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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 5
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 17764)
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 5
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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text
string | meta
dict |
|---|---|
\section{Introduction}
\renewcommand{\baselinestretch}{1.3}
\small \normalsize
A number of investigations (see e.g.~$^{1,2)}$) of the nuclear
properties of neutron rich nuclei close to and at the magic neutron
number $N=28$ has been motivated by the fact that these isotopes may
play a crucial role in explaining the Ca-Ti-Cr-Fe isotopic anomalies
found in meteoritic inclusions. Experimental information is
scarce due to the short half-life of the involved nuclei. However,
$\beta$-decay properties of several isotopes in that region
have been measured~$^{1,3,4)}$ recently. Theoretical
predictions can be improved by utilizing such experimental data.
\bigskip
For the isotopes $^{43}$P, $^{42}$S, $^{44}$S, $^{45}$S, $^{44}$Cl,
$^{45}$Cl, $^{46}$Cl, and $^{47}$Ar, nuclear deformations have been derived
from the measured $\beta$-decay
half-lives in a recent QRPA parameter study~$^{3,4)}$.
Taking these deformations we
calculated the QP-levels below and close to the neutron separation
energy, which are needed as input for the determination of the capture
cross sections. However, we also extended our calculations to more
neutron rich isotopes of S and Ar, and to Cr-, Ti-
and Fe-isotopes by taking the deformations from a theoretical mass
formula~$^{5)}$. With this level information we updated the theoretical
cross sections of the even-even isotopes of $^{40-44}$S, $^{46-50}$Ar,
$^{56-66}$Ti, $^{62-68}$Cr, and $^{72-76}$Fe. The importance of -- and
the accuracy in the reproduction of experimental data by -- direct
capture for the
case of $^{48}$Ca was already shown elsewhere~$^{6)}$.
\section{Method}
Mainly two reaction mechanisms have to be considered for astrophysically
relevant
neutron energies: the Compound Nucleus Mechanism
(CN, Hauser-Feshbach, statistical model) and Direct Capture (DC) to
bound states. For
the majority of neutron-rich intermediate and heavy mass nuclei CN will
dominate. However, the statistical model is only applicable as long as
the level density is sufficiently high (i.e. $\ge 10$ MeV$^{-1}$).
Therefore, DC may dominate for capture on nuclei with low level
densities at the neutron separation energy. This can be the case for
light nuclei, nuclei close to magic neutron numbers, and nuclei close to
the neutron drip line (similar for proton capture on the proton rich
side).
\bigskip
In previous r-process network calculations~$^{1)}$ only CN was considered in the
theoretical neutron capture cross sections for the relevant nuclear
region. Moreover, the level densities were computed from a back-shifted
Fermi-gas formula~$^{7)}$. For our purposes we calculated
quasi-particle levels in a folded-Yukawa potential and with
Lipkin-Nogami pairing~$^{5)}$. The levels derived in such a way were
used as input for the statistical model code SMOKER~$^{8)}$ and for the direct
capture code TEDCA~$^{9)}$.
\bigskip
After the calculation of the nuclear energy levels and the CN cross
sections, the DC contributions for capture on even-even nuclei (to first
order one can assume that the level density in nuclei with one or two
unpaired nucleons will be higher than for even-even nuclei, and that
therefore DC will be most important for capture on even-even targets)
were determined as follows. The theoretical cross section
$\sigma^{\mathrm{th}}$ is given by a sum over each final state $i$~$^{10)}$
\begin{equation}
\label{dc}
\sigma^{\mathrm{th}}=\sum_i C_i^2 S_i \sigma_i^{\mathrm{ DC}} \quad.
\end{equation}
In our case the isospin Clebsch-Gordan coefficients $C_i$ are equal to
unity. The
spectroscopic
factors $S_i$ describe the overlap between the antisymmetrized wave
functions of
target+n and the final state. In the case of one-nucleon capture on
even-even deformed nuclei, the
spectroscopic factor for capture into a state $i$, which has an occupation
probability $v_i^2$ in the target, can be reduced to~$^{11)}$
\begin{equation}
S_i=1-v_i^2\quad.
\end{equation}
The corresponding probabilities $v_i^2$ are found by solving the
Lipkin-Nogami pairing equations~$^{5)}$.
\bigskip
The factors $\sigma_i^{\mathrm{DC}}$ in Eq.~\ref{dc} are essentially
determined by the overlap of the scattering wave function in the
entrance channel, the bound-state wave function and the
multipole-transition operator. The potentials needed for the calculation
of the before-mentioned wave functions are obtained by applying the
folding procedure. In this approach, the nuclear density of the target
$\rho_{\mathrm T}$ is folded with an energy and density dependent
effective nucleon-nucleon interaction $w_{\mathrm{eff}}$~$^{12)}$
\begin{equation}
V(E,R)=\lambda V_{\mathrm{F}}(E,R)=\lambda \int \rho_{\mathrm
T}(\vec{r})w_{\mathrm{eff}}(E,\rho_{\mathrm T},\vert \vec{R}-\vec{r}
\vert)\,d\vec{r}\quad,
\end{equation}
with $\vec{R}$ being the separation of the centers of mass of the two
colliding nuclei. The interaction $w_{\mathrm{eff}}$ is only weakly
energy dependent in the energy range of interest $^{13)}$.
The density distributions $\rho_{\mathrm T}$ were
calculated from the folded-Yukawa wave functions.
\bigskip
The only remaining parameter $\lambda$ was determined by employing a
parametrization of the volume integral $I$
\begin{equation}
I(E)=\frac{4\pi}{A}\int V_{\mathrm{F}}(R,E)R^2\,dR\quad,
\end{equation}
expressed in units of MeV\,fm$^3$, and with the mass number $A$ of the
target nucleus. Recently, the averaged volume integral $I_0$
was fitted to a function of mass number $A$, charge $Z$ and
neutron number $N$ for a set of specially selected nuclei:~$^{14)}$
\begin{equation}
I_0=255.13+984.85A^{-1/3}+9.52\times 10^6 \frac{N-Z}{A^3}\quad.
\end{equation}
Thus, the strength factor $\lambda$ can easily be computed for each
nucleus by using
\begin{equation}
\lambda=\frac{I_0}{I}\quad.
\end{equation}
For the bound states, the parameters $\lambda$ are fixed by the requirement
of a correct
reproduction of the separation energies.
\section{Results and Discussion}
The calculated CN and DC cross sections as well as the deformation
parameters used for the calculation of the single-particle levels are shown in
Table 1. The quoted neutron separation energies in the
final nucleus are taken
from an experimental
compilation~$^{15)}$ where available, otherwise they were calculated from
the theoretical mass formula~$^{5)}$. Furthermore,
$\beta$-decay half-lives are compared to theoretical lifetimes against
neutron capture, computed from our results with a neutron number density
of $3\times10^{19}$ cm$^{-3}$ (S, Ar) and $6\times10^{20}$ cm$^{-3}$
(Ti, Cr, Fe). Shown are experimental
$\beta$-decay properties~$^{1,3,4)}$ and also theoretical values obtained
by using
the QRPA code~$^{16)}$ with folded-Yukawa wave functions and
Lipkin-Nogami pairing for nuclei for which no experimental
$\beta$-decay properties were known.
\begin{table}
\renewcommand{\baselinestretch}{1.0}
\small \normalsize
\caption{Calculated 30 keV (c.m.) Maxwellian averaged neutron capture
cross sections $<\sigma>_{30\,\mathrm{keV}}$ for CN and DC.
The column labeled `\%' gives the portion of direct capture in the total cross
section. Also shown are the deformations $\epsilon_2$ and neutron
separation energies
S$_{\mathrm{n}}$ of the final nucleus target+n. The neutron capture
half-lives T$_{1/2}$(n) were computed with the values from column
`DC+CN' and a neutron number density of $3\times10^{19}$ (S, Ar) and
$6\times10^{20}$ cm$^{-3}$ (Ti, Cr, Fe), respectively.}
\begin{center}
\begin{tabular}{clllllcll}
\hline
Target&\multicolumn{1}{c}{$\epsilon_2$}&\multicolumn{1}{c}{S$_{\mathrm{n}}$}
&\multicolumn{1}{c}{DC}&\multicolumn{1}{c}{CN}&\multicolumn{1}{c}{DC+CN}
&\multicolumn{1}{c}{\%}&\multicolumn{1}{c}{T$_{1/2}$(n)}
&\multicolumn{1}{c}{T$_{1/2}$($\beta$)}\\
& &\multicolumn{1}{c}{[MeV]}&\multicolumn{1}{c}{[mb]}
&\multicolumn{1}{c}{[mb]}&\multicolumn{1}{c}{[mb]}
& &\multicolumn{1}{c}{[s]}&\multicolumn{1}{c}{[s]}\\
\hline
\hline
$^{40}$S&+0.24&3.8238$^{\ddagger}$&0.4246&0.0851&0.5097&83&0.218&8.60\\
$^{42}$S&+0.30$^*$&3.3114$^{\ddagger}$&0.9466&0.0202&0.9668&98
&0.115&0.56$\pm$0.06$^{\dagger}$\\
$^{44}$S&$-$0.20$^*$&1.3344&0.0000&0.0044&0.0044&0
&25.25&0.12$\pm$0.01$^{\dagger}$\\
$^{44}$S$^{\mathrm{a}}$&+0.30&1.3344&0.014&0.0044&0.0184&76
&6.040&0.12$\pm$0.01$^{\dagger}$\\
\hline
$^{46}$Ar&$-$0.18$^*$&4.2590$^{\ddagger}$&0.5295&0.1203&0.6498&81&0.171&7.80\\
$^{48}$Ar&$-$0.22&1.7074&0.0427&0.0144&0.0571&75&1.946&0.11\\
$^{50}$Ar&$-$0.28&1.0804&0.0016&0.0030&0.0046&35&24.15&0.05\\
\hline
$^{56}$Ti&+0.13&2.1936$^{\ddagger}$&0.0147&0.1305&0.1452&10&0.038&0.421\\
$^{58}$Ti&$-$0.10&2.4244&0.0185&0.0797&0.0982&19&0.057&0.152\\
$^{60}$Ti&$-$0.02&2.1194&0.0165&0.0403&0.0568&29&0.098&0.054\\
$^{62}$Ti&$-$0.04&0.5878&0.0068&0.0030&0.0098&69&0.569&0.018\\
$^{64}$Ti&+0.06&0.4094&0.0013&0.0002&0.0015&87&3.561&0.039\\
$^{66}$Ti&+0.14&0.3914&0.0009&0.0002&0.0011&82&4.960&0.013\\
\hline
$^{62}$Cr&+0.30&2.9134&0.0119&0.3846&0.3965&$<$1&0.014&0.349\\
$^{64}$Cr&+0.05&1.8614&0.0170&0.0353&0.0523&33&0.106&0.154\\
$^{66}$Cr&+0.10&2.2374&0.0126&0.0673&0.0799&16&0.071&0.071\\
$^{68}$Cr&+0.16&1.8274&0.0129&0.0268&0.0397&32&0.140&0.026\\
\hline
$^{72}$Fe&+0.14&2.2474&0.0101&0.0572&0.0673&15&0.083&0.089\\
$^{74}$Fe&+0.07&2.0564&0.0104&0.0360&0.0464&22&0.120&0.052\\
$^{76}$Fe&+0.06&0.0584&0.0050&$3\times10^{-6}$&0.0050&$>$99&1.12&0.045\\
\hline
\end{tabular}
\end{center}
$^{\mathrm{a}}$ $\epsilon_2$($^{45}$S)$=$$\epsilon_2$($^{44}$S)$=+0.3^*$
(see text)\\
$^*$ deformation inferred from experimental $\beta$-decay
half-life~$^{3,4)}$\\
$^{\dagger}$ experimental $\beta$-decay half-life~$^{1,3,4)}$\\
$^{\ddagger}$ S$_{\mathrm{n}}$ from experimental values~$^{15)}$
\end{table}
\bigskip
From Table 1 one can see nicely the importance of direct capture when
approaching the magic neutron number $N=28$ (S, Ar), but also the
increasing contribution of DC to the cross section when approaching the
drip line. The latter point can clearly be seen for the Ti isotopes and
coincides well with the drop in level density at the neutron
separation energy. The calculated Cr isotopes are still farther away
from the drip line, therefore the capture cross section is dominated by
CN.
\bigskip
An interesting effect can be seen for $^{44}$S: DC is suppressed totally
when assuming a deformation $\epsilon_2$($^{45}$S)$=-0.2$, as suggested
in order to reproduce the experimental $\beta$-decay
half-life~$^{3,4)}$ of $^{45}$S. Due to the different deformations of
target ($\epsilon_2$($^{44}$S)$=+0.3$) and final nucleus, the level
order is changed and the previously unbound [303]7/2$^-$ level
($E_x$=0.0582 MeV) is shifted well below the Fermi energy
in $^{45}$S. This means that not only is a captured neutron added to one
level of
$^{44}$S, but also that the nucleus is undergoing a reordering process.
Such a process cannot be described by DC and therefore the respective
cross section will vanish. As found in the QRPA parameter
study~$^{3,4)}$, another deformation value consistent with the
experiment would be $\epsilon_2$($^{45}$S)$=+0.125$. However, even with
this value a similar effect can still be seen. Only with a deformation
very close to the deformation of the target nucleus, non-zero DC cross
sections can be obtained. Nevertheless, one has to note that the
calculated QP levels might have an uncertainty which is larger than the
distance of the level in question from the Fermi energy. Therefore, it
might still be reasonable to calculate a DC contribution by assuming the
same deformation for target and final nucleus.
The resulting cross sections for the two cases
$\epsilon_2$($^{45}$S)$=-0.2$ and $\epsilon_2$($^{45}$S)$=+0.3$ are
shown in Table 1.
The level schemes for $^{44}$S and $^{45}$S are shown in Table 2.
The astrophysical consequences are not changed for either deformation
because the neutron capture lifetime is longer by orders of
magnitude than the $\beta$-decay lifetime in both cases.
\begin{table}
\renewcommand{\baselinestretch}{1.0}
\small \normalsize
\caption{Quasi-particle levels for $^{44}$S and $^{45}$S.}
\begin{center}
\begin{tabular}{rl|rl}
\hline
\multicolumn{2}{c|}{$^{44}$S ($\epsilon_2=+0.3$)}&
\multicolumn{2}{c}{$^{45}$S ($\epsilon_2=-0.2$)}\\
\multicolumn{1}{c}{$E_{\mathrm{x}}$ [MeV]}&
\multicolumn{1}{c|}{[Nn$_{\mathrm{z}}$$\Lambda$]$\Omega$}&
\multicolumn{1}{c}{$E_{\mathrm{x}}$ [MeV]}&
\multicolumn{1}{c}{[Nn$_{\mathrm{z}}$$\Lambda$]$\Omega$}\\
\hline
\hline
$-$27.1979& [ 0 0 0] 1/2&$-$26.0041& [ 0 0 0] 1/2\\
$-$20.3364& [ 1 1 0] 1/2&$-$18.2826& [ 1 0 1] 3/2\\
$-$17.8157& [ 1 0 1] 3/2&$-$17.6452& [ 1 1 0] 1/2\\
$-$15.8641& [ 1 0 1] 1/2&$-$15.5487& [ 1 0 1] 1/2\\
$-$12.4255& [ 2 2 0] 1/2&$-$10.3816& [ 2 0 2] 5/2\\
$-$10.3579& [ 2 1 1] 3/2&$-$9.2965& [ 2 1 1] 3/2\\
$-$7.8438& [ 2 0 2] 5/2&$-$8.9932& [ 2 2 0] 1/2\\
$-$7.5158& [ 2 0 0] 1/2&$-$5.8346& [ 2 0 0] 1/2\\
$-$4.1707& [ 2 1 1] 1/2&$-$5.7213& [ 2 0 2] 3/2\\
$-$3.9511& [ 3 3 0] 1/2&$-$3.0690& [ 2 1 1] 1/2\\
$-$3.2493& [ 2 0 2] 3/2&$-$1.7317& [ 3 0 3] 7/2\\
$-$2.3583& [ 3 2 1] 3/2&$-$0.6623& [ 3 1 2] 5/2\\
$-$0.5604& [ 3 1 2] 5/2&$-$0.3329& [ 3 2 1] 3/2\\
0.0000& [ 3 1 0] 1/2&$-$0.1400& [ 3 3 0] 1/2\\
0.0582& [ 3 0 3] 7/2&0.0000& [ 3 0 1] 3/2\\
0.9147& [ 3 2 1] 1/2&0.5831& [ 3 1 0] 1/2\\
&&1.2550& [ 3 0 3] 5/2\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Summary}
It has been shown that it is important to consider DC cross
sections as well as CN cross sections for nuclei close to magic numbers
and close to the drip lines. However, in order to reliably calculate DC
contributions one needs level schemes which can be
calculated from microscopic models. In this context, improved experimental
data is of utmost importance not only for a better knowledge of
$\beta$-decay half-lives, but also to be able to infer deformation parameters.
With our new neutron capture cross sections we obtain ``turning points''
in the r-process path for the mentioned neutron densities at $^{44}$S,
$^{48}$Ar, $^{62}$Ti (with a minor branching at $^{60}$Ti), $^{68}$Cr
($^{66}$Cr), and $^{74}$Fe ($^{72}$Fe). (Note, however, that the
theoretical $\beta$-decay half-lives shown are taken from QRPA
calculations~$^{16)}$. Other models might yield different values).
Full reaction network
calculations with varying neutron densities have to be performed
for a more detailed
determination of the resulting abundances and isotopic ratios.
\medskip
{\bf Acknowledgement:} We are indebted to
P. M\"oller for discussions
and for making his QRPA code available to us. We also thank F.-K.
Thielemann for discussions. This work was supported in part by the
Austrian Science Foundation (S7307--AST), the \"Osterreichische
Nationalbank (project 5054) and the DFG (Kr 806/3).
\section*{References}
\renewcommand{\baselinestretch}{1.0}
\small \normalsize
\newcounter{tomref}
\begin{list}{$^{\arabic{tomref})}$}{\usecounter{tomref}
\itemsep0cm
\parsep0cm}
\item O. Sorlin et al., {\it Phys.\ Rev.\ C} {\bf 47} (1993)
2941;\\
O. Sorlin et al., {\it Proc. Nuclei in the Cosmos III}, eds.\ M. Busso,
R. Gallino, C.M. Raiteri (AIP Press, New York 1995), p.\ 191.
\item K.-L. Kratz, A.C. M\"uller, F.-K. Thielemann, {\it Phys.\
Bl.} {\bf 51} (1995) 183.
\item W. B\"ohmer, P. M\"oller, B. Pfeiffer, K.-L. Kratz, in {\it
IKCMz report}, ed.\ H. Denschlag (IKCMz, Mainz 1995), p.\ 35.
\item W. B\"ohmer et al., to be published.
\item P. M\"oller, J.R. Nix, W.D. Myers, W.J. Swiatecky, {\it
At.\ Data Nucl.\ Data Tables} (1995), in press.
\item A. W\"ohr et al., in {\it Proc. VIII. Int. Symp. on Gamma-Ray
Spectroscopy and Related
Topics, Fribourg, Schweiz}, ed.\ J. Kern (World Scientific:
Singapore 1994), p.\ 762.
\item J.J. Cowan, F.-K. Thielemann, J.W. Truran, {\it Phys.\
Rep.} {\bf 208} (1991) 267.
\item F.-K. Thielemann, M. Arnould, J.W. Truran, in {\it
Advances in Nuclear Astrophysics}, eds.\ E. Vanghioni-Flam et al.,
(\'editions fronti\`eres, Gif-sur-Yvette 1987), p.\ 525.
\item H. Krauss, unpublished.
\item K.H. Kim, M.H. Park, B.T. Kim, {\it Phys.\ Rev.\ C} {\bf 23}
(1987) 363.
\item N.K. Glendenning, {\it Direct Nuclear Reactions} (Academic Press,
New York 1983).
\item A.M. Kobos, B.A. Brown, R. Lindsay, G.R. Satchler, {\it Nucl.\
Phys.} {\bf A425} (1984) 205.
\item H. Oberhummer, G. Staudt, in {\it Nuclei in the Cosmos}, ed.\ H.
Oberhummer (Springer, Berlin 1991), p.\ 29.
\item W. Balogh, diploma thesis, University of Technology, Vienna
1994;\\
W. Balogh et al., to be published.
\item G. Audi, A.H. Wapstra, {\it Nucl.\ Phys.} {\bf A565} (1993) 1.
\item P. M\"oller, J. Randrup, {\it Nucl.\ Phys.} {\bf A514} (1990) 1.
\end{list}
\end{document}
|
{
"timestamp": "2015-04-20T02:05:03",
"yymm": "1504",
"arxiv_id": "1504.04456",
"language": "en",
"url": "https://arxiv.org/abs/1504.04456"
}
|
\section{Introduction}
Let $ {\mathcal H} $ be a Krein space with fundamental symmetry $J$. A pseudo-regular subspace of $ {\mathcal H} $ is a subspace $ {\mathcal S} $ such that $ {\mathcal S} =\mathcal{M}[\dot{+}] {\mathcal S} ^{\circ}$, where $\mathcal{M}$ is a regular subspace of $ {\mathcal H} $ and $ {\mathcal S} ^{\circ}$ is the isotropic part of $ {\mathcal S} $. For instance, subspaces of Pontryagin spaces are always pseudo-regular. Pseudo-regularity appeared as a condition to generalize
results on spectral measures of definitizable operators from Pontryagin spaces to general Krein spaces \cite{jonas, GJ}. It was also useful to
extend the Beurling-Lax theorem for shifts acting on indefinite metric spaces \cite{ball, G98}.
This paper is devoted to investigating the geometric structure of the set of $J$-normal projections, namely
\[ {\mathcal Q} =\{ \, Q \in L( {\mathcal H} ) \, : \, Q^2=Q, \, QQ^{\#}=Q^{\#}Q \, \}, \]
where $L( {\mathcal H} )$ is the algebra of bounded linear operators acting on $ {\mathcal H} $ and $Q^{\#}$ stands for the $J$-adjoint of $Q$. This class of projections is intimately related to the family of closed pseudo-regular subspaces of $ {\mathcal H} $.
In fact, a (closed) subspace $ {\mathcal S} $ is pseudo-regular if and only if $ {\mathcal S} $ is the range of a $J$-normal projection. However, the correspondence between pseudo-regular subspaces and $J$-normal projections is not bijective: there can be infinitely many $J$-normal projections onto the same subspace, see \cite{maestripieri martinez13}.
An operator $U \in L( {\mathcal H} )$ is $J$-unitary if $UU^{\#}=U^{\#}U=I$. The set $\mathcal{U}_J$ of all $J$-unitary operators is a Banach-Lie group endowed with the norm topology of $L( {\mathcal H} )$. It naturally acts by conjugation on the set of $J$-normal projections, i.e. if $U \in \mathcal{U}_J$ and $Q \in {\mathcal Q} $ the action of $U$ on $Q$ is defined by $U \cdot Q=UQU^{\#}$.
In this paper, it is shown that, for each $Q_0 \in {\mathcal Q} $, the orbit $\mathcal{U}_J \cdot Q_0$ is an analytic homogenous space of $\mathcal{U}_J$. Thus, each orbit can be endowed with the quotient topology. On the other hand, $\mathcal{U}_J \cdot Q_0$ has also the topology inherited from $L( {\mathcal H} )$. But it is shown that both topologies coincide. In order to obtain this result, it is proved that the map induced by the action
\begin{equation}\label{mapa action}
p_{Q_0} : \mathcal{U}_J \to \mathcal{U}_J \cdot Q_0, \, \, \, \, \, p_{Q_0}(U)=UQ_0U^{\#},
\end{equation}
has a local continuous cross section (Theorem \ref{section}). In \cite{G}, A. Gheondea found several conditions equivalent to the existence of a $J$-unitary implementing equivalence between two pseudo-regular subspaces. The above mentioned local continuous cross section allows to find a $J$-unitary that (locally) depends continuously on the $J$-normal projections and implements the equivalence between their ranges. As a consequence, it follows that the action of $\mathcal{U}_J$ on $ {\mathcal Q} $ fills connected components. Furthermore, the orbits can be characterized by means of the signatures and cosignatures associated to the range of any of its projections (Proposition \ref{comp q}).
The problem of finding a local continuous cross section for the action is central to develop the differential geometry of infinite dimensional smooth homogeneous spaces which arise in operator theory. Several examples can be found in \cite{beltita}. However, each example usually requires ad-hoc techniques. In particular, the existence of a section for the map given in \eqref{mapa action} relies on two facts. First, the section given in \cite{corach por re93} for the set of projections in $L( {\mathcal H} )$. Second, after noticing that the isotropic subspaces of the range and nullspace of a $J$-normal projection are closed neutral companions \cite{GJ}, the construction of biorthogonal bases of the sum of these subspaces for each projection in the orbit.
Concerning the smooth structure of $ {\mathcal Q} $, it turns out that $ {\mathcal Q} $ is an analytic submanifold of $L( {\mathcal H} )$. In particular, the same result holds for the set of $J$-selfadjoint projections
$$ {\mathcal E} =\{ \, E \in L( {\mathcal H} ) \, : \, E^2=E, \, E^{\#}=E\, \}.$$
These facts allow to understand the relationship between $J$-normal projections and $J$-selfadjoint projections from a geometrical point of view: the map $F: {\mathcal Q} \to {\mathcal E} $ defined by $F(Q)=QQ^\#$ is a real analytic submersion (Theorem \ref{q submanifold lh}). This kind of results can be seen as a contribution to the differential geometry of projections, which has been a subject of study in different settings, see e.g. \cite{corach por re90, corach por re93, pr87, BR07b, andr lar, andruchow ch di 12}.
The last part of this paper deals with a topological description of the set of $J$-normal projections with a prescribed range.
For a fixed pseudo-regular subspace $ {\mathcal S} $ of $ {\mathcal H} $, denote by $ {\mathcal Q} _{ {\mathcal S} }$ the set of $J$-normal projections with range $ {\mathcal S} $, that is,
$$
{\mathcal Q} _{ {\mathcal S} } = \{ \, Q \in {\mathcal Q} \, : \, R(Q)= {\mathcal S} \, \}.
$$
Unless $ {\mathcal S} $ is regular, $ {\mathcal Q} _{ {\mathcal S} }$ has infinitely many elements. If the isotropic part $ {\mathcal S} ^\circ$ is non trivial, each complement $\mathcal{M}$ in the decomposition $ {\mathcal S} =\mathcal{M}[\dot{+}] {\mathcal S} ^{\circ}$ is regular. Thus, $ {\mathcal Q} _{ {\mathcal S} }$ can be decomposed as the disjoint union of the decks
\[ {\mathcal Q} _{ {\mathcal S} , \mathcal{M}}=\{ \,Q \in {\mathcal Q} _{ {\mathcal S} } \, : \, R(QQ^{\#})=\mathcal{M} \,\}, \]
where $\mathcal{M}$ is any (regular) complement of $ {\mathcal S} ^{\circ}$ in $ {\mathcal S} $. The group $ {\mathcal U} _{ {\mathcal S} }$ of all $J$-unitary operators
leaving $ {\mathcal S} $ invariant, acts transitively on $ {\mathcal Q} _{ {\mathcal S} }$ by conjugation. Moreover, the action has the following remarkable property: the restriction to $ {\mathcal U} _{ {\mathcal S} }$ of the map defined in \eqref{mapa action} admits a global continuous cross section (Proposition \ref{sec global qs}). This is the key to prove that $ {\mathcal Q} _{ {\mathcal S} }$ is a covering space of any of the decks $ {\mathcal Q} _{ {\mathcal S} , \mathcal{M}}$ (Theorem \ref{cov spa fixed range}).
The contents of this paper are as follows. Section 2 contains notation and preliminaries on Krein spaces. Section 3 has the construction of the continuous local cross section for the natural action of $\mathcal{U}_J$ on $ {\mathcal Q} $. The differential structure of $ {\mathcal Q} $ is developed in Section 4. Finally, Section 5 presents the covering space structure of the $J$-normal projections with a prescribed range.
\section{Preliminaries}
Let $( {\mathcal H} , \PI{\,}{\,})$ be a complex separable Hilbert space. If $ {\mathcal K} $ is another Hilbert space, $L( {\mathcal H} , {\mathcal K} )$ stands for the vector space of bounded linear operators from $ {\mathcal H} $ to $ {\mathcal K} $. In particular, $L( {\mathcal H} )$ is the algebra of bounded operators on $ {\mathcal H} $.
If $T \in L( {\mathcal H} )$, $T^*$ is the adjoint of $T$. The range and the nullspace of $T$ are denoted by $R(T)$ and $N(T)$, respectively.
The spectrum of $T$ is denoted by $\sigma(T)$.
Throughout this paper, $J$ is a fixed symmetry acting on $ {\mathcal H} $ (i.e. $J=J^*=J^{-1}$), which defines a
fundamental decomposition $ {\mathcal H} = {\mathcal H} _+ \oplus {\mathcal H} _-$ given by $ {\mathcal H} _\pm=N(J\mp I)$. This symmetry induces a Krein space structure $( {\mathcal H} ,\PK{\,}{\,})$, where
\[ \PK{f}{g}=\PI{Jf}{g}, \, \, \, \, \, \, \, f,g \in {\mathcal H} . \]
The orthogonal projection onto $ {\mathcal H} _{\pm}$ is denoted by $P_{\pm}$.
For a detailed exposition of the facts below, and a deeper discussion on Krein spaces see \cite{ando79, azizov iok89, bognar}.
A vector $f \in {\mathcal H} $ is $J$-positive if $\PK{f}{f}>0$. A subspace $ {\mathcal S} $ of $ {\mathcal H} $ is $J$-positive if every nonzero vector $f \in {\mathcal S} $ is $J$-positive. A subspace $ {\mathcal S} $ is called uniformly $J$-positive if there is a constant $c>0$ such that $\PK{f}{f}\geq c \|f\|^2$ for every $f \in {\mathcal S} $.
A $J$-positive (resp. uniformly $J$-positive) subspace is said to be maximal if it is not properly contained in a larger $J$-positive (resp. uniformly $J$-positive) subspace. Similarly, one can define $J$-nonnegative, $J$-neutral, $J$-negative and uniformly $J$-negative subspaces.
For each $J$-positive subspace $ {\mathcal S} $ of $ {\mathcal H} $, the angular operator $K: P_{+} ( {\mathcal S} ) \to {\mathcal H} _-$ is defined by $K(P_+ f)=P_- f$. It is a contraction ($\|K\|\leq 1$) and its graph coincides with $ {\mathcal S} $:
\[
Gr(K)\simeq \{ \, f + Kf \, : \, f \in P_{+} ( {\mathcal S} ) \, \}= {\mathcal S} .
\]
Moreover, $K$ is a uniform contraction ($\|K\|< 1$) if and only if $ {\mathcal S} $ is uniformly $J$-positive. If $ {\mathcal S} $ is maximal (in the corresponding class of subspaces) the $P_+ ( {\mathcal S} )= {\mathcal H} _+$. Observe that the angular operator can also be defined for $J$-negative subspaces in the obvious way.
Let $ {\mathcal S} $ be a subspace of $ {\mathcal H} $. The $J$-orthogonal subspace of $ {\mathcal S} $ in $ {\mathcal H} $ is defined by
\[ {\mathcal S} ^{[\perp]}=\{ \, f \in {\mathcal H} \, : \, \PK{f}{g}=0 \ \text{for every $g \in {\mathcal S} $} \, \}. \]
The isotropic part of $ {\mathcal S} $ is given by $ {\mathcal S} ^{\circ}:= {\mathcal S} \cap {\mathcal S} ^{[\perp]}$. In general, it is a non-trivial subspace. A subspace $ {\mathcal S} $ is $J$-non-degenerate if $ {\mathcal S} \cap {\mathcal S} ^{[\perp]}=\{ 0\}$. Otherwise, it is a $J$-degenerated subspace. If $\mathcal{T}$ is another subspace of $ {\mathcal H} $, $ {\mathcal S} \dot{+}\mathcal{T}$ stands for the direct sum of the subspaces, meanwhile $ {\mathcal S} [\dot{+}]\mathcal{T}$ is the $J$-orthogonal (direct) sum of them.
Given $T \in L( {\mathcal H} )$, the $J$-adjoint operator of $T$ is defined by $T^{\#}=JT^*J$. An operator $T$ is $J$-selfadjoint if $T^{\#}=T$, and it is $J$-antihermitian if $T^{\#}=-T$.
\subsection{The $J$-unitary group}
A $J$-unitary operator $U$ is a surjective isometry respect to the indefinite inner product, i.e. an operator satisfying $\K{Uf}{Uf}=\K{f}{f}$ for every $f\in {\mathcal H} $. Observe that it is possible to find unbounded $J$-unitary operators in Krein spaces, see e.g. \cite{G88} and the references therein. Along this work, only bounded $J$-unitary operators are considered. Then, $U \in L( {\mathcal H} )$ is $J$-unitary if and only if $UU^{\#}=U^{\#}U=I$. The group of all (bounded) $J$-unitary operators is denoted by $\mathcal{U}_J$.
\begin{rem}\label{B L subg} Let $Gl( {\mathcal H} )$ denote the group of invertible operators.
The group of bounded $J$-unitary operators can be rewritten as
\begin{equation}\label{subg simetrias}
\mathcal{U}_J = \{ \, U \in Gl( {\mathcal H} ) \, : \, U^*JU=J \, \}.
\end{equation}
It was mentioned in \cite[Section 23]{upmeier85} that this set is a real Banach-Lie subgroup of
$Gl( {\mathcal H} )$. In fact $\mathcal{U}_J$ turns out to be a real algebraic subgroup of $Gl( {\mathcal H} )$ and, by \cite[Theorem 7.14]{upmeier85}, $\mathcal{U}_J$ is a Banach-Lie group endowed with the operator norm topology.
Its Lie algebra $\mathfrak{u}_J$ can be identified with the subspace of $J$-antihermitian operators, i.e.
\[ \mathfrak{u}_J=\{ \, X \in L( {\mathcal H} ) \, : \, X^\#=-X \, \}. \]
\end{rem}
When the Hilbert space $ {\mathcal H} $ is considered over a general field, subgroups of $Gl( {\mathcal H} )$ defined as in (\ref{subg simetrias}) are not necessarily connected. However, if $ {\mathcal H} $ is a complex Hilbert space, $\mathcal{U}_J$ is connected. This fact seems to be well-known, but no references could be found by the authors. A proof is included below based on the following well-known description of $J$-unitaries acting on Krein spaces, see e.g. \cite{ando79}.
\begin{teo}\label{ando caracteriz}
Let $ {\mathcal S} $ be a maximal uniformly $J$-positive subspace with angular operator $K$. Then, for any choice of unitary operators $V_+$ on $ {\mathcal H} _+$ and $V_-$ on $ {\mathcal H} _-$ the block operator matrix $U$ with respect to the decomposition $ {\mathcal H} = {\mathcal H} _+\oplus {\mathcal H} _-$ given by
\[ U = \left( \begin{array}{ll} (I_+ - K^*K)^{-1/2} V_+ & K^*(I_- - KK^*)^{-1/2}V_- \\ K(I_+ - K^*K)^{-1/2} V_+ & (I_- - KK^*)^{-1/2}V_- \end{array} \right)\]
is $J$-unitary and transforms $ {\mathcal H} _+$ onto $ {\mathcal S} $. Conversely, every $J$-unitary operator that maps $ {\mathcal H} _+$
onto $ {\mathcal S} $ is of this form.
\end{teo}
\begin{prop}\label{connected uj}
The Banach-Lie group $\mathcal{U}_J$ is (arcwise) connected.
\end{prop}
\begin{proof}
Let $U$ be a $J$-unitary operator. It is not difficult to see that $U {\mathcal H} _+$ is a maximal uniformly $J$-positive subspace. By Theorem \ref{ando caracteriz}, $U$ can be written in the form
\[ U = \left( \begin{array}{ll} (I_+ - K^*K)^{-1/2} V_+ & K^*(I_- - KK^*)^{-1/2}V_- \\ K(I_+ - K^*K)^{-1/2} V_+ & (I_- - KK^*)^{-1/2}V_- \end{array} \right), \]
where $K$ is the angular operator of $U {\mathcal H} _+$. Here $V_+$ and $V_-$ are unitary operators on $ {\mathcal H} _+$ and $ {\mathcal H} _-$ respectively. Then, there exist antihermitian operators $X_+$ acting on $ {\mathcal H} _+$ and $X_-$ acting on $ {\mathcal H} _-$ such that $V_{+}=e^{X_+}$ and $V_-=e^{X_-}$. Notice that the operators of the form $e^{tX_{\pm}}$ are unitaries on $ {\mathcal H} _{\pm}$ for $t \in \mathbb{R}$, and $tK$ is a uniform contraction for $t \in [0,1]$. For each $t \in [0,1]$, the uniform contraction $tK$ is uniquely associated to a maximal uniformly $J$-positive subspace, see \cite[Corollary 1.1.2]{ando79}.
Therefore, by Theorem \ref{ando caracteriz}, the curve $\gamma: [0,1]\rightarrow L( {\mathcal H} )$ given by
\[ \gamma(t)= \left( \begin{array}{ll} (I_+ - t^2K^*K)^{-1/2} e^{tX_+} & tK^*(I_- - t^2KK^*)^{-1/2}e^{tX_-} \\ tK(I_+ - t^2K^*K)^{-1/2} e^{tX_+} & (I_- - t^2KK^*)^{-1/2}e^{tX_-} \end{array} \right), \]
takes values on $\mathcal{U}_J$. Moreover, this curve is clearly continuous, and it joins $\gamma(0)=I$ with $\gamma(1)=U$. Thus, every $J$-unitary operator can be joined by means of a continuous curve with the identity.
Hence $\mathcal{U}_J$ is arcwise connected.
\end{proof}
\begin{rem}\label{log cerca de 1}
The exponential map $\exp: \mathfrak{u}_J \to \mathcal{U}_J$ is given by $\exp(X)=e^X$. It is always a local diffeomorphism. A surjectivity radius of the exponential map can be estimated as follows:
Let $U \in \mathcal{U}_J$ such that $\|U-I\|<1$. Consider the principal branch of the logarithm, i.e. $\log: \mathbb{C}\setminus \mathbb{R}^-\rightarrow \mathbb{C}$ given by $\log(z)=\log(|z|) + i\theta$, where $z=|z|e^{i\theta}$, $\theta\in (-\pi,\pi)$. Since every $\lambda \in \sigma(U)$ satisfies $|\lambda - 1|<1$, the logarithm of $U$ can be defined by the analytic functional calculus:
\[
\log(U)=\frac{1}{2\pi i}\int_\Gamma \log(z)(zI-U)^{-1}\ dz,
\]
where $\Gamma$ is a suitable Jordan contour in the resolvent set of $U$ surrounding $\sigma(U)$.
Observe that $\sigma((U^*)^{-1})=\{\overline{\lambda^{-1}}: \ \lambda\in\sigma(U)\}$ is also contained in the right-half plane. Then there exist $0<\varepsilon<M$ and $N>0$ such that $\sigma(U)\cup\sigma((U^*)^{-1})$ is contained in the rectangle $[\varepsilon,M]\times [-N, N]$. Let $\Gamma$ be the border of this rectangle.
Since $U\in\mathcal{U}_J$, it follows that $JU=(U^*)^{-1}J$ and, given $z\in\mathbb{C}$,
$
(zI-U)^{-1}J=(J(zI-U))^{-1}=((zI-(U^*)^{-1})J)^{-1}=J(zI-(U^*)^{-1})^{-1}.
$
Thus,
\begin{eqnarray*}
\log(U)J &=& \frac{1}{2\pi i}\int_\Gamma \log(z)(zI-U)^{-1}J\ dz \\
&=& J\left(\frac{1}{2\pi i}\int_\Gamma \log(z)(zI-(U^*)^{-1})^{-1} dz\right) =J \log((U^*)^{-1}).
\end{eqnarray*}
Note that $f(z):=\log(\tfrac{1}{z})$ is an analytic function in the right-half plane satisfying $f(z)=-log(z)$. Then, $\log(U)J= J\log((U^*)^{-1})=-J\log(U^*)=-J\log(U)^*$.
Set $X=\log(U)$. By the above computation $X$ is $J$-antihermitian and $e^X=U$. Hence, every operator $U$ satisfying $\|U-I\|< 1$ has a logarithm in $\mathfrak{u}_J$.
\end{rem}
\subsection{Regular and pseudo-regular subspaces}
A (closed) subspace $ {\mathcal S} $ of a Krein space $ {\mathcal H} $ is called \textit{regular} if $ {\mathcal S} \, [\dot{+}]\, {\mathcal S} ^{[\perp]}= {\mathcal H} $. Equivalently, $ {\mathcal S} $ is regular if and only if there exists a (unique) $J$-selfadjoint projection $E$ such that $R(E)= {\mathcal S} $ (see e.g. \cite[Ch. 1, Thm. 7.16]{azizov iok89}). Thus, the set of regular subspaces is in bijective correspondence with the set of $J$-selfadjoint projections, namely,
\[ {\mathcal E} =\{ E \in L( {\mathcal H} ) \, : \, E^2=E, \, E^{\#}=E\, \}. \]
The following criterion will be useful: $ {\mathcal S} $ is a regular subspace if and only if $ {\mathcal S} = {\mathcal M} \, [\dot{+}] \, {\mathcal N} $, where
$ {\mathcal M} $ is a uniformly $J$-positive subspace and $ {\mathcal N} $ is a uniformly $J$-negative subspace (see \cite[Theorem 1.3]{ando79}).
A closed subspace $ {\mathcal S} $ of $ {\mathcal H} $ is called \textit{pseudo-regular} if there exists a regular subspace $ {\mathcal M} $ such that $ {\mathcal S} = {\mathcal S} ^\circ \, [\dot{+}]\, {\mathcal M} $. Equivalently, $ {\mathcal S} $ is pseudo-regular if the algebraic sum $ {\mathcal S} \, +\, {\mathcal S} ^{[\perp]}$ is closed. In \cite{maestripieri martinez13} it was shown that a subspace $ {\mathcal S} $ is pseudo-regular if and only if $ {\mathcal S} $ is the range of a $J$-normal projection, i.e. there exists a projection $Q\in L( {\mathcal H} )$ with $R(Q)= {\mathcal S} $ such that $QQ^\#=Q^\#Q$.
The following results also belong to \cite{maestripieri martinez13}. Their statements are included in order to make the paper self-contained.
\begin{prop}\label{desc}
Given a projection $Q \in L( {\mathcal H} )$, $Q$ is $J$-normal if and only if there exist a projection $E \in {\mathcal E} $ and a projection $P \in L( {\mathcal H} )$ satisfying $PP^{\#}=P^{\#}P=0$ such that
\[ Q=E+P.\]
The projections $E$ and $P$ are uniquely determined by $Q$.
\end{prop}
Projections $P\in L( {\mathcal H} )$ satisfying $PP^{\#}=P^{\#}P=0$ were previously considered in \cite{J81,GJ}, in connection with neutral dual companions. If $ {\mathcal S} $ is a fixed (closed) $J$-neutral subspace of $ {\mathcal H} $, a {\it neutral dual companion of $ {\mathcal S} $} is another (closed) $J$-neutral subspace $ {\mathcal T} $ of $ {\mathcal H} $ such that $ {\mathcal H} = {\mathcal S} \dotplus {\mathcal T} ^{[\bot]}$ holds. If $ {\mathcal T} $ is a neutral dual companion of $ {\mathcal S} $ then also $ {\mathcal H} = {\mathcal T} \dotplus {\mathcal S} ^{[\bot]}$ holds. So, the pair of subspaces $( {\mathcal S} , {\mathcal T} )$ is called a {\it neutral dual pair}.
\begin{rem}\label{desc q y 1-q}
In the proof of the above mentioned result, the projections $E$ and $P$ are explicitly computed: $E=QQ^{\#}$ and $P=Q(I-Q^{\#})$.
Furthermore, the decomposition for the $J$-normal projection $I-Q$ is given by
\[ I-Q=F + P^{\#}, \]
where $F=(I-Q)(I-Q)^{\#}$. From these formulas, it is easy to see that $EP=PE=EP^\#=P^\#E=0$ and $FP=PF=FP^\#=P^\#F=0$.
Also, it follows that $R(P + P^\#)=R(Q)^{\circ} \dot{+} N(Q)^{\circ}$ is a regular subspace, and the Krein space $ {\mathcal H} $ can be decomposed as the $J$-orthogonal sum of the following three
regular subspaces:
$$
{\mathcal H} =R(E)[\dotplus] R(P + P^{\#})[\dotplus] R(F).
$$
\end{rem}
In the sequel, given a a $J$-normal projection $Q\in L( {\mathcal H} )$, $E$, $F$ and $P$ stand for the projections $E=QQ^{\#}$, $P=Q(I-Q^{\#})$ and $F=(I-Q)(I-Q)^{\#}$. If $Q_0$ is another $J$-normal projection, $E_0$, $F_0$ and $P_0$ have the obvious meaning.
\section{The orbit of a $J$-normal projection}
\noindent The set of $J$-normal projections is given by
\[ {\mathcal Q} =\{ \, Q \in L( {\mathcal H} ) \, : \, Q^2=Q, \, QQ^{\#}=Q^{\#}Q \, \}. \]
The Banach-Lie group $\mathcal{U}_J$ acts smoothly on $L( {\mathcal H} )$ by conjugation. Clearly, the restriction of this action gives an action of $\mathcal{U}_J$ on $ {\mathcal Q} $ defined by
$$U\cdot Q=UQU^{\#},$$
where $U \in \mathcal{U}_J , \, \, Q \in {\mathcal Q} $. It is worth pointing out that each orbit $\mathcal{U}_J \cdot Q$ is connected in the norm topology (see Proposition \ref{connected uj}).
For the notion of real analytic homogeneous space in the following result see e.g. \cite{beltita, upmeier85}.
\begin{prop}\label{kernel sup}
Given $Q_0 \in {\mathcal Q} $, the orbit $\mathcal{U}_J \cdot Q_0$ is a real analytic homogeneous space of $\mathcal{U}_J$.
\end{prop}
\begin{proof}
Clearly, there is a bijection from $\mathcal{U}_J \cdot Q_0$ onto $\mathcal{U}_J / \mathcal{G}$, where $\mathcal{G}$ is the isotropy group at $Q_0$, i.e.
\[ \mathcal{G}= \{ \, U \in \mathcal{U}_J \, : \, UQ_0=Q_0U \, \}. \]
Observe that the Lie algebra of $\mathcal{G}$ can be identified with
\[ \mathfrak{g}= \{ \, X \in \mathfrak{u}_J \, : \, XQ_0=Q_0X \, \}. \]
Then, the conclusion of this proposition will follow if $\mathcal{G}$ is a Banach-Lie subgroup of $\mathcal{U}_J$.
This last fact is equivalent to show that $\mathcal{G}$ is a Banach-Lie group in the norm topology of $L( {\mathcal H} )$ and $\mathfrak{g}$ is a closed complemented subspace of $ \mathfrak{u}_J$. In this case, $\mathcal{U}_J / \mathcal{G}$ has an analytic manifold structure endowed with the quotient topology (see e.g. \cite[Theorem 8.19]{upmeier85}).
Let $\mathcal{V}=\exp^{-1}(B_1 (I))$, where $B_1 (I)$ is the open unit ball around the identity contained in $\mathcal{U}_J$. Given $U \in \exp(\mathcal{V}) \cap \mathcal{G}$, there exists $X \in \mathcal{V}$ such that $U=e^X$.
Notice that the logarithm $X \in \mathfrak{u}_J$, which is computed in Remark \ref{log cerca de 1}, is unique. Indeed,
if $\| U -I\|<1$ then $\sigma(U)\subset \mathbb{R} + i (-\pi, \pi)$. But in the latter set the complex exponential is bijective, so the exponential in $L( {\mathcal H} )$ is also bijective by well-known properties of the functional analytic calculus. Now recall that $X=\frac{1}{2\pi i}\int_\Gamma \log(z)(zI-U)^{-1}dz$. If the operator $U$ belongs to $\mathcal{G}$, that is $UQ_0=Q_0U$, then by standard arguments one can see that $XQ_0=Q_0X$. Thus, $X \in \mathfrak{g}$. This shows that $\exp(\mathcal{V})\cap \mathcal{G}\subseteq \exp(\mathcal{V} \cap \mathfrak{g})$. Since the reversed inclusion is always trivial, it follows that
$\exp(\mathcal{V})\cap \mathcal{G}=\exp(\mathcal{V} \cap \mathfrak{g})$. Hence $\mathcal{G}$ is a Banach-Lie group in the norm topology of $L( {\mathcal H} )$.
Note that $\mathfrak{g}$ is closed in $ \mathfrak{u}_J$. To prove that $\mathfrak{g}$ is complemented in $ \mathfrak{u}_J$, consider the map
\begin{equation}\label{Pmap}
{\mathcal P} : L( {\mathcal H} )\rightarrow L( {\mathcal H} ), \, \, \, \, {\mathcal P} (X)=E_0XE_0 + P_0XP_0 + P_0^{\#}XP_0^{\#} + F_0XF_0.
\end{equation}
By the relations between the projections $E_0$, $F_0$, $P_0$ and $P_0^{\#}$ pointed out in Remark \ref{desc q y 1-q}, it follows that $ {\mathcal P} $ is a continuous projection satisfying $ {\mathcal P} (\mathfrak{u}_J)\subseteq \mathfrak{u}_J$.
Also, notice that
$
Q_0 {\mathcal P} (X)=E_0XE_0 + P_0XP_0= {\mathcal P} (X)Q_0$.
Then, $ {\mathcal P} (\mathfrak{u}_J)\subseteq \mathfrak{g}$. To prove the reversed inclusion, pick $X\in \mathfrak{g}$, i.e. $X\in\mathfrak{u}_J$ and $XQ_0=Q_0X$. Observe that $X$ also commutes with $Q_0^{\#}$. Therefore, $X$ commutes with $E_0$, $F_0$, $P_0$ and $P_0 ^{\#}$, so that
\[
{\mathcal P} (X)=E_0XE_0 + P_0XP_0 + P_0^\#XP_0^\# + F_0XF_0= (E_0 + P_0 + P_0^\# + F_0)X=X.
\]
The latter means that $X\in {\mathcal P} (\mathfrak{u}_J)$, and consequently, $ {\mathcal P} (\mathfrak{u}_J)= \mathfrak{g}$.
To finish the proof, note that the map $h :L( {\mathcal H} ) \to \mathfrak{u}_J$ given by $h(X)=\tfrac{X-X^\#}{2}$ is a continuous real projection. Therefore the map
$ {\mathcal P} \circ h$ is a continuous real projection onto $\mathfrak{g}$. Hence, $\mathfrak{g}$ is complemented in $L( {\mathcal H} )$.
\end{proof}
According to the above proposition, the orbit $\mathcal{U}_J \cdot Q_0$ has a Banach manifold structure such that
the canonical projection
$$p_{Q_0}:\mathcal{U}_J \to \mathcal{U}_J \cdot Q_0, \, \, \, \, p_{Q_0}(U)=UQ_0U^{\#}$$
is a real analytic submersion. This manifold structure defines on $ \mathcal{U}_J \cdot Q_0 \simeq \mathcal{U}_J / \mathcal{G}$ the quotient topology. On the other hand, $\mathcal{U}_J \cdot Q_0$ is endowed with the relative topology as a subset of $L( {\mathcal H} )$. If one considers the identity map $Id: \mathcal{U}_J \cdot Q_0 \simeq \mathcal{U}_J /\mathcal{G} \to \mathcal{U}_J \cdot Q_0 \subseteq L( {\mathcal H} ) $, it is easy to see that this map is always continuous. However, it may fail to be a homeomorphism.
To see that in this setting it is actually a homeomorphism, it will be sufficient to prove that
$p_{Q_0}$ admits local continuous cross sections when $\mathcal{U}_J \cdot Q_0$ is endowed with the relative topology of $L( {\mathcal H} )$.
To this end, recall that in Remark \ref{desc q y 1-q}, it is stated that $ {\mathcal H} $ can be written as the $J$-orthogonal sum of three regular subspaces
\[ {\mathcal H} =R(E_0)[\dot{+}]R(P_0 + P_0^{\#})[\dot{+}]R(F_0). \]
Let $Q$ be another $J$-normal projection sufficiently close to $Q_0$. The space $ {\mathcal H} $ can also be decomposed as $ {\mathcal H} =R(E)[\dot{+}]R(P + P^{\#})[\dot{+}]R(F)$. Therefore, the problem of finding a $J$-unitary that maps $Q_0$ in $Q$ can be reduced to find $J$-isometric isomorphisms mapping $R(E_0)$ onto $R(E)$, $R(F_0)$ onto $R(F)$ and $R(P_0+P_0^{\#})$ onto $R(P+P^{\#})$. It is worth pointing out that $R(P_0)$ has to be mapped onto $R(P)$, and obviously, the $J$-unitary has to depend continuously on $Q$.
This work is carried out in the next two subsections. The first one deals with the case of $J$-selfadjoint projections, and the second subsection treats the general case.
\subsection{A local continuous cross section. The $J$-selfadjoint case.}
Observe that the group $\mathcal{U}_J$ also acts on $ {\mathcal E} $ by conjugation: $U \cdot E=UEU^{\#}$, where $U \in \mathcal{U}_J$ and $E \in {\mathcal E} $.
\begin{prop}\label{section j seladj}
The map $p_{E_0}:\mathcal{U}_J \to \mathcal{U}_J \cdot {\mathcal E} $ given by $p_{E_0}(U)=UE_0U^{\#}$ has local continuous cross sections.
\end{prop}
\begin{proof} In what follows, a section will be given in a neighborhood of $E_0$; standard arguments can be applied to translate this section to other points.
It will be useful to recall some facts on the geometry of projections in $L( {\mathcal H} )$, see \cite{corach por re93}.
The set of projections in $L( {\mathcal H} )$, namely
\[ \mathbb{Q}=\{ \, Q \in L( {\mathcal H} ) \, : \, Q^2=Q \, \}, \]
is a smooth homogeneous space of the group $Gl( {\mathcal H} )$. Its tangent space at $Q \in {\mathcal Q}$ can be identified with
\[ T_{Q} \mathbb{Q}=\{ \, X \in L( {\mathcal H} ) \, :\, XQ+QX=X \, \}, \]
which are co-diagonal operators with respect to $Q$, i.e. co-diagonal block-operator matrices according to the decomposition $ {\mathcal H} =R(Q)\dotplus N(Q)$. For a fixed projection $Q_0\in \mathbb{Q}$, the exponential map
\[ \exp: T_{Q_0}\mathbb{Q} \to \{ GQ_0G^{-1}: \ G \in Gl( {\mathcal H} )\}, \, \, \, \, \, \exp(X)=e^XQ_0e^{-X}, \]
is a local diffeomorphism at $Q_0$. Therefore, there is a positive radius $r$ (depending on $Q_0$) such that the map $\{ \, Q \in \mathbb{Q} \, : \, \| Q - Q_0\|<r \, \} \to T_{Q_0}\mathbb{Q}$ given by $Q \mapsto X_Q$ is smooth and satisfies
\[
e^{X_Q}Q_0e^{-X_Q}=Q.
\]
Taking into account the facts stated above for the projection $E_0 \in {\mathcal E} $, given a suitable radius $r$, it is possible to define a continuous map
\begin{equation}\label{radio}
s: \{ \, E \in {\mathcal E} \, : \, \|E-E_0\|<r \, \} \to Gl( {\mathcal H} ), \, \, \, \, \, s(E)=e^{X_E}.
\end{equation}
If this map takes values in $\mathcal{U}_J$, it will clearly be the required continuous local cross section for $p_{E_0}$. The following argument to show that $s(E) \in \mathcal{U}_J$
is borrowed and adapted from \cite[Proposition 4.4]{andruchow ch di 12}. It is useful to change from projections to symmetries via the map $E \mapsto R_E=2E-I$. Since $e^{X_E}E_0e^{-X_E}=E$, it follows that $e^{X_E}R_{E_0}e^{-X_E}=R_E$. Next, notice that an operator is co-diagonal with respect to $E_0$ if and only if it anticommutes with $R_{E_0}$. This implies that $R_{E_0}e^{-2X_E}=R_E=e^{2X_E}R_{E_0}$ and
\[
(e^{2X_E})^{\#}=(R_ER_{E_0})^{\#}=R_{E_0} R_E=e^{-2X_E}=(e^{2X_E})^{-1}.
\]
Then, $e^{2X_E} \in \mathcal{U}_J$. Shrinking the radius $r$ if it is necessary, one gets that $\| e^{2X_E} - I \|<1$. By Remark \ref{log cerca de 1}, it follows that $2X_E \in \mathfrak{u}_J$, and consequently, $X_E \in \mathfrak{u}_J$. Hence $e^{X_E} \in \mathcal{U}_J$ and the proof is completed.
\end{proof}
\subsection{A local continuous cross section. The general case.}
Given a neutral dual pair $( {\mathcal S} , {\mathcal T} )$ in $ {\mathcal H} $, in the next lemma a pair of biorthogonal bases for $ {\mathcal S} $ and $ {\mathcal T} $ are constructed. This result was known for finite-dimensional subspaces \cite[Lemma 1.31]{azizov iok89}, but it is original for the general case.
\begin{lem}\label{biort}
If $( {\mathcal S} , {\mathcal T} )$ is a a neutral dual pair in $ {\mathcal H} $, then for any orthonormal basis $\{s_n\}_{n\geq 1}$ of $ {\mathcal S} $ (in the Hilbert space sense) there exists a Riesz basis $\{t_n\}_{n\geq 1}$ of $ {\mathcal T} $ such that
\[
\K{s_i}{t_j}=\delta_{ij}, \ \ \ i,j\geq 1.
\]
\end{lem}
\begin{proof}
Let $P\in L( {\mathcal H} )$ be the projection onto $ {\mathcal S} $ along $ {\mathcal T} ^{[\bot]}$. Then, $P^{\#}$ is the projection onto $ {\mathcal T} $ along $ {\mathcal S} ^{[\bot]}$. For a fixed orthonormal
basis $\{s_n\}_{n\geq 1}$ of $ {\mathcal S} $, define $t_n=P^{\#}Js_n \in {\mathcal T} $, $n \geq 1$. Hence, given $i,j\geq 1$,
\[
\K{s_i}{t_j}=\K{s_i}{P^{\#}Js_j}=\K{P s_i}{Js_j}=\K{s_i}{Js_j}=\PI{s_i}{s_j}=\delta_{ij}.
\]
To prove that $\{t_n\}_{n \geq 1}$ is a Riesz basis, observe that $T=P^{\#}J|_{ {\mathcal S} }: {\mathcal S} \to {\mathcal T} $ is a (continuous) surjective operator since
\[
T( {\mathcal S} )=P^{\#}( J( {\mathcal S} ) + N(P^{\#}))= R(P^{\#})= {\mathcal T} .
\]
On the other hand, if $f\in N(T)$ then $Jf \in {\mathcal S} ^{[\bot]}=J( {\mathcal S} ^{\perp})$. It follows that
$f \in {\mathcal S} ^\bot\cap {\mathcal S} $. Thus, $T$ is injective. Hence, $\{t_n\}_{n \geq 1}$ is the image of an orthonormal basis by an invertible operator, i.e. it is a Riesz basis.
\end{proof}
If $Q$ is a $J$-normal projection, notice that the subspaces $R(Q)^\circ$ and $N(Q)^\circ$ form a neutral dual pair.
\begin{lem}\label{s link}
Let $Q, Q_0\in L( {\mathcal H} )$ be $J$-normal projections. Assume that the isotropic parts of their ranges have the same dimension.
Then, there is a continuous $J$-isometric isomorphism
\[
V: R(Q_0)^\circ \dot{+} N(Q_0)^\circ \to R(Q)^\circ \dot{+} N(Q)^\circ,
\]
satisfying $V(R(Q_0)^\circ)=R(Q)^\circ$ and $V(N(Q_0)^\circ)=N(Q)^\circ$.
\end{lem}
\begin{proof}
According to Lemma \ref{biort}, for fixed orthonormal basis $\{s_n^0\}_{n\geq 1}$ and $\{s_n\}_{n\geq 1}$ of $R(Q_0)^\circ$ and $R(Q)^\circ$, there exist Riesz basis $\{t_n^0\}_{n\geq 1}$ and $\{t_n\}_{n\geq 1}$ of $N(Q_0)^\circ$ and $N(Q)^\circ$, respectively, such that $\K{s_i^0}{t_j^0}=\K{s_i}{t_j}=\delta_{ij}$.
Next, consider the operator $V:R(Q_0)^\circ \dot{+} N(Q_0)^\circ \to R(Q)^\circ \dot{+} N(Q)^\circ$ given by
\[
V\bigg(\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0 \bigg)=\sum_{n} \alpha_n s_n + \sum_m\beta_m t_m .
\]
Since $V$ maps the (Riesz) basis $\{s_n^0\}_{n\geq 1}\cup \{t_n^0\}_{n\geq 1}$ onto the (Riesz) basis $\{s_n\}_{n\geq 1}\cup \{t_n\}_{n\geq 1}$, it follows that it is a continuous operator.
Moreover, $V$ is a $J$-isometry by construction: due to the $J$-biorthogonal\-i\-ty of the bases, it follows that
\begin{flalign*}
&\K{V\bigg(\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0 \bigg)}{V\bigg(\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0 \bigg)} &\\
&= \K{\sum_{n} \alpha_n s_n + \sum_m\beta_m t_m }{\sum_{n} \alpha_n s_n + \sum_m\beta_m t_m } = 2 \sum_{n,m} \text{Re} (\alpha_n \bar{\beta}_m[s_n,t_m]) & \\
& = 2 \sum_{n} \text{Re} (\alpha_n \bar{\beta}_n) = \K{\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0}{\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0},&
\end{flalign*}
where in the second equality, it is taken into account that $\sum_{n} \alpha_n s_n\in R(Q)^\circ$ and $\sum_m\beta_m t_m\in N(Q)^\circ$, and in the last equality, it is used that $\sum_{n} \alpha_n s_n^0\in R(Q_0)^\circ$ and $\sum_m\beta_m t_m^0\in N(Q_0)^\circ$. Hence, $V$ is a $J$-isometric isomorphism.
\end{proof}
The next step is to show that the above $J$-isometric isomorphism $V$ depends continuously on $Q$.
Some basic facts on the geometry of the unitary group and the space of selfadjoint projections will be needed. Let $\mathcal{U}$ be the unitary group of $L( {\mathcal H} )$, and $\mathcal{P}$ be the manifold of selfadjoint projections, i.e.
\[ \mathcal{P}=\{ \, P \in L( {\mathcal H} )\, : \,P=P^2 , \, P=P^* \, \}. \]
The natural action of $\mathcal{U}$ on $\mathcal{P}$ given by $U\cdot P=UPU^*$ has local continuous cross sections. Although this fact was pointed out in \cite[Remark 3.2]{corach por re90}, in the following lemma a short proof is included for the sake of completeness.
The main idea is adapted from a similar context in \cite[Proposition 2.2]{andr lar}.
\begin{lem}\label{section on p}
If $P_0 \in \mathcal{P}$ the map $\mathcal{U} \to \mathcal{P}$, given by $U \mapsto UP_0U^*$, has local continuous cross sections.
\end{lem}
\begin{proof}
Consider the open set
\[ \mathcal{V}=\{ \, P \in \mathcal{P} \, : \, \|P - P_0\|<1 \, \}. \]
For $P \in \mathcal{V}$, set $S=PP_0 + (I-P)(I-P_0)$. Then it is well-known that $\| S - I \| < 1$. Thus, $S$ is invertible.
The unitary part of $S$ given by $U=|S^*|^{-1}S$ is a continuous function of $P$. Notice that $SS^*P=PSS^*$, which implies that
$|S^*|P=P|S^*|$ and $P|S^*|^{-1}=|S^*|^{-1}P$. Therefore, $PU=P|S^*|^{-1}S=|S^*|^{-1}PS=|S^*|^{-1}SP_0=UP_0$, i.e. $UP_0U^*=P$. Hence
$U=U(P)$ is a continuous local cross section. \qedhere
\end{proof}
It will be also useful to state here a well-known result on projections.
\begin{lem}(\cite[Ch. I]{kato76})\label{kato proj}
Let $E_1, E_2 \in L( {\mathcal H} )$ be projections. If $P_{R(E_1)}$ and $P_{R(E_2)}$ are the orthogonal projections onto their ranges, respectively, then
\[ \| P_{R(E_1)} - P_{R(E_2)} \| \leq \| E_1- E_2 \|. \]
\end{lem}
Now, given a fixed $J$-normal projection $Q_0\in L( {\mathcal H} )$, consider the following neighborhood of $Q_0$:
\[
\mathcal{V}_{Q_0}=\bigg\{ \, Q \in \mathcal{Q} \, : \, \| Q - Q_0 \| < \frac{1}{2(1 + \|Q_0\|)} \, \bigg\}.
\]
Then, define a map $V: \mathcal{V}_{Q_0} \rightarrow L( {\mathcal H} )$ such that $V(Q)$ is a $J$-isometric isomorphism between $R(Q_0)^\circ + N(Q_0)^\circ$ and $R(Q_0)^\circ + N(Q_0)^\circ$ as follows:
Given $Q\in \mathcal{V}_{Q_0}$, it is easy to see that $\|Q\|<\|Q_0\|+1$. Recall that $P=Q(I-Q^\#)$ and $P_0=Q_0(I-Q_0^\#)$, then
\begin{equation}\label{q vs p}
\| P - P_0\| \leq \| Q- Q_0 \| + \|QQ^\# - Q_0Q_0^\# \| \leq 2 (1 + \|Q_0\|)\|Q - Q_0\|<1.
\end{equation}
According to Lemma \ref{kato proj}, it follows that
\begin{equation}\label{p vs p ort}
\| P_{R(Q)^\circ} - P_{R(Q_0)^\circ} \| \leq \| P- P_0\| < 1.
\end{equation}
By Lemma \ref{section on p}, there exists a unitary operator $U=U(P_{R(Q)^\circ})$, which depends
continuously on $P_{R(Q)^\circ}$, and satisfies $UP_{R(Q_0)^\circ}U^*=P_{R(Q)^\circ}$.
In particular, this implies that $\dim R(Q)^\circ=\dim R(Q_0)^\circ$.
Moreover, for a fixed orthonormal basis $\{s_n^0\}_{n\geq 1}$ of $R(Q_0)^\circ$, this $U\in\mathcal{U}$ gives a procedure to choose an orthonormal basis of $R(Q)^\circ$: set $s_{n,Q}=Us_n ^0$ for every $n\geq 1$.
According to Lemma \ref{biort}, there are Riesz bases $\{t_{n,Q}\}_{n \geq 1}$ and $\{t_n ^0\}_{n \geq 1}$ of $N(Q)^\circ$ and $N(Q_0)^\circ$, respectively, such that $\K{s_n^0}{t_m^0}=\K{s_{n,Q}}{t_{m,Q}}=\delta_{nm}$. Applying Lemma \ref{s link}, one can construct
a $J$-isometric isomorphism $ V(Q)$ between $R(Q_0)^\circ \dot{+} N(Q_0)^\circ$ and $R(Q)^\circ \dot{+} N(Q)^\circ$. In fact, it will be useful to
extend this linear operator to $ {\mathcal H} $, i.e.
$$ V(Q) f :=\left\{
\begin{array}{cl}
\displaystyle{\sum_{n} \alpha_n s_{n,Q} + \sum_m\beta_m t_{m,Q} } & \text{ if } f=\displaystyle{\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0};\\
\\
0 & \text{ if } f \in (R(Q_0)^\circ \dot{+} N(Q_0)^\circ)^{[\bot]}.
\end{array}\right.
$$
\begin{lem}\label{sec cont lema}
The map $V:\mathcal{V}_{Q_0} \to L( {\mathcal H} )$ defined above is continuous.
\end{lem}
\begin{proof}
Let $\{ Q_k\}_{k\geq 1}$ be a sequence in $\mathcal{V}_{Q_0}$. Assume that $\|Q_k - Q\|\to 0$ for some $Q \in \mathcal{V}_{Q_0}$. Let $U_k=U(P_{R(Q_k)^\circ})$ be the unitary associated with $P_k=Q_k(I-Q_k^\#)$ defined after \eqref{p vs p ort}. Analogously, let $U$ and $P$ be the the corresponding unitary and projection associated with $Q$.
Pick a vector $f=\sum_{n} \alpha_n s_n^0 + \sum_m\beta_m t_m^0 \in R(Q_0)^\circ \dot{+} N(Q_0)^\circ$, where $\{s_n^0\}_{n\geq 1}$ is an orthonormal basis of $R(Q_0)^\circ$ and $\{t_n^0\}_{n\geq 1}$ is the Riesz basis of $N(Q_0)^\circ$ given by Lemma \ref{biort}.
In order to prove the continuity of the map $V$, note that
\begin{align*}
\|(V(Q_k) - V(Q) )f \| & = \|\sum_{n} \alpha_n (s_{n, Q_k} - s_{n, Q}) + \sum_m\beta_m (t_{m, Q_k} - t_{m, Q}) \| \\
& = \| \sum_{n} \alpha_n (U_k - U )s_n^0 + \sum_m\beta_m (P^{\#}_k - P^{\#})Js_m^0 \| \\
& \leq \|U_k - U \| \, \| \sum_{n} \alpha_n s_n^0 \| + \| P^{\#}_k - P^{\#} \| \, \| \sum_m \beta_m s_m^0 \| \\
& \leq \|U_k - U \| \, \|f\| + C \, \| P_k - P \| \, \|f\|,
\end{align*}
for a suitable $C=c_1 c_2>0$, because
$$\| \sum_n \beta_n s_n^0 \|=\left(\sum_n |\beta_n|^2\right)^{1/2}\leq c_1 \| \sum_n \beta_n t_n^0 \|\leq c_1 c_2 \|f\|,$$
where $c_1$ is a constant related to the Riesz basicity of $\{t_n^0\}_{n\geq 1}$ and $c_2$ is the norm of the projection onto $N(Q_0)^\circ$ along $R(Q_0)^\circ$.
Therefore,
\[
\| V(Q_k) - V(Q) \| \leq \|U_k - U \| + C \, \| P_k - P\|.
\]
From (\ref{q vs p}) one gets that $\| P_k - P\| \to 0$, so it remains to show that $ \|U_k - U \|\to 0 $. Lemma \ref{kato proj} implies that $\| P_{R(Q_k)^\circ} - P_{R(Q)^\circ}\|\to 0$ and the map given by $P_{R(Q)} \mapsto U(P_{R(Q)})$ is continuous, so that $\| U_k - U\|\to 0$.
\end{proof}
Now the main result of this section follows. In particular, when $J=I$, one recovers the connected components of the Grassmann manifold of a Hilbert space, and this topological result can be deduced in a different fashion from the submanifold structure proved in \cite{pr87}.
\begin{teo}\label{section}
Let $Q_0 \in \mathcal{Q}$, then the map $p_{Q_0}: \mathcal{U}_J \to \mathcal{U}_J \cdot Q_0$ given by
\[
p_{Q_0}(U)=UQ_0U^{\#},
\]
has local continuous cross sections. In particular, the quotient topology and the topology inherited from $L( {\mathcal H} )$ are equivalent in $\mathcal{U}_J \cdot Q_0$.
\end{teo}
\begin{proof}
Recall that $E_0=Q_0Q_0^\#$ and $F_0=(I-Q_0)(I-Q_0)^\#$. By Proposition \ref{section j seladj}, there is a continuous section $s_1$ of the map $p_{E_0}(U)=UE_0U^\#$. By the same method one can construct another continuous section $s_2$ of the
map $p_{F_0}(U)=UF_0U^\#$.
In fact, these sections are defined in open balls of radii $r_{E_0}$ and $r_{F_0}$, respectively, contained in $ {\mathcal E} $. Set
$$r_{Q_0}= \min \bigg\{ \frac{r_{E_0}}{1 + 2\|Q_0\|}, \frac{r_{F_0}}{1 + 2\|Q_0\|}, \frac{1}{1 + 2\|Q_0\|} \bigg\} .$$
Recall that $V(Q)$ is a continuous function of $Q$ by Lemma \ref{sec cont lema}. Moreover, it satisfies $V(Q)P_0V(Q)^\# =P$. Then the map $s:\{ Q \in \mathcal{Q} : \ \| Q - Q_0 \| < r_{Q_0} \} \to \mathcal{U}_J$ defined by
\[
s(Q)=Es_1(E)E_0 + Fs_2(F)F_0 +(P+ P^\#)V(Q)(P_0+ P_0^\#),
\]
is the required continuous section for $p_{Q_0}$. To show that $s(Q)\in\mathcal{U}_J$ for $Q \in \mathcal{Q}$ with $\| Q - Q_0 \| < r_{Q_0}$, observe that $s(Q)$ can be alternatively written as $s(Q)=s_1(E)E_0 + s_2(F)F_0 + V(Q)(P_0+ P_0^\#)$ or $s(Q)=Es_1(E) + Fs_2(F) +(P+ P^\#)V(Q)$.
Recall that the identity map $Id: \mathcal{U}_J \cdot Q_0 \simeq \mathcal{U}_J /\mathcal{G} \to \mathcal{U}_J \cdot Q_0 \subseteq L( {\mathcal H} ) $ is continuous by definition of the quotient topology. On the other hand, the existence of local continuous cross sections implies that $p_{Q_0}$ is an open map, and consequently, $Id$ is a homeomorphism. This proves the equivalence between both topologies.
\end{proof}
\subsection{Connected components of $ {\mathcal Q} $}
It is necessary to recall some terminology used in \cite{G}. Let $ {\mathcal S} $ be a pseudo-regular subspace of $ {\mathcal H} $. So, there exists a regular subspace $ {\mathcal M} $ such that $ {\mathcal S} = {\mathcal M} [\dot{+}] {\mathcal S} ^{\circ}$. Consider a decomposition
\[
{\mathcal S} = {\mathcal S} ^\circ [\dotplus] \, {\mathcal M} _+ \, [\dot{+}] \, {\mathcal M} _-,
\]
where $ {\mathcal M} _+$ is a uniformly $J$-positive subspace, $ {\mathcal M} _-$ is a uniformly $J$-negative subspace and $ {\mathcal M} = {\mathcal M} _+ [\dot{+}] {\mathcal M} _-$. Then, the numbers $\ka{+}( {\mathcal S} )=\dim {\mathcal M} _+$, $\ka{-}( {\mathcal S} )=\dim {\mathcal M} _-$ and $\ka{0}( {\mathcal S} )=\dim {\mathcal S} ^\circ$ are called the \emph{positive, negative and isotropic signatures} of $ {\mathcal S} $, respectively. It has been shown that these numbers do not depend on the particular decomposition considered (see e.g. \cite[Ch. 1, Thm. 6.7]{azizov iok89}).
If $ {\mathcal S} $ is pseudo-regular then so is $ {\mathcal S} ^{[\bot]}$, and the \emph{positive and negative cosignatures of $ {\mathcal S} $} are defined as $c\ka{+}( {\mathcal S} ):=\ka{+}( {\mathcal S} ^{[\bot]})$ and $c\ka{-}( {\mathcal S} ):=\ka{-}( {\mathcal S} ^{[\bot]})$.
\begin{prop}(\cite[Proposition 4.6]{G})\label{spatial c}
Let $ {\mathcal S} $ and $ {\mathcal T} $ be two pseudo-regular subspaces of $ {\mathcal H} $. The following statements are equivalent:
\begin{enumerate}
\item[i)] $ {\mathcal S} $ and $ {\mathcal T} $ are $J$-unitarily equivalent, i.e. there exists $U\in\mathcal{U}_J$ such that $U( {\mathcal S} )= {\mathcal T} $;
\item[ii)] $ {\mathcal S} $ is $J$-isometrically isomorphic to $ {\mathcal T} $ and $ {\mathcal S} ^{[\bot]}$ is $J$-isometrically isomorphic to $ {\mathcal T} ^{[\bot]}$;
\item[iii)] $\ka{+}( {\mathcal S} )=\ka{+}( {\mathcal T} )$, $\ka{-}( {\mathcal S} )=\ka{-}( {\mathcal T} )$, $c\ka{+}( {\mathcal S} )=c\ka{+}( {\mathcal T} )$, $c\ka{-}( {\mathcal S} )=c\ka{-}( {\mathcal T} )$ and $\ka{0}( {\mathcal S} )=\ka{0}( {\mathcal T} )$.
\end{enumerate}
\end{prop}
With the latter result at hand, it is now straightforward to give a spatial characterization of the orbits.
Moreover, the orbits are the connected components of $ {\mathcal Q} $.
\begin{prop}\label{comp q}
Let $Q_0,Q \in {\mathcal Q} $. The following assertions are equivalent:
\begin{enumerate}
\item[i)] $Q \in \mathcal{U}_J \cdot Q_0$.
\item[ii)] $R(Q)$ and $R(Q_0)$ have the same (three) signatures and the same (two) cosignatures.
\end{enumerate}
Moreover, the connected component of $Q_0$ in $ {\mathcal Q} $ coincides with $\mathcal{U}_J \cdot Q_0$.
\end{prop}
\begin{proof}
If $Q\in {\mathcal Q} $ then $R(Q)=R(Q)^\circ [\dotplus] {\mathcal M} $ and $N(Q)=N(Q)^\circ [\dotplus] {\mathcal N} $, where $ {\mathcal M} $ and $ {\mathcal N} $ are regular subspaces. Then, it is easy to see that $R(Q)^{[\bot]}=R(Q)^\circ [\dotplus] {\mathcal N} $ and
\[
c\ka{\pm} (R(Q))= \ka{\pm} (R(Q)^{[\bot]})=\ka{\pm} (N(Q)).
\]
Hence, the equivalence between $i)$ and $ii)$ follows from applying Proposition \ref{spatial c} to the ranges of two $J$-normal projections.
Let $C_{Q_0}$ be the connected component of $Q_0$. Recall that $\mathcal{U}_J$ is connected (Proposition \ref{connected uj}). Therefore $\mathcal{U}_J \cdot Q_0$ is connected. Hence $\mathcal{U}_J \cdot Q_0 \subseteq C_{Q_0}$. In order to show the converse inclusion, note that the map
\[ Q \mapsto (\,\ka{+}(R(Q)),c\ka{+}(R(Q)),\ka{-}(R(Q)),c\ka{-}(R(Q)),\ka{0}(R(Q))\,) \]
is continuous. In fact, if $\| Q- Q_0\|<r_{Q_0}$, where $r_{Q_0}$ is defined in the proof of Theorem \ref{section}, then there is an operator $U \in \mathcal{U}_J$ such that $Q=UQ_0U^{\#}$. According to the equivalence $i)$-$ii)$, it follows that
the five indices must coincide. This proves that the above map is continuous. Since it takes values on a discrete set, the map has to be constant on $C_{Q_0}$. Now if $Q \in C_{Q_0}$, then the five indices associated to $Q$ are equal to those of $Q_0$. Hence there exists a $J$-unitary such that $Q=UQ_0U^{\#}$.
\end{proof}
The connected components of $ {\mathcal E} $ can be obtained as a particular case of the above result.
\begin{coro}
Let $E_0,E \in {\mathcal E} $. The following assertions are equivalent:
\begin{enumerate}
\item[i)] $E \in \mathcal{U}_J \cdot E_0$.
\item[ii)] $R(E)$ and $R(E_0)$ have the same (two) signatures and the same (two) cosignatures.
\end{enumerate}
Moreover, the connected component of $E_0$ in $ {\mathcal E} $ coincides with $\mathcal{U}_J \cdot E_0$.
\end{coro}
\section{Differential structure of $ {\mathcal Q} $}
The following is a well-known criterion to determine if a proper subset of a manifold is indeed a submanifold, see \cite[Proposition 8.7]{upmeier85}.
\begin{prop}\label{sous varietes}
Consider two Banach manifolds $M$ and $N$. Suppose that $g:M \to N$ is an analytic inmersion and a homeomorphism onto $N'=g(M)$. Then $N'$ is a submanifold of $N$ and the mapping $g:M \to N'$ is bianalytic.
\end{prop}
This criterion will be used to show that $ {\mathcal Q} $ is a submanifold of $L( {\mathcal H} )$. Note that one can restrict to the connected components of $ {\mathcal Q} $ given by the orbits $\mathcal{U}_J \cdot Q_0$, $Q_0 \in {\mathcal Q} $. By Proposition \ref{kernel sup}, $\mathcal{U}_J \cdot Q_0$ has a manifold structure compatible with the quotient topology. Moreover, the map $p_{Q_0}$ is an analytic submersion with this manifold structure. Equivalently, $p_{Q_0}$ admits local analytic cross sections (see \cite[Corollary 8.3]{upmeier85}).
Note that the following diagram commutes
\begin{displaymath}
\xymatrix{
\mathcal{U}_J \ar[r]^{p_{Q_0}} \ar[dr]_{\tilde{p}_{Q_0}} & \mathcal{U}_J \cdot Q_0 \ar[d]^{i} \hspace{-0.5cm}& \hspace{-.5cm} \simeq \mathcal{U}_J /\mathcal{G} \\
& L( {\mathcal H} ) \hspace{-0.5cm} & \hspace{-0.5cm}
}
\end{displaymath}
where $i$ is the inclusion map and $\tilde{p}_{Q_0}(U)=UQ_0U^{\#}$. The map $\tilde{p}_{Q_0}$ is clearly analytic because it consists in multiplication and inversion in $L( {\mathcal H} )$. The inclusion map can be locally written as $i=\tilde{p}_{Q_0} \circ s$, where $s$ is a analytic section of $p_{Q_0}$. Hence, $i$ is analytic.
To prove that $i$ is an inmersion (i.e. its differential map is injective and has complemented range), notice that the range of the differential at $Q\in \mathcal{U}_J \cdot Q_0$ of $i$ is precisely the tangent space $T_Q (\mathcal{U}_J \cdot Q_0)$. The latter is computed as derivatives of smooth curves in the orbit, and it is given by
\[ T_Q (\mathcal{U}_J \cdot Q_0)= \{ \, XQ-QX \, : \, X \in \mathfrak{u}_J \, \}. \]
On the other hand, it was shown that the quotient and the inherited topologies coincide in the orbits (Theorem \ref{section}). Hence, to see that $\mathcal{U}_J \cdot Q_0$ is a submanifold of $L( {\mathcal H} )$ it is sufficient to find a complement of $T_{Q_0} (\mathcal{U}_J \cdot Q_0)$ in $L( {\mathcal H} )$.
To this end, if $Q_0\in \mathcal{Q}$ consider again the decompositions
\[
Q_0=E_0+P_0 \ \ \ \text{and} \ \ \ I-Q_0= F_0 + P_0^\#,
\]
given in Proposition \ref{desc}. Let $XQ_0 - Q_0X$ be a tangent vector of the orbit $\mathcal{U}_J \cdot Q_0$ at the point $Q_0$. Since $X^\#=-X$ and $E_0=E_0^{\#}$, the $J$-selfadjoint and the $J$-antihermitian parts of $XQ_0-Q_0X$ are given by
$$XE_0 - E_0X + \frac{1}{2}(X(P_0+P_0^{\#}) - (P_0+P_0^{\#})X);$$
and
$$
\frac{1}{2i}(X(P_0-P_0^{\#}) - (P_0-P_0^{\#})X),$$
respectively. Clearly, $T_{Q_0}(\mathcal{U}_J \cdot Q_0)$ will be complemented in $L( {\mathcal H} )$ if
\[
\mathcal{L}_s:=\{ \, XE_0 - E_0X + \frac{1}{2}(X(P_0+P_0^{\#}) - (P_0+P_0^{\#})X) \, : \, X^\#=-X \, \}
\]
is complemented in the subspace of $J-$selfadjoint operators $i \mathfrak{u}_J$ and
\[
\mathcal{L}_a:=\{ \, X(P_0-P_0^{\#}) - (P_0-P_0^{\#})X \, : \, X^\#=-X \, \}
\]
is complemented in $ \mathfrak{u}_J$.
\medskip
\begin{lem}\label{sup self}
$\mathcal{L}_s$ is complemented in $i \mathfrak{u}_J$.
\end{lem}
\begin{proof}
Set $ {\mathcal S} _1=R(E_0)$, $ {\mathcal S} _2=R(P_0+P_0^\#)$ and $ {\mathcal S} _3=R(F_0)$ and consider the $J$-orthogonal decomposition $ {\mathcal H} = {\mathcal S} _1 [\dotplus] {\mathcal S} _2 [\dotplus] {\mathcal S} _3$.
If a $J$-antihermitian operator $X$ is represented as a block-operator matrix according to this decomposition:
\[
\begin{blockarray}{lrrrl}
\begin{block}{l(rrr)l}
\, & X_{11} & X_{12} & X_{13} \, & \matindex{$ {\mathcal S} _1$} \\
\, X=&-X_{12}^{\#} & X_{22} & X_{23} \, & \matindex{$ {\mathcal S} _2$} \\
\, &-X_{13}^{\#} & -X_{23}^{\#} & X_{33}\, & \matindex{$ {\mathcal S} _3$} \\
\end{block}
\end{blockarray},
\]
then an operator in the subspace $\mathcal{L}_s$
is represented as
$$XE_0 - E_0X + \frac{1}{2}(X(P_0+P_0^{\#}) - (P_0+P_0^{\#})X)=\left( \begin{array}{ccc}
0 & -\frac{1}{2}X_{12} & -\frac{1}{2}X_{13} \\
-\frac{1}{2}X_{12}^{\#} & 0 & -\frac{1}{2}X_{23} \\
-\frac{1}{2}X_{13}^{\#} & -\frac{1}{2}X_{23}^{\#} & 0 \\
\end{array}
\right).$$
Therefore, the subspace $\mathcal{L}_s$ can be described with three parameters as
\[ \mathcal{L}_s=
\left\{ \,
\left( \begin{array}{ccc}
0 & A & B \\
A^{\#} & 0 & C \\
B^{\#} & C^{\#} & 0 \\
\end{array}
\right)
\, :
\, A \in L( {\mathcal S} _2, {\mathcal S} _1), \, B \in L( {\mathcal S} _3, {\mathcal S} _1), \, C \in L( {\mathcal S} _3, {\mathcal S} _1) \,
\right\}.
\]
From this representation, it is easy to see that $\mathcal{L}_s$ is complemented in the subspace of $J$-selfadjoint operators. In fact, a complement is given by the subspace of $J$-selfadjoint operators which are block-diagonal according to the decomposition considered above.
\end{proof}
As in the previous result, the main idea in the proof of the following lemma is to find an alternative description of $\mathcal{L}_a$ by means of $3 \times 3$ block-operator matrices.
However, the decomposition will be given in terms of different projections.
\begin{lem}\label{sup antih}
$\mathcal{L}_a$ is complemented in $ \mathfrak{u}_J$.
\end{lem}
\begin{proof}
Set $A_0=P_0-P_0^{\#}$. Note that $A_0^2=P_0+P_0^{\#}$ and $A_0^3=A_0$. In particular, $A_0 ^2$ is a $J$-selfadjoint
projection. Now set
\[ R_0=\frac{1}{2}(A_0 ^2 + A_0). \]
From the properties of $A_0$, it follows that
$$R_0^2=\frac{1}{4}(A_0^4 + 2 A_0 ^3 + A_0 ^2)=\frac{1}{4}(2A_0^2 + 2A_0)=R_0,$$
hence $R_0$ is a projection. Then $R_0^{\#}$ is also a projection. Taking into account that $A_0^{\#}=-A_0$, one gets
$R_0 + R_0^{\#}=A_0^2$. Further useful relations between these projections are the following:
\[ \, \, \, A_0R_0=R_0, \, \, \ R_0A_0=R_0 , \, \, \, -R_0^{\#} A_0=R_0^{\#}, \, \, \, - A_0R_0^{\#}=R_0^{\#}\, . \]
Set $ {\mathcal T} _1=R(R_0)$, $ {\mathcal T} _2=R(R_0^{\#})$ and $ {\mathcal T} _3=R(I-A_0 ^2)$. Hence the space $ {\mathcal H} $ can be decomposed as $ {\mathcal H} = {\mathcal T} _1\dotplus {\mathcal T} _2\dotplus {\mathcal T} _3$.
Next, suppose that $X=-X^{\#}$ is represented as
\[
\begin{blockarray}{lrrrl}
\begin{block}{l(rrr)l}
\, & X_{11} & X_{12} & X_{13} \, & \matindex{$ {\mathcal T} _1$} \\
\, X= & X_{21} & X_{22} & X_{23} \, & \matindex{$ {\mathcal T} _2$} \\
\, & X_{31} & X_{32} & X_{33}\, & \matindex{$ {\mathcal T} _3$} \\
\end{block}
\end{blockarray}
\]
From the properties of the projections $R_0$, $R_0^{\#}$ and $I-A_0^2$, it is possible to consider only five parameters to represent $X$ as a block operator-matrix, that is
$$X=\left( \begin{array}{ccc}
X_{11} & X_{12} & X_{13} \\
-X_{12}^{\#} & -X_{11}^{\#} & X_{23} \\
-X_{23}^{\#} & -X_{13}^{\#} & X_{33} \\
\end{array}
\right),$$
\noindent where $X_{33}^{\#}=-X_{33}$. On the other hand, the operator $XA_0-A_0X \in \mathcal{L}_a$ is represented as
\[ XA_0-A_0X =
\left( \begin{array}{ccc}
0 & 2X_{12} & X_{13} \\
-2X_{12}^{\#} & 0 & X_{23} \\
-X_{23}^{\#} & -X_{13}^{\#} & 0 \\
\end{array}
\right).
\]
Therefore,
$$\mathcal{L}_a=\bigg\{ \,
\left( \begin{array}{ccc}
0 & A & B \\
-A^{\#} & 0 & C \\
-C^{\#} & -B^{\#} & 0 \\
\end{array}
\right) \, :
\, A \in L( {\mathcal T} _2 , {\mathcal T} _1),\, B \in L( {\mathcal T} _3 , {\mathcal T} _1),\, C \in L( {\mathcal T} _3 , {\mathcal T} _2) \, \bigg\}. $$
Now the subspace $\mathcal{L}_a$ can be easily complemented in $ \mathfrak{u}_J$ as follows:
\[ \mathfrak{u}_J = \mathcal{L}_a \oplus \bigg\{ \,
\left( \begin{array}{ccc}
Y & 0 & 0 \\
0 & -Y^{\#} & 0 \\
0 & 0 & Z \\
\end{array}
\right)
\, : \, Y \in L( {\mathcal T} _1) ,\, Z \in L( {\mathcal T} _3), \, Z=-Z^{\#} \, \bigg\}. \qedhere \]
\end{proof}
\noindent The main facts on the differential structure of $ {\mathcal Q} $ and $ {\mathcal E} $ are collected in the following result.
\begin{teo}\label{q submanifold lh}
The following assertions hold:
\begin{enumerate}
\item[i)] $ {\mathcal Q} $ is an analytic submanifold of $L( {\mathcal H} )$.
\item[ii)] $ {\mathcal E} $ is an analytic submanifold of $L( {\mathcal H} )$.
\item[iii)] The map $F: {\mathcal Q} \to {\mathcal E} $, $F(Q)=QQ^\#$, is a real analytic submersion.
\end{enumerate}
\end{teo}
\begin{proof}
${\rm i)}$ The assumptions in the criterion stated in Proposition \ref{sous varietes} are verified in each connected component of $ {\mathcal Q} $. Indeed, it has been shown in Theorem \ref{section} that the quotient topology of $\mathcal{U}_J \cdot Q_0$ coincides with the topology inherited from $L( {\mathcal H} )$. In addition, the tangent space $T_{Q_0}(\mathcal{U}_J \cdot Q_0)$ is complemented in $L( {\mathcal H} )$ by Lemmas \ref{sup self} and \ref{sup antih}.
But this says that the range of the differential map of the inclusion $\mathcal{U}_J \cdot Q_0 \hookrightarrow L( {\mathcal H} )$ is complemented in $L( {\mathcal H} )$. So, the proof is completed.
\smallskip
\noindent ${\rm ii)}$ It is analogous to the proof of ${\rm i)}$.
\smallskip
\noindent ${\rm iii)}$ It suffices to prove the statement for a connected component $\mathcal{U}_J \cdot Q_0$ of $ {\mathcal Q} $ and a connected component $\mathcal{U}_J \cdot E_0$ of $ {\mathcal E} $, where $E_0=Q_0Q_0^{\#}$. According to Proposition \ref{sous varietes}, the identity map $\mathcal{U}_J \cdot Q_0 \simeq \mathcal{U}_J / \mathcal{G} \to \mathcal{U}_J \cdot Q_0 \subseteq L( {\mathcal H} )$ is bianalytic. Thus, if one considers the submanifold structure in $\mathcal{U}_J \cdot Q_0$, then the map $p_{Q_0}$ is also an analytic submersion. Analogously, the map $p_{E_0}$ is an analytic submersion when this orbit has the submanifold structure.
Next note that the following diagram commutes
\begin{displaymath}
\xymatrix{
\mathcal{U}_J \ar[r]^{p_{Q_0}} \ar[dr]_{p_{E_0}} & \mathcal{U}_J \cdot Q_0 \ar[d]^{F} \hspace{-0.5cm}& \hspace{-.5cm} \\
& \mathcal{U}_J \cdot E_0 \hspace{-0.5cm} & \hspace{-0.5cm}
}
\end{displaymath}
Since $p_{Q_0}$ is a surjective analytic submersion and $p_{E_0}$ is an analytic submersion, it is a well-known fact that $F$ turns out to be an analytic submersion (see for instance \cite[Corollary 8.4]{upmeier85}).
\end{proof}
\section{Covering space structure of $ {\mathcal Q} _{ {\mathcal S} }$}
For a fixed pseudo-regular subspace $ {\mathcal S} $, consider the set of $J$-normal projections onto $ {\mathcal S} $, i.e.
\[
\mathcal{Q}_{ {\mathcal S} } =\{Q\in \mathcal{Q}:\, R(Q)= {\mathcal S} \}.
\]
Clearly, the group $\mathcal{U}_J$ does not leave $ {\mathcal Q} _{ {\mathcal S} }$ invariant. In order to find a suitable group acting on $ {\mathcal Q} _{ {\mathcal S} }$, one can restrict to
the subgroup of $\mathcal{U}_J$ given by
\[ {\mathcal U} _{ {\mathcal S} }=\{ \, U \in \mathcal{U}_J \, : \, U( {\mathcal S} )= {\mathcal S} \, \}. \]
It is easy to see that if $U\in {\mathcal U} _ {\mathcal S} $ then $U( {\mathcal S} ^{[\bot]})= {\mathcal S} ^{[\bot]}$ and $U( {\mathcal S} ^{\circ})= {\mathcal S} ^{\circ}$.
\medskip
As before, the action of $ {\mathcal U} _ {\mathcal S} $ on $ {\mathcal Q} _ {\mathcal S} $
is defined by $U \cdot Q=UQU^{\#}$, where $U \in {\mathcal U} _{ {\mathcal S} }$ and $Q \in {\mathcal Q} _{ {\mathcal S} }$.
The following result was proved in \cite[Proposition 3.1]{G}. Below there is another proof with an explicit
construction of the $J$-isometric isomorphism. This formula will be helpful later. Along this section, when $ {\mathcal T} _1$, $ {\mathcal T} _2$ are two (closed) subspaces of $ {\mathcal S} $ such that $ {\mathcal T} _1 \dot{+} {\mathcal T} _2= {\mathcal S} $, the projection in $L( {\mathcal S} )$ with range $ {\mathcal T} _1$ and nullspace $ {\mathcal T} _2$ is denoted by $P_{ {\mathcal T} _1 // {\mathcal T} _2}$.
\begin{lem}\label{j isom iso}
Let $ {\mathcal S} $ be a pseudo-regular subspace of $ {\mathcal H} $. If $ {\mathcal M} _1$, $ {\mathcal M} _2$ are two regular subspaces such that $ {\mathcal S} = {\mathcal M} _1 [\dot{+}] {\mathcal S} ^{\circ}= {\mathcal M} _2 [\dot{+}] {\mathcal S} ^{\circ}$, then
$(P_{ {\mathcal M} _2// {\mathcal S} ^{\circ}} )|_{ {\mathcal M} _1}: {\mathcal M} _1 \to {\mathcal M} _2$ is a $J$-isometric isomorphism.
\end{lem}
\begin{proof}
Set $W=(P_{ {\mathcal M} _2// {\mathcal S} ^{\circ}} )|_{ {\mathcal M} _1}\in L( {\mathcal M} _1, {\mathcal M} _2)$. Let $f \in {\mathcal M} _1$ such that $W f=0$. Then, $f \in {\mathcal S} ^{\circ}\cap {\mathcal M} _1=\{0\}$. Thus, $W$ is one-to-one. To show that $W$ is surjective, pick $g \in {\mathcal M} _2$. Then $g=f_{ {\mathcal M} _1}+f_{ {\mathcal S} ^{\circ}}$, where $f_{ {\mathcal M} _1} \in {\mathcal M} _1$ and $f_{ {\mathcal S} ^{\circ}} \in {\mathcal S} ^{\circ}$. Therefore $g=P_{ {\mathcal M} _2// {\mathcal S} ^{\circ}}g=P_{ {\mathcal M} _2// {\mathcal S} ^{\circ}}f_{ {\mathcal M} _1} $. Hence $g=Wf_{ {\mathcal M} _1}$.
Finally, notice that $W$ is a $J$-isometric isomorphism. Indeed, given $f,g \in {\mathcal M} _1$, suppose that $f=f_{ {\mathcal M} _2}+f_{ {\mathcal S} ^{\circ}}$ and $g=g_{ {\mathcal M} _2}+g_{ {\mathcal S} ^{\circ}}$. Since $f_{ {\mathcal S} ^{\circ}}, g_{ {\mathcal S} ^{\circ}} \in {\mathcal S} ^{\circ}$, it follows that
\[ [Wf,Wg]=[f_{ {\mathcal M} _2},g_{ {\mathcal M} _2}]=[f,g]. \]
Hence $W$ is $J$-isometric.
\end{proof}
The next result shows that, given $Q_0\in {\mathcal Q} _ {\mathcal S} $, any other $Q\in {\mathcal Q} _ {\mathcal S} $ can be written as $Q=UQ_0U^\#$ for a suitable $U\in {\mathcal U} _ {\mathcal S} $.
\begin{prop}\label{accion t grup chic}
The group $ {\mathcal U} _{ {\mathcal S} }$ acts transitively on $\mathcal{Q}_{ {\mathcal S} }$.
\end{prop}
\begin{proof}
If $Q, Q_0\in \mathcal{Q}_ {\mathcal S} $, consider the usual associated projections $E$, $F$, $P$ and $E_0$, $F_0$, $P_0$. According to Remark \ref{desc q y 1-q}, $ {\mathcal H} $ can be decomposed as
\[
{\mathcal H} = R(E)[\dotplus] R(P + P^\#) [\dotplus] R(F)= R(E_0)[\dotplus] R(P_0 + P_0^\#) [\dotplus] R(F_0).
\]
Notice that $R(P)=R(P_0)= {\mathcal S} ^\circ$.
Then, by Lemma \ref{s link}, there exists a $J$-isometric isomorphism
$$V: R(P_0 + P_0^\#) \rightarrow R(P + P^\#),$$
which can be defined as the identity operator on $ {\mathcal S} ^{\circ}$.
On the other hand, by Lemma \ref{j isom iso}, there is a $J$-isometric isomorphism $W: R(E_0) \rightarrow R(E)$.
It only remains to show that the ranges of $F$ and $F_0$ are $J$-isometrically isomorphic.
To this end, note that $ {\mathcal S} ^{[\perp]}$ is also a pseudo-regular subspace. Moreover, it follows that $ {\mathcal S} ^{[\perp]}=R(Q)^{[\perp]}=N(Q^{\#})=R(I-Q^{\#})=R(F+P)= R(F)[\dotplus] {\mathcal S} ^{\circ}$. Similarly, one can see that $ {\mathcal S} ^{[\perp]}=R(F_0)[\dot{+}] {\mathcal S} ^{\circ}$.
Therefore, $R(F)$ and $R(F_0)$ are two different regular complements of $ {\mathcal S} ^{\circ}$ in $ {\mathcal S} ^{[\perp]}$. As in the previous paragraph, there is a $J$-isometric isomorphism $W': R(F_0) \rightarrow R(F)$.
Finally, define $U: {\mathcal H} \rightarrow {\mathcal H} $ by $U(f+g+h)=Wf + Vg + W'h$, where $f\in R(E_0)$, $g\in R(P_0 + P_0^\#)$ and $h\in R(F_0)$.
It is easy to see that $U \in {\mathcal U} _{ {\mathcal S} }$ and, by construction, $UQ_0U^\#=Q$.
\end{proof}
Given a pseudo-regular subspace $ {\mathcal S} $ of $ {\mathcal H} $, consider the family of regular complements of $ {\mathcal S} ^\circ$ in $ {\mathcal S} $:
$$ {\mathcal F} =\{ \, {\mathcal M} \text{ is a regular subspace of $ {\mathcal H} $} \, : \, {\mathcal S} = {\mathcal M} [\dot{+}] {\mathcal S} ^{\circ} \, \}.$$
It is not difficult to see that $ {\mathcal Q} _{ {\mathcal S} }$ can be rewritten as the following disjoint union
\begin{equation}\nonumber
{\mathcal Q} _ {\mathcal S} = \bigcup_{ {\mathcal M} \in {\mathcal F} } {\mathcal Q} _{ {\mathcal S} ,\mathcal{M}},
\end{equation}
where
$ {\mathcal Q} _{ {\mathcal S} ,\mathcal{M}}=\{ \, Q \in {\mathcal Q} _{ {\mathcal S} } \, : \, R(QQ^{\#})= {\mathcal M} \, \}$, see \cite[Lemma 6.4]{maestripieri martinez13}.
For each $ {\mathcal M} \in {\mathcal F} $, it is natural to consider the subgroup $ {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}}$ of $ {\mathcal U} _{ {\mathcal S} }$ defined by
\[ {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}}=\{ \, U \in \mathcal{U}_J \, : \, U( {\mathcal M} )= {\mathcal M} , \, U( {\mathcal S} ^{\circ})= {\mathcal S} ^{\circ} \, \}. \]
Clearly, $ {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}}$ acts on $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$ by conjugation. Furthermore,
\begin{prop}\label{accion t grup chic deck}
The group $ {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}}$ acts transitively on $\mathcal{Q}_{ {\mathcal S} , {\mathcal M} }$.
\end{prop}
\begin{proof}
The same proof of Proposition \ref{accion t grup chic} works in this case. Indeed, the $J$-unitary $U$ constructed in that proof leaves $ {\mathcal M} $ invariant whenever $Q,Q_0 \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$.
\end{proof}
In the next result a continuous selection from $ {\mathcal F} $ to $ {\mathcal Q} _{ {\mathcal S} }$ is constructed. The set $ {\mathcal F} $ is endowed with the topology defined by the metric $$d( {\mathcal M} , {\mathcal N} )=\| E_{ {\mathcal M} } - E_{ {\mathcal N} } \|,$$ where $E_{ {\mathcal M} }$ denotes the (unique) $J$-selfadjoint projection onto $ {\mathcal M} $;
meanwhile $ {\mathcal Q} _{ {\mathcal S} }$ is considered with the topology inherited from $L( {\mathcal H} )$.
\begin{lem}\label{cont sel}
There exists a continuous map
$g: {\mathcal F} \to {\mathcal Q} _{ {\mathcal S} }$ such that $g( {\mathcal M} ) \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$.
\end{lem}
\begin{proof}
Let $ {\mathcal M} $ be a regular subspace of
$ {\mathcal H} $ such that $ {\mathcal S} = {\mathcal M} [\dot{+}] {\mathcal S} ^{\circ}$. Consider the following orthogonal decomposition $ {\mathcal H} = {\mathcal S} ^{\circ} \oplus ( {\mathcal S} \ominus {\mathcal S} ^{\circ} )\oplus {\mathcal S} ^{\perp}$. According to \cite[Theorem 6.9]{maestripieri martinez13}, a $J$-normal projection $Q$ belongs to $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$ if and only if $Q$ can be written as
\[
Q=\left( \begin{array}{ccc}
I & 0 & A + (\text{Re}(Bc^*br a^*) - \frac{1}{2}(BdB^*+ar^*b^3ra^*))a + B + ar^*(c+b) \\
0 & I & b^{-1}c + r \\
0 & 0 & 0
\end{array}
\right),
\]
where $r=P_{ {\mathcal S} \ominus {\mathcal S} ^{\circ}} E_{ {\mathcal M} }|_{J( {\mathcal S} ^\circ)}\in L( {\mathcal S} ^\bot, {\mathcal S} \ominus {\mathcal S} ^\circ)$, $A=-A^* \in L( {\mathcal S} ^{\circ})$ and $B \in L( {\mathcal S} ^{\perp}, {\mathcal S} ^{\circ})$ satisfies $J( {\mathcal S} ^{\circ})\subseteq N(B)$. Here the lowercase letters $a,b,c$ and $d$ come from the decomposition of the fundamental symmetry
\[
J= \left( \begin{array}{ccc}
0 & 0 & a \\
0 & b & c \\
a^* & c^* & d
\end{array}
\right).
\]
Clearly, $r: {\mathcal F} \to L( {\mathcal S} ^\bot, {\mathcal S} \ominus {\mathcal S} ^\circ)$ given by $r( {\mathcal M} )=P_{ {\mathcal S} \ominus {\mathcal S} ^{\circ}} E_{ {\mathcal M} }|_{J( {\mathcal S} ^\circ)}$ is a continuous function by the definition of the metric in $ {\mathcal F} $.
To construct the required continuous selection, it is possible to set $A=B=0$ in the above decomposition. Therefore, the map
$$
g: {\mathcal F} \to {\mathcal Q} _{ {\mathcal S} } \, \, \, \text{defined by}\, \, \,
g( {\mathcal M} )=\left( \begin{array}{ccc}
I & 0 & - \frac{1}{2}ar^*b^3ra^*a + ar^*(c+br) \\
0 & I & b^{-1}c + r \\
0 & 0 & 0\\
\end{array}
\right),
$$
satisfies $g( {\mathcal M} ) \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$. The continuity of $g$ follows from that of $r$.
\end{proof}
The map defined locally in Lemma \ref{sec cont lema} can be defined globally in $ {\mathcal Q} _{ {\mathcal S} }$. This allows to prove the existence of a global section for the restriction of $p_{Q_0}$ to $ {\mathcal U} _ {\mathcal S} $.
\begin{prop}\label{sec global qs}
Let $Q_0$ be a projection in $ {\mathcal Q} _ {\mathcal S} $. Then, the map $$p_{Q_0}: {\mathcal U} _{ {\mathcal S} } \to {\mathcal Q} _ {\mathcal S} , \, \, \, \,
p_{Q_0}(U)=UQ_0U^{\#},$$
has a global continuous cross section.
\end{prop}
\begin{proof}
First recall that $ {\mathcal H} = R(E_i) [\dotplus] R(F_i)[\dotplus] R(P_i + P_i^\#)$ for $i =1,2$ and consider $U: {\mathcal Q} _{ {\mathcal S} } \times {\mathcal Q} _{ {\mathcal S} } \to {\mathcal U} _{ {\mathcal S} }$ defined by
\[
U(Q_1,Q_2)=P_{R(E_2)// {\mathcal S} ^{\circ}} E_1 + P_{R(F_2)// {\mathcal S} ^{\circ}} F_1 + V(Q_2)V(Q_1)^{\#}(P_1 + P_1^{\#}).
\]
The map $U$ is a $J$-isometric isomorphism
restricted to each of these three pairs of subspaces. In fact, the map $R(E_1) \to R(E_2)$ given by $f \mapsto P_{R(E_2)// {\mathcal S} ^{\circ}}f$ is a $J$-isometric isomorphism by Lemma \ref{j isom iso}. Similarly, one can see that $R(F_1) \to R(F_2)$ given by $f \mapsto P_{R(F_2)// {\mathcal S} ^{\circ}}f$ is also a $J$-isometric isomorphism.
Also, by Lemma \ref{s link}, $V(Q_2)V(Q_1)^\#$ is a $J$-isometric isomorphism from $R(P_1 + P_1^\#)$ onto $ R(P_2 + P_2^\#)$.
Hence $U(Q_1,Q_2)$ is a $J$-unitary. Moreover, it is the identity operator in $ {\mathcal S} ^{\circ}$, and it leaves $ {\mathcal S} $ invariant. Hence the operator $U(Q_1,Q_2)$ belongs to $ {\mathcal U} _{ {\mathcal S} }$. In addition, by Lemma \ref{j isom iso} and Lemma \ref{s link} it follows that
\[
U(Q_1,Q_2) \, Q_1 \, U(Q_1,Q_2)^{\#}=Q_2.
\]
Note that $E_i=Q_i Q_i ^{\#}$, $F_i=(I-Q_i)(I- Q_i )^{\#}$ and $P_i=Q_i(I-Q_i ^{\#})$ are continuous functions of $Q_i$ for $i=1,2$. On the other hand, the continuity of $V(Q_i)$, $i=1,2$, and consequently, the continuity of $ V(Q_2)V(Q_1)^{\#}$, is proved in
Lemma \ref{s link}. To show that $U$ is a continuous map, it remains to prove that the maps
$ {\mathcal Q} _ {\mathcal S} \to L( {\mathcal S} )$ given by $Q \mapsto P_{R(E)// {\mathcal S} ^{\circ}}$ and $ {\mathcal Q} _ {\mathcal S} \to L( {\mathcal S} ^{[\bot]})$ given by $Q \mapsto P_{R(F)// {\mathcal S} ^{\circ}}$ are continuous.
Let $\{ Q_k \}_{k \geq 1}$ be a sequence in $ {\mathcal Q} _{ {\mathcal S} }$ such that $\| Q_k - Q\| \to 0$.
Again recall that $E=Q Q ^{\#}$ is a continuous function of $Q$.
Thus, by Lemma \ref{kato proj} one finds that
\[ \| P_{R(E_k)} - P_{R(E)} \| \leq \| E_k - E\| \to 0. \]
Applying the formula proved in \cite[Lemma 3.1]{ando11}, the projection $ P_{R(E) // {\mathcal S} ^\circ}$ can
be rewritten as
\[ P_{R(E) // {\mathcal S} ^\circ}=P_{R(E)}(P_{R(E)}+P_{ {\mathcal S} ^\circ})^{-1}. \]
Notice that this formula is a continuous function of the orthogonal projection $P_{R(E)}$. Hence it
follows that $\|P_{R(E_k) // {\mathcal S} ^\circ} - P_{R(E) // {\mathcal S} ^\circ} \| \to 0$. The proof of the continuity of $Q \mapsto P_{R(F)// {\mathcal S} ^{\circ}}$ is similar.
Therefore, the map $s: {\mathcal Q} _ {\mathcal S} \to {\mathcal U} _{ {\mathcal S} }$ defined by $s(Q)=U(Q_0,Q)$ is a global continuous cross section of $p_{Q_0}(U)=UQ_0U^{\#}$.
\end{proof}
In the next result $s$ stands for the global section considered in Proposition \ref{sec global qs}, and $g$ is the continuous selection defined in Lemma \ref{cont sel}.
\begin{teo}\label{cov spa fixed range}
Let $ {\mathcal S} $ be a pseudo-regular subspace and $ {\mathcal M} _0$ be a regular subspace such that
$ {\mathcal S} = {\mathcal M} _0 [\dotplus] {\mathcal S} ^{\circ}$. Let $Q_0$ be a fixed projection in $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} _0}$. Consider the map $ r: {\mathcal Q} _ {\mathcal S} \to {\mathcal Q} _{ {\mathcal S} , {\mathcal M} _0}$ defined by
\[
r(Q)=s(g( {\mathcal M} ))^{\#}\, Q \,s(g( {\mathcal M} )),
\]
whenever $Q \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$. Then $r$ is a covering map.
\end{teo}
The map $r$ has an alternative expression. By Lemma \ref{accion t grup chic deck} there exists $U \in {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}}$ such that $Q=Ug( {\mathcal M} )U^{\#}$. Therefore, note that
\begin{align}
r(Q) &= s(g( {\mathcal M} ))^{\#} \, U g( {\mathcal M} ) U^{\#} \, s(g( {\mathcal M} )) \nonumber \\
&= s(g( {\mathcal M} ))^{\#} U s(g( {\mathcal M} )) \, Q_0 \, s(g( {\mathcal M} ))^{\#} U^{\#} s(g( {\mathcal M} ))\nonumber \\
&= Ad_{s(g( {\mathcal M} ))} (U)\, Q_0 \, (Ad_{s(g( {\mathcal M} ))} (U))^{\#}, \label{another exp}
\end{align}
where $Ad_U: L( {\mathcal H} ) \to L( {\mathcal H} )$ is defined by $Ad_U(X)=U^{\#}XU$, for $U \in \mathcal{U}_J$ and $X \in L( {\mathcal H} )$. This expression does not depend on the choice of $U\in {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^\circ}$.
\begin{proof}
It has been previously noted that $ {\mathcal Q} _ {\mathcal S} $ is the disjoint union of the decks $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$ with $ {\mathcal M} \in {\mathcal F} $. Then, for any
$Q \in {\mathcal Q} _ {\mathcal S} $ there exists a unique $ {\mathcal M} \in {\mathcal F} $ such that $Q \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$. Thus, $r$ is well defined.
Next notice that $r$ is a surjective map. For this purpose it is helpful to use the alternative expression of $r$. Suppose that
\[
g( {\mathcal M} )=E_ {\mathcal M} + P \ \ \ \text{ and } \ \ \ I-g( {\mathcal M} )=F+ P^{\#}.
\]
Note that $V=s(g( {\mathcal M} ))$ satisfies $V( {\mathcal M} _0)= {\mathcal M} $, $V(R(F_0))=R(F)$, $V( {\mathcal S} ^{\circ})= {\mathcal S} ^{\circ}$ and $V(R(P_0^{\#}))=R(P^{\#})$. From this latter fact, it is not difficult to see that $Ad_{V}( {\mathcal U} _{ {\mathcal M} , {\mathcal S} ^{\circ}})= {\mathcal U} _{ {\mathcal M} _0, {\mathcal S} ^{\circ}}$.
According to Lemma \ref{accion t grup chic deck} the group $ {\mathcal U} _{ {\mathcal M} _0, {\mathcal S} ^{\circ}}$ transitively acts on $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} _0}$, and consequently, $r$ turns out to be surjective.
Observe that there is a continuous inverse of the restriction of $r$ to $ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$, which is given by
$$
f:=(r|_{ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }})^{-1}: {\mathcal Q} _{ {\mathcal S} , {\mathcal M} _0} \to {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }, \, \, \, f(Q)=s(g( {\mathcal M} )) \, Q\, s(g( {\mathcal M} ))^{\#} .
$$
Since $s$ is a continuous map by Proposition \ref{sec global qs}, it follows that $f$ is continuous. To show that $f$ is actually the inverse of $r|_{ {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }}$, observe that if $Q\in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$,
\begin{align*}
(f\circ r)(Q) &= f(s(g( {\mathcal M} ))^{\#}\, Q \,s(g( {\mathcal M} ))) \\
&= s(g( {\mathcal M} )) s(g( {\mathcal M} ))^{\#}\, Q \,s(g( {\mathcal M} )) s(g( {\mathcal M} ))^{\#}=Q.
\end{align*}
Also, $f(Q) \in {\mathcal Q} _{ {\mathcal S} , {\mathcal M} }$ and
\begin{align*}
(r\circ f)(Q) &=r(s(g( {\mathcal M} )) \, Q\, s(g( {\mathcal M} ))^{\#})= \\
&= s(g( {\mathcal M} ))^{\#} s(g( {\mathcal M} )) \, Q\, s(g( {\mathcal M} ))^{\#} s(g( {\mathcal M} ))=Q.
\end{align*}
The map $g$ is continuous by Lemma \ref{cont sel}, meanwhile
the map $s$ is continuous by Proposition \ref{sec global qs}.
Thus, $r$ is clearly a continuous map. This completes the proof.
\end{proof}
\subsection*{Acknowledgement}
The authors are indebted with Prof. Aurelian Gheondea for some discussions on the material of this manuscript.
In addition, they wish to thank the referee for his/her careful reading of this work and helpful comments.
|
{
"timestamp": "2015-04-17T02:10:09",
"yymm": "1504",
"arxiv_id": "1504.04253",
"language": "en",
"url": "https://arxiv.org/abs/1504.04253"
}
|
\section{Introduction}
\subsection{Statement of the problem}
We consider $\Omega \subset {\Bbb R}^n$, $n\geq2$, a $\mathcal C^{1,1}$ bounded and connected domain such that ${\mathbb R}^n\setminus\Omega$ is also connected. We set $\Gamma=\partial\Omega$. Let $A\in W^{1,\infty}(\Omega, {\mathbb R}^n)$, $V\in L^\infty(\Omega,{\mathbb R})$ and consider the magnetic Schr\"odinger operator $H=(-{\rm i}\nabla+A)^2+V$ acting on $L^2(\Omega)$ with domain $D(H)=\{v\in H^1_0(\Omega):\ (-{\rm i}\nabla+A)^2v\in L^2(\Omega)\}$.
Let $A_j\in W^{1,\infty}(\Omega, {\mathbb R}^n)$, $V_j\in L^\infty(\Omega,{\mathbb R})$, $j=1,2$, and consider the magnetic Schr\"odinger operators $H_j=H$ for $A=A_j$ and $V=V_j$, $j=1,2$. We say that $H_1$ and $H_2$ are gauge equivalent if there exists $p\in W^{2,\infty}(\Omega,{\mathbb R})\cap H^1_0(\Omega)$ such that $H_2=e^{-{\rm i}p}H_1e^{{\rm i}p}$.
It is well known that $H$ is a selfadjoint operator. By the compactness of the embedding $H^1_0(\Omega) \hookrightarrow L^2(\Omega)$, the spectrum of $H$ is
purely discrete. We note $\{ \lambda_k:\ k\in {\mathbb N}^* \}$ the non-decreasing sequence of eigenvalues of $H$ and $\{ \varphi_k:\ k\in {\mathbb N}^* \}$ an associated Hilbertian basis of eigenfunctions. In the present paper we consider the Borg-Levinson inverse spectral problem of determining uniquely $H$, modulo gauge equivalence, from partial knowledge of the boundary spectral data $\{ (\lambda_k,{\partial_\nu\varphi_k}_{|\Gamma}):\ k\in {\mathbb N}^* \}$ with $\nu$ the outward unit normal vector to $\Gamma$. Namely, we prove that some asymptotic knowledge of $(\lambda_k,{\partial_\nu\varphi_k}_{|\Gamma})$ with respect to $k\in {\mathbb N}^* $ determines uniquely the operator $H$ modulo gauge transformation.
\subsection{ Borg-Levinson inverse spectral problems }
It is Ambartsumian who first investigated in 1929 the inverse spectral problem of determining the real potential $V$ appearing in the Schr\"odinger operator $H=-\Delta+V$, acting in $L^2(\Omega)$, from partial spectral data of $H$. For $\Omega=(0,1)$, he proved in \cite{A} that $V=0$ if the spectrum of the Neumann realization of $H$ equals $\{ k^2:\ k \in {\mathbb N} \}$. For the same operator, but endowed with homogeneous Dirichlet boundary conditions, Borg \cite{B} and Levinson \cite{L} established that the Dirichlet spectrum $\{ \lambda_k:\ k \in {\mathbb N}^*\}$ does not uniquely determine $V$. They showed that additional spectral data, namely $\{ \| \varphi_k \|_{L^2(0,1)}:\ k \in {\mathbb N}^* \}$, where $\{ \varphi_k:\ k \in {\mathbb N}^* \}$ is an $L^2(0,1)$-orthogonal basis of eigenfunctions of $H$ obeying the condition $\varphi_k'(0)=1$, is needed. Gel'fand and Levitan proved in \cite{GL} that uniqueness is still valid upon substituting the terminal velocity $\varphi_k'(1)$ for $\| \varphi_k \|_{L^2(0,1)}$ in the one-dimensional Borg and Levinson theorem.
In 1988, Nachman, Sylvester, Uhlmann \cite{NSU} and Novikov \cite{No} proposed a multidimensional formulation of the result of Borg and Levinson. Namely, they proved that the boundary spectral data $\{ (\lambda_k ,{\partial_\nu \varphi_k}_{\vert\partial\Omega}):\ k \in {\mathbb N}^* \}$, where $\nu$ denotes the outward unit normal vector to $\partial \Omega$ and $(\lambda_k, \varphi_k)$ is the $k^{\rm th}$ eigenpair of $-\Delta+V$, determines uniquely the Dirichlet realization of the operator $-\Delta+V$. The initial formulation of the multidimensional Borg-Levinson theorem by \cite{NSU} and \cite{No} has been improved in several ways by various authors. Isozaki \cite{I} (see also \cite{Ch}) extended the result of \cite{NSU} when finitely many eigenpairs remain unknown, and, recently, Choulli and Stefanov \cite{CS} claimed uniqueness in the determination of $V$ from the asymptotic behavior of $(\lambda_k,{\partial_\nu \varphi_k}_{\vert\Gamma})$ with respect to $k$.
Moreover, Canuto and Kavian \cite{ CK1, CK2} considered the determination of the conductivity $c$, the electric potential $V$ and the weight $\rho$ from the boundary spectral data of the
operator $\rho^{-1}(-div (c \nabla\cdot) + V)$ acting on the weighted space $L^2_\rho(\Omega)$ endowed with either Dirichlet or Neumann boundary conditions. Namely, \cite{ CK1, CK2} proved that the boundary spectral data of $\rho^{-1}(-div (c \nabla\cdot) + V)$ determines uniquely two of the three coefficients $c$, $V$ and $\rho$. The case of magnetic Schr\"odinger operator has been treated by \cite{Ser} who determined both the magnetic field $dA$ and the electric potential $V$ of the operator $H=(-{\rm i}\nabla+A)^2+V$. Here the 2-form $dA$ of a vector valued function $A=(a_1,\ldots,a_n)$ is defined by
\[dA=\sum_{i<j}(\partial_{x_j}a_i-\partial_{x_i}a_j)dx_j\wedge dx_i.\]
All the above mentioned results were obtained with $\Omega$ bounded and operators of purely discrete spectral type. In some recent work \cite{KKS} examined a Borg-Levinson inverse problem stated in an infinite cylindrical waveguide for Schr\"odinger operators with purely absolutely continuous spectral type. More precisely, \cite{KKS} proved that a real potential $V$ which is $2\pi$-periodic along the axis of the waveguide is uniquely determined
by some asymptotic knowledge of the boundary Floquet spectral data of the Schr\"odinger operator $-\Delta+V$ with Dirichlet boundary conditions.
Finally, let us mention for the sake of completeness that the stability issue in the context of Borg-Levinson inverse problems was examined in \cite{ BCY, BF, Ch, CS, KKS} and
that \cite{BK,BF, KK} established related results on Riemannian manifolds. We also precise that \cite{Mo,SU1,SU2} have proved stability estimates in the recovery of coefficients from the hyperbolic Dirichlet-to-Neumann map which is equivalent to the determination of general Schr\"odinger operators from boundary spectral data.
\subsection{Main result}
Let $A_j\in W^{1,\infty}(\Omega, {\mathbb R}^n)$, $V_j\in L^\infty(\Omega,{\mathbb R})$ and consider the magnetic Schr\"odinger operators $H_j=H$ for $A=A_j$ and $V=V_j$, $j=1,2$. Further we note $(\lambda_{j,k},\varphi_{j,k})$, $k\geq1$, the $k^{\rm th}$ eigenpair of $H_j$, for $j=1,2$. Our main result can be stated as follows.
\begin{theorem}
\label{thm-1}
We fix $\Omega_1$ an arbitrary open neighborhood of $\Gamma$ in $\Omega$ $($$\Gamma\subset\overline{\Omega_1}$ and $\Omega_1\subsetneq\Omega$$)$. For $j = 1, 2$, let $V_j \in L^\infty(\Omega,{\mathbb R})$ and let $A_j\in \mathcal C^1(\overline{\Omega},{\mathbb R}^n)$ fulfill
\begin{equation} \label{t1a}A_1(x)=A_2(x),\quad x\in\Omega_1.\end{equation}
Assume that the conditions
\begin{equation}\label{tt2a}
\lim_{k\to+\infty}\abs{\lambda_{1,k}-\lambda_{2,k}}=0,\quad \sum_{k=1}^{+\infty}\norm{\partial_\nu\varphi_{1,k}-\partial_\nu\varphi_{2,k} }_{L^2(\Gamma)}^2<\infty\end{equation}
hold simultaneously. Then, we have $dA_1=dA_2$ and $V_1 = V_2$.
\end{theorem}
Note that condition \eqref{t1a} corresponds to the knowledge of the magnetic potential on a neighborhood of the boundary.
Let us observe that, as mentioned by \cite{CS,KKS}, Theorem \ref{thm-1} can be considered as a uniqueness theorem under the assumption that
the spectral data are asymptotically "very close". Conditions \eqref{tt2a} are similar to the one considered by \cite{KKS} and they are weaker than the requirement that
\[\abs{\lambda_{1,k}-\lambda_{2,k}}\leq Ck^{-\alpha},\quad \norm{\partial_\nu\varphi_{1,k}-\partial_\nu\varphi_{2,k} }_{L^2(\Gamma)}\leq Ck^{-\beta}\]
for some $\alpha>1$ and $\beta>1-{1\over 2n}$, considered in \cite[Theorem 2.1]{CS}. Note also that conditions \eqref{tt2a} are weaker than the knowledge of the boundary spectral data with a finite number of data missing considered by \cite{I}.
Let us remark that there is an obstruction to uniqueness given by the gauge invariance of boundary spectral data for magnetic Shr\"odinger operators. More precisely, let $p\in \mathcal C^\infty_0(\Omega\setminus\overline{\Omega_1})\setminus\{0\}$ and assume that $A_1=\nabla p +A_2\neq A_2$, $V_1=V_2$. Then, we have $H_1=e^{-{\rm i}p}H_2e^{{\rm i}p}$ and one can check that we can choose the spectral data of $H_1$ and $H_2$ in such a way that the conditions
\[{\partial_\nu\varphi_{1,k}}_{|\Gamma}={\partial_\nu\varphi_{2,k}}_{|\Gamma},\quad \lambda_{1,k}=\lambda_{2,k},\quad k\in\mathbb{N}^*\]
are fulfilled. Therefore, conditions \eqref{t1a}-\eqref{tt2a} are fulfilled but $H_1\neq H_2$. Nevertheless, assuming \eqref{t1a} fulfilled, the conditions $dA_1=dA_2$ and $V_1 = V_2$ imply that $H_1$ and $H_2$ are gauge equivalent. Therefore, Theorem \ref{thm-1} is equivalent to the unique determination of magnetic Schr\"odinger operators modulo gauge transformation from the asymptotic knowledge of the boundary spectral data given by conditions \eqref{tt2a}.
We stress out that the problem under examination in this text is a Borg-Levinson inverse problem for the magnetic Schr\"odinger operator $H=(-{\rm i}\nabla+A)^2+V$. To our best knowledge, there are only two multi-dimensional Borg-Levinson uniqueness result for magnetic Schr\"odinger operators available in the mathematical literature, \cite[Theorem B]{KK} and \cite[Theorem 3.2]{Ser} (we refer also to \cite{Ni} for related inverse scattering results). In \cite{KK}, the authors considered general magnetic Schr\"odinger operators with smooth coefficients on a smooth connected Riemannian manifold and they proved unique determination of these operators modulo gauge invariance from the knowledge of the boundary spectral data with a missing finite number of data. In \cite{Ser}, Serov treated this problem on a bounded domain of ${\mathbb R}^n$, and he proved that, for $A\in W^{1,\infty}(\Omega,{\mathbb R}^n)$ and $V\in L^\infty(\Omega,{\mathbb R})$, the full boundary spectral data $\{ (\lambda_k ,{\partial_\nu \varphi_k}_{|\Gamma}):\ k \in {\mathbb N}^* \}$ determines uniquely $dA$ and $V$. In contrast to \cite{KK, Ser}, in the present paper we prove that the asymptotic knowledge of the boundary spectral data, given by the conditions \eqref{tt2a}, is sufficient for the unique determination of $dA$ and $V$. To our best knowledge, conditions \eqref{tt2a} are the weakest conditions on boundary spectral data that guaranty uniqueness of magnetic Schr\"odinger operators modulo gauge transformation. Moreover, our uniqueness result is stated with conditions similar to \cite[Theorem 1.4]{KKS}, which seems to be the most precise Borg-Levinson uniqueness result so far for Schr\"odinger operators without magnetic potential ($A=0$).
An important ingredient in our analysis is a suitable representation that allows to express the magnetic potential $A$ and the electric potential $V$ in terms of Dirichlet-to-Neumann map associated to the equations $(-{\rm i}\nabla +A)^2u+Vu-\lambda u=0$ for some $\lambda\in\mathbb C$. In \cite{I} Isozaki applied a similar approach to the Schr\"odinger operator $-\Delta+V$ with Dirichlet boundary condition\footnote{This argument was inspired by the Born approximation method of the scattering theory.} and \cite{CS,KKS} applied the representation formulas of \cite{I}. Inspired by the construction of complex geometric optics solutions of \cite{BC,FKSU,KLU,KU,NSU1,Sa1,Su} we prove that the approach of \cite{CS,I,KKS} can be extended to magnetic Schr\"odinger operators. More precisely, we derive two representation formulas that allow to recover both the magnetic field and the electric potential of magnetic Schr\"odinger operators which means recovery of both coefficients of order one and zero in contrast to \cite{CS,I,KKS} where only determination of coefficients of order zero is considered. This paper is the first where the extension of the approach developed by \cite{I} to more general coefficients than coefficients of order zero is considered. Note also that our approach make it possible to prove this extension without imposing important assumptions of regularity of the admissible coefficients.
We believe that the approach developed in the present paper can be used for results of stability in the determination of the magnetic field $dA$ and the electric potential $V$ similar to \cite[Theorem 1.3]{KKS}. Indeed, following the strategy set in this paper we expect a stability estimate associated to the the determination of the magnetic field $dA$. The main issue comes from the stability in the determination of the electric potential $V$. Nevertheless, we believe that this problem can be solved by adapting suitably the argument developed in \cite{T} related to the inversion of the $d$ operator on differential forms restricted to the right subspaces.
\subsection{Outline}
This paper is organized as follows. In Section 2 we consider some useful preliminary results concerning solutions of equations of the form $(-{\rm i}\nabla +A)^2u+Vu-\lambda u=0$ for some $\lambda\in\mathbb C\setminus\sigma(H)$. In Section 3 we introduce two representation formulas making the connection between the Dirichlet-Neumann map associated with the previous equations and the couple $(A,V)$ of magnetic and electric potential. Finally, in section 4 we combine all these results and we prove Theorem \ref{thm-1}.
\section{Notations and preliminary results}
\label{sec:Prelim}
In this section we introduce some notations and we give some properties of solution of the equation $(-{\rm i}\nabla +A)^2u+Vu-\lambda u=0$.
We denote by $\langle f,\psi\rangle$ the duality between $\psi \in H^{1/2}(\Gamma)$ and $f$ belonging to the dual $H^{-1/2}(\Gamma)$ of $H^{1/2}(\Gamma)$. However, when in $\langle f,\psi \rangle $ both $f$ and $\psi$ belong to $L^2(\Gamma)$, to make things simpler $\langle \cdot,\cdot\rangle$ can be interpreted as the scalar product of $L^2(\Gamma)$, namely
$$\langle f,\psi\rangle = \int_{\Gamma} f(x)\, \overline{\psi(x)}\,d\sigma(x).$$
\medskip
We introduce the operator $H$ defined as
\begin{equation}\label{eq:Def-A-theta} H u := (-{\rm i}\nabla+A)^2 u + V u,\quad u\in D(H) := \left\{\psi \in H^1_0(\Omega) \; ; \;(-{\rm i}\nabla+A)^2\psi \in L^2(\Omega)\right\}.\end{equation}
Recall that $H$ is associated to the quadratic form $b$ given by
$$b(u,v) = \int_{\Omega}(-{\rm i}\nabla+A) u(x)\cdot\overline{(-{\rm i}\nabla+A) v(x)}\,dx + \int_{\Omega}V(x)\,u(x)\overline{v(x)}\,dx,\quad u,v\in H^1_0(\Omega).$$
Moreover, the spectrum of $H$ is discrete and composed of the non-decreasing sequence of eigenvalues denoted by $\sigma (H) = \{\lambda_{k}\; ; \; k \geq 1\}$. If we write $V = V^+ - V^-$, with $V^\pm := \max(0,\pm V)$, we have that the spectrum $\sigma(H)$ of $H$ is contained into
$ [-\|V^-\|_{L^\infty(\Omega)}, + \infty)$.
According to \cite[Theorem 2.2.2.3]{Gr}, we can show that $D(H)$ embedded continuously into $H^2(\Omega)$. Therefore the eigenfunctions $(\varphi_{k})_{k\geq1}$ of $H$, that form an Hilbertian basis, are lying in $H^2(\Omega)$ and we have ${\partial_\nu \varphi_k}_{|\Gamma}\in H^{1/2}(\Gamma)$.
From now on, we fix $f \in H^{1/2}(\Gamma)$ and $\lambda \in {\Bbb C} \setminus \sigma (H)$ and we consider the problem
\begin{equation} \label{eq1}
\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A)^2 u + Vu -\lambda u & = & 0, & \mbox{in}\ \Omega ,\\
u(x) & = & f(x),& x\in \Gamma.
\end{array}\right.
\end{equation}
We start with two results related to the asymptotic behavior of solutions of \eqref{eq1} as $\lambda\to-\infty$.
\begin{lemma}\label{lem:Resolution}
For any $f \in H^{1/2}(\Gamma)$ and $\lambda \in {\Bbb C} \setminus \sigma (H)$, there exists a unique solution $u \in H^1(\Omega)$ to \eqref{eq1}
which can be written as
\begin{equation}\label{eq:Sol-Series}
u_{\lambda} := u = \sum_{k \geq 1} {\alpha_{k} \over \lambda - \lambda_{k}}\, \varphi_{k},
\end{equation}
where for convenience we set
\begin{equation}\label{eq:Def-psi-alpha}
h_{k} := {\partial_\nu \varphi_{k}}_{|\Gamma} , \qquad
\mbox{and}\qquad
\alpha_{k} :=\langle f,h_k \rangle.
\end{equation}
Moreover, we have
$$\|u_{\lambda}\|^2_{L^2(\Omega)} = \sum_{k \geq 1} {|\alpha_{k}|^2 \over |\lambda - \lambda_{k}|^2} \to 0 \qquad\mbox{as }\, \lambda \to -\infty.$$
\end{lemma}
\begin{proof}
Since $\lambda\notin\sigma(H)$, one can easily check that \eqref{eq1} admits a unique solution $u_\lambda \in H^1(\Omega)$. Moreover, $u_\lambda$ can be written in terms of the eigenvalues and eigenfunctions $\lambda_{k},\varphi_{k}$. Indeed, $u_\lambda \in L^2(\Omega)$ can be expressed in the Hilbert basis $(\varphi_{k})_{k\geq 1}$ as
$$u_\lambda = \sum_{k\geq 1} (u_\lambda|\varphi_{k})\, \varphi_{k}\, $$
with $(\cdot,\cdot)$ the scalar product with respect to $L^2(\Omega)$. Since $u_\lambda\in H^1(\Omega)$ and $\Delta u_\lambda=-2iA\cdot\nabla u_\lambda+(-{\rm i}\ div(A) +|A|^2+V)u_\lambda\in L^2(\Omega)$, we have $\nabla u_\lambda\in H_{div}(\Omega)=\{v\in L^2(\Omega;\mathbb C^n):\ div(v)\in L^2(\Omega)\}$.
Thus, taking the scalar product of the first equation in \eqref{eq1} with $\varphi_{k}$ and applying the Green formula we obtain
$$\left\langle f,h_k\right\rangle = (\lambda - \lambda_{k})\,(u|\varphi_{k}),$$
which yields the expression given by \eqref{eq:Sol-Series}.
The fact that $\|u_{\lambda}\| \to 0$ as $\lambda \to -\infty$ is a consequence of the fact that we may fix $c_{0} > \norm{V}_{L^\infty(\Omega)}$ large enough so that if $\lambda$ is real and such that $\lambda \leq - c_{0}$, we have $|\lambda - \lambda_{k}|^2 \geq |c_0-\lambda_{k}|^2$ for all $k \geq 1$, and thus
$${|\alpha_{k}|^2 \over |\lambda - \lambda_{k}|^2} \leq
{|\alpha_{k}|^2 \over |c_0-\lambda_{k}|^2 } \, ,$$
so that we may apply Lebesgue's dominated convergence as $\lambda \to -\infty$.\end{proof}
\begin{lemma}\label{lem:Resolution:grad}
For all $\lambda<-\norm{V}_{L^\infty(\Omega)}-6\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2$, the solution $u_\lambda$ of \eqref{eq1} satisfies
\begin{equation} \label{lol1} \norm{\nabla u_\lambda}_{L^2(\Omega\setminus \Omega_1)}\leq C\norm{u_\lambda}_{L^2(\Omega_1)}\end{equation}
with $C$ depending only on $\Omega$ and $\Omega_1$.
\end{lemma}
\begin{proof}
Let us denote by $\chi\in\mathcal C^\infty_0(\Omega,{\mathbb R})$ a function satisfying $\chi=1$ on $\Omega\setminus \Omega_1$. Then, since $\nabla u_\lambda\in H_{div}(\Omega)$, multiplying \eqref{eq1} by $\chi^2\overline{u_\lambda}$ and applying the Green formula we obtain
\begin{equation} \label{in}\begin{aligned}0&=\int_\Omega (-{\rm i}\nabla +A)^2 u_\lambda \chi^2\overline{u_\lambda}dx+\int_\Omega(V-\lambda)\chi^2\abs{u_\lambda}^2dx\\
\ &= \int_\Omega |\chi\nabla u_\lambda|^2dx+2\int_\Omega (\chi\nabla u_\lambda)\cdot \overline{\nabla\chi u_\lambda}dx\\
\ &\ \ +\int_\Omega [2{\rm i}(u_\lambda \chi)A\cdot \overline{u_\lambda\nabla\chi }+{\rm i}\chi u_\lambda A\cdot \overline{\chi\nabla u_\lambda} +\chi\nabla u_\lambda\cdot \overline{{\rm i}A\chi u_\lambda}]dx+\int_\Omega(|A|^2+V-\lambda)\chi^2\abs{u_\lambda}^2dx.\end{aligned}\end{equation}
Applying the Cauchy-Schwarz inequality we find
\[\begin{array}{l} \norm{\chi\nabla u_\lambda}_{L^2(\Omega)}^2+(-\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2-\norm{V}_{L^\infty(\Omega)}-\lambda)\norm{\chi u_\lambda}^2_{L^2(\Omega)}\\
\ \\
\leq 2\norm{u_\lambda\nabla \chi }_{L^2(\Omega)}\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}+2\norm{A}_{L^\infty(\Omega)}\norm{\chi u_\lambda}_{L^2(\Omega)}\norm{u_\lambda\nabla \chi }_{L^2(\Omega)}+2\norm{A}_{L^\infty(\Omega)}\norm{\chi u_\lambda}_{L^2(\Omega)}\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}\\
\ \\
\leq 4\norm{u_\lambda\nabla \chi }^2_{L^2(\Omega)}+{\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}^2\over 4} +\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2\norm{\chi u_\lambda}_{L^2(\Omega)}^2+\norm{u_\lambda\nabla \chi }_{L^2(\Omega)}^2+4\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2\norm{\chi u_\lambda}_{L^2(\Omega)}^2\\
\ \\
\ \ \ +{\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}^2\over 4}.\end{array}\]
From this estimate, we deduce
\[{\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}^2\over 2}+(-\norm{V}_{L^\infty(\Omega)}-6\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2-\lambda)\norm{\chi u_\lambda}^2_{L^2(\Omega)}\leq 5\norm{u_\lambda\nabla\chi }^2_{L^2(\Omega)}.\]
Using the fact that $\lambda<-\norm{V}_{L^\infty(\Omega)}-6\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2$, we obtain
\[\norm{\nabla u_\lambda}_{L^2(\Omega\setminus \Omega_1)}^2\leq\norm{\chi\nabla u_\lambda}_{L^2(\Omega)}^2\leq 10\norm{u_\lambda\nabla\chi }^2_{L^2(\Omega)}\leq 10\norm{\nabla\chi}_{L^\infty(\Omega)}^2\norm{ u_\lambda}^2_{L^2( \Omega_1)}\leq C\norm{ u_\lambda}^2_{L^2( \Omega_1)}.\]
From this estimate we deduce \eqref{lol1}.
\end{proof}
It is clear that the series \eqref{eq:Sol-Series} giving $u_{\lambda}$ in terms of $\alpha_{k},\lambda_{k}$ and $\varphi_{k}$, converges only in $L^2(\Omega)$ and thus we cannot deduce an expression of the normal derivative $\partial_\nu u_{\lambda}$ in terms of $\alpha_{k}, \lambda_{k}$ and $h_{k}$. To avoid this difficulty, in a similar way to \cite{KKS}, we have the following lemma:
\begin{lemma}\label{lem:v-lambda-mu}
Let $f \in H^{1/2}(\Gamma)$ be fixed and for $\lambda,\mu \in {\Bbb C} \setminus \sigma (H)$ let $u_{\lambda}$ and $u_{\mu}$ be the solutions given by Lemma \ref{lem:Resolution}. If we set $v := v_{\lambda,\mu} := u_{\lambda} - u_{\mu}$, then
\begin{equation}\label{eq:v-Normal-Deriv}
\partial_\nu v = \sum_{k\geq1}
{(\mu - \lambda)\alpha_{k} \over (\lambda - \lambda_{k})(\mu - \lambda_{k})}\, h_{k}\, ,
\end{equation}
the convergence taking place in $H^{1/2}(\Gamma)$.
\end{lemma}
\begin{proof} Let $v_{\lambda,\mu} := u_{\lambda} - u_{\mu}$; One verifies that $v_{\lambda,\mu}$ solves
\begin{equation}\label{eq:v-lambda}
\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A)^2 v_{\lambda,\mu} + V v_{\lambda,\mu} - \lambda v_{\lambda,\mu} & = & (\lambda - \mu)u_{\mu}, & \mbox{in}\ \Omega ,\\
v_{\lambda,\mu}(x) & = & 0,& x\in \Gamma.
\end{array}\right.
\end{equation}
Since $(u_{\mu}|\varphi_{k}) = \alpha_{k}/(\mu - \lambda_{k})$, it follows that
$$v_{\lambda,\mu} = \sum_{k\geq1}{(\lambda - \mu) \alpha_{k}\over (\lambda_{k} - \lambda)(\mu - \lambda_{k}) }\, \varphi_{k},$$
the convergence taking place in $D(H)$. Since the operator $v \mapsto \partial_\nu v$ is continuous from $D(H)$ into $H^{1/2}(\Gamma)$, the result of the lemma follows.
\end{proof}
The next lemma states essentially that if for $j=1$ or $j=2$ we have two magnetic potentials $A_j$, two electric potentials $V_j$ and $u_j := u_{j,\mu}$ solutions of
\begin{equation}\label{eq:u-m}
\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_j)^2 u_j + V_ju_j -\mu u_j & = & 0, & \mbox{in}\ \Omega ,\\
u_j(x) & = & f(x),& x\in \Gamma,
\end{array}\right.
\end{equation}
then $u_{1,\mu}$ and $u_{2,\mu}$ are {\it close\/} as $\mu \to -\infty$: in some sense the influence of the potentials $A_j$ and $V_{j}$ are dimmed when $\mu \to -\infty$. More precisely we have:
\begin{lemma}\label{lem:z-mu}
Let $V_{j} \in L^\infty(\Omega,{\Bbb R})$ and $A_j\in W^{1,\infty}(\Omega,{\mathbb R}^n)$ be given for $j=1$ or $j=2$, and denote by $H_j$ the corresponding operator defined by \eqref{eq:Def-A-theta}. We assume that condition \eqref{t1a} is fulfilled.
For $f \in H^{1/2}(\Gamma)$ and $\mu\in(-\infty, \mu_*)\subset {\Bbb C} \setminus \sigma (H) $, let $u_{j,\mu} := u_{j}$ be the solution of \eqref{eq:u-m}. Then $z_{\mu} := u_{1,\mu} - u_{2,\mu}\in H^2(\Omega)$ converge to $0$ in $H^{2}(\Omega)$ as $\mu\to-\infty$. In particular $\partial_\nu z_{\mu} \to 0$ in $L^2(\Gamma)$ as $\mu \to -\infty$.
\end{lemma}
\begin{proof}
Since the trace map $v\mapsto \partial_\nu v$ is continuous from $H^{2}(\Omega)$ to $L^2(\Gamma)$, it is enough to show that $z_\mu\in H^2(\Omega)$ and $\|z_{\mu}\|_{H^{2}(\Omega)} \to 0$ when $\mu \to -\infty$. We fix $\mu<\mu_*$ with $\mu_*<-\norm{V}_{L^\infty(\Omega)}-6\norm{A}_{L^\infty(\Omega,{\mathbb R}^n)}^2$ less than the constants given by Lemma \ref{lem:Resolution} for $A=A_j$, $V=V_j$, $j=1,2$. Without lost of generality we assume that $H_j-\mu_*$ is positive, $j=1,2$.
One verifies that $z_{\mu}$ solves the equation
\begin{equation}\label{eq:z-mu}
\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_1)^2 z_{\mu} + V_1z_{\mu} - \mu z_{\mu} & = & -2{\rm i}(A_2-A_1)\cdot\nabla u_{2,\mu}+(p_2-p_1)u_{2,\mu}, & \mbox{in}\ \Omega,\\
z_{\mu}(x) & = & 0,& x\in \Gamma
\end{array}\right.
\end{equation}
with $p_j=-{\rm i}div(A_j)+\abs{A_j}^2+V_j$, $j=1,2$.
That is, denoting by $R_{1,\mu} = (H_1 - \mu I)^{-1}$ the resolvent of the operator $H_1:= (-{\rm i}\nabla+A_1)^2 + V_1$, we have $$z_{\mu} = R_{1,\mu}(-2{\rm i}(A_2-A_1)\cdot\nabla u_{2,\mu}+(p_2-p_1)u_{2,\mu})=\sum_{k=1}^{+\infty}{(w_\mu,\varphi_{1,k})\over (\lambda_{1,k}-\mu)}\varphi_{1,k}$$
with $w_\mu=-2{\rm i}(A_2-A_1)\cdot\nabla u_{2,\mu}+(p_2-p_1)u_{2,\mu}$ and $(\lambda_{1,k})_{k\geq1}$, $(\varphi_{1,k})_{k\geq1}$ respectively the eigenvalues of $H_1$ and an Hilbertian basis of eigenfunctions associated to these eigenvalues. Since $w_\mu\in L^2(\Omega)$, $z_\mu$ is lying in $D(H_1)$ and by the same way in $H^2(\Omega)$. It remains to show that $\|z_{\mu}\|_{H^{2}(\Omega)} \to 0$ when $\mu \to -\infty$. Since $D(H_1)$ embedded continuously into $H^2(\Omega)$ there exists a generic constant $C$ depending on $A_1$, $V_1$ and $\Omega$ such that
\[\norm{z_\mu}_{H^2(\Omega)}^2\leq C\sum_{k=1}^{\infty}\abs{\lambda_{1,k}-\mu_*}^2\abs{(z_\mu,\varphi_{1,k})}^2\leq C\norm{w_\mu}_{L^2(\Omega)}.\]
On the other hand, condition \eqref{t1a} implies
\[\norm{w_\mu}_{L^2(\Omega)}\leq C(\norm{\nabla u_{2,\mu}}_{L^2(\Omega\setminus\Omega_1)}+\norm{ u_{2,\mu}}_{L^2(\Omega)})\]
with $C$ independent of $\mu$. Then, according to Lemma \ref{lem:Resolution} and \eqref{lol1}, we obtain
\[\limsup_{\mu\to-\infty}\norm{w_\mu}_{L^2(\Omega)}\leq C\limsup_{\mu\to-\infty}\norm{ u_{2,\mu}}_{L^2(\Omega)}=0.\]
Thus, we have
$$\limsup_{\mu\to-\infty}\norm{z_\mu}_{L^2(\Gamma)}\leq C \limsup_{\mu\to-\infty}\|z_{\mu}\|_{H^{2}(\Omega)}\leq C\limsup_{\mu\to-\infty}\norm{w_\mu}_{L^2(\Omega)}=0.$$
This completes the proof.
\end{proof}
Armed with these results, we will prove Theorem \ref{thm-1} by using some asymptotic properties of solutions of \eqref{eq1} with respect to $\lambda$. For this purpose, like in \cite{CS,I,KKS} we use representation formulas that will allow us to make a connection between the boundary spectral data and the potentials $A$ and $V$.
\section{Representation formulas}
From now on, for all $x=(x_1,\ldots,x_n)\in{\mathbb C}^n$ and $y=(y_1,\ldots,y_n)\in{\mathbb C}^n$, we denote by $x\cdot y$ the quantity
\[x\cdot y=\sum_{k=1}^nx_ky_k\]
and for all $x\in{\mathbb R}^n$ we denote by $x^\bot$ the subspace of ${\mathbb R}^n$ defined by $\{y\in{\mathbb R}^n:\ y\cdot x=0\}$. Moreover, we set $A_j\in\mathcal C^1(\overline{\Omega},{\mathbb R}^n)$, $V_j\in L^\infty(\Omega,{\mathbb R})$, $j=1,2$, and we assume that condition \eqref{t1a} is fulfilled. For $j=1,2$ and $\lambda\in \mathbb C\setminus {\mathbb R}$, we associate to the problem
\begin{equation} \label{eq:lambda}
\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_j)^2 u_j + V_ju_j -\lambda u_j & = & 0, & \mbox{in}\ \Omega ,\\
u_j(x) & = & f(x),& x\in \Gamma
\end{array}\right.\end{equation}
the Dirichlet-to-Neumann map
\[\Lambda_{j,\lambda}:H^{{1\over 2}}(\partial\Omega)\ni f\mapsto{(\partial_\nu +{\rm i}A_j\cdot\nu)u_{j,\lambda}}_{|\Gamma},\]
where $u_{j,\lambda}$ solves \eqref{eq:lambda}.
The goal of this section is to apply the Dirichlet-to-Neumann maps $\Lambda_{j,\lambda}$ to some suitable ansatzs associated with \eqref{eq:lambda} in order to get two representation formulas involving the magnetic potentials $A_j$ and the electric potentials $V_j$, $j=1,2$. A similar approach was developed by \cite{I} and \cite{CS,KKS} used the representation of \cite{I}. The idea is to establish the link between the electric and magnetic potentials and the boundary spectral data by mean of an expression involving the Dirichlet-to-Neumann maps $\Lambda_{1,\lambda}$, $\Lambda_{2,\lambda}$.
We start with two general representation formulas, stated in the next subsection, where some properties of the ansatzs will not be completely specified. This will allow us to clarify the main goal of these formulas. Then, in Subsection 3.2 we will introduce the remaining properties of our ansatzs and establish some asymptotic properties from our representations which will be one of the main points of our analysis.
\subsection{General representation formulas}
In this subsection we introduce the first formulation of two representation formulas involving respectively the Dirichlet-to-Neumann maps $\Lambda_{1,\lambda}$, $\Lambda_{2,\lambda}$ and some ansatzs associated with problem \eqref{eq:lambda}. In \cite{I}, Isozaki considered such formulas for Schr\"odinger operators $-\Delta+V$ with an electric potential $V$, in other words for Schr\"odinger operators with a variable coefficient of order zero. In our case we need to extend this strategy to Schr\"odinger operators with
both magnetic and electric potentials, which means an extension to Schr\"odinger operators with variable coefficients of order zero and one. In addition, we need to consider ansatzs that allow to recover both the magnetic field and the electric potential. Therefore, we consider some ansatzs, associated with \eqref{eq:lambda}, of the form
\begin{equation} \label{anz}\Phi_{j,\lambda}(x)=e^{\zeta_j\cdot x}g_j(x),\quad \zeta_j\in\mathbb C^n,\ x\in\Omega,\ j=1,2\end{equation}
with $\zeta_j$ satisfying $\zeta_j\cdot\zeta_j=-\lambda$ and with $g_1$ and $g_2$ respectively a solution of
\begin{equation} \label{eqeq}\zeta_1\cdot\nabla g_1+({\rm i}\zeta_1\cdot A_{1,\sharp})g_1=0,\quad \zeta_2\cdot\nabla g_2-({\rm i}\zeta_2\cdot A_{2,\sharp})g_2=0\end{equation}
with $A_{j,\sharp}$ some smooth function close to the magnetic potential $A_j$, $j=1,2$.
More precisely, we fix $\eta_1,\eta_2 \in \mathbb S^{n-1}=\{y \in {\mathbb R}^n,\ \abs{y} = 1 \}$ and we define $A_{j,\sharp}\in \mathcal C_0^\infty({\mathbb R}^n,{\mathbb R}^n)$, $j=1,2$, some smooth approximations on $\overline{\Omega}$ of $A_j$. Then, we set
$\zeta_1={\rm i}\sqrt{\lambda}\eta_1$, $\zeta_2=-{\rm i}\sqrt{\lambda}\eta_2$ and we consider solutions of the transport equations \eqref{eqeq} given by
$$g_1(x):=e^{{\rm i}\psi_1(x)},\quad g_2(x):=b_2(x)e^{-{\rm i}\psi_2(x)},\quad \psi_j(x):=-\int_{-\infty}^0 \eta_j\cdot A_{j,\sharp}(x+s\eta_j)ds,\quad \eta_2\cdot\nabla b_2(x)=0,\quad x\in{\mathbb R}^n.$$
Therefore, we consider ansatzs associated with \eqref{eq:lambda} taking the form
\begin{equation} \label{jt1}
\Phi_{1,\lambda}(x):=e^{{\rm i} \sqrt{\lambda} \eta_1 \cdot x}e^{{\rm i}\psi_1(x)},\quad \Phi_{2,\lambda}(x):=e^{-{\rm i} \sqrt{\lambda} \eta_2 \cdot x}b_2(x)e^{-{\rm i}\psi_2(x)},\quad x\in\Omega.
\end{equation}
We assume in addition that $b_2\in W^{2,\infty}({\mathbb R}^n)$ and we recall that $\psi_j$ solves the equation
\[\eta_j\cdot\nabla\psi_j(x)=-\eta_j\cdot A_{j,\sharp}, \ \ j=1,2,\quad x\in{\mathbb R}^n.\]
For the time being, we consider general ansatzs of the form \eqref{jt1} with the properties describe above. Additional information about the parameter $\lambda$, the function $A_{j,\sharp}$, the vector $\eta_j$, $j=1,2$, and the function $b_2$ will be given in Subsection 3.2. In a similar way to \cite{FKSU,KLU,KU,NSU1,Sa1,Su}, in the construction of our ansatzs we consider some smooth approximations of the magnetic potentials instead of the magnetic potentials to obtain sufficiently smooth functions $\Phi_{j,\lambda}$, $j=1,2$. Using this approach, we can weaken the regularity assumption imposed on admissible magnetic potential from $W^{3,\infty}(\Omega)$ to $\mathcal C^1(\overline{\Omega})$.
Further, for $j=1,2$, we put
\begin{equation} \label{kt1}
S_j(\lambda,\eta_1,\eta_2)=\left \langle \Lambda_{j,\lambda} \Phi_{1,\lambda} , \overline{\Phi_{2,\lambda}}\right\rangle=\int_\Gamma(\Lambda_{j,\lambda} \Phi_{1,\lambda})\Phi_{2,\lambda}(x) d\sigma(x).
\end{equation}
In other words, we apply $\Lambda_{j,\lambda}$, $j=1,2$, to ansatzs of the form \eqref{anz} with $\zeta_1={\rm i} \sqrt{\lambda} \eta_1 $, $\zeta_2=-{\rm i} \sqrt{\lambda}\eta_2$, $g_1=e^{{\rm i} \psi_1}$ and $g_2=b_2e^{-{\rm i} \psi_2}$. We recall that quantities similar to $S_1$ and $S_2$ have also been used by \cite{FKSU,I,KKS,KLU,KU,NSU1,Sa1,Su}. Let us also mention that, like in \cite{I,KKS}, the ansatzs \eqref{jt1} do not depend on the potential $V_1$ and $V_2$ which are coefficients of order zero of the equation \eqref{eq:lambda}. On the other hand, the ansatzs \eqref{jt1} depend on the magnetic potentials $A_1$ and $A_2$ which are coefficients of order one of the equation \eqref{eq:lambda}. By modifying the construction of
\cite{I,KKS} with the new expression $g_j$, $j=1,2$, we will extend the approach of \cite{I,KKS} to Shr\"odinger operators with magnetic potentials.
From now on, for the sake of simplicity we will systematically omit the subscripts $\lambda$ in $\Phi_{j,\lambda}$, $j=1,2$, in the remaining of this text. In view of determining the behavior of $S_1-S_2$, as $\mathfrak I\lambda\to+\infty$, we introduce the following representations associated with $S_1$ and $S_2$.
\begin{proposition}
\label{l1}
For all $\lambda \in {\mathbb C}\setminus{\mathbb R}$ and $\eta_j \in\mathbb S^{n-1}$, $j=1,2$, the scalar products $S_j(\lambda,\eta_1,\eta_2)$ have the following expression
\begin{eqnarray}\label{l1a}
&&S_1(\lambda,\eta_1,\eta_2)\cr
&&= 2\sqrt{\lambda}\int_\Omega \eta_2\cdot (A_1-A_{2,\sharp})e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\cr
&&\ +\int_\Omega (V_1-q_{12})e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\cr
&&\ -{\rm i} \int_{\Gamma} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(b_2\sqrt{\lambda} \eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_1)\cdot \nu d\sigma(x)\cr
&&\ -\int_\Omega \left[(H_1-\lambda)^{-1}\left(2\sqrt{\lambda}\eta_1\cdot(A_1-A_{1,\sharp})+q_{11}\right)\Phi_1\right]\left(2\sqrt{\lambda}\eta_2\cdot (A_1-A_{2,\sharp})b_2+V_1b_2-q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx,\cr &&\end{eqnarray}
\begin{eqnarray}\label{l1b}
&&S_2(\lambda,\eta_1,\eta_2) \cr
&&=\int_\Omega \left[2\sqrt{\lambda}\eta_2\cdot (A_2-A_{2,\sharp})+V_2-q_{22}\right]e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\cr
&&\ -{\rm i} \int_{\partial\Omega} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(b_2\sqrt{\lambda} \eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_2)\cdot \nu d\sigma(x)\cr
&&\ -\int_\Omega \left[(H_2-\lambda)^{-1}\left(2\sqrt{ \lambda}\eta_1\cdot(A_2-A_{1,\sharp})+q_{21}\right)\Phi_1\right](2\sqrt{\lambda}\eta_2\cdot (A_2-A_{2,\sharp})b_2+V_2b_2-q_{22})e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx.\cr &&\end{eqnarray}
Here we denote by $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$ the expressions
\[q_{11}=-{\rm i}div (A_1)+\abs{A_1}^2+V_1(x)+2 A_1\cdot\nabla\psi_1-{\rm i}\Delta \psi_1+\abs{\nabla\psi_1}^2,\]
\[q_{12}=\Delta b_2-2{\rm i}\nabla\psi_2\cdot\nabla b_2-2{\rm i}\nabla b_2\cdot A_1+\left(-{\rm i}\Delta\psi_2-\abs{\nabla\psi_2}^2-2\nabla\psi_2\cdot A_1-{\rm i}div(A_1)-\abs{A_1}^2\right)b_2,\]
\[q_{21}=-{\rm i}div (A_2)+\abs{A_2}^2+V_2(x)+2 A_2\cdot\nabla\psi_1-{\rm i}\Delta \psi_1+\abs{\nabla\psi_1}^2,\]
\[q_{22}=\Delta b_2-2{\rm i}\nabla\psi_2\cdot\nabla b_2-2{\rm i}\nabla b_2\cdot A_2+\left(-{\rm i}\Delta\psi_2-\abs{\nabla\psi_2}^2-2\nabla\psi_2\cdot A_2-{\rm i}div(A_2)-\abs{A_2}^2\right)b_2.\]
Moreover, $H_j$, $j=1,2,$ denotes the selfadjoint operator $(-{\rm i}\nabla+A_j) + V_j$ acting on $L^2(\Omega)$ with domain
\[D(H_j)=\{v\in H^1_0(\Omega):\ (-{\rm i}\nabla+A_j) v\in L^2(\Omega)\}.\]
\end{proposition}
Note that formulas \eqref{l1a}-\eqref{l1b} contain expressions involving the magnetic potentials $A_1$, $A_2$ and the electric potentials $V_1$, $V_2$, expressions on the boundary $\partial\Omega$ and expressions described by the resolvent $(H_j-\lambda)^{-1}$, $j=1,2$. Using condition \eqref{t1a} one can check that the expressions on $\partial\Omega$ of $S_1$ and $S_2$ coincide and applying the decay of the resolvent $(H_j-\lambda)^{-1}$, $j=1,2$, as $\mathfrak I \lambda\to+\infty$ we will show in the next subsection that, for some suitable choice of our ansatzs, the expressions
$$-\int_\Omega \left[(H_1-\lambda)^{-1}\left(2\sqrt{\lambda}\eta_1\cdot(A_1-A_{1,\sharp})+q_{11}\right)\Phi_1\right]\left(2\sqrt{\lambda}\eta_2\cdot (A_1-A_{2,\sharp})b_2+V_1b_2-q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx,$$
$$-\int_\Omega \left[(H_2-\lambda)^{-1}\left(2\sqrt{ \lambda}\eta_1\cdot(A_2-A_{1,\sharp})+q_{21}\right)\Phi_1\right](2\sqrt{\lambda}\eta_2\cdot (A_2-A_{2,\sharp})b_2+V_2b_2-q_{22})e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx,$$
vanish as $\mathfrak I \lambda\to+\infty$. Thus, what will remain in the asymptotic expansion of $S_1-S_2$, as $\mathfrak I\lambda\to+\infty$, will be two expressions involving $A_1-A_2$ and $V_1-V_2$. These two expressions, that will be given in the next subsection, are one of the main ingredients in our proof. The remaining of this subsection will be devoted to the proof of Proposition \ref{l1}.\\
\textbf{Proof of Proposition \ref{l1}.} Let us first remark that the expressions \eqref{l1a}-\eqref{l1b} correspond to some asymptotic expansion of the expression $S_j$, $j=1,2$, with respect to $\sqrt{\lambda}$ \footnote{This statement will be clarified in the next subsection where we will give additional information about the parameter $\lambda$ and the vectors $\eta_1$, $\eta_2$.}. We will prove \eqref{l1a}-\eqref{l1b} by combining properties of the ansatzs \eqref{anz}, with properties of solutions of \eqref{eq:lambda} when $f=\Phi_1$. This proof will be divided into two steps, first for $S_1$ then for $S_2$. We start by showing that for $j=1,2$ and $f=\Phi_1$ problem \eqref{eq:lambda} admits a unique solution $u_j\in H^2(\Omega)$ taking the form
\begin{equation} \label{l1d}
u_1=\Phi_1-(H_1-\lambda)^{-1}\left[2\sqrt{\lambda}\eta_1\cdot (A_1-A_{1,\sharp})+q_{11}\right]\Phi_1,
\end{equation}
\begin{equation} \label{l1i}
u_2=\Phi_1-(H_2-\lambda)^{-1}\left[2\sqrt{ \lambda}\eta_1\cdot(A_2-A_{1,\sharp})+q_{21}\right]\Phi_1.
\end{equation}
Then, combining these formulas with the properties of the ansatzs \eqref{anz} and applying the Green formula, we derive \eqref{l1a}-\eqref{l1b}.
We start with the expression of $S_1(\lambda,\eta_1,\eta_2)$. Let us first prove \eqref{l1d}. Recall that
\[(-{\rm i}\nabla+A_1)^2 u+V_1u-\lambda u=-\Delta u -2{\rm i} A_1\cdot \nabla u+qu -\lambda u\]
with $q(x)=-{\rm i}div (A_1)(x)+\abs{A_1(x)}^2+V_1(x)$. Therefore, in light of \eqref{jt1} we have
\[\begin{array}{l}(-{\rm i}\nabla+A_1)^2 \Phi_1+V_1\Phi_1-\lambda \Phi_1\\
=(\lambda+2\sqrt{\lambda}\eta_1\cdot\nabla\psi_1-{\rm i}\Delta \psi_1+\abs{\nabla\psi_1}^2)\Phi_1+(2\sqrt{\lambda} \eta_1\cdot A_1+2 A_1\cdot\nabla\psi_1)\Phi_1+q\Phi_1-\lambda\Phi_1\\
=2\sqrt{\lambda}(\eta_1\cdot\nabla\psi_1+\eta_1\cdot A_1)\Phi_1+ q_{11}\Phi_1\end{array}\]
with $q_{11}=q+2 A_1\cdot\nabla\psi_1-{\rm i}\Delta \psi_1+\abs{\nabla\psi_1}^2$. On the other hand, since $\psi_1$ satisfies $\eta_1\cdot\nabla\psi_1+\eta_1\cdot A_{1,\sharp}=0$, we deduce that
\begin{equation} \label{l1c}
(-{\rm i}\nabla+A_1)^2 \Phi_1+V_1\Phi_1-\lambda \Phi_1=\left[2\sqrt{\lambda}\eta_1\cdot (A_1-A_{1,\sharp})+q_{11}\right]\Phi_1.
\end{equation}
Now consider $u_1$ the solution of
\[\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_1)^2 u_1 + V_1u_1 -\lambda u_1 & = & 0, & \mbox{in}\ \Omega ,\\
u_1(x) & = & \Phi_1(x),& x\in \partial\Omega.
\end{array}\right.\]
Note that, with our assumptions one can check that $D(H_1)=H^1_0(\Omega)\cap H^2(\Omega)$. In view of \eqref{l1c}, we can split $u_1$ into two terms $u_1=\Phi_1+v_1$ with $v_1$ the solution of
\[\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_1)^2 v_1 + V_1v_1 -\lambda v_1 & = & -\left[2\sqrt{\lambda}\eta_1\cdot (A_1-A_{1,\sharp})+q_{11}\right]\Phi_1, & \mbox{in}\ \Omega ,\\
v_1(x) & = & 0,& x\in \partial\Omega.
\end{array}\right.\]
Then, $u_1\in H^2(\Omega)$ take the form \eqref{l1d}. Using this formula we will complete the proof of \eqref{l1a}. Since
\begin{equation} \label{l1f}
S_1 =\int_{\partial\Omega}(\partial_\nu+{\rm i}A_1\cdot\nu) u_1(x) e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2(x)}d \sigma(x),
\end{equation}
from \eqref{kt1}, applying Green formula,
we get
\begin{equation} \label{l1g}
\begin{aligned}S_1&=
\int_\Omega div\left( (\nabla+{\rm i}A_1(x)) u_1(x) e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2(x)}\right) dx\\
\ &= \int_\Omega (\nabla+{\rm i}A_1)^2u_1e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx+\int_\Omega (\nabla+{\rm i}A_1)u_1\cdot (\nabla-{\rm i}A_1)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx.\end{aligned}
\end{equation}
Doing the same with the second term on the right hand side of this formula, we find out that
\begin{eqnarray*}
& & \int_\Omega (\nabla+{\rm i}A_1)u_1\cdot (\nabla-{\rm i}A_1)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx \\
& = & -{\rm i}\int_{\Gamma} u_1(x) e^{-{\rm i}\sqrt{\lambda}{\eta}_2\cdot x} e^{-{\rm i}\psi_2}(\sqrt{\lambda} b_2\eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_1)\cdot \nu d\sigma(x)\\&\ &
- \int_\Omega u_1(x) (\nabla-{\rm i}A_1)^2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}b_2e^{-{\rm i}\psi_2}dx.
\end{eqnarray*}
In light of \eqref{jt1} and the identity ${u_1}_{\vert\Gamma}=\Phi_{1}$, this entails
\[\begin{array}{l}
\int_\Omega (\nabla+{\rm i}A_1)u_1\cdot (\nabla-{\rm i}A_1)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx \\
= -{\rm i} \int_{\Gamma} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(\sqrt{\lambda} b_2\eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_1)\cdot \nu d\sigma(x)\\
\ \ \ -
\int_\Omega u_1(x) (\nabla-{\rm i}A_1)^2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}b_2e^{-{\rm i}\psi_2}dx.
\end{array}\]
Moreover, one can check that
\[(\nabla-{\rm i}A_1)^2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}b_2e^{-{\rm i}\psi_2}=\left(-\lambda b_2-2\sqrt{\lambda}(\eta_2\cdot\nabla\psi_2+A_1\cdot\eta_2)b_2-2{\rm i}\sqrt{\lambda}\eta_2\cdot \nabla b_2+q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}\]
with $q_{12}=\Delta b_2-2{\rm i}\nabla\psi_2\cdot\nabla b_2-2{\rm i}\nabla b_2\cdot A_1+\left(-{\rm i}\Delta\psi_2-\abs{\nabla\psi_2}^2-2\nabla\psi_2\cdot A_1-{\rm i}div(A_1)-\abs{A_1}^2\right)b_2$. Combining this with the fact that $\psi_2$ satisfies $\eta_2\cdot\nabla\psi_2+\eta_2\cdot A_{2,\sharp}=0$ and $b_2$ solves $\eta_2\cdot\nabla b_2=0$, we deduce that
\[(\nabla-{\rm i}A_1)^2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}b_2e^{-{\rm i}\psi_2}=\left([-\lambda-2\sqrt{\lambda}\eta_2\cdot (A_1-A_{2,\sharp})]b_2+q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}.\]
Therefore, we find
\[\begin{array}{l}
\int_\Omega (\nabla+{\rm i}A_1)u_1\cdot (\nabla-{\rm i}A_1)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx \\
= -{\rm i} \int_{\Gamma} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(\sqrt{\lambda} b_2\eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_1)\cdot \nu d\sigma(x)\\
\ \ \ -
\int_\Omega u_1(x) \left(-\lambda b_2-2\sqrt{\lambda}\eta_2\cdot (A_1-A_{2,\sharp})b_2+q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx.
\end{array}\]
Then, from \eqref{l1d} we get
\begin{eqnarray}\label{l1h}
&&\int_\Omega (\nabla+{\rm i}A_1)u_1\cdot (\nabla-{\rm i}A_1)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx \cr
&&= -{\rm i} \int_{\Gamma} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(\sqrt{\lambda} b_2\eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_1)\cdot \nu d\sigma(x)\cr
&&\ \ \ +\lambda \int_\Omega u_1e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}b_2e^{-{\rm i}\psi_2}dx+2\sqrt{\lambda}\int_\Omega \eta_2\cdot (A_1-A_{2,\sharp})e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\cr
&&\ \ \ -\int_\Omega q_{12}e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\cr
&&\ \ \ -\int_\Omega [(H_1-\lambda)^{-1}\left(2\sqrt{\lambda}\eta_1\cdot (A_1-A_{1,\sharp})+q_{11}\right)\Phi_1]\left(2\sqrt{\lambda}\eta_2\cdot (A_1-A_{2,\sharp})b_2-q_{12}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx.\cr
&&
\end{eqnarray}
Next, taking into account the fact that
$(\nabla+{\rm i}A_1)^2u_1= (V_1 -\lambda) u_1$ in $\Omega$, we obtain
\[\begin{array}{ll}\int_\Omega (\nabla+{\rm i}A_1)^2u_1e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx&=\int_\Omega (V_1 -\lambda) u_1e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx\\
\ \\
\ &=-\lambda\int_\Omega u_1e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx+\int_{\Omega} V_1e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx\\
\ \\
\ &\ \ \ -\int_\Omega V_1\left[(H_1-\lambda)^{-1}\left(2\sqrt{\lambda}\eta_1\cdot (A_1-A_{1,\sharp})+q_{11}\right)\Phi_1\right]e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx.\end{array}\]
Finally, we deduce \eqref{l1a} from \eqref{l1g}-\eqref{l1h}.
Now let us consider \eqref{l1b}. For this purpose, we start by proving formula \eqref{l1i}. In a similar way to \eqref{l1d}, we have
\[(-{\rm i}\nabla+A_2)^2 \Phi_1+V_2\Phi_1-\lambda \Phi_1
=2\sqrt{\lambda}(\eta_1\cdot\nabla\psi_1+A_2\cdot \eta_1)\Phi_1+ q_{21}\Phi_1\]
with $q_{21}=-{\rm i}div (A_2)+\abs{A_2}^2+V_2(x)+2 A_2\cdot\nabla\psi_1-{\rm i}\Delta \psi_1+\abs{\nabla\psi_1}^2$. Then, since $\psi_1$ is a solution of $\eta_1\cdot\nabla\psi_1+\eta_1\cdot A_{1,\sharp}=0$, we deduce that
\[(-{\rm i}\nabla+A_2)^2 \Phi_1+V_2\Phi_1-\lambda \Phi_1=\left(2\sqrt{ \lambda}\eta_1\cdot(A_2-A_{1,\sharp})+q_{21}\right)\Phi_1.\]
Moreover, one can check that the solution $u_2$ of
\[\left\{
\begin{array}{rcll}
(-{\rm i}\nabla+A_2)^2 u_2 + V_2u_2 -\lambda u_2 & = & 0, & \mbox{in}\ \Omega ,\\
u_2(x) & = & \Phi_1(x),& x\in \partial\Omega
\end{array}\right.\]
is given by \eqref{l1i}. Repeating our previous arguments, we deduce
\begin{equation} \label{l1j}S_2= \int_\Omega (\nabla+{\rm i}A_2)^2u_2e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx+\int_\Omega (\nabla+{\rm i}A_2)u_2\cdot (\nabla-{\rm i}A_2)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx.\end{equation}
On the other hand, using the fact that $\psi_2$ is a solution of the equation $\eta_2\cdot\nabla\psi_2+\eta_2\cdot A_{2,\sharp}=0$, we get
\begin{eqnarray}
&&\int_\Omega (\nabla+{\rm i}A_2)u_2\cdot (\nabla-{\rm i}A_2)e^{-{\rm i} \sqrt{\lambda}{\eta}_2\cdot x}b_2e^{-{\rm i}\psi_2}dx \cr
&& = -{\rm i} \int_{\Gamma} e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}(\sqrt{\lambda} b_2\eta_2+b_2\nabla\psi_2+{\rm i}\nabla b_2 +b_2A_2)\cdot \nu d\sigma(x)\cr
&&\ \ \ -
\int_\Omega u_2(x) \left(-\lambda b_2-2\sqrt{\lambda}\eta_2\cdot (A_2-A_{2,\sharp})b_2+q_{22}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx
\end{eqnarray}
with $q_{22}=\Delta b_2-2{\rm i}\nabla\psi_2\cdot\nabla b_2-2{\rm i}\nabla b_2\cdot A_2+\left(-{\rm i}\Delta\psi_2-\abs{\nabla\psi_2}^2-2\nabla\psi_2\cdot A_2-{\rm i}div(A_2)-\abs{A_2}^2\right)b_2$.
Combining this with \eqref{l1i}-\eqref{l1j} and repeating our previous arguments we obtain \eqref{l1b}.\qed
\subsection{Asymptotic properties of $S_1-S_2$ and representation formulas for $A_1-A_2$ and $V_1-V_2$}
In this subsection we will apply formulas \eqref{l1a}-\eqref{l1b} in order to derive two expressions involving $A_1-A_2$ and $V_1-V_2$ from the asymptotic expansion of $S_1-S_2$, as $\mathfrak I\lambda\to+\infty$. For this purpose, we start by specifying our choice for the parameter $\lambda$, the function $A_{j,\sharp}$, the vector $\eta_j$, $j=1,2$, and the function $b_2$ appearing in \eqref{anz}. Let us first define the parameter $\lambda$ and the vectors $\eta_1$, $\eta_2$. We consider an arbitrary $\xi \in {\mathbb R}^n \setminus \{0\}$ and pick $\eta \in \mathbb S^{n-1}$ such that $\eta \cdot \xi=0$. Then, for $\tau>\abs{\xi}$ we put
\begin{equation} \label{BA1}
B_\tau=\sqrt{1-\frac{\abs{\xi}^2}{4\tau^2}},\ \eta_1(\tau)=B_\tau\eta-\frac{\xi}{2\tau},\ \eta_2(\tau)=B_\tau\eta+\frac{\xi}{2\tau}\ \mbox{and}\ \lambda(\tau)=(\tau+i)^2,
\end{equation}
in such a way that
\begin{equation} \label{BA2}\left\{\begin{aligned} \eta_1,\eta_2\in\mathbb S^{n-1},\\
\sqrt{\lambda}(\eta_1-\eta_2)\to-\xi,\quad \textrm{as } \tau\to+\infty,\\
\mathfrak I\lambda\to+\infty,\quad \textrm{as } \tau\to+\infty,\\
\mathfrak I \sqrt{\lambda}\eta_1,\mathfrak I \sqrt{\lambda}\eta_2\ \ \textrm{are bounded with respect to } \tau>\abs{\xi}.\end{aligned}\right.\end{equation}
In order to get a suitable expression of the functions $A_{j,\sharp}$, we first need to extend identically the magnetic potentials $A_j$, $j=1,2$. For this purpose we set $\tilde{\Omega}$ an arbitrary open bounded set of ${\mathbb R}^n$ such that $\overline{\Omega}\subset\tilde{\Omega}$ and we define $\tilde{A_1}\in \mathcal C^1_0(\tilde{\Omega},{\mathbb R}^n)$ such that ${\tilde{A_1}}_{|\Omega}=A_1$. Then, we define $\tilde{A_2}$ by
\[\tilde{A}_2(x)=\left\{\begin{array}{l} A_2(x),\ \textrm{for }x\in\Omega,\\ \tilde{A}_1(x),\ \textrm{for }x\in\tilde{\Omega}\setminus\Omega.\end{array}\right.\]
In view of \eqref{t1a}, it is clear that $\tilde{A}_2\in \mathcal C^1_0(\tilde{\Omega},{\mathbb R}^n)$.
We define the functions $A_{j,\sharp}\in\mathcal C^\infty_0({\mathbb R}^n;{\mathbb R}^n)$, $j=1,2$, by
\[A_{j,\sharp}(x):=\chi_{\delta}*\tilde{A}_j(x)=\int_{{\mathbb R}^n}\chi_\delta(x-y)\tilde{A}_j(y)dy,\]
where $\chi_\delta(x)=\delta^{-n}\chi(\delta^{-1}x)$, with $\delta>0$, is the usual mollifer with $\chi\in\mathcal C_0^\infty({\mathbb R}^n)$, supp$(\chi)\subset\{x\in{\mathbb R}^n:\ |x|\leq 1\}$, $\chi\geq 0$ and $\int_{{\mathbb R}^n}\chi dx=1$. From now on we set $\delta=\tau^{-{1\over 3}}$ and we recall that
\[\psi_j(x)=-\int_{-\infty}^{0}\eta_j\cdot A_{j,\sharp}(x+s\eta_j)ds.\]
We set also
\begin{equation} \label{b2}b_2(x)= e^{{\rm i}\omega\cdot x}y\cdot\nabla \left[ \exp\left( -{\rm i}\int_{\mathbb R} \eta_2\cdot A_\sharp(x+s\eta_2)ds\right)e^{-{\rm i}\omega\cdot x}\right],\end{equation}
where $A_\sharp=A_{2,\sharp}-A_{1,\sharp}$, $\omega=B_\tau\xi-{|\xi|^2\eta\over 2\tau}\in\eta_2^\bot$, $B_\tau=\sqrt{1-{|\xi|^2\over 4\tau^2}}$, and
\[b(x)= e^{{\rm i}x\cdot\xi}y\cdot\nabla \left[ \exp\left( -{\rm i}\int_{\mathbb R} \eta\cdot A(x+s\eta)ds\right)e^{-{\rm i}x\cdot\xi}\right],\quad \psi(x)=\int_{-\infty}^{0}\eta\cdot A(x+s\eta)ds.\]
Here $y\in\mathbb S^{n-1}\cap\eta^{\bot}$, $y\cdot\nabla$ denotes the derivative in the $y=(y_1,\ldots,y_n)$ direction given by
$$y\cdot\nabla=\sum_{j=1}^ny_j\partial_{x_j}$$
and $A$ is the function defined by $A_2-A_1$ on $\Omega$ extended by $0$ outside of $\Omega$. Note that, in view of condition \eqref{t1a} we have $A\in \mathcal C^1_0(\Omega)$.
Since $\tilde{A}_j\in \mathcal C^1_0({\mathbb R}^n,{\mathbb R}^n)$, we find
\begin{equation} \label{mol1}\norm{A_{j,\sharp}-A_j}_{L^\infty(\Omega)}\leq \norm{A_{j,\sharp}-\tilde{A}_j}_{L^\infty({\mathbb R}^n)}\leq C \delta=C \tau^{-{1\over 3}}\end{equation}
with $C$ depending on $\Omega$ and any $ M\geq \underset{j=1,2}{\max}\norm{\tilde{A_j}}_{W^{1,\infty}({\mathbb R}^n)}$. On the other hand, one can check that
\begin{equation} \label{mol2}\norm{\partial_x^\alpha A_{j,\sharp}}_{L^\infty({\mathbb R}^m)}\leq C\delta^{\abs{\alpha}-1}=C\tau^{{\abs{\alpha}-1\over 3}},\quad \alpha\in\mathbb N^n\setminus\{0\},\end{equation}
where $C$ depends on $\Omega$ and any $ M\geq \underset{j=1,2}{\max}\norm{\tilde{A_j}}_{W^{1,\infty}({\mathbb R}^n)}$.
\begin{remark} Let us observe that, our anstazs are related to the principal part of the complex geometric optics solutions of \cite{SUU} and the extension of this construction to magnetic Schr\"odinger operators by \cite{FKSU,KLU,KU,NSU1,Sa1,Su}. Nevertheless, in contrast to the complex geometric optics solutions of \cite{SUU}, the large parameter of the ansatzs \eqref{anz}, that will be send to $+\infty$ for the uniqueness result, is given by $\mathfrak I\lambda$ where the parameter $\lambda$ appears explicitly in \eqref{eq:lambda}. This makes it possible to construct ansatzs bounded with respect to the large parameter and to use the resolvent $(H_j-\lambda)^{-1}$, $j=1,2$, for the construction of a remainder term that admits a decay with respect to the large parameter $\mathfrak I\lambda$. Moreover, in contrast to the geometric optics solutions of \cite{SUU}, whose principal parts take the form \eqref{anz} when $\zeta_j\cdot\zeta_j=0$, our construction is not restricted to dimension $n\geq3$. Indeed, the vector $\zeta_j$, $j=1,2$, that we consider in the present paper are subjected only to the condition \eqref{BA2}, already considered by \cite{I}, which requires only the two orthogonal vectors $\eta$ and $\xi$ appearing in \eqref{BA1}. For this reason, in contrast to the construction of \cite{SUU}, that requires three orthogonal vectors, our construction works also for $n=2$.\end{remark}
From now on, our goal is to derive from \eqref{l1a}-\eqref{l1b} two formulas from some asymptotic properties of $S_1-S_2$ as $\tau\to+\infty$. For this purpose we need the following intermediate result which follows from \eqref{mol1} and \eqref{mol2}.
\begin{lemma}\label{ll2}
Let the condition introduced above be fulfilled. Then, we have
\begin{equation} \label{ll2a} \sup_{\tau>|\xi|+1}\norm{b_2}_{L^\infty({\mathbb R}^n)}<\infty\end{equation}
and
\begin{equation} \label{ll2b}\lim_{\tau\to+\infty} b_2(x)=b(x),\quad \lim_{\tau\to+\infty}\psi_1(x)-\psi_2(x)=\psi(x),\ x\in{\mathbb R}^n.\end{equation} \end{lemma}
\begin{proof} Note first that
\begin{equation} \label{ll2c}b_2(x)=\left(-{\rm i}\omega\cdot y-{\rm i}\int_{\mathbb R} \eta_2\cdot y\cdot\nabla A_\sharp(x+s\eta_2)ds\right)\exp\left( -{\rm i}\int_{\mathbb R} \eta_2\cdot A_\sharp(x+s\eta_2)ds\right).\end{equation}
On the other hand, we have $|\omega|\leq 1+|\xi|$ and, since $\tilde{A}_2-\tilde{A}_1$ is compactly supported and $\tilde{A}_2-\tilde{A}_1\in \mathcal C^1_0({\mathbb R}^n,{\mathbb R}^n)$, we find $y\cdot\nabla A_\sharp=\chi_\delta*\left(y\cdot\nabla (\tilde{A}_2-\tilde{A}_1)\right)$. Therefore, we obtain
\[\norm{b_2}_{L^\infty({\mathbb R}^n)}\leq 1+|\xi|+C\norm{\chi_\delta}_{L^1({\mathbb R}^n)}\norm{y\cdot\nabla (\tilde{A}_2-\tilde{A}_1)}_{L^\infty({\mathbb R}^n,{\mathbb R}^n)}\leq 1+|\xi|+CM\]
with $C$ a generic constant depending only on $\Omega$ and $ M\geq \underset{j=1,2}{\max}\norm{\tilde{A_j}}_{W^{1,\infty}({\mathbb R}^n)}$. From this last estimate we deduce \eqref{ll2a}. Now let us prove \eqref{ll2b}. Since $\tilde{A}_1$ and $\tilde{A}_2$ coincide outside of $\Omega$, we have $\tilde{A}_2-\tilde{A}_1=A$. Therefore, we deduce that $A_\sharp=\chi_\delta* A$ and
\begin{equation} \label{ll2d}\abs{y\cdot\nabla A_\sharp(x+s\eta_2)-y\cdot\nabla A(x+s\eta)}\leq \abs{y\cdot\nabla A_\sharp(x+s\eta_2)-y\cdot\nabla A_\sharp(x+s\eta)}+\abs{y\cdot\nabla A_\sharp(x+s\eta)-y\cdot\nabla A(x+s\eta)}.\end{equation}
The second term on the right hand side of this estimate can be rewritten as
\[y\cdot\nabla A_\sharp(x+s\eta)-y\cdot\nabla A(x+s\eta)=\chi_\delta*[y\cdot\nabla A](x+s\eta)-y\cdot\nabla A(x+s\eta)\]
and since $A\in C^1_0({\mathbb R}^n)$, we get
\begin{equation} \label{ll2e} \lim_{\tau\to+\infty} y\cdot\nabla A_\sharp(x+s\eta)-y\cdot\nabla A(x+s\eta)=0,\quad x\in{\mathbb R}^n,\ s\in{\mathbb R}.\end{equation}
For the first term on the right hand side of \eqref{ll2d}, using the fact that for $\tau$ sufficiently large we have
\[\eta_2=\eta+{\xi\over 2\tau}+ \underset{\tau\to+\infty}{ o}\left({1\over\tau}\right)\]
and applying \eqref{mol2}, we get
\[\abs{y\cdot\nabla A_\sharp(x+s\eta_2)-y\cdot\nabla A_\sharp(x+s\eta)}\leq \norm{A_\sharp}_{W^{2,\infty}({\mathbb R}^n)}\abs{s(\eta-\eta_1)}\leq C\abs{s}\tau^{-{2\over3}}\]
with $C$ depending on $\xi$, $\Omega$, $\tilde{A_1}$ and $\tilde{A_2}$. In view of this estimate we have
\[\lim_{\tau\to+\infty} y\cdot\nabla A_\sharp(x+s\eta_2)-y\cdot\nabla A_\sharp(x+s\eta)=0,\quad x\in{\mathbb R}^n,\ s\in{\mathbb R}.\]
Combining this last result with \eqref{ll2d}-\eqref{ll2e}, we get
\[\lim_{\tau\to+\infty}y\cdot\nabla A_\sharp(x+s\eta)=y\cdot\nabla A(x+s\eta),\quad x\in{\mathbb R}^n,\ s\in{\mathbb R}.\]
Then, using the fact that supp$(A_\sharp)\subset \Omega+\{x\in{\mathbb R}^n:\ |x|\leq \delta\}$ and \eqref{mol2}, by the dominate convergence theorem we get that
\[\lim_{\tau\to+\infty}\int_{\mathbb R} y\cdot\nabla A_\sharp(x+s\eta_2)ds=\int_{\mathbb R} y\cdot\nabla A(x+s\eta)ds,\quad \textrm{$x\in{\mathbb R}^n$}.\]
Putting this together with \eqref{ll2c} and the fact that $\omega\to\xi$, $\eta_2\to\eta$ as $\tau\to+\infty$, we obtain
\[\lim_{\tau\to+\infty}b_2(x)=\left(-{\rm i}\xi\cdot y+-{\rm i}\int_{\mathbb R} \eta\cdot y\cdot\nabla A(x+s\eta)ds\right)\exp\left( -{\rm i}\int_{\mathbb R} \eta\cdot A(x+s\eta)ds\right)=b(x),\ x\in{\mathbb R}^n.\]
Using similar arguments we deduce that
\[\lim_{\tau\to+\infty}\psi_1(x)-\psi_2(x)=\psi(x)=\int_{-\infty}^{ 0} \eta\cdot A(x+s\eta)ds,\quad x\in{\mathbb R}^n.\]
This completes the proof of the lemma.\end{proof}
Applying \eqref{l1a}-\eqref{l1b}, \eqref{mol1}-\eqref{ll2b} and sending $\tau\to+\infty$, we obtain our first formula involving the magnetic potentials $A_1, A_2$.
\begin{proposition}
\label{l2}
Fix $\xi\in{\mathbb R}^n\setminus \{0\}$ and $\eta\in \mathbb S^{n-1}$ such that $\eta\cdot\xi=0$. Let $\lambda$, $\eta_1$ and $\eta_2$ be defined by \eqref{BA1} and let $b_2$ be defined by \eqref{b2}. Then, we have
\begin{equation} \label{l2a}
\lim_{\tau \to+\infty} \frac{S_1-S_2}{\sqrt{\lambda}}=2\int_\Omega \eta\cdot(A_1-A_2)e^{-{\rm i}\xi\cdot x} b e^{{\rm i}\psi(x)}dx.
\end{equation}
\end{proposition}
\begin{proof}
With reference to \eqref{jt1} and \eqref{BA1} we have
$\abs{\Phi_{1}(x)}=e^{-\eta_1 \cdot x}$ and $\abs{e^{-{\rm i}\sqrt{\lambda} \eta_2 \cdot x}}=e^{\eta_2 \cdot x}$ for all $x\in\Omega$,
hence $\norm{\Phi_{1}}_{L^2(\Omega)}^2 = \int_{\Omega} e^{-2 \eta_1 \cdot x} dx \leq | \Omega | e^{2 | \Omega |}$ and
$\norm{e^{-{\rm i}\sqrt{\lambda} \eta_2 \cdot x}}_{L^2(\Omega)}^2 \leq | \Omega | e^{2 | \Omega |}
$ since $|\eta_1|=|\eta_2|=1$. Moreover, in view of \eqref{BA1}, we have the estimate
$$\norm{(H_j-\lambda)^{-1}}_{\mathcal B(L^2(\Omega))} ={1 \over \textrm{dist}(\lambda,\sigma(H_j)) } \leq {1 \over | \mathfrak I \lambda |}= {1\over 2\tau},\quad j=1,2.$$
In addition, in light of \eqref{mol2}, we get
\[\norm{\psi_j}_{W^{2,\infty}(\Omega)}\leq C\delta=C\tau^{{1\over3}},\quad \norm{b_j}_{W^{2,\infty}(\Omega)}\leq C\delta^2=C\tau^{{2\over3}}\]
with $C$ a generic constant depending on $\xi$, $\Omega$ and $\tilde{A_j}$, $j=1,2$.
Putting these estimates together with \eqref{t1a}, \eqref{l1a}-\eqref{l1b} and \eqref{mol1} , we deduce that
\[ \frac{S_1-S_2}{\sqrt{\lambda}}=2\int_\Omega \eta_2\cdot (A_1-A_2)e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} b_2 e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx+\underset{\tau\to+\infty}{\mathcal O}\left(\tau^{-{1\over3}}\right).\]
Combining this with \eqref{BA2}, \eqref{ll2a}-\eqref{ll2b} and applying the dominate convergence theorem we deduce \eqref{l2a}.
\end{proof}
Using similar arguments and assuming that the magnetic potentials are known ($A_1=A_2$), we obtain our second formula involving the electric potentials $V_1,V_2$.
\begin{proposition}
\label{l3}
Assume that $A_1=A_2$. Fix $\xi\in{\mathbb R}^n\setminus \{0\}$ and $\eta\in \mathbb S^{n-1}$ such that $\eta\cdot\xi=0$. Let $\lambda$, $\eta_1$ and $\eta_2$ be defined by \eqref{BA1} and $b_2=1$. Then, we have
\begin{equation} \label{l3a}
\lim_{\tau \to+\infty} S_1-S_2=\int_\Omega (V_1-V_2)e^{-{\rm i}\xi\cdot x} dx.
\end{equation}
\end{proposition}
\begin{proof} Note that for $A_1=A_2$ we have $q_{11}-V_1=q_{21}-V_2$, $q_{12}=q_{22}$, $A_{1,\sharp}=A_{2,\sharp}$. Therefore, we deduce that \eqref{l1a}-\eqref{l1b} imply
\begin{equation} \label{l3b}\begin{aligned}S_1-S_2=&\int_\Omega (V_1-V_2)e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx-\int_\Omega \left[\lambda\left((H_1-\lambda)^{-1}-(H_2-\lambda)^{-1}\right)Q_1\right]Q_2dx\\
\ & -\int_\Omega \left[\sqrt{\lambda}(H_1-\lambda)^{-1}Q_1\right]V_1e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx-\int_\Omega \left[\sqrt{\lambda}(H_1-\lambda)^{-1}V_1\Phi_1\right]Q_2dx\\
\ &-\int_\Omega \left[(H_1-\lambda)^{-1}V_1\Phi_1\right]V_1e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx
+\int_\Omega \left[\sqrt{\lambda}(H_2-\lambda)^{-1}Q_1\right]V_2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx\\
\ &+\int_\Omega \left[\sqrt{\lambda}(H_2-\lambda)^{-1}V_2\Phi_1\right]Q_2dx+\int_\Omega \left[(H_2-\lambda)^{-1}V_2\Phi_1\right]V_2e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}dx,\end{aligned}\end{equation}
where
\[Q_1=2\eta_1\cdot (A_1-A_{1,\sharp})\Phi_1+{(q_{11}-V_1)\Phi_1\over\sqrt{\lambda}},\quad Q_2=\left(2\eta_2\cdot (A_1-A_{1,\sharp})-{q_{12}\over\sqrt{\lambda}}\right)e^{-{\rm i}\sqrt{\lambda}{\eta}_2 \cdot x}e^{-{\rm i}\psi_2}.\]
On the other hand, since $H_2-\lambda=H_1-\lambda-(V_1-V_2)$, for $\tau$ sufficiently large we have
\[\begin{aligned}(H_1-\lambda)^{-1}-(H_2-\lambda)^{-1}&=(H_1-\lambda)^{-1}\left(\textrm{Id}-\left(\textrm{Id}-(V_1-V_2)(H_1-\lambda)^{-1}\right)^{-1}\right)\\
\ &=-(H_1-\lambda)^{-1}\sum_{k=1}^\infty\left((V_1-V_2)(H_1-\lambda)^{-1}\right)^k.\end{aligned}\]
Combining this with the fact that $\mathfrak I\lambda=2\tau$, $\abs{\lambda}\leq |\tau^2-1|+2\tau$, and the fact that
\[\norm{(H_1-\lambda)^{-1}}_{\mathcal B(L^2(\Omega))}+\norm{(V_1-V_2)(H_1-\lambda)^{-1}}_{\mathcal B(L^2(\Omega))}\leq {C\over|\mathfrak I\lambda|}={C\over2\tau}\]
with $C$ depending only on $V_1$, $V_2$ and $\Omega$, we deduce that
\begin{equation} \label{l3c}\sup_{\tau>|\xi|+1}\norm{\lambda\left((H_1-\lambda)^{-1}-(H_2-\lambda)^{-1}\right)}_{\mathcal B(L^2(\Omega))}<\infty.\end{equation}
In addition, \eqref{mol1}-\eqref{mol2} imply
\[\lim_{\tau\to+\infty}\norm{Q_1}_{L^\infty(\Omega)}=\lim_{\tau\to+\infty}\norm{Q_2}_{L^\infty(\Omega)}=0.\]
Putting this result together with \eqref{BA2}, \eqref{l3b}-\eqref{l3c}, we obtain
\[\limsup_{\tau \to+\infty} \abs{(S_1-S_2)-\int_\Omega (V_1-V_2)e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx}=0.\]
On the other hand, repeating the arguments of Lemma \ref{ll2}, we find
\[\lim_{\tau\to+\infty}\psi_1(x)-\psi_2(x)=\psi(x)=\int_{-\infty}^{0} \eta\cdot A(x+s\eta)ds=0\]
since $A_1=A_2$. Thus, applying the dominate convergence theorem we obtain
\[\lim_{\tau\to+\infty}\int_\Omega (V_1-V_2)e^{{\rm i}\sqrt{\lambda}({\eta}_1-{\eta}_2)\cdot x} e^{{\rm i}(\psi_1(x)-\psi_2(x))}dx=\int_\Omega (V_1-V_2)e^{-{\rm i}x\cdot\xi}dx\]
and we deduce \eqref{l3a}.\end{proof}
Armed with formulas \eqref{l2a}-\eqref{l3a}, in the next section we will complete the proof of Theorem \ref{thm-1}.
\section{Proof of the main result}
\label{sec:Proof-main}
This section is devoted to the proof of our main result. In all this section, for $j= 1$ and $j= 2$, we consider two magnetic potentials $A_j$ and electric potentials $V_{j}$ satisfying the assumptions of Theorem \ref{thm-1} and we denote by $H_j$ the associated operators defined by \eqref{eq:Def-A-theta} for $A=A_j$ and $V=V_j$. Let $(\lambda_{j,k},\varphi_{j,k})_{k \geq 1}$ be a sequence of eigenvalues and eigenfunctions of $H_j$. In order to prove Theorem \ref{thm-1}, in light of \eqref{l2a}-\eqref{l3a}, we prove first that the condition
\begin{equation} \label{l4b}\lim_{\tau\to+\infty}{S_1(\lambda(\tau),\eta_1(\tau),\eta_2(\tau))-S_2(\lambda(\tau),\eta_1(\tau),\eta_2(\tau))\over\sqrt{\lambda(\tau)}}=0\end{equation}
implies $dA_1=dA_2$. Then, we show that for $A_1=A_2$ the condition
\begin{equation} \label{l4c}\lim_{\tau\to+\infty}S_1(\lambda(\tau),\eta_1(\tau),\eta_2(\tau))-S_2(\lambda(\tau),\eta_1(\tau),\eta_2(\tau))=0\end{equation}
implies $V_1=V_2$. Finally, we complete the proof by proving that condition \eqref{t1a}-\eqref{tt2a} imply \eqref{l4b}-\eqref{l4c}.
We start by proving that \eqref{l4b} implies $dA_1=dA_2$.
\begin{lemma}
\label{l4} Let $\eta_1(\tau)$, $\eta_2(\tau)$ and $\lambda(\tau)$ be fixed by \eqref{BA1} and $b_2$ be defined by \eqref{b2}. Assume that \eqref{l4b} is fulfilled.
Then, we have $dA_1=dA_2$.\end{lemma}
\begin{proof}
Combining \eqref{l4b} with \eqref{l2a} we deduce that for all $\xi\in{\mathbb R}^n\setminus\{0\}$, $\eta\in\mathbb S^{n-1}$, satisfying $\eta\cdot\xi=0$,
we get
\[\int_\Omega \eta\cdot(A_2-A_1)e^{-{\rm i}\xi\cdot x} b(x) e^{{\rm i}\psi(x)}dx=0.\]
Here $b$ takes the form
\[b(x)=e^{{\rm i}x\cdot\xi}y\cdot\nabla \left[\exp\left(-{\rm i}\int_{\mathbb R}\eta\cdot A(x+s\eta)ds\right)e^{-{\rm i}x\cdot\xi}\right]\]
with $y\in \mathbb S^{n-1}\cap\eta^\bot$. Then, applying Fubini's theorem, we obtain
\[0=\int_{{\mathbb R}^n}\eta\cdot A(x)e^{-{\rm i}\xi\cdot x} b(x) e^{{\rm i}\psi(x)}dx=\int_{\eta^\bot}\int_{\mathbb R} \eta\cdot A(x'+t\eta)e^{{\rm i}\psi(x'+t\eta)}b(x')e^{-{\rm i}\xi\cdot x'}dtdx'.\]
Here we use the fact that $b(x)=b(x-(x\cdot\eta)\eta)$ and $\xi\cdot\eta=0$. On the other hand, for all $x'\in\eta^\bot$ and $t\in{\mathbb R}$, we have
\[\eta\cdot A(x'+t\eta)e^{{\rm i}\psi(x'+t\eta)}=\eta\cdot A(x'+t\eta)\exp\left({\rm i}\int_{-\infty}^t\eta\cdot A(x'+s\eta)ds\right)=-{\rm i}\partial_t \exp\left({\rm i}\int_{-\infty}^t\eta\cdot A(x'+s\eta)ds\right).\]
Therefore, we find
\[\begin{aligned}\int_{{\mathbb R}^n}\eta\cdot A(x)e^{-{\rm i}\xi\cdot x} b(x) e^{{\rm i}\psi(x)}dx&=-{\rm i}\int_{\eta^\bot}\left[\int_{\mathbb R} \partial_t \exp\left({\rm i}\int_{-\infty}^t\eta\cdot A(x'+s\eta)ds\right)dt \right]b(x')e^{-{\rm i}\xi\cdot x'}dx'\\
\ &=-{\rm i}\int_{\eta^\bot} \left[\exp\left({\rm i}\int_{\mathbb R}\eta\cdot A(x'+s\eta)ds\right)-1\right] b(x')e^{-{\rm i}\xi\cdot x'}dx'.\end{aligned}\]
It follows
\begin{equation} \label{t1b}\int_{\eta^\bot} \left[\exp\left({\rm i}\int_{\mathbb R}\eta\cdot A(x'+s\eta)ds\right)-1\right] b(x')e^{-{\rm i}\xi\cdot x'}dx'=0.\end{equation}
We fix $i,j\in\{1,\ldots,n\}$ such that $i< j$ and we assume that $\xi \in\{\xi=(\xi_1,\ldots,\xi_n):\ \xi_i\neq0\}$. We can choose $\eta={\xi_j e_i-\xi_ie_j\over \sqrt{\xi_i^2+\xi_j^2}}$ and $y={\xi_i e_i+\xi_je_j\over \sqrt{\xi_i^2+\xi_j^2}}\in\eta^\bot$. Here $(e_1,\ldots,e_n)$ is the canonical basis of ${\mathbb R}^n$ defined by $e_1=(1,0,\ldots,0),\ldots, e_n=(0,\ldots,0,1)$. Then, \eqref{t1b} implies
\[\int_{\eta^\bot} \left[\exp\left({\rm i}\int_{\mathbb R}\eta\cdot A(x'+s\eta)ds\right)-1\right]y\cdot\nabla \left[\exp\left(-{\rm i}\int_{\mathbb R}\eta\cdot A(x'+s\eta)ds\right)e^{-ix'\cdot\xi}\right]dx'=0.\]
Integrating by parts we get
\[{-{\rm i}\over\sqrt{\xi_i^2+\xi_j^2}}\cdot\int_{{\mathbb R}^n}(\xi_jy\cdot\nabla a_i(x)-\xi_iy\cdot\nabla a_j(x))e^{-{\rm i}x\cdot\xi}dx=-{\rm i}\int_{\eta^\bot} \left( \int_{\mathbb R}\eta\cdot y\cdot\nabla A(x'+s\eta)ds\right)e^{-{\rm i}x'\cdot\xi}dx'=0\]
with $A=(a_1,\ldots,a_n)$. Integrating again by parts, we find
\[\begin{aligned}\int_{{\mathbb R}^n}(\xi_ja_i-\xi_ia_j)e^{-{\rm i}x\cdot\xi}dx&={y\cdot\xi\over\sqrt{\xi_i^2+\xi_j^2}}\int_{{\mathbb R}^n}(\xi_ja_i-\xi_ia_j)e^{-{\rm i}x\cdot\xi}dx\\
\ &={-{\rm i}\over\sqrt{\xi_i^2+\xi_j^2}}\cdot\int_{{\mathbb R}^n}(\xi_jy\cdot\nabla a_i(x)-\xi_iy\cdot\nabla a_j(x))e^{-{\rm i}x\cdot\xi}dx=0\end{aligned}\]
and it follows that for all $\xi \in\{\xi=(\xi_1,\ldots,\xi_n):\ \xi_i\neq0\}$ we have $\mathcal F[\partial_{x_j}a_i-\partial_{x_i}a_j](\xi)=0$. On the other hand, since $\partial_{x_j}a_i-\partial_{x_i}a_j$ is compactly supported, $\mathcal F(\partial_{x_j}a_i-\partial_{x_i}a_j)(\xi)$ is continuous in $\xi\in{\mathbb R}^n$ and it follows $\mathcal F(\partial_{x_j}a_i-\partial_{x_i}a_j)=0$ on ${\mathbb R}^n$. From this last result, we deduce that $\partial_{x_j}a_i-\partial_{x_i}a_j=0$ which implies that $dA_1=dA_2$. \end{proof}
Now assuming that $A_1=A_2$, we show in the next lemma that \eqref{l4c} implies $V_1=V_2$.
\begin{lemma}
\label{l66} Let $\eta_1(\tau)$, $\eta_2(\tau)$ and $\lambda(\tau)$ be fixed by \eqref{BA1} and $b_2=1$. Assume that $A_1=A_2$ and \eqref{l4c} is fulfilled. Then, we have $V_1=V_2$.\end{lemma}
\begin{proof} Fix $\xi\in{\mathbb R}^n\setminus\{0\}$ and choose $\eta\in\mathbb S^{n-1}\cap\xi^\bot$. Fix also $b=1$.
Thus, combining \eqref{l3a} and \eqref{l4c}, we find
\[\int_{{\mathbb R}^n} V(x)e^{-{\rm i}x\cdot\xi}dx=0\]
with $V=V_1-V_2$ extended by $0$ outside of $\Omega$. It follows that $V_1=V_2$.\end{proof}
According to Lemma \ref{l4}, \ref{l66}, the proof of Theorem \ref{thm-1} will be completed if we show that conditions \eqref{tt2a} imply conditions \eqref{l4b}, \eqref{l4c}. For this purpose, we adapt the approach of \cite{KKS} to magnetic Schr\"odinger operators. Let $f\in H^{1\over2}(\Gamma)$ being fixed, with the notations of Lemmas \ref{lem:Resolution} and \ref{lem:v-lambda-mu}, we denote by $v_{j,\lambda,\mu} := u_{j,\lambda} - u_{j,\mu}$ the solution of \eqref{eq:v-lambda} where $V$ is replaced by $V_{j}$ and $A$ by $A_j$. We fix also $h_{j,k} := {\partial_\nu\varphi_{j,k}}_{|\Gamma}$ $\alpha_{j,k} := \langle f,h_{j,k}\rangle$.
Recalling that in Lemma \ref{lem:z-mu} we have set $z_{\mu} = u_{1,\mu} - u_{2,\mu}$, in a similar way to \cite{KKS}, writing the above identity for $j=1$ and $j=2$, applying \eqref{t1a} and then subtracting the resulting equations, we end up with a new relation, namely
\begin{equation}\label{dif}
\begin{aligned}{(\partial_\nu +{\rm i}A_1\cdot\nu) u_{1,\lambda}}_{|\Gamma}-{(\partial_\nu +{\rm i}A_2\cdot\nu) u_{2,\lambda}}_{|\Gamma}&={\rm i}(A_1-A_2)\cdot\nu f+\partial_\nu u_{1,\lambda} -
\partial_\nu u_{2,\lambda} \\
\ &=
\partial_\nu z_{\mu}
+ \partial_\nu v_{1,\lambda,\mu} - \partial_\nu v_{2,\lambda,\mu} .\end{aligned}
\end{equation}
Now let us set
\[ F_j(\lambda,\mu,f):={\partial_\nu v_{j,\lambda,\mu}}_{|\Gamma},\quad j=1,2.\]
According to \eqref{eq:v-Normal-Deriv}, we have
\begin{equation}\label{eq:Def-F-m}
F(\lambda,\mu,f) := F_1(\lambda,\mu,f)-F_2(\lambda,\mu,f)=\sum_{k =1}^{+\infty} \left[{(\mu - \lambda)\alpha_{1,k} \over (\lambda - \lambda_{1,k})(\mu - \lambda_{1,k})} \, h_{1,k}-{(\mu - \lambda)\alpha_{2,k} \over (\lambda - \lambda_{2,k})(\mu - \lambda_{2,k})} \, h_{2,k}\right].
\end{equation}
Consider the following intermediate results.
\begin{lemma}\label{lem-6}
Let $\eta_1,\eta_2,\lambda$ be given by \eqref{BA1}. Consider $\Phi_j$, $j=1,2$, with $\Phi_1$ introduced in the previous section and $\Phi_2= e^{-{\rm i}\sqrt{\lambda} {\eta}_2 \cdot x} b_2e^{-{\rm i}\psi_2}$, where $b_2$ is defined by \eqref{b2} or $b_2=1$. Then, we have
\begin{equation} \label{ll6a}\sup_{\tau>1}\sum_{k=1}^\infty \abs{{\left\langle \Phi_1,h_{j,k}\right\rangle \over \lambda_{j,k}-\lambda }}^2<\infty,\quad \sup_{\tau>1}\sum_{k=1}^\infty \abs{{\left\langle \overline{\Phi_2},h_{2,k}\right\rangle \over \lambda_{2,k}-\lambda }}^2<\infty,\ j=1,2.\end{equation}\end{lemma}
\begin{proof}
We start with the first estimate of \eqref{ll6a} for $j=1$. According to Lemma \ref{lem:Resolution} the solution $u_{1,\lambda}$ of \eqref{eq1} for $f=\Phi_1$, $A=A_1$ and $V=V_1$, is given by
\[u_{1,\lambda}=\sum_{k=1}^\infty {\left\langle \Phi_1,h_{1,k}\right\rangle\over \lambda-\lambda_{1,k}}\varphi_{1,k}.\]
Therefore, we have
\begin{equation} \label{ll6b}\norm{u_{1,\lambda}}_{L^2(\Omega)}^2=\sum_{k=1}^\infty \abs{{\left\langle \Phi_1,h_{1,k}\right\rangle \over \lambda_{1,k}-\lambda }}^2.\end{equation}
On the other hand, in view of \eqref{l1d}, we have
\[\norm{u_{1,\lambda}}_{L^2(\Omega)}\leq \norm{\Phi_1}_{L^2(\Omega)}+\norm{\sqrt{\lambda}(H_1-\lambda)^{-1}\left[2\eta_1\cdot (A_1-A_{1,\sharp})+{q_{11}\over\sqrt{\lambda}}\right]}_{L^2(\Omega)}.\]
Here $q_{11}$ is the expression introduced in Lemma \ref{l1}.
Combining this with the fact that
\[\norm{\sqrt{\lambda}(H_1-\lambda)^{-1}}_{\mathcal B(L^2(\Omega))}\leq {|\tau+{\rm i}|\over |\mathfrak I \lambda|}= {|\tau+{\rm i}|\over 2\tau}\leq 1\]
and the fact that, according to \eqref{mol1}-\eqref{mol2}, we have
\[\lim_{\tau\to+\infty}\norm{\eta_1\cdot (A_1-A_{1,\sharp})}_{L^\infty(\Omega)}=\lim_{\tau\to+\infty} \norm{{q_{11}\over\sqrt{\lambda}}}_{L^\infty(\Omega)}=0\]
we deduce the first estimate of \eqref{ll6a} for $j=1$. In a same way, for $j=2$ using the fact that according to \eqref{mol2} we have
\[(-{\rm i}\nabla +A_2)^2\Phi_1 + V_2\Phi_1 - \lambda \Phi_1=\underset{\tau\to+\infty}{\mathcal O}(\tau)\]
and repeating our previous arguments we deduce the first estimate \eqref{ll6a} for $j=2$. For the second estimate of \eqref{ll6a}, repeating the previous arguments we find
\[(-{\rm i}\nabla +A_2)^2\overline{\Phi_2}+ V_2\overline{\Phi_2} - \overline{\lambda}\ \overline{\Phi_2}=\overline{({\rm i}\nabla +A_2)^2\Phi_2+ V_2\Phi_2 -\lambda\Phi_2}=\underset{\tau\to+\infty}{\mathcal O}(\tau).\]
Combining this estimate with the fact that
\[\abs{{\left\langle \overline{\Phi_2},h_{2,k}\right\rangle \over \lambda_{2,k}-\lambda }}= \abs{{\left\langle \overline{\Phi_2},h_{2,k}\right\rangle \over \lambda_{2,k}-\overline{\lambda }}}\]
since $\lambda_{2,k}\in{\mathbb R}$, we deduce the second estimate of \eqref{ll6a} by repeating the above arguments.\end{proof}
From now on we set
$$\begin{aligned}G(\lambda,\mu,\Phi_1,\Phi_2)&:=\langle F(\lambda,\mu, \Phi_1), \overline{\Phi_2} \rangle\\
\ &=\sum_{k =1}^{+\infty} (\mu - \lambda)\left[{\left\langle \Phi_1,h_{1,k} \right\rangle \left\langle h_{1,k},\overline{\Phi_2}\right\rangle\over (\lambda - \lambda_{1,k})(\mu - \lambda_{1,k})} \, -{\left\langle \Phi_1,h_{2,k} \right\rangle \left\langle h_{2,k},\overline{\Phi_2}\right\rangle \over (\lambda - \lambda_{2,k})(\mu - \lambda_{2,k})} \right]. \end{aligned}$$
Combining estimates \eqref{ll6a} with Lemma 4.3, 4.4, 4.5 of \cite{KKS}, we obtain the following.
\begin{lemma}\label{lem-0} Let the conditions of Theorem \ref{thm-1} be fulfilled and let $\eta_1,\eta_2,\lambda$ be given by \eqref{BA1}. Then,
$G(\lambda,\mu,\Phi_1,\Phi_2)$ converge to $G_*(\lambda,\Phi_1,\Phi_2)$ as $\mu\to-\infty$ and $G_*(\lambda,\Phi_1,\Phi_2)$ converge to $0$ as $\tau\to +\infty$. Here we consider both the case $b_2$ given by \eqref{b2} and the case $b_2=1$.\end{lemma}
Armed with Lemma \ref{lem-0}, we are now in position to complete the proof of Theorem \ref{thm-1}. \\
\ \\
\textbf{Proof of Theorem \ref{thm-1}.} Note first that according \eqref{dif}, for $M=\norm{V_1}_{L^\infty(\Omega)}+\norm{V_2}_{L^\infty(\Omega)}$, we have
\[S_1(\lambda,\eta_1,\eta_2)-S_2(\lambda,\eta_1,\eta_2)=\left\langle \partial_\nu z_{\mu},e^{{\rm i}\overline{\sqrt{\lambda}} {\eta}_2 \cdot x} \overline{b_2}e^{{\rm i}\psi_2} \right\rangle+G(\lambda,\mu,\Phi_1,\Phi_2),\quad \mu\in(-\infty, -M),\]
where $\lambda$, $\eta_1$, $\eta_2$ are fixed by \eqref{BA1}, $b_2$ is given by \eqref{b2} or $b_2=1$ and $z_{\mu} = u_{1,\mu} - u_{2,\mu}$ with $u_{j,\mu}$, $j=1,2$, the solution of \eqref{eq:v-lambda} where $\lambda$ is replaced by $\mu$, $V$ by $V_{j}$, $A$ by $A_j$ and $f$ by $\Phi_1$.
In view of Lemma \ref{lem:z-mu} and Lemma \ref{lem-0}, sending $\mu\to-\infty$ we get
\[S_1(\lambda,\eta_1,\eta_2)-S_2(\lambda,\eta_1,\eta_2)= G_*(\lambda,\Phi_1,\Phi_2).\]
Then, in view of Lemma \ref{lem-0}, conditions \eqref{l4b} and \eqref{l4c} are fulfilled and according to Lemma \ref{l4} we have $dA_1=dA_2$. Therefore, condition \eqref{t1a} implies that for $A=A_2-A_1$ extended by $0$ outside of $\Omega$ we have $dA=0$ on ${\mathbb R}^n$. Thus, there exists $p\in W^{2,\infty}({\mathbb R}^n)$ given by
$$p(x)=\int_0^1x\cdot A(tx)dt$$
such that $A=\nabla p$ on ${\mathbb R}^n$. Since ${\mathbb R}^n\setminus \Omega$ is connected, applying the fact that $A=0$ on ${\mathbb R}^n\setminus \Omega$, upon eventually subtracting a constant we may assume that $p_{|{\mathbb R}^n\setminus \Omega}=0$ which implies that $p_{|\Gamma}=0$. Now let us consider the operator $H_3=(-{\rm i}\nabla+A_1)+V_2$ acting on $L^2(\Omega)$ with Dirichlet boundary condition and let $(\lambda_{3,k},\varphi_{3,k})_{k \geq 1}$ be a sequence of eigenvalues and eigenfunctions of $H_3$. Since $A_1=A_2-\nabla p$ one can check that $H_3=e^{{\rm i}p}H_2e^{-{\rm i}p}$. From this identity we deduce that
$$ \lambda_{3,k}=\lambda_{2,k},\quad k\geq1.$$
Moreover, for all $k\geq1$ we can choose $\varphi_{3,k}=e^{ip}\varphi_{2,k}$ and deduce that the condition
$$\partial_\nu \varphi_{3,k}=\partial_\nu \varphi_{2,k},\quad k\geq1$$
is also fulfilled.
Thus, conditions \eqref{tt2a} imply that
$$\lim_{k\to+\infty}|\lambda_{1,k}-\lambda_{3,k}|=0\quad\textrm{and}\quad \sum_{k=1}^{+\infty}\norm{\partial_\nu\varphi_{1,k} -\partial_\nu\varphi_{3,k}}_{L^2(\Gamma)}^2<\infty.$$
Then, repeating the arguments of Lemma \ref{lem-0} we obtain
$$\lim_{\tau\to+\infty} \tilde{S}_1(\lambda(\tau), \eta_1(\tau),\eta_2(\tau))-\tilde{S}_3(\lambda(\tau), \eta_1(\tau),\eta_2(\tau))=0,$$
where
$$\tilde{S}_j(\lambda,\eta_1,\eta_2)=\left \langle \Lambda_{j,\lambda} \Phi_1 , e^{{\rm i}\overline{\sqrt{\lambda}} {\eta}_2 \cdot x} e^{{\rm i}\tilde{\psi}_2}\right\rangle,\quad j=1,3$$
with
$$\tilde{\psi}_2(x)=\int_{-\infty}^{ x\cdot\eta_2} \eta_2\cdot A_{1,\sharp}(x+(s-x\cdot\eta_2)\eta_2)ds,\quad b_2=1$$
and $\Lambda_{3,\lambda}$ the Dirichlet-to-Neumann map associated to problem \eqref{eq1} for $A=A_1$ and $V=V_2$. Then, in view of Lemma \ref{l66} we have $V_1=V_2$. This completes the proof of Theorem \ref{thm-1}.\qed
\section*{Acknowledgements}
The author would like to thank the anonymous referee for valuable comments and helpful remarks. The author is grateful to Otared Kavian and Eric Soccorsi for their remarks and suggestions.
|
{
"timestamp": "2016-10-14T02:04:26",
"yymm": "1504",
"arxiv_id": "1504.04514",
"language": "en",
"url": "https://arxiv.org/abs/1504.04514"
}
|
"\\section{Introduction}\n\nConsider the following ordinary differential equation (abbreviated as OD(...TRUNCATED)
| {"timestamp":"2015-05-06T02:03:53","yymm":"1504","arxiv_id":"1504.04450","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\n\\medskip\nWe do not need to emphasize that norms and metrics induced by(...TRUNCATED)
| {"timestamp":"2015-04-17T02:07:42","yymm":"1504","arxiv_id":"1504.04149","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\r\nIn the last years, the field of non-additive measures was intensively st(...TRUNCATED)
| {"timestamp":"2015-10-14T02:07:08","yymm":"1504","arxiv_id":"1504.04110","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nLet $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces. $B_X(x,r)$ denotes the c(...TRUNCATED)
| {"timestamp":"2015-04-17T02:10:06","yymm":"1504","arxiv_id":"1504.04250","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction and main results}\n\nThe directed polymer in random environment is a statist(...TRUNCATED)
| {"timestamp":"2015-08-12T02:04:27","yymm":"1504","arxiv_id":"1504.04505","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:intro}\n\n\nIn this paper, we develop an online local adaptivi(...TRUNCATED)
| {"timestamp":"2015-04-20T02:03:34","yymm":"1504","arxiv_id":"1504.04417","language":"en","url":"http(...TRUNCATED)
|
"\\section{Definitions and preliminaries}\n\nAn \\emph{$n$-dimensional dual arc} $\\mathcal D$ in a (...TRUNCATED)
| {"timestamp":"2015-04-17T02:08:10","yymm":"1504","arxiv_id":"1504.04170","language":"en","url":"http(...TRUNCATED)
|
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