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\begin{document}
\begin{titlepage}
\thispagestyle{empty}
\vspace{6cm}
\title {A Simple Model for Predissociation\footnotemark[1]}
\date{}
\author {P.\,Duclos\footnotemark[2]
\and B.\,Meller\footnotemark[2]}
\maketitle
\vskip 1cm
\abstract {We analyse a very simple class of one dimensional two by two matrix
Schr\"odinger operators. Their diagonal part has embedded eigenvalues in the
continuous spectrum which become resonances when the off-diagonal part is turned
on. Our analysis is semiclassical and contains a regular perturbative calculus of
these resonances, asymptotics of the Fermi rule contribution to the width of these
as well as lower bounds on the corresponding life time.}
\vskip 4cm\nid
November 1993\\
CPT-93/P.2968
\vspace{4cm}
\footnoterule\nid
*\
To appear in the proceedings of the conference ``Mathematical results in Quantum
\ Mechanics'', Blossin, Germany, Mai 17-21 1993, M. Demuth, P. Exner, H. Neidhardt,
\ V. Zagrebnov eds., Birkh\"auser. \vsl\nid
\dag\ and PhyMat, Universit\' e de Toulon et du Var, BP 132, F-83957 La
Garde Cedex
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
We start directly exposing the model; the discussion about the content of this
paper is split in several parts which are put at the end of their relevent
sections.
\subsection{The Model} Let
$$
H:= H^d + W,\quad
H^d := \left(\begin{array}{ll}H^1 & 0\\0 & H^2\end{array}\right),\quad
W:= \left(\begin{array}{ll}0 & V^{1,2}\\V^{2,1} & 0\end{array}\right)
$$
be a matrix Schr\"odinger operator acting on
$L^2 (\R) \oplus L^2 (\R)=:{\cal H}$, where
\begin{eqnarray*}
H^k & := & D^2 + V^k, \quad k=1,2,\quad D:=\frac{\hbar}{i}\frac{d}{dx},
\\
V^2 (x) & := & x^2,\quad V^{1,2}:= \hbar (tD+Dt) = -V^{2,1}.
\end{eqnarray*}
Dilation and translation analyticity of the potentials play here an important
role; we assume
\vsl
\hypothesis{1}{ $V^1$ and $it$ are bounded analytic multiplication
operators in $\Sigma_{\beta_0,\eta_0 }:=\{ z\in \C,
|\argu z | < \beta_0\ {\rm or}\ |\im z |<\eta_0\},\ \beta_0, \eta_0$ being both
strictly positive; $V^1$ and $it$ are real on $\R$.}
\vsl
\nid The images of all operators under the scaling $x\rightarrow e^{\theta} x$
will be denoted when necessary by a subcript $\theta$. With these assumptions we
have the following
\vsl
\nid {\bf theorem 1}. $ H_{\theta}, H^{i}_{\theta}$, i=1,2 are selfadjoint
analytic families of type A for all $\theta$ such that $| \im\theta
|<\hat{\beta}_0 :=\min\{ \beta_0,{\pi\over 4}\}$ with domains
\begin{eqnarray*}
{\cal D} (H_\theta)& = &{\cal D}( H^1_\theta)\oplus{\cal D}( H^2_\theta),\\
{\cal D}(H^1_\theta ) &=&{\cal H}^2 (\R),\ \;
{\cal D}( H^2_\theta)={\cal H}^2\cap\hat{\cal H}^2 (\R).
\end{eqnarray*}
\nid The definition of ``type A family'' may be found in \cite[Ch.VII \S 2.1]{Ka}.
${\cal H}^n (\R)$ denotes the usual Sobolev space and $\hat{\cal H}^n $ its
Fourier image. The proof of this theorem is rather standard. For $H^1_\theta$
it is obvious, since $V^1_\theta$ is bounded analytic; for $H^2_\theta$ a
detailed proof can be found in \cite{BCD2}. $H_\theta$ can be treated
perturbatively as in \cite{DES}; in the form sense on ${\cal D} (H^d_\theta)$
one has $$
\vert e^\theta W_\theta\vert^2 \leq \hbar ^2 \{ 8 \Vert t_{\theta}\Vert ^2 D^2
+2 \hbar^2\Vert t'_{\theta}\Vert ^2 \}
\left(\begin{array}{ll}1&0\\0&1\end{array}\right)
$$
which shows that $W_\theta$ is $H^d$-bounded with relative bound zero.
In order to have a reasonable spectrum for $H^1$ we assume
\vsl
\hypothesis{2}{%
$\exists \varepsilon >0,\;\exists v_\infty < 0, \;
\forall 0<\vert\im\theta\vert < \beta_0,\quad
V^1_\theta = v_\infty + \OO(x^{-1-\varepsilon})$, as
$\vert x\vert$ tends to infinity.}
\vsl\nid
This implies in particular that the essential spectrum of $H^1_\theta$
is simply: \\$ \sigma_{ess} (H^1_\theta) = v_\infty + e^{-2\theta}\R_+ $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion and Further Hypothesis}
$H^2$ has only discrete spectrum, $\sigma (H^2) =(2\N +1)\hbar $.
When $\theta$ equals zero, these eigenvalues are embedded in the continuous
spectrum of $H^1$. The effect of the perturbation $W$ is usually to couple
these bound states to the scattering states of $H^1$. If the quantum system
is initially prepared in a state $0\oplus \varphi^2$, $\varphi^2$ being a
bound state of $H^2$, it will eventually turn into a scattering state. The
mechanism behind this effect is very similar to the one which causes the
existence of shape resonances, viz. tunneling through the potential barrier
(see e.g. \cite{CDKS,HeSj}).
If the two potentials $V^1$ and $V^2$ cross one can reduce the
problem for the lowest energies to a shape resonance situation $\cite{K}$;
the relevent effective potential is $\min\{V^1, V^2\}$ which possesses a well,
separated from the escaping regions by a potential barrier.
Here we want to address explicitly the case with no crossing of $V^1$
and $V^2$: we assume
\vsl
\hypothesis{3}{$$\max V^1 < \min V^2 =0.$$}
\vsl
\nid Now, at first glance, one would say there is no barrier. But classically one
can see that for the energy $e_0 =(2n +1)\hbar$ (the $n^{th}$
quantum state of $H^2$)
the allowed momenta for $H^1$ and $H^2$ are separated by a gap of size
$\sqrt{-\max V^1}$ for $\hbar$ small enough. This gap indicates a
classically forbidden region for the hamiltonian $H^d$ in the momentum space
rather than in the configuration space. So we can speak of a {\it dynamical
barrier} being present and of {\it dynamical tunneling} as the reason for the
escape of the bound state (see \cite{AD} for the same discussion in the case of
the reflection over a potential barrier). However assumption (H3) is much
stronger than necessary for this phenomenon to take place. Wilkinson \cite{W}
and Martinez \cite{Ma2} have remarked that it is sufficient to require that the
energy shells do not cross:
$\{H_{cl}^1(q,p)=H_{cl}^2(q,p)=E\}\footnote{$ H_{cl}^i$ denotes the classical
Hamiltonian funtion associated to $H^i$.}=\emptyset$ to get such an effect which
they called respectively {\it tunneling in phase space } and {\it microlocal
tunneling}.
The dynamical tunneling manifests itself in the so-called resonances of the
quantum system. According to the standard machinery (\cite{AgC,RS4}) these
resonances are recognized as complex eigenvalues of $H_\theta$ which are
the perturbed eigenvalues of $H^d _\theta$. Notice that these eigenvalues do not
depend on $\theta$. Since the essential spectrum of $H^1_\theta$ has turned
down in the complex energy plane, the eigenvalues of $H^d_\theta$ to be perturbed
are now isolated provided $H^1_\theta$ does not have eigenvalues too close. This
last requirement is achieved by imposing a nontrapping condition on $V^1$:
\vsl
\hypothesis{4}{ $\exists \,S<0, \quad \forall\, 0\leq \beta <\beta_0 \quad
\im e^{i2\beta}V^1_{i\beta}\leq \beta S$}
\vsl
\nid The perturbation $W_\theta$ being $H^d _\theta$-bounded with relative
bound zero our problem falls into the category of regular perturbation theory.
Section~2 is devoted to this perturbation theory. Once the resonances are
shown to exist we shall give for a restricted model the asymptotics of the
Fermi rule contribution to their width in section~3 and finally estimate this
width in section 4.
Such a model with $V^1=-1$ and $V^{1,2}=V^{2,1}=\hbar ^2$ has been proposed by
J. Asch \cite{A} as a simple model to understand the predissociation phenomenon
in diatomic molecules. $V^1$ and $V^2$ play the role of the electronic curves,
$\hbar ^2$ the inverse of the nucleus mass. We have chosen, here, more realistic
coupling terms. Reducing the complete molecular hamiltonian to a two by two matrix
of this type is the purpose of the Born-Oppenheimer approximation, see e.g.
\cite{CDS} and \cite{Ma1}. In these articles the basic algebraic tool is a
method bearing numerous names: Brilloin-Wigner, Feshbach, Grushin, Schur... not
to forget the Livsic Matrix \cite{Ho}. However by ``putting'' into $H^1$ all the
electronic curves except the second one, it is conceivable to obtain the same
result not using the energy dependent perturbation theory, see e.g. \cite{DES}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[pertu]{Perturbation Expansion of \\the Resonances}
We first prove that under (H1,4) $H^{1}_{\theta}$ has no spectrum
close to a given eigenvalue $e_0$ of $H^2$. It will be sufficient to have
this property in the closed neighbourhood of $e_0$ bounded by the contour
$\Gamma := \{z\in\C, |z-e_0 |=\hbar\}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nontrapping Estimates}
\nid {\bf Lemma 2}. Under hypothesis (H1,4) one has:
$$
\forall\, 0<\beta<\beta_0,\ \exists \,c^{1}_{\beta}:={-2\over \beta S},\
\forall\, 0<\hbar<{1\over c^{1}_{\beta}},\
\forall\, z\in \Gamma,\ \Vert R^{1}_{i\beta}(z)\Vert\leq c^{1}_{\beta}.
$$
\vsl
\nid {\em Sketch of the proof\/}: Such type of result is now rather standard
(cf. \cite{BCD1}). The condition on $\hbar$ (stronger than necessary),
insures that
$$
\Gamma \subset \nu :=\{ z\in\C,\ \im e^{i2\beta}z \geq \beta S \}.
$$
\nid $\nu$ is a set of complex energies that cannot be resonances of $H^1$ due
to (H4). More precisely, since in the form sense on ${\cal D}(H^1)$ one has:
$$
\vert H^{1}_{i\beta}-z\vert \geq
\im e^{i2\beta}(z-H^{1}_{i\beta})=
\im e^{i2\beta}(z-V^{1}_{i\beta})\geq
\im e^{i2\beta}z-\beta S \geq
-\beta {S\over2},
$$
\nid which yields the a priori estimate we need to bound $R^{1}_{i\beta}(z)$.
In the last step we have explictely used that $z$ belongs to $\Gamma$
and the condition on $\hbar$. \QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stability of the Resolvent Set of $H^{d}_{\theta}$}
The previous lemma gives sufficient conditions to insure that $\Gamma$ is
included in $\rho (H^{1}_{\theta})$, the resolvent set of $H^{1}_{\theta}$.
Since it is well known (see e.g. \cite{BCD2}) that the spectrum of
$H^{2}_{\theta}$ is invariant with respect to $\theta$ as long as
$\vert\im\theta\vert<{\pi\over4}$, we conclude that $\Gamma$ is also in $\rho
(H^{d}_{\theta})$ under this extra condition on $\im\theta$. Therefore, by
standard perturbation theory, $\Gamma$ will also be in $\rho (H_{i\beta})$ if
in addition $$
\forall z \in \Gamma,\quad
\Vert R^{1}_{i\beta}(z) V^{1,2}_{i\beta}R^{2}_{i\beta}(z) V^{2,1}_{i\beta}\Vert
<1.
$$
\nid $R^{1}_{i\beta}(z)$ is already estimated by lemma~2. In the next lemma we
shall estimate $V^{1,2}_{i\beta}R^{2}_{i\beta}(z)V^{2,1}_{i\beta}$ and other
quantities needed in the sequel. As in Kato \cite[Ch.II \S 2.1]{Ka}, we use the
notation:
$$
S_{i\beta}^{2 \, (k)} := ({\hat R}^{2}_{i\beta}(e_0))^k ,\,\, \mbox{if} \,\,
k\geq 1 \,\, \mbox{and} \,\, S_{i\beta}^{2 \, (0)}=-P_{i\beta}^{2}
$$
\nid where ${\hat R}^{2}_{i\beta}(e_0)$ is the reduced resolvent of
$H^{2}_{i\beta}$ at $e_0$ and $P_{i\beta}^{2}$ the corresponding spectral
projection.
\vsl
\nid {\bf Lemma 3}. For any $0<\beta<\hat{\beta}_0$ there exists $c^{2}_{\beta}$
such that for any $\hbar>0$,
\begin{eqnarray*}
\forall\, z\in \Gamma,\quad
\Vert V^{1,2}_{i\beta}R^{2}_{i\beta}(z) V^{2,1}_{i\beta}\Vert
& \leq & c^{2}_{\beta}\hbar^2 ,\\
\forall\, k\geq 0,\quad
\Vert V^{1,2}_{i\beta}S_{i\beta}^{2(k)}V^{2,1}_{i\beta}\Vert
&\leq & c^{2}_{\beta}\hbar^{3-k} .
\end{eqnarray*}
\vsl
\nid {\em Sketch of proof\/}:
By the scaling $x\rightarrow \sqrt{\hbar}x $\quad
$V^{1,2}_{i\beta}R^{2}_{i\beta}(z) V^{2,1}_{i\beta}$ is unitarily
equivalent to:
$$
\hbar^{2}
(t_{i\beta}(\hbar^{1\over2}x)\partial_x + c.)
(-e^{-i2\beta}\partial^{2}_{x} + e^{i2\beta}x^2 -\zeta)^{-1}
(\partial_x t_{i\beta}(\hbar^{1\over2}x) + c.)
$$
\nid where $c.$ means `commutated term' and $\zeta$ belongs to the fixed
compact set $\hbar^{-1}\Gamma := \{\zeta \in \C, \vert\zeta -(2n+1)\vert
=1\}$. The first statement follows easily by the continuity in $\zeta$ of the
lhs of the formula above since $t_{i\beta} $ is bounded and $\partial _x$
relatively bounded to $-e^{-i2\beta}\partial^{2}_{x} + e^{i2\beta}x^2$.
For the derivation of the second statement we
use the Cauchy formula and the same scaling trick.
\QED
\vsl\nid
Thus we have obtained the stability of the resolvent set:
\begin{equation}\label{defcBeta}
\left(0<\beta <\hat\beta_0\ {\rm and}\ 0<\hbar<{1\over c_\beta}\right)
\Rightarrow \Gamma\subset\rho(H_{i\beta}),\quad
c_\beta := \max\{c^{1}_\beta , c^{2}_\beta\}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stability of the Spectrum of $H^2$ and Existence of Resonances for $H$}
The preceeding analysis proved that for $\hbar$ small enough $P_{i\beta}$, the
eigenprojection of $H_{i\beta}$ associated to $\Gamma$, is well defined. This
certainly remains true if one replaces $W$ by $\alpha W$ with $0\leq \alpha\leq
1$ thus defining a continuous family of projections interpolating between
$P_{i\beta}\, (\alpha=1)$ and $P_{i\beta}^2 \,(\alpha=0)$.
Consequently
$$
\forall\, 0<\beta <\hat\beta_0 ,\;
\forall\,0<\hbar< {1\over c_\beta},\quad
{\rm dim}\, P_{i\beta} =1.
$$
Standard arguments on resonances (\cite[Ch. XIII.10]{RS4}) insure that the
imaginary part of the eigenvalue associated to $P_{i\beta}$ cannot be positive.
So we have proven the existence of a resonance of $H$ close to each eigenvalue of
$H^2$; however we cannot exclude a vanishing
imaginary part of this resonance.
\vsl
\nid {\bf Remark 1.} $c^{1}_\beta$ depends only on $\beta$ and $S$ whereas
$c^{2}_\beta$ depends on $\beta$ and the quantum number $n$ of the eigenvalue
$e_0$ of $H^2$. So the range of values of $\hbar$ for which the existence of the
resonances is obtained depends on $\beta$, $S$ and $n$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Convergent Expansion of the Resonances}
We denote by $E$ the resonance obtained by perturbation of $e_0=(2n+1)\hbar$.
With the standard formula of regular perturbation theory \cite[Ch.II\S 1]{Ka}
and noticing that $W$ is off-diagonal we get:
\begin{eqnarray}\label{cvse}
E &=& \sum_{m=0}^{\infty}e_m,\\ \nonumber
e_m & = &{1\over2m}\sum_{l=0}^{m-1} \sum_{\sigma}
\tr V^{2,1} (R^1)^{k_1} V^{1,2}(S^2)^{l_1}
\dots V^{2,1}(R^1)^{k_m} V^{1,2}(S^2)^{l_m},\\\nonumber
\sigma &= & \{% l_1+l_2 +\cdots + l_m = l
\mbox{$\sum l_i = l \wedge \sum k_i =2m-1-l$},\ l_i\geq 0,\ k_i\geq 1 \},
\end{eqnarray}
\nid where we have dropped the indices $i\beta$; all the resolvents in the above
formula are evaluated at $z=e_0$. Straightforward combinatorics,
lemmas 2 and 3, the definition (\ref{defcBeta}) of $c_\beta$
and the extra condition $\hbar c_\beta <1$
to simplify the analysis yield
$$
\vert e_m \vert \leq {1\over m c_\beta}{3m-2\choose m-1} (c_\beta \hbar)^{2m
+1}. $$
Using the d'Alembert criterium we arrive at
\vsl
\nid {\bf theorem 4}. Under hypothesis (H1,4) and for $\hbar$ small enough each
eigenvalue of $H^2$ gives rise to a resonance of $H$ of multiplicity one.
Furthermore for all $\beta$ in $(0,\hat\beta_0 )$ let
$c_\beta$ be defined by (\ref{defcBeta}). If
$\hbar$ is in $(0,\hbar_0)$, where
$\hbar_0:={2\over 3\sqrt{3}}{c_\beta}^{-1}$,
then the series (\ref{cvse}) converges to this resonance.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion}
The method of this section follows tightly \cite{DES} with extra niceties due to
the simple form of $H$ and the fact that the perturbation is off diagonal. Also
we have been able to give a critical value of $\hbar$ below which the
convergence of the perturbation series is assured. $c^{1}_\beta $ is easily
estimated in terms of $V^1$ but for $c^{2}_\beta$ we have only an existence
result since we do not yet know
how to estimate the resolvent of the harmonic oscillator
scaled with a complex parameter $\theta$ and for a spectral parameter in the
numerical range. Thus this critical value of $\hbar$ is for the moment
merely theoretical.
We stress that we do not give here expansions of the resonances in
$\hbar$; Martinez \cite{Ma1} has shown that such expansions are
asymptotic, see also \cite{CDS}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fermi-Rule Contribution to the Width
of the Resonance}
\subsection{Asymptotics of $\im e_1$ as $\hbar$ Tends to Zero}
Heuristic arguments \cite[\S 90]{LL} lead to the conclusion that the width
of the resonances of $H$ which we found in section 2 are in fact
exponentially small as $\hbar $ tends to zero. As an indication of this
property we compute below the asymptotics of the imaginary part of
the first order coefficient in the expansion (\ref{cvse}). \\
\vsl\nid
{\bf Definition 1}. Each point $x$ in $\sum_{\beta_0 ,
\eta_0}$ which is a root of $V^1(x)=V^2(x)$ is called a
{\it transition point}.
\vsl\nid
We shall only consider here the restricted model
\vsl
\nid \hypothesis{5}{%
$$V^1 = v_\infty, \quad v_\infty <0.$$}
\vsl\nid
We then conclude immediately that there are two transition points
$$x_\star :=i\sqrt{-v_\infty} \quad\mbox{and}\quad -x_\star.$$
The other important points are the singularities of $V^{1,2}$
or equivalently of $t$. We suppose
\vsl\nid
\hypothesis{6}{%
$\eta_0 > |x_\star|$ which means that $t$ is analytic beyond
the transition points.}
\vsl
\nid
{\bf Theorem 5}.
Under (H1,5,6) and for the resonance associated to $e_0:=(2n+1)
\hbar,\; n\in \N$ one has
\begin{equation}\label{FR1}
\im e_1 = - \frac{2\sqrt{2\pi}}{n!e^{2n+1}}\hbar^2
\left(\frac{4d_\star}{\hbar}\right)^{n+\frac{1}{2}}
\exp (-\frac{2d_\star}{\hbar})
\{|t(x_\star)|^2 + \OO (\hbar)\}
\end{equation}
%\end{theorem}
where
$$
d_\star := \frac{-v_\infty}{2} = \vert\im \int_{0}^{x_\star} \sqrt{-V^2 (y)}
dy\vert .
$$
\vsl
\nid
{\em Proof\/}: Under (H5) $H^1$ obviously has a pair of
generalized eigenvectors at energy $e_0$
$$H^1 \varphi^{1,\nu} =e_0\varphi^{1,\nu},\nu=\pm1,\quad
\varphi^{1,\nu}:=(2\pi\hbar)^{-\frac{1}{2}} e^{\nu i\frac{k}{\hbar}x}
, k:=\sqrt{e_0 -v_\infty}.$$
$e_1$ which is constant with respect to $\theta$ may be computed
for $\theta=0$ using the boundary value of $R^1$ at $e_0 +i0$.
We get \cite[XII.6]{RS4}
$$
\im e_1 = -\frac{\pi}{2k} \sum_{\nu =\pm 1} |(V^{2,1}\varphi^{1,\nu}
,\varphi^{2})|^2,
$$
where
$$\varphi^{2}(x):= \frac{C_n}{\hbar^{ \frac{1}{4} } }
{\cal P}_n(\frac{x}{\sqrt{\hbar}}) e^{-\frac{x^2}{2\hbar}},\quad
C_n:=\pi^{-\frac{1}{4}}(2^n n!)^{-\frac{1}{2}},
$$
${\cal P}_n$ being the $n^{th}$ Hermite polynomial. One derives easily
$$
(V^{2,1}\varphi^{1,\nu},\varphi^{2}) =
-\hbar(f\varphi^{1,\nu},\varphi^{2}),\quad
f:= 2\nu kt +\frac{\hbar}{i}t'.
$$
Since by (H6)
$\R +i\nu k$ is
in $\sum_{\beta_0 ,\eta_0}$, we may take it as the
``contour'' of integration:
$$
(V^{2,1}\varphi^{1,\nu},\varphi^{2}) = -
\frac{C_n}{\sqrt{2\pi} }\hbar^{ \frac{1}{4} }
e^{-\frac{k^2}{2\hbar}}
\int_{\R} f(x +\nu i k){\cal P}_n(\frac{x+\nu ik}{\sqrt{\hbar}})
e^{-\frac{x^2}{2\hbar}}
.$$
It is now easy to compute the asymptotics of the last integral by
first scaling $x$ by $\sqrt{\hbar}$, then expanding
%$$
%f(\nu ik+\sqrt{\hbar}x){\cal P}_n(\nu ik\hbar^{-\frac{1}{2}}+x)=
%(f(\nu ik)+f'(\nu ik)\sqrt{\hbar}x){\cal P}_n(\nu ik\hbar^{-\frac{1}{2}}) +\OO
%(\hbar^{-\frac{n-2}{2}}x^n)
%$$
\begin{eqnarray*}
f(\nu ik\!+\!\sqrt{\hbar}x){\cal P}_n({\nu ik\over\sqrt{\hbar}}+x)\!\!\!&=
&\!\!\!
\left(f(\nu ik)\!+\!f'(\nu ik)\sqrt{\hbar}x\right)\!\!\left({\cal P}_n({\nu
ik\over\sqrt{\hbar}}) + {\cal P}_n '({\nu
ik\over\sqrt{\hbar}})x\right)
\\ & & +\OO
\left(\hbar^{-\frac{n}{2}+1}(1+\vert x\vert^{n+2})\right)
\ ({\rm as}\ \hbar\rightarrow 0\
\mbox{and uniformly in }x),
\end{eqnarray*}
and finally integrating with $ e^{-\frac{x^2}{2}}$.\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion}
The idea of looking at the imaginary part of the first non
real coefficient of the perturbation expansion (\ref{cvse}) has a long history,
see e.g. \cite{RS4,Ya,GMS,DES} not to mention the physics litterature. Methods to
compute the aymptotics of $\im e_1$ using the analyticity of the potential are
all based more or less on the stationary phase or steepest descent methods.
However since it is expected that all coefficients of
(\ref{cvse}) will contribute to the imaginary part of the resonance
$E$ with the same exponential behavior, the prefactor of
(\ref{FR1}) has not the right asymptotics for $\im E$. Such a
phenomenon is discussed for example in \cite{Be}.
%
%We can rewrite (\ref{FR1}) under the following form
%$$
%\im e_1 =\sqrt{\pi} \frac{e^{n+2}}{n!}\hbar^{3\over2}
%\left(\frac{S}{\hbar}\right)^{n+\frac{1}{2}}
%\exp (\frac{S}{\hbar})
%\{|t(x_\star)|^2 + 0(\hbar^{\frac{1}{2}})\}
%$$
%where $S=-v_\infty +e_0$.
Below we give a classical interpretation of
$d_\star$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Exponential Decay of $\im E$ and Classical Action of
Instantons}
The following interpretation of $d_\star$ is a well
known heuristic fact (see \cite{LL}); we would like to illustrate it in detail
with our simple model.
On the complex phase space $\C\times\C$ we consider the two energy shells
$$
\Sigma^{(i)} (e_0) :=\{(q,p)\in\C,\quad H_{cl}^i (q,p)=e_0\},\quad i=1,2,
$$
and their trace on the real phase space $ \Sigma^{(i)}_{\R}, i=1,2$. The
two real energy shells do not intersect, but the complex ones do:
$$\left\{\begin{array}{lcl}p^2 +v_\infty & = & e_0\\
p^2+q^2 & = & e_0\end{array}\right.
\Longleftrightarrow
\left\{\begin{array}{lcl}p & = & \pm\sqrt{e_0-v_\infty}=\pm k\\
q & = & \pm i\sqrt{-v_\infty}=\pm x_\star\end{array}\right.
$$
These points will also be called {\em transition points\/}.
We want to endow the union of the two complex energy shells
$\Sigma(e_0):=\Sigma^{(1)}(e_0)\cup\Sigma^{(2)}(e_0)$ with a (pseudo-) distance
$\delta$ as follows.
Let $A$ and $B$ be two points of $\Sigma(e_0)$, then
$$
A,B\in\Sigma^{(i)}(e_0)\quad\Longrightarrow\quad
\delta (A,B):=|\im s(A,B)|,
$$
where $s(A,B)$ is the minimal (complex) action to join $A$ and $B$ by a (complex)
trajectory on the energy shell $\Sigma^{(i)}(e_0)$:
$$s(A,B) :=\int_{A\rightarrow B}pdq =\int_{A\rightarrow B}
\sqrt{e_0 - V^i (q)}dq.
$$
Such a complex trajectory is usually called an {\em instanton\/}.
Otherwise,
\begin{eqnarray*}& &
A\in\Sigma^{(i)}(e_0),\ B\in\Sigma^{(j)}(e_0),\
i\neq j,\quad \Longrightarrow\\
\delta(A,B) & \!:= &\!\min\{|\im s(A,T_\star) + \im s(T_\star,B)|,\ T_\star\in
\Sigma^{(1)}(e_0)\cap\Sigma^{(2)}(e_0)\}.
\end{eqnarray*}
In other words the distance $\delta (A,B)$ is the minimal imaginary part of the
action of all instantons joining $A$ and $B$.
We want to compute the distance between the two real energy shells:
$\delta(\Sigma^{(1)}_{\R}(e_0),\Sigma^{(2)}_{\R}(e_0))$. We remark that the
distance between two points of the same connected component of
$\Sigma^{(i)}_{\R}(e_0),i=1,2$, of course vanishes. As consequence we may
take any point in each one and compute their distance. So let
$A=(0,k)$ be in $\Sigma^{(1)}_{\R}(e_0)$
and $B= (0,\sqrt{e_o}) $ be in $\Sigma^{(2)}_{\R}(e_0)$.
We choose $T_\star =(x_\star,k)$ among the
transition points. Then we have:
\begin{eqnarray*}
s(A,T_\star)\! + s(T_\star,B) & =& \int_0 ^{x_\star} kdq
+\int_{x_\star} ^0 \sqrt{e_0 -q^2}dq \\
& =& kx_\star + {1\over 2}\left[ q\sqrt{e_0 -q^2} +e_0 \arcsin
({q\over\sqrt{e_0}}) \right] _{x_\star} ^0
\\
& = & %{i\over 2}(-v_\infty) +{\cal O}(\hbar\ln\hbar )
id_\star+{\cal O}(\hbar\ln\hbar )
\end{eqnarray*}
Performing the analoguous calculations for the other transition points, we see
that the above result gives indeed the minimal contribution to the definition
of $\delta (A,B)$. This shows that the distance between the two real energy
shells is $d_\star$ in the limit $\hbar$ tending to zero. But on the other hand
this is nothing else than the rate of exponential decay of the width of the
resonance due to quantum dynamical tunneling between the two real energy
shells.
This tunneling takes place through
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The Dynamical Barrier.}
Looking at the two real energy shells in the real phase space we see three
curves.
$\Sigma^{(2)}_{\R}(e_0)$ is the circle centered at the origin with radius
$\sqrt{e_0}$ and $\Sigma^{(1)}_{\R}(e_0)$ consists of the two straight lines
$p=\pm k$. As already mentioned the curves do not intersect.
More precisely their projections in the configuration space do intersect but
not their projection in the momentum space. The latter are the classically
allowed regions in the momentum space: $\{\pm k\}$ for
$H_{cl}^1$
and $[ -\sqrt{e_0},\sqrt{e_0} ]$ for $H_{cl}^2$. They are
separated by the classically forbidden region:
$(-k,-\sqrt{e_0})\cup(\sqrt{e_0},k)$. In
analogy with tunneling in the configuration space we would like to say that
associated to this classically forbidden region there is a {\em dynamical
barrier\/}. And through this barrier the bound state of $H^2$ has to tunnel to
become a scattering state of $H^1$. The strength of this tunneling, as in the
configuration space, depends on the diameters of the dynamical barriers.
These are measured by the length of each component of the classically forbidden
region in the {\em instanton metric\/}:
\begin{equation}\label{instantonmetric}
\left( (V^2)^{-1} (p^2-e_0)\right)_+ ^2 dp^2 = (p^2-e_0)_+ dp^2.
\end{equation}
Computing this diameter for the barrier of the positive momenta we obtain
$$\int_{\sqrt{e_0}}^{k} \sqrt{(p^2-e_0)_+}dp = d_\star
+{\cal O}(\hbar\ln\hbar) $$
which is in agreement with the result of theorem 5. The reason why $V^1$
does not enter in formula (\ref{instantonmetric}) is due to the fact that the
instanton joining $A$ and $T_\star$ has a constant velocity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exponential Bounds on the Resonance Width}
We recall that in section~2 we proved that $H$ possesses a resonance $E$ in the
vicinity of each eigenvalue $e_0$ of $H^2$ for small enough $\hbar$. In
section 3, for the restricted model $V^1=v_\infty$ (see H5), we have shown
that the imaginary part of the first term $e_1$ of the perturbation expansion
(\ref{cvse}) of $E-e_0$ is actually exponentially small as $\hbar$ tends to
zero. The purpose of this section is to show that this exponential behavior is
also true for the full width of the resonance $E$.
For technical reasons we have to require the following additional condition on
the coupling term $V^{1,2}$:
\vsl\nid
\hypothesis{7}{$\exists \varepsilon >0,\quad t(z)={\cal O}(|z|^{-\varepsilon})$
as z tends to infinity in $\sum_{\beta_0 ,\eta_0}$,}
\vsl\nid
to state our main
\vsl\nid
{\bf theorem 6}. Assume (H1,5,6,7). Then each eigenvalue $e_0$ of $H^2$
gives rise for $\hbar$ small enough to a resonance $E$ of $H$ and:
$$ \forall 1\geq \xi >0,\quad 0\geq \im E = \OO(\xi^{-2}\hbar \exp
(-\frac{2d_\xi}{\hbar}))\quad\mbox{as}\ \hbar\rightarrow 0,$$ where
$$
d_\xi := \int_{0}^{|x_\star|}\sqrt{(V^2(y) -e_0 -\hbar\xi)_+}dy.
$$
\vsl
\nid {\bf Remark 2}. Since the lifetime $\tau_E$ of the resonance $E$ is defined
as ${2\over \im E}$, the above theorem provides a lower bound on $\tau_E$.
One can check easily that
$$
\xi^{-2}\hbar e^{-2 \hbar^{-1}d_\xi}=\OO(\xi^{-2}\hbar^{-n+(1-\xi)/2}
e^{-2 \hbar^{-1}d_\star}).
$$
\nid This shows that the exponential behaviour of the bound of theorem 6 is
the same as the one of
$\im e_1$ (see (\ref{FR1})). However the exponent in the prefactor differs by
the quantity $-1-{\xi\over 2}$. We do not know yet whether the $\hbar$ behavior
of the prefactor of $\im e_1$ differs from the one of $\im E$ or if our
upper bound is not optimal.
\vsl\nid
Since the proof is rather involved we only give a synopsis of it.
The details of the proof will appear somewhere else together with a more
general analysis including non constant $V^1$'s.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Passing to the Fourier Image}
As we explained in \S 3.2, the exponential behaviour of the width of the
resonance $E$ is due to tunneling through the barrier of classically
forbidden momenta between the two energy shells
$\Sigma_{\R}^{(1)}(e_0)$ and $\Sigma_{\R}^{(2)}(e_0)$.
A Fourier transformation of $H$ causes the exchange of $q$ and $p$ in the
classical picture. Consequently the tunneling takes now place in the
configuration space, a situation we are more familiar with. We denote below by
the same symbols the Fourier image of $H^i, i=1,2$ and $ V^{1,2}$:
$$H^1 = x^2 +v_\infty,\quad H^2 = D^2 +x^2, \quad V^{1,2} =\hbar
(t(D)x+xt(D)).$$ The price to pay is that we have to deal with non local
operators as, e.g. $t(D)$ (see \S 4.5 below). We emphasize that this
transformation is only for convenience and in principle not necessary.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Why Exterior Scaling?}
Consider now $(E,\phi_\theta)$ a resonance and the corresponding resonance
function: $H_\theta \phi_\theta = E\phi_\theta$. By a simple algebraic
manipulation one also has
$$
(H^2 _\theta -B_\theta) \phi^{2}_\theta = E\phi^{2}_\theta,
\quad B_\theta := V^{2,1}_\theta R^{1}_\theta (E)V^{1,2}_\theta ,
$$
where $\phi^{i}_\theta$ denotes the $i^{th}$ component of $\phi_\theta$. Thus
\begin{equation}\label{fundamformula}
\im E \Vert\phi^{2}_\theta\Vert ^2 =
(\im (H^2 _\theta -B_\theta)\phi^{2}_\theta,\phi^{2}_\theta).
\end{equation}
In section 2 we have seen that $\phi_\theta$ and therefore $\phi^{2}_\theta$
``converge'' in norm to $\varphi^{2}_{\theta}$, the corresponding eigenfunction
of $H^2 _\theta$. The latter is of course known to decay exponentially
as $x$ goes to infinity and/or $\hbar$ tending to zero. So suppose that the
operator $ \im (H^{2}_\theta -B_\theta)$ is localized on $\Omega_e$ where
$$
\Omega _i := (-\omega,\omega),\quad \Omega_e :=\R \setminus\overline{\Omega}_i,
\quad \omega^2 = -v_\infty,
$$
we would obtain that $\im E$ is roughly $ \Vert\chi_e \phi^{2}_\theta\Vert^2\sim
e^{-\hbar^{-1}\omega^2}$
which is more or less what we are looking for.
$\chi_a$ will denote the sharp characteristic function of
$\Omega_a, a=i,e$.
$\Omega_i$ is nothing but the dynamical barrier for $\hbar =0$ (see \S 3.2.2).
With the usual complex scaling $\im (H^{2}_\theta -B_\theta)$ is certainly not
localized in $\Omega_e$, whereas $ \im H^{2}_\theta$ would be with the exterior
scaling \begin{equation}\label{extscale}
s_\theta (x) := \left\{\begin{array}{lll} x & \mbox{if}& x\in \Omega_i\\
\pm \omega +e^\theta (x\mp\omega ) &
\mbox{if}&\!\! \pm x > \omega .\end{array}\right.
\end{equation}
Thus in this section we choose the above complex
exterior scaling to deform all our operators; a subscript $\theta$ will now mean
the image under (\ref{extscale}).
It is well known that the resonances of $H$ do not depend on the choice of the
complex deformation (see e.g. \cite{Hu}). Here they will be considered as
eigenvalues of $H_\theta$ for a certain complex $\theta$ where, as announced
above, $H_\theta$ is now obtained by (\ref{extscale}).
Making sense out of all our scaled objects is rather standard (see e.g
\cite{CDKS}) except for the non local terms $V^{1,2}_\theta$ and $B_\theta$.
We shall explain briefly in \S 4.5 how we proceed. We have in particular the
analogue of theorem 1:
\vsl\nid{\bf theorem 7}.
$H^i_\theta, i=1,2$ and $H_\theta$ are selfadjoint analytic families for all
$\theta$ such that $|\im\theta |< {\hat\beta}_0$. $H^1 _\theta $ is of type A
with domain ${\cal D}(H^1_\theta)={\cal D}(x^2)$. The domain of $H^2_\theta$
is given by
$$\begin{array}{rcl} u\in {\cal D}(H^2_\theta) & \Longleftrightarrow &
u \in {\cal H}^2 (\Omega_i)\oplus{\cal H}^2 (\Omega_e)\ \mbox{and}\\
& & u(\pm\omega\pm0) = e^{\theta /2} u(\pm \omega \mp 0)\\
& & u'(\pm\omega\pm0) = e^{3\theta /2} u'(\pm \omega \mp 0).\end{array}
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spectral Stability Again}
We shall need in the sequel information on the spectrum of $H^d_\theta$ and
some bounds on its resolvent. Since $H^1_\theta $ is a multiplication operator
its spectrum is nothing but the range of the function $x \mapsto v_\infty +
s^2_\theta (x)$. There are two facts to remark here. First the spectrum of
$H^1_\theta$ consists of the union of the interval $\left[ v_\infty,0\right]$ and
a curve starting from zero, contained in the sector
$\{ |\im \theta |< \mbox{sgn}(\im\theta)\arg (z) < 2|\im \theta |\}$; secondly
the essential spectrum of $H^1$ above zero turns up in the upper half plane,
if $\im\theta > 0$. In order to work with the resonances which have a negative
imaginary part we shall from now on take only $\theta$ with negative imaginary
part. The spectrum of $H^2_\theta$ remains $ (2\N +1)\hbar$ by standard
arguments. So for small enough $\hbar$ the same contour $\Gamma$ as in \S 2 is
contained in the resolvent set of $H^d_\theta$. To prove that it lies also in
$\rho (H_\theta)$ we need the
\vsl\nid
{\bf lemma 8}.
For any $0<\vert\beta\vert <{\hat\beta}_0$ and for any $z$ on $\Gamma$ one has\\
i)
$\Vert%
V^{1,2}_{i\beta}R^2_{i\beta}(z)V^{2,1}_{i\beta}\Vert=\OO(\hbar^2)$. \\
ii)
$\Vert R^1_{i\beta} (z)\Vert,\Vert x R^1_{i\beta} (z)\Vert, \Vert x
R^1_{i\beta} (z) x\Vert $ are all $\OO(\hbar^{-1})$; this is also valid for $z$
inside $\Gamma$.
\vsl\nid
Thus stability of $e_0$ is assured for $\hbar $ small enough.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Estimate of %
$(\im (H^2_\theta\phi^{2}_\theta,\phi^{2}_\theta) $}
We want to show that $(\im (H^2_\theta\phi^{2}_\theta,\phi^{2}_\theta) $ is
exponentially small using the decay properties of $\phi^{2}_\theta $ and the
localisation of $\im H^2_\theta$. The following estimate
is adapted from \cite{Agm}. However there is a novelty with respect
to the standard situation: the operator $H^2_\theta -B_\theta$
contains the non local energy dependent ``potential term'' $B_\theta$. Let
$\exp(-\hbar^{-1}\rho)$ be the expected decay behavior and let $\phi^{2}_{\theta
,\rho}:= \exp(\hbar^{-1}\rho)\phi^{2}_\theta $ be the boosted eigenfunction; more
generally we denote all objects boosted by $\exp(\hbar^{-1}\rho)$ with the
subscript $\rho$. $\phi^{2}_{\theta ,\rho}$ is a solution of
$(H^2_{\theta ,\rho} -B_{\theta ,\rho} -E)\phi^{2}_{\theta ,\rho} =0$ and
therefore
$$
\re ((H^2_{\theta ,\rho} -B_{\theta ,\rho}
-E)\phi^{2}_{\theta,\rho},\phi^{2}_{\theta ,\rho}) =0.
$$
To cope with the $B_{\theta ,\rho}$ problem we use the following property
\vsl\nid
{\bf lemma 9}.
Assume (H6), then there exists $a,b>0$ such that
for any $0<\vert\im\theta\vert<{\hat\beta}_0$ and any real Lifschitz function
$\rho$ one has:
$$ \rho'{}^2 \leq \omega ^2\chi_i\quad \Longrightarrow\quad
-\re B_{\theta ,\rho}\geq -a\hbar \re x^2_\theta -b\hbar ^3.
$$
\vsl\nid
>From this result we deduce that
$$\hbar^2\Vert s'_\theta{} ^{-{1\over 2}}\partial_x\phi^{2}_{\theta ,\rho}\Vert^2
+ \left( ((1-a\hbar)\re x^2_\theta -b\hbar^3 -\re E -\rho '{}^2)
\phi^{2}_{\theta ,\rho},\phi^{2}_{\theta ,\rho}\right)\leq 0.
$$
This motivates the choice of $\rho$:
$$
\rho '{}^2:= \left((1-a\hbar) x^2 -b\hbar^3 -\re E -\hbar\xi\right)_+\chi_i,
\quad \xi >0,
$$
and we obtain the
\vsl\nid
{\bf lemma 10}.
For any $0<| \im\theta|< {\hat \beta}_0$ one has as $\hbar$ tends to zero
and for any $0<\xi\leq 1$:
$$
%%%\begin{array}{rclrcl}
\Vert \phi^{2}_{\theta ,\rho}\Vert ={\cal O}(\xi ^{-{1 \over 2}}),\quad
%%\Vert x\phi^{2}_{\theta ,\rho}\Vert & = & \OO (\hbar^{{1\over 2}}
%%%\xi^{-{1\over 2}})
\Vert \chi_e x\phi^{2}_{\theta ,\rho}\Vert = \OO(\hbar ^{{1 \over2}}),
\quad
\hbar\Vert \chi_e\partial_x\phi^{2}_{\theta,\rho}\Vert =
\OO (\hbar^{{1\over2}}).
%%%\end{array}
$$
\vsl\nid
We can now estimate $ \im (H^2_\theta\phi^{2}_\theta,\phi^{2}_\theta)$:
\begin{eqnarray*}
\im (H^2_\theta\phi^{2}_\theta,\phi^{2}_\theta) &=&
\im (e^{-\hbar^{-1}\rho}H^2_\theta
e^{-\hbar^{-1}\rho}\phi^{2}_{\theta,\rho},\phi^{2}_{\theta,\rho})\\
& = & e^{-2\hbar^{-1}\rho(\omega)}(\im H^2_\theta
\phi^{2}_{\theta,\rho},\phi^{2}_{\theta,\rho}) =
{\cal O}(\hbar e^{-2\hbar^{-1}\rho(\omega)} ).
\end{eqnarray*}
An elementary calculus shows that $e^{-2\hbar^{-1}\rho(\omega)}
=\OO(e^{-2\hbar^{-1}d_\xi})$ so that
this term has the announced behavior. It remains to do the
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Estimate of $\im (B_\theta\phi^{2}_\theta,\phi^{2}_\theta)$.}
We start by explaining how we make sense out of $B_\theta$. We may write
formally $$
B_\theta = -\hbar^2 (2t(D_\theta)x_\theta + i \hbar t'(D_\theta))R^1_\theta
(E) (2x_\theta t(D_\theta) -i\hbar t' (D_\theta)).
$$
The operator $x_\theta$ is controlled by $R^1_\theta (E)$ (see lemma 8) so it
is sufficient to show that $t(D_\theta)$ and $t'(D_\theta)$ are bounded
operators. $D_\theta$ is the image of $D$ under our exterior scaling for which
we know the
\vsl\nid
{\bf theorem 11}.
$\{ D_\theta, \theta \!\in \C\}$ is a selfadjoint analytic family of
operators and:
\\ i)
the domain of $D_\theta$ is defined by
$$\begin{array}{rcl} {\cal D}(D_\theta) \ni\!u & \Longleftrightarrow &
u \!\in\! {\cal H}^1 (\Omega_i)\oplus{\cal H}^1 (\Omega_e)\
\mbox{and}\\
& & u(\pm\omega\pm0) = e^{\theta / 2} u(\pm \omega \mp 0);
\end{array}
$$
ii) the spectrum of $D_\theta$ is just: $\sigma (D_\theta) =
e^{-\theta}\R$.
\\
iii) The following resolvent estimate holds: let $\nu :=\{
z\!\in\C,\, \im s'_\theta z > 0$ or $\im s'_\theta z < 0 \}$; then
for every $z$ in $\nu$, one has
$$
\Vert (D_\theta -z)^{-1}\Vert \leq |\im s'_\theta z |^{-1} =
\mbox{dist}(z,\C\setminus\nu )^{-1}.
$$
\vsl\nid
To define $t(D_\theta)$ we use the Dunford-Taylor integral
$$
t(D_\theta):={1\over 2\pi i}\int_{C} t(\lambda)(D_\theta -\lambda)^{-1}d\lambda
,\quad |\im\theta |< {\hat \beta}_0
$$
where the contour $C$ is taken in $\nu\cap\sum_{\beta_0,\eta_0}$ enclosing
$\sigma (D_\theta)$ and such that dist$(C,\nu)^{-1} =\OO(\lambda^{-1})$ as
$|\lambda |$ tends to infinity. The convergence of the above integral
is due to (H7) and lemma 11 iii). Furthermore we know that
\vsl
\nid
{\bf lemma 12}. $t(D_\theta),t'(D_\theta)$ are bounded selfadjoint analytic
families as long as $|\im\theta | < {\hat \beta}_0$.
\vsl
\nid We now explain how one can define $B_{\theta,\rho}$. Using the same
strategy as for $B_\theta$ and the fact that the boost $e^{\hbar^{-1}\rho}$
commutes with $x_\theta$ and $R^1_\theta (E)$, it suffices to show that
$t(D_{\theta,\rho})$ and $t'(D_{\theta,\rho})$ are bounded.
One can show the analogue of theorem~11 and lemma~12 for them. Moreover
$t(D_{\theta,\rho})$ and $t'(D_{\theta,\rho})$ are uniformly bounded with
respect to $\rho$ as long as $\rho$ obeys: $\rho'{}^2\leq \omega^2\chi_i$. Here
one must use assumption (H6). By a straightforward though rather involved
algebraic calculus one gets a reformulation of (\ref{fundamformula})
%%%
\begin{equation}\label{fundamformulabis} \im E(\Vert
\phi_\theta\Vert^2 +(1-e^{-2\hbar^{-1}\rho(\omega)})\Vert
\phi^1_{\theta,\rho}\Vert^2)= e^{-2\hbar^{-1}\rho(\omega)}\im
((H^2_\theta-B_{\theta,\rho})\phi^2_{\theta,\rho},\phi^2_{\theta,\rho}).
\end{equation}
%%
\nid One has the estimate
$$
\Vert \phi^1_{\theta,\rho}\Vert^2=\Vert R^1_\theta(E)V^{1,2}_{\theta,\rho}\phi^2_{\theta,\rho}\Vert^2=
\OO(\xi^{-1}),
$$
\nid (use lemma 8, 12 and 10). Finally one also has
$$
(B_{\theta,\rho}\phi^2_{\theta,\rho},\phi^2_{\theta,\rho})=\OO(\hbar\xi^{-1}).
$$
Inserting this estimate and the one of \S 4.4 in (\ref{fundamformulabis})
completes the proof of theorem 6.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion}
The use of a complex deformation in the momentum space leaving invariant
$\Omega_{i,\varepsilon}:=\{p\in\R , \vert p\vert <\omega -\varepsilon\},
\varepsilon $ small, i.e. a region which contains approximately the dynamical
barrier (see \S 3.2.2), was brought to our attention by M. Rouleux. In his notes
\cite{Ro} he combines this idea with the Green's formula
\begin{equation}\label{GF}
2\ \im E\int_{\Omega_{i,\varepsilon}}\vert \phi(x)\vert^2 dx =
-\hbar^2\im\phi'\overline{\phi}
\Bigr|_{\partial\Omega_{i,\varepsilon}} ;
\end{equation}
as above $\phi$ stands here for the Fourier image of the original $\phi$.
The last ingredient to get the asymptotics of $\im E$ would be the knowledge of
the behavior of the r.h.s. of (\ref{GF}) since obviously
the norm of $\phi$ on $\Omega_{i,\varepsilon}$ is $1 + o(1)$
as $\hbar$ tends to zero.
The idea of using the Green's formula can be traced back to Herring \cite{Her}
through e.g. \cite{HeSj,AsHa,W}.
The first idea, the exterior complex deformation, is now rather standard when
translated to the shape resonance framework (see e.g. \cite{CDKS,HeSj}).
The choice of $\Omega_{i,\varepsilon}$ with a non zero $\varepsilon$ is a
consequence of the use of a smooth exterior distortion. Here we hope to get
immediately the right exponential decay rate by employing an exterior scaling (\S
4.2).
Also instead of the Green's formula (\ref{GF}) we use a full $L^2$ calculus
(\ref{fundamformula}). With (\ref{fundamformula}) we need only $L^2$-exponential
decay estimates on $\phi$ (see \S 4.4,5) which are simpler to derive than
the pointwise estimates needed in (\ref{GF}). However by using this $L^2$ approach
we are forced to loose a little bit of the exponential decay rate; the correct one
being given by $d_\xi$ with $\xi=0$ (see theorem 6).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgement}
The main part of this work has been done while both of us were paying a visit to
Ph. Blanchard at BiBos, University of Bielefeld.
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