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\title{On a General Born--Oppenheimer\\ Reduction Scheme}
\author{
Andr\'e MARTINEZ
\footnote{Investigation supported by University of Bologna. Funds for
selected
research topics.}
\\
Universit\`a di Bologna\\ Dipartimento di Matematica
\\
Piazza di porta San Donato, 5\\
40127 Bologna - ITALY
}
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\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{abstract} We perform a general reduction scheme that can be
applied in particular to the
spectral study of operators of the type $P=P(x,y,hD_x,D_y)$ as $h$ tends
to zero. This scheme
permits to reduce the study of
$P$ to the one of a semiclassical matrix operator of the type
$A=A(x,hD_x)$. Here, for any fixed
$(x,\xi )\in\R^n$, the eigenvalues of the principal symbol $a(x,\xi )$
of $A$ are eigenvalues of
the operator $P(x,y,\xi ,D_y)$.
\end{abstract}
\section{Introduction}
In the last decade, many efforts have been made by several authors in
order to apply semiclassical
techniques to problems in which extra nonsemiclassical variables occur
(see, e.g., \cite{GMS,
Ha, KMSW, Ne, NeSo, So}). Such efforts have shown that, despite the
presence of these extra
variables, in many situations it is still possible to perform
semiclassical constructions related
to the existence of some hidden effective semiclassical operator.
In particular, this has been completely clarified in the
case of the spectral study of molecules. In this case, the Hamiltonian
can be written in the form
$$
H = -h^2\Delta_x -\Delta_y +V(x,y) = -h^2\Delta_x +H_{\rm el}(x),
$$
where $x\in\R^n$ represents the position of the nuclei, $y\in\R^p$ is
the position of the
electrons,
$h$ is proportional to the inverse of the square-root of the nuclear
mass, and $V(x,y)$ is the
sum of all the interactions. The operator $H_{\rm el}(x)$ is the
so-called electronic Hamiltonian
and its eigenvalues are the so-called electronic levels. Then, by using
the symbolic calculus,
it has been proved in \cite{KMSW} that the spectral study of $H$ on
$L^2(\R^{n+p})$ can be reduced
to that of a semiclassical pseudodifferential matrix operator $H_{eff}
=H_{eff} (x,hD_x)$ on
$L^2(\R^n)^{\oplus N}$, where
$N>0$ depends on the energy level. Moreover, the principal part of
$H_{eff}$ can be
explicitly related to the electronic levels in agreement with the
original intuition of M.~Born
and R.~ Oppenheimer. Let us observe that, actually,
$H_{eff}=H_{eff}^\lambda$ also depends on the
spectral parameter $\lambda$, but this dependence is analytic and
involves only ${\cal O}(h^2)$
terms. In compensation, the reduction is exact in the sense that one has
the following
equivalence (without error-terms):
\begin{equation}
\label{red1}
\lambda\in \sigma (H)\; \Leftrightarrow\; \lambda\in\sigma
(H_{eff}^\lambda ).
\end{equation}
(Here $\sigma$ stands for the spectrum.) If one accepts error-terms of
size ${\cal O}(h^\infty
)$, then other techniques exist that permit to construct a
$\lambda$-independent effective
Hamiltonian (see, e.g., \cite{NeSo, So}). However, when one wants to
study exponentially small
quantitities (such as the tunneling effect), it becomes necessary to use
(\ref{red1}).
The way in which (\ref{red1}) has been proved relies on the construction
of an operator acting on
a greater space (the so-called Grushin operator), by means of the
eigenfunctions of $H_ {\rm
el}(x)$.
In the same spirit, another result of reduction has been proved in
\cite{GMS} for
differential operators of the type $P(x,y, hD_x+D_y)$, where $P(x,y,\xi
)$ is periodic in $y$.
Here again, the idea was to construct a Grushin operator by means of the
eigenfunctions of the
operator $Q(x) := P(x,y,D_y)$.
Although these two constructions seem to be rather different from one
each
other, actually there exists some unified way to see them. Indeed, in
both cases the construction
is based on the eigenfunctions of the operator obtained by substituting
a vector (say, $\xi$) to
the operator $hD_x$ (that is, the same procedure that relates quantum
mechanics to classical
mechanics). In the first case, the operator one obtains is $\xi^2
+H_{\rm el}(x)$ (that has the
same eigenfunctions as $H_{\rm el}(x)$), and in the second case, one
obtains $P(x,y,\xi +D_y)$,
the eigenfunctions of which are deduced from those of $P(x,y,D_y)$ by
conjugating with
$e^{i\xi\cdot y}$. Because of the explicit dependence of these
eigenfunctions with respect to
$\xi$ (indeed, trivial in the first case), the constructions performed
in \cite{KMSW, GMS} could
be done without particular problems.
Here we plan to give a unified version of these reduction schemes, that
can be applied to a
general class of operators of the type $P(x,y,hD_x,D_y)$. In particular,
we have to solve the
additional difficulty that the eigenfunctions of $P(x,y,\xi ,D_y)$ may
depend on $\xi$ in an
essentially arbitrary way. However, our assumtions will permit us to
quantize this dependence,
and to obtain in this way a Grushin problem in the same spirit as in
\cite{KMSW, GMS}. Note that
we consider only time-independent problems here: For related
time-dependent results, one may
consult e.g. \cite{HaJo, MaSo, PST, SpTe}.
\section{Assumptions and Result}
Let ${\cal H}_1$ and ${\cal H}_2$ be two Hilbert spaces such that ${\cal
H}_1\subset {\cal H}_2$
and the natural injection ${\cal H}_1\hookrightarrow {\cal H}_2$ is
continuous. We denote by
${\cal H}_{1,2}:={\cal L}({\cal H}_1,{\cal H}_2)$ the space of bounded
linear operators from
${\cal H}_1$ to ${\cal H}_2$ and we consider a family of operator-valued
functions
$(p_h)_{00$ is some fixed small
number), such that $p_h(x,\xi )=p_0(x,\xi )+hr_h(x,\xi )$ with $p_0$
independent of $h$, and for
every multi-index
$\alpha\in\N^{2n}$,
\begin{equation}
\label{S1}
\left\|\partial^\alpha p_0(x,\xi )\right\|_{{\cal
H}_{1,2}}+\left\|\partial^\alpha r_h(x,\xi
)\right\|_{{\cal H}_{1,2}}={\cal O}(1)
\end{equation}
uniformly with respect to $h\in (0,h_0]$ and $(x,\xi )\in\R^{2n}$. For
any $h>0$ small we define
the pseudodifferential operator $P_h$ with symbol $p_h$ by
$$
\begin{array}{cccc}
P_h\; : & L^2(\R^n,{\cal H}_{1})&\rightarrow& L^2(\R^n,{\cal H}_{2}),\\
{}& u&\mapsto &P_hu,
\end{array}$$
where for almost all $x\in\R^n$, $P_hu(x)\in {\cal H}_2$ is defined by
the oscillatory integral
(the so-called Weyl quantization of $p_h$, see, e.g., \cite{Ma1})
\begin{equation}
\label{quant}
P_hu(x)={\rm Op}_h^W(p_h)u(x):=\frac1{(2\pi h)^n}\int e^{i(x-y)\xi
/h}p_h\left(\frac{x+y}2,\xi
\right)u(y)dy\d\xi .
\end{equation}
Note that $P_h$ maps continuously $L^2(\R^n,{\cal H}_{1})$ into
$L^2(\R^n,{\cal H}_{2})$, thanks
to (a slight generalization of) the Calder\'on--Vaillancourt theorem
(see \cite{Ma1}, Theorem
2.8.1).
We assume that for any $(x,\xi )\in\R^{2n}$, the spectrum $\sigma
(p_0(x,\xi ))$ of $p_0(x,\xi )$
contains a finite subset $\sigma_1 (x,\xi )$ such that the following
conditions hold for all $(x,\xi )\in\R^{2n}$:
\begin{itemize}
\item[(H1)] There exist $\varphi_1,\dots ,\varphi_m\in C_b^\infty
(\R^{2n};{\cal H}_1)$ such that
for all $(x,\xi )\in\R^{2n}$ the family $(\varphi_1(x,\xi ), \dots ,
\varphi_m(x,\xi ))$ forms
an orthonormal basis of the vector space
$\ds {\cal E}(x,\xi ):=\sum_{\lambda\in\sigma_1(x,\xi )}\sum_{k\geq
1}\Ker (p_0(x,\xi
)-\lambda )^k$. (Here $C_b^\infty$ stands for the space of functions
that are uniformly bounded
together with all their derivatives.)
\item[(H2)] The space ${\cal H}_2$ can be split into ${\cal H}_2 = {\cal
E}(x,\xi )\oplus {\cal
F}(x,\xi )$ where ${\cal F}(x,\xi )$ is stable under $p_0(x,\xi )$, in
the sense that $p_0(x,\xi
)$ maps ${\cal F}(x,\xi )\cap {\cal H}_1$ into ${\cal F}(x,\xi )$.
Moreover the two
(nonorthogonal) projections $\Pi_{{\cal E}/{\cal F}}$ and $\Pi_{{\cal
F}/{\cal E}}$
associated with the decomposition ${\cal H}_2 = {\cal E}(x,\xi ) \oplus
{\cal F}(x,\xi )$ are
uniformly bounded and depend continuously on $(x,\xi)$ in $\R^{2n}$.
\end{itemize}
In particular $\sigma_1(x,\xi )$ is included in the discrete spectrum of
$p_0(x,\xi )$ and
consists in the eigenvalues of the
$m\times m$ complex matrix
\begin{equation}
\label{defM}
M(x,\xi ) =\left( \la p_0(x,\xi )\varphi_k(x,\xi ),\varphi_j(x,\xi
)\ra\right)_{1\leq j,k\leq m},
\end{equation}
where $\la\cdot ,\cdot\ra$ stands for the scalar product in ${\cal
H}_2$.
Setting
$$
\sigma_{\cal F}:=\bigcup_{(x,\xi )\in\R^{2n}}\sigma \left( p_0(x,\xi
)\left\vert_{{\cal F}(x,\xi
)\cap {\cal H}_1}\right.\right),
$$
our main result is the following:
\begin{theorem} Assume (\ref{S1}) and (H1)--(H2). Then for any
$z\in\C\backslash
\sigma_{\cal F}$ there exists a
$m\times m$ matrix of $h$-pseudodifferential operators $A_z =\left(
A_z^{j,k}\right)_{1\leq j,k\leq m}$, bounded on $L^2(\R^n)^{\oplus m}$,
with
principal symbol
$M(x,\xi)$, and such that the following equivalence holds:
$$
z\in\sigma (P_h) \;\Longleftrightarrow \; z\in\sigma (A_z ).
$$
Moreover, $A_z$ depends analytically on $z$ in the interior of
$\C\backslash
\sigma_{\cal F}$.
\end{theorem}
{\bf Example}\hskip 0.1cm For $x\in\R^n$, let $Q(x)$ be a (possibly
unbounded)
nonnegative self-adjoint operator on some Hilbert space ${\cal H}_2$
with domain ${\cal H}_1$,
such that $\sigma (Q(x))=\{\lambda_1 (x) ,\dots ,\lambda_m(x)\}\cup
\Sigma (x)$, where
$\lambda_1(x),
\dots ,\lambda_m(x)$ depend continuously on $x$, remain uniformly
separated outside some
compact subset of $\R^n$, and
$\inf
\Sigma (x)\geq\max\{\lambda_1 (x) ,\dots ,\lambda_m(x)\}+\delta$ for
some $\delta >0$ and for all
$x\in\R^n$. Assume also that
$Q(x)$ depends smoothly on
$x$ and is uniformly bounded together with all its derivatives as an
operator from ${\cal H}_1$ to
${\cal H}_2$. Then our result can be applied with
$$
P_h=\left(-h^2\Delta_x + Q(x) +1\right)^{-1}
$$
and
$$
\sigma_1(x,\xi ) =\left\{ \left(\xi^2 + \lambda_j(x)
+1\right)^{-1},\; 1\leq j\leq m\right\}.
$$
Indeed, using the method of \cite{DiSj}, Section 8, we see that both
(\ref{quant}) and (\ref{S1})
are satisfied, and $p_0(x,\xi )=\left(\xi^2 + Q(x) +1\right)^{-1}$.
Moreover, the constructions
made in \cite{KMSW} show the existence of an orthonormal family $\left(
\varphi_1(x),\dots
,\varphi_m(x)\right)$ in
${\cal H}_2$, which depends smoothly on $x\in\R^n$, has all its
derivatives uniformly
bounded in ${\cal H}_2$, and generates $\ds\bigoplus_{j=1}^m\Ker
(Q(x)-\lambda_j(x) )$. Since,
by the spectral mapping theorem,
this latter space is also equal to
$$
\bigoplus_{j=1}^m\Ker \left( \left(\xi^2
+Q(x)+1\right)^{-1}-\left(\xi^2 +\lambda_j(x)+1\right)^{-1} \right),
$$
we see that (H1) is
satisfied, too. Finally, Condition (H2) is automatically satisfied by
taking ${\cal F}(x,\xi )$
as the orthogonal space of ${\cal E}(x,\xi)$, since
$p_0(x,\xi )$ is self-adjoint on ${\cal H}_2$. In this case
$\sigma_{\cal F}= (0,
(1+\lambda_+)^{-1}]$ with
$$\lambda_+ : = \inf_{\R^n}\left( \sigma (Q(x))\backslash
\{\lambda_1(x), \dots ,\lambda_m(x)\}\right),$$
and our result permits to recover the reduction
method used, e.g., in
\cite{Ma2} (see also \cite{KMSW} for the Coulombic case), for studying
the spectrum of the
molecular Schr\"odinger operator
$H:=-h^2\Delta_x + Q(x)$ (typically, $Q(x)=-\Delta_y +V(x,y)$ acts on
$L^2(\R^p_y)$ where $y$
stands for the electronic position variables).
\vskip 0.5cm\noindent
{\bf Remark}\hskip 0.1cm As one can see from the proof, there is no real
problem to generalize
this theorem to unbounded symbols that, instead of (\ref{S1}), satisfy,
e.g., estimates of the
type
$$
\left\|\partial^\alpha p_0(x,\xi )\right\|_{{\cal
H}_{1,2}}+\left\|\partial^\alpha r_h(x,\xi
)\right\|_{{\cal H}_{1,2}}={\cal O}(\la \xi\ra^k)
$$
for some $k\geq 0$. However, as we have illustrated in the previous
example, it is often possible,
in the
applications,
to transform the problem in order to deal with bounded symbols only.
\section{Proof}
The idea of the proof is to reduce the problem to the inversion of some
Grushin problem,
and to use the semiclassical symbolic calculus in order to construct the
inverse.
We denote by $B$ the operator $L^2(\R^n)^{\oplus m}\rightarrow
L^2(\R^n;{\cal H}_2)$ defined by
$$
B(u_1\oplus\dots\oplus u_m) = \sum_{j=1}^m \frac1{(2\pi h)^n}\int
e^{i(x-y)\xi
/h}\varphi_j \left(\frac{x+y}2,\xi \right) u_j(y)dy\d\xi,
$$
and we denote by $B^*$ its adjoint $L^2(\R^n;{\cal H}_2)\rightarrow
L^2(\R^n)^{\oplus m}$, given
by
$$
Bu(x) = \bigoplus_{j=1}^m\frac1{(2\pi h)^n}\int e^{i(x-y)\xi
/h}\la u(y), \varphi_j \left(\frac{x+y}2,\xi \right)\ra_{{\cal
H}_2}dy\d\xi.
$$
Then for $z\in\C$ we consider the following matrix operator (the
so-called Grushin operator):
$$
{\cal P}(z):=
\left(
\begin{array}{cc}
P_h -z & B\\
B^* & 0
\end{array}\right)$$
that maps $L^2(\R^n ;{\cal H}_1)\oplus L^2(\R^n)^{\oplus m}$ into
$L^2(\R^n ;{\cal H}_2)\oplus L^2(\R^n)^{\oplus m}$. In particular, we
see that ${\cal P}(z)$ can
be seen as an $h$-pseudodifferential operator with operator-valued
principal symbol ${\cal
P}_z(x,\xi )$ given by
\begin{equation}
\label{symbgrus}
{\cal P}_z(x,\xi )=
\left(
\begin{array}{cc}
p_0(x,\xi ) -z & b(x,\xi )\\
b^*(x,\xi ) & 0
\end{array}\right)\; :\; {\cal H}_1\oplus\C^n\rightarrow {\cal
H}_2\oplus\C^n
\end{equation}
where $b(x,\xi )(\alpha_1,\dots ,\alpha_m)=\alpha_1\varphi_1
(x,\xi)+\dots+\alpha_m \varphi_m
(x,\xi)$, ($\alpha_1,\dots,\alpha_m\in\C$), and
$b^*(x,\xi )f = \left(\la f,\varphi_1(x,\xi )\ra_{{\cal H}_2} ,\dots \la
v,\varphi_m(x,\xi
)\ra_{{\cal H}_2}\right)$, ($f\in {\cal H}_1$). In order to show that
${\cal P}(z)$ is
invertible, we first prove the following lemma:
\begin{lemma}
\label{lemma1}
Denote by $\Pi_{\cal E}^0$ the orthogonal projection onto ${\cal
E}(x,\xi )$. Then
for all $z\in\C\backslash\sigma_{\cal F}$ and for all
$(x,\xi )\in\R^{2n}$, the operator
${\cal P}_z(x,\xi )$ defined in (\ref{symbgrus}) is invertible and its
inverse is given by
$$
{\cal P}_z(x,\xi )^{-1} =:{\cal Q}_z(x,\xi )=
\left(
\begin{array}{cc}
Q_z(x,\xi ) & Q_z^+(x,\xi )\\
Q_z^-(x,\xi ) & Q_z^\pm (x,\xi )
\end{array}\right)
$$
where, for $g\in {\cal H}_2$ and $\beta =(\beta_1,\dots
,\beta_m)\in\C^n$,
\begin{eqnarray*}
Q_z(x,\xi )g &:=& \left( 1-\Pi^0_{\cal E}\right)( p_0'(x,\xi
)-z)^{-1}\Pi_{{\cal F}/{\cal
E}}g , \\
Q_z^+(x,\xi )\beta &:=& \sum_{j=1}^m \beta_j\varphi_j,\\
Q_z^-(x,\xi )g &:=&\left( \la \left( 1 +(p_0(x,\xi )-z)\Pi^0_{\cal E}(
p_0'(x,\xi
)-z)^{-1}\right)\Pi_{{\cal E}/{\cal F}}g,\varphi_j\ra\right)_{1\leq
j\leq m}\\
Q_z^\pm
(x,\xi )\beta &:=& \left( z-M(x,\xi )\right)\beta .
\end{eqnarray*}
Here we have denoted by $( p_0'(x,\xi )-z)^{-1}$ the inverse of $\left(
p_0(x,\xi ) -
z\right)\left|_{{\cal F}(x,\xi )}\right.$.
\end{lemma}
\proof For $g\in {\cal H}_2$ and $\beta =(\beta_1,\dots
,\beta_m)\in\C^n$ we have to solve the
problem
\begin{equation}
\label{grus1}
{\cal P}_z(x,\xi )(f\oplus \alpha )= g\oplus\beta
\end{equation}
where the unknown $f\oplus \alpha = f\oplus (\alpha_1,\dots ,\alpha_m )$
is in ${\cal
H}_1\oplus\C^n$. We can rewrite (\ref{grus1}) as
$$
\left\{\begin{array}{l}
\left( p_0(x,\xi )-z\right) f + \sum_{j=1}^m\alpha_j\varphi_j (x,\xi) =
g\\
\la f,\varphi_j(x,\xi )\ra = \beta_j \quad (j=1,\dots ,m),
\end{array}
\right.
$$
and, writing $f=f_{\cal E}+f_{\cal F}$ with $f_{\cal E}\in {\cal
E}(x,\xi )$ and $f_{\cal F}\in
{\cal F}(x,\xi )$, we obtain (since $(\varphi_1(x,\xi
),\dots,\varphi_m(x,\xi ))$ is an
orthonormal basis of ${\cal E}(x,\xi )$),
\begin{equation}
\label{grus2}
\left\{\begin{array}{l}
f_{\cal E}=\sum_{j=1}^m \left( \beta_j - \la f_{\cal F},\varphi_j\ra
\right)\varphi_j\\
\left( p_0-z\right) f_{\cal F} + (p_0 - z)f_{\cal
E}+\sum_{j=1}^m\alpha_j\varphi_j = g.
\end{array}
\right.
\end{equation}
(Here we have omitted the dependence on $(x,\xi )$ to simplify the
notation.) Since
both spaces ${\cal E}(x,\xi )$ and ${\cal F}(x,\xi )$ are stable under
$p_0(x,\xi )$ we see that
(\ref{grus2}) is equivalent to
\begin{equation}
\label{grus3}
\left\{\begin{array}{l}
f_{\cal E}=\sum_{j=1}^m \left( \beta_j - \la f_{\cal F},\varphi_j\ra
\right)\varphi_j\\
\left( p_0-z\right) f_{\cal F} = \Pi_{{\cal F}/{\cal E}}g\\
\left( p_0-z\right) f_{\cal E} +\sum_{j=1}^m\alpha_j\varphi_j =
\Pi_{{\cal E}/{\cal F}}g,
\end{array}
\right.
\end{equation}
and thus, since by assumption $\displaystyle (p_0 - z)\left|_{{\cal
F}(x,\xi )}\right.$ is
invertible for all $(x,\xi )\in\R^n$,
\begin{equation}
\label{grus3}
\left\{\begin{array}{l}
f_{\cal F} = \left( p_0'-z\right)^{-1}\Pi_{{\cal F}/{\cal E}}g\\
f_{\cal E}=\sum_{j=1}^m \left( \beta_j - \la f_{\cal F},\varphi_j\ra
\right)\varphi_j\\
\alpha_j = \la \Pi_{{\cal E}/{\cal F}}g -\left( p_0-z\right) f_{\cal
E},\varphi_j\ra\quad
(j=1\dots,m).
\end{array}
\right.
\end{equation}
In particular $f_{\cal F}$, $f_{\cal E}$, $\alpha_1,\dots,\alpha_m$ can
all be determined in
function of $g$ and $\beta$ and, using (\ref{defM}) and the fact that
$\Pi^0_{\cal E}v =
\sum_{j=1}^m\la v,\varphi_j\ra\varphi_j$, we obtain the formulae given
in the
lemma.\hfill$\diamond$
\vskip 0.3cm
The next step is important, since it will permit us to construct the
inverse of ${\cal P}(z)$ by
means of the symbolic calculus of
$h$-pseudodifferential operators (see \cite{Ma1} and the appendix of
\cite{GMS}).
\begin{lemma}
\label{lemma2}
For any $z\in\C\backslash \sigma_{\cal F}$, the application
$$
\begin{array}{cccc}
{\cal Q}_z\; :\; &\R^{2n} &\rightarrow& {\cal L}\left( {\cal H}_2\oplus
\C^n ; {\cal H}_1\oplus
\C^n\right)\\
{} & (x,\xi ) &\mapsto & Q_z(x,\xi )
\end{array}
$$
is $C^\infty$ and uniformly bounded together with all its derivatives on
$\R^{2n}$. Moreover, it
depends analytically on $z$ in the interior of $\C\backslash
\sigma_{\cal F}$.
\end{lemma}
\proof Thanks to Assumptions (H1) and (H2), it is straighforward to
verify that ${\cal
Q}_z(x,\xi)$ is uniformly bounded on $\R^{2n}$ as an operator ${\cal
H}_2\oplus \C^n \rightarrow
{\cal H}_1\oplus
\C^n$. Since, moreover, ${\cal P}_z(x,\xi)$ depends smoothly on
$(x,\xi)$ and is uniformly bounded
together with all its derivatives, the continuity and the
differentialbility follows by writing,
for all $(x,\xi ), (x'\xi')\in\R^{2n}$,
$$
{\cal Q}_z(x,\xi ) -{\cal Q}_z(x',\xi') = {\cal Q}_z(x,\xi )\left ({\cal
P}_z(x',\xi' )-{\cal
P}_z(x,\xi)\right) {\cal Q}_z(x',\xi').
$$
This permits to obtain
$$
\left( \nabla_{x,\xi }{\cal Q}_z\right) (x,\xi ) = -{\cal Q}_z(x,\xi
)\left(\nabla_{x,\xi}{\cal
P}_z\right) (x,\xi) {\cal Q}_z(x,\xi )
$$
and the result follows by differentialting iteratively this equality.
The analyticity of ${\cal
Q}_z(x,\xi )$ with respect to $z$ is also a direct consequence of the
analytic dependence of
${\cal P}_z(x,\xi )$ on $z$.\hfill$\diamond$
\vskip 0.3cm
Thanks to Lemma \ref{lemma2}, we can consider the Weyl
quantization
${\cal Q}(z)={\rm Op}_h^W({\cal Q}_z)$ of
${\cal Q}_z (x,\xi )$, defined on $L^2(\R^n;{\cal H}_2\oplus \C^n)\simeq
L^2(\R^n;{\cal
H}_2)\oplus L^2(\R^n)^{\oplus m}$ by the oscillatory integral
$$
{\cal Q}(z)\psi (x)=\frac1{(2\pi h)^n}\int e^{i(x-y)\xi
/h}{\cal Q}_z \left(\frac{x+y}2,\xi \right) \psi (y)dy\d\xi.
$$
Here $\psi = u\oplus u_1\oplus\dots \oplus u_m\in L^2(\R^n;{\cal
H}_2)\oplus
L^2(\R^n)^{\oplus m}$. By the Calder\'on--Vaillancourt theorem, ${\cal
Q}(z)$ is a bounded
operator from $L^2(\R^n;{\cal H}_2)\oplus
L^2(\R^n)^{\oplus m}$ to $L^2(\R^n;{\cal H}_1)\oplus
L^2(\R^n)^{\oplus m}$, and as a consequence we can perform the two
compositions ${\cal Q}(z){\cal
P}(z)$ and ${\cal P}(z){\cal
Q}(z)$. Moreover, the symbolic calculus permits us to estimate it as
follows:
$$
{\cal Q}(z){\cal
P}(z) ={\rm Op}_h^W({\cal Q}_z{\cal P}_z) + hR_1=I+hR_1
$$
with
$$
\| R_1\|_{{\cal L}(L^2(\R^n;{\cal H}_1)\oplus
L^2(\R^n)^{\oplus m})}={\cal O}(1)
$$
uniformly with respect to $h$. Similarly, we also have
$$
{\cal P}(z){\cal
Q}(z) ={\rm Op}_h^W({\cal P}_z{\cal Q}_z) + hR_2=I+hR_2
$$
with
$$
\| R_2\|_{{\cal L}(L^2(\R^n;{\cal H}_2)\oplus
L^2(\R^n)^{\oplus m})}={\cal O}(1)
$$
uniformly with respect to $h$. As a consequence, for $h$ small enough
${\cal P}(z)$ is invertible
and its inverse is given by the convergent Neuman series
\begin{equation}
\label{inverse}
{\cal P}(z)^{-1} = \left( \sum_{k=1}^{+\infty}h^kR_1^k\right)\circ {\cal
Q}(z) = {\cal
Q}(z)\circ\left(
\sum_{k=1}^{+\infty}h^kR_2^k\right).
\end{equation}
Therefore, we have proved the first part of the following proposition:
\begin{proposition}
\label{prop1}
The operator ${\cal P}(z)\; ;\; L^2(\R^n;{\cal H}_1)\oplus
L^2(\R^n)^{\oplus m}\rightarrow L^2(\R^n;{\cal H}_2)\oplus
L^2(\R^n)^{\oplus m}$ is invertible and its inverse can be written as
$$
{\cal P}(z)^{-1} =\left(
\begin{array}{cc}
Q(z) & Q^+(z)\\
Q^-(z) & Q^\pm (z)
\end{array}
\right)
$$
where $Q(z)$, $Q^+(z)$, $Q^-(z)$, and $Q^\pm (z)$ are
$h$-pseudodifferential operators with
principal symbols $Q_z(x,\xi )$, $Q^+_z(x,\xi )$, $Q^-_z(x,\xi )$, and
$Q^\pm_z(x,\xi )$
respectively. Moreover, we have the following equivalence:
\begin{equation}
\label{equiv}
z\in \sigma (P_h)\Longleftrightarrow 0\in \sigma \left( Q^\pm
(z)\right).
\end{equation}
\end{proposition}
\proof In view of (\ref{inverse}) it remains only to prove that $Q(z)$,
$Q^+(z)$, $Q^-(z)$, and
$Q^\pm (z)$ are $h$-pseudodifferential operators and that the
equivalence (\ref{equiv}) holds.
The first assertion comes from the fact that ${\cal P}(z)^{-1}$ is the
inverse of an elliptic
$h$-pseudodifferential operator, and it can be proved in a standard way
by using the Beals
characterization theorem, see \cite{DiSj}, Proposition 8.3. The second
assertion comes from the
two following series of algebraic identities:
\begin{eqnarray}
(P_h - z)u=v&\Leftrightarrow &{\cal P}(z)(u\oplus 0) = v\oplus
B^*u\Leftrightarrow
{\cal Q}(z)(v\oplus B^*u) = u\oplus 0\nonumber\\
\label{alg1}
&\Leftrightarrow &
\left\{\begin{array}{l}
Q(z)v + Q^+(z)B^*u = u,\\
Q^-(z)v +Q^\pm (z)B^*u = 0,
\end{array}
\right.
\end{eqnarray}
and
\begin{eqnarray}
Q^\pm (z)\alpha =\beta &\Leftrightarrow &{\cal Q}(z)(0\oplus \alpha) =
Q^+(z)\alpha \oplus
\beta\Leftrightarrow {\cal P}(z)(Q^+(z)\alpha \oplus
\beta) = 0\oplus \alpha\nonumber\\
\label{alg2}
&\Leftrightarrow &
\left\{\begin{array}{l}
(P_h-z)Q^+(z)\alpha + B\beta = 0,\\
B^*Q^+(z)\alpha = \alpha.
\end{array}
\right.
\end{eqnarray}
Since we also have $B^*Q^+(z) =1$ (just write explicitly that ${\cal
P}(z){\cal
Q}(z)=I$), we see that if $z\notin \sigma (P_h)$, then (\ref{alg2})
permits to
obtain the following equivalence:
$$
Q^\pm (z)\alpha =\beta\;\Leftrightarrow \;\alpha =
-B^*(P_h-z)^{-1}B\beta .
$$
In particular $0\notin \sigma (Q^\pm (z))$ and
\begin{equation}
Q^\pm (z)^{-1} = -B^*(P_h-z)^{-1}B.
\end{equation}
Conversely, if $0\notin \sigma (Q^\pm (z))$, then (\ref{alg1}) gives the
following equivalence:
$$
(P_h-z)u=v\;\Leftrightarrow \;\left\{\begin{array}{l}
B^*u = -Q^\pm (z)^{-1}Q^-(z)v,\\
u= Q(z)v -Q^+(z)Q^\pm (z)^{-1}Q^-(z)v.
\end{array}
\right.
$$
Moreover, the fact that $B^*Q(z) =0$ and $B^*Q^+(z) =1$ shows that,
actually, the first equation
of the latter system is implied by the second one. As a consequence, in
this case we have
$$
(P_h-z)u=v\;\Leftrightarrow \;
u= Q(z)v -Q^+(z)Q^\pm (z)^{-1}Q^-(z)v,
$$
and thus $z\notin \sigma (P_h)$ and
\begin{equation}
(P_h-z)^{-1}= Q(z) -Q^+(z)Q^\pm (z)^{-1}Q^-(z).
\end{equation}
This completes the proof of the proposition.\hfill$\diamond$
\vskip 0.3cm
To complete the proof of the theorem, it just remains to observe that,
by construction, $Q^\pm
(s)$ is a $h$-pseudodifferential operator on $L^2(\R^n)^{\oplus m}$,
that is, a $m\times m$ matrix
of pseudodifferential operators on $L^2(\R^n)$. Therefore the theorem
follows from Proposition
\ref{prop1} and Lemma \ref{lemma1} by setting $A_z=Q^\pm (z) -
z$.\hfill$\diamond$
\begin{thebibliography}{99}
\bibitem[DiSj]{DiSj} Dimassi, M., Sj\"{o}strand, J.: Spectral
Asymptotics in the Semiclassical
Limit, London Math. Soc. Lecture Notes Series 268, Cambridge University
Press
(1999).
\bibitem[GMS]{GMS} G\'erard, C., Martinez, A., Sj\"{o}strand, J.: A
mathematical approach
to the effective hamiltonian in perturbed periodic problems, (joint
work with C.G\'erard and
J.Sjöstrand), Comm. Math. Physics Vol.142, n°2 (1991).
\bibitem[Ha] {Ha} {\sc G. Hagedorn,} \
{ \it High Order Corrections to the time-independent Born-Oppenheimer
Approximation},
Ann. Inst. H. Poincar\'e 47 (1987) 1-16
\bibitem[HaJo] {HaJo} {\sc G. Hagedorn, A. Joye}, \
{ \it A Time--Dependent Born--Oppenheimer Approximation with
Exponentially Small Error Estimates},
Preprint mp${}_{-}$arc. 00-209 (2000)
\bibitem[Ho]{Ho} H\"{o}rmander, L.: The Analysis of Linear Partial
Differential Operators, Vol. I to IV. Springer Verlag (1985).
\bibitem[KMSW] {KMSW} Klein, M., Martinez, A., Seiler, R., Wang, X.: On
the Born-Oppenheimer Expansion
for Polyatomic Molecules, Comm. Math. Phys. 143, p. 607-639 (1992).
\bibitem[Ma1] {Ma1} Martinez, A.:
An Introduction to Semiclassical and Microlocal Analysis,
Springer-Verlag New--York, UTX Series, 2002
\bibitem[Ma2] {Ma2} Martinez, A.: D\'eveloppements asymptotiques et
effet tunnel
dans l'approximation de
Born-Oppenheimer, Ann.Inst.H.Poincar\'e, Vol.49, n°3, (1989), p.
239-257.
\bibitem[MaSo] {MaSo} Martinez, A., Sordoni, V.: A general reduction
scheme for the
time-dependent Born-Oppenheimer approximation, C.R.Acad.Sci. Paris, Ser.
I 334 (2002) 185-188
\bibitem[Ne] {Ne} {\sc G. Nenciu}, \
{ \it Linear Adiabatic theory: I.Exponential Estimates},
Comm. Math. Phys., 152 (1993).
\bibitem[NeSo] {NeSo} {\sc G.Nenciu, V.Sordoni}, {\it Semiclassical
Limit for Multistate Klein-Gordon
Systems: Almost
Invariant Subspaces and Scattering Theory}, Preprint Bologna (2001)
\bibitem[PST] {PST} Panati, G., Spohn, H., Teufel, S.: Space-Adiabatic
Perturbation Theory,
Preprint Technische Universit\"at M\"unchen (2002)
\bibitem[So] {So} {\sc V.Sordoni}, {\it Scattering for Multistate
Schr\"odinger Systems
in the Semiclassical
Limit}, Preprint Bologna (2001) and to appear in Comm. Part. Diff. Eq.
(2002)
\bibitem[SpTe] {SpTe} Spohn, S., Teufel, S.: Adiabatic Decoupling and
Time -Dependent
Born-Oppenheimer Theory, Comm. Math. Phys. 224 (1) (2001) 113-132
\end{thebibliography}
\end{document}