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schrodinger operator, semi classical limit, spectral shift function,  
resonances, trace formula, Breit-Wigner approximation, Weyl asymptotic
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\title[Meromorphic continuation]
{Meromorphic continuation of the spectral shift function}
\author[V. Bruneau, V. Petkov]{Vincent Bruneau and  Vesselin Petkov}

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\begin{document}

\begin{abstract}
We obtain a  representation of the derivative of the spectral shift
function $\xi(\lambda, h)$ in the framework of semi-classical "black box"
perturbations. Our representation implies a meromorphic continuation of
$\xi(\lambda, h)$ involving the semi-classical resonances. Moreover, we obtain a Weyl type asymptotics of the spectral shift function as well as a Breit-Wigner approximation in an interval $(\lambda - \delta, \lambda + \delta), \:\: 0 < \delta < \epsilon h.$\\

 AMS classification: 35B34, 35P25
\end{abstract}


\maketitle

 

\section{Introduction}

The purpose of this paper is to obtain a meromorphic continuation of the
derivative of the spectral shift function $\xi(\lambda, h)$. This problem
is closely related to the trace formulae (see \cite{GZ}, \cite{Z1},
\cite{Z2} \cite{PZ1}, \cite{PZ3}, \cite{SZ}, \cite{Sj1}, \cite{Sj2}) and to
resonances expansions (\cite{BZ}, \cite{TZ}). For compact perturbations
the function $\xi(\lambda, h)$
coincides with the {\em scattering phase}
\[ \sigma(\lambda, h) = \frac{1}{2 \pi i} \log \det S(\lambda, h), \:\:
\lambda \in \RR \,, \]
where $S(\lambda, h) = I + A(\lambda, h): \: L^2(S^{n-1}) \longrightarrow
L^2(S^{n-1})$ is the scattering operator and for more information about
the spectral shift function we refer to
 \cite{Ya}. In the classical case ($h = 1$) the first result proving a
representation of $\sigma (\lambda) = \sigma(\lambda, 1)$ containing the
resonances $z_j \in \C_{-} = \{z \in \C: \Im z < 0 \}$ was established by
Melrose \cite{Me2} for obstacle scattering in odd dimensions $n \geq 3.$
More precisely, given a function $\chi(t) \in C^{\infty}(\RR)$ such that
$0 \leq \chi(t) \leq 1, \:\: \chi(t) = 1$ for $t \leq 2,\:\:\chi(t) = 0$
for $t \geq 3$, Melrose showed that
\[ \sigma(\lambda) = \sigma_{{\rm{sing}}}(\lambda) +
\sigma_{{\rm{reg}}}(\lambda), \, \]
with
\[ \frac{d}{d\lambda} \sigma_{{\rm{sing}}} (\lambda) = - \frac{1}{\pi}
\sum_{j} \chi\Big( \frac{|z_j|}{\lambda}\Bigr) \frac{\Im z_j}{|\lambda -
z_j|^2}, \:\:\sigma_{{\rm{sing}}} (0) = 0, \:\:\lambda \in \RR \,, \]
\[ \sigma_{{\rm{reg}}}(\lambda) \in S^n(\RR) \,. \]
Since $\sigma(\lambda, h)$ is the logarithmic derivative of the {\em
scattering determinant}
\[ s(\lambda, h) = \det (I + A(\lambda, h)) \,, \]
it is natural to examine the behavior of $s(z, h)$ for $z$ in the
"physical half plane", where we have no resonances. This idea was
developed by Guillop\'e and Zworski \cite{GZ} for the analysis of the
scattering resonances for certain Riemann surfaces and in the classical
case $h =1$, Zworski \cite{Z1}, \cite{Z2} gave an elegant proof of the
trace formula for "black box" compact perturbations based on the
meromorphic continuation of $s(z)$ (see \cite{Z1} for other works on trace
formulae).\\

In \cite{PZ1}, \cite{PZ3} the Breit-Wigner approximation for the
scattering phase has been justified for "black box" scattering with
compact perturbations in the classical and the semi-classical cases. Among
the ideas introduced in \cite{PZ1}, \cite{PZ3}, one of the main point in
\cite{PZ3} was the estimate of the holomorphic function $g(z, h)$,
\begin{equation} \label{eq:1.1}
 |g(z, h)| \leq C(\Omega) h^{-\ns}, \:\: \ns \geq n
\end{equation}
 in the local factorization
\[ s(z, h) = e^{g(z, h)}\frac{\overline{P(\overline{z} , h)}}{P(z, h)},\:
z \in  \Omega \,, \]
where
\[ P(z, h) = \prod_{w \in {\rm{Res}\: L(h)}\: \cap \Omega_{\epsilon} ,
\atop \Im  w \neq 0} (z -w) \,, \]
\[ \Omega = (a, b) + i(-c, c), \:\: 0 < a < b,\:\: c > 0,\:\:
\Omega_{\epsilon} = \{z \in \C: d(\Omega, z) < \epsilon \}, \: \epsilon >
0 \,. \]

Here $L(h)$ is a compactly supported  perturbation of the operator $-h^2
\Delta,\:\: 0 < h \leq h_0,$ and $\ns$ depends on the estimates of the
number of the  eigenvalues of the reference operator.
 The local factorization implies immediately
\begin{equation} \label{eq:1.2}
\partial_z \sigma(z, h) = \frac{1}{2 \pi i}\partial_z g(z, h) +
\frac{1}{2 \pi i}\sum_{w \in {\rm{Res}}\:L(h)\: \cap\: \Omega_{\epsilon},
\atop \Im  w \neq 0} \Bigl( \frac{1}{z - \overline{w}} - \frac{1}{z - w}
\Bigr),\:\: z \in \Omega
\end{equation}
and for $\lambda \in (a, b)$ we obtain an analogue of the formula of Melrose
mentioned above. Combining (\ref{eq:1.2}) with the Birman-Krein formula
one obtains easily the trace formula of \cite{Sj1} exploiting the
meromorphic continuation of $\partial_z \sigma(z, h)$ in $\{ z \in \C: \Im
\leq 0 \}$ (see Theorem 1 in \cite{PZ3}). Moreover, a similar
factorization has been established in \cite{PZ3} in domains $\lambda + h
\Omega$ with an improved estimate for the holomorphic function $g(z, h).$
\\

In the case of "black box" long-range perturbations the existence of
the scattering operator and that of the scattering determinant are far
from apparent. In this direction Sj\"{o}strand \cite{Sj1}, \cite{Sj2}
proposed powerful techniques based on the complex scaling operators,
introduced in \cite{SZ}, and complex analysis. The scattering determinant
is replaced by $D(z,h) = \det (I + \tilde{K}(z))$, where $\tilde{K}(z)$ is
trace class operator which is not uniquely  determined and the resonances
are the zeros of $D(z, h).$ Applying the approach  of Sj\"{o}strand,
J.-F. Bony \cite{Bo1}, \cite{Bo2}, established upper and lower bounds on
the number of the semi-classical resonances in small domains and the
Breit-Wigner approximation has been extended to long-range perturbations
in \cite{BSj}. For a pair of self-adjoint operators $L_j(h),\:\: j =1,2,$
satisfying some assumptions (see Section 2) the spectral shift function
$\xi(\lambda, h)$ is a distribution in ${\mathcal D}'(\RR)$ such that
\[ <\xi'(\lambda, h), f(\lambda) >_{{\mathcal D}'(\RR), {\mathcal D}(\RR)}
= \trb \Bigl(f(L_2(h)) -  f(L_1(h))\Bigr), f(\lambda) \in C_0^{\infty}
(\RR)  \,, \]
where $\trb$ is a generalized trace defined in Section 2. We denote by Res
$L_j(h), j =1, 2$ the set of the resonances $w \in \overline{\C}_{-}$ of
$L_j(h).$\\

In this work we are strongly inspired by the approach in \cite{PZ3}  and
our main goal is to obtain an analogue of (\ref{eq:1.2}) in the cases when
a scattering determinant is not available. We show that the representation
(\ref{eq:1.2}) remains true in the general case of semi-classical "black
box" scattering, replacing $\sigma'(\lambda, h)$ by the "regular part"
  $$\xi'(\lambda, h) - \Bigl[\sum_{w \in {\rm{Res}}\:L_j(h)\: \cap (a, b)}
\delta(\lambda - w)\Bigr]_{j=1}^2,$$
where here and throughout the paper we use the notation $[a_j]_{j=1}^2 =
a_2 - a_1.$ Our principal result is the following.
\begin{thm}
Assume that $L_j(h), \: j =1,2,$ satisfy the assumptions of Section $2$.
Let $\Omega \subset\subset  e^{i]-2\theta, 2 \theta[}]0, +\infty[, \: 0
<\theta < \pi/2$, be an open simply connected set and let $W \subset
\subset \Omega$ be an open simply connected and relatively compact set
which is symmetric with respect to $\RR$.  Assume that $J = \Omega \cap
\RR^{+}, \:\: I = W \cap \RR^{+}$ are intervals. Then for $\lambda \in I$
we have the representation
\begin{equation} \label{eq:1.3}
 \xi'(\lambda, h) = \frac{1}{\pi} \Im r(\lambda, h) + \Bigl[\sum_{w \in
{\rm Res}\:L_j \cap \Omega, \atop \Im w \neq 0} \frac{-\Im w}{\pi |\lambda
- w|^2} +
\sum_{w  \in {\rm Res}\:L_j \cap J} \delta (\lambda - w) \Bigr]_{j=1}^2,
\end{equation}
where $r(z, h) = g_{+}(z, h) - \overline{g_{+}(\overline{z}, h)},\:\:
g_{+}(z,h)$ is a function holomorphic in $\Omega$ and $g_{+}(z, h)$
satisfies the estimate
\begin{equation} \label{eq:1.4}
|g_{+}(z, h)| \leq C(W)h^{-\ns},\:\: z \in W
\end{equation}
with $C(W) > 0$ independent on $h \in ]0, h_0].$
\end{thm}

{\bf Remarks.}
\begin{itemize}
\item { The terms related to the resonances are measures. In fact, the
resonances $w, \: \Im w < 0,$ are related to harmonic measures
\[ \omega_{\C_{-}} ( w, E) = -\frac{1}{\pi} \int_E \frac{\Im w}{|t - w|^2}
dt,\:\: E \subset \RR = \partial \C_{-} \,, \]
while the resonances $w \in \RR^{+}$ coincide with the embedded
eigenvalues of $L_j(h), j =1, 2$. Moreover, in a small neighborhood
$U_{\lambda}(h)$ of every  $\lambda \in I \setminus \cup_{j=1}^2 \{\lambda
\in \RR: \lambda \in \sigma_{pp}(L_j(h)) \}$ the derivative $\xi'(\lambda
, h)$ coincides with a {\em real analytic function} on $U_{\lambda}(h)$.
In particular, if we have no embedded positive eigenvalues of $L_j(h)$ in
$I$, then $\xi'(\lambda, h)$ is real analytic in $I.$}\\

\item{ The representations of $\xi'(\lambda, h)$ obtained in \cite{R2},
\cite{BrPe2} involve the traces of the cut-off resolvents $\chi (L_j -
\lambda \mp i0)^{-1} \chi, \:\: \chi \in C_0^{\infty} (\RR^n),$  and some
regular terms whose meromorphic continuation is far from apparent. The
form of $\xi'(\lambda, h)$ in \cite{R2}, \cite{BrPe2} has been used for
the investigation of the Weyl type asymptotics of $\xi(\lambda, h)$ (see also
\cite{Na}, \cite{Br} for semi-classical asymptotics in the trapping
case).} \\
\end{itemize}


The proof of (\ref{eq:1.3}) relies heavily on the work of Sj\"ostrand
\cite{Sj2}, while the arguments in \cite{PZ3} were self-contained and
based on the semi-classical estimates of the scattering determinant.
Having in mind (\ref{eq:1.3}), we obtain in the general case of "black box" semi-classical scattering several results:\\

I) We establish a Weyl type asymptotics of the spectral shift
function in the general framework of semi-classical ``black box'' perturbations improving our previous result \cite{BrPe2} and working without any assumption on the behavior of the resonances close to the real axis. We generalize the results of Christiansen
\cite{Ch} for compact perturbations and those of Robert \cite{R2} for long-range perturbations. Theorem 1 allows to consider the sum of the harmonic measures related to the resonances $w, \: \Im w \neq 0$, as a monotonic function and to apply a Tauberian argument as in \cite{Me2}.\\

II) We present a new direct and short proof of the recent result of J.-F. Bony and Sj\"{o}strand \cite{BSj} on the  Breit-Wigner approximation in the long-range case (see Theorem 3). For this purpose the Weyl asymptotics obtained in Theorem 2 plays an essential role. Moreover, Theorem 2 and Theorem 3 are established under the "black box" assumptions in Section 2 and the condition (5.1). Thus we have an unified approach to these problems. Next, assuming the existence of free resonances domain, we obtain a Breit-Wigner approximation involving only the resonances $w$ lying in small ``boxes''
$$\{ w \in \C: |\Re w - \lambda| \leq R(h),\:\: |\Im w| \leq R_1(h)\}$$
 with $R(h) = \sqrt{hR_1(h)} = {\mathcal O}(h^{\infty}).$\\

III) In the same way as in \cite{PZ3}, we obtain the local  trace formula of Sj\"{o}strand \cite{Sj1}, \cite{Sj2} in a slightly stronger version (see Section 7). Moreover, we prove a trace formula involving the unitary groups $e^{-i\frac{t}{h}L_j(h)},\:j =1,2$ (see Theorem 5) which is a semi-classical version of the classical trace formulae.\\

 We expect that the  approach of our work could be useful in other situations
as in the analysis of periodic potentials \cite{DZ} or the study of matrix Schr\"{o}dinger operators \cite{Ne} if a representation like (\ref{eq:1.3}) is established.\\


 The plan of the paper is the following. In Section 2 we introduce the
"black box" scattering assumptions and in Section 3 we obtain a formula
for $\xi'(\lambda,h)$ involving the limits of the functions
$\sigma_{\pm}(z)$ as $\Im z \to 0.$ Theorem 1 is proved in Section 4 and
in Section 5 we establish a Weyl type asymptotics for the spectral shift
function $\xi(\lambda, h)$. The semi-classical Breit-Wigner
approximation is established in Section 6 together with a stronger
approximation based on some recent results of Stefanov \cite{St}. In
Section 7 we prove some trace formulae combining (\ref{eq:1.3}) with the
arguments of \cite{PZ3}. In particular, we obtain a trace formula
involving the unitary groups $e^{-ih^{-1}L_j}$. Finally, in Section 8 the
Breit-Wigner approximation is applied to establish the existence of
clusters of resonances close to the real axis.\\

{\bf Acknowledgments.} The authors are grateful to J. Sj\"{o}strand and
M. Zworski for many  helpful discussions.

\section{Preliminaries}


We start by the abstract ``black box'' scattering assumptions introduced
in  \cite{SZ}, \cite{Sj1} and \cite{Sj2}. The operators $L_j(h) = L_j, j
=1,2, \:\: 0 < h \leq h_0,$ are defined in domains ${\cal D}_j \subset
{\cal H}_j$ of a complex Hilbert space ${\cal H}_j$ with an orthogonal
decomposition
$${\cal H}_j = {\cal H}_{R_0,j} \oplus L^2({\RR}^n \setminus
B(0,R_0)),\:B(0,R_0) = \{x \in {\RR}^n: |x| \leq R_0 \},\:\: R_0 > 0,
\:\:n \geq 2. $$
Below $h > 0$ is a small parameter and we suppose the assumptions
satisfied for $j = 1, 2.$ We suppose that ${\cal D}_j$ satisfies
\begin{equation} \label{eq:2.1}
{\Bbbone }_{{\RR}^n \setminus B(0,R_0)}{\cal D}_j = H^2({\RR}^n \setminus
B(0,R_0)),
\end{equation}
uniformly with respect to $h$ in the sense of \cite{Sj1}. More precisely,
equip $H^2({\RR}^n \setminus B(0,R_0))$ with the norm
$\|<hD>^2u\|_{L^2},\:<hD>^2 = 1 + (hD)^2$, and equip ${\cal D}_j$ with the
norm $\|(L_j+i)u\|_{{\cal H}_j}.$ Then we require that ${\Bbbone
}_{{\RR}^n \setminus B(0,R_0)}: {\cal D}_j \longrightarrow H^2({\RR}^n
\setminus B(0,R_0))$ is uniformly bounded with respect to $h$ and this map
has a uniformly bounded right inverse.

Assume that
\begin{equation}  \label{eq:2.2}
{\Bbbone}_{B(0,R_0)}(L_j+i)^{-1} \hbox{is compact}
\end{equation}
and
\begin{equation}\label{eq:2.3}
 (L_j u)\vert_{{\RR}^n \setminus  \overline{B(0,R_0)}} = Q_j\Bigl(
u\vert_{{\RR}^n \setminus \overline{B(0,R_0)}}\Bigr),
\end{equation}
where $Q_j$ is a formally self-adjoint differential operator
\begin{equation}\label{eq:2.4}
 Q_j u = \sum_{| \nu | \leq 2} a_{j,\nu} (x;h) (hD_x)^\nu u,
\end{equation}
with $ a_{j,\nu} (x;h)=  a_{j,\nu} (x)$ independent of $h$ for $| \nu | =
2$
 and $a_{j,\nu} \in C_b^\infty(\RR^n)$ uniformly bounded with respect to
$h$.

We assume also the following properties:

 There exists $C>0$ such that
\begin{equation} \label{eq:2.5}
 l_{j,0}(x,\xi) =\sum_{| \nu | = 2} a_{j,\nu} (x) \xi^\nu \geq C |\xi|^2,
\end{equation}

\begin{equation} \label{eq:2.6}
 \sum_{|\nu| \leq 2}a_{j,\nu} (x;h) {\xi}^{\nu} \longrightarrow
|\xi|^2,\:\: |x| \longrightarrow \infty
\end{equation}
uniformly with respect to $h$.

 There exists  $\overline{n} > n$ such that we have
\begin{equation}\label{eq:2.7}
 \Bigl| a_{1,\nu} (x;h) - a_{2,\nu} (x;h) \Bigl|
 \leq {\mathcal O}(1)  {\langle x \rangle}^{-\overline{n}}
\end{equation}
uniformly with respect to $h$.
This assumption will guarantee that for every  $f \in C_0^\infty(\RR)$ the
operator $f(L_1) - f(L_2)$ is ``trace class near infinity''.

There exist $\theta_0 \in ]0, \frac{\pi}{2}[,\:\epsilon > 0$ and $R_1 >
R_0$ so that the coefficients $a_{j,\nu}(x;h)$ of $Q_j$ can be extended
holomorphically in $x$ to
\begin{equation} \label{2.8}
\Gamma = \{r\omega;\:\omega \in {\C}^n,\: {\rm dist}\:(\omega, S^{n-1}) <
\epsilon, \: r \in \C, r \in e^{i[0, \theta_0]} ]R_1, +\infty[ \}
\end{equation}
and (2.6), (2.7) extend to $\Gamma$.

Let $R > R_0,\:T = ({\RR}/\tilde{R}{\Z})^n,\: \tilde{R} > 2R.$ Set
$${\cal H}_j^{\#} = {\cal H}_{R_0,j} \oplus L^2(T \setminus B(0, R_0))$$
and consider a differential operator
$$Q_j^{\#} = \sum_{|\nu| \leq 2} a_{j,\nu}^{\#}(x;h)(hD)^{\nu}$$
on $T$ with $a_{j,\nu}^{\#}(x;h) = a_{j,\nu}(x;h)$ for $|x| \leq R$
satisfying (2.3), (2.4), (2.5) with $\RR^n$ replaced by $T$. Consider a
self-adjoint operator
$L_j^{\#}: {\cal H}_j^{\#} \longrightarrow {\cal H}_j^{\#}$
 defined by
$$L_j^{\#}u = L_j \varphi u +Q_j ^{\#}(1-\varphi)u, \: u \in {\cal
D}_j^{\#},$$
with domain
$${\cal D}_j^{\#} = \{u \in {\cal H}_j^{\#}: \: \varphi  u \in {\cal D}_j,
\: (1-\varphi)u \in H^2 \},$$
where  $\varphi \in C^{\infty}_0(B(0,R); [0,1])$ is equal to 1 near
$\overline{B(0,R_0)}.$

 Denote by $N(L_j^{\#}, [-\lambda, \lambda])$ the number of eigenvalues of
$L_j^{\#}$ in the interval $[-\lambda, \lambda]$. Then we assume that
\begin{equation}\label{eq:2.9}
N(L_j^{\#}, [-\lambda, \lambda]) = {\mathcal
O}(\Bigl(\frac{\lambda}{h^2}\Bigr)^{n_{j}^{\#}/2}),\: n_j^{\#} \geq n,\:
\lambda \geq 1. \label{eq:1.12}
\end{equation}
Finally,  we suppose that with some constant $C \geq 0$ independent on $h$
we have
\begin{equation} \label{eq:2.10}
{\rm{sp}}\:\: L_j(h) \subset [-C, \infty[, \:\: j =1,2,
\end{equation}
where sp $(L)$ denotes the spectrum of $L.$ This condition is a technical
one and we expect that by a more fine version of Proposition 1 we could
cover the general case.\\

Given $f \in C_0^{\infty}(\RR)$ independent on $h$ and $\chi \in
C_0^{\infty}(\RR^n)$ equal to 1 on $\overline{B(0, R_0)}$ we can define
$\trb [f(L_j)]_{j=1}^2$, as in \cite{Sj1}, \cite{Sj2}, by the equality
\[ \trb \Bigl(f(L_2) - f(L_1)\Bigr) = [\tr (\chi f(L_j)\chi + \chi f(L_j)
(1 - \chi) + (1 - \chi)  f(L_j)\chi) ]_{j=1}^2 \, \]
\[  + \tr [(1-\chi)f(L_j)(1- \chi)]_{j=1}^2 \,.\]
Following \cite{Sj1}, \cite{Sj2}, we can define the resonances $w \in
\overline{\C}_{-}$ by the complex scaling method as the eigenvalues of the
complex scaling operators $L_{j, \theta},\: j =1,2$. We denote by
${\rm{Res}}\: L_j(h), \: j =1,2,$ the set of resonances and set $\ns =
\max \{n_1^{\#}\:,n_2^{\#} \}.$



\section{Representation of the derivative of the spectral shift function}
 Consider the resolvents
$$R_j(\lambda \pm i\epsilon) = i\int_0^{\pm \infty} e^{it\lambda}
e^{-it(L_j \mp i\epsilon)}dt,\:\: \lambda \in \RR, \:\: \epsilon > 0.$$
$$R_j(\lambda - i\epsilon) =-i\int_{-\infty}^0 e^{it\lambda} e^{-it(L_j
+i\epsilon)}dt.$$
Given a function $f(\lambda) \in C_0^{\infty}(\RR)$, we have
$$\frac{1}{2 \pi i} \int R_j(\lambda + i\epsilon)f(\lambda)d\lambda =
\frac{1}{2 \pi }\int_0^{\infty} \hat{f}(-t) e^{-itL_j - t\epsilon} dt,$$
$$ - \frac{1}{2 \pi i}\int R_j(\lambda - i \epsilon) f(\lambda) d\lambda =
\frac{1}{2 \pi}\int_{-\infty}^0 \hat{f}(-t) e^{-itL_j + t\epsilon} dt,$$
where $\hat{f}$ denotes the Fourier transform of $f.$
Choose $z_0 \in \RR^{-}$ which is away from sp $(L_j),\:\: j = 1,2,$ and
set $g(\lambda) = (\lambda - z_0)^{m} f(\lambda)$, where the integer $m >
n/2$ will be taken sufficiently large and independent on $h$. Applying the
above formula, we obtain
\begin{equation}
 \label{eq:3.1}
\begin{split}
\frac{1}{2 \pi i} \trb \int \Bigl[ (L_j - z_0)^{-m} \Bigl((\lambda +
i\epsilon - z_0)^m R_j(\lambda + i \epsilon) - (\lambda - i\epsilon -
z_0)^m R_j(\lambda - i \epsilon)\Bigr) \Bigr]_{j=1}^2 f(\lambda) d\lambda
\\
 = \frac{1}{2 \pi} \trb \Bigl[ (L_j - z_0)^{-m} \Bigl( \int_0^{\infty}
e^{-\epsilon t-it L_j} (\hat{g}(-t)+ i \epsilon G_{+, \epsilon}(t))dt \\
+ \int_{-\infty}^0 e^{\epsilon t -tL_j}(\hat{g}(-t)+ i\epsilon G_{-,
\epsilon}(t)) dt \Bigl) \Bigr]_{j=1}^2.
\end{split}
\end{equation}
Here $G_{\pm, \epsilon}(t)$ are some functions in ${\mathcal S}(\RR)$
related to the Fourier transform of   $\lambda^k f(\lambda), \:\: 0 \leq k
\leq m - 1,$ which are uniformly bounded with respect to $0 < \epsilon <
1.$ To justify the limit $\epsilon \downarrow 0$ in (\ref{eq:3.1}), we
need to establish the estimates of the trace uniformly with respect to
$\epsilon > 0$. To do this we will prove the following.

\begin{lem}
For any $t \in \RR$, the trace $\trb \Bigl[  (L_j - z_0)^{-m} e^{- itL_j}
\Bigr]_{j=1}^2$ is well defined, and
$$\trb \Bigl[  (L_j - z_0)^{-m} e^{- itL_j}  \Bigr]_{j=1}^2 = {\cal
O}(h^{-\ns} (1 + |t|)).$$
\end{lem}
\begin{pf}
Let $\chi \in C_0^\infty(\RR^n)$ be equal to 1 near $\overline{B(0,R_1)},
\:\: R_1 > R_0$. Since the operators $\chi (L_j-z_0)^{-m}$ and
$(L_j-z_0)^{-m}\chi$ are
trace class (see \cite{Sj1}) and $e^{- itL_j}$ is uniformly bounded
with respect to $t$, it is clear that $\chi (L_j - z_0)^{-m}
e^{- itL_j}$ and  $(L_j - z_0)^{-m} e^{- itL_j}\chi$ are trace class ones
with trace bounded by ${\mathcal O}(h^{-\ns})$. To be more precise let us
note that in \cite{Sj2} the condition (2.10) is not assumed and we can
formally apply the results of \cite{Sj2} for $z_0 \in \C \setminus \RR.$
In our case $z_0 \in \RR^{-}$ and according to the resolvent equation we
have
\[ (L_j - z_0)^{-m} = (L_j - z_1)^{-m}\Bigl(I + (z_0 - z_1)(L_j -
z_0)^{-1}\Bigr)^m \,. \]
So taking $z_1 \in \C \setminus \RR$, we obtain the trace class properties
mentioned above.

 Now consider  the operator

$$\Bigl[(1- \chi) (L_j - z_0)^{-m} e^{- itL_j} (1-\chi)\Bigr]_{j=1}^2.$$
By Duhamel formula we obtain
\[ (1- \chi) (L_j - z_0)^{-m} e^{- itL_j} (1-\chi) = e^{-itQ_j}(1 -\chi)
(L_j - z_0)^{-m} (1- \chi) \, \]
\[ + i \int_0^t e^{-i(t-s)Q_j} [\chi, L_j] (L_j - z_0)^{-m}e^{-isL_j} ds
\,. \]
The integrand is a trace class operator with trace bounded by ${\mathcal
O}(h^{-\ns})$ and it remains to study the operator
\[\Bigl[ e^{-itQ_j}(1 - \chi)(L_j - z_0)^{-m} (1 - \chi)\Bigr ]_{j=1}^2
\,. \]
For $R_1 > R_0, \:\: \chi_0 \in C_0^{\infty}(\RR^n)$ equal to 1 near
$\overline{B(0, R_1)}$ and $\chi_0 \prec \chi$ we have
\[ (L_j - z_0)^{-1} (1- \chi) = (1 - \chi_0) (Q_j - z_0)^{-1} (1- \chi) +
(L_j - z_0)^{-1}[Q_j, \chi_0] (Q_j - z_0)^{-1} (1- \chi) \,. \]
Here and below the notation $\varphi \prec \psi$ means that $\psi = 1$ on
supp $\varphi.$
Choose cut-off functions $\theta_N \prec ... \prec \theta_1 \prec \chi$ so
that $\theta_N = 1$ on $\overline{B(0, R_0)}$ and apply the telescopic
formula
\[ (L_j - z_0)^{-1}[Q_j , \chi_0](Q_j - z_0)^{-1} (1- \chi)  \, \]
\[ = (L_j - z_0)^{-1}[Q_j, \chi_0](Q_j - z_0)^{-1}[Q_j , \theta_{N}](Q_j -
z_0)^{-1}[Q_j, \theta_{N-1}]...[Q_j, \theta_1 ] (Q_j - z_0)^{-1} (1 -
\chi) \,. \]
For $N > n/2$ this operator is trace class. In fact, for $\tilde{\chi} \in
C_0^{\infty}$ equal to 1 on supp $\theta_N$ the operator
\[ \tilde{\chi}(Q_j - i)^{-N/2}(Q_j - i)^{N/2}[Q_j, \theta_N](Q_j -
z_0)^{-1}...[Q_j , \theta_1 ] (Q_j - z_0)^{-1} (1 - \chi) \, \]
is trace class, while $(L_j - z_0)^{-1}[Q_j, \chi_0](Q_j - z_0)^{-1}$ is
bounded. Here we have used the fact that $Q_j$ are elliptic operators and
$$(Q_j - z_0)^{-1} = {\mathcal O}(1): H^{N}(\RR^n) \longrightarrow H^{N +
2}(\RR^n),\:\: \forall N \in \N.$$
Repeating this procedure, we obtain modulo trace class operators
\[ e^{-itQ_j}(L_j - z_0)^{-m}(1- \chi) \, \]
\[ = e^{-itQ_j}(1- \theta_m)(Q_j - z_0)^{-1}...(1-\theta_1)(Q_j -
z_0)^{-1}(1-\chi) \,. \]

In the same way, since $\theta_k \prec \theta_{k-1},$ each term
$\theta_k(Q_j - z_0)^{-1}(1- \theta_{k-1})$ in the above product is trace
class operator and modulo a trace class operator we  are going to study
$$\Bigl[e^{-itQ_j}(Q_j - z_0)^{-m}(1- \chi)\Bigr]_{j=1}^2.$$

Consider the difference
$$(Q_2 - z_0)^{-m} e^{-itQ_2} - (Q_1 - z_0)^{-m} e^{-itQ_1} $$
$$= e^{-itQ_2}\Bigl((Q_2 - z_0)^{-m} - (Q_1 - z_0)^{-m}\Bigr) +
\Bigl(e^{-itQ_2} - e^{-itQ_1}\Bigr) (Q_1 - z_0)^{-m}.$$
For the first term at the right hand side observe that the operator
$(Q_2 - z_0)^{-m} - (Q_1 - z_0)^{-m}$ for $m > n/2 $ is a trace class one
(see \cite{DS}, \cite{R1}, \cite{Sj1}). To handle the second term, notice
that
$$\Bigl(e^{-itQ_2} - e^{-itQ_1}\Bigr) (Q_1 - z_0)^{-m} = i\int_{0}^t
e^{-i(t-s)Q_2} (Q_1 - Q_2)(Q_1 - z_0)^{-m} e^{-isQ_1}ds$$
and use the fact that $(Q_1 - Q_2)(Q_1 - z_0)^{-m}$ is trace class for $m
> \frac{n}{2} + 1.$
\end{pf}

 According to Lemma 1, in the equation (3.1)  we can take the limit
$\epsilon \downarrow 0$ with respect to the norm in the space of trace
class operators and taking into account the definition of $\trb ( . )$, we
get
$$\lim_{\epsilon \downarrow 0} \frac{1}{2 \pi} \trb \Bigl[ (L_j -
z_0)^{-m} \Bigl(\int_0^{\infty} e^{-\epsilon t - it L_j} (\hat{g}(-t) +
i\epsilon G_{+, \epsilon}(t)) dt $$
$$ + \int_{-\infty}^0 e^{\epsilon t - itL_j} (\hat{g}(-t) + i\epsilon
G_{-, \epsilon}(t)) dt\Bigr)\Bigr]_{j=1}^2$$
$$= \frac{1}{2\pi} \trb \Bigl[ (L_j - z_0)^{-m} \int_{-\infty}^{\infty}
e^{-itL_j} \hat{g}(-t) dt \Bigr]_{j=1}^2$$
$$= \trb \Bigl[ (L_j - z_0)^{-m} g(L_j)\Bigr ]_{j=1}^2 = \trb \Bigl(f(L_1)
- f(L_2)\Bigr)= <\xi'(\lambda, h), f(\lambda)>_{{\mathcal D}'(\RR),
{\mathcal D}(\RR)}.$$


 Thus we have proved the following.
\begin{prop}  We have
\begin{equation} \label{eq:3.2}
\xi'(\lambda, h) = \frac{1}{2 \pi i} \lim_{\epsilon \downarrow  0}
\tr_{\rm bb} \Bigl[ \Bigl((\lambda + i\epsilon - z_0)^m(L_j -\lambda - i
\epsilon)^{-1}
\end{equation}
\[ - (\lambda - i\epsilon - z_0)^m(L_j -\lambda + i
\epsilon)^{-1}\Bigr)(L_j - z_0)^{-m} \Bigl]_{j=1}^2 \,, \]
where the limit is taken in the sense of distributions ${\mathcal
D}'(\RR).$
\end{prop}

Introduce the functions
\begin{equation}\label{eq:sigma}
\sigma_{\pm} (z) = {(z - z_0)^m}
 \tr_{\rm bb} \Bigl[ (L_j - z)^{-1} (L_j - z_0)^{-m} \Bigl]_{j=1}^2, \:\:
\pm \Im z > 0.
\end{equation}
which are well defined (see \cite{Sj2} and Proposition 2 below). The
relation
\[ \trb \Bigl[(L_j -(\lambda - i \epsilon))^{-1}(L_j -
z_0)^{-m}\Bigr]_{j=1}^2 = \overline{\trb \Bigl[(L_j -(\lambda + i
\epsilon))^{-1}(L_j - z_0)^{-m}\Bigr]_{j=1}^2} \,, \]
 implies immediately
\begin{equation} \label{eq:cong}
\sigma_{-}(z) = \overline{\sigma_{+}(\overline{z})}, \:\: \Im z < 0.
\end{equation}
The equality (\ref{eq:cong}) plays a crucial role in the proof of
(\ref{eq:1.3}) and our choice of real $z_0$ is related to the above
relation.


\section{Meromorphic continuation of the spectral shift function}


In this section we prove our principal result given in Theorem 1. Taking
$0 < \theta \leq \theta_0 < \pi/2$, consider the complex scaling operators
$L_{j, \theta}$ related to $L_j, \: j = 1,2,$ introduced by Sj\"ostrand
and Zworski (see  \cite{SZ}, \cite{Sj1} and Section 2 in \cite{Sj2}). More
precisely, given $\epsilon_0 > 0, \:  R_1 > R_0$, consider a function
$$f_{\theta}(t): \: ]0, \frac{\pi}{2}[ \times [0, \infty[ \ni (\theta, t)
\mapsto \C$$
which is injection for every $\theta$ and has the properties:
\[ f_{\theta}(t) = t\:\: {\rm{for}}\:\: 0 \leq t \leq R_1 \,, \]

\[  0 \leq \arg f_{\theta}(t) \leq \theta, \: \partial_t f_{\theta} \neq 0
\,, \]

\[ \arg f_{\theta}(t) \leq \arg \partial_t f_{\theta}(t) \leq \arg
f_{\theta} + \epsilon_0 \,, \]

\[ f_{\theta}(t) = e^{i \theta} t,\:\: {\rm{for}}\: t \geq T_0 \,, \]
where $T_0$ depends on $\epsilon_0$ and $R_1.$ Next consider the map
$$\kappa_{\theta}: \RR^n \ni x = t\omega \mapsto  f_{\theta}(t) \omega \in
\C^n, \: t = |x|$$
and introduce $\Gamma_{\theta} = \kappa_{\theta}(\RR^n)$ which coincides
with $\RR^n$ along $B(0, R_1).$ We define
$${\mathcal H}_{j,\theta} = {\mathcal H}_{R_0, j} \oplus
L^2(\Gamma_{\theta} \setminus B(0, R_0))$$
 and $L_{j,\theta} : {\mathcal H}_{j,\theta} \longrightarrow {\mathcal
H}_{j, \theta}$ with domain ${\mathcal D}_j$ as the operator
\[ L_{j, \theta} u = L_j(\chi_1 u) + Q_j\vert_{\Gamma_{\theta}}(1
-\chi_1)u \,, \]
$\chi_1 \in C_0^{\infty} (B(0, R_1))$ being a function equal to 1 near
$\overline{B(0, R_0)}$.\\

Let $\Omega \subset  e^{i]-2\theta, 2 \theta[}]0, +\infty[$ be a simply
connected open relatively compact set  such that $\Omega \cap \RR^{+} = J$
is an interval. The spectrum of $L_{j, \theta}$ outside of
$e^{-2i\theta}[0, +\infty[$ consists of the negative eigenvalues of $L_j$
and the eigenvalues in $e^{-i[0, 2 \theta[}]0, +\infty[$ (see \cite{Sj1}).
Since the spectrum of $L_j$ is  bounded from below, we may choose $z_0 \in
\RR^{-}, \: z_0 \notin \overline{\Omega}$, so that $z_0$ is away from sp
$(L_j)$ and sp $(L_{j,\theta}), \:j = 1,2.$
Given a positive number $\delta > 0$,  we can apply Proposition 4.1 of
Sj\"{o}strand \cite{Sj2},  saying that for all $z \in \Omega \cap \{z :
\Im z \geq \delta\}$ we have
\begin{equation} \label{eq:3.4}
\trb \Bigl[(L_j -z)^{-1}(L_j - z_0)^{-m}\Bigr]_{j=1}^2 =
\trb \Bigl[(L_{j, \theta} -z)^{-1}(L_{j,\theta} - z_0)^{-m}\Bigr]_{j=1}^2
,
\end{equation}
where in the definition of  the complex scaling operators $L_{j, \theta}$
the parameter $\epsilon_0$ is chosen small enough. Notice that the choice
of $z_0 \in e^{i[3\epsilon_0, \:\: \min (\pi,\: 2\pi - 2\theta - 3
\epsilon_0)]}]0, +\infty[$ in \cite{Sj2} says that we may take $z_0 \in
\RR^{-}$ , assuming $\theta < \frac{\pi}{2} - \frac{3}{2}\epsilon_0.$\\

Below we assume $\delta$ and $\theta$ fixed and we will drop in the
notations $L_j$ the index $j$ writing $L_{.}$ when the properties are
satisfied for both operators $L_j, \: j = 1,2.$ Following \cite{Sj2},
Section 4, there exists an operator $\hL_{.,\theta}: \: {\mathcal D_{.}}
\longrightarrow {\mathcal H}_{.}$
so that
$$K_{.,\theta} = \hL_{., \theta} - L_{.,\theta} \:{\rm has}\:\: {\rm
rank}\: {\mathcal O}(h^{-\ns})$$
 and for all $N, \: M \in \N$ we have
 $$K_{. , \theta} = {\mathcal O}(1):  {\mathcal D}(L_.^N) \longrightarrow
{\mathcal D}(L_.^M).
$$
Secondly, $K_{. , \theta}$ is compactly supported, that is if $\chi \in
C_0^{\infty} (\RR^n)$ is equal to 1 on $B(0, R)$ for
$R \geq R_0$ large enough, we have $K_{. , \theta} = \chi K_{. , \theta}
\chi$ and, finally, for every $N \in \N$ we have
$$(\hL_{., \theta} - z)^{-1} = {\mathcal O}(1):\:{\mathcal D}(L_{.}^N)
\longrightarrow {\mathcal D}(L_{.}^{N+1}),$$
uniformly for $z \in \overline{\Omega}.$ These properties imply for $z \in
\Omega \cap \{\Im z > 0\}$ the representation
\begin{equation}\label{3.3}
(L_{. , \theta} - z )^{-1} = (\hL_{. ,\theta} - z)^{-1} + (L_{. ,\theta} -
z )^{-1}K_{. ,\theta}(\hL_{. , \theta} - z)^{-1}.
\end{equation}

The contributions related to the resolvent $(\hL_{. ,\theta} - z)^{-1}$
are examined in the following.
\begin{prop}
There exists a function $a_{+}(z,h)$ holomorphic in $\Omega$ such that for
$z \in \Omega \cap \{\Im z > 0 \}$ we have
\begin{equation} \label{eq:pr2}
\sigma_{+}(z) =  \tr \Bigl[ (L_{j,\theta} - z)^{-1} K_{j,\theta}(\hL_{j,
\theta} - z)^{-1}\Bigr]_{j=1}^2 +  a_{+}(z,h).
\end{equation}
Moreover,
\begin{equation}
|a_{+}(z, h)| \leq C(\Omega) h^{-\ns}, \:\: z \in \Omega
\end{equation}
with a constant $C(\Omega)$ independent on $h \in ]0, h_0].$
\end{prop}

{\bf Remark.} The singularities of $\sigma_{+}(z)$ for $\Im z \downarrow
0$ are independent on $z_0 \in \RR^{-}$ and $m \in \N.$\\

{\em Proof.}
According to (\ref{3.3}), for $z \in \Omega \cap \{\Im z \geq \delta\}$ we
have
\begin{eqnarray} \label{eq:3.6}
\sigma_{+}(z) = (z-z_0)^m \trb \Bigl[  (\hL_{j, \theta} - z)^{-1}(L_{j,
\theta} - z_0)^{-m}\Bigr]_{j=1}^2\\ \label{eq:3.7}
 + (z-z_0)^m \Bigl[ \tr \Bigl((L_{j, \theta} - z)^{-1} K_{j,
\theta}(\hL_{j, \theta} - z)^{-1}(L_{j, \theta} -
z_0)^{-m}\Bigr)\Bigr]_{j=1}^2.
\end{eqnarray}

>From the resolvent equation we obtain
$$ (z-z_0)^m   (L_{j, \theta} - z_0)^{-m}(L_{j, \theta} - z)^{-1}=
(L_{j, \theta} - z)^{-1} - \sum_{k=1}^m (z-z_0)^{k-1}   (L_{j, \theta} -
z_0)^{-k}.$$
To treat (\ref{eq:3.7}) we use the cyclicity of the trace and the above
equality and conclude that this term is equal to $\tr \Bigl[ (L_{j,\theta}
- z)^{-1} K_{j,\theta}(\hL_{j, \theta} - z)^{-1}\Bigr]_{j=1}^2$ modulo a
function  holomorphic in $\Omega$ and bounded by ${\mathcal O}(h^{-\ns}).$

Now we pass to the analysis of (\ref{eq:3.6}). Our purpose is to show that
(\ref{eq:3.6}) is holomorphic in $\Omega$ and bounded by ${\mathcal
O}(h^{-\ns}).$
By construction, $(\hL_{j, \theta} - z)^{-1}$ is holomorphic on $\Omega$
and for any cut-off function $\chi \in C_0^{\infty}(\RR^n),\: \chi =1$ on
$\overline{B(0, R_0)}$ with supp $\chi \subset B(0, R_1)$ the operators
$\chi  (L_{j, \theta} - z_0)^{-m}, \: (L_{j, \theta} - z_0)^{-m} \chi$
are  trace class ones. Hence the function $\tr \Bigl((\hL_{j, \theta} -
z)^{-1} (L_{j, \theta} - z_0)^{-m}\chi \Bigr)$ is holomorphic in $\Omega$.
On the other hand,
\begin{equation} \label{eq:3.11}
 (L_{j, \theta}- z_0)^{-m}(\hL_{j, \theta} - z)^{-1} - (\hL_{j, \theta} -
z)^{-1}(L_{j, \theta} - z_0)^{-m}
\end{equation}
$$= (L_{j, \theta} - z_0)^{-m}(L_{j,\theta} - z)^{-1}K_{j, \theta}
(\hL_{j, \theta} - z)^{-1}-
(L_{j, \theta} - z)^{-1} K_{j, \theta} (\hL_{j, \theta} - z)^{-1}(L_{j,
\theta} - z_0 )^{-m}.$$
 Consequently, for $\Im z > 0$ if $\chi_1 \in C_0^{\infty}(\RR^n)$ is a
cut-off function and $\chi_1 \prec \chi$, applying the cyclicity of the
trace once more, we get
$$\tr \Bigl(\chi_1(\hL_{j, \theta} - z)^{-1}(L_{j, \theta} - z_0)^{-m}(1
-\chi)\Bigr) = 0.$$
Thus it remains to examine
$$ \tau_{+}(z) = \tr \Bigl[  (1-\chi_1)(\hat{L}_{j, \theta} - z)^{-1}(1 -
\chi)(L_{j, \theta} - z_0)^{-m} (1-\chi)\Bigr]_{j=1}^2.$$

Consider the operator $Q_{.,\theta} = Q_. \vert_{\Gamma_{\theta}}$
and note that for $\psi \in C^{\infty}$ supported away from $B(0, R_1)$ we
have $L_{.,\theta} \psi = Q_{., \theta} \psi.$ Repeating the construction
of $\hL_{.,\theta}$ in  Section 4, \cite{Sj2}, we can find  an operator
$\hQ_{.,\theta}: \: H^2(\Gamma_{\theta}) \longrightarrow
L^2(\Gamma_{\theta}) $
so that
$$ \hQ_{., \theta} - Q_{.,\theta} \:{\rm has}\:\: {\rm rank}\: {\mathcal
O}(h^{-n}),$$
the operator $ \hQ_{., \theta} - Q_{.,\theta}$ is  compactly supported and
for $ z \in  \overline{\Omega}$ we have
 $$( \hQ_{., \theta}-z)^{-1} = {\mathcal O}(1) : D(Q_.^N)\longrightarrow
D(Q_.^{N+1}),\:\: \forall N \in \N.$$
 Moreover,  for $\psi \in C^{\infty}$ supported away from $ B(0,R_1)$ we
have $\hL_{.,\theta}\psi = \hat{Q}_{., \theta} \psi$ and for $\chi \in
C_0^\infty(\Gamma_\theta)$ equal to $1$ on a sufficiently large set, $z
\in \Omega$ and $\chi_1 \prec \chi_0 \prec \chi$ we obtain
$$(\hL_{., \theta}-z)^{-1} (1- \chi) = (1- \chi_0) (\hQ_{.,
\theta}-z)^{-1}(1-\chi)$$
$$+(\hL_{., \theta}-z)^{-1} [\hQ_{.,\theta}, \chi_0] (\hQ_{.,
\theta}-z)^{-1}(1- \chi).$$
As above, we assume that $z_0 \in \RR^{-}$ is chosen so that $z_0 \notin
{\rm sp}\:(Q_j),\: z_0 \notin {\rm sp}\:(Q_{j,\theta}), \:\: j =1,2.$ For
simplicity of the notations below we omit the index $\theta$ and we get
$$\tau_{+}(z) =  \tr \Bigl[ (1-\chi_0) (\hQ_j - z)^{-1} (1-\chi)(L_{j} -
z_0)^{-m}(1 - \chi)\Bigr]_{j=1}^2$$
$$+  \tr \Bigl[  (1-\chi_1)(\hL_j - z)^{-1}[\hQ_j, \chi_0] (\hQ_j -z)^{-1}
(1-\chi)(L_{j} - z_0)^{-m}(1 - \chi)\Bigr]_{j=1}^2.$$
Obviously, $[\hQ_j , \chi_0] = [Q_j, \chi_0] + M_j$ with a trace class operator $M_j$. To show that the operator $[Q_j,
\chi_0] (\hQ_j - z)^{-1}(1 - \chi)$ is a trace class one, we apply the
telescopic formula choosing cut-off functions $\theta_N \prec \theta_{N-1}
\prec ... \prec \theta_1 \prec \chi$ and write
$$[Q_j, \chi_0](\hQ_j - z)^{-1}(1- \chi) = [Q_j, \chi_0](\hQ_j
-z)^{-1}\chi(Q_j - i)^{-m} $$
$$\times \Bigl [(Q_j - i)^m
 [\hQ_j, \theta_{N}] (\hQ_j - z)^{-1}[\hQ_j , \theta_{N-1}]...[\hQ_j ,
\theta_1] (\hQ_j - z)^{-1} (1 - \chi)\Bigr]$$
with $N \geq 2m > n.$
 The operator in the brackets $[...]$ and $[Q_j, \chi_0] (\hQ_j - z)^{-1}$
are bounded, while $\chi (Q_j - i)^{-m}$ is trace class. Thus the term
involving $[\hQ_j, \chi_0]$ is holomorphic in $\Omega$ and bounded by
${\mathcal O}(h^{-\ns}).$\\

 As in the proof of Proposition 1, we have
$$\| (1-\chi) (L_{j} - z_0)^{-m}(1-\chi) - (1-\chi) (Q_{j} -
z_0)^{-m}(1-\chi)\|_{\tr} = {\cal O}(h^{-\ns}).$$
Moreover, $(Q_j - z_0)^{-m}\chi$ is trace class and, consequently, there
exists a function $b(z,h)$ holomorphic in $\Omega$ and bounded by ${\cal
O}(h^{-\ns})$ so that
\begin{equation} \label{3.9}
\tau_{+}(z)= b(z, h) + \tr \Bigl[  (1-\chi) (\hQ_j - z)^{-1}(Q_{j} -
z_0)^{-m} (1-\chi)\Bigr]_{j=1}^2.
\end{equation}

We write
\[(\hQ_2 - z)^{-1}(Q_2 - z_0)^{-m} - (\hQ_1 - z)^{-1}(Q_1 - z_0)^{-m} \]
\[ = (\hQ_2 - z)^{-1}\Bigl[ (Q_2 - z_0)^{-m} - (Q_1 - z_0)^{-m} \Bigr] +
\Bigl[(\hQ_2 - z)^{-1} - (\hQ_1 - z)^{-1}\Bigr] (Q_1 - z_0)^{-m} = I + II
\,. \]

According to \cite{Sj1}, \cite{Sj2}, the  operator $(Q_2 - z_0)^{-m} -
(Q_1 - z_0)^{-m}$ is trace class one and the contribution of I is
holomorphic and bounded by ${\cal O}(h^{-\ns})$.  For II we obtain the
representation
\[ II  = (\hQ_2 - z)^{-1} (\hQ_1 - \hQ_2) (\hQ_1 - z )^{-1}( Q_1 -
z_0)^{-m} \,. \]
It is clear that $\hQ_1 - \hQ_2 = Q_1 - Q_2 + K_{1,2}$ with a finite rank
operator $K_{1,2}$, and modulo a trace class operator we have
\[ II =  (\hQ_2 - z )^{-1}\Bigl((Q_1 - Q_2) (Q_2 -
z_0)^{-m}\Bigr)\Bigl((Q_2 - z_0)^m(\hQ_1 -z)^{-1}(Q_1 - z_0)^{-m} \Bigr)
\,. \]
The second factor is a trace class operator, while the first and the third
ones are bounded operators. Consequently, II has the same property as I.
Combining the above results, we conclude that $\tau_{+}(z)$ is holomorphic
in $\Omega$ and bounded by ${\cal O}(h^{-\ns})$.\\

To establish (\ref{eq:pr2}), notice that the right hand side of this
equality is holomorphic for $z \in \Omega \cap \{\Im z > 0 \}.$ The left
hand side is also holomorphic in this domain since we may apply
(\ref{eq:3.4}) with different $\delta > 0, \: \epsilon_0 > 0$ and $0 <
\theta < \frac{\pi}{2} - \frac{3}{2}\epsilon_0.$ By analytic continuation
we deduce (\ref{eq:pr2}) and the proof of Proposition 2 is complete.
\hfill{$\Box$}\\

 {\em Proof of Theorem $1$.} To obtain a meromorphic continuation of
$\sigma_{+}(z)$ through the real axis, it suffices to do this for the
trace involving $K_{j, \theta}.$ Next we will follow closely the argument
of Sj\"ostrand \cite{Sj2} and since $\theta$ is fixed,  we will omit it in
the notations. Setting $\tilde{K}_.(z) = K_.(z - \hL_.)^{-1},$  from
(4.31) in \cite{Sj2} we get the representation
$$- \tr((L_. - z)^{-1}K_.(\hL_. - z)^{-1}) = \tr \Bigl((1 +
\tilde{K}_.(z))^{-1}\frac{\partial}{\partial z} \tilde{K}_.(z)\Bigr) \, \]

\[  =  \partial_z \log \det (1 + \tilde{K}_. (z)) \, \]
and the resonances of $L_.$ are precisely the zeros of the function
\begin{equation}
 D(z, h) = \det(1 + \tilde{K}_.(z)) = {\mathcal O}(1) \exp(Ch^{-\ns}).
\end{equation}
 Notice that the multiplicities of the resonances and the zeros coincide.
Below in the notations we omit the subscript . since the argument does not depend on $j
= 1,2$. Let Res $(L)$ be the resonances of $L$ and let
\[ D(z, h) = G(z, h) \prod_{w \in {\rm{Res}}\: (L) \cap \Omega}(z - w) \,,
\]
where $G(z, h)$ and $\frac{1}{G(z, h)}$ are holomorphic in $\Omega$ and
the resonances in the product are repeated following their multiplicity.
Obviously,
\[ \partial_z \log D(z, h) = \partial_z \log G(z, h) + \sum_{w \in
{\rm{Res}} \:(L) \cap \Omega} \frac{1}{z - w} \, \]
 and according to the estimate (4.54) in \cite{Sj2}, we get
\begin{equation} \label{eq:3.13}
 \Bigl|\frac{\partial}{\partial z} \log G(z, h)\Bigr| \leq
C(\tilde{\Omega})h^{-\ns}, \:\: z \in \tilde{\Omega},
\end{equation}
where $\tilde{\Omega} \subset \subset \Omega$ is an arbitrary open simply
connected domain and $C(\tilde{\Omega})$ is independent on $h \in ]0,
h_0].$\\

Going back to  the representation (\ref{eq:3.2}) and taking into account
(\ref{eq:cong}), we observe that for $\lambda \in I \subset \RR^{+},\:\:
\Im w \neq 0,$ we have
\[- \frac{1}{2 \pi i}\lim_{\epsilon \downarrow 0} \Bigl(\frac{1}{\lambda +
i\epsilon - w} - \frac{1}{\lambda - i\epsilon -\overline{w}}\Bigr) =
\frac{- \Im w}{\pi |\lambda - w|^2} \,, \]
while for $w \in \RR$ we get
\[ - \frac{1}{2 \pi i}\lim_{\epsilon \downarrow 0} \Bigl(\frac{1}{\lambda
+ i\epsilon - w} - \frac{1}{\lambda - i\epsilon - w}\Bigr) = \delta
(\lambda - w) \,, \]
where both limits are taken in the sense of distributions. Combining
Propositions 1, 2 and the above arguments we complete the proof of Theorem
1.
\hfill{$\Box$}\\

The representation (\ref{eq:1.3}) shows that modulo a constant the
spectral shift function $\xi(\lambda, h)$ coincides with the distribution
\[ \xi(\lambda, h) =  \frac{1}{\pi} \bigl[ \sum_{w \in
{\rm{Res}}\:L_j(h)\atop \Im w \neq 0} \int_{\lambda_0}^{\lambda} \brw d\mu
\bigr]_{j=1}^2 \, \]
\[ + \Bigl[ \# \{ \mu \in [\lambda_0, \lambda]: \: \mu \in
\sigma_{pp}(L_j(h)) \} \Bigr]_{j=1}^2 + \frac{1}{\pi}
\int_{\lambda_0}^{\lambda} \Im r(\mu, h) d\mu,\:\: \lambda_0 > 0, \:
\lambda_0 \notin I \,. \]
In particular, for $\lambda \in I \setminus \cup_{j=1}^2 \{  \lambda \in
\RR: \lambda \in {\rm sp}_{pp}(L_j(h)) \}$ the distribution $\xi(\lambda,
h)$ is continuous and the function
$$\eta(\lambda, h) = \xi(\lambda, h) - \Bigl[ \# \{ \mu \in [\lambda_0,
\lambda]: \: \mu \in {\rm sp}_{pp}(L_j(h)) \} \Bigr]_{j=1}^2$$
is real analytic in $I$.\\


\section{Weyl asymptotics}

In this section we obtain a Weyl type asymptotics for the spectral shift
function. We generalize the results of Christiansen \cite{Ch} and Robert
\cite{R2}
covering the "black box" long-range perturbations of the Laplacian and we
improve our previous result (see Theorem 2 in \cite{BrPe2}) working
without any condition on the behavior of the resonances close to the real
axis.

 We will say that $\lambda \in \RR$ is a {\em non-critical  energy level} for $Q$ if for all  $(x,\xi) \in \Sigma_\la=\{(x,\xi) \in \RR^{2n}:
l(x,\xi) =\lambda\}$ we have  $\nabla_{x,\xi} l(x,\xi) \neq 0,\: l(x,
\xi)$ being the principal symbol of $Q.$ Given a Hamiltonian $l(x,\xi)$,
denote by
 $$\exp(tH_l)(x_0, \xi_0) =  (x(t,x_0,\xi_0),\: \xi(t, x_0, \xi_0))$$
 the trajectory of the Hamilton flow $\exp(tH_{l})$ passing through $(x_0,
\xi_0) \in \Sigma_{\lambda}.$ Recall that  $\la \in J$ is a {\em
non-trapping energy level} for $l(x,\xi)$  if for every $R>0$ there exists
$T(R) > 0$ such that
for $(x_0,\xi_0) \in \Sigma_\la$, $|x_0| <  R$, the $x$-component of the
trajectory of $\exp(tH_l)$ passing through $(x_0,\xi_0)$ satisfies
$$ \quad |x(t, x_0,\xi_0)| > R,\:\:\forall | t | > T(R). $$

 Denote by $N(L_j^\#,I)$ the number of eigenvalues of $L_j^\#$ in
the interval $I$. From the assumptions (\ref{eq:2.5}) and (\ref{eq:2.10}) we deduce easily
that there exists a constant $C^\#$ such that the spectrums of
$L_j^\#$, $j=1,2$, do not intersect the interval $]-\infty,
-C^\#]$ and consequently  $N(L_j^{\#}, ]-\infty,  -C^\#])=0$.
In fact, let $\chi_0$, $\chi$, $\chi_1 \in C_0^\infty(B(0,R);[0,1])$ be
equal to $1$ on $\overline{B(0,R_0)}$ and let $\chi_1 \succ \chi \succ
\chi_0$. Using the resolvent equality we get
\begin{eqnarray}
(L_j^\# -z)^{-1}& =&  (L_j^\# -z)^{-1}\chi +  (L_j^\#
-z)^{-1}(1-\chi)\nonumber \\
        & = & \chi_1 (L_j -z)^{-1}\chi - (L_j^\# -z)^{-1} [Q_j^\#,
\chi_1] (L_j -z)^{-1}\chi \nonumber\\
& & +(1- \chi_0)(Q_j^\# -z)^{-1}(1-\chi) + (L_j^\#
-z)^{-1}[Q_j^\#, \chi_0] (Q_j^\# -z)^{-1}(1-\chi).\nonumber
\end{eqnarray}
Then
$$(L_j^\# -z)^{-1} \Big( 1 +[Q_j^\#, \chi_1] (L_j -z)^{-1}\chi-
[Q_j^\#, \chi_0] (Q_j^\# -z)^{-1}(1-\chi) \Big)$$
$$= \chi_1 (L_j -z)^{-1}\chi +(1- \chi_0)(Q_j^\# -z)^{-1}(1-\chi).$$
According to the   assumptions (\ref{eq:2.5}) and (\ref{eq:2.10}) there
exists $C^\#$ such that spectrums of $L_j$, $Q_j^\#,$ $j=1,2$, do
not intersect the interval
$]-\infty, -C^\#]$, hence for $z \in ]-\infty, -C^\#]$, the
resolvents $(L_j -z)^{-1}$, $(Q_j^\# -z)^{-1}$ are bounded and we
obtain immediately
$$ [Q_j^\#, \chi_1] (L_j -z)^{-1}\chi- [Q_j^\#, \chi_0]
(Q_j^\# -z)^{-1}(1-\chi) = {\cal O}(h).$$
Consequently, for $h$ small enough and  $z \in ]-\infty, -C^{\#}]$,
the resolvent $(L_j^\# -z)^{-1} $ is bounded and $z \notin$
sp$(L_j^\#)$. In the following we will use the notation
\[ N(L_j^{\#}, \lambda) = N(L^{\#}_j, ]-C^{\#}, \lambda]), \:\: j =1,2 \,. \]

The spectral shift function $\xi(\lambda, h)$ is determined modulo a constant
and from (\ref{eq:2.10}) we deduce that $\xi(\lambda, h)$ is constant on
$]-\infty, -C_1]$ for $C_1$ sufficiently large. In the following, without
loss of the generality, we may choose $\xi(\lambda, h)$ so that
$\xi(\lambda, h)  =0$ on $]-\infty, -C^{\#}].$ Moreover, in this section we consider $\xi(\lambda, h) = \lim_{\epsilon \downarrow 0} \xi(\lambda + \epsilon, h)$ as a function continuous from the right. The main result in this section is a Weyl type asymptotics for the spectral shift function.

\begin{thm}\label{weyl}
Assume that $L_j, \: j =1,2$ satisfy the assumptions of Section $2$. Let $0 < E_0 < E_1$ and suppose  that each $\lambda \in[E_0, E_1]$ is a non-critical energy level for $Q_j,\:Q_j^{\#},\: j=1, 2$. Assume that there exist positive constants  $B,\:\epsilon_1,\:C_1,\: h_1$ such that for any $\lambda \in [E_0 - \epsilon_1, E_1 + \epsilon_1],\:\:h/B \leq \delta \leq B$ and $h \in ]0, h_1]$ we have
\begin{equation} \label{eq:5.1}
N(L^{\#}, [\lambda - \delta, \lambda + \delta]) \leq C_1 \delta h^{-\ns},\:\; j= 1,2.
\end{equation}
Then there exist $\omega (\lambda) \in C^0(\RR),\: h_0 > 0$ such that
\begin{equation}\label{eq:5.2}
\xi(\lambda, h) = \left[ N(L_j^{\#}, \lambda]) \right]_{j=1}^2   +
\omega(\lambda)h^{-n} + {\cal O}(h^{1- \ns})
\end{equation}
uniformly with respect to $\lambda \in[E_0,E_1]$ and  $h \in ]0, h_0]$.
\end{thm}

{\bf Remark.}
Notice that if $\lambda$ is a  non-critical energy level, then for
$\epsilon >0$ small enough  each $\mu \in ]\lambda-\epsilon,
\lambda+\epsilon[$ is also  non-critical one. Consequently,
$(\ref{eq:5.2})$ remains valid on some interval $[E_0-\alpha,E_1+\alpha],
\: \alpha > 0$. Recall that the operators $L_j^{\#},\: j =1,2,$ have been defined in Section 2 by using the operators $Q_j^{\#},\: j=1,2, $ whose coefficients satisfy $a_{j,\nu}^{\#}(x;h) = a_{j,\nu}(x;h)$ for $|x| \leq R, \: R > R_0.$ If the principal symbol $l_j(x.\xi)$ of $Q_j$ is non-critical for $\lambda \in [E_0, E_1],$ we can extend $a_{j,\nu}^{\#}(x;h)$ for $|x| > R$ in a such way that $\lambda \in [E_0, E_1]$ become non-critical for $Q_j^{\#}$. This continuation changes the operator $L_j^{\#}$ but as it has been proved by J.-F. Bony \cite{Bo1}, the assumption (5.1) does not depend on the continuation of $a_{j,\nu}^{\#}(x;h)$.\\

To prove Theorem 2, we will introduce intermediate operators exploiting
the following result of J.-F. Bony (see also \cite{Sj}).

\begin{prop}\label{prop:bony} (\cite{Bo2})
 Assume that $L$  satisfy the
assumptions of Section $2$ and suppose  that each  $\lambda \in[E_0,E_1]$
is a  non-critical energy level for $Q$. Given a fixed $\lambda \in [E_0, E_1]$, there exists a differential operator $\tL$, such that\\

$(a)$ The pair $(L,\tL)$ satisfies the assumptions of Section $2$, with ${\overline n}= n+1$,\\

$(b)$ There exists an interval $I_0 \ni \lambda$, such that each $\mu \in
I_0$ is non-trapping and non-critical energy level for $\tL$,\\

$(c)$ The operator $\tL$ has no resonances in a complex neighborhood
$\Omega_0$ of $I_0$ and $\Omega_0$  is independent on $h$.
\end{prop}

Now denote by $\xi(\lambda; A, B)$ the spectral shift function related to
the operators $A$ and $B$. Using the above proposition for each operator
$L_j$, $j=1,2$, we can construct operators $\tL_j,\: j =1,2,$ and decompose the
spectral shift function  $\xi(\lambda; L_1,L_2)$ as follows
$$\xi(\lambda; L_1,L_2) = \xi(\lambda; L_1,\tL_1)+\xi(\lambda;
\tL_1,\tL_2) - \xi(\lambda; L_2,\tL_2).$$
Moreover, if every $\lambda \in [\alpha, \beta]$ is  non-trapping and
non-critical energy level for $\tL_j,\: j = 1, 2$, we have a complete
asymptotic expansion of $\xi(\lambda; \tL_1,\tL_2)$ and its derivatives
(see \cite{R2}) uniformly with respect to $\lambda \in [\alpha, \beta]$. Here we may estimate the difference $\tL_1 - \tL_2 = (\tL_1 - L_1) + (L_1 - L_2) + (L_2 - \tL_2)$ by applying our assumptions on $Q_1 - Q_2 .$
Thus it is sufficient to prove the theorem for $\lambda \in I_2 \subset I_0$
and the  pair $(L_1,L_2)$ with $L_2=Q_2$ being a differential operator
having no resonances in a complex neighborhood $\Omega_0$ of $I_0$ and such that every $\lambda \in I_0$ is non-trapping and non-critical energy level for $L_2$. Then the assertion
follows by applying the local result and covering the compact interval
$[E_0, E_1]$ by small intervals.\\

We denote $ \xi(\lambda, h)$ the spectral shift function for the operators
$(L_1,L_2)$. Applying Theorem 1 in the domain $\Omega_0$, we deduce that there
exists a function $g_+(z,h)$ holomorphic in $\Omega_0$ such that for $\lambda \in I_0 =
W_0 \cap \RR$, $W_0 \subset \subset \Omega_0$ we have
\begin{equation} \label{eq:5.3}
 \xi'(\lambda, h) = \frac{1}{\pi} \Im g_+(\lambda, h) + \sum_{w \in {\rm Res}\:L_1
\cap \Omega_0, \atop \Im w \neq 0} \frac{-\Im w}{\pi |\lambda - w|^2} +
\sum_{w  \in {\rm Res}\:L_1 \cap I_0} \delta (\lambda - w),
\end{equation}
where  $g_{+}(z, h)$ satisfies the estimate
\begin{equation} \label{eq:1.4bis}
|g_{+}(z, h)| \leq C(W_0)h^{-\ns},\:\: z \in W_0
\end{equation}
with $C(W_0) > 0$ independent on $h \in ]0, h_0].$\\


In the following, we fix  an open interval $I_0 \subset \RR^+$ so that
each $\mu \in I_0$ is a non-critical energy level for $Q_j$, $j=1,2,$ and
we introduce open intervals $I_2  \subset \subset I_1 \subset \subset
I_0$. It is convenient to decompose $\xi(\lambda, h)$ for $\lambda \in I_2$ into a
sum of a term independent on $\lambda$ and a second one localized in $I_0$
where (\ref{eq:5.3}) holds.

\begin{lem}\label{lem2}
Let $C^{\#} > 0$ be such that the spectrums of $L_j$ and $L^\#_j, \: j =1,2,$
do not intersect the interval $ [- \infty, -C^{\#}]$. Let $\varphi_1$,
$\varphi_2 \in C^\infty_0(\RR; \RR^+)$ be  such that supp $\varphi_1 \subset (-\infty, \gamma_1),\:supp\: \varphi_2 \subset I_1$, $\varphi_2 = 1$ on $I_2 = (\gamma_1, \: \gamma_2)$ and
$\varphi_1 + \varphi_2 = 1$ on $[-C^{\#}- \eta_0, \gamma_2], \: \eta_0 > 0$. Then for $\lambda \in I_2$ we have
\begin{equation} \label{eq:xi}
\xi(\lambda, h) = \trb \Bigl[ \varphi_1(L_j)\Bigr]_{j=1}^2 + G_{\varphi_2}(\lambda) +
M_{\varphi_2}(\lambda),
\end{equation}
where
$$G_{\varphi_2}(\lambda) = \frac{1}{\pi} \int_{]-\infty,\lambda]}
\Im g_+(\mu, h) \varphi_2(\mu) d \mu, $$
\begin{equation} \label{eq:repr}
M_{\varphi_2}(\lambda) =  \sum_{w \in {\rm Res}\:L_1 \cap \Omega_0, \atop \Im
w \neq 0} \int_{]-\infty,\lambda]}\frac{-\Im w}{\pi |\mu -
w|^2}{\varphi_2}(\mu) d \mu + \sum_{w  \in {\rm Res}\:L_1 \cap ]-C^{\#},
\lambda]} \varphi_2 (w)
\end{equation}
and we omit in $M_{\varphi_2}$ and $G_{\varphi_2}$ the dependence of $h$. 
\end{lem}
\begin{pf}
Roughly speaking, for $\lambda \in I_2$, if we express the action of the distributions as integrals, we must have
$$\xi(\lambda, h) = \int_{-\infty}^{\lambda} \varphi_1(\mu) \xi'(\mu, h) d\mu +
\int_{-\infty}^{\lambda} \varphi_2(\mu) \xi'(\mu, h) d\mu.$$
Since $ \varphi_1$ vanishes on $I_2$, the first term is independent on
$\lambda \in I_2$ and equal to $\trb \Bigl[\varphi_1(L_j)\Bigr]_{j=1}^2$. For the second one we may apply (\ref{eq:5.3}) since $\varphi_2$ is supported in $I_1\subset I_0$.

For a rigorous proof of the above representation, take $f \in
C_0^\infty(I_2)$ and  introduce
$$F(\lambda)= ( \varphi_1 + \varphi_2)(\lambda) \: \int_{\lambda}^{+
\infty} f(\mu) d\mu$$
which is compactly supported.  Since supp $f \subset I_2$ and
$\varphi_1+\varphi_2 =1$ on $I_2$, we have
$$F'(\lambda) = -  f(\lambda) + ( {\varphi}'_1 + {\varphi'}_2)(\lambda) \:
\int_{\lambda}^{+ \infty} f(\mu) d\mu,$$
where the second term vanishes on $[-C^{\#}- \eta_0, +\infty[$. Our choice of
$\xi(\lambda, h)  = 0$ on $]-\infty, -C^{\#}]$ makes possible to write
$$\langle \xi, f \rangle_{{\cal D}', {\cal D}}= - \langle \xi, F'
\rangle_{{\cal D}', {\cal D}}= \langle \xi', F \rangle_{{\cal D}', {\cal
D}}.$$
Next the equality $\varphi_1  \int_{\lambda}^{+ \infty} f= \varphi_1
\int_\RR f$ yields
$$\langle \xi',\varphi_1  \int_{\lambda}^{+ \infty} f  \rangle_{{\cal D}',
{\cal D}}= \Bigl(\int_\RR f\Bigr) \langle \xi',\varphi_1   \rangle_{{\cal
D}', {\cal D}} = \Bigl(\int_\RR f\Bigr)
\trb \Bigl[ \varphi_1(L_j)\Bigr]_{j=1}^2.$$
For the term involving $\varphi_2$, we apply (\ref{eq:5.3}) and we get
$$\langle \xi',\varphi_2  \int_{\lambda}^{+ \infty} f  \rangle_{{\cal D}',
{\cal D}}=
\langle G_{\varphi_2}',\psi  \int_{\lambda}^{+ \infty} f  \rangle_{{\cal
D}', {\cal D}}+
\langle M_{\varphi_2}',\psi  \int_{\lambda}^{+ \infty} f  \rangle_{{\cal
D}', {\cal D}}$$
for  $\psi \in C^\infty(\RR)$ equal to $1$ on $\RR^+$ and vanishing on
$]-\infty, -1]$.
The above relations imply (\ref{eq:xi}) in the sense of distributions
since $G_{\varphi_2}\psi'=M_{\varphi_2}\psi'=0$ and $\psi f=f$.
\end{pf}


To prove Theorem 2, we will apply a Tauberian argument for the increasing
function $M_{\varphi_2}(\lambda)$. Consider a function $\theta (t) \in
C_0^\infty(]-\delta_1,\delta_1[)$, $\theta(0)=1$,
$\theta(-t)=\theta(t)$, such that the Fourier transform $\hat{\theta}$ of $\theta$
satisfies ${\hat \theta}(\lambda) \geq 0$ on $\RR$ and assume that there
exist $0 <\epsilon_0 < 1$, $\delta_0 >0$ so that ${\hat \theta}(\la) \geq
\delta_0 > 0$ for $\mid \la \mid \leq \epsilon_0$. Next introduce
$$\Bigl({\cal F}_h^{-1} \theta\Bigr)(\lambda) = (2\pi h )^{-1} \int e^{it
\lambda/h}\theta (t) dt = (2\pi h )^{-1}{\hat \theta}(- h ^{-1}\lambda).$$

{\bf Remark.} It is obvious that the Lemma 2 holds if we take a partition of unity
$\varphi_1^2 + \varphi_2^2$ over $[-C^{\#} - \eta_0, \gamma_2]$ with cut-off functions $\varphi_j,\: j=1,2$.\\

The next lemma permits to establish a connection between the asymptotics
of the functions $M_{\varphi_2}$ and $N^{\#}_{\varphi_2}.$

\begin{lem}\label{lem3}
Let $\varphi_2  \in C^\infty_0(I_1; \RR^+)$ and let
$N_{\varphi_2}^{\#}(\lambda)= \tr \Big( \varphi_2(L_1^\#){ \bf
1}_{]-C^{\#}, \lambda]}(L_1^\#) \Big)$. Then there exists
$\omega_{\varphi_2}(\lambda) \in C^0_0(I_0)$ such that for any $\lambda \in \RR$ we have
\begin{equation}\label{eq:5.5}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * M_{\varphi_2}
)(\lambda)=\frac{d}{d \lambda}({\cal F}_h^{-1} \theta
*N_{\varphi_2}^{\#})(\lambda)-  G_{\varphi_2}'(\lambda)
+ \omega_{\varphi_2}(\lambda)h^{-n} +  {\mathcal O}(h^{1-n^{\#}}),
\end{equation}
where ${\mathcal O}(h^{1-n^{\#}})$ is  uniform with respect to $\lambda \in \RR.$ Moreover,
 we have
\begin{equation} \label{eq:mrep}
M_{\varphi_2}(\lambda) = ({\cal F}_h^{-1} \theta *  M_{\varphi_2} )(\lambda) +
{\cal O}(h^{1-n^\#})
\end{equation}
$$= ({\cal F}_h^{-1} \theta *  N^\#_{\varphi_2} )(\lambda)-
G_{\varphi_2}(\lambda) + \int_{-\infty}^{\lambda}\omega_{\varphi_2}(\mu)d\mu  h^{-n} +  {\cal O}(h^{1-n^\#})$$
uniformly with respect to $\lambda \in I_0$.
\end{lem}

\begin{pf} For simplicity of the notations we omit the subscript
$\varphi_2$ and denote by $M$, $G$, $N^\#$, $\omega$ the functions
$M_{\varphi_2}$, $G_{\varphi_2}$, $N^\#_{\varphi_2}$,
$\omega_{\varphi_2}$.
According to (\ref{eq:repr}) and (\ref{eq:5.3}), for any $\lambda \in \RR$ we
have
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * M )(\lambda)= ({\cal F}_h^{-1}
\theta * M')(\lambda) = ({\cal F}_h^{-1} \theta * \varphi_2 \xi' )(\lambda) -
({\cal F}_h^{-1} \theta * G' )(\lambda).$$
Using the Cauchy inequalities, it follows easily that $G'(\lambda)=  {\cal
O}(h^{-n^\#})$ and $G''(\lambda)=  {\cal O}(h^{-n^\#})$ and we
obtain immediately
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * G )(\lambda)=G'(\lambda) +
{\cal O}(h^{1-n^\#})$$
uniformly with respect to $\lambda \in \RR$.


It remains to examine
$$({\cal F}_h^{-1} \theta * \varphi_2 \xi' )(\lambda)=\trb \Bigl[ ({\cal
F}_h^{-1} \theta)\Bigl(\lambda- L_j \Bigr) {\varphi_2}(L_j) \Bigr]_{j=1}^2$$
$$\quad \qquad = \frac{1}{2\pi h} \int e^{it\lambda h^{-1}} \theta(t)
 \trb \Bigl[ e^{-it h^{-1} L_j} {\varphi_2}(L_j) \Bigr]_{j=1}^2 dt.$$
We will prove that
\begin{equation}\label{eq:5.6}
({\cal F}_h^{-1} \theta * {\varphi_2} \xi' )(\lambda)=\frac{d}{d
\lambda}({\cal F}_h^{-1} \theta *N_{}^{\#})(\lambda)+ \omega(\lambda)h^{-n}
+  {\mathcal O}(h^{1-n}), \:\: \lambda \in \RR,
\end{equation}
where $\omega(\lambda)\in C_0^0(I_0)$ has compact support and ${\mathcal O}(h^{1-n})$ is uniform with respect to $\lambda \in \RR.$ As in Section 2, define the operator  $L_1^{\#}$  on the torus $T_{\tilde{R}} = ({\RR}/\tilde{R}{\Z})^n$ with $ \tilde{R} > 2R >  2 R_0$
and  introduce  $\chi \in C_0^\infty (\{x: |x| \leq \tilde{R}\})$  equal
to 1 for $|x| \leq 2 R > 2 R_0$. We have
$$ \trb \Bigl[ e^{-it h^{-1} L_j} {\varphi_2}(L_j) \Bigr]_{j=1}^2 =  \Bigl[
\tr\Big(\chi e^{-it h^{-1} L_j} {\varphi_2}(L_j) \chi\Big) \Bigr]_{j=1}^2
                                   +  \trb \Bigl[ e^{-it h^{-1} L_j}
{\varphi_2}(L_j)(1-\chi^2) \Bigr]_{j=1}^2.$$
Applying the Duhamel formula and the semi-classical Egorov theorem (see
Section 6 of \cite{BrPe2} for more details), for $|t|$ sufficiently small
we obtain
$$\trb \Bigl[ e^{-it h^{-1} L_j} {\varphi_2}(L_j)(1-\chi^2) \Bigr]_{j=1}^2
=  \tr \Bigl[ e^{-it h^{-1} Q_j} {\varphi_2}(Q_j)(1-\chi^2) \Bigr]_{j=1}^2
+ {\cal O}(h^{\infty}),$$
\begin{eqnarray}
 \tr\Big(\chi e^{-it h^{-1} L_1} {\varphi_2}(L_1) \chi\Big)& =&
\tr\Big(\chi e^{-it h^{-1} L^\#_1} {\varphi_2}(L^\#_1) \chi\Big) +
{\cal O}(h^{\infty})\nonumber\\
&= & \tr\Big( e^{-it h^{-1} L^\#_1} {\varphi_2}(L^\#_1) \Big) -
\tr \Bigl( e^{-it h^{-1} Q^\#_1} {\varphi_2}( Q^\#_1)(1-\chi^2)
\Bigr)  + {\cal O}(h^{\infty}),\nonumber
\end{eqnarray}
where  $Q_1^{\#}$ is a differential operator
$$Q_1^{\#} = \sum_{|\nu| \leq 2} a_{1,\nu}^{\#}(x;h)(hD)^{\nu}$$
on the torus $T_{\tilde{R}}$ introduced in Section 2 and $a_{1,\nu}^{\#}(x;h) = a_{1,\nu}(x;h)$ for $|x| < r_0$, $r_0>  2  R_0$. 
 Using  the classical constructions of a parametrix for small $|t|$ for
the unitary groups $e^{-ith^{-1}Q_1^{\#}},\: e^{-ith^{-1}L_2},$ combined with the fact that $\lambda \in I_0$ is non-critical  for $Q_1^\#,\:\: L_2$  we deduce for $\lambda \in I_0$
$$\tr \Bigl( ({\cal F}_h^{-1} \theta) \Bigl(\lambda- Q_1^\# \Bigr)
{\varphi_2}(Q_1^\#)(1-\chi^2) \Bigr) = \omega_{1}(\lambda)h^{-n} +  {\cal O}(h^{1-n}),$$
$$\tr \Bigl(\chi ({\cal F}_h^{-1} \theta) \Bigl( \lambda- L_2 \Bigr)
{\varphi_2}(L_2)\chi \Bigr) = \omega_{2}(\lambda)h^{-n} +  {\cal O}(h^{1-n}),$$
with functions $\omega_{1}$, $\omega_{2} \in C^0_0(I_1)$ and ${\mathcal O}(h^{1-n})$ uniform with respect to  $\lambda \in I_0.$ The problem can be reduced to the application of the stationary phase method to some integrals where the integration is over a compact set. We refer to Chapter 10, \cite{DS}, for more details. Since $\hat{\theta} \in {\mathcal S}(\RR)$, we can extend the above relations to all $\lambda \in \RR$ with ${\mathcal O}(h^{1- n})$ uniform with respect to $\lambda \in \RR.$\\

For the trace involving $Q_j,\: j =1,2,$ we have for $\lambda \in I_0$
\begin{equation} \label{eq:as}
\tr \Bigl[ ({\cal F}_h^{-1} \theta) \Bigl(\lambda- Q_j\Bigr)
{\varphi_2}(Q_j)(1-\chi^2) \Bigr]_{j=1}^2 = \omega_{ext}(\lambda)h^{-n} +{\cal O}(h^{1-n})
\end{equation}
with  $\omega_{ext} \in C_0^0(I_0)$ and ${\mathcal O}(h^{1-n})$ uniform with respect to $\lambda \in I_0.$ The proof of (\ref{eq:as}) is more technical since we must integrate over a non-compact domain. In fact, it is similar to the calculation of the traces in Section 4 in \cite{Bo2} and  for the sake of completeness we present a proof in Appendix. Moreover, we show in the Appendix that we can extend (\ref{eq:as}) to all $\lambda \in \RR$ with ${\mathcal O}(h^{1-n})$ uniform with respect to $\lambda \in \RR.$
Taking together the asymptotics of the traces and the above relations, we obtain
(\ref{eq:5.6}) and  (\ref{eq:5.5}).

Now we will apply a Tauberian theorem (see for example, Theorem V-13 of
\cite{R1}) for the increasing function $M_{\varphi_2}(\lambda)$. For this purpose
we need the estimates
\begin{equation}\label{eq:5.9b}
M_{\varphi_2}(\lambda) = {\cal O}(h^{-n^{\#}}),\:\:\frac{d}{d
\lambda}({\cal F}_h^{-1} \theta * M_{\varphi_2})(\lambda) =  {\cal O}(h^{-n^\#}),\:\: \forall \lambda \in \RR.
\end{equation}
The first one follows easily from (\ref{eq:repr}). To establish the
second one, we apply the equality  (\ref{eq:5.5}). Thus it suffices to prove the estimate
\begin{equation} \label{eq:5.7}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N_{\varphi_2}^{\#})(\lambda)
= (2 \pi h)^{-1} \tr\Bigl( \hat{\theta}(\frac{ L_1^\# - \lambda}{h})
\varphi_2(L_1^\#) \Bigr) = {\cal O}(h^{-n^\#}),\:\: \forall \lambda \in \RR.
\end{equation}
To do this, assume first that $\lambda \in [E_0 - \epsilon_1, E_1 + \epsilon_1]$. Taking into account (5.1), we obtain
$$ \Bigl |\tr \Big( \hat{\theta}(\frac{ L_1^\# - \lambda}{h})
\varphi_2(L_1^\#) \Big)\Bigr| = \Bigl | \sum_{\mu \in
\hbox{sp}\:(L_1^\#)\cap \hbox{supp}\: \varphi_2}
 {\hat \theta}(\frac{ \mu - \lambda}{h}) \varphi_2(\mu)\Bigl |$$
$$\leq  \sum_{k=0}^{C/h}\; \;  \sum_{\frac {kh}{B} \leq |\mu - \lambda| \leq \frac{(k+1)h}{B}}\Bigl |{\hat \theta}(\frac{ \mu - \lambda}{h}) \varphi_2(\mu)\Bigr | \leq C\Big(h^{1-\ns} + \sum_{k=1}^{C/h} \frac{(k+ 1)h^{1-n^\#}}{k^3}\Big) \leq Ch^{1-n^\#},$$
where we have used the inequality $|\hat{\theta}(\mu)| \leq C(1 + |\mu|)^{-3}$. On the other hand, for $\lambda \notin [E_0 - \epsilon_1, E_1 + \epsilon_1]$ and $\mu \in {\rm supp}\:\:\varphi_2$, we have $|\mu - \lambda| \geq \delta_2 > 0$ and the term (5.11) is estimated by ${\mathcal O}(h^{\infty}).$ Now a Tauberian argument implies the first assertion in (\ref{eq:mrep}).
The second one is obtained by integration of (\ref{eq:5.5}) over $[$inf
$I_0, \lambda]$ combined with the equalities
$$M_{\varphi_2}(\mu)= G_{\varphi_2}(\mu)= N^\#_{\varphi_2}(\mu)=0,
\:\:\mu \leq \inf I_1$$
and  the fact that  $\hat{\theta}(t) \in {\cal S}(\RR)$.
\end{pf}

{\em Proof of Theorem $2$.}
As mentioned above, it remains to show that
\begin{equation}\label{eq:5.8}
\xi (\lambda, h) = \xi(\lambda; L_1,L_2) = N(L^\#_1, \lambda) +
\omega(\lambda) h^{-n} + {\cal O}(h^{1-n^\#})
\end{equation}
for a differential operator  $L_2=Q_2$ having no resonances in $\Omega_0$
and such that each $\lambda \in I_0$ is non-trapping and non-critical energy
level for $L_2$.
According to Lemma \ref{lem2} and Lemma \ref{lem3}, for $\lambda \in I_2$ we
have
$$\xi (\lambda, h) = \trb \Bigl[\varphi_1(L_j)\Bigr]_{j=1}^2 + ({\cal F}_h^{-1}
\theta *  N^\#_{\varphi_2} )(\lambda) + \int_{-\infty}^{\lambda}\omega_{\varphi_2}(\mu) d\mu h^{-n} +
{\cal O}(h^{1-n^\#}).$$
Given a function $\chi\in C_0^\infty(\RR^n)$, $\chi=1$ on
${\overline{B(0,R_0)}}$, exploiting the functional calculus for smooth
functions and the estimates for the trace (see \cite{Sj2}),  we obtain
\begin{eqnarray}
\trb \Bigl[\varphi_1(L_j)\Bigr]_{j=1}^2 & =&\Bigl[ \tr\Bigl(\chi
\varphi_1(L_j) \chi\Bigr) \Bigr]_{j=1}^2 +
\trb \Bigl[\varphi_1(L_j)(1-\chi^2) \Bigr]_{j=1}^2 \nonumber\\
 & = & \tr \Big(\chi \varphi_1(L^{\#}_1) \chi\Big) - \tr\Big(\chi
\varphi_1(L_2) \chi\Big) + \tr\Bigl[\varphi_1(Q_j)(1-\chi^2)
\Bigr]_{j=1}^2 + {\cal O}(h^{\infty}) \nonumber \\
&  = & \tr\Big( \varphi_1(L^\#_1) \Big)  + C(\varphi_1) h^{-n} +
{\cal O}(h^{1-n}) \nonumber,
\end{eqnarray}
where $C(\varphi_1)$ is a constant depending on $\varphi_1.$\\

On the other hand, applying a Tauberian theorem for $N^\#_{\varphi_2}(\lambda)
= {\cal O}(h^{-n^{\#}})$, we deduce
$$ N^\#_{\varphi_2} (\lambda) = ({\cal F}_h^{-1} \theta * N^\#_{\varphi_2} )(\lambda)
+  {\cal O}(h^{1-n^\#}), \:\: \forall \lambda \in \RR.$$
Consequently, for $\lambda \in I_2$ we get
$$\xi (\lambda, h) =  \tr\Big( \varphi_1(L^\#_1) \Big) +  \tr\Big(
\varphi_2(L^\#_1) {\bf 1}_{]- C^{\#},\lambda]}(L^\#_1)  \Big)
+ \Big(C(\varphi_1) + \int_{-\infty}^{\lambda}\omega_{\varphi_2}(\mu)d\mu\Big) h^{-n} +  {\cal
O}(h^{1-n^\#}).$$
  By construction we have
$$\varphi_1(L_j^\#) + \varphi_2(L_j^\#)  {\bf
1}_{]-C^{\#},\lambda]}(L^\#_j) =   {\bf 1}_{]-C^{\#},\lambda]}(L^\#_j), \:\: \forall \lambda \in I_2$$
and this implies (\ref{eq:5.8}).\\

To obtain (5.2), we construct a covering of the interval $[E_0, E_1] \subset \cup_{\nu = 1}^M J_{\nu}$ by small open intervals $J_{\nu}$ so that for every $J_{\nu}$ we can find an operator $Q_{\nu}$ with the properties of Proposition 3, where $I_0$ is replaced by $J_{\nu}$. Next we introduce a partition of unity
\[ \sum_{\nu = 1}^M \varphi_{\nu}(x) = 1 \:\: {\rm on}\: [E_0, E_1],\:\:\varphi_{\nu} \in C_0^{\infty}( J_{\nu}; \RR^{+}) \, \]
and we apply the above argument. This completes the proof of Theorem 2. 

\hfill{$\Box$}\\


\section{Breit-Wigner approximation}

In this section we consider small domains of width $h$ and we prove a
semi-classical analogue of the Breit-Wigner approximation for
$\xi(\lambda, h)$ (see \cite{PZ1}, \cite{PZ3},  \cite{BSj} for similar results, \cite{GMR} for the case of a potential having the form of an "well in the island" and \cite{FR} for the one dimensional critical case). In the following $\eta(\lambda, h)$ denotes the real analytic function defined by
$$\eta(\lambda, h) = \xi(\lambda, h) - \Bigl[ \# \{ \mu \in [E_0,
\lambda]: \: \mu \in {\rm sp}_{pp}(L_j(h)) \} \Bigr]_{j=1}^2.$$ 

\begin{thm}
 Assume that $L_j(h),\: j= 1,2$  satisfy the assumptions of Theorem $2$. Then for any $\lambda \in [E_0, E_1]$, any
$0 < \delta < h/B,\:\:0 <  B_1 < B,$ and  $h$ sufficiently small we have
\begin{equation} \label{eq:4.13}
 \eta(\lambda + \delta, h) - \eta(\lambda - \delta, h) = \Bigl[ \sum_{w
\in {\rm{Res}}\: L_j(h), \atop \Im w \neq 0,\: |w - \lambda| < h/B_1}
\omega_{\C_{-}} ( w, [\lambda - \delta, \lambda + \delta]) \Bigr]_{j=1}^2
+ {\mathcal O}(\delta) h^{-\ns},
\end{equation}
where $B > 0$ is the constant introduced in Theorem $2$.

\end{thm}

{\bf Remark.} Following the recent result of J.-F. Bony \cite{Bo1} the assumption (5.1) implies the existence of positive  constants $D,\: \epsilon_2,\: C_2, \: h_2 > 0$ such that for any $\lambda \in [E_0 - \epsilon_2, E_1 + \epsilon_2],\: h/D \leq \delta \leq D$ and $h \in ]0, h_2]$  we have
\begin{equation} \label{eq:poles}
\# \{ z \in \C: z \in {\rm Res}\:L_j(h), \: |z - \lambda| \leq \delta\} \leq C_2\delta h^{-\ns}, \:\: j =1,2. 
\end{equation}

\begin{pf}
We apply Theorem 1 in the interval $I_0 \supset (\lambda - \delta, \lambda + \delta),
\: 0 < \delta \leq h/B_1,$ and introduce the function
\[ F(z, h) =  \Bigl[\sum_{w \in {\rm{Res}}\: L_j,\: \Im w \neq 0, \atop h/B_1
\leq |w - \lambda| \leq C_4} \Bigl(\frac{1}{z - w} - \frac{1}{z -
\overline{w}} \Bigr)\Bigr]_{j=1}^2,\:\:  z \in D(\lambda, h/B)
\,. \]
 It is sufficient to show that
\begin{equation} \label{eq:est}
 |F(z, h)| \leq Ch^{-\ns},\:\: |z - \lambda| \leq h/B.
\end{equation}
We have
\[ \partial_z F(z, h) =  \Bigl[\sum_{w \in {\rm{Res}}\: L_j,\: \Im w \neq
0, \atop h/B_1 \leq |w - \lambda| \leq C_4} \frac{1}{(z - \overline{w})^2} -
\frac{1}{(z - w)^2} \Bigr]_{j=1}^2 \,. \]
Let $l_0 \in \N$ be an integer such that $D \leq 2^{l_0 - 1}B$. Following the argument in \cite{PZ3} and applying (\ref{eq:poles}), for any  $z \in D(\lambda, h/B)$ we  obtain
\[\sum_{w \in {\rm{Res}}\: L_j,\:\Im w \neq 0, \atop h/B_1 \leq |w- \lambda|
< C_4} \frac{1}{|z - w|^2} \leq \sum_{w \in {\rm{Res}}\: L_j,\:\Im w \neq 0, \atop h/B_1 \leq |w- \lambda| \leq \frac{2^{l_0}h}{D}} \frac{1}{|z - w|^2} \, \]
\[ +\sum_{k=l_0}^{C \log (1/h)} \sum_{\frac{2^k h}{D}
\leq |w - \lambda| \leq \frac{2^{k+1} h}{D}} \frac{1}{|z - w|^2} \, \]
\[ \leq C 2^{l_0}D^{-1}h^{-1-\ns} +  C\sum_{k = l_0}^{C \log (1/h)} \frac{(2^{k+1}h)h^{-\ns}}{(2^k h)^2} \leq Ch^{-1 -\ns} \,. \]
Here and below we denote by $C > 0$ different constants which may change
from line to line and which are independent on $h$ and the choice of
$\lambda$ in the interval $[E_0, E_1].$ Thus we get the estimate
\[ |\partial_z F(z, h)| \leq Ch^{-\ns - 1}, \:\: z \in D(\lambda,
 h/B) \,. \]
It remains to find an estimate of $|F(\mu_0, h)| = |\Im F(\mu_0, h)|$ at a
suitable point $\mu_0 = \mu_0(h).$ \footnote{There is some similarity
between the proof of the existence of $\mu_0(h)$ and that of the existence
of a suitable point $z_0(h), \:\: \Im z_0(h) \geq \delta > 0$ in Section 4
in \cite{PZ3} so that $\log |\det S(z_0(h), h)| \geq -Ch^{-\ns}.$}
Set $\nu = \frac{h}{B} < \frac{h}{B_1}$ and suppose that for all $\mu \in \RR,\: |\mu - \lambda| \leq \nu$, we have $|\Im F(\mu, h)| \geq Mh^{-\ns},\: M > 0.$
 The continuity of the function $\Im F(\mu, h)$ implies
that $\Im F(\mu, h)$ is either positive or negative in $[\lambda - \nu,
\lambda + \nu].$ Assuming $\Im F(\mu, h)$  positive, we get
\[ \frac{M  h^{-\ns +1}}{ B\pi} \leq \frac{1}{2 \pi} \int_{\lambda-
\nu}^{\lambda + \nu} \Im F(\mu, h) d\mu  \leq \frac{1}{\pi} \int_{\lambda-
\nu}^{\lambda + \nu} \Bigl[ \sum_{w \in {\rm{Res}} \: L_j,\:\Im w \neq 0
\atop |w - \lambda| \leq C} \brw \Bigr]_{j=1}^2 d\mu \, \]
\[ + \frac{1}{\pi} \sum_{j=1}^2 \int_{\lambda - \nu}^{\lambda + \nu}
\sum_{w \in {\rm{Res}}\: L_j,\: \Im w \neq 0, \atop \: |w - \lambda| < h/B_1}
\brw d\mu \, \]
\[ \leq  |\eta(\lambda + \nu, h) - \eta(\lambda - \nu, h)| + Ch^{1-\ns} \,. \]
Here we have used the inequality
$$\int_{\lambda - \nu}^{\lambda + \nu} \brw d\mu \leq
\int_{-\infty}^{\infty} \brw d\mu \leq \pi$$
and  (\ref{eq:poles}) to estimate the number of resonances in $\{w: |w -\lambda| < h/B_1\}$. Notice that if $D \leq B_1$, we have $\{w: |w -\lambda| < h/B_1\} \subset \{w: |w -\lambda| < h/D \}.$
Next the  assumption (5.1) combined with  Theorem 2 yield the  estimate
 $$|\xi(\lambda + \nu, h) -\xi(\lambda - \nu, h)| \leq Ch^{1-\ns}.$$ 
 Thus,
\[ |\eta(\lambda + \nu, h) - \eta(\lambda - \nu, h)| \leq |\xi(\lambda + \nu, h) - \xi(\lambda - \nu, h)| \, \]
\[ + \sum_{j =1}^2 \sharp \{\mu \in {\rm sp}_{pp}(L_j): \: |\mu - \lambda| \leq \nu \} \leq Ch^{1-\ns} \,, \]
where for the second inequality we have used once more (\ref{eq:poles}), observing that the positive eigenvalues of $L_j$ coincide with the resonances on $\RR^{+}.$ Consequently, we obtain a bound for $M$. Hence there exists a constant
$C > 0$ and $\mu_0 \in [\lambda - \nu, \lambda + \nu]$ so that
\begin{equation} \label{eq:6.1}
| F(\mu_0, h)| \leq C h^{-\ns}.
\end{equation}
Writing
\[ F(z, h) =  F(\mu_0, h) + \int_{\mu_0}^z \partial_z  F(z, h) dz,\:\: |z
- \lambda| \leq h/B \,,  \]
we obtain (\ref{eq:est}).
The case $\Im F(\mu, h) < 0$ can be treated by the same argument
exploiting the inequality $- \Im F(\mu, h) \geq M h^{-\ns}, \: |\mu -
\lambda| \leq \nu.$
 By an integration over the interval $(\lambda - \delta, \lambda + \delta),$
 we complete the proof of (6.1).
\end{pf}
 {\bf Remark.} Our proof goes without a factorization in small domains
$\{z \in \C: |z - \lambda| \leq Ch \}$ and a suitable trace formula (see
Lemma 6.2 in \cite{PZ3} and Theorem 1.3 in \cite{BSj}). The above argument
can be applied to simplify the proof of Lemma 6.2 in \cite{PZ3}.\\

Next, the estimate (\ref{eq:est}) of $F(z,h)$ yields immediately the
following.

\begin{cor}
Under the assumptions of Theorem $3$ for $\mu \in \RR,\: |\mu - \lambda| <
h/B$ we have the representation
\begin{equation} \label{eq:loc}
 \xi'(\mu, h) = \frac{1}{\pi} \Im q(\mu, h) + \Bigl[\sum_{w \in {\rm
Res}\:L_j,\:|w - \lambda| < h/B_1 \atop \Im w \neq 0} \frac{-\Im w}{\pi |\mu -
w|^2} +
\sum_{w  \in ({\rm Res}\:L_j \cap \RR), \atop |w - \lambda|< h/B } \delta (\mu - w) \Bigr]_{j=1}^2,
\end{equation}
where $q(z, h) = p(z, h) - \overline{p(\overline{z}, h)},\:\: p(z,h)$ is
holomorphic in $D(\lambda, h/B)$ and $p(z, h)$ satisfies the
estimate
\[ |p(z, h)| \leq C h^{-\ns},\:\: z \in D(\lambda, h/B) \, \]
with $C > 0$ independent on $h \in ]0, h_0]$ and $\lambda \in [E_0, E_1].$

\end{cor}

We may slightly improve Theorem 3, noting that for every $0 < \epsilon < 1$ and $|\mu - \lambda| \leq \frac{\epsilon h}{B}$
we have
\[ \sum_{w \in {\rm{Res}}\:L(h),\: \atop\epsilon h/B_1 \leq |w - \lambda| \leq h/B_1}
\frac{|\Im w|}{|\mu  - w|^2} \leq \frac{h}{\epsilon^2 h^2}Ch^{1-\ns} =
{\mathcal O}_{\epsilon}(h^{-\ns}) \,. \]
Thus for $0 < \delta \leq \frac{\epsilon h}{B}$ the right hand part in (6.1) can be replaced by
\[ \Bigl[ \sum_{w \in {\rm{Res}} L_j(h),\:\: \Im w \neq 0, \atop |w -
\lambda| \leq \epsilon h/B_1} \omega_{\C_{-}} (w, [\lambda - \delta, \lambda +
\delta])\Bigr]_{j=1}^2 + {\mathcal O}_{\epsilon}(\delta)h^{-\ns} \,. \]

To obtain a stronger version involving the resonances in smaller "boxes", we
need some additional information for the distribution of the resonances in
$\{ w \in \C: |w - \lambda| \leq \epsilon h \}.$
In the case of the Schr\"{o}dinger operator $L(h) = - h^2\Delta + V(x)$
with $V(x) \in C_0^{\infty}(\RR^n)$ real valued this is possible applying
the recent result of Stefanov \cite{St}. Set $a_0(x, \xi) = |\xi|^2 +
V(x)$ and let $0 < E_0 < E_1$ be non-critical values of $a_0(x, \xi).$ Let
\[ a_0^{-1} [E_0, E_1] = W_{\rm{int}} \cup W_{\rm{ext}} \,, \]
where $W_{\rm{ext}}$ is the {\em unbounded} connected component, while
$W_{\rm{int}}$ is the union of bounded ones if there are such connected
components. Assume that all points in $W_{\rm{ext}}$ are non-trapping (see
\cite{St} for a precise definition). Then, according to Theorem 6.1 in
\cite{St}, there exists a function $0 < R_1(h) = {\mathcal O}(h^{\infty})$
such that for any $M \in \N$ the operator  $L(h)$ has no resonances in the
set
\begin{equation} \label{eq:fr}
 \Omega_M(\lambda, h) = [E_0, E_1] + i [-Mh, - R_1(h)], \:\: 0 < h \leq
h(M).
\end{equation}
Setting $0 < R(h) = \sqrt{h R_1(h)} = {\mathcal O}(h^{\infty})$, an
elementary argument shows that for $\lambda \in [E_0, E_1]$ and $|\mu - \lambda| \leq R(h)/2$ we have
\[ \sum_{w \in {\rm{Res}}\: L(h), \: |\Im w| \leq R_1(h) \atop R(h) \leq
|Re w - \lambda| \leq h} \frac{|\Im w|}{|\mu - w|^2} \leq Ch^{-\ns} \,. \]

In the next result we treat a formally symmetric  differential operator
\[ L_1(h) =  \sum_{|\alpha| \leq 2} a_{\alpha} (x, h) (hD_x)^{\alpha} \, \]
on $L^2(\RR^n)$ satisfying the assumptions of Section 2. Given a fixed $\lambda \in ]E_0, E_1[$, as in the previous section, we may  construct an operator $L_2(h)$ having the properties (a) - (c) of Proposition 3. Applying Theorem 3
for $L_j(h), \: j = 1,2,$ and $\{z \in \C: |z - \lambda| \leq h/B_1\} \subset
W$, and assuming that we have a free resonances domain, we obtain the
following improvement of Corollary 1.

\begin{cor}
 Let $E_0 < \lambda < E_1$ be fixed.  Let $L_2(h)$ be chosen so that $L_j(h), \: j =1,2,$
satisfy the assumptions of Theorem $3$ and $L_2(h)$ has no resonances in
the disk $\{ z \in \C: |z - \lambda| \leq h/B_1 \}.$ Suppose that there exists a
function $0 < R_1(h) = {\mathcal O}(h^{\infty})$ such that $L_1(h)$ has no
resonances in the set
\[ [E_0, E_1] +i [-\epsilon h, - R_1(h)], \: \epsilon > 0, \:\: 0 < h \leq
h(\epsilon) \,. \]
 Then for $|\mu - \lambda| < \frac{R(h)}{2}$ and $h$ sufficiently small we have
\begin{equation}
\xi'(\mu, h) = \frac{1}{\pi} \Im q(\mu, h) + \sum_{w \in {\rm
Res}\:L_1,\:|\Re w - \lambda| <R(h) \atop 0 < |\Im w| \leq R_1(h)} \frac{-\Im
w}{\pi |\mu - w|^2} +
\sum_{w  \in {\rm Res}\:L_1 \cap \RR, \atop |w - \lambda| <R(h)/2}
\delta (\mu - w)
\end{equation}
with $R(h) = \sqrt{hR_1(h)} = {\mathcal O}(h^{\infty})$ and $q(\mu,h)$ as
in Corollary $1$.

\end{cor}



\section{Local trace formula}

In this section we prove a local trace formula which is a slightly
stronger version  of that in \cite{Sj1}, \cite{Sj2} (see \cite{PZ3} for
compactly supported perturbations). Exploiting Theorem 1, we repeat with
trivial modifications the argument of Section 5, \cite{PZ3}, to get the
following.

\begin{thm}
Assume that $L_j(h)$ satisfy the assumptions of Section $2$. Let $\Omega
\subset e^{i]-2\theta_0, 2\theta_0[} ]0, \infty[$ be an open, simply
connected,  relatively compact set such that $I = \Omega \cap \RR$ is an
interval. Suppose that $f$ is holomorphic on a neighborhood of $\Omega$
and that $\psi \in C_0^{\infty}(\RR)$ satisfies
\[ \psi ( \lambda ) = \left\{ \begin{array}{ll} 0, & d  ( I , \lambda ) >
2 \epsilon, \\
1, & d (I , \lambda ) < \epsilon, \end{array} \right.
\, \]
where $ \epsilon > 0 $ is sufficiently small. Then
\begin{equation}
\label{eq:4.1}
 \trb \Bigl[( \psi f ) ( L_j( h ) )\Bigr]_{j=1}^2  =
\Bigl[\sum_{ z \in {\rm{Res} \;} ( L_j ( h )) \cap \Omega } f ( z
)\Bigr]_{j=1}^2
+ E_{ \Omega, f , \psi}  ( h )
\end{equation}
with
\[ | E_{ \Omega, f , \psi} ( h )| \leq M( \psi, \Omega )
 {\rm{sup} } \;  \{ |f ( z ) | \; : \; 0 \leq  d (\Omega, z ) \leq  2
\epsilon \,, \ \Im z \leq 0 \} h^{-\ns} \,. \]
\end{thm}
\begin{pf}
Choose an almost analytic extension $\tilde{\psi}$ of $\psi$ so that
$\tilde{\psi} \in {\mathcal C}_c^{\infty} (\C)$, $\tilde{\psi} = 1$ on
$\Omega$ and
$$ {\rm{supp}}\:\: \overline{\partial}_z \tilde{\psi} \subset \{ z \in \C:
\epsilon \leq d(\Omega, z) \leq 2\epsilon \}.$$
Setting $\Omega_{\epsilon} = \{ z \in \C: d(\Omega, z) \leq \epsilon \},$
we have
\[ \trb \Bigl[( \psi f ) ( L_j( h ) )\Bigr]_{j=1}^2 = <\xi'(\lambda, h),
(\psi f)(\lambda) > \, \]
\[  = \Bigl[\sum_{w \in {\rm{Res}}\:(L_j(h)) \cap\:\: {\rm{supp}}\: \psi}
(\psi f)(w)\Bigr]_{j=1}^2  + \frac{1}{2 \pi i}\int (\psi f)(\lambda)
r(\lambda, h) d\lambda \, \]
\[ + \frac{1}{2 \pi i} \int (\psi f)(\lambda) \Bigl[ \sum_{ w \in
{\rm{Res}}\:(L_j(h)) \cap \Omega_{2\epsilon},  \atop \Im w \neq 0}
\Bigl(\frac{1}{\lambda - \overline{w}} - \frac{1}{\lambda - w}
\Bigr)\Bigr]_{j=1}^2 d\lambda \,. \]
 The integral involving $r(\lambda, h)$ can be estimated using
(\ref{eq:1.4}) with $W = \Omega_{2\epsilon}.$ For the integral containing
the resonances we apply Green formula and we get the term
\[ \Bigl[\sum_{z \in {\rm{Res}}\: L_j,\: \Im z \neq 0} (\tilde{\psi}f)(z)
\Bigr]_{j=1}^2 \, \]
\[ + \frac{1}{\pi} \int_{\C_{-}} (\overline{\partial}_z\tilde{\psi})(z)
f(z) \Bigl[ \sum_{ w \in {\rm{Res}}\:(L_j(h)) \cap \Omega_{2\epsilon},
\atop \Im w \neq 0} \Bigl(\frac{1}{z - \overline{w}} -
\frac{1}{z - w} \Bigr)\Bigr]_{j=1}^2 {\mathcal L}(dz) \,, \]
where ${\mathcal L}(dz)$ is the Lebesgue measure on $\C.$ As in the proof
of Theorem 1 in \cite{PZ3}, we apply the inequality
\[ \int_{\Omega_1} \frac{1}{|z - w|} {\mathcal L}(dz) \leq 2 \sqrt{2 \pi
|\Omega_1|} \, \]
and an upper bound for the number of the resonances in
$\Omega_{2\epsilon}$ to obtain the result.
\end{pf}

Since we have no restrictions on the behavior of the holomorphic function
$f(z)$ on $\Omega \cap \: \{ \Im z > 0 \}$, we may apply the above
argument choosing $f(z) = e^{-itz/h}, \:\: t \in \RR$, to get the
following.
\begin{thm}
Let $\Omega$ and $\psi$ be as in Theorem $4$ and let $\tilde{\psi} \in
{\mathcal C}_c^{\infty}(\C)$ be an almost analytic extension of $\psi$
supported in $\Omega_{2\epsilon}$. Then for any $0 < \delta < 1$ and $t
\geq h^{\delta}$ we have
\begin{equation}
\trb\Bigl[ \psi(L_j(h)) e^{-i\frac{t}{h} L_j(h)} \Bigr]_{j=1}^2 = \Bigl[
\sum_{ w \in {\rm{Res} \;} (L_j(h)) \cap \Omega_{2\epsilon}}
\tilde{\psi}(w) e^{-itw/h}\Bigr]_{j=1}^2 + {\mathcal
O}_{\delta}(h^{\infty}).
\end{equation}
Moreover, for $t \geq \epsilon > 0$ and $N \in \N$ there exists $h_N > 0$
such that for $0 < h \leq h_N$ we have
\begin{equation}
\trb\Bigl[ \psi(L_j(h)) e^{-i\frac{t}{h} L_j(h)} \Bigr]_{j=1}^2  =\Bigl[
\sum_{ w \in {\rm{Res} \;} (L_j(h)) \cap \Omega_{2\epsilon} \atop |\Im w|
\leq - N h \log h } \tilde{\psi}(w) e^{-itw/h}\Bigr]_{j=1}^2 +  {\mathcal
O}_{\epsilon}(h^{N\epsilon- \ns}).
\end{equation}
\end{thm}

\begin{pf}
Choose an almost analytic extension $\tilde{\psi}$ of $\psi$ as in Theorem
4. Applying Green formula, we must examine the integrals
\[ \int_{\C_{-}} \overline{\partial}_z \tilde{\psi}(z) e^{-itz/h} r(z, h)
{\mathcal L}(dz) \,, \]
\[ \int_{\C_{-}} \overline{\partial}_z \tilde{\psi}(z) e^{-itz/h}
\Bigl[\sum_{w \in {\rm{Res}}\:L_j(h) \cap \Omega_{2\epsilon}} \Bigl(
\frac{1}{z - w} - \frac{1}{z -\overline{w}}\Bigr)\Bigr]_{j=1}^2 {\mathcal
L}(dz) \,. \]
Choose $\mu >0, \:\: 0 < \delta + \mu < 1.$ For $-h^{\mu} \leq \Im z \leq
0$ we have
\[ |\overline{\partial}_z \tilde{\psi} | \leq C_N |\Im z |^N \leq C_N
h^{\mu N},\: \forall N \in \N \, \]
and the integration over $-h^{\mu} \leq \Im z \leq 0$ combined with the
argument of the proof of Theorem 4 yield a term bounded by ${\mathcal
O}(h^{\infty}).$ On the other hand, for $t \geq h^{\delta},\:\: \Im z \leq
- h^{\mu}$ we get
\[ |e^{-itz/h}| \leq e^{-th^{\mu - 1}} \leq e^{-h^{\delta + \mu -1}} =
{\mathcal O}_{\delta}(h^{\infty}) \, \]
and this implies (7.2). For the second assertion we have
\[ \Bigl | e^{-itw/h} \Bigr | \leq e^{tN \log h} \leq h^{N\epsilon} \, \]
for $|\Im w | \geq - N h \log h$ and this completes the proof.
\end{pf}

{\bf Remark.} For non-trapping compactly supported perturbations $L(h)$  (see \cite{TZ}, \cite{B}) and for non-trapping long-range perturbations $L(h) = -h^2\Delta + V(x)$ of the Laplacian (see \cite{Ma}) there are no resonances of
$L(h)$ in the domain
\[ - N h \log \frac{1}{h} \leq \Im z \leq 0,\:\: 0 < h \leq h_N \,. \]
For such perturbations the left hand side of (7.3) is equal to ${\mathcal
O}_{\epsilon}(h^{N\epsilon- \ns})$ and we obtain an analogue of the classical trace formula for non-trapping perturbations.

\section{Existence of resonances close to the real axis}

In this section we consider the operator $L(h) = - h^2 \Delta_g + V(x),$
where $\Delta_g$ is symmetric Laplace-Beltrami operator on $L^2(\RR^n)$
associated to a metric $g(x) = \{g_{i,j}(x)\}_{1\leq i, j \leq n}$ and
$V(x) \in C^{\infty}(\RR^n)$ is a real valued function. We assume that
there exists $\rho > n$ so that
\begin{equation} \label{eq:7.1}
 |\partial_x^{\alpha} (g_{i,j}(x) - \delta_{i,j})| + |\partial_x^{\alpha}
V(x)| \leq C_{\alpha}<x>^{-\rho - |\alpha|}, \:\: 1 \leq i,j \leq n,
\:\:\forall \alpha.
\end{equation}
Moreover, we assume that the coefficients $\{g_{i,j}(x)\}$ and $V(x)$ can
be extended holomorphically in $x$ to the domain given in (2.8) and the
estimate (\ref{eq:7.1}) holds in this domain.\\

Consider the symbol
\[ a_0(x, \xi) = < g(x)^{-1}\xi, \xi > + V(x) \, \]
and denote by $H_{a_0}$ the Hamilton vector field associated to $a_0$ and
by $\Phi^t = \exp (t H_{a_0})$ the Hamilton flow. Given $\lambda > 0,$ let
$\Sigma_{\lambda} = \{(x,\xi) \in \RR^n: a_0(x, \xi) = \lambda \}$ be the
energy surface and let $\nabla a_0(x, \xi) \neq 0$ on $\Sigma_{\lambda}.$ A
point $\nu \in \Sigma_{\lambda}$ is called {\em periodic}, if there exists
$T > 0$ such that $\Phi^T(\nu) = \nu$ and the smallest $T > 0$ with this
property is called period $T(\nu)$ of $\nu.$ Given a periodic point $\nu$, consider
the trajectory
\[ \gamma(\nu) = \{ \Phi^t (\nu): 0 \leq t \leq T(\nu) \} = \{ (x(t),
\xi(t)) : 0 \leq t \leq T(\nu) \} \, \]
 and define the {\em action} $S(\nu)$ along $\gamma(\nu)$ by
\[ S(\nu) = \int_{\gamma(\nu)} \xi dx = \int_0^{T(\nu)} \xi(t) x'(t) dt
\,. \]
Next we denote by $m(\nu) \in \Z_4$ the Maslov index related to
$\gamma(\nu)$ and set $q(\nu) = - \frac{\pi}{2} m(\nu).$ Let $\Pi$ be the
set of all periodic points on $\Sigma_{\lambda}$ and let
\begin{equation} \label{eq:q}
 Q(h, r) = (2\pi)^{-n} \int_{\Pi} \Bigl [ \pi - h^{-1}S(\nu) + q(\nu) -
rT(\nu) \Bigr]_{2\pi} T(\nu)^{-1} d\nu ,
\end{equation}
where $d\nu$ is the Liouville measure on $\Sigma_{\lambda}$ and the
residue $-\pi < [z]_{2\pi}\leq \pi$ is defined so that $ z = [z]_{2\pi} +
2\pi k, \: k \in \Z.$ The set $\Pi$ is bounded, the integrand in
(\ref{eq:q}) is a measurable function and $T(\nu) \geq T_0 > 0,\:\: \forall
\nu \in \Pi.$ The oscillatory function $Q(h, r)$ has been  introduced in
\cite{PP} for the analysis of the semi-classical behavior of the eigenvalues
and it is a semi-classical analogue of the oscillating function defined by
Guriev and Safarov \cite{GS} and Safarov \cite{Sa}. Notice that the limits
$Q(h, r \pm 0) = \lim_{\epsilon \downarrow 0} Q(h, r \pm \epsilon)$ exist
for each $r$ and $0 < h \leq h_0$ and, moreover,
\[ Q(h, r + 0) - Q(h, r - 0) = (2\pi)^{1-n} \int_{\Omega_{h,r}}
\frac{d\nu}{T(\nu)} \,, \]
where $\Omega_{h,r} = \{ \nu \in \Pi: h^{-1}S(\nu) - q(\nu) + rT(\nu)
\equiv 0 (2 \pi) \}$. Following the arguments in Section 6, \cite{PZ1}, we
will prove the following.

\begin{thm} Let $L(h) = -h^2\Delta_g + V(x)$, where the metric $g(x)$ and
$V(x)$ satisfy the estimates  $(\ref{eq:7.1})$ and let $\nabla a_0(x, \xi)
\neq 0$ on $\Sigma_{\lambda},\:\lambda > 0.$ Assume that there exist an
integer $p \in \Z$ and a subset $\Pi_0 \subset \Pi$ with positive
Liouville measure $\mu(\Pi_0) > 0$ so that
\[ \Bigl( \Bigl[ q(\nu) - h^{-1}S(\nu) \Bigr]_{2\pi} + 2\pi p \Bigr)
T(\nu)^{-1} = r(h),\: 0 < h \leq h_0 \, \]
does not depend on $\nu \in \Pi_0.$ Then for $0 < h \leq h_1$ we have
\begin{equation} \label{eq:7.2}
\# \{ w \in {\rm{Res}}\: L(h) :  | w - \lambda - r(h)h| \leq h \} \geq
\frac{(2\pi)^{1-n}}{2}h^{1-n} \int_{\Pi_0} \frac{d\nu}{T(\nu)} .
\end{equation}

\end{thm}

{\bf Remark.}  Clearly, $|r(h)| \leq \max \{ |2p - 1|, \: |2p + 1| \}\pi
(T_0)^{-1}$ and  applying the Remark after Theorem 3, we conclude that
\[ \# \{ w \in {\rm{Res}}\: L(h):\: |w - \lambda -r(h)h| \leq h \} \leq A
h^{1-n} \,. \]
This shows that the order $h^{1-n}$ in our estimate is optimal. On the other hand, it is clear that we may apply Theorem 3 with $B > 1$ and $B_1 = 1.$\\
\begin{pf}
Consider the scattering phase $\sigma(\lambda, h) = \frac{1}{2 \pi i} \det
S(\lambda, h)$, where the scattering operator $S(\lambda, h)$ is related
to $L(h)$ and $L_0(h) = -h^2 \Delta.$ According to Birman-Krein theory
(see for instance \cite{Ya}), the scattering phase can be identified with
the spectral shift function and, under our assumptions, we have not
embedded positive eigenvalues. Following Theorem 2.1  in \cite{Br}, and
taking $|r(h)| \leq r_0, \: 0 < \epsilon \leq \epsilon_0, \: 0 < h \leq
h_0$ and $\lambda > 0$ we have
\[ \sigma \Bigl(\lambda + (r(h) + \epsilon) h, h\Bigr) - \sigma
\Bigl(\lambda + (r(h) - \epsilon)h, h\Bigr) \, \]
\[ \geq h^{1-n} \Bigl [ Q\Bigl(h, r(h) + \epsilon/2 \Bigr) - Q\Bigl(h,
r(h) - \epsilon/2\Bigr) \Bigr ] + 2 \epsilon \gamma'_0(\lambda) h^{1-n} -
C_0 \epsilon h^{1-n} - o_{\epsilon}(h^{1-n}) \,, \]
where
\[ \gamma_0(\lambda) = (2 \pi)^{-n} \int_{\RR^n}\Bigl( \int_{a_0(x, \xi)
\leq \lambda} d\xi - \int_{|\xi|^2 \leq \lambda} d\xi \Bigr) dx \,, \]
$C_0 > 0$ is independent on $r(h), \: \epsilon$ and $h$ and
$o_{\epsilon}(h^{1-n})$ means that for any fixed $\epsilon > 0$ we have
$$ \lim_{h \downarrow 0} \frac{o_{\epsilon }(h^{1-n})}{h^{1-n}} = 0.$$
On the other hand, for small $\epsilon > 0$ an application of  Theorem 3
with $\delta = \epsilon h$ yields the estimate
\[ \sigma \Bigl(\lambda + (r(h) + \epsilon) h, h\Bigr) - \sigma
\Bigl(\lambda + (r(h) - \epsilon)h, h\Bigr) \, \]
\[  \leq \# \{ w \in {\rm{Res}}\: L(h) : \: | w - \lambda - r(h)h| \leq h
\} + C_1 \epsilon h^{1-n}, \: 0 < h \leq h_2 \leq h_0 \, \]
with $C_1 >0$ independent on $\epsilon, r(h)$ and $h$.
We claim that
\begin{equation} \label{eq:7.4}
 Q(h, r(h) + \epsilon/2) - Q(h, r(h) - \epsilon/2) \geq
-(2\pi)^{-n}\epsilon \mu(\Pi) + (2 \pi)^{1-n} \int_{\Pi_0} \frac{d
\nu}{T(\nu)}.
\end{equation}
In fact, according to the representation of the oscillatory function $Q(h,
r)$ (see for instance, Proposition 1, \cite{Sa}), we have
\[ Q(h, r(h) + \epsilon/2) - Q(h, r(h) - \epsilon/2) = -
\epsilon(2\pi)^{-n} \mu(\Pi) \, \]
\[ + (2 \pi)^{1-n} \int_{\Pi} T^{-1}(\nu) \sum_{k \in \Z}
\chi^{\epsilon}_{h, k}(\nu) d\nu \,,\]
where $\chi^{\epsilon}_{h, k}$ is the characteristic function of the set
\[ \Omega^{\epsilon}_{h, k} = \{ \nu \in \Pi: -\epsilon T(\nu) \leq
h^{-1}S(\nu) - q(\nu) + r(h)T(\nu) - 2k \pi < \epsilon T(\nu) \} \,. \]
Obviously, for any $\nu \in \Pi_0$ we get
\[ h^{-1}S(\nu) - q(\nu) + r(h) T(\nu) + 2 M(\nu, h) \pi - 2p \pi  = 0 \,
\] with some $M(\nu, h) \in \Z$. Consequently,
\[ \nu \in \Pi_0 \Longrightarrow  \sum_{k \in \Z} \chi^{\epsilon}_{h,k}
(\nu) \geq 1 \, \]
and we obtain (\ref{eq:7.4}). Choosing $\epsilon > 0$ small enough, we
arrange the inequality
\[ -\epsilon (2\pi)^{-n}\mu(\Pi) - \epsilon (C_0 + C_1) + 2\epsilon
\gamma'_0(\lambda) \geq -\frac{\alpha_0}{4} \, , \]
with ${\alpha_0}=  (2 \pi)^{1-n} \int_{\Pi_0} \frac{d
\nu}{T(\nu)}$.
Next we fix $\epsilon > 0$ and choose $0 < h_1 \leq h_2$ so that for $0 <
h \leq h_2$ we have
\[ | o_{\epsilon}(h^{1-n}) | \leq \frac{\alpha_0}{4} h^{1-n} \,.\]
Combining the above estimates for the difference $\sigma \Bigl(\lambda +
(r(h) + \epsilon) h, h\Bigr) - \sigma \Bigl(\lambda + (r(h) - \epsilon)h,
h\Bigr), $ we complete the proof.

\end{pf}

{\bf Example} (see Section 7 in \cite{Br}). Let $L(h) = -h^2\Delta + V(x)$
with
\[ V(x) = \Phi_a(x - y_0)\Bigl( |x - y_0|^2 + b \Bigr) \,, \]
where $a > 0, \:\: b > 0$ and $y_0 \in \RR^n$ are fixed and $\Phi_a(x) \in
C_0^{\infty}(\RR^n),\:\: \Phi_a(x) = 1$ for $|x| \leq 2a.$ Let $ 0 < \epsilon < a/2$, $|\eta_0| = \sqrt{\lambda -
b}$ and  let $\lambda
\in ]b, b + a^2 [$ be a non-critical energy level for $a_0(x,\xi) = | \xi |^2 + V(x)$.  Therefore the set
\[ \Pi_0 = \{ (x, \xi) \in \Sigma_{\lambda}: |\xi - \eta_0|^2 + |x -
y_0|^2 \leq \epsilon^2 \} \, \]
has a positive Liouville measure and $\Pi_0 \subset \Pi.$ Moreover, for
every $\nu \in \Pi_0$ we have
\[ T(\nu) = \pi, \:\: S(\nu) = (\lambda - b) \pi, \:\: q(\nu) =
\frac{\pi}{2} m \, \]
with $m \in \Z$ independent on $\nu.$ We may apply Theorem 6 with $r_p(h) = \frac{1}{\pi}\Bigl[\frac{\pi}{2}
m - h^{-1}(\lambda - b) \pi\Bigr]_{2\pi} + 2p,\:\: p \in \Z$, to conclude
that
\[ \# \{w \in {\rm{Res}}\: L(h): | w - \lambda - r_p(h)h| < h \} \geq (2\pi)^{-n}\mu(\Pi_0) h^{1-n} \,. \]
 On the other hand, for $p \neq j$ and $0 < h \leq h_0$ we have
\[ \{w : |\Re w - \lambda - r_p(h)h| < h \} \cap \{ w: |\Re w -
\lambda - r_j(h)h| < h \} = \emptyset \, \]
and the clusters related to $p \neq j$ produce different resonances.
Choosing $\delta > 0$ so that $]\lambda - \delta, \lambda + \delta [
\subset ]b, b + a^2[$, one obtains easily
\[ \# \{ w \in {\rm{Res}}\:L(h) : | w - \lambda| \leq \delta \} \geq \alpha \delta (2\pi)^{-n} \mu(\Pi_0) h^{-n} \, \]
with $\alpha > 0$ independent on $\delta.$ A stronger asymptotic for the
number of the resonances in $$[b, b + a^2] + i[-R(h), 0]$$
 has been obtained by Stefanov \cite{St}. Notice that in the above result we count
only the resonances lying in clusters.


\section{Appendix}

In this Appendix we present a proof of (\ref{eq:as}). Following the Remark after Lemma 2, we will assume that $\varphi_2 = \psi^2,\: \psi \in C_0^{\infty}(I_1; \RR^{+}),\:\:I_1 \subset I_0$. Recall that $\lambda \in I_0,\: \supp\: \theta(t) \subset [-\delta_1, \delta_1]$ and $\chi(x) = 1$ for $|x| \leq 2R,\: R > R_0.$ It is easy to see that
\[ \tr  \Bigl[\frac{1}{2 \pi h}\int e^{it(\lambda - Q_j) h^{-1}} \theta(t)\psi^2(Q_j)(1- \chi^2) dt \Bigr]_{j=1}^2 \, \]
\[ =  \frac{1}{2 \pi h}\int e^{it\lambda h^{-1}} \theta(t) \tr \Bigl( \Big[\psi^2(Q_j)\Bigr]_{j=1}^2  e^{-itQ_2/h} (1- \chi^2)\Bigr) dt \, \]
\[ +  \frac{1}{2 \pi h} \int  e^{it\lambda h^{-1}}\: \theta(t)\: \tr \Bigl(\psi^2(Q_1) \Bigl[ e^{-itQ_j/h}\Bigr]_{j = 1}^2(1 - \chi^2) \Bigr)dt  = {\mathcal A} + {\mathcal B} \,. \]
This representation is justified by applying Lemma 4.1 in \cite{Bo2} saying that
\[ \| \Bigl[\psi^2(Q_j)\Bigr]_{j=1}^2 \|_{\tr} = {\mathcal O}(h^{-n}),\:\: \|\psi^2(Q_1)\Bigl[ e^{-itQ_j/h}\Bigr]_{j = 1}^2 \|_{\tr} = {\mathcal O}(h^{-1-n}) \,. \]
We treat below ${\mathcal A}$ following closely the analysis of J.-F. Bony in Section 4.2, \cite{Bo2}. Put ${\mathcal A} = {\mathcal A}_1 + {\mathcal A}_2,$ where
\[ {\mathcal A}_1 = \frac{1}{2 \pi h}\int e^{it\lambda h^{-1}} \theta(t) \tr \Bigl((\psi(Q_1) - \psi(Q_2))e^{-itQ_2/h}\psi(Q_2) (1- \chi^2)\Bigr)dt \,, \]
\[ {\mathcal A}_2 = \frac{1}{2 \pi h}\int e^{it\lambda h^{-1}} \theta(t) \tr \Bigl (\psi(Q_1)(\psi(Q_1) - \psi(Q_2))e^{-itQ_2/h} (1 - \chi^2)\Bigr) dt \,. \]
We deal with the analysis of ${\mathcal A}_1$ only, since that of ${\mathcal A}_2$ is similar (see also Section 4.2,  \cite{Bo2}). First, we find a pseudodifferential operator $Q$ with symbol in $S^0(1)$ so that
\[ {\mathcal A}_1 = \frac{1}{2 \pi h}\int e^{it\lambda h^{-1}} \theta(t) \tr \Bigl( e^{-itQ_2/h} \psi(Q_2)Q(Q_1 - Q_2)\tilde{\psi}(Q_2)\Bigr) dt \,, \]
where $\tilde{\psi} \in C_0^{\infty}(\RR)$ is such that $\tilde{\psi} = 1$ on $\supp\:\: \psi.$ We use the notations of \cite{DS} for h-pseudodifferential operators and set $\langle x \rangle = (1 + |x|^2)^{1/2}$. Moreover, modulo a term in $S^N(1)$, the symbol of $Q$ is supported in $\{(x,\xi):\: |x | > 2R\}$.  Secondly, we obtain the existence of a pseudodifferential operator $S$ with symbol 
$$s(x,y,\xi;h) \in S^0 \Bigl( {\langle x \rangle}^{-n-1} {\langle \xi \rangle}^{-N}\Bigr),\:\: \forall N \in \N,$$
 having compact support in $\xi$ and $(x-y)$ and support in $\{ (x, \xi): |x | > 2R,\:\: (x,\xi) \in l_2^{-1}(I_1)\}$ so that
\[ {\mathcal A}_1 = \frac{1}{2 \pi h} \tr \Bigl(\int e^{it\lambda h^{-1}} \theta(t) e^{-itQ_2/h} S dt\Bigr) + {\mathcal O}(h^{\infty}) \,.\]
Applying Theorem 2 in \cite{Bo2}, we obtain the existence of a Fourier integral operator ${\mathcal U}_t$ such that for $|t| \leq \delta_1$ and $\delta_1$ sufficiently small we have 
$$\|{\mathcal U}_t - e^{-itL_2/h}S \|_{ \tr} = {\mathcal O}(h^{\infty}).$$
Next, we write the kernel of the operator $\int e^{it\lambda h^{-1}} \theta(t) {\mathcal U}_t dt$ in the form
$$ K(x,y; h) = \frac{1}{(2\pi h)^n} \int\int e^{i\Bigl(t\lambda +\Phi(t, x,\xi) - y.\xi \Bigr)/h} \theta(t) A(t,x,y,\xi; h) dt d\xi$$
and deduce that
$${\mathcal A}_1 = \frac{1}{(2 \pi h)^{n+1}} \int \int\int e^{i\Bigl(t\lambda + \Phi(t, x,\xi) - x.\xi\Bigr)/h} \theta(t) A(t,x, x, \xi ; h) dt\: dx\: d\xi + {\mathcal O}(h^{\infty}) \,.\] Here $\Phi(t,x,\xi)$ is the solution of the eikonal equation
$$\cases \partial_t \Phi + l_2(x, \partial_x \Phi) = 0, \cr
\Phi(0,x, \xi) = x.\xi, \endcases $$ 
$l_j(x,\xi)$ being the principal symbol of $Q_j,\:\ j= 1,2$, and all derivatives $\partial_t^{\alpha}\partial_x^{\beta}\partial_{\xi}^{\gamma}\Bigl(\Phi(t,x,\xi) - x.\xi\Bigr)$ are uniformly bounded for $(t,x,\xi) \in [-\delta_1, \delta_1] \times \RR^n \times B(0, C_1)$ and $(\alpha, \beta, \gamma) \neq (0,0,0).$ Moreover, the symbol $A(t,x,x,\xi)$ has support in  $\{(x,\xi):  |x | > 2R,\: |\xi| \leq C_1,\:\: (x,\xi) \in l_2^{-1}(I_1)\}$ so that for all $\alpha$ and $|t| \leq \delta_1$ we have
\begin{equation}
 |\partial^{\alpha} A(t,x,x,\xi)| \leq C_{\alpha} {\langle x \rangle}^{-n-1}. 
\end{equation}
The last estimate enables us to calculate ${\mathcal A}_1$ by using an infinite partition of unity
\[ \sum_{\alpha \in \N^n} \Psi (x - \alpha) = 1,\:\: \forall x \in \RR^n \,, \]
$\Psi \in C_0^{\infty}(K),\: \Psi(x) \geq 0$, $K$ being a neighborhood of the unit cube.
Consequently, for every fixed $h \in ]0,h_0]$ we have
\[ {\mathcal A}_1 =  \frac{1}{(2 \pi h)^{n+1}} \lim_{m \to \infty} \int \int\int e^{i\Bigl(t\lambda + \Phi(t, x,\xi) - x.\xi\Bigr)/h} \theta(t) \, \]
\[ \times \sum_{|\alpha| \leq m} \Psi(x -\alpha)A(t,x, x, \xi ; h) dt\: dx\: d\xi + {\mathcal O}(h^{\infty}) = \lim_{m \to \infty} I_m + {\mathcal O}(h^{\infty}) \, \]
and we reduce the problem to the analysis of the integrals $I_m$ over  a compact set in $(t, x, \xi).$ Concerning the phase function, we observe that
\[ t\lambda + \Phi(t, x,\xi) - x.\xi = t\Bigl( \lambda - l_2(x,\xi) + {\mathcal O}(t)\Bigr) \,, \]
where ${\mathcal O}(t)$ is uniformly bounded on the support of $\theta(t)A(t,x,x,\xi)$ since the derivatives of $\Bigl(\Phi(t,x,\xi) - x.\xi\Bigr)$ are bounded on this set. Finally, to have an uniform bound for the remainder with respect to $m \to \infty$, notice that
\begin{equation}
|\partial_{x, \xi} l_2(x,\xi)| \geq \delta_2 > 0
\end{equation}
for $|\xi| \leq C_1,\:\: (x,\xi) \in l_2^{-1}( \lambda),\:\: \lambda \in I_0.$
The last condition follows easily from the form of the principal symbol
$$l_2(x,\xi) = |\xi|^2 + \sum_{|\alpha| = 2} b_{\alpha,R}(x) \xi^{\alpha} + \sum_{|\alpha| \leq 1} b_{\alpha, R}(x) \xi^{\alpha}$$
of the operator  $Q_2$, constructed in \cite{Bo2}, and the fact that
 $$|b_{\alpha, R}(x)| +  |\partial_x b_{\alpha, R}(x)| \leq \epsilon_1(R)$$
 with $\epsilon_1(R) \longrightarrow 0$ as $R \longrightarrow +\infty$ (see Section 2.3 in \cite{Bo2} for more details). Taking $R \gg 1$ sufficiently large, we arrange (9.2) uniformly with respect to $|\xi| \leq C_1$ and $(x,\xi) \in l_2^{-1}(\lambda).$ Now the critical points of the phase function $(t\lambda + \Phi(t, x,\xi) - x.\xi)$ become $t = 0,\: l_2(x,\xi) = \lambda$ and by the stationary phase method we obtain 
\[ I_m = \frac{1}{(2 \pi h)^n} \psi(\lambda)\int_{l_2(x,\xi) = \lambda} \sum_{|\alpha| \leq m} \Psi(x - \alpha)A_1(0,x,\xi,\lambda)(1 - \chi^2)(x) L_{\lambda}(d\omega) + {\mathcal O}(h^{1-n}) \,, \]
where $L_{\lambda}(d\omega)$ is the Liouville measure on ${l_2(x,\xi) = \lambda}$ and the remainder ${\mathcal O}(h^{1-n})$ is uniform with respect to $\lambda \in I_0$ and $m \in \N.$ Taking the limit  $m \to \infty$, we obtain an asymptotics of ${\mathcal A}_1.$ \\

For the analysis of ${\mathcal B}$ we use the representation

\[ \Bigl[e^{-itQ_j/h}\Bigr]_{j=1}^2 = \frac{t}{ih} \int_0^1 e^{-istQ_1/h} (Q_1 - Q_2)e^{-i(1-s)tQ_2/h} ds \,. \]
 Following the argument in Section 4.3, \cite{Bo2}, we find pseudodifferential operators $$Q \in {\rm Op}_{h} \Bigl(S^0({\langle x \rangle}^{-n-1} {\langle \xi \rangle}^{-N})\Bigr),\: \tilde{Q} \in {\rm Op}_{h} \Bigl(S^0( {\langle \xi \rangle}^{-N})\Bigr)$$
 with symbols $q(x,y,\xi; h),\: \tilde{q}(x,y,\xi; h)$ having compact support in $\xi$ and $(x-y)$ so that
\[ {\mathcal B} = \frac{1}{2\pi h^2} \tr \Bigl( \int e^{i t\lambda /h} t \theta(t) \int_0^1 e^{-istQ_1/h} Q  e^{-i(1-s)tQ_2/h} \tilde{Q} ds dt \Bigr) + {\mathcal O}(h^{\infty}) \,. \] 
 Moreover, modulo a term in $S^N(1)$, the symbol of $\tilde{Q}$ is supported in $\{(x,\xi):\: |x | > 2R\}$.
 Applying an approximation of the unitary groups $e^{-istQ_1/h},\: e^{-i(1-s)tQ_2/h}$ by Fourier integral operators, we are reduced to study the integral
\[ J  =  \frac{1}{(2 \pi h)^{2n + 2}} \int \int_0^1 \int e^{i t\lambda/h} t\theta(t) e^{i\Bigl(\Phi_1(st, x, \xi) - z.\xi\Bigr)/h}e^{i\Bigl(\Phi_2((1-s)t, z, \eta) - x.\eta\Bigr)/h} \, \]
\[ \times  B(t,s,X) dt ds dX \,, \]
where $X = (x, z, \xi, \eta)$ and the phase functions $\Phi_1(t,x,\xi),\:\: \Phi_2(s, z, \eta)$ are related to the eikonal equations with symbols $l_1(x,\xi)$ and $l_2(z,\eta)$, respectively. The amplitude $B(t,s,X)$ has a compact support with respect to $(\xi, \eta)$ and its support with respect to $x$ is included in the set $\{(x,\xi):\: |x| \geq 2R \}.$ Moreover, $\partial^{\alpha}B(t,s, X)$ satisfy decreasing estimates with respect to $(x, z)$ like those in (9.1).\\

In the same way, as in \cite{Bo2}, we check that the critical points of the phase in the  integral $J$ are related to the closed trajectories composed as union of a curve 
$$\{\exp\Bigl(\tau H_{l_1}\Bigr)(\rho):\: 0 \leq \tau \leq st \}$$
 of the Hamilton field $H_{l_1}$ starting at same point $\rho \in \{(x,\xi) \in \RR^n: |x| > 2R\} $ and a curve 
$$\{\exp\Bigl(\tau H_{l_2}\Bigr)(\sigma):\: 0 \leq \tau \leq (1-s)t \},\:\:\sigma = \exp(st H_{l_1})(\rho)$$
 of the Hamilton field $H_{l_2}$. For $ 0 < t \leq \delta_1$, $\delta_1$ sufficiently small and $R > 0$ large enough, there are no such closed trajectories and the critical points are obtained for $t = 0$, only. We write the phase function in the form
\[ t\Bigl[ \lambda - sl_1(x,\xi) - (1-s)l_2(z,\eta) + {\mathcal O}(t)\Bigr] + (x- z)(\xi - \eta) \, \]
and the critical points become
\[ t = 0,\: sl_1(x,\xi)+ (1-s)l_2(x,\xi) = \lambda,\: x = z, \: \xi = \eta \,. \]
For $|x| \geq 2R$ and $0 \leq s \leq 1$, according to (2.6), we deduce
\[m_s(x, \xi) = sl_1(x,\xi) + (1-s)l_2(x,\xi) = |\xi|^2 + \eta_0(R)|\xi|^2 \, \]
\[ = l_1(x,\xi) + \eta_1(R)|\xi|^2 = l_2(x,\xi) + \eta_2(R)|\xi|^2 \, \]
with $\eta_i(R) \longrightarrow 0$ as $R \to +\infty,\:\: i = 0,1,2.$ Thus for $\lambda \in I_0$ and $R$ large enough the energy surface 
$$\Sigma_s(\lambda) = \{(x,\xi): m_s(x, \xi) = \lambda,\: |x| \geq 2R \}$$
 is non-degenerate. Repeating the argument used for ${\mathcal A}_1$, and applying the stationary phase method, we get an asymptotics
\[ J = \frac{1}{(2 \pi h)^n} b(\lambda) \int_0^1 \int_{m_s(x,\xi) = \lambda} B_1(s, x, \xi, \lambda) L_{s, \lambda}(d\omega)ds + {\mathcal O}(h^{1-n}) \,, \]
where $L_{s,\lambda}(d\omega)$ is the Liouville measure on $\Sigma_s(\lambda).$
 Notice that the first term with power $h^{-1-n}$ vanishes because we have the factor $t\theta(t)$ and the term involving $h^{-n}$  yields the contribution to the leading term in (5.10). Moreover, $b(\lambda)$ has support in a small neighborhood of $I_1$ and taking $R > 0$ large,  we may assume that $b(\lambda) \in C_0^0(I_0).$ This completes the proof of (5.10).\\

The above argument shows that for $\lambda \notin I_0$ the phase functions in $I_m$ and $J$ have no critical points over the support of the integrand. Consequently, by an integration by parts, we obtain 
\[ \tr \Bigl[ ({\cal F}_h^{-1} \theta) \Bigl(\lambda- Q_j\Bigr)
{\varphi_2}(Q_j)(1-\chi^2) \Bigr]_{j=1}^2 =  {\mathcal O}(h^{\infty}) \, \]
uniformly with respect to $\lambda \notin I_0.$


{\footnotesize


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\bigskip

\noindent
{\sc D\'epartement de Math\'ematiques Appliqu\'ees,
Universit\'e Bordeaux I, \\
351, Cours de la Lib\'eration, 33405  Talence, FRANCE}\\
\email{vbruneau@@math.u-bordeaux.fr}\\
\email{petkov@@math.u-bordeaux.fr}\\
\medskip

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