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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 14
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 45389)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 14
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
Vibrational modes of atomic nuclei provide crucial information about
nuclear structure. In particular, collective low-lying states contain information about the nucleonic shell structure, pairing correlations,
and nuclear deformations \cite{Bohr-MottelsonV2,Ring-Schuck}. Giant resonances
tell us about global properties of nuclear matter, such as compressibility and symmetry energy \cite{Lip89,Har01}.
Electromagnetic strength plays an important role in nuclear reactions involving photo-nuclear
processes, including astrophysical reactions \cite{Arnould200797,PhysRevC.86.034328} and the transmutation of nuclear waste \cite{Bea12}.
The random-phase approximation (RPA) and its superfluid extension, the quasiparticle random-phase approximation (QRPA), are well-established microscopic theories
describing excitations of many-body systems \cite{Bohr-MottelsonV2,Ring-Schuck}. QRPA can be viewed as a small-amplitude approximation of the time-dependent density functional theory \cite{Bla86,Nakatsukasa12}.
By using nuclear energy density functionals (EDFs) applicable to a large portion of the nuclear landscape,
a variety of excited modes can be described by using QRPA.
Recently, there has been a considerable progress in the area of fully self-consistent QRPA calculations based on the nuclear density functional theory.
Due to advances in high performance computing, deformed QRPA frameworks have been developed that can address properties of well-bound and loosely-bound nuclei
\cite{PhysRevC.82.034324,PhysRevC.83.021304,PhysRevC.82.034326,PhysRevC.84.014332,PhysRevC.81.064307,PhysRevC.83.014314,peru:044313,PhysRevC.83.034309}.
The traditional QRPA methodology is based on a generalized eigenvalue problem involving a QRPA matrix containing the residual two-quasiparticle interaction.
Because of a large number of quasiparticle states involved, the dimension of the QRPA matrix is typically quite
large, especially when spherical symmetry is broken. This makes the problem computationally challenging.
Therefore, in order to reduce the dimension of the two-quasiparticle basis, additional cutoffs are imposed on the configuration space of
QRPA. Such truncations result in inconsistencies between the model spaces of Hartree-Fock-Bogoliubov (HFB) and QRPA calculations, and can result in breaking self-consistency and appearance of spurious modes \cite{PhysRevC.71.034310}.
To circumvent these problems, efficient methods to solve RPA have been formulated in the framework of the linear response
theory and time-dependent HFB. One of these methods is the finite amplitude method (FAM) proposed in Ref.~\cite{nakatsukasa:024318}.
Within FAM, the strength function of an arbitrary one-body transition operator can be calculated without actually constructing and
diagonalizing the full (Q)RPA matrix. Instead, the fields induced by the one-body transition (driving) operator are calculated and the linear
response problem is solved iteratively. The practical implementation of the FAM requires minor extensions to the
existing HFB codes to calculate the induced fields and, therefore, is fairly straightforward.
Systematic calculations with the FAM have been performed for the electric giant dipole resonances
and low-lying dipole strength, illustrating computational advantages of the method \cite{PhysRevC.80.044301,PhysRevC.84.021302}.
The FAM has also been extended to the superfluid systems, both
spherical \cite{PhysRevC.84.014314} and deformed \cite{PhysRevC.84.041305}.
Since the FAM equations are solved by introducing a small width, an imaginary part of the QRPA frequency,
the method is very effective for describing excited modes in a region of high density of states. However, until now,
a direct application of FAM to discrete low-lying excitations has not been fully accomplished.
Quite recently, an efficient method to evaluate the QRPA matrix using FAM has been reported \cite{PhysRevC.87.014331} that
significantly reduces the computational effort, also
enabling computations of low-lying discrete QRPA modes.
A disadvantage of this approach is that
a large memory is required to store the huge QRPA matrix, which subsequently needs to be diagonalized.
An alternative technique to solve the linear response problem is based on the iterative Arnoldi diagonalization method
\cite{PhysRevC.81.034312}. This method was first implemented for spherical systems without pairing
and then further extended to spherical superfluid nuclei \cite{PhysRevC.86.024303}.
Because the Arnoldi diagonalization algorithm solves the QRPA eigenvalue problem in a smaller Krylov-space, the discrete excitations are within the scope of this method~\cite{PhysRevC.86.014307}.
The goal of this work is to derive a method to calculate the discrete low-lying QRPA modes within the FAM framework. We shall refer to this new technique as FAM-QRPA in the following.
Starting from the linear response theory, we show in Secs.~\ref{sec:fam} and \ref{sec:famdiscrete} that a contour integration in the complex frequency plane around a QRPA root provides
the QRPA eigenvectors.
A similar technique was proposed in Ref.~\cite{Sakurai2003119} to solve generalized eigenvalue problems.
We devise several techniques to compute and assess
the accuracy of QRPA modes.
Next, in Sec.~\ref{sec:result}, we numerically demonstrate that the discrete FAM-QRPA solution for the low-lying states
reproduces the modes obtained within the conventional matrix formulation of QRPA (MQRPA) and we apply FAM-QRPA
to collective modes in deformed Er and Yb nuclei.
Finally, the conclusions of our work are given in Sec.~\ref{sec:conclusion}.
\section{Finite amplitude method} \label{sec:fam}
In this section we recapitulate the derivation of the FAM equations
for superfluid systems following Sec.~II of Ref.~\cite{PhysRevC.84.014314}.
In the FAM formalism, the polarization of the system is induced by an external time-dependent field $\Fhat(t)$ with a frequency $\omega$:
\begin{equation}\label{eq:F}
\Fhat(t) = \eta \left\{ \Fhat e^{-i\omega t} + \Fhat^\dagger e^{i\omega t} \right\},
\end{equation}
where
\begin{equation}\label{eq:Fhatb}
\Fhat = \frac{1}{2} \sum_{\mu\nu}
\left\{ F^{20}_{\mu\nu} \Abdag_{\mu\nu} + F^{02}_{\mu\nu} \Ab_{\mu\nu} + F^{11}_{\mu\nu} \Bb_{\mu\nu} \right\},
\end{equation}
and $\Abdag_{\mu\nu}=\ahatdag_\mu \ahatdag_\nu$ and $\Bb_{\mu\nu}=\ahatdag_\mu \ahat_\nu$ are two-quasi\-par\-ti\-cle operators.
The parameter $\eta$ is a (small) real number to expand particle and pair HFB densities to the first order. Contrary to Ref.~\cite{PhysRevC.84.014314},
we assume that $\Fhat$ is $\omega$-independent in all applications in this work.
However, our scheme can be easily extended to the case where $\Fhat$ depends on $\omega$.
The time-evolution of quasiparticle operators under the external field $\Fhat(t)$ is determined by the time-dependent HFB (TDHFB) equation:
\begin{align}
i \frac{\del}{\del t}\ahat_{\mu}(t) = [ \Hhat(t) + \Fhat(t), \ahat_{\mu}(t) ] \, , \label{eq:TDHFB}
\end{align}
where time-dependent oscillation of quasiparticle operators is:
\begin{subequations}\begin{align}
\ahat_{\mu}(t) & = \left\{ \ahat_\mu + \delta \ahat_\mu(t)\right\} e^{iE_\mu t} \, ,\\
\delta \ahat_\mu(t) & = \eta \sum_{\nu} \ahatdag_\nu \left\{ X_{\nu\mu}(\omega) e^{-i\omega t} + Y^\ast_{\nu\mu}(\omega) e^{i\omega t}\right\} \, ,
\end{align}\label{eq:a}\end{subequations}
where $E_\mu$ is the one-quasiparticle energy and $X_{\mu\nu}(\omega)$ and $Y_{\mu\nu}(\omega)$ are the FAM amplitudes.
In terms of time-dependent quasiparticles, the TDHFB Hamiltonian can be written as $\Hhat(t)= \Hhat_0 + \delta \Hhat(t)$,
where:
\begin{equation}
\Hhat_0 = \sum_{\mu} E_\mu \Bb_{\mu\mu}
\end{equation}
is the HFB Hamiltonian and
\begin{equation}\label{eq:H}
\delta \Hhat (t) = \eta \left\{ \delta \Hhat(\omega) e^{-i\omega t} + \delta \Hhat^\dagger (\omega) e^{i\omega t}\right\}
\end{equation}
with
\begin{equation}
\delta \Hhat (\omega) = \frac{1}{2} \sum_{\mu\nu} \left\{
\delta H^{20}_{\mu\nu}(\omega) \Abdag_{\mu\nu} + \delta H^{02}_{\mu\nu}(\omega) \Ab_{\mu\nu} \right\}
\end{equation}
represents a small-amplitude oscillation.
Inserting (\ref{eq:F}), (\ref{eq:a}), and (\ref{eq:H}) into (\ref{eq:TDHFB}) results in the FAM equations:
\begin{subequations}\begin{align}
(E_{\mu} + E_{\nu} - \omega) X_{\mu\nu}(\omega) + \delta H^{20}_{\mu\nu}(\omega) & = -F^{20}_{\mu\nu} \, ,\\
(E_{\mu} + E_{\nu} + \omega) Y_{\mu\nu}(\omega) + \delta H^{02}_{\mu\nu}(\omega) & = -F^{02}_{\mu\nu} \, .
\end{align}\label{eq:FAM}\end{subequations}
By expanding $\delta H^{20}(\omega)$ and $\delta H^{02}(\omega)$ in terms of $X(\omega)$ and $Y(\omega)$, one obtains:
\begin{subequations}\begin{align}
\delta H^{20}_{\mu\nu}(\omega) & = \sum_{\mu'<\nu'}
\left\{ A_{\mu\nu,\mu'\nu'} - (E_\mu + E_\nu) \delta_{\mu\mu'} \delta_{\nu\nu'} \right\} X_{\mu'\nu'}(\omega) \nonumber \\
&+ \sum_{\mu'<\nu'} B_{\mu\nu,\mu'\nu'} Y_{\mu'\nu'}(\omega) \, , \\
\delta H^{02}_{\mu\nu}(\omega) &=
\sum_{\mu'<\nu'}\left\{ A^\ast_{\mu\nu,\mu'\nu'} - (E_\mu + E_\nu) \delta_{\mu\mu'}\delta_{\nu\nu'}
\right\} Y_{\mu'\nu'}(\omega) \nonumber \\
&+ \sum_{\mu'<\nu'} B^{\ast}_{\mu\nu,\mu'\nu'} X_{\mu'\nu'}(\omega) \, ,
\end{align}\label{eq:dHAB}\end{subequations}
where $A$ and $B$ are the usual QRPA matrices \cite{Ring-Schuck}.
The advantage of the FAM formulation is that the $A$ and $B$ matrices do not have to be computed explicitly.
By substituting (\ref{eq:dHAB}) into the FAM equations (\ref{eq:FAM}),
the linear response equation becomes:
\begin{align}
\left[
\begin{pmatrix} A & B \\ B^\ast & A^\ast \end{pmatrix}
- \omega
\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\right]
\begin{pmatrix} X(\omega) \\ Y(\omega) \end{pmatrix}
= -
\begin{pmatrix} F^{20} \\ F^{02} \end{pmatrix}, \label{eq:linres}
\end{align}
where the sum over two quasiparticle space is restricted to quasiparticle indices $\mu<\nu$. The FAM equations are thus equivalent to the linear response formalism. Furthermore,
the left hand side of (\ref{eq:linres}) yields the QRPA equations when the right-hand side is set to zero.
The FAM equations (\ref{eq:FAM}) are solved by using complex frequencies $\omega_\gamma=\omega+i\gamma$, where
the imaginary part $\gamma$ corresponds to a smearing width.
In terms of the FAM amplitudes $X(\omega_\gamma)$ and $Y(\omega_\gamma)$,
the strength function $dB(\omega ;F)/d\omega$ for the operator $\Fhat$ can be written as:
\begin{align}
\frac{dB(\omega;F)}{d\omega} & = -\frac{1}{\pi}{\rm Im} S(F;\omega_\gamma), \\
S(F;\omega_\gamma) & = \sum_{\mu<\nu} \left\{
F^{20\ast}_{\mu\nu} X_{\mu\nu}(\omega_\gamma) + F^{02\ast}_{\mu\nu} Y_{\mu\nu}(\omega_\gamma)
\right\}. \label{eq:FAMstrength}
\end{align}
\section{FAM for discrete QRPA modes} \label{sec:famdiscrete}
The objective of this work is to formulate a FAM capable of describing low-lying discrete QRPA modes.
We start by introducing the $2N\times 2N$ matrices \cite{Ring-Schuck}:
\begin{align}\label{matrices}
{\cal S}=\begin{pmatrix} A & B \\ B^\ast & A^\ast \end{pmatrix}, \,
{\cal N}=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \,
{\cal X}=\begin{pmatrix} X & Y^\ast \\ Y & X^\ast \end{pmatrix},
\end{align}
where $N$ is the dimension of the two-quasiparticle space, and the matrix elements
$X_{\mu\nu}^i$ and $Y_{\mu\nu}^i$ of ${\cal X}$ are the QRPA amplitudes of the $i$-th mode with a positive eigenfrequency $\Omega_i$.
There also exists a counterpart QRPA solution ($Y^{i\ast}, X^{i\ast}$) with a negative eigenfrequency $-\Omega_i$.
We assume here that all the QRPA frequencies are real, that is, ${\cal S}$ is positive definite.
In terms of matrices (\ref{matrices}), the QRPA equation can be expressed as:
\begin{align}
{\cal S}{\cal X} = {\cal N}{\cal X}{\cal O}, \label{eq:QRPA}
\end{align}
where ${\cal O}$ is a $2N\times 2N$ diagonal matrix containing the QRPA eigenfrequencies
($\Omega_1, \ldots, \Omega_N, -\Omega_1, \ldots, -\Omega_N$).
The orthonormalization condition for the QRPA eigenvectors is:
\begin{align}
{\cal X}{\cal N}{\cal X}^\dagger = {\cal N}. \label{eq:orthonormalization}
\end{align}
By applying Eqs.~(\ref{eq:QRPA}) and (\ref{eq:orthonormalization}), the matrix on the left-hand side of (\ref{eq:linres})
can be inverted, which yields the FAM amplitudes \cite{Ring-Schuck}:
\begin{align}
\begin{pmatrix}X(\omega_\gamma) \\ Y(\omega_\gamma) \end{pmatrix}
& = -R(\omega_\gamma) \begin{pmatrix} F^{20} \\ F^{02} \end{pmatrix} \nonumber \\
& = -{\cal X}({\cal O} - \omega_\gamma {\cal I})^{-1} {\cal N}{\cal X}^\dagger \begin{pmatrix} F^{20} \\ F^{02} \end{pmatrix},\label{eq:FAMXY1}
\end{align}
where ${\cal I}$ is a $2N\times 2N$ unit matrix and $R(\omega_\gamma)$ is the response function.
The explicit form of $R(\omega_\gamma)$ is:
\begin{widetext}
\begin{equation}
R_{\mu\nu\mu'\nu'}(\omega_\gamma)
= \sum_i
\begin{bmatrix}
\displaystyle\frac{X_{\mu\nu}^i X_{\mu'\nu'}^{i\ast}}{\Omega_i - \omega_\gamma} + \frac{Y_{\mu\nu}^{i\ast} Y_{\mu'\nu'}^{i}}{\Omega_i + \omega_\gamma}
&
\displaystyle\frac{X_{\mu\nu}^i Y_{\mu'\nu'}^{i\ast}}{\Omega_i - \omega_\gamma} + \frac{Y_{\mu\nu}^{i\ast} X_{\mu'\nu'}^{i}}{\Omega_i + \omega_\gamma}
\\
\displaystyle\frac{Y_{\mu\nu}^i X_{\mu'\nu'}^{i\ast}}{\Omega_i - \omega_\gamma} + \frac{X_{\mu\nu}^{i\ast} Y_{\mu'\nu'}^{i}}{\Omega_i + \omega_\gamma}
&
\displaystyle\frac{Y_{\mu\nu}^i Y_{\mu'\nu'}^{i\ast}}{\Omega_i - \omega_\gamma} + \frac{X_{\mu\nu}^{i\ast} X_{\mu'\nu'}^{i}}{\Omega_i + \omega_\gamma}
\end{bmatrix}. \label{eq:resp}
\end{equation}
\end{widetext}
Substitution of Eq.~(\ref{eq:resp}) into Eq.~(\ref{eq:FAMXY1}) provides the relation between the FAM amplitudes and QRPA amplitudes
\begin{subequations}
\begin{align}
X_{\mu\nu}(\omega_\gamma) & = -\sum_i \left\{
\frac{ X^i_{\mu\nu} \bra{i}\Fhat\ket{0}} {\Omega_i - \omega_\gamma} + \frac{ Y^{i\ast}_{\mu\nu} \bra{0}\Fhat\ket{i}}{\Omega_i + \omega_\gamma}
\right\}, \\
Y_{\mu\nu}(\omega_\gamma) & = -\sum_i \left\{
\frac{ Y^i_{\mu\nu} \bra{i}\Fhat\ket{0}} {\Omega_i - \omega_\gamma} + \frac{ X^{i\ast}_{\mu\nu} \bra{0}\Fhat\ket{i}}{\Omega_i + \omega_\gamma}
\right\},
\end{align}
\label{eq:FAMXY2}
\end{subequations}
where
\begin{align}
\bra{i}\Fhat\ket{0} & = \bra{\Phi_0}[\Ohat_i,\Fhat]\ket{\Phi_0} \nonumber \\
& = \sum_{\mu<\nu}(X^{i\ast}_{\mu\nu} F^{20}_{\mu\nu} + Y^{i\ast}_{\mu\nu} F^{02}_{\mu\nu})\, ,\label{eq:strengthfromQRPA} \\
\bra{0}\Fhat\ket{i} & = \bra{\Phi_0}[\Ohat^\dagger_i,\Fhat]\ket{\Phi_0} \nonumber \\
& = \sum_{\mu<\nu}(F^{02}_{\mu\nu} X^{i}_{\mu\nu} + F^{20}_{\mu\nu} Y^{i}_{\mu\nu}),
\end{align}
are the QRPA transition strengths between the QRPA ground state $\ket{0}$ and $i$-th excited state $\ket{i}$,
$\ket{\Phi_0}$ is the HFB state, the operator:
\begin{align}
\Ohat^\dagger_i = \sum_{\mu<\nu} \{ X^i_{\mu\nu} \Abdag_{\mu\nu} - Y^i_{\mu\nu} \Ab_{\mu\nu} \}
\end{align}
is the QRPA phonon operator, and $X^i_{\mu\nu}$ and $Y^i_{\mu\nu}$ are the QRPA amplitudes of a state $i$.
Equation (\ref{eq:FAMXY2}) shows that the FAM amplitudes $X(\omega_\gamma)$ and $Y(\omega_\gamma)$ have
first-order poles on the real axis at $\omega_\gamma=\Omega_i$ and $-\Omega_i$. By calculating
the standard FAM strength function, approximate positions of the poles of the low-lying states of interest
can be located. This allows one to define a closed contour $C_i$ in the complex energy plane that encloses the $i$-th positive pole $\Omega_i$.
According to Cauchy's integral formula, the contour integration of the FAM amplitudes (\ref{eq:FAMXY2}) along $C_i$ gives the residue at the $i$-th pole:
\begin{subequations}\begin{align}
\frac{1}{2\pi i} \oint_{C_i} X_{\mu\nu}(\omega_\gamma) d\omega_\gamma & =
\Res(X_{\mu\nu},\Omega_i)=
X^i_{\mu\nu}\bra{i}\Fhat\ket{0}, \\
\frac{1}{2\pi i} \oint_{C_i} Y_{\mu\nu}(\omega_\gamma) d\omega_\gamma & =
\Res(Y_{\mu\nu},\Omega_i) = Y^i_{\mu\nu}\bra{i}\Fhat\ket{0}.
\end{align}\label{eq:FAMXYcontour}
\end{subequations}
The absolute value of the transition strength for the $i$-th QRPA mode can then be expressed as:
\begin{align}
|\bra{i}\Fhat\ket{0}|^2 & = \sum_{\mu<\nu} \left\{
\Big| \frac{1}{2\pi i} \oint_{C_i} X_{\mu\nu}(\omega_\gamma)d\omega_\gamma\Big|^2 \right. \nonumber \\
&- \left.
\Big| \frac{1}{2\pi i} \oint_{C_i} Y_{\mu\nu}(\omega_\gamma)d\omega_\gamma\Big|^2
\right\}, \label{eq:strengthfromXY}
\end{align}
where we have used the normalization condition (\ref{eq:orthonormalization}) for the QRPA amplitudes. The individual QRPA amplitudes
$X_{\mu\nu}^i$ and $Y_{\mu\nu}^i$ can thus be calculated as:
\begin{subequations}\begin{align}
X_{\mu\nu}^i & = e^{-i\theta} |\bra{i}\Fhat\ket{0}|^{-1} \frac{1}{2\pi i}\oint_{C_i} X_{\mu\nu}(\omega_\gamma)d\omega_\gamma, \\
Y_{\mu\nu}^i & = e^{-i\theta} |\bra{i}\Fhat\ket{0}|^{-1} \frac{1}{2\pi i}\oint_{C_i} Y_{\mu\nu}(\omega_\gamma)d\omega_\gamma.
\end{align}\label{eq:QRPAdiscreteXY}\end{subequations}
The common phase $e^{i\theta}=\bra{i}\Fhat\ket{0}/\vert\bra{i}\Fhat\ket{0}\vert$
cannot be determined and remains arbitrary.
The information about the exact value of the QRPA eigenfrequency is not necessary to perform the contour integration as long
as the corresponding pole is located inside the contour. However, it can be calculated from the integration of the induced fields. Indeed, from
Eqs.~(\ref{eq:dHAB}) and (\ref{eq:QRPA}), one obtains:
\begin{subequations}\begin{align}
\frac{1}{2\pi i} \oint_{C_i} \delta H^{20}_{\mu\nu}(\omega_\gamma) d\omega_\gamma & =
\bra{i}\Fhat\ket{0} X^i_{\mu\nu} \left\{ \Omega_i - (E_\mu + E_\nu)\right\}, \\
\frac{1}{2\pi i} \oint_{C_i} \delta H^{02}_{\mu\nu}(\omega_\gamma) d\omega_\gamma & =
\bra{i}\Fhat\ket{0} Y^i_{\mu\nu} \left\{-\Omega_i - (E_\mu + E_\nu)\right\}.
\end{align} \label{eq:omega1}\end{subequations}
These $2N$ equations can be used to compute $\Omega_i$,
but this method is prone to large numerical errors when amplitudes $X^i_{\mu\nu}$ or $Y^i_{\mu\nu}$ are very small. To this end, a better way
of determining the QRPA eigenfrequencies is
through an expression derived from Eq.~(\ref{eq:omega1}):
\begin{align}\label{eq:omega2}
\Omega_i^2 & = \sum_{\mu<\nu} ( |\Omega_i X^i_{\mu\nu}|^2 - |\Omega_i Y^i_{\mu\nu}|^2) = \frac{1}{|\bra{i}\Fhat\ket{0}|^2} \sum_{\mu<\nu} \nonumber \\
& \left\{
\Big| \frac{1}{2\pi i} \oint_{C_i} \left((E_\mu + E_\nu) X_{\mu\nu}(\omega_\gamma) + \delta H^{20}_{\mu\nu}(\omega_\gamma)\right)d\omega_\gamma \Big|^2\right. \nonumber \\
-& \left.
\Big| \frac{1}{2\pi i} \oint_{C_i} \left((E_\mu + E_\nu) Y_{\mu\nu}(\omega_\gamma) + \delta H^{02}_{\mu\nu}(\omega_\gamma)\right)d\omega_\gamma \Big|^2\right\}.
\end{align}
The formalism presented above allows one to establish an explicit connection between
the FAM strength function and the smeared QRPA strength function. By
substituting Eq.~(\ref{eq:FAMXY2}) into Eq.~(\ref{eq:FAMstrength}) we obtain:
\begin{align}
S(F,\omega_\gamma) & = -\sum_i \left( \frac{|\bra{i}\Fhat\ket{0}|^2}{\Omega_i - \omega - i\gamma}
+ \frac{|\bra{i}\Fhat\ket{0}|^2}{\Omega_i + \omega + i\gamma} \right), \label{eq:FAMstrength2} \\
\frac{dB}{d\omega}(F,\omega) & = -\frac{1}{\pi} {\rm Im} S(F,\omega_\gamma) \nonumber \\
& = \frac{\gamma}{\pi}
\sum_i \left\{
\frac{|\bra{i}\Fhat\ket{0}|^2}{(\Omega_i -\omega)^2 + \gamma^2}
- \frac{|\bra{i}\Fhat\ket{0}|^2}{(\Omega_i + \omega)^2 + \gamma^2} \right\}.
\end{align}
According to Eq.~(\ref{eq:FAMstrength2}), the discrete QRPA transition strength can be directly computed
from the FAM strength function (\ref{eq:FAMstrength}):
\begin{align}
|\bra{i}\Fhat\ket{0}|^2 = \frac{1}{2\pi i} \oint_{C_i} S(F,\omega)d\omega \, . \label{eq:strengthfromFAMS}
\end{align}
In summary, as discussed above, there exist several techniques,
based on the residue at the QRPA pole, to calculate discrete transition strengths within the FAM-QRPA formalism:
\begin{enumerate}[A:]
\item The contour integration of the FAM amplitudes $X_{\mu\nu}(\omega)$ and $Y_{\mu\nu}(\omega)$ as in Eq.~(\ref{eq:strengthfromXY});
\item The contour integration of the FAM strength function as in Eq.~(\ref{eq:strengthfromFAMS});
\item Individual QRPA amplitudes $X^{i}_{\mu\nu}$ and $Y^{i}_{\mu\nu}$ can be found using (\ref{eq:QRPAdiscreteXY})
to obtain the transition matrix element (\ref{eq:strengthfromQRPA});
\item The QRPA amplitudes $X^{i}_{\mu\nu}$ and $Y^{i}_{\mu\nu}$ found with technique C are independent of the choice of the
external field used in FAM-QRPA. Therefore, for example, the isoscalar strength associated with the field $\Fhat'$ can be computed using the
QRPA amplitudes obtained in FAM-QRPA with the isovector external field $\Fhat$.
\end{enumerate}
Although all of these strategies are formally equivalent, the technique B is the easiest to implement in the current FAM codes. By virtue of D, once the discrete QRPA amplitudes have been found for a given state, they can be used to calculate a transition matrix element for any transition operator.
If assigned incorrectly, the integration contour $C'$ could include secondary unwanted poles. (For example, there could
be two states: a collective one carrying a strong transition strength and a nearby-lying non-collective one with a negligible contribution
to the total transition strength.)
Since the FAM amplitudes $X_{\mu\nu}(\omega)$ and $Y_{\mu\nu}(\omega)$ (\ref{eq:FAMXY2}) are sums of the residua,
the right hand side of Eq.~(\ref{eq:FAMXYcontour}) contains contributions from all the poles included inside $C'$.
The calculated transition strength then becomes:
\begin{align}
B(C';F) = \sum_{i\in C'}|\bra{i}\Fhat\ket{0}|^2.
\end{align}
Because of the orthogonality of QRPA amplitudes $X_{\mu\nu}^{i}$ and $Y_{\mu\nu}^{i}$, the interference terms between
different states cancel out. Therefore,
if $C'$ encircles two or more poles, the transition strengths from all those poles contribute to the total strength without the interference term when techniques A and B are used. Within C, calculated discrete amplitudes
$X^{i}_{\mu\nu}$ and $Y^{i}_{\mu\nu}$ contain a mixture of all states inside the contour. However, when applied to Eq.~(\ref{eq:strengthfromQRPA}), the same transition strength as with techniques A and B is obtained.
However, in the method D, due to the incorrect amplitudes $X^{i}_{\mu\nu}$ and $Y^{i}_{\mu\nu}$, the final strength
\begin{align}
B_{\rm D}(C';F') = \frac{ \vert\sum_{i\in C'}\bra{i}\Fhat'\ket{0}\bra{i}\Fhat\ket{0}^*\vert^2 }
{ \sum_{i\in C'} \vert\bra{i}\Fhat\ket{0}\vert^2 },
\end{align}
would be incorrect.
As will be demonstrated in Sec.~\ref{sec:result}, we have checked numerically that
when the contour includes multiple poles, techniques A-C
indeed yield the total summed strength while D does not. This apparent deficiency of D can be used to our advantage to verify that the selected contour $C_{i}$ includes only one pole.
One can also find a clue to correct the assignment of the contour by calculating the QRPA eigenfrequency $\Omega^2$ using Eq.~(\ref{eq:omega2}):
\begin{equation}
\Omega_{C'}^{2} = \sum_{i\in C'} \left(\vert\bra{i}\Fhat\ket{0}\vert^2\Omega_i^2\right) \big/ \sum_{i\in C'}\vert\bra{i}\Fhat\ket{0}\vert^2.
\end{equation}
When the contour encloses one collective and one non-collective QRPA root with respect to an external field $\Fhat$,
Eq.~(\ref{eq:omega2}) yields the approximate energy of the collective state.
\section{Numerical results} \label{sec:result}
To validate the FAM-QRPA formalism discussed in the previous section,
we carried out numerical computations using the FAM framework developed in Ref.~\cite{PhysRevC.84.041305} to evaluate
the transition strength and the corresponding residua.
Our FAM-QRPA method is based on the HFB code \pr{HFBTHO} \cite{Stoitsov200543,Stoitsov20131592}, which solves
the Skyrme-HFB equations in the (transformed) harmonic oscillator basis assuming axial and mirror symmetries.
The FAM equations are solved iteratively by using the modified Broyden's procedure \cite{PhysRevB.38.12807,baran:014318},
which offers a rapid and stable convergence, which weakly depends on the magnitude of the imaginary frequency $\gamma$.
\subsection{Test case: monopole strength in $^{24}$Mg}
To compare with full MQRPA, we consider the same case of the monopole strength in $^{24}$Mg as discussed in Ref.~\cite{PhysRevC.84.041305}.
We use SLy4 Skyrme EDF \cite{Chabanat1998231} and a contact volume pairing
with a 60\,MeV quasiparticle energy cutoff and the pairing strength $V_0=-125.20\,{\rm MeV\, fm}^{-3}$ for both neutrons and protons. In order to perform exact comparison without any truncation at the MQRPA level, we take
the single-particle basis consisting of $N_{\rm sh}=5$ oscillator shells
~\cite{PhysRevC.84.041305}.
The oblate-deformed HFB minimum of $^{24}$Mg was obtained at the
quadrupole mass deformation $\beta=-0.163$. In this configuration, both neutrons and protons are in the superfluid phase,
with pairing gaps $\Delta_{\rm n}=0.666$\,MeV and $\Delta_{\rm p}=0.654$\,MeV, respectively.
In the FAM calculation, we used the value of the parameter $\eta=10^{-7}$, which was found to provide the best accuracy~\cite{PhysRevC.84.041305}. For the convergence criterion of FAM iterations, defined in terms of the maximum difference between collective FAM amplitudes in two consecutive iterations,
we used the value of $10^{-5}$; this accuracy
is typically reached after about 40 iterations.
As for $\Fhat$, we consider the isoscalar monopole (ISM) and isovector monopole (IVM) operators:
\begin{subequations}
\begin{align}
\Fhat^{\rm ISM} & = \frac{eZ}{A}\sum_{i=1}^A r_i^2, \\
\Fhat^{\rm IVM} & = \frac{eZ}{A}\sum_{i=1}^N r_i^2 - \frac{eN}{A}\sum_{i=1}^Z r_i^2.
\end{align}
\end{subequations}
For the integration contours we take circles with radii 0.02\,MeV, centered close to
MQRPA frequencies. The contour integration is discretized with 11 points, unless
stated otherwise.
\begin{figure}[htbp]
\includegraphics[width=1.0\columnwidth]{24Mg-ISM.eps}
\caption{(Color online)
The low-lying isoscalar monopole strength at the oblate HFB minimum of $^{24}$Mg
calculated with the conventional FAM-QRPA using three
values of smearing width $\gamma$ (in MeV).
\label{fig:24Mg-ISM}}
\end{figure}
Figure~\ref{fig:24Mg-ISM} shows the isoscalar monopole strength function at the oblate configuration of $^{24}$Mg calculated
with the conventional FAM by using three values of $\gamma$.
The strength function obtained with $\gamma=0.5$\,MeV shows a very smooth distribution with the broad bumps carrying the largest strength. By going to
smaller values of $\gamma$, one reveals the detailed structure of QRPA modes. For example, to
separate the smaller first peak at $\Omega_1=1.32$\,MeV from the second one at $\Omega_2=1.37$\,MeV, a
very small $\gamma$ -- of the order of 1\,keV -- is required.
\begin{table*}[htbp]
\caption{\label{table:24MgIS}
Low-lying $K=0$ QRPA energies $\Omega_i$ and isoscalar monopole strength $|\bra{i}\Fhat\ket{0}|^2$
calculated with MQRPA and FAM-QRPA for the oblate configuration of $^{24}$Mg.
All the modes with $\Omega_i < 7.5$\,MeV are listed. The transition strength
was computed using the techniques A-D described in Sec.~\ref{sec:famdiscrete}.
The isoscalar monopole strength FAM-D is calculated from the FAM-QRPA amplitudes
generated by the external isovector monopole field. The numbers in parentheses denote powers of 10.}
\begin{ruledtabular}
\begin{tabular}{cc|lllll}
\multicolumn{2}{c|}{$\Omega_i$ (MeV)} & \multicolumn{5}{c}{$|\bra{i}\Fhat^{\rm ISM}\ket{0}|^2$ ($e^2\,{\rm fm}^4$)} \\
MQRPA & FAM & MQRPA & FAM-A
& FAM-B & FAM-C
& FAM-D \\ \hline
1.3185 & 1.3183 & 5.729(-4) & 5.771(-4) & 5.773(-4) & 5.776(-4) & 5.781(-4) \\
1.3731 & 1.3731 & 1.539(-2) & 1.511(-2) & 1.511(-2) & 1.510(-2) & 1.511(-2) \\
2.4582 & 2.4581 & 0.1796 & 0.1780 & 0.1782 & 0.1784 & 0.1783 \\
2.5998 & 2.5975 & 2.957(-3) & 3.056(-3) & 3.058(-3) & 3.060(-3) & 3.057(-3) \\
3.6687 & 3.6657 & 0.5776 & 0.5755 & 0.5771 & 0.5788 & 0.5788 \\
5.1185 & 5.1212 & 3.539(-4) & 3.744(-4) & 4.040(-4) & 4.360(-4) & 4.345(-4) \\
7.4108 & 7.4084 & 0.4900 & 0.4820 & 0.4834 & 0.4848 & 0.4848 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}[htbp]
\caption{\label{table:24MgIV}
Similar as in Table~\ref{table:24MgIS} but for the isovector monopole modes.}
\begin{ruledtabular}
\begin{tabular}{cc|lllll}
\multicolumn{2}{c|}{$\Omega_i$ (MeV)} & \multicolumn{5}{c}{$|\bra{i}\Fhat^{\rm IVM}\ket{0}|^2$ ($e^2\,{\rm fm}^4$)} \\
MQRPA & FAM & MQRPA & FAM-A
& FAM-B & FAM-C
& FAM-D \\ \hline
1.3185 & 1.3183 & 1.557(-3) & 1.547(-3) & 1.547(-3) & 1.547(-3) & 1.547(-3) \\
1.3731 & 1.3731 & 5.771(-5) & 5.810(-5) & 5.818(-5) & 5.827(-5) & 5.824(-5) \\
2.4582 & 2.4581 & 1.968(-6) & 1.643(-6) & 1.896(-6) & 2.188(-6) & 2.047(-6) \\
2.5998 & 2.5975 & 8.978(-5) & 8.870(-5) & 8.894(-5) & 8.919(-5) & 8.907(-5) \\
3.6687 & 3.6657 & 1.555(-5) & 8.681(-6) & 1.140(-5) & 1.498(-5) & 1.515(-5) \\
5.1185 & 5.1212 & 3.907(-2) & 3.885(-2) & 3.899(-2) & 3.914(-2) & 3.914(-2) \\
7.4108 & 7.4084 & 1.388(-5) & 2.926(-5) & 2.228(-5) & 1.697(-5) & 1.622(-5) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
In Tables~\ref{table:24MgIS} and \ref{table:24MgIV} we show the energies and transition strength of the low-lying $K=0$ QRPA modes, calculated with MQRPA and FAM.
Although the centers of the contours in the complex $\omega_\gamma$-plane are only approximately chosen, the low-lying QRPA eigenfrequencies calculated with FAM are in good agreement with MQRPA results.
The agreement is excellent for the lowest-lying states, which are usually of more interest. The FAM transition strength was obtained using the techniques A-D described in Sec.~\ref{sec:famdiscrete}. It is gratifying
to see that the four methods generally agree at least up to two decimal places, except for the modes carrying very small strength ($\sim 10^{-4}\,e^2$fm$^4$).
The nice agreement between FAM-C and FAM-D results demonstrates the consistency between the two sets of QRPA amplitudes calculated from the isoscalar
and isovector external monopole fields.
The difference between the strengths obtained by the MQRPA and FAM is consistent with the convergence criteria used in the FAM iterations.
\begin{figure}[htbp]
\includegraphics[width=0.9\columnwidth]{convergenceXY.eps}
\caption{(Color online)
Convergence of the orthogonality of the states as a number of discretization points of a circular contour with radius 0.02\,MeV.
The first three low-lying states labeled as 1 (1.32\,MeV), 2 (1.37\,MeV), and 3 (2.46\,MeV) are shown.\label{fig:convergence}}
\end{figure}
Figure~\ref{fig:convergence} demonstrates the convergence of the QRPA amplitudes
against the number of discretization points $N_{\rm disc}$ used in the contour integration. Specifically, it shows
the orthogonality of the QRPA amplitudes for the three lowest QRPA states.
The orthogonality between the first and third state, and between the second and third state, is
achieved already at $N_{\rm disc}=4$, while the convergence for the pair of first and second states is slower. This is not surprising as
the energies of the first and second QRPA roots differs only by 0.05\,MeV; hence,
and a finer integration mesh is required to remove the contribution from the unwanted pole outside of the contour.
Our results show that to obtain the convergence for the contour integration, consistent with the accuracy required during the regular FAM iterations,
taking 11 points is fully sufficient, at least for the two lowest modes.
Lastly, we discuss an example of an incorrect contour assignment following the discussion in Sec.~\ref{sec:famdiscrete}.
A contour of radius of 0.2\,MeV, centered at 1.3\,MeV, includes the first two QRPA solutions at 1.32\,MeV and 1.37\,MeV.
Such an incorrect choice would be made if the contour were determined
from the isoscalar strength function calculated with a resolution of $\gamma=0.02$\,MeV shown in Fig.~\ref{fig:24Mg-ISM}.
The calculated isoscalar (isovector) monopole transition strength determined
according to A-C is $1.57\times 10^{-2}$ ($1.61\times 10^{-3}$) $e^2\,{\rm fm}^4$, which is precisely the sum of the two QRPA strengths.
However, method D completely fails, yielding the values of $6\times 10^{-8}\,e^2\,{\rm fm}^4$ (isoscalar) and
$3\times 10^{-9}\,e^2\,{\rm fm}^4$ (isovector), that are clearly off from those obtained with the procedures A-C.
The QRPA frequency calculated from Eq.~(\ref{eq:omega2}) with the isoscalar (isovector) monopole external field is 1.371\,MeV (1.320\,MeV).
Since the isoscalar (isovector) monopole strength of the second (first) QRPA state is larger than that of the first (second) QRPA state
by two orders of magnitude, the QRPA frequency calculated using the contour, which encloses both poles, is close to the energy of the collective state.
Therefore, the consistency between the results obtained in methods C and D, together with the value of weighted frequency, can be used to find the contour that encloses a single QRPA pole.
\subsection{Low-lying QRPA modes in deformed rare-earth nuclei}
To demonstrate the feasibility of the FAM-QRPA formalism to describe the low-lying collective modes of deformed nuclei,
we have performed FAM calculations for the low-lying $K=0$ strength of
$^{166,168,172}$Yb, and $^{170}$Er, which were previously studied with MQRPA in Refs.~\cite{PhysRevC.82.034326,PhysRevC.84.014332}.
The calculations were carried out using SkM* Skyrme EDF \cite{Bartel198279} with the volume pairing. The pairing strengths have been adjusted
to reproduce the
odd-even binding energy difference in $^{172}$Yb evaluated with the three-point expression. They are:
$V_{\rm n}=-176\,{\rm MeV\,fm}^{-3}$ and $V_{\rm p}=-218\,{\rm MeV\,fm}^{-3}$ for the quasiparticle energy cutoff $E_{\rm cut}=60$\,MeV and
$V_{\rm n}=-150\,{\rm MeV\,fm}^{-3}$ and $V_{\rm p}=-177.5\,{\rm MeV\,fm}^{-3}$ for $E_{\rm cut}=200$\,MeV.
To obtain QRPA amplitudes in FAM, we applied the isoscalar quadrupole $K=0$ external field~\cite{PhysRevC.71.034310}, and
the electric reduced matrix elements $B(E2)$ for the excitational modes discussed in Refs.~\cite{PhysRevC.82.034326,PhysRevC.84.014332}
are computed using the technique D described in Sec.~\ref{sec:famdiscrete}.
The transformed harmonic oscillator basis with 20 major oscillator shells was employed. To compute residua, we used the circular contours
with radii 0.1\,MeV for $^{166}$Yb, $^{172}$Yb, and $^{170}$Er,
and with radii 0.01\,MeV for $^{168}$Yb.
The locations of the contour centers estimated from the conventional FAM calculations are:
1.40\,MeV, 1.75\,MeV, 1.30\,MeV, and 1.30\,MeV for $^{166}$Yb, $^{168}$Yb, $^{172}$Yb, and $^{170}$Er, respectively.
The results were compared with the MQRPA calculations of Refs.~\cite{PhysRevC.82.034326,PhysRevC.84.014332} employing a different HFB solver and an additional cutoff associated
with the occupation probabilities of canonical states.
\begin{table}[htbp]
\begin{threeparttable}[b]
\caption{FAM-QRPA energies and $B(E2)$ values of the low-lying $K=0$ states in $^{166}$Yb, $^{168}$Yb, $^{172}$Yb, and $^{170}$Er at $E_{\rm cut}=200$\,MeV compared to the MQRPA results of
Ref.~\cite{PhysRevC.84.014332}. The additional result for $^{172}$Yb
corresponding $E_{\rm cut}=60$\,MeV is compared to the MQRPA values obtained in Ref.~\cite{PhysRevC.82.034326}.
\label{table:Yb2}}
\begin{ruledtabular}
\begin{tabular}{ccccc}
\multirow{2}{*}{nucleus}
& \multicolumn{2}{c}{$\Omega_i$ (MeV)} & \multicolumn{2}{c}{$B(E2)$ ($e^2$b$^2$)} \\
& MQRPA & FAM & MQRPA & FAM \\
\hline \\[-6pt]
$^{166}$Yb & 1.802 & 1.422 & 0.0398 & 0.0327 \\
$^{168}$Yb & 2.039 & 1.747 & 0.0343 & 0.0186 \\
$^{172}$Yb & 1.605 & 1.306 & 0.0049 & 0.0088 \\
$^{170}$Er & 1.596 & 1.322 & 0.0030 & 0.0047 \\ [4pt]
$^{172}$Yb$^a$ & 1.390 & 1.319 & 0.0050 & 0.0092
\end{tabular}
\end{ruledtabular}
\begin{flushleft}
$^a$$E_{\rm cut}=60$\,MeV
\end{flushleft}
\end{threeparttable}
\end{table}
Table~\ref{table:Yb2} displays the results for excitation energies and $B(E2)$ rates of the $K=0$ QRPA modes.
For $^{172}$Yb the calculations were carried out with two quasiparticle cutoffs:
$E_{\rm cut}=60$\,MeV and 200\,MeV.
The FAM-QRPA excitation energy is close to the MQRPA value
with $E_{\rm cut}=60$\,MeV~\cite{PhysRevC.82.034326}, but this agreement does not hold when $E_{\rm cut}$ is increased. Indeed,
while our FAM-QRPA values weakly depend on $E_{\rm cut}$,
a 15\% increase of the MQRPA energy for $^{172}$Yb was reported in
Ref.~\cite{PhysRevC.84.014332} when going to $E_{\rm cut}=200$\,MeV.
Interestingly, the $B(E2)$ values weakly depend on energy cutoff in both methods. However, the $B(E2)$ values obtained in FAM-QRPA are
twice as large as the MQRPA results.
For $^{166}$Yb, $^{168}$Yb, and $^{170}$Er,
the excitation energies obtained in MQRPA are larger by 0.3-0.4\,MeV than those in FAM-QRPA.
The agreement between $B(E2)$ values is good in $^{166}$Yb, but
gets worse in the other cases studied. It is difficult to speculate what is the origin of those differences. We note, however, that (i) the HFB solvers used in both calculations are different (see benchmarking results in Ref.~\cite{Pei08}), and (ii) there are additional canonical energy cutoffs in in MQRPA \cite{PhysRevC.71.034310} that are not present in our FAM-QRPA method.
\begin{table}[htbp]
\caption{
Isoscalar and isovector quadrupole strength (in $e^2\,{\rm fm}^4$) of the low-lying $K=0$ states in $^{166}$Yb, $^{168}$Yb, $^{172}$Yb, and $^{170}$Er shown in
Table~\ref{table:Yb2} with $E_{\rm cut}=200$\,MeV. The isoscalar (isovector) strength in FAM-C is calculated
using the isoscalar (isovector) quadrupole external field. The isoscalar (isovector)
quadrupole strength in FAM-D is calculated using the QRPA amplitudes obtained from the FAM calculation using the isovector (isoscalar)
quadrupole external field. \label{table:ISQIVQ}}
\begin{ruledtabular}
\begin{tabular}{ccccc}
\multirow{2}{*}{nucleus} & \multicolumn{2}{c}{ISQ} & \multicolumn{2}{c}{IVQ} \\
& FAM-C & FAM-D & FAM-C & FAM-D \\ \hline \\[-6pt]
$^{166}$Yb & 299.854 & 299.856 & 0.585519 & 0.585520 \\
$^{168}$Yb & 160.126 & 160.127 & 0.969114 & 0.969124 \\
$^{172}$Yb & 93.2710 & 93.2735 & 0.081406 & 0.081404 \\
$^{170}$Er & 56.2932 & 56.2913 & 0.460285 & 0.460254 \\
\end{tabular}
\end{ruledtabular}
\end{table}
The isoscalar
quadrupole (ISQ) and isovector quadrupole (IVQ) strengths
are displayed
in Table~\ref{table:ISQIVQ} for the deformed nuclei shown in
Table~\ref{table:Yb2}.
The strengths obtained from the QRPA amplitudes derived from two external quadrupole fields agree excellently.
This result clearly shows that the contours used in these calculations enclose only a single QRPA pole.
\section{Conclusions} \label{sec:conclusion}
We have formulated and tested the FAM-QRPA method for efficient computations of discrete QRPA modes. The new framework is based on the application of Cauchy's integral formula to the FAM amplitudes defined in the complex frequency plane. The method is fully self-consistent and does not require any configuration-space truncations at the QRPA level.
The method is particularly useful when applied to the isolated collective QRPA modes.
For the description of the transition strength carried by densely distributed modes, the conventional FAM formulation is more
appropriate.
The FAM-QRPA method has been benchmarked and tested
by comparing with MQRPA results for an oblate configuration of
$^{24}$Mg.
Illustrative examples of large-scale calculations have been presented for the $K=0$ isoscalar and isovector quadrupole modes of
deformed rare-earth nuclei $^{166}$Yb, $^{168}$Yb, $^{172}$Yb, and $^{170}$Er.
Our results demonstrate that the proposed formulation of FAM-QRPA can be used as an efficient tool to calculate discrete QRPA modes of heavy, deformed, and superfluid nuclei. Once the contour around the mode of interest is specified,
the FAM-QRPA method allows one to perform a fully self-consistent QRPA calculation employing {\it the same} model space as in HFB.
Thanks to the rapid convergence achieved with Broyden's method used in our implementation, FAM-QRPA is amenable to high-performance parallel computing. This offers promise of systematic calculations of various kinds of low-lying excitations and decays over the entire nuclear landscape. Of particular importance are QRPA studies of low-energy dipole and quadrupole states, $\beta$ decays, and $\beta\beta$ decays.
The work on extending the FAM-QRPA formalism to $K\ne 0$ and charge-exchange modes is in progress.
\begin{acknowledgments}
Useful discussions with J. Dobaczewski and T. Nakatsukasa
are gratefully acknowledged. This work was
supported by
the U.S. Department of Energy under
Contract Nos.\ DE-FG02-96ER40963
(University of Tennessee),
DE-SC0008499 (NUCLEI SciDAC Collaboration),
by JUSTIPEN (Japan-U.S. Theory Institute for Physics with Exotic Nuclei) under grant
number No.\ DEFG02-06ER41407 (University of Tennessee),
by the Academy of Finland under the Centre of Excellence Programme 2012--2017
(Nuclear and Accelerator Based Physics Programme at JYFL), and FIDIPRO programme. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
|
{
"timestamp": "2013-04-16T02:02:48",
"yymm": "1304",
"arxiv_id": "1304.4008",
"language": "en",
"url": "https://arxiv.org/abs/1304.4008"
}
|
\section{Introduction}
The connection between Yangians and finite $W$-algebras of type A was first noticed by mathematical physicists Briot, Ragoucy and Sorba in \cite{RS, BR1} under some restrictions, and then constructed in general cases by Brundan and Kleshchev explicitly in \cite{BK}. In this way, a realization of finite $W$-algebra in terms of truncated Yangian is obtained, and this provides a useful tool for the study of the representation theory of finite $W$-algebras. In this note, we establish such a connection between finite $W$-superalgebras and super Yangians of type A where the nilpotent element $e$ is {\em rectangular} (cf. \textsection 3 for the precise definition).
Let $Y_{m|n}$ be the super Yangian for $\mathfrak{gl}_{m|n}$ and $Y_{m|n}^{\ell}$ be the truncated super Yangian of level $\ell$ for some non-negative integer $\ell$. Our main result is that there exists an isomorphism of filtered superalgebras between $Y_{m|n}^{\ell}$ and $\mathcal{W}_e$, the finite $W$-superalgebra associated to a rectangular $e\in \mathfrak{gl}_{m\ell|n\ell}$. Such a connection was firstly obtained in \cite{BR3}. In this article, we provide a new proof in a different approach, which is similar to \cite{BK}.
In a recent paper \cite{BBG}, such a connection between the super Yangian $Y_{1|1}$ and the finite $W$-superalgebra associated to a {\em principal} nilpotent element $e$ is obtained. In particular, the specialization of our result when $m=n=1$ coincides with the special case in \cite{BBG} when $e$ is both principal and rectangular.
\section{Super Yangian $Y_{m|n}$ and its truncation $Y_{m|n}^{\ell}$}
Let $m$ and $n$ be non-negative integers. Let $\mathfrak{b}$ be an $\epsilon$-$\delta$ sequence of $\mathfrak{gl}_{m|n}$ introduced in \cite{FSS1,LS,CW}. To be precise, $\mathfrak{b}$ is a sequence consisting of exactly $m$ $\delta$'s and $n$ $\epsilon$'s, both indistinguishable.
For example, $\delta\epsilon\delta\epsilon\ep$ is such a sequence of $\mathfrak{gl}_{2|3}$.
Note that the set of such sequences is in one-to-one correspondence with the classes of the simple systems of $\mathfrak{gl}_{m|n}$ modulo the Weyl group action, and each $\epsilon$-$\delta$ sequence $\mathfrak{b}$ gives rise to a distinguished Borel subalgebra, which will be also denoted by $\mathfrak{b}$. In particular, the sequence $\mathfrak{b}^{st}:=\stackrel{m}{\overbrace{\delta\ldots\delta}}\,\stackrel{n}{\overbrace{\epsilon\ldots\epsilon}}$ corresponds to the standard Borel subalgebra consisting of upper triangular matrices.
\begin{remark}
Contrary to the $\mathfrak{gl}_n$ case, there are many inequivalent simple root systems and hence many inequivalent Dynkin diagrams for $\mathfrak{gl}_{m|n}$. It is also true that each $\epsilon$-$\delta$ sequence corresponds to exactly one of the Dynkin diagrams. As an example, the following are two inequivalent Dynkin diagrams for $\mathfrak{gl}_{2|3}$ (equivalently, $\mathfrak{sl}_{2|3}$), where their corresponding simple systems and $\epsilon$-$\delta$ sequences are listed:
\begin{center}
\begin{equation*}
\hskip -3cm \setlength{\unitlength}{0.16in}
\begin{picture}(20,1)
\put(0,0){\makebox(0,0)[c]{$\bigcirc$}}
\put(3.5,0){\makebox(0,0)[c]{$\bigotimes$}}
\put(7,0){\makebox(0,0)[c]{$\bigcirc$}}
\put(10.5,0){\makebox(0,0)[c]{$\bigcirc$}}
\put(17,0){\makebox(0,0)[c]{$\bigotimes$}}
\put(20.5,0){\makebox(0,0)[c]{$\bigotimes$}}
\put(24,0){\makebox(0,0)[c]{$\bigotimes$}}
\put(27.5,0){\makebox(0,0)[c]{$\bigcirc$}}
\put(0.5,0){\line(1,0){2.5}}
\put(4,0){\line(1,0){2.5}}
\put(7.5,0){\line(1,0){2.5}}
\put(17.5,0){\line(1,0){2.5}}
\put(21,0){\line(1,0){2.5}}
\put(24.5,0){\line(1,0){2.5}}
\put(0,-1.3){\makebox(0,0)[c]{\tiny$\delta_1-\delta_2$}}
\put(3.5,-1.3){\makebox(0,0)[c]{\tiny$\delta_2-\epsilon_1$}}
\put(7,-1.3){\makebox(0,0)[c]{\tiny$\epsilon_1-\epsilon_2$}}
\put(10.5,-1.3){\makebox(0,0)[c]{\tiny$\epsilon_2-\epsilon_3$}}
\put(17,-1.3){\makebox(0,0)[c]{\tiny$\delta_1-\epsilon_1$}}
\put(20.5,-1.3){\makebox(0,0)[c]{\tiny$\epsilon_1-\delta_2$}}
\put(24,-1.3){\makebox(0,0)[c]{\tiny$\delta_2-\epsilon_2$}}
\put(27.5,-1.3){\makebox(0,0)[c]{\tiny$\epsilon_2-\epsilon_3$}}
\put(5,1.5){\makebox(0,0)[c]{$\mathfrak{b}=\mathfrak{b}^{st}=\delta\del\epsilon\ep\epsilon$}}
\put(22,1.5){\makebox(0,0)[c]{$\mathfrak{b}=\delta\epsilon\delta\epsilon\ep$}}
\end{picture}
\end{equation*}
\vskip 0.8cm
\end{center}
Here $\bigcirc$ denotes an even simple root, $\bigotimes$ denotes an odd simple root, $\delta_i$ and $\epsilon_j$ are elements of $\mathfrak{h}^*$ sending a matrix to its $i$-th and $(2+j)$-th diagonal entry, respectively. Such a phenomenon can also be observed in other types of Lie superalgebras. We refer the reader to \cite{FSS2,CW} for the detail.
\end{remark}
For each $1\leq i\leq m+n$, define a number $|i|\in\mathbb{Z}_2$, called the parity of $i$, as follows:
\begin{equation}\label{parity}
|i|:= \left\{
\begin{array}{ll}
\overline{0} &\hbox{if the $i$-th position of $\mathfrak{b}$ is $\delta$,}\\
\overline{1} &\hbox{if the $i$-th position of $\mathfrak{b}$ is $\epsilon$.}
\end{array}
\right.
\end{equation}
For a given $\mathfrak{b}$, the super Yangian $Y_{m|n}$ (cf. \cite{Na}) is the associative $\mathbb{Z}_2$-graded algebra (i.e., superalgebra) over $\mathbb{C}$ with generators
\begin{equation}\label{RTTgen}
\left\lbrace t_{ij}^{(r)}\,| \; 1\le i,j \le m+n; r\ge 1\right\rbrace,
\end{equation}
subject to certain relations.
To write down the relations, we firstly define the parity of $t_{ij}^{(r)}$ to be $|i|+|j|$, and $t_{ij}^{(r)}$ is called an even (odd, respectively) element if its parity is $\overline{0}$ ($\overline{1}$, respectively).
The defining relations are
\begin{equation}\label{RTT}
[t_{ij}^{(r)}, t_{hk}^{(s)}] = (-1)^{|i|\,|j| + |i|\,|h| + |j|\,|h|}
\sum_{t=0}^{\mathrm{min}(r,s) -1} \Big( t_{hj}^{(t)}\, t_{ik}^{(r+s-1-t)} - t_{hj}^{(r+s-1-t)}\, t_{ik}^{(t)} \Big),
\end{equation}
where the bracket is understood as the supercommutator.
By convention, we set $t_{ij}^{(0)}:=\delta_{ij}$.
In the case when $m=0$ or $n=0$, it reduces to the usual Yangian. The generators in (\ref{RTTgen}) are called the RTT generators while the relations (\ref{RTT}) are called the RTT relations. As in the case of $\mathfrak{gl}_{m|n}$, $Y_{m|n}$ are isomorphic for all $\mathfrak{b}$. Note that the original definition in \cite{Na} corresponds to the case when $\mathfrak{b}=\mathfrak{b}^{st}$.
For all $1\leq i,j\leq m+n$, we define the formal power series
\[
t_{ij}(u):= \sum_{r\geq 0} t_{ij}^{(r)}u^{-r}.
\]
It is well-known (cf. \cite{Na}) that $Y_{m|n}$ is a Hopf-algebra, where the comultiplication
$\Delta:Y_{m|n}\rightarrow Y_{m|n}\otimes Y_{m|n}$ is defined by
\begin{equation}\label{Del}
\Delta(t_{ij}^{(r)})=\sum_{s=0}^r \sum_{k=1}^{m+n} t_{ik}^{(r-s)}\otimes t_{kj}^{(s)},
\end{equation}
and one has the evaluation homomorphism $\operatorname{ev}:Y_{m|n}\rightarrow U(\mathfrak{gl}_{m|n})$ defined by
\begin{equation}\label{ev}
\operatorname{ev}\big(t_{ij}(u)\big):= \delta_{ij} + (-1)^{|i|} e_{i,j},
\end{equation}
where $e_{i,j}$ denotes the elementary matrix in $\mathfrak{gl}_{m|n}$.
\begin{definition}
Let $\ell$ be a non-negative integer and $I_\ell$ be the 2-sided ideal of $Y_{m|n}$ generated by the elements $\{ t_{ij}^{(r)}|1\leq i,j\leq m+n, r>\ell \,\}$. The {\em truncated super Yangian of level $\ell$}, denoted by $Y_{m|n}^{\ell}$, is defined to be the quotient $Y_{m|n}/I_\ell$.
\end{definition}
Note that $Y_{m|n}$, $I_\ell$ and the quotient $Y_{m|n}^{\ell}$ are all independent of the choice of $\mathfrak{b}$. The image of $t_{ij}^{(r)}$ in the quotient $Y_{m|n}^{\ell}$ will be denoted by $t_{ij;\mathfrak{b}}^{(r)}$, since it will be identified with certain element depending on $\mathfrak{b}$ later; see (\ref{tdef}).
Define $\kappa_1=\operatorname{ev}$, and for any integer $\ell\geq 2$, we define the following homomorphism
\begin{equation}\label{kappa}
\kappa_\ell:=(\overbrace{\operatorname{ev}\otimes\cdots\otimes\operatorname{ev}}^{\ell-copies})\circ \Delta^{\ell-1}:Y_{m|n}\rightarrow U(\mathfrak{gl}_{m|n})^{\otimes \ell},
\end{equation}
then we have
\begin{equation}\label{kpimg}
\kappa_\ell(t_{ij}^{(r)})=\sum_{1\leq s_1<\cdots<s_r\leq \ell}\,\,\sum_{1\leq i_1,\ldots, i_{r-1}\leq m+n}
(-1)^{|i|+|i_1|+\cdots+|i_{r-1}|}
e_{i,i_1}^{[s_1]}e_{i_1,i_2}^{[s_2]}\cdots e_{i_{r-1},j}^{[s_r]},
\end{equation}
where $e_{i,j}^{[s]}:= 1^{\otimes(s-1)}\otimes e_{i,j}\otimes 1^{\otimes (\ell-s)}$.
The following proposition is a PBW theorem for $Y_{m|n}^{\ell}$ and $Y_{m|n}$.
\begin{proposition}\cite[Theorem 1]{Go}\label{PBWSY}
Let $\kappa_\ell$, $I_\ell$ be as above.
\begin{enumerate}
\item[(1)] The kernel of $\kappa_\ell$ is exactly $I_{\ell}$.
\item[(2)]
The set of all supermonomials in the elements of $Y_{m|n}^{\ell}$
\[
\left\lbrace t_{ij;\mathfrak{b}}^{(r)}| 1\leq i,j\leq m+n, 1\leq r\leq \ell\right\rbrace
\] taken in some fixed order
forms a basis for $Y_{m|n}^{\ell}$.
\item[(3)]
The set of all supermonomials in the elements of $Y_{m|n}$
\[
\left\lbrace t_{ij}^{(r)}| 1\leq i,j\leq m+n, r\geq 1 \right\rbrace
\] taken in some fixed order
forms a basis for $Y_{m|n}$.
\end{enumerate}
\end{proposition}
As a consequence, we may identify $Y_{m|n}^{\ell}$ with the image of $Y_{m|n}$ under the map $\kappa_\ell$. In particular, the induced homomorphism
\begin{equation}\label{kappainj}
\kappa_\ell:Y_{m|n}^{\ell}\rightarrow U(\mathfrak{gl}_{m|n})^{\otimes \ell}
\end{equation}
is injective.
Define the canonical filtration of $Y_{m|n}$
\[
F_0Y_{m|n}\subseteq F_1Y_{m|n}\subset F_2Y_{m|n}\subseteq \cdots
\]
by deg $t_{ij}^{(r)}:=r$, i.e., $F_dY_{m|n}$ is the span of all supermonomials in the generators $t_{ij}^{(r)}$ of total degree $\leq$ d. It is clear from (\ref{RTT}) that the associated graded algebra $\operatorname{gr} Y_{m|n}$ is supercommutative. We also obtain the canonical filtration on $Y_{m|n}^{\ell}$ induced from the natural quotient map $Y_{m|n}\twoheadrightarrow Y_{m|n}^{\ell}$.
\section{Finite $W$-superalgebras}
Let $M$ and $N$ be non-negative integers and let $\mathfrak{g}=\mathfrak{gl}_{M|N}$. Every elements of $\mathfrak{g}$ in the context are considered $\mathbb{Z}_2$-homogeneous unless mentioned specifically. Recall that $\mathfrak{g}$ acts on $\mathbb{C}^{M|N}$ via the standard representation, which we will denote by $\psi:\mathfrak{g}\rightarrow \operatorname{End}(\mathbb{C}^{M|N})$. Let $(\, \cdot \, ,\,\cdot\,)$ denote the non-degenerate even supersymmetric invariant bilinear form on $\mathfrak{g}$ defined by
\begin{equation}\label{strform}
(x,y):=str\big(\psi(x)\circ\psi(y)\big),\,\, \forall x,y \in \mathfrak{g},
\end{equation}
where $str$ means the super trace. See \cite{CW,FSS2,Ka} for more comprehensive studies about Lie superalgebras.
\begin{remark}
There exists another well-known even supersymmetric invariant bilinear form on $\mathfrak{g}$, called the {\em Killing form}, defined by
\begin{equation*}
<x,y>:= str\big( \Psi(x)\circ \Psi(y) \big), \,\,\forall x,y\in\mathfrak{g},
\end{equation*}
where $\Psi:\mathfrak{g}\rightarrow \operatorname{End} \mathfrak{g}$ denotes the adjoint representation of $\mathfrak{g}$. The Killing form is non-degenerate for all $M\neq N$ and it plays an important rule in the classification of simple Lie superalgebras \cite{Ka}; however, it is degenerate if $M=N$ and so it is not suitable for our purpose. Note that the form (\ref{strform}) is indeed non-degenerate even when $M=N$.
\end{remark}
Let $e$ be an even nilpotent element in $\mathfrak{g}$. It can be shown (cf. \cite{Wa,Ho}) that there exists (not uniquely) a semisimple element $h\in\mathfrak{g}$ such that $\operatorname{ad} h:\mathfrak{g}\rightarrow\mathfrak{g}$ gives a {\em good $\mathbb{Z}$-grading of $\mathfrak{g}$ for $e$}, which means the following conditions are satisfied:
\begin{enumerate}
\item[(1)] $\operatorname{ad} h(e)=2e$,
\item[(2)] $\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}} \mathfrak{g}(j)$, where $\mathfrak{g}(j):=\{x\in\mathfrak{g}|\operatorname{ad} h(x)=jx\}$,
\item[(3)] the center of $\mathfrak{g}$ is contained in $\mathfrak{g}(0)$,
\item[(4)] $\operatorname{ad} e:\mathfrak{g}(j)\rightarrow\mathfrak{g}(j+2)$ is injective for all $j\leq -1$,
\item[(5)] $\operatorname{ad} e:\mathfrak{g}(j)\rightarrow\mathfrak{g}(j+2)$ is surjective for all $j\geq -1$.
\end{enumerate}
In this article we only care about the case where the grading is {\em even}; that is, we always assume that $\mathfrak{g}(i)=0$ for all $i\notin 2\mathbb{Z}$.
Define the nilpotent subalgebras of $\mathfrak{g}$ as follows:
\begin{equation}\label{mpdef}
\mathfrak{p}:=\bigoplus_{j\geq 0}\mathfrak{g}(j), \quad \mathfrak{m}:=\bigoplus_{j<0}\mathfrak{g}(j).
\end{equation}
Let $\chi\in\mathfrak{g}^*$ be defined by $\chi(y):=(y,e)$, for all $y\in\mathfrak{g}$. Then the restriction of $\chi$ on $\mathfrak{m}$ gives a one dimensional $U(\mathfrak{m})$-module. Let $I_\chi$ denote the left ideal of $U(\mathfrak{g})$ generated by $\{a-\chi(a)|a\in\mathfrak{m}\}$. By PBW theorem of $U(\mathfrak{g})$, we have $U(\mathfrak{g})=U(\mathfrak{p})\oplus I_\chi$. Let $\operatorname{pr}_\chi:U(\mathfrak{g})\rightarrow U(\mathfrak{p})$ be the natural projection and we can identify $U(\mathfrak{g})/I_\chi \cong U(\mathfrak{p})$. Furthermore we define the $\chi$-twisted action of $\mathfrak{m}$ on $U(\mathfrak{p})$ by \[a\cdot y := \operatorname{pr}_\chi([a,y]) \text{ for all }a\in\mathfrak{m} \text{ and } y\in U(\mathfrak{p}).\] An element $y\in U(\mathfrak{p})$ is said to be $\mathfrak{m}$-invariant if $a\cdot y=0$ for all $a\in\mathfrak{m}$.
The {\em finite} W{\em -superalgebra} (which we will omit the term ``finite" from now on) is defined to be the subspace of $\mathfrak{m}$-invariants of $U(\mathfrak{p})$ under the $\chi$-twisted action; that is,
\begin{align*}
\mathcal{W}_{e,h}:=U(\mathfrak{p})^\mathfrak{m}=&\{y\in U(\mathfrak{p})| \operatorname{pr}_\chi ([a,y])=0, \forall a\in\mathfrak{m}\}\\
=&\{y\in U(\mathfrak{p})| \big(a-\chi(a)\big)y\in I_\chi, \forall a\in\mathfrak{m}\}.
\end{align*}
At this point, the definition of $W$-superalgebra depends on the nilpotent element $e$ and a semisimple element $h$ which gives a good $\mathbb{Z}$-grading for $e$.
\begin{example}
If we take the nilpotent element $e=0$, then $\chi=0$, $\mathfrak{g}=\mathfrak{g}(0)=\mathfrak{p}$, $\mathfrak{m}=0$, $\mathcal{W}_{e,h}=U(\mathfrak{p})=U(\mathfrak{g})$.
\end{example}
Now we introduce certain combinatorial objects called {\em $(m,n)$-colored rectangles} \cite{Ho, BBG} (which are in fact special cases of the so called {\em pyramids}). These objects provide a diagrammatic way to record the information needed to define $W$-superalgebras.
Let $\pi$ be a rectangular Young diagram with $m+n$ boxes as its height and $\ell$ boxes as its base. We choose arbitrary $m$ rows and color the boxes in these rows by $+$, while we color the other $n$ rows by $-$. Such a diagram is called an ($m,n$)-{\em colored rectangle}, or a {\em rectangle} for short. For example,
\[
\pi={\begin{picture}(90, 65)%
\put(15,-10){\line(1,0){60}}
\put(15,5){\line(1,0){60}}
\put(15,20){\line(1,0){60}}
\put(15,35){\line(1,0){60}}
\put(15,50){\line(1,0){60}}
\put(15,65){\line(1,0){60}}
\put(15,-10){\line(0,1){75}}
\put(30,-10){\line(0,1){75}}
\put(45,-10){\line(0,1){75}}
\put(60,-10){\line(0,1){75}}
\put(75,-10){\line(0,1){75}}
\put(18,-5){$-$}\put(33,-5){$-$}\put(48,-5){$-$}\put(63,-5){$-$}
\put(18,10){$-$}\put(33,10){$-$}\put(48,10){$-$}\put(63,10){$-$}
\put(18,25){$+$}\put(33,25){$+$}\put(48,25){$+$}\put(63,25){$+$}
\put(18,40){$-$}\put(33,40){$-$}\put(48,40){$-$}\put(63,40){$-$}
\put(18,55){$+$}\put(33,55){$+$}\put(48,55){$+$}\put(63,55){$+$}
\end{picture}}\\[4mm]
\]
Set $M=m\ell$ and $N=n\ell$. We enumerate the \begin{young}(+)\end{young} boxes by the numbers $\{\overline{1},\ldots, \overline{M}\}$ down columns from left to right, and enumerate the \begin{young}(-)\end{young} boxes by the numbers $\{1,\ldots,N\}$ in the same fashion. In fact, we may enumerate the boxes by an arbitrary order as long as the parities are preserved so we just choose the easiest way according to our purpose. Moreover, we image that each box of $\pi$ is of size $2\times 2$ and $\pi$ is built on the $x$-axis where the center of $\pi$ is exactly on the origin. For example,
\begin{equation}\label{exp}
\pi={\begin{picture}(90, 110)%
\put(15,-10){\line(1,0){80}}
\put(15,10){\line(1,0){80}}
\put(15,30){\line(1,0){80}}
\put(15,50){\line(1,0){80}}
\put(15,70){\line(1,0){80}}
\put(15,90){\line(1,0){80}}
\put(15,-10){\line(0,1){100}}
\put(35,-10){\line(0,1){100}}
\put(55,-10){\line(0,1){100}}
\put(75,-10){\line(0,1){100}}
\put(95,-10){\line(0,1){100}}
\put(23,-3){$3$}\put(43,-3){$6$}\put(63,-3){$9$}\put(80,-3){$12$}
\put(23,16){$2$}\put(43,16){$5$}\put(63,16){$8$}\put(80,16){$11$}
\put(23,35){$\overline{2}$}\put(43,35){$\overline{4}$}\put(63,35){$\overline{6}$}\put(82,35){$\overline{8}$}
\put(23,55){$1$}\put(43,55){$4$}\put(63,55){$7$}\put(80,55){$10$}
\put(23,75){$\overline{1}$}\put(43,75){$\overline{3}$}\put(63,75){$\overline{5}$}\put(82,75){$\overline{7}$}
\put(0,-20){\line(1,0){110}}
\put(53,-23){$\bullet$}
\put(63,-35){$1$}\put(83,-35){$3$}
\put(35,-35){$-1$}\put(15,-35){$-3$}
\end{picture}}\\[10mm]
\end{equation}
We now explain how to read off an even nilpotent $e(\pi)$ and a semisimple $h(\pi)$ in $\mathfrak{g}=\mathfrak{gl}_{M|N}$ which gives a good $\mathbb{Z}$-grading of $\mathfrak{g}$ for $e$ from a given rectangle $\pi$. Let $J=\{\overline{1}<\ldots<\overline{M}<1<\ldots<N\}$ be an ordered index set and let $\{e_i|i\in J\}$ be a basis of $\mathbb{C}^{M|N}$. We identify $\mathfrak{gl}_{M|N}\cong$ End $(\mathbb{C}^{M|N})$ with the $(M+N)\times(M+N)$ matrices over $\mathbb{C}$ by this basis of $\mathbb{C}^{M|N}$ with respect to the order $e_i<e_j$ if $i<j$ in $J$.
Define the element
\begin{equation}\label{edef}
e(\pi):=\sum_{\substack{i,j\in J}}e_{i,j}\in \mathfrak{gl}_{M|N},
\end{equation}
summing over all adjacent pairs $\young(ij)$ of boxes in $\pi.$ It is clear that such an element $e(\pi)$ is even nilpotent.
Let $\widetilde{\text{col}}(i)$ denote the $x$-coordinate of the box numbered with $i\in J$. For instance, in our example (\ref{exp}),
$\widetilde{\text{col}}(\overline{1})=-3$ and $\widetilde{\text{col}}(8)=1$.
Then we define the following diagonal matrix
\begin{equation}\label{hdef}
h(\pi):=-\text{diag}\big(\widetilde{\text{col}}(\overline{1}), \ldots, \widetilde{\text{col}}(\overline{M}), \widetilde{\text{col}}(1),\ldots, \widetilde{\text{col}}(N)\big)\in\mathfrak{g}.
\end{equation}
One may check directly that $\operatorname{ad} h(\pi)$ gives a good $\mathbb{Z}$-grading of $\mathfrak{g}$ for $e(\pi)$.
\begin{remark}\label{choiceofh}
In general, there are other even good $\mathbb{Z}$-gradings for $e$. However, if our $e=e(\pi)$ is obtained by a rectangle $\pi$ according to (\ref{edef}), then such a grading is unique (cf. \cite[Theorem 7.2]{Ho}).
\end{remark}
Now we characterize those $e$ obtained from (\ref{edef}) for some rectangle $\pi$. Consider
\begin{equation}\label{edecomp}
e=e_M\oplus e_N\in \operatorname{End} \mathbb{C}^{M|N},
\end{equation}
where $e_M$ and $e_N$ are the restrictions of $e$ to $\mathbb{C}^{M|0}$ and $\mathbb{C}^{0|N}$, respectively. Let $\mu=(\mu_1,\mu_2,\ldots)$ and $\nu=(\nu_1,\nu_2,\ldots)$ be the partitions representing the Jordan type of $e_M$ and $e_N$, respectively.
\begin{definition}
An element $e$ is called {\em rectangular} if it is even nilpotent and the Jordan blocks of $e_M$ and $e_N$ are all of the same size $\ell$, i.e., $\mu=(\,\overbrace{\ell,\ldots,\ell}^{m-copies}\,)$ and $\nu=(\,\overbrace{\ell,\ldots,\ell}^{n-copies}\,)$ for some for some non-negative integers $\ell,m,n$.
\end{definition}
Clearly, $e$ is of the form (\ref{edef}) for some rectangle $\pi$ if and only if $e$ is rectangular. Assume now $e$ is rectangular. We define a new partition $\lambda$ by collecting all parts of $\mu$ and $\nu$ together and reorder them by an arbitrary order. Since all the parts are the same number $\ell$, we use $\overline{\ell}$ to denote the parts obtained from $\mu$.
For example, consider $\mathfrak{gl}_{8|12}$, $\mu=(\overline{4},\overline{4})$ and $\nu=(4,4,4)$. Then one possible $\lambda$ is $\lambda=(\overline{4},4,\overline{4},4,4)$.
Next we read off an $\epsilon$-$\delta$ sequence $\mathfrak{b}$ from $\lambda:$ if the $i$-th position of $\lambda$ is $\ell$ (respectively, $\overline{\ell}$), then the $i$-th position of $\mathfrak{b}$ is $\epsilon$ (respectively, $\delta$). For example, the $\lambda$ above corresponds to the $\epsilon$-$\delta$ sequence $\mathfrak{b}=\delta\epsilon\delta\epsilon\ep$.
Then we color the rectangle of height $m+n$ base $\ell$ with respect to $\mathfrak{b}$: we color the $i$-th row of $\pi$ by + (respectively, $-$) if the $i$-th position of $\mathfrak{b}$ is $\delta$ (respectively, $\epsilon$) where the rows are counted from top to bottom. After coloring the rows, we enumerate the boxes of $\pi$ by exactly the same fashion explained in the paragraph before (\ref{exp}).
Therefore, we have a bijection between the set of $(m,n)$-colored rectangles of base $\ell$ and the set of pairs $(e,\mathfrak{b})$ where $e$ is a rectangular element in $\mathfrak{gl}_{m\ell|n\ell}$ and $\mathfrak{b}$ is an $\epsilon$-$\delta$ sequence containing exactly $m$ $\delta$'s and $n$ $\epsilon$'s.
Let $\pi$ be a fixed $(m,n)$-colored rectangle and $e(\pi)$ denote the rectangular element associates to $\pi$. We will denote by $\mathcal{W}_\pi:=\mathcal{W}_{e(\pi)}$ the $W$-superalgebra associated to $e(\pi)$. Note that we may omit $h(\pi)$ in our notation with no ambiguity due to Remark \ref{choiceofh}.
\begin{remark}\label{Wind}
An interesting observation is that $\mathcal{W}_{\pi}$ is independent of the choices of the sequence $\mathfrak{b}$ because any other sequence $\mathfrak{b}^\prime$ yields to the same $e$ and hence the same $W$-superalgebra. This is parallel to the fact that $Y_{m|n}^\ell$ is independent of the choice of $\mathfrak{b}$.
\end{remark}
Now we label the columns of $\pi$ from left to right by $1,\ldots, \ell$, and for any $i\in J$ we let col($i$) denote the column where $i$ appears. Define the {\em Kazhdan filtration} of $U(\mathfrak{gl}_{M|N})$
\[
\cdots\subseteq F_dU(\mathfrak{gl}_{M|N}) \subseteq F_{d+1}U(\mathfrak{gl}_{M|N})\subseteq \cdots
\]
by declaring
\begin{equation}\label{degdef}
\deg (e_{i,j}):= \text{col}(j)-\text{col}(i)+1
\end{equation}
for each $i,j \in J$ and $F_dU(\mathfrak{gl}_{M|N})$ is the span of all supermonomials $e_{i_1,j_1}\cdots e_{i_s,j_s}$ for $s\geq 0$ and $\sum_{k=1}^s$ deg $(e_{i_k,j_k})\leq d$. Let $\operatorname{gr} U(\mathfrak{gl}_{M|N})$ denote the associated graded superalgebra. The natural projection $\mathfrak{gl}_{M|N}\twoheadrightarrow\mathfrak{p}$ induces a grading on $\mathcal{W}_\pi$.
On the other hand, let $\mathfrak{g}^e$ denote the centralizer of $e$ in $\mathfrak{g}=\mathfrak{gl}_{M|N}$ and let $S(\mathfrak{g}^e)$ denote the associated supersymmetric algebra. We define the Kazhdan filtration on $S(\mathfrak{g}^e)$ by the same setting (\ref{degdef}). The following proposition was observed in \cite{Zh}, where the mild assumption on $e$ there is satisfied when $e$ is rectangular.
\begin{proposition}\label{dimprop}\cite[Remark 3.9]{Zh}
$\operatorname{gr} \mathcal{W}_{\pi}\cong S(\mathfrak{g}^e)$ as Kazhdan graded superalgebras.
\end{proposition}
The following proposition is a well-known result about $\mathfrak{g}^e$. As remarked in \cite{BBG}, the result is similar to the Lie algebra case $\mathfrak{gl}_{M+N}$ since $e$ is even.
\begin{proposition}\label{counting2}
Let $\pi$ be an (m,n)-colored rectangle and $e=e(\pi)$ be the associated rectangular nilpotent element. For all $1\leq i,j\leq m+n$ and $r>0$, define
\[
c_{i,j}^{(r)}:=
\sum_{\substack{1\leq h,k\leq m+n \\ \text{row}(h)=i, \text{row}(k)=j\\ \text{col}(k)-\text{col}(h)=r-1}} (-1)^{|i|}e_{h,k}\in \mathfrak{g}=\mathfrak{gl}_{M|N}.
\]
Then the set of vectors $\lbrace c_{i,j}^{(r)}|1\leq i,j\leq m+n, 1\leq r\leq \ell \rbrace$ forms a basis for $\mathfrak{g}^e$.
\end{proposition}
\begin{corollary}\label{dimcoro}
Consider $Y_{m|n}^{\ell}$ with the canonical filtration and $S(\mathfrak{g}^e)$ with the Kazhdan filtration. Let $F_dY_{m|n}^{\ell}$ and $F_dS(\mathfrak{g}^e)$ denote the associated filtered algebras, respectively. Then for each $d\geq 0$, we have $\dim F_dY_{m|n}^{\ell} = \dim F_dS(\mathfrak{g}^e)$.
\end{corollary}
\begin{proof}
Follows from Proposition \ref{PBWSY}, Proposition \ref{counting2} and induction on $d$.
\end{proof}
\section{Isomorphism between $Y_{m|n}^{\ell}$ and $\mathcal{W}_\pi$}
Let $\pi$ be a given $(m,n)$-colored rectangle with base $\ell$ and $\mathfrak{b}$ be the $\epsilon$-$\delta$ sequence determined by the colors of rows of $\pi$. We now define some elements in $U(\mathfrak{p})$. It turns out later that they are $\mathfrak{m}$-invariant, i.e., belong to $\mathcal{W}_\pi$.
For each $1\leq r\leq \ell$, set
\begin{equation}\label{rhodef}
\rho_r := -(\ell-r)(m-n).
\end{equation}
For $1\leq i,j\leq m+n$, define
\begin{equation}\label{etil}
\tilde e_{i,j} := (-1)^{\text{col}(j)-\text{col}(i)}
(e_{i,j} + \delta_{ij} (-1)^{|i|}\rho_{\text{col}(i)}),
\end{equation}
where $|i|$ is determined by $\mathfrak{b}$ as in (\ref{parity}).
One may check that \begin{multline}\label{etilrel}
[\tilde e_{i,j}, \tilde e_{h,k}]
=(\tilde{e}_{i,k} - \delta_{ik} (-1)^{|i|} \rho_{\text{col}(i)})\delta_{hj}\\
- (-1)^{(|i|+|j|)(|h|+|k|)}
\delta_{i,k} (\tilde e_{h,j} - \delta_{hj} (-1)^{|j|}\rho_{\text{col}(j)}).
\end{multline}
Also, for any $1\leq i,j\leq m+n$, we have
\begin{equation}\label{chidef}
\chi (\tilde e_{i,j}) = \left\{
\begin{array}{ll}
(-1)^{|i|+1}&\hbox{if $\text{row}(i)=\text{row}(j)$ and $\text{col}(i) = \text{col}(j)+1$;}\\
0&\hbox{otherwise.}
\end{array}\right.
\end{equation}
For $1\leq i,j\leq m+n$,
we let
$t_{ ij;\mathfrak{b}}^{(0)} := \delta_{i,j}$
and for $1\leq r \leq \ell$, define
\begin{equation}\label{tdef}
t_{ij;\mathfrak{b}}^{(r)}
:=
\sum_{s = 1}^r
\sum_{\substack{i_1,\dots,i_s\\j_1,\dots,j_s}}
(-1)^{|i_1|+\cdots+|i_s|}
\tilde e_{i_1,j_1} \cdots \tilde e_{i_s,j_s}
\end{equation}
where the second sum is over all $i_1,\dots,i_s,j_1,\dots,j_s\in J$
such that
\begin{itemize}
\item[(1)] $\deg(e_{i_1,j_1})+\cdots+\deg(e_{i_s,j_s}) = r$;
\item[(2)] $\text{col}(i_t) \leq \text{col}(j_t)$ for each $t=1,\dots,s$;
\item[(3)] $\text{col}(j_t) < \text{col}(i_{t+1})$ for each $t=1,\dots,s-1$;
\item[(4)] $\text{row}(i_1)=i$, $\text{row}(j_s) = j$;
\item[(5)]
$\text{row}(j_t)=\text{row}(i_{t+1})$ for each $t=1,\dots,s-1$.
\end{itemize}
First note that these elements depends on the choice of $\mathfrak{b}$. Also, the restrictions (1) and (2) imply that $t_{ij;\mathfrak{b}}^{(r)}$ belongs to $\mathrm{F}_r U(\mathfrak p)$ with respect to the Kazhdan grading. Define the following series for all $1\leq i,j\leq m+n$:
\begin{equation}\label{tseries}
t_{ij;\mathfrak{b}}(u) := \sum_{r = 0}^\ell t_{ij;\mathfrak{b}}^{(r)} u^{-r}
\in U(\mathfrak p) [[u^{-1}]].
\end{equation}
Now we prove that these distinguished elements in $U(\mathfrak{p})$ are in fact $\mathfrak{m}$-invariant under the $\chi$-twisted action. Let $T(Mat_\ell)$ be the tensor algebra of the $\ell\times \ell$ matrices space over $\mathbb{C}$ and $\mathfrak{g}=\mathfrak{gl}_{M|N}$ where $M=m\ell$, $N=n\ell$.
For all $1\leq i,j\leq m+n$, define a $\mathbb{C}$-linear map $t_{ij;\mathfrak{b}}:T(Mat_\ell)\rightarrow U(\mathfrak{g})$ inductively by
\[
t_{ij;\mathfrak{b}}(1):=\delta_{i,j},\qquad\quad t_{ij;\mathfrak{b}}(e_{a,b}):=(-1)^{|i|}e_{i\star a,j\star b},
\]
\begin{equation}\label{uptij1}
t_{ij;\mathfrak{b}}(x_1\otimes x_2\otimes\ldots \otimes x_r):= \sum_{1\leq i_1,i_2,\ldots,i_{r-1}\leq m+n} t_{ii_1;\mathfrak{b}}(x_1)t_{i_1i_2;\mathfrak{b}}(x_2)\cdots t_{i_{r-1}j;\mathfrak{b}}(x_r),
\end{equation}
for $1\leq a,b\leq \ell$, $r\geq 1$ and $x_1,\ldots,x_r\in Mat_\ell$, where $i\star a$ is defined to be the number in the $(i,a)$-th position of $\pi$, where we label the rows and columns from top to bottom and from left to right. For example, let $\pi$ be the rectangle in (\ref{exp}), then $t_{23;\mathfrak{b}}(e_{2,4})=(-1)^{|2|}e_{2\star 2, 3\star 4}=-e_{4,\overline{8}}$. For an indeterminate $u$, we extend the scalars from $\mathbb{C}$ to $\mathbb{C}[u]$ in the obvious way.
\begin{lemma}
For each $1\leq i,j,h,k\leq m+n$ and $x,y_1,\ldots, y_r\in Mat_\ell$, we have
\begin{align}\notag
[t_{ij;\mathfrak{b}}(x),t_{hk;\mathfrak{b}}(y_1\otimes\cdots\otimes y_r)]&=\\
\notag (-1)^{|i|\,|j|+|i|\,|h|+|j|\,|h|}\big(& \sum_{s=1}^r t_{hj;\mathfrak{b}}(y_1\otimes\cdots\otimes y_{s-1})t_{ik;\mathfrak{b}}(xy_s\otimes\cdots \otimes y_r)\\
&\,-t_{hj;\mathfrak{b}}(y_1\otimes\cdots\otimes y_{s}x)t_{ik;\mathfrak{b}}(y_{s+1}\otimes\cdots\otimes y_{r})
\big),\label{detcomp1}
\end{align}
where the products $xy_s$ and $y_sx$ on the right are the matrix products in $Mat_\ell$.
\end{lemma}
\begin{proof}
By (\ref{uptij1}) and induction on $r$.
\end{proof}
Introducing the following matrix $A(u)$ with entries in the algebra $T(Mat_\ell)[u]$:
$$
A(u) =
\left(
\begin{array}{ccccc}
u+e_{1,1}+\rho_1 & e_{1,2} & e_{1,3} & \cdots & e_{1,\ell}\\
1 & u+e_{2,2}+\rho_2 & e_{2,3} & &e_{2,\ell}\\
0&1&\ddots& &\vdots\\
\vdots& & 1 & u+e_{\ell-1,\ell-1}+\rho_{\ell-1} & e_{\ell-1,\ell}\\
0&\cdots & 0 & 1 & u+e_{\ell,\ell}+\rho_\ell
\end{array}
\right).\:
$$
A key observation is that for all $1\leq i,j\leq m+n$ and $0\leq r\leq \ell$, the element $t_{ij;\mathfrak{b}}^{(r)}$ of $U(\mathfrak{p})$ defined by (\ref{tdef}) equals to the coefficient of the term $u^{\ell-r}$ in $t_{ij;\mathfrak{b}}\big(\operatorname{rdet} A(u)\big)$, where
\[
\operatorname{rdet} A:= \sum_{\tau\in S_l}\operatorname{sgn}(\tau)a_{1,\tau(1)}a_{2,\tau(2)}\cdots a_{l,\tau(l)},
\]
for a matrix $A=(a_{i,j})_{1\leq i,j\leq \ell}$. We also let $A_{p,q}(u)$ stand for the submatrix of $A(u)$ consisting only of rows and columns numbered by $p,\ldots,q$.
\begin{proposition}\label{recminv}
For all $1\leq i,j\leq m+n$, $0\leq r\leq \ell$ and a fixed $\mathfrak{b}$, the elements $t_{ij;\mathfrak{b}}^{(r)}$ of $U(\mathfrak{p})$ are $\mathfrak{m}$-invariant under the $\chi$-twisted action.
\end{proposition}
\begin{proof}
Firstly we observe that the statement is trivial when $\ell\leq 1$, hence we may assume $\ell\geq 2$. Note that $\mathfrak{m}$ is generated by elements of the form $t_{ij;\mathfrak{b}}(e_{c+1,c})$, hence it suffices to show that
\[
\operatorname{pr}_\chi\Big( \big[t_{ij;\mathfrak{b}}(e_{c+1,c}), t_{hk;\mathfrak{b}}\big(\operatorname{rdet} A(u)\big)\big] \Big)=0
\]
for all $1\leq i,j,h,k\leq m+n$ and $1\leq c\leq \ell-1$. In this proof, we omit the fixed $\mathfrak{b}$ in our notation which shall cause no confusion.
By (\ref{detcomp1}), up to an irrelevant sign, we have
\begin{align*}
\Big[t_{ij}&\big(e_{c+1,c}\big), t_{hk}\big(\operatorname{rdet} A(u)\big)\Big]=\\
&t_{hj}\big(\operatorname{rdet} A_{1,c-1}(u)\big) t_{ik}(\operatorname{rdet} \left(
\begin{array}{cccc}
e_{c+1,c} & e_{c+1,c+1} & \cdots & e_{c+1,\ell}\\
1 & u+e_{c+1,c+1}+\rho_{c+1} & \cdots &e_{c+1,\ell}\\
\vdots & &\ddots &\vdots\\
0& \cdots & 1 & u+e_{\ell,\ell}+\rho_{\ell}
\end{array}
\right))\\
&-t_{hj}(\operatorname{rdet} \left(
\begin{array}{cccc}
u+e_{1,1}+\rho_1 & \cdots & e_{1,c} & e_{1,c}\\
1 & \ddots & &\vdots\\
\vdots & & u+e_{c,c}+\rho_c & e_{c,c}\\
0& \cdots & 1 & e_{c+1,c}
\end{array}
\right)) t_{ik}\big(\operatorname{rdet} A_{c+2,\ell}(u)\big).
\end{align*}
A crucial observation is that for $1\leq i,j\leq m+n$ and $1\leq c\leq \ell-1$, we have
\begin{equation}\label{crue}
t_{ij}\big(e_{c+1,c}(u+e_{c+1,c+1}+\rho_{c+1})\big)-t_{ij}\big(u+e_{c+1,c+1}+\rho_c\big) \equiv 0 \quad (\text{mod} \,\,I_\chi).
\end{equation}
Indeed, it is enough to check (\ref{crue}) when $\ell=2$. The trick here is to rewrite the quadratic terms in $U(\mathfrak{g})$ by supercommtator relations. Then the term $\rho_c$ gives exactly the required constant so that the left-hand side of (\ref{crue}) can be expressed by a linear combination of elements in $I_{\chi}$.
Therefore,
\begin{align*}
\Big[t_{ij}&\big(e_{c+1,c}\big), t_{hk}\big(\operatorname{rdet} A(u)\big)\Big]\equiv\\
&t_{hj}\big(\operatorname{rdet} A_{1,c-1}(u)\big) t_{ik}(\operatorname{rdet} \left(
\begin{array}{cccc}
1 & e_{c+1,c+1} & \cdots & e_{c+1,\ell}\\
1 & u+e_{c+1,c+1}+\rho_{c} & \cdots &e_{c+1,\ell}\\
\vdots & &\ddots &\vdots\\
0& \cdots & 1 & u+e_{\ell,\ell}+\rho_{\ell}
\end{array}
\right))\\
&-t_{hj}(\operatorname{rdet} \left(
\begin{array}{cccc}
u+e_{1,1}+\rho_1 & \cdots & e_{1,c} & e_{1,c}\\
1 & \ddots & &\vdots\\
\vdots & & u+e_{c,c}+\rho_c & e_{c,c}\\
0& \cdots & 1 & 1
\end{array}
\right)) t_{ik}\big(\operatorname{rdet} A_{c+2,\ell}(u)\big)
\end{align*}
modulo $I_\chi$. Making the obvious row and column operations gives that
\[
\operatorname{rdet} \left(
\begin{array}{cccc}
1 & e_{c+1,c+1} & \cdots & e_{c+1,\ell}\\
1 & u+e_{c+1,c+1}+\rho_{c} & \cdots &e_{c+1,\ell}\\
\vdots & &\ddots &\vdots\\
0& \cdots & 1 & u+e_{\ell,\ell}+\rho_{\ell}
\end{array}
\right)=(u+\rho_c)\operatorname{rdet} A_{c+2,\ell}(u),
\]
\[
\operatorname{rdet} \left(
\begin{array}{cccc}
u+e_{1,1}+\rho_1 & \cdots & e_{1,c} & e_{1,c}\\
1 & \ddots & &\vdots\\
\vdots & & u+e_{c,c}+\rho_c & e_{c,c}\\
0& \cdots & 1 & 1
\end{array}
\right)=(u+\rho_c)\operatorname{rdet} A_{1,c-1}(u).
\]
Substituting these back shows that
$\operatorname{pr}_\chi \Big( \big[t_{ij}(e_{c+1,c}), t_{hk}\big(\operatorname{rdet} A(u)\big)\big] \Big) =0$.
\end{proof}
Set a new notation
$t_{ij;\mathfrak{b}}(e_{r,r}) = (-1)^{|i|} e_{i,j}^{[r]}\in U(\mathfrak{h})$, where $\mathfrak{h}\cong ({\mathfrak{gl}_{m|n}})^{\oplus \ell}$.
Clearly, $\mathfrak{h}$ has a basis $\{ e_{i,j}^{[r]}|1\leq r\leq \ell, 1\leq i,j\leq m+n\}$.
Define \[\eta:U(\mathfrak{h})\rightarrow U(\mathfrak{h}), \quad e_{i,j}^{[r]}\mapsto e_{i,j}^{[r]}-\delta_{ij}\rho_r.\]
Let $\xi:U(\mathfrak{p})\rightarrow U(\mathfrak{h})$ be the algebra homomorphism induced from the natural projection $\mathfrak{p}\twoheadrightarrow \mathfrak{h}$. Define the map
$\mu:= \eta \circ \xi :U(\mathfrak{p})\rightarrow U(\mathfrak{h})\cong U(\mathfrak{gl}_{m|n})^{\otimes \ell}$.
\begin{lemma}\label{muimg}
For $1\leq i,j\leq m+n$ and $r>0$,
\[
\mu(t_{ij;\mathfrak{b}}^{(r)})=\sum_{1\leq s_1<\cdots<s_r\leq \ell}\,\,\sum_{1\leq i_1,\ldots, i_{r-1}\leq m+n}
(-1)^{|i|+|i_1|+\cdots+|i_{r-1}|}
e_{i,i_1}^{[s_1]}e_{i_1,i_2}^{[s_2]}\cdots e_{i_{r-1},j}^{[s_r]}.
\]
\end{lemma}
\begin{proof}
Applying the map $e_{r,s}\mapsto \delta_{r,s} (e_{r,r}-\rho_r)$ to the matrix $A(u)$ gives a diagonal matrix with determinant $(u+e_{1,1})(u+e_{2,2})\cdots (u+e_{\ell,\ell})$, where its $u^{\ell-r}$-coefficient equals to
$\sum_{1\leq s_1<\cdots<s_r\leq \ell} e_{s_1,s_1}e_{s_2,s_2}\cdots e_{s_r,s_r}$. Now the lemma follows from (\ref{uptij1}).
\end{proof}
Now we are ready to state and prove our main result of this article.
\begin{theorem}\label{main}
Let $\pi$ be an $(m,n)$-colored rectangle and $\mathfrak{b}$ be the $\epsilon$-$\delta$ sequence determined by the rows of $\pi$. Then there exists an isomorphism $Y_{m|n}^{\ell}\cong \mathcal{W}_\pi$ of filtered superalgebras such that the generators \[\{t_{ij;\mathfrak{b}}^{(r)}|1\leq i,j\leq m+n, 1\leq r\leq \ell\}\] of $Y_{m|n}^{\ell}$ are sent to the elements of $\mathcal{W}_\pi$ with the same names defined by (\ref{tdef}).
\end{theorem}
\begin{proof}
Again, the result is trivial when $\ell\leq 1$ so we assume that $\ell\geq 2$. By Proposition \ref{PBWSY}, the set of all supermonomials in the elements \[\{t_{ij;\mathfrak{b}}^{(r)}|1\leq i,j\leq m+n, 1\leq r\leq \ell\}\] of $Y_{m|n}^{\ell}$ taken in some fixed order and of total degree $\leq d$ forms a basis for $F_d Y_{m|n}^{\ell}$ (with respect to the canonical filtration). Since $\kappa_\ell$ is injective, by Corollary \ref{dimcoro} we have
\[
\dim \kappa_\ell(F_dY_{m|n}^{\ell}) = \dim F_d Y_{m|n}^{\ell} = \dim F_d S(\mathfrak{g}^e),
\]
where $S(\mathfrak{g}^e)$ is equipped with the Kazhdan grading.
Let $X_d$ denote the subspace of $U(\mathfrak{p})$ spanned by all supermonomials in the elements $\{t_{ij;\mathfrak{b}}^{(r)}|1\leq i,j\leq m+n, 1\leq r\leq \ell\}$ defined by (\ref{tdef}) taken in some fixed order and of total degree $\leq d$. By Lemma \ref{muimg} and the discussion above, we have $\mu(X_d)=\kappa_\ell(F_dY_{m|n}^{\ell})$.
Proposition \ref{recminv} assures that $X_d\subseteq F_d\mathcal{W}_\pi$. Together with Proposition \ref{dimprop}, we have
\[
\dim F_dS(\mathfrak{g}^e)=\dim \mu(X_d)\leq \dim X_d\leq \dim F_d\mathcal{W}_\pi\leq \dim F_dS(\mathfrak{g}^e).
\]
Thus equality holds everywhere and hence $X_d=F_d\mathcal{W}_\pi$. Moreover, $\mu$ is injective and $\mu(t_{ij;\mathfrak{b}}^{(r)})=\kappa_\ell(t_{ij;\mathfrak{b}}^{(r)})$ for all $1\leq i,j\leq m+n$ and $0\leq r\leq \ell$. The composition $\mu^{-1}\circ \kappa_\ell:Y_{m|n}^{\ell}\rightarrow \mathcal{W}_\pi$ gives the required isomorphism.
\end{proof}
\begin{remark}
The connection between $W$-algebras of $so(n)$ and $sp(n)$ and their corresponding (twisted) Yangians has not been studied in full generality. Some partial results can be found in \cite{BR2,Ra}; see also \cite{Br} for an approach similar to this article.
\end{remark}
\subsection*{Acknowledgements}
The author would like to thank the anonymous reviewers
for their valuable comments and suggestions to improve the
quality of this article. The author is supported by
post-doctoral fellowship of the
Institution of Mathematics, Academia Sinica, Taipei, Taiwan.
|
{
"timestamp": "2013-08-16T02:05:48",
"yymm": "1304",
"arxiv_id": "1304.3913",
"language": "en",
"url": "https://arxiv.org/abs/1304.3913"
}
|
\section{Introduction}
\label{S:intro}
We consider a compact periodic surface of revolution $X = {\mathbb S}^1_x \times {\mathbb S}^1_\theta$, equipped with a metric of the form
\[
ds^2 = d x^2 + A^2(x) d \theta^2,
\]
where $A \in {\mathcal C}^\infty$ is a smooth function, $A \geq \epsilon>0.$ Our
analysis is microlocal, so applies also to any compact surface of revolution with no
boundary, and to certain surfaces of revolution with boundary under
mild assumptions, however we will concentrate on the toral case for
ease of exposition.
From such a metric, we get the volume form
\[
d \text{Vol} = A(x) dx d \theta,
\]
and the Laplace-Beltrami operator acting on $0$-forms
\[
\Delta f = (\partial_x^2 + A^{-2} \partial_\theta^2 + A^{-1}
A' \partial_x) f.
\]
We are concerned with quasimodes, which are the building blocks from
which eigenfunctions are made, however we need to define the most
basic kind of quasimodes, which we will call {\it irreducible
quasimodes}, meaning the quasimodes which cannot be decomposed as a
sum of two or more nontrivial quasimodes. In order to make our
definitions, we recall first that the geodesic flow on $T^*X$
is the Hamiltonian system associated to the principal symbol of the
Laplace-Beltrami operator:
\[
p(x, \xi, \theta, \eta) = \xi^2 + A^{-2}(x) \eta^2.
\]
A fixed energy level $p = \text{ const.}$ consists of all the
geodesics of that constant ``speed''.
For the case of the geodesic Hamiltonian system on
$T^*X$, there are two conserved quantities, the total energy and the
angular momentum $\eta^2$. The {\it moment map} is the map sending
points of $T^*X$ to their associated conserved quantities, that is
\[
M(x, \xi, \theta, \eta) = \left( \begin{array}{c} \xi^2 + A^{-2}(x)
\eta^2 \\ \eta^2 \end{array} \right).
\]
When the gradient of $M$ has rank $2$, then $M$ defines a submersion,
so each connected component of the preimage is a $2$-manifold.
Points in $T^*X$ where $M$ has rank $1$ or $0$ are called critical points,
and points
$(P,Q) \in {\mathbb R}^2$ such that $\{ M = (P,Q) \}$ contains critical
points are called critical values. Critical points
correspond to
latitudinal periodic geodesics, which can also carry quasimode mass,
and critical values have preimages which may have infinitely many
latitudinal periodic geodesics.
The semiclassical wavefront set is always a closed invariant subset of the
energy surface, so our definition of irreducible quasimode will be one which has wavefront mass confined to the closure of one of
these two kinds of sets, distinguished by rank of $M$.
\begin{definition}
An {\it irreducible} quasimode is a quasimode whose semiclassical
wavefront set is contained in the closure of a single connected
component in $T^*X$ where the moment map has constant rank.
\end{definition}
We also will require a limit on the geodesic complexity by assuming
there are only a finite number of connected regions of latitudinal
periodic geodesics. This will not preclude having infinitely many
periodic latitudinal geodesics, but merely having accumulation points
of connected components of latitudinal geodesics. We therefore will
assume that the moment map has a finite number of critical values,
each of which has a preimage of finitely many non-empty connected
components. Note this allows intervals of latitudinal periodic
geodesics, but does not allow accumulation of such sets. For an
example, see Figure \ref{fig:ex-pot}.
\begin{figure}
\hfill
\centerline{\input{ex-pot}}
\caption{\label{fig:ex-pot} The reduced phase space of a toral surface of revolution with many
periodic latitudinal geodesics.}
\hfill
\end{figure}
Finally, we will require a certain $0$-Gevrey regularity on the manifold,
which in a sense says our manifold is not too far from being
analytic.
Such a $0$-Gevrey assumption nevertheless allows for
non-trivial functions which are constant on intervals, so this is a
very general class of manifolds. Of course this includes analytic
manifolds, for which we have a stronger estimate. See Subsection \ref{SS:gevrey} for the precise definitions.
\begin{theorem}
\label{T:T1}
Let $X$ be as above, for a generating curve in the $0$-Gevrey class $A(x)
\in \mathcal{G}^0_\tau( {\mathbb R})$ for some $\tau < \infty$. Assume the moment
map has finitely many critical values, with preimages consisting of
finitely many connected components. Suppose $u$ is a (weak)
irreducible quasimode satisfying $\| u \| = 1$ and
\[
(-\Delta - \lambda^2) u = {\mathcal O}( \lambda^{-\beta_0} ),
\]
for some fixed $\beta_0 >0$. Let $\Omega \subset
X$ be a rotationally invariant neighbournood, $\Omega = (a,b)_x \times {\mathbb S}_\theta^1$. Then either
\begin{enumerate}
\item
\[
\| u \|_{L^2(\Omega)} = {\mathcal O}( \lambda^{-\infty} ),
\]
or
\item
for any $\epsilon>0$, there exists $C = C_{\epsilon, \Omega, \beta_0}>0$ such that
\begin{equation}
\label{E:lower-bound}
\| u \|_{L^2(\Omega)} \geq C \lambda^{-1 - \epsilon}.
\end{equation}
\end{enumerate}
\end{theorem}
\begin{remark}
The proof will show that a more or less straightforward
commutator/contradiction argument gives a lower bound of
$\lambda^{-1-\beta_0}$. The difficulty comes in trying to beat this
lower bound.
\end{remark}
In the analytic category, we have a significant improvement. Of
course in the case of an analytic manifold, there can be no infinitely
degenerate critical elements, nor can there be any accumulation points
of sets of
latitudinal periodic geodesics, so we do not need to make the
assumption about finite geodesic complexity.
\begin{corollary}
\label{C:C1}
Let $X$ be as above, and assume $X$ is analytic. Suppose $u$ is a (weak)
irreducible quasimode satisfying $\| u \| = 1$ and
\[
(-\Delta - \lambda^2) u = {\mathcal O}( 1 ).
\]
Then for any open rotationally invariant neighbourhood $\Omega \subset
X$, either
\begin{enumerate}
\item
\[
\| u \|_{L^2(\Omega)} = {\mathcal O}( \lambda^{-\infty} ),
\]
or
\item
there exists a fixed $\delta>0$ and a constant $C = C_{\Omega}>0$ such that
\[
\| u \|_{L^2(\Omega)} \geq C \lambda^{-1 + \delta}.
\]
\end{enumerate}
\end{corollary}
\begin{remark}
The assumption that $\Omega \subset X$ is a rotationally invariant neighbourhood of
the form $\Omega = (a,b)_x \times {\mathbb S}_\theta^1$ is necessary for this level
of generality. To see this, consider the case where $X$ has part of a
2-sphere embedded in it. Then there are many periodic geodesics close
to the latitudinal one. But these geodesics can be rotated in
$\theta$ without changing the angular momentum. Each one of these is
elliptic and can carry a Gaussian beam type quasimode. Hence one can
create an irreducible quasimode as a superposition of these Gaussian
beams. The resulting ``band'' of quasimodes need not have nontrivial
mass except in a rotationally invariant neighbourhood. See Figure \ref{fig:sph}.
\end{remark}
\begin{figure}
\hfill
\centerline{\input{sph}}
\caption{\label{fig:sph} A surface of revolution with a piece of
${\mathbb S}^2$ embedded. Also sketched are two ``isoenergetic'' periodic
geodesics which are $\theta$ rotations of each other. One can
construct pathological quasimodes which are continuous, compactly
supported
superpositions of isoenergetic quasimodes associated to such geodesics.}
\hfill
\end{figure}
\section{Preliminaries}
In this section we review some of the definitions and preliminary
computations necessary for Theorem \ref{T:T1}, as well as recall the
spectral estimates we will be using.
\subsection{The $0$-Gevrey class of functions}
\label{SS:gevrey}
For this paper, we use the
following
$0$-Gevrey classes of functions with respect to order of vanishing,
introduced in \cite{Chr-inf-deg}.
\begin{definition}
For $0 \leq \tau <
\infty$, let $\mathcal{G}^0_\tau (
{\mathbb R} )$ be the set of all smooth functions $f : {\mathbb R} \to {\mathbb R}$
such that, for each $x_0 \in {\mathbb R}$, there exists a neighbourhood $U \ni
x_0$ and a constant $C$ such that, for all $0 \leq s \leq k$,
\[
| \partial_x^k f(x) -\partial_x^k f(x_0) | \leq C (k!)^C | x - x_0
|^{-\tau (k-s) } | \partial_x^s f(x) - \partial_x^s f(x_0) |, \,\,\, x
\to x_0 \text{ in } U.
\]
\end{definition}
This definition says that the order of vanishing of derivatives of a
function is only polynomially worse than that of lower derivatives.
Every analytic function is in one of the 0-Gevrey classes $\mathcal{G}^0_\tau$ for some $\tau < \infty$, but many more functions are as well.
For example, the function
\[
f(x) = \begin{cases} \exp (-1/x^p), \text{ for } x >0, \\ 0, \text{ for }x \leq 0
\end{cases}
\]
is in $\mathcal{G}^0_{p+1}$, but
\[
f(x) = \begin{cases} \exp (-\exp(1/x)), \text{ for } x >0, \\ 0, \text{ for }x \leq 0
\end{cases}
\]
is not in any $0$-Gevrey class for finite $\tau$.
\subsection{Conjugation to a flat problem}
We observe that we can conjugate
$\Delta$ by an isometry of metric spaces and separate variables so
that spectral analysis of $\Delta$ is equivalent to a one-variable
semiclassical problem with potential. That is, let $T : L^2(X, d
\text{Vol}) \to L^2(X, dx d \theta)$ be the isometry given by
\[
Tu(x, \theta) = A^{1/2}(x) u(x, \theta).
\]
Then $\widetilde{\Delta} = T \Delta T^{-1}$ is essentially self-adjoint on $L^2 (
X, dx d \theta)$. A simple calculation gives
\[
-\widetilde{\Delta} f = (- \partial_x^2 - A^{-2}(x) \partial_\theta^2 + V_1(x) )
f,
\]
where the potential
\[
V_1(x) = \frac{1}{2} A'' A^{-1} - \frac{1}{4} (A')^2 A^{-2}.
\]
If we now separate variables and write $\psi(x, \theta) = \sum_k
\varphi_k(x) e^{ik \theta}$, we see that
\[
(-\widetilde{\Delta}- \lambda^2) \psi = \sum_k e^{ik \theta} P_k \varphi_k(x),
\]
where
\[
P_k \varphi_k(x) = \left(-\frac{d^2}{dx^2} + k^2 A^{-2}(x) + V_1(x) -
\lambda^2 \right)
\varphi_k(x).
\]
Setting $h = |k|^{-1}$ and rescaling, we have the semiclassical operator
\begin{equation}
\label{E:separation}
P(z,h) \varphi(x) = (-h^2 \frac{d^2}{dx^2} + V(x) -z) \varphi(x),
\end{equation}
where the potential is
\[
V(x) = A^{-2}(x) + h^2 V_1(x)
\]
and the spectral parameter is $z = h^2 \lambda^2$. In Section
\ref{S:proofs} we will at first let $h = \lambda^{-1}$ be our
semiclassical parameter for the whole quasimode, but then switch to $h
= |k|^{-1}$ to estimate the parts of the quasimode microsupported
where the critical elements are located. The relevant microlocal
estimates near critical elements are summarized in the following Subsection.
\subsection{Spectral estimates for weakly unstable critical sets}
In this subsection we summarize the spectral estimates we will use for weakly
unstable critical elements obtained in \cite{Chr-NC,Chr-NC-erratum,Chr-QMNC,ChWu-lsm,ChMe-lsm,Chr-inf-deg}.
\begin{definition}
Let $(P,Q)$ be a critical value of the moment map. Then there are
points in $M^{-1}(P,Q)$ where the moment map has rank $1$ (or $0$, but
these points are easy to handle (see below)). For these points, there are
latitudinal periodic geodesics. If the principal part of the
potential, $A^{-2}(x)$, for the reduced Hamiltonian $\xi^2 +
A^{-2}(x)$ has an ``honest'' minimum at $x_0$ in the sense that if
$[a,b]$ is the maximal closed interval containing $x_0$ with
$
A^{-2}(x) = A^{-2}(x_0)$ on it, then $(A^{-2})' <0$ for $x < a$ in
some small neighbourhood, and $(A^{-2} )' >0$ for $x >b$ in some other
small neighbourhood, then we say this critical element is {\it weakly
stable}. In all other cases, we say the critical element is {\it
weakly unstable}.
\end{definition}
In the following subsections, we review the microlocal estimates from
\cite{Chr-inf-deg} for weakly unstable critical elements. Taken
together, they imply the following theorem.
\begin{theorem}
\label{T:inf-deg-est}
Let $\Lambda$ be a weakly unstable critical element in the reduced
phase space $T^*{\mathbb S}^1_x$, and
assume $u$ has $h$-wavefront set sufficiently close
$\Lambda$. Then for any $\epsilon>0$, there exists $C = C_\epsilon$
such that
\[
\| u \| \leq C h^{-2-\epsilon} \| ((hD)^2 + V(x) -z) u \|,
\]
for any $z \in {\mathbb R}$.
\end{theorem}
\subsubsection{Unstable nondegenerate critical elements}
\label{SS:unst-nd}
A nondegenerate unstable critical element exists where the principal
part of the potential $V_0(x) =
A^{-2}(x)$ has a nondegenerate maximum. To say that $x = 0$ is a nondegenerate maximum means that
$x = 0$ is a critical point of
$V_0(x)$ satisfying $V_0'(0) = 0$, $V_0''(0) < 0$.
The following result as stated can be read off from \cite{Chr-NC,Chr-QMNC}, and has also been
studied in slightly different contexts in \cite{CdVP-I,CdVP-II} and
\cite{BuZw-bb}, amongst many others.
\begin{lemma}
\label{L:ml-inv-0}
Suppose $x = 0$ is a nondegenerate local maximum of the principal part
of the potential $V_0$,
$V_0(0) = 1$. For
$\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |(x,\xi) |\leq \epsilon\}$. Then there
exists $C_\epsilon>0$ such that
\begin{equation}
\label{E:ml-inv-0}
\| P(z,h) \varphi^w u \| \geq C_\epsilon \frac{h}{\log(1/h)} \|
\varphi^w u \|, \,\,\, z \in [1-\epsilon, 1 + \epsilon].
\end{equation}
\end{lemma}
\begin{remark}
This estimate is known to be sharp, in the sense that the logarithmic
loss cannot be improved (see, for example, \cite{CdVP-I}).
\end{remark}
\subsubsection{Unstable finitely degenerate critical elements}
\label{SS:unst-fin}
In this subsection, we consider an isolated critical point at an
unstable but finitely degenerate maximum. That is, we now assume
that $x = 0$ is a degenerate maximum for the function $V_0(x) = A^{-2}(x)$ of order
$m \geq 2$. If we again assume $V_0(0) = 1$, then this means that near
$x = 0$, $V_0(x) \sim 1 - x^{2m}$. Critical points of this form were
first
studied in \cite{ChWu-lsm}.
This Lemma and the proof are given in \cite[Lemma 2.3]{ChWu-lsm}.
\begin{lemma}
\label{L:ml-inv-1}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |(x,\xi) |\leq \epsilon\}$. Then there
exists $C_\epsilon>0$ such that
\begin{equation}
\label{E:ml-inv-1}
\| P(z,h) \varphi^w u \| \geq C_\epsilon h^{2m/(m+1)} \|
\varphi^w u \|, \,\,\, z \in [1-\epsilon, 1 + \epsilon].
\end{equation}
\end{lemma}
\begin{remark}
This estimate is known to be sharp, in the sense that the exponent
$2m/(m+1)$ cannot be improved (see \cite{ChWu-lsm}).
\end{remark}
\subsubsection{Finitely degenerate inflection transmission critical elements}
We next study the case when the principal part of the potential has an inflection point of
finitely degenerate type. That is, let us assume the point $x = 1$ is
a finitely degenerate inflection point, so that locally near $x = 1$,
the potential $V_0(x) = A^{-2}(x)$ takes the form
\[
V_0(x) \sim C_1^{-1} -c_2(x-1)^{2m_2 + 1}, \,\, m_2 \geq 1
\]
where $C_1>1$ and $c_2>0$. Of course the constants are arbitrary
(chosen to agree with those in \cite{ChMe-lsm}), and $c_2$ could be
negative without changing much of the analysis. This Lemma and the
proof are in \cite{ChMe-lsm}.
\begin{lemma}
\label{L:ml-inv-2}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |(x-1,\xi) |\leq \epsilon\}$. Then there
exists $C_\epsilon>0$ such that
\begin{equation}
\label{E:ml-inv-2}
\| P(z,h) \varphi^w u \| \geq C_\epsilon
h^{(4m_2+2)/(2m_2+3)} \|
\varphi^w u \|, \,\,\, z \in [C_1^{-1}-\epsilon, C_1^{-1} + \epsilon].
\end{equation}
\end{lemma}
\begin{remark}
This estimate is also known to be sharp in the sense that the exponent
$(4m_2+2)/(2m_2+3)$ cannot be improved (see \cite{ChMe-lsm}).
\end{remark}
\subsubsection{Unstable infinitely degenerate and cylindrical critical elements}
\label{SS:unst-inf}
In this subsection, we study the case where the principal part of the
potential $V(x) = A^{-2}(x) + h^2 V_1(x)$ has
an infinitely degenerate maximum, say, at the point $x = 0$. Let
$V_0(x) = A^{-2}(x)$. As
usual, we again assume that $V_0(0) = 1$, so that
\[
V_0(x) = 1 - {\mathcal O}(x^\infty)
\]
in a neighbourhood of $x = 0$. Of course this is not very precise, as
$V_0$ could be constant in a neighbourhood of $x = 0$ and still satisfy
this.
So let us first assume that $V_0(0) = 1$, and
$V_0'(x)$ vanishes to infinite order at $x = 0$, however, $\pm V_0'(x) <0$
for $\pm x >0$. That is, the critical point at $x = 0$ is infinitely
degenerate but isolated.
\begin{lemma}
\label{L:ml-inv-3a}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |(x,\xi) |\leq \epsilon\}$. Then for any
$\eta>0$, there
exists $C_{\epsilon,\eta}>0$ such that
\begin{equation}
\label{E:ml-inv-3a}
\| P(z,h) \varphi^w u \| \geq C_{\epsilon, \eta} {h^{2+\eta}} \|
\varphi^w u \|, \,\,\, z \in [1-\epsilon, 1 + \epsilon].
\end{equation}
\end{lemma}
For our next result, we consider the case where there is a whole
interval at a local maximum value. That is, we assume the principal part
of the effective potential
$V_0(x)$ has a maximum $V_0(x) \equiv 1$ on an interval, say $x \in
[-a,a]$, and that $\pm V_0'(x) <0$ for $\pm x > a$ in some neighbourhood.
\begin{lemma}
\label{L:ml-inv-3b}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |x| \leq a + \epsilon,\, |\xi |\leq \epsilon\}$. Then for any
$\eta>0$, there
exists $C_{\epsilon,\eta}>0$ such that
\begin{equation}
\label{E:ml-inv-3b}
\| P(z,h) \varphi^w u \| \geq C_{\epsilon, \eta} {h^{2+\eta}} \|
\varphi^w u \|, \,\,\, z \in [1-\epsilon, 1 + \epsilon].
\end{equation}
\end{lemma}
\subsubsection{Infinitely degenerate and cylindrical inflection
transmission critical elements}
\label{SS:inf-deg-infl}
In this subsection, we assume the effective potential has a critical
element
of infinitely degenerate or cylindrical inflection
transmission type. This is very similar to Subsection \ref{SS:unst-inf},
but now the potential is assumed to be monotonic in a neighbourhood of
the critical value.
We begin with the case where the potential has an isolated
infinitely degenerate critical point of inflection transmission type.
As in the previous subsection, we write $V(x) = A^{-2}(x) + h^2
V_1(x)$ and denote $V_0(x) = A^{-2}(x)$ to be the principal part of
the potential.
Let us assume the point $x = 1$ is
an infinitely degenerate inflection point, so that locally near $x = 1$,
the potential takes the form
\[
V_0(x) \sim C_1^{-1} - (x-1)^{\infty},
\]
where $C_1>1$. Of course the constant is arbitrary
(chosen to again agree with those in \cite{ChMe-lsm}). Let us assume
that our potential satisfies $V_0'(x) \leq 0$ near $x = 1$, with
$V_0'(x)<0$ for $x \neq 1$ in some neighbourhood so that the critical point $x = 1$ is isolated.
\begin{lemma}
\label{L:ml-inv-4a}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |(x-1,\xi) |\leq \epsilon\}$. Then for
any $\eta>0$, there
exists $C = C_{\epsilon,\eta}>0$ such that
\begin{equation}
\label{E:ml-inv-4a}
\| P(z,h) \varphi^w u \| \geq C_\epsilon {h^{2 + \eta}} \|
\varphi^w u \|, \,\,\, z \in [C_1^{-1}-\epsilon, C_1^{-1} + \epsilon].
\end{equation}
\end{lemma}
On the other hand, if $V_0'(x) \equiv 0$ on an interval, say $x-1 \in
[-a,a]$ with $V_0'(x) < 0$ for $x-1 < -a$ and $x -1> a$, we do not expect
anything better than Lemma \ref{L:ml-inv-4a}. The
next lemma says that this is exactly what we do get. To fix an energy
level, assume $V_0 \equiv
C_1^{-1}$ on $[-a,a]$.
\begin{lemma}
\label{L:ml-inv-4b}
For $\epsilon>0$ sufficiently small, let $\varphi \in {\mathcal S}(T^* {\mathbb R})$
have compact support in $\{ |x-1| \leq a + \epsilon,/, |\xi| \leq \epsilon\}$. Then for
any $\eta>0$, there
exists $C = C_{\epsilon,\eta}>0$ such that
\begin{equation}
\label{E:ml-inv-4b}
\| P(z,h) \varphi^w u \| \geq C_\epsilon {h^{2 + \eta}} \|
\varphi^w u \|, \,\,\, z \in [C_1^{-1}-\epsilon, C_1^{-1} + \epsilon].
\end{equation}
\end{lemma}
\section{Proof of Theorem \ref{T:T1} and Corollary \ref{C:C1}}
\label{S:proofs}
Recall the conjugated Laplacian is
\[
-\widetilde{\Delta} = - \partial_x^2 - A^{-2}(x) \partial_\theta^2 + V_1(x),
\]
where $V_1(x)$ has been computed above. We will do some analysis and
reductions now {\it before} separating variables. If we are
considering quasimodes
\[
(-\widetilde{\Delta} - \lambda^2) u = E(\lambda) \| u \|,
\]
where
\[
E(\lambda) = {\mathcal O}( \lambda^{-\beta_0})
\]
for some $\beta_0 >0$,
then we begin by rescaling. Set $h = \lambda^{-1}$ so that
\[
(-h^2 \partial_x^2 - h^2 A^{-2}(x) \partial_\theta^2 + h^2 V_1(x) -1) u = \widetilde{E}(h) \|
u \|,
\]
where $\widetilde{E}(h) = h^2 E( h^{-1} ) = {\mathcal O}( h^{2 + \beta_0})$. With $\xi,
\eta$ the dual variables to $x, \theta$ as usual, the semiclassical
symbol of this operator is
\[
p = \xi^2 + A^{-2}(x) \eta^2 + h^2 V_1(x)-1,
\]
and the semiclassical principal symbol is
\[
p_0 = \xi^2 + A^{-2}(x) \eta^2-1.
\]
It is worthwhile to point out that at this point our semiclassical
parameter is $h = \lambda^{-1}$. After separating variables later in
the proof, we will let $h = |k|^{-1}$, where $k$ is the angular
momentum parameter. However, in the regime where we so take $h$,
$|k|$ and $\lambda$ will be comparable, so it is merely a choice of convenience.
It is important to keep in mind for the remainder of this paper what the various parameters
represent. Here, the variable $\eta$ represents $h D_\theta$. As we will
eventually be decomposing in Fourier modes in the $\theta$ direction,
this means that the variable $\eta$ takes values in $h {\mathbb Z}$.
We next record that a standard $h$-parametrix argument tells us that
any quasimode is concentrated on the energy surface where $\{p_0 = 0
\}$. The proof is standard.
\begin{lemma}
\label{L:char-surf}
Suppose $u$ satisfies
\[
(-h^2 \partial_x^2 - h^2 A^{-2}(x) \partial_\theta^2 + h^2 V_1(x) -1) u = \widetilde{E}(h) \|
u \|,
\]
where $\widetilde{E}(h) = h^2 E( h^{-1} ) = {\mathcal O}( h^{2 + \beta_0})$, and $\Gamma \in
{\mathcal S}^0$ satisfies $\Gamma \equiv 1$ in a small fixed neighbourhood of $\{p_0 = 0
\}$. Then
\[
(1-\Gamma^w) u = {\mathcal O}( h^{2 + \beta_0} ).
\]
\end{lemma}
Hence we will restrict our attention to the characteristic surface
where $\{ p_0 = 0 \}$.
Using our moment map idea, we know that $\eta$ is invariant under the
classical flow. Hence if $\eta$ is very large, our operator will be
elliptic, while if $\eta$ is very small, the parameter $\xi$ will be
bounded away from zero, and hence we will have uniform propagation
estimates. Let us make this more precise. Let $A_0 = \min (A(x))$
and $A_1 = \max (A(x))$, and let
\[
1 = \psi_0(\eta) + \psi_1(\eta) + \psi_2(\eta)
\]
be a partition of unity satisfying
\[
\psi_0 \equiv 1 \text{ on } \{ | \eta |^2 \leq \frac{1}{2} A_0^2 \}
\]
with support in $\{ | \eta |^2 \leq \frac{3}{4} A_0^2 \}$;
\[
\psi_2 \equiv 1 \text{ on } \{ | \eta |^2 \geq {2} A_1^2 \}
\]
with support in $\{ | \eta |^2 \geq \frac{3}{2} A_1^2 \}$.
Then, on $\mathrm{supp}\, \psi_0$, we have
\[
\eta^2 A^{-2}(x) \leq \eta^2 A_0^{-2} \leq \frac{3}{4},
\]
and on $\mathrm{supp}\, \psi_2$, we have
\[
\eta^2 A^{-2}(x) \geq \eta^2 A_1^{-2} \geq \frac{3}{2}.
\]
Now for our quasimode $u$, write
\[
u = u_0 + u_1 + u_2 + u_3 := \psi_0^w \Gamma^w u + \psi_1^w \Gamma^w u
+ \psi_2^w \Gamma^w u + (1 - \Gamma^w)u.
\]
Since $hD_\theta$ commutes with $-\widetilde{\Delta}$ and we can choose $\Gamma =
\Gamma(p_0)$ so that $[p_0^w, \Gamma^w] = {\mathcal O}(h^3)$, each of these $u_j$ are
also quasimodes of the same order as $u$ (but of course may have small
or even trivial $L^2$ mass).
\subsubsection{Estimation of $u_0$}
\label{SSS:u0}
Observe that on the support of $\psi_0$, since $\eta$ is invariant, we
have $| \xi |^2 \geq 1/4 - {\mathcal O}(h^2)$, which means the propagation speed in the
$x$-direction is bounded below. We claim this implies
\[
\| u_0 \|_{L^2_{x, \theta} } \leq c_0 \| u_0 \|_{L^2([a,b]_x \times
{\mathbb S}_\theta ) }
\]
for some $c_0 >0$. In other words, $u_0$ is {\it uniformly}
distributed in the sense that the mass cannot be vanishing in $h$ on
any set.
The claim follows by propagation of
singularities.
The standard propagation of singularities result applies whenever the
classical flow propagates singularities from one region to another in
phase space. Since we are analyzing the region where $\xi \neq 0$, we
have uniform propagation in the $x$ direction.
A general statement is given in the following Lemma (a refinement of
H\"ormander's original result \cite{Hor-sing}). For a proof in this context, see,
for example, \cite[Lemma 6.1]{Chr-NC} and \cite[Lemma 4.1]{BuZw-bb}.
\begin{lemma}
\label{wf-lemma}
Suppose $V_0 \Subset T^*X$, $p$ is a symbol, $T >0$, $A$ an operator, and $V \Subset T^*X$ a neighbourhood of $\gamma$ satisfying
\begin{eqnarray}
\left\{ \begin{array}{l}
\forall \rho \in \{ p_0^{-1}(0) \} \setminus V, \,\,\, \exists \, 0 <t<T \,\, \text{and} \,\, \epsilon = \pm 1 \,\,\,
\text{such that} \\
\exp(\epsilon s H_{p_0})(\rho) \subset \{ p_0^{-1}(0) \} \setminus V \,\, \text{for} \,\, 0 < s < t, \,\, \text{and} \\
\exp(\epsilon t H_{p_0})(\rho) \in V_0; \\
\end{array} \right.
\end{eqnarray}
and $A$ is microlocally elliptic in $V_0 \times V_0 $. If $B \in \Psi^{0,0}(X, \Omega_X^{\frac{1}{2}})$ and $\mathrm{WF}_h (B) \subset T^*X
\setminus V$, then
\begin{eqnarray}
\label{E:prop-00}
\left\| Bu \right\| \leq C \left( h^{-1} \left\| Pu \right\| + \| Au \| \right) + {\mathcal O} (h^\infty) \|u\|.
\end{eqnarray}
\end{lemma}
Fix two non-empty intervals in the $x$ direction, $(a,b)$ and
$(c,d)$ and assume $u = u_0$ is $L^2$ normalized.
Now using that $Pu = {\mathcal O}(h^{2 + \beta_0}) \| u \|$, we have
\begin{align*}
\| u \|_{L^2((c,d)\times {\mathbb S}^1} & \leq C h^{-1} \|P u \| + C_2 \| u
\|_{L^2((a,b)\times {\mathbb S}^1)} \\
& \leq C h^{1 + \beta_0} \| u \|_{L^2({\mathbb S}^1 \times {\mathbb S}^1)} + C_2 \| u
\|_{L^2((a,b)\times {\mathbb S}^1)},
\end{align*}
for some $C_2>0$.
For $h>0$ sufficiently small, this implies if $u$ has mass bounded
below independent of $h$ in any $x$ neighbourhood $(c,d)$, then
\[
\| u \|_{L^2((a,b)\times {\mathbb S}^1)} \geq c'>0
\]
independent of $h$. Rescaling in terms of $u_0$ if $u_0$ is not normalized, we recover
\[
\| u_0 \|_{L^2((a,b)\times {\mathbb S}^1)} \geq c' \| u_0 \|.
\]
Since the interval $(a,b)$ is arbitrary, we have shown that the
$L^2$-mass on any rotationally invariant neighbourhood is positive independent of $h$. Thus
\eqref{E:lower-bound} holds with a lower bound independent of $h = \lambda^{-1}$.
\subsubsection{Estimation of $u_2$}
On the other hand, on the support of $\psi_2$, we have the principal
symbol satisfies
\[
| p_0 | \geq \frac{1}{2},
\]
so we claim that an elliptic argument shows
\[
\| u_2 \|_{L^2} = {\mathcal O}(h^\infty) \| u_2 \|_{L^2}.
\]
That is, since $| p_0 | \geq \frac{1}{2}$ on support of $\psi_2$,
there is an $h$-parametrix for $P$ there: there exists $Q$ such that
\[
Q P \psi_2^w = \psi_2^w + {\mathcal O}(h^\infty),
\]
and further $Q$ has bounded $L^2$ norm.
Hence
\begin{align*}
\| u_2 \| & = \| QP u_2 \| + {\mathcal O}(h^\infty) \| u_2 \| \\
& \leq C \| P u_2 \| + {\mathcal O}(h^\infty) \| u_2 \| \\
& = {\mathcal O}(h^{2 + \beta_0} ) \| u_2 \|.
\end{align*}
This implies $u_2 = {\mathcal O}(h^\infty)$.
\subsubsection{Estimation of $u_1$}
\label{SSS:u1}
In order to consider the final part $u_1$, which is microsupported
where all the critical points are, we will employ one further
reduction. Since $u_1$ is microsupported in a region where $| \eta |$
is bounded between two constants, say, $a_0 \leq | \eta | \leq a_1$,
and $\eta = h k$ for some integer $k$,
a priori the number of angular momenta $k$ in the wavefront set of
$u_1$ is comparable to $h^{-1}$. We can do better than that. Using
the semiclassical calculus, we will next show that there exists $k_0
\in {\mathbb Z}$ such that for any $\epsilon>0$, we have
\[
u_1 = \sum_{|k - k_0| \leq h^{-\epsilon} } e^{i k \theta} \varphi_k(x) +
{\mathcal O}(h^\infty) \| u_1 \|.
\]
That is, we claim that the Fourier decomposition of $u_1$ can actually
only have ${\mathcal O}(h^{-\epsilon})$ non-trivial modes. To prove this claim,
fix $k_0 \in {\mathbb Z}$ and any $\epsilon>0$, and choose a $k_1 \in {\mathbb Z}$
satisfying
\[
| k_1 - k_0 | \geq h^{-\epsilon}.
\]
We will show that we can decompose $u_1$ into (at least) two pieces with disjoint
microsupport, one near $hk_0$ and one near $hk_1$. Evidently, these
two pieces correspond to different angular momenta $\eta$, so have
wavefront sets associated to different level sets of the moment map.
Of course, level sets sufficiently close (in an $h$-dependent set) may
contribute to a single irreducible quasimode, but the point is to
quantify how far away from a single level set one needs to go before
leaving the microsupport of an irreducible quasimode.
In order to make this rigorous, let $\eta_j = h k_j$ for $j = 0, 1$,
and choose $\chi(r) \in {\mathcal C}^\infty_c ( {\mathbb R}) $ satisfying
\[
\chi(r) \equiv 1 \text{ for } | r | \leq 1 ,
\]
with support in $\{ | r | \leq 2 \}$. For $j = 0, 1$, let
\[
\chi_j( \eta, h) = \chi \left( \frac{ \eta - \eta_j }{h^{1 -
\epsilon/2}} \right).
\]
As semiclassical symbols, the $\chi_j$ are in a harmless $h^{1/2 -
\epsilon/4}$ calculus, and moreover they only depend on $\eta$ (not on
$\theta$) and commute with $-\widetilde{\Delta}$.
On the support of each of the $\chi_j$, we have
\[
\left| \frac{ \eta - \eta_j }{h^{1 - \epsilon/2} } \right| = \left|
\frac{ h k - h k_j }{h^{1 - \epsilon/2} }\right| = \left| \frac{ k -
k_j }{h^{- \epsilon/2}} \right| \leq 2.
\]
This implies
\[
| k - k_j | \leq 2 h^{-\epsilon/2},
\]
so as $h \to 0+$, $\chi_0$ and $\chi_1$ have disjoint supports. This means the functions $\chi_1 u_1$ and
$\chi_2 u_1$ have disjoint $h$-wavefront sets, so they are almost
orthgonal:
\[
\left\langle \chi_1 u_1, \chi_2 u_1 \right\rangle = {\mathcal O}(h^\infty).
\]
Hence if each of these functions has nontrivial $L^2$ mass, then $u$
was not an irreducible quasimode.
Finally, we analyse the function $u_1$, but spread
over at most ${\mathcal O}(h^{-\epsilon})$ Fourier modes.
Throughout the remainder of this section, let $\lambda$ be large and
fixed. Let us consider a single Fourier mode confined
to a single angular momentum $k$.
The case of $u_1$ is the most interesting case, as the microsupport of
$u_1$ contains all
of the critical elements.
Now recalling again the separated equation \eqref{E:separation} with the
potential
\[
V(x) = A^{-2}(x) + h^2 V_1(x),
\]
let $A_0$ and $A_1$ again be the min/max respectively of $A(x)$. Our
spectral parameter now is $z = h^2 \lambda^2$.
We are localized where
\[
\frac{1}{2} A_0^2 \leq (\lambda^{-1} k)^2 \leq 2 A_1^2,
\]
or
\[
\frac{1}{2} A_1^{-2} \leq z \leq 2 A_0^{-2}.
\]
This of course implies that $\lambda$ and $k$ are
comparable. Let $(a,b) \subset {\mathbb S}^1$ be a non-empty interval. We
need to show that if $u$ is a weak irreducible quasimode,
\[
( (hD)^2 + V(x) -z) u = {\mathcal O}( h^{2 + \beta_0}) \| u \|,
\]
with $\| u \|= 1$,
then either $\| u \|_{L^2(a,b)} = {\mathcal O}(h^\infty) =
{\mathcal O}(\lambda^{-\infty})$, or $\| u \|_{L^2(a,b)} \geq C_\epsilon h^{1 + \epsilon}$
for any $\epsilon>0$. Let us assume that $u$ is nontrivial so that
$\|u \|_{L^2(a,b)} \geq c h^N$ for some $N$.
There are a number of subcases to consider here. We observe that,
according to Lemma \ref{wf-lemma}, we
can always microlocalize further to a set close to the energy level of
interest. That is, for $P(z,h) = (hD)^2 + V(x) -z$, if $P(z,h) u =
{\mathcal O}(h^{2 + \beta_0})$, then if $\psi(r) \in {\mathcal C}^\infty_c( {\mathbb R})$ satisfies
$\psi \equiv 1$ for $r$ near $0$, we have for any $\delta>0$
\[
\psi^w ( ( \xi^2 + V(x) - z) / \delta ) u = u + {\mathcal O}(h^{2 + \beta_0} ).
\]
For the rest of this section, we write $\psi^w$ for this energy cutoff.
{\bf Case 1:}
Next, assume $z$ is in a small neighbourhood of a critical energy level, and assume $A'(x)
\neq 0$ somewhere on $(a,b)$. Then let $(a', b')$ be a non-empty
interval with
\[
(a', b') \Subset \{ A' \neq 0\} \cap (a,b),
\]
and let $(\alpha', \beta) \supset (a', b')$ be the maximal connected
interval with $A'(x) \neq 0$ on $(\alpha', \beta)$. Now $A'$ has constant sign on $(\alpha', \beta)$, so at least one of
$\alpha'$ or $\beta$ is part of a weakly unstable critical element
(see Figure \ref{fig:fig3b-1}).
\begin{figure}
\hfill
\centerline{\input{fig3b-1}}
\caption{\label{fig:fig3b-1} The function $A^{-2}(x)$ and the weakly
unstable critical point $\beta$.}
\hfill
\end{figure}
Without loss in generality, assume $A' <0$ on $(\alpha', \beta)$ so
that at least $\beta$ lies in a weakly unstable critical element.
That is, the principal part of the potential $A^{-2}(x)$ increases as
$x \to \beta-$, and takes the value, say $A^{-2}(\beta) = A_2$. Let
$(\alpha, \beta)$ be the maximal open interval containing $(\alpha',
\beta)$ where $A^{-2}(x) < A_2$ on $(\alpha, \beta)$. As $A^{-2}(x) <
A^{-2}(\alpha)$ for $x \in (\alpha, \beta)$ and $A^{-2}(\alpha) = A_2$, we have $(A^{-2}(x) ) '
<0$ for $x \in (\alpha, \beta)$ sufficiently close to $\alpha$. That
means that either $\alpha$ is part of a weakly unstable critical
element, or $A'(\alpha) \neq 0$. We break the analysis into the two
separate subsubcases, beginning with $A'(\alpha) \neq 0$.
{\bf Case 1a:}
If $A'(\alpha) \neq 0$, then the weakly unstable/stable manifolds
associated to $(A^{-2})'(\beta) = 0$
are
homoclinic to each other (see Figure \ref{fig:fig3b-2}), and in particular, propagation of
singularities can be used to control the mass along this whole
trajectory, as long as we stay away from the right hand endpoint
$\beta$. That is, propagation of singularities implies for any
$\eta>0$ independent of $h$,
\begin{align*}
\| \psi^w u \|_{L^2(\alpha, \beta - \eta)} & \leq C_\eta (h^{-1} \|
((hD)^2 + V - z ) \psi^w u \| + \| \psi^w u \|_{L^2(a', b') } \\
& \leq C_\eta h^{1 + \beta_0} \| u \| + \| \psi^w u \|_{L^2(a', b') } .
\end{align*}
Hence by taking $h>0$ sufficiently small, we need to bound $\| \psi^w
u \|_{L^2(\alpha, \beta - \eta)}$ from below in terms of $\| u \|$.
\begin{figure}
\hfill
\centerline{\input{fig3b-2}}
\caption{\label{fig:fig3b-2} If $A'(\alpha) \neq 0$, the unstable
manifold from $\beta$ flows into the stable manifold at $\beta$
(homoclinicity). The interval indicates a region with propagation
speed uniformly bounded below.}
\hfill
\end{figure}
Let
$[\beta, \kappa]$ be the maximal connected interval containing $\beta$
on which $A' = 0$ (we allow $\kappa = \beta$ if the critical point is isolated).
Let $\tilde{\chi} \equiv 1$ on $[\beta, \kappa]$ with
support in a small neighbourhood thereof, and let $\chi \equiv 1$ on
$\mathrm{supp}\, \tilde{\chi}$ with support in a slightly smaller set so that $(1 -
\tilde{\chi}) \geq (1 - \chi)$ and $(1 - \tilde{\chi}) \geq c | \chi'|$. Then
writing $P(z,h) = (hD)^2 + V - z$, we have from Theorem
\ref{T:inf-deg-est} (for any $\epsilon>0$)
\begin{align*}
\| u \| & \leq \| \chi u \| + \| ( 1 - \chi) u \| \\
& \leq C_\epsilon h^{-2-\epsilon} \| P(z,h) \chi u \| + \| (1 - \tilde{\chi}
) u \| \\
& \leq C_\epsilon h^{-2-\epsilon} (\| \chi P(z,h) u \| + \| [P(z,h) ,
\chi ] u \| ) + \| (1 - \tilde{\chi}
) u \| \\
& \leq C_\epsilon' ( h^{\beta_0 - \epsilon} \| u \| +
h^{-1-\epsilon}\| (1 - \tilde{\chi}) u \| ) + \| (1 - \tilde{\chi}
) u \|
\end{align*}
Rearranging and taking $h>0$ sufficiently small and $\epsilon <
\beta_0$, we get
\begin{equation}
\label{E:tchi-1}
\| (1 - \tilde{\chi}) u \| \geq C_\epsilon h^{1 + \epsilon} \| u \|.
\end{equation}
Now either the wavefront set of $u$ is contained in the closure of the
lift of
$(\alpha, \beta)$ or it isn't. In the latter case there is nothing to
prove. In the former case, we conclude that $u = {\mathcal O}(h^\infty)$ on any
open subset whose closure does not meet the set $[\alpha, \beta]$. We
appeal to propagation of singularities one more time. Since
$A'(\alpha) \neq 0$, propagation of singularities applies in a
neighbourhood of $\alpha$, so that (shrinking $\eta>0$ if necessary)
for some $c_1>0$,
\[
\| u \|_{L^2(\alpha, \beta - \eta)} \geq c_1 \| u \|_{L^2( \alpha -
\eta, \beta - \eta)}.
\]
Since we have assumed $u = {\mathcal O}(h^\infty)$ on $(\alpha - \eta, \beta +
\eta)^c$,
this estimate, together with \eqref{E:tchi-1} and \eqref{E:prop-00} allows us to conclude
\[
\| u \|_{L^2(\alpha, \beta - \eta)} \geq C \| (1 - \tilde{\chi}) u \| \geq C_\epsilon h^{1 + \epsilon}
\| u \|.
\]
{\bf Case 1b:}
We now consider
the possibility that $A'(\alpha) = 0$ as well as $A'(\beta) = 0$ (see Figure \ref{fig:fig3b-3}). In this case,
propagation of singularities fails at both endpoints of $(\alpha,
\beta)$, so we can only conclude that for any $\eta>0$
independent of $h$,
\[
\| u \|_{L^2(\alpha + \eta, \beta - \eta)} \leq C_\eta (h^{-1} \|
((hD)^2 + V - z ) u \| + \| u \|_{L^2(a', b') }.
\]
\begin{figure}
\hfill
\centerline{\input{fig3b-3}}
\caption{\label{fig:fig3b-3} If $A'(\alpha) = 0$, the unstable
manifold from $\beta$ flows into the stable manifold at $\alpha$ and
vice versa. The interval indicates a region with propagation
speed uniformly bounded below.}
\hfill
\end{figure}
Hence now it suffices to prove that for some $\eta>0$ small but
independent of $h$, we have the estimate
\[
\| u \|_{L^2(\alpha + \eta, \beta - \eta)} \geq C_\epsilon h^{1 +
\epsilon} \| u \|
\]
for any $\epsilon>0$.
Let
$[\beta, \kappa]$ be the maximal connected interval containing $\beta$
on which $A' = 0$, and let $[\omega, \alpha]$ be the maximal connected
interval containing $\alpha$ on which $A' = 0$.
Let $\tilde{\chi} \equiv 1$ on $[\beta, \kappa] \cup [\omega, \alpha]$ with
support in small neighbourhoods thereof, and let $\chi \equiv 1$ on
$\mathrm{supp}\, \tilde{\chi}$ with support in a slightly smaller set so that $(1 -
\tilde{\chi}) \geq (1 - \chi)$ and $(1 - \tilde{\chi}) \geq c | \chi'|$. Since both
$[\omega, \alpha]$ and $[\beta, \kappa]$ are weakly unstable, we can apply
Theorem \ref{T:inf-deg-est} and the same argument as above to finish
this case.
{\bf Case 2:}
Finally, we assume $(a,b) \subset \{ A' = 0 \}$.
Again, if $A^{-2} \equiv A_3$ on $(a,b)$ and $z \neq A_3$, we can use
propagation of singularities to control $\| u \|_{L^2(a,b)}$ from
below by its mass on the connected component in $\{ p = z \}$
containing $(a,b)$ (as in the case of $u_0$ above). Hence we are interested in the
case where $z$ is in a small neighbourhood of $A_3$.
If $u = {\mathcal O}(h^\infty)$ on $(a,b)$ there is nothing to prove, so assume
not. Then if $[\alpha, \beta] \supset (a,b)$ is the
maximal connected interval where $A^{-2}(x) \equiv A_3$, the wavefront
set of $u$ is contained in a small neighbourhood of $[\alpha, \beta]$, so that for
$\delta>0$ as small as we like by taking a sufficiently localized
energy cutoff, we have
\[
\| u \|_{L^2( [\alpha - \delta, \beta + \delta]^c )} = {\mathcal O}_\delta (h^\infty).
\]
That means that, either
\[
\| u \|_{L^2([a,b])} \geq c >0, \,\,\,
\| u \|_{L^2([\alpha - \delta, a])} \geq c >0, \text{ or }
\| u \|_{L^2([b, \beta +
\delta])} \geq c >0.
\]
If the first estimate is true, we're done, so assume without loss in
generality that $\| u \|_{L^2([b, \beta +
\delta])} \geq c >0$. Assume for contradiction that there exists
$\epsilon_0>0$ such that $\| u \|_{L^2(a,b)} \leq C h^{1 +
\epsilon_0}$. Let $\chi \in {\mathcal C}^\infty_c$ be a smooth function such that
$\chi \equiv 1$ on $[b, \beta + \delta]$ with support in $(a, \beta
+ 2 \delta)$. Write $\tilde{u} = \chi u$. If $[\alpha, \beta]$ is a
weakly stable critical element, modify $A^{-2}(x)$ on the support of
$1-\chi$ so that $[\alpha, \beta]$ is weakly unstable. That is, if
$(A^{-2}(x))' <0$ for $x < \alpha$ in some neighbourhood, replace
$A$ with a locally defined function $\tilde{A}$ satisfying $\tilde{A} \equiv A$
on $\mathrm{supp}\, \chi$ but $(\tilde{A}^{-2}(x))' >0$ for $x < \alpha$ in some
neighbourhood. If $[\alpha, \beta]$ is weakly unstable, then let
$\tilde{A} \equiv A$ (see Figure \ref{fig:fig3c}. We apply Theorem \ref{T:inf-deg-est} once again
(for any $\epsilon>0$):
\begin{align*}
\| \chi u \| & \leq C_\epsilon h^{-2 - \epsilon} \| ((hD)^2 + \tilde{A}^{-2} + h^2 V_1 -z )
\chi u \| \\
& = C_\epsilon h^{-2 - \epsilon} \| ((hD)^2 + A^{-2} + h^2 V_1 -z )
\chi u \| \\
& \leq C_\epsilon h^{-2 - \epsilon} ( \| P(z,h) u \| + \| [P(z,h),
\chi] u \| ) \\
& \leq C_\epsilon' (h^{\epsilon_0 - \epsilon} \| u \| + h^{-1 -
\epsilon} \| u \|_{L^2(a,b)} ) + {\mathcal O}(h^\infty),
\end{align*}
where the ${\mathcal O}(h^\infty)$ error comes from the part of the commutator
$[P(z,h), \chi]$ supported outside a neighbourhood of $[\alpha,
\beta]$ (the other part contributing the integral over $(a,b)$). But
our contradiction assumption implies that the right hand side is
$o(1)$ as $h \to 0$ provided $\epsilon< \epsilon_0$. As $\| \chi u \|
\geq c >0$, this is a contradiction.
\begin{figure}
\hfill
\centerline{\input{fig3c}}
\caption{\label{fig:fig3c} The setup for Case 2. Here if the
quasimode is small in $(a,b)$, we cut off to the right of $(a,b)$
and modify $A^{-2}$ to the left to be weakly unstable. We then arrive at a contradiction.}
\hfill
\end{figure}
\subsection{Finishing up the proof}
We now put together the estimates of $u_0, u_1, u_2, u_3$.
Since $u_3 = {\mathcal O}( h^{2 + \beta_0} )$ and $u_2 = {\mathcal O}(h^\infty)$, for
$h>0$ sufficiently small, at least one of $u_0$ and $u_1$ must have $L^2$
mass bounded below independent of $h$. If $u_0$ has $L^2$ mass
bounded below independent of $h$ we're done by the propagation of
singularities argument in Subsection \ref{SSS:u0}. Hence we need to
conclude Theorem \ref{T:T1} assuming $u_0$ is small and $u_1$ carries most of
the $L^2$ mass.
Fix $(a,b)$ as considered in Subsection \ref{SSS:u1} and recall we
know that for any $\epsilon>0$
\[
u_1 = \sum_{|k - k_0| \leq h^{-\epsilon} } e^{i k \theta} \varphi_k(x) +
{\mathcal O}(h^\infty) \| u_1 \|.
\]
We use the notation $\Omega = (a,b)_x \times {\mathbb S}^1_\theta$ as in the
statement of Theorem \ref{T:T1}.
Each $\varphi_k$ satisfies either
\[
\| \varphi_k \|_{L^2(a,b)} = {\mathcal O}(|k|^{-\infty})
\]
or for any $\epsilon>0$,
\[
\| \varphi_k \|_{L^2 (a,b) } \geq c_2 |k|^{-1-\epsilon} \| \varphi_k \|_{L^2
({\mathbb S}^1_x)}.
\]
In the first case, these $\varphi_k$s have disjoint wavefront sets from
the $\varphi_k$s in the latter case, so leaving them in the sum would
mean our quasimode was not irreducible. Removing these from the sum
and reindexing if necessary, we conclude
\begin{align*}
\| u_1 \|_{L^2(\Omega)}^2 & = \sum_{|k - k_0| \leq k_0^{\epsilon} } \|\varphi_k(x)\|_{L^2(a,b)}^2 +
{\mathcal O}(h^\infty) \| u_1 \|^2 \\
& \geq c_2' k_0^{-2-2\epsilon} \sum_{|k - k_0| \leq k_0^{\epsilon} }
\|\varphi_k(x)\|_{L^2({\mathbb S}^1_x )}^2 -
{\mathcal O}(h^\infty) \| u_1 \|^2 \\
& = c_2'' \lambda^{-2-2\epsilon} \| u_1 \|_{L^2({\mathbb S}^1_x \times {\mathbb S}^1_\theta)}^2 - {\mathcal O}(
\lambda^{-\infty}) \| u_1 \|^2.
\end{align*}
This concludes the proof of Theorem \ref{T:T1}.
\qed
\bibliographystyle{alpha}
|
{
"timestamp": "2013-04-16T02:04:25",
"yymm": "1304",
"arxiv_id": "1304.4178",
"language": "en",
"url": "https://arxiv.org/abs/1304.4178"
}
|
\section{Introduction}
In this paper, module means a left module, $K$ is a
field of characteristic zero and $K^*$ is its group of units, and the following notation is fixed:
\begin{itemize}
\item $P_n:= K[x_1, \ldots , x_n]=\bigoplus_{\alpha \in \N^n}
Kx^{\alpha}$ is a polynomial algebra over $K$ where
$x^{\alpha}:=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$,
\item $G_n:=\Aut_K(P_n)$ is the group of automorphisms of the polynomial algebra $P_n$,
\item $\der_1:=\frac{\der}{\der x_1}, \ldots , \der_n:=\frac{\der}{\der
x_n}$ are the partial derivatives ($K$-linear derivations) of
$P_n$,
\item $D_n:=\Der_K(P_n) =\bigoplus_{i=1}^nP_n\der_i$ is the Lie
algebra of $K$-derivations of $P_n$ where $[\der , \d ]:= \der \d -\d \der $,
\item $\d_1:=\ad (\der_1), \ldots , \d_n:=\ad (\der_n)$ are the inner derivations of the Lie algebra $D_n$ determined by the elements $\der_1, \ldots , \der_n$ (where $\ad (a)(b):=[a,b]$),
\item $\mG_n:=\Aut_{{\rm Lie}}(D_n)$ is the group of automorphisms of the Lie algebra $D_n$,
\item $\CD_n:=\bigoplus_{i=1}^n K\der_i$,
\item $\CH_n :=\bigoplus_{i=1}^n KH_i$ where $H_1:=x_1\der_1, \ldots , H_n:=x_n\der_n$,
\item $A_n:= K \langle x_1, \ldots
, x_n , \der_1, \ldots , \der_n\rangle =\bigoplus_{\alpha , \beta
\in \N^n} Kx^\alpha \der^\beta$ is the $n$'th {\em Weyl
algebra},
\item for each natural number $n\geq 2$, $\ggu_n :=
K\der_1+P_1\der_2+\cdots +P_{n-1}\der_n$ is the {\em Lie algebra of triangular polynomial derivations} (it is a Lie subalgebra of the Lie algebra $D_n$) and $ \Aut_K(\ggu_n)$ is its group of automorphisms.
\end{itemize}
The aim of the paper is to prove the following theorem.
\begin{theorem}\label{11Mar13
$\mG_n = G_n$.
\end{theorem}
{\it Structure of the proof}. (i) $G_n\subseteq \mG_n$ via the group monomorphism (Lemma \ref{b11Mar13}.(3))
$$G_n\ra \mG_n, \;\; \s \mapsto \s : \der \mapsto \s (\der ):=\s \der \s^{-1}.$$
(ii) Let $\s \in \mG_n$. Then $\der_1':=\s (\der_1), \ldots , \der_n':=\s (\der_n)$ are commuting, locally nilpotent derivations of the polynomial algebra $P_n$ (Lemma \ref{c13Mar13}.(1)).
$\noindent $
(iii) $\bigcap_{i=1}^n \ker_{P_n}(\der_i') = K$ (Lemma \ref{c13Mar13}.(2)).
$\noindent $
(iv)(crux) There exists a polynomial automorphism $\tau \in G_n$ such that $\tau \s \in \Fix_{\mG_n}(\der_1, \ldots , \der_n)$ (Corollary \ref{b13Mar13}).
$\noindent $
(v) $\Fix_{\mG_n}(\der_1, \ldots , \der_n)=\Sh_n$ (Proposition \ref{B11Mar13}.(3)) where
$$\Sh_n:=\{ s_\l \in G_n\, | \, s_\l (x_1)=x_1+\l_1, \ldots , s_\l (x_n) = x_n+\l_n\}$$ is the {\em shift group} of automorphisms of the polynomial algebra $P_n$ and $\l = (\l_1, \ldots , \l_n)\in K^n$.
$\noindent $
(vi) By (iv) and (v), $\s \in G_n$, i.e. $\mG_n = G_n$. $\Box $
$\noindent $
{\bf An analogue of the Jacobian Conjecture is true for $D_n$}. The Jacobian Conjecture claims that {\em certain} monomorphisms of the polynomial algebra $P_n$ are isomorphisms: {\em Every algebra endomorphism $\s $ of the polynomial algebra $P_n$ such that $\CJ (\s ):= \det (\frac{\der \s (x_i)}{\der x_j})\in K^*$ is an automorphism.} The condition that $\CJ (\s )\in K^*$ implies that the endomorphism $\s$ is a monomorphism.
$\noindent $
{\bf Conjecture}. {\em Every homomorphism of the Lie algebra $D_n$ is an automorphism.}
\begin{theorem}\label{10Mar12
\cite{Bav-Lie-Un-MON} Every monomorphism of the Lie algebra $\ggu_n$ is an automorphism.
\end{theorem}
{\it Remark}. Not every epimorphism of the Lie algebra $\ggu_n$ is an automorphism. Moreover, there are countably many distinct ideals $\{ I_{i\o^{n-1}}\, | \, i\geq 0\}$ such that $$I_0=\{0\}\subset I_{\o^{n-1}}\subset I_{2\o^{n-1}}\subset \cdots \subset I_{i\o^{n-1}}\subset \cdots$$ and the Lie algebras $ \ggu_n/I_{i\o^{n-1}}$ and $\ggu_n$ are isomorphic (Theorem 5.1.(1), \cite{Bav-Lie-Un-GEN}).
Theorems \ref{10Mar12} and Conjecture have bearing of the Jacobian Conjecture and the Conjecture of Dixmier \cite{Dix}
for the Weyl algebra $A_n$ over a field of characteristic zero that claims: {\em every homomorphism of the Weyl algebra is an automorphism}. The Weyl algebra $A_n$ is a simple algebra, so every algebra endomorphism of $A_n$ is a monomorphism. This conjecture is open since 1968 for all $n\geq 1$. It is stably equivalent to the Jacobian Conjecture for the polynomial algebras as was shown by Tsuchimoto
\cite{Tsuchi05}, Belov-Kanel and Kontsevich \cite{Bel-Kon05JCDP},
(see also \cite{JC-DP} for a short proof which is based on the author's new inversion formula for polynomial automorphisms \cite{Bav-inform}).
$\noindent $
{\bf An analogue of the Conjecture of Dixmier is true for the algebra $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ of polynomial
integro-differential operators}.
\begin{theorem}\label{11Oct10
{\rm (Theorem 1.1, \cite{Bav-cdixintdif})} Each algebra endomorphism of $\mI_1$ is an automorphism.
\end{theorem}
In contrast to the Weyl algebra $A_1=K\langle x, \frac{d}{dx} \rangle$, the algebra of polynomial
differential operators, the algebra $\mI_1$ is neither a left/right Noetherian algebra nor a simple algebra. The left localizations, $A_{1,\der}$ and $\mI_{1, \der}$, of the algebras $A_1$ and $\mI_1$ at the powers of the element $\der= \frac{d}{dx}$ are isomorphic. For the simple algebra $A_{1,\der} \simeq \mI_{1, \der}$, there are algebra endomorphisms that are not automorphisms \cite{Bav-cdixintdif}.
$\noindent $
{\bf The group of automorphisms of the Lie algebra $\ggu_n$}.
In \cite{Bav-Lie-Un-AUT}, the group of automorphisms $\Aut_K(\ggu_n)$ of the Lie algebra $\ggu_n$ of triangular polynomial derivations is found ($n\geq 2$), it is isomorphic to an iterated semi-direct product (Theorem 5.3, \cite{Bav-Lie-Un-AUT}),
$$\mT^n\ltimes (\UAut_K(P_n)_n\rtimes( \mF_n' \times \mE_n )) $$
where $\mT^n$ is an algebraic $n$-dimensional torus, $\UAut_K(P_n)_n$ is an explicit factor group of the group $\UAut_K(P_n)$ of unitriangular polynomial automorphisms, $\mF_n'$ and $\mE_n$ are explicit groups that are isomorphic respectively to the groups $\mI$ and $\mJ^{n-2}$ where
$\mI := (1+t^2K[[t]], \cdot )\simeq K^{\N}$ and
$\mJ := (tK[[t]], +)\simeq K^\N$. Comparing the groups $G_n$ and $\Aut_K(\ggu_n)$ we see that the group $(\UAut_K(P_n)_n$ of polynomial automorphisms is a {\em tiny} part of the group $\Aut_K(\ggu_n)$ but in contrast $\mG_n = \Aut_K(P_n)$.
It is shown that the {\em adjoint group} of automorphisms $\CA (\ggu_n)$ of the Lie algebra $\ggu_n $ is equal to the group $\UAut_K(P_n)_n$ (Theorem 7.1, \cite{Bav-Lie-Un-AUT}). Recall that the {\em adjoint group} $\CA (\CG )$ of a Lie algebra $\CG$ is generated by the elements $ e^{ \ad (g)}:=\sum_{i\geq 0}\frac{\ad (g)^i}{i!}\in \Aut_K(\CG )$ where $g$ runs through all the locally nilpotent elements of the Lie algebra $\CG$ (an element $g$ is a {\em locally nilpotent element} if the inner derivation $\ad (g):= [g, \cdot ]$ of the Lie algebra $\CG$ is a locally nilpotent derivation).
\section{Proof of Theorem \ref{11Mar13} }\label{PPPAAA
This section can be seen as a proof of Theorem \ref{11Mar13}. The proof is split into several statements that reflect `Structure of the proof of Theorem \ref{11Mar13}' given in the Introduction.
{\bf The Lie algebra $D_n$ is $\Z^n$-graded}. The Lie algebra
\begin{equation}\label{xadbd}
D_n =\bigoplus_{\alpha\in \N^n} \bigoplus_{i=1}^n Kx^\alpha \der_i
\end{equation}
is a $\Z^n$-graded Lie algebra
$$D_n = \bigoplus_{\beta\in \Z^n} D_{n , \beta}\;\; {\rm where}\;\; D_{n,\beta}=\bigoplus_{\alpha -e_i=\beta}Kx^\alpha \der_i,$$
i.e. $[D_{n,\alpha} , D_{n,\beta }] \subseteq D_{n, \alpha +\beta}$ for all $\alpha ,\beta \in \N^n$ where $e_1:=(1, 0 , \ldots , 0), \ldots , e_n:=(0, \ldots , 0 , 1)$ is the canonical free basis for the free abelian group $\Z^n$. This follows from the commutation relations
\begin{equation}\label{xadbd1}
[x^\alpha\der_i, x^\beta \der_j]= \beta_i x^{\alpha+\beta - e_i} \der_j-\alpha_j x^{\alpha + \beta - e_j} \der_i.
\end{equation}
Clearly, for all $i,j=1, \ldots , n$ and $\alpha \in \N^n$,
\begin{equation}\label{xadbd2}
[H_j, x^\alpha \der_i]=\begin{cases}
\alpha_j x^{\alpha} \der_i & \text{if }j\neq i ,\\
(\alpha_i-1)x^{\alpha} \der_i& \text{if }j=i, \\
\end{cases}
\end{equation}
\begin{equation}\label{xadbd3}
[\der_j, x^\alpha \der_i]=\alpha_j x^{\alpha -e_j} \der_i.
\end{equation}
The {\em support} $\Supp (D_n):=\{ \beta \in \Z^n\, | \, D_{n,\beta}\neq 0\}$ is a submonoid of $\Z^n$. Let us find the support $\Supp (D_n)$, the graded components $D_{n,\beta}$ and their dimensions $\dim_K\, D_{n,\beta}$. For each $i=1, \ldots , n$, let $\N^{n,i}:=\{ \alpha \in \N^n \, | \, \alpha_i=0\}$ and $P_n^{\der_i}:=\ker_{P_n}(\der_i)$. It follows from the decompositions
$P_n = P_n^{\der_i}\oplus P_nx_i$ for $i=1, \ldots , n$ that
$$D_n = \bigoplus_{i=1}^n (P_n^{\der_i}\oplus P_nx_i)\der_i =\bigoplus_{i=1}^n P_n^{\der_i}\der_i \oplus \bigoplus_{i=1}^nP_nH_i, $$
\begin{equation}\label{Dnb2}
D_n =\bigoplus_{i=1}^n P_n^{\der_i}\der_i \oplus \bigoplus_{\alpha \in \N^n} x^\alpha \CH_n.
\end{equation}
Hence,
\begin{equation}\label{Dnb}
\Supp (D_n) =\coprod_{i=1}^n (\N^{n,i}-e_i) \coprod\N^n.
\end{equation}
\begin{equation}\label{Dnb1}
D_{n,\beta} =\begin{cases}
x^\alpha\der_i& \text{if }\beta = \alpha - e_i\in \N^{n,i}-e_i,\\
x^\beta \CH_n& \text{if }\beta \in \N^n.
\end{cases}
\end{equation}
$$\dim_K\, D_{n,\beta} =\begin{cases}
1& \text{if }\beta = \alpha - e_i\in \N^{n,i}-e_i,\\
n& \text{if }\beta \in \N^n.
\end{cases}$$
Let $\CG$ be a Lie algebra and $\CH$ be its Lie subalgebra. The {\em centralizer} $C_\CG (\CH ) := \{ x\in \CG \, | \, [ x, \CH ] =0\}$ of $\CH$ in $\CG$ is a Lie subalgebra of $\CG$. In particular, $Z(\CG ) := C_{\CG }(\CG ) $ is the {\em centre} of the Lie algebra $\CG$. The {\em normalizer} $N_\CG (\CH ) :=\{ x\in \CG \, | \, [ x, \CH ] \subseteq \CH\}$ of $\CH$ in $\CG$ is a Lie subalgebra of $\CG$, it is the largest Lie subalgebra of $\CG$ that contains $\CH $ as an ideal.
Let $V$ be a vector space over $K$. A $K$-linear map $\d : V\ra V$
is called a {\em locally nilpotent map} if $V=\bigcup_{i\geq 1} \ker
(\d^i)$ or, equivalently, for every $v\in V$, $\d^i (v) =0$ for
all $i\gg 1$. When $\d$ is a locally nilpotent map in $V$ we
also say that $\d$ {\em acts locally nilpotently} on $V$. Every {\em nilpotent} linear map $\d$, that is $\d^n=0$ for some $n\geq 1$, is a locally nilpotent map but not vice versa, in general. Let
$\CG$ be a Lie algebra. Each element $a\in \CG$ determines the
derivation of the Lie algebra $\CG$ by the rule $\ad (a) : \CG
\ra \CG$, $b\mapsto [a,b]$, which is called the {\em inner
derivation} associated with $a$. The set $\Inn (\CG )$ of all the
inner derivations of the Lie algebra $\CG$ is a Lie subalgebra of
the Lie algebra $(\End_K(\CG ), [\cdot , \cdot ])$ where $[f,g]:=
fg-gf$. There is the short exact sequence of Lie algebras
$$ 0\ra Z(\CG ) \ra \CG\stackrel{\ad}{\ra} \Inn (\CG )\ra 0,$$
that is $\Inn (\CG ) \simeq \CG / Z(\CG )$ where $Z(\CG )$ is the {\em centre} of the Lie algebra $\CG$ and $\ad ([a,b]) = [
\ad (a) , \ad (b)]$ for all elements $a, b \in \CG$. An element $a\in \CG$ is called a {\em locally nilpotent element} (respectively, a {\em nilpotent element}) if so is the inner derivation $\ad (a)$ of the Lie algebra $\CG$.
$\noindent $
{\bf The Cartan subalgebra $\CH_n$ of $D_n$}. A nilpotent Lie subalgebra $C$ of a Lie algebra $\CG$ is called a {\em Cartan subalgebra} of $\CG$ if it coincides with its normalizer. We use often the following obvious observation: {\em An abelian Lie subalgebra that coincides with its centralizer is a maximal abelian Lie subalgebra}.
\begin{lemma}\label{a11Mar13
\begin{enumerate}
\item $\CH_n$ is a Cartan subalgebra of $D_n$.
\item $\CH_n=C_{D_n}(\CH_n)$ is a maximal abelian subalgebra of $D_n$.
\end{enumerate}
\end{lemma}
{\it Proof}. Statements 1 and 2 follows from (\ref{Dnb}) and (\ref{Dnb1}). $\Box $
$\noindent $
{\bf $P_n$ is a $D_n$-module}. The polynomial algebra $P_n$ is a (left) $D_n$-module: $D_n \times P_n\ra P_n$, $(\der, p)\mapsto \der *p$. In more detail, if $\der = \sum_{i=1}^n a_i\der_i$ where $a_i\in P_n$ then
$$\der * p = \sum_{i=1}^n a_i\frac{\der p}{\der x_i}.$$
The field $K$ is a $D_n$-submodule of $P_n$ and
\begin{equation}\label{IkerdiK}
\bigcap_{i=1}^n \ker_{P_n}(\der_i)= K.
\end{equation}
\begin{lemma}\label{xa11Mar13
The $D_n$-module $P_n/K$ is simple with $\End_{D_n}(P_n/K)=K\id $ where $\id$ is the identity map.
\end{lemma}
{\it Proof}. Let $M$ be a nonzero submodule of $P_n/K$ and $0\neq p\in M$. Using the actions of $\der_1, \ldots , \der_n$ on $p$ we obtain an element of $M$ of the form $\l x_i$ for some $\l\in K^*$. Hence, $x_i\in M$ and $x^\alpha = x^\alpha \der_i *x_i \in M$ for all $0\neq \alpha \in \N^n$. Therefore, $M = P_n /K$. Let $f\in \End_{D_n}(P_n/K)$. Then applying $f$ to the equalities $\der_i*(x_1+K)=\d_{i1}$ for $i=1, \ldots , n$, we obtain the equalities
$$ \der_i*f(x_1+K)=\d_{i1} \;\; {\rm for }\;\; i=1, \ldots , n.$$ Hence, $f(x_1+K)\in \bigcap_{i=2}^n \ker_{P_n/K}(\der_i) \cap \ker_{P_n/K}(\der_i^2) = (K[x_1]/K)\cap \ker_{P_n/K}(\der_i^2) =K(x_1+K)$. So, $f(x_1+K) = \l ( x_1+K)$ and so $f=\l \id$, by the simplicity of the $D_n$-module $P_n/K$.
$\Box $
$\noindent $
{\bf The $G_n$-module $D_n$}. The Lie algebra $D_n$ is a $G_n$-module,
$$ G_n\times D_n\ra D_n, \;\; (\s , \der ) \mapsto \s (\der ) := \s \der \s^{-1}.$$
Every automorphism $\s \in G_n$ is uniquely determined by the elements
$$x_1':=\s (x_1), \; \ldots , \; x_n':=\s (x_n).$$
Let $M_n(P_n)$ be the algebra of $n\times n$ matrices over $P_n$. The matrix $J(\s) := (J(\s )_{ij}) \in M_n(P_n)$, where $J(\s )_{ij} =\frac{\der x_j'}{\der x_i}$, is called the {\em Jacobian matrix} of the automorphism (endomorphism) $\s$ and its determinant $\CJ (\s ) :=\det \, J(\s)$ is called the {\em Jacobian} of $\s$. So, the $j$'th column of $J(\s )$ is the {\em gradient} $\grad \, x_j':=(\frac{\der x_j'}{\der x_1}, \ldots , \frac{\der x_j'}{\der x_n})^T$ of the polynomial $x_j'$. Then the derivations
$$\der_1':= \s \der_1\s^{-1}, \; \ldots , \; \der_n':= \s\der_n\s^{-1}$$ are the partial derivatives of $P_n$ with respect to the variables $x_1', \ldots , x_n'$,
\begin{equation}\label{ddp=dxi}
\der_1'=\frac{\der}{\der x_1'}, \; \ldots , \; \der_n'=\frac{\der}{\der x_n'}.
\end{equation}
Every derivation $\der \in D_n$ is a unique sum $\der = \sum_{i=1}^n a_i\der_i$ where $a_i = \der *x_i\in P_n$. Let $\der := (\der_1, \ldots , \der_n)^T$ and $ \der' := (\der_1', \ldots , \der_n')^T$ where $T$ stands for the transposition. Then
\begin{equation}\label{dp=Jnd}
\der'=J(\s )^{-1}\der , \;\; {\rm i.e.}\;\; \der_i'=\sum_{j=1}^n (J(\s )^{-1})_{ij} \der_j\;\; {\rm for }\;\; i=1, \ldots , n.
\end{equation}
In more detail, if $\der'=A\der $ where $A= (a_{ij})\in M_n(P_n)$, i.e. $\der_i=\sum_{j=1}^n a_{ij}\der_j$. Then for all $i,j=1, \ldots , n$,
$$\d_{ij}= \der_i'*x_j'=\sum_{k=1}^na_{ik}\frac{\der x_j'}{\der x_k}$$
where $\d_{ij}$ is the Kronecker delta function. The equalities above can be written in the matrix form as $AJ(\s) = 1$ where $1$ is the identity matrix. Therefore, $A= J(\s )^{-1}$.
Suppose that a group $G$ acts on a set $S$. For a nonempty subset $T$ of $S$, $\St_G(T):=\{ g\in G\, | \, gT=T\}$ is the {\em stabilizer} of the set $T$ in $G$ and $\Fix_G(T):=\{ g\in G\, | \, gt=t$ for all $t\in T\}$ is the {\em fixator} of the set $T$ in $G$. Clearly, $\Fix_G(T)$ is a {\em normal} subgroup of $\St_G(T)$.
$\noindent $
{\bf The maximal abelian Lie subalgebra $\CD_n$ of $D_n$}.
\begin{lemma}\label{b11Mar13
\begin{enumerate}
\item $C_{D_n}(\CD_n) =\CD_n$ and so $\CD_n$ is a maximal abelian Lie subalgebra of $D_n$.
\item $\Fix_{G_n}(\CD_n) = \Fix_{G_n}(\der_1, \ldots , \der_n) = \Sh_n$.
\item $D_n$ is a faithful $G_n$-module, i.e. the group homomorphism $G_n\ra \mG_n$, $ \s\mapsto \s : \der\mapsto \s \der \s^{-1}$, is a monomorphism.
\item $\Fix_{G_n}(\der_1, \ldots , \der_n , H_1, \ldots , H_n)=\{ e\}$.
\end{enumerate}
\end{lemma}
{\it Proof}. 1. Statement 1 follows from (\ref{xadbd1}).
2. Let $\s \in \Fix_{G_n}(D_n)$ and $J(\s ) = (J_{ij})$. By (\ref{dp=Jnd}), $\der = J(\s ) \der $, and so, for all $i,j=1, \ldots , n$,
$\d_{ij} = \der_i*x_j=J_{ij}$,
i.e. $J(\s ) = 1$, or equivalently, by (\ref{IkerdiK}),
$$x_1'=x_1+\l_1, \ldots , x_n'=x_n+\l_n$$
for some scalars $\l_i\in K$, and so $\s\in \Sh_n$.
3 and 4. Let $\s\in \Fix_{G_n}=(\der_1, \ldots , \der_n , H_1, \ldots , H_n)$. Then $\s \in \Fix_{G_n}(\der_1, \ldots , \der_n)=\Sh_n$, by statement 2. So, $\s (x_1) = x_1+\l_1, \ldots , \s (x_n) = x_n+\l_n$ where $\l_i\in K$. Then $x_i\der_i = \s (x_i\der_i) = (x_i+\l_i) \der_i$ for $i=1, \ldots , n$, and so $\l_1=\cdots = \l_n=0$. This means that $\s = e$. So,
$\Fix_{G_n}=(\der_1, \ldots , \der_n , H_1, \ldots , H_n)=\{ e\}$ and $D_n$ is a faithful $G_n$-module. $\Box $
$\noindent $
By Lemma \ref{b11Mar13}.(3), we identify the group $G_n$ with its image in $\mG_n$.
\begin{lemma}\label{c11Mar13
\begin{enumerate}
\item $D_n$ is a simple Lie algebra.
\item $Z(D_n)=\{ 0\}$.
\item $[D_n, D_n]=D_n$.
\end{enumerate}
\end{lemma}
{\it Proof}. 1. Let $0\neq a\in D_n$ and $\ga = (a)$ be the ideal of the Lie algebra $D_n$ generated by the element $a$. We have to show that $\ga = D_n$. Using the inner derivations $\d_1, \ldots , \d_n$ we see that $\der_i\in \ga$ for some $i$. Then $\ga = D_n$ since
$$ x^\alpha \der_j = (\alpha_i+1)^{-1} [ \der_i, x^{\alpha +e_i}\der_j]\in \ga$$ for all $\alpha$ and $j$.
2 and 3. Statements 2 and 3 follow from statement 1.
$\Box $
$\noindent $
\begin{proposition}\label{B11Mar13
\begin{enumerate}
\item $\Fix_{\mG_n} (\der_1, \ldots , \der_n, H_1, \ldots , H_n) = \{ e\}$.
\item Let $\s, \tau \in \mG_n$. Then $\s = \tau $ iff $\s (\der_i) = \tau (\der_i)$ and $\s (H_i) = \tau (H_i)$ for $i=1, \ldots , n$.
\item $\Fix_{\mG_n} (\der_1, \ldots , \der_n) = \Sh_n$.
\end{enumerate}
\end{proposition}
{\it Proof}. 1. Let $\s\in F:= \Fix_{\mG_n} (\der_1, \ldots , \der_n, H_1, \ldots , H_n)$. We have to show that $\s = e$. Since $\s \in \Fix_{\mG_n} (H_1, \ldots , H_n)$, the automorphism $\s$ respects the weight decomposition of $D_n$. By (\ref{Dnb1}),
$\s (x^\alpha \der_i) = \l_{\alpha , i} x^\alpha \der_i$ for all $\alpha \in \N^{n,i}$ and $i=1, \ldots , n$ where $\l_{\alpha , i}\in K$.
Clearly, $\l_{0 , i}=1$ for $i=1, \ldots , n$. Since $\s \in \Fix_{\mG_n} (\der_1, \ldots , \der_n)$, by applying $\s$ to the relations $\alpha_jx^{\alpha-e_j}\der_i=[\der_j, x^\alpha\der_i]$, we get the relations
$$\alpha_j \l_{\alpha-e_j,i} x^{\alpha-e_j} \der_i= [ \der_j, \l_{\alpha, i} x^\alpha\der_i]= \alpha_j\l_{\alpha , i}x^{\alpha - e_j}\der_i.$$
Hence $\l_{\alpha , i} = \l_{\alpha - e_j, i}$ provided $\alpha_j\neq 0$. We conclude that all the coefficients $\l_{\alpha , i}$ are equal to one of the coefficients $\l_{e_i, j}$ where $i,j=1, \ldots , n$ and $i\neq j$. The relations $\der_j=[\der_i, x_i\der_j]$ implies the relations $\der_j= [\der_i, \l_{e_i, j}x_i\der_j]=\l_{e_i, j}\der_j$, hence all the coefficients $\l_{e_i, j}$ are equal to 1.
So, $\oplus_{i=1}^n P_n^{\der_i} \der_i\subseteq \CF := \Fix_{D_n}(\s ) :=\{ \der\in D_n \,| \, \s (\der ) = \der\}$. To finish the proof of statement 1 it suffices to show that $x^\alpha H_i\in \CF$ for all $\alpha \in \N^n$ and $i=1, \ldots , n$, see (\ref{Dnb2}) and (\ref{Dnb}). We use induction on $|\alpha |:=\alpha_1+\cdots + \alpha_n$. If $|\alpha |=0$ the statement is obvious as $\s\in F$. Suppose that $|\alpha |>0$. Using the commutation relations
\begin{equation}\label{djxaHj}
[\der_j, x^\alpha H_i] = \begin{cases}
\alpha_jx^{\alpha - e_j}H_i& \text{if }j\neq i,\\
(\alpha_i+1) x^\alpha \der_i& \text{if }j=i, \\
\end{cases}
\end{equation}
the induction and the previous case, we see that
$$ [\der_j, \s (x^\alpha H_i) - x^\alpha H_i]=0 \;\; {\rm for }\;\; i=1, \ldots , n. $$
Therefore, $ \s (x^\alpha H_i) - x^\alpha H_i\in C_{D_n} (\CD_n) = \CD_n$. Since the automorphism $\s$ respects the weight decomposition of $D_n$, we must have $ \s (x^\alpha H_i) - x^\alpha H_i\in x^\alpha \CH_n \cap \CD_n=\{ 0 \}$. Hence, $x^\alpha H_i\in \CF$, as required.
2. Statement 2 follows from statement 1.
3. Clearly, $\Sh_n \subseteq F=\Fix_{\mG_n} (\der_1, \ldots , \der_n)$. Let $\s \in F$ and $H_i':=\s (H_i), \ldots , H_n':=\s (H_n)$. Applying the automorphism $\s$ to the commutation relations $[\der_i, H_j]=\d_{ij}\der_i$ gives the relations $[\der_i, H_j']=\d_{ij}\der_i$. By taking the difference, we see that $[\der_i, H_j'-H_j]=0$ for all $i$ and $j$. Therefore, $H_i'=H_i+d_i$ for some elements $d_i\in C_{D_n}(\CD_n) = \CD_n$ (Lemma \ref{b11Mar13}.(1)), and so $d_i=\sum_{j=1}^n \l_{ij}\der_j$ for some elements $\l_{ij}\in K$. The elements $H_1', \ldots , H_n'$ commute, hence
$$ [ H_j, \der_i]= [H_i, \der_j] \;\; {\rm for \; all}\;\; i,j, $$
or equivalently,
$$ \l_{ij}\der_j= \l_{ji}\der_i \;\; {\rm for \; all}\;\; i,j. $$
This means that $\l_{ij}=0$ for all $i\neq j$, i.e. $$H_i'= H_i+\l_{ii}\der_i= (x_i+\l_{ii})\der_i= s_\l (H_i)$$
where $s_\l \in \Sh_n$, $s_\l (x_i) = x_i+\l_{ii}$ for all $i$. Then $s_\l^{-1}\s \in \Fix_{\mG_n} (\der_1, \ldots , \der_n, H_1, \ldots , H_n) = \{ e\}$ (statement 2), and so $\s = s_\l \in \Sh_n$. $\Box $
\begin{lemma}\label{c13Mar13
Let $\s \in \mG_n$ and $\der_1':=\s (\der_1), \ldots , \der_n':= \s (\der_n)$. Then
\begin{enumerate}
\item $\der_1', \ldots , \der_n'$ are commuting, locally nilpotent derivations of $P_n$.
\item $\bigcap_{i=1}^n\ker_{D_n}(\der_i')=K$.
\end{enumerate}
\end{lemma}
{\it Proof}. 1. The derivations $\der_1', \ldots , \der_n'$ commute since $\der_1, \ldots , \der_n$ are commute. The inner derivations $\d_1, \ldots , \d_n$ of the Lie algebra $D_n$ are commuting and locally nilpotent. Hence, inner derivations
$$\d_1':= \ad (\der_1'), \ldots , \d_n':= \ad (\der_n')$$
of the Lie algebra $D_n$ are commuting and locally nilpotent. The vector space $P_n\der_i'$ is closed under the derivations $\d_j'$ since
$$ \d_j'(P_n\der_i') = [ \der_j', P_n\der_i']= (\der_j' *P_n)\cdot \der_i' \subseteq P_n\der_i'.$$
Therefore, $\der_1', \ldots , \der_n'$ are locally nilpotent derivations of the polynomial algebra $P_n$.
2. Let $\l \in \bigcap_{i=1}^n \ker_{P_n}(\der_i')$. Then
$$\l \der_1' \in C_{D_n}(\der_1', \ldots , \der_n')=\s (C_{D_n}(\der_1, \ldots , \der_n))= \s (C_{D_n}(\CD_n))= \s (\CD_n) = \s (\bigoplus_{i=1}^n K\der_i) = \bigoplus_{i=1}^n K\der_i', $$
since $C_{D_n}(\CD_n)=\CD_n$, Lemma \ref{b11Mar13}.(1). Then $\l \in K$ since otherwise the infinite dimensional space $\bigoplus_{i\geq 0} K\l^i \der_1'$ would be a subspace of a finite dimensional space $\s (\CD_n)$. $\Box $
$\noindent $
The following lemma is well-known and it is easy to prove.
\begin{lemma}\label{Aslice
Let $\der$ be a locally nilpotent derivation of a commutative $K$-algebra $A$ such that $\der (x) =1$ for some element $x\in A$. Then $A= A^\der [x]$ is a polynomial algebra over the ring $A^\der := \ker (\der )$ of constants of the derivation $\der$ in the variable $x$.
\end{lemma}
The next theorem is the most important point in the proof of Theorem \ref{11Mar13} and, roughly speaking, the main reason why Theorem \ref{11Mar13} holds.
\begin{theorem}\label{A13Mar13
Let $\der_1', \ldots , \der_n'$ be commuting, locally nilpotent derivations of the polynomial algebra $P_n$ such that $\bigcap_{i=1}^n \ker_{P_n}(\der_i')=K$. Then there exist polynomials $x_1', \ldots , x_n'\in P_n$ such that
\begin{equation}\label{con*}
\der_i'*x_j'=\d_{ij}.
\end{equation}
Moreover, the algebra homomorphism
$$\s : P_n\ra P_n , \;\; x_1\mapsto x_1', \ldots , x_n\mapsto x_n'$$ is an automorphism such that $\der_i'= \s \der_i \s^{-1} = \frac{\der}{\der x_i'}$ for $i=1, \ldots , n$.
\end{theorem}
{\it Proof}. Case $n=1$: By Lemma \ref{c13Mar13}, the derivation $\der_1'$ of the polynomial algebra $P_1$ is a locally nilpotent derivation with $K_1':=\ker_{P_1}(\der_1') = K$. Hence, $\der_1'*x_1'=1$ for some polynomial $x_1'\in P_1$. By Lemma \ref{Aslice}, $K[x_1]= K_1'[x_1']= K[x_1']$, and so $\s : K[x_1]\ra K[x_1]$, $ x\mapsto x_1'$, is an automorphism such that $ \der_1' = \frac{d}{d x_1'} = \s \frac{d}{d x_1}\s^{-1}$.
Case $n\geq 2$. Let $K_i':= \ker_{P_n}(\der_i')$ for $i=1, \ldots, n$. Clearly, $ K\subseteq K_i'$.
(i) $K_i'\neq K$ {\em for} $i=1, \ldots , n$: If $K_i'= K$ for some $i$ then by the same argument as in the case $n=1$ there exists a polynomial $x_i'\in P_n$ such that $\der_i'*x_i'=1$, and so $P_n= K_i'[x_i'] = K[x_i]$, a contradiction.
(ii) Let $m $ be the maximum of ${\rm card} (I)$ such $\emptyset \neq I \subseteq \{ 1, \ldots , n-1\}$ and $\bigcap_{i\in I} K_i'\neq K$. By (i), $2\leq m \leq n-1$. Changing (if necessary) the order of the derivations $\der_1',\ldots , \der_n'$ we may assume that $A:= \bigcap_{i=1}^m K_i'\neq K$. Then the algebra $A$ is infinite dimensional (since $K\neq A \subseteq P_n$) and invariant under the action of the derivations $\der_j'$ for $j=m+1, \ldots , n$. By the choice of $m$,
$$ A^{\der_j'}= K_j'\cap \bigcap_{i=1}^m K_i' = K\;\; {\rm for}\;\; j=m+1, \ldots , n$$
and the derivations $\der_j'$ acts locally nilpotently on the algebra $A^{\der_j'}$. Therefore, for each index $j=m+1, \ldots , n$, there exists an element $x_j'\in A$ such that $\der_j'*x_j'=1$, and so (Lemma \ref{Aslice})
\begin{equation}\label{Adjr}
A= A^{\der_j'} [ x_j']= K[x_j']\;\; {\rm for} \;\; j=m+1, \ldots , n.
\end{equation}
(ii)(a) Suppose that $m=n-1$, i.e. $\der_i'*x_n'=\d_{in}$ for all $i=1, \ldots , n$. By Lemma \ref{Aslice}, $P_n = K_n'[x_n']$. The algebra $K_n'$ admits the set of commuting, locally nilpotent derivations
$$ \der_1'':= \der_1'|_{K_n'}, \ldots , \der_{n-1}'':= \der_{n-1}'|_{K_n'}$$
with $\bigcap_{i=1}^{n-1} \ker_{K_n'} (\der_i'') = K_n'\cap \bigcap_{i=1}^{n-1} K_i' = K.$
(ii)(b) Suppose that $m<n-1$. By (\ref{Adjr}),
$$ K^*x_{m+1}' +K= K^*x_{m+2}' +K=\cdots = K^*x_n ' +K, $$
and so $\l_j := \der_j'*x_n'\in K$ for $j=m+1, \ldots , n-1$. Hence, $(\der_j'-\l_j\der_n') *x_n'=0$ for $j=m+1, \ldots , n-1$. A linear combination of commuting, locally nilpotent derivations is a locally nilpotent derivation (the proof boils down to the case $\der +\d$ of two commuting, locally nilpotent derivations, then the result follows from $(\der +\d )^m = \sum_{i=0}^m {m \choose i} \der^i \d^{m-i}$ and $ \der^i \d^{m-i}= \d^{m-i} \der^i$).
Using the set of commuting, locally nilpotent derivations $\der_1', \ldots , \der_n'$ that satisfy (\ref{con*}) we obtain the set of commuting, locally nilpotent derivations
$$\d_1':=\der_1',\; \ldots , \; \d_m':=\der_m',\; \d_{m+1}':=\der_{m+1}'-\l_{m+1}\der_n', \; \ldots , \; \d_{n-1}':=\der_{n-1}'-\l_{n-1}\der_n',\; \d_n':=\der_n$$
that satisfy (\ref{con*}) with
$$ \d_i'*x_n'=\d_{in}\;\; {\rm for}\;\; i=1, \ldots , n.$$ Then repeating the arguments of (ii)(a), we see that $P_n = K_n'[x_n']$. The algebra $K_n'$ admits the set of commuting, locally nilpotent derivations
$$ \der_1'':= \d_1'|_{K_n'},\; \ldots , \; \der_{n-1}'':= \d_{n-1}'|_{K_n'}$$
with $$\bigcap_{i=1}^{n-1} \ker_{K_n'} (\der_i'') = K_n'\cap \bigcap_{i=1}^{n-1}\ker_{P_n}(\d_i')= K_n'\cap \bigcap_{i=1}^{n-1}\ker_{P_n}(\der_i')=
\bigcap_{i=1}^n K_i' = K.$$
(iii) Using the cases (ii)(a) and (ii)(b) $n-1$ more times we find polynomials $x_1', \ldots , x_n'$ and commuting set of locally nilpotent derivations of $P_n$, say, $\D_1, \ldots , \D_n$ that satisfy (\ref{con*}) and such that
$(\alpha)$ $\D_i*x_j'=\d_{ij}$ for all $i,j=1, \ldots , n$;
$(\beta )$ the $n$-tuple of derivations $\D = (\D_1, \ldots , \D_n)^T$ is obtained from the
$n$-tuple of derivations $\der'= (\der_1' , \ldots , \der_n')^T$ by unitriangular (hence invertible) scalar matrix $\L = (\l_{ij})\in M_n(K)$ such that $ \D = \L \der'$; and
$(\gamma )$ (where $K_1'':=\ker_{P_n}(\D_1), \ldots , K_n'':=\ker_{P_n}(\D_n)$)
\begin{eqnarray*}
P_n &=& K_n''[x_n']= (K_{n-1}''\cap K_n'')[x_{n-1}', x_n']= \cdots = (\bigcap_{i=s}^n K_i'') [x_s',\ldots , x_n']= \cdots \\
&=& (\bigcap_{i=1}^n K_i'') [x_1',\ldots , x_n']
=K[x_1',\ldots , x_n'].
\end{eqnarray*}
(iv) Replacing the row $x'= (x_1', \ldots , x_n')$ by the row $x'\L$ gives the required elements of the theorem. Indeed, by $(\alpha )$, $\L \cdot (\der_i'*x_j') = 1$, the identity $n\times n$ matrix. Hence, $ (\der_i'*x_j') \cdot \L = 1$, as required.
(v) Let $x_1' , \ldots , x_n'$ be the set of polynomials as in the theorem. Then $\s $ is an algebra automorphism (see $(\gamma )$ and (iv)) such that $\der_i'= \s \der_i\s^{-1} = \frac{\der}{\der x_i'}$ for $i=1, \ldots , n$. $\Box$
\begin{corollary}\label{b13Mar13
Let $\s \in \mG_n$. Then $\tau \s \in \Fix_{\mG_n} (\der_1, \ldots , \der_n)$ for some $\tau \in G_n$.
\end{corollary}
{\it Proof}. By Lemma \ref{c13Mar13}, the elements $\der_1':= \s (\der_1), \ldots , \der_n':= \s (\der_n)$ satisfy the assumptions of Theorem \ref{A13Mar13}. By Theorem \ref{A13Mar13}, $\der_1':= \tau^{-1} (\der_1), \ldots , \der_n':= \tau^{-1} (\der_n)$ for some $\tau \in G_n$. Therefore, $ \tau \s \in \Fix_{\mG_n} (\der_1, \ldots , \der_n)$. $\Box $
$\noindent $
{\bf Proof of Theorem \ref{11Mar13}}. Let $\s \in \mG_n$. By Corollary \ref{b13Mar13}, $\tau \s \in \Fix_{\mG_n} (\der_1, \ldots , \der_n)=\Sh_n$ (Proposition \ref{B11Mar13}.(3)). Therefore, $\s \in G_n$, i.e. $\mG_n = G_n$. $\Box$
$${\bf Acknowledgements}$$
The work is partly supported by the Royal Society and EPSRC.
\small{
|
{
"timestamp": "2013-04-16T02:01:17",
"yymm": "1304",
"arxiv_id": "1304.3836",
"language": "en",
"url": "https://arxiv.org/abs/1304.3836"
}
|
"\\section{Introduction}\n\n Twenty seven years ago Matsui and Satz \\cite{Matsui:1986dk} proposed t(...TRUNCATED)
| {"timestamp":"2013-04-16T02:04:12","yymm":"1304","arxiv_id":"1304.4154","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nThe Antiproton Decelerator (AD) facility of CERN \\cite{Maury:97,Pavel:0(...TRUNCATED)
| {"timestamp":"2013-04-16T02:00:07","yymm":"1304","arxiv_id":"1304.3721","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nIntegrated luminosity measurement at a future linear collider will be pe(...TRUNCATED)
| {"timestamp":"2013-08-22T02:04:07","yymm":"1304","arxiv_id":"1304.4082","language":"en","url":"https(...TRUNCATED)
|
"\n\n\\section{Introduction}\\label{sec:intro}\nConsider two parties, Alice and Bob, connected by an(...TRUNCATED)
| {"timestamp":"2013-04-15T02:02:31","yymm":"1304","arxiv_id":"1304.3658","language":"en","url":"https(...TRUNCATED)
|
"\\section{\\texorpdfstring{Introduction: Structure, application, data,\r\noutline.}{Introduction: S(...TRUNCATED)
| {"timestamp":"2013-04-16T02:03:23","yymm":"1304","arxiv_id":"1304.4066","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nIn quantum theory Hamilton operators with Fermi-interactions\nhave a lon(...TRUNCATED)
| {"timestamp":"2013-07-29T02:03:29","yymm":"1304","arxiv_id":"1304.3991","language":"en","url":"https(...TRUNCATED)
|
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