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Universit  d'Oran Es-S nia 
Facult  des Sciences 
D partement de Math matiques 
E-mail: bmessirdi@yahoo.fr, senoussaoui_abdou@yahoo.fr
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The Born-Oppenheimer approximation, Resonances, Pseudodifferential Operators
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\begin{center}
{\LARGE Resonances for a General Hamiltonian}

{\LARGE in the Born-Oppenheimer Approximation}\\[0pt]
\bigskip {\small by}\\[0pt]
\bigskip \textbf{Messirdi Bekkai and Senoussaoui Abderrahmane}

Universit\'{e} d'Oran Es-S\'{e}nia

D\'{e}partement de Math\'{e}matiques

Facult\'{e} des Sciences

B.P 1524\ El Mnaouer Oran 31000, ALGERIA
\end{center}

\bigskip \textbf{Abstract:} {\small We study the discrete spectrum of a
general class of Born-Oppenheimer Hamiltonians of the type: }%
\begin{equation*}
H=-h^{2}\Delta _{x}+P\left( x,y,D_{y}\right) \text{ on }L^{2}\left( \mathbb{R%
}_{x}^{n}\times \mathbb{R}_{y}^{p}\right) ,n,p\in \mathbb{N}^{\ast }
\end{equation*}%
{\small when }$h${\small \ tends to }$0^{+}${\small , here }$P\left(
x,y,D_{y}\right) ${\small \ is a pseudodifferential operator on }$%
L^{2}\left( \mathbb{R}_{y}^{p}\right) .$ {\small In the case where the first
eigenvalue }$\lambda _{1}\left( x\right) ${\small \ of }$P\left(
x,y,D_{y}\right) ${\small \ on }$L^{2}\left( \mathbb{R}_{y}^{p}\right) $%
{\small \ admits one non degenerate point-well, we obtain WKB-type
expansions for all order in }${\small h}^{{\small 1/2}}${\small \ of
eigenvalues (in the interval }$[0,C_{0}h],${\small \ }$C_{0}>0)${\small \
and associated normalized eigenfunctions of }$H,${\small \ and this for all
orders in }$h^{1/2}$.

\textbf{AMS classifications:}{\small {\ }35P15, 35Q20, 35P99, 35S99.}%
\bigskip \newline
\textbf{0 \ \ \ Introduction}\newline
The Born-Oppenheimer approximation is a method introduced in \cite{BoOp} to
analyse the spectrum of molecules. It consists in studying the behavior of
the associate Hamiltonian when the nuclear mass tends to infinity. This
Hamiltonian can be written in the form:%
\begin{equation*}
P=-h^{2}\Delta _{x}-\Delta _{y}+V\left( x,y\right) 
\end{equation*}%
where $x\in \mathbb{R}^{n}$ represents the position of the nuclei, $y\in 
\mathbb{R}^{p}$ is the position of the electrons, $h$ is proportional to the
inverse of the square-root of the nuclear mass and $V\left( x,y\right) $ is
the interaction potential.

In the last decade, many efforts have been made in order to study in the
semiclassical limit the spectrum of $P$ ( see e.g. \cite{GMS}, \cite{KMSW}, 
\cite{Ma2}, \cite{MaMe}, \cite{MeSe}, \cite{MSD},...). These authors have
shown that in many situations it is still possible to perform, by Grushin's
method, semiclassical constructions related to the existence of some hidden
effective semiclassical operator.

It has been proved, both for smooth potentials \cite{Ma2} and for the
physically interesting case of Coulomb interaction potentials (see , \cite%
{KMSW,MeSe}), the existence for the operator $P$ of asymptotic expansions
for eigenvalues and associated eigenfunctions of the types:%
\begin{equation*}
\sum\limits_{j\geq 0}\alpha _{j}h^{j/2}\text{ and }e^{-\psi \left( x\right)
/h}\left( \sum\limits_{j\geq 0}a_{j}\left( x,y\right) h^{j/2}\right) \text{,}
\end{equation*}%
where $\psi \left( x\right) $ is the Agmon distance between $x$ and the
potential well.

Here we plan to give a unified version of the two results in \cite{Ma2} and 
\cite{KMSW}, which can be applied to the general class of operators of the
type $H=-h^{2}\Delta _{x}+P\left( x,y,D_{y}\right) ,$ where $P\left(
x,y,D_{y}\right) $ is a pseudodifferential operator on $\mathcal{H}%
_{2}=L^{2}\left( \mathbb{R}_{y}^{p}\right) $ (the so-called electronic
Hamiltonian and its eigenvalues are the so-called electronic levels).

By using the $h$-pseudodifferential operators with operator-valued symbol
and the general Feshbach reduction scheme (see \cite{Ba,Se}), the spectral
study of $H$ on $L^{2}\left( \mathbb{R}_{x}^{n}\times \mathbb{R}%
_{y}^{p}\right) $ is reduced to that of a matrix of $h$-pseudodifferential
operators $F\left( \lambda \right) $ on $\left( L^{2}\left( \mathbb{R}%
_{x}^{n}\right) \right) ^{\oplus m}$ (the so-called effective Hamiltonian)
with principal symbol the diagonal matrix $diag\left( \xi ^{2}+\lambda
_{j}\left( x\right) \right) _{1\leq j\leq m}$ where $m>0$ depends on the
energy level and $\left( \lambda _{j}\left( x\right) \right) _{1\leq j\leq
m} $ are the electronic levels. In particular, we obtain the following
equivalence:%
\begin{equation*}
\lambda \in Sp\left( H\right) \Longleftrightarrow \lambda \in Sp\left(
F\left( \lambda \right) \right)
\end{equation*}%
(here $Sp$ stands for the spectrum).

The general theory of Helffer and Sj\"{o}strand in \cite{HeSj} can be
applied to the operator $F\left( \lambda \right) $ and shows the existence
of formal WKB-type expansions for eigenfunctions of this operator. This
finally gives the formal WKB-type expansions for the operator $H$ itself.

The argument of Martinez in \cite{Ma2} gives a justification to the formal
WKB-constructions by showing that, modulo an error of size $\mathcal{O}%
\left( h^{\infty }\right) ,$ the formal eigenfunctions approximate correctly
the true eigenfunctions of $H.$

The plan of this paper is the following. In the first section we introduce
our assumptions and give preliminaries. The spectral reduction of the
problem is given in the second section. The third section is devoted to
state our main result and apply the reduction theorem obtained in the
section 2 to establish the proof.

\section{Assumptions and preliminaries}

We study the resonaces of a general class of Born-Oppenheimer Hamiltonian of
the type:%
\begin{equation*}
P\left( h\right) =-h^{2}\Delta _{x}+P\left( x,y,D_{y}\right) \text{ on }%
L^{2}\left( \mathbb{R}_{x}^{n}\times \mathbb{R}_{y}^{p}\right) ,n,p\in 
\mathbb{N}^{\ast }
\end{equation*}%
when $h$ tends to $0^{+},$ $P\left( x,y,D_{y}\right) $ is a
pseudodifferential operator on $L^{2}\left( \mathbb{R}_{y}^{p}\right) $ with 
$x$-independent domain.

We assume that:

$(H1)$ For every $x\in \mathbb{R}^{n},$ $P\left( x,y,D_{y}\right) $ is
selfadjoint and bounded from below on $L^{2}\left( \mathbb{R}_{y}^{p}\right)
.$ $P\left( x,y,D_{y}\right) $ can be analytically extended on the complex
strip 
\begin{equation*}
D_{\delta }=\left\{ x\in \mathbb{C}^{n},\text{ }\left\vert \func{Im}%
x\right\vert \leq \delta <\func{Re}x>\right\} ,\text{ }\delta >0.
\end{equation*}

$(H2)$ The spectrum of the pseudodifferential operator $P\left(
x,y,D_{y}\right) $ has two disjoint components for every $x\in \mathbb{R}%
^{n} $:%
\begin{equation*}
Sp\left( P\left( x,y,D_{y}\right) \right) =\left\{ \lambda _{1}\left(
x\right) ,\lambda _{2}\left( x\right) ,...,\lambda _{M}\left( x\right)
\right\} \cup \sigma \left( x\right)
\end{equation*}%
where $\lambda _{1}\left( x\right) ,\lambda _{2}\left( x\right) ,...,\lambda
_{M}\left( x\right) $ are eigenvalues of $P\left( x,y,D_{y}\right) $ depend
continuously on $x\in \mathbb{R}^{n}$ and can be analytically extended on $%
D_{\delta }.$ There is a gap between the two components: 
\begin{equation*}
\inf\limits_{\lambda \in \sigma \left( x\right) ,j\in \left\{
1,...,M\right\} }\left\vert \lambda _{j}\left( x\right) -\lambda \right\vert
\geq \delta .
\end{equation*}%
In particular, this implies that the spectral projector $\pi \left( x\right) 
$ of $P\left( x,y,D_{y}\right) $ associated to $\left\{ \lambda _{1}\left(
x\right) ,\lambda _{2}\left( x\right) ,...,\lambda _{M}\left( x\right)
\right\} $ is $C^{2}$-regular with respect to $x$ (see \cite{CoSe}).

$(H3)$ We also assume that $\lambda _{1}\left( x\right) ,\lambda _{2}\left(
x\right) ,...,\lambda _{M}\left( x\right) $ are separated at the infinity:%
\begin{equation*}
\exists \widetilde{C}>0,\text{ }\inf_{\substack{ j\neq k  \\ \left\vert
x\right\vert \geq C}}\left\vert \lambda _{j}\left( x\right) -\lambda
_{k}\left( x\right) \right\vert \geq \widetilde{C},\text{ }C>0.
\end{equation*}%
This last assumption is essential in our work to obtain a good behavior of
the spectral projectors of $P\left( x,y,D_{y}\right) $ where $\left\vert
x\right\vert \rightarrow +\infty .$ This is because our technique stand
strongly on pseudodifferential calculus, which requires a lot of regularity
with respect to $x.$

$\left( H4\right) $ $P\left( x,y,D_{y}\right) \in C_{b}^{\infty }\left( 
\mathbb{R}^{n},\mathcal{L}\left( \mathcal{D}\left( P\left( x,y,D_{y}\right)
\right) ,L^{2}\left( \mathbb{R}_{y}^{p}\right) \right) \right) ,$ here $%
C_{b}^{\infty }$ denotes the space of $C^{\infty }$-functions that have
their derivatives of any order uniformly bounded.

\textbf{Examples:}

\begin{itemize}
\item The operator $P\left( x,y,D_{y}\right) =-\frac{d^{2}}{dy^{2}}+\left(
1+x^{2}\right) ^{2l}y^{2},$ $x\in \mathbb{R},l\in \mathbb{R}$ satisfies the
assumptions (H1) to (H3) with domain%
\begin{gather*}
\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) =H^{2}\left( \mathbb{R}%
_{y}\right) \cap \left\{ \varphi \in L^{2}\left( \mathbb{R}_{y}\right) ;%
\text{ }y^{2}\varphi \in L^{2}\left( \mathbb{R}_{y}\right) \text{ }\right\} ,
\\
\lambda _{j}\left( x\right) =\left( 2j+1\right) \left( 1+x^{2}\right) ^{l};%
\text{ }j=1,...,M\text{ and} \\
\text{ }\Gamma \left( x\right) =\left\{ \left( 2j+1\right) \left(
1+x^{2}\right) ^{l};\text{ }j\geq M+1\right\} .
\end{gather*}

\item A second example is the Born-Oppenheimer Hamiltonian (see e.g \cite%
{MeSe,MSD,KMSW}) for the differential operator%
\begin{equation*}
P\left( x,y,D_{y}\right) =-\Delta _{y}+V\left( x,y\right) ,
\end{equation*}%
where $V\left( x,y\right) $ is the Coulomb interaction potential. For the
study of resonances of $P\left( h\right) $ in this example see the works of
Martinez-Messirdi \cite{MaMe} and those of Messirdi-Senoussaoui-Djellouli 
\cite{MSD}.
\end{itemize}

\section{Preliminaries and main result}

In this paper we characterize the resonances of $P\left( h\right) $ by using
the analytic dilation introduced by Hunziker \cite{Hu}. More precisely, for $%
\theta $ real small enough we consider the transformation $xe^{\theta }$ and
the associated dilation operator $U_{\theta }$ defined by:%
\begin{equation*}
U_{\theta }\varphi \left( x,y\right) =e^{n\theta /2}\varphi \left(
xe^{\theta },y\right) ,\text{ \ }\varphi \in C_{0}^{\infty }\left( \mathbb{R}%
_{x}^{n}\times \mathbb{R}_{y}^{p}\right)
\end{equation*}%
$U_{\theta }$ is an unitary operator on $L^{2}\left( \mathbb{R}%
_{x}^{n}\times \mathbb{R}_{y}^{p}\right) .$ Now let the dilation $P_{\theta
}\left( h\right) =U_{\theta }P\left( h\right) U_{\theta }^{-1}$ of the
operator $P\left( h\right) $: 
\begin{equation*}
P_{\theta }\left( h\right) =-h^{2}e^{-2\theta }\Delta _{x}+P\left(
xe^{\theta },y,D_{y}\right) .
\end{equation*}%
Then using the assumption $\left( H_{1}\right) $ the familly $P_{\theta
}\left( h\right) $ can be extended to small enough complex values of $\theta 
$ as analytic family of type $A$ (see e.g \cite{Ka,ReSi}).

\begin{definition}
\bigskip We say that a complex number $\rho $ is a resonance of $P\left(
h\right) $ if $\func{Re}\rho >\inf \sigma _{ess}\left( P\left( h\right)
\right) $ and there exists $\theta $ small enough, $\func{Im}\theta >0,$
such that $\rho \in \sigma _{disc}\left( P\left( h\right) \right) .$ $\sigma
_{ess}$ and $\sigma _{disc}$ are respectively the essential and the discrete
spectrum.
\end{definition}

\begin{notation}
We denote by $\Gamma \left( h\right) $ the set of resonances of the operator 
$P\left( h\right) $ .
\end{notation}

A family of unbounded operators $A\left( h\right) $ on $L^{2}\left( \mathbb{R%
}^{n}\right) ,$ with fixed domain $H^{k_{0}}\left( \mathbb{R}^{n}\right) $ $%
k_{0}\geq 0,$ is said to be $h$-pseudodifferential if there exists a
sequence $\left( a_{j}\left( x,\xi \right) \right) _{j\in \mathbb{N}}$ of $%
C^{\infty }$-functions on $\mathbb{R}^{2n}$ satisfying:%
\begin{equation*}
\forall j\in \mathbb{N}\text{, }\forall \alpha ,\beta \in \mathbb{N}^{n},%
\text{ }\left\vert \partial _{x}^{\alpha }\partial _{\xi }^{\beta
}a_{j}\left( x,\xi \right) \right\vert =\mathcal{O}\left( <\xi
>^{k_{0}-\left\vert \beta \right\vert }\right)
\end{equation*}%
uniformly on $\mathbb{R}^{n},$ with $<\xi >=\left( 1+\left\vert \xi
\right\vert ^{2}\right) ^{1/2}$ and for any $N\in \mathbb{N}$ large enough, $%
A\left( h\right) $ can be written%
\begin{equation*}
A(h)=\overset{N}{\underset{j=0}{\sum }}h^{j}Op_{h}^{w}(a_{j})+h^{N}R_{N}(h)
\end{equation*}%
where $R_{N}(h)$ is uniformly bounded on $L^{2}\left( \mathbb{R}^{n}\right) $
as $h\rightarrow 0^{+},$ and $Op_{h}^{w}$ denotes the Weyl $h$-quantization
of symboles:%
\begin{equation*}
Op_{h}^{w}\left( a_{j}\right) \varphi \left( x\right) =\left( 2\pi h\right)
^{-n}\int\limits_{\mathbb{R}^{2n}}e^{\frac{i}{h}<x-y,\xi >}a_{j}\left( \frac{%
x+y}{2},\xi \right) \varphi \left( y\right) dyd\xi .
\end{equation*}%
The function $a_{0}$ is called the principal symbol of $A\left( h\right) .$

Denote now $\lambda _{0}=\inf \left\{ Sp\left( P\left( x,y,D_{y}\right)
\right) \backslash \left\{ \lambda _{1}\left( x\right) ,\lambda _{2}\left(
x\right) ,...,\lambda _{M}\left( x\right) \right\} \right\} .$

Our main result is:

\begin{theorem}
\label{MainTheorem}\textit{Under assumptions }$(H1)$\textit{\ to }$(H4)$%
\textit{, and for any }$z$\textit{\ complex close enough to }$\lambda _{0}$%
\textit{, there exists a family of }$M\times M$\textit{-matrixes }$A_{\theta
}^{-+}(z),$ $\theta $ complex small enough, of $h$\textit{%
-pseudodifferential operators on }$\mathbb{R}^{n}$\textit{\ depending
analytically on }$\theta $\textit{\ such that: }%
\begin{equation*}
z\in \Gamma (h)\Leftrightarrow \exists \theta \in \mathbb{C},\text{ }\func{Im%
}\theta >0,0\in \sigma _{disc}(A_{\theta }^{-+}(z)).
\end{equation*}%
In particular, $F\left( z\right) =z-A_{\theta }^{-+}(z)$ has the diagonal
matrix $diag\left( \xi ^{2}+\lambda _{j}\left( x\right) \right) _{1\leq
j\leq M}$ as principal symbol.
\end{theorem}

\section{\protect\bigskip The dilation Feshbach method}

The Feshbach reduction is a way to construct an effective Hamiltonian of the
spectral problem of $P_{\theta }\left( h\right) .$ To get this construction
in the context of $h$-pseudodifferential calculus we make use of a so-called
Grushin problem involving a convenient choice of sections of $Ran\pi \left(
x\right) ,$ where $\pi \left( x\right) $ denotes the orthogonal projector
onto the eigenspace of $P\left( x,y,D_{y}\right) $ associated to $\left\{
\lambda _{1}\left( x\right) ,\lambda _{2}\left( x\right) ,...,\lambda
_{M}\left( x\right) \right\} .$

In fact, since we are interested in the resonances of $P\left( h\right) $,
we make all these constructions for the analytic dilation $P_{\theta }\left(
h\right) $ of $P\left( h\right) .$

Using the constructions made in \cite{Me1}, we have the following lemma:

\begin{lemma}
\label{Lemma1}\textit{Under }$\left( \mathit{H1}\right) $ to $\left(
H4\right) $\textit{, there exists an orthonormal family }$\left\{
v_{1}^{\theta }\left( x\right) ,v_{2}^{\theta }\left( x\right)
,...,v_{M}^{\theta }\left( x\right) \right\} $\textit{\ in }$\mathcal{D}%
\left( P\left( x,y,D_{y}\right) \right) $ \textit{depending analytically
with respect to }$\theta $ complex small enough such that\textit{:\ }

\begin{description}
\item[1.] $v_{j}^{\theta }\left( x\right) \in C_{b}^{\infty }\left( \mathbb{R%
}^{n},\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) \right) $ for all $%
j\in \left\{ 1,...,M\right\} ,$

\item[2.] $\left\{ v_{1}^{\theta }\left( x\right) ,...,v_{M}^{\theta }\left(
x\right) \right\} $ generate the space $\bigoplus\limits_{j=1}^{M}\ker
\left( P\left( xe^{\theta },y,D_{y}\right) -\lambda _{j}\left( xe^{\theta
}\right) \right) .$
\end{description}
\end{lemma}

If $\underset{j=1}{\overset{M}{\oplus }}\psi _{j}=\left( \psi _{1},...,\psi
_{M}\right) \in \left( L^{2}\left( \mathbb{R}^{n}\right) \right) ^{\oplus M}$
and $\varphi \in L^{2}\left( \mathbb{R}^{n},\mathcal{D}\left( P\left(
x,y,D_{y}\right) \right) \right) $ then we define the two following
operators $R_{\theta }^{\pm }$ by:%
\begin{equation*}
\begin{array}{ccc}
R_{\theta }^{-}:\tbigoplus\limits_{j=1}^{M}L^{2}\left( \mathbb{R}^{n}\right)
& \longmapsto & L^{2}\left( \mathbb{R}^{n},\mathcal{D}\left( P\left(
x,y,D_{y}\right) \right) \right) \\ 
\psi =\underset{j=1}{\overset{M}{\oplus }}\psi _{j} & \longrightarrow & 
R_{\theta }^{-}\psi =\dsum\limits_{j=1}^{M}\psi _{j}v_{j}^{\theta }\left(
x\right)%
\end{array}%
\end{equation*}%
and 
\begin{equation*}
\begin{array}{ccc}
R_{\theta }^{+}=\left( R_{\theta }^{-}\right) ^{\ast }:L^{2}\left( \mathbb{R}%
^{n},\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) \right) & 
\longrightarrow & \tbigoplus\limits_{j=1}^{M}L^{2}\left( \mathbb{R}%
^{n}\right) \\ 
\varphi & \longmapsto & R_{\theta }^{+}\varphi =\underset{j=1}{\overset{m}{%
\oplus }}<\varphi ,v_{j}^{\theta }\left( x\right) >_{L^{2}\left( \mathbb{R}%
^{n}\right) }.%
\end{array}%
\end{equation*}%
We then consider a Grushin problem that will lead to the Feshbach reduction.
For $z\in \mathbb{C}$, we consider the following matrix operator:%
\begin{equation*}
\mathcal{P}_{\theta }\left( z\right) =\left( 
\begin{array}{cc}
P_{\theta }\left( h\right) -z & R_{\theta }^{-} \\ 
R_{\theta }^{-} & 0%
\end{array}%
\right) \text{ on }L^{2}\left( \mathbb{R}^{n},\mathcal{D}\left( P\left(
x,y,D_{y}\right) \right) \right) \oplus \left( L^{2}\left( \mathbb{R}%
^{n}\right) \right) ^{\oplus M}.
\end{equation*}%
Then we have:

\begin{theorem}
\textit{Assume }$\mathit{(H1)}$ to $\mathit{(H4)}$\textit{. Then for any }$z$
complex such that $\func{Re}z<\lambda _{0},$\textit{\ the Grushin operator }$%
\mathcal{P}_{\theta }\left( \lambda \right) :$\textit{\ }$H^{2}\left( 
\mathbb{R}^{n},\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) \right)
\oplus \left( L^{2}\left( \mathbb{R}^{n}\right) \right) ^{\oplus
M}\longrightarrow $\textit{\ }$L^{2}\left( \mathbb{R}^{n},L^{2}\left( 
\mathbb{R}^{p}\right) \right) \oplus \left( L^{2}\left( \mathbb{R}%
^{n}\right) \right) ^{\oplus M},$\textit{\ if we write it's inverse as:}%
\begin{equation*}
\left( \mathcal{P}_{\theta }\left( z\right) \right) ^{-1}=\left( 
\begin{array}{rr}
A_{\theta }\left( z\right) \;\; & A_{\theta }^{+}\left( z\right) \\ 
A_{\theta }^{-}\left( z\right) \; & A_{\theta }^{-+}\left( z\right)%
\end{array}%
\right)
\end{equation*}%
then $A_{\theta }\left( z\right) ,$ $A_{\theta }^{\pm }\left( z\right) $%
\textit{\ and }$A_{\theta }^{-+}\left( z\right) $\textit{\ are }$h$\textit{%
-pseudodifferential operators. Moreover, we have the following spectral
reduction:}%
\begin{equation}
z\in Sp\left( P_{\theta }\left( h\right) \right) \Longleftrightarrow z\in
Sp\left( F_{\theta }\left( z\right) \right) ,\text{ }\func{Im}\theta >0 
\tag{2.1}
\end{equation}%
\textit{where }$F_{\theta }\left( z\right) =z-A_{\theta }^{-+}\left(
z\right) $\textit{\ is a }$M\times M$\textit{\ matrix of }$h$\textit{%
-pseudodifferential operators on }$\left( L^{2}\left( \mathbb{R}^{n}\right)
\right) ^{\oplus M}$\textit{\ with the diagonal matrix }$diag\left( \xi
^{2}+\lambda _{j}\left( xe^{\theta }\right) \right) _{1\leq j\leq M}$\textit{%
\ as principal symbol}.
\end{theorem}

\begin{proof}
We can consider the Grushin operator $\mathcal{P}_{\theta }\left( z\right) $
as an $h$-pseudodifferential operator with operator-valued symbol $p_{\theta
}\left( x,\xi ;z\right) $ given by:%
\begin{equation}
p_{\theta }\left( x,\xi ;z\right) =\left( 
\begin{array}{cc}
\xi ^{2}+P\left( xe^{\theta },y,D_{y}\right) -z\;\ \ \ \; & R_{\theta }^{-}
\\ 
R_{\theta }^{+} & 0%
\end{array}%
\right) .  \tag{2.2}
\end{equation}%
Using the fact that for any $z\in \mathbb{C}$ such that $\func{Re}z<\lambda
_{0}$ and $x\in \mathbb{R}^{n}$,%
\begin{equation}
\func{Re}\left( \widehat{\pi }_{\theta }\left( x\right) P\left( xe^{\theta
},y,D_{y}\right) \widehat{\pi }_{\theta }\left( x\right) -z\right) >0 
\tag{2.3}
\end{equation}%
the symbol $p_{\theta }\left( x,\xi ;z\right) $ is invertible and its
inverse $q_{\theta }\left( x,\xi ;z\right) $ is given by:%
\begin{equation}
q_{\theta }\left( x,\xi ;z\right) =\left( 
\begin{array}{cc}
\widehat{\pi }_{\theta }\left( x\right) \left( \xi ^{2}+\widehat{\pi }%
_{\theta }\left( x\right) P\left( xe^{\theta },y,D_{y}\right) \widehat{\pi }%
_{\theta }\left( x\right) -z\right) ^{-1}\widehat{\pi }_{\theta }\left(
x\right) \;\ \ \  & R_{\theta }^{-} \\ 
R_{\theta }^{+}\,\,\,\;\;\text{\thinspace \thinspace \thinspace \thinspace
\thinspace \thinspace }\;\;\;\;\;\;\;\;\; & \left( z-\xi ^{2}-\lambda
_{j}\left( xe^{\theta }\right) \right) _{1\leq j\leq M}%
\end{array}%
\right)  \tag{2.4}
\end{equation}%
where $\widehat{\pi }_{\theta }\left( x\right) =1-\pi _{\theta }\left(
x\right) ,$ $\pi _{\theta }\left( x\right) $ denotes the orthogonal
projection on the space $\bigoplus\limits_{j=1}^{M}\ker \left( P\left(
xe^{\theta },y,D_{y}\right) -\lambda _{j}\left( xe^{\theta }\right) \right)
. $

Due to $(H4)$ and $\left( 2.3\right) ,$ we can consider the Weyl
quantification $Q_{\theta }\left( z\right) =Op_{h}^{w}\left( q_{\theta
}\left( x,\xi ;z\right) \right) :$ $L^{2}\left( \mathbb{R}^{n},L^{2}\left( 
\mathbb{R}^{p}\right) \right) \oplus \left( L^{2}\left( \mathbb{R}%
^{n}\right) \right) ^{\oplus M}$ $\longrightarrow $ $H^{2}\left( \mathbb{R}%
^{n},\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) \right) \oplus
\left( L^{2}\left( \mathbb{R}^{n}\right) \right) ^{\oplus M}.$\linebreak The
symbolic calculus and especially the composition theorem of $h$%
-pseudodiff-erential operators allows us to obtain,%
\begin{equation*}
\left\{ 
\begin{array}{l}
\mathcal{P}_{\theta }\left( z\right) Q_{\theta }\left( z\right) =I+hR_{1};%
\text{\ }\left\Vert R_{1}\right\Vert _{\mathcal{L}\left( L^{2}\left( \mathbb{%
R}^{n},L^{2}\left( \mathbb{R}^{p}\right) \right) \oplus \left( L^{2}\left( 
\mathbb{R}^{n}\right) \right) ^{\oplus M}\right) }=\mathcal{O}\left( 1\right)
\\ 
Q_{\theta }\left( z\right) \mathcal{P}_{\theta }\left( z\right) =I+hR_{2};%
\text{\ }\left\Vert R_{2}\right\Vert _{\mathcal{L}\left( H^{2}\left( \mathbb{%
R}^{n},\mathcal{D}\left( P\left( x,y,D_{y}\right) \right) \right) \oplus
\left( L^{2}\left( \mathbb{R}^{n}\right) \right) ^{\oplus M}\right) }=%
\mathcal{O}\left( 1\right)%
\end{array}%
\right. .
\end{equation*}%
Here, the estimates of $\left\Vert R_{1}\right\Vert $ and $\left\Vert
R_{2}\right\Vert $ are uniform with respect to $h.$ As a consequence, for $h$
small enough, $\mathcal{P}_{\theta }\left( z\right) $ is invertible and its
inverse is given by the Neumann series:%
\begin{equation}
\left( \mathcal{P}_{\theta }\left( z\right) \right) ^{-1}=Q_{\theta }\left(
z\right) \left( I+\sum_{k=1}^{+\infty }h^{k}R_{1}^{k}\right) =\left(
I+\sum_{k=1}^{+\infty }h^{k}R_{2}^{k}\right) Q_{\theta }\left( z\right) . 
\tag{2.5}
\end{equation}%
In view of $\left( 2.5\right) $ and the expression of the symbol $q_{\theta
}\left( x,\xi ;z\right) $ it remains to prove the equivalence $\left(
2.1\right) .$ This comes from the two following algebraic identities: 
\begin{eqnarray*}
\left( \left( P_{\theta }\left( h\right) -z\right) u=v\right)
&\Leftrightarrow &\mathcal{P}_{\theta }\left( z\right) \left( u\oplus
0\right) =v\oplus (<u,\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta
}\left( x\right) >_{L^{2}\left( \mathbb{R}^{p}\right) }) \\
&\Leftrightarrow &\left( u\oplus 0\right) =\left( \mathcal{P}_{\theta
}\left( z\right) \right) ^{-1}(v\oplus (<u,\underset{j=1}{\overset{M}{\oplus 
}}v_{j}^{\theta }\left( x\right) >_{L^{2}\left( \mathbb{R}^{p}\right) }))
\end{eqnarray*}%
\begin{equation}
\left( \left( P_{\theta }\left( h\right) -z\right) u=v\right)
\Leftrightarrow \left\{ 
\begin{array}{l}
u=A_{\theta }\left( z\right) \;v+A_{\theta }^{+}\left( z\right) (<u,\underset%
{j=1}{\overset{M}{\oplus }}v_{j}^{\theta }\left( x\right) >_{L^{2}\left( 
\mathbb{R}^{p}\right) }) \\ 
0=A_{\theta }^{-}\left( z\right) v+A_{\theta }^{-+}\left( z\right) (<u,%
\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta }\left( x\right)
>_{L^{2}\left( \mathbb{R}^{p}\right) })%
\end{array}%
\right.  \tag{2.6}
\end{equation}%
and%
\begin{eqnarray*}
\left( A_{\theta }^{-+}\left( z\right) \alpha =\beta \right)
&\Leftrightarrow &\left( \mathcal{P}_{\theta }\left( z\right) \right)
^{-1}\left( 0\oplus \alpha \right) =\left( A_{\theta }^{+}\left( z\right)
\alpha \right) \oplus \beta \\
&\Leftrightarrow &0\oplus \alpha =\mathcal{P}_{\theta }\left( z\right)
(\left( A_{\theta }^{+}\left( z\right) \alpha \right) \oplus \beta )
\end{eqnarray*}%
\begin{equation}
\left( A_{\theta }^{-+}\left( z\right) \alpha =\beta \right) \Leftrightarrow
\left\{ 
\begin{array}{l}
0=\left( P_{\theta }\left( h\right) -z\right) \left( A_{\theta }^{+}\left(
z\right) \alpha \right) +\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta
}\left( x\right) \beta \\ 
\alpha =<A_{\theta }^{+}\left( z\right) \alpha ,\underset{j=1}{\overset{M}{%
\oplus }}v_{j}^{\theta }\left( x\right) >_{L^{2}\left( \mathbb{R}^{p}\right)
}%
\end{array}%
\right. .  \tag{2.7}
\end{equation}%
If $z\notin Sp\left( P_{\theta }\left( h\right) \right) ,$ then from $\left(
2.7\right) $ we deduce:%
\begin{equation*}
A_{\theta }^{-+}\left( z\right) \alpha =\beta \Leftrightarrow \left\{ 
\begin{array}{l}
A_{\theta }^{+}\left( z\right) \alpha =-\left( P_{\theta }\left( h\right)
-z\right) ^{-1}(\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta }\left(
x\right) \beta ) \\ 
\alpha =<-\left( P_{\theta }\left( h\right) -z\right) ^{-1}(\underset{j=1}{%
\overset{M}{\oplus }}v_{j}^{\theta }\left( x\right) .),\underset{j=1}{%
\overset{M}{\oplus }}v_{j}^{\theta }\left( x\right) >_{L^{2}\left( \mathbb{R}%
^{p}\right) }\beta%
\end{array}%
\right. .
\end{equation*}%
In particular,%
\begin{equation*}
0\notin Sp\left( A_{\theta }^{-+}\left( z\right) \right) \text{ and }\left(
A_{\theta }^{-+}\left( z\right) \right) ^{-1}=-<\left( P_{\theta }\left(
h\right) -z\right) ^{-1}(\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta
}\left( x\right) .),\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta
}\left( x\right) >_{L^{2}\left( \mathbb{R}^{p}\right) }.
\end{equation*}%
Conversely, if $0\notin Sp\left( A_{\theta }^{-+}\left( z\right) \right) ,$
then \ $\left( 2.6\right) $ gives: 
\begin{equation*}
\left( P_{\theta }\left( h\right) -z\right) u=v\Leftrightarrow \left\{ 
\begin{array}{l}
<u,\underset{j=1}{\overset{M}{\oplus }}v_{j}^{\theta }\left( x\right)
>_{L^{2}\left( \mathbb{R}^{p}\right) }=-\left( A_{\theta }^{-+}\left(
z\right) \right) ^{-1}(A_{\theta }^{-}\left( z\right) v) \\ 
u=A_{\theta }\left( z\right) v-A_{\theta }^{+}\left( z\right) \left(
A_{\theta }^{-+}\left( z\right) \right) ^{-1}A_{\theta }^{-}\left( z\right) v%
\end{array}%
\right. .
\end{equation*}%
As a consequence, 
\begin{equation*}
z\notin Sp\left( P_{\theta }\left( h\right) \right) \text{ and }\left(
P_{\theta }\left( h\right) -z\right) ^{-1}=A_{\theta }\left( z\right)
-A_{\theta }^{+}\left( z\right) \left( A_{\theta }^{-+}\left( z\right)
\right) ^{-1}A_{\theta }^{-}\left( z\right) 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
\end{equation*}
\end{proof}

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\end{document}
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