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magnetic bottles, eigenvalue, semi-classical, Schroedinger, superconductivity 
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%\title{Magnetic bottles in connection with  superconductivity}

%\author{ Bernard Helffer$^{1,3}$ and Abderemane Morame $^{2}$ }



\pagestyle{myheadings} 
\markright{B. Helffer, A. Morame: Magnetic bottles ...} 




 
        
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\begin{document}

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%title 
\centerline{\Large \bf 
{Magnetic bottles in connection with  superconductivity}} 

\vskip 0.5cm 


%authors 
\centerline{\Large \bf 
{ Bernard Helffer$^{1,3}$ and Abderemane Morame $^{2}$}}

\centerline{\bf {November 13, 2000}} 

\vskip 0.5cm 



%address 


\noindent $^1${\small   D\'epartement de Math\'ematiques, UMR CNRS 8628, 
B\^at. 425, 
Universit\'e Paris-Sud,\\ 
 F-91405 Orsay Cedex, France. \\ 
E.Mail: Bernard.Helffer@math.u-psud.fr}\\ 
$^2${\small  
D\'epartement de Math\'ematiques, UMR  CNRS 6629 , Universit\'e de Nantes,\\  
2, rue  de la Houssini\`ere, B.P. 92208, 44322 Nantes Cedex 3, France.\\  
E.Mail: morame@math.univ-nantes.fr}\\
\noindent $^3${\small This work has been partially supported by the European Union under the TMR Program EBRFXT 960001.}\\
%
%RESUME
%
\begin{abstract}
Motivated by the theory of superconductivity and more precisely by the problem
 of the onset of superconductivity in dimension two, a 
lot of papers  devoted to the analysis in a semi-classical regime 
 of the lowest eigenvalue of the Schr\"odinger operator 
with magnetic field have appeared recently. Here we would like to mention the works by 
Bernoff-Sternberg, Lu-Pan and Del Pino-Felmer-Sternberg.
This recovers
 partially questions analyzed in a different context 
by the authors around the question of the so called magnetic bottles. 
Our aim 
 is to analyze the former results, to treat them in a more systematic way
 and to improve them by giving sharper estimates
 of the remainder. In particular, we improve significatively the lower bounds
 and as a byproduct we solve a  conjecture proposed by Bernoff-Sternberg  concerning the localization of the 
 ground state inside the boundary in the case with constant magnetic fields.
 \end{abstract} 
% %FIN DE RESUME %
\tableofcontents
\newpage

\section{Introduction}
We would like to discuss  the spectrum of various selfadjoint realizations of the Schr\"odinger operator~:
$$
P_{h, A,\Omega} = (h \, D_{x_1} - A_1)^2 + (h \,  D_{x_2} - A_2)^2\;,
$$
in an open set $\Omega\subset \rz^2$, where $h >0$ is a small parameter. Note that
 in the case of the problems considered in superconductivity the parameter
 $h$ is proportional to the inverse of the intensity of some external field but
 the question which is posed in this theory  is actually a semi-classical question, that
 is the question of analyzing  the behavior of the spectrum as $h \ar 0$ (more specifically the asymptotic behavior
 of the ground state energy and the  localization of the ground state).\\
We have  shown in \cite{HeMo2} (note that the former spelling of the second author
 was  Abderemane  Mohamed), that, in the case when $\Omega=\rz^2$, and in the  semiclassical regime, the potential 
 $h |B(x)|$ 
 plays the role of an effective electric  potential. By this we mean that, very roughly, the analysis of the
 Schr\"odinger operator~: $- h^2 \Delta + h |B(x)|$, could give some good insight
 on the localization of the ground state.\\
 For PDE specialists, this
 is strongly related to Garding-Melin-H\"ormander
 lower bounds (see \cite{Mel}, \cite{HeRo}) for the operator~: 
$$
P_{h , A D_t} = (h \, D_{x_1} - A_1 D_t)^2 + (h \,  D_{x_2} - A_2 D_t)^2\;,
$$
in $\Omega\times \rz$.\\
Although the case with boundary is
 not explicitely discussed in the former literature (\cite{HeMo2}), one gets similar results (in the case with boundary), {\bf for the Dirichlet
 problem}, at least
 under the additional  condition 
\begin{equation}\label{w1}
b <b'\;,
\end{equation}
where we have used the notations~:
\begin{equation}\label{defbb'}
\inf_{x\in {\bar \Omega}} |B(x)| = b\;,\; \inf_{x\in {\pa \Omega}} |B(x)| = b'\;.
\end{equation}

\begin{theorem}.\\
Under the condition (\ref{w1}), the lowest eigenvalue $\lambda^{(1)}(h)$ of the Dirichlet realization $P_{h,A,\Omega}^D$ of $P_{h,A,\Omega}$ satisfies~:
\begin{equation}
\frac{ \lambda^{(1)}(h)} {h} = b  + o(1)\;.
\end{equation}
\end{theorem}

The points where the minima of $|B|$ are obtained are called the magnetic wells.
The decay outside the minima of $|B|$ is related to the Agmon distance
  associated with $|B| - b $.\\
We recall that the estimate is quite easy to prove, when the dimension is $2$
 and when $B$ has a constant sign due to the inequality~:
\begin{equation}\label{1.3}
\pm h  \int B(x) |u(x)|^2 \leq \langle  P_{h,A} u\;|\; u \rangle\;,\; \forall
 u\in C_0^\infty(\Omega)\;.
\end{equation}
Here $\langle \cdot \;|\;\cdot\rangle$ denotes the scalar product in $L^2(\Omega)$ and $||\;\cdot\;||$ will denote the corresponding $L^2$ norm.\\
When $B$ has not a constant sign, a similar estimate is obtained but with 
 a remainder estimate (See \cite{HeMo2}, Theorem~3.1) in $\Og (h^\frac 54)$.
The most difficult case when the dimension is greater than $2$ is also treated.

As in the case when $B=0$, where we discussed various results according to the
 properties of $V$ near the minimum, one can discuss various results
 according to the properties of $|B(x)|$ near the minimum (see \cite{HeMo2}, \cite{Mon}).

As we shall see later, {\bf this property is no more 
true in the case of the Neumann problem}. The infimum of $|B(x)|$ over $\bar \Omega$
 is no more the good measure for analyzing the bottom of the spectrum, also
 when (\ref{w1}) is satisfied.\\
When the condition (\ref{w1}) is not satisfied, the situation is less clear.\\
First we observe  by comparison between Neumann and Dirichlet that, 
 when (\ref{w1}) is satisfied, we have, for the lowest eigenvalue $\mu^{(1)}(h)$ of the Neumann realization $P_{h,A,\Omega}^N$ of $P_{h,A,\Omega}$, the same upper bound
as for $\lambda^{(1)}(h)$. 
But this is  also true when the condition is not satisfied (if one comes back for example 
 to the proof 
 by Helffer-Mohamed \cite{HeMo2}, which is done, after a suitable gauge transform,
 by construction of gaussian
 quasimodes). This  was also stated more recently
 by Lu-Pan \cite{LuPa2} (Lemma~6.1) with a similar proof.
\begin{theorem}\label{ThHMLP}.\\
 The lowest eigenvalue $\mu^{(1)}(h)$ of the Neumann realization $P_{h,A,\Omega}^N$ of $P_{h,A,\Omega}$ satisfies~:
\begin{equation}
\frac{ \mu^{(1)}(h)} {h} \leq  b  + o(1)\;.
\end{equation}
\end{theorem}
 The problem
 is that the corresponding lower bound is no more always  true.\\

The aim of this paper is to discuss all these results and give some improvements
 in the case of the Dirichlet realization and of the Neumann realization. We shall
 in particular
 show how the technique of the Agmon estimates developed in \cite{HeSj1} and \cite{HeMo2}
 will give the localization of the groundstate in a rather more accurate
 form than in the previous works of Lu-Pan \cite{LuPa2} 
and Del Pino-Felmer-Sternberg \cite{PiFeSt}. May be the more significant result is the proof (cf Theorems \ref{Tlocal} and \ref{tbetter}) of the following conjectured
 property~:
\begin{theorem}\label{MainTh}.\\
Let us assume that the magnetic field is constant and not zero. Then any normalized  groundstate of the Neumann realization of $P_{h,A,\Omega}$ 
 is exponentially localized as $h\ar 0$ in the neighborhood of the points of the boundary with maximal curvature.
\end{theorem}

\paragraph{Contents}.\\
\noindent The paper is organized as follows.\\
In Section \ref{Section2}, we establish rough results under very weak assumptions on the potential. This extends a result established for the case with electric field
 \cite{BeHeVe} and was partially motivated by a question of S. Zelditch.
In Sections \ref{Sectionbasmod} and \ref{Section4}, we analyze
 the reference models corresponding to the half plane and to the disk. Section \ref{Section5} continues the general description of the 
 state of the art and exhibits new questions. In Section \ref{Section6}, we recall the method of the Agmon estimates in the most basic cases and show how it
 gives at a low price results concerning the concentration of the eigenvalues near the boundary (under specific conditions) or in the interior. We pursue
 in this section the analysis of the decay inside the boundary when some general condition on the magnetic field is satisfied.
Section \ref{Section7} treats the Dirichlet case or the Neumann case, when one can a priori show that the groundstate is localized  inside a compact of the open set $\Omega $. We 
 in particular give a two-term expansion of the groundstate enrgy, improving a one term expansion given in \cite{HeMo2}. Section \ref{br} discusses the Dirichlet problem when the boundary
 plays an important role, that is when the minimum of the magnetic field is attained at the boundary.
Section \ref{Section9} comes back to the analysis of the Neumann problem, we recover with a better control  error term  the upper bounds of \cite{BeSt}, \cite{LuPa2}, \cite{PiFeSt}
(this is not simply an academic problem but could have consequence in the superconductivity problems) and improve the lower bound of \cite{LuPa1} in the generic case.
The two last sections \ref{Section10} and \ref{Section11} are devoted to the case of the constant magnetic case and to the complete proof of the conjecture of \cite{BeSt}.\\
Finally, the two appendices recall now well known results which are frequently used along the paper.


\noindent \paragraph{Acknowledgements}~:\\
Preliminary versions of some of  these results were presented 
by the first author 
at various seminars or meetings~: Paris XIII, Leyden, Berlin, Jerusalem, San Feliu de Guixols, Singapore  during the academic years 1999-2000 and 2000-2001. We would like
 to thank all the organizers of these fruitful meetings.
 Many thanks also to P.~Sternberg and X-B.~Pan
 for communicating to us interesting preprints, to M.~Dutour,  L.~Erd\"os, X-B.~Pan and  U.~Smilansky for useful discussions
 and to K.~Hornberger for the nice pictures he has transmitted to us.


\section{Very rough lower bounds.}\label{Section2}
Let us start the analysis of the question with very rough estimates.
In the case of Dirichlet and if $B(x)\neq 0$ (say for example $B(x) >0$), we can use\footnote{ S.~Zelditch
 was transmitting to one of us (July~1999) a question of this type
 coming from geometry.}
 (\ref{1.3})
 which gives a comparison between selfadjoint operators in the form (for any $\rho \in [0,1]$)
\begin{equation}
P_{h,A}^D \geq \rho (P_{h,A}^D ) + (1- \rho) h  B(x)\;.
\end{equation}
In order to find a lower bound
 for the smallest eigenvalue of the Dirichlet realization, it is enough to 
 apply for a suitable $\rho$ a rough lower bound for the operator~:

$$
\rho (P_{h,A}^D ) + (1- \rho) h  B(x)\;.
$$
Using Kato's inequality (cf for example \cite{CFKS}), we can look for a lower bound for the operator~:
$$
- \rho h^2 \Delta_\Omega^D + (1- \rho) h  B(x)\;.
$$
We recall that this inequality gives, for any real potential $V$,  the comparison~:
\begin{equation}\label{diamD}
\inf \;\sp \left(P_{h,A,\Omega}^D + V \right)  \geq \inf \;\sp \left( -h^2 \Delta_\Omega^D + V \right)\;,
\end{equation}
and that a similar result is true in the case of Neumann (see \cite{HHOO})~:
\begin{equation}\label{diamN}
\inf \;\sp \left(P_{h,A,\Omega}^N + V \right) \geq \inf \;\sp \left( -h^2 \Delta_\Omega^N + V \right)\;,
\end{equation}

We shall show  as quite preliminary result the following proposition~:
\begin{proposition}.\\
 Under the condition that
$x\mapsto B(x)$ is $\geq 0$,  non constant and  analytic, then there exists $\theta \in ]0, \frac 12]$ and $C >0$ such that~:
\begin{equation}\label{min}
\lambda^{(1)}(h) -  b h \geq  \frac{1}{C} 
h^{\frac{1}{\theta} }\;.
\end{equation}

\end{proposition}

{\bf Proof~:}\\
Using well known lower bounds for the Schr\"odinger operator $- \epsilon \Delta + V$ (see \cite{BeHeVe}) (related to the Cwickel-Lieb-Rosenblyum estimate)
with $\epsilon = \rho h$ and $V (x)= \frac 12 (B(x) - b)$, 
we start from the property, that there exists $\theta \in ]0,\frac 12]$ and $C$ such that,
 $\forall \rho \in ]0, \frac 12]$,
$$
\lambda^{(1)}(h) - (1-\rho) h b \geq \frac{1}{C} (\rho h^2)
 (\rho h )^{-\theta}\;.
$$
This can be rewritten in the form~:
$$
\lambda^{(1)}(h) - h b \geq \frac{1}{C} \rho^{1-\theta} 
h^{2-\theta} - b  \rho h\;,
$$
or
$$
\lambda^{(1)}(h) - h b \geq \rho^{1-\theta} h
 \left(
\frac {1}{C}  h  ^{1-  \theta} - b  \rho^\theta \right)\;.
$$
If we take $\rho = \gamma h^ {\frac{1-\theta}{\theta}}$ and
 $\gamma b $ small enough, we get (\ref{min})
for $h$ small enough.\\

\begin{remark}.\\
{\rm 
The optimality of this inequality will be discussed later in  particular cases. In particular, we will
 discuss the case when $B(x)= b $ and the case when $B(x) - b $ has a non degenerate minimum.}
\end{remark}
\begin{remark}.\\
{\rm 
When $b =0$, we can take $\rho = \frac 12$, and get, for some $\theta >0$~: $$
\lambda^{(1)}(h) \geq \frac{1}{C}h^{2- \theta}\;.
$$
Results in \cite{HeMo2}, \cite{Mon},  \cite{Ue2} or \cite{LuPa1} show that it is optimal.}
\end{remark}

\begin{remark}.\\
{\rm Note finally  that if $B\equiv 0$ in a simply connected open set $\omega$ 
(with ${\bar \omega}\subset \Omega$), then~:
$$
\lambda^{(1)}(h) \leq  {\tilde \lambda}^{(1)}(\omega) h^{2}\;,
$$
where ${\tilde \lambda}^{(1)}(\omega)$ is the lowest eigenvalue of the Dirichlet Laplacian in $\omega$.}
\end{remark}



\section{The basic models in the half-plane}\label{Sectionbasmod}

Let us consider in a regular domain $\Omega$ of $\rz^2$ the Neumann (or Dirichlet)
 realization of the operator~:
\begin{equation}\label{bm1}
P_{h,b A_0,\Omega}:= ( h  \, D_{x_1} - \frac b 2  x_2)^2 + (h\, D_{x_2} +  \frac b  2 x_1)^2\;,
\end{equation}
with
\begin{equation}
A_0(x_1,x_2) = (\frac 12 x_2, -\frac 12 x_1)\;.
\end{equation}
Note that, by Neumann realization, we mean the condition~:
\begin{equation}
\nu \cdot (h \nabla - i b A_0) u= 0 \mbox{ on } \pa \Omega\;,
\end{equation}
where $\nu$ is the external normal at the boundary.\\
The parameter $h$ is the semiclassical parameter and $b$ is the constant magnetic field.
 Of course, as we have seen already,  replacing $(h,b)$
 by $(1, \frac{b}{h}) $ does not really change the problem. We have indeed~:
\begin{equation}\label{bm2}
P_{h,b A_0} = h^2 P_{1,b A_0/h}\;.
\end{equation}
Here $b$ is assumed to be different from $0$ and, without loss of generality, we can assume~:
\begin{equation}\label{bm3}
b >0\;.
\end{equation}
So the problem of analyzing (for fixed $b$) the semiclassical limit~: $h \ar 0$
 is the same as the problem of analyzing (for fixed $h >0$) the large field limit
 $b \ar + \infty$.\\
Let us denote by $\mu^{(1)} ( h, b, \Omega)$ and 
$\lambda^{(1)} ( h , b , \Omega )$ the bottoms
 of the spectrum of the Neumann or Dirichlet  realization of $P_{h, b A_0}$ in $\Omega$. 
Depending on $\Omega$,
 this may correspond to an eigenvalue (if $\Omega$ is bounded) or to a point in the essential spectrum
 (for example if $\Omega = \rz^2$ or $\Omega = \rz^2_+$).
Let us recall that~:


\begin{equation}\label{bm4}
\mu^{(1)} ( h, b, \rz^2)= \lambda^{(1)} ( h , b , \rz^2) = h  b \;.
\end{equation}
$b$  is an eigenvalue with infinite multiplicity.\\

We note that in the Dirichlet case, we have by monotonicity the lower bound~:
\begin{equation}\label{bm5}
\lambda^{(1)} ( h , b, \Omega)\geq  \lambda^{(1)} ( h , b, \rz^2) = h  b\;.
\end{equation}
Such a lower bound is false in the case of the Neumann problem and this makes the problem
 more interesting.

The analysis in $\rz^2_+ = \{ (x_1,x_2) \in \rz^2 | x_2 >0\}$ is less known and
 will play an important role in our discussion.\\

First we observe by construction of quasimodes and (\ref{bm5}) that~:
\begin{equation}\label{bm6}
\lambda^{(1)} (h , b, \rz^2_+) = h  b\;.
\end{equation}

Let us now analyze the Neumann case.\\
We take $h =1$ and first perform a gauge transformation which leads to the analysis
 of the operator~:
\begin{equation}\label{bm7}
{\widetilde S_{b}} = D_{x_2}^2 + (D_{x_1} - b  x_2)^2\;,
\end{equation} 
with Dirichlet or Neumann conditions. Note that we now get the standard Neumann condition
 on $x_2 = 0$.\\
In order to make the spectral analysis of the operator, we can perform
 a partial Fourier transform and we get~:
\begin{equation} \label{bm8}
{\widehat S_b} (t, \sigma, D_t; B) = D_t^2 + ( \sigma - b \,t )^2
\end{equation}
considered as an operator on $\rz^2_+ = \{ (t, \sigma ) \in \rz^2\; |\; t>0 \}$ with Dirichlet
 or Neumann conditions on $t=0$.\\
A scaling shows that it is enough to consider the case $b =1$ and then we recover
 the general case by multiplication by $b$.\\
The analysis of the spectrum is then reduced to the spectral analysis of
 an $\xi$-family of differential operators 
\begin{equation}\label{mod1}
H(t, D_t;\xi):= D_t^2 + (t-\xi)^2\;.
\end{equation}
We shall consider the Dirichlet and Neumann realizations of this operator in $\rz^+$
 respectively denoted by $H^{D,\xi}$ and $H^{N,\xi}$.
Let us observe that the operator is unitary equivalent to the corresponding realization of
 the harmonic oscillator $D_t^2 + t^2$ in $]-\xi, + \infty[$.\\
The Dirichlet spectrum of $P_{1, A_0}$ is obtained by
\begin{equation}\label{bm9}
\sp (P_{1,A_0}^D) = \{ \lambda \in \rz^+ | \exists \xi \in \rz \mbox{ s. t. } \lambda
 \in \sp (H^{D,\xi}) \} \;,
\end{equation}
and similarly for Neumann.
The eigenvalues of $H^{D,\xi}$ and $H^{N,\xi}$ are analytic with respect to $\xi$ and not constant. Hence  one obtains easily (See Reed-Simon, Vol.~IV \cite{ReSi}) that we get a priori a continuous  band spectrum in the two cases.\\
The analysis of the bottom of the spectrum is an immediate consequence of the
 analysis of the lowest eigenvalue of $H^{D,\xi}$ or $H^{N,\xi}$ as a function of $\xi$.\\
Here the answer is quite different for Dirichlet and Neumann.\\
For Dirichlet, the  lowest eigenvalue ${\hat \lambda}^{(1)} (\xi)$ decays monotonically 
from $+ \infty$
 to $1$ as $\xi$ goes from $- \infty$ to $+ \infty$.\\
For Neumann, the  lowest eigenvalue ${\hat \mu} ^{(1)} ( \xi)$ decays monotonically 
from $+ \infty$
 to $1$ as $\xi$ goes from $- \infty$ to $0$. Then one can show \cite{DaHe} that there exists
 $\xi_0 >0$ such that  ${\hat \mu}^{(1)} ( \xi )$ continues to decay monotonically  till
 some value 
\begin{equation}\label{Dau}
\Theta_0:={\hat \mu}^{(1)} (\xi_0)<1
\end{equation}
 and increases again monotonically 
to $1$ as $ \xi \ar + \infty$ (see Appendix \ref{AppA} for more details) 
 and we get 
\begin{proposition}\label{Propositionspectre}.\\
The spectrum of the Dirichlet realization  in $\rz^2_+$ is~:
\begin{equation}\label{bm10}
\sp (P_{h, bA_0, \rz_+^2}^D ) = [bh, + \infty [\;.
\end{equation}
The spectrum of the Neumann realization  in $\rz^2_+$ is~:
\begin{equation}\label{bm11}
\sp (P_{h, bA_0, \rz_+^2}^N ) = [ b  \Theta_0 h , + \infty [\;,
\end{equation}
with
\begin{equation}\label{bm12}
0<\Theta_0 <1\;.
\end{equation}
\end{proposition}


The fact that the bottom of the spectrum of the Neumann problem is strictly lower
 that the bottom of the problem in $\rz^2$ will play a quite important role in the discussion.

\begin{remark}.\\
{\rm The analysis of the Dirichlet problem in the half-plane appears
 also in the study of the quantum Hall effect. We refer to \cite{DePu}
 and to \cite{FrGrWa} for more general domains.}
\end{remark}

\begin{remark}.\\
{\rm 
The case of a domain with a corner was recently analyzed in \cite{Ja} and \cite{Pan}. It
 is in particular proved that in domains with right angles, we have~:

$$ \mu^{(1)} (b, \Omega) \leq  \Theta_1  b
+ o(b ) \;,
$$
for some $\Theta_1$ satisfying~:
$$ 0 < \Theta_1 < \Theta_0.$$
This inequality is obtained by the choice of a suitable quasimode.
This $\Theta_1$ is the lowest eigenvalue of the Neumann realization
 in~: $$\rz_{++}^2 = \{ (x_1,x_2)\in \rz^2\;|\; x_1 >0, x_2 >0\}\;.$$
The localization of the ground state in the corners is also obtained by this author.}
\end{remark}

\section{The case of  the disc in the constant magnetic case.}\label{Section4}

We first  present the results by L.~Erd\"os (\cite{Er1}, \cite{Er2})  and give an alternative proof for improving some of the statement.
When considering lower bounds, it is useful to mention the following  result obtained by L.~Erd\"os~:
\begin{proposition}.\\
For any planar domain $\Omega$ and $b  >0$, we have~:
\begin{equation}\label{ineq}
\lambda^{(1)}(b , \Omega) \geq \lambda^{(1)} (b , D(0,R))\;,
\end{equation}
where $D(0,R)$ is the disk with same area as $\Omega$~:
 $$\pi R^2 = Area(\Omega)\;,$$ 
and $\lambda^{(1)}(b ,\Omega)$ is the lowest
 eigenvalue  of the Dirichlet
realization of the Schr\"odinger operator with constant magnetic field $b  >0$.\\
Moreover the equality in (\ref{ineq})  occurs if and  only if $\Omega = D (0,R)$.
\end{proposition}

\noindent In the case of the disk, L.~Erd\"os observes that
 $\lambda^{(1)}(b , D (0,R))$ is equal to the lowest eigenvalue
 of the Dirichlet realization of the two dimensional   harmonic oscillator~: 
$-\Delta + \frac{r^2}{4}$ in $D(0,R)$.\\
This one is equal to $ 2 {\hat \lambda}^{(1)}(]-R,+R[)$  where ${\hat \lambda}^{(1)}(]-R,+R[)$
 is the groundstate energy of the  one dimensional harmonic oscillator $-\frac{d^2}{dx^2}  + \frac 14 x^2 $
 in the interval $]-R,+R[$ with Dirichlet condition.\\

 
In the one-dimensional case, we can then apply semi-classical
 techniques developped in this context 
in \cite{Bo}, \cite{BoHe} for getting the following improvement of \cite{Er1} 
and \cite{Er2}~:
\begin{proposition}.\\
In the case of Dirichlet conditions in $]-R,+R[$, the lowest eigenvalue
 of the Dirichlet realization of $-d^2/dx^2 + x^2/4$ satisfies, in the limit $R\ar +\infty$~:
\begin{equation}\label{DimD}
{\hat \lambda}^{(1)}(]-R,+R[) - \frac 12  \sim 2^\frac 12   \pi^{-\frac 12}  R \exp (- \frac 12 R^2 )\;.
\end{equation}
\end{proposition}

The proof is quite similar to the proof given in \cite{Bo} (pages 250-254)
for treating the Neuman case
 \cite{Bo} (Formula (2.23)) and \cite{BoHe}~:

\begin{proposition}.\\
In the case of Neumann  conditions in $]-R,+R[$, the lowest eigenvalue
 of the Neumann  realization of $-d^2/dx^2 + x^2/4$ satisfies, in the limit $R\ar +\infty$~:
\begin{equation}\label{DimN}
 {\hat \mu}^{(1)}(]-R,+R[) - \frac 12  \sim - 2^\frac 12  \pi^{-\frac 12} R \exp (- \frac 12  R^2) \;. 
\end{equation}
\end{proposition}

As is expected, we observe that the boundary effect is very small as $R\ar +\infty$~:
$$
{\hat \lambda}^{(1)}(]-R,+R[)  - {\hat \mu}^{(1)}(]-R,+R[)  \sim
 2^\frac 32  \pi^{-\frac 12} R \exp (- \frac 12  R^2)\;,
$$
as $R$ tends to $\infty$.

Coming back to our initial problem, this gives using the scaling invariance~:
\begin{proposition}.\\
In the regime  $R\sqrt{b }$ large, we have the asymptotic~:
\begin{equation}\label{scl}
\lambda^{(1)} (b , D(0,R)) - b  \sim  2^\frac 32  \pi^{-\frac 12} b^\frac 32  R \exp ( - \frac {b   R^2}{2}) \;. 
\end{equation}
\end{proposition}

\begin{remark}.\\
{\rm 
It is a natural question to ask for  an isoperimetric inequality in the case of Neumann, that is if the inequality
 $\mu^{(1)}(b,\Omega)\geq \mu^{(1)} (b,D(0,R))$ is true. But the recent results by Sternberg and coauthors \cite{BeSt}, \cite{PiFeSt} that we shall discuss here, 
 compared with those of Bauman-Phillips-Tang \cite{BaPhTa},  contradict this
 result. Playing with the maximal curvature at the boundary and keeping fixed the area, one can indeed (in the semiclassical regime) get a contradiction with this isoperimetric inequality (see Theorem \ref{Thster} in this paper) in the regime  $b$ large.}\\
$\bullet$

\end{remark}
\begin{remark}.\\
{\rm In the paper by Baumann-Phillips-Tang \cite{BaPhTa} (Theorem 6.1, p.~24)
 (see also \cite{PiFeSt}) the authors propose the following asymptotic as $b$ is large~:
\begin{equation}\label{fPh}
 \mu^{(1)} (b, D(0,R)) = \Theta_0 b  - 2 M_3 \, \frac 1 R \, b ^\frac 12 +
\Og(1)\;.
\end{equation}
Here we recall that 
$\Theta_0$ is introduced in (\ref{Dau})  and $M_3 >0 $ is a universal constant
which will be explicited in Appendix  \ref{AppA} (cf (\ref{7ct2}) 
and (\ref{M3})).  We shall give an alternative proof in Section \ref{Section11}.}\\
$\bullet$
\end{remark}
\begin{remark}.\\
{\rm 
Another interesting result is the case of the exterior of the disk. One 
 can first observe that the bottom of the essential spectrum is equal to $b$.
One can show that, when $b$ is large, there is at least one eigenvalue
 below $b$. Moreover the groundstate energy will satisfy~:
\begin{equation}
 \mu^{(1)} (b, D(0,R)^c) = \Theta_0 b  +  2 M_3  \, \frac 1 R \, b ^\frac 12 +
\Og(1)\;.
\end{equation}
This last result, which seems new, will be proven in Section \ref{Section11}.
}\\
$\bullet$
\end{remark}
Finally, let us observe that, due to the homogeneity, we get for free the following semi-classical corollaries
 of respectively (\ref{scl}) and (\ref{fPh}).
\begin{proposition}\label{PphiD}.\\
The lowest eigenvalue of $P_{h,A_0,D(0,R)}^D$ satisfies as $h\ar 0$~:
\begin{equation}
\lambda^{(1)}(h,b,D(0,R)) -  b h  \sim 2^\frac 32 \pi^{-\frac 12} 
b^\frac 32 h^\frac 12 R \exp ( - \frac{ b R^2}{2h}) \;.
\end{equation}
\end{proposition}

\begin{proposition}\label{Pphil}.\\
The lowest eigenvalue of $P_{h,A_0,D(0,R)}^N$ satisfies as $h\ar 0$~:
\begin{equation}
\mu^{(1)}(h,b,D(0,R)) =  \Theta_0 b h  - \frac 2 R  \, M_3 \, b^\frac 12 \,  h^\frac 32 + \Og(  h^2)\;.
\end{equation}
\end{proposition}


\section{Semiclassical questions: former results}\label{Section5}
We now consider the questions which are considered in superconductivity. 
In the case of a bounded regular open set, del Pino, Felmer and Sternberg  \cite{PiFeSt}
 obtain the following results.

\begin{theorem}\label{Thster0}.\\
Let $\Omega\subset \rz^2$ be a bounded, open, simply connected domain with a $C^\infty$ boundary 
 $\pa \Omega$. Let $A$ be a potential in $\Omega$ such that $B(x)=b >0$. 
If $u_h$ is a sequence of groundstates for the Neumann realization
 of $ P_{h,A,\Omega}^N $ in $\Omega$ associated with the lowest eigenvalue $\mu^{(1)}(h)$.
Then, there exist $h_0 >0$ and constants $c_1 >0$ and $c_2 >0$ such that~:
\begin{equation}
|u_h (x) | \leq c_1\; ||u_h ||_{L^\infty(\Omega)} \; \exp ( - c_2 h^{-\frac 12} \mbox{dist } (x, \pa \Omega)~)\;,
\end{equation}
for all $x\in \Omega$.\\
Moreover if $\Omega$ is not a disc, then we have~:
\begin{equation}\label{5.2}
\lim_{\begin{array}{c}
h\ar 0\\
h>0
\end{array} } \left( \min_{x\in \pa \Omega} |u_h (x) |\right) = 0\;.
\end{equation}
\end{theorem}
The second theorem proved by Bernoff-Sternberg \cite{BeSt} (see also \cite{LuPa3}) gives an upper bound of $\mu^{(1)}(h)$. The localization  (\ref{5.2}) is a weak version of a much stronger result which
 will be proved in Section \ref{Section11}.
\begin{theorem}\label{Thster}.\\
Under the assumptions of Theorem \ref{Thster0}, we have~:
\begin{equation}\label{5.3}
\mu^{(1)}(h) \leq \Theta_0  h - 2 \kappa_{max} M_3  h^\frac 32 + o(h^\frac 32)\;.
\end{equation}
\end{theorem} 
Here $\kappa_{max}$ is the value of the maximal
 curvature which is assumed to be strictly 
positive and $M_3$ is a universal constant (cf (\ref{M3})).\\
The proof  is essentially a consequence of 
the construction of matched solutions. Let us just mention that in suitable coordinates 
 $(s,t)$
 ($t =0$ defining the boundary)(see the appendix \ref{AppB})
 in a neighborhood of a point of maximal curvature ($s=0$ defining this point), the solution
 takes the form~:\\
\begin{equation}
u:= \phi (h^{-\frac 12} t ) \exp - \gamma h^{-\frac 14} s^2 \,\cdot\,\exp i h^{-1} [ \Phi_0 (s,t) + h^{\frac 12}
 \Phi_1(s,t)]
\end{equation}
Note that it is coherent with
 the result given, in the case of the disc, by 
 Bauman-Phillips-Tang \cite{BaPhTa} which was already discussed in (\ref{fPh}).
 We will come back to  these two results
 in Sections \ref{Section10} and \ref{Section11}.\\


This result is completed by a theorem by Lu-Pan \cite{LuPa2} which is valid for general
 magnetic fields~: 
\begin{theorem}\label{ThLuPa2}.\\
\begin{equation}
\lim_{h\ar + 0}  \frac{\mu^{(1)}(h)}{h} = \min \left( \Theta_0
b'\;,\;b \right)\;.
\end{equation}
\end{theorem}
\begin{remark}.\\
{\rm 
1. Although it is not completely
 proved in these previous contributions, the localization of the 
Neumann groundstate is  far from the boundary when
$$ b < \Theta_0 b'\;.$$
This will be shown in this paper.\\
2. When $b > \Theta_0 b' $, one will show that 
the groundstate is localized near the boundary. This is for example the
 case when the magnetic field is constant (which is analyzed in \cite{PiFeSt}).\\
3.  It is 
 interesting to have a better localization (than for example (\ref{5.2}) in the constant magnetic field 
case) 
inside the boundary. One can wait for example for exponential decay outside
 the minima of $|B|_{/\pa \Omega}$ assuming they are isolated.\\
Preliminary arguments are given by \cite{PiFeSt} in the case $B=\const$ but we shall come back to the problem
 in Sections \ref{Section10}  and \ref{Section11}. \\
4. When $b' >0$ and $b  =0$, the analysis
 by \cite{HeMo2} is relevant. The case when $b' =0$
 and $ |B(x)| >0$ in $\Omega$ is treated in \cite{LuPa2}.\\
5. Note that Giorgi-Phillips \cite{GiPh} and Ueki \cite{Ue2} 
give earlier weaker lower bounds  which have independent interest.}
\end{remark}

\section{IMS type decomposition and Agmon's estimates}\label{Section6}
\subsection{Preliminaries}
We recall in this section the technique of the Agmon estimates and show
 how they give rather easily rough results which will be improved later.
In the semi-classical context, this technique was mainly developed by
 Helffer-Sj\"ostrand \cite{HeSj1}, \cite{HeSj4} but the inspiration comes from former results
 due to many authors~: Lithner, Agmon \cite{Ag}, Simon \cite{Si}
 between others.\\

In the Dirichlet case, the inequality (\ref{1.3}) was (at least when the condition $B(x) >0$ is satisfied)
the starting point of the analysis of the decay. This is no more the case when Neumann boundary conditions
 are assumed, but we can keep the general strategy as developped in \cite{HeMo2}.\\
We assume that $\Omega$ is a bounded, regular open set and that
\begin{equation}\label{A1}
B(x) >0\;.
\end{equation}
One could also consider unbounded domains but in this case one  has to add
 some assumption at $\infty$, permitting to localize the essential spectrum.
\subsection{Upper bounds}
Using suitable quasimodes, one can  get~:
\begin{equation}\label{LuPa3}
\mu^{(1)} (h) \leq   \min (b, \Theta_0 b') \,h  + C\,  h^{\frac 54}\;,
\end{equation}
We will show improved version of this property later but note that Lu-Pan give
 at least the weaker~:
\begin{equation}\label{LuPa3w}
\mu^{(1)} (h) \leq  \min (b, \Theta_0 b')\,h + o(h)\;,
\end{equation}
which is enough for our analysis of the decay. Note also that the upper bound involving $\inf B$
 can also be obtained by using \cite{HeMo2}.\\
Further improvements of (\ref{LuPa3}) will be given in Theorem \ref{Thnm1}.
\subsection {Lower bounds}
Let $0 \leq \rho \leq 1$. We first claim that there exists $C$ such that, for any $\epsilon_0 >0$, we can,  by scaling a standard partition of unity of $\rz^2$, and by restricting it 
to $\overline{\Omega}$, find a partition of unity $\chi^h_j$ satisfying in $\Omega$,
\begin{equation}\label{U1}
\sum_j |\chi^h_j|^2 =1\;,
\end{equation}
\begin{equation}\label{U2}
\sum_j|\nabla \chi_j^h|^2 \leq C \, \epsilon_0^{-2}\, h^{-2 \rho}\;,
\end{equation}
and 
\begin{equation}\label{U3}
\supp (\chi^h_j) \subset Q_{j} = B(z_j, \epsilon_0 \, h^\rho)\;,
\end{equation}
where $B(c,r)$ denotes the open disc in $\rz^2$ of center $c$ and radius $r$.
Moreover, we can add the property that~:\\
\begin{equation}\label{U4}
\mbox{ either } \supp \chi_j \cap \partial \Omega = \emptyset\;,\; \mbox{ either } z_j \in \partial \Omega\;.
\end{equation}
According to the two alternatives in (\ref{U4}), we can decompose the sum in (\ref{U1}) in the form~:
$$ \sum = \sum_{int} + \sum_{bnd}\;,$$
where ``int'' is in reference to the $j$'s such that $z_j \in \Omega$ and ``bnd'' is
 in reference to the $j$'s such that $z_j \in \pa  \Omega$.\\

The second point is to implement this partition of unity in the following way~:
\begin{equation} \label{IMS1}
q_h^N (u) = \sum_j q_h (\chi_j^h u) - h^2 \sum_j || \;|\nabla \chi_j^h|\, u \; ||^2\;,\; \forall u \in H^1 (\Omega)\;.
\end{equation}
Here $q_h^N$ (or $q_{h,A}^N$, if we want to keep the reference to the magnetic potential) denotes the quadratic form~:
\begin{equation} \label{IMS2}
q_{h,A}^N (u) = \int_\Omega |h  \nabla u  - i A u |^2 \; dx \;,
\end{equation}
and we recall that $||\;\cdot\;||$ denotes the $L^2$-norm in $\Omega$.\\
This formula is usually called IMS formula (see \cite{CFKS}) but is actually
 much older (see for example \cite{Mel}).\\
If $a_{h,A}^N$ is the associated sesquilinear form, (\ref{IMS1}) is the consequence of the identity, for any function $\chi \in C^\infty({\bar \Omega})$ and
 any $u\in H^1(\Omega)$~:
\begin{equation}\label{inta}
q_{h,A}^N (\chi u) = \Re  a_{h,A}^N (u, \chi^2 u) + h^2 || \, |\nabla \chi| u \,||_{L^2(\Omega)}^2\;.
\end{equation}
We will also use later the property that,
for any function $\chi \in C^\infty({\bar \Omega})$ and
 any $u$ in the domain of $P_{h,A,\Omega}^N$, that is
 for any $u$ in the space \break 
$D (P_{h,A,\Omega}^N):=\{ v\in H^2(\Omega)\;|\; \nu\cdot(\pa -i A) u_{/\pa \Omega}=0 \}$~:
\begin{equation}\label{intb}
q_{h,A}^N (\chi u) = \Re  \langle P_{h,A,\Omega}^N\, u\;|\;  \chi^2 u \rangle_{L^2(\Omega)} + h^2 || \, |\nabla \chi|\,  u \,||_{L^2(\Omega)}^2\;.
\end{equation}

We can rewrite the right hand side of (\ref{IMS1}) as the sum of three (types of) terms.\\

\begin{equation} \label{IMS3}
q_h (u) = \sum_{int} q_h (\chi_j^h u) + \sum_{bnd} q_h (\chi_j^h u)  - h^2 \sum_j || \;|\nabla \chi_j^h|\, u \; ||^2\;,\; \forall u \in H^1 (\Omega)\;.
\end{equation}

For the last term in the right hand side of (\ref{IMS3}), we get using (\ref{U2})~:
\begin{equation}\label{IMS4}
 h^2 \sum_j || \;|\nabla \chi_j^h|\, u \; ||^2 \leq C\,h^{2- 2 \rho}\, \epsilon_0^{-2} \, || u ||^2\;.
\end{equation}
This measures the price to pay when using a fine partition of unity~: If $\rho$ is large,
 the error is big as $h^{2- 2 \rho}$. We shall see later what could be the best choice of $\rho$ or of $\epsilon_0$
 for our various problems (note that the play with $\epsilon_0$ large will be only interesting when
 $\rho = \frac 12$).\\

The first term in the right hand side of (\ref{IMS3}) can be estimated  from below by using (\ref{1.3}). The support of
 $\chi_j^h u$ is indeed contained in $\Omega$. So we have~:
\begin{equation}\label{IMS5}
 \sum_{int} q_h (\chi_j^h u) \geq h\, \sum_{int} \int B(x) |\chi_j^h u|^2 \; dx \;.
\end{equation}

The second term in the right hand side of (\ref{IMS3}) is the more delicate and corresponds to the specificity of the 
Neumann problem. We have to find a lower bound for $q_h (\chi_j^h u)$ for some $j$ such that $z_j \in \partial \Omega$.
We emphasize that $z_j$ depends on $h$, so we have to be careful in the control of the uniformity.\\
Let $z$ be a point in $\pa \Omega$. The boundary being regular, we can, by a change of coordinates in a small neighborhood
 of this point, rewrite the form $q_{h,A}$ for $u$'s with support in this neighborhood of $z$~:
$$
q_{h,A} (u) = \int_{{\tilde x}_2>0}   \sum g^{k,\ell}({\tilde x}) (ih \pa_{{\tilde x}_k}{\tilde u} + A_k({\tilde x}){\tilde u})
\cdot \overline{(ih \pa_{{\tilde x}_\ell}{\tilde u} + A_\ell({\tilde x}){\tilde u})} \det (g({\tilde x}))\;  d {\tilde x}\;.
$$

Here we can assume that the new cordinates of $z$ are $(0,0)$ and we can also assume that the matrix
 $g$ is the identity at $z$~:
$$
g^{k,\ell} (0) = \delta_{k,\ell}\;.
$$
Of course $g$ depends on $z$, but all the estimates we could need on the derivatives of $g$ will be uniform
in  $z$.\\
The game is now to compare for $u$'s with support in a ball of the type $B(z,2  C \epsilon_0 h^\rho)$ $q_{h,A}$ with
 the quadratic form~:
$$
q_{h,{\tilde A}} ({\tilde u}) = \int_{x_2 >0} | (ih \pa_{x_1} -  \frac 12 B(z)  x_2) u|^2 + | (ih \pa_{x_2} + 
  \frac 12 B(z)  x_1) u|^2 \, dx\;.
$$
We have omitted for simplicity the tilde's in the right hand side.
The comparison is not direct but as an intermediate step, we have to use a gauge transformation (multiplication by $\exp - i \frac{\phi_j}{h}$) associated
 to a $C^\infty$ function $\phi_j$ such that~:
$$
\omega_A = \omega_{A_{new,j}} - d\phi_j\;,
$$
with
$$ A_{new,j}(z_j)=0\;,$$
$$ |A_{new,j} (x) - \frac 12 (B(z_j) (-x_2,x_1))|\leq C |x|^2\; .
$$
In this formula, $\omega_A$ is the one-form attached to the vector field $A$~: $\omega_A:= A_1 dx_1 + A_2 dx_2$. Let us  emphasize that $C$ is independent of $j$. Let us also introduce for the next formula~:
 $A_j^{lin}:= \frac 12 (B(z_j) (-x_2,x_1))$. \\
Following line by line the computations of \cite{HeMo2}, we get~:
$$
\begin{array}{ll}
q_{h,A} (\chi_j^h u) &\geq (1- C h^{2 \theta} \epsilon^2 - C \epsilon_0 h^\rho) q^h [A_{j}^{lin}]
 (\exp - \frac ih \phi_j \chi_j^h u)  - C h^{-2 \theta} \epsilon^{-2}
 || |x|^2 \chi_j^h u||^2\\
&\geq (1-  C h^{2 \theta} \epsilon^2 - C \epsilon_0 h^\rho) q^h [A_{j}^{lin}]
 (\exp - \frac ih \phi_j \chi_j^h u) - C h^{4 \rho -2 \theta} \epsilon^{-2}
 || \chi_j^h u||^2\;.
\end{array}
$$
We can now use the result concerning the half -plane in order to get~:
\begin{equation}\label{IMS6}
q_{h,A} (\chi_j^h u) \geq
 (1- C h^{2 \theta} \epsilon^2 - C \epsilon_0 h^\rho) h \Theta_0
\int B(z_j) |\chi_j^h u|^2 \, dx -  C h^{4 \rho -2 \theta} \epsilon^{-2}
 || \chi_j^h u||^2\;.
\end{equation}
We now put together all the estimates and obtain~:
\begin{equation}\label{IMS7}
\begin{array}{ll}
q_{h,A} (u) & \geq h \sum_{int}\int  B(x)| \chi_j^h u|^2 dx\\
&\quad  + (1- C h^{2 \theta} \epsilon^2 - C \epsilon_0 h^\rho) h \Theta_0
\sum_{bnd} \int B(z_j) |\chi_j^h u|^2 \, dx \\
&\quad - C h^{ 4 \rho -2 \theta} \epsilon^{-2} \; \sum_{bnd}  
 || \chi_j^h u||^2 \\
& \quad - C \epsilon_0^{-2}\, h^{2 - 2 \rho} ||u||^2\;.
\end{array}
\end{equation}

We have  now  to optimize our choices of $\rho$, $\theta$
 and $\epsilon$, $\epsilon_0$. If we just want to get a lower bound
 of the spectrum, we can first write~:
$$
\begin{array}{ll}
q_{h,A} (u)& \geq h \min \left( b,
 \Theta_0 b' \right)|| u||^2\\
&\quad  -\left( C h^{2 \theta +1} \epsilon^2 + C \epsilon_0 h^{\rho +1}  +  C h^ { 4 \rho -2 \theta} \epsilon^{-2} +
 C \epsilon_0^{-2} h^{2 - 2 \rho}\right)  ||u||^2\;.
\end{array}
$$
Taking $\rho = \frac 3 8 $, $\theta = \frac {1}{8}$, $\epsilon =\epsilon_0 =1$, we get~:
\begin{equation}\label{comp}
q_{h,A} (u) \geq \left( \min (b ,
 \Theta_0 b') h
 - C h^{\frac 54} \right)  ||u||^2\;.
\end{equation}
So, taking $u=u_h^1$, where $u_h^1$ is a groundstate, we obtain from (\ref{comp})~:  
\begin{proposition}\label{PropLuPa}.\\
There exist constants $C >0$ and $h_0 >0$ such that, for all $h\in]0,h_0]$~:
\begin{equation}\label{LuPa1}
\mu^{(1)}(h) \geq 
 \left (\min ( b ,
 \Theta_0 b') \, \right) h
 - C h^{\frac 54} \;.
\end{equation}
\end{proposition}

This  result is not optimal. We shall show in Theorem \ref{Thnm1} a stronger
 estimate of the remainder in $\Og (h^\frac 32) $.  In the case of a constant magnetic field, we shall actually determine
 the coefficient of $h^\frac 32$ as suggested by the computations of \cite{BeSt} and \cite{PiFeSt}. This will be done in Sections \ref{Section10} and \ref{Section11}.

But for the control of the decay, we need also to take in (\ref{IMS7}) 
$\rho = \frac 12$, $\theta = \frac 18 $, $\epsilon=1$  and $\epsilon_0$ large. This gives an estimate which may look weaker but will be more efficient.
\begin{proposition}\label{Propdec}.\\
There exists $C$ and $h_0$ and, for all $\epsilon_0 >0$, there exists $C(\epsilon_0)$ such that, for $h\in ]0,h_0]$, the following inequality~:
\begin{equation}\label{dec1}
\begin{array}{ll}
q_{h,A} (u) & \geq h \sum_{int}\int  B(x)| \chi_j^h u|^2 dx\\
&\quad   - C(\epsilon_0) h 
\sum_{bnd} \int  |\chi_j^h u|^2 \, dx \\
& \quad - \frac{C h }{\epsilon_0^2} \sum_{int} \int | \chi_j^h u|^2 dx \;.
\end{array}
\end{equation}
is satisfied,   for all $u\in H^1(\Omega)$.
\end{proposition}

\subsection{Agmon's estimates}
We continue to follow the proof given in \cite{HeMo2} and will explain the modifications needed
 in order to treat Neumann problems.\\
We first observe that if $\Phi$ is a real and uniformly Lipschitzian function and if $u$ is in the
 domain of the Neuman realization of $P_{h,A}$, then we have by a simple integration by part (see (\ref{inta}))~:
\begin{equation}\label{Agm1}
\begin{array}{l}
\Re\langle P_{h,A} u, \exp  \frac{2 \Phi}{h^\frac 12} u \rangle \\
 = \Re  \langle (\frac hi \nabla - A) u, (\frac hi \nabla - A) \exp  \frac{2 \Phi}{h^\frac 12} u \rangle \\
= \langle (\frac hi \nabla - A) \exp \frac{\Phi}{h^\frac 12} u, (\frac hi \nabla - A) \exp \frac{\Phi}{h^\frac 12} u \rangle
 - h || |\nabla \Phi|\,  \exp  \frac{\Phi}{h^\frac 12} u ||^2\\
= q_h[A] (  \exp  \frac{\Phi}{h^\frac 12} u)  - h || |\nabla \Phi|\,  \exp  \frac{\Phi}{h^\frac 12} u ||^2\;.
\end{array}
\end{equation}

We now take $u=u_h$ an eigenfunction attached to the lowest eigenvalue $\mu^{(1)}(h)$. This gives~:
\begin{equation}\label{Agm2}
\mu^{(1)} (h) ||  \exp  \frac{\Phi}{h^\frac 12} u ||^2 =  q_{h,A} (  \exp  \frac{\Phi}{h^\frac 12} u)  - h || |\nabla \Phi|\,  \exp  \frac{\Phi}{h^\frac 12} u ||^2\;.
\end{equation}

It remains to reimplement the previous inequality in this new one and  to use the upper bound (\ref{LuPa3}).

Let us take $\Phi(x) = \alpha \max (d(x,\pa \Omega) , h^\frac 12)$, where $\alpha >0$ has  to be determined.
Let us use Proposition \ref{Propdec}. We first write~:
\begin{equation}\label{Agm4}
\begin{array}{ll}
q_{h,A} (\exp \frac{\Phi}{h^\frac 12} u) & \geq h \sum_{int}\int  B(x)| \exp \frac{\Phi}{h^\frac 12} \chi_j^h u|^2 dx\\
&\quad   - C(\epsilon_0) h 
\sum_{bnd} \int  |\chi_j^h \exp \frac{\Phi}{h^\frac 12} u|^2 \, dx \\
& \quad - \frac{C h }{\epsilon_0^2} \sum_{int} \int | \exp \frac{\Phi}{h^\frac 12} \chi_j^h u|^2 dx\;.
\end{array}
\end{equation}
Let us first consider the case $B(x) =b$. The inequality (\ref{LuPa3}) becomes~:
\begin{equation}\label{Agm6}
\mu^{(1)} (h) \leq \Theta_0\,b\,h  + C\, h^{\frac 54}\;.
\end{equation}
Using (\ref{Agm1}), we finally obtain~:
\begin{equation}\label{Agm7}
( b (1-\Theta_0) - C h^\frac 14 - \frac {C}{\epsilon_0^2} - \alpha^2)  
\sum_{int} \int | \exp \frac{\Phi}{h^\frac 12} \chi_j^h u|^2 dx \leq  C(\epsilon_0) 
 \sum_{bnd} \int  |\chi_j^h u|^2 \, dx\;.
\end{equation}
Taking $\epsilon_0$ large enough and
 $$ \alpha < \sqrt{b(1-\Theta_0)}$$
we finally obtain the estimate~:
\begin{equation}\label{Agm8}
|| \exp \alpha \frac {d(x, \pa \Omega)}{h^\frac 12} u_h || \leq C || u_h||\;,
\end{equation}
for some new constant $C>0$.\\


Note that we just need a weak upper bound of $\mu^{(1)}(h)$ in order to make the argument correct. In particular, the
 remainder $C\,  h^\frac 14$ in (\ref{Agm7}) can be replaced by $o(1)$ without changing the argument.
This is the upper bound obtained  in \cite{LuPa2}.\\


Let us now show how to go from an $L^2$-estimate to an $L^\infty$-estimate.
Using (\ref{Agm2}), we first get~:
\begin{equation}\label{Agm2a}
 q_{h,A} (  \exp  \frac{\Phi}{h^\frac 12} u)  \leq C\,  h || \exp  \frac{\Phi}{h^\frac 12} u ||^2\;.
\end{equation}
This gives, together with (\ref{Agm8}), an
estimate in $H^1$ norm.  Coming back to the second order differential equation satisfied
 by $\exp \frac{\Phi}{h^\frac 12} u$ and using that $\Phi$ is constant near the boundary\footnote{Actually,
 we need first to use
 a regularized $\Phi$ in order to make the argument rigorous. Another way, would be to use $L^p$ estimates.}, we can use the regularity of the Neumann problem for getting  a control in $H^2$. This gives finally (through the Sobolev
 injection theorem) the proof in the constant magnetic field case of the following theorem~:
\begin{theorem}\label{Proppoint}.\\
Let us assume that the condition~:
\begin{equation}\label{lp1}
\Theta_0 b' < b\;,
\end{equation}
is satisfied.\\
There exists $C >0$, $\alpha >0$ and $\nu \in \rz$, such that if $u_h$ is the ground state of $P_{A,h,\Omega}^N$,  then~:
\begin{equation}\label{point}
 \exp \alpha \frac {d(x, \pa \Omega)}{h^\frac 12} |u_h|  \leq C h^{- \nu} || u_h||_{L^2}\;.
\end{equation}\end{theorem}
The condition (\ref{lp1}) is always satisfied when $B$ is constant because $b=b'$ and $\Theta_0 <1$. In this case,
this  is essentially similar to the result of 
\cite{PiFeSt}, which was recalled in  Theorem \ref{Thster0}  modulo possibly a negative power of $h$.
 These authors use indeed a normalization by the $L^\infty$-norm.\\

The proof  when $B$ is not constant is essentially the same. The inequality (\ref{Agm7}) becomes~:
\begin{equation}\label{Agm7a}
\left( b -\Theta_0 b' - o(1)  - \frac {C}{\epsilon_0^2} - \alpha^2\right)  
\sum_{int} \int | \exp \frac{\Phi}{h^\frac 12} \chi_j^h u|^2 dx \leq  C(\epsilon_0) 
 \sum_{bnd} \int  |\chi_j^h u|^2 \, dx\;.
\end{equation}
We can then conclude in the same way (modulo the proof of (\ref{LuPa3})
 which will be given later  or using the result of Lu-Pan (\ref{LuPa3w}).


\begin{remark}.\\
{\rm 
 On the contrary, 
when $b < \Theta_0 b'$ the ground state decays exponentially outside neighborhoods of points
 where $B(x) = b$.}
\end{remark}

\begin{remark}\label{RemarkJadallah}.\\
{\rm 
In the case when the boundary has right corners (i.e. there exists near these points of the boundary, a local diffeomorphism,
whose differential is a rotation at the corner), similar results can be  proved\footnote{See \cite{Ja}. This is true at least for the $H^1$ estimates. In the proof we describe above, one has probably to be careful with the regularity theorems in domain with corners.}
  using for example
 the distance to the corner points instead of the distance at the boundary. We only need to show that~:
$\Theta_1 < \Theta_0 $ in the case of a constant magnetic field in order to have the following condition
 satisfied~:
\begin{equation}
\Theta_1  b'' < \min (b, \Theta_0  b')\;,
\end{equation}
where $b''$ is the infimum of $B(x)$ over the various right corner points.
Other results  devoted to the case of three dimensional domains with regular 
boundaries \cite{LuPa5}
 or  with edges and corners  \cite{Pan} appear recently. These results were also announced in \cite{LuPa4}.}
\end{remark}

\subsection{Decay inside the boundary.}
We first treat the ``easy'' case when (\ref{lp1}) is satisfied and when $B(x)$ restricted to $\pa \Omega$ has only
 isolated minima. What we would like is to recover in a more precise way the statements of \cite{LuPa2} (Proposition 7.2 and 
Theorem 7.3). 
Let us first remember what we got from (\ref{IMS7}) by taking $\rho=\frac 38$ and  $\theta = \frac 18$.
\begin{equation}\label{IMS7a}
\begin{array}{ll}
q_{h,A}^N (u) & \geq h \sum_{int} \int B(x)| \chi_j^h u|^2 dx\\
&\quad  + (1- C h^{\frac 14}) h \Theta_0
\sum_{bnd} \int B(z_j) |\chi_j^h u|^2 \, dx \\
&\quad - C\,    h^{ \frac 54}  \sum_{bnd}
 || \chi_j^h u||^2 \\
& \quad -   C h^{ \frac 54} \sum_{int}
 || \chi_j^h u||^2 \;.
\end{array}
\end{equation}
We now apply this inequality with $u = \exp \frac{\Phi_h}{h^\frac 12} u_h$ where $u_h$ is the ground state.
This gives (cf (\ref{Agm2}))~:
\begin{equation}
\mu^{(1)}(h) ||  \exp \frac{\Phi_h}{h^\frac 12} u_h ||^2
 \geq q_{h,A} (  \exp \frac{\Phi_h}{h^\frac 12} u_h) - || |\nabla \Phi|  \exp \frac{\Phi_h}{h^\frac 12} u_h ||^2
\end{equation}

Combining the two estimates and (\ref{LuPa3}), we obtain~:
\begin{equation}\label{IMS7b}
\begin{array}{ll}
0 & \geq h \sum_{int}\int \left( B(x) - \Theta_0 b' - C h^\frac 14 - |\nabla \Phi|^2\right)\;| \chi_j^h \exp \frac{\Phi_h}{h^\frac 12} u|^2 dx\\
&\quad  + (1- C h^{\frac 14}) h \Theta_0 
\sum_{bnd} \int \left(B(x) - b' - C\,  h^\frac 14 - |\nabla \Phi|^2\right) \; |\chi_j^h \exp \frac{\Phi_h}{h^\frac 12} \, u|^2 \, dx \;.
\end{array}
\end{equation}

We have just to choose a suitable $\Phi$ for getting the right estimate. Let us  remember that
 we have already obtained a good control inside $\Omega$ under the condition $\Theta_0 b' < b$. 
 So it is enough to give a control in a neighborhood $\Ng$ of $\partial \Omega$ of size $\sim h^\frac 38$.
Outside this neighborhood, we know indeed already that the ground state is exponentially small.
In this neighborhood, we can assume that we have local coordinates $(t,s)$
 introduced in the appendix \ref{AppB} (the neighborhood being described by 
$\{ (t,s)\;|\; 0 \leq t \leq C h^\frac 38\}$).
Changing possibly the decomposition $\sum_{int} + \sum_{bnd}$, we can assume that $\sum_{bnd} (\chi_{j}^h)^2  \geq 1$
 on this neighborhood and vanishes outside  a neighborhood of the same type $\Ng'$. 
By taking $\Phi = \alpha_1 \chi (t) d(s)$ where $d$ is some  distance inside $\partial \Omega$ to the set of minima's attached to the degenerate Agmon metric $(B(s,0)- b') ds^2$ 
and $\alpha_1$ sufficiently small and $\chi(t)$ localizes near the boundary, 
 we obtain, for some constant $C>0$,   the inequality~:
\begin{equation}
0 \geq \sum_{bnd} \int  (B(s,0) - b' - C h^\frac 14 - C |\phi'(s)|^2) |\chi_j^h \exp \frac{\Phi}{h^\frac 12} \, u_h|^2 \, dx \;.
\end{equation}
This leads to  a decay\footnote{We do not try to be optimal in the measure of the decay.} in the form~:
\begin{equation}
 \sum_{bnd} \int  |\chi_j^h \exp \frac{\Phi}{h^\frac 12} \, u_h|^2 \, dx \leq C_\epsilon \exp \frac{\epsilon}{h^\frac 12}||u_h||_{L^2}^2\;,
\end{equation}
for any $\epsilon >0$,
or
\begin{equation}\label{6.36}
 \int_{\Ng}  | \exp \frac{\Phi}{h^\frac 12} \, u_h|^2 \, dx \leq C_\epsilon \exp \frac{\epsilon}{h^\frac 12}||u_h||_{L^2}^2\;.
\end{equation}
Actually, we can extend this inequality to an $h$-independent tubular neighborhood of $\pa \Omega$. Combining with the decay outside  $\pa \Omega$, this
 leads to the following localization theorem~:
\begin{theorem}.\\
Let
\begin{equation}
 p(\pa \Omega) = \{ \omega \in \pa \Omega\;|\; B(\omega) = b'\}\;.
\end{equation}
Let us assume that $\Theta_0 b' < b$.
Then, there exists $\alpha_1 >0$ and, for any $\epsilon >0$,  constants $C(\epsilon)$ and $h_0(\epsilon)$ such that a normalized  groundstate $u_h$ satisfies~:
\begin{equation}
|| \exp \frac{\alpha_1  d(z, p(\pa \Omega))}{h^\frac 12} u_h||\leq C(\epsilon)\exp \frac{\epsilon}{h^\frac 12}\;,
\end{equation}
for all $h\in ]0,h_0(\epsilon)]$.
\end{theorem}


As in \cite{HeSj1}, assuming  the non degeneracy of 
the minima of $s\mapsto B(s,O)$, one can probably improve the estimate and get the
 existence of $C>0$ such that~:
\begin{equation}
|| \exp \frac{\alpha_1  d(z , p(\pa \Omega))}{h^\frac 12} u_h||\leq 
C\, h^{-\nu}\;,
\end{equation}
for some $\nu \in \rz$.\\


\begin{remark}.\\
{\rm This theorem gives an accurate version of the result given in \cite{LuPa2}.\\
Of course these estimates are not sufficient  to treat the problem
 considered in \cite{PiFeSt}, when $B=\const >0$, where we would like to prove
 localization near the points of maximal curvature.
When $B$ is constant, the effect of localization is more difficult to show. 
We shall see that 
the effective potential creating the localization is $- h^\frac 32 \kappa(s)$ where
 $\kappa(s)$ is the curvature at the point $s$ of the boundary. One hopes to show a decay in $\exp - \frac{\psi}{h^\frac 14}$
 for some $\psi$ defined on $\partial \Omega$ and related to the metric $(\kappa_{max} - \kappa(s)) g(s) ds^2$,  where $g$ is some strictly positive density.
But this supposes finer estimates with remainders in $o(h^\frac 32)$, which will be given in Section \ref{Section10}.}
\end{remark}

\section{A sharp  semi-classical 
estimate of the groundstate energy for the  two dimensional magnetic Dirichlet problem } \label{Section7}

\subsection{Main theorems}

Let $\ \Omega \; \subset \; \rz ^2$ be a regular bounded open set and 
let $\ A=(A_1,A_2)$  in $ C^1(\overline{\Omega};\rz ^2)$ 
 such that 
\begin{equation}\label{Hb0} 
B(z):=\frac{\pa A_2}{\pa x}(z)-\frac{\pa A_1 }{\pa y}(z)\; \geq \; 0. 
\end{equation} 
For any $\ h\; \in \; ]0,1],\ \ P^D_{h,A,\Omega }$ denotes 
the Dirichlet self-adjoint operator, 
defined  on $\ L^2(\Omega )$, associated to $\ (hD-A)^2;\ \ (D=D_z=(D_x;D_y)).$ \\ 
The associated sesquilinear form will be denoted by $\ q_{h,A,\Omega }^D:$ 
\begin{equation}\label{sq} 
q_{h,A,\Omega }^D(u)=\| hDu-A u\| ^{2}_{L^2(\Omega )},\ \forall \; u\; \in \; 
H^{1}_{0}(\Omega )\;.
\end{equation} 
The associated self-adjoint operator on $\ L^2(\Omega )$ is 
\begin{equation}\label{ph} 
P_{h,A,\Omega }u=(hD-A)^2u ,\ \forall \; u\; \in \; 
H^2(\Omega )\cap  H^{1}_{0}(\Omega ). 
\end{equation}  
 When $\ \Omega =\rz ^2,$ this self-adjoint operator on 
 $\ L^2(\rz ^2)$  will be denoted by 
$\ P_{h,A}.$ We recall that $\ -i[hD_x-A_1\;,\; hD_y-A_2]=hB\geq 0$ leads to the estimate 
\begin{equation}\label{Es0} 
h\| B^{1/2}u\| ^{2}_{L^2(\Omega )}
\leq q_{h,A,\Omega }(u),\ \forall \; u\; 
\in \; H^{1}_{0}(\Omega ).
\end{equation} 

\begin{theorem}\label{Th1}.\\  
If $\ A\; \in \; C^1(\overline{\Omega} ;\rz ^2)$ and if $\ B\geq 0,$ then 
for any $\ \tau \; \in \; ]0,h^{-1}[,$ 
\begin{equation}\label{1Th1} 
h\;  \inf \; \sp (-\tau h^2\Delta^D_{\Omega }+(1-\tau h)B)\; \leq \; 
\inf \; \sp(P^D_{h,A,\Omega }) 
\end{equation} 
\end{theorem}
Here $-\Delta^D_{\Omega }=P_{1,0,\Omega }^D$ is the Laplace operator 
on $\Omega $ with 
Dirichlet boundary condition and, for an operator 
$\ P$, $\sp(P)$ denotes its spectrum. We will omit the superscript $D$ when there is no possible confusion.\\

\noindent 
 {\bf Proof} \\
 The estimate  (\ref{Es0}) leads to 
\begin{equation}\label{interpol} 
\tau h P_{h,A,\Omega }+(1-\tau h)hB\leq P_{h,A,\Omega },\ \ 
\forall \ \tau \; \in \; [0,h^{-1}]. 
\end{equation}
Then we use (\ref{diamD}) and 
this gives (\ref{1Th1}).\\
$\bullet $ \\


We will be interested in the special case when $B(z)\; \in \; C^{3+M}({\overline {\Omega }})$  satisfies, 
for some $\ z_0\; \in \; \Omega$~:
\begin{equation}\label{Hb1} 
\ B(z)\; >\; b:=B(z_0)>0,\ \forall \; z\; \in \; 
{\overline {\Omega }}\setminus \{ z_0\} , 
\end{equation}
and we assume that the minimum is non degenerate~:
\begin{equation}\label{Hb1a}
 \ \Hess B(z_0)>0\;.
\end{equation} 
We introduce in this case the notation~:
\begin{equation}\label{defa}
 a={\rm {Tr}}\left(\frac{1}{2}\Hess B(z_0)\right)^{1/2}\;.
\end{equation}

If we use the estimate (\ref{1Th1}) 
with $\ \tau =\frac{a^2}{4b^2}$,  
then the assumptions (\ref{Hb1}) and (\ref{Hb1a}) and the well known 
result giving the semi-classical estimate of the  
ground state of the Schr\"odinger operator\ 
$$-\tau h^2\Delta _{\Omega }+(1-\tau h)B,$$ 
(see for example \cite{HeSj1}, \cite{Si} or \cite{He2}),  
guarantee that 
\begin{equation}\label{Schro1} 
\begin{array}{l}
\inf \;\sp(-\tau h^2\Delta _{\Omega }+(1-\tau h)B)\\
\quad \quad =(1-\tau h)b+h\tau ^{1/2}a
+\Og(h^2)=b+\frac{a^2}{4b} h +\Og(h^2)\;.
\end{array}
\end{equation}

So from (\ref{Schro1}) and (\ref{1Th1}), we obtain the existence of a  constant $\ C>0$ such that~:
\begin{equation}\label{lowb} 
h\, b+\frac{a^2}{4b}h^2 - C \, h^3 \leq \inf \; \sp(P_{h,A,\Omega }) 
\end{equation} 
This estimate is actually not optimal and will be improved by the following theorem. 

\begin{theorem}\label{Th2}.\\  
If $\ A\; \in \; C^{4+M }(\overline{\Omega};\rz ^2)$,
with $\ M\geq 0$,   and 
if the hypotheses (\ref{Hb1}) and (\ref{Hb1a}) 
 are  satisfied, then there 
exists a constant $\ C>0$ such that 
\begin{equation}\label{1Th2} 
-C\, h^{19/8} \leq \inf \; \sp(P_{h,A,\Omega })\; -\; 
[b+\frac{a^2}{2b}h]h \leq C\, h^{5/2}\;. 
\end{equation} 
\end{theorem} 
The upper bound  in (\ref{1Th2})  follows 
from the following theorem.  

\begin{theorem}\label{Th3}.\\  
Under 
the assumptions (\ref{Hb1}) and (\ref{Hb1a}), then  the lowest
eigenvalue $\ \lambda^{(1)}(h)$ of 
$\ P_{h,A,\Omega }^D$ satisfies 
\begin{equation}\label{1Th3} 
\lambda^{(1)}(h) \leq b h+\frac{a^2}{2b}h^2 +\Og(h^{5/2}),
\end{equation}
as $\ h\to 0^{+}$.\\
Here $a$ and $b$ are introduced in (\ref{Hb1}) and (\ref{defa}).
\end{theorem} 

\noindent The left-hand side of the estimate (\ref{1Th2}) will follow as in  
(\ref{lowb}) from (\ref{interpol}) but without using (\ref{diamD}). The magnetic 
field near $\ z_0$ will not be neglected.  So in (\ref{1Th1}) we 
will  take  
$\ P_{h,A^1,\Omega }$ instead of 
$\ -h^2\Delta _{\Omega }$, 
where  $\ A^1(z)=(-\frac{1}{2}by,\frac{1}{2}bx)$.\\
For that, we will use the following theorem. 
\begin{theorem}\label{Th4} . \\ 
Let $\ A\; \in \; C^3({\overline {\Omega }};\rz ^2)$ 
and $\ V\; \in \; C^3({\overline {\Omega }};\rz )$ 
such that, for some $\ z_0\; \in \; \Omega $, 
\begin{equation}\label{1Th4} 
B(z)=\frac{\partial A_2}{\partial x} (z)- 
\frac{\partial A_1 }{\partial y} (z)\geq B(z_0)=b>0,\ 
\forall \; z\; \in \; 
{\overline {\Omega }} 
\end{equation} 
\begin{equation}\label{2Th4} 
\begin{array}{l}
V(z)>V(z_0),\ \forall \; z\; \in \; 
{\overline {\Omega }}\setminus \{ z_0\} , \\
\Hess V(z_0)\; >\; 0\;.
\end{array}
\end{equation} 
Let $\ \rho \; \in \; [0,1]$ be given and  
let us  consider the Dirichlet operator 
$$P_{h,A,V,\Omega }^D =P_{h,A,\Omega }^D -hB(z)+h^{\rho }V(z)\;. $$
Then   there 
exists a constant $\ C=C_{\rho }>0$ such that 
\begin{equation}\label{CTh4}
|\inf \; \sp (P_{h,A,V,\Omega }^D)- h^{\rho} V(z_0) +
h^{1+\rho }\frac {a^{2}_{V}}{2b}| 
\leq C\,  h^{d(\rho )}\; ; 
\end{equation} 
with~:\\
 \begin{equation}\label{drho}
d(\rho )\; =\; \inf \left( \frac{3}{2}+\frac{3}{4}\rho, 2, 1 + 2 \rho \right) \; 
\end{equation}
\begin{equation}\label{av}
 a_V =\Tr(\frac{1}{2}\Hess V(z_0))^{1/2} ,  
\end{equation}

\end{theorem} 

As a matter of fact, if we take in (\ref{interpol})  
$\ \tau =h^{-1/2}$,  then we get the following corollary.
\begin{corollary} 
$$h^{1/2}\inf \; \sp\left(P_{h,A,\Omega}^D -hB(z)+h^{1/2}V(z)\right)
\; \leq \; \inf \; \sp(P_{h,A,\Omega }^D )$$ 
with $\ V(z)=B(z).$ 
\end{corollary}
So (\ref{1Th4}) and (\ref{2Th4}) are satisfied, 
 when (\ref{Hb1}) and (\ref{Hb1a}) are
satisfied. Formula  (\ref{CTh4}) with $\ \rho =\frac{1}{2}$  gives  
the lower bound of (\ref{1Th2}). 

\begin{remark}\label{rmk1} . \\ 
{\rm 
1. Theorem \ref{Th4} is valid for a constant magnetic field.\\  
2. Under the assumptions of Theorem \ref{Th2}, 
Theorem \ref{Th4} can also  be applied to
$\ P_{h,A,\Omega }$ by taking 
$\ \rho =1$ and $\ V(z)=B(z).$  \\ 
But in this case we get  that 
$$ \inf \sp(P_{h,A,\Omega })=hb+\Og(h^2).$$ 
}
\end{remark} 

\begin{remark}\label{rmk2} . \\ 
{\rm 
Theorem \ref{Th4} is valid if we change the definition of 
$\ P_{h,A,V,\Omega }$ 
into~: 
$$\ P_{h,A,V,\Omega }=P_{h,A,\Omega }-b\,h\, +h^{\rho }V(z).$$ 
}
\end{remark} 

\begin{remark}\label{rmk3} . \\
{\rm  
The results of Theorem \ref{Th4}, when $\ \rho \; \in \; [0,1[$, 
  are valid for the modified Neumann operator 
$\ \displaystyle P^{N, mod}_{h,A,V,\Omega }$ 
associated to the quadratic form, defined on $H^1(\Omega)$ by~:
$$q^{N}_{h,A,V,\Omega }(u)=\int _{\Omega } 
\{ |[hD_x-A_1(z)+i(hD_y-A_2(z))]u(z)|^2+
h^{\rho }V(z)|u(z)|^2\} dz\;.$$ 

It  is also the case   for  the Neumann operator 
$\ \displaystyle P^{N}_{h,A,\Omega }+h^{\rho }V(z)-hb$\\ 
associated to the quadratic form 
$$q^{N}_{h,A,\Omega }(u)=\int _{\Omega } 
\{ |hD \, u(z)-A(z)\, u(z)|^2+
[h^{\rho }V(z)-hb]|u(z)|^2\} dz, \ \ 
\forall \; u\; \in \; H^1(\Omega ).$$ 
But the reader should not forget that (\ref{Es0}) is not valid for Neumann operator. }
\end{remark} 
Let us prove this last remark. Let  $\ \chi $ be  a cut-off 
function supported in $\ \Omega $, such that 
$\ \chi (z)=1$ in a neighborhood of $\ z_0.$\\  
Let us consider the partition of unity 
$\ (\vartheta _j(z)),\ j\in \{ 0,1\} :$\\ 
$$\vartheta _0(z)=\chi (z)[\chi ^2(z)+(1-\chi (z))^2]^{-1/2},\ \ 
\vartheta _1(z)=(1-\chi (z))[\chi ^2(z)+(1-\chi (z))^2]^{-1/2}.$$ 
Then, by the so called IMS Formula (see for example \cite{HeMo2} or the discussion in Section \ref{Section6}  (\ref{IMS1} or (\ref{inta})~), there exists  
a constant  $\ C_1>0$,  such that  
$$|q^{N}_{h,A,V,\Omega }(u)-[q^{N}_{h,A,V,\Omega }(\vartheta _0u)+
q^{N}_{h,A,V,\Omega }(\vartheta _1u)]|
\leq C_1\,h^2\| u\| ^{2}_{L^2(\Omega )},\ 
\forall \; u\; \in \; H^1(\Omega ),$$ 
and, thanks to  (\ref{2Th4})), there exists $\delta >0$ such that~:
$$ q^{N}_{h,A,V,\Omega }(\vartheta _1u)-
h^{\rho }V(z_0)\| \vartheta _1 u\| ^{2}_{L^2(\Omega)}
\geq \delta h^{\rho } \| \vartheta _1 u\| ^{2}_{L^2(\Omega)},\ 
\forall \; u\; \in \; H^1(\Omega).$$ 
Combining with the theorem, this gives the lower bound.\\
But the upper bound is an immediate consequence  of the min-max principle. 
We get indeed  easily that 
$$\inf \; \sp(P^{N}_{h,A,V,\Omega })\; \leq \; 
\inf \; \sp(P_{h,A,V,\Og })$$ 
for any open subset  $\ \Og $ of $\Omega $  
containing $z_0$. 

The proof remains the same for the operator 
$\ \displaystyle P^{N}_{h,A,\Omega }+h^{\rho }V(z)-hb. $\\ 
Later on, we will assume that $\ z_0=0$  when giving the proof of the 
 above Theorems \ref{Th2} and \ref{Th4}. 




\subsection{Some estimates on eigenfunctions} 

We take the hypotheses of Theorem \ref{Th4}. 
The quadratic form associated to $\ P_{h,A,V,\Omega }$ is defined on $H^1_0(\Omega)$ by~:
\begin{equation}\label{sq2}
\begin{array}{ll}  
q_{h,A,V,\Omega }(u)& =q_{h,A,\Omega }+ \int _{\Omega } 
[-hB(z)+h^{\rho }V(z)]|u(z)|^2\; dz\\ 
&=\int _{\Omega } 
\{ |[hD_x-A_1(z)+i(hD_y-A_2(z))]u(z)|^2+
h^{\rho }V(z)|u(z)|^2\}\; dz\;.
\end{array}
\end{equation}
Let $\ \lambda (h)$ be an eigenvalue of $\  P_{h,A,V,\Omega }$ 
and $\ u_h$ be  an associated eigenfunction. 
Let $\ \phi _h(z)$ denotes the Agmon distance of $\ z$ 
to the well 
$$U_h=\{ w\in \Omega ;\ h^{\rho }V(w)-\lambda (h)\leq 0\} ,$$ 
associated to the Agmon metric 
$\ [h^{\rho }V(w)-\lambda (h)]_{+}dz^2$.

\begin{proposition}\label{Pro1} .\\ 
For any $\ \epsilon \; \in \; ]0,1],$ 
\begin{equation}\label{1Pro1} 
\begin{array}{l}
(1-\epsilon ^2)\int _{\Omega }
[h^{\rho }V(z)-\lambda (h)]_{+}e^{2\epsilon \phi_h(z)/h}
|u_h(z)|^2\;dz \\
\quad \leq \int _{\Omega }[h^{\rho }V(z)-\lambda (h)]_{-} |u_h(z)|^2\;dz.
\end{array}
\end{equation}
\end{proposition} 
{\bf Proof} \\
As in \cite{HeMo2}, we use the formula 
(already used in (\ref{Agm2})), 
\begin{equation}\label{2Pro1} 
q_{h,A,V,\Omega }(e^{\phi }u_h)=\lambda (h)
\| e^{\phi }u_h\| ^{2}_{L^2(\Omega )} 
+h^2\| e^{\phi }u_h\nabla \phi \| ^{2}_{L^2(\Omega )},
\end{equation} 
for any real Lipschitz function $\ \phi$. \\ 

Observing that  $\ \displaystyle \ |\nabla \phi _h(z)|^2\leq 
[h^{\rho }V(z)-\lambda (h)]_{+}$ and using  (\ref{Es0}) and (\ref{2Pro1}),
 we get 
$$\int _{\Omega }\{ h^{\rho }V(z)-\lambda (h)-
\epsilon ^2[h^{\rho }V(z)-\lambda (h)]_{+}\} 
e^{2\epsilon \phi _h(z)/h}|u_h(z)|^2dz\leq 0,$$ 
which is exactly (\ref{1Pro1}), if we observe that, in  the support 
of $[h^{\rho }V(z)-\lambda (h)]_-$, we have  $\phi_h(z)=0$.\\
  $\ \bullet $ \\


\vskip 0.5cm 

\noindent {\bf  A rough upper bound of the ground state energy}\\ 

\noindent 
We take a magnetic potential $\ A$ such that 
\begin{equation}\label{ov1} 
A(z)=\frac{1}{2}(-by,bx)+\Og(|z|^3). 
\end{equation} 
This is always possible thanks to (\ref{1Th4}) (see for example \cite{HeMo1}). 

Let us consider the  test function $\ u^{0}_{h}(z)=\chi (z)e^{-b|z|^2/(2h)},$\\ 
where  $\ \chi $ a cut-off function with support in $\ \Omega $ 
and $\ \chi (z)=1$ in a neighborhood of $\ z_0=0$.  
We get easily using  (\ref{2Th4}) and (\ref{ov1}) that 
$$q_{h,A,V,\Omega }(u^{0}_{h})-
h^{\rho }V(0)\| u^{0}_{h}\| ^{2}_{L^2(\Omega )}
\leq Ch^{1+\rho }\| u^{0}_{h}\| ^{2}_{L^2(\Omega )}, $$ 
so 
\begin{equation}\label{ov2} 
\inf \; \sp(P_{h,A,V,\Omega })-h^{\rho }V(0)\leq C\, h^{1+\rho }. 
\end{equation} 
So we have proved that, if $\lambda^{(1)}(h)$ is a ground state energy, then
\begin{equation}\label{Hl1} 
\lambda^{(1)} (h)=h^{\rho }V(0)+\Og(h^{1+\rho }). 
\end{equation} 
Then, the assumption (\ref{2Th4}) ensures, by comparing 
the Agmon metric with the one associated to the metric  
$\  |z|^2\,dz^2,$ that,  for $\ C$ large enough, 
\begin{equation}\label{phas1} 
C\, h^{\rho /2}|z|^2 \leq \phi _h(z),\ \ {\rm {if}}\ \ h^{1/2}C\leq |z|. 
\end{equation} 
A direct consequence of (\ref{1Pro1}) and (\ref{phas1}) 
is the following lemma. 
\begin{lemma}\label{lem1}.\\  
For any fixed real  $\ k\; \geq \; 0,\ \ 
\displaystyle \| \; |z|^ku_h\| _{L^2(\Omega )}\leq 
C(h^{k/2}+h^{(k(2-\rho )+\rho )/4})
\| u_h\| _{L^2(\Omega )}.$ 
\end{lemma} 
{\bf Proof of Lemma \ref{lem1}.} \\
 Let $\ C_1>1$ 
be fixed and large enough, 
 and let us define the following 
domains 
\begin{equation}\label{lm110} 
\Omega _{0,h}=\{ z\; \in \; \Omega \;|\; |z| \; \leq C_1\; h^{1/2}\} ,
\end{equation} 
\begin{equation}\label{lm111} 
\Omega _{1,h}=\{ z\; \in \; \Omega \;|\; C_1\,  h^{1/2}\; \leq \; |z| \; 
\leq \; 2 C_1\, h^{(2-\rho )/4}\} , 
\end{equation}  
and 
\begin{equation}\label{lm112} 
 \Omega _{2,h}=\{ z\; \in \; \Omega \;|\;  
2 C_1\, h^{(2-\rho )/4} \; \leq \; |z| \} . 
\end{equation} 
By (\ref{Hl1}), (\ref{phas1}), (\ref{lm111}) and 
(\ref{lm112}),  if $C_1$ is large enough, then 
\begin{equation}\label{lm12} 
h^{\rho }|z|^2/C\leq h^{\rho }V(z)-\lambda^{(1)}(h),
\ \ \forall \; z\; \in \; 
\Omega _{1,h}\cup \Omega _{2,h} 
\end{equation} 
and 
\begin{equation}\label{lm13} 
h^{\rho }V(z)-\lambda^{(1)} (h)\leq C\,  h^{\rho }|z|^2 \,,\ \ \forall \; z\; \in \; 
\Omega _{1,h}\;.
\end{equation} 
Hence, by immediate comparison, we have for some constant $C >0$, 
\begin{equation}\label{lm14} 
\phi _h(z)\leq C\, h^{\rho /2}|z|^2\,,\ \ \forall \; z\; \in \; 
\Omega _{1,h} . 
\end{equation} 
In particular this implies for some constant $C >0$, 
\begin{equation}\label{lm14a} 
\phi _h(z)\leq C\, h\,,\ \ \forall \; z\; \in \; 
\Omega _{1,h} . 
\end{equation} 
Using (\ref{lm12}) and (\ref{1Pro1}) with $\; \epsilon = 1/2$, we first get~:
$$
\begin{array}{ll} 
h^{\rho }\int _{\Omega _{1,h}}|z|^2|u_h(z)|^2\; dz
\; &\leq \; C \int _{\Omega_{1,h} }[h^{\rho }V(z)-\lambda^{(1)} (h)] |u_h(z)|^2\; dz\\
&\leq \; C \int _{\Omega_{1,h} }[h^{\rho }V(z)-\lambda^{(1)}(h)] \exp \phi_h(z)/h\, |u_h(z)|^2\; dz\\
&\leq \; C \int _{\Omega }[h^{\rho }V(z)-\lambda^{(1)}(h)]_+ \exp \phi_h(z)/h\, |u_h(z)|^2\; dz\\
&\leq \;2  C \int _{\Omega }[h^{\rho }V(z)-\lambda^{(1)}(h)]_- \, |u_h(z)|^2\; dz\;.
\end{array}
$$ 
Now, we observe that the support of $[h^{\rho }V(z)-\lambda^{(1)}(h)]_-$ is contained in $\Omega_{0,h}$, and using 
 the behavior of $V$ in $\Omega_{0,h}$ near $0$ and (\ref{Hl1}), we get, for $z\in \Omega_{0,h}$ ~:
$$
|h^\rho V(z) - \lambda^{(1)}(h)|\leq h^{\rho} |V(z) - V(0)| + C\, h^{1+\rho} \leq {\tilde C} h^{1+ \rho}\;.
$$
So we have proved, for some suitable constant $C$, 
$$
h^{\rho }\int _{\Omega _{1,h}}|z|^2|u_h(z)|^2\; dz
 \leq C \; h^{1+\rho }\;\int _{\Omega }|u_h(z)|^2\; dz \;. 
$$
Dividing by $h^\rho$, we obtain~:
\begin{equation}\label{lm14b}
\int _{\Omega _{1,h}}|z|^2|u_h(z)|^2\; dz
 \leq C \; h^{1}\;\int _{\Omega }|u_h(z)|^2\; dz \;. 
\end{equation}
This estimate and (\ref{lm111}) leads  to 
\begin{equation}\label{lm15}
\begin{array}{ll} 
\int _{\Omega _{1,h}}|z|^{2k}|u_h(z)|^2dz
\;&  \leq \;C\,  h^{(k-1)(2-\rho )/2}\int _{\Omega _{1,h}}|z|^2|u_h(z)|^2\; dz\\
&\leq \;{\tilde C} h^{1+(k-1)(2-\rho )/2} \int _{\Omega }|u_h(z)|^2\; dz \;.
\end{array}
\end{equation}
For the integral on $\Omega _{2,h}$, we use also 
(\ref{1Pro1}) with $\; \epsilon =1/2.$  
If 
$$ \ \displaystyle \tau (h)=
\sup \{ |z|^{2k}[h^{\rho }V(z)-\lambda^{(1)}(h)]_{+}^{-1}
e^{-\phi _h(z)/h} ;\ z\; \in \; \Omega _{2,h} \} ,$$ 
then 
$$
\begin{array}{ll}
\int _{\Omega _{2,h}}|z|^{2k}|u_h(z)|^2dz\; & \leq \; 
C\, \tau (h)\,\int _{\Omega _{2,h}}[h^{\rho }V(z)-\lambda h)]_{+}
e^{\phi _h(z)/h}|u_h(z)|^2\; dz\\
&\leq \;{\tilde C}\, \tau (h)\, \int _{\Omega }[h^{\rho }V(z)-
\lambda h)]_{-}|u_h(z)|^2\, dz\\
&\leq \;{\hat C}\, h^{1+\rho }\tau (h) \, \int _{\Omega }|u_h(z)|^2\, dz.
\end{array}
$$ 
For the estimate of $\ \tau (h)$, 
we use (\ref{Hl1}), (\ref{phas1}), (\ref{lm112}) and (\ref{lm12}). This leads to~:
$$
\begin{array}{ll}
 \tau (h)\; & \leq \; C\, h^{-\rho }
\sup \{ t^{k-1}e^{-(h^{-1+\rho /2}t)/C};\ h^{-1+\rho /2}t\; 
\geq \; 4C_{1}^{2}\} \;\\
& \leq \;{\tilde C} \;  h^{-\rho +(k-1)(2-\rho )/2}\;.
\end{array}
$$
So there exists a constant $C >0$ such that~:
\begin{equation}\label{lm16} 
\int _{\Omega _{2,h}}|z|^{2k}|u_h(z)|^2dz\; \leq \; 
C\, h^{1+(k-1)(2-\rho )/2}\, \int _{\Omega }|u_h(z)|^2\; dz. 
\end{equation} 
For the integral on $\Omega _{0,h}$, we use (\ref{lm110}) and get~:
\begin{equation}\label{lm17} 
\int _{\Omega _{0,h}}|z|^{2k}|u_h(z)|^2\;dz\; \leq \; C \,
h^k\, \int _{\Omega _{0,h}}|u_h(z)|^2\; dz\;.
\end{equation} 
Hence  Lemma \ref{lem1} comes from  (\ref{lm15})--(\ref{lm17}).\\
 $\ \bullet $\\

\noindent {\bf Proof of Theorem  \ref{Th4} }\\

\noindent Let us introduce  
$$A^1(z)=\frac{1}{2}(-by,bx)$$ and 
$$ V_2(z)=V(0)+\frac{1}{2}\, z\cdot (\Hess V(0))\cdot z\ .$$ 
We can suppose that $\ V(0)=0$,  
so $$\ P_{h,A,V,\Omega }\geq 0\;.$$
 Note that we have also 
\begin{equation}\label{1pth4} 
V(z)-V_2(z)=\Og(|z|^3). 
\end{equation} 
So using  (\ref{sq2}),  (\ref{ov1}) and (\ref{1pth4}), 
  there exists  a constant $\ C>0,$ such that, 
 for any $\ h\; \in \; ]0,1]$ and 
any $\ u\; \in \; H^{1}_{0}(\Omega ),$ 
\begin{equation}\label{sq2b} 
|q_{h,A,V,\Omega }(u)-q_{h,A^1,V_2,\Omega }(u)|
\end{equation} 
$$\leq 
C\left [(q_{h,A,V,\Omega }(u))^{1/2}\| \; |z|^3u\| _{L^2(\Omega ) }+ ||\,|z|^3 u\,||^2 +
h^{\rho }\|\; |z|^{3/2}u\| ^{2}_{L^2(\Omega )}\right ] \ . $$
The estimate (\ref{Hl1}) (with $\ V(0)=0)$ 
of the ground state energy  
$\lambda^{(1)}(h)$ of  $\ P_{h,A,V,\Omega }$, Lemma~\ref{lem1} applied to 
the associated ground state  $\ u_h$
 and (\ref{sq2b}) show that 
\begin{equation}\label{sq3}
\begin{array}{l} 
|q_{h,A^1,V_2,\Omega }(u_h)-\lambda^{(1)}(h)\| u_h\| ^{2}_{L^2(\Omega) }| \\ 
\quad \quad 
\leq C\, \left[ (\lambda^{(1)}(h))^{1/2}\| u_h\| _{L^2(\Omega) }
\times \|\, |z|^3u_h\| _{L^2(\Omega ) } + ||\, |z|^3 u_h  \,||^2  
+h^{\rho }\| \, |z|^{3/2}u_h \, \| ^{2}_{L^2(\Omega )} \right] \\ 
\quad \quad 
\leq {\tilde C} \,\left [ h^2+h^{3(2+\rho )/4}\right ] \| u_h\| ^{2}_{L^2(\Omega )}\; . 
\end{array} 
\end{equation} 
So 
$$\ \inf \; \sp(P_{h,A^1,V_2,\Omega })\; \leq \; 
\inf \; \sp(P_{h,A,V,\Omega })+Ch^{d_0(\rho )},$$ 
with
$$
 d_0(\rho)= \inf (\frac 32 + \frac 34 \rho, 2)\;.
$$
As the potentials $\ A^1(z)$ and $\ V_2(z)$ 
satisfy also the assumptions of 
Theorem~\ref{Th4}, we have in the same way~: 
$$ \ \inf \; \sp(P_{h,A,V,\Omega })\; \leq \; 
\inf \; \sp(P_{h,A^1,V_2,\Omega })+Ch^{d(\rho )}\;.
$$ 
Hence~:
\begin{equation}\label{sq4} 
|\inf \; \sp(P_{h,A,V,\Omega })-\inf \; \sp(P_{h,A^1,V_2,\Omega })|
\leq C\,h^{d(\rho )}. 
\end{equation} 
But Proposition~\ref{Pro1} and Lemma~\ref{lem1} are also 
available for 
 the operator $\ P_{h,A^1,V_2}$ in  $\rz ^2$.  
The ground state  of 
$\  P_{h,A^1,V_2}$ has an exponential decay. \\ 
 Lemma~\ref{lem1} applied to the ground states 
of $\ P_{h,A^1,V_2}$ and  $\ P_{h,A^1,V_2,\Omega }$ leads 
also to 
\begin{equation}\label{sq5} 
|\inf \; \sp(P_{h,A^1,V_2})-\inf \; \sp(P_{h,A^1,V_2,\Omega })|
\leq Ch^{k},\ \ \forall \; k>0. 
\end{equation} 
But one can explicitely compute the spectrum in 
the case of Schr\"odinger 
operator with constant magnetic field and positive quadratic 
potential, see for example \cite{Mel} (Theorem~2.4,  p.~121)
for the lower bound of the spectrum,   or \cite{MatUe} 
(Theorem~2.2, p.~222) and \cite{Par}. 
In particular we get 
\begin{equation}\label{sq5a}  
\inf \; \sp(P_{h,A^1,V_2})=\mu _h(\rho ),
\end{equation}  
where~:
\begin{equation}
\begin{array}{rl}
 \mu _h(\rho )&=\;
h^{\rho }V(z_0)-hb+\frac{h}{\sqrt 2}
\left [ h^{\rho }t_V+b^2-[(h^{\rho }t_V+b^2)^2-4h^{2\rho }d_V]^{1/2}
\right ]^{1/2}\\
&\quad  +  \frac{h}{\sqrt 2}
\left [ h^{\rho }t_V+b^2+[(h^{\rho }t_V+b^2)^2-4h^{2\rho }d_V]^{1/2}
\right ]^{1/2},
\end{array}
\end{equation}
with
\begin{equation} 
t_V =\; {\rm {Tr}}(\frac{1}{2}\Hess V(z_0)),
\end{equation}
\begin{equation}\label{dv}
d_V =\; {\rm {det}}(\frac{1}{2}\Hess V(z_0))\;.
\end{equation}  
A simple computation shows that~:
\begin{equation}\label{RTh4} 
 \mu _h(\rho )=h^{\rho} V(z_0) +
h^{1+\rho }\frac {a^{2}_{V}}{2b}+\Og(h^{1+2\rho }).
\end{equation} 
So (\ref{CTh4}) comes from (\ref{sq4}), 
(\ref{sq5}) and (\ref{sq5a}). \\
$\ \bullet $ 


\subsection{Polynomial magnetic approximation } 
\noindent We begin now the first step of the proof of 
Theorem \ref{Th3}. 

\noindent We recall that, with $\ \rho =1$ and $\ V(z)=B(z)$,  we can write  
$$\ P_{h,A,\Omega }=P_{h,A,V,\Omega }\ .$$
We assume that the hypotheses of Theorem \ref{Th3} are satisfied. 

\noindent Let $\ \lambda^{(1)}(h)$ be a lowest  eigenvalue of 
$\ P_{h,A,\Omega }$ and $\ u_h$ an associated 
eigenfunction. 
Let us remember that we have proved  (\ref{Hl1})
 which becomes in our case 
\begin{equation}\label{Hl11} 
\lambda^{(1)} (h)=hb+\Og(h^2). 
\end{equation} 
The estimate of Lemma \ref{lem1} becomes 
\begin{equation}\label{lem01} 
\| \; |z|^ku_h\| ^{2}_{L^2(\Omega )}\; \leq \; 
C(h^{k/2}+h^{(k+1)/4}) \| u_h\| ^{2}_{L^2(\Omega )}, 
\ \forall \; k\geq 0. 
\end{equation}
We can assume that the magnetic potential satisfies 
\begin{equation}\label{HA1} 
\div(A)=0\ \ {\rm {and}}\ \ A(0)=0.
\end{equation} 

Let $\ m\; \in \; \nz $ a fixed integer, $\ 1<m< 4+M,$ 
and let us denote by
$\ A^m$ the Taylor expansion (up to order $m$) of the magnetic potential: 
\begin{equation}\label{Am} 
A^m(z)=\sum _{|\alpha |\leq m}\frac{z^{\alpha }}{\alpha !} 
\frac{\pa ^{\alpha } A}{\pa z^{\alpha }}(0). 
\end{equation} 

\begin{lemma}\label{lem2}.\\   
Let $\ \chi \; \in \; C^{\infty }_{0}(\rz ^2)$ 
be a cut-off function such that 
$$ \supp \chi \; \subset \; \Omega $$ and 
$$\ 0\; \notin \; \supp (1-\chi ).$$ 
Then there exists $C$ and $h_0$ such that~:
$$ \| (P_{h,A^m}-\lambda^{(1)} (h))\chi u_h\| \leq C\,h^{(2m+3)/4} 
\| \chi u_h\| \;,\; \forall h\in h_0\;,
$$
where $\| \; .\; \| = \| \; .\; \| _{L^2(\rz ^2)}$. 
\end{lemma} 
 {\bf Proof} \\
 Let us first prove that, if 
$\ k\; \in \; \nz ^{\star }$ is  fixed, then 
\begin{equation}\label{1lem2} 
\| \; |z |^k(hD-A)u_h\| _{L^2(\Omega )}\leq \, C\,  h^{(2k+3)/4}
\| u_h\| _{L^2(\Omega )}. 
\end{equation} 
Using that 
$$ 
(hD-A)[|z|^{2k}(hD-A)u_h]=\lambda^{(1)}(h)|z|^{2k}u_h
-2ihk|z|^{2k-2}[z(hD-A)]u_h\;,
$$ 
and  (\ref{lem01}), we get that 
$$
\begin{array}{ll}
\| \; |z|^k(hD-A)u_h\| ^{2}_{L^2(\Omega )}& \leq 
C\, \lambda^{(1)}(h)h^{(2k+1)/2}
\| u_h\| ^{2}_{L^2(\Omega )}\\
&\quad + C\,h\, \| \; |z |^k(hD-A)u_h\| _{L^2(\Omega )}
\| |z|^{k-1}u_h\| _{L^2(\Omega )}.
\end{array}
$$ 
So we deduce 
$$\| \; |z |^k(hD-A)u_h\| ^{2}_{L^2(\Omega )}\leq 
C\,\lambda^{(1)}(h)h^{(2k+1)/2}
\| u_h\| ^{2}_{L^2(\Omega )}+ C\, h^2
\| |z|^{k-1}u_h\| ^{2}_{L^2(\Omega )}.$$ 
 Using (\ref{Hl11}) and once more  (\ref{lem01}) 
 we get  (\ref{1lem2}). \\ 
 
 
\noindent Let us now write, (with $\ R^m=A-A^m).$ 
 As $\div A =0,$ then $\div R^m = 0$ and  
 $$ P_{h,A^m}=P_{h,A}+2R^m(hD-A)+|R^m|^2.$$ 
 As $\ |z|^{-m-1}R^m(z)$ is bounded in $\ \Omega$,
 (\ref{lem01}) and 
 (\ref{1lem2}) show  that 
 $$ \| (P_{h,A^m}-\lambda^{(1)} (h))\chi u_h\| \leq C\, h^{(2m+3)/4}
\| u_h\| _{L^2(\Omega )}.$$ 
We finally recall that a consequence of 
 (\ref{lem01}) is that, for any $k\in \nz$,
\begin{equation}\label{2lem2} 
\| \chi u_h\| =(1+\Og(h^k))\| u_h\| _{L^2(\Omega)} \;.
\end{equation} 
This ends the proof of Lemma \ref{lem2}.\\
 $\bullet$ \\

 Lemma \ref{lem2} ensures that 
\begin{equation}\label{distL} \\ 
{\rm {distance}}(\lambda^{(1)}(h),\; \sp(P_{h,A^m}))\leq C\; h^{(2m+3)/4}\;. 
\end{equation} 

\noindent Let us introduce~: 
\begin{equation}\label{Bm} 
B^m(z)=\sum _{|\alpha |\leq m-1}\frac{z^{\alpha }}{\alpha !} 
\frac{\pa ^{\alpha } B }{\pa z^{\alpha }}(0)=
\frac{\pa A^{m}_{2}}{\pa x}(z)-
\frac{\pa A^{m}_{1}}{\pa y}(z)\;. 
\end{equation} 
Assumption (\ref{Hb1}) ensures that, if $\ m>2,$ then 
\begin{equation}\label{BmInfi} 
\sum _{|\alpha |\leq m-1}|\frac{\pa ^{\alpha }B^m}{\pa z^{\alpha }}(z)|
\ \to \ +\infty \ \ {\rm {as}}\ \ |z|\ \to \ \infty . 
\end{equation} 
So, a general result in the representation theory  of nilpotent 
 Lie algebra of \cite{HeNo1} says that 
$\ P_{h,A^m}$ has compact resolvent, (see \cite{MoNo} or 
\cite{HeMo1} for an analytic proof). More precisely, there 
exists a constant $\ C$ such that, for all $\  v\; \in 
\; C^{\infty }_{0}(\rz ^2),$ 
\begin{equation}\label{HeNou} 
\sum _{|\alpha |\leq m-1}h^{(2|\alpha |+2)/(|\alpha |+2)}
\| |\frac{\pa ^{\alpha }B^m}{\pa z^{\alpha }}(z)|^{1/(|\alpha |+2)}
v\| ^{2}\leq C\| (hD-A^m)v\| ^2.
\end{equation} 
Conversely, if $\ \lambda_m (h)$ is an eigenvalue of 
$\ P_{h,A^m}$ satisfying 
(\ref{Hl11}), and if $\ u_h^m$ is an associated eigenfunction 
satisfying  (\ref{lem01}),  then we will get as for (\ref{distL}), that 
\begin{equation}\label{compa}
 {\rm {distance}}(\lambda_m (h),\; \sp(P_{h,A,\Omega })
\leq C\, h^{(2m+3)/4}\;.
\end{equation}
So, if we want to analyze the bottom of the 
spectrum of $P_{h,A,\Omega}$  modulo an error of order  $\Og (h^{(2m+3)/4})$, we just have to study the bottom of the 
spectrum of $ P_{h,A^m}$. 
In order to determine the coefficient of $h^2$ in the expansion, we will choose in the next subsections~:  $\ m\geq 4$.

\subsection{Some simplifications.} 


We make some simplifications which do not change the spectrum of 
$\ P_{h,A,\Omega }$ nor $\ P_{h,A^m}.$ 

We can always take orthonormal coordinates in $\ \rz ^2$ such that, 
in a neighborhood of $\ z_0=0,$   
\begin{equation}\label{simp1} 
 B(z)=b+\alpha x^2+\beta y^2+\Og(|z|^3). 
\end{equation} 
We can also choose a gauge $\ A(z)$ such that 
\begin{equation}\label{simp2} 
A_1(z)=0\ \ {\rm {and}}\ \ A_2(z)=bx+
\frac{\alpha }{3}x^3+\beta xy^2+\Og(z^4). 
\end{equation} 
So $\ A^{m}_{1}=0$ and we can write 
\begin{equation}\label{simp3} 
A^{m}_{2}(z)=bx+\sum _{j=3}^{m}S_j(z)\;,
\end{equation}
with
$$ 
S_j(z)=\sum _{|\gamma |=j}S_{j,\gamma }z^{\gamma }, 
$$
and in particular~: 
$$S_3(z)=\frac{\alpha }{3}x^3+\beta xy^2\;.$$ 
By scaling we get that 
\begin{equation}\label{simp4} 
\sp(P_{h,A^m})=h\; \sp(Q_h(z,D_z)), 
\end{equation} 
$$ Q_h(z,D_z)=D_{x}^{2}+
[D_y-bx-h\sum _{j=3}^{m}h^{(j-3)/2}S_j(z)]^2.$$

We are now going to apply  some metaplectic transformations.
We recall that 
these transformations are unitary and consequently  preserve the spectrum of 
the operators. Note also that they preserve  the functions  
of the form $\ f(z)=p(z)e^{iq_1(z)-q_2(z)}$ with 
$\ p$ and the $\ q_j$ polynomial, 
the $\ q_j$ real and homogeneous of degree 
$\ j, \ q_2(z)>0$ if $\ z\neq 0.$ 
 
Using Fourier-transform and translation and  writing 
$$ Q_h(z,D_z)=Q_h(x,y,D_x,D_y)\;,$$
 we get that 
\begin{equation}\label{simp5}
\begin{array}{ll} 
\sp(Q_h(z,D_z))&=\sp(Q_h(x,-D_{\xi },D_x,\xi ))\\
& =\sp(Q_h(x+\frac{\xi }{b},-D_{\xi }+\frac{1}{b}D_x,D_x,\xi ));.
\end{array} 
\end{equation}
So 
\begin{equation}\label{simp6} 
\sp(P_{h,A^m})=h\cdot \sp(H_h(x,\xi ,D_x,D_{\xi })), 
\end{equation} 
$$H_h(x,\xi ,D_x,D_{\xi })=D_{x}^{2}+[bx+
h\sum _{j=3}^{m}h^{(j-3)/2}T_j(x,\xi ,D_x,D_{\xi })]^2, $$
with $\ \displaystyle \ T_j(x,\xi ,D_x,D_{\xi })=
S_j(x+\frac{\xi }{b},-D_{\xi }+\frac{1}{b}D_x).$ 

The differential operator $\ T_3$ has the following form: 
\begin{equation}\label{simp7} 
\begin{array}{ll}
T_3(x,\xi ,D_x,D_{\xi })&=xL(\xi ,D_{\xi })+M_0(\xi ,D_{\xi })+
M_1(x,D_x)+M_2(\xi ,D_x,D_{\xi })\\
&\quad +M_3(x,D_x,D_{\xi })+ M_4(x,\xi ,D_x), 
\end{array}
\end{equation}
where $\ L(\xi ,D_{\xi })$ an operator with compact resolvent, 
$$
\begin{array}{rl}
L(\xi ,D_{\xi })& = \,\frac{\alpha }{b^2}\xi ^2+\beta D_{\xi }^{2}\;,\\ 
M_0(\xi ,D_{\xi })&=\, \frac{\alpha }{3b^3}\xi ^3+
\frac{\beta }{b}\xi D_{\xi }^{2},\\ 
M_1(x,D_x)& = \, \frac{\alpha }{3}x^3+\frac{\beta }{b^2}xD_{x}^{2}\;, \\ 
M_2(\xi ,D_x,D_{\xi }) & = \, -2\frac{\beta }{b^2}D_x\xi D_{\xi }\;,\\
M_3(x,D_x,D_{\xi })&=\, -2\frac{\beta }{b}xD_xD_{\xi }\;,\\ 
M_4(x,\xi ,D_x)& = \, \frac{\alpha }{b} x^2\xi +
\frac{\beta }{b^3}\xi D_{x}^{2}\;.
\end{array}
$$
Using the notation  $\ w=(x,\xi )$, we can write 
\begin{equation}\label{simp8} 
H_h(w,D_w) =H^{0}_{h}(w,D_w)+W_h(w,D_w)\;,
\end{equation}
with
\begin{equation} \label{simp8a}
H^{0}_{h}(w,D_w) =D_{x}^{2}+x^2[b+hL(\xi ,D_{\xi })]^2 \;,
\end{equation} 
and 
\begin{equation}\label{simp8b}
\begin{array}{l}
W_h(w,D_w)=\\
\qquad +  h\displaystyle 
\sum _{\ell =0}^{4}[x(b+hL(\xi ,D_{\xi }))M_\ell (w,D_w)+
M_\ell (w,D_w)x(b+hL(\xi ,D_{\xi }))]\\
 \qquad + 
 h^{3/2}\sum _{j=4}^{m}h^{(j-4)/2}[
x(b+hL(\xi ,D_{\xi }))T_j(w,D_w)+
T_j(w,D_w)x(b+hL(\xi ,D_{\xi }))]\\
\qquad + 
 h^2[\sum _{\ell =0}^{4}M_\ell (w,D_w)]^2 \\
\qquad + h^\frac 52 \sum_{\ell, j} h^{(j-4)/2} [M_\ell(w,D_w)\, T_j (w,D_w)\,+\, T_j(w,D_w) \, M_\ell(w,D_w)]\\
\qquad +h^{3}
[\sum _{j=4}^{m}h^{(j-4)/2}T_j(w,D_w)]^2  \;.
\end{array}
\end{equation}
The operator $\ H^{0}_{h}(w,D_w)$ has compact resolvent and 
\begin{equation}\label{simp9} 
\sp(H^{0}_{h}(w,D_w))=\{ \mu _{j,k}(h)=
(b+h\mu _j)(2k-1); \ (j,k)\; \in \; (\nz ^{\star })^2\} ,
\end{equation} 
with 
\begin{equation}\label{simp10} 
\{ \mu _j=\frac{d}{b}(2j-1);\ j\; \in \; \nz ^{\star }\} 
=\sp \left(L(\xi ,D_{\xi }\right)\;,
\end{equation} 
and
\begin{equation}\label{simp10a}
d=(\alpha \beta )^{1/2}\; . 
\end{equation}
The eigenfuction of $\ H^{0}_{h}(w,D_w)$ 
associated to the eigenvalue $\ \mu _{j,k}(h)$ 
is 
\begin{equation}\label{simp11}
\psi ^{h}_{j,k}(w)=(\mu _1)^{1/2}(\mu _{j,1}(h))^{1/2}
\varphi_k ((\mu _{j,1}(h))^{1/2}x)
\varphi_j ((\mu _1)^{1/2}\xi ), 
\end{equation} 
where the  $\ \varphi_j (x)$ are the normalized eigenfunctions 
of the harmonic oscillator 
$\ D_{x}^{2}+x^2$, $$ \varphi _j(x)=p_j(x)e^{-x^2/2},$$
and  $\ p_j(x)$ is the Hermite  polynomial 
function of order $j-1$. We recall in particular that~:
$$ p_1(x)=\pi ^{-1/2}.$$ 

\subsection{End of the proof of Theorem  \ref{Th3} }
Then, by (\ref{simp8})--(\ref{simp11}), we have~: 
\begin{equation}\label{1pth3} 
H_h(w,D_w)\psi ^{h}_{1,1}(w)=\mu _{1,1}(h)
\psi ^{h}_{1,1}(w)+W_h(w,D_w)\psi ^{h}_{1,1}(w), 
\end{equation}  
and we can write, using the expansion of 
$W_h(w,D_w)$ given above in (\ref{simp8b})~:
\begin{equation}\label{2pth3} 
\begin{array}{l}
W_h(w,D_w)\psi ^{h}_{1,1}(w)= \\
\qquad h\,b[xM(w,D_w)+M(w,D_w)x]
\psi ^{h}_{1,1}(w)+ \Og(h^\frac 32)\;,
\end{array}
\end{equation}
in $L^2(\rz^2)$, 
where  $$ \ M(w,D_w)=\sum _{\ell =0}^{4}M_\ell (w,D_w))\;.$$
Because we need only an upper bound, it is enough to consider~:
$$ \delta \lambda:= \langle hb[xM(w,D_w)+M(w,D_w)x]\psi ^{h}_{1,1}(w)\;|\;\psi ^{h}_{1,1}(w)\rangle\;.$$ 

By (\ref{simp11}), the function 
$\psi ^{h}_{1,1}(1,\xi )$ 
is even in the $\ x$ variable, so  the expansion of 
the $\ M_\ell (w,D_w)$ given below (\ref{simp7}) entails 
\begin{equation}\label{4pth3}
\begin{array}{l}
\delta \lambda \\
 \quad  =
 b<[xM_1(w,D_w)+
M_1(w,D_w)x]\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\\
\qquad +
b<[xM_2(w,D_w)+M_2(w,D_w)x]\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\\  
\quad  =b<[\frac{2\alpha }{3}x^4+
\frac{\beta }{b^2}x^2D_{x}^{2}+
\frac{\beta }{b^2}xD_{x}^{2}x]\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\\
\qquad -
b<[\frac{2\beta }{b^2}xD_x\xi D_{\xi }+
\frac{2\beta }{b^2}D_xx\xi D_{\xi }]
\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\\ 
\quad  =2b<[\frac{\alpha }{3}x^4+
\frac{\beta }{b^2}x^2D_{x}^{2}\\
\qquad -i\frac{\beta }{b^2}xD_{x}]\psi ^{h}_{1,1}-
[\frac{2\beta }{b^2}xD_{x}\xi D_{\xi }-
i\frac{\beta }{b^2}\xi D_{\xi }]
\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\;.
\end{array}
\end{equation}
As $\ D_{x}^{2}\psi ^{h}_{1,1}=\mu _{1,1}(h)
(1-\mu _{1,1}(h)x^2)\psi ^{h}_{1,1}$ 
and $\ xD_x\psi ^{h}_{1,1}=i\mu _{1,1}(h)x^2\psi ^{h}_{1,1},$ \\ 
we get from (\ref{4pth3}) 
\begin{equation}\label{5pth3} 
\begin{array}{l}
\delta \lambda \\
\quad =
2b<[(\frac{\alpha }{3}-
\frac{\beta }{b^2}\mu ^{2}_{1,1}(h))x^4+\frac{2\beta }{b^2}
\mu _{1,1}(h)x^2]\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>\\
\qquad- 
i\frac{2\beta }{b}<(2\mu _{1,1}(h)x^2-1)\xi D_{\xi }
\psi ^{h}_{1,1}|\psi ^{h}_{1,1}>.
\end{array}
\end{equation}
Let us recall the formulas 
$$ 2\int x^2e^{-x^2}dx=\pi \ 
{\rm {and}} \ 4\int x^4e^{-x^2}dx=3\pi .$$ 
From the first one and (\ref{simp11}) with $\ k=1$, we get 
immediately that,  for any differential operator $\ U(\xi ,D_{\xi }),$ 
$$<(2\mu _{1,1}(h)x^2-1)U(\xi ,D_{\xi })\psi ^{h}_{1,1}
|\psi ^{h}_{1,1}>=0.$$
Using again the formulas,    
we get from (\ref{5pth3}) 
\begin{equation}\label{6pth3} 
\begin{array} 
{ll}
\delta \lambda & =    
 \frac{3b}{2\mu ^{2}_{1,1}(h)}
(\frac{\alpha }{3}-\frac{\beta }{b^2}\mu ^{2}_{1,1}(h))
+2\frac{\beta }{b}\\ 
&
=\frac{1}{2b}(\alpha +\beta )+\Og(h)\;.
\end{array} 
\end{equation}  
Here we have used that $\ \mu _{1,1}(h)=b+h\mu _1$. \\
Returning to the original metaplectic coordinates, we have found $u_{1,h}$ in $\Sg(\rz^2)$ and of norm $1$ in $L^2(\rz^2)$
 such that~:
\begin{equation}\label{14pth3}
\begin{array}{l} 
\langle (P_{h,A^m}-h\mu _{1,1}(h)-\frac{h^2}{2b}(\alpha +\beta )]
u_{h,1}\;|\; u_{h,1}\rangle \leq C\, h^{5/2} \;.
\end{array}
\end{equation}
Keeping in mind  that (\ref{simp9}) and (\ref{simp10}) give that 
$$h\mu _{1,1}(h)+\frac{h^2}{2b}(\alpha +\beta )=
hb+h^2\frac{a^2}{2b},$$
with $a=(\alpha )^{1/2}+(\beta )^{1/2}$,  
(\ref{14pth3}) and (\ref{compa}) show that~:
$$ 
\inf \sp(P_{h,A,\Omega }))\leq hb+h^2\frac{a^2}{2b} + C \, h^{5/2}\ \;. $$ 
The theorem is proved.\\
 $\bullet$



\section{ On  the Dirichlet problem 
when the minimum of the magnetic field 
is achieved on the boundary}\label{br}
\subsection{Introduction}
In this section, instead of the assumption (\ref{Hb1}),  we consider the case when there exists $z_0\in \pa \Omega$
 such that~:
\begin{equation}\label{br1} 
0<b :=B(z_0)\leq B(z)\;,\; \forall \; z\; \in \; {\overline {\Omega }}, 
\end{equation} 
is satisfied.\\
 In particular, we have $b=b'$. \\
We assume $\ \Omega $ is a bounded open set of class $\ C^3$ 
and that the magnetic potential is  
of class $\ C^3.$ 
Let $\ \lambda^{(1)}(h)$ be the ground state energy of the Dirichlet operator 
and let  $\ u_h(z)\in H^2(\Omega) \cap H_0^1(\Omega)$ be an associated eigenfunction: 
\begin{equation}\label{br2}
\begin{array}{c} 
 \lambda^{(1)}(h)=\inf \; \sp (P_{h,A,\Omega }^D)\;,\\ 
P_{h,A,\Omega }^D\; u_h=\lambda^{(1)}(h) \;  u_h \;.
\end{array}
\end{equation} 
We know, thanks to (\ref{Es0}) and (\ref{br1}), that 
\begin{equation}\label{br4} 
hb \leq \lambda^{(1)} (h)
\end{equation} 
and by (\ref{1Pro1}) (with $\rho =1$ and $ V(z)=B(z)$), we know 
also that  
\begin{equation}\label{br5}
\begin{array}{l}
(1-\epsilon ^2)\int _{\Omega }
[h B(z)-\lambda^{(1)}(h)]_{+}e^{2\epsilon \phi_h (z)/h}
|u_h(z)|^2\;dz \\
\quad \quad \leq \int_{\Omega} [h B(z)-\lambda^{(1)}(h)]_{-}|u_h(z)|^2\; dz, 
\end{array}
\end{equation} 
for any $\ \epsilon \; \in \; ]0,1],$  
where $\  \phi _h(z)$ is the Agmon distance of 
$\ z$ to the well  
$$\ U_h=\{ \omega \in {\overline {\Omega }};\ hB(z)-\lambda^{(1)} (h) \leq 0\} ,$$ 
(associated to the metric $\ [hB(z)-\lambda^{(1)}(h)]_+\, dz^2).$ 

The purpose  of this subsection  is to get  bounds  of 
$\ \lambda^{(1)}(h)$ and then to exploit this upper bound and   (\ref{br5})
 for  a more explicit   asymptotic localization 
of the eigenfunction.   
\subsection{Rough lower bounds}
Let us start with lower bounds. The following estimate
 give the probably correct behavior as $h\ar 0$.
\begin{proposition}\label{Rmbr02}.\\
Let us assume that the inequality~:
\begin{equation}\label{br100} 
0<b <B(z)\;,\; \forall \; z\; \in \; \Omega\;,
\end{equation} 
is satisfied. Then, 
if 
$\ \displaystyle \nabla B(z_0)\neq 0$, 
for any  $\ z_0 \in \pa \Omega$ such that $B(z_0)=b$, 
 then there exists $\ h _0>0$ and
 $\ \eta _0>0$ such that
\begin{equation}\label{br30}
hb +h^{3/2}\eta _0\leq \inf \; \sp (P_{h,A,\Omega }^D),
\ \ \forall \; h\; \in \; ]0,h_0].
\end{equation}
\end{proposition}
As a matter of fact, (\ref{interpol}) with $\tau = \rho h^{-1/2}$
 leads to
\begin{equation}\label{br31}
\rho h^{1/2} P^D_{h,A,\Omega } +(1-\rho h^{1/2})hB\leq P^D_{h,A,\Omega },\ \
\forall \ \rho  \; \in \; [0,h^{-1/2}].
\end{equation}
If  we use  Kato's inequality (\ref{diamD})
 we get   that~:
\begin{equation}\label{br33}
h\;  \inf \; \sp\left(-\rho h^{3/2}\Delta^D_{\Omega }+(1-\rho h^{1/2})B\right)\; \leq \;
\inf \; \sp(P_{h,A,\Omega })\;.
\end{equation}
Here $-\Delta^D_{\Omega }=P_{1,0,\Omega }$ is the Laplace operator
on $\Omega $ with
Dirichlet boundary condition.

\noindent But the assumptions of Proposition \ref{Rmbr02} lead to
 existence of $c_0 >0$ such that~:
\begin{equation}\label{br34}
c_0 \, t(z)+b \leq B(z)\ 
\end{equation}
for all $ z$ in a neighborhood of $\pa \Omega$ in $\Omega$, 
where  $t(z)=\dist (z,\partial \Omega )$.\\
It is enough to take~:
$$ 0< c_0 < \beta_1 = \min_{x\in m(\pa \Omega)} \pa_\nu B (x)\;, $$
where
\begin{equation}\label{br34a}
  m(\pa \Omega):= \{x\in \pa \Omega\;|\; B(x) = b\}\;.
\end{equation}
So, if $\ \rho \leq h^{-1/2}/2$,
\begin{equation}\label{br35}
h\;  \inf \; \sp(-\rho h^{3/2}\Delta^D _{\Omega }+\frac{c_0}{2}t(z))
\; +\ h(1-\rho h^{1/2})b \; \leq \;
\inf \; \sp(P_{h,A,\Omega }^D).
\end{equation}
But it is easy to see, (see for example \cite{Mai}), that, for $h$ small enough, 
\begin{equation}\label{br36}
-C_0\rho h^{3/2}+(\rho h^{3/2})^{1/3}(\frac{c_0}{2})^{2/3}\omega _0
\leq
\inf \; \sp(-\rho h^{3/2}\Delta^D_{\Omega }+\frac{c_0}{2}t(z)),
\end{equation}
where  $\ \omega _0$ is the first eigenvalue of the Dirichlet problem
on $\ L^2(\rz _{+})$,
 associated to the Airy differential operator~:
\begin{equation}\label{airy}
A(t, D_t):= -\frac{d^2}{dt^2}+t\;.
\end{equation}
The constant $C_0$ depends only on $\ c_0$ and does not
 depend on $\ \rho$ and $h$.

The estimates (\ref{br35}) and (\ref{br36}), with for example
$$ \rho =(\omega _0(\frac{c_0}{2})^{2/3}(3b )^{-1})^{3/2},$$
give the proof of  (\ref{br30}).

\subsection{Rough upper bounds}

The purpose  of this subsection  is to get an upper bound of 
$ \lambda^{(1)}(h)$ and then to exploit this upper bound and   (\ref{br5})
 for  a more explicit   asymptotic localization 
of the eigenfunction.   

\begin{theorem}\label{Thbr0}  .\\ 
If  $\ \Omega \; \subset \; \rz ^2$ is a bounded open connected 
 set of class $\ C^3$ 
and if  
 $A$ belongs to $C^3({\overline {\Omega }},\rz ^2)$, 
then, under the assumption (\ref{br1}), there exists 
a constant $\ C$ and $h_0$ such that 
\begin{equation}\label{br0e} 
b\,h \leq \inf \; \sp (P_{h,A,\Omega }^D)\leq b\,h + 
C\,h^{3/2}(\ln(h))^2\;,\ \ \forall \; h\; \in ]0, h_0[. 
\end{equation} 
\end{theorem} 

The difficulty to get (\ref{br0e}) without the logarithmic  term comes 
from the following fact. The model operator at a point of the boundary where $B$ is minimal is 
$\ P^{+}_{h,A^0}:=P_{h,A^0,\rz _{+}\times \rz }$ 
with $\ A^{0}(z)=\frac{b }{2} (-y,x)$.
It is easy to see that 
$$ \inf \; \sp ( P^{+}_{h,A^0})= b h \inf _{\xi \in \rz }\; 
\inf \; \sp (H^{D,\xi })= b\, h $$
where $\ H^{D,\xi }$ is the Dirichlet operator on $\ \rz _{+}$ 
associated to 
$$  -\frac{d ^2}{d u^2}+(\xi - u)^2\,.$$
Unlike  the Neumann realization  $H^{N,\xi }$, 
the infimum in $\ \xi $    of 
the ground state energy of $\ H^{D,\xi }$, which is equal to $1$,  is not 
achieved for finite $\xi$.
 So, in order to construct a quasimode,  we have to take $\ \xi $ large
 and in this case 
the groundstate eigenfunction of  $H ^{D,\xi }$ is localized 
exponentially near $\ u =\xi$.
But, in  our model, $\  \Omega $ is bounded. So we can only work  in $\Omega$ and we will work (due to the scaling)
in a domain such that  $ \ 0< h^{1/2} u <\epsilon (\Omega )$.\\ 
Here $\epsilon (\Omega)$ is a geometrical constant such that the distance to the boundary
 in the domain $\{ z\in{\bar \Omega}\;|\; d(z,\pa \Omega) \leq \epsilon(\Omega)\}$
 is regular.\\

Let us recall from \ref{AppA} that, for the Neumann 
realization,  the infimum  was obtained  for some $\xi_0$ in $]0,+\infty[$. 

\vskip 0.5cm 

{\bf {Proof of Theorem \ref{Thbr0}}} \\

\noindent Let us work in the system of coordinates recalled in Appendix \ref{AppB}. The assumption (\ref{br1}) and the identities (\ref{br10})
and (\ref{br13a}) show that, in the right gauge~: 
\begin{equation}\label{br14} 
|\frac{\partial{\widetilde {A}_1}} {\partial t}
+b (1-t\kappa (0))+
t\frac{\partial {\hat B} }{\partial t}(0,0)|\leq C\, (s^2+t^2),
\end{equation} 
so we can assume that in $\ K$,   
\begin{equation}\label{br15} 
|{\widetilde {A}}_1(w)+tb (1-\frac{t}{2}\kappa (0))
+\frac{t^2}{2}\frac{\partial {\hat B} }{\partial t} (0,0)|
\leq C\, t(s^2+t^2). 
\end{equation} 
Let us take the test function 
\begin{equation}\label{br16} 
u^{0}_{h}(z)=e^{-i\xi_h s/h}a^{-1/2}(w) \, v^{0}_{h}(w),\ 
\end{equation} 
with $ (w=(s,t)=\psi (z))$,  $\ \xi _h\: \in \; \rz $ and 
$\ v^{0}_{h}\: \in \; C^{2}_{0}(K;\rz )$  
(to be chosen suitably). \\ 
So by (\ref{br8}) - - (\ref{br13}), 
\begin{equation}\label{br17} 
q_{h,A,\Omega }(u^{0}_{h})=
\int _K\{ h^2[(\frac{\partial v^{0}_{h} }{\partial t})^2+
(1-t\kappa (s))^{-2}(\frac{\partial v^{0}_{h} }{\partial s})^2] 
+[a^{-2}(\xi _h +{\widetilde {A}}_1)^2+h^2W](v^{0}_{h})^2\} dw. 
\end{equation} 
Let $\ C_0>0$ be fixed large enough. 
Let $\ \chi \; \in \; C^{\infty }_{0}(\rz ;\rz )$ be a cut-off function 
such that $\ \chi (x)=1$ if $\ |x|<C_0$ and 
$\ \chi (x)=0$ if $\ |x|>2C_0$.\\
We choose 
\begin{equation}\label{br18} 
\xi _h = 2C_0^{1/2} h^{1/2}|\ln(h)| 
\end{equation} 
and, with 
$$ \ \theta _h(t)=b ^{1/2}h^{-1/2}(t-b ^{-1}\xi _h) $$
\begin{equation}\label{br19} 
v^{0}_{h}(w)=b ^{1/4}h^{-3/8}
\varphi _1(\theta _h(t))
\chi (|\ln(h)|^{-1}\theta _h(t))
\times f(h^{-1/4}s)\;.
\end{equation} 
Here $\varphi_1$ is defined by~:
 $$\ \varphi _1(x)=\pi ^{-1/2}e^{-x^2/2} $$
and $\ f$ is a function in $ C^{\infty }_{0}(]-\frac{1}{2},\frac{1}{2}[;\rz )$ 
 such that $\ \int f(x)^2dx=1.$ \\ 
So, if $\ h$ is small enough, 
 $\ v^{0}_{h}$ belongs to  $ C^{\infty }_{0}(K;\rz )$ 
and it is easy to check from 
(\ref{br15}), (\ref{br17}) - - (\ref{br19}) 
that
 there exist  
$h_0 \in  ]0,1/e[$ and $ \ C_1>0$  such that 
\begin{equation}\label{br20} 
|q_{h,A,\Omega }(u^{0}_{h})-h\minB 
\| u^{0}_{h}\| ^{2}_{L^2(\Omega )} | 
\leq C_1\, h^{3/2}(\ln(h))^2,\ \forall \; h\; \in \; ]0,h_0[.
\end{equation} 
Moreover,
\begin{equation}\label{br21} 
u^{0}_{h} \; \in \; H^{1}_{0}(\Omega ),\ \ {\rm {and}}\ \ 
|\, | u^{0}_{h}\| _{L^2(\Omega )}-1|\leq C\,h^2\;.
\end{equation} 
The properties (\ref{br21}) and  (\ref{br20}) 
prove (\ref{br0e}).\\
 $\ \bullet $\\
 
\begin{remark}\label{Rmbr00} .\\
{\rm  
If instead of (\ref{br1}) we have, for some $z_0\in \partial \Omega$, assumption 
(\ref{br100}), 
and if, for any such 
$  z_0$, we have~: $\nabla  B(z_0)\neq 0$, 
then we get easily from (\ref{br5}) and (\ref{br0e}) 
 the following localization of the groundstate 
eigenfunction. There exist  $\ \delta \; \in \; ]0,1[$ 
$\ C>0$ and $h_0$ such that, for all $h\in ]0,h_0]$, 
$$ \begin{array}{l}
|| \exp {\delta t(z)^{3/2}/h^{1/2}}|u_h(z)|\,||\\ 
\quad \quad \leq C\, h^{-C} ||u_h||\;,
\end{array}
$$
where we recall that
$t(z)=\dist (z,\partial \Omega )$.}
\end{remark}




\section{On  the ground state energy of the  Neumann realization} \label{Section9}

\subsection{The upper  bound of the ground state energy } 

Let $\ \Omega $ and $\ A$ be as in Section \ref{br}. 
Let $\ P^N_{h,A,\Omega }$ be the Neumann operator associated 
to the quadratic form 
\begin{equation}\label{nm1} 
q^N_{h,A,\Omega }(u)=\int _{\Omega } 
|(hD_z-A(z))u(z)|^2\; dz,\ \ \forall \; u\; \in \; H^1(\Omega ). 
\end{equation} 
Let us recall that we have defined 
\begin{equation}\label{nm2} 
\mu^{(1)}(h)=\inf \; \sp (P^N_{h,A,\Omega }) 
\end{equation} 
and 
\begin{equation}\label{nm3} 
\Theta _0 = \inf _{\xi \in \rz }\; \inf \; \sp (H^{N,\xi }).
\end{equation} 
The operator $H^{N,\xi }$ is the Neuman operator on $\ L^2(\rz _{+})$ 
associated to the differential operator 
$\ D^{2}_{x}+(x-\xi )^2$.  We recall from Section \ref{Sectionbasmod}  and Appendix \ref{AppA} that 
$\ \Theta _0$ is the lower bound of 
Neumann operator on $\ \rz_+ \times \rz $ associated 
to $\ D_{x}^{2}+(D_y\pm x)^2.$ \\

Let us first prove a version of the first part of the theorem by Lu-Pan
 recalled in Theorem \ref{ThLuPa2}  with a better control of the remainder.

\begin{theorem}\label{Thnm1} .\\ 
Under the assumptions of Theorem \ref{Thbr0}, but  with 
(\ref{br1}) replaced by~:
\begin{equation}\label{nm4} 
0<b' =\min _{\omega \in \partial \Omega }B(\omega ), 
\end{equation} 
then there exists $\ C_0>0$ such that 
\begin{equation}\label{nm5} 
\mu^{(1)}(h)\leq \Theta_0 \, b'\, h  +C_0\, h^{3/2},\ \ \forall \; h\; \in \; ]0,1] . 
\end{equation} 

\end{theorem} 
{\bf  {Proof}}.\\
The proof is similar to the one of Theorem \ref{Thbr0}.  The adapted coordinates near the boundary 
of the appendix \ref{AppB} are still valid.  
We can assume (\ref{br13}). Let 
\begin{equation}\label{nm8} 
z_0\in \partial \Omega \ {\rm {such \ that}} \ 
B(z_0)=b' \;.
\end{equation} 
Then  (\ref{br14}) is  valid 
thanks to (\ref{nm4}), so  we can  assume (\ref{br15}). 

As in the proof of  Theorem \ref{Thbr0}, we can take\footnote{Similar computations can be found in \cite{BeSt}.} 
a test function  $\ u^{0}_{h}(z)$ 
of the form (\ref{br16}), where~:
\begin{equation}\label{nm10} 
v^{0}_{h}(w)=(b')^{1/4}h^{-3/8}
\varphi_0\left((b')^{1/2} h^{-1/2}t\right)\chi (t)\times  f(h^{-1/4}s)\;, 
\end{equation} 
\begin{equation}\label{nm9} 
\xi _h=(b')^{1/2}h^{1/2}\xi_0  \;,
\end{equation} 
the function  $\ f$ is   as in (\ref{br19}) and 
$\ \chi (t)$ is a cut-off function such that~:\\ 
$\ \chi (t)=1$ if $\ 0\leq t\leq \epsilon _0/2$ 
and $\ \chi (t)=0$ if $\ t \geq \epsilon _0$. \\ 
The eigenfunction $\phi_0 = \phi_{\xi_0}$ is introduced in Appendix \ref{AppA}.
Using  formulas  (\ref{br8}), (\ref{br08})
 and (\ref{br15}), it is then easy to check 
that (\ref{nm6}) and  (\ref{nm10}) imply the existence of $C>0$ such that~:
\begin{equation}\label{nm11} 
|q^N_{h,A,\Omega }(u^{0}_{h})- \Theta_0\, b'\, h 
\| u^{0}_{h}\| ^{2}_{L^2(\Omega )} |\leq C\,h^{3/2},  
\end{equation} 
$$|\ \| u^{0}_{h}\| _{L^2(\Omega )}-1|\leq C\, h^{3/2} \;. $$ 
So (\ref{nm11}) proves  (\ref{nm5}).\\
 $\ \bullet $


  
\subsection{Rough lower bound estimates} \label{Subsection9.2}

Our aim  is to find an equivalent of the estimate (\ref{Es0}) 
for the Neumann quadratic form. We will follow the 
methods of \cite{HeMo2} . 


\noindent Let $\ (\chi _{\gamma }(z))_{\gamma \in \Gamma }$ 
be a partition of unity of $\ \rz ^2.$ 
For example we can take 
$$
 \Gamma =\zz ^2,\ \chi _{\gamma }\in C^{\infty }(\rz ^2;\rz )\ 
{\rm {and}}\ \supp (\chi _{\gamma })
\subset \gamma +[-1,1]^2,\ \forall \gamma \in \Gamma , 
$$
\begin{equation}\label{low1} 
\sum _{\gamma }\chi _{\gamma }^{2}(z)=1 
\ {\rm {and}}\ 
\sum _{\gamma } |\bigtriangledown \chi _{\gamma }(z)|^2\; < \; \infty . 
\end{equation} 
 If $\ \tau (h)$ is a function of $\ h$ such that 
$\ \tau (h)\in ]0,\epsilon(\Omega)[,$ 
($\epsilon (\Omega)$ is the  geometric constant related to $\ \partial \Omega$, and introduced in Section \ref{br}),  we will define the functions 
\begin{equation}\label{low2} 
\chi _{\gamma ,\tau (h)}(z)=\chi _{\gamma }(z/\tau (h)),\ 
\ \forall \; \gamma \; \in \; \Gamma \;.
\end{equation} 
So we get a new partition of unity such that 
\begin{equation}\label{low3} 
\sum _{\gamma }\chi _{\gamma ,\tau (h)}^{2}(z)=1,  
\ \ \ 
 \sum _{\gamma} |\nabla \chi _{\gamma ,\tau (h)}(z)|^2\leq C\, \tau (h) ^{-2}\;,
\end{equation}  
and $\ \ \displaystyle \supp (\chi _{\gamma ,\tau (h)})\subset 
\tau (h)\gamma +[-\tau (h),\tau (h)]^2 . $\\ 
Then, for any $\ u\; \in \; H^1(\Omega )$, (see (\ref{IMS1})), we have~:
\begin{equation}\label{low4} 
q^N_{h,A,\Omega }(u)=\sum _{\gamma } 
\left[q^N_{h,A,\Omega }(\chi _{\gamma ,\tau (h)}u)
-h^2\| |\nabla 
\chi _{\gamma ,\tau (h)}| \,u \; \| ^{2}_{L^2(\Omega )}\right]\;. 
\end{equation} 
Let us define 
\begin{equation}\label{low5} 
\begin{array} 
{l} 
\Gamma _{\tau (h)}(\Omega )=
\{ \gamma \in \Gamma ;\ \supp (\chi _{\gamma ,\tau (h)})\cap  \Omega 
\neq \emptyset \} \\ 
\Gamma ^{0}_{\tau (h)}(\Omega )=
\{ \gamma \in \Gamma _{\tau (h)}(\Omega );\ 
\dist (\supp (\chi _{\gamma ,\tau (h)}),\partial \Omega )>\tau (h)
 \} \\ 
\Gamma ^{1}_{\tau (h)}(\Omega )=
\{ \gamma \in \Gamma _{\tau (h)}(\Omega );\ 
\dist (\supp (\chi _{\gamma ,\tau (h)}),\partial \Omega )\leq \tau (h)
 \} 
\end{array} 
\end{equation} 
We assume that 
\begin{equation}\label{low6} 
B(z)>0,\ \ \forall\ z\; \in \; {\overline {\Omega }}. 
\end{equation} 
Then, for any $\ u\; \in \; H^1(\Omega ),$ 
\begin{equation}\label{low7} 
q^N_{h,A,\Omega }(\chi _{\gamma ,\tau (h)}u)\geq h 
\| B^{1/2} \chi _{\gamma ,\tau (h)}u \| ^{2}_{L^2(\Omega )}, 
\ \forall \; \gamma \; \in \; \Gamma ^{0}_{\tau (h)}(\Omega )\;. 
\end{equation} 
For any $\ \gamma \; \in \; \Gamma ^{1}_{\tau (h)}(\Omega )$, 
we use the adapted coordinates 
and  formulas  (\ref{br8}) and (\ref{br08}).\\  
As in \cite{HeMo2}, from the identities (\ref{br10}) and (\ref{br13a}), 
 it is easy to find a gauge such that 
\begin{equation}\label{low8} 
 |{\widetilde A}(w)+(B(z_{\gamma ,\tau (h)})t,0)|\leq C\;\tau (h)^2,
\end{equation} 
for all $w=w(z)$ such that $\  z\; \in \; \supp (\chi _{\gamma ,\tau (h)})$.\\  
Here $\ z_{\gamma ,\tau (h)}$ is chosen in 
$\  \supp (\chi _{\gamma ,\tau (h)})$ 
such that  $\ B(z_{\gamma ,\tau (h)})$ is maximum on 
$\ \supp (\chi _{\gamma ,\tau (h)}).$\\ 
Then using that $\ \Theta _0$ is a lower bound of the Neumann 
operator associated 
to\\ 
 $\ (D_s+t)^2+D_{t}^{2}$, so  
\begin{equation}\label{low9} 
\int _{\rz \times \rz _{+}} 
[|(D_s+t)v| ^{2}+|D_tv|^2- \Theta _0|v|^2]\, dw\geq 0, 
\ \forall v\in C^{1}_{0}(\rz \times {\overline {\rz _{+}}})\;.
\end{equation} 
Using also the property, deduced from (\ref{br08}), that 
\begin{equation}\label{low09} 
|\int |v|^2\,dz-\int |v|^2dw|\leq C\, \tau (h)\, \int |v|^2\,dz,\ 
{\rm {if}} \ \supp (v)\subset \Omega _{4\tau (h)}, 
\end{equation}  
we get easily 
from (\ref{low8}) and (\ref{low9}) that,  
$\ \forall  \; \gamma \; \in \; \Gamma ^{1}_{\tau (h)}(\Omega ),
\ \forall \; \epsilon \; \in \; ]0,1/2],$ and 
$\ \forall \; u\; \in \; H^1(\Omega ),$ 
$$
\begin{array}{l}
 (1+\epsilon )[1+C\tau (h)]
q^N_{h,A,\Omega }(\chi _{\gamma ,\tau (h)}u)\\ 
\qquad \geq 
\int [|(hD_s+tB(z_{\gamma ,\tau (h)})\chi _{\gamma ,\tau (h)}u|^2
+h^2|D_t\chi _{\gamma ,\tau (h)}u|^2]dw \\ 
\qquad\quad -C\,\epsilon ^{-1}(\tau (h))^4 
\|  \chi _{\gamma ,\tau (h)}u \| ^{2}_{L^2(\Omega )},
\end{array}
$$ 
for any $\ z_{\gamma ,\tau (h)}
\; \in \; \supp (\chi _{\gamma ,\tau (h)})
\cap {\overline {\Omega }}$.\\ 
So, there exists $C>0$, such that, for any $\epsilon >0$,
\begin{equation}\label{low10} 
\begin{array} 
{l}
(1+\epsilon )[1+C\tau (h)]
q^N_{h,A,\Omega }(\chi _{\gamma ,\tau (h)}u)\\ 
\qquad \geq h \Theta _0[1-C\tau (h)]
\| B^{1/2} \chi _{\gamma ,\tau (h)}u \| ^{2}_{L^2(\Omega )}\\ 
\qquad \quad -C\epsilon ^{-1}(\tau (h))^4 
\|  \chi _{\gamma ,\tau (h)}u \| ^{2}_{L^2(\Omega )}.
\end{array}
\end{equation} 
We can now get the following proposition. 
\begin{proposition}\label{Prnm1} .\\ 
Under the assumptions of Theorem \ref{Thbr0},  with 
(\ref{br1}) replaced by (\ref{low6}), then 
\begin{equation}\label{ePrnm1} 
q^N_{h,A,\Omega }(u)\geq \int _{\Omega }W_h(z)|u(z)|^2dz,\ 
\forall \; u\; \in \; H^1(\Omega ),
\end{equation} 
 for any $\ h\; \in \; ]0,1].$ \\
Here $W_h$ is defined, for some constant $C_0 >0$, 
by~:
$$
\begin{array}{lll}
W_h(z)&=h B(z) 
&{\rm if}  \dist (z;\partial \Omega )>2h^{3/8} \\ 
&=h\Theta_0\,B(z)-C_0\, h^{5/4}
&{\rm if} \ \dist (z;\partial \Omega )\leq 2h^{3/8},
\end{array}
$$
and 
$\Theta_0\; \in \; ]0,1[$ is defined by (\ref{nm3}). 
\end{proposition} 
{\bf  {Proof}}. \\ 
We take a partition of unity on 
$\ \rz$ such that~:
\begin{equation}\label{part} 
\psi ^{2}_{0,\tau (h)}(t)+\psi ^{2}_{1,\tau (h)}(t)=1,\ \ 
|\psi'_{j,\tau (h)}(t)|\leq C/\tau (h)\;,\; {\rm for} \; j=0, 1\;,\end{equation} 
and
 \begin{equation}\label{parta} 
\supp (\psi ^{2}_{0,\tau (h)})\subset [\tau (h)/20, +\infty [, 
\  \supp (\psi ^{2}_{1,\tau (h)})\subset ]-\infty ,\tau (h)/10].
\end{equation}
Then, for $\ t=t(z)=\dist (z;\partial \Omega )$ in 
(\ref{part}) we get as (\ref{low4}),  
\begin{equation}\label{low40} 
q^N_{h,A,\Omega }(u)=\sum _{j=0}^{1} 
\left[q^N_{h,A,\Omega }\left((\psi _{j,\tau (h)}\circ t)\, u\right)
-h^2\|  |\nabla (\psi _{j,\tau (h)}\circ t)\;|\; u\; \| ^{2}_{L^2(\Omega )}\right],  
\end{equation} 
for any $\ u\; \in \; H^1(\Omega )$,
and as in  (\ref{low7}) we have also 
\begin{equation}\label{low70} 
q^N_{h,A,\Omega }\left((\psi _{0,\tau (h)}\circ t) u\right)\geq h 
\| B^{1/2} (\psi _{0,\tau (h)}\circ t) \,  u \| ^{2}_{L^2(\Omega )}. 
\end{equation} 
Moreover by (\ref{low3}) and (\ref{low5}), we get~:  
\begin{equation}\label{low41}  
\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\chi ^{2}_{\gamma ,\tau (h)}(z)=1,\ \ \forall \; z\; \in \; 
\supp (\psi _{1,\tau (h)}\circ t). 
\end{equation} 
So, for any $\ u\; \in \; H^1(\Omega ),$ 
\begin{equation}\label{low42} 
\begin{array}{l}
q^N_{h,A,\Omega }((\psi _{1,\tau (h)}\circ t)\; u)\\
\quad =
\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
[q^N_{h,A,\Omega }(\chi _{\gamma ,\tau (h)}(\psi _{1,\tau (h)}\circ t)\; u) \\
\quad \quad -h^2
\|(\psi _{1,\tau (h)}\circ t)\; 
|\nabla  \chi _{\gamma ,\tau (h)}|\,u\; \| ^{2}_{L^2(\Omega )}]. 
\end{array}
\end{equation} 
Taking $\ \tau (h)=h^{3/8}$ and in (\ref{low10}) $\ \epsilon =h^{1/4}$, 
we deduce the estimate (\ref{ePrnm1}) of 
 Proposition \ref{Prnm1}, from (\ref{low3}), 
 (\ref{low10}) and (\ref{part}) - - (\ref{low42}).\\ 
$\ \bullet $ \\


With the Proposition  \ref{Prnm1} and Theorem \ref{Thnm1}, 
we can recover a more accurate version of the result of \cite{LuPa2}. We recall that~:
$$b:=\inf _{z\in \Omega }\; B(z) \;\mbox{ and }  
 b'  :=\inf _{\omega \in \partial \Omega }\; B(\omega )\;.
$$

\begin{corollary}\label{Cornm1} .\\ 
Under the assumptions of Proposition \ref{Prnm1} then 
the first eigenvalue $\ \mu^{(1)}(h)$ of the Neumann 
operator $\ P^N_{h,A,\Omega }$ satisfies the following estimate. \\ 
 {\bf {Case 1}}~:\\
 If 
\begin{equation} b < \Theta_0 b'\;,
\end{equation}
there exists $\ C_1>0$ and $\ h_0>0$ such that 
\begin{equation}\label{e1Cornm1} 
- C h^2\leq \mu^{(1)}(h)- b h \leq C_1\; h^2,\ \ \forall \; h\; \in \; ]0,h_0]\;.
\end{equation} 
 {\bf {Case 2}}~:\\
 If 
\begin{equation}
  b >  \Theta_0 b'\;,
\end{equation}
there exists $\ C_2>0$ and $h_0$ such that 
\begin{equation}\label{e2Cornm1} 
-C_1h^{5/4}\leq  \mu^{(1)}(h)- \Theta _{0}b'h \leq C_1\, h^{3/2}, 
\ \ \forall \; h\; \in \; ]0,h_0]\;.
\end{equation} 
 {\bf {Case 3}}~:\\
 If \begin{equation}
  b =  \Theta_0 b'\;,
\end{equation}
there exists $\ C_2>0$ and $h_0$ such that 
\begin{equation}\label{e2Cornm1b} 
-C_1h^{5/4}\leq  \mu^{(1)}(h)- \Theta _{0}b'h \leq C_1\, h^{2}, 
\ \ \forall \; h\; \in \; ]0,h_0]\;.
\end{equation} 
\end{corollary} 
\begin{remark}.\\
{\rm 
Using Agmon's estimates (see the next proposition),   one can actually 
improve in Case~1  the left hand side  of (\ref{e1Cornm1})
 into~:
\begin{equation}\label{e1Cornm1a} 
- C \exp - \frac{\delta}{h}\leq \mu^{(1)}(h)- b\, h ,
\end{equation}
for some $\delta >0$ and for all $ h$ in $ ]0,h_0]$.\\
In a particular case, this is related to (\ref{DimN}).\\
Note also  that for the proof of Case 3, we can use the proof of the upper bound
 used in Case 1. In Case $3$, the localization of the groundstate is, as $h\ar 0$, in the union of
 a small neighborhood of the boundary and of the set where $B =b$.}
\end{remark}


With the  estimate (\ref{ePrnm1}) of 
 Proposition \ref{Prnm1}  we can get also the localization in 
energy of the groundstate eigenfunction $\ u^{1}_{h}(z)$ of 
$\ P^N_{h,A,\Omega }$ as Proposition \ref{Pro1}.

\begin{proposition}\label{Prnm2} .\\ 
Under the assumptions of Proposition \ref{Prnm1}, 
for any $\ \epsilon \; \in \; ]0,1],$ 
\begin{equation}\label{ePrnm2} 
\begin{array}{l}
(1-\epsilon ^2)\int _{\Omega }
[W_h(z)-\mu^{(1)} (h)]_{+}e^{2\epsilon \phi _h(z)/h}
|u^{1}_{h}(z)|^2\;dz \\
\quad \leq \int _{\Omega }[W_h(z)-\mu^{(1)} (h)]_{-}|u^{1}_{h}(z)|^2\;dz.
\end{array}
\end{equation} 
for any real Lipschitz  function $\phi_h$ on
 $\ {\overline {\Omega }}$ such that~
$$ |\nabla  \phi _h(z)|^2\leq [W_h(z)-\mu^{(1)} (h)]_{+}.$$ 
\end{proposition} 

In the case 1 of Corollary \ref{Cornm1}, the estimate 
(\ref{ePrnm2})  and the one (\ref{1Pro1}) associated to the 
ground state of Dirichlet operator show that the 
first eigenvalue of the Neumann operator 
$\ P^N_{h,A,\Omega }$ is exponentially closed 
to the one of Dirichlet operator 
$\ P_{h,A,\Omega }.$ 

In  the case 2 of Corollary \ref{Cornm1}, we 
can improve the localization of the ground state 
eigenfunction (\ref{ePrnm2}) by the following 
proposition. 
\begin{proposition}\label{Prnm3} .\\ 
Under the assumptions of Proposition \ref{Prnm1} and  if\footnote{ The condition (\ref{ePrnm2a}) involves only the infimum of $B$ in a compact of $\Omega$. We recall indeed
from (\ref{Dau})  that $\Theta_0 <1$.}
\begin{equation}\label{ePrnm2a}
\Theta _0 b'
< b\; ,
\end{equation} 
then, for any $\ k\; \in \; \nz$,  there exists 
$\ C_k>0$ such that, $\ \forall \; h\; \in \; ]0,1]$,   
\begin{equation}\label{ePrnm3}  
\int _{\Omega }t(z)^k\, |u^{1}_{h}(z)|^2\,dz 
\leq C_k\, h^{k/2}\,\| u^{1}_{h}\|^{2}_{L^2(\Omega )}; 
\end{equation} 
and
\begin{equation}\label{ePrnm3a}  
\int _{\Omega }t(z)^k\,|(hD_z-A)u^{1}_{h}(z)|^2\,dz 
\leq C_k\,  h^{1+k/2}\, \| u^{1}_{h}\| ^{2}_{L^2(\Omega )}; 
\end{equation} 
where $t(z)=\dist (z;\partial \Omega )$ 
and   $\ u^{1}_{h}$ denotes an  eigenfunction associated to the 
first eigenvalue of the Neumann operator 
$\ P^N_{h,A,\Omega }.$ 
\end{proposition} 
{\bf {Proofs}} \\
 We can always find a function 
$\ {\widetilde t}(z)\; \in \; C^1({\overline {\Omega }})$, 
such that $t(z)/{\widetilde t}(z)$ and 
$\ {\widetilde t}(z)/t(z)$ are bounded on 
$\ \Omega $. Actually, we can in addition take $t={\tilde t}$ in the neighborhood of $\pa \Omega$.\\
 So, forgetting the tilde  for simplicity, we can assume 
that $\ t(z)\; \in \; C^1({\overline {\Omega }})$.\\
We can actually propose two proofs. The first one consists in applying  Theorem~\ref{Proppoint}
 together with (\ref{Agm2a}), observing in addition that $ u^k \leq k! \exp u$, for $u\geq 0$.
Let us now give an independent and more direct proof.
We also observe that the proposition is true for $k=0$. Let us now consider
 the case when $k\geq 1$.\\
As $\ t(\omega )= 0,\ \forall \; \omega \; \in \; \partial \Omega$,
we   have  the formula 
\begin{equation}\label{e30} 
\begin{array} 
{l}
h\int _{\Omega }t(z)^k\, B(z)| u^{1}_{h}|^2 \; dz =\\
\qquad i\int _{\Omega }t(z)^k\left[(hD_y-A_2)u^{1}_{h}
{\overline {(hD_x-A_1)u^{1}_{h}}}-(hD_x-A_1)u^{1}_{h}
{\overline {(hD_y-A_2)u^{1}_{h}}}\right]\; dz\\ 
\quad \quad 
-hk\int _{\Omega }t(z)^{k-1}[\frac{\partial t }{\partial x}(hD_y-A_2)u^{1}_{h}
-\frac{\partial t }{\partial y} (hD_x-A_1)u^{1}_{h}]
{\overline u^{1}_{h}} \, dz\;. 
\end{array} 
\end{equation} 
We deduce from (\ref{e30})  that 
\begin{equation}\label{e31} 
\begin{array} 
{l}
 h\int _{\Omega } t(z)^k   B(z)  | u^{1}_{h}|^2 \, dz\leq 
\int _{\Omega }t(z)^k|(hD_z-A) u^{1}_{h}|^2\, dz\\ 
\quad \quad 
+C\,h \| t^{(k-1)/2}(hD_z-A) u^{1}_{h}\| _{L^2(\Omega )}
\times \| t^{(k-1)/1} u^{1}_{h}\| _{L^2(\Omega )}.
\end{array} 
\end{equation} 
Writing that 
$$
\begin{array}{ll}
\int _{\Omega }t(z)^k|(hD_z-A) u^{1}_{h}|^2\, dz& = 
\int _{\Omega }(hD_z-A) u^{1}_{h}.{\overline {(hD_z-A)t^ku^{1}_{h}}}\, dz\\  
& \quad -ihk\int _{\Omega }t(z)^{k-1}\, [\nabla t \cdot 
(hD_z-A) u^{1}_{h}]{\overline u^{1}_{h}}\, dz,
\end{array} 
$$
or
$$
\begin{array}{ll}
\int _{\Omega }t(z)^k|(hD_z-A) u^{1}_{h}|^2\, dz & = 
\int _{\Omega } P_{h,A,\Omega} u^{1}_{h}.{\overline {t^k\,u^{1}_{h}}}\, dz \\ 
&\quad -ihk\int _{\Omega }t(z)^{k-1}\, [\nabla t \cdot 
(hD_z-A) u^{1}_{h}]{\overline u^{1}_{h}}\, dz,
\end{array}
$$ 
we get   
\begin{equation}\label{e32} 
\begin{array} 
{l}
|\int _{\Omega }t(z)^k[|(hD_z-A) u^{1}_{h}|^2-
\mu^{(1)}(h)|u^{1}_{h}|^2]dz|\leq \\ 
\quad \quad 
C\, h\| t^{(k-1)/2}(hD_z-A) u^{1}_{h}\| _{L^2(\Omega )}
\times \| t^{(k-1)/2}u^{1}_{h}\| _{L^2(\Omega )}. 
\end{array}  
\end{equation} 
So we get from (\ref{e31}) and (\ref{e32}) that 
\begin{equation}\label{e33} 
\begin{array}{l}
\int _{\Omega }\,t(z)^k \, \left(B(z)-\frac{\mu^{(1)}(h)}{h}\right)\,| u^{1}_{h}|^2 dz\\
\leq 
\qquad C\, \| t^{(k-1)/2}(hD_z-A) u^{1}_{h}\| _{L^2(\Omega )}
\times \| t^{(k-1)/2}u^{1}_{h}\| _{L^2(\Omega )}.  
\end{array}
\end{equation} 
The upper bound in  (\ref{e2Cornm1}), the assumption (\ref{ePrnm2a}) which gives
 a lower bound for $B(x) - \frac{\mu^{(1)}(h)}{h}$ and (\ref{e33}) 
 give the existence of 
$ h_0>0$ such that, for all $ h \in \; ]0,h_0]$, 
\begin{equation}\label{e34} 
 \| t^{k/2}u^{1}_{h}\| ^{2}_{L^2(\Omega )}\leq 
C\, \| t^{(k-1)/2}(hD_z-A) u^{1}_{h}\| _{L^2(\Omega )}
\times \| t^{(k-1)/2}u^{1}_{h}\| _{L^2(\Omega )}.  
\end{equation} 


\noindent Now we can  prove (\ref{ePrnm3})  by recursion.\\

\noindent  If $\ k=1$, we use   that 
$$  \| (hD_z-A) u^{1}_{h}\| ^{2}_{L^2(\Omega )}
=\mu^{(1)}(h)\|  u^{1}_{h}\| ^{2}_{L^2(\Omega )}
\leq C\,  h \, \|  u^{1}_{h}\| ^{2}_{L^2(\Omega )}\;.
$$
We get from (\ref{e34}) that 
\begin{equation}\label{e34a}
 \| t^{1/2}u^{1}_{h}\| _{L^2(\Omega )}\leq C\,
h^{1/4}\,  \| u^{1}_{h}\| _{L^2(\Omega )}\;.
\end{equation}
In the same way, (\ref{e32}) and (\ref{e34a}) give 
(\ref{ePrnm3}) for $\ k=1$. \\


\noindent Suppose now that $\ m\; \in \; \nz ,\ m>1$ and that 
(\ref{ePrnm3}) is valid for $\ k=m-1.$ 
Then (\ref{e34}) with $\ k=m$ and 
(\ref{ePrnm3})  for $\ k=m-1$ give 
$$ \| t^{m/2}u^{1}_{h}\| _{L^2(\Omega )}\leq 
C\, h^{m/4}\,  \| u^{1}_{h}\| _{L^2(\Omega )}.$$ 
This estimate, together with (\ref{e32}) and (\ref{ePrnm3}) 
for  $\ k=m-1$, prove the estimate (\ref{ePrnm3})
 for $\ k=m $.\\
$ \bullet $
\section{The case of constant magnetic field in domains with positive curvature}\label{Section10}

The case of constant magnetic field has been intensively studied. 
In this case, the upper bound in (\ref{e2Cornm1}) was established 
in \cite{BeSt}, the lower bound in \cite{BeSt} is 
less precise than our one in (\ref{e2Cornm1}). 
But  we can still improve 
the lower bound in  (\ref{e2Cornm1}) 
when the  magnetic field is constant. 

\subsection{Bounds modulo $\Og(h^\frac 32)$.} 


\begin{theorem}\label{Thnm2} .\\ 
Suppose that $\ \Omega $ is a bounded open and connected set 
of $\ \rz ^2.$ 
If  the magnetic field is constant~: $B=b $,
then there exists a constant $\ C_0>0$ and $h_0 >0$ such that 
$\ \mu^{(1)}(h),$ the first eigenvalue of 
the Neumann operator associated to $\ (hD_z-A)^2$  satisfies 
\begin{equation}\label{eThnm2} 
-C_0\, h^{3/2}\leq \mu^{(1)}(h)-\Theta_0 b \, h \leq C_0\, h^{3/2},\ 
\ \forall \; h\; \in \; ]0,h_0]\;, 
\end{equation} 
where $\Theta_0 \; \in \; ]0,1[$ is defined by (\ref{Dau}).
\end{theorem} 
{\bf  {Proof}}.\\
The upper bound was already obtained in (\ref{e2Cornm1}). For the 
 lower bound, we use the notations of the proof 
of Proposition \ref{Prnm1}. Let $u^1_h$ a normalized groundstate. 
We take 
\begin{equation}\label{0ct} 
 \tau (h)=h^{1/4}. 
\end{equation} 
So (\ref{part})-(\ref{parta}) and  (\ref{inta}) 
show  that 
\begin{equation} 
\begin{array}{l}
|q^{N}_{h,A,\Omega }((\psi _{1,\tau (h)}\circ t)\; u^{1}_{h})- 
\mu^{(1)}(h)\| (\psi _{1,\tau (h)}\circ t) \,u^{1}_{h}\| ^{2}_{L^2(\Omega )}|\\
\qquad \leq \, h^2\,  \| |\nabla (\psi _{1,\tau (h)}\circ t)|\,u^{1}_{h}\| ^{2}_{L^2(\Omega )}\;.
\end{array}
\end{equation} 
On the other hand,  if 
$$ \| u^{1}_{h}\| _{L^2(\Omega )}=1\;,$$
 we consequently obtain, 
 thanks to (\ref{0ct}), the condition on the support (\ref{part}) and (\ref{ePrnm3}), 
\begin{equation}\label{2ct}  
 \| (\psi _{1,\tau (h)} \circ t)\,u^{1}_{h}\| _{L^2(\Omega )}=
1+\Og(h^k),\ \forall \; k>0. 
\end{equation} 
Similarly, one can show that~:
\begin{equation}
\| |\nabla (\psi _{1,\tau (h)}\circ t)|\,u^{1}_{h}\| ^{2}_{L^2(\Omega )}
 \leq C_k\,  h^k\;,\; \forall k>0\;.
\end{equation}
This leads to the existence of $C>0$ such that~:
\begin{equation}\label{1ct} 
|q^{N}_{h,A,\Omega }((\psi _{1,\tau (h)}\circ t)\; u^{1}_{h})- 
\mu^{(1)}(h)\| (\psi _{1,\tau (h)}\circ t) \,u^{1}_{h}\| ^{2}_{L^2(\Omega )}|
\leq C\, h^2\;.
\end{equation}  

The assumption that the magnetic  field is constant  allows us 
to  choose a gauge (in the adapted coordinates introduced in the appendix \ref{AppB}), 
such that in the formula (\ref{br8}) (cf (\ref{hatB})), 
we have  
\begin{equation}\label{3ct} 
{\widetilde A}(w)=-b t(1-\frac{t}{2}\kappa (s),0), 
\end{equation}
which implies~:
\begin{equation}
 |{\widetilde A}(w)+(b t,0)|\leq C\, t^2\;. 
\end{equation} 

 Using the formula (\ref{br8}), (\ref{br08}) 
and (\ref{ePrnm3})) for $\ k=1$ and $\ k= 4$,   
we get, from (\ref{0ct}) - -   (\ref{3ct}), and 
the fact that $\ \mu^{(1)}(h)=\Og(h)$,  
 the  existence of $C>0$ and  $\ h_0>0$ such that, 
for any $\ h\; \in \; ]0,h_0],$  
\begin{equation}\label{4ct} 
\begin{array}{l}
|\int _{\rz \times \rz _{+}}[|(hD_s+b t)
(\psi _{1,\tau (h)}\; u^{1}_{h}) |^2
+h^2 \, |D_t (\psi_{1,\tau (h)} \; u^{1}_{h})|^2 -
\mu^{(1)}(h)|\,\psi _{1,\tau (h)}\, u^{1}_{h}\,|^2]\, dw|\\
\qquad \leq C\, h^{3/2} \;.
\end{array}
\end{equation} 
As 
\begin{equation}\label{5ct} 
\begin{array}{l}
\int _{\rz \times \rz _{+}}[|(hD_s+b t)
(\psi _{1,\tau (h)}(t) u^{1}_{h}(s,t))|^2
+h^2 |D_t (\psi _{1,\tau (h)}(t) u^{1}_{h}(s,t))|^2]\, dw \\
\qquad \geq \Theta _0 b \, h\, \int _{\rz \times \rz _{+}}
|\psi _{1,\tau (h)}( t)  u^{1}_{h}(s,t)|^2 \, dw,
\end{array}
\end{equation} 
the estimates (\ref{2ct}),  (\ref{4ct}) and (\ref{5ct}) 
give the existence of $C_0 >0$ such  that $$ \ \mu^{(1)}(h)\geq \Theta_0 b \,h  -C_0\, h^{3/2}\;.$$
$ \ \ \bullet $ \\

\subsection{Refined lower bounds}\label{Subsection10.2}

We are now able to prove the refined lower bound. We will add 
 two elements in comparison with the previous proof.\\
The first point is that we shall use the lower bound
 obtained in the case of the disk which is recalled 
in Proposition \ref{Pphil}. An alternative will be presented
 in the next section using the techniques of the
 previous subsection. The second point will be to improve the remainder
 estimates.\\

Let us first come back to the case of the disk in the semi-classical context.
Proposition \ref{Pphil} (due to \cite{BaPhTa}) gives the existence of $C >0$ such that~:
\begin{equation}\label{inphil}
\mu^{(1)}(h,b,D(0,R)) \geq   \Theta_0 b h  - \frac 2 R  \, M_3 \, b^\frac 12 \,  h^\frac 32 - C\,  h^2 \;.
\end{equation}
We observe, using Agmon's estimates, that the same inequality
 is true, for any $0<R_0<R$,  for the problem
 in $\Omega (R_0,R) = \{ z\in D(0,R)\;|\; |z| >R_0\}$ with the Neumann 
 condition at the external boundary  and the Dirichlet condition 
 at the internal boundary.\\

This is this second problem which  is easier to compare with the general one.
So one can reduce rather easily the general problem to the analysis of
 the problem in a tubular neighborhood of $\pa \Omega$ where the coordinates $(t,s)$
 analyzed in the appendix \ref{AppB} are defined.\\
In the case of the disk, we observe the relation~:\\
$$
r = R - t\;, \theta = s/ R\;,
$$
which is coherent with~:
$$
r dr\wedge d\theta = (1 - \frac 1R s) ds \wedge dt\;.
$$
In one case we have $\kappa=\const$ and in general $\kappa$ is not constant.
So we have mainly to compare the two situations. For this we shall introduce near the boundary of $\Omega$ a 
partition of unity, and compare in each ball meeting the boundary the general case with the case with constant curvature.
So with the notation of Subsection \ref{Subsection9.2}, we introduce  $\tau(h) = h^\rho\;$,
 and a corresponding lattice $\Gamma_{\tau(h)}$. For the moment, we just introduce the condition $\rho < \frac 14$ in order
 that the error term due to the partition of unity in the IMS formula has the right order.


Let us consider the balls near the boundary. This means that in the  sums over $\gamma$ which will be considered in the next considerations, we shall only
 sum over the points of the lattice $\Gamma_{\tau(h)}$ such that the corresponding
 balls are contained in a fixed neighborhood of $\pa \Omega$ where we can use the special coordinates. Let us denote by $\Gamma'_{\tau(h)}$ this sublattice. 
We would like to use a localized version of  Proposition \ref{Prnm3} in order to control the comparison with the model. This is the aim of the following proposition.

\begin{proposition}\label{Prnm3bis} .\\ 
Under the assumptions of Proposition \ref{Prnm1}, 
then, for any $\ k\; \in \; \nz$,  $\ \forall \; h\; \in \; ]0, h_0]$,  there exist  sequences
$(a_k(\gamma,h))_\gamma$  and $(b_k(\gamma,h))_\gamma$ uniformly bounded  in $\ell^2$  with respect to $h$,  such that,  $\forall \gamma\in \Gamma'_{\tau(h)} $,  
\begin{equation}\label{ePrnm3x}  
\int _{\Omega }t(z)^k\, |\chi_{\gamma,\tau(h)} u^{1}_{h}(z)|^2\,dz 
\leq a_k(\gamma,h)^2 \, h^{k/2}\,\| u^{1}_{h}\|^{2}_{L^2(\Omega )}; 
\end{equation} 
and
\begin{equation}\label{ePrnm3y}  
\int _{\Omega }t(z)^k\,|(hD_z-A) \chi_{\gamma, \tau(h)}u^{1}_{h}(z)|^2\,dz 
\leq b_k(\gamma,h)^2 \,  h^{1+k/2}\, \| u^{1}_{h}\| ^{2}_{L^2(\Omega )}\;. 
\end{equation} 
\end{proposition} 
{\bf Proof}\\
The assertion (\ref{ePrnm3x}) is a consequence of (\ref{ePrnm3}) and 
of the property that there exists $C$ such that~: 
$$ \sum_{\gamma\in \Gamma'_{\tau(h)}} \chi_{\gamma,\tau(h)}^2 \leq C\;.$$
For the second one we observe that~:
$$
\begin{array}{l}
\int _{\Omega }t(z)^k\,|(hD_z-A) \chi_{\gamma, \tau(h)}u^{1}_{h}(z)|^2\,dz \\
\leq 2  \int _{\Omega }t(z)^k\,|\chi_{\gamma, \tau(h)} (h D_z-A) u^{1}_{h}(z)|^2\,dz\\
\quad  + 2  h \int _{\Omega }t(z)^k\,|\nabla \chi_{\gamma, \tau(h)}u^{1}_{h}(z)|^2\,dz 
\end{array}
$$
The first term of the right hand side is immediate to control using (\ref{ePrnm3a}). 
The second term is controlled  by $h^{2- 2 \rho + \frac k2} c(\gamma,h)^2$
 with $ (c(\gamma,h))$  uniformly  in $\ell^2$, that is satisfying
 $ \sum_{\gamma\in \Gamma'_{\tau(h)}} |c(\gamma,h)|^2$ is bounded independently of $h\in ]0,h_0]$. So this is of the right order when $\rho < \frac 12$.\\
$\bullet$\\



Let us now consider the various errors which we have  to control when comparing with the case with constant curvature. We work 
near the boundary in the adapted coordinates introduced in Appendix \ref{AppB}. We have
 to compare more carefully, for each $\gamma$, the expressions $\int  |(hD_z- A) \chi_{\gamma, \tau(h)}\, u_h^1|^2  (1- t \kappa(s)) ds dt\;$
 and $\int  |(hD_z- A^\gamma) \chi_{\gamma, \tau(h)}\, u_h^1|^2  (1- t \kappa(s_\gamma)) ds dt\;$, 
 where (using the notations of Appendix \ref{AppB})
$${\tilde A}  = - bt (-1- \frac t2 \kappa(s), 0)\;,\;  {\tilde A}^\gamma = - bt (-1- \frac t2 \kappa(s_\gamma), 0)\;.
$$  
These errors are of the following type. The first
 term to consider is.
$$
r_1(\gamma,h)=\Og(h^\rho) \int t(z) |(hD_z- A) \chi_{\gamma, \tau(h)}\, u_h^1|^2 dz\;,
$$
where we have used that $\kappa(s) - \kappa (s_\gamma) =\Og(h^\rho)$.\\
This should lead (using (\ref{ePrnm3y}) with $k=1$) to a remainder of order~:
$$
r_1(\gamma,h) = \Og(h^{\rho + \frac 32} )\,  a_1(\gamma,h)^2\;,
$$
with $\sum_\gamma a_1(\gamma,h)^2 < C$, for some $C$ independent of $h$.\\
\noindent The next type of term is like~:
$$
r_2(\gamma,h) = \Og(h^\rho) \int t^2 |(hD_z - A) \chi_{\gamma,\tau(h)} u_h^1)|\cdot 
 |\chi_{\gamma, \tau(h)}\; u_h^1| dz\;.
$$
 or like
$$
r_3(\gamma, h) =  \int t^3 |(hD_z - A) \chi_{\gamma,\tau(h)} u_h^1)|\cdot |\chi_{\gamma, \tau(h)} u_h^1| dz\;.
$$
For $r_2$, we can use Cauchy-Schwarz and (\ref{ePrnm3y}) with $k=4$.
This leads to~:
$$
r_2(\gamma, h) = \Og (h^{\rho + \frac 32}) a_1(\gamma,h) b_4(\gamma,h).
$$
For $r_3$,  we can use Cauchy-Schwarz and (\ref{ePrnm3y}) with $k=6$.
This leads to~:
$$
r_3(\gamma,h) = \Og (h^{2})a_1(\gamma,h) b_6 (\gamma,h) \;.
$$

 
\noindent The last  type of remainder is
$$
r_4 (\gamma, h) = \Og(h^{2 \rho}) \int t^4 |\chi_{\gamma, \tau(h)}\; u_h^1 |^2\; dz\;,
$$
which can be treated by (\ref{ePrnm3x}) and leads to~:
$$
r_4(\gamma, h) = \Og(h^{2\rho+2}) a_4(\gamma,h)^2 \;.
$$


Summing up over $\gamma\in \Gamma'_{\tau(h)}$, this shows  that we will have
 the general case, if we have the case of the disk with an error given by~:
$\Og (h^{\rho + \frac 32}) + \Og(h^{2 - 2 \rho})$.
The optimal $\rho$ seems to be obtained when~: $\rho + \frac 32 = 2 - 2 \rho$.
This leads to the choice $\rho = \frac 16$ and to an error in $\Og(h^{\frac 53})$. .

So  we have proved~:
\begin{theorem}\label{Tmin}.\\
If $B=b$ and if $\kappa(\omega) >0$ for all $\omega \in \pa \Omega$, then we have~:
\begin{equation}\label{7ctBis}
\mu^{(1)} (h) \geq  \Theta_0 b h  - 2 M_3 \kappa_{max} b^{\frac 12} h^\frac 32 + \Og ( h^{\frac 53})\;.
\end {equation}
\end{theorem}


\begin{remark}.\\
{\rm 
We are obliged for the moment to assume that $\kappa (s)> 0$. This is not natural. We shall eliminate this artificial condition in the next section.}
\end{remark}

\subsection{Localization of the ground state}
\begin{proposition}\label{Prnm1am} .\\ 
Under the assumptions of Theorem \ref{Tmin}, we have
\begin{equation}\label{ePrnm1am} 
q^N_{h,A,\Omega }(u)\geq \int _{\Omega }W^1_h(z)|u(z)|^2dz,\ 
\forall \; u\; \in \; H^1(\Omega ),
\end{equation} 
 for any $\ h\; \in \; ]0,1].$ \\
Here $W^1_h$ is defined, for some constant $C_0 >0$, 
by~:
$$
\begin{array}{lll}
W^1_h(z)&= b h 
&{\rm if}  \dist (z;\partial \Omega )>2h^{\frac 16} \\ 
&=\Theta_0 b h - 2 M_3 b^\frac 12 \kappa (s) h^\frac 32 -C_0\, h^{5/3}
&{\rm if} \ \dist (z;\partial \Omega )\leq 2h^\frac 16\;.
\end{array}
$$
\end{proposition}

In particular, we have, using also the upper bound of $\mu^{(1)}(h)$~:
\begin{equation}
\begin{array}{ll} 
q^N_{h,A,\Omega }(u_h^1) - \mu^{(1)}(h)&\geq 
\int _{\Omega }(W^1_h(z) -\mu^{(1)}(h))|u_h^1(z)|^2\;dz,\\ 
& \geq 2 M_3 b^\frac 12 h^\frac 32 \int_{\Omega_{\tau(h)}} 
(\kappa_{max} - \kappa(s) - C_0 h^\frac 16) |u_h^1|^2 dz\;.
\end{array}
\end{equation}

As for the proof of Proposition \ref{Proppoint} (cf also (\ref{6.36})) we get~:
\begin{theorem}\label{Tlocal}.\\
Under the assumptions of Theorem \ref{Tmin}, one
 has the following localization. There exists $\delta >0$ and for any $\epsilon >0$, $C_\epsilon >0$ and $h_\epsilon >0$ such that, for all $h\in ]0, h_\epsilon]$~:
\begin{equation}
|| \exp \delta \frac{{\rm dist} (x, n(\pa \Omega))}{h^\frac 14}\; u_h^1 || \leq C_\epsilon \exp \frac{\epsilon}{h^\frac 14}
\end{equation}
Here $ n(\pa \Omega)$ is the set~:
\begin{equation}
n(\pa \Omega) =\{ z\in \pa \Omega\;|\; \kappa (z)= \kappa_{max}\}
\end{equation}
 of the points of maximal curvature.
\end{theorem}

Very roughly, the behavior of the ground state in the neighborhood
 of the boundary is determined by some  Agmon distance
 associated with the effective potential 
$h^\frac 32 (\kappa_{max} - \kappa(s)))$.

It is actually enough according to the results obtained already far from $\pa \Omega$ to choose instead of $d(x, n(\pa \Omega))$  a function 
$\phi (s,z) = \phi(s)$ for $d\leq \eta(\epsilon)$ (with $\eta(\epsilon) \ar 0$
 as $\epsilon \ar 0$ and equal to $0$  when $d \geq 2 \eta(\epsilon)$.\\

The proof is then similar to the proof given in Section \ref{Section6} (see also the course \cite{He2}, chapter 3).

\subsection{Upper bounds modulo $\Og(h^\frac 74)$.}

Let us recall in this section the result obtained in \cite{BeSt}, and established more precisely in \cite{PiFeSt}. 
\begin{proposition}\label{RmMajor} .\\ 
When $\ \Omega $ is of class $\ C^4$, there exists a constant $\ C_0$ such that\begin{equation}\label{7ct}
\mu^{(1)}(h)\leq \Theta_0 b h - 2 M_3 \kappa _{max} b^{1/2} h^{3/2}
  + C_0\, h^{7/4}\;. 
\end{equation}  
 Here $\ \kappa _{max}$ is the maximum 
of the scalar curvature of 
$\ \partial \Omega $ and $ M_3$ is defined in (\ref{7ct2}).
\end{proposition}

\begin{remark}.\\
{\rm 
In \cite{BeSt}, the authors propose a formal expansion modulo $o(h^2)$.
In \cite{PiFeSt}, the authors give a proof of the remainder in $\Og(h^2)$
 under the stronger assumption that there is one point where the curvature
 is maximal is non degenerate. In \cite{LuPa1}, another estimate of the remainder is obtained in their appendix. Our proof gives a better  remainder
 estimate than in \cite{LuPa1}, weaker than in  \cite{PiFeSt} but  does not use an assumption of non degeneracy.}
\end{remark}

{\bf Proof of Proposition \ref{RmMajor}}.\\
To prove  (\ref{7ct}), we take in the proof of Theorem \ref{Thnm1}
a point $\ z_0\; \in \; \partial \Omega $ 
such that $\ \kappa (s(z_0))=\kappa _{max}$.
We take instead of  (\ref{nm10}), with the notations of Appendix \ref{AppA}, 
\begin{equation}\label{13ct}  
v^{0}_{h}(w)= b^{1/4}\,h^{-5/16}\, g^h\left(b^{1/2}\,h^{-1/2}\,t \right)\chi (t)
\cdot f(h^{-1/8}\, s), 
\end{equation} 
with 
\begin{equation}
 g^h(x)=\varphi _0(x)-h^{1/2}(b )^{-1/2}
\kappa_{max}\psi _1(x)\;.
\end{equation}
The functions  
$\chi $ and $\ f$ are as in (\ref{nm10}), and 
\begin{equation}\label{13ct1} 
\psi _1= {\tilde R}^{N,\xi_0}
[H_1\varphi _0-K_3\varphi _0], 
\end{equation} 
with $$\ H_1u(x)=(x-\xi_0 )^3u(x)-\xi _{0}^{2}
(x-\xi_0 )u(x)+u^{\prime }(x),$$
and  
$$
 K_3=\int _{\rz _{+}}(H_1\varphi _0)\varphi_0\; dx \;.
$$
Here ${\tilde R}^{N,\xi_0}$ is the regularized resolvent which vanishes
 on $\varphi_0$ and is equal to  
$[H^{N,\xi_0 }-\Theta _0]^{-1}$ on $\{\varphi_0\}^{\perp}$.\\
  Using the lemma, one gets~:
$$
K_3 = - 2 M_3\;.
$$

Then $\ \psi _1(x)$ is a well defined real function. It is standard to show 
that 
$\ \displaystyle x^k\psi _1(x)\; \in \; L^2(\rz _{+})\;,\;
\forall \; k\; \in \; \nz .$\\ 
 Let us introduce
 \begin{equation}\label{15ct}
H^h=a^{-2}_{0}(t) [b t(1-\frac{t}{2}\kappa_{max})
-h^{1/2}b ^{1/2}\xi_0 ]^2+
h^2a^{-1}_{0}(t)D_t[a_0(t)D_tu].   
\end{equation}
with $\ a_0(t)=1-\kappa_{max}t$.\\
We get easily that 
\begin{equation}\label{14ct} 
|q^{N}_{h,A,\Omega }(v^{0}_{h})-h^{-1/2}(b )^{1/2}\int _{\rz _{+}}
H^h(U^h(g^h(t)))U^h(g^h)(t)\chi ^2(t)\, dt |\leq C\,h^{7/4},  
\end{equation} 
and that~:
\begin{equation}
|\ \| u^{0}_{h}\| _{L^2(\Omega )}-1|\leq C\, h^{1/2} \;,
\end{equation}
with 
$$ U^h(g)(t)= (b )^{1/4}h^{-1/4}
g(b ^{1/2}h^{-1/2}t),\ \forall \; g\; \in \; L^2(\rz _{+}).$$
Then, for any $\ t\; \in \; [0,\epsilon ]$ and 
for any $\ g\; \in \; H^2(\rz _{+}),$  
\begin{equation}\label{16ct}
\begin{array}{l} 
|\left(H^h g-H^{h}_{0}g(t)-H^{h}_{1}g\right) (t)|\leq 
C\,\left[t^2(b t
-h^{1/2}\,b^{1/2}\xi_0 )^2+t^3|(b t
-h^{1/2}\,b^{1/2}\xi_0 )|+t^4+ht|
(b t
-h^{1/2}\,b^{1/2}\xi_0 )|+ht^3\right]|g(t)|\;,
\end{array}
\end{equation} 
with 
$$
H^{h}_{0}g=(b t
-h^{1/2}b ^{1/2}\xi_0 )^2+h^2D_{t}^{2}g
$$ 
and 
$$
H^{h}_{1}g =
2t\kappa_{max}(b t
-h^{1/2}b ^{1/2}\xi_0 )^2-b \kappa_{max}t^2(b t
-h^{1/2}b ^{1/2}\xi_0 )+ih^2\kappa_{max}D_tg \; .
$$ 
So
 $$\ (U^h)^{\star }H^{h}_{0}U^h= bh  [L+\Theta _0]\;,
$$ 
and
 $$
 (U^h)^{\star }H^{h}_{1}U^h=\,b^{1/2}\kappa_{max}\,h^{3/2}\, H_1\;.
$$ 
The estimates (\ref{14ct}) and (\ref{16ct}) (applied to $g=g^h$)  prove 
easily the existence of $C>0$ such that~: 
$$|q^{N}_{h,A,\Omega }(u^{0}_{h})-
 \left[\Theta_0 b h - 2 M_3  \,\kappa_{max}\, b^{1/2} \,  h^{3/2} \right]
\| u^{0}_{h}\| ^{2}_{L^2(\Omega )}|
\leq C\,  h^{7/4} \| u^{0}_{h}\| ^{2}_{L^2(\Omega )},$$ 
and (\ref{7ct}) follows.\\
 $\ \bullet $ 

\section{An  alternative  proof and extension to non-convex domains.}\label{Section11}
The previous proof was based on a result by 
Baumann-Phillips-Tang \cite{BaPhTa}(see also \cite{PiFeSt} 
for an alternative proof). We
 give now  a more direct proof which permits to avoid the unnatural 
technical condition $\kappa(\omega) >0$. This could in particular be applied to
 the Neumann problem in the exterior of the disk.
\begin{theorem}\label{tbetter}.\\
When $\ \Omega $ is of class $\ C^4$, 
Theorems \ref{Tmin} and \ref{Tlocal} are true without the assumption
 that the curvature is always strictly positive.
\end{theorem} 

\noindent We begin as in the proof of the left-hand side 
of (\ref{eThnm2}). Instead of (\ref{0ct}), we take 
\begin{equation}\label{lw1} 
\tau (h)=h^{\delta }\ \ \ {\rm {with}}\ \ 
\frac{1}{6}\leq \delta <\frac{1}{2}, 
\end{equation} 
($\delta $ will be fixed later), 
so (\ref{1ct}) and (\ref{2ct}) are still valid. We can no more use the result of \cite{BaPhTa} and it will not be enough to localize
in balls (or squares) and we choose to localize in rectangles of size $h^\delta$
 in the normal direction and of size $h^\frac 16$ in the tangential direction.

\noindent We use the partition of unity defined in 
(\ref{low1}) - - (\ref{low3}) and we get 
as (\ref{low42}) 
\begin{equation}\label{lw2} 
\begin{array}{ll}
q^N_{h,A,\Omega }(\psi _{1,\tau (h)}u^{1}_{h})& = 
\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
q^N_{h,A,\Omega }(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})\; \\
&\quad 
-\; h^2\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\| |\nabla \chi _{\gamma ,h^{1/6}}|
\psi _{1,\tau (h)}u^{1}_{h}\| ^{2}_{L^2(\Omega )}.
\end{array} 
\end{equation} 
We deduce from (\ref{lw2}) that 
\begin{equation}\label{lw3} 
|q^N_{h,A,\Omega }(\psi _{1,\tau (h)}u^{1}_{h})- 
\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
q^N_{h,A,\Omega }(\chi _{\gamma ,h^{1/6}}
\psi_{1,\tau (h)} u^{1}_{h})|
\leq C\, h^{5/3}\,  \| 
\psi _{1,\tau (h)}u^{1}_{h}\| ^{2}_{L^2(\Omega )}. 
\end{equation} 
But it is easy to see from (\ref{6ct}) that, for any 
$\ \gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )$, 
\begin{equation}\label{lw4} 
\begin{array}{l}
|q^N_{h,A,\Omega }(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})\;\\ - \int _{K(\gamma ,h)}
a_{\gamma }(t)[h^2|D_t(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2\\
\qquad\qquad\qquad +(1+2\kappa _{\gamma }t)
|(hD_s-A^{\gamma }(t))(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2]dsdt \\
\leq C\, h^{1/6}\,\int _{\Omega }
[t|(hD-A)(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2+(t^3+t^4)
|\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h}|^2]dz. 
\end{array}
\end{equation}
Here  $\ K(\gamma ,h)=]-2h^{1/6}+s_{\gamma },s_{\gamma }+2h^{1/6}[
\times ]0,h^{\delta }[$, 
where 
$\ s_{\gamma }=s(z_{\gamma })$ for some $
z_{\gamma }\; \in \; \partial \Omega $ 
such that 
$\ \forall \; z\; \in \supp (\chi _{\gamma ,h^{1/6}}),\ 
|s-s_{\gamma }|\leq \frac{3}{2}h^{1/6}$.\\ 
Moreover
$$
\begin{array}{ll}
\kappa _{\gamma }&=\kappa (s_{\gamma }),\\
a_{\gamma }(t)& =1-\kappa _{\gamma }t\\
 A^{\gamma }(t)& =- b  t(1-\frac{\kappa _{\gamma }}{2}t). 
\end{array}
$$
We proceed like  in Subsection \ref{Subsection10.2} for the control of this remainder. 
Summing (\ref{lw4}) and taking into account 
(\ref{low41}),(\ref{lw3}), (\ref{ePrnm3}) (with 
$k=1,\ 2)$ and  (\ref{2ct}) or alternatively using Proposition \ref{Prnm3bis},  we get that 
\begin{equation}\label{lw5}  
\begin{array}{l}
|q^N_{h,A,\Omega }(\psi _{1,\tau (h)}u^{1}_{h})\;\\
 -\;
\sum _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\int _{K(\gamma ,h)}
a_{\gamma }(t)\left[h^2|D_t(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2\; \right.\\
\qquad\qquad \left.+  (1+2\kappa _{\gamma }t)
|(hD_s-A^{\gamma }(t))(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2\right]\;dsdt|\\
\leq C  \, h^{5/3} \| 
\psi _{1,\tau (h)}u^{1}_{h}\| ^{2}_{L^2(\Omega )}.
\end{array}
\end{equation} 
Our result being  semi-classical ($h\ar 0$), 
we assume that $\ 2h^{1/6}\leq \pi $ for simplicity.\\ 
Let us consider, for some $\gamma \in \Gamma^1_{\tau(h)}$, 
 the term appearing in  the left hand side of (\ref{lw5})~:
$$
\begin{array}{l}
\int _{K(\gamma ,h)}
a_{\gamma }(t)\\
\quad \left[h^2|D_t(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2 +  (1+2\kappa _{\gamma }t)
|(hD_s-A^{\gamma }(t))(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2\right]\;dsdt\;.
\end{array}
$$
We get, by taking the partial Fourier transform $s\mapsto k$, that,  for any 
$\ \gamma$ in $\Gamma ^{1}_{\tau (h)}(\Omega )$,  
\begin{equation}\label{lw6}
\begin{array}{l}
\int _{K(\gamma ,h)}
a_{\gamma }(t)\\
\left[h^2|D_t(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2+(1+2\kappa _{\gamma }t)
|(hD_s-A^{\gamma }(t))(\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})|^2\right] \, dsdt \\
\quad 
\geq \mu _{1}^{\gamma }(h)\int _{K(\gamma ,h)} 
a_{\gamma }(t)|\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h}|^2\;dsdt,
\end{array} 
\end{equation} 
with 
\begin{equation}\label{lw6Bis} 
\mu _{1}^{\gamma }(h)=\inf_{k\in \zz }\; 
\inf \sp (H^{N,D}_{h,\gamma ,k}).
\end{equation} 
Here  $\ H^{N,D}_{h,\gamma ,k}$ is the self-adjoint 
operator on $\ L^2(]0,h^{\delta }[;a_{\gamma }(t)dt)$\\ 
associated to the differential operator 
\begin{equation}\label{lw7} 
H_{h,\gamma ,k}  =h^2\, a^{-1}_{\gamma }(t)
D_t(a_{\gamma }(t)D_t\; .) + (1+2\kappa _{\gamma }t)
(hk-A^{\gamma }(t))^2 , 
\end{equation} 
with domain 
\begin{equation}\label{lw8} 
D(H^{N,D}_{h,\gamma ,k})=\{ v\in H^2(]0,h^{\delta }[);\ 
v'(0)=v(h^{\delta })=0\} . 
\end{equation} 
As 
\begin{equation}\label{lw9}
\begin{array}{l}
  |\;  \| \chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h})\| ^{2}_{L^2(\Omega )}-
\int _{K(\gamma ,h)} 
a_{\gamma }(t)|\chi _{\gamma ,h^{1/6}}
\psi _{1,\tau (h)}u^{1}_{h}|^2dsdt| \\
 \quad \leq C\, h^{1/6} \, \| t^{1/2}\chi _{\gamma ,h^{1/6}}
\psi_{1,\tau (h)}\, u^{1}_{h})\| ^{2}_{L^2(\Omega )} , 
\end{array}
\end{equation} 
we get from (\ref{lw5}),  (\ref{lw6}) 
and  (\ref{lw9}) that 
\begin{equation}\label{lw10} 
\begin{array}{l}
q^N_{h,A,\Omega }(\psi _{1,\tau (h)}u^{1}_{h})\qquad\\
\geq \left( \inf _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\mu _{1}^{\gamma }(h)\right) \times \\ \qquad \times 
\left [\| \psi _{1,\tau (h)}u^{1}_{h})\| ^{2}_{L^2(\Omega )}
\quad - C \, h^{1/6}\, \| t^{1/2}\psi _{1,\tau (h)}u^{1}_{h})\| ^{2}_{L^2(\Omega )}\right]\\
\qquad - C\, h^\frac 53\, ||\psi_{1,\tau(h)} u_h^1||^2\;.
\end{array}
\end{equation} 
Then (\ref{ePrnm3}) (with $k=1$), 
(\ref{2ct}) and (\ref{lw10}) show the existence of $C$ and $h_0$ such that, for all $h\in ]0,h_0]$,
\begin{equation}\label{lw11} 
\begin{array}{l}
q^N_{h,A,\Omega }(\psi _{1,\tau (h)}u^{1}_{h})\\
\qquad \geq \left(\inf _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\mu _{1}^{\gamma }(h)\right)\, [1- C\,h^{2/3}]
\| \psi _{1,\tau (h)}u^{1}_{h})\| ^{2}_{L^2(\Omega )} - C h^\frac 53 \| \psi _{1,\tau (h)}u^{1}_{h})\| ^{2}_{L^2(\Omega )}\;. 
\end{array}
\end{equation} 
The estimates (\ref{1ct}), (\ref{2ct}) and 
(\ref{lw11}) prove that 
\begin{equation}\label{lw12}  
\mu _1(h)\geq \inf _{\gamma \in \Gamma ^{1}_{\tau (h)}(\Omega )}
\mu _{1}^{\gamma }(h)[1- C\, h^{2/3}] - C\, h^\frac 53. 
\end{equation} 
But it is easy to see, noting that $\gamma$ appears only through 
$\kappa (s_\gamma)$ which stays in a compact interval,  that there exists $C>0$ such that, for all $\gamma$,
\begin{equation}\label{up}
 \mu _{1}^{\gamma }(h)\leq C\, h \;. 
\end{equation} 
So  (\ref{lw12}) and (\ref{up}) give us the existence of $C>0$, such that~:
\begin{equation}\label{lw13} 
\mu^{(1)}(h)\geq \inf _{\gamma \in \Gamma^{1}_{\tau (h)}(\Omega )}
\mu _{1}^{\gamma }(h) \; -\;C\, h^{5/3}\;. 
\end{equation} 
Therefore it remains to analyze $\mu_1^\gamma (h)$.\\
By scaling, that is using the change of variables $t = h^\frac 12 b^{-\frac 12}{\tilde t}$, we have just to study, for any 
$\ \xi \in \rz $ 
and for any 
$\ \alpha \in [-C,C]$,    the ground state energy 
of the self-adjoint operator $H^{N,D,\xi}_{h}$, which is defined 
as an unbounded operator on 
$L^2(]0, b^\frac 12 h^{\delta -1/2}[;\; (1-h^{1/2}\alpha t)dt)$,  
by
associating  to the differential operator 
\begin{equation}\label{lw14}
\begin{array}{ll} 
 H_{h}^\xi
& =D_{t}^{2}+(t-\xi )^2\\
&\quad 
+ih^{1/2}\alpha (1-h^{1/2}\alpha t)^{-1}D_t
+2h^{1/2}\alpha t(t-\xi -h^{1/2}\alpha \frac{t^2}{2})^2
-h^{1/2}\alpha t^2(t-\xi )\\
&\quad +h\alpha ^2\frac{t^4}{4}, 
\end{array}
\end{equation} 
the realization whose domain is~:
\begin{equation}\label{lw15} 
D(H^{N,D,\xi}_{h})=\{ v\in H^2(]0,b^\frac 12 h^{\delta -1/2}[);\ 
v'(0)=v(b^\frac 12 h^{\delta -1/2})=0\}\;. 
\end{equation}
Note that  
$H^{N,D,\xi}_{0}$ still depend on $h$ by its domain  and that $H_h^\xi$
 depend actually on $\alpha h^\frac 12$.

If 
$\alpha = b ^{-1/2}\kappa _{\gamma }$ and
 $\xi =-h^{1/2} b ^{-1/2}k$, then
\begin{equation}\label{lw17} 
 \sp (H^{N,D}_{h,\gamma ,k })=
bh \; \sp (H^{N,D,\xi}_{h}). 
\end{equation} 
We choose 
\begin{equation}
\delta \; \in \; ]\frac{1}{4},\frac{1}{2}[\;.
\end{equation} 
We observe that this si compatible with (\ref{lw1})  
In this case, it is easy to see that 
\begin{equation}\label{lw18} 
|\mu _j( H^{N,D,\xi }_{h})-\mu _j( H^{N , D, \xi}_{0})|
\leq C\, h^{2\delta -\frac{1}{2}} (1+\mu _j( H^{N,D,\xi }_{0}))\;. 
\end{equation} 
For  a bounded from below  self-adjoint operator $\ T$ with 
compact resolvent, $\ (\mu _j(T))$ denotes its increasing 
sequence of eigenvalues. \\ 
But the mini-max principle (by the same argument as for a Dirichlet problem)
 says that 
\begin{equation}\label{lw19} 
\mu_1( H^{N,D,\xi }_{0})\geq \mu (\xi )=
\mu_1( H^{N,\xi }), 
\end{equation} 
where $ H^{N,\xi }$ is the Neumann operator on 
$L^2(\rz _{+})$  
defined in (\ref{nm3}). 

The exponential decay at infinity of the eigenfunctions of 
$ H^{N,\xi }$  and the uniform decay of the lowest eigenfunction for
 $(h,\xi) \in ]0,h_0]\times [-J, + J]$
 of $H^{N,D,\xi}_0$
show  also that, for any fixed $\ j\in \nz $ 
and $J>0$,  there exists $\ C_{j,J}>0$ such that 
\begin{equation}\label{lw19Bis} 
|\mu_j ( H^{N,D,\xi }_{0})-\mu_j ( H^{N ,\xi })|
\leq C_{j,J}\, h^2,\ \forall (h,\xi)  \in \; ]0,h_0]\times [-J,J]. 
\end{equation} 

Now, let us remind (see Appendix \ref{AppA}) that the calculus of \cite{DaHe} 
is valid for $\  H^{N,\xi }$.

Therefore (\ref{lw18}), (\ref{lw19}), (\ref{lw19Bis}) 
 and (\ref{lw21}) prove 
that there exists $\ \eta >0$ such that 
\begin{equation}\label{lw22} 
\mu_1( H^{N,\xi}_{h})\geq \Theta_0 + h^{2\delta -\frac{1}{2}}, 
\ \forall \; \xi ,\ |\xi -\xi_0 |\geq \eta h^{\delta -\frac{1}{4}}. 
\end{equation} 
Now, if $\ \ |\xi -\xi_0 |\leq \eta h^{\delta -\frac{1}{4}}$,   
we use the asymptotic expansion of eigenvalues for 
the  perturbation of a self-adjoint operator in  
Chapter 1 of \cite{MoPa} to determine  $\mu (\xi )$ and 
$\varphi_{\xi }$ modulo $|\xi -\xi_0 |^3$.\\
Let us recall how the method works. We expand $H^{N,\xi}$, the
 eigenvalue $\mu(\xi)$ and the eigenvector $\varphi_{\xi}$
 around $\xi_0$ in powers of $(\xi-\xi_0)$. This leads to formal
 expansions  and we keep as approximate eigenvector the expansion of order
 2. So we take~:
\begin{equation}\label{lw23} 
\begin{array}{l}
f_{\xi }(t)=\varphi _{\xi_0 }(t)+2(\xi -\xi_0 )
{\tilde R}^{N,\xi_0}[(t-\xi_0 )\varphi _{\xi_0 }(t)] 
\\
 + 4(\xi -\xi_0 )^2{\tilde R}^{N,\xi_0}
\{ (t-\xi_0 ){\tilde R}^{N,\xi_0}
[(t-\xi_0 )\varphi _{\xi_0 }(t)]-d_2\varphi _{\xi_0 }(t) \} \end{array}
\end{equation}
with $$ 
 d_2=\int _{\rz _{+}} (t-\xi_0 ){\tilde R}^{N,\xi_0}
[(t-\xi_0 )\varphi _{\xi_0 }(t)]\times \varphi _{\xi_0 }(t)dt\;.$$ 
We get easily that 
\begin{equation}\label{lw24} 
\| [D_{t}^{2}+(t-\xi )^2]f_{\xi }(t)
\; -\; [\Theta _0+(\xi -\xi_0 )^2(1-4d_2)]f_{\xi }(t)
\| _{L^2(\rz _{+})} \leq C\, |\xi -\xi_0 |^3\,. 
\end{equation}
Let us now consider the construction of an approximate eigenvector 
for $H_h^{N,\xi}$. We first observe   that 
$$\int_{\rz _{+}}[iD_t+2t(t-\xi_0 )^2-t^2(t-\xi_0 )]
\varphi_{\xi_0 }(t)\times \varphi _{\xi_0 }(t)dt=
-2\int _{\rz _{+}}(t-\xi_0 )^3
\varphi_{\xi_0 }^{2}(t)dt=-2M_3\;.$$ 
We now introduce~:
\begin{equation}\label{lw26} 
\begin{array}{ll}
f^{h}_{\xi }(t)&=\chi^h(t)\{ f_{\xi }(t)\\
 &\quad - h^{1/2}\alpha 
{\tilde R}^{N,\xi_0}\{ [iD_t+2t(t-\xi_0 )^2-t^2(t-\xi_0 )
+2M_3]\varphi_{\xi_0 }(t)\} \} \;.
\end{array}
\end{equation}
Here $\ \chi ^h(t)=\chi (h^{-\delta +\frac{1}{2}}t)$ 
where $\chi$ is  a smooth cutoff function  satisfying~:\\
$\ \chi (t)=1$, if $\ t<1/2$,  and 
$\ \chi (t)=0$,  if $t>1$.\\   

In the same way as for the proof of (\ref{lw24}), we  get  from (\ref{lw14}) and (\ref{lw23})~:
\begin{equation}\label{lw25} 
\begin{array}{l}
||\;[H^{N,\xi }_{h}\; -\; [\Theta _0+(\xi -\xi_0 )^2(1-4d_2)
-2\alpha  h^{1/2} M_3]f^{h}_{\xi }(t)\| _{L^2(\rz _{+})} \\
\quad \quad \leq C\, [|\xi -\xi_0 |^3+h^{1/2} |\xi -\xi_0 |], 
\end{array}
\end{equation} 
Note that  we have used the exponential decay  of 
$\ \varphi_{\xi_0 }$ that gives also 
\begin{equation}\label{lw26Bis} 
|\; \| f^{h}_{\xi }\| _{L^2(\rz _{+})}-1|
\leq C[|\xi -\xi_0 |+h^{1/2}]. 
\end{equation} 
The estimates (\ref{lw25}) and  (\ref{lw26Bis})  show that, 
if $\ |\xi -\xi_0 |\leq \eta h^{\delta -\frac{1}{4}},$ 
then 
\begin{equation}\label{lw27} 
\dist ([\Theta _0+(\xi -\xi_0 )^2(1-4d_2)-2\alpha  h^{1/2} M_3];\; 
\sp (H^{N,\xi }_{h}))\leq C[|\xi -\xi_0 |^3+h^{1/2} |\xi -\xi_0 |]. 
\end{equation} 
The localization of the eigenvalues given by 
  (\ref{lw18}) and (\ref{lw19Bis}) shows  that 
the nearest eigenvalue of $\ H^{N,\xi }_{h}$ 
minimizing (\ref{lw27}) is 
$\ \mu _1(H^{N,\xi }_{h})$,  so we have  
\begin{equation}\label{lw27Bis} 
|\mu _1(H^{N,\xi }_{h})-
[\Theta _0+(\xi -\xi_0 )^2(1-4d_2)-2\alpha  h^{1/2} M_3]| 
\leq C[|\xi -\xi_0 |^3+h^{1/2} |\xi -\xi_0|]. 
\end{equation} 
But the asymptotic expansion of 
$\ \mu (\xi )$ in powers of $(\xi -\xi_0)$ shows that 
\begin{equation}\label{lw28} 
2(1-4d_2)=\mu''(\xi_0 ) >0 \;. 
\end{equation} 
Therefore  (\ref{lw21}), (\ref{lw27Bis}) and (\ref{lw28}) 
prove that, if $\ |\xi -\xi_0 |\leq \eta h^{\delta -\frac{1}{4}},$ 
then  
\begin{equation}\label{lw29} 
\mu _1(H^{N,\xi }_{h})\geq \Theta _0+(\xi -\xi_0 )^2
\frac{1-4d_2}{2}-2\alpha h^{1/2} M_3-Ch^{1/2}|\xi -\xi_0 |. 
\end{equation} 
So (\ref{lw21}), (\ref{lw22}) and (\ref{lw29}) 
show that, for any 
$\ \delta \; \in \; ]\frac{1}{4},\frac{1}{2}[$,    
\begin{equation}\label{lw30} 
\mu _1(H^{N,\xi }_{h})\geq \Theta_0 - 2\alpha h^{1/2} M_3
-Ch^{\delta +\frac{1}{4}}. 
\end{equation} 
We conclude by observing that (\ref{lw6Bis}), 
(\ref{lw13}), (\ref{lw17})  
and (\ref{lw30}) (with the choice of  
$\delta =5/12$)  give the lower bound (\ref{7ctBis}).\\ 
$\ \bullet $

\section{Conclusion}
We have completed the results obtained by Lu-Pan \cite{LuPa1},  \cite{LuPa2}, 
Baumann-Phillips-Tang \cite{BaPhTa},  Bernoff-Sternberg \cite{BeSt} and 
Del Pino-Felmer-Sternberg \cite{PiFeSt}. We have in particular achieved the
 proof that, in the case
 of a constant magnetic field, the ground state  
is localized near the points of maximal curvature. This leaves open the question of localization
 of the groundstate in the case where there is more than one isolated point of maximal curvature. We have
 also completed the analysis of the Dirichlet probem initiated
 in our paper \cite{HeMo2}, whose intial motivation was a paper by Montgomery \cite{Mon}.

The analysis of the intermediate boundary conditions between
 Dirichlet and Neumann could be interesting (see \cite{LuPa3}, \cite{HoSm1} and \cite{HoSm2}) and is physically relevant.

Let us mention for connected results (relative to excited states)
the heuristic results by K.~Hornberger and  U.~Smilansky \cite{HoSm1} and
 \cite{HoSm2} 
 concerning bulk states and edge states.

As a result of the analysis we have presented, let us also mention
 that it is interesting to look at  the exterior problem. In particular, in the
 constant magnetic field case, one can show that there is in an interval
 of the form to $[ \Theta_0 \,b \,h  + \Og(h^\frac 32)\; , \; h\, b[$, we have
 a discrete spectrum corresponding to states localized near the boundary (edge states).

Moreover  the case of piecewise $C^\infty$ domains has to be achieved in order 
 to complete the results of Jadallah \cite{Ja} (see \cite{LuPa4} and references therein).

The analysis of the  case of dimension 3 has  been less analyzed 
(although see \cite{LuPa4} and \cite{LuPa5}) and it would be interesting
 to see what replace the curvature in this case.

As suggested by J.~Sj\"ostrand (and also used in the numerical computations of
 \cite{HoSm1}), it could be interesting to develop a theory of reduction to the boundary.

Computations performed by K.~Hornberger and U.~Smilansky\footnote{Many thanks
 to U.~Smilansky and K.~Hornberger for accepting to analyze
 our problem with their numerical program.} for special
 domains confirm (for $b=1$ and $h=0.1$) the semi-classical
 prediction.

Finally, in the spirit of Lu-Pan, the analysis of the non-linear problem
 in superconductivity 
 motivating this analysis  has to be continued.

\newpage




 


\appendix

{\bf APPENDICES.}
\section{On an important family of ordinary differential equations}\label{AppA}
Let us recall for the comfort of the reader
 the main properties (mainly due to \cite{DaHe} and \cite{BeSt}),
  concerning
 the Neumann realization
 of $H^{N,\xi}$ in $L^2(\rz^+)$ associated to $\ D_{x}^{2}+(x-\xi )^2$. We denote by ${\hat \mu}^{(1)}(\xi)$
 the lowest eigenvalue of $H^{N,\xi}$ and by $\varphi_\xi$ the corresponding strictly positive normal
 eigenvalue. More simply we will write $\mu(\xi)$ instead of ${\hat \mu}^{(1)}(\xi)$ in this appendix.
\noindent It has been proved  that the infimum $\inf_{\xi\in \rz} \inf \Sp ( H^{N,\xi} )$ introduced in (\ref{nm3}) is actually
 a minimum \cite{DaHe}. Then one can show \cite{DaHe} that there exists
 $\xi_0 >0$ such that  $ \mu ( \xi )$ continues to decay monotonically  till
 some value $\Theta_0 <1$ and is then increasing monotonically  and tending to $1$ at $+\infty$.
So it was obtained~:
\begin{equation}\label{nm6} 
\Theta _0= \inf \; \sp (H^{N,\xi_0 }), 
\end{equation}
and moreover (see \cite{DaHe} or the proof of Lemma \ref{Mom} below) that~:
\begin{equation}\label{Dada}
\Theta_0 = \xi_0^2\;.
\end{equation} 
It is indeed proved in \cite{DaHe}  that 
\begin{equation}\label{lw20} 
\mu' (\xi )=-[\mu (\xi )-\xi ^2]\varphi_{\xi }(0) ^{2}\;. 
\end{equation} 
To get (\ref{lw20}), we observe  that, if $\ \tau >0$,  then 
$$
\begin{array}{l}
0=\int _{\rz _{+}}
[D^{2}_{t}\varphi _{\xi }(t)+(t-\xi )^2\varphi _{\xi }(t)]
\varphi _{\xi +\tau }(t+\tau )dt \\ 
\quad =
-\varphi _{\xi }(0)\varphi ^{\prime } _{\xi +\tau  }(\tau )
+(\mu (\xi +\tau )-\mu (\xi ))\int _{\rz _{+}} 
\varphi _{\xi }(t)\varphi _{\xi +\tau }(t+\tau )\,dt\;.
\end{array}
$$ 
We then take the limit $\tau \ar 0$ to get the formula.\\
From (\ref{lw20}), it comes that 
\begin{equation}\label{lw21} 
\mu''(\xi_0  )=2\xi_0 \varphi ^{2}_{\xi_0 }(0)>0\;. 
\end{equation} 
Here we recall that 
$\mu (\xi_0 )=\xi _{0}^{2}=\Theta _0$ is the strict minimum 
of $\xi \mapsto  \mu (\xi)$ on $\ \rz $~. \\ 
Let $\ \varphi _{\xi_0}(x)$ be the normalized strictly positive eigenfunction of 
$\ H^{N,\xi_0 }$ associated to the eigenvalue $\ \Theta _0$.\\
{\bf  We use more simply the notation  $ \varphi_0$ instead of $\varphi_{\xi_0}$.
}\\
It is easy to see that $\ \varphi _0(x)$  belongs to $\Sg(\overline{\rz^+})$. In particular we have 
\begin{equation}\label{nm7} 
\varphi _0\; \in \; H^2(\rz _{+}),\ \varphi'_0 (0)=0 \ 
{\rm {and}}\ x^k\varphi _0(x)\; \in \; L^2(\rz _{+}),\ 
 \forall \; k\; \in \; \nz .
\end{equation} 
We now described some formulas appearing in \cite{BeSt}.
Let $M_k$ denote the centred moment, of order $\ k$ 
of the probability measure $\ \varphi _{0}^{2}(x)\,dx$~:
\begin{equation}\label{7ct2}
M_k=\int _{\rz _{+}}(x-\xi_0 )^k\varphi _{0}^{2}(x)dx
\end{equation}
We begin by recalling how one can calculate 
the moments $\ M_k$, 
 as done in \cite{BeSt}. 
\begin{lemma}\label{Mom}.\\
The moments can be expressed by the following formulas~:
\begin{equation}
M_0 =1\;,\; M_1 = 0\;,
\end{equation}
 \begin{equation}\label{11ct}  
M_2=\frac{\Theta _0}{2}\;,
\end{equation}
\begin{equation}\label{M3}
M_3=\frac{\varphi _{0}^{2}(0)}{6}>0. 
\end{equation} 
More generally, if $\ k>3$, we have  
\begin{equation}\label{12ct} 
4kM_k=(k-1)\{ 4\xi _{0}^{2}M_{k-2}+(k-2)[(-\xi_0 )^{k-3}\varphi _{0}^{2}(0)
+(k-3)M_{k-4}]\} . 
\end{equation}
\end{lemma} 
{\bf Proof of Lemma \ref{Mom}}.\\

\noindent We use the arguments of \cite{BeSt}. Let $L$ be defined by~:
$$
L = H^{N,\xi_0} - \Theta_0\;.
$$
We first observe the following identity~:
\begin{equation}\label{8ct} 
L(2p\varphi _{0}^{\prime }-p^{\prime }\varphi _0)=\varphi _0
[p^{(3)}-4((x-\xi_0 )^2-\Theta _0)p^{\prime }-4(x-\xi_0 )p], 
\end{equation} 
for $\ p\; \in \; C^2({\overline {\rz _{+}}})$,  and 
\begin{equation}\label{9ct} 
\int_0^{+\infty} \varphi _0Lvdx=\varphi _0(0)v^{\prime }(0),\ 
\forall \;  v\; \in \; H^2(\rz _{+})\cap L^2(\rz _{+};x^2dx). 
\end{equation} 
So, for any polynomial function $\ p,$ 
\begin{equation}\label{10ct} 
\int_0^{+\infty} \varphi _0L(2p\varphi _{0}^{\prime }-p^{\prime }\varphi _0)dx
=\varphi _{0}^{2}(0)[2p(0)(\xi _{0}^{2}-\Theta _0)-p^{(2)}(0)]. 
\end{equation} 

\indent 
When $\ p$ is the constant polynomial ($=1$), (\ref{8ct}) and  (\ref{10ct}) show that 
$\ M_1= -\varphi _{0}^{2}(0)(\xi _{0}^{2}-\Theta _0)/2,$\\ 
as we know that $\ M_1=0$ and $\ \varphi _0(0)\neq 0,$ we 
can recover  (\ref{Dada}):  $\ \xi_0 =\Theta _{0}^{1/2}.$ 

For $\ p(x)=x-\xi_0 $ and then for $\ p(x)=(x-\xi_0 )^2,$ 
 we get in the same way the value of $M_2$ and $M_3$.
The general case is obtained by considering $p(x) = (x-\xi_0)^{k-1}$.\\
$\bullet$



\section{Coordinates near the boundary.} \label{AppB}

For the most accurate estimates,  we need to introduce rather standard  adapted coordinates\footnote{See for example \cite{PiFeSt}} near the boundary. 
Let  $\ell $ be the length of the 
boundary $\   \partial \Omega $ and 
$\ I=]-\frac{\ell}{2}, \frac{\ell}{2}].$ 
Let $\ M\; \in \; C^3(I;\partial \Omega )$ be a 
parametrization of $\  \partial \Omega $
  such that 
$\ M(0)=z_0$ and $s$ is the  distance inside $\partial \Omega$  between $M(s)$ and $z_0$.
We denote by $$ T(s):= M'(s) $$  the unit tangent vector of 
$\  \partial \Omega $ at $\ M(s)$ and the scalar curvature by  $\kappa (s)$, which can
be defined by
$$ T'(s)=\kappa (s)\, N(s) $$ where  
$\ N(s)$ is  the interior normal unit vector of 
$\  \partial \Omega $ at $\ M(s)$.\\  
Moreover the parametrization is chosen positive~:  
$$ 
{\rm {det}}\; (T(s),N(s))=1,\ \forall \; s\; \in \; I.$$

From $N\cdot T =0$ and $N\cdot N =1$, we get first $N'. T + N. T' = 0$ and $N\cdot N' =0$.\\
Consequently, we get~:
$$
N'(s) = - \kappa(s) T(s)\;.
$$
For any $\ z\; \in \; {\overline { \Omega }},$ 
we denote by $\ t(z)$ the standard distance of $\ z$ to 
$\  \partial \Omega :$ 
$$t(z)=\inf _{\omega \in \partial \Omega }\; |z-\omega |.$$ 
So, there exists $\ \epsilon _0>0$ and a 
diffeomorphism of class $\ C^3:$ 
$$\psi :\Omega _{\epsilon _0}\; \to \; 
S^{1}_{\ell/(2\pi )} \times ]0,\epsilon _0[ $$
uch that $\  \psi (z)=w=(s(z),t(z))$ and 
$\ |z-M(s(z))|=t(z)$. \\ 
We have denoted, for small enough $\epsilon$, by $\ \Omega _{\epsilon }$ the tubular neighborhood of $\partial \Omega$~:
$$ \Omega_{\epsilon}~:=\{ z\in \Omega ;\ \dist (z, \partial \Omega )<\epsilon \} $$ 
and $\ S^{1}_{r}$ is the circle of radius $\ r$ 
is identified  with $[-\pi r,\pi r[$.\\ 
So we have the identity 
\begin{equation}\label{br6} 
z=M(s(z))+t(z)N(s(z)),\ \forall \; z\; \in \; \Omega _{\epsilon _0}. 
\end{equation} 
From this equality, it is easy to check that 
\begin{equation}\label{br7} 
T(s(z))=[1-t(z)\kappa (s(z))]\nabla  s(z)
\ \ {\rm {and}}\ \ N(s(z))=\nabla  t(z). 
\end{equation} 
So for all $ u\; \in \; H^{1}(\Omega )$ such that $ \supp (u)\; 
\subset  \Omega _{\epsilon _0}$,  
\begin{equation}\label{br8} 
\begin{array} 
{l} 
\int _{\omega } |(hD_z-A)u|^2\, dz=\\ 
\qquad \int _K[|(hD_t-{\widetilde {A}}_2)v|^2
+(1-t\kappa (s))^{-2}|(hD_s-{\widetilde {A}}_1)v|^2]
(1-t\kappa (s))\, dw
\end{array} 
\end{equation} 
and 

\begin{equation}\label{br08}
\int _{\omega }  |u|^{2}\, dz=\int _K|v|^2(1-t\kappa (s))\, dw
\end{equation} 
with $\ v(w)=u(\psi ^{-1}(w))$, 
$K=I\times ]0,\epsilon _0[$, $ w=(s,t)$ and $ dw=dsdt$.\\  
The magnetic potential  $\ {\widetilde {A}}$ satisfies 
$${\widetilde {A}}_1\, ds+{\widetilde {A}}_2\, dt = A_1\, dx+A_2\, dy\;.$$ 
So 
\begin{equation}\label{br9} 
[\frac{\partial \widetilde {A}_2  }{\partial s}(w)-
\frac{\partial \widetilde {A}_1 }{\partial t}(w)]\; ds\wedge dt 
=B(z)\; dx\wedge dy={\hat B}(w)[1-t\kappa (s)]\; ds\wedge dt \;,
\end{equation} 
with $\psi (z)=w$ and ${\hat B}$ is defined by~:
\begin{equation}\label{hatB}
{\hat B}(w)= B(z)\;.
\end{equation}
This gives~: 
\begin{equation}\label{br10} 
\frac{\partial \widetilde {A}_2}{\pa s}(w)-
\frac{\partial \widetilde {A}_1}{\pa t}(w)
=B(\psi^{-1}(w))[1-t\kappa (s)]={\hat B}(t,s) (1-t \kappa(s))\;. 
\end{equation} 
Then we get the identity between   differential operators 
\begin{equation}\label{br11} 
(hD_z-A)^2=a^{-1}[(hD_s-{\widetilde {A}}_1)a^{-1}
(hD_s-{\widetilde {A}}_1)+
(hD_t- {\widetilde {A}}_2)a(hD_t- {\widetilde {A}}_2)],  
\end{equation} 
where $a(w)=1-t\kappa (s)$. \\
The usual Hilbert space $\ L^2(\Omega _{\epsilon _0})$ is transformed 
to $\ L^2(K;a\,dw)$. \\
The unitary transform 
$ U$ from $L^2(\Omega _{\epsilon _0};dz)$  into $ 
L^2(K;dw)$, defined by~:  
\begin{equation}\label{br11a}
  U(u)(w)=a^{1/2}(w)u(\psi ^{-1}(w))\;,
\end{equation}
allows us to work in $ L^2(K)$ and then we get the new identity 
between  differential operators 
\begin{equation}\label{br12} 
a^{1/2}(hD_z-A)^2a^{-1/2}=(hD_s-{\widetilde {A}}_1)a^{-2}
(hD_s-{\widetilde {A}}_1)+
(hD_t- {\widetilde {A}}_2)^2+h^2 W \;,
\end{equation}
where $W$ is a scalar function given by~:
$$  W(w)=a^{- \frac 12} \frac{\partial ^2}{\partial t^2}a^{1/2}
+a^{-5/2}\frac{\partial ^2}{\partial s^2}a^{1/2}
-4a^{-3}[\frac{\partial }{\partial s}a^{1/2}]^2\;.$$
A small computation gives~:
$$
W(s,t) = - \frac 14 a^{-2} \kappa^2 - \frac t2 a^{-3} \kappa'' - \frac 54 t^2 a^{-4} (\kappa')^2\;.
$$

In the new coordinates and using a gauge transform,  we can always assume that the magnetic potential 
has no normal component in a neighbourhood of 
$\ \partial \Omega :$ 
\begin{equation}\label{br13} 
 {\widetilde {A}}_2=0. 
\end{equation} 
In this case, we have~:
\begin{equation}\label{br13a}
\partial_t  {\widetilde {A}}_1 = {\hat B}(t,s) (1- t \kappa(s))\;,
\end{equation}
where ${\hat B}$ was introduced in (\ref{hatB}).\\

These change may be useful for analyzing the situation near the boundary.\\
For example, in the Dirichlet case, with the additional condition that
 the functions are supported near the boundary, we get the identity~:
\begin{equation}
q_{h,A,\Omega}^D (u) = {\tilde q}_{h}^D (v)\;,
\end{equation}
with $ v= U u$, $u\in H_0^1(\Omega)$, $\supp u \subset \Omega_{\epsilon(\Omega)}$
and
\begin{equation}
\begin{array}{ll}
{\tilde q}_h^D (v)& = \int a^{-2} | h D_s v|^2 dt ds \\
& \quad + \int | (h D_t - {\tilde A}_2) v|^2 dt ds\\
& \quad + h^2 \int W(t,s) |v|^2 dt ds\;.
\end{array}
\end{equation}

But one has to be careful in the Neumann case, because $U$ does not respect the Neumann
 condition. In any case, one can use  the identity~:
\begin{equation}
q_{h,A,\Omega}^N(u) = {\tilde q}_{h}^D (v)\;,
\end{equation}
with $ v= U u$, $u\in H^1(\Omega)$, $\supp u \subset \Omega_{\epsilon(\Omega)}$
and
\begin{equation}\label{normf}
\begin{array}{ll}
{\tilde q}_h^N (v)& = \int a^{-2} | h D_s v|^2 dt ds \\
& \quad + \int | (h D_t - {\tilde A}_2) v|^2 dt ds\\
& \quad + h^2 \int W(t,s) |v|^2 dt ds\\
& - \frac 12 \kappa h^2 \int_{t=0} |v(s,0)|^2 ds \;.
\end{array}
\end{equation}

It could be dangerous to forget the last term (boundary term) 
in Formula (\ref{normf}), when analyzing  the asymptotic behavior as $h\ar 0$ of the ground state energy. Its contribution
 is indeed of order $\Og (h^\frac 32)$. On the contrary, the contribution of the third term
 appears only in the remainder with the order $\Og(h^2)$. We emphasize that the minimizer
 of the last functional does not satisfy the usual Neumann condition but
 the distorted Neumann condition~: 
\begin{equation} \label{distorted}
(\pa_t v)(s,0) = - \frac 12 \kappa (s) v(s,0)\;.
\end{equation}
This is actually quite natural if we think of the relation $v= U u$.
\paragraph{The constant magnetic field case}.\\
In the neighborhood of 
$\ \partial \Omega ,\ \Omega _{\epsilon _0}$,  
we have (cf  (\ref{br11}))~:
\begin{equation}\label{6ct} 
P^{N}_{h,A,\Omega }u=a^{-1}\{ (hD_s+b t(1-\frac{t}{2}\kappa (s)))
[a^{-1}
(hD_s+b t(1-\frac{t}{2}\kappa (s)))u]+
h^2D_t(aD_tu)\} \;.  
\end{equation} 
\newpage 

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\end{thebibliography}

\end{document}
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