ \ \ \ \,
{\cal G}_3^2 = 35r^3-60r^2+30r-4,\ \ \ \ \,
{\cal G}_4^2 = 126r^4-280r^3+210r^2-60r+5, \label{Gl2}
\\
{\cal G}_1^3 = 4r-3,\ {\cal G}_2^3 = 15r^2-20r+6,\ \
{\cal G}_3^3 = 56r^3-105r^2+60r-10,\ \
{\cal G}_4^3 = 210r^4-504r^3+420r^2-140r+15. \label{Gl3}
\end{eqnarray} \end{mathletters}
Easy, but slightly tedious manipulations validate the following recursion
relations,
\begin{mathletters} \begin{eqnarray}
n\, {\cal G}_n^1 &=&
(2n-1)\, (2r-1)\, {\cal G}_{n-1}^1 - (n-1)\, {\cal G}_{n-2}^1\, , \\
(n+1)\, (2n-1)\, {\cal G}_n^2 &=&
2\, [\,(4n^2-1)r-2n^2\,]\, {\cal G}_{n-1}^2 - (n-1)\, (2n+1)\,
{\cal G}_{n-2}^2\, , \\
n^2\, (n+2)\, {\cal G}_n^3 &=&
(2n+1)\, [\, 2n(n+1)r - (n^2+n+1)\, ]\, {\cal G}_{n-1}^3 - (n-1)\, (n+1)^2\,
\, {\cal G}_{n-2}^3\, .
\end{eqnarray} \end{mathletters}
and the differential equation,
\begin{equation}
r\, (r-1)\, {\cal G}_n^{d\, \prime \prime} + [\, (d+1)\, r-d\,]\,
{\cal G}_n^{d\, \prime} - n\, (n+d)\, {\cal G}_n^d = 0.
\end{equation}
Finally the normalization of the polynomials reads,
\begin{equation}
\gamma_n^d \equiv \int_0^{\infty} dr\ r^{d-1}\, [{\cal G}_n^d(r)]^2 =
1/(2n+d)\, .
\end{equation}
We show in Figure 2 the plots of ${\cal G}_n^d$ for $n=1,2$ and $d=1,2,3.$
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol2.eps}
}
\caption{Modified Legendre polynomials ${\cal G}_1^1,$ ${\cal G}_2^1$
(full lines), ${\cal G}_1^2,$ ${\cal G}_2^2$ (long dashes), ${\cal G}_1^3,$
${\cal G}_2^3$ (dashes).}
\end{figure}
\section{Polynomials for centrifuges}
The case of centrifuges is worth a short comment. As soon as the matter under
centrifugation is compressible, the density becomes much larger at the outer
edge, $r=1,$ than at the rotation axis, $r=0.$ Let $h$ be the height
of the centrifuge. Assume, for the sake for the argument, that one studies
fluctuations about a reference density of the form,
$\rho(r)=\rho_c\, e^{Kr^2},$ where the parameter $K$ contains
all informations about the angular velocity, compressibility, etc. of the
process. The factor,
$\rho_c = M\, \left[h\, \int_0^1 dr\, r\, \rho(r)\right]^{-1} =
M\, h^{-1}\, 2\, K\, \left[e^K-1\right]^{-1},$
ensures the conservation of the mass $M$ included in the vessel. If a cause
for fluctuations of $\rho$ is an instability of $K,$ the first order for
density change is,
\begin{equation}
\frac{\partial \rho}{\partial K}(r)=2\, \frac{K\, r^2\,
\left[e^K-1\right]+ e^K - K e^K -1}{\left[e^K-1\right]^2}\, e^{Kr^2}\, , \ \ \
\ \ \ \int_0^1 dr\, r\, \frac{\partial \rho}{\partial K}(r)=0,
\end{equation}
namely a polynomial of order 2 multiplied by $e^{Kr^2}.$ Higher derivatives
with respect to $K$ will generate similar, even order polynomials, with the
same property, $\int_0^1 dr\, r\, \partial^n \rho/ \partial K^n(r)=0.$ An
orthogonalization, under a metric $\propto e^{2Kr^2},$ might be useful.
This new set of polynomials will depend on $K,$ however, since $r$ is already
scaled to a radius $1$ for the cylinder and thus $K$ cannot be scaled away.
Because of this $K$ dependence we do not elaborate further on this issue.
For a large list of {\it ad hoc} polynomials and integration weights, see
\cite{JPB}.
\section{Projector on the constrained subspace}
For the sake of the discussion and short notations, set first $d=1,$
$\mu(r)=e^{-\frac{1}{2}r},$ and temporarily include normalization to unity
factors into both the Laguerre polynomials $L_n$ and the constrained
$G_n^1.$ This summarizes as,
\begin{equation}
\int_0^{\infty} dr\ [\mu(r)]^2\, L_m(r)\, L_n(r)=\delta_{mn},\ \ \ \ \
\int_0^{\infty} dr\ [\mu(r)]^2\, G_m^1(r)\, G_n^1(r)=\delta_{mn},
\ \ \ \ \
\int_0^{\infty} dr\ \mu(r)\, G_n^1(r)=0,
\label{scheme}
\end{equation}
Then the kets and bras defined by
$\langle r | w_n \rangle = \langle w_n | r \rangle = w_n(r) =
\mu(r)\, G_n^1(r)$ and
$\langle r | z_n \rangle = \langle z_n | r \rangle = z_n(r) =
\mu(r)\, L_n(r)$ provide two ``truncation'' projectors,
${\cal P}_N = \sum_{n=1}^N | w_n \rangle \langle w_n | $ and
${\cal Q}_N = \sum_{n=0}^N | z_n \rangle \langle z_n | ,$ available
for subspaces where polynomial orders do not exceed $N.$ Their respective
ranks $N$ and $N+1,$ and the embedding and commutation relation,
$\left[{\cal P}_N,{\cal Q}_N\right]={\cal P}_N,$ are obvious. Obvious also is
the limit, $\lim_{N \rightarrow \infty} {\cal Q}_N =1.$ The role of the rank
one $| \sigma_N \rangle \langle \sigma_N | $ difference
${\cal P}_N - {\cal Q}_N$ is to subtract from any test state,
$| \tau \rangle = \sum_{n=0}^N \tau_n | z_n \rangle,$
that part which violates the condition of vanishing average. We shall show
that the elementary ansatz,
\begin{equation}
| \sigma_N \rangle = \left( \sum_{m=0}^N\, \langle z_m \rangle^2
\right)^{-\frac{1}{2}}\,
\sum_{n=0}^N\, \langle z_n \rangle\ | z_n \rangle,\ \ \ \ \ \
\langle z_n \rangle = \int_0^{\infty} dr\, \langle r | z_n \rangle,
\label{ansatz}
\end{equation}
defines the proper ``subtractor'' operator
$| \sigma_N \rangle \langle \sigma_N |.$ Indeed, from
\begin{equation}
\left(\, {\cal Q}_N - | \sigma_N \rangle \langle \sigma_N |\, \right)\,
| \tau \rangle = \sum_{n=0}^N \tau_n\, | z_n \rangle -
\left( \sum_{m=0}^N \langle z_m \rangle^2 \right)^{-1}\,
\left(\, \sum_{n=0}^N \langle z_n \rangle\, | z_n \rangle\, \right)\,
\left( \sum_{p=0}^N \langle z_p \rangle\, \tau_p \right),
\end{equation}
one obtains
\begin{equation}
\int_0^{\infty} dr\, \langle r |\left( {\cal Q}_N -
| \sigma_N \rangle \langle \sigma_N | \right)\, | \tau \rangle =
\sum_{n=0}^N \tau_n\, \langle z_n \rangle -
\left( \sum_{m=0}^N \langle z_m \rangle^2 \right)^{-1}\,
\left(\, \sum_{n=0}^N \langle z_n \rangle\, \langle z_n \rangle\, \right)\,
\left( \sum_{p=0}^N \langle z_p \rangle\, \tau_p \right)=0.
\end{equation}
Hence ${\cal Q}_N - | \sigma_N \rangle \langle \sigma_N |$ is the projector
${\cal P}_N.$ Incidentally, the Laguerre result for $\sigma_N$ is very simple,
because $\langle z_m \rangle = 2,\ \forall m.$ But the ansatz for $\sigma_N,$
Eq.(\ref{ansatz}), generalizes to all cases. For instance, with Hermite
polynomials, odd orders already satisfy the constraint when integrated
from $-\infty$ to $\infty,$ naturally, and thus do not
contribute to $\sigma_N.$ Even orders contribute, and it is easy to verify,
upon integrating from $-\infty$ to $\infty$ again, that
$\langle z_{2p} \rangle^2 = \pi^{\frac{1}{2}}\, 2^{1-p}\, (2p-1)!!/p!$.
It may be pointed out that the condition, $\int dr\, \mu(r)\, f(r)=0,$ for
functions $f$ orthogonalized, like our polynomials, by a metric
$[\mu(r)]^2,$ might be interpreted as an orthogonality condition,
$\int dr\, f(r)\, [\mu(r)]^2\, g(r)=0,$ with $g(r)=[\mu(r)]^{-1}.$ This makes
$g$ a candidate for the subtractor form factor $\sigma.$ This
is of some interest for the centrifuge case, where a state function such as,
for instance, $e^{-Kr^2},$ remains finite when $0\le r \le 1.$ But there
is little need to stress that, when the support of $\mu$ extends to $\infty,$
then $\mu^{-1}$ does not belong to the Hilbert space and cannot be used for
$\sigma.$
More interesting is the limiting process, $N \rightarrow \infty,$ as
illustrated by Figures 2-5. Figs. 2 and 3 show the shapes, in terms of $r,$
of $\langle 2 | {\cal P}_N | r \rangle$ and
$\langle 10 | {\cal P}_N | r \rangle,$ respectively, when the projectors
are made of the modified Laguerre polynomials $G_n^1.$ The build up of an
approximate $\delta$-function when $N$ increases from $N=50$ (short dashes)
to $N=100$ (long ones) and $N=150$ (full lines) is transparent, although the
convergence is faster when peaks are closer to the origin, compare Figs. 2
and 3. The slower convergence in Fig. 3 is due to the cut-off
imposed by exponential weights as long as $N$ is finite. Given $N,$ there is
a ``box effect'', the range of the box being of order $\sim N.$ A similar
build up is observed for our other families of constrained polynomials, with
slightly different details of minor importance such as, for instance, a box
range $\sim \sqrt N$ for the Hermite case.
The box effect is even more transparent in Figs. 4 and 5, which show the
shapes of subtractors
$\langle 10 | \sigma_N \rangle \langle \sigma_N | r \rangle$ and
$\langle 0 | \sigma_N \rangle \langle \sigma_N | r \rangle$ deduced
from constrained polynomials of the Laguerre (Fig. 4) and Hermite (Fig. 5)
type, respectively. (For graphical convenience, the polynomials
$\Gamma_n^1$ and $H_n$ used for the Hermite case, Fig. 5, are tuned to
a weight $e^{-r^2}$ rather than $e^{-\frac{1}{2}r^2},$ but this detail is
not critical.)
It seems safe to predict that, given an effective length $\Lambda(N)$ for the
box, the wiggles of the subtractor will smooth out when $N \rightarrow \infty$
and that only a background $\sim -1/\Lambda(N)$ will then remain.
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol3.eps}
}
\caption{Shapes of projectors made of polynomials $G_n^1.$ Full line,
$\langle 2|{\cal P}_{150}|r \rangle,$ long dashes,
$\langle 2|{\cal P}_{100}|r \rangle,$ short dashes
$\langle 2|{\cal P}_{50}|r \rangle.$}
\end{figure}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol4.eps}
}
\caption{Shapes of projectors made of polynomials $G_n^1.$ Full line,
$\langle 10|{\cal P}_{150}|r \rangle,$ long dashes,
$\langle 10|{\cal P}_{100}|r \rangle,$ short dashes
$\langle 10|{\cal P}_{50}|r \rangle.$}
\end{figure}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol5.eps}
}
\caption{Subtractors made of $G_n^1.$ Shapes centered at $r=10.$
Short dashes, $N=10,$ long dashes, $N=20,$ full line, $N=30.$}
\end{figure}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol6.eps}
}
\caption{Subtractors made of $\Gamma_n.$ Shapes centered at $r=0.$
Stronger wiggles, shorter cut-off, dashed line, $N=50.$ Weaker wiggles, larger
cut-off, full line, $N=100.$}
\end{figure}
\section{Illustrative example: trajectories in density space}
We return here to the toy model discussed in \cite{GiMeWe} and the
corresponding, modified Hermite polynomials. The model consists of Z
non interacting, spinless fermions, driven by a one dimensional harmonic
oscillator $H_0=\frac{1}{2}(-d^2/dr^2+r^2).$ The ground state density
from the $Z$ lowest orbitals reads,
$\rho(r)=\sum_{i=1}^Z [\psi_i(r)]^2.$ Let $i=1,..,Z$ and $I=Z+1,...\,\infty$
label ``hole'' and ``particle'' orbitals, respectively. Add a perturbation
$\delta u(r)$ to the initial potential $r^2/2.$ The first order variation of
the density is,
\begin{equation}
\delta \rho(r)= 2 \sum_{iI} \, \psi_i(r) \psi_I(r) \,
\frac{\langle I | \delta u | i \rangle}{i-I} \, .
\label{prtrbG}
\end{equation}
If we expand $\delta u$ and $\delta \rho$ in that basis $\{w_n\}$ provided by
the new polynomials, the formula, Eq. (\ref{prtrbG}), becomes,
\begin{equation}
\delta \rho_m= 2 \sum_{iI\, n} {\cal D}_{m\, iI} \,
\frac{ 1 } { i- I } \, {\cal D}_{n\, iI} \, \delta u_n,\ \ \ \ \
{\cal D}_{n\, iI} \equiv \int dr \, w_n(r) \, \psi_i(r) \psi_I(r),
\label{expansionG}
\end{equation}
where ${\cal D}$ denotes both a particle-hole matrix element of a potential
perturbation and the projection of a particle-hole product of
orbitals upon the basis $\{w_n\}.$ In \cite{GiMeWe} we briefly studied the
eigenvalues and eigenvectors of this symmetric matrix,
${\cal F}={\cal D}\, (E_0-H_0)^{-1}\, \tilde {\cal D},$
where $(E_0-H_0)^{-1}$ is a short notation to account for the denominators
and the particle-hole summation, and the tilde indicates transposition. It
is clear that the invertible ${\cal F}$ represents the functional derivative
$\delta \rho_m / \delta u_n$ and is suited for {\it infinitesimal}
perturbations. We shall now take advantage of the representation provided by
$\{ w_n \}$ to study {\it finite} trajectories $\rho(u).$
For this, we consider a variable Hamiltonian,
${\cal H}_m(\lambda)=H_0+\lambda\, w_m(r),$
made of the initial harmonic oscillator, but with a finite perturbation
$\Delta u$ along one ``mode'' $w_m.$ It is trivial to diagonalize
${\cal H}_m(\lambda)$ with an excellent numerical accuracy and thus obtain,
given $Z,$ the ground state density $\rho(r,\lambda).$ Then it is trivial
to expand the finite variation, $\Delta \rho=\rho(r,\lambda)-\rho(r,0),$
in the basis $\{w_n\}.$ This defines coordinates $\Delta \rho_n(\lambda;m)$
for trajectories, parametrized by the intensity of the chosen mode $m$ for
$\Delta u.$
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol7.eps}
}
\caption{Coordinates of the perturbation density $\Delta \rho$ created by a
perturbing potential $\Delta u=\lambda_4\, w_4.$ Full line:
$2\, \Delta \rho_2.$ Long dashes: $\Delta \rho_4.$ Moderate dashes:
$2\, \Delta \rho_6.$ Short dashes: $4\, \Delta \rho_8.$ Very short dashes:
$8\, \Delta \rho_{10}.$}
\end{figure}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol8.eps}
}
\caption{Same as Fig. 7, but now $\Delta u=\lambda_6\, w_6.$ Full line:
$4\, \Delta \rho_2.$ Long dashes: $2\, \Delta \rho_4.$ Moderate ones:
$\Delta \rho_6.$ Short ones: $2\, \Delta \rho_8.$ Very short
dashes: $4\, \Delta \rho_{10}.$}
\end{figure}
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol9.eps}
}
\caption{3D trajectory in density space. $\Delta \rho_4,$ $\Delta \rho_6$ and
$\Delta \rho_8$ taken from Fig. 7, the latter two coordinates blown $4$
times.}
\end{figure}
In Figures 7 and 8 we show, with $Z=4,$ results from
${\cal H}_4=H_0 + \lambda_4\, 2\, (2\pi)^{-\frac{1}{4}}\, 15^{-\frac{1}{2}}\,
(8r^4-14r^2+1)\, e^{-r^2}$ and
${\cal H}_6=H_0 + \lambda_6\, (2\pi)^{-\frac{1}{4}}\, 105^{-\frac{1}{2}}\,
(32r^6-128r^4+94r^2-11)\, e^{-r^2},$ respectively.
The case,
${\cal H}_2=H_0 + \lambda_2\, 2\, (2/\pi)^{\frac{1}{4}}\, 3^{-\frac{1}{2}}\,
(2r^2-1)\, e^{-r^2},$
makes almost a harmonic oscillator and is probably of academic interest only;
anyhow we verified that its confirms the results with ${\cal H}_4$ and
${\cal H}_6.$ We use a basis $\{ w_n \}$ containing a factor $e^{-r^2}$ rather
than $e^{-\frac{1}{2}r^2}$ to better match the same factor $e^{-r^2}$ created
by products of harmonic oscillator orbitals in the calculation of matrix
elements $\langle z_p | \Delta u | z_q \rangle,$ but this technicality is not
important for the physics.
The main result to be observed seems to be the lack of ``collectivity'' for
such modes and for such elementary Hamiltonians. Indeed, for $\lambda_4=2,$
the first five coordinates of $\Delta \rho$ read
$\{0.016, -0.267, -0.055, 0.023, 0.018\},$ with a strong dominance of
$\Delta \rho_4,$ while for $\lambda_6=2,$ these read
$\{-0.013, -0.041, -0.376, 0.008, 0.040\},$ with a strong dominance of
$\Delta \rho_6.$ To clarify Figs. 7 and 8, we had indeed to blow up
each $\Delta \rho_n$ by a factor $2^{|n-m|},$ where $m$ is the index of the
driver mode in potential space. Other modes than $m=4$ and $m=6$ show the
same property: in the density space, a trajectory driven by
$\Delta u=\lambda w_m$ stays close to the same $w_m$ axis in that density
space, although curvatures effects, while somewhat modest, are not absent.
Such non linearity, slight curvatures are seen in Figs. 7-8, and also in
Figure 9, where the three $\Delta \rho_4, \Delta \rho_6, \Delta \rho_8$
sets of data shown by Fig. 7 are converted into a parametric plot for a
trajectory. For graphical purposes again, $ \Delta \rho_6$ and $\Delta \rho_8$
are blown up $4$ times to create Fig. 9. It can be concluded, temporarily,
that the ``flexibility'' matrix ${\cal F}$ is not too far from being
diagonal in the $\{ w_n \}$ basis, or in other words, that the $w_n$ modes
indicate an approximately natural hierarchy in both the potential and the
density spaces.
A subsidiary question pops up: that of the positivity of $\rho.$ Indeed,
while the space of potentials is basically a linear space, with arbitrary
signs for $u(r)$ when the position $r$ changes, densities $\rho(r)$ must
remain positive for every $r.$ This creates severe constraints for any
linear parametrization of $\Delta \rho$ in terms of the basis $\{ w_n \}.$
In our toy model, it turns out that
$\rho(r,0)=\pi^{-\frac{1}{2}} (8r^6-12r^4+18r+9)\, e^{-r^2}\, /6.$ Hence, if
we truncate $\Delta \rho$ to have two components only, $w_2$ and $w_4$ for
instance, then $\rho$ is the product of $e^{-r^2}$ and a polynomial
${\cal P}(r),$
\begin{equation}
6\, \pi^{\frac{1}{2}}\, {\cal P}(r) = 8r^6-12r^4+18r^2+9 +
\Delta \rho_2\, 12\, (2 \pi)^{\frac{1}{4}}\, 3^{-\frac{1}{2}}\, (2r^2-1) +
\Delta \rho_4\, 12\, (\pi/2)^{\frac{1}{4}}\, 15^{-\frac{1}{2}}\,
(8r^4-14r^2+1).
\end{equation}
Rescale out inessential factors, for a simpler polynomial,
$\bar {\cal P}=8r^6-12r^4+18r^2+9 + \Delta R_2 (2r^2-1) + \Delta R_4
(8r^4-14r^2+1).$
Eliminate $r$ between $\bar {\cal P}$ and $d \bar {\cal P}/dr.$
The resultant ${\cal R}(\Delta R_2,\Delta R_4),$ when it vanishes,
gives the border of the convex domain of parameters $\Delta R_2, \Delta R_4$
where $\bar {\cal P}$ remains positive definite. This domain contains the
origin, because of $\rho(r,0).$ The precise form of ${\cal R}$ is
a little cumbersome and does not need to be published here. But the resulting
border is shown in Figure 10. Generalizations to more $\Delta \rho$
parameters are obvious, with more cumbersome resultants ${\cal R}.$
\begin{figure}[htb] \centering
\mbox{ \epsfysize=100mm
\epsffile{fignwpol10.eps}
}
\caption{Domain of values of $\Delta R_2$ and $\Delta R_4$ acceptable
for the positivity of the density of the toy model. The domain sits
inside the full line curve and left of the straight line. It contains the
origin.}
\end{figure}
\section{Discussion and Conclusion}
The subject of orthogonal polynomials has been so treated and overtreated that
any claim to novelty must contain much more than a change of the integration
measure. We took therefore a different approach, motivated by a law of physics
and/or chemistry, matter conservation. This means a constraint of a vanishing
average for the states described by weighted polynomials.
For a support $[0,\infty[$ and a simple exponential weight such as
$e^{-\frac{1}{2}r},$ a non trivial generalization of Laguerre polynomials
occurs. This extends the generalization of Hermite polynomials described
in \cite{GiMeWe} with the support $]-\infty,\infty[$ and Gaussian weights
such as $e^{-\frac{1}{2}r^2}.$
We also took care of cylindrical and spherical geometries, by replacing
$\int dr$ with $\int dr\, r$ and $\int dr\, r^2,$ respectively. The new sets
of constrained polynomials are clearly sensitive to the geometry.
For finite supports such as $[0,1]$ and constant weights, the constraint is
already satisfied by the usual brand of orthogonal polynomials as soon as
their order is $\ge 1.$ In that sense, we did not find significantly original
generalizations of Legendre polynomials, although we generated polynomials
fitted to the cylindrical and spherical geometries. The cause of the failure
is transparent: when the weight $\mu(r)$ is a constant, there is no difference
between the orthogonality metric $\mu^2$ and the constraint weight $\mu.$
For each set of new polynomials we found a recursion relation and a
differential equation. There seems to be a systematic property for those
cases where the constraint generates truly original polynomials, namely
when $\mu^2 \ne \mu.$ In such cases, recursion and differentiation seem to be
necessarily entangled. This does not happen for traditional orthogonal
polynomials, indeed, and this ``entanglement'' may deserve some future
attention.
Constrained polynomials expressing matter conservation in centrifuges do make
an original set if the fluid under centrifugation is compressible; a non
constant reference weight $\mu$ is indeed in order there. But the set depends
on the precise form of $\mu$ via potentially many physical parameters. We
found it difficult to design, through scaling, a sufficiently ``universal''
set. ``Centrifuge polynomials'' will have to be calculated specifically
for each practical situation.
For those new polynomials generalizing the Hermite and Laguerre ones, we
found a description of the subspace accounting for their defect of
completeness. A codimension 1 is the consequence of the constraint,
expressed at first by the obvious lack of a polynomial of order $n=0.$
Finally the use of such polynomials was illustrated by a toy model for
the Hohenberg-Kohn functional. A slightly surprising result was found: our
polynomials, those of low order at least, define potential perturbations
which are reflected by density perturbations having almost the same shapes.
This occurs despite the delocalization created by the kinetic energy operator,
hints at short ranges in effective interactions and validates the
localization spirit of the Thomas-Fermi method. Whether such hints are
good when the full zoology of the density functional is investigated
is, obviously, an open question; for a review of the richness of the
functional, we refer to\cite{PK}. If long range forces are active, a
significant amount of delocalization between the ``potential cause'' and the
``density effect'' is not excluded. It would be interesting indeed to discover
collective degrees of freedom in this connection between potential and density.
In any case, our main conclusion may be that the new polynomials provide, for
the context of matter conservation, a discrete and full set of modes and
coordinates, hence a systematic and constructive representation of phenomena.
\bigskip
It is a pleasure to thank Y. Abe, J.-P. Boujot, B. Eynard, C. Normand,
R. Peschanski and A. Weiguny for stimulating discussions.
|
{
"timestamp": "2005-03-25T15:10:30",
"yymm": "0503",
"arxiv_id": "math-ph/0503060",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503060"
}
|
\section{Introduction}\label{s:1}
Nonlocality, {\em i.e.} the existence of correlations which cannot
be explained by any local hidden variable model, is perhaps
the most debated implication of quantum mechanics. During the last
decade other aspects of nonlocality, in addition to
generating nonlocal correlations, have been discovered. For example, the
possibility of teleporting and effectively encoding information,
as well as the ability to perform certain computations exponentially
faster than any classical device.
\par
Realistic implementations of quantum information protocols
require the investigation of nonlocality properties of quantum
states in a noisy environment. In particular, the robustness of
nonlocality should be addressed, as well as the design of protocols
to preserve and possibly enhance nonlocality in the presence of noise.
\par
The evolution of nonlocality for a twin-beam state of radiation (TWB) in a
thermal environment was studied in Ref.~\cite{jeong:noise} by means of the
displaced parity test \cite{bana}, whereas in Ref.~\cite{filip:PRA:66} its
nonlocality was investigated using the pseudospin operators
\cite{chen:PRL:88} when only dissipation occurs.
\par
In Ref.~\cite{ips:PRA:67} we have suggested a conditional measurement
scheme on TWB leading to a non-Gaussian entangled mixed state, which
improves fidelity of teleportation of coherent states. This process,
termed inconclusive photon subtraction (IPS), is based on mixing each mode
of the TWB with the vacuum in an unbalanced beam splitter and then
performing inconclusive photodetection on both modes, {\em i.e.} revealing
the reflected beams without discriminating the number of the detected
photons. IPS states have the following properties: they improve the
teleportation fidelity for coherent states \cite{ips:PRA:67} and show
enhanced nonlocal correlations in the phase space \cite{ips:PRA:70} in
ideal conditions, namely in the absence of noise. Motivated by these
results and by the recent experimental generation of IPS states
\cite{wenger:PRL:04}, in this paper we extend the previous studies on the
TWB and consider the nonlocality of the IPS state in the presence of noise.
\par
The paper is structured as follows. In Sec.~\ref{s:lossy} we address the
evolution of the TWB in a noisy channel where both dissipation and thermal
noise are present, whereas in Sec.~\ref{s:IPS} we briefly review the IPS
process. In Secs.~\ref{s:DP}, \ref{s:HD} and \ref{s:PS} we investigate the
nonlocality of TWB and IPS by means of three different tests: displaced
parity, homodyne detection, and pseudospin test, respectively. Finally,
Sec.~\ref{s:remarks} closes the paper with some concluding remarks.
\section{Dynamics of TWB in noisy channels} \label{s:lossy}
The so called twin-beam state of radiation (TWB), {\em i.e.}
$\dket{\Lambda} = \sqrt{1-\lambda^2}\sum_k \lambda^2 \ket{k}\otimes\ket{k}$
with $\lambda=\tanh r$, $r$ being the TWB squeezing parameter.
$\dket{\Lambda}$ is obtained by parametric down-conversion of the vacuum,
$\dket{\Lambda} = \exp\{ r(a^\dag b^\dag - ab) \}\ket{0}$, $a$ and $b$
being field operators, and it is described by the Gaussian Wigner function
\begin{equation}
W_{0}(\alpha,\beta) =
\frac{\exp\{
-2 \widetilde{A}_0 (|\alpha|^2+|\beta|^2)
+ 2 \widetilde{B}_0 (\alpha\beta + \alpha^*\beta^*) \}}
{4\pi^2\sqrt{{\rm Det}[\boldsymbol \sigma_0]}}\,,
\label{twb:wig}
\end{equation}
with
\begin{equation}
\widetilde{A}_0 = \frac{A_0}{16 \sqrt{{\rm Det}[\boldsymbol \sigma_0]}}\,,\qquad
\widetilde{B}_0 = \frac{B_0}{16 \sqrt{{\rm Det}[\boldsymbol \sigma_0]}}\,,
\end{equation}
where $A_0 \equiv A_0(r) = \cosh(2 r)$,
$B_0 \equiv B_0(r) = \sinh (2 r)$ and $\boldsymbol \sigma_0$ is the covariance matrix
\begin{equation}\label{cvm:twb}
\boldsymbol \sigma_0 = \frac14
\left(
\begin{array}{cc}
A_0\, \mathbbm{1}_2 & B_0\, \boldsymbol \sigma_3\\[1ex]
B_0\, \boldsymbol \sigma_3 & A_0\, \mathbbm{1}_2
\end{array}\right)\:,
\end{equation}
$\mathbbm{1}_2$ being the $2 \times 2$ identity matrix and $\boldsymbol \sigma_3 =
{\rm Diag}(1,-1)$. Using a more compact form, Eq.~(\ref{twb:wig}) can
also be rewritten as
\begin{equation}\label{gauss:form}
W_{0}(\boldsymbol X) =
\frac{\exp\left\{ -\frac12\, \boldsymbol X^{T}\,\boldsymbol \sigma_{0}^{-1}\,\boldsymbol X \right\}}
{4 \pi^2 \sqrt{{\rm Det}[\boldsymbol \sigma_0]}}\,,
\end{equation}
with $\boldsymbol X = (x_1,y_1,x_2,y_2)^{T}$, $\alpha=x_1+iy_1$ and
$\beta=x_2+iy_2$, and $(\cdots)^{T}$ denoting the transposition operation.
\par
When the two modes of the TWB interact with a noisy environment, namely in the
presence of dissipation and thermal noise, the evolution of the Wigner
function (\ref{twb:wig}) is described by the following Fokker-Planck equation
\cite{wm:quantopt:94,binary,seraf:PRA:69}
\begin{equation}\label{fp:eq:cmp}
\partial_t W_{t}(\boldsymbol X) = \frac12 \Big(
\partial_{\boldsymbol X}^T {\rm I}\!\Gamma \boldsymbol X + \partial_{\boldsymbol X}^T
{\rm I}\!\Gamma \boldsymbol \sigma_{\infty} \partial_{\boldsymbol X} \Big) W_{t}(\boldsymbol X)\,,
\end{equation}
with $\partial_{\boldsymbol X} =
(\partial_{x_1},\partial_{y_1},\partial_{x_2},\partial_{y_2})^{T}$.
The damping matrix is given by ${\rm I}\!\Gamma = \bigoplus_{k=1}^2\,
\Gamma_k \mathbbm{1}_2$, whereas
\begin{eqnarray}
\boldsymbol \sigma_{\infty} &= \bigoplus_{k=1}^{2}\, \boldsymbol \sigma_{\infty}^{(k)} =
\left(
\begin{array}{cc}
\boldsymbol \sigma_{\infty}^{(1)} & \boldsymbol{0} \\[1ex]
\boldsymbol{0} & \boldsymbol \sigma_{\infty}^{(2)}
\end{array}
\right)\,,
\end{eqnarray}
where $\boldsymbol{0}$ is the $2 \times 2$ null matrix and
\begin{equation}
\boldsymbol \sigma_{\infty}^{(k)} =
\frac14
\left(
\begin{array}{cc}
1 + 2 N_{k} & 0\\[1ex]
0 & 1 + 2 N_k
\end{array}
\right)\,.
\end{equation}
$\Gamma_k$, $N_k$ denotes the damping rate and the average number of
thermal photons of the channel $k$, respectively. $\boldsymbol \sigma_{\infty}$
represents the covariance matrix of the environment and, in turn, the
asymptotic covariance matrix of the evolved TWB. Since the environment is
itself excited in a Gaussian state, the evolution induced by
(\ref{fp:eq:cmp}) preserves the Gaussian form (\ref{gauss:form}). The
covariance matrix at time $t$ reads as follows
\cite{seraf:PRA:69,FOP:napoli:05}
\begin{equation}
\boldsymbol \sigma_t = \mathbbm{G}_t^{1/2}\,\boldsymbol \sigma_0\,\mathbbm{G}_t^{1/2}
+ (\mathbbm{1} - \mathbbm{G}_t)\,\boldsymbol \sigma_{\infty}\,,
\end{equation}
where $\mathbbm{G}_t = \bigoplus_{k=1}^2\,e^{-\Gamma_k t}\,\mathbbm{1}_2$.
The covariance matrix $\boldsymbol \sigma_t$ can be also written as
\begin{equation}\label{evol:cvm:12}
\boldsymbol \sigma_t = \frac 14
\left(
\begin{array}{cc}
A_t(\Gamma_1,N_1)\, \mathbbm{1}_2& B_t(\Gamma_1)\,\boldsymbol \sigma_3 \\[1ex]
B_t(\Gamma_2)\, \boldsymbol \sigma_3 & A_t(\Gamma_2,N_2)\, \mathbbm{1}_2
\end{array}
\right)
\end{equation}
with
\begin{equation}
\label{AtBt}
\eqalign{
&A_t(\Gamma_k,N_k) = A_0\,e^{-\Gamma_k t}
+ \left(1-e^{-\Gamma_k t}\right) (1 + 2 N_k)\,,\\
&B_t(\Gamma_k) = B_0\,e^{-\Gamma_k t}\,.
}
\end{equation}
\par
Let us now consider channels with the same damping rate $\Gamma$ but
different number of thermal photons, $N_1$ and $N_2$: using the density
matrix formalism, the state corresponding to the covariance matrix
(\ref{evol:cvm:12}) has the following form
\begin{equation}\label{rho:t:evol}
\varrho_t = S_2(\xi)\,\mu_1\otimes\mu_2\,S_2^{\dag}(\xi)\,,
\end{equation}
where $\mu_k$ is the thermal state
\begin{equation}
\mu_k = \frac{1}{1 + M_k}
\left( \frac{M_k}{1+M_k} \right)^{a^{\dag}_k a_k}
\end{equation}
$a_k$, $k=1,2$ being the mode operators. The average number of photons are
given by
\begin{eqnarray}
M_1 &=
\frac14 \left[ \sqrt{A_{+}^2 - 16 B_t} -
(2 - A_{-}) \right]\label{m1:evol}\,,\\
M_2 &=
\frac14 \left[ \sqrt{A_{+}^2 - 16 B_t} -
(2 + A_{-}) \right]\label{m2:evol}\,,
\end{eqnarray}
$A_{\pm} = A_{1,t} \pm A_{2,t}$, $A_{k,t}\equiv A_{t}(\Gamma,N_k)$ and
$B_t=B_t(\Gamma)$. In Eq.~(\ref{rho:t:evol}) $S_2(\xi)= \exp\{ \xi
a_1^{\dag}a_2^{\dag} - \xi^* a_1 a_2 \}$ denotes the two-mode squeezing
operator, with parameter $\xi \in \mathbb{C}$
\begin{eqnarray}
&|\xi| = \sinh^{-1}\left( \sqrt{
\frac{A_{+}}
{2(A_{+}^2 - 16 B_t)^{1/2}}-\frac12}\right)\label{x1:evol}\,,\\
&\arg[\xi] = \pi/2\label{arg:xi:evol}\,.
\end{eqnarray}
Eq.~(\ref{rho:t:evol}) says that the quantum state of a TWB, after
propagating in a noisy channel, is the same of a state obtained by
parametric down-conversion from a noisy background \cite{FOP:napoli:05}.
Their properties, and in particular entanglement and nonlocality, can be
addressed in an unified way using Eq.~(\ref{rho:t:evol}) or, equivalently,
Eqs.~(\ref{evol:cvm:12}) and (\ref{AtBt}).
\par
Finally, if we assume $\Gamma_1 = \Gamma_2 = \Gamma$ and $N_1 = N_2 = N$,
then the covariance matrix (\ref{evol:cvm:12}) becomes formally identical
to (\ref{cvm:twb}) and the corresponding Wigner function reads
\begin{equation}
W_{t}(\alpha,\beta) =
\frac{\exp\{
-2 \widetilde{A}_t (|\alpha|^2+|\beta|^2)
+ 2 \widetilde{B}_t (\alpha\beta + \alpha^*\beta^*)\}}
{4\pi^2\sqrt{{\rm Det}[\boldsymbol \sigma_t]}}\,,
\label{twb:wig:noise}
\end{equation}
with
\begin{equation}
\widetilde{A}_t = \frac{A_t(\Gamma,N)}{16\sqrt{{\rm Det}[\boldsymbol \sigma_t]}}\,,
\qquad
\widetilde{B}_t = \frac{B_t(\Gamma)}{16\sqrt{{\rm Det}[\boldsymbol \sigma_t]}}\,,
\end{equation}
whereas the density matrix, {\em mutatis mutandis}, is still given by
Eq.~(\ref{rho:t:evol}).
\section{De-Gaussification and noise}\label{s:IPS}
\begin{figure}
\begin{center}
\includegraphics[scale=.8]{ips_scheme.eps}
\end{center}
\vspace{-.3cm}
\caption{\label{f:IPS:scheme} Scheme of the IPS process.}
\end{figure}
When thermal noise and dissipation affect the propagation of an entangled
state, its nonlocal properties are reduced and, finally, destroyed
\cite{binary,seraf:PRA:69,rossi:JMO:04}. Therefore it is of interest to
look for some technique in order to preserve, at least in part, such
correlations, or to enhance the nonlocality of the state which will face
the lossy transmission line. Since it has been shown that the
de-Gaussification of a TWB can enhance its entanglement in the ideal case
and since non-Gaussian states can be produced using the current technology
\cite{wenger:PRL:04}, in this and the following Sections we will
investigate whether or not this process can be useful also in the presence
of noise.
\par
The de-Gaussification of a TWB can be achieved by subtracting photons from
both modes \cite{ips:PRA:67,opatr:PRA:61,coch:PRA:65}. In
Ref.~\cite{ips:PRA:67} we referred to this process as to inconclusive
photon subtraction (IPS) and showed that the resulting state, the IPS
state, can be used to enhance the teleportation fidelity of coherent states
for a wide range of the experimental parameters. Moreover, in
Ref.~\cite{ips:PRA:70}, we have shown that, in the absence of any noise
during the transmission stage, the IPS state has nonlocal correlations
larger than those of the TWB irrespective of the IPS quantum efficiency
(see also Refs.~\cite{nha:PRL:93,garcia:PRL:93}).
\par
First of all we briefly recall the IPS process, whose scheme is sketched in
Fig.~\ref{f:IPS:scheme}. The two modes, $a$ and $b$, of the TWB are mixed
with the vacuum (modes $c$ and $d$, respectively) at two unbalanced beam
splitters (BS) with equal transmissivity; the modes $c$ and $d$ are then
revealed by avalanche photodetectors (APDs) with equal efficiency, which
can only discriminate the presence of radiation from the vacuum: the IPS
state is obtained when the two detectors jointly click. The mixing with the
vacuum at a beam splitter with transmissivity $T$ followed by the on/off
detection with quantum efficiency $\eta$ is equivalent to mixing with an
effective transmissivity $\tau$ \cite{ips:PRA:67}
\begin{equation}
\tau \equiv \tau(T,\eta) = 1 - \eta (1-T)\,,
\end{equation}
followed by an ideal ({\em i.e.} efficiency equal to $1$) on/off detection.
Using the Wigner formalism, when the input state arriving at the two beam
splitters is the TWB $W_{0}(\alpha,\beta)$ of Eq.~(\ref{twb:wig}), the
state produced by the IPS process reads as follows (see
Ref.~\cite{ips:PRA:70} for the details about the calculation and about the
de-Gaussification map for the density matrix and Wigner function in the
case of a TWB)
\begin{equation}\label{ips:wigner}
W_{0}^{\rm (IPS)}(\alpha,\beta) =
\frac{1}{\pi^2\,p_{11}(r,\tau)}
\sum_{k=1}^4 {\cal C}_k(r,\tau)\,
W_{r,\tau}^{(k)}(\alpha,\beta)\,,
\end{equation}
where
\begin{equation}\label{ips:probability}
p_{11}(r,\tau) = \sum_{k=1}^4 \frac{{\cal C}_k(r,\tau)}{
(b-f_k)(b-g_k)-(2 \widetilde{B}_0 \tau + h_k)^2}\,
\end{equation}
is the probability of a click in both the APDs. In Eqs.~(\ref{ips:wigner})
and (\ref{ips:probability}) we introduced
\begin{equation}
\fl
W_{r,\tau}^{(k)}(\alpha,\beta) =
\exp\{ -(b-f_k) |\alpha|^2 -(b-g_k) |\beta|^2
+ (2 \widetilde{B}_0 \tau + h_k) (\alpha\beta + \alpha^*\beta^*)\}\,,
\end{equation}
and defined
\begin{equation}
{\cal C}_k(r,\tau)=
\frac{C_k}
{ \sqrt{{\rm Det}[\boldsymbol \sigma_0]}\,[x_k y_k - 4 \widetilde{B}_0^2 (1-\tau)^2]}\,,
\end{equation}
where $C_1 = 1$, $C_2 = C_3 = -2$, $C_4 = 4$; $x_k \equiv x_k(r,\tau)$, and
$y_k \equiv y_k(r,\tau)$ are
\begin{eqnarray}
&x_1 = x_3 = y_1 = y_1 = a \nonumber\\
&x_2 = x_4 = y_3 = y_4 = a +2 \nonumber
\end{eqnarray}
with $a \equiv a(r,\tau) = 2 [\widetilde{A}_0 (1-\tau) +
\tau]$,
$b \equiv b(r,\tau) = 2 [\widetilde{A}_0 \tau + (1-\tau)]$; finally, $f_k$, $g_k$,
and $h_k$ depend on $r$ and $\tau$ and are given by
\begin{eqnarray}
f_k & = {\cal N}_k
\, [x_k \widetilde{B}_0^2 + 4 \widetilde{B}_0^2 (1-\widetilde{A}_0) (1-\tau) + y_k (1-\widetilde{A}_0)^2]\,,\\
g_k &= {\cal N}_k
\, [x_k (1-\widetilde{A}_0)^2 + 4 \widetilde{B}_0^2 (1-\widetilde{A}_0) (1-\tau) + y_k \widetilde{B}_0^2]\,,\\
h_k &= {\cal N}_k
\, \{(x_k + y_k) \widetilde{B}_0 (1-\widetilde{A}_0) + 2 \widetilde{B}_0 [\widetilde{B}_0^2 + (1-\widetilde{A}_0)^2]
(1-\tau)\}\,,\\
{\cal N}_k &\equiv {\cal N}_k(r,\tau) = {\displaystyle
\frac{4 \tau\, (1-\tau)}{x_k y_k - 4 \widetilde{B}_0^2
(1-\tau)^2}\,.
}
\end{eqnarray}
The state corresponding to Eq.~(\ref{ips:wigner}) is no longer a Gaussian
state and its nonlocal properties, in ideal conditions, were studied in
Ref.~\cite{ips:PRA:70}.
\par
Here we are interested in the case when the IPS process is performed on a
TWB evolved in a noisy environment with both the channels having the same
damping rate and thermal noise. The Wigner function of the state arriving
at the beam splitters is now given by Eq.~(\ref{twb:wig:noise}), and the
output state is still described by Eq.~(\ref{ips:wigner}), but with the
following substitutions
\begin{equation}\label{sostituzioni}
\widetilde{A}_0 \to \widetilde{A}_t \,,\quad
\widetilde{B}_0 \to \widetilde{B}_t \,,\quad
\boldsymbol \sigma_0 \to \boldsymbol \sigma_t\,.
\end{equation}
We will denote with $W_{\Gamma,N}^{\rm (IPS)}(\alpha,\beta)$ the Wigner
function of this degraded IPS state.
\par
In the next Sections we will analyze the nonlocality of the IPS state in
the presence of noise by means of Bell's inequalities.
\section{Nonlocality in the phase space} \label{s:DP}
Parity is a dichotomic variable and thus can be used to establish
Bell-like inequalities \cite{CHSH}.
The displaced parity operator on two modes is defined as \cite{bana}
\begin{equation}
\hat{\Pi}(\alpha,\beta) =
D_a(\alpha)(-1)^{a^\dag a}D_a^\dag(\alpha)
\otimes D_b(\beta)(-1)^{b^\dag b}D_b^\dag(\beta)\,,
\end{equation}
where $\alpha, \beta \in {\mathbb C}$, $a$ and $b$ are mode operators and
$D_a(\alpha)=\exp\{\alpha a^\dag - \alpha^* a\}$ and $D_b(\beta)$ are
single-mode displacement operators. Since the two-mode Wigner function
$W(\alpha,\beta)$ can be expressed as \cite{FOP:napoli:05}
\begin{equation}
W(\alpha,\beta) = \frac{4}{\pi^2}\, \Pi(\alpha,\beta)\,,
\end{equation}
$\Pi(\alpha,\beta)$ being the expectation value of $\hat\Pi(\alpha,\beta)$,
the violation of these inequalities is also known as nonlocality in the
phase-space. The quantity involved in such inequalities can be written as
follows
\begin{equation}\label{bell:general}
{\cal B}_{\rm DP} = \Pi(\alpha_1,\beta_1)+ \Pi(\alpha_2,\beta_1)
+ \Pi(\alpha_1,\beta_2)-\Pi(\alpha_2,\beta_2)\,,
\end{equation}
which, for local theories, satisfies $|\mathcal{B}_{\rm DP}|\le 2$.
\par
Following Ref.~\cite{bana}, one can choose a particular set of
displaced parity operators, arriving at the following combination
\cite{ips:PRA:70}
\begin{equation}
\fl
{\cal B}_{\rm DP}({\cal J}) =
\Pi(\sqrt{\cal J},-\sqrt{\cal J})+ \Pi(-3\sqrt{\cal J},-\sqrt{\cal J})
+ \Pi(\sqrt{\cal J},3\sqrt{\cal J})-\Pi(-3\sqrt{J},3\sqrt{\cal J})\,,
\label{bell:ale}
\end{equation}
which, for the TWB, gives a maximum ${\cal B}_{\rm DP} = 2.32$,
greater than the value $2.19$ obtained in Ref.~\cite{bana}. Notice that,
even in the infinite squeezing limit, the violation is never maximal, {\em
i.e.} $|\mathcal{B}_{\rm DP}| < 2\sqrt{2}$ \cite{jeong1}.
\par
In Ref.~\cite{ips:PRA:70} we studied Eq.~(\ref{bell:ale}) for both the TWB
and the IPS state in an ideal scenario, namely in the absence of
dissipation and noise; we showed that, using IPS, the maximum violation
is achieved for $\tau \to 1$ and for values of $r$ smaller than for the
TWB.
\par
\begin{figure}
\vspace{-1.5cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,100)(0,0)
\put(10,0){\includegraphics[width=6cm]{DPTWBandIPS.eps}}
\put(42,42){$r$}
\put(-2,65){${\cal B}_{\rm DP}^{\rm (TWB)}$}
\put(42,0){$r$}
\put(-2,23){${\cal B}_{\rm DP}^{\rm (IPS)}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameters ${\cal B}_{\rm DP}$ for the TWB (top)
and IPS (bottom); we set ${\cal J}=1.6 \times 10^{-3}$ and $\tau = 0.9999$.
The dashed lines refer to the absence of noise ($\Gamma t = N = 0$),
whereas, for both the plot, the solid lines are ${\cal B}_{\rm DP}$ with
$\Gamma t = 0.01$ and, from top to bottom, $N=0, 0.05, 0.1,$ and $0.2$. In
the ideal case the maxima are ${\cal B}_{\rm DP}^{\rm (TWB)}=2.32$ and
${\cal B}_{\rm DP}^{\rm (IPS)}=2.43$, respectively.} \label{f:DP}
\end{figure}
Now, by means of the Eq.~(\ref{ips:wigner}) and the substitutions
(\ref{sostituzioni}), we can study how noise affects ${\cal B}_{\rm
DP}$. The results are showed in Fig.~\ref{f:DP}: as one may expect, the
overall effect of noise is to reduce the violation of the Bell's
inequality. When dissipation alone is present ($N=0$), the maximum of
violation is achieved using the IPS for values of $r$ smaller than for the
TWB, as in the ideal case. On the other hand, one can see that the presence
of thermal noise mainly affects the IPS results. In fact, for $\Gamma t =
0.01$ and $N=0.2$, one has $|{\cal B}_{\rm DP}^{\rm (TWB)}|>2$ for a range
of $r$ values, whereas $|{\cal B}_{\rm DP}^{\rm (IPS)}|$ falls below the
threshold for violation.
\par
We conclude that, considering the displaced parity test in the presence
of noise, the IPS is quite robust if the thermal noise is below a threshold
value (depending on the environmental parameters) and for small values of the
TWB parameter $r$.
\section{Nonlocality and homodyne detection} \label{s:HD}
In principle there are two approaches how to test the Bell's inequalities
for bipartite state: either one can employ some test for continuous variable
systems, such as that described in Sec.~\ref{s:DP}, or one can convert the
problem to Bell's inequalities tests on two qubits by mapping the
two modes into two-qubit systems. In this and the following Section we
will consider this latter case.
\par
The Wigner function $W_{0}^{\rm (IPS)}(\alpha,\beta)$ given in
Eq.~(\ref{ips:wigner}) is no longer positive-definite and thus
it can be used to test the violation of Bell's
inequalities by means of homodyne detection, {\em i.e.} measuring the
quadratures $x_{\vartheta}$ and $x_{\varphi}$ of the two IPS modes $a$ and
$b$, respectively, as proposed in Refs.~\cite{nha:PRL:93,garcia:PRL:93}.
In this case, one can dichotomize the measured quadratures assuming as
outcome $+1$ when $x \ge 0$, and $-1$ otherwise. The nonlocality of
$W_{0}^{\rm (IPS)}(\alpha,\beta)$ in ideal conditions has been studied in
Ref.~\cite{ips:PRA:70} where we also discussed the effect of the homodyne
detection efficiency $\eta_{\rm H}$.
\par
Let us now we focus our attention on
$W_{\Gamma,N}^{\rm (IPS)}(\alpha,\beta)$, namely the state produced
when the IPS process is applied to the TWB evolved through the noisy
channel. After the dichotomization of the
homodyne outputs, one obtains the following Bell parameter
\begin{equation}\label{bell:homo}
{\cal B}_{\rm HD} =
E(\vartheta_1,\varphi_1) + E(\vartheta_1,\varphi_2)
+ E(\vartheta_2,\varphi_1) - E(\vartheta_2,\varphi_2)\,,
\end{equation}
where $\vartheta_k$ and $\varphi_k$ are the phases of the two
homodyne measurements at the modes $a$ and $b$, respectively, and
\begin{equation}
E(\vartheta_h,\varphi_k) =
\int_{\mathbb{R}^2} d x_{\vartheta_h}\,d x_{\varphi_k}\,
{\rm sign}[x_{\vartheta_h}\, x_{\varphi_k}]\,
P(x_{\vartheta_h}, x_{\varphi_k})\,,
\end{equation}
$P(x_{\vartheta_h}, x_{\varphi_k})$ being the joint
probability of obtaining the two outcomes
$x_{\vartheta_h}$ and $x_{\varphi_k}$ \cite{garcia:PRL:93}. As usual,
violation of Bell's inequality is achieved when $|{\cal B}_{\rm HD}|>2$.
\par
\begin{figure}[tb]
\vspace{-1cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,100)(0,0)
\put(4,0){\includegraphics[width=6.5cm]{HDetaIPS.eps}}
\put(37,45){$r$}
\put(-2,70){${\cal B}_{\rm HD}$}
\put(37,-1){$r$}
\put(-2,24.5){${\cal B}_{\rm HD}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameter ${\cal B}_{\rm HD}$ for the IPS states
for two different values of the homodyne detection efficiency: $\eta_{\rm
H} = 1$ (top), and $\eta_{\rm H}=0.9$ (bottom). We set $\tau = 0.99$. The
dashed lines refer to the absence of noise ($\Gamma t = N = 0$), whereas,
for both the plots, the solid lines are ${\cal B}_{\rm HD}$ with $\Gamma t
= 0.05$ and, from top to bottom, $N=0, 0.05, 0.1$ and $0.2$.} \label{f:HD}
\end{figure}
In Fig.~\ref{f:HD} we plot ${\cal B}_{\rm HD}$ for $\vartheta_1 = 0$,
$\vartheta_2 = \pi/2$, $\varphi_1 = -\pi/4$ and $\varphi_2 = \pi/4$: as for
the ideal case \cite{ips:PRA:70,garcia:PRL:93}, the Bell's inequality is
violated for a suitable choice of the squeezing parameter $r$. Obviously,
the presence of noise reduces the violation, but we can see that the effect
of thermal noise is not so large as in the case of the displaced parity
test addressed in Sec.~\ref{s:DP} (see Fig.~\ref{f:DP}).
\par
Notice that the high efficiencies of this kind of detectors
allow a loophole-free test of hidden variable theories
\cite{gil}, though the violations obtained are quite small.
This is due to the intrinsic information loss of the binning
process, which is used to convert the continuous homodyne data in
dichotomic results \cite{mun1}.
\section{Nonlocality and pseudospin test} \label{s:PS}
Another way to map a two-mode continuous variable system into a two-qubit
system is by means of the pseudospin test: this consists in measuring
three single-mode Hermitian operator $S_k$ satisfying the Pauli matrix algebra
$[S_h,S_k]=2i\varepsilon_{hkl}\,S_l$, $S_k^2 = {\mathbb I}$, $h,k,l=1,2,3$,
and $\varepsilon_{hkl}$ is the totally antisymmetric tensor with
$\varepsilon_{123}=+1$ \cite{filip:PRA:66,chen:PRL:88}. For the sake of
clarity, we will refer to $S_1$, $S_2$ and $S_3$ as $S_x$, $S_y$ and $S_z$,
respectively. In this way one can write the following correlation function
\begin{equation}
E({\bf a},{\bf b}) = \langle ({\bf a}\cdot{\bf S})\,
({\bf b}\cdot{\bf S})\rangle\,,
\end{equation}
where ${\bf a}$ and ${\bf b}$ are unit vectors such that
\begin{equation}
\eqalign{
{\bf a}\cdot{\bf S} &= \cos \vartheta_a\, S_z +
\sin \vartheta_a\, (e^{i \varphi_a} S_{-} + e^{-i \varphi_a} S_{+})\,,\\
{\bf b}\cdot{\bf S} &= \cos \vartheta_b\, S_z +
\sin \vartheta_b\, (e^{i \varphi_b} S_{-} + e^{-i \varphi_b} S_{+})\,,
}
\end{equation}
with $S_{\pm} = \frac12 (S_x \pm S_y)$. In the following, without loss of
generality, we set $\varphi_k = 0$. Finally, the Bell parameter reads
\begin{equation}\label{bell:PS}
{\cal B}_{\rm PS} = E({\bf a}_1,{\bf b}_1)+E({\bf a}_1,{\bf b}_2)
+E({\bf a}_2,{\bf b}_1)-E({\bf a}_2,{\bf b}_2)\,,
\end{equation}
corresponding to the CHSH Bell's inequality $|{\cal B}_{\rm PS}|\le 2$. In
order to study Eq.~(\ref{bell:PS}) we should choose a specific
representation of the pseudospin operators; note that, as pointed out in
Refs.~\cite{revzen, ferraro:3:nonloc}, the violation of Bell inequalities
for continuous variable systems depends, besides on the orientational
parameters, on the chosen representation, since different $S_k$ leads to
different expectation values of ${\cal B}_{\rm PS}$. Here we consider the
pseudospin operators corresponding to the Wigner functions \cite{revzen}
\begin{eqnarray}
W_x(\alpha)&=\frac{1}{\pi}\,{\rm sign}\big[\Re{\rm e}[\alpha]\big]\,,\quad
W_z(\alpha)= -\frac{1}{2}\,\delta^{(2)}(\alpha)\,,\label{PS:W:xz}\\
&W_y(\alpha)=-\frac{1}{2\pi}\, \delta\big(\Re{\rm e}[\alpha] \big)\,
{\cal P} \frac{1}{\Im{\rm m}[\alpha]}\,,
\end{eqnarray}
where ${\cal P}$ denotes the Cauchy's principal value. Thanks to
(\ref{PS:W:xz}) one obtains
\begin{equation}
E_{\rm TWB}({\bf a},{\bf b}) = \cos\vartheta_a \cos\vartheta_b
+ \frac{2\sin\vartheta_a \sin\vartheta_b}{\pi}\,
\arctan\big[ \sinh(2r) \big]\,,
\end{equation}
for the TWB, and, for the IPS,
\begin{equation}
\fl
E_{\rm IPS}({\bf a},{\bf b}) =
\sum_{k=1}^4 \frac{{\cal C}_k(r,\tau)}{p_{11}(r,\tau)}
\Bigg[
\frac{\cos\vartheta_a \cos\vartheta_b}{4}
+ \frac{2 \sin\vartheta_a \sin\vartheta_b}{\pi{\cal A}_k}\,
\arctan\left( \frac{2 \widetilde{B}_0 \tau + h_k}{\sqrt{{\cal A}_k}} \right)
\Bigg]
\end{equation}
where $ {\cal A}_k=(b-f_k)(b-g_k)-(2 \widetilde{B}_0 \tau + h_k)^2$,
and all the other quantities have been defined in Sec.~\ref{s:IPS}.
\par
\begin{figure}[tb]
\vspace{-1cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,50)(0,0)
\put(4,0){\includegraphics[width=6.5cm]{PSidTWBandIPS.eps}}
\put(40,-3){$r$}
\put(-2,24.5){${\cal B}_{\rm PS}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameter ${\cal B}_{\rm PS}$ in ideal case
($\Gamma t = N = 0$): the dashed line refers to the TWB, whereas the solid
lines refer to the IPS with, from top to bottom, $\tau = 0.9999, 0.99,
0.9$, and $0.8$. There is a threshold value for $r$ below which IPS gives a
higher violation than TWB. Note that there is also a region of small
values of $r$ for which the IPS state violates the Bell's inequality while
the TWB does not. The dash dotted line is the maximal violation value
$2\sqrt{2}$.} \label{f:PS:id}
\end{figure}
In Fig.~\ref{f:PS:id} we plot ${\cal B}_{\rm PS}$ for the TWB and IPS in
the ideal case, namely in the absence of dissipation and thermal noise. For
all the Figures we set $\vartheta_{a_1}=0$, $\vartheta_{a_2}=\pi/2$, and
$\vartheta_{b_1}=-\vartheta_{b_2}=\pi/4$. As
usual the IPS leads to better results for small values of $r$. Whereas
${\cal B}_{\rm PS}^{\rm (TWB)} \to 2\sqrt{2}$ as $r\to \infty$,
${\cal B}_{\rm PS}^{\rm (IPS)}$ has a maximum and, then, falls below the
threshold $2$ as $r$ increases. It is interesting to note that there is a
region of small values of $r$ for which ${\cal B}_{\rm PS}^{\rm (TWB)}\le
2 < {\cal B}_{\rm PS}^{\rm (IPS)}$, {\em i.e.} the IPS process can increases
the nonlocal properties of a TWB which does not violates the Bell's
inequality for the pseudospin test, in such a way that the resulting state
violates it. This fact is also present in the case of the displaced parity
test described in Sec.~\ref{s:DP}, but using the pseudospin test the effect
is enhanced. Notice that the maximum violations for the IPS occur for a
range of values $r$ experimentally achievable.
\par
\begin{figure}[tb]
\vspace{-1cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,50)(0,0)
\put(4,0){\includegraphics[width=6.5cm]{PStauTWBandIPS.eps}}
\put(40,-3){$r$}
\put(-2,24.5){${\cal B}_{\rm PS}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameter ${\cal B}_{\rm PS}$ for $\Gamma t =
0.01$: the dashed line refers to the TWB, whereas the solid lines refer to
the IPS with, from top to bottom, $\tau = 0.9999, 0.99, 0.9$, and $0.8$.
The same comments as in Fig.~\ref{f:PS:id} still hold.} \label{f:PS:tau}
\end{figure}
In Fig.~\ref{f:PS:tau} we consider the presence of the dissipation alone
and vary $\tau$. We can see that IPS is effective also when the
effective transmissivity $\tau$ is not very high.
We take into account the effect of dissipation and thermal noise
in Figs.~\ref{f:PS:gamma}, and \ref{f:PS:th}: we can conclude that
IPS is quite robust with respect to this sources of noise and, moreover,
one can think of employing IPS as a useful resource in order to reduce the
effect of noise.
\begin{figure}[tb]
\vspace{-1cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,50)(0,0)
\put(4,0){\includegraphics[width=6.5cm]{PSgammaTWBandIPS.eps}}
\put(40,-3){$r$}
\put(-2,24.5){${\cal B}_{\rm PS}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameter ${\cal B}_{\rm PS}$ for different
values of $\Gamma t$ and in the absence of thermal noise ($N = 0$): the
dashed lines refer to the TWB, whereas the solid ones refer to the IPS with
$\tau = 0.9999$; for both the TWB and IPS we set, from top to bottom,
$\Gamma t = 0, 0.01, 0.05$, and $0.1$. The dash dotted line is the maximal
violation value $2\sqrt{2}$.} \label{f:PS:gamma}
\end{figure}
\begin{figure}[tb]
\vspace{-1cm}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,50)(0,0)
\put(4,0){\includegraphics[width=6.5cm]{PSthTWBandIPS.eps}}
\put(40,-3){$r$}
\put(-2,24.5){${\cal B}_{\rm PS}$}
\end{picture}
\end{center}
\caption{Plots of the Bell parameter ${\cal B}_{\rm PS}$ for $\Gamma t =
0.01$ and different values $N = 0$: the dashed lines refer to the TWB,
whereas the solid ones refer to the IPS with $\tau = 0.9999$; for both the
TWB and IPS we set, from top to bottom, $N = 0, 0.01, 0.1$, and $0.2$.}
\label{f:PS:th}
\end{figure}
\section{Concluding remarks} \label{s:remarks}
We have addressed three different nonlocality tests, namely, displaced
parity, homodyne detection and pseudospin test, on TWB and IPS in the
presence of noise. We have shown that the IPS process on TWB enhances
nonlocality not only in ideal cases, but also when noise (dissipation and
thermal noise) affects the propagation. As in the ideal situation, the
enhancement is achieved when the TWB energy is not too high (small
squeezing parameter $r$), depending on the environmental parameters.
Moreover, in the case of the pseudospin test, we have seen that there is a
region of small $r$ for which the TWB itself does not violates the Bell's
inequality, wheres after the IPS process it does.
\par
Finally, we mention that the enhanced nonlocality also in the presence of
noise makes the IPS states useful resources for continuous variable quantum
information processing.
\ack
Stimulating and useful discussions with M.~S.~Kim, A.~Ferraro and
A.~R.~Rossi are gratefully acknowledged.
\section*{References}
|
{
"timestamp": "2005-03-10T14:51:39",
"yymm": "0503",
"arxiv_id": "quant-ph/0503104",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503104"
}
|
\section{Introduction}
In Newton's conception, there was an absolute background space. While matter, forces, energy and the like were actors acting out in time. So also the law of gravitation was an action at a distance theory. Every material particle exerted the force of gravitation instantly on every other material particle.\\
In the next century, Coulomb discovered the law of electric, more precisely electrostatic interaction. It had the same form of an inverse square dependence on distance, as gravitation. This too was an action at a distance force. While the action at a distance gravitational law worked satisfactorily, in the nineteenth century the Coulomb law encountered difficulties when it was discovered by Ampere, Faraday and others that moving charges behaved differently. The stage was being set for Maxwell's electrodynamics. Maxwell could unify the experimental laws of Faraday, Ampere and others in a Field Theory \cite{jackson}. Already in the seventeenth century itself Olaf Romer had noticed that light travels with a finite speed and does not reach us instantly. He could conclude this by observing the eclipses of the satellites of Jupiter. Christian Huygens took the cue and described the motion of light in the form of waves. The analogy with ripplies moving outwards on the surface of a pool was clear.\\
Maxwell utilised these ideas in interpreting the experimentally observed laws of electricity and magnetism. Thus a moving charge would cause a ripple in an imaginary medium or field, and that ripple would be propagated further till it hit and acted upon another charge. This was a dramatic departure from the action at a distance concept because the effect of the movement of the charge would be felt at a later time by another charge. Maxwell even noticed that the speed at which these disturbances would propagate through the field was the same as that of light. Already the stage was being set for Einstein's Theory of Special Relativity \cite{einstein}. Even so, it must be mentioned that in the earlier formulation of the old action at a distance theory which resembled closely Maxwell's Field Theory, in mathematical form.\\
At this stage it was clear that two closely related concepts were important-- Locality and Causality. We will return to this shortly but broadly what is meant is that parts of the universe could be studied in isolation and further, that an event at a point $A$ cannot influence an event at a point $B$ which cannot be reached by a ray of light during this interval. Roughly speaking, all events within this light radius would be causally connected, but not so events beyond this radius.
\section{Action at a Distance Electrodynamics}
From a classical point of view a charge that is accelerating radiates energy which dampens its motion. This is given by the well known Maxwell-Lorentz equation, which in units $c = 1$, is \cite{hoyle}
\begin{equation}
m \frac{d^2x^\imath}{dx^2} = e F^{\imath k} \frac{dx^k}{dt} + \frac{4e}{3} g_{\imath k} \left(\frac{d^3x^\imath}{dx^3} \frac{dx^1}{dt} - \frac{d^3x^1}{dx^3} \frac{dx^\imath}{dt}\right) \frac{dx^k}{dt},\label{e1}
\end{equation}
The first term on the right is the usual external field while the second term is the damping field which is added ad hoc by the requirement of the energy loss due to radiation. In 1938 Dirac introduced instead of (\ref{e1}),
\begin{equation}
m \frac{d^2x^\imath}{dx^2} = e \left\{F^\imath_k + R^\imath_k\right\} \frac{dx^k}{dt}\label{e2}
\end{equation}
where
\begin{equation}
R^\imath_k \equiv \frac{1}{2} \left\{F^{\mbox{ret}\imath}_k - F^{\mbox{adv}\imath}_k\right\}\label{e3}
\end{equation}
In (\ref{e3}), $F^{\mbox{ret}}$ denotes the retarded field and $F^{\mbox{adv}}$ the advanced field. While the former is the causal field where the influence of a charge at $A$ is felt by a charge at $B$ at a distance $r$ after a time $t = \frac{r}{c}$, the latter is the advanced acausal field which acts on $A$ from a future time.
In effect what Dirac showed was that the radiation damping term in (\ref{e1}) or (\ref{e2}) is given by (\ref{e3}) in which an antisymmetric difference of the advanced and retarded fields is taken, which of course seemingly goes against causality as the advanced field acts from the future backwards in time. It must be mentioned that Dirac's prescription lead to the so called runaway solutions, with the electron acquiring larger and larger velocities in the absense of an external force. This he related to the infinite self energy of the point electron.\\
As far as the breakdown of causality is concerned, this takes place within a period $\sim \tau$, the Compton time. It was at this stage that Wheeler and Feynman reformulated the above action at a distance formalism in terms of what has been called their Absorber Theory. In their formulation, the field that a charge would experience because of its action at a distance on the other charges of the universe, which in turn would act back on the original charge is given by
\begin{equation}
R_e = \frac{2e^2}{3} \vec{x}\label{e4}
\end{equation}
The interesting point is that instead of considering the above force in (\ref{e4}) at the charge $e$, if we consider the responses in its neighbourhood, in fact a neighbourhood at the Compton scale, as was argued recently \cite{iaad}, the field would be precisely the Dirac field given in (\ref{e2}) and (\ref{e3}). The net force emanating from the charge is thus given by
\begin{equation}
F^{\mbox{ret}} = \frac{1}{2} \left\{ F^{\mbox{ret}} + F^{\mbox{adv}}\right\} + \frac{1}{2} \left\{F^{\mbox{ret}} - F^{\mbox{adv}}\right\}\label{e5}
\end{equation}
which is the causal acceptable retarded field. The causal field now consists of the time symmetric field of the charge $e$ together with the Dirac field, that is the second term in (\ref{e5}), which represents the response of the rest of the charges. Interestingly in this formulation we have used a time symmetric field, viz., the first term of (\ref{e5}) to recover the retarded field with the correct arrow of time.\\
There are two important inputs which we can see in the above formulation. The first is the action of the rest of the universe at a given charge and the other is spacetime intervals which are of the order of the Compton scale. Infact we can push the above calculations further. The work done on a charge $e$ at $O$ by the charge at $P$ a distance $r$ in causing a displacement $x$ is given by
\begin{equation}
\frac{e^2x}{r^2} dx\label{e6}
\end{equation}
Now the number of particles at distance $r$ from $O$ is given by
\begin{equation}
n(r) = \rho(r) \cdot 4\pi^2 drUcrcR\label{e7}
\end{equation}
where $\rho(r)$ is the density of particles. So using (\ref{e7}) in (\ref{e6}) the total work is given by
\begin{equation}
E = \int \int \frac{e^2}{r^2} cr \rho 4\pi^2 dr\label{e8}
\end{equation}
which can be shown to be $\sim mc^2$. We thus recover in (\ref{e8}) the inertial energy of the particle in terms of its electromagnetic interactions with the rest of the universe in an action at a distance scheme. Interestingly this can also be deduced in the context of gravitation: The work done on a particle of mass $m$ which we take to be a pion, a typical elementary particle, by the rest of the particles (pions) in the universe is given by
\begin{equation}
\frac{Gm^2N}{R}\label{e9}
\end{equation}
It is known that in (\ref{e9}) $N \sim 10^{80}$ while $R \sim \sqrt{N}l$, the well known Weyl-Eddington formula. Whence the gravitational energy of the pion is given by
\begin{equation}
\frac{Gm^2\sqrt{N}}{l} = \frac{e^2}{l} \sim mc^2\label{e10}
\end{equation}
where in (\ref{e10}) we have used the fact that
\begin{equation}
Gm^2 \sim \frac{e^2}{\sqrt{N}}\label{e11}
\end{equation}
(It must be mentioned that though the Eddington formula and (\ref{e11}) were empirical, they can infact be deduced from theory \cite{cu}, as we will see shortly.)
\section{The Machian Universe}
This dependence of the mass of a particle on the rest of the universe was argued by Mach in the nineteenth century itself in what is now famous as Mach's Principle \cite{mwt,jv}. The Principle is counter intuitive in that we consider the mass which represents the quantity of matter in a particle to be an intrisic property of the particle. But the following statement of Mach's Principle shows it to be otherwise.\\
If there were no other particles in the universe, then the force acting on the particle $P$ would vanish and so we would have by Newton's second law
\begin{equation}
m\vec{a} = O\label{e12}
\end{equation}
Can we conclude that the acceleration of the particle vanishes? Not if we do not postulate the existence of an absolute background frame in space. In the absense of such a Newtonian absolute space frame, the acceleration $\vec{a}$ would infact be arbitrary, because we could measure this acceleration with respect to arbitrary frames. Then (\ref{e12}) implies that $m = 0$. That is, in the absense of any other matter in the universe, the mass of a material particle would vanish. From this point of view the mass of a particle depends on the rest of the material content of the universe. This has been brought out by the above calculations in (\ref{e8}) and (\ref{e10}).\\
Though Einstein was an admirer of Mach's ideas, his Special Theory of Relativity went counter to them. He subscribed to the concept of Locality according to which information about a part of the universe can be obtained by dealing with that part without taking into consideration the rest of the universe at the same time. In his words, \cite{singh} ``But one one supposition we should, in my opinion absolutely hold fast: the real factual situation of the system $S_2$ is independent of what is done with the system $S_1$ which is spatially separated from the former.''Further, Causality is another cornerstone in Einstein's Physics.
\section{The Quantum Universe}
The advent of Quantum Mechanics however threw up several counter intuitive ideas and Einstein could not reconcile to them. One of these ideas was the wave particle duality. Another was that of the collapse of the wave function in which process Causality becomes a casuality. To put it simply, if the wave function is a super position of the eigen states of an observable then a measurement of the observable yields one of the eigen values no doubt, but it is not possible to predict which one. Due to the observation, the wave function instantly collapses to any one of its eigen states in an acausal manner. To put it another way, the wave function obeys the causal Schrodinger equation, for example, till the instant of observation at which point, causality ceases.\\
Another important counter intuitive feature of Quantum Mechanics is that of non locality. In fact Einstein with Podolsky and Rosen put forward in 1935 his arguments for the incompleteness of Quantum Mechanics on this score \cite{singh,EPR}. This has later come to be known as the EPR paradox. To put it in a simple way, without sacrificing the essential concepts, let us consider two elementary particles, for example two protons kept together somehow. They are then released and move in opposite directions. When the first proton reaches the point $A$ its momentum is measured and turns out to be say, $\vec{p}$. At that instant we can immediately conclude, without any further measurement that the momentum of the second proton which is at the point $B$ is $-\vec{p}$. This follows from the Conservation of Linear Momentum, and is perfectly acceptable in Classical Physics, in which the particles possess a definite momentum at each instant.\\
In Quantum Physics, the difficulty is that we cannot know the momentum at $B$ until and after a measurement is actually performed, and then that value of the momentum is unpredictable. What the above experiment demonstrates is that the proton at $B$ instantly came to have the value $-\vec{p}$ for its momentum when the momentum of the proton at $A$ was measured. This ``instant'' or ``spooky action at a distance'' feature was unacceptable to Einstein.\\
In Quantum Theory however this is legitimate because of another counter intuitive feature which is called Quantum Nonseparability. That is if two systems interact and then separate to a distance, they still have a common state vector. This goes against the concept of Locality and Causality, because it implies instantaneous interaction between distant systems. So in the above example, even though the protons at $A$ and $B$ may be separated, they still have a common wave function which collapses with the measurement of the momentum of any one of them and selfconsistently provides an explanation. This Nonseparability has been characterised by Schrodinger in the following way: ``I would not call that \underline{one}, but rather \underline{the} characteristic of Quantum Mechanics.'' For Einstein however this was like spooky action at a distance. All this has been experimentally verified since 1980 which sets at rest Einstein's objections.\\
However this ``entanglement'' as it is called these days, between distant objects in the universe, does not really manifest itself. An explanation for this was given by Schrodinger himself who argued in effect that entanglement is perfectly legitimate and observable in a universe that consists of let us say just two particles. But a measurement destroys the entanglement. Now in the universe as there are so many particles and correspondingly a huge amount of interference, the entanglement is considerably weakened. What is these days called decoherence works along these lines. This is infact the explanation of the famous ``Schrodinger's Cat'' paradox.\\
This paradox can be explained in the following simple terms: A cat is in an enclosure along with, let us say a microscopic amount of radioactive material. If this material decays, emitting let us say an electron, the electron would fall on a vial of cyanide, releasing it and killing the cat in the process. Let us say that there is a certain probability of such an electron being emitted. So there is the same probability for the cat to be killed. There is also a probability that the electron is not emitted, so that there is the same probability for the cat to remain alive. The cat is therefore in a state which is a superposition of the alive and dead states. It is only when an observer makes an observation that this superposed wave function collapses into either the dead cat state or the alive and kicking cat state, and this happening is acausal. So it is only on an observation being made that the cat is killed or saved, and that too in an unpredictable manner. Till the observation is made the cate is described by the superposed wave function and is thus neither alive nor dead.\\
The resolution of this paradox is of course quite simple. The paradox is valid if the system consists of such few particles and at such distances that they do not interact with each other. Clearly in the real world this idealization is not possible. There are far too many particles and interferences taking place all the time and the superposed wave function would have collapsed almost instantly. This role of the environment has come to be called decoherence. We will return to this point shortly.\\
The important point is that all of Classical and Quantum Physics is based on such idealized laws as if there were no interferences present, that is what may be called a two body scenario is implicit. Clearly this is not a real life scenario.
\section{The Zero Point Field}
Another counter intuitive concept which Quantum Theory introduces is that of the Zero Point Field or Quantum Vacuum. If there were a vacuum, in which at a given point the momentum (and energy) would vanish, then by the Heisenberg Principle, the point itself becomes indeterminate. More realistically, in the vacuum the average energy vanishes but there are fluctuations-- these are the Zero Point Fluctuations. A more classical way of looking at this is that the source free vacuum electromagnetic equations have non zero solutions, in addition to the zero solutions. Interestingly we can argue that the Zero Point Field leads to a minimum interval at the Compton scale \cite{def}.\\
The manifestation of the Zero Point Field has been experimentally tested in what is called the Lamb Shift, which is caused by the fact that the Zero Point Field buffets an ordinary electron in an atom. It has also been verified in the famous Casimir effect \cite{mes,mdef}. The Zero Point Field in this case manifests itself as an attractive force between two parallel plates.\\
Interestingly based on such a Quantum Vacuum and the minimum spacetime intervals the author had proposed a cosmological model in 1997 which predicted an accelerating universe and a small cosmological constant. In addition, several so called large number relations which had been written off as inexplicable empirical coincidences were shown to follow from the theory \cite{ijmpa}. At that time the prevailing cosmological model was one of dark matter and a decelerating universe. Observational confirmation started coming for the new predictions from 1998 itself while the observational discovery of dark energy, which displaces dark matter, was the scientific Breakthrough of the Year 2003 of the American Association for Advancement of Science \cite{science}.\\
It may be observed that the idea of the Zero Point Field was introduced as early as in 1911 by Max Planck himself to which he assigned an energy $\frac{1}{2} \hbar \omega$. Nernst, a few years later extended these considerations to fields and believed that there would be several interesting consequences in Thermodynamics and even Cosmology.\\
Infact later authors argued that there must be fluctuations of the Quantum Electromagnetic Flield, as required by the Heisenberg Principle, so that we have for an extent $\sim L$
\begin{equation}
(\Delta B)^2 \geq \hbar c/L^4\label{e13}
\end{equation}
Whence from (\ref{e13}), the dispersion in energy in the entire volume $\sim L^3$ is given by
\begin{equation}
\Delta E \sim \hbar c/L\label{e14}
\end{equation}
(It should be noticed that if $L$ is the Compton wavelength, then (\ref{e14}) gives us the energy of the particle.)Interestingly Braffort and coworkers deduced the Zero Point Field from the Absorber Theory of Wheeler and Feynman, which we encountered earlier. In the process they found that the spectral density of the vacuum field was given by \cite{depena}
\begin{equation}
\rho (\omega) = \mbox{const}\cdot \omega^3\label{e15}
\end{equation}
There have been other points of view which converge to the above conclusions. In any case as we have seen a little earlier, it would be too much of an idealization to consider an atom or a charged particle to be an isolated system. It is interacting with the rest of the universe and this produces a random field.\\
It has also been shown that the constant of proportionality in (\ref{e15}) is given by (Cf.ref.\cite{depena})
$$\frac{\hbar}{2\pi^2 c^3}$$
Interestingly such a constant is implied by Lorentz invariance.\\
From the point of view of Quantum Electrodynamics we reach conclusions similar to those seen above. As Feynman and Hibbs put it \cite{feynman} ``Since most of the space is a vacuum, any effect of the vacuum-state energy of the electromagnetic field would be large. We can estimate its magnitude. First, it should be pointed out that some other infinities occuring in quantum-electrodynamic problems are avoided by a particular assumption called the \underline{cutoff rule}. This rule states that those modes having very high frequencies (short wavelength) are to be excluded from consideration. The rule is justified on the ground that we have no evidence that the laws of electrodynamics are obeyed for wavelengths shorter than any which have yet been observed. In fact, there is a good reason to believe that the laws cannot be extended to the short-wavelength region.\\
``Mathematical representations which work quite well at longer wavelengths lead to divergences if extended into the short wavelength region. The wavelengths in question are of the order of the Compton wavelength of the proton; $1/2\pi$ times this wavelength is $\hbar/mc \simeq 2 \times 10^{-14}cm$.\\
``For our present estimate suppose we carry out sums over wave numbers only up to the limiting value $k_{max} = mc/\hbar$. Approximating the sum over states by an integral, we have, for the vacuum-state energy per unit volume,
$$\frac{E_e}{\mbox{unit \, vol}} = 2 \frac{\hbar c}{2(2\pi)^3} \int^{k_{max}}_0 k 4\pi k^2 dk - \frac{\hbar c k^4_{max}}{8\pi^2}$$
``(Note the first factor $2$, for there are two modes for each $k$). The equivalent mass of this energy is obtained by dividing the result by $c^2$. This gives
$$\frac{m_0}{\mbox{unit \, vol}} = 2 \times 10^{15} g/cm^3$$
Such a mass density would, at first sight at least, be expected to produce very large gravitational effects which are not observed. It is possible that we are calculating in a naive manner, and, if all of the consequences of the general theory of relativity (such as the gravitational effects produced by the large stresses implied here) were included, the effects might cancel out; but nobody has worked all this out. It is possible that some cutoff procedure that not only yields a finite energy density for the vacuum state but also provides relativistic invariance may be found. The implications of such a result are at present completely unknown.\\
``For the present we are safe in assigning the value zero for the vacuum-state energy density. Up to the present time no experiments that would contradict this assumption have been performed.''\\
However the high density encountered above is perfectly meaningful if we consider the Compton scale cut off: Within this volume the density gives us back the mass of an elementary particle like the pion. All this can be put into perspective in the following way. It has been shown in detail by the author that the universe can be considered to have an underpinning of ZPF oscillators at the Planck scale \cite{bgsfpl}. Indeed in all recent approaches towards a unified formulation of gravitation and electromagnetism (including String Theory), the differentiable spacetime manifold of Classical Physics and Quantum Physics has been abandoned and we consider the minimum Planck scale $\sim 10^{-33}cms$ and $10^{-42}secs$ \cite{uof}. We can then show that the universe is a coherent mode of $\bar{N} \sim 10^{120}$ Planck oscillators, spaced a distance $l_P$ apart, that is at the Planck scale. Then the spatial extent is given by
\begin{equation}
R = \sqrt{\bar{N}}l_P\label{e16}
\end{equation}
The mass of the universe is given by
\begin{equation}
M = \sqrt{\bar{N}} m_P\label{e17}
\end{equation}
where $m_P$ is the Planck mass. Moreover we can show that a typical elementary particle like the pion is the ground state of $n \sim 10^{40}$ oscillators and we have (Cf.ref.\cite{bgsfpl})
\begin{equation}
m = \frac{m_P}{\sqrt{n}}\label{e18}
\end{equation}
\begin{equation}
l = \sqrt{n} l_P\label{e19}
\end{equation}
There are $N \sim 10^{80}$ such elementary particles in the universe. Whence we have
\begin{equation}
M = Nm\label{e20}
\end{equation}
We note that equations like (\ref{e16}) and (\ref{e19}) have the Brownian Random Walk characters. At this stage we see asymmetry between equations (\ref{e17}), (\ref{e18}) and (\ref{e20}). The reason is that the universe is an excited state of $\bar{N}$ oscillators whereas an elementary particle is a stable ground state of $n$ Planck oscillators. Furthermore, let us denote the state of each Planck oscillator by $\phi_n$; then the state of the universe can be described in the spirit of entanglement discussed earlier by
\begin{equation}
\psi = \sum_{n} c_n \phi_n,\label{e21}
\end{equation}
$\phi_n$ can be considered to be eigen states of energy with eigen values $E_n$. It is known that (\ref{e21}) can be written as \cite{bgscsf}
\begin{equation}
\psi = \sum_{n} b_n \bar{\phi}_n\label{e22}
\end{equation}
where $|b_n|^2 = 1 \, \mbox{if}\, E < E_n < E + \Delta$ and $= 0$ otherwise under the assumption
\begin{equation}
\overline{(c_n,c_m)} = 0, n \ne m\label{e23}
\end{equation}
(Infact $n$ in (\ref{e23}) could stand for not a single state but for a set of states $n_\imath$, and so also $m$). Here the bar denotes a time average over a suitable interval. This is the well known Random Phase Axiom and arises due to the total randomness amongst the phases $c_n$. Also the expectation value of any operator $O$ is given by
\begin{equation}
< O > = \sum_{n} |b_n|^2 (\bar{\phi}_n, O \bar{\phi}_n)/\sum_{n} |b_n|^2\label{e24}
\end{equation}
Equations (\ref{e22}) and (\ref{e24}) show that effectively we have incoherent states $\bar{\phi}_1, \bar{\phi}_2,\cdots$ once averages over time intervals for the phases $c_n$ in (\ref{e23}) vanish owing to their relative randomness. In the light of the preceding discussion of random fluctuations, we can interpret all this meaningfully: We can identify $\phi_n$ with the ZPF. The time averages are the same as Dirac's zitterbewegung averages over intervals $\sim \frac{\hbar}{mc^2}$ (Cf.ref.\cite{cu}). We then get disconnected or incoherent particles from a single background of vacuum fluctuations exactly as before. The incoherence arises because of the well known random phase relation (\ref{e23}), that is after averating over the suitable interval. Here the entanglement is weakened by the interactions and hence we have (\ref{e20}) for elementary particles, rather than (\ref{e17}).\\
How do we characterize time in this scheme? To consider this problem, we observe that the ground state of $\bar{N}$ Planck oscillators considered above would be, exactly as in (\ref{e18}),
\begin{equation}
\bar {m} = \frac{m_P}{\sqrt{\bar{N}}} \sim 10^{-65}gms\label{ex2}
\end{equation}
The universe is an excited state and consists of $\bar{N}$ such ground state
levels and so we have, from (\ref{ex2})
$$M = \bar{m} \bar{N} = \sqrt{\bar{N}} m_P \sim 10^{55}gms,$$ as required, $M$
being the mass of the universe. Interestingly, the Compton wavelength and time of $\bar{m}$ turn out to be the radius and age of the universe.\\
Due to the fluctuation $\sim
\sqrt{n}$ in the levels of the $n$ oscillators making up an elementary
particle, the energy is, remembering that $mc^2$ is the ground state,
$$\Delta E \sim \sqrt{n} mc^2 = m_P c^2,$$
and so the indeterminacy time is
$$\frac{\hbar}{\Delta E} = \frac{\hbar}{m_Pc^2} = \tau_P,$$ as indeed
we would expect.\\
The corresponding minimum indeterminacy length
would therefore be $l_P$. We thus recover the Planck scale. One of the consequences of the minimum
spacetime cut off is that the Heisenberg Uncertainty
Principle takes an extra term. Thus we have,
\begin{equation}
\Delta x \approx \frac{\hbar}{\Delta p} + \alpha \frac{\Delta
p}{\hbar},\, \alpha = l^2 (\mbox{or}\, l^2_P)\label{ex6}
\end{equation}
where $l$ (or $l_P$) is the minimum interval under consideration (Cf.\cite{cu,uof}).
The first term gives the usual Heisenberg Uncertainty
Principle.\\
Application of the time analogue of (\ref{ex6}) for the
indeterminacy time $\Delta t$ for the fluctuation in energy $\Delta
\bar{E} = \sqrt{N} mc^2$ in the $N$ particle states of the universe
gives exactly as above,
$$\Delta t = \frac{\Delta E}{\hbar} \tau^2_P =
\frac{\sqrt{N}mc^2}{\hbar} \tau^2_P = \frac{\sqrt{N}
m_Pc^2}{\sqrt{n}\hbar} \tau^2_P = \sqrt{n} \tau_P = \tau ,$$
In other words, for the fluctuation
$\sqrt{N}$, the time is $\tau$. It must be re-emphasized that the
Compton time $\tau$ of an elementary particle, is an interval within
which there are unphysical effects like zitterbewegung - as pointed
out by Dirac, it is only on averaging over this interval, that we
return to meaningful Physics. This gives us,
\begin{equation}
dN/dt = \sqrt{N}/\tau\label{ex3}
\end{equation}
On the other hand
due to the fluctuation in the $\bar{N}$ oscillators
constituting the universe, the fluctuational energy is similarly given
by
$$\sqrt{\bar {N}} \bar {m} c^2,$$ which is the same as (\ref{ex2})
above. Another way of deriving (\ref{ex3}) is to observe that as
$\sqrt{n}$ particles appear fluctuationally in time $\tau_P$ which is,
in the elementary particle time scales, $\sqrt{n} \sqrt{n} = \sqrt{N}$
particles in $\sqrt{n} \tau_P = \tau$. That is, the rate of the
fluctuational appearance of particles is
$$
\left(\frac{\sqrt{n}}{\tau_P}\right) = \frac{\sqrt{N}}{\tau} = dN/dt$$
which is (\ref{ex3}). From here by integration,
$$T = \sqrt{N} \tau$$ $T$ is the time elapsed from $N = 1$ and $\tau$
is the Compton time. This gives $T$ its origin in the fluctuations -
there is no smooth ``background'' (or ``being'') time - the root of
time is in ``becoming''. It is the time of a Brownian Wiener
process: A step $l$ gives a step in time $l/c \equiv \tau$ and
therefore the Brownian relation $\Delta x = \sqrt{N} l$ gives $T = \sqrt{N} \tau$ (Cf.refs.\cite{bgsfpl} and \cite{uof}). Time is
born out of acausal fluctuations which are random and therefore
irreversible. Indeed, there is no background time. Time is
proportional to $\sqrt{N}$, $N$ being the number of particles which
are being created spontaneously from the ZPF by fluctuations to the higher energy states of the coherent $\bar{N}$ Planck oscillators.
\section{The Underpinning of the Universe}
So our description of the universe at the Planck scale is that of an entangled wave function as in (\ref{e21}). However we percieve the universe at the elementary particle or Compton scale, where the random phases would have weakened the entanglement, and we have the description as in (\ref{e22}) or (\ref{e24}). Does this mean that $N$ elementary particles in the universe are totally incoherent in which case we do not have any justification for treating them to be in the same spacetime? We can argue that they still interact amongst each other though in comparison this is ``weak''. For instance let us consider the background ZPF whose spectral frequency is given by (\ref{e15}). If there are two particles at $A$ and $B$ separated by a distance $r$, then those wavelengths of the ZPF which are atleast $\sim r$ would connect or link the two particles. Whence the force of interaction between the two particles is given by, remembering that $\omega \propto \frac{1}{r}$,
\begin{equation}
\mbox{Force}\, \quad \propto \int^\infty_r \omega^3 dr \propto \frac{1}{r^2}\label{e27}
\end{equation}
Thus from (\ref{e27}) we are able to recover the familiar Coulomb Law of interaction. The background ZPF thus enables us to recover the action at a distance formulation. Infact a similar argument can be given \cite{fisch} to recover from QED the Coulomb Law--here the carriers of the force are the virtual photons, that is photons whose life time is within the Compton time of uncertainty permitted by the Heisenberg Uncertainty Principle.\\
It is thus possible to synthesize the field and action at a distance concepts, once it is recognized that there are minimum spacetime intervals at the Compton scale \cite{iaad}. Many of the supposed contradictions arise because of our characterization in terms of spacetime points and consequently a differentiable manifold. Once the minimum cut off at the Planck scale is introduced, this leads to the physical Compton scale and a unified formulation free of divergence problems. We now make a few comments.\\
We had seen that the Dirac formulation of Classical Electrodynamics needed to introduce the acausal advanced field in (\ref{e3}). However the acausality was again within the Compton time scale. Infact this fuzzy spacetime can be modelled by a Wiener process as discussed in \cite{uof}(Cf. also \cite{nottale}). The point here is that the backward and forward time derivatives for $\Delta t \to 0^-$ and $0^+$ respectively do not cancel, as they should not, if time is fuzzy. So we automatically recover from the electromagnetic potential the retarded field for forward derivatives and the advanced fields for backward derivatives. In this case we have to consider both these fields. Causality however is recovered as in (\ref{e5}). This is a transition to intervals which are greater in magnitude compared to the Compton scale.\\
It must also be mentioned that a few assumptions are implicit in the conventional theory using differentiable spacetime manifolds. In the variational problem we use the conventional $\delta$ (variation) which commutes with the time derivatives. So such an operator is constant in time. So also the energy momentum operators in Dirac's displacement operators theory are the usual time and space derivatives of Quantum Theory. But here the displacements are ``instantaneous''. They are valid in a stationary or constant energy scenario, and it is only then that the space and time operators are on the same footing as required by Special Relativity \cite{davydov}. Infact it can be argued that in this theory we neglect intervals $\sim 0(\delta x^2)$ but if $\delta x$ is of the order of the Compton scale and we do not neglect the square of this scale, then the space and momentum coordinates become complex indicative of a noncommutative geometry which has been discussed in detail \cite{bgscst,bgsknmetric,uof}. What all this means is that it is only on neglecting $0(l^2)$ that we have the conventional spacetime of Quantum Theory, including relativistic Quantum Mechanics and Special Relativity, that is the Minkowski spacetime. Coming to the conservation laws of energy and momentum these are based on translation symmetries \cite{roman}-- what it means is the operators $\frac{d}{dx} \, \mbox{or}\, \frac{d}{dt}$ are independent of $x$ and $t$. There is here a homogeneity property of spacetime which makes these laws non local. This has to be borne in mind, particularly when we try to explain the EPR paradox.\\
The question how a ``coherent'' spacetime can be extracted out of the particles of the universe could be given a mathematical description along the following lines: Let us say that two particles $A$ and $B$ are in a neighbourhood, if they interact at any time. We also define a neighbourhood of a point or particle $A$ as a subset of all points or particles which contains $A$ and at least one other point. If a particle $C$ interacts with $B$ that is, is in a neighbourhood of $B$, then we would say that it is also in the neighbourhood of $A$. That is we define the transitivity property for neighbourhoods. We can then assume the following property \cite{bgsaltaisky}:\\
Given two distinct elements (or even subsets) $A$ and $B$, there is a neighbourhood $N_{A_1}$ such that $A$ belongs to $N_{A_1}$, $B$ does not belong to $N_{A_1}$ and also given any $N_{A_1}$, there exists a neithbourhood $N_{A_\frac{1}{2}}$ such that $A \subset N_{A_\frac{1}{2}} \subset N_{A_1}$, that is there exists an infinite sequence of neithbourhoods between $A$ and $B$. In other words we introduce topological ``closeness''. Alternatively, we could introduce the reasonable supposition that these are a set of Borel subsets.\\
From here, as in the derivation of Urysohn's lemma \cite{simmons}, we could define a mapping $f$ such that $f(A) = 0$ and $f(B) = 1$ and which takes on all intermediate values. We could now define a metric, $d(A,B) = |f(A) - f(B)|$. We could easily verify that this satisfies the properties of a metric.\\
It must be remarked that the metric turns out to be again, a result of a global or a series of large sets, unlike the usual local picture in which is is the other way round.
|
{
"timestamp": "2005-03-29T14:59:39",
"yymm": "0503",
"arxiv_id": "physics/0503220",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503220"
}
|
\section{Introduction}
\subsection{Reminder on Moyal product.}\label{reminder} Let $V$ be a finite
dimensional vector space equipped with a nondegenerate
bivector $\pi\in\wedge^2V$. Associated with $\pi$ is
a Poisson bracket $f,g\mapsto \{f,g\}:=\langle df\wedge dg,\pi\rangle$
on $\mathbf{k}[V],$ the polynomial algebra on $V$.
The usual commutative product $m: \mathbf{k}[V]\otimes \mathbf{k}[V]\to\mathbf{k}[V]$
and the Poisson bracket $\{-,-\}$ make $\mathbf{k}[V]$ a Poisson algebra.
This Poisson algebra has a well-known {\em Moyal-Weyl quantization}
(\cite{M}, see also \cite{CP}).
This is
an associative star-product
depending on a formal quantization parameter $\mathrm{h}$, defined by the formula
\begin{equation}\label{star}
f *_\mathrm{h} g:=m{{}_{\,{}^{^\circ}}} e^{\frac{1}{2} \mathrm{h} \pi} (f\otimes g)\in \mathbf{k}[V][\mathrm{h}],\quad
\forall f,g\in\mathbf{k}[V][\mathrm{h}].
\end{equation}
To explain the meaning of this formula, view elements of ${\text{Sym\ }} V$
as
constant-coefficient differential operators on $V$. Hence,
an element of ${\text{Sym\ }} V\otimes {\text{Sym\ }} V$ acts as a
constant-coefficient differential operator on
the algebra $\mathbf{k}[V]\otimes\mathbf{k}[V]=\mathbf{k}[V\times V].$ Now, identify
$\wedge^2V$ with the subspace of skew-symmetric tensors in $V\otimes V$.
This way, the bivector $\pi\in\wedge^2V\subset V\otimes V$ becomes
a second order constant-coefficient differential operator
$\pi: \mathbf{k}[V]\otimes\mathbf{k}[V]\to\mathbf{k}[V]\otimes\mathbf{k}[V].$ Further, it is clear that
for any element $f\otimes g\in\mathbf{k}[V]\otimes\mathbf{k}[V]$ of total degree
$\leq N$, all terms with $d>N$ in the
infinite sum $e^{\mathrm{h}\cdot \pi}(f\otimes g)=\sum_{d=0}^\infty
\frac{\mathrm{h}^d}{d!} \pi^d(f\otimes g)$
vanish, so the sum makes sense.
Thus,
the
symbol $m{{}_{\,{}^{^\circ}}} e^{\mathrm{h}\cdot \pi}$ in the right-hand side
of formula \eqref{star} stands for the composition
$$ \mathbf{k}[V]\otimes\mathbf{k}[V]\stackrel{e^{\mathrm{h}\cdot \pi}}{\;{-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}\;} \mathbf{k}[V]\otimes\mathbf{k}[V]\otimes\mathbf{k}[\mathrm{h}]
\stackrel{m\otimes\mathrm{Id}_{\mathbf{k}[\mathrm{h}]}}{\;{-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}\;} \mathbf{k}[V]\otimes\mathbf{k}[\mathrm{h}],
$$
where
$e^{\mathrm{h}\cdot \pi}$
is an infinite-order formal differential operator.
In down-to-earth terms, choose coordinates
$x_1, \ldots, x_n, y_1, \ldots, y_n$ on $V$ such that
the bivector $\pi$, resp., the Poisson bracket $\{-,-\}$, takes
the canonical form
\begin{equation}\label{pois}
\pi = \sum_i \frac{\partial}{\partial x_i} \otimes \frac{\partial}{\partial
y_i} - \frac{\partial}{\partial y_i} \otimes \frac{\partial}{\partial x_i},
\quad\text{resp.,}\quad
\{f,g\}=\sum_i \frac{\partial f}{\partial x_i}\frac{\partial g}{\partial
y_i} - \frac{\partial f}{\partial y_i} \frac{\partial g}{\partial x_i}.
\end{equation}
Thus, in canonical coordinates
$x=(x_1, \ldots, x_n), y=(y_1, \ldots, y_n),$ formula \eqref{star} for the Moyal
product
reads
\begin{align}\label{star1}
(f *_\mathrm{h} g)(x,y)&=\sum_{d=0}^\infty
\frac{\mathrm{h}^d}{d!}\left(
\sum_i \frac{\partial}{\partial
x'_i} \frac{\partial}{\partial
y''_i} - \frac{\partial}{\partial y'_i}
\frac{\partial}{\partial x''_i}
\right)^df(x',y') g(x'',y'')\Big|_{{x'=x''=x}\atop{y'=y''=y}}\nonumber\\
&=\sum_{\mathbf{j},\mathbf{l}\in{\mathbb Z}^n_{\geq 0}}
(-1)^{\mathbf{l}|}\frac{\mathrm{h}^{|\mathbf{j}|+|\mathbf{l}|}}{\mathbf{j}!\,\mathbf{l}!}\cdot
\frac{\partial^{\mathbf{j}+\mathbf{l}}f(x,y)}{\partial x^\mathbf{j}\partial y^\mathbf{l}}
\cdot
\frac{\partial^{\mathbf{j}+\mathbf{l}}g(x,y)}{\partial y^\mathbf{j}\partial x^\mathbf{l}},
\end{align}
where for $\mathbf{j}=(j_1, \ldots,j_n)\in {\mathbb Z}^n_{\geq 0}$
we put $|\mathbf{j}|=\sum_i j_i$ and given
$\mathbf{j},\mathbf{l}\in {\mathbb Z}^n_{\geq 0},$ write
$$\frac{1}{\mathbf{j}!\,\mathbf{l}!}\frac{\partial^{\mathbf{j}+\mathbf{l}}}
{\partial x^{\mathbf{j}}\partial y^{\mathbf{l}}}:=
\frac{1}{j_1!\ldots
j_n!l_1!\ldots l_n!}\cdot\frac{\partial^{|\mathbf{j}|+|\mathbf{l}|}}
{\partial x_1^{j_1}\ldots\partial x_n^{j_n}\partial
y_1^{l_1}\ldots\partial y_n^{l_n}}.
$$
A more conceptual approach to formulas \eqref{star}--\eqref{star1}
is obtained by introducing the {\em Weyl algebra} $A_\mathrm{h}(V)$.
This is a $\mathbf{k}[\mathrm{h}]$-algebra defined by the quotient
$$
A_\mathrm{h}(V):= (TV^*)[\mathrm{h}]/I(u\otimes u' - u'\otimes u-\mathrm{h}\cdot\langle \pi,
u\otimes u'\rangle)_{u,u'\in V^*},
$$
where $TV^*$ denotes the tensor algebra of the vector space $V^*$,
and $I(\ldots)$ denotes the two-sided ideal generated by the
indicated set.
Now, a version of the Poincar\'e-Birkhoff-Witt theorem
says that the natural {\em symmetrization map}
yields a $\mathbf{k}[\mathrm{h}]$-linear bijection
$\phi_W: \mathbf{k}[V][\mathrm{h}]{\;\stackrel{_\sim}{\to}\;} A_\mathrm{h}(V)$.
Thus, transporting the multiplication map in the Weyl algebra $A_\mathrm{h}(V)$ via
this bijection, one obtains an associative product
$$\mathbf{k}[V][\mathrm{h}]\otimes_{\mathbf{k}[\mathrm{h}]}\mathbf{k}[V][\mathrm{h}]\to\mathbf{k}[V][\mathrm{h}],
\quad f\otimes g\mapsto \phi_W^{-1}(\phi_W(f)\cdot \phi_W(g)).
$$
It is known that this associative product is equal to the one
given by formulas \eqref{star}--\eqref{star1}.
\subsection{The quiver analogue.} The goal of this paper
is to extend the constructions outlined above to noncommutative
symplectic geometry. Specifically, following
an original idea of Kontsevich \cite{K}, to any quiver, one
associates canonically a certain Poisson algebra
(\cite{G}, \cite{BLB}). Then,
we will produce a quantization of that Poisson algebra
given by an explicit formula analogous to
formulas \eqref{star}--\eqref{star1}.
In more detail, fix a quiver with vertex set $I$ and edge set $Q,$ and
let $\overline{Q}$ be the double of $Q$ obtained by adding
reverse edge $e^*\in\overline{Q}$ for each edge $e\in Q$.
Let $P$ be the {\em path algebra} of $\overline{Q}$. The commutator
quotient space $P/[P,P]$ may be identified naturally
with the space $L$ spanned by cyclic paths (forgetting which was
the initial edge), sometimes called
{\em necklaces}. Letting $\text{pr}_L: P \rightarrow P/[P,P] = L$ be he
projection, there is a natural bilinear
pairing
\begin{equation}\label{pair}\{-,-\}:\
L\otimes L\to L,\quad
f\otimes g\mapsto\{f,g\}:= \text{pr}_L \biggl( \sum_{e \in Q} \frac{\partial f}{\partial e}
\frac{\partial g}{\partial e^*} - \frac{\partial f}{\partial e^*}
\frac{\partial g}{\partial e} \biggr).
\end{equation}
Interpreting $\frac{\partial}{\partial e}, \frac{\partial}{\partial e^*}$
appropriately as maps $L \rightarrow P, P \rightarrow P$, this formula,
which is a quiver analogue of \eqref{pois}, provides $L$
with a Lie algebra structure, first studied in \cite{G}, \cite{BLB}.
More recently, the second author showed in \cite{S}
that there is also a natural Lie {\em cobracket}
on $L$.
To explain this, write $a_1\cdots a_p\in P$ for a path
of length $p$ and let
$1_i$ denote the trivial (idempotent) path at the
vertex $i\in I$. Further, for
any edge $e\in \overline{Q}$ with head $h(e)\in I$ and tail
$t(e)\in I$,
let
$D_e: P\to P\otimes P$ be
the derivation defined by the assignment
$$D_e:\
P\to P\otimes P,\quad
a_1\cdots a_p\mapsto\sum_{a_r=e}
a_1\cdots a_{r-1}1_{t(e)}\o1_{h(e)} a_{r+1}\cdots a_p.
$$
The map $D_e$ is a derivation. Moreover, the
following map, cf. \cite[(1.7)-(1.8)]{S}:
\begin{equation}\label{delta}
\delta: L\to L\wedge L,
\quad
f\mapsto \delta(f)= (\text{pr}_L \otimes \text{pr}_L)
\biggl( \sum_{e \in Q} D_e(\frac{\partial f}{\partial e^*})
- D_{e^*}(\frac{\partial f}{\partial e}) \biggr)
\end{equation}
(that is, in a sense, dual to \eqref{pair})
makes the Lie algebra $L$ a Lie {\em bialgebra},
to be referred to as the {\em necklace Lie bialgebra}.
The necklace Lie bialgebra admits a very interesting quantization.
Specifically, the main construction of \cite{S} produces
a Hopf $\mathbf{k}[\mathrm{h}]$-algebra
$A_\mathrm{h}(Q)$ equipped with an algebra
isomorphism $A_\mathrm{h}(Q)/\mathrm{h}\cdot A_\mathrm{h}(Q){\;\stackrel{_\sim}{\to}\;} {\text{Sym\ }} L,\,f\mapsto\operatorname{pr}{f}.$
The algebra $A_\mathrm{h}(Q)$ is a quantization of the
Lie bialgebra $L$ in the sense that $A_\mathrm{h}(Q)$ is flat over $\mathbf{k}[\mathrm{h}]$ and,
for any $a,b\in A_\mathrm{h}(Q),$ one has
$$
\operatorname{pr}\left(\frac{ab-ba}{\mathrm{h}}\right)=\{\operatorname{pr}{a},\operatorname{pr}{b}\},\quad
\text{and}\quad
\operatorname{pr}\left(\frac{\Delta(a)-\Delta^{op}(a)}{\mathrm{h}}\right)=\delta(\operatorname{pr}(a)),
$$
where $\Delta: A_\mathrm{h}(Q)\to A_\mathrm{h}(Q)\otimes_{\mathbf{k}[\mathrm{h}]}A_\mathrm{h}(Q)$ denotes the
coproduct in the Hopf algebra $A_\mathrm{h}(Q)$,
and where $\Delta^{op}$ stands for the map
$\Delta$ composed with the flip of the two factors in
$A_\mathrm{h}(Q)\otimes_{\mathbf{k}[\mathrm{h}]}A_\mathrm{h}(Q).$
\subsection{Moyal quantization for quivers.} In \cite{S}, the Hopf algebra
$A_\mathrm{h}(Q)$ was defined, essentially, by generators
and relations. Thus, the algebra
$A_\mathrm{h}(Q)$ may be viewed, roughly, as a quiver analog
of the Weyl algebra $A_\mathrm{h}(V)$. One of the main
results proved in \cite{S} is a version of
Poincar\'e-Birkhoff-Witt (PBW) theorem. The PBW theorem
insures
that $A_\mathrm{h}(Q)$ is isomorphic
to $({\text{Sym\ }} L)[\mathrm{h}]$ as a $\mathbf{k}[\mathrm{h}]$-module, in particular, it is
flat over $\mathbf{k}[\mathrm{h}]$.
The goal of the present paper is to
provide an alternative construction of the Hopf algebra
$A_\mathrm{h}(Q)$. Instead of defining the algebra
by generators and relations, we define
a multiplication $m$ and comultiplication
$\Delta$ on the vector space
$({\text{Sym\ }} L)[\mathrm{h}]$ by explicit formulas which
are both analogous to
formula \eqref{star} for the Moyal star-product. In fact, suitably interpreted,
they will be written as $f *_\mathrm{h} g = m \circ e^{\frac{1}{2} \mathrm{h} \pi}(f \otimes g)$
and $\Delta_h(f) = e^{\frac{1}{2} \mathrm{h} \pi} f$.
We directly check
associativity, coassociativity
and compatibility of $m$ and $\Delta$.
Thus, the present approach is (up to some difficulties
involving the antipode) independent of that used in \cite{S}.
Further, in complete analogy with the case of Moyal-Weyl
quantization, we construct a symmetrization
map $\Phi: ({\text{Sym\ }} L)[\mathrm{h}]\to A_\mathrm{h}(Q)$. This map is
a bijection, and we show that Hopf algebra structure
on $({\text{Sym\ }} L)[\mathrm{h}]$ defined in this paper may
be obtained by transporting the Hopf algebra structure
on $A_\mathrm{h}(Q)$ defined in \cite{S} via $\Phi$.
\subsection{Representations for the Moyal quantization.}
In \cite{G}, an interesting representation of the necklace Lie algebra is presented
which is quantized in \cite{S}. Namely, for any representation of the
double quiver $\overline{Q}$ assigning to each arrow $e \in \overline{Q}$ the
matrix $M_e: V_{t(e)} \rightarrow V_{h(e)}$, we can consider the map $L \rightarrow
\mathbf{k}$ given by $e_1 e_2 \cdots e_m \mapsto {\text{tr}}(M_{e_1} M_{e_2} \cdots M_{e_m})$. More
generally, if $\mathbf{l} \in {\mathbb Z}_{\geq 0}^I$, then we can consider the representation
space $\mathrm{Rep}_{\mathbf{l}}(\overline{Q})$ of representations with dimension vector $\mathbf l$,
meaning that $\mathrm{dim}\ V_i = l_i$. Then this is a vector space of dimension
$\sum_{e \in \overline{Q}} l_{t(e)} l_{h(e)}$. It has a natural bivector $\pi((M_{e})_{ij},
(M_{f})_{kl}) = \delta_{il} \delta_{jk} [e,f]$, where $[e,f] = 1$ if $e \in Q, f = e^*$
and $[e,f] = -1$ if $f \in Q, e = f^*$, with $[e,f]=0$ otherwise. We then have
the Poisson algebra homomorphism
\begin{equation}\label{trrep}
{\text{tr}}_{\mathbf{l}}: {\text{Sym\ }} L \rightarrow \mathbf{k}[\mathrm{Rep}_{\mathbf{l}}(\overline{Q})], \quad
{\text{tr}}_{\mathbf{l}}(e_1 e_2 \cdots e_m)(\psi) = {\text{tr}}(M_{e_1} M_{e_2} \cdots M_{e_m}).
\end{equation}
In \cite{S}, this representation was quantized by a representation $\rho_{\mathbf{l}}: A \rightarrow
\mathcal D(\mathrm{Rep}_{\mathbf{l}}(Q))$, where the latter is the space of differential operators
with polynomial coefficients on $\mathrm{Rep}_{\mathbf{l}}(Q)$. We may modify the $\rho_{\mathbf{l}}$
and $A$ slightly to obtain $\rho_{\mathbf{l}}^\mathrm{h}, A_\mathrm{h}$ so that we have the following diagram:
\begin{equation}
\xymatrix{
{\text{Sym\ }} L \ar[rr]_-{\mathrm{asympt. inj.}}^{{\text{tr}}_{\mathbf{l}}} & & \mathbf{k}[{\mathrm{Rep}}_{\mathbf{l}}(\overline{Q})] \\
A_\mathrm{h} \ar[u] \ar[rr]_-{\mathrm{asympt. inj.}}^{\rho_{\mathbf{l}}^\mathrm{h}} & &
D_Q
\ar[u]}
\end{equation}
Here, $A_\mathrm{h}$ is obtained from $A$ by modifying (3.3) in \cite{S} so that the right-hand
side has an $\mathrm{h}$ just like (3.4). [Note: More generally, it makes sense to consider
the space where (3.3) has an independent formal parameter $\hbar$; for the Moyal
version, we want the two to be the same.] Then, the representations $\rho_{\mathbf{l}}^\mathrm{h}$
send elements $(e_1, 1) (e_2, 2) \cdots (e_m, m) \in A_\mathrm{h}$ (see \cite{S}: this is one
lift of $e_1 e_2 \cdots e_m \in L$)
to operators $\sum_{i_1, i_2, \cdots, i_m}
\iota(e_1)_{i_1 i_2} \iota (e_2)_{i_2 i_3} \cdots \iota(e_m)_{i_m i_1}$,
where $\iota(e)$ is the matrix $M_e$ if $e \in Q$, and $\iota(e^*) = M_{e^*}$ for
$e \in Q$, where $M_{e^*}$ is the matrix given by
$(M_{e^*})_{ij} = -\mathrm{h} \frac{\partial}{\partial (M_e)_{ji}}$. Then, the space $D_Q \subset
\mathcal D(\mathrm{Rep}_{\mathbf{l}}(Q))$ is just generated by $e_{ij},
-\mathrm{h} \frac{\partial}{\partial e_{ji}}$.
The diagram indicates that the representations are ``asymptotically injective'' in
the sense that the kernels of the representations $\rho_{\mathbf{l}}, {\text{tr}}_{\mathbf{l}}$ have zero
intersection, and moreover, for any finite-dimensional vector subspace $W$ of the
algebra $A$, there is a vector ${\mathbf{l}} \in N^I$ such that for each
${\mathbf{l}}' \geq {\mathbf{l}}$ (i.e.~such that $l_i' \geq l_i, \forall i$,
we have that $W \cap \text{Ker }{\text{tr}}_{\mathbf{l}} = 0$ (and similarly for $\rho$).
By construction of the map $\Phi_W$,
the Moyal quantization fits into a diagram as follows:
\begin{equation} \label{2d}
\xymatrix{ {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \ar[rr]_{\mathrm{asympt. inj.}}^{{\text{tr}}_{\mathbf{l}}[\mathrm{h}]} \ar[d]
\ar@/_5pc/[dd]^{\Phi_W}_{\sim} & & \mathbf{k}[\mathrm{h}][{\mathrm{Rep}}_{\mathbf{l}}(\overline{Q})]_{\mathrm{Moyal}} \ar[d]
\ar@/^5pc/[dd]^{\phi_W}_{\sim} \\ {\text{Sym\ }} L
\ar[rr]_-{\mathrm{asympt. inj.}}^{{\text{tr}}_{\mathbf{l}}} & & \mathbf{k}[{\mathrm{Rep}}_{\mathbf{l}}(\overline{Q})] \\ A_\mathrm{h} \ar[u]
\ar[rr]_-{\mathrm{asympt. inj.}}^{\rho^h_{\mathbf{l}}} & & D_Q \ar[u] }
\end{equation}
Here, we denote by $\mathbf{k}[\mathrm{h}][{\mathrm{Rep}}_{\mathbf{l}}(\overline{Q})]_{\mathrm{Moyal}}$ the Moyal quantization
of $\mathbf{k}[{\mathrm{Rep}}_{\mathbf{l}}(\overline{Q})]$ using the bivector $\pi$, and by ${\text{Sym\ }}
L[\mathrm{h}]_{\mathrm{Moyal}}$ the quiver version to be defined in this article. Because
of the asymptotic injectivity, to prove that a Moyal quantization
exists completing the diagram, all that is necessary is the map
$\Phi_W$; then the definitions of the product, coproduct, and antipode
follow. However, the definitions are interesting in their own right.
\subsection{Organization of the article.}
The article is organized as follows: In Section \ref{mps}, we
will define the Moyal product $*_\mathrm{h}$ on ${\text{Sym\ }} L[\mathrm{h}]$. In Section \ref{phiws},
we define the map $\Phi_W$. Next, in Section \ref{mpts}, we show
that this transports the product on
$A_\mathrm{h}$ to the product $*_\mathrm{h}$. Finally, in Section \ref{ass}, we directly prove the
associativity of $*_\mathrm{h}$.
In Section \ref{cps} we define
the Moyal coproduct $\Delta_\mathrm{h}$. Then, in Section \ref{cpts}, we show that $\Delta_\mathrm{h}$
is obtained by transporting the coproduct from $A_\mathrm{h}$ using $\Phi_W$.
Section \ref{casss} proves directly that $\Delta_\mathrm{h}$ is coassociative, and
Section \ref{bas} shows directly that $*_\mathrm{h}, \Delta_\mathrm{h}$ are compatible, inducing
a bialgebra structure on ${\text{Sym\ }} L[h]_{\mathrm{Moyal}}$.
In Section \ref{as} we give the definition
of antipode $S$, which clearly is the one obtained from $A_\mathrm{h}$ by transportation.
This makes ${\text{Sym\ }} L[h]_{\mathrm{Moyal}}$ a Hopf algebra satisfying $S^2 = \mathrm{Id}$. The eigenvectors
of $S$ are just products of necklaces, with eigenvalue $\pm 1$ depending on the parity
of the number of necklaces.
We will make use of the following tensor convention throughout:
\begin{ntn}
If $S, T$ are $\mathbf{k}[\mathrm{h}]$-modules, then we will always mean by $S \otimes T$ the
tensor product over $\mathbf{k}[\mathrm{h}]$ (never over $\mathbf{k}$).
\end{ntn}
\subsection{Acknowledgements}
The authors would like to thank Pavel Etingof for useful discussions.
The work of both authors was partially supported by the NSF.
\section{The Moyal product}
\subsection{Definition of the Moyal product $*_\mathrm{h}$.}\label{mps}
To define the product $*_\mathrm{h}$ on ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$, we proceed
by analogy: let $\pi = \sum_{e \in Q} \frac{\partial}{\partial e} \otimes
\frac{\partial}{\partial e^*} - \frac{\partial}{\partial e^*} \otimes
\frac{\partial}{\partial e}$. For each $n \geq 0$, we define an
operator $\pi^n: {\text{Sym\ }} L \otimes {\text{Sym\ }} L \rightarrow {\text{Sym\ }} L$, and hence
$e^{\frac{1}{2} \mathrm{h} \pi}: {\text{Sym\ }} L[\mathrm{h}] \otimes {\text{Sym\ }} L[\mathrm{h}] \rightarrow {\text{Sym\ }} L[\mathrm{h}]$ as
follows. We define the action of each
\begin{equation} \label{act}
T = \frac{\partial}{\partial a_1} \frac{\partial}{\partial a_2}
\cdots \frac{\partial}{\partial a_m} \otimes \frac{\partial}{\partial
a_1^*} \frac{\partial}{\partial a_2^*} \cdots \frac{\partial}{\partial
a_m^*}, \quad a_i \in \overline{Q}, (e^*)^* := e;
\end{equation}
and extend by linearity. This action is best described by considering
monomials in ${\text{Sym\ }} L$ to be collections of closed paths in $\overline{Q}$.
Each closed path corresponds to a single cyclic monomial of $L$, so a
collection of closed paths corresponds to a symmetric product of the
corresponding cyclic monomials, giving an element of ${\text{Sym\ }} L$. Such
elements generate all of ${\text{Sym\ }} L$.
Take any operator of the form \eqref{act}, and two elements $P, R \in
{\text{Sym\ }} L$, which are symmetric products (i.e.~collections) of closed
paths. Then the element $T$ of \eqref{act} acts on $P \otimes R$ by
summing over all ordered choices of distinct instances of edges $e_1,
e_2, \cdots, e_m$ in the graph of $P$ such that $e_i$ is identical
with $a_i$ as elements of $\overline{Q}$, and over all ordered choices of
distinct instances of edges $f_1, f_2, \cdots, f_m$ in the graph of
$R$ such that $f_i$ is identical with $e_i^*$ as elements of $\overline{Q}$,
and adding the following element: Delete each $e_i$ from $P$ and each
$f_i$ from $R$, and join $P$ and $R$ at each $h(e_i) = t(f_i)$ and
each $h(f_i) = t(e_i)$. The result is some element $Z \in {\text{Sym\ }} L$
obtained from $P \otimes R$, which is some new collection of closed paths (or
isolated vertices, which correspond to idempotents).
So, $T(P \otimes R)$ is the sum of all such elements $Z$ (some of them can
be identical, of course; we are summing over the element $Z$ we get
for each choice of instances of the given edges in $P$ and $R$).
Let us more precisely define this deletion and gluing process (as
in \cite{S}). We can
define an ``abstract edge'' of an element
\begin{multline} \label{pform}
P = a_{11} a_{12} \cdots a_{1 l_1} \& a_{21} a_{22} \cdots a_{2 l_2}
\& \cdots \& a_{k1} a_{k2} \cdots a_{k l_k} \\ \& v_1 \& v_2 \& \cdots
\& v_q \in {\text{Sym\ }} L
\end{multline}
to be an index $(i,j)$ where $1 \leq i \leq k$ and $1 \leq j \leq
l_k$. Here we note that $v_i \in I$, the set of vertices of the
quiver, which act as idempotents in the path algebra of the double
quiver. These indices are considered as edges, just where we keep
track of which occurrence of the edge of $\overline{Q}$ we are considering. Let
$X$ be the set of abstract edges of such an element $P$; then there is
a natural map $\mathrm{pr}_X: X \rightarrow \overline{Q}$ which gives the element of
$\overline{Q}$ defined by the given edge.
To cut and glue for a single such element $P$, we need a set of
``cutting edges,'' $I \subset X$, along with a (fixed-point free)
involution $\phi: I \rightarrow I$ such that $\mathrm{pr}_X \circ \phi = *
\circ \mathrm{pr}_X$, where $*: \overline{Q} \rightarrow \overline{Q}$ is the edge reversal
operation. Then we can define a map $f: X \rightarrow X$ which takes
each edge $(i,j) \notin I$ to the next edge, $(i,j+1)$ (where $j+1$ is
taken modulo $l_i$); and takes each edge $(i,j) \in I$ to $\phi(i,j) +
1$, where the ``$+1$'' operation is just $(i,j) + 1 = (i,j+1)$, again
where $j+1$ is taken mod $l_i$. The map $f$ is bijective, and
each orbit of $X$ under $f$ is of the form $(x_1, x_2, \ldots, x_p)$
where $f(x_i) = x_{i+1}$, taken modulo $p$. Each such orbit defines a
cyclic monomial or idempotent as follows: for each $x_i$, let
$\mathrm{pr}'(x_i) = \mathrm{pr}_X(x_i)$ if $x_i \notin I$, and $\mathrm{pr}'(x_i) = t(x_i)$,
the starting vertex idempotent, if $x_i \in I$. So $\mathrm{pr}'$ extends
to $\mathrm{pr}'(x_1, x_2, \ldots, x_p) = \mathrm{pr}'(x_1) \mathrm{pr}'(x_2) \cdots \mathrm{pr}'(x_p)
\in L$, which gives us the desired cyclic monomial or vertex
idempotent. Then the result of cutting and gluing along the edges $I$
is simply the symmetric product of $\mathrm{pr}'$ applied to all orbits of $X$
under $f$, symmetric-multiplied by $v_1 \& v_2 \& \cdots \& v_q$ (the
original vertex idempotents are ``untouched'' by cutting and gluing at
edges).
Given two elements $P, R$ of the form \eqref{pform} (except for
different indices $i_l, k, m$, and different edges $a_{ij}$ etc.), we
can cut and glue $P$ and $R$ in an analogous way as follows: Let $X,
Y$ be the sets of abstract edges of $P$ and $R$, and $\mathrm{pr}_X, \mathrm{pr}_Y$
the projections to $\overline{Q}$. Then we can cut and glue along subsets $I_X
\subset X, I_Y \subset Y$ equipped with a bijection $\phi: I_X
\rightarrow I_Y$ such that $\mathrm{pr}_Y \circ \phi = * \circ \phi$, much in
the same way as the above: first, extend $\phi$ by $\phi^{-1}$ to
$I_Y$ to get an involution on $I_X \sqcup I_Y$. Then we take $X \sqcup
Y$, look at orbits of this under the map $f$ defined just as above,
and then define the map $\mathrm{pr}'$ just as above (except that we need to
use $\mathrm{pr}_Y$ instead of $\mathrm{pr}_X$ on edges of $Y$), and
symmetric-multiply the result with any vertex idempotents appearing in
the original formulas for $P$ and $R$.
It is this latter operation which is what we precisely meant when we
spoke of ``cutting along edges and gluing the endpoints'' in the
definition of \eqref{act}. In that case, we are summing over all
ordered choices of distinct elements $x_1, x_2, \ldots, x_m \in X$ and
$y_1, y_2, \ldots, y_m \in Y$, such that $\mathrm{pr}_X(x_i) = e_i$ and
$\mathrm{pr}_X(y_i) = e_i^*$. Then we let $I_X = \{x_1, \ldots, x_m\}$ and
$I_Y = \{y_1, \ldots, y_m\}$ and $\phi(x_i) = y_i$, and perform
cutting and gluing (multiplying in some coefficient in $\mathbf{k}[\mathrm{h}]$).
Now that we have defined the action of \eqref{act}, we can extend
linearly over $\mathbf{k}$ to obtain the action of $\pi^n: {\text{Sym\ }} L \otimes {\text{Sym\ }} L
\rightarrow {\text{Sym\ }} L$ for any $n$, and by linearity over $\mathbf{k}[\mathrm{h}]$, also
$e^{\frac{1}{2} \mathrm{h} \pi}: {\text{Sym\ }} L[\mathrm{h}] \otimes {\text{Sym\ }} L[\mathrm{h}] \rightarrow {\text{Sym\ }}
L[\mathrm{h}]$. (Note that only polynomials in $\mathrm{h}$ are required since the
application of any differential operator of degree greater than the
total number of edges appearing in a given $P \otimes R$ is zero).
Now, we define $*_\mathrm{h}: {\text{Sym\ }} L[\mathrm{h}] \otimes {\text{Sym\ }} L[\mathrm{h}] \rightarrow {\text{Sym\ }} L[\mathrm{h}]$ by
\begin{equation}
P *_\mathrm{h} R = e^{\frac{1}{2} \mathrm{h} \pi} (P \otimes R).
\end{equation}
This defines the necessary product which allows us to define ${\text{Sym\ }}
L[\mathrm{h}]_{\mathrm{Moyal}}$.
We can describe this more directly as follows: again let $P, R$ be of
the form \eqref{pform} with sets of abstract edges $X, Y$, respectively,
and maps $\mathrm{pr}_X: X \rightarrow \overline{Q}, \mathrm{pr}_Y: Y \rightarrow \overline{Q}$.
Then
\begin{equation}
P *_\mathrm{h} R = \sum_{(I_X, I_Y, \phi)} \frac{\mathrm{h}^{\#(I_X)}}{2^{\#(I_X)}}
s(I_X, I_Y, \phi)
PR_{I_X, I_Y, \phi},
\end{equation}
where $(I_X, I_Y, \phi)$ is any triple of a subset $I_X \subset X, I_Y
\subset Y$ and a bijection $\phi: I_X \rightarrow I_Y$ satisfying
$\mathrm{pr}_Y \circ \phi = * \circ \mathrm{pr}_X$, and $PR_{I_X, I_Y, \phi}$ is the
result of cutting and gluing $P$ and $R$ along this triple as
described previously. The sign $s(I_X, I_Y, \phi)$ is defined by
$s(I_X, I_Y, \phi) = (-1)^{\#(I_Y \cap \mathrm{pr}_Y^{-1}(Q))}$. This follows
because $e^{\frac{1}{2} \mathrm{h} \pi} = \sum_{N \geq 0} \frac{\mathrm{h}^N}{2^N}
\frac{\pi^N}{N!}$, and each $\pi^N$ involves a sum over all cuttings and
gluings of $P$ and $R$ along $N$ edges counting each ordering and
multiplying in $-1$ for each time the $\frac{\partial}{\partial e}$
appears in the second component for $e \in Q$; dividing by $N!$ means
we don't count orderings of $I_X$ so that it is only over subsets that
we sum.
In general, elements $P, R \in {\text{Sym\ }} L[\mathrm{h}]$ are linear combinations over
$\mathbf{k}[\mathrm{h}]$ of such collections of necklaces, so the element $P *_\mathrm{h} R$
is given by summing over each choice of necklace collections in $P$
and $R$, of the product of the coefficients of the two necklace
collections and the element described in the previous paragraph. In
other words, we sum over all ways to take the product of terms from $P$
and $R$, not just by the usual product in ${\text{Sym\ }} L[\mathrm{h}]$, but also by
$\frac{\mathrm{h}^p}{2^p}$ times the ways in which we can cut out $p$ matching
edges from each term and join them together (while just multiplying
the $\mathbf{k}[\mathrm{h}]$-coefficients).
\subsection{Definition of the symmetrization map $\Phi_W$.} \label{phiws}
Now, we define $\Phi_W: {\text{Sym\ }} L[\mathrm{h}] \rightarrow A_\mathrm{h}$. To do this, we
need to define the notion of ``height assignments''. Let's consider a
collection of necklaces $P$ of the form \eqref{pform}. Let $X$ be the
set of abstract edges of $P$, say $\#(X) = N$. Then, a \textsl{height
assignment} for $P$ is defined to be a bijection $H: X \rightarrow
\{1, 2, \ldots, N\}$.
We have the
element $P_H \in A_\mathrm{h}$ obtained by assigning heights to the edges in
$X$ by $H$, that is
\begin{multline}
P_H = (a_{11}, H(1,1)) \cdots (a_{1 l_1},
H((1, l_1)) \& \cdots \\ \& (a_{k 1}, H(k, 1)) \cdots (a_{k l_k},
H(k, l_k)) \& v_1 \& v_2 \& \cdots \& v_q.
\end{multline}
Note that we could also think of $H$ as an element of $S_N$ with some
modifications to the formula.
The element $\Phi_W$ involves taking an average over all
height assignments:
\begin{equation}
\Phi_W(P) = \frac{1}{N!} \sum_{H} P_H,
\end{equation}
where $H$ ranges over all height assignments. Following is the alternative
description in terms of permutations $S_N$:
Let $\theta(i,j) = j + \sum_{p = 1}^{i-1} l_p$ so that $\theta(k,
l_k) = N$. Then
\begin{multline}
\Phi_W(a_{11} \cdots a_{1 l_1} \& a_{21} \cdots
a_{2 l_2} \& \cdots \& a_{k 1} \cdots a_{k l_k} \& v_1 \& v_2 \& \cdots \& v_q) \\ = \sum_{\sigma
\in S_N} \frac{1}{N!} (a_{11}, \sigma(\theta(1,1))) \cdots (a_{1 l_1},
\sigma(\theta(1, l_1))) \& \cdots \\ \& (a_{k 1}, \sigma(\theta(k, 1))) \cdots
(a_{k l_k}, \sigma(\theta(k, l_k))) \& v_1 \& v_2 \& \cdots \& v_q.
\end{multline}
\subsection{Proof that $*_\mathrm{h}$ is obtained from $\Phi_W$.}\label{mpts}
Let's show that $\Phi_W$ makes the diagram \eqref{2d} commute. We know
that $\Phi_W$ is an isomorphism of free $\mathbf{k}[\mathrm{h}]$-modules (using PBW
for $A_\mathrm{h}$, or the fact that $\rho_{\mathbf{l}}$ is asymptotically injective and
the fact that the Weyl symmetrization map is an isomorphism on the
right-hand side of \eqref{2d}). So, once we show commutativity of the
diagram, it will follow that $\Phi_W$ induces some multiplicative
structure on ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$ making the $\Phi_W$ an isomorphism of
$\mathbf{k}[\mathrm{h}]$-algebras. We will then want to show that this structure is
the one we have just defined, i.e.~to show that $\Phi_W$ is a
homomorphism of rings using our $*_\mathrm{h}$ structure.
We need to show that $\rho_{\mathbf{l}} \circ \Phi_W = \phi_W \circ {\text{tr}}$. This
follows immediately from the definitions, because $\rho_{\mathbf{l}} \circ
\Phi_W$ involves summing over the symmetrization of polynomials in
$(M_e)_{ij}, \frac{\partial}{\partial (M_e)_{ji}}, e \in Q$ where
$(M_e)_{ij}$ are the coordinate functions of the matrix corresponding
to the vertex $e$; also, ${\text{tr}}$ takes an element of ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$
and gives the element of $\mathbf{k}[\mathrm{h}][Rep_{\mathbf{l}}(\overline{Q})]$ corresponding to the
trace of the (cyclic noncommutative) polynomial, which upon
substituting $(M_{e^*})_{ij} \mapsto -h \frac{\partial}{\partial (M_e)_{ji}}$
and symmetrizing (which we needed to do for this to be well-defined,
since the $(M_{e^*})_{ij}, (M_e)_{ij}$ commuted), gives the same element.
Next, let us show that the ring structure obtained from $\Phi_W$, making
$\Phi_W$ an isomorphism of rings, is exactly the product $*_\mathrm{h}$ we
have described in detail.
\begin{equation}
\label{pqe}
\Phi_W(P *_\mathrm{h} R) = \Phi_W(P) \Phi_W(R).
\end{equation}
Now we prove \eqref{pqe}. Let's take $P = P_1 \& P_2 \& \cdots \&
P_n$, as before, to be a collection of necklaces, and similarly for $R
= R_1 \& R_2 \& \cdots \& R_m$. (We can forget about the idempotents
such as in \eqref{pform}, since they won't affect what we have to
prove.) Let $X$ be the set of abstract edges of $P$ and $Y$ the set
of abstract edges of $R$. We will use $H_P: X \rightarrow \{1,
\ldots, |X|\}$ to denote a height assignment for $P$ and $H_R: Y
\rightarrow \{1, \ldots, |Y|\}$ to denote a height assignment for $R$.
Let us say that a height assignment $H_{PR}: X \sqcup Y \rightarrow
\{1, \ldots, |X|+|Y|\}$ \textsl{extends} height assignments $H_P, H_R$
if $H_{PR}$ restricted to $P$ is equivalent to $H_P$ and $H_{PR}$
restricted to $R$ is equivalent to $H_R$. In other words, $H_{PR}(x_1)
< H_{PR}(x_2)$ iff $H_P(x_1) < H_P(x_2)$ for all $x_1, x_2 \in X$, and
similarly $H_{PR}(y_1) < H_{PR}(y_2)$ iff $H_R(y_1) < H_R(y_2)$ for
all $y_1, y_2 \in Y$.
Now, we know that
\begin{equation}
\Phi_W(P *_\mathrm{h} R) - \Phi_W(PR) = \sum_{N = 1}^\infty \frac{\mathrm{h}^N}{2^N}
\Phi_W(\frac{\pi^N}{N!} (P \otimes R)),
\end{equation}
and also that
\begin{multline}
\Phi_W(P) \Phi_W(R) - \Phi_W(PR) \\ = \frac{1}{(|X|+|Y|)!}
\sum_{H_P, H_R} \sum_{H_{PR} \text{\ extending\ }H_P, H_R} (P_{H_P} R_{H_R}
- PR_{H_{PR}}).
\end{multline}
We are left to show, using the relations which define $A_\mathrm{h}$, that
\begin{equation} \label{ltsiso}
\sum_{N = 1}^\infty \frac{\mathrm{h}^N}{2^N} \Phi_W(\frac{\pi^N}{N!} (P \otimes R)) =
\sum_{H_{PR} \text{\ extending\ }H_P, H_R} (P_{H_P} R_{H_R}
- PR_{H_{PR}})
\end{equation}
To prove this, let us fix a particular $H_P, H_R$, and $H_{PR}$, and
expand $P_{H_P} R_{H_R} - PR_{H_{PR}}$ using
the relations that define $A_\mathrm{h}$. We do this by expressing this as a sum
of commutators obtained by commuting a single edge coming from $R$
with a single edge coming from $P$. We get
\begin{equation}
P_{H_P} R_{H_R} - PR_{H_{PR}} = \sum_{\underset{\mathrm{pr}_X(x) = \mathrm{pr}_Y(y)^*}{x \in X, y \in Y \text{\ such that\
} H_P(x) > H_R(y),} } [\mathrm{pr}_X(x), \mathrm{pr}_Y(y)] \mathrm{h} PR'_{x,y},
\end{equation}
where $PR'_{x,y}$ corresponds to taking $PR$, deleting $x$ and $y$ and
joining the endpoints, and using the height assignment which is
(equivalent to the choice of heights) identical to $H_P$ on elements
$x' \in X$ such that $H_P(x') < H_P(x)$, and equal to $H_P(x) +
H_{PR}(z)$ for all other $z \in X \sqcap Y \setminus \{x,y\}$. Here
we say ``equivalent to the choice of heights'' in parentheses to mean
that the given assignment won't be an assignment to $\{1, \ldots,
|X|+|Y|-2\}$ as we strictly defined height assignments, but we could
extend the definition of height assignments to include any injective
map to ${\mathbb Z}$, and say that two are equivalent if the ordering is the
same ($H \equiv H'$ if $H(z) < H(z')$ iff $H'(z) < H'(z')$); so really
we are looking for the height assignment mapping to $\{1, \ldots,
|X|+|Y|-2\}$ which is equivalent to the assignment we described. Also
note here that $[e,e^*] = 1$ if $e \in Q$ and $-1$ if $e^* \in Q$.
By applying the relations repeatedly we get that
\begin{multline} \label{hpre}
P_{H_P} R_{H_R} - PR_{H_{PR}} \\ = \sum_{\underset{\text{\ such that\ }
H_P(x_i) > H_R(y_i), \mathrm{pr}_X(x_i)
= \mathrm{pr}_Y(y_i)^*}{x_1, \ldots, x_k \in X, y_1,
\ldots, y_k \in Y } } [\mathrm{pr}_X(x_1), \mathrm{pr}_Y(y_1)] \\ \cdots [\mathrm{pr}_X(x_k),
\mathrm{pr}_Y(y_k)] \mathrm{h}^k PR''_{(x_1, y_1), \ldots, (x_k, y_k)},
\end{multline}
where $PR''_{(x_1, y_1), \ldots, (x_k, y_k)}$ involves taking $PR$ and
deleting the pairs of edges and gluing at their respective endpoints;
and this time assigning heights by restricting $H_{PR}$ to $X \sqcup Y
\setminus \{x_1, \ldots, x_k, y_1, \ldots, y_k\}$ (and changing to an
equivalent height assignment which has image $\{1, \ldots,
|X|+|Y|-2k\}$).
Now, let's look at the sum again (no longer fixing $H_P, H_R$, and
$H_{PR}$). We see that for any given choice of pairs $(x_1, y_1),
\ldots, (x_k, y_k)$ with $\mathrm{pr}_X(x_i) = \mathrm{pr}_Y(y_i)^*$, the summands
that involve deleting these pairs and gluing their endpoints are the
same in number for each choice of height assignment for the deleted
pairs. The coefficient for each height is just
$\frac{\mathrm{h}^k}{(|X|+|Y|)!}$ times the number of height assignments
$H_{PR}$ that restrict to the given height assignment, and also
satisfy $H_{PR}(x_i) > H_{PR}(y_i)$ for all $1 \leq i \leq k$. In
other words, this is $\mathrm{h}^k$ times the probability of picking a height
assignment randomly of $PR$ that has $x_i$ greater in height than
$y_i$ for all $i$, and is identical with the given height assignment
on all $x, y \notin \{x_1, \ldots, x_n, y_1, \ldots, y_n\}$. So we
get that the coefficient is $\frac{\mathrm{h}^k}{2^k (|X|+|Y|-2k)!}$.
But this is exactly what we would expect, desiring that \eqref{ltsiso}
hold. That is because the left-hand side, as described previously in
our discussion of $\frac{\pi^N}{2^N}$, just involves summing over all
$N$ of $\frac{\mathrm{h}^k}{2^k}$ times $\Phi_W$ of the collection of
necklaces described for each choice of pairs $(x_1, y_1), \ldots,
(x_N, y_N)$ with some choice of sign; and then $\Phi_W$ just sums over
$\frac{1}{(|X|+|Y|-2N)!}$ times each choice of height assignment for
this collection of necklaces. The sign choice just matches exactly with
the sign $\prod_{i} [\mathrm{pr}_X(x_i), \mathrm{pr}_Y(y_i)]$ appearing in \eqref{hpre},
since each commutator is $-1$ just in the case that $\mathrm{pr}_Y(y_i) \in Q$.
This proves \eqref{ltsiso} and hence that $\Phi_W$ is an isomorphism
of $\mathbf{k}[\mathrm{h}]$-algebras, using $*_\mathrm{h}$ as the ring structure on ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$.
\subsection{Associativity of $*_\mathrm{h}$.} \label{ass}
Although we already know from commutativity of the diagram and associativity
of $A_\mathrm{h}$ that $*_\mathrm{h}$ is associative, it is easy to prove directly. We
prove
\begin{equation} \label{tpass}
(P *_\mathrm{h} R) *_\mathrm{h} S = P *_\mathrm{h} (R *_\mathrm{h} S)
\end{equation}
where $P, R$, and $S$ are collections of necklaces of the form \eqref{pform}
(with different indices).
First we describe the left-hand side of \eqref{tpass}
Let $X, Y$, and $Z$ be the sets of abstract edges
of $P, R,$ and $S$, and let $\mathrm{pr}_X: X \rightarrow \overline{Q}, \mathrm{pr}_Y: Y
\rightarrow \overline{Q}$, and $\mathrm{pr}_Z: Z \rightarrow \overline{Q}$ be the projections
from occurrences of edges to edges of $\overline{Q}$.
We sum over all sets of pairs $\{(x_1, y_1), \ldots, (x_N, y_N)\}
\subset X \times Y$, such that $y_i = x_i^*$ for each $i$ (and we
assume that the $x_i$ and the $y_i$ are all distinct). Summing over
$\frac{\mathrm{h}^N}{2^N}$ times the necklaces we get by cutting out these
pairs of edges and gluing their endpoints, we get $P *_\mathrm{h} R$ as
described in the previous section.
To get $(P *_\mathrm{h} R) *_\mathrm{h} S$, we will first be summing over choices of
pairs $\{(x_1, y_1), \ldots (x_N, y_N)\}$, and then over pairs
$\{(w_1, z_1), \ldots, (w_M, z_M)\} \subset W \times Z$, where $W = (X
\setminus \{x_1, x_2, \ldots, x_N\}) \sqcup (Y \setminus \{y_1, y_2,
\ldots, y_N\})$, and performing a similar operation. We can also
describe this as summing over pairs $(x_1, y_1), \ldots, (x_N, y_N),
(x_{N+1}, z_1), (x_{N+2}, z_2), \ldots, (x_{N+M_1}, z_{M_1}),$ \\
$(y_{N+1}, z_{M_1+1}), (y_{N+2}, z_{M_1+2}), \ldots, (y_{N+M_2},
z_{M_1+M_2})$, again where all $x_i, y_i,$ and $z_i$ are distinct, and
in each pair, one edge is the reverse of the other. This description,
along with signs and coefficients, is exactly the same as what we
could obtain in the same way from $P *_\mathrm{h} (R *_\mathrm{h} S)$, proving
associativity.
\section{The Moyal coproduct}
\subsection{Definition of $\Delta_\mathrm{h}$.}\label{cps}
There is a nice formula for the coproduct on ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$ compatible
with the the $*_\mathrm{h}$ product. The formula is actually surprisingly
similar to the one for $*_\mathrm{h}$. We will be giving the coproduct which
makes the diagram \eqref{2d} consist of coalgebra homomorphisms
(namely, the maps ${\text{tr}}$ and $\Phi_W$ involving ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$); the map
$\Phi_W$ will then be an isomorphism of bialgebras. The coproduct can
be described as follows: We need to define an operator $e^{\frac{1}{2}
\mathrm{h} \pi}: {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \rightarrow {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \otimes {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$. To do
this, we set
\begin{equation}
\pi = \sum_{e \in Q} \frac{\partial}{\partial e} \frac{\partial}{\partial e^*}
\end{equation}
and we define operators
\begin{multline} \label{bco}
\frac{\partial}{\partial e_1} \frac{\partial}{\partial e_1^*} \frac{\partial}{\partial e_2} \frac{\partial}{\partial e_2^*} \cdots \frac{\partial}{\partial e_N}\frac{\partial}{\partial e_N^*}: \\ {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \rightarrow {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \otimes {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}.
\end{multline}
The operator \eqref{bco} acts as follows: Taking a collection of
necklaces $P = P_1 \& P_2 \& \cdots \& P_n$ \\ $\& v_1 \& v_2 \& \cdots \& v_q
\in {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$, where
each $P_i \in L$ is a cyclic monomial (i.e.~a necklace), let $X$ be
the set of abstract edges of $P$ and $\mathrm{pr}_X: X \rightarrow \overline{Q}$ the
projection (cf. Section \ref{mps}). Then we sum over all choices of
pairs $(x_1, y_1), \ldots, (x_N, y_N)$ such that the $x_i$ and $y_i$
are all distinct (the set $\{x_1, y_1, \ldots, x_N, y_N\}$ has $2N$
elements), and $\mathrm{pr}_X(x_i) = \mathrm{pr}_X(y_i)^*$ for all $i$. We delete the
edges and glue the endpoints, obtaining another collection of
necklaces. More precisely, the cutting and gluing is done as
described in the previous section, for $I = \{x_1, y_1, x_2, y_2,
\ldots, x_N, y_N\}$ and $\phi(x_i) = y_i$ for all $i$. Now, the only
difficult part is figuring out what components to assign to each
necklace (the first or second), and what sign to attach to each
choice.
We sum over all component assignments of the resulting chain of
necklaces: suppose that the above procedure yields the collection $R_1
\& R_2 \& \cdots \& R_m \in {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$ (this includes the original
idempotents $v_1, v_2, \ldots, v_q$); then the contribution to the
result of \eqref{bco} applied to $P$ is the following:
\begin{equation} \label{cass}
\sum_{\mathbf{c} \in \{1,2\}^m} s(\mathbf c, I, \phi)
R_1^{c_1} \& R_2^{c_2} \& \cdots \& R_m^{c_m},
\end{equation}
where $R_i^{c_i}$ denotes $R_i \otimes 1$ if $c_i = 1$ and $1 \otimes R_i$ if
$c_i = 2$, and the symmetric product in ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \otimes {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$
is the expected $(X \otimes Y) \& (X' \otimes Y') = (X \& X') \otimes (Y \& Y')$,
with $1 \& X = X \& 1 = X$ for all $X$. The term $s(\mathbf c, I, \phi)$
is a sign which is determined as follows:
\begin{equation}
s(\mathbf c, I, \phi) = s_1 s_2 \cdots s_n,
\end{equation}
where $s_i = 1$ if the component assigned to the start of
arrow $x_i$ is $1$ and the component assigned to the target of arrow
$x_i$ is $2$; $s_i = -1$ if the component assigned to the start of
arrow $x_i$ is $2$ and the component assigned to the target of arrow
$x_i$ is $1$; and $s_i = 0$ if the start and target are assigned the
same component.
Let's more precisely define what it means to say ``the component
assigned to the start/target of an arrow'' which is deleted from $P$.
What we mean by this is simply the orbit of the arrow $x_i$ in $X$
under $f$. Each orbit corresponds to one of the $R_i$. So, there is
a map $g: X \rightarrow \{1, 2, \ldots, m\}$, which corresponds to
which $R_i$ the ``start'' of each edge is assigned to. The
``targets'' are the same as the ``starts'' of the next edge, so that
$g(x+1)$ gives the component that the ``target'' of $x$ is assigned
to. Here the ``$+1$'' operation is once again $(i,j)+1=(i,j+1)$ mod
$l_i$, or in other words, $x+1$ is the edge succeeding $x$.
We then have that
\begin{equation} \label{scc}
s_i = \begin{cases} 1 & c_{g(x_i)} < c_{g(x_i)+1}, \\
0 & c_{g(x_i)} = c_{g(x_i)+1}, \\
-1 & c_{g(x_i)} > c_{g(x_i)+1}.
\end{cases}
\end{equation}
This assignment of signs has a combinatorial flavor because it is
essentially what the ``colorings'' of \cite{S} reduce to. There does
not seem to be a way to avoid this complication in choosing signs,
because the sign is what prevents the coproduct from being
cocommutative.
As before, we extend linearly to powers $\pi^N$ and to $e^{\frac{1}{2} \mathrm{h} \pi}$.
Then, the coproduct is given by
\begin{equation}
\Delta_\mathrm{h} = e^{\frac{1}{2} \mathrm{h} \pi}: {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \rightarrow {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \otimes \
{\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}},
\end{equation}
and as before we can describe this action on our element $P$ as
\begin{equation}
\Delta_\mathrm{h}(P) = \sum_{(I, \phi)} \frac{\mathrm{h}^{\#(I)/2}}{2^{\#(I)/2}} P_{I, \phi}, \\
\end{equation}
where the sum is over all $I \subset X$ with involution $\phi$ such
that $\mathrm{pr}_X \circ \phi = * \circ \mathrm{pr}_X$, and the element $P_{I, \phi}$
is given from the result of the cuttings and gluings by summing over
component assignments as described in \eqref{cass}.
\subsection{Proof that $\Delta_\mathrm{h}$ is obtained from $\Phi_W$.} \label{cpts}
Let's prove that this coproduct $\Delta_\mathrm{h}$ makes the diagram \eqref{2d}
consist of coalgebra homomorphisms. It suffices to prove that
$\Phi_W$ is a coalgebra homomorphism.
Take an element $P$ of the form \eqref{pform} with set of abstract
edges $X$ and projection $\mathrm{pr}_X: X \rightarrow \overline{Q}$. Now, let's
consider what the element $\Delta(\Phi_W(P))$ is in $A$. We know that
for each height assignment $H_P$ of $P$, $\Delta(P_{H_P})$ involves
summing over all pairs $(I, \phi)$ with $I \subset X$ and $\phi: I
\rightarrow I$ an involution satisfying $\mathrm{pr}_X \circ \phi = * \circ
\mathrm{pr}_X$, cutting and gluing as before. Then we sum over all component
assignments such that if $x, y \in I$ with $\phi(x) = y$, and the
heights satisfy $H(x) < H(y)$, then the component assigned to the
start of $x$ is $1$ and the component assigned to the target of $x$ is
$2$. When the components cannot be assigned in this way, this pair
$(I, \phi)$ cannot be used. These notions are all explained more
precisely in the preceding section.
Then we multiply in a sign $s(I, \phi, H)$ and a power of $\mathrm{h}$ determined
as follows: for each pair $x, y \in I$ with $\phi(x) = y, H(x) <
H(y)$, we multiply a $+1$ if $x \in Q, y \in Q^*$ and a $-1$ if $x \in
Q^*, y \in Q$. We also multiply in $\mathrm{h}^{\#(I)/2}$ (note: this power
of $\mathrm{h}$ is different from the one in \cite{S} simply because we are
describing the structure for $A_\mathrm{h}$, not $A$: it is easy to see in
general how the relations for the algebra and the formula for
coproduct change if we introduce a new formal parameter $\hbar$ into
(3.3) of \cite{S}).
So we find that $\Delta(P_H)$ is just a sum over cuttings and gluings,
and over component choices $\mathbf c$ compatible with the heights;
our sign choice satisfies $s(I, \phi, H) = s(\mathbf c, I, \phi)$,
where $I = \{x_1, y_2, \ldots, x_m, y_m\}$, and for all $i$, $x_i \in
Q$ and $\phi(x_i) = y_i$; finally, we multiply in $\mathrm{h}^m$ for cuttings
and gluings involving $\#(I)=2m$.
Hence, $\Delta(\Phi_W(P))$ is just given by a sum over all cuttings and
gluings $(I, \phi)$ together with component choice $\mathbf c$,
multiplying in $\mathrm{h}^{\#(I)/2}$, the sign $s(\mathbf c, I, \phi)$, and
the coefficient $\frac{1}{\#(P)!}$ where $\#(P)$ is the number of edges
in $P$, i.e.~the total number of height assignments.
Each summand in $\Delta(\Phi_W(P))$ is clearly given by a height
assignment of the term in $\Delta_\mathrm{h}(P)$ corresponding to the same $(I,
\phi, \mathbf c)$. For each term in $\Delta_\mathrm{h}(P)$, the coefficients of
the height-assigned terms in $\Delta(\Phi_W(P))$ are all the same. So
we see that $\Delta(\Phi_W(P)) = (\Phi_W \otimes \Phi_W)(P')$, for some $P'
\in {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}} \otimes {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$, where $\Phi_W \otimes \Phi_W (P \otimes R) =
\Phi_W(P) \otimes \Phi_W(R)$.
The element $P'$ can be computed just as we were computing $\Delta(\Phi_W(P))$, but instead of multiplying in $\frac{1}{\#(P)!}$, we need to multiply by the
fraction of all height choices compatible with this
component choice. But clearly, each pair $x, y \in I, \phi(x) = y$
induces a single restriction on the choice of heights, namely that
$H(x) < H(y)$ if the component assigned to the start of $x$ is $1$
and the component assigned to the target of $x$ is $2$, and $H(y) > H(x)$
if the opposite is true (the start of $x$ is assigned component $2$ and
the target assigned $1$). Note that the component assigned to the start
and target of $x$ cannot be the same for there to exist any compatible
height choices.
We see then that, provided a compatible height choice exists, we have
$\#(I)/2$ restrictions, each of which occur with independent probabilities
$\frac{1}{2}$. Hence the coefficient is just $\frac{1}{2^{\#(I)/2}}$. This
shows that $P' = \Delta_\mathrm{h}(P)$, proving that $\Phi_W$ is a coalgebra homomorphism
and hence an isomorphism of bialgebras. (In fact we have now proved that
$({\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}, *_\mathrm{h}, \Delta_\mathrm{h})$ is in fact a bialgebra).
\subsection{Coassociativity of $\Delta_\mathrm{h}$.} \label{casss}
Using the coassociativity of $A_\mathrm{h}$ from \cite{S}, we already know from
the fact that $\Phi_W$ is an isomorphism that the product $\Delta_\mathrm{h}$ is
coassociative, but it is not difficult to prove directly. We do that here
by proving
\begin{equation} \label{coasse}
(1 \otimes \Delta_\mathrm{h}) \Delta_\mathrm{h} (P) = (\Delta_\mathrm{h} \otimes 1) \Delta_\mathrm{h} (P),
\end{equation}
where once again $P$ is of the form \eqref{pform}.
The left-hand side can be described by summing over choices of cutting
pairs and components $(I, \phi, \mathbf c)$ for $P$, and then cutting
pairs and components for the first component of the result, $(I',
\phi', \mathbf {c'})$, and gluing, assigning the components, etc., and
multiplying by a sign and power of $\frac{\mathrm{h}}{2}$. We see that this
is the same as choosing just once the triple $(I'', \phi'', \mathbf
{c''})$, where $\mathbf {c''}$ assigns each necklace to one of three
components, $1, 2,$ or $3$, $I'' = I \cup I'$, and $\phi'' |_I = \phi,
\phi''|_{I'} = \phi'$. Then we can cut and glue just one time to get a
tensor in ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}^{\otimes 3}$; the sign and power of $\frac{\mathrm{h}}{2}$
can be determined by using \eqref{scc} where now the two sides of the
inequality have values in $\{1,2,3\}$.
For the same reason, the right-hand side of \eqref{coasse} can be
described in the preceding way, proving \eqref{coasse} and hence
coassociativity.
\subsection{The bialgebra condition for ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$.} \label{bas}
Using the fact that $A_\mathrm{h}$ is a bialgebra (proved in \cite{S}), we
know that ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$ is a bialgebra. But it is not difficult to
prove directly, which we do in this section.
We need to show, for collections of necklaces
$P = P_1 P_2 \cdots P_n, R = R_1 R_2 \cdots R_m$ of the form \eqref{pform},
that
\begin{equation} \label{bac}
\Delta(P *_\mathrm{h} R) = \Delta(P) *_\mathrm{h} \Delta(R),
\end{equation}
where we use the notation $(A \otimes B) *_\mathrm{h} (C \otimes D) = (A *_\mathrm{h} B) \otimes (C *_\mathrm{h} D)$.
First, define $X$ to be the set of abstract edges of $P$ and $Y$ to be
the set of abstract edges for $R$. Define the projections $\mathrm{pr}_X: X
\rightarrow \overline{Q}, \mathrm{pr}_Y: Y \rightarrow \overline{Q}$. Now, let us first take a
closer look at the right-hand side of \eqref{bac}. We can expand it
by the following sum over pairs. We first pick $I_X \subset X, I_Y
\subset Y,$ and involutions $\phi_X: I_X \rightarrow I_X, \phi_Y: I_Y
\rightarrow I_Y$ such that $\mathrm{pr}_X \circ \phi_X = * \circ \mathrm{pr}_X, \mathrm{pr}_Y
\circ \phi_Y = * \circ \mathrm{pr}_Y$. Pick component choices $\mathbf{c}$
for the result of cutting and gluing $P$ along $(I_X, \phi_X)$, and
$\mathbf{c'}$ for the result of cutting and gluing $R$ along $(I_Y,
\phi_Y)$. As before, we define signs $s(I_X, \phi_X, \mathbf c),
s(I_Y, \phi_Y, \mathbf {c'})$. For example, $s(I_X, \phi_X, \mathbf c)$
is defined by multiplying in all the $\pm 1$ or $0$ contributions from
each $x \in I_X$ such that $\mathrm{pr}_X(x) \in Q$, according to \eqref{scc}.
Next, we cut and glue both $P$ and $R$ by $(I_X, \phi_X, \mathbf c)$
and $(I_Y, \phi_Y, \mathbf {c'})$, respectively, and multiply the first
by $s(I_X, \phi_X, \mathbf c)\frac{\mathrm{h}^{\#(I_X)/2}}{2^{\#(I_X)/2}}$ and
the second by $s(I_Y, \phi_Y, \mathbf {c'}) \frac{\mathrm{h}^{\#(I_Y)/2}}
{2^{\#(I_Y)/2}}$ to obtain elements $P', R' \in {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$.
This will include all the summands we need for the coproducts of $P$ and
$R$, respectively.
For each such summand, we need to take care of contributions from
multiplying these together. So, we need to pick $J_X \subset X
\setminus I_X, J_Y \subset Y \setminus I_Y$, and a bijection $\psi:
J_X \rightarrow J_Y$ such that $\mathrm{pr}_Y \circ \psi = * \circ \mathrm{pr}_X$ and
also the extra condition that $\psi$ preserves components: that is, if
$\psi(x) = y$ and $x, y$ live in necklaces assigned components $c_i,
c_j'$, respectively, then $c_i = c_j'$.
To be more precise, the cutting and gluing $P \mapsto T_1 \& T_2 \&
\cdots \& T_p$ along $(I_X, \phi_X)$ induces a map $\mu_X: X \setminus
I_X \rightarrow \{1, 2, \ldots, p\}$ depending on which necklace each
edge not cut out ends up in. So each edge $x \in X \setminus I_X$ is
assigned a component $c_{\mu_X(x)}$. Similarly we can define $\mu_Y$.
The condition above is that $\psi(x) = y$ implies that $c_{\mu_X(x)} =
c'_{\mu_Y(y)}$.
Given each such choice of $(I_X, \phi_X, \mathbf c), (I_Y, \phi_Y,
\mathbf {c'}),$ and $(J_X, J_Y, \psi)$, we get the following
contribution to the expression $\Delta(P) *_\mathrm{h} \Delta(R)$: We cut and
glue $P'$ and $R'$ together along $(J_X, J_Y, \psi)$, and multiply
in the sign $s(J_X, J_Y, \psi) = (-1)^{\#(J_Y \cap \mathrm{pr}_Y^{-1}(Q))}$
and the coefficient $\frac{\mathrm{h}^{\#(J_X)/2}}{2^{\#(J_X)/2}}$. We sum
this contribution over all such triples of triples to get the right-hand
side of \eqref{bac}.
Now, let's compare this with the left-hand side of \eqref{bac}. Let the
map $\mathrm{pr}: X \sqcup Y \rightarrow \overline{Q}$ be given by $\mathrm{pr}(x) = \mathrm{pr}_X(x),
\mathrm{pr}(y) = \mathrm{pr}_Y(y)$ for $x \in X, y \in Y$. Then, the left-hand side
involves first a sum over $(J_X, J_Y, \psi)$ with $J_X \subset X, J_Y
\subset Y,$ and $\psi: J_X \rightarrow J_Y$ a bijection such that $\mathrm{pr}_Y
\circ \psi = * \circ \mathrm{pr}_X$. Then we sum over $(I, \phi, \mathbf {c''})$
such that $I \subset (X \setminus J_X) \sqcup (Y \setminus J_Y)$ and
$\phi: I \rightarrow I$ is an involution satisfying $\mathrm{pr} \circ \phi = *
\circ \mathrm{pr}$, and $\mathbf {c''}$ is a component choice of $PR_{J_X, J_Y,
\psi}$, the result of
cutting and gluing $P$ and $R$ along $(J_X, J_Y, \psi)$ and then along
$(I, \phi)$. For each such choice of triples, we multiply a coefficient
of $\frac{\mathrm{h}^{\#(J_X) + \#(I)/2}}{2^{\#(J_X) + \#(I)/2}}$, and a sign of
$(-1)^{\#(J_Y \cap \mathrm{pr}_Y^{-1}(Q))} s(I, \phi, \mathbf {c''})$ where the
$s(I, \phi, \mathbf {c''})$ is calculated just as in the definition of
$\Delta(PR_{J_X, J_Y, \psi})$.
If $\phi(I \cap X) \subset X$ and $\phi(I \cap Y) \subset Y$, then
the summand thus obtained will be identical with the summand
corresponding to $(I \cap X, \phi|_X, \mathbf c), (I \cap Y, \phi|_Y,
\mathbf {c'}), (J_X, J_Y, \psi)$ for the choice of $\mathbf c, \mathbf
{c'}$ such that $c''_{\mu_{X \sqcup Y \setminus (J_X \cup J_Y)}(z)}$
equals $c_{\mu_X(z)}$ if $z \in X$ and $c'_{\mu_Y(z)}$ if $z \in Y$,
where $\mu_{X \sqcup Y \setminus (J_X \cup J_Y)}$ is defined just as
$\mu_X, \mu_Y$ were, in the context of cutting and gluing $PR_{J_X, J_Y,
\psi}$ along $(I, \phi)$. The power of $\frac{\mathrm{h}}{2}$ will clearly be
the same. The sign will also be the same, since $s(I, \phi, \mathbf
{c''}) = s(I \cap X, \phi|_X, \mathbf c) s(I \cap Y, \phi|_Y, \mathbf
{c'})$ in this case.
All that remains is to show that all summands from the left-hand side
not of this form cancel. Summands which are not of the above form must
either include $x \in I \cap X, y \in I \cap Y$ such that $\phi(x) = y$,
or else must have some component choice such that $c''_i \neq c''_j$
even though the necklaces $i$ and $j$ would be joined if we had omitted
some $x$ from $J_X$ and $\psi(x)$ from $J_Y$. The latter comes from the
fact that $c''_i = c''_j$ is exactly what is required for $\mathbf{c''}$
to be compatible with some $\mathbf{c}, \mathbf{c'}$ such that
$c_{\mu_X(x)} = c'_{\mu_Y(y)}$.
Let us make the definitions
\begin{multline}
J_X' =
\{x \in J_X \mid c''_i \neq c''_j, \\ \text{\ where $i$ and $j$ would
be joined by omitting $x, \phi(x)$ from $I$}\},
\end{multline}
\begin{equation}
I_X' = \{x \in I \cap X \mid \phi(x) \in Y\}.
\end{equation}
For each $x \in J_X'$,
we can obtain a similar summand by removing $x$ from $J_X$ and $y = \psi(x)$
from $J_Y$, and adding $x, y$ to $I$, setting $\phi(x) = y, \phi(y) =
x$: so $x$ ends up in $I_X'$.
We get the same resulting necklaces and can consider the
$\mathbf{c''}$ which makes the same assignments to the corresponding
necklaces (where necklaces correspond if they come from the same edges
in $X$ or vertex idempotents in the expression for $P$). The only
change is perhaps a change of sign; the sign changes iff $c''_{g(x)} >
c''_{g(x)+1}$ where $g$ is defined as in \eqref{scc} (and $g(x) +1$ is
the edge following $x$ in $P$): we are saying that
the sign changes iff the component assigned to the start of $x$ is $2$
and the component assigned to the target of $x$ is $1$.
Similarly, for each such summand, we can perform the operation of
removing an $x \in I_X'$ and $y = \phi(x) \in (I \cap Y)$, and
adding $x$ to $J_X$ and $y$ to $J_Y$ in such a way that the component
assignments remain the same: in this case, $x$ ends up in $J_X'$. Again,
we get a sign change just in the event that $c''_{g(x)} > c''_{g(x)+1}$.
So, if we sum up all summands which can be obtained from each other by
applying the above two operations, we will get zero unless all sign
changes in the above two paragraphs are positive. But, this cannot
happen if $J_X'$ or $I \cap X$ was originally nonempty for the following
reason:
\begin{equation} \label{fbap}
0 = \sum_{x \in X} (c''_{g(x)} - c''_{g(x)+1}) = \sum_{x \in I_X'
\cup J'_X} c''_{g(x)} - c''_{g(x)+1},
\end{equation}
so that one of the summands on the right-hand side must be negative
(since $J'_X \cup I_X' \neq \emptyset$ shows that the last summand is
nonempty, and each summand is $\pm 1$ in that last summand). The
justification for passing from the first to the second summation in
\eqref{fbap} is that the only nonzero terms we have eliminated in doing
so are those that correspond to $x \in J_X \cup I \cap X$ such that $x$
is paired by $\phi$ or $\psi$ with another $x' \in X$, so that the first
summation includes both $c''_{g(x)} - c''_{g(x) + 1}$ and $c''_{g(x')} -
c''_{g(x') + 1}$, which cancel pairwise.
We have proven that all summands in the left-hand side of \eqref{bac}
either correspond in a bijective fashion with a summand from the
right-hand side, or else lie in a set of summands with nonempty $I_X'$
and $J_X'$, whose contributions cancel. The proof of \eqref{bac} is
finished, so that ${\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$ is a bialgebra.
\section{The antipode} \label{as}
Using $\Phi_W$ and the formula for the antipode in \cite{S}, it
immediately follows that our antipode $S: {\text{Sym\ }} L[\mathrm{h}]_{\mathrm{Moyal}}$
is given by the formula
\begin{equation} \label{man}
S(P_1 \& P_2 \& \cdots \& P_m) = (-1)^m P_1 \& P_2 \& \cdots \& P_m,
\end{equation}
where each $P_i \in L$ is a necklace (i.e.~a cyclic monomial or vertex
idempotent). It is immediate that $S^2 = \mathrm{Id}$. Indeed, $S$ is
diagonalizable with eigenvalues $\pm 1$ and eigenvectors which are
collections of necklaces of the form \eqref{pform}.
Unfortunately, a direct proof that \eqref{man} is the antipode for ${\text{Sym\ }}
L[\mathrm{h}]_{\mathrm{Moyal}}$ turned out to be too difficult. The authors are
interested in any good proof of this fact from purely the Moyal point of
view.
|
{
"timestamp": "2005-03-20T23:52:15",
"yymm": "0503",
"arxiv_id": "math/0503405",
"language": "en",
"url": "https://arxiv.org/abs/math/0503405"
}
|
\section{Introduction}
\setcounter{equation}{0}
Let $M$ be a connected complex manifold and $\hbox{Aut}(M)$ the group of holomorphic automorphisms of $M$. If $M$ is Kobayashi-hyperbolic, $\hbox{Aut}(M)$ is a Lie group in the compact-open topology \cite{Ko}, \cite{Ka}. Let $d(M):=\hbox{dim}\,\hbox{Aut}(M)$. It is well-known (see \cite{Ko}, \cite{Ka}) that $d(M)\le n^2+2n$, and that $d(M)= n^2+2n$ if and only if $M$ is holomorphically equivalent to the unit ball $B^n\subset{\Bbb C}^n$, where $n:=\hbox{dim}_{{\Bbb C}}M$. In \cite{IKra} we studied lower automorphism group dimensions and showed that, for $n\ge 2$, there exist no hyperbolic manifolds with $n^2+3\le d(M)\le n^2+2n-1$, and that the only manifolds with $n^2<d(M)\le n^2+2$ are, up to holomorphic equivalence, $B^{n-1}\times\Delta$ (where $\Delta$ is the unit disc in ${\Bbb C}$) and the 3-dimensional Siegel space (the symmetric bounded domain of type $(\hbox{III}_2)$ in ${\Bbb C}^3$). Further, in \cite{I1} all manifolds with $d(M)=n^2$ were determined (for partial classifications in special cases see also \cite{GIK} and \cite{KV}). The classification in this situation is substantially richer than that for higher automorphism group dimensions.
Observe that a further decrease in $d(M)$ almost immediately leads to unclassifiable cases. For example, no good classification exists for $n=2$ and $d(M)=2$, since the automorphism group of a generic Reinhardt domain in ${\Bbb C}^2$ is 2-dimensional (see also \cite{I1} for a more specific statement). While it is possible that there is some classification for $d(M)=n^2-2$, $n\ge 3$ as well as for particular pairs $d(M)$, $n$ with $d(M)< n^2-2$ (see \cite{GIK} in this regard), the case $d(M)=n^2-1$ is probably the only remaining candidate to investigate for the existence of a reasonable classification for every $n\ge 2$. It turns out that all hyperbolic manifolds with $d(M)=n^2-1$, $n\ge 2$ indeed can be explicitly described and that the case $n=2$ substantially differs from the case $n\ge 3$. In this paper we obtain a classification for $d(M)=n^2-1$, $n\ge 3$ and give examples that demonstrate some of the specifics of the case $n=2$. Our main result is the following theorem.
\begin{theorem}\label{main}\sl Let $M$ be a connected hyperbolic manifold of dimension $n\ge 3$ with $d(M)=n^2-1$. Then $M$ is holomorphically equivalent to one of the following manifolds:
\vspace{0cm}\\
\noindent (i) $B^{n-1}\times S$, where $S$ is a hyperbolic Riemann surface with $d(S)=0$;
\vspace{0cm}\\
\noindent (ii) the tube domain
$$
\begin{array}{ll}
T:=\Bigl\{(z_1,z_2,z_3,z_4)\in{\Bbb C}^4:&(\hbox{Im}\,z_1)^2+(\hbox{Im}\,z_2)^2+
\\&(\hbox{Im}\,z_3)^2-(\hbox{Im}\,z_4)^2<0,\,\hbox{Im}\,z_4>0\Bigr\}.
\end{array}
$$
(here $n=4$).
\end{theorem}
For $n=2$ in addition to the direct products specified in (i) of Theorem \ref{main} many other manifolds occur. They arise, in particular, from gluing together certain homogeneous strongly pseudoconvex real hypersurfaces in 2-dimensional complex manifolds with 3-dimensional groups of\linebreak $CR$-automorphisms. All such hypersurfaces were determined by E. Cartan \cite{C}, and our considerations for $n=2$ required an appropriate interpretation of Cartan's results (see \cite{I2}). Obtaining the classification for $n=2$ is quite lengthy, and therefore the author has decided to publish it in a separate paper. Some non-trivial examples of hyperbolic domains in ${\Bbb C}^2$ and ${\Bbb C}{\Bbb P}^2$ with 3-dimensional automorphism groups are given in Section \ref{examples23}.
The proof of Theorem \ref{main} is organized as follows. In Section \ref{dimorbits} we determine the dimensions of the orbits of the action on $M$ of $G(M):=\hbox{Aut}(M)^c$, the connected component of the identity of $\hbox{Aut}(M)$. It turns out that, unless $M$ is homogeneous, every $G(M)$-orbit is either a real or complex hypersurface in $M$, every real hypersurface orbit is spherical and every complex hypersurface orbit is holomorphically equivalent to $B^{n-1}$ (see Proposition \ref{dim}). Note that Proposition \ref{dim} also contains some information about $G(M)$-orbits for $n=2$, in particular, it allows in this case for some real hypersurface orbits to be either Levi-flat or Levi non-degenerate non-spherical, and for some 2-dimensional orbits to be totally real rather than complex submanifolds of $M$. It turns out that such orbits indeed exist; the corresponding examples are given in Section \ref{examples23}.
Next, in Section \ref{sectspher} we show that real hypersurface orbits in fact cannot occur (see Proposition \ref{nospher}). First, we prove that there may be three possible kinds of such orbits and that the presence of an orbit of a particular kind determines $G(M)$ as a Lie group. Further, when we attempt to glue real hypersurface orbits together, it turns out that for any resulting hyperbolic manifold $M$, the dimension $d(M)$ is always greater than $n^2-1$. Hence all orbits are in fact complex hypersurfaces unless the manifold in question is homogeneous. Parts of the arguments in Section \ref{sectspher} apply in the case $n=2$ as well.
In Section \ref{sectcomplex} we prove Theorem \ref{main} in the non-homogeneous case and obtain manifolds in (i) of Theorem \ref{main} (see Proposition \ref{theoremcomplex}).
In Section \ref{homog} homogeneous manifolds are considered. We show that in this case $n=4$ and obtain the tube domain in (ii) of Theorem \ref{main} (see Proposition \ref{homogeneous}). Note that Proposition \ref{homogeneous} holds for any $n\ge 2$, hence no additional homogeneous manifolds occur when $n=2$.
\section{Dimensions of Orbits}\label{dimorbits}
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The action of $G(M)=\hbox{Aut}(M)^c$ on $M$ is proper (see Satz 2.5 of \cite{Ka}), and therefore for every $p\in M$ its orbit $O(p):=\{f(p):f\in G(M)\}$ is a closed submanifold of $M$ and the isotropy subgroup $I_p:=\{f\in G(M): f(p)=p\}$ of $p$ is compact (see \cite{Ko}, \cite{Ka}). In this section we will obtain an initial classification of the $G(M)$-orbits.
Let $L_p:=\{d_pf: f\in I_p\}$ be the linear isotropy subgroup, where $d_pf$ is the differential of a map $f$ at $p$. The group $L_p$ is a compact subgroup of $GL(T_p(M),{\Bbb C})$ isomorphic to $I_p$ by means of the isotropy representation
$$
\alpha_p:\, I_p\rightarrow L_p, \quad \alpha_p(f)=d_pf
$$
(see e.g. Satz 4.3 of \cite{Ka}). We will now prove the following proposition.
\begin{proposition}\label{dim} \sl Let $M$ be a connected hyperbolic manifold of dimension $n\ge 2$ with $d(M)=n^2-1$, and $p\in M$. Then the following holds:
\vspace{0cm}\\
\noindent (i) Either $M$ is homogeneous, or $O(p)$ is a real or complex closed hypersurface in $M$, or, for $n=2$, the orbit $O(p)$ is a totally real 2-dimensional closed submanifold of $M$.
\vspace{0cm}\\
\noindent (ii) If $O(p)$ is a real hypersurface, the identity component $I_p^c$ of the isotropy subgroup $I_p$ is isomorphic to $SU_{n-1}$, and $I_p$ is isomorphic to a subgroup of ${\Bbb Z}_2\times U_{n-1}$ by means of the isotropy representation $\alpha_p$. If $n\ge 3$, the orbit $O(p)$ is spherical and $I_p$ is isomorphic to a subgroup of $U_{n-1}$. If $n=2$ and $O(p)$ is strongly pseudoconvex, then it is spherical, provided $I_p$ contains more than two elements; if $n=2$ and $O(p)$ is Levi-flat, it is foliated by complex curves holomorphically equivalent to the unit disk $\Delta$.
\vspace{0cm}\\
\noindent (iii) If $O(p)$ is a complex hypersurface, it is holomorphically equivalent to $B^{n-1}$. If $n\ge 3$, then $I_p^c$ is isomorphic, by means of the isotropy representation $\alpha_p$, to the group $H_{k_1,k_2}^n$ of all matrices of the form
\begin{equation}
\left(\begin{array}{cc}
a & 0\\
0 & B
\end{array}\right),\label{grouphk}
\end{equation}
where $B\in U_{n-1}$ and $a\in
(\det B)^{\frac{k_1}{k_2}}$,
for some $k_1,k_2\in{\Bbb Z}$, $(k_1,k_2)=1$, $k_2\ne 0$. If $n=2$, then either $I_p^c$ is isomorphic, by means of the isotropy representation $\alpha_p$, to the group $H_{k_1,k_2}^2$ for some $k_1,k_2\in{\Bbb Z}$, or $L_p^c$ acts trivially on the tangent space to $O(p)$ at $p$ and $I_p^c$ is isomorphic to $U_1$ by means of the isotropy representation $\alpha_p$. If $I_p^c$ is isomorphic to $H_{k_1,k_2}^n$ for some $k_1\ne 0$, there is a real hypersurface orbit in $M$.
\vspace{0cm}\\
\noindent (iv) if $n=2$ and $O(p)$ is totally real, then $I_p^c$ is isomorphic to $SO_2({\Bbb R})$ by means of the isotropy representation $\alpha_p$.
\end{proposition}
\noindent {\bf Proof:} Let $V\subset T_p(M)$ be the tangent space to $O(p)$ at $p$. Clearly, $V$ is $L_p$-invariant. We assume now that $O(p)\ne M$ (and therefore $V\ne T_p(M)$) and consider the following three cases.
\smallskip\\
{\bf Case 1.} $d:=\hbox{dim}_{{\Bbb C}}(V+iV)<n$.
\smallskip\\
Since $L_p$ is compact, one can choose coordinates in $T_p(M)$ such
that $L_p\subset U_n$. Further, the action of $L_p$ on $T_p(M)$
is completely reducible and the subspace $V+iV$ is invariant under this
action. Hence $L_p$ can in fact be embedded in $U_{n-d}\times
U_d$. Since $\hbox{dim}\,O(p)\le 2d$, it follows that
$$
n^2-1\le (n-d)^2+d^2+2d,
$$
and therefore either $d=0$ or $d=n-1$.
If $d=0$, then $p$ is a fixed point for the action of $G(M)$ on $M$. Then $I_p=G(M)$ and $L_p$ is isomorphic to $G(M)$. Since $\hbox{dim}\,L_p=n^2-1$, we have $L_p=SU_n$. The group $SU_n$ acts transitively on directions in $T_p(M)$. Since $d(M)>0$, the manifold $M$ is non-compact. Then, by \cite{GK}, $M$ is holomorphically equivalent to $B^n$, which is clearly impossible.
Suppose that $d=n-1$. Then we have
$$
n^2-1=\hbox{dim}\,L_p+\hbox{dim}\,O(p)\le n^2-2n+2+\hbox{dim}\,O(p).
$$
Hence $\hbox{dim}\,O(p)\ge 2n-3$, that is, either $\hbox{dim}\,O(p)=2n-2$, or $\hbox{dim}\,O(p)=2n-3$.
Suppose first that $\hbox{dim}\,O(p)=2n-2$. In this case we have $iV=V$, hence $O(p)$ is a complex hypersurface. Then $\hbox{dim}\, L_p=(n-1)^2$. It now follows from the proof of Lemma 2.1 of \cite{IKru1} that $L_p^c$ is either $U_1\times SU_{n-1}$, or, for some $k_1$, $k_2$, the group $H_{k_1,k_2}^n$ defined in (\ref{grouphk}).
Therefore, if $n\ge 3$ or $n=2$ and $L_p^c=H_{k_1,k_2}^2$ for some $k_1$, $k_2$, then $L_p$ acts transitively on directions in $V$, and \cite{GK} implies that $O(p)$ is holomorphically equivalent to $B^{n-1}$.
Let $n\ge 3$ and $L_p^c=U_1\times SU_{n-1}$. It then follows (see, for example, Satz 4.3 of \cite{Ka}) that $I_p':=\alpha_p^{-1}(U_1)$ is the kernel of the action of $G(M)$ on $O(p)$, in particular, $I_p'$ is normal in $G(M)$. Therefore, the factor-group $G(M)/I_p'$ acts effectively on $O(p)$. Clearly, $\hbox{dim}\,G(M)/I_p'=n^2-2$. Thus, the group $\hbox{Aut}(O(p))$ is isomorphic to $\hbox{Aut}(B^{n-1})$ (in particular, its dimension is $n^2-1$) and has a codimension 1 (possibly non-closed) subgroup. However, the Lie algebra ${\frak {su}}_{n-1,1}$ of the group $\hbox{Aut}(B^{n-1})$ does not have codimension 1 subalgebras, if $n\ge 3$ (see, e.g., \cite{EaI}). Thus, we have shown that if $n\ge 3$, then $L_p^c=H_{k_1,k_2}^n$ for some $k_1,k_2$.
Next, if $n=2$ and $L_p^c=U_1\times SU_1=U_1$, then the above argument shows that $O(p)$ is a hyperbolic 1-dimensional manifold with automorphism group of dimension at least 2. Hence $O(p)$ is holomorphically equivalent to $\Delta$ if $L_p^c=U_1$ as well.
Suppose that $I_p^c$ is isomorphic to $H_{k_1,k_2}^n$ where $k_1\ne 0$. Then $L_p^c$ acts as $U_1$ on the orthogonal complement to $V$. Therefore, in this case there are real hypersurface orbits in $M$ arbitrarily close to $O(p)$.
Suppose now that $\hbox{dim}\,O(p)=2n-3$. In this case
$\hbox{dim}\,I_p=n^2-2n+2$. Since $L_p$ can be embedded in $U_1\times U_{n-1}$, we obtain $L_p=U_1\times U_{n-1}$. In particular, $L_p$ acts transitively on directions in $V+iV$. This is, however, impossible since $V$ is of codimension 1 in $V+iV$ and is $L_p$-invariant.
\smallskip\\
{\bf Case 2.} $T_p(M)=V+iV$ and $r:=\hbox{dim}_{{\Bbb C}}(V\cap iV)>0$.
\smallskip\\
As above, $L_p$ can be embedded in $U_{n-r}\times U_r$ (clearly, we have
$r<n$). Moreover,
$V\cap iV\ne V$ and since $L_p$ preserves $V$, it follows that
$\hbox{dim}\,L_p<r^2+(n-r)^2$. We have $\hbox{dim}\,O(p)\le 2n-1$, and
therefore
$$
n^2-1<(n-r)^2+r^2+2n-1,
$$
which shows that either $r=1$, or $r=n-1$. It then follows that
$\hbox{dim}\,L_p<n^2-2n+2$. Therefore, we have
$$
n^2-1=\hbox{dim}\,L_p+\hbox{dim}\,O(p)<n^2-2n+2+\hbox{dim}\,O(p).
$$
Hence $\hbox{dim}\,O(p)>2n-3$. Thus, $\hbox{dim}\,O(p)=2n-1$, or
$\hbox{dim}\,O(p)=2n-2$.
Suppose that $\hbox{dim}\,O(p)=2n-1$. Let $W$ be the orthogonal complement to $V\cap iV$ in $T_p(M)$. Clearly, in this case $r=n-1$ and $\hbox{dim}_{{\Bbb C}}\,W=1$. The group $L_p$ is a subgroup of $U_n$ and preserves $V$, $V\cap iV$, and $W$; hence it preserves the line $W\cap V$. Therefore, it can act only as $\pm\hbox{id}$ on $W$, that is, $L_p\subset{\Bbb Z}_2\times U_{n-1}$. Since $\hbox{dim}\,L_p=(n-1)^2-1$, we have $L_p^c=SU_{n-1}$. In particular, $L_p$ acts transitively on directions in $V\cap iV$, if $n\ge 3$. Hence, the orbit $O(p)$ is either Levi-flat or strongly pseudoconvex for all $n\ge 2$.
Suppose first that $n\ge 3$ and $O(p)$ is Levi-flat. Then $O(p)$ is foliated by connected complex manifolds. Let $M_p$ be the leaf passing through $p$. Denote by ${\frak g}$ the Lie algebra of vector fields on $O(p)$ arising from the action of $G(M)$, and let ${\frak l}_p\subset{\frak g}$ be the subspace consisting of all vector fields tangent to $M_p$ at $p$. Since vector fields in ${\frak l}_p$ remain tangent to $M_p$ at each point in $M_p$, the subspace ${\frak l}_p$ is in fact a Lie subalgebra of ${\frak g}$. It follows from the definition of ${\frak l}_p$ that $\hbox{dim}\,{\frak l}_p=n^2-2$. Denote by $H_p$ the (possibly non-closed) connected subgroup of $G(M)$ with Lie algebra ${\frak l}_p$. It is straightforward to verify that the group $H_p$ acts on $M_p$ by holomorphic transformations and that $I_p^c\subset H_p$. If some non-trivial element $g\in H_p$ acts trivially on $M_p$, then $g\in I_p$, and corresponds to the non-trivial element in ${\Bbb Z}_2$ (recall that $L_p\subset {\Bbb Z}_2\times U_{n-1}$). Thus, either $H_p$ or $H_p/{\Bbb Z}_2$ acts effectively on $M_p$ (the former case occurs if $g_p\not\in H_p$, the latter if $g_p\in H_p$). The group $L_p$ acts transitively on directions in the tangent space $V\cap iV$ to $M_p$, and it follows from \cite{GK} that $M_p$ is holomorphically equivalent to $B^{n-1}$. Therefore, the group $\hbox{Aut}(M_p)$ is isomorphic to $\hbox{Aut}(B^{n-1})$ (in particular, its dimension is $n^2-1$) and has a codimension 1 (possibly non-closed) subgroup.
However, as we noted above, the Lie algebra of $\hbox{Aut}(B^{n-1})$ does not have codimension 1 subalgebras, if $n\ge 3$. Thus, $O(p)$ is strongly pseudoconvex. Hence, $L_p$ acts trivially on $W$ and therefore $L_p\subset U_{n-1}$. Since $L_p^c=SU_{n-1}$, the dimension of the stability group of $O(p)$ at $p$ is greater than or equal to $(n-1)^2-1$, which for $n\ge 3$ implies that $p$ is an umbilic point of $O(p)$ (see e.g. \cite{EzhI}). The homogeneity of $O(p)$ now yields that $O(p)$ is spherical, if $n\ge 3$. For $n=2$ the above argument shows that $O(p)$ is foliated by connected hyperbolic complex curves with automorphism group of dimension at least 2, that is, by complex curves holomorphically equivalent to $\Delta$.
If $n=2$, the orbit $O(p)$ is Levi non-degenerate and $I_p$ contains more than two elements, then arguing as in the proof of Lemma 3.3 of \cite{IKru2}, we obtain that $O(p)$ is spherical. Alternatively, this fact can be derived from the classification in \cite{C}.
Suppose now that $\hbox{dim}\,O(p)=2n-2$. Since $T_p(M)=V+iV$, the orbit $O(p)$ is not a complex hypersurface. Therefore, $r=n-2$, which is only possible for $n=3$ (recall that we have either $r=1$, or $r=n-1$). In this case $\hbox{dim}\,L_p=4$ and, arguing as in the proof of Lemma 2.1 of \cite{IKru1}, we see that $L_p$ acts transitively on directions in the orthogonal complement $W$ to $V\cap iV$ in $T_p(M)$. This is, however, impossible since $L_p$ must preserve $W\cap V$.
\smallskip\\
{\bf Case 3.} $T_p(M)=V\oplus iV$.
\smallskip\\
In this case $\hbox{dim}\, V=n$ and $L_p$ can be embedded in the real orthogonal group $O_n({\Bbb R})$,
and therefore
$$
\hbox{dim}\,L_p+\hbox{dim}\, O(p)\le \frac{n(n-1)}{2}+n.
$$
Hence, for $n\ge 3$, we have $\hbox{dim}\,L_p+\hbox{dim}\, O(p)<n^2-1$
which is impossible.
Assume now that $n=2$. If $\hbox{dim}\,L_p=0$, we get a contradiction as above. Hence $\hbox{dim}\,L_p=1$ and $L_p^c=SO_2({\Bbb R})$.
The proof of the proposition is complete.{\hfill $\Box$}
\smallskip\\
\section{Real Hypersurface Orbits}\label{sectspher}
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In this section we will deal with real hypersurface orbits and eventually show that they do not occur. Our goal is to prove the following proposition.
\begin{proposition}\label{nospher}\sl Let $M$ be a connected hyperbolic manifold of dimension $n\ge 3$ with $d(M)=n^2-1$. Then no orbit in $M$ is a real hypersurface.
\end{proposition}
\noindent {\bf Proof:} Recall that every real hypersurface orbit is spherical. First, we narrow down the class of all possible spherical orbits.
\begin{lemma}\label{spherorbitsprop}\sl Let $M$ be a connected hyperbolic manifold of dimension $n\ge 3$ with $d(M)=n^2-1$. Assume that for a point $p\in M$ its orbit $O(p)$ is spherical. Then $O(p)$ is $CR$-equivalent to one of the following hypersurfaces:
\begin{equation}
\begin{array}{ll}
\hbox{(i)}& \hbox{a lens manifold ${\cal L}_m:=S^{2n-1}/{\Bbb Z}_m$ for some $m\in{\Bbb N}$},\\
\hbox{(ii)} & \sigma:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}:\hbox{Re}\,z_n=|z'|^2\right\},\\
\hbox{(iii)} & \delta:=\hbox{$\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z_n|=\exp\left(|z'|^2\right)\right\}$},\\
\hbox{(iv)} & \omega:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}:|z'|^2+\exp\left(\hbox{Re}\,z_n\right)=1\right\},\\
\hbox{(v)} & \varepsilon_{\alpha}:=\hbox{$\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+|z_n|^{\alpha}=1,\, z_n\ne 0\right\}$,}\\
&\hbox{for some $\alpha>0$.}
\end{array}\label{classificationspherorb}
\end{equation}
\end{lemma}
\noindent {\bf Proof of Lemma \ref{spherorbitsprop}:} The proof is similar to that of Proposition 3.1 of \cite{I1}. For a connected Levi non-degenerate $CR$-manifold $Q$ denote by $\hbox{Aut}_{CR}(Q)$ the Lie group of its $CR$-automorphisms. Let $\tilde O(p)$ be the universal cover of $O(p)$. The connected component of the identity $\hbox{Aut}_{CR}(O(p))^c$ of $\hbox{Aut}_{CR}(O(p))$ acts transitively on $O(p)$ and therefore its universal cover $\widetilde{\hbox{Aut}}_{CR}(O(p))^c$ acts transitively on $\tilde O(p)$. Let $G$ be the (possibly non-closed) subgroup of $\hbox{Aut}_{CR}(\tilde O(p))$ that consists of all $CR$-automorphisms of $\tilde O(p)$ generated by this action. Observe that $G$ is a Lie group isomorphic to the factor-group of $\widetilde{\hbox{Aut}}_{CR}(O(p))^c$ by a discrete central subgroup. Let $\Gamma\subset \hbox{Aut}_{CR}(\tilde O(p))$ be the discrete subgroup whose orbits are the fibers of the covering $\tilde O(p)\rightarrow O(p)$. The group $\Gamma$ acts freely properly discontinuously on $\tilde O(p)$, lies in the centralizer of $G$ in $\hbox{Aut}_{CR}(\tilde O(p))$ and is isomorphic to $H/H^c$, with $H=\pi^{-1}(I_p)$, where $\pi:\widetilde{\hbox{Aut}}_{CR}(O(p))^c\rightarrow\hbox{Aut}_{CR}(O(p))^c$ is the covering map.
The manifold $\tilde O(p)$ is spherical, and there is a local $CR$-isomorphism $\Pi$ from $\tilde O(p)$ onto a domain $D\subset S^{2n-1}$. By Proposition 1.4 of \cite{BS}, $\Pi$ is a covering map. Further, for every $f\in\hbox{Aut}_{CR}(\tilde O(p))$ there is $g\in\hbox{Aut}(D)$ such that
\begin{equation}
g\circ \Pi=\Pi\circ f.\label{liftspher}
\end{equation}
Since $\tilde O(p)$ is homogeneous, (\ref{liftspher}) implies that $D$ is homogeneous as well, and $\hbox{dim}\,\hbox{Aut}_{CR}(\tilde O(p))=\hbox{dim}\,\hbox{Aut}_{CR}(D)$.
Clearly, $\dim\hbox{Aut}_{CR}(O(p))\ge n^2-1$ and therefore we have $\dim\hbox{Aut}_{CR}(D)\ge n^2-1$. All homogeneous domains in $S^{2n-1}$ are listed in Theorem 3.1 in \cite{BS}. It is not difficult to exclude from this list all the domains with automorphism group of dimension less than $n^2-1$. This gives that $D$ is $CR$-equivalent to one of the following domains:
$$
\begin{array}{ll}
\hbox{(a)}& S^{2n-1},\\
\hbox{(b)}& S^{2n-1}\setminus\{\hbox{point}\},\\
\hbox{(c)}& S^{2n-1}\setminus\{z_n=0\}.
\end{array}
$$
Thus, $\tilde O(p)$ is respectively one of the following manifolds:
$$
\begin{array}{ll}
\hbox{(a)}& S^{2n-1},\\
\hbox{(b)}& \sigma,\\
\hbox{(c)}& \omega.
\end{array}
$$
If $\tilde O(p)=S^{2n-1}$, then by Proposition 5.1 of \cite{BS} the orbit $O(p)$ is $CR$-equivalent to a lens manifold as in (i) of (\ref{classificationspherorb}).
Suppose next that $\tilde O(p)=\sigma$. The group $\hbox{Aut}_{CR}(\sigma)$ consists of all maps of the form
\begin{equation}
\begin{array}{lll}
z' & \mapsto & \lambda Uz'+a,\\
z_n & \mapsto & \lambda^2z_n+2\lambda\langle Uz',a\rangle+|a|^2+i\alpha,
\end{array}\label{thegroupsphpt}
\end{equation}
where $U\in U_{n-1}$, $a\in{\Bbb C}^{n-1}$, $\lambda\in{\Bbb R}^*$, $\alpha\in{\Bbb R}$, and $\langle\cdot\,,\cdot\rangle$ is the inner product in ${\Bbb C}^{n-1}$. It then follows that $\hbox{Aut}_{CR}(\sigma)=CU_{n-1}\ltimes N$, where $CU_{n-1}$ consists of all maps of the form (\ref{thegroupsphpt}) with $a=0$, $\alpha=0$, and $N$ is the Heisenberg group consisting of the maps of the form (\ref{thegroupsphpt}) with $U=\hbox{id}$ and $\lambda=1$.
Further, description (\ref{thegroupsphpt}) implies that $\hbox{dim}\,\hbox{Aut}_{CR}(\sigma)=n^2+1$, and therefore $n^2-1\le \hbox{dim}\,G\le n^2+1$. If $\hbox{dim}\,G=n^2+1$, then we have $G=\hbox{Aut}_{CR}(\sigma)^c$, and hence $\Gamma$ is a central subgroup of $\hbox{Aut}_{CR}(\sigma)^c$. Since the center of $\hbox{Aut}_{CR}(\sigma)^c$ is trivial, so is $\Gamma$. Thus, in this case $O(p)$ is $CR$-equivalent to the hypersurface $\sigma$.
Assume now that $n^2-1\le\hbox{dim}\,G\le n^2$. Since $G$ acts transitively on $\sigma$, we have $N\subset G$. Furthermore, since $G$ is of codimension 1 or 2 in $\hbox{Aut}_{CR}(\sigma)$, it either contains the subgroup $SU_{n-1}\ltimes N$, or $n=3$ and $G$ contains a subgroup of the form $L\ltimes N$, where $L$ is conjugate to $U_1\times U_1$ in $U_2$. By Proposition 5.6 of \cite{BS}, we have $\Gamma\subset U_{n-1}\ltimes N$. The centralizer of $SU_{n-1}\ltimes N$ in $U_{n-1}\ltimes N$ and that of $L\ltimes N$ in $U_2\ltimes N$ consist of all maps of the form
\begin{equation}
\begin{array}{lll}
z' &\mapsto & z',\\
z_n &\mapsto & z_n+i\alpha,
\end{array}\label{center}
\end{equation}
where $\alpha\in{\Bbb R}$. Since $\Gamma$ acts freely properly discontinuously on $\sigma$, it is generated by a single map of the form (\ref{center}) with $\alpha=\alpha_0\in{\Bbb R}^*$. The hypersurface $\sigma$ covers the hypersurface
\begin{equation}
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z_n|=\exp\left(\frac{2\pi}{\alpha_0}|z'|^2\right)\right\}\label{intermediate1}
\end{equation}
by means of the map
\begin{equation}
\begin{array}{lll}
z' & \mapsto & z',\\
z_n & \mapsto & \exp\left(\displaystyle\frac{2\pi}{\alpha_0} z_n\right),
\end{array}\label{coverrr}
\end{equation}
and the fibers of this map are the orbits of $\Gamma$. Hence $O(p)$ is $CR$-equivalent to hypersurface (\ref{intermediate1}). Replacing if necessary $z_n$ by $1/z_n$ we obtain that $O(p)$ is $CR$-equivalent to the hypersurface $\delta$.
Suppose finally that $\tilde O(p)=\omega$. First, we will determine the group $\hbox{Aut}_{CR}(\omega)$. The general form of a $CR$-automorphism of $S^{2n-1}\setminus\{z_n=0\}$ is given by the formula
$$
\begin{array}{lll}
z'&\mapsto&\displaystyle\frac{Az'+b}{cz'+d},\\
\vspace{0mm}&&\\
z_n&\mapsto&\displaystyle
\frac{e^{i\beta}z_n}{cz'+d},
\end{array}
$$
where
$$
\left(\begin{array}{cc}
A& b\\
c& d
\end{array}
\right)
\in SU_{n-1,1},\quad\beta\in{\Bbb R},
$$
and the covering map $\Pi$ by the formula
$$
\begin{array}{l}
z'\mapsto z',\\
z_n\mapsto \exp\left(\displaystyle\frac{z_n}{2}\right).
\end{array}
$$
Using (\ref{liftspher}) we then obtain the general form of a $CR$-automorphism of $\omega$ as follows
\begin{equation}
\begin{array}{lll}
z'&\mapsto&\displaystyle\frac{Az'+b}{cz'+d},\\
\vspace{0mm}&&\\
z_n&\mapsto&\displaystyle z_n-2\ln(cz'+d)+i\beta,
\end{array}\label{autgrpcov}
\end{equation}
where
$$
\left(\begin{array}{cc}
A& b\\
c& d
\end{array}
\right)
\in SU_{n-1,1},\quad\beta\in{\Bbb R}.
$$
In particular, $\hbox{Aut}_{CR}(\omega)$ is a connected group of dimension $n^2$, and therefore $n^2-1\le\hbox{dim}\,G\le n^2$.
Thus, either $G=\hbox{Aut}_{CR}(\omega)$, or $G$ coincides with the subgroup of $\hbox{Aut}_{CR}(\omega)$ given by the condition $\beta=0$ in formula (\ref{autgrpcov}). In either case, the centralizer of $G$ in $\hbox{Aut}_{CR}(\omega)$ consists of all maps of the form (\ref{center}). Hence $\Gamma$ is generated by a single such map with $\alpha=\alpha_0\in{\Bbb R}$. If $\alpha_0=0$, the orbit $O(p)$ is $CR$-equivalent to $\omega$.
Let $\alpha_0\ne 0$. The hypersurface $\omega$ covers the hypersurface
\begin{equation}
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+|z_n|^{\frac{\alpha_0}{2\pi}}=1,\, z_n\ne 0\right\}\label{intermediate2}
\end{equation}
by means of map (\ref{coverrr}). Since the fibers of this map are the orbits of $\Gamma$, it follows that $O(p)$ is $CR$-equivalent to hypersurface (\ref{intermediate2}). Replacing if necessary $z_n$ by $1/z_n$, we obtain that $O(p)$ is $CR$-equivalent to the hypersurface
$\varepsilon_{\alpha}$ for some $\alpha>0$.
The proof of Lemma \ref{spherorbitsprop} is complete.{\hfill $\Box$}
\smallskip\\
\begin{remark}\label{spher2}\rm For $n=2$ there is an additional possibility for $D$ that has to be taken into the account. Namely, $S^3\setminus {\Bbb R}^2$ has a 3-dimensional automorphism group arising from the natural transitive action of $O^c_{2,1}({\Bbb R})$ by fractional-linear transformations (see Section \ref{examples23}).
\end{remark}
We will now show that in most cases the presence of a spherical orbit of a particular kind in $M$ determines the group $G(M)$ as a Lie group. Suppose that for some $p\in M$ the orbit $O(p)$ is spherical, and let ${\frak m}$ be the manifold from list (\ref{classificationspherorb}) to which $O(p)$ is $CR$-equivalent (we say that ${\frak m}$ is the {\it model}\, for $O(p)$). Since $G(M)$ acts effectively on $O(p)$, the $CR$-equivalence induces an isomorphism between $G(M)$ and a (possibly non-closed) connected $(n^2-1)$-dimensional subgroup $R_{\frak m}$ of $\hbox{Aut}_{CR}({\frak m})$ (this subgroup a priori depends on the choice of the $CR$-equivalence).
We need the following lemma.
\begin{lemma}\label{groupsdeterm} \sl${}$\linebreak
\noindent(i) $R_{S^{2n-1}}$ is conjugate to $SU_n$ in $\hbox{Aut}(B^n)$, and $R_{{\cal L}_m}=SU_n/(SU_n\cap{\Bbb Z}_m)$ for $m>1$;
\noindent (ii) $R_{\sigma}=SU_{n-1}\ltimes N$;
\noindent (iii) $R_{\delta}$ consists of all maps of the form
$$
\begin{array}{lll}
z' & \mapsto & Uz'+a,\\
z_n & \mapsto &e^{i\beta}\exp\Bigl(2\langle Uz',a\rangle+|a|^2\Bigr)z_n,
\end{array}
$$
where $U\in SU_{n-1}$, $a\in{\Bbb C}^{n-1}$, $\beta\in{\Bbb R}$;
\noindent (iv) $R_{\omega}$ consists of all maps of the form (\ref{autgrpcov}) with $\beta=0$;
\noindent (v) $R_{\varepsilon_{\alpha}}$ consists of all maps of the form
\begin{equation}
\begin{array}{lll}
z'&\mapsto&\displaystyle\frac{Az'+b}{cz'+d},\\
\vspace{0mm}&&\\
z_n&\mapsto&\displaystyle
\frac{z_n}{(cz'+d)^{2/\alpha}},
\end{array}\label{repsilon}
\end{equation}
where
$$
\left(\begin{array}{cc}
A& b\\
c& d
\end{array}
\right)
\in SU_{n-1,1}.
$$
\end{lemma}
\noindent{\bf Proof of Lemma \ref{groupsdeterm}:} Suppose first that ${\frak m}={\cal L}_m$, for some $m\in{\Bbb N}$. Then $O(p)$ is compact and, since $I_p$ is compact as well, it follows that $G(M)$ is compact. Assume first that $m=1$. In this case $R_{S^{2n-1}}$ is a subgroup of $\hbox{Aut}_{CR}(S^{2n-1})=\hbox{Aut}(B^n)$.
Since $R_{S^{2n-1}}$ is compact, it is conjugate to a subgroup of $U_n$, which is a maximal compact subgroup in $\hbox{Aut}(B^n)$. Since both $R_{S^{2n-1}}$ is $(n^2-1)$-dimensional, it is conjugate to $SU_n$. Suppose now that $m>1$. It is straightforward to determine the group $\hbox{Aut}_{CR}\left({\cal L}_m\right)$ by lifting $CR$-automorphisms of ${\cal L}_m$ to its universal cover $S^{2n-1}$. This group is $U_n/{\Bbb Z}_m$ acting on ${\Bbb C}^n\setminus\{0\}/{\Bbb Z}_m$ in the standard way. Since $R_{{\cal L}_m}$ is of codimension 1 in $\hbox{Aut}_{CR}\left({\cal L}_m\right)$, we obtain $R_{{\cal L}_m}=SU_n/(SU_n\cap{\Bbb Z}_m)$.
Assume now that ${\frak m}=\sigma$. The group $\hbox{Aut}_{CR}(\sigma)$ consists of all maps of the form (\ref{thegroupsphpt}) and has dimension $n^2+1$.
Since $R_{\sigma}$ acts transitively on $\sigma$, it contains the subgroup $N$ (see the proof of Proposition \ref{spherorbitsprop}).
Furthermore, $R_{\sigma}$ is a codimension 2 subgroup of $\hbox{Aut}_{CR}(\sigma)$, and thus either is the group $SU_{n-1}\ltimes N$, or, for $n=3$, $R_{\sigma}\cap (U_2\ltimes N)=L\ltimes N$, where $L$ is conjugate to $U_1\times U_1$ in $U_2$. By (ii) of Proposition \ref{dim}, $I_p^c$ is isomorphic to $SU_{n-1}$, hence the latter case in fact does not occur.
Next, the group $\hbox{Aut}_{CR}(\delta)$ can be determined by considering the universal cover of $\delta$ (see the proof of Proposition \ref{spherorbitsprop}) and consists of all maps of the form
\begin{equation}
\begin{array}{lll}
z' & \mapsto & Uz'+a,\\
z_n & \mapsto &e^{i\beta}\exp\Bigl(2\langle Uz',a\rangle+|a|^2\Bigr)z_n,
\end{array}\label{gdelta}
\end{equation}
where $U\in U_{n-1}$, $a\in{\Bbb C}^{n-1}$, $\beta\in{\Bbb R}$. This group has dimension $n^2$, and hence $R_{\delta}$ is of codimension 1 in $\hbox{Aut}_{CR}(\delta)$. Since $R_{\delta}$ acts transitively on $\delta$, it consists of all maps of the form (\ref{gdelta}) with $U\in SU_{n-1}$.
Assume now that ${\frak m}=\omega$. The only codimension 1 subgroup of $\hbox{Aut}_{CR}(\delta)$ is given by maps with $\beta=0$ in formula (\ref{autgrpcov}).
Let finally ${\frak m}=\varepsilon_{\alpha}$. The group $\hbox{Aut}_{CR}(\varepsilon_{\alpha})$ consists of all maps of the form
\begin{equation}
\begin{array}{lll}
z'&\mapsto&\displaystyle\frac{Az'+b}{cz'+d},\\
\vspace{0mm}&&\\
z_n&\mapsto&\displaystyle
\frac{e^{i\beta}z_n}{(cz'+d)^{2/\alpha}},
\end{array}\label{autvarepsilon}
\end{equation}
where
$$
\left(\begin{array}{cc}
A& b\\
c& d
\end{array}
\right)
\in SU_{n-1,1},\quad\beta\in{\Bbb R},
$$
and its only codimension 1 subgroup is given by $\beta=0$.
The proof of Lemma \ref{groupsdeterm} is complete.{\hfill $\Box$}
We will now finish the proof of Proposition \ref{nospher}. Our argument is similar to that in Section 4 of \cite{I1}. For completeness of our exposition, we will repeat it here in detail.
Suppose that for some $p\in M$ the orbit $O(p)$ is $CR$-equivalent to a lens manifold ${\cal L}_m$. In this case $G(M)$ is compact, hence there are no complex hypersurface orbits and the model for every orbit is a lens manifold. Assume first that $m=1$. Then $M$ admits an effective action of $SU_n$ by holomorphic transformations and therefore is holomorphically equivalent to one of the manifolds listed in \cite{IKru2}. However, none of the manifolds on the list in \cite{IKru2} with $n\ge 3$ is hyperbolic and has $(n^2-1)$-dimensional automorphism group.
Assume now that $m>1$. Let $f: O(p)\rightarrow {\cal L}_m$ be a $CR$-isomorphism. Then we have
\begin{equation}
f(gq)=\varphi(g)f(q),\label{equivar}
\end{equation}
where $q\in O(p)$, for some Lie group isomorphism $\varphi: G(M)\rightarrow SU_n/(SU_n\cap{\Bbb Z}_m)$. The $CR$-isomorphism $f$ extends to a biholomorphic map from a neighborhood $U$ of $O(p)$ in $M$ onto a neighborhood $W$ of ${\cal L}_m$ in ${\Bbb C}^n\setminus\{0\}/{\Bbb Z}_m$. Since $G(M)$ is compact, one can choose $U$ to be a connected union of $G(M)$-orbits. Then property (\ref{equivar}) holds for the extended map, and therefore every $G(M)$-orbit in $U$ is taken onto an $SU_n/(SU_n\cap{\Bbb Z}_m)$-orbit in ${\Bbb C}^n\setminus\{0\}/{\Bbb Z}_m$ by this map. Thus, $W=S_r^R/{\Bbb Z}_m$ for some $0\le r<R<\infty$, where $S_r^R:=\left\{z\in{\Bbb C}^n: r<|z|<R\right\}$ is a spherical shell.
Let $D$ be a maximal domain in $M$ such that there exists a biholomorphic map $f$ from $D$ onto $S_r^R/{\Bbb Z}_m$ for some $r,R$, satisfying (\ref{equivar}) for all $g\in G(M)$ and $q\in D$. As was shown above, such a domain $D$ exists. Assume that $D\ne M$ and let $x$ be a boundary point of $D$. Consider the orbit $O(x)$. Let ${\cal L}_k$ for some $k>1$ be the model for $O(x)$ and $f_1:O(x)\rightarrow{\cal L}_k$ a $CR$-isomorphism satisfying (\ref{equivar}) for $g\in G(M)$, $q\in O(x)$ and an isomorphism $\varphi_1:G(M)\rightarrow SU_n/(SU_n\cap{\Bbb Z}_k)$ in place of $\varphi$. The map $f_1$ can be holomorphically extended to a neighborhood $V$ of $O(x)$ that one can choose to be a connected union of $G(M)$-orbits. The extended map satisfies (\ref{equivar}) for $g\in G(M)$, $q\in V$ and $\varphi_1$ in place of $\varphi$. For $s\in V\cap D$ we consider the orbit $O(s)$. The maps $f$ and $f_1$ take $O(s)$ into some surfaces $r_1S^{2n-1}/{\Bbb Z}_m$ and $r_2S^{2n-1}/{\Bbb Z}_k$, respectively, with $r_1,r_2>0$.
Hence $F:=f_1\circ f^{-1}$ maps $r_1S^{2n-1}/{\Bbb Z}_m$ onto $r_2S^{2n-1}/{\Bbb Z}_k$. Since ${\cal L}_m$ and ${\cal L}_k$ are not $CR$-equivalent for distinct $m$, $k$, we obtain $k=m$. Furthermore, every $CR$-isomorphism between $r_1S^{2n-1}/{\Bbb Z}_m$ and $r_2S^{2n-1}/{\Bbb Z}_m$ has the form $[z]\mapsto [r_2/r_1Uz]$, where $U\in U_n$, and $[z]\in{\Bbb C}^n\setminus\{0\}/{\Bbb Z}_m$ denotes the equivalence class of a point $z\in{\Bbb C}^n\setminus\{0\}$. Therefore, $F$ extends to a holomorphic automorphism of ${\Bbb C}^n\setminus\{0\}/{\Bbb Z}_m$.
We claim that $V$ can be chosen so that $D\cap V$ is connected and\linebreak $V\setminus(D\cup O(x))\ne\emptyset$. Indeed, since $O(x)$ is strongly pseudoconvex and closed in $M$, for $V$ small enough we have $V=V_1\cup V_2\cup O(x)$, where $V_j$ are open connected non-intersecting sets. For each $j$, $D\cap V_j$ is a union of $G(M)$-orbits and therefore is mapped by $f$ onto a union of the quotients of some spherical shells. If there are more than one such factored shells, then there is a factored shell such that the closure of its inverse image under $f$ is disjoint from $O(x)$, and hence $D$ is disconnected which contradicts the definition of $D$. Thus, $D\cap V_j$ is connected for $j=1,2$, and, if $V$ is sufficiently small, then each $V_j$ is either a subset of $D$ or is disjoint from it. If $V_j\subset D$ for $j=1,2$, then $M=D\cup V$ is compact, which is impossible since $M$ is hyperbolic and $d(M)>0$. Therefore, for some $V$ there is only one $j$ for which $D\cap V_j\ne\emptyset$. Thus, $D\cap V$ is connected and $V\setminus(D\cup O(x))\ne\emptyset$, as required.
Setting now
\begin{equation}
\tilde f:=\Biggl\{\begin{array}{l}
f\hspace{1.7cm}\hbox{on $D$}\\
F^{-1} \circ f_1\hspace{0.45cm}\hbox{on $V$},
\end{array}\label{extens}
\end{equation}
we obtain a biholomorphic extension of $f$ to $D\cup V$. By construction, $\tilde f$ satisfies (\ref{equivar}) for $g\in G(M)$ and $q\in D\cup V$. Since $D\cup V$ is strictly larger than $D$, we obtain a contradiction with the maximality of $D$. Thus, we have shown that in fact $D=M$, and hence $M$ is holomorphically equivalent to $S_r^R/{\Bbb Z}_m$. However, in this case $d(M)=n^2$, which is impossible.
The orbit gluing procedure utilized above can in fact be applied in a very general setting. We will now describe it in full generality (see also \cite{I1}), assuming that every orbit in $M$ is a real hypersurface. The procedure comprises the following steps:
\vspace{0cm}\\
\noindent (1). Start with a real hypersurface orbit $O(p)$ with model ${\frak m}$ and consider a real-analytic $CR$-isomorphism $f:O(p)\rightarrow{\frak m}$ that satisfies (\ref{equivar}) for all $g\in G(M)$ and $q\in O(p)$, where $\varphi:G(M)\rightarrow R_{\frak m}$ is a Lie group isomorphism.
\vspace{0cm}\\
\noindent (2). Verify that for every model ${\frak m}'$ the group $R_{{\frak m}'}$ acts by holomorphic transformations with real hypersurface orbits on a domain ${\cal D}\subset{\Bbb C}^n$ that contains ${\frak m}'$ and that every orbit of the action is $CR$-equivalent to ${\frak m}'$.
\vspace{0cm}\\
\noindent (3). Observe that $f$ can be extended to a biholomorphic map from a $G(M)$-invariant connected neighborhood of $O(p)$ in $M$ onto an $R_{\frak m}$-invariant neighborhood of ${\frak m}$ in ${\cal D}$. First of all, extend $f$ to some neighborhood $U$ of $O(p)$ to a biholomorphic map onto a neighborhood $W$ of ${\frak m}$ in ${\Bbb C}^n$. Let $W'=W\cap {\cal D}$ and $U'=f^{-1}(W')$. Fix $s\in U'$ and $s_0\in O(s)$. Choose $h_0\in G(M)$ such that $s_0=h_0s$ and define $f(s_0):=\varphi(h_0)f(s)$. To see that $f$ is well-defined at $s_0$, suppose that for some $h_1\in G(M)$, $h_1\ne h_0$, we have $s_0=h_1s$, and show that $\varphi(h)$ fixes $f(s)$, where $h:=h_1^{-1}h_0$. Indeed, for every $g\in G(M)$ identity (\ref{equivar}) holds for $q\in U_g$, where $U_g$ is the connected component of $g^{-1}(U')\cap U'$ containing $O(p)$. Since $h\in I_s$, we have $s\in U_h$ and the application of (\ref{equivar}) to $h$ and $s$ yields that $\varphi(h)$ fixes $f(s)$, as required. Thus, $f$ extends to $U'':=\cup_{q\in U'}O(q)$. The extended map satisfies (\ref{equivar}) for all $g\in G(M)$ and $q\in U''$.
\vspace{0cm}\\
\noindent (4). Consider a maximal $G(M)$-invariant domain $D\subset M$ from which there exists a biholomorphic map $f$ onto an $R_{\frak m}$-invariant domain in ${\cal D}$ satisfying (\ref{equivar}) for all $g\in G(M)$ and $q\in D$. The existence of such a domain is guaranteed by the previous step. Assume that $D\ne M$ and consider $x\in\partial D$. Let ${\frak m}_1$ be the model for $O(x)$ and let $f_1:O(x)\rightarrow{\frak m}_1$ be a real-analytic $CR$-isomorphism satisfying (\ref{equivar}) for all $g\in G(M)$, $q\in O(x)$ and some Lie group isomorphism $\varphi_1: G(M)\rightarrow R_{{\frak m}_1}$ in place of $\varphi$.
Let ${\cal D}_1$ be the domain in ${\Bbb C}^n$ containing ${\frak m}_1$ on which $R_{{\frak m}_1}$ acts by holomorphic transformations with real hypersurface orbits $CR$-equivalent to ${\frak m}_1$. As in (3), extend $f_1$ to a biholomorphic map from a connected $G(M)$-invariant neighborhood $V$ of $O(x)$ onto an $R_{{\frak m}_1}$-invariant neighborhood of ${\frak m}_1$ in ${\cal D}_1$. The extended map satisfies (\ref{equivar}) for all $g\in G(M)$, $q\in V$ and $\varphi_1$ in place of $\varphi$. Consider $s\in V\cap D$. The maps $f$ and $f_1$ take $O(s)$ onto an $R_{\frak m}$-orbit in ${\cal D}$ and an $R_{{\frak m}_1}$-orbit in ${\cal D}_1$, respectively. Then $F:=f_1\circ f^{-1}$ maps the $R_{\frak m}$-orbit onto the $R_{{\frak m}_1}$-orbit. Since all models are pairwise $CR$ non-equivalent, we obtain ${\frak m}_1={\frak m}$.
\vspace{0cm}\\
\noindent (5). Show that $F$ extends to a holomorphic automorphism of ${\cal D}$.
\vspace{0cm}\\
\noindent (6). Show that $V$ can be chosen so that $D\cap V$ is connected and\linebreak $V\setminus(D\cup O(x))\ne\emptyset$. This follows from the hyperbolicity of $M$ and the existence of a neighborhood $V'$ of $O(x)$ such that $V'=V_1\cup V_2\cup O(x)$, where $V_j$ are open connected non-intersecting sets. The existence of such $V'$ follows from the strong pseudoconvexity of ${\frak m}$.
\vspace{0cm}\\
\noindent (7). Use formula (\ref{extens}) to extend $f$ to $D\cup V$ thus obtaining a contradiction with the maximality of $D$. This shows that in fact $D=M$ and hence $M$ is biholomorphically equivalent to an $R_{\frak m}$-invariant domain in ${\cal D}$. In all the cases below the determination of $R_{\frak m}$-invariant domains will be straightforward.
\vspace{0cm}\\
Assume first that every orbit in $M$ is a real hypersurface. Let first ${\frak m}=\sigma$. Clearly, the group $R_{\sigma}$ acts with real hypersurface orbits on all of ${\Bbb C}^n$, so in this case ${\cal D}={\Bbb C}^n$. The $R_{\sigma}$-orbit of every point in ${\Bbb C}^n$ is of the form
$$
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}:\hbox{Re}\,z_n=|z'|^2+r\right\},
$$
where $r\in{\Bbb R}$, and every $R_{\sigma}$-invariant domain in ${\Bbb C}^n$ is given by
$$
{\frak S}_r^R:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: r+|z'|^2<\hbox{Re}\,z_n<R+|z'|^2\right\},
$$
where $-\infty\le r<R\le\infty$. Every $CR$-isomorphism between two $R_{\sigma}$-orbits is a composition of a map of the form (\ref{thegroupsphpt}) and a translation in the $z_n$-variable. Therefore, $F$ in this case extends to a holomorphic automorphism of ${\Bbb C}^n$. Now our gluing procedure implies that $M$ is holomorphically equivalent to ${\frak S}_r^R$ for some $-\infty\le r<R\le\infty$. Therefore, $M$ is holomorphically equivalent either to the domain
$$
{\frak S}:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: -1+|z'|^2<\hbox{Re}\,z_n<|z'|^2\Bigr\},
$$
or (for $R=\infty$) to $B^n$. The latter is clearly impossible; the former is impossible either since $d({\frak S})=n^2$ (see e.g. \cite{I1}).
Assume next that ${\frak m}=\delta$. Again, we have ${\cal D}={\Bbb C}^n$. The $R_{\delta}$-orbit of every point in ${\Bbb C}^n$ has the form
$$
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z_n|=r\exp\left(|z'|^2\right)\right\},
$$
where $r>0$, and hence every $R_{\delta}$-invariant domain in ${\Bbb C}^n$ is given by
$$
D_r^R:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: r\exp\left({|z'|^2}\right)<|z_n|<R\exp\left({|z'|^2}\right)\Bigr\},
$$
for $0\le r<R\le\infty$. Every $CR$-isomorphism between two $R_{\delta}$-orbits is a composition of a map from of the form (\ref{gdelta}) and a dilation in the $z_n$-variable. Therefore, $F$ extends to a holomorphic automorphism of ${\Bbb C}^n$. Hence, we obtain that $M$ is holomorphically equivalent to $D_r^R$ for some $0\le r<R\le\infty$ and therefore either to
$$
D_{r/R,\,1}:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: r/R\exp\left({|z'|^2}\right)<|z_n|<
\exp\left({|z'|^2}\right)\Bigr\},
$$
or (for $R=\infty$) to
$$
D_{0,-1}:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: 0<|z_n|<\exp\left({-|z'|^2}\right)\Bigr\}.
$$
This is, however, impossible since $d(D_{r/R,\,1})=d(D_{0,-1})=n^2$ (see e.g. \cite{I1}).
Assume now that ${\frak m}=\omega$. In this case ${\cal D}$ is the cylinder\linebreak ${\cal C}:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}:|z'|<1\right\}$. The $R_{\omega}$-orbit of every point in ${\cal C}$ has the form
$$
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+r\exp\left(\hbox{Re}\,z_n\right)=1\right\},
$$
where $r>0$, and any $R_{\omega}$-invariant domain in ${\cal C}$ is of the form
$$
\begin{array}{lll}
\Omega_r^R:&=&\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\\
&&r(1-|z'|^2)<\exp\left(\hbox{Re}\,z_n\right)<R(1-|z'|^2)\Bigr\},
\end{array}
$$
for $0\le r<R\le\infty$. Every $CR$-isomorphism between two $R_{\omega}$-orbits is a composition of a map from of the form (\ref{autgrpcov}) and a translation in the $z_n$-variable. Therefore, $F$ extends to a holomorphic automorphism of ${\cal C}$. In this case $M$ is holomorphically equivalent to $\Omega_r^R$ for some $0\le r<R\le\infty$, and hence either to
$$
\begin{array}{ll}
\Omega_{r/R,1}:=&\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\,r/R(1-|z'|^2)<\\
&\exp\left(\hbox{Re}\,z_n\right)<(1-|z'|^2)\Bigr\},
\end{array}
$$
or (for $R=\infty$) to
$$
\Omega_{0,-1}:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\,\exp\left(\hbox{Re}\,z_n\right)<(1-|z'|^2)^{-1}\Bigr\}.
$$
As before, this is impossible since $d(\Omega_{r/R,\,1})=d(\Omega_{0,-1})=n^2$ (see e.g. \cite{I1}).
Assume now that ${\frak m}=\varepsilon_{\alpha}$ for some $\alpha>0$. Here ${\cal D}$ is the domain ${\cal C}':={\cal C}\setminus\{z_n=0\}$.
The $R_{\varepsilon_{\alpha}}$-orbit of every point in ${\cal C}'$ is of the form
$$
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+r|z_n|^{\alpha}=1,\, z_n\ne 0\right\},
$$
where $r>0$, and every $R_{\varepsilon_{\alpha}}$-invariant domain in ${\cal C}'$ is given by
$$
\begin{array}{lll}
{\cal E}_{r,1/\alpha}^R:&=&\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\\
&&r(1-|z'|^2)^{1/\alpha}<|z_n|<R(1-|z'|^2)^{1/\alpha}\Bigr\},
\end{array}
$$
for $0\le r<R\le\infty$. Since every $CR$-isomorphism between $R_{\varepsilon_{\alpha}}$-orbits is a composition of a map of the form (\ref{autvarepsilon}) and a dilation in the $z_n$-variable, the map $F$ extends to an automorphism of ${\cal C}'$. Thus, we have shown that $M$ is holomorphically equivalent to ${\cal E}_{r,1/\alpha}^R$ for some $0\le r<R\le\infty$, and hence either to
$$
\begin{array}{ll}
{\cal E}_{r/R,1/\alpha}:=&\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\, r(1-|z'|^2)^{1/\alpha}<\\
&\hspace{1.3cm}|z_n|<(1-|z'|^2)^{1/\alpha}\Bigr\},
\end{array}
$$
or (for $R=\infty$) to
$$
{\cal E}_{0,-1/\alpha}:=\Bigl\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\, 0<
|z_n|<(1-|z'|^2)^{-1/\alpha}\Bigr\}.
$$
As above, this is impossible since $d({\cal E}_{r/R,1/\alpha})=d({\cal E}_{0,-1/\alpha})=n^2$ (see e.g. \cite{I1}).
Assume now that both real and complex hypersurface orbits are present in $M$. Since the action of $G(M)$ on $M$ is proper, it follows that the orbit space $M/G(M)$ is homeomorphic to one of the following: ${\Bbb R}$, $S^1$, $[0,1]$, $[0,1)$ (see \cite{M}, \cite{B-B}, \cite{AA1}, \cite{AA2}), and thus there can be no more than two complex hypersurface orbits in $M$. It follows from (iii) of Proposition \ref{dim} and Lemma \ref{groupsdeterm} that the model for every real hypersurface orbit is $\varepsilon_{\alpha}$ for some $\alpha>0$, $\alpha\in{\Bbb Q}$. Let $M'$ be the manifold obtained from $M$ by removing all complex hypersurface orbits. It then follows from the above considerations that $M'$ is holomorphically equivalent to ${\cal E}_{r,1/\alpha}^R$ for some $0\le r<R\le\infty$.
Let $f:M'\rightarrow {\cal E}_{r,1/\alpha}^R$ be a biholomorphic map satisfying
(\ref{equivar}) for all $g\in G(M)$, $q\in M'$ and some isomorphism $\varphi: G(M)\rightarrow R_{\varepsilon_{\alpha}}$. The group $R_{\varepsilon_{\alpha}}$ in fact acts on all of ${\cal C}$, and the orbit of any point in ${\cal C}$ with $z_n=0$ is the complex hypersurface
$$
c_0:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\,z_n=0\right\}.
$$
For a point $s\in{\cal C}$ denote by $J_s$ the isotropy subgroup of $s$ under the action of $R_{\varepsilon_{\alpha}}$. If $s_0\in c_0$ and $s_0=(z'_0,0)$, $J_{s_0}$ is isomorphic to $H^n_{k_1,k_2}$, where $k_1/k_2=2/\alpha n$ and consists of all maps of the form (\ref{repsilon}) for which the transformations in the $z'$-variables form the isotropy subgroup of the point $z_0'$ in $\hbox{Aut}(B^{n-1})$.
Fix $s_0=(z_0',0)\in c_0$ and let
$$
N_{s_0}:=\left\{s\in {\cal E}_{r,1/\alpha}^R:J_s\subset J_{s_0}\right\}.
$$
We have
$$
N_{s_0}=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: z'=z'_0,\,r(1-|z_0'|^2)^{1/\alpha}<|z_n|<R(1-|z_0'|^2)^{1/\alpha}\right\}.
$$
Thus, $N_{s_0}$ is either an annulus (possibly, with an infinite outer radius) or a punctured disk. In particular, $N_{s_0}$ is a complex curve in ${\cal C}'$. Since $J_{s_0}$ is a maximal compact subgroup of $R_{\varepsilon_{\alpha}}$, $\varphi^{-1}(J_{s_0})$ is a maximal compact subgroup of $G(M)$. Let $O$ be a complex hypersurface orbit in $M$. For $q\in O$ the subgroup $I_q$ is compact and has dimension $(n-1)^2=\hbox{dim}\,J_{s_0}$. Therefore, $\varphi^{-1}(J_{s_0})$ is conjugate to $I_q$ for every $q\in O$ (in particular, $I_q$ is connected), and hence there exists $q_0\in O$ such that $\varphi^{-1}(J_{s_0})=I_{q_0}$. Since the isotropy subgroups in $R_{\varepsilon_{\alpha}}$ of distinct points in $c_0$ do not coincide, such a point $q_0$ is unique.
Let
$$
K_{q_0}:=\left\{q\in M': I_q\subset I_{q_0}\right\}.
$$
Clearly, $K_{q_0}=f^{-1}(N_{s_0})$. Thus, $K_{q_0}$ is a $I_{q_0}$-invariant complex curve in $M'$ equivalent to either an annulus or a punctured disk. By Bochner's theorem there exist a local holomorphic
change of coordinates $F$ near $q_0$ on $M$ that identifies an $I_{q_0}$-invariant neighborhood $U$ of $q_0$ with an $L_{q_0}$-invariant neighborhood of the origin in $T_{q_0}(M)$ such that $F(q_0)=0$ and $F(gq)=\alpha_{q_0}(g)F(q)$ for all $g\in I_{q_0}$ and $q\in U$ (here $L_{q_0}$ is the linear isotropy group and $\alpha_{q_0}$ is the isotropy representation at $q_0$). In the proof of Proposition \ref{dim} (see Case 1) we have seen that $L_{q_0}$ has two invariant subspaces in $T_{q_0}(M)$. One of them corresponds in our coordinates to $O$, the other to a complex curve $C$ intersecting $O$ at $q_0$. Observe that near $q_0$ the curve $C$ coincides with $K_{q_0}\cup\{q_0\}$. Therefore, in a neighborhood of $q_0$ the curve $K_{q_0}$ is a punctured analytic disk. Further, if a sequence $\{q_n\}$ from $K_{q_0}$ accumulates to $q_0$, the sequence $\{f(q_n)\}$ accumulates to one of the two ends of $N_{s_0}$, and therefore we have either $r=0$ or $R=\infty$. Since both these conditions cannot be satisfied simultaneously due to hyperbolicity of $M$, we conclude that $O$ is the only complex hypersurface orbit in $M$.
Assume first that $r=0$. We will extend $f$ to a map from $M$ onto the domain
\begin{equation}
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+\frac{1}{R}|z_n|^{\alpha}<1\right\}\label{ellr}
\end{equation}
by setting $f(q_0)=s_0$, where $q_0\in O$ and $s_0\in c_0$ are related as specified above. The extended map is one-to-one and satisfies (\ref{equivar}) for all $g\in G(M)$, $q\in M$. To prove that $f$ is holomorphic on all of $M$, it suffices to show that $f$ is continuous on $O$. It will be more convenient for us to show that $f^{-1}$ is continuous on $c_0$. Let first $\{s_j\}$ be a sequence of points in $c_0$ converging to $s_0$. Then there exists a sequence $\{g_j\}$ of elements of $R_{\varepsilon_{\alpha}}$ converging to the identity such that $s_j=g_js_0$ for all $j$. Then $f^{-1}(s_j)=\varphi^{-1}(g_j)q_0$, and, since $\left\{\varphi^{-1}(g_j)\right\}$ converges to the identity, we obtain that $\{f^{-1}(s_j)\}$ converges to $q_0$. Next, let $\{s_j\}$ be a sequence of points in ${\cal E}_{0,\alpha}^R$ converging to $s_0$. Then we can find a sequence $\{g_j\}$ of elements of $R_{\varepsilon_{\alpha}}$ converging to the identity such that $g_js_j\in N_{s_0}$ for all $j$. Clearly, the sequence $\{f^{-1}(g_js_j)\}$ converges to $q_0$, and hence the sequence $\{f^{-1}(s_j)\}$ converges to $q_0$ as well. Thus, we have shown that $M$ is holomorphically equivalent to domain (\ref{ellr}) and hence to the domain
$$
E_{\alpha}:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|^2+|z_n|^{\alpha}<1\right\}.
$$
This is, however, impossible since $d(E_{\alpha})\ge n^2$.
Assume now that $R=\infty$. Observe that the action of the group $R_{\varepsilon_{\alpha}}$ on ${\cal C}$ extends to an action on $\tilde{\cal C}:=B^{n-1}\times{\Bbb C}{\Bbb P}^1$ by holomorphic transformations by setting $g(z',\infty):=(a(z'),\infty)$ for every $g\in R_{\varepsilon_{\alpha}}$, where $a$ is the corresponding automorphism of $B^{n-1}$ in the $z'$-variables (see formula (\ref{repsilon})). Now arguing as in the case $r=0$, we can extend $f$ to a biholomorphic map between $M$ and the domain in $\tilde{\cal C}$
$$
\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}: |z'|<1,\,|z_n|>r(1-|z'|^2)^{1/\alpha}\right\}\cup \left(B^{n-1}\times\{\infty\}\right).
$$
This domain is holomorphically equivalent to
$$
{\cal E}_{-1/\alpha}:=\left\{(z',z_n)\in{\Bbb C}^{n-1}\times{\Bbb C}:|z'|<1,\, |z_n|<(1-|z'|^2)^{-1/\alpha}\right\},
$$
and so is $M$. This is, however, impossible since $d({\cal E}_{-1/\alpha})=n^2$.
The proof of Proposition \ref{nospher} is complete.{\hfill $\Box$}
\section{The Case of Complex Hypersurface Orbits}\label{sectcomplex}
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We will now assume that all orbits in $M$ are complex hypersurfaces. As we have shown above, this is always the case for $n\ge 3$, unless $M$ is homogeneous. We will prove the following proposition.
\begin{proposition}\label{theoremcomplex}\sl Let $M$ be a connected hyperbolic manifold of dimension $n\ge 3$ with $d(M)=n^2-1$, and such that for every $p\in M$ its orbit $O(p)$ is a complex hypersurface in $M$. Then $M$ is holomorphically equivalent to $B^{n-1}\times S$, where $S$ is a hyperbolic Riemann surface with $d(S)=0$.
\end{proposition}
\noindent {\bf Proof:} Fix $p\in M$. It then follows from (iii) of Proposition \ref{dim} that $I_p^c$ is isomorphic to $U_{n-1}$, moreover, one can choose coordinates $(w_1,\dots,w_n)$ in $T_p(M)$ so that $L_p^c$ consists of all matrices of the form
\begin{equation}
\left(
\begin{array}{ll}
1 & 0\\
0 & B
\end{array}
\right),\label{formisotropy1}
\end{equation}
where $B\in U_{n-1}$ and $T_p(O(p))=\{w_1=0\}$. Arguing as in the proof of Lemma 4.4 of \cite{IKru1} we obtain that the full group $L_p$ consists of all matrices of the form
\begin{equation}
\left(
\begin{array}{ll}
\alpha & 0\\
0 & B
\end{array}
\right),\label{formisotropy}
\end{equation}
where $B\in U_{n-1}$ and $\alpha^m=1$ for some $m\ge 1$. It then follows (see e.g. Satz 4.3 of \cite{Ka}) that the kernel of the action of $G(M)$ on $O(p)$ is $J_p:=\alpha_p^{-1}({\Bbb Z}_m)$, where we identify ${\Bbb Z}_m$ with the subgroup of $L_p$ that consists of all matrices of the form (\ref{formisotropy}) with $B=\hbox{id}$. Thus, $G(M)/J_p$ acts effectively on $O(p)$. Since $O(p)$ is holomorphically equivalent to $B^{n-1}$ and $\hbox{dim}\,G(M)=n^2-1=\hbox{dim}\,\hbox{Aut}(B^{n-1})$, we obtain that $G(M)/J_p$ is isomorphic to $\hbox{Aut}(B^{n-1})$. It then follows that $I_p$ is a maximal compact subgroup in $G(M)$ since its image under the projection $G(M)\rightarrow\hbox{Aut}(B^{n-1})$ is a maximal compact subgroup of $\hbox{Aut}(B^{n-1})$. However, every maximal compact subgroup of a connected Lie group is connected whereas $I_p$ is not if $m>1$. Thus, $m=1$, hence $G(M)$ is isomorphic to $\hbox{Aut}(B^{n-1})$. In particular, $L_p$ fixes every point of the orthogonal complement $W_p$ to $T_p(O(p))$ in $T_p(M)$. Observe that the above arguments apply to every point in $M$.
Define
$$
N_p:=\left\{s\in M: I_s=I_p\right\}.
$$
Clearly, $I_p$ fixes every point in $N_p$ and $N_{gp}=gN_p$ for all $g\in G(M)$. Further, since for two distinct points $s_1,s_2$ lying in the same orbit we have $I_{s_1}\ne I_{s_2}$, the set $N_p$ intersects every orbit in $M$ at exactly one point. By Bochner's theorem there exist a local holomorphic
change of coordinates $F$ near $p$ on $M$ that identifies an $I_p$-invariant neighborhood $U$ of $p$ with an $L_p$-invariant neighborhood $V$ of the origin in $T_p(M)$ such that $F(p)=0$ and $F(gq)=\alpha_p(g)F(q)$ for all $g\in I_p$ and $q\in U$. Since $L_p$ coincides with the group of matrices of the form (\ref{formisotropy1}), $N_p\cap U=F^{-1}(W_p\cap V)$. In particular, $N_p$ is a complex curve near $p$. Since the same argument can be carried out at every point of $N_p$, we obtain that $N_p$ is a closed complex hyperbolic curve in $M$.
We will now construct a biholomorphic map $\Phi: M\rightarrow B^{n-1}\times N_p$. Let $\Psi: O(p)\rightarrow B^{n-1}$ be a biholomorphism. For $q\in M$ let $r$ be the (unique) point where $N_p$ intersects $O(q)$. Let $g\in G(M)$ be such that $q=gr$. Then we set $\Phi(q):=(F(gp),r)$. By construction, $\Phi$ is biholomorphic. Since $M$ is holomorphically equivalent to $B^{n-1}\times N_p$, we have $d(N_p)=0$.
The proof is complete.{\hfill $\Box$}
\section{The Homogeneous Case}\label{homog}
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In this section we will prove the following proposition.
\begin{proposition}\label{homogeneous} \sl If $M$ is a homogeneous connected hyperbolic manifold of dimension $n\ge 2$ with $d(M)=n^2-1$, then $n=4$ and $M$ is holomorphically equivalent to the tube domain
$$
\begin{array}{ll}
T=\Bigl\{(w_1,w_2,w_3,w_4)\in{\Bbb C}^4:&(\hbox{Im}\,w_1)^2+(\hbox{Im}\,w_2)^2+
\\&(\hbox{Im}\,w_3)^2-(\hbox{Im}\,w_4)^2<0,\,\hbox{Im}\,w_4>0\Bigr\}.
\end{array}
$$
\end{proposition}
\noindent{\bf Proof:} The proof is similar to that of Proposition 5.1 of \cite{I1}. Since $M$ is homogeneous, by \cite{N}, \cite{P-S}, it is holomorphically equivalent to a Siegel domain $U$ of the second kind in ${\Bbb C}^n$. For $n=2$, this gives that $M$ is equivalent to either $B^2$ or $\Delta^2$, which is impossible since $d(B^2)=8$ and $d(\Delta^2)=6$. For $n=3$ we obtain that $M$ is equivalent to one of the following domains: $B^3$, $B^2\times\Delta$, $\Delta^3$, $S$, where $S$ is the 3-dimensional Siegel space. None of these domains has an automorphism group of dimension 8.
Assume now that $n\ge 4$. The domain $U$ has the form
$$
U=\left\{(z,w)\in{\Bbb C}^{n-k}\times{\Bbb C}^k: \hbox{Im}\,w-F(z,z)\in C\right\},
$$
where $1\le k\le n$, $C$ is an open convex cone in ${\Bbb R}^k$ not containing an entire affine line and $F=(F_1,\dots,F_k)$ is a ${\Bbb C}^k$-valued Hermitian form on ${\Bbb C}^{n-k}\times{\Bbb C}^{n-k}$ such that $F(z,z)\in\overline{C}\setminus\{0\}$ for all non-zero $z\in{\Bbb C}^{n-k}$.
We will show first that in most cases we have $k\le 2$. As we noted in \cite{IKra}
\begin{equation}
d(U)\le 4n-2k+\hbox{dim}\,{\frak g}_0(U).\label{form1}
\end{equation}
Here ${\frak g}_0(U)$ is the Lie algebra of all vector fields on ${\Bbb C}^n$ of the form
$$
X_{A,B}=Az\frac{\partial}{\partial z}+Bw\frac{\partial}{\partial w},
$$
where $A\in{\frak{gl}}_{n-k}({\Bbb C})$, $B$ belongs to the Lie algebra ${\frak g}(C)$ of the group $G(C)$ of linear automorphisms of the cone $C$, and the following holds
\begin{equation}
F(Az,z)+F(z,Az)=BF(z,z),\label{form3}
\end{equation}
for all $z\in{\Bbb C}^{n-k}$. By the definition of Siegel domain, there exists a positive-definite linear combination $R$ of the components of the Hermitian form $F$. Then, for a fixed matrix $B$ in formula (\ref{form3}), the matrix $A$ is determined at most up to a matrix that is skew-Hermitian with respect to $R$. Since the dimension of the algebra of matrices skew-Hermitian with respect to $R$ is equal to $(n-k)^2$, we have
\begin{equation}
\hbox{dim}\,{\frak g}_0(U)\le (n-k)^2+\hbox{dim}\,{\frak g}(C).\label{estg}
\end{equation}
In Lemma 3.2 of \cite{IKra} we showed that
\begin{equation}
\hbox{dim}\,{\frak g}(C)\le\frac{k^2}{2}-\frac{k}{2}+1.\label{lemmaestim}
\end{equation}
It now follows from (\ref{estg}) and (\ref{lemmaestim}) that the following holds
$$
\hbox{dim}\,{\frak g}_0(U)\le\frac{3k^2}{2}-k\left(2n+\frac{1}{2}\right)+n^2+1,
$$
which together with (\ref{form1}) for gives
\begin{equation}
d(U)\le\frac{3k^2}{2}-k\left(2n+\frac{5}{2}\right)+n^2+4n+1.\label{form2}
\end{equation}
It is straightforward to check that the right-hand side of (\ref{form2}) is strictly less than $n^2-1$ if $k\ge 3$ for $n\ge 5$, and does not exceed 15 for $n=4$. Furthermore, for $n=4$ the right-hand side of (\ref{form2}) is equal to 15 only if $k=3$ or $k=4$ and $\hbox{dim}\,{\frak g}(C)=k^2/2-k/2+1$.
Suppose that $n=4$ and the right-hand side of (\ref{form2}) is equal to 15. In this case for every point $x_0\in C$ there exist coordinates in ${\Bbb R}^k$ such that the isotropy subgroup of $x_0$ in $G(C)$ contains $SO_{k-1}({\Bbb R})$ (see the proof of Lemma 3.2 in \cite{IKra}). Then after a linear change of coordinates the cone $C$ takes the form
$$
\left\{x=(x_1,\dots,x_k)\in{\Bbb R}^k:\left\langle x,x\right\rangle <0,\,x_k>0\right\},
$$
where $\left\langle x,x\right\rangle:=x_1^2+\dots+x_{k-1}^2-x_k^2$.
In these coordinates the algebra ${\frak g}(C)$ is generated by
the subalgebra of scalar matrices in ${\frak {gl}}_k({\Bbb R})$ and the algebra of pseudo-orthogonal matrices ${\frak o}_{k-1,1}({\Bbb R})$. Assume first that $k=3$. Then we have $F=(v_1|z|^2,v_2|z|^2,v_3|z|^2)$ for some vector $v:=(v_1,v_2,v_3)\in C$. It follows from (\ref{form3}) that $v$ is an eigenvector of the matrix $B$ for every $X_{A,B}\in{\frak g}_0(U)$, which implies that $\hbox{dim}\,{\frak g}_0(U)=3$. Hence by (\ref{form1}) we have $\hbox{dim}\,\hbox{Aut}(U)\le 13$, which is impossible.
Suppose now that $k=4$. In this case $U$ is holomorphically equivalent to the tube domain $T$. Let ${\frak g}(T)$ be the Lie algebra of $\hbox{Aut}(T)$. It follows from the results of \cite{KMO} that ${\frak g}(T)$ is a graded Lie algebra
$$
{\frak g}(T)={\frak g}_{-1}(T)\oplus{\frak g}_0(T)\oplus{\frak g}_1(T),
$$
where ${\frak g}_{-1}$ is spanned by $i\partial/\partial w_j$, $j=1,2,3,4$, and $\hbox{dim}\,{\frak g}_1(T)\le 4$. Clearly, ${\frak g}_0(T)$ is isomorphic to ${\Bbb R}\oplus{\frak o}_{3,1}({\Bbb R})$ and thus has dimension 7. The component ${\frak g}_1(T)$ also admits an explicit description (see e.g. p. 218 in \cite{S}). It follows from this description that ${\frak g}_1(T)$ consists of all vector fields of the form
$$
\begin{array}{ll}
Z_{\alpha,\beta,\gamma,\delta}:=&\displaystyle\Bigl(\alpha(w_1^2-w_2^2-w_3^2+w_4^2)+2(\beta w_1w_2+\gamma w_1w_3 +\delta w_1w_4)\Bigr)\frac{\partial}{\partial w_1}+\\
\vspace{0cm}&\\
&\displaystyle\Bigl(\beta(-w_1^2+w_2^2-w_3^2+w_4^2)+2(\alpha w_1w_2+\gamma w_2w_3 +\delta w_2w_4)\Bigr)\frac{\partial}{\partial w_2}+\\
\vspace{0cm}&\\
&\displaystyle\Bigl(\gamma(-w_1^2-w_2^2+w_3^2+w_4^2)+2(\alpha w_1w_3+\beta w_2w_3 +\delta w_3w_4)\Bigr)\frac{\partial}{\partial w_3}+\\
\vspace{0cm}&\\
&\displaystyle\Bigl(\delta(w_1^2+w_2^2+w_3^2+w_4^2)+2(\alpha w_1w_4+\beta w_2w_4 +\gamma w_3w_4)\Bigr)\frac{\partial}{\partial w_4},
\end{array}
$$
where $\alpha,\beta,\gamma,\delta\in{\Bbb R}$, and thus has dimension 4. Therefore, $\hbox{dim}\,\hbox{Aut}(T)=15$. It is also clear that $T$ is homogeneous under affine automorphisms.
Assume now that $n\ge 4$ is arbitrary and $k\le 2$. If $k=1$, the domain $U$ is equivalent to $B^n$ which is impossible. Hence $k=2$. It follows from (\ref{form3}) that the matrix $A$ is determined by the matrix $B$ up to a matrix $L\in{\frak {gl}}_{n-2}({\Bbb C})$ satisfying
$$
F(Lz,z)+F(z,Lz)=0,
$$
for all $z\in{\Bbb C}^{n-2}$. Let $s$ be the dimension of the subspace of all such matrices $L$. Then
$$
\hbox{dim}\,{\frak g}_0(U)\le s+\hbox{dim}\,{\frak g}(C),
$$
and (\ref{lemmaestim}) yields
$$
\hbox{dim}\,{\frak g}_0(U)\le s+2,
$$
which together with (\ref{form1}) implies
\begin{equation}
s\ge n^2-4n+1.\label{form4}
\end{equation}
By the definition of Siegel domain, there exists a positive-definite linear combination of the components of $F$, and we can assume that $F_1$ is positive-definite. Further, applying an appropriate linear transformation of the $z$-variables, we can assume that $F_1$ is given by the identity matrix and $F_2$ by a diagonal matrix.
Suppose first that the matrix of $F_2$ is scalar. If $F_2\equiv 0$, then $U$ is holomorphically equivalent to $B^{n-1}\times\Delta$ which is impossible. If $F_2\not\equiv 0$, then $U$ is holomorphically equivalent to the domain
$$
V:=\left\{(z,w)\in{\Bbb C}^{n-2}\times{\Bbb C}^2:\hbox{Im}\, w_1-|z|^2>0,\,\hbox{Im}\,w_2-|z|^2>0\right\}.
$$
It was shown in \cite{IKra} that $d(V)\le n^2-2n+3$ and hence $d(V)<n^2-1$. Thus, the matrix of $F_2$ is not scalar. Inequality (\ref{form4}) now yields that the matrix of $F_2$ can have at most one pair of distinct eigenvalues, and therefore $n=4$ and $U$ is holomorphically equivalent to $B^2\times B^2$. This is clearly impossible, and the proof of the proposition is complete.{\hfill $\Box$}
\section{Examples for the Case $n=2$, $d(M)=3$}\label{examples23}
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In this section we give examples of families of hyperbolic domains in ${\Bbb C}^2$ and ${\Bbb C}{\Bbb P}^2$ with automorphism groups of dimension 3 whose orbit structure is different from that observed above for $n\ge 3$. Define
$$
\Omega_t:=\left\{(z,w)\in{\Bbb C}^2: |z|^2+|w|^2-1< t |z^2+w^2-1|\right\},
$$
where $0<t\le 1$. Clearly, $\Omega_t$ is bounded if $0<t<1$. Further, $\Omega_1$ is hyperbolic since it is contained in the hyperbolic product domain
$$
\left\{(z,w)\in{\Bbb C}^2: z,w\not\in(-\infty,-1]\cup[1,\infty)\right\}.
$$
The group $\hbox{Aut}(\Omega_t)$ for every $t$ consists of the maps
$$
\left(
\begin{array}{c}
z\\
w
\end{array}
\right) \mapsto \displaystyle\frac{\left(\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)\left(
\begin{array}{c}
z\\
w
\end{array}\right)+\left(
\begin{array}{c}
b_1\\
b_2
\end{array}
\right)}{c_1z+c_2w+d},
$$
where
\begin{equation}
Q:=\left(\begin{array}{ccc}
a_{11} & a_{12} & b_1\\
a_{21} & a_{22} & b_2\\
c_1& c_2 &d
\end{array}\label{matq}
\right)
\in SO_{2,1}({\Bbb R}),
\end{equation}
and thus is 3-dimensional. The group $\hbox{Aut}(\Omega_t)$ has two connected components (that correspond to the connected components of $SO_{2,1}({\Bbb R})$), and its identity component $G(\Omega_t)$ is given by the condition $a_{11}a_{22}-a_{12}a_{21}>0$. The orbits of $G(\Omega_{t})$ on $\Omega_t$ are as follows:
$$
\begin{array}{ll}
O^{\Omega}_{\alpha}:=&\left\{(z,w)\in{\Bbb C}^2: |z|^2+|w|^2-1=\alpha|z^2+w^2-1|\right\}\setminus\\
&\left\{(x,u)\in{\Bbb R}^2:x^2+u^2=1\right\},\quad -1<\alpha<t,\\
\vspace{0cm}&\\
\Delta_{{\Bbb R}}:=&\left\{(x,u)\in{\Bbb R}^2:x^2+u^2<1\right\}.
\end{array}
$$
Note that $O^{\Omega}_0$ is the only spherical real hypersurface orbit in $\Omega_t$ and that $\Delta_{{\Bbb R}}$ is a totally real orbit. All the orbits are pairwise $CR$ non-equivalent.
The next family of domains is associated with a different action of $SO_{2,1}({\Bbb R})$ on a part of ${\Bbb C}^2$. Define
$$
D_t:=\left\{(z,w)\in{\Bbb C}^2: 1+|z|^2-|w|^2> t |1+z^2-w^2|,\,\hbox{Im}\,z(1+\overline{w})>0\right\},
$$
where $t\ge 1$. All these domains lie in the hyperbolic product domain
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Im}\,z>0,\,w\not\in(-\infty,-1]\cup[1,\infty)\right\},
$$
hence they are hyperbolic as well. For every matrix $Q\in SO^c_{2,1}({\Bbb R})$ as in (\ref{matq}) consider the map
$$
\left(
\begin{array}{c}
z\\
w
\end{array}
\right) \mapsto \displaystyle\frac{\left(\begin{array}{cc}
a_{22} & b_2\\
c_2 & d
\end{array}
\right)\left(
\begin{array}{c}
z\\
w
\end{array}\right)+\left(
\begin{array}{c}
a_{21}\\
c_1
\end{array}
\right)}{a_{12}z+b_1w+a_{11}}.
$$
The group $\hbox{Aut}(D_t)=G(D_t)$ for every $t$ consists of all such maps. The orbits of $G(D_t)$ on $D_t$ are the following non-spherical hypersurfaces
$$
\begin{array}{ll}
O^D_{\alpha}:=&\Bigl\{(z,w)\in{\Bbb C}^2: 1+|z|^2-|w|^2=\alpha|1+z^2-w^2|,\\
&\hbox{Im}\,z(1+\overline{w})>0\Bigr\},\quad \alpha>t.
\end{array}
$$
All the orbits are pairwise $CR$ non-equivalent.
The next family of domains is associated with an action of $SO_3({\Bbb R})$ on ${\Bbb C}{\Bbb P}^2$. Define
$$
E_t:=\left\{(z:w:\zeta)\in{\Bbb C}{\Bbb P}^2: |z|^2+|w|^2+|\zeta|^2< t |z^2+w^2+\zeta^2|\right\},
$$
where $t>1$. The domain $E_t$ is hyperbolic for each $t$ since it is covered in a 2-to-1 fashion by the manifold
$$
\Bigl\{(z,w,\zeta)\in{\Bbb C}^3: |z|^2+|w|^2+|\zeta|^2<t,\,z^2+w^2+\zeta^2=1\Bigr\},
$$
which is clearly hyperbolic; the covering map is $(z,w,\zeta)\mapsto (z:w:\zeta)$. The group $\hbox{Aut}(E_t)=G(E_t)$ for every $t$ is given by applying matrices from $SO_3({\Bbb R})$ to vectors of homogeneous coordinates. The action of the group $G(E_t)$ on $E_t$ has the totally real orbit ${\Bbb R}{\Bbb P}^2$, and the rest of the orbits are the following non-spherical hypersurfaces
$$
O^E_{\alpha}:=\left\{(z:w:\zeta)\in{\Bbb C}{\Bbb P}^2: |z|^2+|w|^2+|\zeta|^2=\alpha |z^2+w^2+\zeta^2|\right\}, \quad 1<\alpha<t.
$$
All the orbits are pairwise $CR$ non-equivalent.
Next, define
$$
S_t:=\left\{(z,w)\in{\Bbb C}^2: \left(\hbox{Re}\,z\right)^2+\left(\hbox{Re}\,w\right)^2<t\right\},
$$
where $t>0$. All these domains are clearly hyperbolic and the group $\hbox{Aut}(S_t)$ for every $t$ consists of all maps of the form
$$
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)\mapsto C
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)+i
\left(
\begin{array}{l}
p\\
q
\end{array}
\right),
$$
where $C\in O_2({\Bbb R})$ and $p,q\in{\Bbb R}$. The group $G(S_t)$ is given by matrices $C\in SO_2({\Bbb R})$. The action of the group $G(S_t)$ on $S_t$ has the totally real orbit
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,z=0,\,\hbox{Re}\,w=0\right\},
$$
and the rest of the orbits are the following non-spherical tube hypersurfaces
$$
O^S_{\alpha}:=\left\{(z,w)\in{\Bbb C}^2: \left(\hbox{Re}\,z\right)^2+\left(\hbox{Re}\,w\right)^2=\alpha\right\},\quad 0<\alpha<t.
$$
Every non-spherical orbit is clearly $CR$-equivalent to $O^S_1$.
Now fix $a\in{\Bbb R}$ such that $|a|>1$, $a\ne 1,2$, and consider the following family of tube domains
$$
R_{a,t}:=\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,z<t\left(\hbox{Re}\,w\right)^a,\,\hbox{Re}\,w>0\right\},
$$
where $t>0$. All these domains are obviously hyperbolic and the group $\hbox{Aut}(R_{a,t})=G(R_{a,t})$ consists of all the maps
$$
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)\mapsto
\left(
\begin{array}{l}
\lambda^a z\\
\lambda w
\end{array}
\right)+i
\left(
\begin{array}{l}
p\\
q
\end{array}
\right),
$$
where $\lambda>0$ and $p,q\in{\Bbb R}$. The action of this group on $R_{a,t}$ has the Levi-flat orbit
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,z=0,\,\hbox{Re}\,w>0\right\},
$$
which is foliated by the half-planes
$$
\left\{(z,w)\in{\Bbb C}^2: z=ic,\,\hbox{Re}\,w>0\right\}, \quad c\in{\Bbb R}.
$$
All other orbits are the following non-spherical hypersurfaces
$$
O^R_{a,\alpha}:=\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,z=\alpha\left(\hbox{Re}\,w\right)^a,\,\hbox{Re}\,w>0\right\},\quad\alpha<t,\,\alpha\ne 0.
$$
Every non-spherical orbit is $CR$-equivalent to $O^R_{a,1}$.
Further, define
$$
U_t:=\left\{(z,w)\in{\Bbb C}^2:\hbox{Re}\,z<\hbox{Re}\,w\cdot\ln\left(t\hbox{Re}\,w\right),\,\hbox{Re}\,w>0\right\},
$$
where $t>0$. All these domains are clearly hyperbolic and the group $\hbox{Aut}(U_t)=G(U_t)$ consists of all the maps
$$
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)\mapsto
\left(
\begin{array}{l}
\lambda z+(\lambda\ln\lambda)w\\
\lambda w
\end{array}
\right)+i
\left(
\begin{array}{l}
p\\
q
\end{array}
\right),
$$
where $\lambda>0$ and $p,q\in{\Bbb R}$. The orbits of $G(U_t)$ on $U_t$ are the following non-spherical hypersurfaces
$$
O^U_{\alpha}:=\left\{(z,w)\in{\Bbb C}^2:\hbox{Re}\,z=\hbox{Re}\,w\cdot\ln\left(\alpha\hbox{Re}\,w\right),\,\hbox{Re}\,w>0\right\},\quad 0<\alpha<t.
$$
Every orbit is $CR$-equivalent to $O^U_1$.
Finally, fix $a>0$ and consider
$$
V_{a,t,s}:=\left\{(z,w)\in{\Bbb C}^2:se^{a\varphi}<r<te^{a\varphi}\right\},
$$
where $t>0$, $e^{-2\pi a}t<s<t$, and $(r,\varphi)$ denote the polar coordinates in the $(\hbox{Re}\,z,\hbox{Re}\,w)$-plane with $\varphi$ varying from $-\infty$ to $\infty$ (thus, the boundary of $V_{a,t,s}$ consists of two infinite spirals). All these domains are hyperbolic and $\hbox{Aut}(V_{a,t,s})=G(V_{a,t,s})$ consists of all maps of the form
$$
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)\mapsto e^{a\beta}
\left(
\begin{array}{rr}
\cos\beta & \sin\beta\\
-\sin\beta & \cos\beta
\end{array}
\right)
\left(
\begin{array}{l}
z\\
w
\end{array}
\right)+i
\left(
\begin{array}{l}
p\\
q
\end{array}
\right),
$$
where $\beta,p,q\in{\Bbb R}$. The orbits under the action of $G(V_{a,t,s})$ on $V_{a,t,s}$ are the following non-spherical hypersurfaces
$$
O^V_{a,\alpha}:=\left\{(z,w)\in{\Bbb C}^2:r=\alpha e^{a\varphi}\right\},\quad s<\alpha<t.
$$
Clearly, every orbit is $CR$-equivalent to $O^V_{a,1}$.
The orbits $O^{\Omega}_{\alpha}$ with $-1<\alpha<1$ and $\alpha\ne 0$, $O^D_{\alpha}$ with $\alpha>1$, $O^E_{\alpha}$ with $\alpha>1$, $O^S_1$, $O^R_{a,1}$ with $|a|>1$ and $a\ne 1,2$, $O^U_1$, $O^V_{a,1}$ with $a>0$ are part of E. Cartan's classification of homogeneous hypersurfaces in the non-spherical case (see \cite{C}). They are pairwise $CR$ non-equivalent, both locally and globally, and give a complete classification from the local point of view. To obtain a global classification, one has to additionally consider all possible covers of these hypersurfaces (see \cite{I2}).
We will now give an example of a hyperbolic domain in ${\Bbb C}^2$, for which almost every orbit is spherical. Define
$$
W_t:=\Bigl\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,w>|z|^2+t\left(\hbox{Re}\,z\right)^2,\,\hbox{Re}\,z>0\Bigr\},
$$
where $t\in{\Bbb R}$. This domain is hyperbolic since for $t\ge -2$ it is equivalent to a subdomain of the hyperbolic product domain
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,z>0,\,\hbox{Re}\,w>0\right\},
$$
and for $t<-2$ it is equivalent to the hyperbolic domain
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,w<|z|^2,\,\hbox{Re}\,z>0\right\}.
$$
The group $\hbox{Aut}(W_t)=G(W_t)$ consists of the maps
$$
\begin{array}{lll}
z & \mapsto & \lambda z+ia,\\
w & \mapsto & \lambda^2w-2i\lambda a z+a^2+i\beta,
\end{array}
$$
where $\lambda>0$, $a,\beta\in{\Bbb R}$ (cf. (\ref{thegroupsphpt})). The action of this group on $W_t$ has the Levi-flat orbit
$$
\left\{(z,w)\in{\Bbb C}^2: \hbox{Re}\,(w+z^2)=0,\,\hbox{Re}\,z>0\right\},
$$
which is foliated by the complex curves
$$
\left\{(z,w)\in{\Bbb C}^2: w+z^2=ic,\,\hbox{Re}\,z>0\right\}, \quad c\in{\Bbb R}.
$$
All other orbits are the following spherical hypersurfaces
$$
O^W_{\alpha}:=\Bigl\{(z,w)\in{\Bbb C}^2:\hbox{Re}\,(w-\alpha z^2/2 )=|z|^2(1+\alpha/2),\,\hbox{Re}\,z>0\Bigr\},\quad \alpha>t.
$$
Clearly, every spherical orbit is $CR$-equivalent to $O^W_0$.
|
{
"timestamp": "2005-09-22T02:04:09",
"yymm": "0503",
"arxiv_id": "math/0503471",
"language": "en",
"url": "https://arxiv.org/abs/math/0503471"
}
|
\section{\label{}Introduction}
In the last years a widespread attention has been devoted to the role
played by the isospin degree of freedom in the heavy--ion reaction
physics. The interest on this subject is twofold: the knowledge
of the symmetry term in the Equation of State (EOS) of
asymmetric nuclear matter, which is a fundamental ingredient in
astrophysical investigations \cite{Lat01}, and the
thermostatistical properties both at equilibrium and out of equilibrium
of systems with two strongly interacting components
\cite{Mue95,Bao97,Bar98,Lar99,Bot01,Bar01,Bar02}. Both the interests concern
systems faraway from the physical conditions of ordinary nuclear matter.
\par
Thanks to the availability of high--performance $4\pi$--detectors
for the investigations of heavy--ion collisions at intermediate energy
\cite{Xu00,Tsa01,Xu02,Ger04},
recent experimental results can provide new insights about isospin
effects on the nuclear dynamics. In particular, for multifragmentation
processes we can obtain information about highly excited
two--component systems and their subsequent decomposition.
Statistical models have been extensively applied to the description
of experimental data, also for isospin observables \cite{Tan03},
and some conclusions have been drawn on the behavior of charge
asymmetric systems. These models, however, imply the
achievement of the statistical equilibrium for the nuclear system.
Then, it would be highly desireable to have some
insight on the path followed by the system to attain equilibrium,
if this occurs. Further, it would be of great advantage to envisage
some observable, which preserves memory of the dynamical processes
occurred during the fragmentation.
\par
In this paper we present an analytical description of the
disassembly of excited nuclear systems
formed during the collision of heavy ions,
in terms of the occurrence of nuclear matter instabilities. Our
approach accounts for the source of the density fluctuations occurring
when the system enters the spinodal instability region
of the density--temperature phase diagram, and describes
the growth of the fluctuations with time until they cause the
decomposition of the system.
This approach is a generalization to include the isospin
degree of freedom, of the model developed in Refs. \cite{Mat00,Mat03}
for symmetric nuclear matter basically. This gives rise to a
substantial improvement of the model, with new valuable results.
Such extension allows us to investigate separately fluctuations of the
neutron and proton densities and their interplay. Following the procedure
introduced in Ref. \cite{Mat00}, we identify the pattern of the domains
containing correlated density fluctuations, with the fragmentation
pattern, and can make predictions on
the isotopic distributions of the fragments. Moreover, we include
in the present treatment the Coulomb force according to the approach
outlined in Ref. \cite{Fab98}. Its effects on the isotopic
distributions turn out to be sizeable.
\par
Our results essentially refer to the distributions of the fragments
just after the early break--up of the system.
So our approach can be considered
complementary to dynamical model calculations
based upon semiclassical kinetic equations
for one--body phase--space density, (for a review on dynamical models
see, e.g., Refs. \cite{Das01,Bor02,Cho04}),
as far as the description of the early fragmentation mechanism
is concerned. The advantage here is that one can make significant
predictions on observables of experimental
interest on an analytical basis. This allows us to directly relate the
results obtained to the EOS properties
and the features of the spinodal mechanism.
In our scheme the onset and the growth of the fluctuations about
the mean phase--space density in unstable situations, are
self--consinstently treated. The self--consistency condition is provided
by the fluctuation--dissipation theorem. Whereas all the processes, which take
place before the system enters the spinodal instability region
and after the break--up, are beyond our approach. Therefore
the mean values of density, temperature and
asymmetry of the nuclear medium when the system starts to break up
are taken from calculations performed within dynamical models.
On the other side, a dynamical model, which appropriately
incorporates the effects of the
fluctuations, might give a detailed description of the whole history
of a collision between heavy ions.
Therefore, it can be of interest to compare the results of our approach
with those obtained by numerical solutions of microscopic
transport equations, also to connect the results of the simulations to
what is expected in a pure spinodal decomposition scenario.
The comparison will be done with the isotopic
distributions for the primary fragments, calculated in
the dynamical stochastic
mean--field (SMF) approach of Refs. \cite{Bar02,Col98}. In particular,
we will consider the ratio, for a given value of the proton number,
between the isotope yields from two different reactions.
This quantity represents a straightforward mean to compare isotopic
distributions, since it is experimentally found to obey a simple
relationship (isoscaling), as a function of the proton number and
neutron number \cite{Xu00,Tsa01,Tan01,Tsa101}.
We will also discuss the dependence of the isoscaling parameters
on the EOS considered.
\par
In Sec. II we outline the extension of the formalism developed in
Ref. \cite{Mat00} only for isoscalar density fluctuations,
to include the isospin degree of freedom. In
Sec. III we discuss the results of our calculations and their comparison
with the calculations performed in Ref. \cite{Liu04} within the
SMF approach. Finally, in Sec. IV a brief summary and conclusions are given.
\section{\label{AA}Formalism}
\subsection{Time evolution of density fluctuations}
We study the density fluctuations by introducing a self--consistent
stochastic field acting on the constituents of the system.
The time evolution of the fluctuations is described by a kinetic equation,
within a linear approximation for the stochastic field.
The growth of fluctuations is
essentially dominated by the unstable mean field. Thus
we focus our attention on the behavior of the mean field and
neglect the collision term in the kinetic equation.
Collisions would mainly add a damping to the growth rate
of the fluctuations and should not change the main results of our calculations,
at least at a qualitative level.
\par
The additional stochastic mean field, which we assume having
a vanishing mean, will induce
fluctuations of the proton and neutron densities,
$\delta\varrho_i({\bf r},t)$, with respect to their uniform mean values
$\varrho_i$ (~$i=1,2$ for protons and neutrons respectively~).
We assume that at the time $t=0$, given density fluctuations
$\delta\varrho_i({\bf r},t=0)$ are present in the system.
The equations for the Fourier coefficients of $\delta\varrho_{i}({\bf r},t)$
for $t>0$ are given by a generalization of the equation for the
isoscalar density fluctuations of Ref. \cite{Mat00,Lalime}. They read
\begin{eqnarray}
\delta\varrho_i({\bf k},t)=&&
\delta\varrho_i({\bf k},t=0)-\Sigma_{j,l}\,\delta\varrho_l({\bf k},t=0)
D_{j,l}^{-1}(k,\omega=0 )\,
\int_{0}^{t}D_{i,j}(k,t-t^\prime)\,dt^\prime
\nonumber\\
&&+\Sigma_j\int_{0}^{t}D_{i,j}(k,t,t^\prime)dW_j({\bf k},t^\prime)\,.
\label{wiener}
\end{eqnarray}
where the $2\times 2$ matrix in the isospin space,
$D_{i,j}(k,t-t^\prime)$, is the density--density
response function and $D_{i,j}(k,\omega)$ its time Fourier transform. For
symmetry reasons the response function and its Fourier transform depend
only on the magnitude of the wave vector. In the last integral
$dW_j({\bf k},t^\prime)$ gives the contribution of the $j$--component of
the stochastic field in the interval $dt^\prime$. Since the stochastic
field is real $W_i^{*}({\bf k},t)=W_i({-\bf k},t)$. The real and imaginary
parts of the Fourier coefficients $W_{i}({\bf k},t)$ are
indipendent components of a multivariate stochastic process \cite{Gard},
with
\begin{equation}
<\int_{0}^t\,dW_i({\bf k},t^\prime)\int_{0}^t\,dW_j(-{\bf k},t^{\prime\prime})>
=\int_{0}^tdt^{\prime}dt^{\prime\prime}\,
B_{i,j}({\bf k},t^\prime,t^{\prime\prime})\,
\label{wiener1}
\end{equation}
defining the correlator for the stochastic field. Angular brackets denote
ensemble averaging.
\par
In the mean--field approximation the response function obeys the
following set of equations
\begin{equation}
D_{i,j}(k,\omega)=\,D^{(0)}_i(k,\omega)\delta_{i,j}+\Sigma_lD^{(0)}_i(k,\omega)
{\cal A}_{i,l}(k)D_{l,j}(k,\omega)\,
\label{resp}
\end{equation}
where $D^{(0)}_i(k,\omega)$ is the non--interacting particle--hole
propagator and ${\cal A}_{i,l}(k)$ is the Fourier transform of the
nucleon--nucleon effective interaction.
\par
In Ref. \cite{Mat00} it has been shown that, in the case
of isoscalar fluctuations in symmetric
nuclear matter, a white--noise hypothesis for the stochastic field
can be retained for values of temperature and density
sufficiently close to the borders of the spinodal
region. In such situations the imaginary part of the response function
displays a sharp peak dominating the particle--hole background
at a value of $\omega\ll kv_F$. This is due to the occurrence of a pole on the
imaginary axis of $\omega$, that corresponds to isoscalar fluctuations,
at a distance from the origin that is much smaller
than the values of $kv_F$. The position of this pole determines
the time scale characteristic of the response function.
However, when one wants to investigate the properties of
neutron and proton distributions, as we do in the present study, one should
consider also the effects due to the isovector fluctuations. Even though
isoscalar modes are the dominant ones, since they are unstable, isovector
fluctuations contribute to the width of the isotopic distributions of the
fragments formed in the spinodal decomposition process.
In asymmetric nuclear matter isovector and isoscalar fluctuations are coupled.
However one can still distinguish oscillations with neutrons and protons
moving in phase (isoscalar-like) or out of phase (isovector-like).
Let us first concentrate on the properties of the isoscalar-like modes.
\subsubsection {Isoscalar-like fluctuations}
The position of the pole $\omega=i\Gamma_k$
for the unstable isoscalar-like mode
is given by
the imaginary root of the equation
\begin{equation}
{\rm det}|\delta_{i,j}-D^{(0)}_i(k,\omega){\cal A}_{i,j}(k)|=0\,.
\label{rate}
\end{equation}
The quantity $\Gamma_k$ is the damping or growth rate (depending on its
sign) of the density fluctuations. In evaluating it, we use the
expression of $D^{(0)}_i(k,\omega)$ for $\omega\ll kv_F$ \cite{Mat00}
\[
D_{i}^{(0)}(k,\omega)\simeq -\frac{\partial \varrho_i}{\partial \tilde\mu_i}
-i\frac{1}{2\pi}m^2F(\beta \tilde\mu_i)\frac{\omega}{k}\, ,
\]
where the effective chemical potential $\tilde\mu_i$ of neutrons or
protons is measured with respect to the uniform mean field
$U_i(\varrho_1,\varrho_2)$ of the unperturbed initial state and
$F(\beta \tilde\mu_i)$ is the function
\[F(\beta \tilde\mu_i)=\,\frac{1}{e^{-\beta \tilde\mu_i}+1}\,,\]
with $\beta =1/T$ being the inverse temperature (we use units
such that $\hbar=~c=~k_B=1$).
\par
Substituting into Eq. (\ref{wiener}) the response function
$D_{i,j}(k,t-t^\prime)$ calculated with these approximations, the equation
for the fluctuations $\delta\varrho_i({\bf k},t)$ becomes
\begin{eqnarray}
\delta\varrho_i({\bf k},t)=&&\delta\varrho_i({\bf k},t=0)+
\Sigma_{j,l}C_{i,l}(k)D_{l,j}^{-1}(k,\omega=0)
\delta\varrho_j({\bf k},t=0)\frac{1}{\Gamma_k}(e^{\Gamma{_kt}}-1)
\nonumber\\
&&+\Sigma_j\,C_{i,j}(k)e^{\Gamma{_kt}}\int_0^te^{-\Gamma{_kt^\prime}}
\,dW_j({\bf k},t^\prime)\, ,
\label{ornul}
\end{eqnarray}
where $C_{i,j}(k)$ are the residues, times $(-i)$, of the components of the
response function at the pole $\omega=i\Gamma_k$. They have the
relevant property
\begin{equation}
{\rm det}|C_{i,j}(k)|=0\,.
\label{det}
\end{equation}
The explicit expression of the inverse of the response function
for $\omega=0$ is
\[D_{i,j}^{-1}(k,\omega=0)=-\Big[\frac{\partial \tilde\mu_j}
{\partial \varrho_i}+{\cal A}_{i,j}(k)\Big]\,.\]
\par
For isoscalar-like fluctuations $W_j({\bf k},t^\prime)$ represents a
Gaussian white noise \cite{Mat00}.
The probability distribution of density fluctuations,
$P[\delta\varrho_i({\bf k},t)]$, is given by a product of Gaussian
distributions. Each single factor
corresponds to the stochastic process of Eq. (\ref{ornul})
for a given wave number $k$ \cite{Mat00,Mat03}, with
the covariance matrix
\begin{equation}
\sigma^2_{i,j}(k,t)=\Sigma_{l,m}C_{i,l}(k)B_{l,m}({\bf k},t)C_{m,j}(k)
\frac{1}{2\Gamma_k}\Big(e^{2\Gamma_kt}-1\Big)\,.
\label{variance0}
\end{equation}
For simplicity, we have assumed that the initial fluctuations are
negligible $\sigma^2_{i,j}(k,t)\simeq 0$. Whenever it is necessary, a
nonvanishing covariance can be easily introduced.
\par
The probability distribution $P[\delta\varrho_i({\bf k},t)]$ is completely
determined once the covariance matrix $\sigma^2_{i,j}(k,t)$
is known. According to the
procedure usually followed when treating instabilities by exploiting
the fluctuation--dissipation theorem, see e.g. Refs. \cite{Gunt83,Hoff95},
we determine the coefficients $B_{i,j}({\bf k},t)$
as functions of $\varrho_1$, $\varrho_2$ and $T$ for the
system at equilibrium, then we extend the
expressions so found to non--equilibrium cases.
Since the relevant
values of the wave vector $k$ turn out to be such that
the quantity $kv_F$ is of the same order of magnitude as $T$,
the limit $\omega/kv_F\ll 1$ also implies $\omega/T\ll 1$. In such
case, the classical limit $\omega/T\ll 1$ (or $|\Gamma_k(t)|/T\ll 1$)
can be taken when evaluating both sides of the
fluctuation--dissipation relation. Then, we get
\begin{equation}
\frac{\partial}{\partial t}
<\delta\varrho_i({\bf k},t)\delta\varrho_j({-\bf k},t^\prime)>=
-TD_{i,j}(k,t-t^\prime)\,.
\label{fdt}
\end{equation}
The equation for the equilibrium fluctuations can
be obtained from Eq. (\ref{wiener}) by shifting the initial time
$t=0$ to $-\infty$.
By exploiting Eq. (\ref{fdt}) we can obtain
the following relation between the coefficients
$B_{i,j}({\bf k},t)$ and the functions $C_{i,j}(k)$:
\begin{equation}
\Sigma_{l,m}C_{i,l}(k)B_{l,m}({\bf k},t)C_{m,j}(k)=\,
-2TC_{i,j}(k)\,.
\label{fdt1}
\end{equation}
From this equation we can see that $B_{i,j}$ are constant and
depend only on the magnitude
$k$ of the wave vector, as it is expected for symmetry reasons.
Following Refs. \cite{Gunt83,Hoff95} (see also the discussion in
Ref. \cite{Mat00} on this point) we assume that the relation (\ref{fdt1})
is valid also in instability situations. In such a way, the covariance
matrix (\ref{variance0}) acquires the form
\begin{equation}
\sigma^2_{i,j}(k,t)=-TC_{i,j}(k)
\frac{1}{\Gamma_k}\Big(e^{2\Gamma_kt}-1\Big)\,,
\label{variance}
\end{equation}
and is completely determined both for stable and
unstable situations.
We notice that, for the isoscalar-like mode, $\sigma^2_{1,2}(k) =
\sigma^2_{2,1}(k)$ is positive. In fact proton and neutron
densities oscillate in phase, although with different amplitudes
in general. However, the ratio between amplitudes,
$\sigma^2_{1,1}(k)/ \sigma^2_{1,2}(k)$, is found to be larger
than the initial proton to neutron ratio, thus leading to
the formation of more symmetric fragments, the so-called isospin
distillation effect \cite{Bar98}.
\subsubsection {Isovector-like fluctuations}
Now we turn to consider the isovector-like modes.
In this case the frequency of the modes, $\omega^{iv}_k$ is real,
i.e. we have stationary oscillations. The position of the pole is given
by the other solution of Eq. (\ref{det}). However,
we add a small negative imaginary part $-\Gamma^{iv}_k$
to the position of the pole, taking into account that here
we are neglecting nucleon-nucleon collisions and finite size effects.
Correspondingly the imaginary part of the response function acquires the
width $\Gamma^{iv}_k$. \par
The contribution of isovector-like fluctuations to
the covariance matrix $\sigma_{i,j}^2(k,t)$ can be written
as it follows:
\begin{eqnarray}
\sigma^2_{i,j}(k,t)&&=4\,\Sigma_{l,m}C_{i,l}^{iv}(k)C_{m,j}^{iv}(k)
e^{-2\Gamma^{iv}_k t}
\nonumber\\
&&\times\int_0^tdt_1dt_1^{\prime}\Big[e^{\Gamma^{iv}_k(t_1+t_1^{\prime})}
B_{l,m}^{iv}({\bf k},t_1,t_1^{\prime})
\sin\big(2\omega^{iv}_k(t-t_1)\big)
\sin\big(2\omega^{iv}_k(t-t_1^{\prime})\big)\Big]\,,
\label{fluc_iv}
\end{eqnarray}
where $C_{i,j}^{iv}(k)$ are the residues at the pole and
$B_{l,m}^{iv}({\bf k},t_1,t_1^{\prime})$ denote the contributions from the
isovector--like fluctuations to the stochastic field.
\par
To determine the amplitude of the stochastic field we essentially
follow again the derivation presented above.
By exploiting the fluctuation-dissipation theorem, now
in the limit $\omega/T>>1$
(since the frequency of the isovector vibrations is rather large with respect
to the relevant values of $T$), we obtain for values of
$\omega$ close to the pole the relation:
\begin{equation}
\Sigma_{l,m}C_{i,l}^{iv}(k)B_{l,m}^{iv}({\bf k},\omega)C_{m,j}^{iv}(k)=\,
2\Gamma^{iv}_k C_{i,j}^{iv}(k)\Big(\,\frac{2(\Gamma^{iv}_k)^2}{(\omega-
\omega^{iv}_k)^2+(\Gamma^{iv}_k)^2}\Big)\,,
\label{fdt1_iv}
\end{equation}
where we have added a Lorentzian factor to the right hand side in order to
restrict to a small region about $\omega^{iv}_k$ the contribution
from the isovector--like pole to the time Fourier transform of
$B_{l,m}^{iv}({\bf k},\omega)$. In this way
the correlator $B_{l,m}^{iv}({\bf k},t_1-t_1^{\prime})$
for the stochastic field results to be proportional to
$e^{-\Gamma^{iv}_k|t_1-t_1^{\prime}|}$. This means that the isovector--like
stochastic field is given by a coloured noise, at variance with the
isoscalar case.
\par
Substituting the time Fourier transform of Eq. (\ref{fdt1_iv}) into
Eq. (\ref{fluc_iv}),
and retaining only the leading term of the expansion in powers of
$(\Gamma^{iv}_k/\omega^{iv}_k)$, we obtain for the covariance matrix
the expression
\begin{equation}
\sigma^2_{i,j}(k,t)= C_{i,j}^{iv}(k)\Big(1-e^{-2\Gamma^{iv}_kt}
-2\Gamma^{iv}_kt\,e^{-2\Gamma^{iv}_kt}\Big)+
O\big((\frac{\Gamma^{iv}_k}{\omega^{iv}_k})^2\big)\,,
\label{var_isov1}
\end{equation}
whose asymptotic value is given by
\begin{equation}
\sigma^2_{i,j}(k) = C_{i,j}^{iv}(k)\,.
\label{var_isov}
\end{equation}
We notice that, for isovector-like fluctuations,
$\sigma^2_{1,2}(k) = \sigma^2_{2,1}(k)$ is negative. Indeed neutron and proton
densities oscillate out of phase. \par
The covariance matrix of Eq. (\ref{var_isov}) refers to equilibrium
fluctuations at given values of density and charge asymmetry. It can be
directly obtained by means of the fluctuation--dissipation relation in the
case of a purely real pole (~$\Gamma^{iv}_k\rightarrow 0$~).\par
We finally remark that the covariance matrix of Eq. (\ref{var_isov}) is
obtained in the limit $T\rightarrow 0$ and, in addition, it does not depend
on the width $\Gamma^{iv}_k$ of the isovector--like resonance. This implies
that the density fluctuations of isovector--like nature, we are considering,
have a quantum origin.
\subsection{Size distributions}
Now we describe the procedure to determine the distribution for the size
of the correlation domains.
We closely follow the derivation given in
Ref. \cite{Mat00} for isoscalar density fluctuations, and we limit
ourselves to outline the steps relevant to the present more general
treatment. We distinguish the fluctuations of the proton density from
those of the neutron density.
\subsubsection {Correlation lengths}
The probability distribution for the sizes of the domains
where the fluctuations are correlated, $b_1$ and $b_2$ for protons and
neutrons respectively, can be obtained by means of the functional integral
\begin{eqnarray}
P(b_1,b_2,t)=&&\,\int d[\delta\varrho_i({\bf r},t)]\,\delta\bigg(b_1
-\int d{\bf r}d{\bf r}^\prime\delta\varrho_1({\bf r},t)f_1({\bf r})
\delta\varrho_1({\bf r}^\prime,t)f_1({\bf r}^\prime)\bigg)
\nonumber
\\
&&\,\delta\bigg(b_2
-\int d{\bf r}d{\bf r}^\prime\delta\varrho_2({\bf r},t)f_2({\bf r})
\delta\varrho_2({\bf r}^\prime,t)f_2({\bf r}^\prime)
\bigg)P[\delta\varrho_i({\bf r},t)]\, ,
\label{pbi0}
\end{eqnarray}
where $P[\delta\varrho_i({\bf r},t)]$ is the probability distribution
for the density fluctuations and $f_i({\bf r})$ are suitable weight
functions.
Moreover, we assume that the dynamical correlation lengths
for proton and neutron density fluctuations, $<b_1>$ and $<b_2>$, coincide
\begin{equation}
L(t)=\,\int\frac{d{\bf k}}{(2\pi)^3}\sigma^2_{1,1}(k,t)|f_1(k)|^2=\,
\int\frac{d{\bf k}}{(2\pi)^3}\sigma^2_{2,2}(k,t)|f_2(k)|^2\,,
\label{corrl}
\end{equation}
where $f_i(k)$ are the Fourier transforms of the weight functions.
In this way we assume that, on average, neutrons and protons are correlated
within the same domain.
We will see in the following how this can be related
to the average isospin distillation effect in the formation of fragments.
\par
Following the procedure used in Ref. \cite{Mat00}
we obtain for the probability distribution $P(b_1,b_2,t)$ the equation
\begin{eqnarray}
P(b_1,b_2,t)=&&\frac{1}{2\pi}\frac{1}{L(t)}\,\frac{1}
{[b_1+b_2]}\frac{1}
{\sqrt{\gamma(t)}}{\rm exp}\bigg(-\frac{[b_1+b_2]}
{4L(t)}\bigg)
\nonumber\\
&&\times{\rm exp}\bigg(-\frac{1}{4L(t)\gamma(t)}\frac{
[b_1-b_2]^2}{[b_1+b_2]}
\bigg)\,,
\label{distrb}
\end{eqnarray}
\par
where the parameter $\gamma(t)$ is given by
\begin{equation}
\gamma(t)=1-\frac{\int d{\bf k}\sigma^2_{1,2}(k,t)|f_1(k)|^2
\int d{\bf k}\sigma^2_{1,2}(k,t)|f_2(k)|^2}
{\int d{\bf k}\sigma^2_{1,1}(k,t)|f_1(k)|^2
\int d{\bf k}\sigma^2_{2,2}(k,t)|f_2(k)|^2}\,.
\label{gamma0}
\end{equation}
At variance with the case of isoscalar fluctuations,
the distribution $P(b_1,b_2,t)$ depends on the weight
functions $f_i(k)$. These functions, to some extent, are
arbitrary, the only requirement is
that the integrals containing them should converge.
For simplicity, we assume $|f_i(k)|^2=a_i|f(k)|^2$.
For the functional form of $|f(k)|^2$ we choose the simplest one:
$|f(k)|^2=1/k^2$. This choice is also supported by the fact that
for equilibrium fluctuations
the integral of the variance weighted with $1/k^2$
gives the correct value of the correlation length \cite{Mat00}.
In addition, we have found that for the physical situations
considered in this paper, the value of the
parameter $\gamma(t)$ to a large extent is insensible to the particular
form of the weight function $|f(k)|^2$.
\par
From the probability distribution of the domain sizes we can obtain
the distribution of the numbers of correlated protons $Z$ and neutrons $N$,
assuming the correlation domains to be spherical.
The relations between $Z$ and $b_1$, and $N$ and $b_2$
can be expressed as
$b_1=2r_{01}Z^{1/3}$ and $b_2=2r_{02}N^{1/3}$, where $r_{0i}$ is
the mean interparticle spacing for nucleons of the $i$--species,
calculated at the actual values of asymmetry and density (when fragments are
formed),
that are different from asymmetry and density of the initial matter.
The fact that the fragment size is related to the correlation length
can be considered as a reasonable assumption in situations where
isoscalar-like modes are the dominant ones, as in fragmentation processes.
So, since on average $b_1$ is equal to $b_2$, we obtain:
$r_{01}/r_{02}=(\rho_2/
\rho_1)^{1/3}=<N^{1/3}>/<Z^{1/3}>$, where $\rho_i$ are the densities
calculated at the time fragments are formed.
In this way the ratio $r_{01}/r_{02}$ can be
related to the average asymmetry of the liquid (fragment) phase,
obtained after the distillation process has occurred.
One can consider, for instance, as average fragment
asymmetry, values extracted from dynamical SMF simulations
for primary fragments \cite{Bar02}.
Then, the probability distribution of $Z$ protons and $N$ neutrons
contained in a correlation domain, acquires the form
\begin{eqnarray}
P(Z,N,t)=&&\frac{1}{9\pi}\frac{r_0}{L(t)}\,\frac{\lambda_1\lambda_2}
{[\lambda_1Z^{1/3}+\lambda_2N^{1/3}]}\frac{1}{(ZN)^{2/3}}\frac{1}
{\sqrt{\gamma(t)}}{\rm exp}\bigg(-\frac{r_0}{2L(t)}
[\lambda_1Z^{1/3}+\lambda_2N^{1/3}]\bigg)
\nonumber\\
&&\times{\rm exp}\bigg(-\frac{r_0}{2L(t)}\frac{1}{\gamma(t)}\frac{
[\lambda_1Z^{1/3}-\lambda_2N^{1/3}]^2}{[\lambda_1Z^{1/3}+\lambda_2N^{1/3}]}
\bigg)\,
\label{distrzn}
\end{eqnarray}
with $\lambda_i=r_{0i}/r_0$, where $r_0$ is the mean interparticle
spacing for nucleons of both species.
\subsubsection {Correlation volumes}
One may also assume that the size of fragments is directly
related to a correlation volume $V$, instead of a correlation length.
Equation (\ref{distrb}) can be rewritten for
the correlation volumes, just replacing $b_1$ and $b_2$ with
$V_1$ and $V_2$. Then the probability distribution, after some algebra,
reads:
\begin{eqnarray}
P(Z,N,t)=&&\frac{1}{2\pi}\frac{1}{{\bar V}(t)}\,\frac{1}
{[\rho_2Z+\rho_1N]}\frac{1}
{\sqrt{\gamma(t)}}{\rm exp}\bigg(-\frac{1}{4{\bar V}(t)}
[Z/\rho_1+N/\rho_2]\bigg)
\nonumber\\
&&\times{\rm exp}\bigg(-\frac{1}{4{\bar V}(t)}\frac{1}{\gamma(t)}\frac{
[Z/\rho_1-N/\rho_2]^2}{[Z/\rho_1+N/\rho_2]}
\bigg)\,
\label{distrzn_volume}
\end{eqnarray}
where ${\bar V}$ is the average correlation volume for nucleons of both
species.
For not too large asymmetries, this can be rewritten in the following form:
\begin{eqnarray}
P(Z,N,t)=&&\frac{1}{\pi A {\bar A}}\,\frac{1}
{\sqrt{\gamma(t)}}{\rm exp}\bigg(-\frac{A}{2{\bar A}}
\bigg)
\nonumber\\
&&\times{\rm exp}\bigg(-\frac{A}{2{\bar A}}\frac{1}{\gamma(t)}
\Big[\frac{N-Z}{A} - \alpha\Big]^2
\bigg)\,
\label{distrzn_stat}
\end{eqnarray}
where $\alpha= (\rho_2-\rho_1)/(\rho_2+\rho_1)$
represents the average asymmetry of fragments and ${\bar A}$ is
the average mass.
\section{\label{BB}Results}
In our calculations we have adopted a schematic Skyrme--like effective
interaction, that can be expressed as a sum of two terms
\[
{\cal A}_{i,j}(k)= {\cal A}(k)+{\cal S}_{i,j}(k)\,.\]
For the symmetric term ${\cal A}(k)$ we use the finite--range effective
interaction introduced in Ref. \cite{ColA94}:
\begin{equation}
{\cal A}(k)=\Big(A\frac{1}{\varrho_{eq}}+(\sigma+1)\frac{B}
{\varrho_{eq}^{\sigma+1}}\varrho^{\sigma}\Big)e^{-c^2\,k^2/2}\,,
\label{inters}
\end{equation}
with $\varrho=\varrho_1+\varrho_2$ and
\[
A=-356.8\,{\rm MeV},~~B=303.9\,{\rm MeV},~~\sigma=\,\frac{1}{6}\,.
\]
These values reproduce the binding energy
($15.75\,{\rm MeV}$) of symmetric nuclear matter at saturation
($\varrho_{eq}=0.16\,{\rm fm}^{-3}$) and give an
incompressibility modulus of $201\,{\rm MeV}$.
The width of the Gaussian in Eq. (\ref{inters}) has been chosen in
order to reproduce the surface-energy term as prescribed in Ref. \cite{Mye66}.
\par
The isospin--dependent part, ${\cal S}_{i,j}(k)$, contains three different
terms
\begin{equation}
{\cal S}_{i,j}(k)=\frac{\partial^2{\cal E}_{symm}}{\partial\varrho_i
\partial\varrho_j}+\tau_i\tau_jDk^2+\frac{1+\tau_i}{2}V_C(k)\delta_{i,j}
\,,
\label{interv}
\end{equation}
with $\tau_1=1$ and $\tau_2=-1$. The double derivative
of the potential part of the symmetry energy density,
${\cal E}_{symm}$, is calculated in the unperturbed initial state.
For the coefficient of the isovector surface term
we use the value $D=40\,{\rm MeV\cdot fm}^5$ \cite{Bay71}.
Concerning the Coulomb
interaction, a mean--field exchange contribution
\[V_C^{ex}=-\frac{1}{3}\Big(\frac{3}{\pi}\Big)^{1/3}e^2\varrho_1^{-2/3}
\]
is added to the bare Coulomb force.
\par
In order to stress the effects of the asymmetry of the nuclear medium,
we will present results obtained with two different parametrizations
of the symmetry energy: one with a stronger density dependence
(~``superstiff'' asymmetry term~) and the other one with a weaker
density dependence (~``soft'' asymmetry term~).
In both cases the density dependence of the symmetry energy
can be expressed by
\[{\cal E}_{symm}(\varrho_1,\varrho_2)=S(\varrho)(\varrho_2-\varrho_1)^2
\,,\]
with
\begin{equation}
S(\varrho)=\frac{2d}{\varrho_{eq}^2}\frac{\varrho}{1+\varrho/\varrho_{eq}}
\,,
\label{stiff}
\end{equation}
where $d=19\,{\rm MeV}$ \cite{Bao00}, for the ``superstiff'' case, and
\begin{equation}
S(\varrho)=d_1-d_2\varrho
\,,
\label{soft}
\end{equation}
where $d_1=240.9\,{\rm MeV\cdot fm}^3$ and $d_2=819.1\,{\rm MeV\cdot fm}^6$
\cite{ColA98}, for the ``soft'' case.
\begin{figure}
\includegraphics{fig_sigmatot}
\caption{\label{fig1}The variance for the unstable modes as a function
of $k$ at four different times: from bottom to top $t=30,100,125,150
\,{\rm fm}/c$. The values of $\varrho$, ${\rm T}$, and $\alpha$ are
$\varrho=0.3\varrho_{eq}$, ${\rm T}=4.5\,{\rm MeV}$,
and $\alpha=0.2$.
}
\end{figure}
\par
The inclusion of the Coulomb interaction presents sizeable
effects on the stability conditions of nuclear matter. It gives rise
to an overall decrease of the growth rate of density
fluctuations with a corresponding contraction of the
instability region in the ($\varrho,T$) phase diagram \cite{Fab98,
ColonnaPRL}.
Moreover, it can be observed that, when the
Coulomb force is included, the growth rate vanishes for
sufficiently low values of the wave vector $k$
($k_{min}\simeq 0.2{\rm fm}^{-1}$) \cite{Fab98}.
\par
In the integrals of Eqs. (\ref{corrl}) and (\ref{gamma0}), which
determine the relevant parameters $L(t)$ and $\gamma(t)$
for the distribution $P(Z,N,t)$, we
consider only the contributions from the unstable modes. To this
purpose, we put the weight function $f(k)$ equal to zero for $k$
larger than the value beyond which the rate $\Gamma_k$ becomes
negative. However,
to evaluate the total value of the covariance matrix,
we will consider the sum
of the asymptotic value of the contribution due to isovector-like fluctuations,
Eq. (\ref{var_isov}) and the contribution due to the isoscalar-like modes,
Eq. (\ref{variance0}), that grows exponentially.
The variance for the unstable fluctuations of
the isoscalar density,
$\sigma^2(k) = \sigma^2_{1,1}(k) + \sigma^2_{2,2}(k) + 2 \sigma^2_{1,2}(k)$,
is displayed in Fig.~\ref{fig1} at
four different times. We only report the results obtained with the
``superstiff'' symmetry term. For the isoscalar fluctuations
the ``soft'' asymmetry term gives almost undistinguishable curves.
The values chosen for the density $\varrho=0.3\varrho_{eq}$ and for
the temperature $T=4.5\,{\rm MeV}$ are in the range expected for the
multifragmentation process \cite{Cho04,Tam98}. For the asymmetry
we choose a value of $\alpha=0.2$. Figure \ref{fig1}
shows that the variance becomes a more and more peaked function
about the most unstable mode with increasing time. It is worth noticing
that the values of the variance of our calculations quite well
compare with those obtained in Ref. \cite{Ayi96} within a different
approach including the effects of the nucleon-nucleon collisions.
This supports the
suggestion that the development and the growth of the fluctuations
are essentially determined by the instabilities of the mean field,
while the seeds are provided by the thermal agitation of the system.
\par
We now turn to evaluate fragment isotopic distributions.
In order to take into account that $Z$ and $N$ are discrete variables
we express the probability of finding a correlation domain
containing $Z$ protons and $N$ neutrons, $Y(Z,N,t)$, through the integral
\begin{equation}
Y(Z,N,t)=\,\int _{Z-1}^{Z}dZ\int _{N-1}^{N}dN\,P(Z,N,t)\,.
\label{probzn}
\end{equation}
For large $Z$ and $N$, $Y(Z,N,t)$ tends to coincide with $P(Z,N,t)$.
We first consider Eq.(\ref{distrzn}) to calculate
the distribution $P(Z,N,t)$ and the probability $Y(Z,N,t)$. They are
determined once the ratio $r_0/L(t)$ and the parameter
$\gamma(t)$ have been calculated for given values of $\varrho$, $T$
and average asymmetry $\alpha$ of the system at the break--up.
The length $L(t)$ characterizes the decrease
of the correlation function with distance. The procedure to determine
its value has been extensively discussed in Refs. \cite{Mat00,Mat03}.
Here, we focus our attention on the calculation
of the parameter $\gamma(t)$
characterizing the widths of the isotopic distributions.
\par
This can be evaluated by rewriting Eq. (\ref{gamma0}) with the assumptions
about the weight functions introduced in Sec.~\ref{AA}:
\begin{equation}
\gamma(t)=1-\frac{\int dk\sigma^2_{1,2}(k,t)|f(k)|^2
\int dk\sigma^2_{1,2}(k,t)|f(k)|^2}
{\int d k\sigma^2_{1,1}(k,t)|f(k)|^2
\int dk\sigma^2_{2,2}(k,t)|f(k)|^2}\,.
\label{gamma}
\end{equation}
\par
Since the magnitude of the isospin--distillation effect, i.e. the ratio
$\sigma_{1,2}^2(k)/\sigma_{1,1}^2(k) = \sigma_{2,2}^2(k)/\sigma_{1,2}^2(k)$,
depends on the wave
number $k$, even considering only the contribution of the isoscalar-like
modes to $\sigma_{i,j}^2(k)$, one obtains a non vanishing value of the width
$\gamma$. Considering also the contribution of isovector-like fluctuations,
the width $\gamma$ increases, as we will show in the following.
For values of the asymmetry $\alpha$ of nuclear interest, the parameter
$\gamma(t)$ turns out to be about $10^{-3}$ for both the considered
asymmetry terms in the nucleon--nucleon interaction
( ``soft'' and ``superstiff'' ).
As a general trend, the parameter $\gamma(t)$ increases
with increasing asymmetry and density of the decomposing system, and
decreases with the time.
\begin{figure}
\includegraphics{fig_yield6}
\caption{\label{fig2}Calculated isotopic yields of $Z=6$--fragment with
the ``superstiff'' symmetry term (diamonds) and the ``soft''
symmetry term (triangles).
The circles represent the results obtained neglecting the contribution
of isovector-like fluctuations in the ``soft'' case.
The values of $\varrho$, ${\rm T}$, $L$, and $t$ are
$\varrho=0.3\varrho_{eq}$, ${\rm T}=4.5\,{\rm MeV}$,
$L=1.3\,r_0$, and $t=125\,{\rm fm}/c$. Top panel: $\alpha=0.1$,
the value of the parameter $\gamma(t)$ is $\gamma(t)=1.02\,10^{-3}$
for the ``superstiff'' symmetry term, for the ``soft'' symmetry term
$\gamma(t)=0.69\,10^{-3}$ and $\gamma(t)=0.37\,10^{-3}$, with and without
the contributions from the isovector--like fluctuations, respectively.
Bottom panel: $\alpha=0.2$, the value of the
parameter $\gamma(t)$ is $\gamma(t)=1.62\,10^{-3}$
for the ``superstiff'' symmetry term, for the ``soft'' symmetry term
$\gamma(t)=0.89\,10^{-3}$ and $\gamma(t)=0.56\,10^{-3}$, with and without
the contributions from the isovector--like fluctuations, respectively.
}
\end{figure}
\par
In Fig.~\ref{fig2} we report the isotopic yields
of $Z=6$--fragment, calculated according to
Eqs. (\ref{distrzn}) and (\ref{probzn})
for two different values of the asymmetry:
$\alpha=0.1$ and $\alpha=0.2$.
The used values of the parameters ${\rm T}=4.5\,{\rm MeV}$,
$\varrho=0.3\varrho_{eq}$ and $t=125\,{\rm fm}/c$, where $t$ is the time that
the system spends in the instability region, are compatible with the
analogous values obtained within the SMF approach of Ref. \cite{Bar02}.
For the dynamical correlation length we have chosen the value
$L=1.3\,r_0$. This value corresponds to the effective exponent
$\tau_{eff}=1.65$ of the power law $Y(Z)=Y_0Z^{-\tau_{eff}}$
for fragment distribution \cite{Mat03}.
In the figure we display the results obtained with the ``superstiff''
asymmetry term and with the ``soft'' asymmetry term of the
nucleon--nucleon interaction.
Moreover we compare also the relative contribution of isoscalar-like
and isovector-like fluctuations to the width.
In the "superstiff" case isovector--like oscillations are suppressed
for the considered values of $\varrho$, $T$ and $\alpha$,
i.e. Eq. (\ref{rate}) has only one pole,
so the width comes essentially from the dispersion of the chemical effect
in the isoscalar-like fluctuations (diamonds).
In the ``soft'' case, the full calculation is represented by triangles, while
the result obtained taking into account only the contribution from
the isoscalar-like modes is
represented by circles. Comparing diamonds and circles,
we observe that the ``superstiff''
asymmetry term gives rise to a wider isotopic distribution.
This is due to the fact that the ``soft'' asymmetry term, at the
considered density, is more effective to drive fragments
closer to the average asymmetry value, with respect to an asymmetry
term with a stronger density dependence.
Indeed we find that, in spite of the competition with Coulomb and surface
effects, the isospin distillation mechanism does not change much with
the wave number $k$, in the ``soft'' case.
The counterpart in our formalism is that in this case the behaviors
of the components of the covariance matrix, as
functions of $k$, are more similar each other
reducing the width of the isotopic distribution.
However, adding the contributions due to the isovector-like fluctuations,
the total width obtained in the ``soft'' case (triangles) becomes closer
to the ``superstiff'' results.
It is also possible to observe that the contribution of the
isovector-like fluctuations to the full width is more important at
smaller asymmetry. This is because isovector-like
fluctuations become weaker
when increasing the asymmetry of the matter.
\par
Figure \ref{fig2} also shows that the width of the isotopic yields
increases with asymmetry. This corresponds to the general property
that for more neutron--rich systems the density--density response
function of neutrons is enhanced with respect to that of protons.
In addition, we can see that
the more neutron--rich system ($\alpha=0.2$) produces the more
neutron--rich isotopes, as expected.
\par
It is worth to remark that both the overall behavior and the widths
of the distributions of Fig.~\ref{fig2}
favourably compare with the corresponding distributions for primary
fragments calculated within the SMF approach \cite{Liu04}.
\par
\begin{figure}
\includegraphics{fig_volayield6_20.eps}
\caption{\label{fig2_new}Isotopic distributions calculated according to the
correlation volume prescription (Eq. \ref{distrzn_stat}).
The values of the parameters and the symbols are the same as in Fig~\ref{fig2}.
}
\end{figure}
In Fig.~\ref{fig2_new} we present isotopic distributions obtained using the
correlation volume prescription (Eq. \ref{distrzn_stat}),
with ${\bar A} = 20$. This value corresponds to the average size
of intermediate mass fragments,
as obtained in the considered conditions of density and temperature.
As one can see by comparing Figs.~\ref{fig2} and \ref{fig2_new},
results are not very different with the two prescriptions.\par
\begin{figure}
\includegraphics{fig_isoscaling03_soft_supstiff}
\caption{\label{fig3}Isotopic ratio
$R_{21}(N,Z)=Y_{\alpha=0.2}(N,Z)/Y_{\alpha=0.1}(N,Z)$
calculated with the ``superstiff'' symmetry term (solid lines )
and with the ``soft'' symmetry term (dashed lines).
Lines correspond to different values of $Z$, $Z=3-8$ from left to
right. The values of remaining parameters are the same as in Fig~\ref{fig2}.
}
\end{figure}
The ratio between isotopic yields observed in two different reactions,
$R_{21}(N,Z)=Y_{\alpha_2}(N,Z)/Y_{\alpha_1}(N,Z)$,
shows a very simple behavior. As a function of $Z$ and $N$, it
can well be fitted by an exponential law (the so called
isoscaling relationship) \cite{Xu00,Tsa01,Tan01,Tsa101}.
In addition, the isoscaling relationship
has been reproduced by SMF--model calculations also for the distributions
of primary fragments \cite{Liu04}. This particular feature
of the isotopic distributions can represent an effective tool
to compare isotopic distributions from systems with different
$N/Z$ ratios.
\par
The isotopic ratio $R_{21}(N,Z)$ calculated in our approach,
according to Eqs. (\ref{distrzn}) and (\ref{probzn}),
for two different values of the asymmetry parameter, $\alpha_2=0.2$ and
$\alpha_1=0.1$, is displayed in Figs.~\ref{fig3} and \ref{fig4}.
In Fig.~\ref{fig3} we compare the values of $R_{21}(N,Z)$
as a function of $N$, obtained
with the ``superstiff'' symmetry term and with the ``soft'' symmetry term.
The linear behavior, in logaritmic scale, with the same slope
for every $Z$ is reproduced in both cases within a satisfying
approximation. Because of the smaller value of the width
parameter $\gamma(t)$, the ``soft'' symmetry term
gives a steeper slope with respect to the ``superstiff'' term.
The average values of the slope approximatively are $2.2\pm 0.2$
and $1.5\pm 0.15$ for the ``soft'' case and the ``superstiff'' case
respectively. \par
\begin{figure}
\includegraphics{fig_isoscaling03_04supstiff}
\caption{\label{fig4}Same as in Fig.~\ref{fig3} but using
only the ``superstiff'' symmetry term and
for two different values of the density:
solid lines correspond to $\varrho=0.3\,\varrho_{eq}$
and $t=125\,{\rm fm}/c$, dashed lines correspond to
$\varrho=0.4\,\varrho_{eq}$ and $t=150\,{\rm fm}/c$.
}
\end{figure}
In Fig.~\ref{fig4} the ratio $R_{21}(N,Z)$ is displayed for two values
of the density of the system at the break--up. In order
to obtain fluctuations of similar magnitude
in the two cases, two different times
the system spends in the instability region are considered.
Nevertheless, a behavior with a steeper slope
is observed in the more unstable case. This is
due to a smaller value of $\gamma(t)$ in this case, since, for a given
charge asymmetry, the response functions of protons and of neutrons
tend to be more similar with decreasing density.
\par
We now perform a more quantitative comparison between predictions of our
approach and results for primary fragments of the SMF--model
calculations of Ref. \cite{Liu04}. To this purpose we adopt for the
average asymmetry of fragments the values predicted
by the SMF model for semicentral collisions of $^{112}{\rm Sn}+^{112}
{\rm Sn }$ and $^{124}{\rm Sn}+^{124}{\rm Sn}$ \cite{Bar02,Liu04}:
$\alpha_1=0.13$ and $\alpha_2=0.195$, respectively.
In both the approaches the same ``superstiff''
symmetry term for the effective interaction is used.
Also the values of density
$\varrho=0.3\varrho_{eq}$, temperature ${\rm T}=4.5\,{\rm MeV}$, and
time spent at the break--up $t=125\,{\rm fm}/c$ are chosen according to the
results of SMF--model calculations. Figure~\ref{fig5} shows the isotopic
ratio $R_{21}(N,Z)$ calculated with our approach
and the curves obtained in Ref.~\cite{Liu04} by fitting the results
of the SMF model with an exponential law. We observe a remarkable
agreement between the results of our nuclear matter calculations
and the simulations of the SMF--model.
\begin{figure}
\includegraphics{fig_isoscalingsup03_125}
\caption{\label{fig5}Comparison of calculated isotopic ratio
$R_{21}(N,Z)=Y_{\alpha=0.195}(N,Z)/Y_{\alpha=0.13}(N,Z)$ (diamonds)
with the fit for primary fragments of Ref.~\cite{Liu04} (solid lines).
From left to right $Z=3,4,5,6,7,8$. Calculations are done with the
``superstiff'' symmetry term. The values of density $\varrho$,
temperature ${\rm T}$, time $t$, and ratio $L/r_0$ are the same as
in Fig.~\ref{fig2}.
}
\end{figure}
\section{Conclusions}
In this article we discuss relevant observables of multifragmentation
processes in charge asymmetric nuclear matter, such
as the isotopic distribution of
intermediate--mass fragments, as obtained within the spinodal decomposition
scenario, on the basis of an analytical approach.
Fragmentation happens due to the development of isoscalar-like unstable
modes, i.e. unstable density oscillations with also a chemical component,
leading to the formation of more symmetric fragments. We
find that the isotopic distributions are peaked at a value given by the
average distillation effect, while the width is determined by
the dispersion of the chemical effect among the relevant unstable modes
and by isovector-like fluctuations present in the matter that undergoes
spinodal decomposition.
The size of this dispersion is mostly due to the
competition between symmetry energy
effects (that favour the formation of symmetric fragments)
and the Coulomb repulsion, that acts against the concentration of protons
in large density domains, expecially for modes with large wavelength.
Clearly the net result of this competion
also depends on the EOS used. Smaller widths are
obtained with a ``soft'' symmetry energy term. However,
the contribution due to isovector-like fluctuations is more important
in the ``soft'' case, indeed in the ``superstiff'' case isovector oscillations
are suppressed. Hence finally the isotopic distributions are quite similar
when using the two parameterizations of the symmetry energy.
In particular, we find that, when considering two systems with different
asymmetry, the isotopic (or isotonic) yields obey an approximate
isoscaling, with a slope connected to the difference betwen the asymmetries
of the two systems and to the differences between the widths
of the isotopic distributions.
Hence isoscaling properties can be recovered in a dynamical picture.
We notice that isoscaling has been found in dynamical simulations
of heavy ion collisions, such as
stochastic mean field \cite{Liu04} and antisymmetrized molecular dynamics
calculations \cite{Ono}.
The isoscaling parameters are also connected to the properties
of the symmetry term in the EOS. Indeed we have seen that
a stiffer behavior of the symmetry energy
term yields larger isotopic widths, leading to smaller values of the
slope (see Fig. \ref{fig3}).
However, as reported in Ref. \cite{Bar02}, we also
observe that in collisions of charge asymmetric systems,
pre-equilibrium emission
is less neutron rich when using a stiffer parametrization of the
symmetry term (thus leading to more asymmetric fragments), with respect to the
``soft'' case. Therefore, in the isoscaling analysis, there could be
a compensation between the average asymmetry of fragments (larger in the
``stiff'' case) and the width of the distribution
(also larger in the ``stiff'' case).
In fact, for the systems considered in Ref. \cite{Liu04},
similar values of the slope are obtained for the two parameterizations
considered for the symmetry energy.
It may also be interesting to notice that the values obtained in our
calculations are larger than the predictions
of statistical multifragmentation models, see Ref. \cite{Tsa101}.
Of course this picture can be modified by the secondary de-excitation
process, that reduces the asymmetry of fragments and, consequently,
the slopes deduced from isoscaling.
Hence the final distributions can be quite different from the primary ones.
A more detailed study, aiming to extract information on the primary
distributions and on the fragmentation mechanism,
would require the introduction of more sophisticated observables,
probably based on an event by event analysis, in line with the
recent investigations of correlations between intermediate--mass
fragments \cite{Bor02}.
|
{
"timestamp": "2005-03-08T19:37:21",
"yymm": "0503",
"arxiv_id": "nucl-th/0503018",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503018"
}
|
\section{Introduction}
There is a well known similarity between the two-dimensional models of the
planetary atmosphere and the magnetized plasma. In the absence of
dissipation the models can be reduced to differential equations having the
same structure: the Charney equation for the nonlinear Rossby waves , in the
physics of the atmosphere \cite{Charney}; and the Hasegawa-Mima equation for
drift wave turbulence, in plasma physics \cite{HM}. They are similar with
the Navier-Stokes equation because they have two conserved quantities, the
energy and the enstrophy. This in principle allows states of negative
temperatures, or, equivalently, these models support a trend to organised
vortical flow. It results the possibility to have as solutions coherent
structures (vortices) besides the turbulent states characterised by spectral
cascade.
These analytical models have led to a serious advancement of our knowledge
in both fields. However the stationary states appear to be described within
these models by a reduced equation having a too wide generality,
representing actually something as a constraint with weak ability to
identify unequivocally the real solutions: it simply states that at
stationarity the advection of the vorticity by the velocity vector field
vanishes. In reality, numerical simulations show that the stationary states
reached in relaxation are very regular and persist for a long time period
and that this set of asymptotic states is not the huge space of functions
able to fulfill the constrained mentioned above. The fluid evolves at
relaxation toward a reduced subset of functions, characterized by regular
shape of the streamfunction \cite{HTK}, \cite{KMcWT}, \cite{KTMcWP}, \cite
{HH} (and references therein). At the oposite limit the turbulent regime can
be treated with renormalization group methods \cite{DiamondKim}.
It is well-known that the same phenomenon exists in the case of the ideal
fluid described by the Euler equation. By experiments and numerical
simulation it has been shown that the ideal fluid evolves at relaxation
toward a very ordered flow pattern, consisting of two (positive and
negative) vortices and that this state persists for very long times, being
limitted by only the effect of some residual dissipation. From numerical
simulations it has also been inferred the form of the flow function. It has
been found that the streamfunction obeys, in these states, the \emph{sinh}%
-Poisson equation. Montgomery and his collaborators have developed a
theoretical statistical model which explains the appearence of this equation
in this context \cite{Montgomery1}, \cite{Montgomery2}, \cite
{KraichnanMontgomery}, \cite{Montg2}, \cite{Montg3}, \cite{Joyce}, \cite
{Smith}. Later, the equation has also been derived by formulating the
continuum version of point-like vortices as a field theoretical model of
interacting gauge and matter fields in the adjoint representation of $%
SU\left( 2\right) $ \cite{FlorinMadi1}. The essential point of the latter
derivation was the self-duality of the relaxation states of the fluid.
No equation (similar to the \emph{sinh}-Poisson equation in the Euler fluid
case) has been found for the Charney-Hasegawa-Mima (CHM) equation, despite a
considerable effort \cite{Seyler}, \cite{Montg4}. However, as mentioned
before, there are convincing experimental and numerical indications that the
fluids (atmosphere and plasma) evolve to a reduced subset of states.
\bigskip
We have developed a field theoretical model for the point-like vortices with
short range interaction, based on Chern-Simons action for the gauge field in
interaction with the nonlinear matter field, again in $SU\left( 2\right) $
algebra. It is then possible to derive the energy as a functional that
becomes extremum on a subset of stationary states and presents particular
properties. The general characterization of this family of states is their
\emph{self-duality}, which here means that the energy functional becomes
minimum because the square terms are all vanishing, leaving as lower bound a
quantity with topological meaning. A very detailed account of the derivation
is in Refs. \cite{FlorinMadi2}, \cite{Toki2003}.
The result is a set of equations parametrized by the solutions of the
Laplacean equation in two-dimensions.
The simplest of these equations is
\begin{equation}
\Delta \psi +\frac{1}{2p^{2}}\sinh \psi \left( \cosh \psi -p\right) =0
\label{eq}
\end{equation}
(where $p$ is a positive constant). There are already some confirmations
that this is the equation governing the asymptotic stationary states of the
CHM fluids : the scatterplots of $\left( \psi ,\omega \right) $ =
(streamfunction, vorticity) obtained in experiments \cite{expgeo} and the
scatterplots obtained in numerical simulations \cite{Seyler} are very
similar to the nonlinear term of Eq.(\ref{eq}).
The objective of this work is to provide the first elements resulting from a
numerical investigation of this equation.
The results are summarised here. This differential equation is able to
reproduce the main two-dimensional features of the typhoon vortical flow. In
the physics of the atmosphere, it seems that other examples, like the
tropical cyclones, can be reproduced by solutions of this equation. The
following are the features we consider as very particular to the typhoon
morphology (in $2D$) \ \cite{Andrew}, \cite{cycrev}, \cite{ReMont}, \cite
{mesov}:
\begin{enumerate}
\item The very narrow dip of the azimuthal velocity (mean tangential wind)
in the center of the vortex, compared with the very large extension in
space. This is characterized by the ``radius of the maximum tangential
wind'' and this radius, as mentioned, is much smaller than the diameter of
the vortex. Our equation is able to generate solutions with this structure.
\item The slow decay of the magnitude of the azimuthal velocity toward the
periphery, compared with the very fast decay toward the center; this is
reproduced by the solutions of this equation.
\item The very low magnitude (almost vanishing) of the vorticity over most
of the vortex (approx. from the radius of maximum wind to the periphery),
while the magnitude in a narrow central region is extremely high. This
feature is also reproduced by the equation.
\item quantitatively, we obtain for the diameter of the typhoon's eye a
relatively good magnitude. The vorticity is higher than in observations but
not far from the realistic range.
\end{enumerate}
\bigskip
We have very encouraging results of studies on plasma vortices, but they are
not reported here. In plasma physics, the symmetrical, stable, vortical
structures observed in experiments in the linear machine seem to belong to
the class of solutions of this equation. We have also obtained several
solutions that are very similar to the crystals of vortices, known from
experiments.
\section{Numerical studies of the equation}
The numerical solution of this equation appears to be very difficult. This
may be explained by the fact that the exponentials of the two functions $%
\sinh $ and $\cosh $ are very rapidly-varying functions and any perturbation
is amplified and propagated in the solution.
In addition, the Laplace operator has spurious solutions with exponential
behavior that have to be eliminated by the numerical procedure.
The paper of McDonald \cite{McDonald} on the numerical integration of the
\emph{sinh}-Poisson equation is very helpful in understanding the problems
related to a numerical treatment of our equation. However the approach
proposed in that paper requires to use a small mesh, specifically for
excluding the spurious modes of the Laplacean. In the case of our equation,
the vortices require a reasonable detailed description and this needs larger
meshes. Then the problem of the precision of integration procedure arises
and, if the initialization happens to be far from one of the solution, the
number of iteration of the solver is high and the errors accumulate, leading
to lack of convergence. It may be supposed that the solutions would be
similar to those of the \emph{sinh}-Poisson equation, but structures with
sharp spatial variation may be possible \cite{MontgPriv}.
\bigskip
The structure of the function space representing the union of attractors for
the various solutions of this equation appears to be very complex. This
immediately translates into serious obstacles in the attempt to reach one of
the presumed solution. The main instrument is, naturally, the
initialization, \emph{i.e.} to start the integration in the right subspace,
representing the attractor of that solution. Since there is no available
analytical description of this space, the search is simply a problem of
guessing a reasonable initial function and to repeat as many times as
necessary. One of the specific behaviors is the tendency of driving the
solution toward the constant value
\begin{equation}
\psi =\psi _{b}^{\left( 1,2\right) } \label{psi12b}
\end{equation}
(see Eq.(\ref{bcon})) which trivially verifies the equation. This seems to
imply that there is a large attractor in the function space around these
constant solutions. The solution which is larger in absolute magnitude is
less stable since any fluctuation around the constant generates high
vorticity. We underline that the integrations described here are \textbf{not}
radial (\emph{i.e.} unidimensional).
With all the difficulties of getting a right initial positioning in the
integration procedure we note however that the solution with the \emph{%
typhoon} morphology appears instistently from a wider class of initial
shapes.
\subsection{The numerical code}
We use the code ``\textbf{GIANT} A software package for the numerical
solution of very large systems of highly nonlinear systems'' written by U.
Nowak and L. Weimann \cite{giant}. The code belongs to the numerical
software library \emph{CodeLib} of the \textbf{Konrad Zuse Zentrum fur
Informationstechnik Berlin}. The meaning of the abbreviation is: GIANT =
Global Inexact Affine Invariant Newton Techniques and corresponds to the
implementation of the method proposed by Deuflhard (for many references see
\cite{giant}).
This code solves nonlinear problems
\begin{eqnarray}
F\left( x\right) &=&0 \label{gian} \\
\text{initial guess of solution, }x &=&x_{0} \notag
\end{eqnarray}
The global affine invariant Newton schemes requires the solution of linear
problems. For higher accuracy meshes the linear problems are solved by
iterative methods. The balance between numerical requirements of the Newton
iteration (called \emph{outer} iteration) and the iterative linear solver (%
\emph{inner}) means that the solution of the linear problem will be
approximative. Two packages of linear solvers can be used, GMRES
(generalized minimum residual : Brown, Hindmarsh, Seager) and GBIT1 (fast
secant method using the \emph{Good-Broyden} updates : Deuflhard, Freund and
Walter).
All necessary description of the method, of the code and many studies of the
numerical precision and computer efficiency are presented by Nowak and
Weimann in the documentation of the code.
The code has been implemented and the tests have been performed with
successful results (we are grateful to Dr. Weimann for his kind help in this
problem).
\subsection{Boundary conditions}
The boundary conditions are dependent on the value of $p$. The physical
model imposes that the scalar function $\psi $ remains nonzero at infinity
for $p>1$. This means that we must require that the boundary condition is
one of the roots of the algebraic equation
\begin{equation}
\cosh \psi -p=0 \label{coshp}
\end{equation}
which can give the vanishing of the physical vorticity at infinity. Then we
impose
\begin{eqnarray}
\text{boundary condition }\psi \left( r\rightarrow \infty \right) &=&\psi
_{b}^{\left( 1,2\right) } \label{bcon} \\
&=&\ln \left( p\pm \sqrt{p^{2}-1}\right) \notag
\end{eqnarray}
\subsection{Initialization}
In general the initial profiles has been of two types: symmetric profiles
with maximum centered on $\left( 0,0\right) $ and initializations with
functions expressed as product of trigonometric functions.
The symmetric profiles has been chosen as Gaussian functions, or various
annular shapes.
For may runs, as suggested by the experiments for the \emph{sinh}-Poisson
equation (paper by McDonald \cite{McDonald}), the initial function is taken
as a product of trigonometric functions in both directions, $x$ and $y$. We
need to prepare the initial function in the sense that the values that are
obtained in for the vorticity, \emph{i.e.} the Laplacean of the initial
distribution should not be too different of what is obtained by simply
inserting the initial function in the nonlinear term. For this we take a
coefficient $\psi _{in}$ of the product of the trigonometric functions as a
parameter to be determined.
The initial function is taken as
\begin{equation}
\psi \left( x,y\right) =\psi _{b}^{\left( 1\right) }+\psi _{in}\sin \left(
k\pi \frac{x-x_{\min }}{x_{\max }-x_{\min }}\right) \sin \left( k\pi \frac{%
y-y_{\min }}{y_{\max }-y_{\min }}\right) \label{trig}
\end{equation}
where $k$ is the periodicity of the profile and $\psi _{in}$ is the
amplitude. We insert in the equation and we require approximative equality
of the two parts, the vorticity and the nonlinearity. This is obtained by
choosing a point $\left( x,y\right) $ where the initial function is maximum
and it results a condition on only the amplitude, $\psi _{in}$.
\begin{eqnarray}
\Delta \psi &=&\psi _{in}\left[ 2\left( k\pi \right) ^{2}\right]
\label{gues} \\
&\simeq &\frac{1}{2p^{2}}\sinh \psi _{in}\left( \cosh \psi _{in}-p\right)
\notag
\end{eqnarray}
This equation is solved and one of the roots is selected as the amplitude of
the initial function.
\bigskip
The experiments with simple $\sin $ functions frequently lead to
difficulties of convergence. Looking at the function's form (either partial
evolutions during iterations or good, converged, results) we notice that the
two-signed values are less tolerated and only one of the signs survives.
This led us to adopt forms expressed as square of the trigonometric
functions.
\section{Results of the numerical integration}
\subsection{The typhoon morphology}
The value of the parameter is $p=1$. The domain is
\begin{equation*}
\left( x,y\right) \in \left[ -0.5,0.5\right] \times \left[ -0.5,0.5\right]
\end{equation*}
with\ $\left[ 101,101\right] $ mesh points. The boundary value is
\begin{equation*}
\psi _{b}^{\left( 1\right) }=\ln \left( p-\sqrt{p^{2}-1}\right) =0
\end{equation*}
and the initial function is
\begin{equation*}
\psi \left( x,y\right) =\psi _{b}^{\left( 1\right) }+4.2\times \sin \left(
4\pi \frac{x-x_{\min }}{x_{\max }-x_{\min }}\right) \sin \left( 4\pi \frac{%
y-y_{\min }}{y_{\max }-y_{\min }}\right)
\end{equation*}
It takes $501$ calls to the function and Jacobian. The accuracy is $%
0.257\times 10^{-3}$. This run has been executed with several mesh
dimensions: $\left[ 31\times 31\right] $, $\left[ 51\times 51\right] $, $%
\left[ 71\times 71\right] $. The results are very close, but higher accuracy
shows much clearer the details.
The results are shown. The Figure (\ref{alfa_k4_7}) shows the choice of the
amplitude of the initialization and Fig.(\ref{exp_7b}) shows the initial
function $\psi $.
The solution has an apparent cylindrical symmetry around the center and for
this reason we present a section along $x$ of the streamfunction $\psi
\left( x,y\right) $ (Fig.(\ref{exp_7c})). A section along $x$ axis of the
vorticity $\omega \left( x,y\right) $ is presented in Fig.(\ref{omega_7}).
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{alfa_k4_7.eps}}
\caption{The procedure to find an approximation to a good initialization.}
\label{alfa_k4_7}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{initial_7a.eps}}
\caption{The initial function, trigonometric profiles.}
\label{exp_7b}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{solsec_7.eps}}
\caption{The section along $x$ of the solution $\protect\psi(x,y)$.}
\label{exp_7c}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{omegasec_7.eps}}
\caption{The vorticity, calculated from $\protect\psi (x,y)$ obtained by
integration.}
\label{omega_7}
\end{figure}
In order to quantify the accuracy of integration we collect in all the
domain $\left( x,y\right) $ the pairs $\left( \psi ,\omega \right) $ and
represent them together with the line of the nonlinear term in our equation,
Fig.(\ref{comparatie_A_7}). In Fig.(\ref{comparatie_C_7}) we show the ratio
of the two quantities the nonlinear term and $\omega $, as resulted from the
calculated $\psi $. This ratio should be $1$. There are points close to the
value $0$ where this ratio is not $1$ but, if we normalize adding an
arbitrary constant to remove the possible singular cases, we notice a very
good clustering of the points around the line $1$. In addition, we represent
the scatterplot of the pairs ($\omega $, magnitudes of nonlinear term for
the $\psi $'s) and notice the close clustering around the diagonal. Other
tests are possible and they indicates that the integration is very good on
most of the region and good within the imposed accuracy in the regions where
the quantities reach values close to $0$.
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{comparatie_A_7.eps}}
\caption{The scatter plot $(\protect\psi ,\protect\omega )$, for $p=1$.}
\label{comparatie_A_7}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{comparatie_C_7.eps}}
\caption{The ratio of $\protect\omega $ and the nonlinear term.}
\label{comparatie_C_7}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{comparatie_E_7.eps}}
\caption{Scatterplot $(\protect\omega ,$ the nonlinear term), compared with
the diagonal.}
\label{comparatie_E_7}
\end{figure}
The contour plot of the solution is shown in Fig.(\ref{vel_7}) on the same
graph with the velocity field (we have used a reduced set of data due to
limitations on the EPS file). We must note that this two-dimensional
integration gives a radial component of the velocity which at maximum is
about $20$ times lower than the tangential one.
The tangential component of the velocity is shown in two figures (\ref
{vth_7a}) and (\ref{vth_7c}) with the purpose of making easier the
observation of the central region. The narrow dip in the center is clearly
visible and its radial extension can be compared with the extension of the
whole domain.
We have represented in Fig.(\ref{vthlin_7}) a section along the $x$ axis of
the amplitude of the azimuthal component of the velocity.
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vel_7.eps}}
\caption{The contours of the scalar streamfunction $\protect\psi (x,y)$ and
the vector field $(v_{x}.v_{y})$.}
\label{vel_7}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vth_7a.eps}}
\caption{The tangential component $v_{\protect\theta }(x,y)$ of the velocity
vector field $(v_{x}.v_{y})$, with center at $(0,0)$.}
\label{vth_7a}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vth_7b.eps}}
\caption{The tangential component $v_{\protect\theta }(x,y)$ of the velocity
vector field $(v_{x}.v_{y})$, with center at $(0,0)$ (same as \ref{vth_7a}).}
\label{vth_7b}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vth_7c.eps}}
\caption{The tangential component $v_{\protect\theta }(x,y)$ of the velocity
vector field $(v_{x}.v_{y})$, with center at $(0,0)$ (same as \ref{vth_7a}).}
\label{vth_7c}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vthlin_7.eps}}
\caption{The magnitude of the tangential component $v_{\protect\theta }(x,y)$%
, seen along a radial line. The central fast decay is clearly visible.}
\label{vthlin_7}
\end{figure}
\subsubsection{Episodic structure of two vortices}
It is worth to mention that in a numerical experiment we have identified a
state where two vortices have been formed, placed in symmetrical positions
along the diagonal of the square domain $\left[ -0.5,0.5\right] \times \left[
-0.5,0.5\right] \;$with a mesh of\ $\left[ 31,31\right] $. The value of the
parameter is $p=1$. The initial function is trigonometric with $k=2$ in Eq.(%
\ref{trig}) with a coefficient $\psi _{lin}=3.8$. It takes longer to obtain
the solution with $0.84\times 10^{-4}$ accuracy, $389$ calls to the
function. The result is in Fig.(\ref{vel_4}).
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{vel_4.eps}}
\caption{The contours of the scalar streamfunction $\protect\psi (x,y)$ and
the vector field $(v_{x}.v_{y}$ for a two-vortices approximative solution.}
\label{vel_4}
\end{figure}
This state has been reexamined with much higher accuracy. It has taken long
time to see that the final solution was again the centered vortex shown
before. Therefore from the point of view of the numerical experience this
state of two vortices is irrelevant. However, the persistence of this state
inside the iterative search may indicate that it is close to a solution,
possible less stable. We have not investigated this further. Instead we will
show below a solution with four vortices.
\subsubsection{Four vortices}
The calculations are done for $p=1$ on meshes with various levels of
details: $31$, $61$, $101$. The initial function is trigonometric with $k=3$.
The results show clearly the formation of four vortices, as shown by Fig.(%
\ref{vel_3}). Each of them has a structure that is similar to the one
presented in Fig.(\ref{exp_7c}). It is interesting to note that again the
vorticity is almost zero everywhere on the domain, except a strict region
around the four vortices, where it reaches very high values.
To the accuracy we have used unitl now we cannot say if the local tangential
velocity presents the same very fast decay to the center of the vortex.
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{solution_3.eps}}
\caption{The scalar streamfunction $\protect\psi (x,y)$ for a four-vortices
solution.}
\label{solution_3}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vel_3.eps}}
\caption{The contours of the scalar streamfunction $\protect\psi (x,y)$ and
the vector field $(v_{x}.v_{y})$ for a four-vortices solution.}
\label{vel_3}
\end{figure}
\subsubsection{Four vortices obtained at $p>1$}
For $p=3$ it is also possible to obtain the four-vortex solution. The
initial function is here a trigonometric combination for, $k=2$ and squared
such that only positive (four) maxima are initially present, with an
amplitude of about $\psi = 4$.
\subsubsection{The central strong decay of the tangential velocity, at $p>1$}
The numerical integration is done for $p=3$ , using an initialization by a
centered peak from an trigonometric function.
\begin{figure}[tbph]
\centerline{\includegraphics[height=10cm]{vthlin_10.eps}}
\caption{The magnitude of the tangential component $v_{\protect\theta }(x,y)$%
, seen along a radial line. The central dip is visible but significantly
narrower than at $p=1$.}
\label{vthlin_10}
\end{figure}
We note from Fig.(\ref{vthlin_10}) that for larger values of the parameter $%
p $ there is a even more narrow zone where there is the strong decay of the
tangential velocity.
\subsection{Relevance of the solutions for the physics of the atmosphere}
In general the space variables of the CHM equation are normalized to the
intrinsic typical length of the model. In this case (atmospheric physics)
are scaled with $\rho _{g}$, the Rossby radius. We note in passing,
(especially for plasma physicists) that there is a major difference compared
with the plasma case. In plasma, perturbations with lengths less or
comparable with an ion Larmor gyroradius $k^{-1}\gtrsim \rho _{i}$ cannot be
described by fluid models.
In the physics of atmosphere, the wavelengths can be much smaller
\begin{equation*}
k\rho _{g}\gg 1
\end{equation*}
At very large $k\rho _{g}$ the description becomes governed by the Euler
equation (see \cite{HH}).
For the numerical studies we choose
\begin{eqnarray*}
\left( x,y\right) &\in &\left[ x_{\min },x_{\max }\right] \times \left[
y_{\min },y_{\max }\right] \\
&=&\left[ -0.5,0.5\right] \times \left[ -0.5,0.5\right]
\end{eqnarray*}
This means that the full domain (the side of the rectangle) is a single unit
length $\rho _{g}$.
\bigskip
In the following we make few consideration about what we can expect as
results, in the case of the atmosphere problem.
As we will notice from numerical solution, the equation produces functions
with very clear similarity with the \emph{typhoon} morphology. The
characteristic aspect is (within the precision of these first integrations)
a sharp extremum of the vorticity on $\left( 0,0\right) $ which means a
localised maximum of the tangential velocity $v_{\theta }$ in close
proximity of the center. Since
\begin{equation*}
v_{\theta }=\frac{d\psi }{dr}
\end{equation*}
the maximum at
\begin{equation*}
r=a
\end{equation*}
means
\begin{equation*}
\frac{dv_{\theta }}{dr}=\frac{d^{2}\psi }{dr^{2}}=0
\end{equation*}
The equation is
\begin{equation*}
\frac{d^{2}\psi }{dr^{2}}+\frac{1}{r}\frac{d\psi }{dr}=\left( -\frac{1}{%
2p^{2}}\right) \sinh \psi \left( \cosh \psi -p\right)
\end{equation*}
We multiply by $r\,\ $and we make a derivation to $r$%
\begin{eqnarray*}
&&\frac{d^{2}\psi }{dr^{2}}+r\frac{d^{2}}{dr^{2}}\frac{d\psi }{dr}+\frac{%
d^{2}\psi }{dr^{2}} \\
&=&\left( -\frac{1}{2p^{2}}\right) \left\{ \sinh \psi \left( \cosh \psi
-p\right) +\right. \\
&&\left. +r\left( \frac{d\psi }{dr}\right) \left[ \cosh ^{2}\psi -p\cosh
\psi +\sinh ^{2}\psi \right] \right\}
\end{eqnarray*}
We calculate this expression and the equation in the point $r=a$ defined as
the point of the maximum of the tangential velocity. This means
\begin{eqnarray*}
r &=&a \\
\left( \frac{d^{2}\psi }{dr^{2}}\right) _{a} &=&0 \\
\left. \frac{d^{2}}{dr^{2}}\left( \frac{d\psi }{dr}\right) \right| _{a}
&\equiv &\left. \frac{d^{2}v_{\theta }}{dr^{2}}\right| _{a}=-\alpha \;\text{%
where\ }\alpha >0 \\
\psi \left( r=a\right) &\equiv &\psi _{0}
\end{eqnarray*}
where we have introduced a notation for the value, $-\alpha <0$ of the
second derivative of the tangential velocity at its maximum. For a \emph{%
very qualitative} estimation, used in predicting shapes of solutions, we
will take this as a parameter. At the point $r=a$ the equation becomes
\begin{equation*}
\frac{1}{a}\left( \frac{d\psi }{dr}\right) _{a}=\left( -\frac{1}{2p^{2}}%
\right) \sinh \psi _{0}\left( \cosh \psi _{0}-p\right)
\end{equation*}
In the equation derivated at $r$ we replace $d\psi /dr$ with its value from
the above equation and also introduce the parameter $\alpha $. Then we have
\begin{eqnarray*}
&&a\left( -\alpha \right) \\
&=&\left( -\frac{1}{2p^{2}}\right) \left\{ \sinh \psi _{0}\left( \cosh \psi
_{0}-p\right) \right. \\
&&+a^{2}\left( -\frac{1}{2p^{2}}\right) \sinh \psi _{0}\left( \cosh \psi
_{0}-p\right) \\
&&\left. \times \left( 2\cosh ^{2}\psi _{0}-p\cosh \psi _{0}-1\right)
\right\}
\end{eqnarray*}
or
\begin{equation*}
\frac{a\alpha \left( 2p^{2}\right) }{\sinh \psi _{0}\left( \cosh \psi
_{0}-p\right) }=1-\frac{a^{2}}{2p^{2}}\left( 2\cosh ^{2}\psi _{0}-p\cosh
\psi _{0}-1\right)
\end{equation*}
This equation may serve to make some estimates if additional informations
(or simply hints from experiments) are available. This is illustrated below.
Consider the case $p=1$%
\begin{equation}
\frac{2a\alpha }{\sinh \psi _{0}\left( \cosh \psi _{0}-1\right) }=1-\frac{%
a^{2}}{2}\left( \cosh \psi _{0}-1\right) \left( 2\cosh \psi _{0}+1\right)
\label{aalfpsi}
\end{equation}
A \emph{short and dirty} approximation should start by using the suggestion
from results of lucky simulations, where $\psi _{0}$ is few units, and $a$
is of the order $0.1$ on a domain of length $1$ in both $x$ and $y$. The
second derivative of the tangential velocity must be high, shown by the
plots of $v_{\theta }$. This means that it may exist a difference of
magnitude of the terms, with the second term in the right hand side
appearing less important. Therefore we try
\begin{equation*}
2a\alpha \sim \sinh \psi _{0}\left( \cosh \psi _{0}-1\right)
\end{equation*}
In addition, we can suppose that the exponentials of negative argument are
less important than those of positive argument, and simplify to
\begin{equation*}
\exp \left( 2\psi _{0}\right) \sim 8a\alpha
\end{equation*}
or
\begin{equation*}
\psi _{0}\sim \frac{1}{2}\ln \left( \alpha a\right) +1
\end{equation*}
For an order of magnitude we may take
\begin{equation*}
\alpha \sim \frac{\psi _{0}}{a^{3}}
\end{equation*}
and then
\begin{eqnarray*}
\psi _{0} &\sim &\frac{1}{2}\ln \left( \frac{\psi _{0}}{a^{2}}\right) +1 \\
&\sim &\frac{1}{2}\ln \psi _{0}-\ln a+1
\end{eqnarray*}
We obtain
\begin{equation*}
\psi _{0}-1-\frac{1}{2}\ln \psi _{0}\sim -\ln a
\end{equation*}
\begin{eqnarray*}
\frac{1}{a} &\sim &\exp \left( \psi _{0}-1-\frac{1}{2}\ln \psi _{0}\right) \\
&\sim &\frac{1}{\sqrt{\psi _{0}}}\exp \left( \psi _{0}-1\right)
\end{eqnarray*}
or
\begin{equation*}
a\sim \frac{1}{e}\sqrt{\psi _{0}}\exp \left( -\psi _{0}\right)
\end{equation*}
We can see that the results are consistent, since if we take from numerical
solution
\begin{equation*}
\psi _{0}\sim 3
\end{equation*}
we obtain from the estimation
\begin{equation*}
a\sim 0.032
\end{equation*}
which is not far from
\begin{equation*}
a^{num}\sim 0.04
\end{equation*}
We must remember that the domain of integration is of length $1$ and the
fact that ``the radius of maximum wind'' is so small, $a\sim 0.04$ , means
that high accuracy is needed to describe correctly what happens close to the
center. This is due to the other constraint, that the solution
streamfunction $\psi \left( r\right) $ needs sufficient space to go to the
constant value at ``infinity'' (large $r$). Any restriction of the domain of
integration which would be aimed to the better description of the central
region would require boundary conditions that are unknown.
\bigskip
There is another benefit from these very rough estimations. We can use them
to determine the spatial domain that would be adequate for the search of the
solution, for particular physical situations.
In order to use this rough estimation we must introduce physical units. In
the following all quantities with physical dimensions have an superscript $%
phy$.
In \emph{atmosphere} the distances are measured in $\rho _{g}$%
\begin{equation*}
a=\frac{a^{phy}}{\rho _{g}}
\end{equation*}
and the streamfunction is normalised with
\begin{equation*}
\psi =\frac{\psi ^{phy}}{\rho _{g}^{2}\left\langle f\right\rangle }
\end{equation*}
where $\left\langle f\right\rangle $ is the Coriolis parameter. This means
\begin{equation*}
\frac{a^{phy}}{\rho _{g}}\sim \frac{1}{e}\sqrt{\frac{\psi _{0}^{phy}}{\rho
_{g}^{2}\left\langle f\right\rangle }}\exp \left[ -\frac{\psi _{0}^{phy}}{%
\rho _{g}^{2}\left\langle f\right\rangle }\right]
\end{equation*}
The physical parameters are (taken from \cite{HH})
The depth of the atmosphere
\begin{equation*}
H_{0}=8\times 10^{3}\;\left( m\right)
\end{equation*}
The Coriolis parameter
\begin{equation*}
\left\langle f\right\rangle =1.6\times 10^{-4}\;\left( s^{-1}\right)
\end{equation*}
From these parameters it results
The Rossby radius (the unit of space)
\begin{eqnarray*}
\rho _{g} &=&\frac{\left( gH\right) ^{1/2}}{\left\langle f\right\rangle } \\
&=&2\times 10^{6}\,\left( m\right)
\end{eqnarray*}
The unit for the streamfunction is
\begin{equation*}
\rho _{g}^{2}\left\langle f\right\rangle =6.4\times 10^{8}\;\left(
m^{2}/s\right)
\end{equation*}
The unit for vorticity
\begin{equation*}
\left\langle f\right\rangle =1.6\times 10^{-4}\;\left( s^{-1}\right)
\end{equation*}
For example, using these parameters, it results that we have integrated on a
spatial domain of length $L$ (in other words: we have imposed that the
streamfunction becomes equal to $\psi _{b}^{\left( 1,2\right) }$ on the
boundaries of a square with side length $L$)
\begin{eqnarray*}
L &\equiv &x_{\max }-x_{\min }=1 \\
L^{phy} &=&1\times \rho _{g}\sim 2\times 10^{6}\;\left( m\right)
=2000\;\left( km\right)
\end{eqnarray*}
and the diameter $d$ of the \emph{eye} of the \emph{typhoon} results
\begin{eqnarray*}
d &=&2\times a=0.08 \\
d^{phy} &\sim &0.08\rho _{g}=128\;\left( km\right)
\end{eqnarray*}
In the Ref.\cite{mesov} it is reproduced a plot of an observation made on the
profile of the vorticity, in Fig.1a. The plot indicates a maximum value of
about $250\times 10^{-4}\left( s^{-1}\right) $. The vorticity we obtain is
larger (of the order of $1000 \times 10^{-4}$).
This shows that the absence of the third dimension
in our model and of the viscous effects have a serious influence on the
physical quantities. They should be somehow accounted for by renormalizing
the two-dimensional model at the initial stage. For example, in the case of
the plasma vortex, a change of the space scale results from the presence
of a translational motion combined with the density gradient. This remains to be
studied.
\section{Summary}
We again underline that this equation is very difficult to solve, although
it requires reasonable computer resources. The main problem is the
complexity of the space of solutions and the need to explore carefully much
of this space in order to establish the basins of attraction. We are not
able at this moment to connect in some practically useful way the sharp
transitions between the attractors with the \emph{stability} of the
solutions.
\bigskip
It seems that the solution where the streamfunction $\psi \left( x,y\right) $
is approximately radially symmetric, strongly peaked in origin, is a
significant attractor, at the level of this very sensitive equation. It
presents the particularity that the vorticity is practically zero for almost
all spatial domain and is strongly localised, almost singular, close to the
maximum. The aspect of this solution is very similar to the two-dimensional
image of a \emph{typhoon}.
We have several arguments in favor of the conclusion that our equation (\ref
{eq}) may represent the hydrodynamic part of the atmospheric vortex. We
mention some of them.
\begin{enumerate}
\item The profile of the magntitude of the tangential velocity, as
represented in Fig.2 of Ref. \cite{cycrev} is very similar to our Fig.\ref
{vthlin_7}. This is also confirmed by similarity with the Fig.1a from Ref(
\cite{ReMont});
\item The profile of the vorticity $\omega $ shown in our Fig.\ref{omega_7}
is very similar to Fig.1a from Ref.\cite{mesov};
\item We note that in \ a series of reported numerical simulations, the
tendency of the fields is to evolve toward profiles that are very close to
those shown in our figures \ref{exp_7c}, \ref{omega_7} and \ref{vthlin_7}.
For example, the Fig.7a and b of Ref.\cite{mesov} show the evolution of the
azimuthal mean of the vorticity and tangential velocity from initial
profiles which correspond to a narrow ring of vorticity to profiles that
show clear ressemblance with our figures \ref{omega_7} and \ref{vthlin_7} or
\ref{vth_7a}. The same striking evolution to profiles similar to ours
appears in Figs.7 a and b of the same Reference. We have investigated
whether a radially annular profile of vorticity can be a solution of our
equation (\ref{eq}). The result is negative, which may explain why such an
initial profile evolves to either a set of vortices (vortex-crystal) or to a
centrally peaked structure as in Fig.\ref{vth_7a}.
\item The four vortices represented in Figure 4a of the Ref. \cite{mesov}
as the late stage of the evolution obtained from numerical simulation of
vorticity, is clearly similar to our figure \ref{vel_3}.
\item We obtain a good consistency between our quantitative results for an
atmospheric vortex (using most elementary input information) and the values
measured or obtained in numerical simulations, at least for some of the
quantities.
\end{enumerate}
A large database on typhoons can be found in \cite{sitety}. The similarity
is striking and it suggests that further work with this equation is worth to
be done.\bigskip
The numerical simulations we have taken as a comparison are very complex. In
general, the physics of the \emph{typhoons} is very complex and includes
hydrodynamics and thermic aspects, with many additional elements:
precipitation, viscosity, etc. In no way we do not claim that this equation
can represent this complexity. It appears however useful as a description of
the regimes where the hydrodynamical processes are dominating and have
reached stationarity.
\bigskip
\textbf{Acknowledgments}. We are very grateful to Professor David Montgomery
for many discussions on a wide spectrum of problems. We thank Dr. L. Weimann
for his kind help on the GIANT code.
This work has been partly supported by a grant from the Japan Society for
the Promotion of Science. The authors are very grateful for this support and
for the hospitality of Professor S.-I. Itoh and of Professor M. Yagi.
\section{Appendix A : The structure of a radial solution near $r=0$ and $%
r=\infty $}
The equation we discuss is
\begin{equation*}
\Delta \psi +\frac{1}{2p^{2}}\sinh \psi \left( \cosh \psi -p\right) =0
\end{equation*}
Other members of the family of equations (parametrized by solutions of the $%
2D$ Laplace equation) will be examined separately. Their importance stems
from the fact that they can provide, in principle, azimuthal trigonometric
variation, as for example the Larichev-Reznik modon.
\subsection{The behavior near $r=0$}
Close to the origin, in a purely radial form, it is
\begin{equation*}
\frac{d^{2}\psi }{dr^{2}}+\frac{1}{r}\frac{d\psi }{dr}+\left( \frac{1}{2p^{2}%
}\right) \sinh \psi \left( \cosh \psi -p\right) =0
\end{equation*}
where $r$ is measured in $\rho _{s}$.
We take an expansion with only even powers of $r$ close to the origin
\begin{equation*}
\psi \sim a_{0}+a_{2}r^{2}+a_{4}r^{4}+a_{6}r^{6}+...
\end{equation*}
Then, for small $r$;
\begin{eqnarray*}
\frac{d\psi }{dr} &=&2a_{2}r+4a_{4}r^{3}+6a_{6}r^{5}... \\
\frac{1}{r}\frac{d\psi }{dr} &=&2a_{2}+4a_{4}r^{2}+6a_{6}r^{4}...
\end{eqnarray*}
\begin{equation*}
\frac{d^{2}\psi }{dr^{2}}=2a_{2}+12a_{4}r^{2}+30a_{6}r^{4}+...
\end{equation*}
\begin{eqnarray*}
&&\sinh \left( a_{0}+a_{2}r^{2}+a_{4}r^{4}+...\right) \\
&=&\sinh a_{0} \\
&&+\left( a_{2}r^{2}+a_{4}r^{4}+...\right) \cosh a_{0} \\
&&+\frac{1}{2}\left( a_{2}^{2}r^{4}+...\right) \sinh a_{0}+...
\end{eqnarray*}
\begin{eqnarray*}
&&\cosh \left( a_{0}+a_{2}r^{2}+a_{4}r^{4}+...\right) \\
&=&\cosh a_{0} \\
&&+\left( a_{2}r^{2}+a_{4}r^{4}+...\right) \sinh a_{0} \\
&&+\frac{1}{2}\left( a_{2}^{2}r^{4}+...\right) \cosh a_{0}+...
\end{eqnarray*}
Introducing the notations
\begin{eqnarray*}
U &\equiv &a_{2}r^{2}+a_{4}r^{4}+... \\
V &\equiv &\frac{1}{2}\left( a_{2}^{2}r^{4}+...\right)
\end{eqnarray*}
\begin{eqnarray*}
&&\sinh \psi \left( \cosh \psi -p\right) \\
&=&\left( \sinh a_{0}+U\cosh a_{0}+V\sinh a_{0}\right) \\
&&\times \left( \cosh a_{0}-p+U\sinh a_{0}+V\cosh a_{0}\right) \\
&=&\sinh a_{0}\left( \cosh a_{0}-p\right) \\
&&+U\left( \sinh ^{2}a_{0}+\cosh ^{2}a_{0}-p\cosh a_{0}\right) \\
&&+V\left( 2\cosh a_{0}\sinh a_{0}-p\sinh a_{0}\right) \\
&&+U^{2}\left( \sinh a_{0}\cosh a_{0}\right) \\
&&+V^{2}\left( \sinh a_{0}\cosh a_{0}\right) \\
&&+UV\left( \cosh ^{2}a_{0}+\sinh ^{2}a_{0}\right) \\
&&+...
\end{eqnarray*}
We collect the various degrees of $r^{\alpha }$%
\begin{equation*}
q_{0}+q_{2}r^{2}+q_{4}r^{4}+...
\end{equation*}
\begin{equation*}
q_{0}=\sinh a_{0}\left( \cosh a_{0}-p\right)
\end{equation*}
\begin{equation*}
q_{2}=a_{2}\left( \sinh ^{2}a_{0}+\cosh ^{2}a_{0}-p\cosh a_{0}\right)
\end{equation*}
\begin{eqnarray*}
q_{4} &=&a_{4}\left( \sinh ^{2}a_{0}+\cosh ^{2}a_{0}-p\cosh a_{0}\right) \\
&&+\frac{1}{2}a_{2}^{2}\left( 2\cosh a_{0}\sinh a_{0}-p\sinh a_{0}\right) \\
&&+a_{2}^{2}\left( \sinh a_{0}\cosh a_{0}\right)
\end{eqnarray*}
Returning to the differential operator
\begin{eqnarray*}
&&\frac{d^{2}\psi }{dr^{2}}+\frac{1}{r}\frac{d\psi }{dr} \\
&=&2a_{2}+12a_{4}r^{2}+30a_{6}r^{4}+... \\
&&+2a_{2}+4a_{4}r^{2}+6a_{6}r^{4}... \\
&=&4a_{2}+16a_{4}r^{2}+36a_{6}r^{4}+...
\end{eqnarray*}
We now identify the expressions corresponding to the same degrees of $r$,
\begin{eqnarray*}
&&4a_{2}+16a_{4}r^{2}+36a_{6}r^{4}+... \\
&&+\left( \frac{1}{2p^{2}}\right) \left(
q_{0}+q_{2}r^{2}+q_{4}r^{4}+...\right) \\
&=&0
\end{eqnarray*}
with the equalities
\begin{equation*}
4a_{2}+\left( \frac{1}{2p^{2}}\right) q_{0}=0
\end{equation*}
\begin{equation*}
16a_{4}+\left( \frac{1}{2p^{2}}\right) q_{2}=0
\end{equation*}
\begin{equation*}
36a_{6}+\left( \frac{1}{2p^{2}}\right) q_{4}=0
\end{equation*}
The equations from which we derive the coefficients of the expansion become
\begin{equation*}
4a_{2}+\left( \frac{1}{2p^{2}}\right) \sinh a_{0}\left( \cosh a_{0}-p\right)
=0
\end{equation*}
\begin{equation*}
16a_{4}+\left( \frac{1}{2p^{2}}\right) a_{2}\left( \sinh ^{2}a_{0}+\cosh
^{2}a_{0}-p\cosh a_{0}\right) =0
\end{equation*}
\begin{eqnarray*}
&&36a_{6}+\left( \frac{1}{2p^{2}}\right) \left[ a_{4}\left( \sinh
^{2}a_{0}+\cosh ^{2}a_{0}-p\cosh a_{0}\right) \right. \\
&&+\frac{1}{2}a_{2}^{2}\left( 2\cosh a_{0}\sinh a_{0}-p\sinh a_{0}\right) \\
&&\left. +a_{2}^{2}\left( \sinh a_{0}\cosh a_{0}\right) \right] \\
&=&0
\end{eqnarray*}
We see that if we take
\begin{equation*}
a_{0}=0
\end{equation*}
then this will vanish all the other coefficients
\begin{eqnarray*}
a_{2} &=&0 \\
a_{4} &=&0 \\
a_{6} &=&0,...
\end{eqnarray*}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{a2_1.eps}}
\caption{Coefficient $a_{2}$ for $p=1$.}
\label{a2_1}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{a4_1.eps}}
\caption{Coefficient $a_{4}$ for $p=1$.}
\label{a4_1}
\end{figure}
\begin{figure}[tbph]
\centerline{\includegraphics[height=5cm]{a6_1.eps}}
\caption{Coefficient $a_{6}$ for $p=1$.}
\label{a6_1}
\end{figure}
Consider the value of the constant
\begin{equation*}
p=1
\end{equation*}
and we choose the main coefficient of the expansion close to $r=0$ to be
\begin{equation*}
a_{0}=1
\end{equation*}
Then
\begin{eqnarray*}
a_{2} &=&-0.0798 \\
a_{4} &=&0.0055 \\
a_{6} &=&-0.000439
\end{eqnarray*}
But the coefficients, as shown in the Figures, are very rapidly growing in
absolute value.
We conclude that any attempt to identify the solution starting from few
terms expansion around $r=0$ will be imprecise.
\subsection{The behavior at infinity}
At $r\rightarrow \infty $ we expect that the function approaches zero in the
case where $p=1$ or approaches one of the roots of the equation
\begin{equation}
\cosh \psi -p=0 \label{psibeq}
\end{equation}
for $p>1$. The case where $\psi \rightarrow 0$ will be treated below. We
note, for the case $p>1$ that the solutions of the Eq.(\ref{psibeq}) are
\begin{eqnarray*}
\psi _{b}^{\left( 1\right) } &=&\ln \left( p+\sqrt{p^{2}-1}\right) \\
\psi _{b}^{\left( 2\right) } &=&\ln \left( p-\sqrt{p^{2}-1}\right)
\end{eqnarray*}
\subsubsection{The case $p=1$}
This requires that $\psi \rightarrow 0$ at $r\rightarrow \infty $.
Change the variable
\begin{equation*}
r\rightarrow \frac{1}{x}
\end{equation*}
\begin{equation*}
\frac{d}{dr}=\frac{dx}{dr}\frac{d}{dx}=-\frac{1}{r^{2}}\frac{d}{dx}=-x^{2}%
\frac{d}{dx}
\end{equation*}
\begin{eqnarray*}
\frac{d^{2}}{dr^{2}} &=&\frac{d}{dr}\left( \frac{d}{dr}\right) =-x^{2}\frac{d%
}{dx}\left( -x^{2}\frac{d}{dx}\right) \\
&=&-x^{2}\left( -2x\frac{d}{dx}-x^{2}\frac{d^{2}}{dx^{2}}\right) \\
&=&2x^{3}\frac{d}{dx}+x^{4}\frac{d^{2}}{dx^{2}}
\end{eqnarray*}
The function
\begin{equation*}
\psi \rightarrow 0
\end{equation*}
\begin{eqnarray*}
\left( \frac{1}{2p^{2}}\right) \sinh \psi \left( \cosh \psi -p\right)
&\rightarrow &\left( \frac{1}{2p^{2}}\right) \left( \psi -\frac{\psi ^{3}}{6}%
\right) \left( 1-p-\frac{\psi ^{2}}{2}\right) \\
&=&\frac{1-p}{2p^{2}}\psi \\
&&+\frac{1}{2p^{2}}\left( -\frac{1}{2}-\frac{1-p}{6}\right) \psi ^{3}+...
\end{eqnarray*}
For
\begin{eqnarray*}
p &=&1 \\
\left( \frac{1}{2p^{2}}\right) \sinh \psi \left( \cosh \psi -p\right)
&\rightarrow &-\frac{1}{4}\psi ^{3}
\end{eqnarray*}
Then the equation becomes
\begin{eqnarray*}
&&\left( 2x^{3}\frac{d}{dx}+x^{4}\frac{d^{2}}{dx^{2}}\right) \psi \\
&&+\left( -x^{2}\frac{d}{dx}\right) \psi \\
&&+\frac{1-p}{2p^{2}}\psi +\frac{1}{2p^{2}}\left( -\frac{1}{2}-\frac{1-p}{6}%
\right) \psi ^{3} \\
&=&0
\end{eqnarray*}
This can be approximated at
\begin{equation*}
x\rightarrow 0
\end{equation*}
\begin{equation*}
-x^{2}\frac{d\psi }{dx}=\alpha \psi +\beta \psi ^{3}
\end{equation*}
or
\begin{equation*}
\frac{d\psi }{\alpha \psi +\beta \psi ^{3}}=-\frac{dx}{x^{2}}=d\left( \frac{1%
}{x}\right) =dr
\end{equation*}
For
\begin{equation*}
p=1
\end{equation*}
\begin{eqnarray*}
\alpha &=&0 \\
\beta &=&-\frac{1}{4}
\end{eqnarray*}
then
\begin{equation*}
\left( -4\right) \frac{d\psi }{\psi ^{3}}=dr
\end{equation*}
\begin{equation*}
\psi \sim \sqrt{\frac{2}{r}}
\end{equation*}
We note however that in this case the vorticity is
\begin{eqnarray*}
\omega &=&\Delta \psi \\
&\sim &r^{-5/2}
\end{eqnarray*}
We would like to have a vanishing vorticity at infinity with a faster decay.
The above calculations seem to suggest that for purely radial structure we
need to consider the differential equation which is derived for a different
choice of the Laplacean equation, as it is explained in the main text.
\subsubsection{The case $p>1$}
One possibility, for
\begin{equation*}
p>1
\end{equation*}
\begin{equation*}
\alpha \equiv \frac{1-p}{2p^{2}}<0
\end{equation*}
\begin{equation*}
\psi \sim \exp \left( -\left| \alpha \right| r\right)
\end{equation*}
This gives
\begin{eqnarray*}
\omega &=&\Delta \psi \\
&\sim &\left( -\left| \alpha \right| \right) \frac{\exp \left( -\left|
\alpha \right| r\right) }{r}+\alpha ^{2}\exp \left( -\left| \alpha \right|
r\right)
\end{eqnarray*}
with a fast decay. This situation is worth to be examined numerically.
\section{Appendix B : various forms of the initial conditions}
\subsection{The ring-type}
The initial form of the function has the form
\begin{equation*}
\psi _{0}=A\exp \left( -sr^{2}\right) \left[ 1-\kappa \exp \left(
-qr^{4}\right) \right]
\end{equation*}
We look for the maximum
\begin{eqnarray*}
\frac{d\psi _{0}}{dr} &=&\left( -2sr\right) \exp \left( -sr^{2}\right) \left[
1-\kappa \exp \left( -qr^{4}\right) \right] \\
&&+\exp \left( -sr^{2}\right) \left( 4qr^{3}\right) \kappa \exp \left(
-qr^{4}\right) \\
&=&0
\end{eqnarray*}
and we take the maximum to be placed at
\begin{equation*}
r=a
\end{equation*}
which is considered to approximate the center line of the ring. The equation
becomes
\begin{equation*}
\left( 2s\kappa +4\kappa qa^{2}\right) \exp \left( -qa^{4}\right) =2s
\end{equation*}
\begin{equation*}
\kappa \exp \left( -qa^{4}\right) =\frac{1}{1+2a^{2}\left( q/s\right) }
\end{equation*}
The other condition is that the maximum of the function $\psi _{0}$ at $r=a$
equals a prescribed value,
\begin{eqnarray*}
\psi _{0}\left( r=a\right) &=&\psi _{c} \\
A\exp \left( -sa^{2}\right) \left[ 1-\kappa \exp \left( -qa^{4}\right) %
\right] &=&\psi _{c}
\end{eqnarray*}
The initial condition is introduced in the following way. We take $q$, $a$, $%
\kappa $ and $\psi _{c}$ as input parameters and determine the other two, $s$
and $A$ from the equations
\begin{equation*}
s=\frac{2a^{2}q}{\kappa \exp \left( -qa^{4}\right) -1}
\end{equation*}
\begin{equation*}
A=\frac{\psi _{c}}{\exp \left( -sa^{2}\right) \left[ 1-\kappa \exp \left(
-qa^{4}\right) \right] }
\end{equation*}
Now the initial function will be
\begin{eqnarray*}
\psi _{initial}\left( r\right) &=&\psi _{0}+\psi _{b}^{\left( 1,2\right) } \\
&=&\psi _{b}^{\left( 1,2\right) }+ \\
&&+A\exp \left( -sr^{2}\right) \left[ 1-\kappa \exp \left( -qr^{4}\right) %
\right]
\end{eqnarray*}
\emph{i.e.} the function just determined is placed on the constant
background of the value at the boundary, calculated form the condition that
the vorticity is zero at infinity.
This class of initial functions is characterised by an annular shape, with
exponential decay for $r\rightarrow \infty $, with a minimum in the region
around $r=0$ of depth that can be fixed by varying $\kappa $. For $\kappa =1$
the function is zero on the symmetry axis and rises slowly (due to $r^{4}$)
toward the maximum at $r=a$.
In order to narrow the space of parameters we require the approximative
equality between the vorticity amplitude at the ring with the nonlinear term
\begin{eqnarray*}
\omega &\sim &-\frac{2}{\delta ^{2}}\psi _{c} \\
&\sim &-\frac{1}{2p^{2}}\sinh \left( \psi _{c}+\psi _{b}^{\left( 1,2\right)
}\right) \left[ \cosh \left( \psi _{c}+\psi _{b}^{\left( 1,2\right) }\right)
-p\right]
\end{eqnarray*}
(Here $\delta $ is the width of the ring shape). These two quantities are
compared in graphical plot for a range of values of the parameter $\psi _{c}$%
, using a Matlab script. This is far from an exact procedure but helps to
generate reasonable ranges for the input parameters.
\bigskip
The conclusion after many trials using this procedure and its initial
function forms can be described as follows.
In most of the cases the central region is corrected and shifted to a
maximum. In the cases $p=1$ the central region which is started with a
deppressed level is rised and a strong peaked form is generated, as in the
cases where the initialization consists of a maximum on center (for example
a Gaussian form). For $p>1$ the run evolves in some cases to the formation
of separate maxima placed symmetrically on a ring, having sharp maxima. The
central region is decreased in amplitude to a somehow flat region. The
region outside the ring is evolving to a state which corresponds with very
good precision, to
\begin{equation*}
\omega \sim 0
\end{equation*}
on the rest of the domain to the periphery.
\subsection{Flat central region for $\protect\psi \left( r\right) $}
We take the central region
\begin{equation*}
0<r<r_{flat}
\end{equation*}
with a fixed, constant value
\begin{equation*}
\psi \left( r\right) =\psi _{c}
\end{equation*}
where $\psi _{c}$ is one of the roots of the equation $\cosh \psi -p=0$. At
the edge we take another fixed value,
\begin{equation*}
\psi =\psi _{b}
\end{equation*}
with $\psi _{b}$ the other, smaller root of the equation.
In between, we take
\begin{equation*}
\psi \left( r\right) =\psi _{1}-A\ln \left( r\right)
\end{equation*}
\begin{equation*}
A=\frac{\psi _{c}-\psi _{b}}{\ln \left( r_{flat}/r_{c}\right) }
\end{equation*}
\begin{equation*}
\psi _{1}=\psi _{c}+A\ln \left( r_{flat}\right)
\end{equation*}
The value $r_{c}$ is
\begin{equation*}
r_{flat}<r_{c}<0.5
\end{equation*}
represents the value where where we stop the decay of the function with
logarithm profile and put $\psi =\psi _{b}$. This is
\begin{eqnarray*}
r_{flat} &=&0.1 \\
r_{c} &=&0.35\cdots 0.45
\end{eqnarray*}
The parameter $p=1.3$.
The result of these calculations is as follows. For small mesh, the
evolution is clearly toward the suppression of the smoothly decaying part,
letting a sort of cylinder in the center, with radius $r_{flat}$, with the
high value equal to $\psi _{c}$ and the rest seems to go progressively to $%
\psi =\psi _{b}$. The vorticity is singular, around $r=r_{flat}$. The
vorticity is positive and negative, with high values, singular in a narrow
ring.
For this cylindrical-rod profile of the streamfunction $\psi \left( r\right)
$, the velocity is very localised, as a very narrow ring, all its values are
positive. The velocity grows from zero, keeps always the same direction on $%
\theta $ and then decays to zero value, after the width of the ring. The
vorticity is also sharply limitted here, but it has positive and negative
values on interior half of the ring and respectively on the exterior half of
the ring.
The same shrinking to the cylindrical column happens when we take the
maximum of $\psi $ (in the central flat region) as
\begin{equation*}
\psi \left( r\right) =0
\end{equation*}
which is the other possibility that the equation is verified for constant
value of $\psi $.
|
{
"timestamp": "2005-03-31T04:34:52",
"yymm": "0503",
"arxiv_id": "physics/0503155",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503155"
}
|
\section{Introduction}
In [\ref{C}], the author studied the problem of finding ``almost squares" in short intervals, namely:
\begin{question}
\label{old}
For $0 \leq \theta < 1/2$, what is the least $f(\theta)$ such
that, for some $c_1, c_2 > 0$, any interval $[x - c_1 x^{f(\theta)}, x
+ c_1 x^{f(\theta)}]$ contains an integer $n$ with $n = ab$, where
$a$, $b$ are integers in the interval $[x^{1/2} - c_2 x^\theta,
x^{1/2} + c_2 x^\theta]$? Note: $c_1$ and $c_2$ may depend on
$\theta$.
\end{question}
A similar question is the following:
\begin{question}
\label{new}
For $0 \leq \theta < 1/2$, what is the least $g(\theta)$ such
that, for some $c_1, c_2 > 0$, any interval $[x - c_1 x^{g(\theta)}, x
+ c_1 x^{g(\theta)}]$ contains an integer $n$ with $n = a_1 b_1 = a_2 b_2$, where
$a_1 < a_2 \leq b_2 < b_1$ are integers in the interval $[x^{1/2} - c_2 x^\theta, x^{1/2} + c_2 x^\theta]$? Note: $c_1$ and $c_2$ may depend on
$\theta$.
\end{question}
Note: Actually, the author first considered Question \ref{new} and then turned to Question \ref{old} which has connection to the problem on the distribution of $n^2 \alpha \pmod 1$ and the problem on gaps between sums of two squares.
In [\ref{C}], we showed that $f(\theta) = 1/2$ when $0 \leq \theta < 1/4$, $f(1/4) = 1/4$ and $f(\theta) \geq 1/2 - \theta$. We conjectured that $f(\theta) = 1/2 - \theta$ for $1/4 < \theta < 1/2$ and gave conditional result when $1/4 < \theta < 3/10$. To Question \ref{new}, we have the following
\begin{thm}
\label{theorem1}
For $0 < \theta < 1/4$, $g(\theta)$ does not exist (i.e. all possible products of pairs of integers in $[x^{1/2} - c_2 x^\theta, x^{1/2} + c_2 x^\theta]$ are necessarily distinct for large $x$).
\end{thm}
\begin{thm}
\label{theorem2}
For $1/4 \leq \theta < 1/2$, $g(\theta) \geq 1 - 2\theta$.
\end{thm}
\begin{thm}
\label{theorem3}
For $1/4 \leq \theta \leq 1/3$, $g(\theta) \leq 1 - \theta$.
\end{thm}
We believe that the lower bound is closer to the truth and conjecture
\begin{conj}
\label{conj1}
For $1/4 \leq \theta < 1/2$, $g(\theta) = 1 - 2\theta$.
\end{conj}
\section{Preliminaries and $0 \leq \theta < 1/4$}
Suppose $n = a_1 b_1 = a_2 b_2$ with $x^{1/2} - c_2 x^\theta \leq a_1 < a_2 \leq b_2 < b_1 \leq x^{1/2} + c_2 x^\theta$. Let $d_1 = (a_1, a_2)$ and $d_2 = (b_1, b_2)$ be the greatest common divisors. Then we must have $d_1, d_2 > 1$. For otherwise, say $d_1 = 1$, then $a_2$ divides $b_1$ which implies $x^{1/2} + c_2 x^\theta \geq b_1 \geq 2 a_2 \geq 2 x^{1/2} - 2 c_2 x^\theta$. This is impossible for large $x$ as $\theta < 1/2$. Now, let $a_1 = d_1 e_1$, $a_2 = d_1 e_2$, $b_1 = d_2 f_1$ and $b_2 = d_2 f_2$. Here $(e_1, e_2) = 1 = (f_1, f_2)$. Then
$$n = d_1 e_1 d_2 f_1 = d_1 e_2 d_2 f_2 \mbox{ gives } e_1 f_1 = e_2 f_2.$$
Due to co-primality, $e_2 = f_1$ and $e_1 = f_2$. Therefore,
\begin{equation}
\label{form}
n = (d_1 e_1) (d_2 e_2) = (d_1 e_2) (d_2 e_1)
\end{equation}
with $1< d_1 < d_2$, $e_1 < e_2$ and $(e_1, e_2) = 1$.
\smallskip
Now, from $a_2 - a_1 \leq 2c_2 x^\theta$, $d_1 \leq d_1 e_2 - d_1 e_1 \leq 2c_2 x^\theta$. Similarly, one can deduce that $d_2, e_1, e_2 \leq 2c_2 x^\theta$. Moreover, as $d_1 e_1 = a_1 \geq x^{1/2} - c_2 x^\theta$, we have $d_1, e_1 \geq \frac{1}{2c_2} x^{1/2 - \theta} - \frac{1}{2}$. Similarly, $d_2, e_2 \geq \frac{1}{2c_2} x^{1/2 - \theta} - \frac{1}{2}$. Summing up, we have
\begin{equation}
\label{range}
\frac{1}{2c_2} x^{1/2 - \theta} - \frac{1}{2} \leq d_1, d_2, e_1, e_2 \leq 2c_2 x^\theta.
\end{equation}
From (\ref{range}), we see that no such $n$ exists for $0 \leq \theta < 1/4$ and hence Theorem \ref{theorem1}.
\section{Lower bound for $g(\theta)$}
From (\ref{form}) and (\ref{range}), we see that an integer $n = a_1 b_1 = a_2 b_2$, satisfying the conditions for $a_1, a_2, b_1, b_2$ in Question \ref{new}, must be of the form:
$$n = (d_1 e_1) (d_2 e_2) \mbox{ with } \frac{1}{2c_2} x^{1/2 - \theta} - \frac{1}{2} \leq d_1, d_2, e_1, e_2 \leq 2c_2 x^\theta$$
and $x^{1/2} - c_2 x^\theta \leq d_1 e_1 < d_1 e_2, \, d_2 e_1 < d_2 e_2 \leq x^{1/2} + c_2 x^\theta$. In particular, $e_2 d_2 - e_2 d_1 \leq 2c_2 x^\theta$ which implies $e_2 - e_1 \leq 2c_2 x^\theta / d_2$. Similarly, $d_2 - d_1 \leq 2c_2 x^\theta / e_2$. Thus, the number of such quartuple $(d_1, d_2, e_1, e_2)$ is bounded by
$$\ll \mathop{\sum_{x^{1/2 - \theta} \ll d_2, e_2 \ll x^\theta}}_{x^{1/2} - c_2 x^\theta \leq d_2 e_2 \leq x^{1/2} + c_2 x^\theta} \frac{x^\theta}{e_2} \frac{x^\theta}{d_2} \ll \frac{x^{2\theta}}{x^{1/2}} x^\theta x^\epsilon = x^{3\theta - 1/2 + \epsilon}$$
for any $\epsilon > 0$ as the number of divisor function $d(n) \ll n^\epsilon$. It follows that there are at most $O(x^{3\theta - 1/2 + \epsilon})$ such integers $n$ in the interval $[x - c_2 x^{1/2+\theta}/3, x + c_2 x^{1/2+\theta}/3]$. Therefore, some two consecutive such $n$'s have gap
$$\gg \frac{x^{1/2 + \theta}}{x^{3\theta - 1/2 + \epsilon}} = x^{1 - 2\theta - \epsilon}.$$
Pick $y$ to be the midpoint between these two integers. Then, for some constant $c > 0$, the interval $[y - c y^{1-2\theta-\epsilon}, y + c y^{1-2\theta-\epsilon}]$ does not contain any integer $n = a_1 b_1 = a_2 b_2$ with $y^{1/2} - c_2 y^\theta/2 \leq a_1 < a_2 \leq b_2 < b_1 \leq y^{1/2} + c_2 y^\theta/2$ as $x - c_2 x^{1/2 + \theta}/3 \leq y \leq x + c_2 x^{1/2 + \theta}/3$. Consequently, for any constant $c, c' > 0$, there is arbitrarily large $y$ such that the interval $[y - c y^{1-2\theta-2\epsilon}, y + c y^{1-2\theta-2\epsilon}]$ does not contain any integer $n = a_1 b_1 = a_2 b_2$ with $y^{1/2} - c' y^\theta \leq a_1 < a_2 \leq b_2 < b_1 \leq y^{1/2} + c' y^\theta$. Therefore, $g(\theta) \geq 1 - 2\theta - 2\epsilon$ which gives Theorem \ref{theorem2} by letting $\epsilon \rightarrow 0$.
\section{Upper bound for $g(\theta)$}
Proof of Theorem \ref{theorem3}: For any large $x$, set $N = [x^{1/4}]$ and $\xi = \{x^{1/4}\}$, the integer part and fractional part of $x^{1/4}$ respectively. Based on (\ref{form}), we are going to pick, for $0 \leq \epsilon \leq 1/2$,
\begin{equation}
\label{de}
d_1 = q N + r_1, \; d_2 = q N + r_2, \; e_1 = \frac{N + s_1}{q}, \; e_2 = \frac{N + s_2}{q}
\end{equation}
for some $1 \leq q \leq N^\epsilon$, $0 \leq r_1, r_2 < N$ and $s_1, s_2 \ll q$ with $N \equiv -s_1 \equiv -s_2 \pmod q$. Our goal is to make
$$x = (N+\xi)^4 = N^4 + 4N^3 \xi + O(N^2) \approx (qN+r_1) \frac{N + s_1}{q} (qN+r_2) \frac{N + s_2}{q}.$$
The right hand side above is
\begin{equation}
\label{approx}
\begin{split}
=& \Bigl[N^2 + \Bigl(\frac{r_1}{q} + s_1\Bigr)N + \frac{r_1 s_1}{q} \Bigr] \Bigl[N^2 + \Bigl(\frac{r_2}{q} + s_2\Bigr)N + \frac{r_2 s_2}{q} \Bigr] \\
=& N^4 + \Bigl(\frac{r_1 + r_2}{q} + s_1 + s_2 \Bigr)N^3 + \Bigl[\frac{r_1 s_1}{q} + \frac{r_2 s_2}{q} + \Bigl(\frac{r_1}{q} + s_1\Bigr) \Bigl(\frac{r_2}{q} + s_2\Bigr) \Bigr] N^2 \\
&+ \Bigl[\frac{r_1 s_1}{q} \Bigl(\frac{r_2}{q} + s_2\Bigr) + \frac{r_2 s_2}{q} \Bigl(\frac{r_1}{q} + s_1\Bigr)\Bigr] N + \frac{r_1 s_1 r_2 s_2}{q^2}
\end{split}
\end{equation}
By Dirichlet's Theorem on diophantine approximation, we can find integer $1 \leq q \leq N^\epsilon$ such that
$$\Big| 4\xi - \frac{p}{q} \Big| \leq \frac{1}{q N^\epsilon}$$
for some integer $p$. Fix such a $q$. Then, pick $s_1 < s_2 < 0$ to be the largest two integers such that $N \equiv -s_1 \equiv -s_2 \pmod q$. Clearly, $s_1, s_2 \ll q$. Then, one simply picks some $0 < r_1 < r_2 \ll q^2$ such that $\frac{r_1 + r_2}{q} + s_1 + s_2 = \frac{p}{q}$. With these values for $q, r_1, r_2, s_1, s_2$, (\ref{approx}) is
$$=N^4 + 4N^3 \xi + O(N^{3-\epsilon}) + O(q^2 N^2) + O(q^3 N) + O(q^4).$$
Hence, we have just constructed an integer $n = d_1 e_1 d_2 e_2$ which is within $O(N^{3-\epsilon}) + O(N^{2+2\epsilon}) = O(x^{3/4 - \epsilon/4}) + O(x^{1/2 + \epsilon/2}) = O(x^{3/4 - \epsilon/4})$ from $x$ if $\epsilon \leq 1/3$. One can easily check that $a_1 = d_1 e_1$, $b_1 = d_2 e_2$, $a_2 = d_1 e_2$ and $b_2 = d_2 e_1$ are in the interval $[x^{1/2} - C x^{1/4 + \epsilon/4}, x^{1/2} + C x^{1/4 + \epsilon/4}]$ for some constant $C > 0$. Set $\theta = 1/4 + \epsilon/4$. We have, for some $C' > 0$, $n = a_1 b_1 = a_2 b_2$ in the interval $[x - C'x^{1 - \theta}, x+C'x^{1 - \theta}]$ such that $a_1 < a_2, b_2 < b_1$ are integers in $[x^{1/2} - Cx^\theta, x^{1/2} + Cx^\theta]$ provided $1/4 \leq \theta \leq 1/4 + 1/12 = 1/3$. This proves Theorem \ref{theorem3}.
\section{Open questions}
Conjecture \ref{conj1} may be too hard to prove in the moment. As a starting point, can one show that $g(1/4) = 1/2$? Or even $g(1/4) < 3/4$? Another possibility would be trying to get some results conditionally like [\ref{C}]. Also, one may consider $g(\theta)$ when $\theta$ is near to $1/2$. This leads to the problem about gaps between integers that have more than one representation as a sum of two squares.
\bigskip
{\bf Acknowledgement} The author would like to thank the American Institute of Mathematics for providing a stimulating environment to work at.
|
{
"timestamp": "2005-03-24T18:42:42",
"yymm": "0503",
"arxiv_id": "math/0503438",
"language": "en",
"url": "https://arxiv.org/abs/math/0503438"
}
|
\section{introduction}
In relativistic heavy ion collisions a localized high energy density
domain, a fireball, is created. The study of the properties of this
hot and dense matter is the main objective of the experiments being
conducted at RHIC and as of 2007 at LHC.
Event-by-event particle fluctuations are the observables
subject to intense current
theoretical~\cite{fluct1,fluct2,fluct3,fluct4,fluct4b,
fluct5,fluct6,fluct7,fluct8},
and experimental ~\cite{starfluct,starfluct2,phefluct} interest.
Fluctuation measurements are important since they can be used:
(i) as a consistency check for existing models, e.g. within statistical
particle production models~\cite{fluct2,fluct3},
(ii) as a way to search for new physics, including QGP
\cite{fluct4,interm2,interm3}
(iii) as a test of particle equilibration \cite{fluct2,fluct8}.
The statistical
hadronization model (SHM), introduced by Fermi in 1950
\cite{Fer50,Pom51,Lan53}, has been used extensively in recent years
in the study of strongly interacting particle production. In this model,
the properties of the final state particles are determined by requiring that
the final state maximizes entropy given the physical properties of the
fireball (energy, baryon content, etc.).
When the full spectrum of hadronic
resonances is included \cite{Hag65}, the SHM
turns into a quantitative model
capable of describing
in detail
the abundances of all hadronic particles.
Fluctuations in conserved quantum numbers (such as charge,
baryon number, strangeness, or
equivalently the net multiplicities of
up, down and strange quarks)
can be studied only
in the Grand Canonical (GC) ensemble,
since
in the micro-canonical and
canonical ensembles these quantities are fixed.
We also mention here
that fluctuations of non-conserved observables, e.g. other
hadron multiplicities, differ for different
ensembles even in the thermodynamic limit \cite{nogc1,nogc2}.
In this paper we will discuss the use of fluctuations as a phenomenological
tool within the framework of the statistical model, and illustrate some issues
pertinent in analyzing
fluctuations data. In section \ref{obschoice}
we will motivate the choice of charge fluctuations
as a useful experimental probe. After demonstrating, in section \ref{secshm},
how the statistical model implies a scaling between
fluctuations and yields, we show (section \ref{noneq}) that a measurement
of both particle yields and charge fluctuations can distinguish between
an equilibrium high temperature statistical freeze-out from a super-cooled
over-saturated freeze-out from a high entropy phase.
Finally, in section \ref{secacceptance} we discuss issues related to detector
acceptance which impact the fluctuation measurement even in a boost-invariant
azimuthally symmetric limit. We quantitatively demonstrate how such limited
acceptance effects can be taken into account and the freeze-out temperature
and non-equilibrium parameters extracted from experimental data.
\section{\label{obschoice} GC Observables}
A study of GC SHM fluctuations of conserved
quantities is of considerable interest at RHIC.
Since the detectors at RHIC (except for the PHOBOS detector) only
see small portions of the final phase space,
using the grand-canonical approach is justified in the following sense:
Provided the fireball is indeed locally thermalized,
we can take
the experimentally observed source to be a subsystem in contact with a
larger reservoir.
The situation is of particular interest for
reactions at RHIC that exhibit a sizable
central plateau in the (pseudo-)rapidity spectrum,
since
a limited (pseudo)rapidity acceptance window
selects a suitable subset of the
source particles. Specifically,
it can be shown (sections \cite{cleymans}. The reasoning used there can be generalized to Fermi-Dirac and Bose-Einstein statistics) that
the rapidity spectrum of a boost invariant system
could be related to the multiplicity in a static GC system with
the same temperature and chemical potentials
\begin{eqnarray}
{\ave{dN_i/dy}_{\rm b.i.} \over \ave{dN_j/dy}_{\rm b.i.}}
=
{\ave{N_i}_{\rm GC}\over \ave{N_j}_{\rm GC}}
\label{eq:boost_inv}
\end{eqnarray}
where $i$ and $j$ are species labels and the subscripts ${\rm b.i.}$ and
${\rm GC}$ denote the boost invariant system and the grand canonical
system, respectively.
The derivation in \cite{cleymans} can be applied to fluctuations at hadronization (before resonance decays) to show
\begin{eqnarray}
{\ave{d{\Delta N_i^2}/dy}_{\rm b.i.} \over \ave{dN_j/dy}_{\rm b.i.}}
=
{\ave{\Delta N_i^2}_{\rm GC}\over \ave{N_j}_{\rm GC}}
\end{eqnarray}
where we denote the variance (fluctuation) of any quantity $X$ as
$\ave{\Delta X^2} = \ave{X^2}{-}\ave{X}^2$.
Given this, SHM average yields and yield fluctuations can be calculated by a textbook
method \cite{huang}, as per section
\ref{secshm}.
When studying finite systems the consideration
of fluctuations in {\em extensive} quantities such
as of particle yield has to address also
volume fluctuations when the volume cannot be fixed
by experimental conditions.
In our case volume fluctuations
can arise due to initial reaction
effects, impact parameter variations, as well as from
fluctuations due to dynamics of the expanding fireball.
It is difficult to arrive at a reliable description of all these effects.
Therefore it is important to select fluctuation observables in which
volume fluctuation effects are sub-dominant.
Among extensive quantities,
the net charge fluctuation stands out as it is relatively easy to measure
and can be shown to be nearly independent of the volume fluctuations
\cite{fluct1}.
In light of the above considerations
we concentrate our effort on the
following net charge fluctuation measure:
\begin{eqnarray}
\label{vqdef}
v(Q) \equiv \left< \Delta Q^2 \right>/\left<N_{\rm ch}\right>
\end{eqnarray}
(where $N_{\rm ch}=N_+ + N_-$)
proposed in the past as a probe of the QGP formation
\cite{fluct4}. First results for $v(Q)$ are also available from RHIC
experiments
\cite{starfluct2,phefluct}.
In the SHM,
the charged particle multiplicity is given by summing
all final state (stable) charged particle multiplicities.
These can be computed by adding the direct
yield and all resonance decay feed-downs.
The total yield of a stable particle $\alpha$ is
\begin{eqnarray}
\label{resoyield}
\langle N_\alpha\rangle_{\rm total} & = &
\langle N_\alpha\rangle_{\rm GC} + \sum_{j\ne \alpha}
B_{j \rightarrow \alpha} \langle N_j \rangle_{\rm GC}
\label{fluctdef}
\end{eqnarray}
where $j$ labels resonances.
$B_{j \rightarrow \alpha}$ is the probability (branching ratio)
for the decay products of $j$ to include $\alpha$.
The charged particle multiplicity is given by the sum of all
charged stable particles.
The net charge fluctuation is given by
\begin{eqnarray}
\ave{\Delta Q^2}_{\rm GC} = \sum_{i} q_i^2 \ave{\Delta N_i^2}_{\rm GC}
\label{eq:DQ2}
\end{eqnarray}
where $q_i$ is the particle charge and
$i$ labels {\em all} particles {\em before} resonance
decays since net charge is conserved \cite{fluct3}.
To use Eq.(\ref{eq:DQ2}) quantitatively,
however, the experimental rapidity window
must be large enough to encompass all
decay particles of the resonances, yet small enough for the GC ensemble to
maintain it's validity. See section \ref{secacceptance} for a discussion of
the validity of this assumption, and how to incorporate deviations from it in
realistic experimental
situations.
\section{\label{secshm}Statistical hadronization}
For a hadron with an energy
$E_{p} = \sqrt{p^2+m^2}$, the GC partition function for each species is
given by
\begin{eqnarray}
\label{partition_function}
\ln Z_i =
(\mp) V g_i\int {d^3p\over (2\pi)^3}
\ln \left(1 \pm \lambda_i e^{-E_i/T} \right)
\end{eqnarray}
where
$g_i$ is the degeneracy factor and the upper sign is for bosons and
the lower sign is for fermions.
Here $\lambda_i$ is the particle fugacity, related to particle chemical
potential $\mu_i=T\ln \lambda_i$.
The yield average and fluctuation is then given by:
\begin{eqnarray}
\label{yield_formula}
\langle N_i\rangle_{\rm GC}
& = & \frac{\partial \ln Z_i }{\partial \lambda_i}
= g_iV\int {4 \pi p^2 dp \over (2\pi)^3}\, n_{i}(E_p),
\\
\label{fluct_formula}
\ave{\Delta N_i^2}_{\rm GC}
& =& \frac{\partial^2 \ln Z_i }{\partial \lambda_i^2} \nonumber \\ &=&
g_iV\int {4 \pi p^2 dp \over (2\pi)^3}\, n_{i}(E_p) \left(1 \mp
n_i(E_p)\right).
\end{eqnarray}
and
\begin{equation}
n_{i}(E_p) = {1\over \lambda_i^{-1} e^{E_p\beta}\pm 1},
\end{equation}
We note that $\lambda_i$ enters the {\em partition function} in
Eq.(\ref{partition_function}). Hence, the validity of
Eqs.(\ref{yield_formula})
and (\ref{fluct_formula}) depends on weather
Eq.(\ref{partition_function}) can be
used as a {\em generating function} for the probability distribution of states.
It is important to underline this as in a dynamical system the value of $\lambda_i$ is
not determined solely in terms of entropy maximization, but is subject
to chemical conditions prevailing in the system, and here importantly, includes
effects related to chemical non-equilibrium.
Where Eq. \ref{partition_function} represents a generating function but the system is not in chemical equilibrium, the fugacity $\lambda_i$, is not anymore a Lagrange multiplier but a parameter
characterizing the quantum number density.
In a scenario where freeze-out occurs as a break-up of a
{\em chemically equilibrated} hadron gas,
the fugacity of the hadron $i$ is given
by the product of the fugacities of conserved quantum numbers.
\begin{equation}
\lambda_i^{\mathrm{eq}} = \lambda_{q}^{q-\overline{q}} \lambda_{s}^{s -
\overline{s}}
\lambda_{I_3}^{I_3} \;, \; \lambda_{\overline{i}}^{\mathrm{eq}} =
(\lambda_i^{\mathrm{eq}})^{-1},
\label{eqlam}
\end{equation}
where $\overline{q},q$ is the number of light anti-quarks and quarks,
respectively and
$\overline{s},s$ is the number of strange anti-quarks and quarks,
respectively and $I_3$ is the isospin.
This formula implies that the fugacity for the
antiparticle is, in full chemical equilibrium,
the inverse of the fugacity for the particle,
and the fugacity for a hadron carrying vanishing
conserved quantum numbers is 1.
In our approach, we do not assume that that the chemical equilibrium is
reached~\cite{JJBook,Rafelski:2003ju}.
Hence Eq.(\ref{eqlam}) no longer applies.
The deviation from chemical equilibrium can be
parametrized by a phase space occupancy factor
$\gamma_q$ (for $u,\bar{u}, d, \bar{d}$
in hadrons)
and $\gamma_s$ (for $s$ and $\bar{s}$).
In this {\em chemical nonequilibrium} case the fugacity becomes
\begin{equation}
\label{chemneq}
\lambda_i = \lambda_i^{\mathrm{eq}}
\gamma_q^{q+\overline{q}} \gamma_s^{s+\overline{s}}
\end{equation}
where $\lambda_i^{\mathrm{eq}}$ is given by Eq.(\ref{eqlam})
(Note that $\gamma_i = \gamma_{\overline{i}}$).
A system undergoing collective expansion is unlikely to be in chemical
equilibrium, since collective expansion and cooling will make it impossible for
endothermic and exothermic reactions, or for creation and destruction
reactions of a rare particle, to be balanced. However, since inelastic
collisions have in general a slower relaxation time than elastic ones, an
approximately
perfect fluid can still have $\gamma \ne 1$ (it's evolution will be a
non-trivial function of time, since $\gamma$ does not commute with the
Hamiltonian).
Furthermore, light quark chemical nonequilibrium is well motivated in a
scenario
where an entropy rich deconfined state quickly hadronizes
\cite{Rafelski:2000by}.
In this scenario, mismatch of entropies between the two phases requires
$\gamma_q>1$.
Despite the lack of equilibrium and entropy maximization w.r.t. conserved
quantum numbers, we will argue that the Eqs. \ref{fluct_formula} and
\ref{yield_formula}
apply in such a situation, with $\gamma$ s contributing to the chemical
potential via Eq.(\ref{chemneq}).
The validity of Eq.(\ref{fluct_formula})
and (\ref{yield_formula}) depend on the extent that
Eq.(\ref{partition_function}) represents a probability generating
function for the statistically hadronizing system.
Within a statistical hadronization scenario where hadrons are
formed in proportion to their phase space
weight given (not necessarily equilibrated) densities
\cite{Danos}, this is indeed the case provided the dynamics
behind $\gamma$ does not generate
additional, non-statistical fluctuations.
For an instance where the last issue is a concern, fluctuations of a quantum number produced
mostly in initial-state processes (such as charm
\cite{thews,becattini_charm}) will likely be dominated not by
the statistical hadronization contribution but to fluctuations in initial
abundance.
Given that in the considered model non-equilibrium arises due to the rapid
hadronization of the collectively expanding system \cite{Rafelski:2000by}, and
since the observable charged particles are produced not in in the initial state
but during the final break-up of a locally thermalized system, such
non-statistical fluctuations should not be significant for the observable we
are considering. Similarly, as we have argued in the previous section,
initial-state
volume fluctuations give a negligible contribution to the observable under
consideration.
However, it is possible that additional sources of irreducible two-particle
correlations and fluctuations
could arise near a phase transition.
These effects go beyond the scope of this work. We will however argue that
the applicability of our scenario, and the absence of further correlations can
be {\em tested} by {\em requiring} that the same temperature and $\gamma$ s
describe both the yields and the fluctuations of {\em all} soft hadronic
observables.
As we will show, this is a very stringent requirement.
If it turns out that a single set of $T$, $\lambda^{\rm eq}$ and
$\gamma_q$ and $\gamma_s$ is capable of describing all yields and
fluctuations,
then it certainly is a strong indication that Eq.(\ref{partition_function})
can be interpreted as a generating function of the probabilities.
The goal of this paper is then to find a way to experimentally determine the additional
parameter $\gamma_q$ which can be then used to compare
the SHM calculation of yields and fluctuations to the experimental measurements.
\section{\label{noneq}fluctuations in chemical non-equilibrium}
Chemical nonequilibrium is of a particular interest since
it can result in a large pion fugacity which influences fluctuations much
more severely than the yields.
If $\gamma_q$ becomes large enough so that $\lambda_{\pi}$
approaches $e^{m_\pi/T}$, then the pion yield and the fluctuations
behave like
(c.f.~Eqs.(\ref{yield_formula},\ref{fluct_formula}))
\begin{equation}
\label{divergence}
\lim_{\epsilon\rightarrow 0} \langle N\rangle \propto \epsilon^{-1},
\qquad
\lim_{\epsilon\rightarrow 0} (\Delta N)^2 \propto \epsilon^{-2}.
\end{equation}
where $\epsilon = 1 - \lambda_\pi e^{-m_\pi/T}$. The fluctuation grows much
faster than the yield as mentioned above.
Some studies of yield ratios have indeed found the value of
$\gamma_q$ that can potentially make $\epsilon$
small~\cite{observing,Rafelski:2003ju,Rafelski:2004dp,gammaq_energy}.
However, other studies of yield ratios~\cite{Becattini:2003wp}
concluded that $\gamma_{q}$ is not necessarily large due to the fact
parameters in such fits are highly correlated.
In this case, adjusting other parameters such as
the temperature can accommodate current
data without having $\gamma_q \ne 1$, but with much reduced statistical
significance.
Since such conflict is common when only the {\em yields} are considered,
it becomes necessary to study
fluctuations as an additional constraint
to determine the occupation factor $\gamma_q$ more convincingly.
We now discuss our specific analysis results. We used the public
domain SHM suite of programs SHARE \cite{share}, expanded to include
the fluctuations \cite{share2}. We evaluate
yields and fluctuations, allowing for production of
hadron resonances, their decay, and
a possible absence of chemical equilibrium.
In the rest of this paper, we set
$\lambda_{I_3}^{\rm eq}=1,
\lambda_q^{\rm eq}=e^{\mu_B/3T}=1.05$ and $\lambda_s^{\rm eq}=1.027$ in
accordance with \cite{Rafelski:2004dp}.
However, the two observables we consider, the net charge
fluctuations and the $\Lambda/{\rm K}^-$
particle yield ratio, are nearly independent of
these quantities as will be shown below.
\begin{figure}[!tb]
\psfig{width=8.cm,clip=,figure=pdatfluct_nofit_boltz.eps}
\caption{(Color online)\label{datfluctboltz}
$v(Q)$ as function of $\gamma_q$ (solid lines).
Dot-dashed lines, no resonance decays;
dashed lines, Boltzmann fluctuations.
Ellipses (blue) indicate the expected result areas for
the equilibrium ($\gamma_q=1$, solid) and non-equilibrium
($\gamma_q\ne 1$, dashed) models.
}
\end{figure}
Fig.~\ref{datfluctboltz} shows the variation in $v(Q)$ as a function of
$\gamma_q$ for $T=140, 170$ MeV. The solid lines
show $v(Q)$ including the resonance
decays, dot-dashed lines comprise only the direct effect
of pion fluctuations.
As the temperature increases (solid lines from top to bottom)
the number of resonances increases. This in turn
increases the unlike-sign charge correlations
and hence reverses
the temperature dependence of the pure pion case (dot-dashed
lines). The short
dashed lines show results for Boltzmann statistics.
Boltzmann charge
fluctuations are nearly constant as function of $\gamma_q$ and primarily
depend
on chemical mix of the directly produced and secondary decay particles,
which dominantly depend on the temperature $T$.
The solid and dot-dashed lines in Fig.~\ref{datfluctboltz}
terminate
when the fluctuations start to diverge as in
Eq.(\ref{divergence}).
To determine both $T$ and $\gamma_q$ values we
require an additional observable.
In this work, we choose the yield ratio $\Lambda/K^-$.
This ratio depends linearly on $\gamma_q$,
and is nearly independent of $\lambda_s^{\rm eq}$ and $\gamma_s$ as
$\Lambda = (sdu)$ and $K^- = (s\bar{u})$.
In Fig.~\ref{datyld}
we show how the relative yield depends on $\gamma_q$ and $T$.
The $\Lambda$ yield we wish to consider
does not include weak decay feed from $\Xi$ but
it includes the electromagnetic decay of $\Sigma^0$ and the strong decays.
$K^-$ excludes feed-down from $\phi$, but includes $K^*$ and higher resonances.
It is important to exclude the $\Xi$ and $\phi$ cascading in order to
eliminate the dependence on $\gamma_s$ and $\lambda_s^{\rm eq}$.
Fortunately, this is experimentally feasible.
A similar ratio, which is
experimentally easier to correct
for, is $\Xi/\phi$, also dependent on temperature and $\gamma_q$ only. See
\cite{ourfluct2} for the equivalent discussion in
terms of $\Xi/\phi$.
\begin{figure}[!tb]
\psfig{width=8.cm,clip=,figure=pdat_particles.eps}
\caption{(Color online)\label{datyld}
Particle yield ratio $\Lambda/K^-$ as a function of $T$ (right panel) and
$\gamma_q$ (left panel)
The $\Lambda$ yield does not include $\Xi \rightarrow \Lambda$ and
the ${\rm K^-}$ yield is without the contribution of $\phi
\rightarrow {\rm K^+}{\rm K^-}$ decays.
Ellipses (blue) indicate the expected result areas for
the equilibrium ($\gamma_q=1$, solid) and non-equilibrium
($\gamma_q\ne 1$, dashed) models.
}
\end{figure}
We now combine results in Figs.~\ref{datfluctboltz} and \ref{datyld}
into our main result Fig.~\ref{datyldfluct}.
Every point in this plane of $v(Q)$ and
$\Lambda/K^-$ corresponds to a specific set of $T$ and $\gamma_q$
as indicated by the grid. Note that
some domains in this plane are not allowed
since they lie in the region where the (generating, GC) partition function
cannot be defined.
The two highlighted regions indicate the
expected chemical equilibrium (solid line ellipse at small $v(Q)$,
corresponding to
$\gamma_q=1$ and $T=170$ MeV) and nonequilibrium
parameter domains (dashed line ellipse at larger $v(Q)$, corresponding to
$\gamma_q=1.62$ and $T=140$ MeV). When particle yields and
fluctuations are considered, the separation of these two
domains confirms that we have found a sensitive method to determine
both $\gamma_q$ and $T$.
\begin{figure}[t]
\psfig{width=8.cm,clip=,figure=pdat_fold_gams_sparse.eps}
\caption{(Color online)\label{datyldfluct}
Particle ratio $\Lambda/K^-$ and particle fluctuation
$v(Q)$ plane: a point in plane
corresponds to a set of values $\gamma_q, T$.
Black Lines correspond to results at fixed $T=200$ (top),
170, 140, 100 MeV (bottom). The
red dashed lines are for $\gamma_q=0.8,1,1.4,1.6,1.8$ from left to right.
Thick lines correspond to $\gamma_s=2.5$, thin lines correspond to
$\gamma_s=1$.
Ellipses (blue) indicate the expected result areas for
the equilibrium ($\gamma_q=1$, solid) and non-equilibrium
($\gamma_q\ne 1$, dashed) models.}
\end{figure}
The results of having two extreme values,
$\gamma_s=1$ and $\gamma_s=2.5$, are also
shown in Fig.~\ref{datyldfluct}. The $\gamma_s$ values corresponds to the
equilibrium \cite{bdm} and non-equilibrium \cite{Rafelski:2003ju} best fits.
Their difference, as seen in Fig.~\ref{datyldfluct}, is small and
well below the experimental error.
The largest remaining systematic deviation is due to the baryon chemical
potential $e^{\mu_B/3 T} = \lambda_q^{\rm eq}$.
It's contribution to $v(Q)$ is negligible,
but this is not true for the case of $\Lambda/K^-$.
Generally the value of $\lambda_q^{\rm eq}$ is well determined
by baryon to antibaryon yield ratios in a model independent way.
To transform the diagram in Fig.~\ref{datyldfluct} (or $\Xi/\phi$ in
\cite{ourfluct2}) to an
equivalent result applicable to lower reaction energy where
$\lambda_q^{\rm eq}$ is greater, one has to allow for this
change: We note that $\Lambda/K^-\propto (\lambda_q^{\rm eq})^3$,
and thus we need to multiply the
axis in Figs.~\ref{datyld} and \ref{datyldfluct} by
$(\lambda_q^{\rm eq})^3/1.05^3$. One can actually use the
$\Lambda/K$ ratio in this. Since
${\Lambda K^+}/{\overline{\Lambda} K^-}\propto (\lambda_q^{\rm eq})^6$,
the axis rescaling would be done with
$({\Lambda K^+}/{\overline{\Lambda} K^-})^{1/2}/1.05^3$
($\Lambda,K$ corrected for $\Xi$ and $\phi$ feed-down).
\section{\label{secacceptance}Issues related to detector acceptance}
The main phenomenological issue that prevents the straight-forward extraction
of parameters from graphs such as Fig. \ref{datyldfluct} are effects relating
to the detector acceptance.
First of all, it has long been known that $v(Q)$ is not a ``robust''
observable, but in general depends on the detector's kinematic (rapidity and $p_T$) cuts.
This difficulty, however, can be lessened via mixed
event background subtraction. It can be shown \cite{fluct6} that observables corrected
this way are in certain limits ``robust'' w.r.t. kinematic cuts and detector response.
We have discussed how to
generalize the methods described in this paper to robust observables
elsewhere \cite{ourfluct1,ourfluct2,ourfluct3}, and hence will not dwell on
this topic, beyond noting that,
while diagrams such as Fig. \ref{datyldfluct} need to be re-thought since
dynamical observables generally also depend on the (average) system volume, the
{\em sensitivities} of the fluctuation and yield observables to the
statistical model parameters follow the pattern described by this paper.
Hence, generalizing the methods described by this paper to dynamical
observables (whether via fits, as was done in \cite{ourfluct3} or
three-dimensional diagrams), is not a difficult task.
An issue that needs to be addressed separately, however, is the acceptance
dependence of particle {\em correlations}. If the detector's pseudo-rapidity
coverage is too large, than the small volume assumption
required for the Grand-Canonical ensemble becomes untenable, and long-range
correlations (such as global conservation laws) can modify fluctuations.
If the detector's pseudo-rapidity coverage is too small, correlations due to
resonance decays acquire a rapidity-dependent correction (which is {\em not}
eliminated by mixed-event subtraction since it
corrects {\em two-particle correlations}). We will address these issues in
the next sub-sections.
\subsection{\label{secconserv}Influence of conservation laws on fluctuations}
If the detector can capture the full phase space of the system than, barring
dramatic departure from standard model physics, the net charge of the event can
not fluctuate.
More generally, if the phase space size of the detected system becomes
comparable to the total system size, observables will not anymore be given by
the Grand-Canonical ensemble.
If the system is a fluid (or in general not in {\em global} equilibrium) {\em
no} ensemble is expected to provide a good description of fluctuations beyond the small volume Grand Canonical limit, since
the observable region of phase space will include many locally equilibrated
volume elements exchanging energy and quantum numbers via hydrodynamic flow.
While yields could still be approximated by some ensemble, the long range
correlations and global non-equilibrium should break all simple scaling of
fluctuations with yields.
Hence, the configuration space coverage needed for a statistical description
needs to be appropriately small for the corrections to the GC ensemble to be
kept under control.
To investigate these corrections quantitatively, consider the
Taylor-expansion of the
entropy of the ``reservoir'':
\begin{eqnarray}
\label{gcdef}
S(N_{\rm tot}-N) & \approx & S(N_{\rm tot})
-N \left. \frac{\partial S}{\partial N} \right|_{N_{\rm tot}}
\nonumber \\
& & {}
+ \frac{1}{2} N^2 \left.
\frac{\partial^2 S}{\partial N^2} \right|_{N_{\rm tot}} + ...
\end{eqnarray}
where $N_{\rm tot}$ is the total number of particles in the reservoir and
the small subsystem, and $N$ is the number of particles in the subsystem.
The first and second terms result in the usual Grand-Canonical
ensemble result \cite{huang} through the identification of the equilibrium chemical potential
$\mu = -T (\partial S/\partial N)$.
The third term gives the first correction;
The Grand-Canonical
ensemble is therefore a valid approximation when
\begin{equation}
\label{gccorr}
\zeta_{GC} = \frac{\ave{N}}{2}
\frac{({\partial^2 S}/{\partial N^2})_{N_{\rm tot}}}
{({\partial S}/{\partial N})_{N_{\rm tot}}}
\ll 1
\end{equation}
This quantity can be easily related to more common thermodynamic quantities
\begin{equation}
\zeta_{GC} = \frac{1}{2} \frac{T \ave{N}}{\mu}{k_{Vtot}} \end{equation}
where $\ave{N}$ is the average multiplicity of the {\em observed volume} and
$k_{Vtot}$ is the susceptibility of the {\em total volume}.
For the relativistic ideal gas, this is given by
\begin{equation}
\zeta_{GC}= \frac{V}{2 V_{tot}} \ \left[ \frac{\sum_{n=0}^{\infty}
\lambda^n m^2 T K_2 \left( \frac{n m}{T} \right)}{ \ln \lambda
\sum_{n=0}^{\infty} \lambda^n m^2 \frac{T}{n} K_2 \left( \frac{n m}{T} \right)}
\right]
\end{equation}
and, as shown in section \ref{obschoice}
\[\ \frac{V}{V_{tot}}=\frac{\Delta \eta}{(\Delta \eta)_{tot}} \]
where $\Delta \eta$ is the detector's (pseudo)rapidity coverage and
$(\Delta \eta)_{tot}$ is the system's rapidity interval.
Thus, we discover that the larger the susceptibility is, the smaller the system
size $V/V_{\rm tot}$ has to be for the Grand-Canonical limit to hold.
In fact, the physics determining the departure from this limit is {\em
precisely the same} as the physics determining the divergence of fluctuations
within an over-saturated pion gas.
This is unsurprising, since over-saturation is argued for as a signature of a
phase transition, and in finite systems undergoing phase transitions it is the
finite size of the system
that gives a cut-off for fluctuations.
The pion chemical potential of the system created at RHIC, however, is kept
below divergence, so it is hoped that one unit of rapidity, corresponding to
$V/(2V_{\rm tot}) \sim 7 \%$, provides a safe limit for the
Grand Canonical ensemble.
In such a small rapidity interval,
however, correlations due to resonances need to be
suitably accounted for. The next
sub-section shows how to do that.
\subsection{\label{correso}Disappearance of resonance correlations at small
$\Delta \eta$}
If charge fluctuations are calculated {\em after} all resonances have decayed,
then Eq. \ref{eq:DQ2} becomes
\begin{equation}
\ave{(\Delta Q)^2} = \ave{(\Delta N_+)^2} + \ave{(\Delta N_-)^2} - 2
\ave{\Delta N_+ \Delta N_-}
\label{fluctdefcorrel1}
\end{equation}
where the last term accounts for unlike-sign charge
correlations coming from the decay of neutral resonances.
For a conserved charge, and full acceptance of all resonances, this expression
is equivalent to Eq.(\ref{eq:DQ2}), with the
correlation term exactly balancing out the amplification of resonance abundance
fluctuations through the greater multiplicity of resonance decay products.
within a hadron gas the correlation term will be given by decays of the
resonance $j$ into $N_+$ and $N_-$
\begin{equation}
\label{correctioncorr}
\ave{\Delta N_+ \Delta N_-} = \sum_j b_{j\rightarrow + -} \ave{N_j}
\end{equation}
while the fluctuation of each stable $N_{\pm}$ has to be augmented by
contributions to it from resonance decays \cite{fluct1}
\begin{eqnarray}
\label{correctionres}
\ave{(\Delta N_{\pm})^2}&=&\sum_i \ave{(\Delta N_{\pm})^2}_{i}+\\
&&\hspace*{-2cm}+ \left( \sum_j b_{j \rightarrow i}(1-b_{j \rightarrow i})
\ave{N_j}
+ b_{j \rightarrow i}^2 \ave{(\Delta N_j)^2} \right)
\nonumber
\end{eqnarray}
For a finite acceptance window in general not all resonances
produced can be reconstructed, even if the efficiency of the
detector were 100\%. Hence these contributions must be weighted
with acceptance weight factors, and this applies here in particular
to the limited rapidity acceptance.
For a neutral resonance $j$ decaying
into $n_+$ positive particles and $n_-$ negative particles,
three such coefficients are needed:
Two will be the fractions of the positively charged and
the negatively charged decay products which land in
the acceptance window, and the third
will give the fraction of
the $+-$ {\em pairs}
that will land in the window.
These coefficients will
modify
the branching ratios $b_{j \rightarrow i}$
in Eq.(\ref{correctionres}) and $b_{j\rightarrow +-}$
in Eq.(\ref{correctioncorr}).
If boost-invariance is a good symmetry, the first two coefficients can be fixed
to unity, since particles coming \textit{out} of the acceptance region
are exactly balanced by particles coming \textit{in}. However, this is not
true for the number of detectable pairs.
If the resonance is out of the detector's acceptance window it is impossible
for {\em all} of it's decay products to be in a window. Hence,
Eq.(\ref{fluctdefcorrel1}) will have to include a
term giving the percentage of resonances whose decay products are both within
the detector's acceptance region.
\begin{eqnarray}
\ave{(\Delta Q)^2} &=& \ave{(\Delta N_+)^2} + \ave{(\Delta N_-)^2} \nonumber \\
&-& 2 R_F(T,\Delta y) \ave{\Delta N_+ \Delta N_-}
\label{fluctdefcorrel2}
\end{eqnarray}
The dependence of the observed fluctuations on $R_F$ is shown in Fig.
\ref{expcorrel}, left panel.
We note two effects not considered here and believed to be unimportant:\\
1) the rescattering after formation is unlikely to alter $R_F$, since the
typical
momentum exchange in each collision the exchanged momentum
$\ave{q} \sim T/3$ tends to be considerably
softer
than what is required to bring particles outside the acceptance region (in most
decays, the characteristic
momentum of the decay products in a resonance's rest frame $p^*$ tends to be
significantly
larger than this value);\\
2) The higher-momentum pseudo-elastic ``regeneration'' processes, where
detectable
resonances would be created, are also unlikely to modify $R_F$ since, by local
thermal
equilibrium, two particles coming into the acceptance region through
kinematically allowed
pseudo-elastic interactions will be balanced out by two particles originally in
the
acceptance region which come out as a result of the re-interaction.\\
Thus, a measurement of fluctuations can still be relied upon to gauge the
number of
resonances present at {\em chemical} freeze-out. This underscores the
importance of
fluctuations as a probe for freeze-out
dynamics.
We now obtain $R_F$ for a azimuthally symmetric perfect detector having a
pseudo-rapidity
coverage $\Delta \eta$. We shall follow the formalism in \cite{Ani85}
to relate the resonance's
rest frame (denoted by $*$) to the lab frame.
For both particles $+$ and $-$ to be within
the detector's acceptance region, $-\Delta \eta/2 < \eta_{+},\eta_{-}< \Delta
\eta/2$ where
\begin{equation}
\label{pseudorap}
\eta_{\pm}= \frac{1}{2} \log \left( \frac{\sqrt{E_{\pm}^2 - m_{\pm}^2} -
p_{L\pm}}{\sqrt{E_{\pm}^2 - m_{\pm}^2} + p_{L\pm}} \right) = \ln \left[ \cot
\left( \frac{\theta_{\pm}}{2} \right) \right]
\end{equation}
If all angular dependence in the resonance's decay matrix elements is neglected
(a valid approximation if many resonances are produced, with an approximately
azimuthally invariant distribution)the fraction of detectable $+-$ pairs will
then be simply given by a phase space integral \begin{equation}
\Omega_{+-} (\eta_R,p_{TR}) = \int \frac{d^3 p^*_+}{E^*_+} \frac{d^3
p^*_-}{E^*_-} \prod_i \frac{d^3 p^*_i}{E^*_i} \Theta_{+-}
\label{correlpt}
\end{equation}
where:
\[\ \Theta_{+-} = \Theta \left[\eta_{+}- \frac{\Delta \eta}{2} \right]
\Theta \left[\eta_{+}+ \frac{\Delta \eta}{2} \right] \Theta \left[\eta_{-}- \frac{\Delta \eta}{2} \right]
\Theta \left[\eta_{-}+ \frac{\Delta \eta}{2} \right] \]
and the function $\Theta(z)$ is the usual step function
\[\
\begin{array}{cc}
\Theta(z)=0 & z<0\\
\Theta(z)=1 & z>0\\
\end{array} \]
Now, for two body decays this reduces to
\begin{equation}
\Omega_{+-} (\eta_R,p_{TR}) = \frac{1}{4 \pi} \int_0^{2 \pi} d \phi \int_0^1 d
\left( \frac{p_L^*}{p^*} \right) \Theta_{+-}
\label{2bodypt}
\end{equation}
while for three body decays we use the Monte-Carlo routine MAMBO \cite{mambo}
to generate points in phase space.
\begin{figure*}[!tb]
\psfig{width=8.cm,clip=,figure=pdatfluct_correl.eps}
\psfig{width=8.cm,clip=,figure=accept_all.eps}
\caption{(Color online) \label{expcorrel}
Left: Sensitivity of the charge fluctuation measure on $R_F$, the fraction of
resonance decay products which remains in the detector acceptance
window ({\it c.f.}\ Eq.(\ref{fluctdefcorrel2})).
Thin black lines denote $T=170$ MeV, thick
red lines $T=140$ MeV. Right: Acceptance fraction for different resonance
decays as a function of the inverse slope $b$ ({\it c.f.}\
Eq.(\ref{slope})) and the
detector pseudo-rapidity acceptance $\Delta \eta$ (NB: $\eta$ in this context
means the pseudo-rapidity. Not to be confused with the decay of the $\eta$
particle, shown on the right panel of the image). Acceptance regions of
$\Delta \eta=6,4,2,1,0.5,0.1$ are considered, top to bottom in descending order
}
\end{figure*}
To calculate $\eta_+$ and $\eta_-$ from the resonance rest frame kinematic
variables we Lorentz-transform to the lab frame, and get \cite{Ani85}
\begin{eqnarray}
\label{plpm}
p_{L\pm} = \pm p^*_{L \pm} + \frac{p_{LR}}{m_R} \left( E^*_{\pm} +
\frac{\vec{p^*}.\vec{p_R}}{E_R+m_R} \right)\\
\label{ptpm}
p_{T\pm} = \pm p^*_{T \pm} + \frac{p_{TR}}{m_R} \left( E^*_{\pm} +
\frac{\vec{p^*}.\vec{p_R}}{E_R+m_R} \right)
\end{eqnarray}
To get an over-all fraction of accepted resonances which will enter
Eq.(\ref{fluctdefcorrel2})
, one has to convolute Eq.(\ref{correlpt}) with a resonance distribution
function in momentum space
\begin{equation}
\label{eqrf}
R_F = \int_0^{\infty} d p_{TR} \int_{-\Delta \eta/2}^{\Delta \eta/2} d \eta_R
P(\eta_R,p_{TR}) \Omega_{+-} ( \eta_R,p_{TR})
\end{equation}
where $P(\eta_R,p_{TR})$ is a suitable distribution function for resonances
\textit{normalized to unity}. A suitable function in the low energy region at
mid-rapidity is
\begin{equation}
\label{slope}
P(\eta_R,p_{TR}) = \frac{m_{TR}^\alpha e^{- b m_{TR}}}{\Delta \eta_R
\int_m^\infty d m_{TR} m_{TR}^\alpha e^{-b m_{TR}}}
\end{equation}
We have performed this integral using a Monte-Carlo method. The result is
shown in the right panel of Fig. \ref{expcorrel}.
We note that the most abundant resonance decays for charge fluctuations do not
depend strongly on the
inverse slope parameter $b^{-1}$: Going from $b^{-1}=200$ MeV to $b^{-1}=300$
MeV while staying in the same rapidity bin changes the $\rho \rightarrow \pi
\pi$ correction by at most 5 $\%$, and the less abundant but more sensitive
$\eta \rightarrow \pi^+ \pi^- \pi^0$ correction by no more than 20$\%$.
Thus, $\Delta \eta$ should be
as small as possible, statistics permitting, due to
the not easily controllable corrections described in section \ref{secconserv}.
A subsequent SHM analysis of the experimental data can than calculate
$R_F$ for each resonance decay important for charge fluctuations.
Hence, a $v(Q)$ , properly corrected for experimental acceptance,
can be computed from SHM parameters via Eqs.(\ref{vqdef}) and
(\ref{fluctdefcorrel2}), and fed into
Fig. \ref{datfluctboltz} and similar
figures or fits \cite{ourfluct1,ourfluct2,ourfluct3}.
The computational tools needed to perform such an analysis have been
published separately as open-source software \cite{share2}.
It is important to underline that to perform this analysis it is not
necessary to understand the full freeze-out dynamics of the fireball (local
temperature, flow field, hadronization hypersurface). It is enough to have a
sensible parametrization of $b^{-1}$ in terms of particle mass. This function
is commonly obtained from particle spectra at {\em thermal} freeze-out
\cite{slopemass}, and is approximately linear in particle mass.
The question is weather we can extrapolate $b^{-1}$ to {\em chemical
freeze-out} conditions with enough precision
in a model-independent way. The relatively mild dependence of $R_F$ on
$b^{-1}$, together with the fact that
hadronic re-interaction decreases the temperature and increases the flow and
the high viscosity of the hadron gas \cite{hadvisc} makes us confident that
we can do it.
\section{Summary and conclusions}
We have studied in this work how a simultaneous
measurement of charge fluctuations and a ratio such as $\Lambda/K^-$
can differentiate
between chemical equilibrium and non-equilibrium
freeze-out, and to constrain the magnitude of the deviation from
equilibrium as well as the freeze-out temperature. Our results show that it is
possible to distinguish
the chemical equilibrium freeze-out condition
$\gamma_q=1$ \cite{Rafelski:2004dp} with $T=170$ MeV \cite{bdm})
from the chemical non-equilibrium freeze-out condition
$\gamma_q=1.6$ \cite{Rafelski:2003ju,Rafelski:2004dp}. This is mainly due to
the increase in the
fluctuations inherent to an oversaturated Bose gas, see Eq.(\ref{divergence}).
We have further discussed the dependence of two-particle correlations on the
detector acceptance region, and have shown that it can be
calculated to a reasonable precision in a model-independent way. The ``right''
experimental detector acceptance for a detailed study of
fluctuations, therefore, is one that is appropriately small yet sizable to
ensure the appropriate ensemble under study is Grand-Canonical, provided that
acceptance corrections to resonance decays are properly taken into
account using the methods described in section \ref{correso}. Quantitative
corrections to Grand Canonical yield/fluctuation relations
for the best fit parameters can be estimated quantitatively via
Eq.(\ref{gccorr})
Provided the detector acceptance region for a given fluctuation measurement
is published, Eq.(\ref{eqrf})
can be used to calculate a correction coefficient $R_F$ to the
$\ave{N_+ N_-}$ correlation for each decay of a neutral resonance.
Using a calculated $R_F$ for each resonance decay, together with the
statistical model parameters, the charge fluctuation variable $v(Q)$ can be
calculated from
Eqs.(\ref{vqdef}) and (\ref{fluctdefcorrel2}).
This $v(Q)$ will still retain
the sensitivities to temperature and $\gamma_q$ demonstrated in
section \ref{noneq}, since $\gamma_q$ impacts the primordial fluctuation terms rather than the correlation. It can therefore be used, together with a measurement
such as $\Lambda/K^-$ as in Fig. \ref{datyldfluct}, or within a fit
as in \cite{ourfluct1,ourfluct2,ourfluct3}, to test the validity of the
statistical model, unambiguously constrain its parameters, and differentiate
between the high-temperature equilibrium and supercooled over-saturated
freeze-out scenarios.
It is our intent to perform a complete data analysis as outlined here,
including consideration of acceptance corrections and of resonance decays,
once final RHIC fluctuation data becomes available.
\subsection{Acknowledgments}
GT thanks C. Gale, L. Shi,
V. Topor Pop, A. Bourque, Wojciech Broniowski, Wojciech Florkowski and Mark Gorenstein for stimulating discussions and the Tomlinson foundation for support.
S.J.~thanks RIKEN BNL Center
and U.S. Department of Energy [DE-AC02-98CH10886] for
providing facilities essential for the completion of this work.
Work supported in part by grants from
the U.S. Department of Energy (J.R. by DE-FG02-04ER41318),
the Natural Sciences and Engineering research
council of Canada, the Fonds Nature et Technologies of Quebec.
|
{
"timestamp": "2006-04-29T17:03:29",
"yymm": "0503",
"arxiv_id": "nucl-th/0503026",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503026"
}
|
\section{INTRODUCTION}
\label{sectoin1}
The proton separation energies of nuclei lying in the domain beyond the proton drip line are negative. Consequently these proton rich nuclei have positive $Q$ values for proton emissions with a natural tendency to shed off excess protons and are spontaneous proton emitters. The phenomenon of proton emission from nuclear ground states limits the possibilities of the creation of more exotic proton rich nuclei which are usually produced by fusion-evaporation nuclear reactions. Apart from providing the limit to the proton dripline, the one proton radioactivity may be used as a tool to obtain spectroscopic information because the decaying proton is the unpaired proton not filling its orbit. These decay rates are sensitive to the $Q$ values and the orbital angular momenta which in turn help to determine the orbital angular momenta of the emitted protons.
Since the observation of proton radioactivity is comparatively recent, only few theoretical attempts have been made to study this exotic process \cite{r1,r2,r3,r4}. In the energy domain of radioactivity, proton can be considered as a point charge having highest probability of being present in the parent nucleus. It has the lowest Coulomb potential among all charged particles and mass being smallest it suffers the highest centrifugal barrier, enabling this process suitable to be dealt within WKB barrier penetration model. In the existing theoretical models \cite{r1,r2} for proton radioactivity, Saxon-Woods type potential has been used for the nuclear interaction. In another recent work \cite{r4}, a unified fission model with proximity potential for nuclear force has been used. In the present work, quantum mechanical tunneling probability is calculated within the WKB approximation using microscopic proton-nucleus interaction potentials. These potentials have been obtained by single folding the densities of daughter nuclei with a realistic effective interaction supplemented by a zero-range pseudo-potential for exchange along with density dependence. Calculations using such potentials provide excellent estimates for lifetimes of the exotic decay process of proton radioactivity.
A well-defined effective nucleon-nucleon (NN) interaction in the nuclear medium is important not only for different structure models but also for the microscopic calculation of the nucleon-nucleus and nucleus-nucleus potentials used in the analysis of the nucleon and heavy-ion scattering. Effective NN interaction can be best constructed from a sophisticated G-matrix calculation. This interaction has been derived by fitting its matrix elements in an oscillator basis to those elements of the G-matrix obtained with the Reid-Elliott soft-core NN interaction \cite{r5}. The ranges of the M3Y forces were chosen to ensure a long-range tail of the one-pion exchange potential as well as a short range repulsive part simulating the exchange of heavier mesons. Such an effective NN interaction has been shown to provide a more realistic shape of the scattering potentials of the nucleon or heavy ion optical potentials obtained by folding in the density distribution functions of two interacting nuclei with the effective NN interaction \cite{r6}.
The density dependent M3Y (DDM3Y) effective NN interaction has been used to determine the incompressibility of infinite nuclear matter \cite{r7}. The equilibrium density of the nuclear matter has been determined by minimising the energy per nucleon. The density dependence parameters have been extracted by reproducing the saturation energy per nucleon and the saturation density of spin and isospin symmetric cold infinite nuclear matter. Result of such calculations also provide a reasonable value of nuclear incompressibility. In nuclear matter calculations, the calculation of potential energy per nucleon involves folding of interaction of one nucleon with the rest of the nuclear matter. It is therefore used in single folding model description for nuclear matter calculations and thus density dependence parameters obtained from nuclear matter calculations may be used as it is in describing nucleon-nucleus interaction potentials where single folding model comes into play. Such nucleon-nucleus interaction potentials have been used successfully to the analysis of elastic and inelastic scattering of protons \cite{r8}.
In the present work we provide estimates for the proton radioactivity lifetimes of the spherical proton emitters from the ground and the isomeric states using the same nucleon-nucleus interaction potentials obtained microscopically by single folding the daughter nuclei density distributions with a realistic DDM3Y effective interaction whose density dependence parameters have been extracted from the nuclear matter calculations.
\section{FORMALISM}
\label{section2}
The microscopic nuclear potentials $V_N(R)$ have been obtained by single folding the density of the daughter nucleus with the finite range realistic DDM3Y effective interacion as
\begin{equation}
V_N(R) = \int \rho (\vec{r}) v[|\vec{r} - \vec{R}|] d^3r
\label{seqn1}
\end{equation}
\noindent
where $\vec{R}$ and $\vec{r}$ are, respectively, the co-ordinates of the emitted proton and a nucleon belonging to the residual daughter nucleus with respect to its centre. The density distribution function $\rho$ used for the daughter nucleus, has been chosen to be of the spherically symmetric form given by
\begin{equation}
\rho(r) = \rho_0 / [ 1 + exp( (r-c) / a ) ]
\label{seqn2}
\end{equation} \noindent
where
\begin{equation}
c = r_\rho ( 1 - \pi^2 a^2 / 3 r_\rho^2 ), ~~ r_\rho = 1.13 A_d^{1/3} ~~ and ~~ a = 0.54 ~ fm
\label{seqn3}
\end{equation}
\noindent
and the value of $\rho_0$ is fixed by equating the volume integral of the density distribution function to the mass number $A_d$ of the residual daughter nucleus. The distance s between a nucleon belonging to the residual daughter nucleus and the emitted proton is given by
\begin{equation}
s = |\vec{r} - \vec{R}|
\label{seqn4}
\end{equation}
\noindent
while the interaction potential between any such two nucleons $v(s)$ appearing in eqn.(1) is given by the DDM3Y effective interaction. The total interaction energy $E(R)$ between the proton and the residual daughter nucleus is equal to the sum of the nuclear interaction energy, the Coulomb interaction energy and the centrifugal barrier. Thus
\begin{equation}
E(R) = V_N(R) + V_C(R) + \hbar^2 l(l+1) / (2\mu R^2)
\label{seqn5}
\end{equation}
\noindent
where $\mu = M_p M_d/M_A$ is the reduced mass, $M_p$, $M_d$ and $M_A$ are the masses of the proton, the daughter nucleus and the parent nucleus respectively, all measured in the units of $MeV/c^2$. Assuming spherical charge distribution (SCD) for the residual daughter nucleus, the proton-nucleus Coulomb interaction potential $V_C(R)$ is given by
\begin{eqnarray}
V_C(R) =&& Z_d e^2/ R~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~for~~~~R \geq R_c \nonumber\\
=&&( Z_d e^2/ 2R_c).[ 3 - (R/R_c)^2]~~~~~~~~~~for~~~~R\leq R_c
\label{seqn6}
\end{eqnarray}
\noindent
where $Z_d$ is the atomic number of the daughter nucleus. The touching radial separation $R_c$ between the proton and the daughter nucleus is given by $R_c = c_p+c_d$ where $c_p$ and $c_d$ have been obtained using eqn.(3). The energetics allow spontaneous emission of protons only if the released energy
\begin{equation}
Q = [ M_A - ( M_p + M_d ) ] c^2
\label{seqn7}
\end{equation}
\noindent
is a positive quantity.
In the present work, the half life of the parent nucleus decaying via proton emission is calculated using the WKB barrier penetration probability. The assault frequency $\nu$ is obtained from the zero point vibration energy $E_v = (1/2)\hbar\omega = (1/2)h\nu$. The decay half life $T$ of the parent nucleus $(A, Z)$ into a proton and a daughter $(A_d, Z_d)$ is given by
\begin{equation}
T = [(h \ln2) / (2 E_v)] [1 + \exp(K)]
\label{seqn8}
\end{equation}
\noindent
where the action integral $K$ within the WKB approximation is given by
\begin{equation}
K = (2/\hbar) \int_{R_a}^{R_b} {[2\mu (E(R) - E_v - Q)]}^{1/2} dR
\label{seqn9}
\end{equation}
\noindent
where $R_a$ and $R_b$ are the two turning points of the WKB action integral determined from the equations
\begin{equation}
E(R_a) = Q + E_v = E(R_b)
\label{seqn10}
\end{equation}
\noindent
From a fit to the experimental data on cluster emitters a law given by eqn.(5) of reference \cite{r9}, which relates $E_v$ with $Q$, was found. For the present calculations same law extended to protons is used for the zero point vibration energies. The shell effects of proton radioactivity is implicitly contained in the zero point vibration energy due to its proportionality with the $Q$ value.
\section{CALCULATIONS}
\label{section3}
The M3Y interaction is based upon a realistic G-matrix. Since the G-matrix was constructed in an oscillator representation, it is effectively an average over a range of nuclear densities and therefore the M3Y has no explicit density dependence. For the same reason there is also an average over energy and the M3Y has no explicit energy dependence either. The only energy dependent effects that arises from its use is a rather weak one contained in an approximate treatment of single-nucleon knock-on exchange. The success of the extensive analysis \cite{r6} indicates that these two averages are adequate for the real part of the optical potential for heavy ions at energies per nucleon of $< 20MeV$. However, it is important to consider the density and energy dependence explicitly for the analysis of $\alpha$-particle scattering at higher energies ($>100 MeV$) where the effects of a nuclear rainbow are seen and hence the scattering becomes sensitive to the potential at small radii. Such cases were studied introducing suitable and semirealistic explicit density dependence \cite{r10,r11} into the M3Y interaction which was then called the DDM3Y and was very successful for interpreting consistently the high energy elastic $\alpha$ and heavy-ion scattering data. Present calculations have been performed using $v(s)$, inside the integral of eqn.(1) for the single folding procedure, as the DDM3Y effective \cite{r8} interaction given by
\begin{equation}
v(s,\rho,E) = t^{\rm M3Y}(s,E)g(\rho,E)
\label{seqn11}
\end{equation}
\noindent
where $t^{\rm M3Y}$ is the same M3Y interaction supplemented by a zero range pseudo-potential is
\begin{equation}
t^{\rm M3Y} = 7999 \frac{e^{ - 4s}}{4s} - 2134\frac{e^{- 2.5s}}{2.5s} + J_{00}(E) \delta(s)
\label{seqn12}
\end{equation}
\noindent
where the zero-range pseudo-potential representing the single-nucleon exchange term is given by
\begin{equation}
J_{00}(E) = -276 (1 - 0.005E / A_p ) (MeV.fm^3)
\label{seqn13}
\end{equation}
\noindent
where $E$ and $A_p$ are the laboratory energy and projectile mass number respectively. In the present case of proton radioactivity it can be shown that $E/A_p=Q.m/\mu$ where m and $\mu$ are the nucleonic mass and reduced mass of the $p+A_d$ system, respectively, in units of $MeV/c^2$. The density dependent part has been taken to be \cite{r11}
\begin{equation}
g(\rho, E) = C (1 - \beta(E)\rho^{2/3})
\label{seqn14}
\end{equation}
\noindent
which takes care of the higher order exchange effects and the Pauli blocking effects. Constants of this interaction $C$ and $\beta$ when used in single folding model description, can be determined from the nuclear matter calculations \cite{r7} as 2.07 and 1.624 fm$^2$ respectively.
The two turning points of the action integral given by eqn.(9) have been obtained by solving eqns.(10) using the microscopic single folding potential given by eqn.(1) along with the Coulomb potential given by eqn.(6) and the centrifugal barrier described in eqn.(5). Then the WKB action integral between these two turning points has been evaluated numerically using eqn.(1), eqn.(5), eqn.(6), eqn.(7) and eqn.(5) of reference \cite{r9}. Finally the half lives have been obtained using eqn.(8).
\section{RESULTS AND DISCUSSIONS}
\label{section4}
In this work, the same set of experimental data of reference \cite{r4} for the proton decay half lives have been chosen for comparison with the present theoretical calculations. Experimentally measured values of the released energy $Q$ (given by eqn.(7)), which is one of the crucial quantity for quantitative predictions of the decay half lives, have been used for the calculations. The proton emitters and the experimental values for their logarithmic half lives have been presented in Table-I. The corresponding results of the present calculations with microscopic potentials are also presented along with the results of the modified preformed cluster model (PCM) called the unified fission model (UFM) calculations \cite{r4}. The three turning points $R_1$, $R_2=R_a$ and $R_3=R_b$ obtained by solving eqn.(10) have been listed in the Table-I.
Experimentally measured and theoretically calculated half-lives of spherical proton emitters have been provided in Table-I. Positions of the turning points are very sensitive to the Coulomb barrier. Comparing the results for ground and isomeric states of same proton emitters it can be observed that the positions of the turning points are quite sensitive to the centrifugal barriers. Results of the present calculations with DDM3Y have been found to predict the general trend of the experimental data very well. The quantitative agreement with experimental data is good. The discrepancy between the results of present calculation and the experimental values for some cases may be due to the uncertainty in the measurements of the $Q$ values to which the results are quite sensitive due to its proportionality with the zero point vibration energies. The degree of reliability of the present estimates for the proton decay lifetimes are equivalent to the very recent UFM estimates. Changing the value of density dependence parameter $\beta$ to 1.668 fm$^2$ \cite{r12}, obtained from nuclear matter calculations using saturation energy per nucleon obtained from fitting the masses of Audi-Wapstra-Thibault \cite{r13} mass table, causes insignificant changes in the second decimal places of logarithmic half lives in some cases.
\begin{table}
\caption{Comparison between experimentally measured and theoretically calculated half-lives of spherical proton emitters. The asterisk symbol (*) denotes the isomeric state. The experimental $Q$ values, half lives and $l$ values are taken from reference [4]. The results of the present calculations have been compared with the experimental values and with the results of UFM estimates [4]. Experimental errors in $Q$ [14] values and corresponding errors in calculated half-lives are given within parentheses. }
\begin{tabular}{ccccccccc}
Parent &Angular &Released &1st turning &2nd turning &3rd turning &Expt. &Present calc.&UFM \\
nuclei &momentum& Energy & point(fm)& point(fm)&point(fm)& & &\\ \hline
& $l(\hbar)$ & $Q(MeV)$ &$R_1$&$R_2=R_a$&$R_3=R_b$ &$log_{10}T(s)$ &$log_{10}T(s)$& $log_{10}T(s)$ \\ \hline
&&&&&&&&\\
$^{105}Sb$&2&0.491(15)&1.55&6.58&134.30&2.049&1.97(46)&2.085\\
$^{145}Tm$&5&1.753(10)&3.49&6.40&56.27&-5.409&-5.14(6)&-5.170\\
$^{147}Tm$&5&1.071(3)&3.51&6.40&88.65&0.591&0.98(4)&1.095\\
$^{147}Tm^*$&2&1.139(5)&1.58&7.15&78.97&-3.444&-3.39(5)&-3.199\\
$^{150}Lu$&5&1.283(4)&3.50&6.44&78.23&-1.180&-0.58(4)&-0.859\\
$^{150}Lu^*$&2&1.317(15)&1.59&7.20&71.79&-4.523&-4.38(15)&-4.556\\
$^{151}Lu$&5&1.255(3)&3.51&6.49&78.41&-0.896&-0.67(3)&-0.573\\
$^{151}Lu^*$&2&1.332(10)&1.59&7.22&69.63&-4.796&-4.88(9)&-4.715\\
$^{155}Ta$&5&1.791(10)&3.51&6.55&57.83&-4.921&-4.65(6)&-4.637\\
$^{156}Ta$&2&1.028(5)&1.61&7.23&94.18&-0.620&-0.38(7)&-0.461\\
$^{156}Ta^*$&5&1.130(8)&3.52&6.53&90.30&0.949&1.66(10)&1.446\\
$^{157}Ta$&0&0.947(7)&0.00&7.42&98.95&-0.523&-0.43(11)&-0.126\\
$^{160}Re$&2&1.284(6)&1.62&7.30&77.67&-3.046&-3.00(6)&-3.109\\
$^{161}Re$&0&1.214(6)&0.00&7.48&79.33&-3.432&-3.46(7)&-3.231\\
$^{161}Re^*$&5&1.338(7)&3.52&6.63&77.47&-0.488&-0.60(7)&-0.458\\
$^{164}Ir$&5&1.844(9)&3.54&6.68&59.97&-3.959&-3.92(5)&-4.193\\
$^{165}Ir^*$&5&1.733(7)&3.52&6.69&62.35&-3.469&-3.51(5)&-3.428\\
$^{166}Ir$&2&1.168(8)&1.61&7.35&87.51&-0.824&-1.11(10)&-1.160\\
$^{166}Ir^*$&5&1.340(8)&3.56&6.70&80.67&-0.076&0.21(8)&0.021\\
$^{167}Ir$&0&1.086(6)&0.00&7.54&91.08&-0.959&-1.27(8)&-0.943\\
$^{167}Ir^*$&5&1.261(7)&3.53&6.72&83.82&0.875&0.69(8)&0.890\\
$^{171}Au$&0&1.469(17)&0.00&7.60&69.09&-4.770&-5.02(15)&-4.794\\
$^{171}Au^*$&5&1.718(6)&3.52&6.77&64.25&-2.654&-3.03(4)&-2.917\\
$^{177}Tl$&0&1.180(20)&0.00&7.62&88.25&-1.174&-1.36(25)&-0.993\\
$^{177}Tl^*$&5&1.986(10)&3.53&6.89&57.43&-3.347&-4.49(6)&-4.379\\
$^{185}Bi$&0&1.624(16)&0.00&7.77&65.71&-4.229&-5.44(13)&-5.184\\
\end{tabular}
\end{table}
For an interesting comparison, the entire calculations have been redone with the recent global optical model potential (GOMP) for protons \cite{r15}. The real central part of the GOMP for protons is given by
\begin{equation}
V_{GOMP}(R) = -V_p(E) f(R)
\label{seqn15}
\end{equation}
\noindent
where the form factor $f(R)$ is given by
\begin{equation}
f(R) = 1/(1+exp[(R-R_V)/a_V]),~~ R_V=1.3039A_d^{1/3}-0.4054,~~a_V=0.6778-1.487 \times 10^{-4}A_d,
\label{seqn16}
\end{equation}
\noindent
and the depth of the potential $V_p(E) $ is given by
\begin{equation}
V_p(E) = v^p_1[1-v^p_2(E-E^p_f)+v^p_3(E-E^p_f)^2-v^p_4(E-E^p_f)^3]+\Delta V_C(E)
\label{seqn17}
\end{equation}
\noindent
where the Coulomb correction term $\Delta V_C(E)$ is given by
\begin{equation}
\Delta V_C(E) = \bar V_C v^p_1 [v^p_2-2v^p_3(E-E^p_f)+3v^p_4(E-E^p_f)^2]
\label{seqn18}
\end{equation}
\noindent
with $v^p_1=59.3+21.0(A_d-2Z_d)/A_d-0.024A_d$, $v^p_2=0.007067+4.23\times10^{-6}A_d$, $v^p_3=1.729\times10^{-5}+1.136\times10^{-8}A_d$, $v^p_4=7\times10^{-9}$, $E^p_f=-8.4075+0.01378A_d$, $ \bar V_C=1.73Z_d/[r_cA_d^{1/3}]$, $r_c=1.198+0.697A_d^{-2/3}+12.994A_d^{-5/3}$. The lab energy $E=Q$ for the proton decay process. This GOMP $V_{GOMP}(R)$ in place of $V_N(R)$ of eqn.(5) along with the centrifugal and Coulomb potentials, with $R_c$ of eqn.(6) taken equal to $[r_cA_d^{1/3}]$ for the Coulomb potential, have been used to evaluate the action integral. Results of these calculations have been presented in Table-II.
The isovector or the symmetry component of the DDM3Y folded potential $V^{Lane}_N(R)$ \cite{r16} has been added to the isoscalar part of the folded potential whose results have already been presented in Table-I. The nuclear potential $V_N(R)$ of eqn.(5), therefore, has been replaced by $V_N(R)+V^{Lane}_N(R)$ \cite{r17} where
\begin{equation}
V^{Lane}_N(R) = \int \int [\rho_{1n}(\vec{r_1})-\rho_{1p}(\vec{r_1})] [\rho_{2n}(\vec{r_2})-\rho_{2p}(\vec{r_2})] v_1[|\vec{r_2} - \vec{r_1} + \vec{R}|] d^3r_1 d^3r_2
\label{seqn19}
\end{equation}
\noindent
where the subscripts 1 and 2 denote the daughter and the emitted nuclei respectively while the subscripts n and p denote neutron and proton densities respectively. With simple assumption that $\rho_{1p}=[\frac{Z_d}{A_d}]\rho$ and $\rho_{1n}=[\frac{(A_d-Z_d)}{A_d}]\rho$, and for the emitted particle being proton $\rho_{2n}(\vec{r_2})- \rho_{2p} (\vec{r_2})=-\rho_2(\vec{r_2})=-\delta(\vec{r_2})$, the Lane potential becomes
$ V^{Lane}_N(R) = -[\frac{(A_d-2Z_d)}{A_d}] \int \rho (\vec{r}) v_1 [|\vec{r} - \vec{R}|] d^3r $ where $v_1(s)=t^{\rm M3Y}_1(s,E)g(\rho,E)$ and for the isovector part $t^{\rm M3Y}_1$ \cite{r6} is given by
\begin{equation}
t^{\rm M3Y}_1 = -[4886 \frac{e^{ - 4s}}{4s} - 1176\frac{e^{- 2.5s}}{2.5s}] + 228 (1 - 0.005 Q.m/\mu ) \delta(s).
\label{seqn16}
\end{equation}
\noindent
The inclusion of this Lane potential causes insignificant changes in the lifetimes as can be seen from Table-II. Although the lifetimes obtained using GOMP are rather close to that using isoscalar folded potentials with isovector Lane potentials (FMPL) but the GOMP and FMPL are quite different at 1st and 2nd turning points while at 3rd turning points only Coulomb potentials and centrifugal barriers are effective and nuclear potentials are negligibly small.
\begin{table}
\caption{Comparison between theoretically calculated half-lives of spherical proton emitters using the GOMP [15] and FMPL respectively. The asterisk symbol (*) denotes the isomeric state. Experimental $Q$ values and $l$ values used are taken from reference [4]. Errors in calculated half-lives arising out of experimental errors in $Q$ [14] values are given within parentheses. The overall normalization constant C=2.07 is not included in FMPL listed below at the turnings points and they should be multiplied by C to obtain their values used in the calculations or comparing them with the GOMP. }
\begin{tabular}{ccccccccccccc}
Parent &1st &Nuclear& 2nd &Nuclear& 3rd & GOMP &1st &Nuclear&2nd &Nuclear&3rd & FMPL \\
&turning&GOMP&turning&GOMP&turning&&turning&FMPL&turning&FMPL&turning&\\
nuclei&point(fm) &at $R_1$& point(fm) &at $R_2$& point(fm) & &point(fm) &at $R_1$& point(fm) &at $R_2$& point(fm) & \\ \hline
&$R_1$&MeV&$R_2=R_a$&MeV&$R_3=R_b$&$log_{10}T(s)$&$R_1$&MeV&$R_2=R_a$&MeV&$R_3=R_b$&$log_{10}T(s)$ \\ \hline
$^{105}Sb$&1.72&-60.5&6.58&-13.2&134.30&1.97(45)&1.52&-36.1&6.61&-6.4&134.30&1.95(46)\\
$^{145}Tm$&3.89&-59.9&6.56&-27.4&56.27&-5.23(6)&3.43&-35.7&6.47&-13.5&56.27&-5.18(6)\\
$^{147}Tm$&3.90&-60.5&6.60&-27.7&88.65&0.86(3)&3.41&-36.1&6.46&-14.0&88.65&0.94(4)\\
$^{147}Tm^*$&1.77&-61.6&7.25&-14.2&78.97&-3.44(5)&1.54&-36.5&7.19&-7.1&78.97&-3.41(5)\\
$^{150}Lu$&3.93&-60.4&6.64&-27.7&78.23&-0.70(4)&3.44&-35.9&6.51&-13.9&78.23&-0.63(4)\\
$^{150}Lu^*$&1.78&-61.5&7.27&-14.7&71.79&-4.43(14)&1.55&-36.3&7.24&-7.0&71.79&-4.40(15)\\
$^{151}Lu$&3.91&-60.6&6.66&-27.7&78.41&-0.78(3)&3.44&-36.0&6.52&-13.9&78.41&-0.70(3)\\
$^{151}Lu^*$&1.79&-61.6&7.29&-14.7&69.63&-4.93(9)&1.56&-36.5&7.25&-7.0&69.63&-4.90(9)\\
$^{155}Ta$&3.91&-60.5&6.75&-26.9&57.83&-4.77(6)&3.44&-36.0&6.62&-13.5&57.83&-4.70(6)\\
$^{156}Ta$&1.81&-61.9&7.33&-15.2&94.18&-0.44(7)&1.57&-36.7&7.27&-7.5&94.18&-0.41(7)\\
$^{156}Ta^*$&3.92&-60.9&6.73&-27.9&90.30&1.53(10)&3.45&-36.2&6.60&-14.0&90.30&1.61(10)\\
$^{157}Ta$&0.00&-62.1&7.52&-12.5&98.95&-0.49(11)&0.00&-35.8&7.48&-6.0&98.95&-0.46(11)\\
$^{160}Re$&1.79&-61.8&7.40&-15.1&77.67&-3.06(6)&1.59&-36.8&7.33&-7.4&77.67&-3.02(6)\\
$^{161}Re$&0.00&-62.0&7.58&-12.4&79.33&-3.52(7)&0.00&-35.8&7.51&-6.1&79.33&-3.48(7)\\
$^{161}Re^*$&3.93&-61.0&6.84&-27.1&77.47&-0.73(7)&3.45&-36.3&6.70&-13.6&77.47&-0.64(7)\\
$^{164}Ir$&3.95&-60.8&6.88&-27.0&59.97&-4.06(5)&3.44&-36.2&6.74&-13.5&59.97&-3.97(6)\\
$^{165}Ir^*$&3.93&-61.0&6.89&-27.1&62.35&-3.65(5)&3.45&-36.3&6.76&-13.5&62.35&-3.56(5)\\
$^{166}Ir$&1.81&-62.1&7.49&-15.1&87.51&-1.18(10)&1.57&-36.8&7.39&-7.6&87.51&-1.13(10)\\
$^{166}Ir^*$&3.93&-61.3&6.91&-27.2&80.67&0.07(8)&3.45&-36.5&6.77&-13.6&80.67&0.16(8)\\
$^{167}Ir$&0.00&-62.3&7.64&-12.8&91.08&-1.34(8)&0.00&-36.0&7.57&-6.3&91.08&-1.30(8)\\
$^{167}Ir^*$&3.94&-61.4&6.92&-27.3&83.82&0.55(8)&3.43&-36.7&6.79&-13.6&83.82&0.64(8)\\
$^{171}Au$&0.00&-62.2&7.70&-12.7&69.09&-5.08(15)&0.00&-35.9&7.63&-6.2&69.09&-5.04(15)\\
$^{171}Au^*$&3.94&-61.3&7.01&-26.4&64.25&-3.18(4)&3.46&-36.5&6.87&-13.2&64.25&-3.09(5)\\
$^{177}Tl$&0.00&-62.5&7.76&-13.2&88.25&-1.44(25)&0.00&-36.1&7.69&-6.4&88.25&-1.39(26)\\
$^{177}Tl^*$&3.92&-61.5&7.10&-26.5&57.43&-4.63(6)&3.43&-36.8&6.96&-13.2&57.43&-4.54(6)\\
$^{185}Bi$&0.00&-62.7&7.88&-13.1&65.71&-5.52(13)&0.00&-36.3&7.81&-6.3&65.71&-5.47(13)\\
\end{tabular}
\end{table}
\section{SUMMARY AND CONCLUSIONS}
\label{section5}
The half lives for proton-radioactivity have been analyzed with microscopic nuclear potentials obtained by the single folding the DDM3Y effective interaction whose energy dependence parameters have been obtained from nuclear matter calculations. This procedure of obtaining nuclear interaction potentials are based on profound theoretical basis. The results of the present calculations are in good agreement over a wide range of experimental data. It is worthwhile to mention that using the realistic microscopic nuclear interaction potentials, the results obtained for the proton radioactivity lifetimes are noteworthy and are comparable to the best available theoretical calculations. It is therefore observed that the DDM3Y effective interaction provides unified descriptions of cluster radioactivity \cite{r18}, scatterings of $\alpha$ and heavy ions \cite{r11} when used in a double folding model, and nuclear matter \cite{r7,r12} and elastic and inelastic scattering of protons \cite{r8} when used in a single folding model. We find that it also provides reasonably good description of proton radioactivity.
|
{
"timestamp": "2005-11-17T07:06:55",
"yymm": "0503",
"arxiv_id": "nucl-th/0503007",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503007"
}
|
\section{Introduction}
\indent \hspace{7mm}
Random variables having orders-of-magnitude large values, but with
correspondingly orders-of-magnitude small probabilities for their
occurrence, are known to give non-Gaussian statistics for their
fluctuations - the L\'{e}vy statistics \cite{Paul}. For
these {\it larger-than-rare} events the variance diverges, and a single
large event may typically dominate the sum of large number of such
random events. Many physical examples of
L\'{e}vy statistics, or the L\'{e}vy flights, are realized in
nature, for example, strange kinetics \cite{Shlesinger}, anomalous diffusion
in living polymers \cite{Ott}, subrecoil laser
cooling \cite{BardouPRL}, rotating fluid flow \cite{Swinney} and
interstellar scintillations \cite{Boldyrev}.
In the present work a Random Amplifying Medium (RAM) is shown
to provide yet another
example of L\'{e}vy statistics of some physical interest. In a RAM,
light scattering, which is usually
considered detrimental to laser action, can infact, lead to enhanced
amplification and hence to lasing. Here we report
some analytical and experimental results on the anomalous fluctuations
of emission from a RAM pumped beyond a threshold of gain.
More specifically, we show that in the classical diffusive regime,
as obtaining in our systems, there is a crossover from the Gaussian
to the L\'{e}vy statistics for the emission intensity over the
ensemble of realizations of the random medium. Also, the
associated L\'{e}vy exponent decreases with the increasing gain. An
interesting finding is that the L\'{e}vy-statistical
fluctuations are enhanced by embedding an amplifying fiber into the
particulate RAM. We also briefly discuss the nature of these
fluctuations as distinct from those observed in transport
through passive random media.
A RAM normally consists of an active
bulk medium, like an optically pumped laser dye solution (for example
Rhodamine in methanol) in which point-like (particulate) scatterers
(rutile (TiO$_{2}$) or polystyrene microspheres) are randomly
suspended [7-17].
Unlike the case of a conventional laser with external cavity mirrors
providing resonant feedback, in a RAM it is the multiple scattering
of light that provides a non-resonant distributed feedback (Fig 1), and hence
the mirrorless lasing. The enhanced path-lengths within the
random medium may arise due to classical diffusion resulting from
incoherent scattering (dilute suspension of scatterers in a dye) [8-12,16],
or due to incipient wave localization with strong coherent
scattering (for example semiconductor powder ZnO, GaN) [13-15].
In general, in a RAM operating in the incoherent diffusive regime,
greater the refractive-index mismatch, greater is the
diffusive path-length enhancement, and hence greater the amplification.
A clear signature of lasing in a RAM is the drastic spectral narrowing
(gain narrowing) of the emission from the system above a well defined
threshold of pump power. In the dye-scatterer system, the threshold
of the pump power, at which the emission linewidth collapses from a
few tens of nanometer to a few nanometers, is almost two orders of
magnitude smaller in the system with scatterers than the one
without. Further, with the increase in the scatterer concentration, both,
the linewidth and the pumping threshold are observed to decrease
drastically. The selection of the lasing wavelength, however, arises
here as a result of an optimization involving, for example, the
wavelength-dependent diffusion coefficient (or the localization
length scale) and the spectral profile of the pumped dye.
An important aspect of the random lasing is that for a high gain
(pumping) the randomness of the amplifying medium makes the emission
fluctuate strongly over the different microscopic realizations
(complexions) of the randomness of the medium. This shows up as non-self
averaging fluctuations of the observed lasing intensity as the medium
is varied over its random realizations, for example, by tapping the
cuvette containing the RAM.
(This, of course, is quite different from the inherent photon
statistics of fluctuations in time observed for a given complexion
\cite{Zacharakis}). Normally, i.e., for passive random media,
these ``sample-to-sample'' fluctuations are Gaussian in nature.
In this work we will be concerned with fluctuations in a RAM only.
To fix the idea, consider a RAM with the scatterers dispersed densely
and randomly in an amplifying continuum. A spontaneously emitted photon
is expected to diffuse with a diffusion constant $D = (1/3) c\ell$,
where $\ell$ is the elastic mean free path and $c$ is the speed of light in the
medium. We assume classical diffusion as $\ell/\lambda \gg 1$ in our
case, where, $\lambda$ is the optical wavelength. As the photon diffuses
and eventually escapes, it undergoes
amplification, or gain (multiplication) due to the optical pumping, and
the associated stimulated emission, which, of course, does not affect $D$.
Assuming for simplicity, a spherical RAM (radius `$a$'),
illuminated uniformly by a short pump-pulse at time $t = 0$, the
probability of escape of a photon from the surface at $r = a$, per unit time
at time $t$ is given by (the first-passage probability density)
\begin{equation}
p_I(t) = - \frac{\partial}{\partial t} \int_0^a \rho(r,t) 4\pi r^2 dr ~,
\end{equation}
where, $\rho(r,t)$ is the probability density of the diffusing photon,
emitted spontaneously at time $t=0$ anywhere within the sample with a uniform
initial probability density ($\rho_0$). Simple solution for the diffusion
problem (with the absorbing boundary condition at $r = a$) gives
\begin{equation}
\rho(r,t) = \rho_0 \sum_{m=1}^{\infty} \big(\frac{2a}{\pi m}\big ) (-1)^{m+1}
\cdot \frac{sin (\pi m r/a)}{r} e^{-\frac{\pi^2m^2}{a^2}Dt}
\end{equation}
giving straightforwardly
\begin{equation}
p_I(t) = \rho_0 \sum_{m=1}^{\infty} 8aD e^{-\frac{\pi^2m^2}{a^2}Dt}
\end{equation}
Now, the arc path-length traversed in the diffusion time $t$
is $ct$ giving a gain factor $g = e^{ct/\ell_g}$, where $\ell_g$ is
the gain length for the RAM. This at once gives, with change of
variable, the probability distribution for the gain $p_g(g)$ as
\begin{equation}
p_g(g) = \sum_{m=1}^{\infty} \big(\frac{\rho_0 8 a D \ell_g}{c}\big )
\frac{1}{g^{1+\alpha_m}} \equiv \sum_{m=1}^{\infty}
\big(\frac{8 \rho_0}{3}\big )
(a\ell \ell_g) \frac{1}{g^{1+\alpha_m}}
\end{equation}
with $\alpha_m = m^2 \big(\frac{\pi^2\ell\ell_g}{a^2}\big )$
$\equiv $ the m$^{th}$ L\'{e}vy exponent.
Thus, with increasing pumping (decreasing gain length $\ell_g$), the
exponent $\alpha_m$ decreases, the tail becomes fatter, and the
variance of $g$ diverges for $\alpha_m < 2$, that happens first
for $m=1$, i.e., for $(\frac{\pi^2\ell \ell_g}{a^2}) < 2$.
This leads to the crossover from a finite variance (Gaussian) to a
divergent variance (L\'{e}vy) limit. This essentially describes the onset
of L\'{e}vy fluctuations as we increase optical pumping. It is
idealized in that only the photons spontaneously emitted at time
$t=0$ are considered. These are amplified most anyway, and dominate
the intensity at time $t$ observed, for large gains (high pump powers).
Further, in our granular random media with grain size $\gg \lambda$,
the random scattering is best described as random refractions at the
interfaces. This can give rise to random closed loops that can trap
and enhance light as in a resonance. Also, inasmuch as the escape
rate is linked to the diffusion constant, one can expect the classical
Ruelle-Pollicott resonances giving pronounced structure to the
fluctuation statistics. We have not addressed these issues here.
Before we proceed further (with experiments), let us clarify the
meaning of {\it ``fluctuations''} once more in our context.
These are statistical
fluctuations over the ensemble of realizations of the randomness (i.e.,
macroscopically identical RAMs).
Of course, we can invoke physically the idea of ergodicity and identify
these fluctuations as unfolding in different parametric contexts. Statistical
fluctuations of transmission/conductance though passive random media are,
of course, well known \cite{Kumar}, where, for a macroscopic sample,
the classical fluctuations are small relative to the wave-mechanical
(or quantum)
fluctuations due to coherent interference effects. In the present case of
strictly classical diffusion ($\ell \gg \lambda$), the anomalously
large fluctuations are due entirely to the amplification inherent to a RAM.
The system that we have experimentally studied is a novel RAM, which
we term the F-RAM (Fiber-Random Amplifying Medium), inasmuch
as the active medium is a random aggregation of segments of dye-doped
amplifying (one-dimensional) fibers (Bicron, red
fluorescent optical fiber) in a passive
medium of air, granular starch etc. (Fig 2). These plastic
fibers fluoresce in the
orange-red when pumped by green light that enters the fibers through
their cylindrical surfaces anywhere along their lengths. The emitted
fluorescent light is mainly guided along the length, and it emerges
from either end amplified by a factor that increases exponentially
with the length
of travel through the fiber. While the random aggregation of the
amplifying fibers itself provides some scattering, the latter
was enhanced in our experiments by the addition of passive scatterers
like non-active fiber pieces or granular starch. Thus, the diffusion
proceeds via random scattering and wave-guidance.
Our initial experiments studied the emission from an F-RAM,
made of amplifying fibers crushed to sub-millimeter sizes, both with
and without long pieces of amplifying fibers embedded in it.
Additionally, these were compared with an
F-RAM consisting of long pieces of amplifying fibers embedded in a passive
scattering medium. These experiments and the observations are described
in section 2. The Arrhenius cascade model as also the L\'{e}vy
microscope \cite{BardouArxiv}, to which the observed
statistics of
intensity fluctuations bear relevance, is described in section 3.
The experimental realization of L\'{e}vy lasers, i.e., F-RAMs with tailored
length distribution, exhibiting the sample-to-sample L\'{e}vy intensity
fluctuations, in the dilute and the dense limits, is described in
section 4. Section 5 concludes the work.
\section{Experiments in F-RAM}
\indent \hspace{7mm}
An F-RAM consisting of amplifying fibers crushed to
sub-millimeter sizes
(which serve both to amplify and scatter the light),
was contained in a glass cuvette of size 1 cm$\times$1 cm
$\times$5 cm. This was pumped by 10 ns, 26 mJ pulses at 532 nm
from a frequency doubled Nd:YAG laser (Spectra Physics). Part of the
pump beam was split off by a beam-splitter to monitor the
pump intensity that was maintained constant. The emission from the
sample was collected transverse to the pump beam and the spectrum
analyzed on a PC based spectrometer (Ocean Optics). The schematic
of the experimental set-up is shown in Fig 3. Lasing action
was seen from this system above a pump threshold of 22 mJ (Fig 4).
The complexion of the system was altered i.e. the sample was
agitated, so that different random configurations were obtained, and
the resulting emission spectra were recorded. In order to obtain good
statistics, this was repeated till the emission spectra for 420
different complexions of the sample were obtained.
A histogram was then constructed; that is
the probability $P(I)$ \footnote {$P(I)$ is the number of times
an intensity was recorded normalized to
the total number of spectra.} of obtaining intensity $I$
was plotted as a
function of the intensity. The histograms shown
for $\lambda = 620~nm$ (emission peak) and $\lambda = 590~nm$ (off-peak)
are both observed to be Gaussian (Fig 5(a,b)). The intensity
as a function of complexion for these wavelengths
show small fluctuations (Fig 5(c,d)).
Ten long pieces (length 6 mm) of amplifying
fibers were then added to the above F-RAM, and in a similar fashion the spectra
for 420 different complexions of the sample recorded. A typical spectrum
of this F-RAM is shown in Fig 6. Unlike the earlier case, the histogram at
$\lambda = 640~nm$ (peak) shows a marked departure
from the Gaussian in the form of a long fat tail (Fig 7(a)).
In addition, the intensity as a function of
complexion showed sudden large fluctuations (Fig 7(c)). In contrast, at
$\lambda = 590~nm$ (off-peak), the intensity fluctuations
remained small (Fig 7(d)) and the histogram Gaussian (Fig 7(b)).
The departure from the normally observed Gaussian statistics and
the sudden large intensity fluctuations at the peak emission
wavelength ($640~nm$) can be explained as
arising from the few long pieces of amplifying fiber, that,
in some complexions of the sample, provide large
gain resulting in the fat tail.
This was verified by studying another system that consisted of a passive
scattering bulk medium (white fiber pieces, length $\sim$ 1 mm), in which
five pieces of amplifying fiber (length 6 mm) were embedded,
at pump energy of $\sim$ 12 mJ. The presence of
the pieces of amplifying fiber, though not visually apparent, is evident
from the intensity statistics of the emitted spectra
as a long tail in the histogram at $\lambda = 640~nm$
(Fig 8(a)) and corresponding large intensity fluctuations over
different complexions (Fig 8(c)).
On the other hand, the histogram and the intensity fluctuations
at $\lambda = 590~nm$ (Figs 8(b,d))
show Gaussian statistics. It is thus clear that
a few long pieces of amplifying fiber
dominate the emission by their large, but rare, amplification
so much so that the presence of a few long amplifying pieces
hidden inside a bulk aggregate of small pieces (active or passive)
can be inferred from the sample-to-sample fluctuations in the emission
from the system. This feature may be used to probe a
relatively long piece of amplifying fiber hidden inside a RAM thus L\'{e}vy
microscope \footnote{The term ``L\'{e}vy microscope'' will become
clearer after section 3}.
\section{The Arrhenius cascade}
\indent \hspace{7mm}
As the above experiments on F-RAMs indicate that a few large events
dominate the emission statistics, we are led to the related problem
of the Arrhenius cascade, which we discuss in brief. The Arrhenius
cascade studies the time of descent of a particle down an incline
that has a series of potential wells of varying random depths, $U$,
occurring with probability
${p_{U}(U) = {\frac{1}{U_o}}~exp({\frac{-U}{U_o}})}$
(${U_{o}}$ is the mean depth) (Fig 9(a): dotted). In a
well of depth $U$, the particle spends a time $t$, with
${\tau = {\frac{t}{t_o}} = {exp({\frac{U}{kT}})}}$
(Fig 9(a): solid). Thus, though deep wells are exponentially improbable,
their presence increases the residence time exponentially.
It can be shown that in the asymptotic limit, the total time of
descent follows the power law
${p_{\tau}(\tau) \sim \tau^{(-1-\alpha)}}$ where,
${{\alpha} = {\frac{kT}{U_o}}}$.
For high temperature ($T$), or for $\alpha \geq 2$, the particle has
a fast descent and the resulting distribution ${p_{\tau}(\tau)}$
is Gaussian. For $0 < \alpha < 2$, corresponding to intermediate
or low temperatures, the distribution is L\'{e}vy (Fig 9(b)) and
the Central Limit Theorem is violated.
To exploit the fact that two functions, one exponentially
increasing and the other exponentially falling, can combine to
give rise to Gaussian or L\'{e}vy statistics depending on the
relative values of the two exponents, we tailored our F-RAM
system, such that the probability distribution of the lengths of the fibers
was ${p_{\ell}(\ell) = {\frac{1}{\ell_o}}~exp({\frac{-\ell}{\ell_o}})}$,
as shown in Fig 10. (Note that this tailored F-RAM is different from
those described in section 2 where all long amplifying fiber pieces
were of same length). The amplification within an active fiber results
in an intensity ${I(\ell) = I_{o}~exp({\frac{\ell}{\ell_g}})}$, or gain
${g_\ell = \frac{I(\ell)}{I_o} = exp({\frac{\ell}{\ell_g}})}$.
Thus, long fibers, though exponentially
rare, provide exponentially high gain \footnote{Note that the parameters
${\ell_{o}}$ and ${\ell_{g}}$ in the tailored
F-RAM correspond to ${U_{o}}$ and $kT$ respectively
in the Arrhenius cascade}.
It can be shown that the probability
distribution of the resultant gain acquired by the photon is given as
${p_g(g) \sim g^{(-1-\nu)}}$ where,
${{\nu} = {\frac{\ell_g}{\ell_o}}}$. It is thus expected
that $0< \nu < 2$
gives L\'{e}vy intensity statistics and $\nu \geq 2$
Gaussian. We demonstrate experimentally, in the next section, the
crossover from Gaussian to L\'{e}vy as $\ell_g$ is reduced.
\section{Experiments with tailored F-RAM (L\'{e}vy Laser)}
\indent \hspace{7mm}
Experiments were conducted on tailored F-RAMs with N pieces
(N = 350, 800) of amplifying fibers in passive
scattering media provided by suspension of polystyrene microspheres
in water (BangsLabs, mean diameter = 0.13 $\mu m$, number density =
$9.357 \times 10^{12}/cc$), granular starch or pieces of white optical
fiber (non-amplifying, length $\sim$ 0.5 mm to 1 mm). In all
three systems (contained in glass cuvettes of size 30 mm$\times$ 30 mm
$\times$ 60 mm) were studied in which, the lengths of the
amplifying fibers ranged from 1 mm to 20 mm and followed an exponential
distribution with $\ell_{o}$ = 5 mm.
As described in section 2, spectra (at pump energies $\sim$ 6-9 mJ)
for $\sim$ 360 different complexions of
each of the systems were obtained and analyzed. The intensity fluctuations and
the corresponding histograms are given in Figs 11 to 13.
These are shown for $\lambda = 645~nm$ and $590~nm$,
the former corresponding to the peak emission wavelength, where the gain is
maximum ($\ell_g$ is minimum), and the latter to off-peak
wavelength ($\ell_g$ is large). The histograms of all three systems
show a L\'{e}vy-like
fat tail at the peak emission wavelength; therefore these tailored
F-RAMs are termed L\'{e}vy lasers.
In contrast, at off-peak wavelengths, the histograms show Gaussian statistics
consistent with the larger value of $\nu$.
The intensity at the peak wavelength ($645~nm$) as a function
of complexion showed sudden large jumps, typical of L\'{e}vy flights.
This feature was absent at off-peak wavelengths.
We now distinguish between the ``dilute'' and the ``dense'' limits
of the L\'{e}vy Laser. The dilute L\'{e}vy Laser contains a few
pieces of amplifying fibers. A photon originating within a
given piece of amplifying fiber gains in intensity as it traverses
the fiber. Upon exiting the fiber, it diffuses through the passive surrounding
medium and exits the sample with a negligible probability of
encountering another amplifying fiber (Fig 14(a)). The intensity
collected in the experiment is the sum of various such
intensities - the {\it additive gain}. As discussed earlier, it gives
a power-law for the gain i.e., ${p_g(g) \sim g^{-1-\nu}}$.
Of the systems studied, the case
with N = 350 amplifying fibers in polystyrene
scattering medium corresponds to a dilute system.
The tail of the histogram can be fitted to a power law function
($g^{-1 -\nu}$) with exponent 1 + $\nu$ = 2.69 i.e., $\nu$ = 1.69.
In the dense L\'{e}vy laser, on the other hand, a photon, upon
exiting an amplifying
fiber, has a high probability of entering another amplifying fiber and
getting further amplified before finally exiting the sample (Fig 14(b)).
In such a case, the total intensity (or gain) is {\it multiplicative}
rather than {\it additive}
i.e. ${G = \Pi_{i}~ g_{\ell_i} = \Pi_{i}~exp({\frac{\ell_i}
{\ell_g}})}$,
where, the index, i, runs over all fibers that a given photon traverses
through, from which we get ${p_x(x) \sim g^{-\nu}}$
where, ${x = ln~g_{\ell_i}}$.
Thus, the dense system with multiplicative gain also gives rise to
a L\'{e}vy distribution, but with a tail that falls off slower than
the dilute system.
Cases with N = 800 amplifying fibers in passive scattering media
are realizations of dense L\'{e}vy lasers.
The tails of the histograms can be fitted to the power
law function ($g^{-\nu}$) with exponents $\nu$ = 0.62 and 1.68,
for systems with passive scattering medium as
non-active white fiber pieces and granular starch respectively.
\section{Conclusions}
\indent \hspace{7mm}
We have demonstrated a new RAM, namely the F-RAM, that is notably
different from a conventional RAM in several aspects.
As opposed to the RAM that has a bulk active medium
(dye solution) with suspended passive point-like scatterers,
an F-RAM, has an active medium that is one-dimensional
(pieces of amplifying fiber) and is suspended in the passive bulk
medium. Further, unlike the conventional RAM,
during its traversal through the passive bulk medium in an F-RAM
the photon does not get amplified.
Consequently, a greater refractive index mismatch between the active
(fiber) and the passive (bulk) media,
which in the case of RAM leads to greater amplification
due to increased path-length, is likely to
result under some conditions in just the opposite in an F-RAM,
as it enhances scattering
off the active fiber. We term an F-RAM with a tailored distribution of
fiber lengths, where long amplifying pieces are exponentially rare, a
``L\'{e}vy Laser'', because the sample-to-sample intensity fluctuations
exhibit L\'{e}vy statistics. The ``larger than rare'' amplification in
such systems
makes feasible a ``L\'{e}vy microscope'' that can pick out the presence
of, and study the characteristics of a long piece of amplifying
fiber embedded in a bulk of smaller (active or passive) pieces.
|
{
"timestamp": "2005-03-08T12:47:39",
"yymm": "0503",
"arxiv_id": "physics/0503059",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503059"
}
|
\section{Introduction}
A nearest particle system
on $S= \{1,2,\cdots,N\}$ is a continuous time Markov chain with
the state space $\{ A : A \subset S \}$. The jump rates are
specified as follows:
$$
\begin{array}{ll}
q (A, A \setminus \{x\}) = 1 & {\rm if} \ x \in A; \\
q(A, A \cup \{x\}) = \beta(l_x(A), r_x(A)) & {\rm if} \ x \in S
\setminus A; \\
q (A, B) = 0 & {\rm otherwise}.
\end{array}
$$
Here $l_x(A)$ and $r_x(A)$ are the distances from $x$ to the
nearest points in $A$ to the left and right respectively, with
convention that $l_x(A)$ (or $r_x(A)$) is $\infty$ if $y> x$ (or
$y< x$,
respectively) for all $y\in A$. We assume that\\
1. $\beta (l, r) = \beta (r, l)$;\\
2. $\beta (l,r)$ is decreasing in $l$ and in $r$;\\
3. $\beta(\infty, \infty) = 0, \beta(l,\infty) > 0$;\\
4. $\sum_l \beta(l, \infty) < \infty$.
There are many choices of $\beta(\cdot, \cdot)$ satisfying the
above assumptions. \\
{\bf Example 1}. (The 1-dim contact process) $\beta (1,1) = 2
\lambda$, $\beta(1,r) = \beta(l,1) = \lambda$ for $l,r >1$, and
$\beta (l,r)
= 0$ otherwise. \\
{\bf Example 2}. (The uniform birth rate) $\beta(l, r) = \lambda/(l+r-1)$.\\
{\bf Example 3}. (The reversible case)
$$
\beta(l,r) = \lambda \frac {\psi(l)\psi(r)}{\psi(l+r)}, \ \ \
\beta(l,\infty) = \beta(\infty,l) = \lambda \psi(l),
$$
where
\begin{equation}\label{rev}
\psi(\cdot)> 0,\ \ \sum_{n=1}^\infty \psi(n) =1, \mbox{ and }
\frac { \psi(n)}{\psi(n+1)} \searrow 1. \end{equation}
Assume further that
\begin{equation}\label{momn}
\sum_n n^2 \psi(n) < \infty.
\end{equation}
For example, $\psi(n) = c n^{-\alpha}$ for
some $\alpha
> 3$ satisfies the above requirement.
It is helpful to associate a subset $A$ of $S$ with an element
$\xi$ of $\{0,1\}^S$ and use them interchangeably: $\xi(x) = 1$ if
and only if $x\in A$. Configuration $\xi$ will be given an
occupancy interpretation. We say there is a particle in $x$ if
$\xi(x)= 1$, and we say the site is vacant if $\xi(x) =0$. Then
the above transition mechanisms can be interpreted as follows:
Each particle disappears at rate 1 independently, and a particle
is born at vacant site $x$ at rate $\beta(l_x(A), r_x(A))$.
The transition mechanisms also make sense if we replace $\{1, 2,
\cdots, N\}$ with the integer lattice $\mathbb{Z}$. However, because of
Assumption 4, the
state space $\{0, 1\}^\mathbb{Z}$ consists of four disjoint parts: \\
1) all finite subsets of $\mathbb{Z}$;\\
2) all subsets of $\mathbb{Z}$ with infinite many particles both to the
left and to the right of the origin; \\
3) all infinite subsets of $\mathbb{Z}$ with finite many particles to the
right of the origin; and \\
4) all infinite subsets of $\mathbb{Z}$ with finite many particles to the
left of the origin.\\
The first two cases are extensively studied, and are
called {\it finite} and {\it infinite} nearest particle systems respectively. A comprehensive
account can be found in Chapter 7 of Liggett (1985).
The last two cases share many properties of the first two cases,
and are indispensable in some occasions, e.g., Lemma 4.1.
For interacting particle systems people are most concerned with
the existence of phase transition and the critical value. For the
infinite nearest particle system with the uniform birth rate
(Example 2), the critical value is 1, see Mountford (1992). For
the reversible nearest particle system (Example 3), the critical
value is also 1. For the contact process (Example 1), the critical
value is unknown but is between 1.5 and 2, and is denoted as
$\lambda_c$ throughout this paper.
Can the critical value of an infinite model be detected by the
counterpart on a finite interval? This interplay was first
explored for the contact process in a series paper by Durrett {\it
et al}. The main results are summerized as follows. Let $\{
\zeta^N_t : t \ge 0 \}$ be the contact process on $\{ 1, 2,
\cdots, N \}$ with the parameter $\lambda$ starting from all sites
occupied, and $\tau_N$ be the first time it hits the empty set.
\begin{thm}\label{c2}
{\rm (i)} If $\lambda < \lambda_c$, then there is a constant
$\gamma_1 (\lambda) > 0 $ so that as $N \rightarrow \infty$, $
\tau_N / \log N \rightarrow 1 / \gamma_1 (\lambda)$ in probability
{\rm (Durrett and Liu (1988), Theorem 1)}.
{\rm (ii)} If $\lambda > \lambda_c$, then there is a constant
$\gamma_2 (\lambda) > 0 $ so that as $N \rightarrow \infty$,
$(\log \tau_N) / N \rightarrow \gamma_2 (\lambda)$ in probability
{\rm (Durrett and Schonmann (1988), Theorem 2)}.
{\rm (iii)} If $\lambda = \lambda_c$ and $a,b \in (0, \infty)$,
then $P \( a N \le \tau_N \le b N^4 \) \rightarrow 1$ as $N
\rightarrow \infty$ {\rm (Durrett {\it et al} (1989), Theorem
1.6)}.
\end{thm}
We believe that these statements hold for a large class of
interacting particle systems. In this paper we like to study the
asymptotical behavior of the hitting time $\sigma_N$ of the
reversible nearest particle systems (Example 3) on a finite
interval, as the length of interval increases. The results read
as follows. Let $\{ C_N : N \ge 1 \}$ be any sequence of
increasing numbers such that $\lim_{N \rightarrow \infty} C_N =
\infty$.
\begin{thm} \label{shang2} Suppose the initial state is $\{1,2,
\cdots, N\}$.\\
(1) If $$\lambda < \min\{1, \ \min_n \frac
{\psi(n)}{\sum_{l+r=n}\psi(l) \psi(r)} \},$$ then $E\sigma_N \leq
C \log N$ for some constant $C$ which is independent of $N$, and
$$
\lim_{N \rightarrow \infty} P ( \sigma_N \leq C_N \log N ) = 1;
$$
(2) If $\lambda > 1$, then there is a constant $\gamma > 0$ such
that
$\lim_{N\rightarrow\infty} P \( \sigma_N \geq e^{\gamma N} \) = 1.$
\end{thm}
\noindent{\it Remarks}: It is not difficult to establish estimates
of the opposite direction, see Theorems \ref{shang} and
\ref{shang1}. Together we have shown that $\sigma_N$ increases
logarithmically if $\lambda$ is small enough and exponentially if
$\lambda > 1$.
For any non-empty set $A =\{x_1, x_2 \cdots,
x_k\}$, we assume without loss of generality that $x_1 < x_2 <
\cdots < x_k$ and define
$$
\nu_\psi(A) = \left\{ \begin{array}{ll}
\psi(x_2-x_1) \psi (x_3-x_2) \cdots \psi(x_k-x_{k-1}) & {\rm if} \
k > 1;
\\
1 & {\rm if} \ k = 1.
\end{array} \right.
$$
Let
$
{\cal S}_N = \{ 0,1 \}^{\{ 1, \cdots, N \}}$, $K_N = \sum_{A\in
{\cal S}_N \setminus \{ \emptyset \}} \nu_\psi (A)$, {\rm and}
$\pi (A) =\nu_\psi(A) /K_N.
$
Then $\pi$ is a probability measure on ${\cal S}_N$.
\begin{thm}\label{cth}
Suppose that $\lambda = 1$ and the initial distribution is $\pi$.
Then
$$
\lim_{N \rightarrow + \infty} P \( \frac N {C_N} \leq \sigma_N
\leq C_N N^2 \) = 1.
$$
\end{thm}
We now proceed to prove Theorems \ref{shang2} and \ref{cth} by
three different approaches.
\section{Comparison by Coupling }
We will prove the first part of Theorem \ref{shang2} by
establishing a more general conclusion (Theorem \ref{shang3}). Let
$\{X_t : t \ge 0\}$ be a birth and death process on $\{ 0, 1,
\cdots, N \}$ with
$$
\begin{array}{llll}
\mbox{ death rate}: \ & a_i = & i, \quad &\mbox{ for } i = 1,
\cdots,
N; \\
\mbox{ birth rate}: \ & b_i = & (i+1) \alpha ,\quad & \mbox{ for
} i = 0, \cdots, N-1.
\end{array}
$$
Let $\tau = \inf \{ t > 0 : X_t = 0 \}$ be the first time that
$\{X_t : t \ge 0\}$ hits $0$. Let $E^N$ be the conditional
expectation on $X_0= N$.
\begin{lem}\label{ex}
Suppose that $X_0= N$. For
large $N$,
$$
E^N \tau \leq \left\{ \begin{array}{ll}
(2\log N)/(1-\alpha) & {\rm if} \ \alpha < 1;
\\
2N \log N & {\rm if} \ \alpha = 1;
\\
\alpha^N \alpha /(\alpha-1)^2 & {\rm if} \ \alpha > 1.
\end{array} \right.
$$
Furthermore
\begin{equation} \label{zia}
E^N \tau^2 \leq 2 \( E^N \tau \)^2.
\end{equation}
\end{lem}
\noindent{\bf Proof}. Let $P^i$ be the conditional probability
distribution on the initial state $i$, $E^i$ be the expectation
with respect to $P^i$, and $m_i = E^i \tau$ for $i = 0, \cdots,
N$. It is shown in Wang (1980) that
$$
E^N \tau = \sum_{i=1}^N e_i, \ \ \ E^N \tau^2 = \sum_{i=1}^N
\varepsilon_i,
$$
where
\begin{eqnarray}
e_i
& = &
\frac{1}{a_i}+ \sum_{k=0}^{N-1-i}\frac{b_ib_{i+1} \cdots
b_{i+k}}{a_ia_{i+1} \cdots a_{i+k}a_{i+k+1}}\\
& = & \frac 1 i (1 + \alpha + \alpha^2 + \cdots \alpha^{N-i}). \label{E.ei} \\
\varepsilon_i
& = &
\frac{2 m_i}{a_i} + \sum_{k=0}^{N-1-i} \frac{2 b_i b_{i+1} \cdots
b_{i+k} m_{i+k+1} } {a_i a_{i+1} \cdots a_{i+k}
a_{i+k+1}}\nonumber
\end{eqnarray}
Notice that $ m_i \leq
m_N $ for any $i \leq N$. It follows that $ \varepsilon_i \le 2
m_N e_i$. Therefore,
$$
E^N \tau^2 = \sum_{i=1}^N \varepsilon_i \le 2 m_N \sum_{i=1}^N
e_i \leq 2 m_N E^N {\tau} = 2 \( E^N \tau \)^2.
$$
If $\alpha= 1$, by (\ref{E.ei}), $e_i = (N-i+1)/i$, and for large
$N$,
$$
E^N \tau = \sum_{i=1}^N e_i \le N \sum_{i=1}^N i^{-1} \le 2 N \log
N. $$
If $\alpha < 1$,
$
E^N \tau = \sum_{i=1}^N e_i \le ( 1- \alpha)^{-1} \sum_{i=1}^N
i^{-1} \leq (2 \log N)/( 1- \alpha ); $
If $\alpha > 1$,
then
$
E^N \tau = \sum_{i=1}^N e_i \le ( \alpha -1 )^{-1} \sum_{i=1}^N
\alpha^{N-i+1} \leq \alpha^{N+1} /( \alpha -1 )^2$.
\hfill $\Box$
Consider a nearest particle system $\{ \xi^N_t : t \ge 0 \}$ on
$\{1,2,\cdots,N\}$ starting from
$\{1,2,\cdots, N\}$ (not necessarily reversible). Let $\sigma_N$
be the first hitting time of the empty set by $\xi^N_t$, and
$$
M = \max \{\max_n \sum_{l+r = n} \beta(l, r),\quad \sum_l \beta(l,
\infty)\}.$$
\begin{thm} \label{shang3} Suppose the initial state is $\{1,2,
\cdots, N\}$. If $M< 1$, then $E \sigma_N \leq (2\log N)/(1-M)$; and
for any sequence $\{ C_N : N \ge 1 \}$ of increasing numbers such
that $\lim_{N \rightarrow \infty} C_N = \infty$, $ \lim_{N
\rightarrow \infty} P \big( \sigma_N \leq C_N log N\big) = 1$.
\end{thm}
\medskip
\noindent {\bf Proof.} Let $|A|$ be the cardinality of set $A$.
For any configuration $\xi$ such that $|\xi| = i$, there are at
most $i+1$ intervals of consecutive vacant sites, separated by
occupied sites; the rate that a new particle in each interval is
born is no more than $ M$. Hence the rate that $|\xi^N_t|$
increases by 1 is no more than $(i+1) M$. On the other hand, when
$|\xi_t| = i$, the rate that $| \xi_t |$ decreases by 1 is equal
to $i$, the total number of particles. Compare $|\xi_t|$ with a
birth and death process $X_t$ with parament $\alpha = M$. Since
initially $X_0 = |\xi^N|$, there is a coupling of $\{ X_t : t \ge
0 \}$ and $\{ \xi^N_t : t \ge 0 \}$ such that
\begin{equation} \label{control}
P^{N, \xi^N} \( X_t \geq |\xi^N_t|, \ \forall \ t \geq 0 \) = 1,
\end{equation}
where $P^{N,\xi^N}$ is the coupling measure with the initial state
$(N,\xi^N)$. By (\ref{control}), $\sigma_N$ is stochastically
dominated by $\tau$,
$i.e$., for any $t \ge 0$,
\begin{equation} \label{E.estimate1}
P (\sigma_N > t) \le P^N (\tau \geq t).
\end{equation}
By the Chebyshev inequality and (\ref{zia}), for any $c_N > 0$,
\begin{equation} \label{E.estimate2}
P (\sigma_N > c_N E^N \tau) \le P^N \left( \tau \geq c_N E^N \tau
\right)
\le \frac
{E^N \tau^2 } {\(c_N E^N \tau \)^2} \le \frac 2 {c_N^2}.
\end{equation}
For any sequence $C_N \rightarrow \infty$ as $N \rightarrow
\infty$, choose $c_N = C_N (1-M)/2$. Then an upper estimate of
$\sigma_N$ may be taken as $c_N E^N \tau$, and the claims in
Theorem \ref{shang3} hold by (\ref{E.estimate2}) and Lemma 2.1.
\hfill $\Box$
By the same argument it is not difficult to establish following
estimates, though a renormalization argument is used in the proof
of the second part of Theorem \ref{shang1}. We will skip the
proof, since they are not needed in proving Theorems \ref{shang2}
and \ref{cth}.
\begin{thm} \label{shang} Suppose the initial state is $\{1,2,
\cdots, N\}$. \\
(1) If $M =1$, then $E \sigma_N \leq N \log N$; and
$
\lim_{N \rightarrow \infty} P \big( \sigma_N \leq C_N N log N\big)
= 1; $\\
(2) If $ M >1$, then $E \sigma_N \leq M^{N+1}/(M-1)^2$; and there is a constant $\gamma_1 > 0$ such that
$$
\lim_{N\rightarrow\infty} P \( \sigma_N \leq e^{\gamma_1 N} \) =
1.
$$
\end{thm}
\begin{thm} \label{shang1} Suppose the initial state is $\{1,2,
\cdots, N\}$.\\
(1) For any $ \varepsilon > 0$, $ \lim_{N \rightarrow \infty} P
\big( \sigma_N > (1-\varepsilon) \log N \big) = 1$;\\
(2) If $\max_n \min\{ \frac 1 2 \sum_{l = n}^{2n} \beta(l, 3n-l),
\quad \sum_{l=n}^{2n} \beta(l, \infty)\}$
is larger than the critical value of the contact process on $\mathbb{Z}$,
then there is a constant $\gamma > 0$ such that
$$
\lim_{N\rightarrow\infty} P \( \sigma_N \geq e^{\gamma N} \) = 1.
$$
\end{thm}
\section{A Lower Estimate of $\sigma_N$}
We first extend the notation introduced before Theorem \ref{cth}.
For any non-empty set $A =\{x_1, x_2 \cdots, x_k\}$, $x_1 < x_2 <
\cdots < x_k$, define
$$
\nu_{\psi,\lambda}(A) = \left\{ \begin{array}{ll}
\lambda^{k-1}\psi(x_2-x_1) \psi (x_3-x_2) \cdots \psi(x_k-x_{k-1})
& {\rm if} \ k > 1;
\\
1 & {\rm if} \ k = 1.
\end{array} \right.
$$
Let $ {\cal S}_N = \{ 0,1 \}^{\{ 1, \cdots, N \}}$, $K_N (\lambda)
= \sum_{A\in {\cal S}_N \setminus \{ \emptyset \}}
\nu_{\psi,\lambda} (A)$, {\rm and} $\pi (A) =\nu_{\psi,\lambda}(A)
/K_N(\lambda)$. Then $\pi$ is a probability measure on ${\cal
S}_N$.
\begin{lem}\label{KN} $K_N (\lambda) \ge C N^2 e^{\gamma(\lambda)
N}$ for $\lambda \geq 1$, where $\gamma (1) = 0$ and $\gamma
(\lambda) > 0$ if $\lambda > 1$.
\end{lem}
\medskip\noindent {\bf Proof.}
\begin{equation} \label{E.Kn}
K_N (\lambda) = \sum_{\xi \in \mathcal{S}_N \setminus \{ \emptyset
\}} \nu_{\psi,\lambda} (\xi) \ge \sum_{x=0}^{[N/3]} \sum_{y= [2N /
3]}^N \sum_{\xi \in S_N (x, y)}\lambda^{|\xi|-1} \nu_\psi (\xi),
\end{equation}
where
$$ S_N (x, y) = \left\{ \xi\in {\cal S}_N : \xi (x) = \xi
(y) = 1, \xi (z)=0, \ \forall \ 1\leq z < x, {\rm \ or \ } y< z
\leq N \right\}.
$$
In light of (\ref{momn}), by the Renewal Theorem,
$\nu_\psi (S_N (x,y)) \ge 1/(2\sum_n n\psi(n))$ whenever $y-x$ is large enough.
If $\lambda =1$, then $K_N \ge C N^2$ when $N$ is large, and we are done.
If $\lambda > 1$, we can choose constant $\delta >0$ such that
$$\nu_\psi \big(\{\xi\in S_N (x,y); |\xi| \geq \delta |y-x|\}\big)\geq
\frac {\nu_\psi (S_N (x,y))} 2.$$
This together with (\ref{E.Kn}) implies the desired conclusion.
In particular we may choose $\gamma(\lambda) = (\delta/3)\log\lambda$. \hfill $\Box$
We now use an idea in proving Theorem 7.1.20 of Liggett (1985) to
prove
\begin{equation} \label{gg2}
\lim_{N\to\infty} P^\pi \( \sigma_N \ge \frac {K_N}{C_N N} \) =
1.
\end{equation}
The first half of Theorem \ref{cth} readily follows from
(\ref{gg2}) and Lemma \ref{KN}. Notice that the hitting time of
the nearest particle system starting from $\{1,2,\cdots, N\}$ is
stochastically larger than that starting from the initial
distribution $\pi$. Therefore the second part of Theorem
\ref{shang2} also follows, with a little change in $\gamma$.
\medskip\noindent {\bf Proof of (\ref{gg2}).}
The reversible nearest particle system $\{ \xi^N_t : t \ge 0 \}$
is a Markov process taking values in $\mathcal{S}_N$ with jump
rate
$$
q (A, B) = \left\{
\begin{array}{ll}
1 & {\rm if} \ x\in A,\ B = A\setminus \{x\};
\\
\lambda \frac{\psi(l_x (A))\psi(r_x (A))}{\psi(l_x (A)+ r_x (A))}
& {\rm if} \ x \notin A,\ B = A \cup \{x\};
\\
0 & {\rm otherwise.}
\end{array} \right.
$$
It is reversible with respect to $\pi$ in the sense that
$\pi(A)q(A, B) = \pi (B) q(B, A)$ for $A, B \in \mathcal{S}_N
\setminus \{ \emptyset \}$.
Let $\{ \widetilde{\xi^N_t} : t \ge 0 \}$ be a Markov process on
$\mathcal{S}_N$, which is a modification of $\{ \xi^N_t : t \ge 0
\}$ so that particles can be born from the empty set. More
specifically, the transition rates of $\{ \widetilde{\xi^N_t} : t
\ge 0 \}$ is defined as follows.
$$
\tilde{q} (A, B) = \left\{
\begin{array}{ll}
q (A, B) & {\rm if} \ A \neq \emptyset;
\\
q & {\rm if} \ A = \emptyset \ {\rm and} \ |B| = 1;
\\
0 & {\rm otherwise,}
\end{array} \right.
$$
where $q > 0$ is a constant to be determined later. Let $K_N$
stand for $K_N(\lambda)$,
$$
\nu_\psi \( \{\emptyset\} \) = q^{-1}, \ \ \mbox{and } \ \ \
\tilde\pi = \nu_\psi / \( K_N + q^{-1} \).$$
Then $\{ \widetilde{\xi^N_t} : t \ge 0 \}$ is reversible with
respect to $\tilde \pi$ in the sense that $\tilde \pi(A) q(A, B) =
\tilde \pi (B) q(B, A)$ for any $A, B\in \mathcal{S}_N$.
Let $\tilde{P}$ be the distribution of $\{ \widetilde{\xi^N_t} : t \ge
0 \}$ with initial distribution $\tilde \pi$, and $\tilde{E}$ be
the expectation with respect to $\tilde{P}$. Notice that $\{
\widetilde{\xi^N_t} : t \ge 0 \}$ is stationary under $\tilde{P}$.
For any $t > 0$,
$$
2 t \tilde{\pi} (\{\emptyset\})
=
\tilde{E} \int_0^{2t} 1_{ \{ \widetilde{\xi^N_s} = \emptyset \} }
ds.
$$
Introduce the stopping time
$
\tau = \inf \{ t \geq 0 : \widetilde{\xi^N_t} = \emptyset \}.
$
By the Strong Markovian Property, the right side above equals
\begin{eqnarray*}
& &
\tilde{E} \tilde{E} \( \left. \int_0^{2t}
1_{\{\widetilde{\xi^N_s} = \emptyset\}} d s \right|
\mathcal{F}_\tau \)
\ge
\tilde{E} \tilde{E} \( \left. 1_{\{\tau<t\}} \int_0^{2t}
1_{\{\widetilde{\xi^N_s} = \emptyset\}} d s \right|
\mathcal{F}_\tau \)
\\ & \ge &
\tilde{E} \tilde{E} \( 1_{\{ \tau < t \}} \left. \int_\tau^{\tau
+ t} 1_{\{\widetilde{\xi^N_s} = \emptyset\}} d s \right|
\mathcal{F}_\tau \)
=
\tilde{P} (\tau<t) \tilde{E} \( \left. \int_0^t
1_{\{\widetilde{\xi^N_s} = \emptyset\}} d s \right|
\widetilde{\xi^N_0} = \emptyset \)
\end{eqnarray*}
Denote by $\sigma$ the first time $\{\widetilde{\xi^N_t} : t \ge 0
\}$ jumps. Then
\begin{eqnarray*}
\tilde{E} \( \left. \int_0^t 1_{\{\widetilde{\xi^N_s} =
\emptyset\}} d s \right| \widetilde{\xi^N_0} = \emptyset \)
& \ge &
\tilde{E} \( \sigma 1_{\{ \sigma \le t \}} | \widetilde{\xi^N_0} =
\emptyset \)
=
\int_{0}^{t} s \tilde{q}_\emptyset e^{ - \tilde{q}_\emptyset s } d
s,
\end{eqnarray*}
where $\tilde{q}_\emptyset = \sum_{\xi} \tilde{q} (\emptyset, \xi)
= N q$. Hence
\begin{equation} \label{2t}
\tilde{P} (\tau<t) \le\frac { 2t \tilde{\pi} (\{\emptyset\})}{
\int_{0}^{t} s \tilde{q}_\emptyset e^{ - \tilde{q}_\emptyset s } d
s } = \frac{ 2 t q^{-1}}{ K_N + q^{-1}} \cdot \frac{Nq}{
1-e^{-Nqt} - Nqt e^{-Nqt}}.
\end{equation}
On the other hand,
\begin{eqnarray*}
\tilde{P} (\tau<t)
& \geq &
\tilde{P} \( \tau<t,\ \widetilde{\xi^N_0} \neq \emptyset \)
=
\tilde{P} \( \widetilde{\xi^N_0} \neq\emptyset \) \tilde{P} \(
\tau<t|\widetilde{\xi^N_0}\neq\emptyset \)
\\ & = &
\frac {K_N}{ K_N + q^{-1} } P (\sigma_N<t) .
\end{eqnarray*}
This together with $(\ref{2t})$ yields that
$$
P \( \sigma_N < t \) \le \frac {2 t N} {K_N \( 1-e^{-Nqt} - Nqt
e^{-Nqt} \)} \ .
$$
Let $q \rightarrow \infty$, then
$$
P \( \sigma_N < t \) \le \frac {2 t N } {K_N} \ .
$$
This implies (\ref{gg2}), by choosing $t = K_N/(C_N N)$. \hfill $\Box$
\section{The Critical Case}
In this section we will prove the second half of Theorem
\ref{cth}, $i.e.$, when $\lambda =1$,
\begin{equation} \label{gg1}
\lim_{N\to\infty} P^\pi \( \sigma_N \le C_N N^2 \) = 1.
\end{equation}
Let $\{\eta_t : t \ge 0 \}$ be an infinite reversible nearest
particle system on $\mathbb{Z}$ with finite many particles to the right of
the origin (The third case on page 2); and $r_t$ the rightmost
particle in $\{ \eta_t : t \ge 0 \}$, i.e.
$r_t : = \sup \{ x : \eta_t (x) = 1 \}$.
The properties of $r_t$ of the critical nearest particle system
are studied in Schinazi (1992). For a recent survey, see Mountford
(2003).
\begin{lem}\label{edge} {\rm (Schinazi (1992), Theorem 1)}
Let $\{\eta_t : t \ge 0 \}$
be the critical reversible nearest particle system on $\mathbb{Z}$.
Suppose the initial configurations have a particle at the origin
and no particle to the right of the origin, and follows the
renewal measure $Ren(\psi)$ with density $\psi(\cdot)$. Then, as
$a\rightarrow\infty$, $r_{a^2t}/a$ converges in distribution to a
Brownian motion with diffusion constant $D > 0$ in the Skorohod
space.
\end{lem}
\noindent {\bf Proof of (\ref{gg1}).}
Partition the configuration space ${\cal S}_N$ according to the
position of the rightmost particle. Namely, let
$$
A_x = \left\{ \xi \in {\cal S}_N : \xi (x) = 1, \mbox{and}\ \xi(y)
= 0 \ \mbox{ for any }\ y > x \right\}
$$
be the set of configurations whose rightmost particle is at $x$.
Denote by $P$ the distribution of $\{ \xi^N_t : t \ge 0 \}$ with
initial distribution $\pi$, and by $P_{N,x}$ the conditional
distribution of the nearest particle system on $\{1,2, \cdots,
N\}$ whose initial configurations are in $A_x$. Then
\begin{equation}\label{E.PartitionP}
P = \sum_{x=0}^N P(A_x) P_{N,x}.
\end{equation}
Denote by ${\bf P}$ the distribution of the nearest particle system on
$\mathbb{Z}$ with the initial distribution in Lemma \ref{edge}, and ${\bf P}_x$
the translation of ${\bf P}$ by $x$. Thanks to the attractive property,
there is a coupling of ${\bf P}_x$ and $P_{N,x}$ such that for all $t
> 0$ and all $i\in \mathbb{Z}$,
\begin{equation}\label{xiao}
\xi^N_t (i) \le \eta_t (i).
\end{equation}
Then under this coupling, $\xi^N_t \equiv \emptyset$ once $r_t <
1$, hence $ \sigma_N \le \inf \{t : r_t < 1 \}$.
Suppose that $\lim_{N \rightarrow \infty} C_N = \infty$. For any
$C > 0$ and large $N$,
\begin{eqnarray*}
P_{N,x} \( \sigma_N \le C_N N^2 \) & \ge &P_{N,x} \( \sigma_N \le
C (x-1)^2 \) \\
& \ge &
{\bf P}_x \( \exists \ t \le C (x-1)^2 {\rm \ s.t.} \ r_t < 1 \)\\
& = &
{\bf P} \( \exists \ t \le C (x-1)^2 {\rm \ s.t.} \ r_t < - (x-1) \)
\\ & = &
{\bf P} \( \exists \ t \le C {\rm \ s.t.} \ r_{ (x-1)^2 t} / (x-1) < -
1 \).
\end{eqnarray*}
Here the first equality holds because ${\bf P}_x$ is the translation of
${\bf P}$ by $x$. This together with Lemma \ref{edge} implies that
$$
\liminf_{N, x \rightarrow + \infty} P_{N,x} \( \sigma_N \le C_N
N^2 \) \ge {\bf P} \( \exists \ t \le C {\rm \ s.t.} \ B_t < - 1 \), \
\ \ \forall \ C > 0,
$$
where $\{ B_t : t \ge 0 \}$ is a Brownian motion with diffusion
constant $D$. Let $C \rightarrow + \infty$, the right side of the
above equation converges to 1. Hence
$$
\lim_{N,x \rightarrow + \infty} P_{N,x} \( \sigma_N \le C_N N^2 \)
= 1.
$$
Consequently, for any $\varepsilon > 0$, there exists $N_0
> 0$ such that for any $N \ge x \ge N_0$
$$
P_{N,x} \( \sigma_N \le C_N N^2 \) > 1 - \varepsilon.
$$
This together with (\ref{E.PartitionP}) implies that
\begin{equation} \label{E.P}
P \( \sigma_N \le C_N N^2 \) = \sum_{x=1}^N P(A_x) P_{N,x} \(
\sigma_N \le C_N N^2 \) \ge (1 - \varepsilon) \sum_{x = N_0}^N
P(A_x).
\end{equation}
On the other hand,
$$
\sum_{x = 1}^{ N_0 - 1} \nu_\psi (A_x) \le \sum_{x = 1}^{ N_0 - 1}
\sum_{y = 1 }^x \nu_\psi (S_N (y,x)) \le N_0^2.
$$
Therefore, as $N \rightarrow \infty$,
$$
\sum_{x = N_0}^N P(A_x) \ge 1 - N_0^2 / (C N^2) \rightarrow 1.
$$
This together with (\ref{E.P}) implies that
$
\liminf_{N \rightarrow \infty} P \( \sigma_N \le C_N N^2 \) \ge 1
- \varepsilon.
$
Let $\varepsilon \rightarrow 0$ and the result follows.\hfill $\Box$
\vspace{0.5cm}
\small
\baselineskip=0.7\baselineskip
\noindent {\bf References}
\noindent Durrett, R. and Liu, X. F. (1988). The contact process
on a finite set. {\em Ann. Probab.} {\bf 16} 1158--1173.
\noindent Durrett, R. and Schonmann, R. H. (1988). The contact
process on a finite set II. {\em Ann. Probab.} {\bf 16}
1570--1583.
\noindent Durrett, R., Schonmann, R. H. and Tanaka, N. I. (1989).
The contact process on a finite set III: The critical case. {\em
Ann. Probab.} {\bf 17} 1303--1321.
\noindent Liggett, T. M. (1985). {\em Interacting particle
systems.} New York, Springer-Verlag.
\noindent Mountford, T.S. (1992). A critical value for the uniform
nearest particle system, {\em Ann. Probab.} {\bf 20} 2031--2042.
\noindent Mountford, T.S. (2003). Critical reversible attractive
nearest particle systems, In {\em Topics in Spatial Stochastic
Processes, Lecture Notes in Mathematics \bf 1802}, Springer,
Berlin.
\noindent Schinazi, R. (1992). Brownian fluctuations of the edge
for critical reversible nearest particle systems. {\em Ann.
Probab.} {\bf 20} 194--205.
\noindent Wang Z. K. (1980). {\em Birth and Death Processes and
Markov Chains} (in Chinese). Beijing, Science Publishing House.
\vspace{0.5cm} \noindent LMAM, School of Mathematical Sciences,
Peking University, Beijing 100871, China
\end{document}
\noindent
E-mail: dayue@math.pku.edu.cn\quad zhangfxi@math.pku.edu.cn\\
Dayue Chen
\noindent Juxin Liu
affiliation: School of Mathematical Sciences, Peking University
addresses: Box 364, 6335 Thunderbird Crescent
\hspace{1.8cm} Vancouver, BC, V6T 2G9, Canada
\vspace{0.5cm}
\noindent Fuxi Zhang (corresponding author)
School of Mathematical Sciences, Peking University
Beijing 100871, China
\vspace{0.5cm}
\vspace{1cm}
running title: Nearest Particle System on an Interval
\newpage
Given $k$,
let $A_+$, $A_-$ and $\partial A_+$ be the set such that
$$
\sum _{x \in I_k} \xi^N (x) \ge \varsigma^{[N / n_0]} (k), \ \ \
\sum _{x \in I_k} \xi^N (x) < \varsigma^{[N / n_0]} (k), \ \ \
\sum _{x \in I_k} \xi^N (x) = \varsigma^{[N / n_0]} (k),
$$
respectively. In order that the system, restricted in the $k$-th
block, stay in the set $A_+$ all the time, we prevent the system
from falling into the set $A_-$. Notice that $A_-$ could be
accessed only through $\partial A_+$. We give the coupling such
that the system always change to $A_+$ from $\partial A_+$, where
the parameter of the contact process is given simultaneously.
Notice that
$$
\lambda^\prime_c < \min \left\{ \lambda \sum_{l=n_0}^{2n_0} \frac
{\psi(l) \psi(3n_0-l)}{2 \psi (3n_0)}, \ \lambda
\sum_{l=n_0}^{2n_0} \psi(l) \right\}.
$$
We can choose $\lambda^\prime > \lambda^\prime_c$ such that
(\ref{E.xixi}) holds.
To show the lower estimate of $\sigma_N$, we
compare $\{ \xi^N_t : t \ge 0 \}$ with $\{ \eta_t : t \ge 0 \}$, a
nearest particle system on $\mathbb{Z}$ which have the same family of
parameters. As to the latter, the following Lemma is quoted from
Let $\{\eta_t : t \ge 0
\}$ be a critical nearest particle system on $\mathbb{Z}$. Suppose the
initial configurations have a particle at the origin and no
particle on the left of the origin, and follows the renewal
measure $Ren(\beta)$ with density $\beta(\cdot)$. Then The last
inequality is from the following lemma.
\begin{lem}\label{Lemma.Sn}
Let
Then and there is a constant $C > 0$ such that $\nu_\psi (S_N
(x,y)) \ge C$
\end{lem}
{\bf Proof.
Let $X_n$ be the time until the first renewal $\ge n$. Then $\{
X_n : n \ge 0 \}$ is a Markov chain with transition probability $p
(0,n) = \beta (n+1)$, $p(n+1, n) = 1$ for all $n \ge 0$. Since
$\mu : = \sum_{n=1}^\infty n \beta (n) < \infty$, $\{ X_n : n \ge
0 \}$ has an invariant distribution $\pi$, and $\pi (0) = 1 /
\mu$. Thus $P(X_n = 0)$ converges to $1/ \mu$ as $n \rightarrow +
\infty$. Notice that $X_n = 0$ if and only if $n$ is a renewal
time. Hence, there exists $n_0 > 0$ such that $ P (X_n = 0) > (2
\mu)^{-1}$ for any $n \ge n_0$. Then
$$
\nu_\psi (S_N (x, y)) = P (X_{y-x} = 0) > (2 \mu)^{-1}, \ \ \
{\rm if \ } y - x > n_0.
$$
It is not difficult to check $\nu_\psi (S_N (x, y)) = P (X_{y-x} =
0) \le 1$. \hfill $\Box$
{\bf Proof of (\ref{xia}). } To be self-contained, we give a proof
of (\ref{xia}). Let $\{ \gamma^N_t : t \ge 0 \} $ be a spin system
on $\{ 1, 2, \cdots, N \}$ starting from all sites occupied, in
which particles die independently with rate 1 and no new particles
are born. Then there is a coupling such that $ P \( \gamma^N_t
\leq \xi^N_t, \ \forall \ t>0 \) = 1$. This implies that
$$
P (\xi^N_t \neq \emptyset) \geq P (\gamma^N_t \neq \emptyset), \ \
\ \forall \ t \ge 0.
$$
Notice that $ P \( \gamma^N_t (x) = 1 \) \geq e^{-t}$ for any $x
= 1, \cdots, N$, and $\gamma^N_t (x)$ are mutually independent. So
$$
P \big( \sigma_N \ge \alpha(N) \big) \ge 1 - \( 1-e^{-\alpha(N)}
\)^N, \ \ \ \forall \ \alpha (N) \ge 0.
$$
Choose $\alpha (N)$ such that $\( 1-e^{-\alpha(N)} \)^N$ converges
to zero as $N \rightarrow \infty$. This gives the lower estimate
of $\sigma_N$. Especially, let $\alpha(N)=(1 - \varepsilon) \log
N$, where $\varepsilon>0$. Then (\ref{xia}) follows. \hfill $\Box$
Suppose
\begin{equation}\label{sj}
M \stackrel{\triangle}{=} \sup_{n} \sum_{l+r=n} \frac {\psi(l)
\psi(r)} {\psi(n)} < \infty.
\end{equation}
Then, for any $\{ C_N : N \ge 1 \}$ such that $\lim_{N \rightarrow
\infty} C_N = \infty$,
$$
\lim_{N \rightarrow \infty} P \big( \sigma_N \leq C_N f_\lambda
(N) \big) = 1,
$$
where
$$
f_\lambda (N) = \left\{ \begin{array}{ll}
\log N & {\rm if} \ \lambda M < 1;
\\
N \log N & {\rm if} \ \lambda M = 1;
\\
(\lambda M)^N & {\rm if} \ \lambda M > 1.
\end{array} \right.
$$
\begin{thm}\label{shang1}
Suppose there exists $n_0$ such that
where $\lambda^\prime_c$ is the critical value for the contact
process on $\mathbb{Z}$. Then
Notice that $\emptyset$ is the unique absorbing state. The
particles will all be healthy at last. Let $\sigma_N$ be the first
time that all of the particles are healthy, i.e. the first time
$\{ \xi^N_t : t \ge 0 \}$ hits the empty set. Then $\sigma_N$ is
finite almost surely for any $N$. On the other hand, $\sigma_N$
should increase in some sense in $N$ since more particles are
intent to be infected when $N$ increases. It is intuitive that for
small $\lambda$, the infection rate is small, hence the particles
will all be healthy after a short time, i.e. $\sigma_N$ is small.
Otherwise, $\sigma_N$ is large. Let $\{ \xi^N_t : t \ge 0 \}$
start from all sites occupied if we do not give a specific initial
distribution. We estimate $\sigma_N$, and the results read as
follows. equivalently, by Theorem VI.1.2 of Liggett (1985)
Suppose, in addition, that the system is
Moreover, the parameters in the lower estimate and the upper
estimate should be amended to the same.
For the {\it reversible} system of Example 2,
The inequality
(\ref{sj}) is not very restrictive. For example, let $\psi(n) = c
n^{-\alpha}$, where $\alpha > 1$. Theorem \ref{shang} and
(\ref{xia}) imply that $\sigma_N$ has a logarithmic increasing
rate as $\lambda$ is small enough. By Theorem and Theorem
\ref{shang1}, $\sigma_N$ has an exponential increasing rate as
$\lambda$ is large enough. Theorem \ref{cth} tells us that
$\sigma$ has a polynomial increasing rate as $\lambda = 1$, the
critical point of the NPS on $\mathbb{Z}$.
Theorems \ref{shang}, \ref{shang1} and \ref{cth} are proved in
Sections 2, 3 and 4 in turn. In Section 3, we give a proof of
(\ref{xia}) for the completeness.
such that $P^{\xi,\zeta} \( \xi_t \le \zeta_t \) = 1$ for all $\xi
\le \zeta$ and all $t \geq 0$. This together with Theorem \ref{c2}
enlightens us to compare a NPS with a contact process by the {\it
renormalization} argument. Assumption (\ref{xxldy}) implies that
we can choose the infection rate $\lambda^\prime$ of the contact
process $\{ \varsigma^L_t : t \ge 0 \}$ to satisfy the following
inequality.
$$
\lambda^\prime_c < \lambda^\prime \le \min \left\{\lambda
\sum_{l=n_0}^{2n_0} \frac {\psi(l) \psi(3n_0-l) }{ 2 \psi (3n_0)},
\ \sum_{l=n_0}^{2n_0} \lambda \psi(l) \right\}.
$$
Then there is $P_{\lambda^\prime}$, a coupling
________________________________________________________________________
\medskip \noindent {\bf Proof of Theorem \ref{shang1}. }
The first statement, included for completeness, can be easily
derived by comparing the nearest particle system with an
independent system without birth. For the second part,
choose $n_0$ such that $m = \min\{ \frac 1 2 \sum_{l =
n_0}^{2n_0} \beta(l, 3n_0-l), \ \sum_{l=n_0}^{2n_0} \beta(l,
\infty)\}$ is maximum.
Let $L= [N/n_0]$ be the integer part
of $N/n_0$, and divide $\{ 1, 2, \cdots, L n_0 \}$ into
subintervals
$$
I_k = \big\{ (k-1) n_0 + 1, (k-1) n_0 + 2, \cdots, k n_0 \big\}, \
\ \ k = 1, 2, \cdots, L.
$$
We compare $\{ \xi_t^N : t \ge 0 \}$ with a contact process $\{
\varsigma^L_t : t \ge 0 \}$ on $\{ 1, \cdots, L \}$, whose initial
state is
\begin{equation*}
\varsigma^L_0 (k)=
\begin{cases}
1 & {\rm if} \ \sum_{x \in I_k} \xi^N_0 (x) \geq 1;
\\
0 & {\rm otherwise}.
\end{cases}
\end{equation*}
We claim that, by setting the infection parameter of $\{
\varsigma^L_t : t \ge 0 \}$ to be $m$,
\begin{equation} \label{xixi*}
\sum _{x \in I_k} \xi^N_t (x) \ge \varsigma^L_t (k), \ \ \ \forall
\ t \ge 0, \ k = 1, \cdots, L.
\end{equation}
Let $\{\xi^N_t : t \ge 0 \}$ and $\{ \varsigma^L_t : t \ge 0 \}$
evolve independently until $\sum _{x \in I_k} \xi^N_t (x)=
\varsigma^L_t (k)$ for some $k$ and $t>0$.
There are two cases.
{\it Case 1.} $\sum _{x \in I_k} \xi^N (x) = \varsigma^L (k) = 1$.
In this case, there is only one particle in the $k$-th subinterval
in the configuration $\xi$ of the nearest particle system and the
individual at site $k$ is infected in the configuration
$\varsigma$ of the contact process. Because both death rates are
1, we may couple two particles to die at the same time.
{\it Case 2.} $\sum _{x \in I_k} \xi^N (x) = \varsigma^L (k) = 0$.
In this case, there are no particles in $I_k$ and the individual
at site $k$ of $\varsigma$ is healthy. Consider the birth rates of
both processes. If $k = 1$, by Assumption 2, the total birth rate
in $I_1$ is at least $\sum_{l=n_0}^{2n_0} \beta(\infty, l)\geq m$
if there are particles in $I_2$. The case $k=L$ is similar. If $1
< k < L$, the total birth rate in $I_k$ is at least $
\sum_{l=n_0}^{2n_0} \beta(l,\infty)\geq m$ if there is at least
one particle in $I_{k-1}$ and no particle in $I_{k+1}$, or vice
versa. If there are particles in both $I_{k-1}$ and $I_{k+1}$,
then the total birth rate in $I_k$ is at least
$\sum_{l=n_0}^{2n_0} \beta(l, 3n_0-l) \geq 2m$.
By Theorem III.1.5 of Liggett (1985), there is a coupling of $\{
\xi_t^N : t \ge 0 \}$ and $\{ \varsigma^L_t : t \ge 0 \}$ such
that the inequality (\ref{xixi*}) is preserved.
Consequently, for any $t \ge 0$,
$$
P (\sigma_N \geq t) \ge P \( \sum _{x \in I_k} \xi^N_t (x) \neq
\emptyset \) \geq P \( \varsigma^L_t \neq \emptyset \) \ge P \(
\tau_L > t \),
$$
where $ \tau_L = \inf \{ t: \zeta^L_t = \emptyset \}$. This
together with part (ii) of Theorem \ref{c2} implies that
$$
\liminf_{N \rightarrow \infty} P( \sigma_N \geq e^{ \gamma_2
(m)L/2}) \ge \lim_{N \rightarrow \infty} P \( \tau_L
> e^{\gamma_2 (m) L/2} \) = 1.$$ Let $\gamma =
\gamma_2 ( m ) / 4 n_0$, then the result follows. \hfill $\Box$
$$
P_{N,x} \( \sigma_N \le C_N N^2 \)
\ge
P_{N,x} \( \sigma_N \le C (x-1)^2 \)
\ge
{\bf P}_x \( \exists \ t \le C (x-1)^2 {\rm \ s.t.} \ r_t < 1 \).
$$
Notice that It holds that
$$
{\bf P}_x \( \exists \ t \le C (x-1)^2 {\rm \ s.t.} \ r_t < 1 \)
=
{\bf P} \( \exists \ t \le C (x-1)^2 {\rm \ s.t.} \ r_t < - (x-1) \).
$$
Hence
|
{
"timestamp": "2005-03-21T05:48:36",
"yymm": "0503",
"arxiv_id": "math/0503409",
"language": "en",
"url": "https://arxiv.org/abs/math/0503409"
}
|
\section{Introduction}
During the past 20 years a number of methods has been devised for
state selective preparation and manipulation of discrete-level
quantum systems
\cite{paramonov1983,chelkowski1990,kaluza1993,bergmann1998,rabitz2003}.
However, simple population oscillations, induced by a resonant
driving pulse have received negligible attention as a prospective
population manipulation method. This might be attributed to two
reasons. The first is that Rabi theory is based on the rotating
wave approximation (RWA), and all attempts to generalize it
without RWA (e.g. \cite{shariar2002.1,barata2000,fujii2003}) are
mathematically very involved. The second is that no attempt has
been made to analytically generalize the original Rabi theory
beyond the two-level systems.
In this paper an analytic extension of Rabi theory to transitions
in many-level systems is presented. The aim is to 'design' a
driving pulse of the form:
\begin{equation}
F(t)= F_{\rm 0} \; m(t) \; \cos{(\omega(t) \; t)}
\label{pulse}
\end{equation}
by establishing analytical optimization relations between its
parameters: maximum pulse amplitude $F_{0}$, pulse envelope shape
$m(t)$, and time dependent carrier frequency $\omega(t)$. The goal
of this enterprize is twofold: the first is to achieve as complete
as possible transfer of population between two selected states of
the system; the second is to make this transfer as rapid as
possible. These two requirements, however, are conflicting:
population transfer can be accelerated by using a more intense
drive, but at the same time a stronger drive increases involvement
of remaining system levels in population dynamics hence
deteriorating population transfer between a selected pair of
levels.
In the previous paper on this topic \cite{bonacci2003.2} it was
shown how, for a pulse of arbitrary shape and duration, the drive
frequency can be analytically optimized to maximize the amplitude
of the population oscillations between the selected two levels in
a general many level quantum system. It was shown how the standard
Rabi theory can be extended beyond the simple two-level systems.
Now, in order to achieve the quickest and as complete as possible
population transfer between two pre-selected levels, driving pulse
should be tailored so that it produces only a single
half-oscillation of the population. In this paper, this second
(and final) step towards the controlled population transfer using
modified (i.e. many level system) Rabi oscillations is discussed.
The results presented herein can be regarded as an extension of
the standard $\pi$-pulse theory (see e.g. \cite{holthaus1994}) -
also strictly valid only in the two level systems - to the
coherently driven population oscillations in general many level
systems.
\section{Theoretical analysis}
All the calculations in this section are done in a system of units
in which $\hbar=1$.
\subsection{Calculation setup}
A quantum system with N discrete stationary levels with energies
$E_i \ (i=1,...,N)$ is considered. The system is driven by a time
dependent perturbation given in Eq. (\ref{pulse}). In the
interaction picture, the dynamics of the system obeys the
Schroedinger equation:
\begin{equation}
\frac{d}{dt}\mathbf{a}(t)=-i \oper{V}(t) \mathbf{a}(t) ,
\label{schrodinger}
\end{equation}
where $\mathbf{a}(t)$ is a vector of time-dependent expansion
coefficients $a_1(t),..., a_N(t)$. The N$\times$N Matrix
$\oper{V}(t)$ describes interaction between the system and
perturbation. Explicitly, its elements are given by:
\begin{equation}
V_{ij}(t) \equiv \frac{F_0 \mu_{ij}}{2} m(t)(e^{i s_{ij}
(\omega(t)-\omega_{ij}) t}+e^{-i s_{ij} (\omega(t)+\omega_{ij}) \; t}).
\end{equation}
$\mu_{ij}$ is transition moment between the i-th and the j-th
levels induced by the perturbation. $s_{ij} \equiv sign(E_i-E_j)$
and $\omega_{ij} \equiv |E_i-E_j|$ are respectively the sign and
the magnitude of the resonant frequency for the transition between
the i-th and the j-th level.
The aim is to induce population transfer between two arbitrarily
selected levels, designated by $\alpha$ and $\beta$, directly
coupled by the perturbation (i.e. such that $\mu_{\alpha \beta}\ne
0$). To simplify equations, the time variable t is re-scaled to
$\tau$, with transformation between the two given by:
\begin{equation}
d\tau \equiv \frac{F_0 \mu_{\alpha \beta}}{2} m(t) dt .
\label{definition tau}
\end{equation}
Then with following substitutions:
\begin{eqnarray}
f_{ij}(\tau)&\equiv&s_{ij} \frac{2}{F_0 \mu_{\alpha \beta}} (\omega(t)-\omega_{ij}) \label{definition f} \\
g_{ij}(\tau)&\equiv&s_{ij} \frac{2}{F_0 \mu_{\alpha \beta}} (\omega(t)+\omega_{ij}) \label{definition g} \\
x(\tau)&\equiv& \frac{F_0 \mu_{\alpha \beta}}{2} t(\tau) \label{definition x} \\
R_{ij} &\equiv& \frac{\mu_{ij}} {\mu_{\alpha \beta}} \label{definition R}
\end{eqnarray}
Eq. (\ref{schrodinger}) transforms into:
\begin{equation}
\frac{d}{d \tau}\mathbf{a}(\tau)=-i \oper{W}(\tau)\mathbf{a}(\tau),
\label{n level}
\end{equation}
where:
\begin{equation}
W_{ij}(\tau) \equiv R_{ij}(e^{i f_{ij}(\tau) x(\tau)}+e^{-i g_{ij}(\tau)x(\tau)}).
\label{wovi}
\end{equation}
Initial conditions for the problem of selective population
transfer comprise complete population initially (at $t=\tau=0$)
contained in only one of the selected levels, either $\alpha$ or
$\beta$. The other selected level, as well as all the remaining
N-2 'perturbing' levels of the system are unpopulated at this
time.
Population evolution $\Pi_i(t)$ of the i-th level is determined
from $\Pi_i(t)=|a_i(t)|^2$.
\subsection{Rabi-like population transfer in a three level system}
It was demonstrated in the previous paper on this topic
\cite{bonacci2003.2} that the analytical extension of the Rabi
oscillations theory beyond two-level systems is anchored in the
analysis of the simplest of the 'many-level' systems - a three
level one. Hence, in this section the impact of the single
additional level on the population transfer period is discussed:
beyond the 'selected' levels $\alpha$ and $\beta$, the system now
contains one additional 'perturbing' level, designated with index
\textit{p}. The only requirements on the system internal structure
are that $\mu_{\alpha \beta}, \mu_{\beta p} \ne 0$ and
$\mu_{\alpha p}=0$. While the first two requirements are
necessary, the last one does not reduce the generality of the
final results to any significant extent and is introduced for
calculational convenience exclusively.
For the observed three-level system, the dynamical equation
(\ref{n level}) reduces to:
\begin{eqnarray}
\label{3 level}
\frac{d}{d \tau}
\begin{bmatrix}
a_\alpha(\tau) \\
a_\beta(\tau) \\
a_p(\tau) \\
\end{bmatrix}
&=& \\
-&i&
\begin{bmatrix}
0 & e^{i f_{\alpha \beta}(\tau) x(\tau)} + e^{-i g_{\alpha \beta}(\tau) x(\tau)} & 0 \\
e^{-i f_{\alpha \beta}(\tau) x(\tau)} + e^{i g_{\alpha \beta}(\tau) x(\tau)} & 0
& R_{\beta p}(e^{-i f_{\beta p}(\tau) x(\tau)} + e^{i g_{\beta p}(\tau) x(\tau)}) \\
0 & R_{\beta p}(e^{i f_{\beta p}(\tau) x(\tau)} + e^{-i g_{\beta p}(\tau) x(\tau)}) & 0 \\
\end{bmatrix}
\begin{bmatrix}
a_\alpha(\tau) \\
a_\beta (\tau) \\
a_p(\tau) \\
\end{bmatrix} \nonumber
\end{eqnarray}
\subsubsection{Recapitulation: minimizing the impact of the perturbing level}
As it was shown in \cite{bonacci2003.2}, Eq. (\ref{3 level}), when
the driving frequency is near the resonant value for the
transition $\alpha \leftrightarrow \beta$, the following
expression can be obtained for the population dynamics of level
$\textit{p}$:
\begin{equation}
\label{apert}
a_p(\tau) \approx - a_{\beta} (\tau)\Big( \sigma_{\beta p}
\frac{ m(t(\tau)) }{1-\Delta_{\beta p}(\tau)} \Big) \; e^{ i f_{\beta
p}(\tau) x(\tau)}
\end{equation}
where
\begin{eqnarray}
\label{delta i Delta}
\sigma_{\beta p} \equiv \frac{ F_0 \mu_{\beta p}}{2(\omega_{\alpha \beta} - \omega_{\beta p})} \, \nonumber \\
\Delta_{\beta p}(\tau) \equiv \frac {\omega(\tau)- \omega_{\alpha \beta}} {\omega_{\beta p} - \omega_{\alpha \beta}} \ .
\end{eqnarray}
Put in words, with the conditions mentioned, the dynamics of the
level \textit{p} parametrically depends on the dynamics of the
level $\beta$ to which it is coupled. The relation between the
amplitudes of the population oscillations for levels \textit{p}
and $\beta$ follows directly from the above expression:
\begin{eqnarray}
\label{beta and p}
\Pi_{p} &\approx& \epsilon_{\beta p}(\tau) \Pi_{\beta}(\tau)
\end{eqnarray}
where:
\begin{equation}
\epsilon_{\beta p}(\tau) \equiv \Big(\sigma_{\beta p}
\frac{m(t(\tau))}{1-\Delta_{\beta p}(\tau)}\Big) ^2 \ .
\end{equation}
Note that, as $\Delta_{\beta p}(\tau)$ is generally very small and
$|m(\tau)|\leq1$, that parameter $\sigma_{\beta p}$ actually
determines the effective strength of applied perturbation: if
$\sigma_{\beta p}^2<<1$, then dynamical impact of level p is
negligible and perturbation may be considered weak; if
$\sigma_{\beta p}^2\sim 1$, perturbation is very strong.
Further, the requirement of the minimization of the dynamical
impact of the perturbing level $\textit{p}$ on the transition
$\alpha \leftrightarrow \beta$ leads to the following equation for
the optimized dynamics of the ($\alpha$,$\beta$) subsystem:
\begin{equation}
\label{beta dynamics}
\frac {d }{d \tau}
\begin{bmatrix}
b_\alpha(\tau) \cr
b_\beta(\tau)
\end{bmatrix}
= - i
\begin{bmatrix}
0 & 1 \cr
1 & 0 \cr
\end{bmatrix}
\begin{bmatrix}
b_\alpha(\tau) \cr
b_\beta(\tau)
\end{bmatrix}
\end{equation}
where the two-level state vector $\big(
b_\alpha(\tau),b_\beta(\tau)\big)$ is merely the unitary
transformed vector of the subsystem ($\alpha$,$\beta$):
\begin{equation}
\label{transformation}
\begin{bmatrix}
b_\alpha(\tau) \\
b_\beta(\tau) \\
\end{bmatrix}
= e^{-i \oper{\Lambda} (\tau)}
\begin{bmatrix}
a_\alpha(\tau) \\
a_\beta(\tau) \\
\end{bmatrix}
\end{equation}
Note that the precise form of the real transformation matrix
$\hat{\mathbf{\Lambda}}(\tau)$ is irrelevant here as it has no
impact on the population dynamics.
The optimization procedure produces the analytic expression for
the chirp of the driving frequency, which in the lowest order of
approximation (suitable for all but the most intensive
perturbations) amounts:
\begin{equation}
\label{chirp}
\omega(t)=
\omega_{\alpha \beta} +
(s_{\beta \alpha} s_{\beta p})( \omega_{\beta p}- \omega_{\alpha
\beta})\frac{2 \omega_{\beta p} }{ \omega_{\alpha \beta} +\omega_{\beta p}}
\sigma_{\beta p}^2 \frac{1}{t} \int_0^t \big( m(t') \big) ^2 dt'
\end{equation}
and from the Eq (\ref{beta dynamics}) it is found that the time
$\Theta$ required for a single population transfer between levels
$\alpha$ and $\beta$, determined from the fundamental relation of
the $\pi$-pulse theory:
\begin{equation}
\label{pi pulse theory}
\int_0^\Theta d\tau = \frac{\pi}{2}
\end{equation}
equals (in units of $\tau$):
\begin{equation}
\label{period pi}
\Theta=\frac{\pi}{2}
\end{equation}
As was shown in \cite{bonacci2003.2}, the 'exact' numerical
solution to the Eq. (\ref{3 level}) indeed maximizes the
population oscillations in the $\alpha-\beta$ subsystem. However,
as will be discussed below, the predicted value for the period of
the population oscillations (Eq. (\ref{period pi})) is somewhat
smaller than the correct one, with the discrepancy increasing with
the increasing population leak to the level \textit{p}. In the
following section this issue is resolved and the corrected
analytical expression for determination of the population transfer
(or oscillation) period is obtained.
\subsubsection{Patching the population conservation of the total system}
To start the following discussion, notice that the optimized
solution for the population transfer between levels $\alpha$ and
$\beta$ (described by the Eq. (\ref{beta dynamics})) is
unfortunately too good to be true. Namely, its serious drawback
lays in the fact that the leak of the population from the
($\alpha$,$\beta$) subsystem into the perturbing level \textit{p}
goes by completely unnoticed!
Formally, the root of the problem hides in the fact that the
mathematical trick which enabled decoupling of the level
\textit{p} dynamics from the rest of the system (i.e. the step
between Eq.(18) and Eq.(19) in \cite{bonacci2003.2}) destroys the
unitarity of the full dynamical equation for the three level
system, Eq.(\ref{3 level}). The consequence is that the dynamical
equation for the $\alpha-\beta$ subsystem, Eq.(\ref{beta
dynamics}) itself claims to be unitary, keeping the population of
that subsystem fully conserved. This is clearly impossible, as
level \textit{p} does indeed capture some population - the exact
amount given by Eq.(\ref{beta and p}).
This malfunction caused by the decoupling procedure unfortunately
cannot be remedied within the decoupling procedure itself - the
patch has to be provided by an independent approach. To do this,
the following argument is used: since it is the equation for the
dynamics of level $\beta$ that changes due to the decoupling
procedure and consequently causes the breakdown of the population
conservation, it is only the dynamical equation for the level
$\beta$ that has to be modified; then, as the dynamics of the
level $\beta$ is governed by the elements in the lower row of the
dynamical matrix in Eq.(\ref{beta dynamics}), a particular ansatz
intervention precisely into these elements might help rectify the
overall population dynamics of the whole three level system.
Hence, the correction is sought in the following form:
\begin{equation}
\label{two level corrected}
\frac{d}{d\tau}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix}
= - i
\begin{bmatrix}
0 & 1 \\
\zeta(\tau) & i \xi(\tau) \\
\end{bmatrix}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix}
\end{equation}
where $\zeta[\tau]$ and $\xi[\tau]$ are real non-negative
functions. Such an ansatz does not interfere with the
phase-fitting effect of the driving frequency optimization and Eq.
(\ref{chirp}) - forged by the decoupling procedure - which
maximizes the population oscillations in the $\alpha-\beta$
subsystem, is left unharmed. Instead, it merely enables
phase-independent modification of the \textbf{amplitudes} of
$\alpha$ and $\beta$ populations.
Expressing requirement of population conservation in the total
three-level system as:
\begin{equation}
d|b_{\alpha}(\tau)|^2+d|b_{\beta}(\tau)|^2+d|b_{p}(\tau)|^2=0 \ ,
\end{equation}
splitting the phase and amplitude contributions of the three wave
function projections on the three stationary states:
\begin{eqnarray}
b_{\alpha}(\tau) &\equiv& B_{\alpha}(\tau) e^{i \phi_{\alpha}(\tau)}\ , \nonumber \\
b_{\beta}(\tau) &\equiv& B_{\beta}(\tau) e^{i \phi_{\beta}(\tau)}\ , \\
b_{p}(\tau) &\equiv& B_{p}(\tau) e^{i \phi_{p}(\tau)} \ \nonumber ,
\end{eqnarray}
and using the known relation between the populations of levels
$\beta$ and \textit{p}, Eq.(\ref{beta and p}), the following
result is obtained:
\begin{equation}
\label{cons condition}
B_{\alpha}(\tau)\; dB_{\alpha} + (1+\epsilon_{\beta p}(\tau))
B_{\beta}(\tau)\; dB_{\beta} + \frac{1}{2} B_{\beta}(\tau)^2 \; d\epsilon_{\beta p} =
0\ .
\end{equation}
Now the corrected equation for the $\alpha-\beta$ subsystem,
Eq.(\ref{two level corrected}), can be used to eliminate
$dB_{\alpha}$ and $dB_{\beta}$ and introduce $\zeta[\tau]$ and
$\xi[\tau]$ in their stead:
\begin{eqnarray}
\label{eqn condition}
dB_{\alpha}&=& -i B_{\beta} \; Im \big(
e^{i(\phi_{\beta}(\tau)-\phi_{\alpha}(\tau))}\big)
\; d{\tau} \ , \nonumber \\
dB_{\beta}&=& -i \Big(\zeta(\tau) \;B_{\alpha}(\tau) \; Im \big( e^{-i(\phi_{\beta}(\tau)-\phi_{\alpha}(\tau))}\big) + \xi(\tau) \; B_{\beta}(\tau) Im \big(e^{-i\phi_{\beta}(\tau)}\big) \Big) \;
d{\tau} \ .
\end{eqnarray}
Finally, taking together Eq.(\ref{cons condition}) and
Eq.(\ref{eqn condition}) it is found that:
\begin{equation}
\label{final condition}
B_{\alpha}(\tau) \Big(\zeta(\tau)-\frac{1}{1+\epsilon_{\beta p}(\tau)}\Big)
\ sin \big( \phi_{\beta}(\tau)-\phi_{\alpha}(\tau) \big)
+ B_{\beta}(\tau) \Big( \xi(\tau) + \frac{d }{d\tau}\ln
\big(1+\epsilon_{\beta p}(\tau)\big)^{\frac{1}{2}} \Big) sin \big(
\phi_{\beta}(\tau) \big)=0 .
\end{equation}
Since for population oscillations $B_{\beta}$ and $B_{\alpha}$ are
$180^o$ out of phase, this condition can be satisfied only if:
\begin{eqnarray}
\label{zeta xi}
\zeta(\tau) &=& \frac{1}{1+\epsilon_{\beta p}(\tau)} \nonumber \ , \\
\xi(\tau) &=& - \frac{d }{d\tau}\ln
\big(1+\epsilon_{\beta p}(\tau)\big)^{\frac{1}{2}} \ .
\end{eqnarray}
\subsubsection{Population oscillation period modified}
Hence, the correct dynamical equation for the $\alpha-\beta$
subsystem which both maximizes the population oscillation
amplitudes of these two levels as well as properly conserves the
overall population of the three-level ($\alpha-\beta-p$) system
is:
\begin{equation}
\label{two level final}
\frac{d}{d\tau}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix}
= - i
\begin{bmatrix}
0 & 1 \\
\frac{1}{1+\epsilon_{\beta p}(\tau)} & - i \frac{d }{d\tau}\ln
\big(1+\epsilon_{\beta p}(\tau)\big)^{\frac{1}{2}} \\
\end{bmatrix}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix} \ .
\end{equation}
A simple extension of this result to the general many-level system
(in which level $\alpha$ also has some perturbing levels - jointly
designated by q - attached to it) yields the total corrected
dynamical equation for such a system:
\begin{equation}
\label{two level final}
\frac{d}{d\tau}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix}
= - i
\begin{bmatrix}
- i \frac{d }{d\tau}\ln \big(1+\epsilon_{\alpha q}(\tau)\big)^{\frac{1}{2}} & \frac{1}{1+ \epsilon_{\alpha q}(\tau)} \\
\frac{1}{1+\epsilon_{\beta p}(\tau)} & - i \frac{d }{d\tau}\ln
\big(1+\epsilon_{\beta p}(\tau)\big)^{\frac{1}{2}} \\
\end{bmatrix}
\begin{bmatrix}
b_{\alpha}(\tau) \\
b_{\beta}(\tau) \\
\end{bmatrix} \ .
\end{equation}
Now to finalize the calculation of the corrected population
transfer period the following procedure is administered. First,
the time variable is transformed $\tau \rightarrow \varphi$
according to:
\begin{equation}
\label{var transform}
d\tau = \kappa(\varphi) d\varphi
\end{equation}
The goal of this variable transformation is to produce, in the new
time variable $\varphi$, the closed dynamical equations for
$b_\alpha(\varphi)$ and $b_\beta(\varphi)$ describing the dynamics
which is as close as possible to the simple harmonic oscillation.
Second, and to that end, the transformation Eq.(\ref{var
transform}) is introduced into Eq.(\ref{two level final}), the
resulting relation is differentiated with respect to $\varphi$ and
all but the lowest order terms in the small parameters
$\epsilon_{\alpha q}(\tau)$ and $\epsilon_{\beta p}(\tau)$ are
kept. Hence the following result is established:
\begin{eqnarray}
\label{many level time correction}
\frac{d^2 b_{\alpha}(\varphi)}{d\varphi^2} +
\frac{1}{2} \frac{d}{d\varphi}\Big(\ln \frac{\big(1+\epsilon_{\alpha q}(\varphi)\big)^3
\big(1+\epsilon_{\beta p}(\varphi)\big)}{\kappa^2} \Big)\;\frac{d b_{\alpha}(\varphi)}{d\varphi}
+\frac{\kappa^2}{\big(1+\epsilon_{\alpha q}(\varphi)\big)
\big(1+\epsilon_{\beta p}(\varphi)\big)}b_{\alpha}(\varphi)= 0
\nonumber
\\
\frac{d^2 b_{\beta}(\varphi)}{d\varphi^2} +
\frac{1}{2} \frac{d}{d\varphi}\Big(\ln \frac{\big(1+\epsilon_{\alpha q}(\varphi)\big)
\big(1+\epsilon_{\beta p}(\varphi)\big)^3}{\kappa^2} \Big)\;\frac{d b_{\beta}(\varphi)}{d\varphi}
+\frac{\kappa^2}{ \big(1+\epsilon_{\alpha q}(\varphi)\big)
\big(1+\epsilon_{\beta p}(\varphi)\big)}b_{\beta}(\varphi)= 0
\end{eqnarray}
In the third and final step, the appropriate value of the free
parameter $\kappa(\varphi)$ is selected:
\begin{equation}
\kappa(\varphi)^2 \equiv \big(1+\epsilon_{\alpha
q}(\varphi)\big)
\big(1+\epsilon_{\beta p}(\varphi)\big)
\end{equation}
which transforms Eq.(\ref{many level time correction}) into:
\begin{eqnarray}
\label{approx equations}
\frac{d^2 b_{\alpha}(\varphi)}{d\varphi^2}+ \frac{d}{d\varphi}(\ln(1+\epsilon_{\alpha q}(\varphi)))\frac{db_{\alpha}(\varphi)}{d\varphi}+
b_{\alpha}(\varphi)=0 \nonumber \\
\frac{d^2 b_{\beta}(\varphi)}{d\varphi^2}+ \frac{d}{d\varphi}(\ln(1+\epsilon_{\beta p}(\varphi)))\frac{db_{\beta}(\varphi)}{d\varphi}+
b_{\beta}(\varphi)=0
\end{eqnarray}
Both these equations are similar to the damped oscillator
equation. For negligible damping ($\epsilon_{\alpha q}(\varphi),
\epsilon_{\beta p}(\varphi)<<1$), they reduce to the harmonic
oscillator equations, in which case the population transfer time
of $\pi/2$ is obtained in the variable $\varphi$. In the the
original time coordinate, $\tau$, the corrected time $\Theta$
required for a single population transfer between the levels
$\alpha$ and $\beta$ is then obtained from:
\begin{equation}
\label{transfer time corrected}
\int_0^\Theta \frac{d\tau}{\sqrt{(1+\epsilon_{\alpha q}(\tau))(1+\epsilon_{\beta p}(\tau))}} =
\frac{\pi}{2}
\end{equation}
Note the difference between this result, and the result in Eq.
(\ref{pi pulse theory}) obtained from the standard $\pi$-pulse
theory: in the lowest order approximation, the corrected
population transfer time is shorter then the one obtained from Eq.
(\ref{pi pulse theory}) by an order of
$\frac{1}{2}(\epsilon_{\alpha q}(\tau)+\epsilon_{\beta p}(\tau))$.
On the other hand, taking into consideration the damping factor in
the Eq.(\ref{approx equations}), approximating:
\begin{equation}
\label{corrected period}
\frac{d}{d\varphi}\ln(1+\epsilon_{\alpha q, \beta
p}(\varphi))\approx \frac{d}{d\varphi}\epsilon_{\alpha q, \beta
p}(\varphi)
\end{equation}
and using the damped oscillator theory \cite{Kent1996}(p.246), an
increase in the damping (expressed as an increase in
$\epsilon_{\alpha q}(\tau)$ and $\epsilon_{\beta p}(\tau)$) leads
to the an increase in the population time transfer by an order of
$\frac{1}{2}\frac{d}{d\tau}\epsilon_{\alpha q, \beta p}^2(\tau)$.
As this correction is an order of magnitude smaller than the
correction obtained from Eq.(\ref{transfer time corrected}), it
can be neglected for all practical purposes. Furthermore, the
additional corrections to the transfer period due to the neglected
higher order elements in the Eq.(\ref{many level time correction})
are of the same order of magnitude as this correction due to the
first derivative component, and as they are impossible to obtain
analytically, the value of this whole second order correction for
the case of strong perturbations is somewhat shaky. However, this
is not an issue, as the whole optimization theory developed in
\cite{bonacci2003.2} - and on whose applicability the results of
the above analysis hinge - assumes rather modest perturbations,
and is not even expected to work properly for the extreme values.
\section{Numerical simulations}
\begin{figure}
\includegraphics[width=8cm]{BonacciFig1.eps}\\
\caption{In all of the numerical examples the same pulse form was
used - $m(t)= \sin(\Omega t)^2$, but with different values of
maximum intensity parameter ($F_0$) and different total pulse
duration, $T$. The respective values are quoted in each particular
example.}
\label{fig1}
\end{figure}
In this section, numerical simulations of system dynamics for
unoptimized and fully optimized driving pulse of the form of Eq.
(\ref{pulse}) are presented and compared. Here, unoptimized
driving pulse is the one with driving frequency equal to the pure
resonant frequency between the two levels selected for population
transfer ($\omega(t)=\omega_{\alpha \beta}$) and with pulse
duration $T$ determined according to the standard $\pi$-pulse
theory relation, Eq.(\ref{pi pulse theory}). On the other hand,
the parameters of the fully optimized pulse are determined from
Eq.(\ref{chirp}) and Eq.(\ref{transfer time corrected}).
In all cases, a three-level system is considered, with the
following system parameters ($a.u.\equiv atomic \;units$):
$\omega_{\beta \alpha}=0.017671 \;a.u.$, $s_{\beta \alpha}=1$,
$\mu_{\beta \alpha}=0.073 \;a.u.$; $\omega_{\beta p}=0.017611
\;a.u.$, $s_{\beta p}=-1$, $\mu_{\beta p}=0.098\; a.u.$. These
system parameters correspond to the three ro-vibrational levels of
the HF molecule in the ground electronic state: $\alpha \equiv
(v=0,j=2,m=0)$, $\beta \equiv (v=1,j=1,m=0)$, $p \equiv
(v=2,j=2,m=0)$. The pulse shape in all of the examples is $m(t)=
\sin(\Omega t)^2$ as shown in Fig 1.
\subsection{Population oscillations}
\begin{figure}
\includegraphics[width=12cm]{BonacciFig2.eps}\\
\caption{Significance of the total analytical correction (in
driving frequency and total pulse duration) for the dynamics of a
mildly disturbed system. In all plots, major oscillations are the
populations of the two targeted levels ($\alpha$ and $\beta$)
whereas the minor oscillations are the population of the
perturbing level ($p$). For the value of perturbation strength
parameter $\sigma_{\beta p}^2=0.05$, the loss of the final
unoptimized population transfer amplitude amounts about 2\%.
Optimization reduces this loss to below 0.05\%. The difference
between the pulse duration obtained from standard $\pi$-pulse
theory and the optimized value is 1.6\%. Right hand-side plots
present the details from the left hand-side plots.}
\label{fig2}
\end{figure}
As was demonstrated in \cite{bonacci2003.2}, frequency
optimization minimizes the impact of the perturbing levels on the
\textit{amplitude} of the population oscillations. In this
subsection, the necessity of the inclusion of additional
correction Eq.(\ref{transfer time corrected}) for the
\textit{population transfer time} - alongside the correction for
the driving frequency - will be demonstrated. Also, the validity
and the limitations of the analytically obtained expression for
this correction will be discussed.
\subsubsection{Legitimate perturbation}
Fig.2 presents the dynamics of the system subjected to the
external drive of limiting intensity, $\sigma_{\beta p}^2=0.05$,
corresponding to the $F_0=2.80534 \ast 10^{-4} a.u.$. It is just
strong enough to noticeably (albeit not significantly) distort the
pure resonant oscillations, but at the same time weak enough so
that the theory developed in \cite{bonacci2003.2} and further in
this paper provides the full and precise quantitative corrections.
Pulse duration determined according to the standard $\pi$-pulse
theory expression, Eq.(\ref{pi pulse theory}) is $T_{\pi}=3077832
a.u.$, whereas the optimized one, obtained from Eq.(\ref{transfer
time corrected}) is $T_{opt}=3126029 a.u.$. The pulse is aimed at
producing five complete population oscillations.
Three cases of dynamics are presented: Fig. 2.a shows the
unoptimized dynamics; Fig 2.b shows the 'semi-optimized' dynamics,
with optimized driving frequency, but unoptimized population
transfer period; finally, Fig. 2.c shows the fully optimized
dynamics. Observe that in the unoptimized case, the population
oscillations end somewhat short of the complete cycle, and the
initially populated level never achieves complete depopulation.
Optimizing only the driving frequency does indeed maximize the
population oscillations by inducing the complete depopulation of
the initially populated level during oscillations, but at the same
time the final population oscillation stops even further from the
full cycle than in the unoptimized case. Finally, introducing the
population transfer period correction alongside the driving
frequency correction yields the required result: complete cycle of
maximized population oscillations.
\subsubsection{Strong perturbation}
Increasing the driving perturbation intensity to somewhat greater
value, $\sigma_{\beta p}^2=0.25$ ($F_0=6.11409 \ast 10^{-4} \
a.u.$.), the limitations of the analytical theory clearly emerge.
This is shown in Fig. 3: Fig. 3.a - Fig. 3.c respectively show the
unoptimized, analytically optimized (according to Eq.(\ref{chirp})
and Eq.(\ref{transfer time corrected})) and 'manually optimized'
dynamics. The corresponding pulse duration times, aimed at
producing three complete population oscillations, are
$T_{\pi}=847324 \ a.u.$, $T_{opt}=901075 \ a.u.$ and
$T_{man}=884300 \ a.u.$.
Notice that in the analytically optimized case, Fig. 3.b, the
initially populated level still fully depopulates, which indicates
that even for this rather strong perturbation, the frequency
correction Eq.(\ref{chirp}) still stands strong. However, the
corrected period, although closer to the correct value than in the
unoptimized case, is still somewhat removed from the correct
value. Unfortunately, this 'optimization error' cannot be remedied
analytically. Remember that the analytical result
Eq.(\ref{transfer time corrected}) is obtained using only the
first order approximation (Eq.(\ref{many level time correction}))
to the full dynamical equations Eq.(\ref{two level final}). With
perturbation as strong as in this case, the dynamical impact of
the neglected elements of that equation begin to show. However, as
shown in Fig. 3.c, the full cycle of oscillations can still be
produced, but this additional correction to the pulse duration had
to be found by hand, using the trial and error method.
\begin{figure}
\includegraphics[width=12cm]{BonacciFig3.eps}\\
\caption{Limitations of the analytical optimization theory. Again,
major oscillations are the population of the two targeted levels
whereas the minor oscillations are the population of the
perturbing level. For the value of perturbation strength parameter
is now $\sigma_{\beta p}^2=0.25$, which is just beyond the
limiting value for the full applicability of the presented
optimization procedure. Although the analytical correction
improves the final population transfer from 92\% to 97\% (with
pulse duration correction of 6\%), the theory presented in this
paper cannot account for an additional 1.6\% correction in the
duration of the pulse that further increases the final population
transfer to over 99.99\%.}
\label{fig3}
\end{figure}
\subsection{Population transfer}
The two final examples demonstrate the application of the
developed optimization theory to the most interesting dynamical
case regarding the coherent control: that of the single population
transfer between the two targeted levels $\alpha$ and $\beta$. As
the validity and the limitations of the whole theory were already
explored in the previous two examples, the following examples will
only demonstrate the improvements to the population transfer that
can be produced using the above results.
\subsubsection{Legitimate perturbation}
Again as in the previous section, the first example (Fig. 4)
presents the dynamics of the system subjected to the external
drive of limiting intensity. In this case, the perturbation
strength parameter amounts $\sigma_{\beta p}^2=0.1$, corresponding
to the $F_0=4.07606 \ast 10^{-4} \ a.u.$. Calculated population
transfer times are $T_{\pi}=211831 \ a.u.$ and $T_{opt}=218483 \
a.u.$.
The unoptimized (dotted line) and the optimized dynamics (solid
line) are plotted on the same graph to facilitate the comparison
between the two. Only the dynamics of the two target levels is
shown - the plot of the perturbing level's ($p$) dynamics is
omitted for the sake of clarity of the overall graph. Although the
loss of the population transfer in the unoptimized case is not
great, it nevertheless is noticeable. On the other hand,
introducing the corrections for pulse frequency and pulse duration
clearly improves the population transfer bringing it very close to
100\%.
\begin{figure}
\includegraphics[width=12cm]{BonacciFig4.eps}\\
\caption{Applicability of the analytical optimization procedure to
the maximization of the population transfer. Yet again, major
oscillations are the population of the two targeted levels whereas
the minor oscillations are the population of the perturbing level.
For this limiting value of perturbation strength parameter of
$\sigma_{\beta p}^2=0.10$, the optimization almost completely
eradicates the loss of the population transfer due to the
dynamical impact of the perturbing level, increasing the
population transfer from 96\% to 99.7\%. Optimized pulse lasts 3\%
longer than the one obtained from the standard $\pi$-pulse
theory.}
\label{fig4}
\end{figure}
\subsubsection{Extreme perturbation}
The final example - presented in Fig. 5. - is qualitative, rather
than quantitative, but even as such it is quite indicative of the
overall usefulness of the whole optimization theory. The
perturbation is now extreme, with strength parameter
$\sigma_{\beta p}^2=1$ corresponding to the $F_0=1.22282 \ast
10^{-3}\ a.u.$. Calculated population transfer times are
$T_{\pi}=70610 \ a.u.$ and $T_{opt}=82816 \ a.u.$.
Again, the unoptimized and the optimized dynamics are plotted on
the same graph. Although optimization now clearly does not lead to
the complete population transfer, the improvement from the
unoptimized dynamics is significant demonstrating that even for
this perturbation intensity the developed optimization theory
qualitatively works quite nicely.
\begin{figure}
\includegraphics[width=12cm]{BonacciFig5.eps}\\
\caption{Breakdown of the quantitative optimization, but
qualitatively the theory is still applicable. Half way through the
pulse, in the unoptimized case now the population of the
perturbing level surpasses the population of the initially
unpopulated level. Introduction of the optimized driving frequency
and pulse duration significantly - although not fully - rectifies
the population transfer from below 40\% to almost 90\%. Optimized
pulse is now almost 20\% longer then the one obtained from the
ordinary $\pi$-pulse theory.} \label{fig5}
\end{figure}
\section{Conclusion}
The aim of research that led to this paper was to explore the
possibility of using 'old fashioned' and rather simple phenomenon
of Rabi oscillations for the controlled manipulation of the
population in general many level system. This paper rounds up the
topic of analytical optimization of pulse parameters (frequency
chirp and pulse duration), opened in the author's previous work
(\cite{bonacci2003.2}) that would lead to maximizing the
population transfer between two targeted levels of the system. The
theory developed provides the exact quantitative predictions of to
what extent the dynamical impact of the remainder of the many
level system (beyond the two levels selected for the population
transfer) begins to interfere with the targeted population
transfer. It also provides the closed (albeit recursive)
analytical expressions for the optimization of pulse parameters.
Although the major correction to the population transfer is
achieved by optimizing the driving pulse's frequency chirp (given
in \cite{bonacci2003.2}), this paper provides the additional fine
tuning by establishing similarly simple analytical expression for
the determination of the optimal pulse duration. It demonstrates
that the standard formula of the $\pi$-pulse theory, Eq. (\ref{pi
pulse theory}) begins to fail as the perturbation increases to and
beyond the well defined limiting value. It also provides some
remedy to this failure.
The whole theory presented in \cite{bonacci2003.2} and this paper
deals with only single laser pulse driving one particular
transition in the many level system. The further research
currently under way considers the possibility of applying a number
of distinct but simultaneous optimized pulses to drive the
population through the chain of transitions through the system,
hence producing as clean as possible transfer between the two
levels not coupled by the single photon transition. The
preliminary results indicate that an analytical optimization
formula can be developed even for such a case.
|
{
"timestamp": "2005-03-25T00:19:11",
"yymm": "0503",
"arxiv_id": "quant-ph/0503197",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503197"
}
|
\section{Introduction}
The century old Riemann hypothesis \cite{WaWh} states that the only nontrivial zeros of the
zeta function,
\begin{eqnarray}
\zeta(s) = \sum_{n=1}^\infty {1\over n^s} = \prod (1-p^{-s})^{-1} \ ,
\end{eqnarray}
are on the set of points $s={1\over 2}+it$. Tremendous numerical computations support
this conjecture. The purpose of this article is to identify that under certain conditions
imposed on the ${\cal N}=4$ amplitude, the zeros of the Riemann zeta function are found
in a formal sense with the zeros of these amplitudes.
In precise terms, after an identification of the real parts of a sequence of derived
dimensions, all gauge theory amplitudes vanish when the zeta function has zeros on
the real axis $s=1/2 + it$. (The Riemann zeta function on this axis has some similarities
with the vanishing of the partition function of certain condensed matter theories as a
function of couplings, i.e. Lee-Yang zeros.)
\section{Review of the S-duality derivative expansion}
The ${\cal N}=4$ spontaneously broken theory is examined in this work. The Lagrangian is,
\begin{eqnarray}
{\cal S}={1\over g^2} {\rm Tr}~ \int ~ d^4x \bigl[ F^2 + \phi \Box \phi +
\psi {\slash D} \psi + \left[ \phi,\phi\right]^2 \bigr] \ .
\end{eqnarray}
The quantum theory is believed to have a full S-duality, which means that the gauge
amplitudes are invariant: under $A\rightarrow A_D$ and $\tau\rightarrow (a\tau+b)/
(c\tau+d)$ the functional form of the amplitude is invariant. The series supports a
tower of dyonic muliplets satisfying the mass formula $m^2=2\vert n^i a_i +m^i a_{d,i}\vert^2$
with $a_i$ and $a_{d,i}$ the vacuum values of the scalars and their duals; $a_{d,i}=\tau
a_i$. The two couplings parameterizing the simplest SU(2)$\rightarrow$U(1) theory is,
\begin{eqnarray}
{\theta\over 2\pi} + {4\pi i\over g^2}=\tau=\tau_1+i\tau_2 \ ,
\end{eqnarray}
taking values in the Teichmuller space of the keyhole region in the upper half plane,
i.e. $\vert\tau\vert\geq 1/2$ and $\vert \tau_1\vert \leq 1/2$. The S-duality invariant
scattering within the derivative expansion is constructed in \cite{Chalmers1}. Derivative
expansions in general are examined in \cite{Chalmers1}-\cite{Chalmers10}.
The full amplitudes of ${\cal N}=4$ theory may be constructed either in a gauge coupling
perturbative series, i.e. the usual diagrammatic expansion formulated via unitarity methods,
or as an expansion in derivatives, with the latter approach being nonperturbative in
coupling. Both expansions are equivalent, found from a diagram by diagram basis.
The full set of operators to create a spontaneously broken ${\cal N}=4$ gauge theory
amplitude is found from
\begin{eqnarray}
{\cal O}= \prod_{j=1} {\rm Tr} F^k_j \ ,
\end{eqnarray}
with possible $\ln^{m_1}(\Box) \ldots \ln^{m_n})$ (from the massless sector) and
combinations with the covariant derivative; the derivatives are gauge covariantized
and the tensor contractions are implied. The dimensionality of the operator is
compensated by a factor of the
vacuum expectation value, $\langle\phi^2\rangle^m$. The generic tensor has been
suppressed in the combination, and we did not include the fermions of scalars as in
\cite{Chalmers1} because the gauge vertices are only required (the coefficients of
course are found via the sewing, involving the integrations \cite{Chalmers1},
\cite{Chalmers3}-\cite{Chalmers5}).
The generating function of the gauge theory ${\cal N}=4$ four-point amplitude is
given
\begin{eqnarray}
{\cal S}_4 =\sum ~ \int d^dx~ h_n(\tau,\bar\tau) {\cal O}_n \ ,
\end{eqnarray}
with the ring of functions spanning $h_n(\tau,\bar\tau)$ consisting of the elements,
\begin{eqnarray}
\prod E_{s_j}^{(q_j,-q_j)} (\tau,\bar\tau) \ ,
\end{eqnarray}
and their weights
\begin{eqnarray}
\sum_j s_j = n/2 \ , \qquad \sum_j q_j = 0 \ ,
\end{eqnarray}
with $s=m/2+1$, and $n$ the number of gauge bosons. The general covariant term in
the effective theory has terms,
\begin{eqnarray}
\prod_{i=1}^{n_\partial} \nabla_{\mu_{\sigma'(i)}} \qquad
\prod^{m_i^A} A_{\mu_{\sigma(i),a_{\sigma(i)}}} \prod_{j=1}^{n_i^\phi}
\phi_{a_{\rho(j)}} \prod^{m_i^\psi} \psi_{a_{\kappa(j)}} \ ,
\end{eqnarray}
with the derivatives placed in various orderings (multiplying fields and products of
combinations of fields; this is described in momentum space in \cite{Chalmers1}). The
multiplying Eisenstein series possessing weights,
\begin{eqnarray}
s=n_A+n_\phi+n_\psi/2 + n_\partial/2+2 \qquad q=n_\psi/2 \ .
\end{eqnarray}
These terms span the general operator ${\cal O}$ in the generating functional. The
non-holomorphic weight $q$ is correlated with the R-symmetry.
The perturbative coupling structure, for the gauge bosons as an example, has the
form,
\begin{eqnarray}
g^{n-1} (g^2)^{n_{\rm max}/2} \Bigl[ \bigl({1\over g^2}\bigr)^{{\rm max}/2} ,
\ldots , \bigl({1\over g^2}\bigr)^{-n_{\rm max}/2+1} \Bigr] \ .
\label{couplingexp}
\end{eqnarray}
The factor in brackets agrees with the modular expansion of the Eisenstein series
pertinent to the scattering amplitudes, and the prefactor may be absorbed by a field
redefinition,
\begin{eqnarray}
A\rightarrow g^{-2} A \qquad x\rightarrow g x \ ,
\end{eqnarray}
which maps the gauge field part of the Lagrangian into
\begin{eqnarray}
\int d^4x~ {1\over g^2} ~ {\rm Tr}\left( \partial A + {1\over g} A^2\right)^2 \ .
\end{eqnarray}
This field redefinition, together with the supersymmetric completion, agrees with the
${\cal N}=4$ S-duality self-mapping in a manifest way (the factor in front may be
removed by a Weyl rescaling).
Fermionic (and mixed) amplitudes would have a non-vanishing $q_j$ sum. The Eisenstein
functions have the representation
\begin{eqnarray}
E_{s_j}^{(q_j,-q_j)} (\tau,\bar\tau) = \sum_{(p,q)\neq (0,0)} {\tau_2^s\over
(p+q\tau)^{s-q} (p+q\bar\tau)^{s+q}} \ ,
\end{eqnarray}
with an expansion containing two monomial terms and an infinite number of exponential
(representing instanton) terms,
\begin{eqnarray}
E_s(\tau,\bar\tau) = 2\zeta(2s) \tau_2^s + {\sqrt\pi} {\Gamma(s-1/2)\over \Gamma(s)}
\zeta(2s-1) \tau_2^{1-2s} + {\cal O}(e^{-2\pi\tau}) \ldots
\end{eqnarray}
with a modification in the non-holomorphic counterpart, $E_s^{(q,-q)}$, but with the
same zeta function factors. The latter terms correspond to gauge theory instanton
contributions to the amplitude; via S-duality all of the instantonic terms are available
from the perturbative sector. (At $s=0$ or $s={1\over 2}$ the expansion is finite:
$\zeta(0)=-1$ and both $\zeta(2s-1)\vert_{s=0}$ and $\Gamma(s)\vert_{s=0}$ have simple
poles.) The $n$-point amplitudes, with the previously discussed modular weight, are
\begin{eqnarray}
\langle A(k_1) \ldots A(k_n)\rangle = \sum_q h_q^{(n)}(\tau,\bar\tau) f_q(k_1,\ldots,
k_n) \ ,
\end{eqnarray}
where the modular factor is h (with the weights $n_A/2+2$) and the kinematic structure
of the higher derivative term $f_q$. The $n_{\rm max}$ follows from the modular
expansion $n_A/2+n_\partial/2+2$, and corresponds to a maximum loop contribution of
$n_A+n_\partial+1$.
We shall not review in detail the sewing relations that allow for a determination
of the coefficients of the modular functions at the various derivative orders. This
is discussed in detail in \cite{Chalmers3}-\cite{Chalmers5}.
\section{Rescaling of coupling}
A rescaling of the coupling constant via $g\rightarrow g^{1+\epsilon}$ changes the
expansion in \rf{couplingexp} to,
\begin{eqnarray}
(g^2)^{2+\epsilon} (g^2)^{(n_{\rm max}/2)(1+\epsilon)} \bigl[
(g^2)^{(n_{\rm max}/2)(1+\epsilon)}, \ldots, (g^2)^{(-n_{\rm max}/2+1)(1+\epsilon)}
\bigr] \ .
\end{eqnarray}
The rescaling of the couplings into the metric and the gauge fields would naively
generate a derivative expansion with modular functions labeled by $E_{s(1+\epsilon)}$,
and hence different coefficients for the expansion. These terms can always be
supersymmetrized to obtain the remaining couplings involving the fermions and scalars.
Within the loop expansion the zeta function takes values in accord with the dimension
of the loop integrals, which suggests that the theory is in a different dimension from
$4$ to $4(1+\epsilon/2)$; comparison with the loop expansion is required to determine
this (note that the tree-level terms found from the first term in \rf{couplingexp} are
invariant after including the gauge field rescaling; this is true for the scatteing
after changing dimension).
Note that for $\epsilon=-1$ the entire scattering has no coupling dependence; gauge
theory in $d=2$ is topological, and the gauge field and coupling may be gauged away
in a background without topology. The self-consistency via the sewing knocks out
the coefficients of the covariant gauge field operators and one is left with the
scalar interactions; the fermionic terms vanish as they only couple to the gauge field.
The dimension changes as $4(1+\epsilon/2)$, or rather to a dimension of $4(1+(d-4)/2)=
4(-1+d/2)=-4+2d$.
In the altered theory the ring of functions consists of
\begin{eqnarray}
\prod E_{s_i(1+\epsilon)}(\tau,\bar\tau) \qquad \sum s_i = s \ ,
\end{eqnarray}
with $s=n/2+1$, and $n$ being the number of external gauge bosons. The expansion at
$\epsilon=-1$ has finite coefficients.
\section{Amplitudes and zeros of the Riemann function}
The arguments of the Riemann zeta function for a given derivative term of the gauge
theory scattering amplitude are $2s$ and $2s-1$. In terms of $s=(n+2)/2$ the
arguments of the zeta function are
\begin{eqnarray}
2(-2+d)(n+2) \qquad {\rm and} \qquad 2(-2+d)(n+2)-1 \ .
\end{eqnarray}
If all of the real parts of the dimensions
\begin{eqnarray}
d_R = {1\over 4(n+2)}+2 \ , \qquad d_R = {3\over 4(n+2)} + 2
\end{eqnarray}
are identified then the arguments of the zeta functions are on the real $s=1/2$ axis.
These series have $d=2$ as a limit point, with a maximum dimension of $2+1/8=2.125$.
The gauge sector vanishes for $d=2$, i.e. at the limit point.
If the amplitudes vanished via the identification on the $s=1/m$ axis, then the real
part of the dimension would be
\begin{eqnarray}
d_R={1\over 2m(n+2)}+2 \ , \qquad d_R={3\over 2m(n+2)} + 2 \ .
\label{dimensions}
\end{eqnarray}
Example dimensions pertaining to the Riemann hypothesis, $m=2$ in \rf{dimensions}, are
\begin{eqnarray}
d_R = 2+1/12=25/12, \qquad 2+1/16=33/16, \qquad 2+1/20=41/20 \ ,
\end{eqnarray}
\begin{eqnarray}
d_R = 2+1/4=9/4 , \qquad 2+3/16=35/16 , 2+3/20=43/20 \ .
\end{eqnarray}
The identification can be thought of as toroidal compactification with the dimensions
identified, or as a series of identified four-manifolds.
\section{Discussion}
${\cal N}=4$ supersymmetric gauge theory amplitudes, including the nonperturbative
corrections, are examined as a function of complex dimension. The zeros of the Riemann
zeta function enforce the vanishing of the four-point gauge theory amplitudes. More
precisely, the Riemann hypothesis is equivalent to the vanishing of the amplitudes
of ${\cal N}=4$ four-point functions when the theory is dimensionally reduced on
identified tori of dimension $d$, with $d=id_I+d_R$,
\begin{eqnarray}
d_R={1\over 2m(n+2)}+2 \qquad {\rm and} \qquad d_R={3\over 2m(n+2)}+2 \ .
\end{eqnarray}
The real parts of these dimensions range from $2$ to $2.125$, with $d=2$ ($d_I=0$) special
from the point of the triviality of the gauge field (pure gauge).
\vfill\break
|
{
"timestamp": "2005-03-16T21:21:47",
"yymm": "0503",
"arxiv_id": "physics/0503141",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503141"
}
|
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