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Schrodinger operator, semiclassical limit, scattering phase, spectral shift function, Weyl asymptotic
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%If  $\ssf_\chi$ satisfies the conditions
%\begin{equation} \label{eq:6.5} \ssf_\chi(\lambda) =  {\cal O}(h^{-n}),
%\end{equation}
%\begin{equation} \label{eq:6.6}
%\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda) = {\cal O}(h^{-n}), 
%\end{equation}
%applying a Tauberian argument (see Theorem IV. 13 in \cite{R1}), we get the representation
%$$ \ssf_\chi(\lambda) = ({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda) + {\cal O}(h^{-n+1}).$$
 
%The estimate (\ref{eq:6.5}) is a consequence of the functional calculus (see \cite{Sj1}) and th%e fact that the function ${\bf 1}_{[\mu_0,\lambda]}\varphi(\lambda)$ is bounded from above by a% smooth function $\Psi(\lambda) $ equal to $1$ on ${\rm supp}\:{\bf 1}_{[\mu_0,\lambda]}\varphi%(\lambda)$. 
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%If  $\ssf_\chi$ satisfies the conditions
%\begin{equation} \label{eq:6.5} \ssf_\chi(\lambda) =  {\cal O}(h^{-n}),
%\end{equation}
%\begin{equation} \label{eq:6.6}
%\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda) = {\cal O}(h^{-n}), 
%\end{equation}
%applying a Tauberian argument (see Theorem IV. 13 in \cite{R1}), we get the representation
%$$ \ssf_\chi(\lambda) = ({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda) + {\cal O}(h^{-n+1}).$$
 
%The estimate (\ref{eq:6.5}) is a consequence of the functional calculus (see \cite{Sj1}) and th%e fact that the function ${\bf 1}_{[\mu_0,\lambda]}\varphi(\lambda)$ is bounded from above by a% smooth function $\Psi(\lambda) $ equal to $1$ on ${\rm supp}\:{\bf 1}_{[\mu_0,\lambda]}\varphi%(\lambda)$. 
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\title[Spectral shift function]
{Representation of the spectral shift function and spectral asymptotics for trapping perturbations}
\author[V. Bruneau, V. Petkov]{Vincent Bruneau and Vesselin Petkov}

%------------------------lettres grecs -----------------------

\def\e{\varepsilon}
\def\phi {\varphi}
\def \la {{\lambda}}
\def \a {{\alpha}}
\def\t{\theta}

%-------------------------domaine---------------------
\newcommand{\omd}{\Omega_{\delta}}

 %analyse
\def\CC{{\cal C}}
\def\lap{\bigtriangleup}
\def\mul{\int_{\mu_0}^{\lambda}}

%-----------------------------tilde-------------------
\newcommand{\tL}{\tilde L}
\newcommand{\tl}{\tilde l}
\newcommand{\tP}{\tilde P}
\newcommand{\tR}{\tilde R}
\newcommand{\tchi}{\tilde \chi}
\newcommand{\tpsi}{\tilde \psi}
\newcommand{\tE}{\tilde E}
\newcommand{\tssf}{\tilde \ssf}

%----------------------------pointe--------------------
\newcommand{\pL}{L_.}

%----------------------------SSF----------------------
\newcommand{\ssf}{\xi}
\newcommand{\ssfr}{\xi_\rho}

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 \def  \rn{{{\RR}^n}}

\begin{document}    

\begin{abstract}

We obtain in the semi-classical setup of "black box" long-range perturbations a representation for the derivative of spectral shift function $\xi(\lambda)$ related to two self-adjoint operators $L_j(h), \: j = 1,2. $ We show that the derivative $\xi'(\lambda)$ is estimated by the norms of the cut-off resolvents of the operators $L_j(h)$. Finally, we establish a Weyl type formula for the spectral shift function $\xi(\lambda)$ generalizing the results of Robert \cite{R3} and Christiansen \cite{Ch}.
\end{abstract}



   
\maketitle   
 


\section{Introduction}
 
The purpose of this paper is to obtain a semi-classical representation for the derivative of the spectral shift function (SSF) related to two self-adjoint operators $L_1=L_1(h),\: L_2 = L_2(h)$ on $\RR^n$, $n \geq 2$, depending on $h \in ]0,h_0]$,  and satisfying long-range "black box" assumptions (2.1)- (2.10) given  in Section 2.  Let $f \in {\cal S}(\RR)$ and let $\chi \in C_0^\infty (\{x: |x| \leq \rho_1\})$ be equal to 1 for $|x| \leq 2\rho_0 < \rho_1.$  The constants $\rho_0, \:\: \rho_1, \:\: 0 < R_0 < \rho_0 $ depends on the behaviour of the Hamiltonian trajectories of $L_1$ and $L_2$ (see Section 2). Under weaker assumptions J. Sj\"ostrand \cite{Sj1} proved that the operators
 $$\chi f(L_j),\:  f(L_j)\chi,\: j =1,2,\:  (1-\chi) f(L_2)(1-\chi)-(1-\chi) f(L_1)(1-\chi) $$
  are trace class and we can define
$$ \trb (f(L_2) - f(L_1))= [\tr \chi f(L_j)\chi + \tr(1- \chi) f(L_j)\chi +
\tr \chi f(L_j)(1-\chi) ]_{j=1}^2$$
$$ + \tr [(1-\chi) f(L_j)(1-\chi)]_{j=1}^2.$$
This expression is independent of the choice of $\chi$ and here and below we write $[a_j]_{j=1}^2=a_2-a_1$. We introduce the {\em spectral shift function} $\xi(\lambda) \in S'(\RR)$ as the distribution
$$ \langle \xi , f' \rangle_{{\cal S}',{\cal S}} =- \trb(f(L_2) - f(L_1)).$$
In the following to simplify the notations we will often omit the subscript $j = 1,2$ and simply write $L_.$ or $L$. Given a fixed interval  $J = ]\mu_0, \mu_1[ \subset \subset \RR^+$, we can find two self-adjoint operators  $\tL_1$, $\tL_2$ on $L^2(\RR^n)$ such that $\tL_.$ is a compactly supported perturbation of $L_.$ and $J$ is non trapping and non-critical for $\tL_. $ (see \cite{BrPe}). The spectral projections $E_.'(\lambda)$ and $\tE_.'(\lambda)$ are given by the formula 
$$E_.'(\lambda) = \frac{1}{2\pi i}( R_.(\lambda + i0) - R_.(\lambda -i0)),\:\tE_.'(\lambda) = \frac{1}{2\pi i}( \tR_.(\lambda + i0) -\tR_.(\lambda -i0)), \: \lambda \in J,$$
where the operators $R(\lambda \pm i0)$ for fixed $h > 0$ are determined as the limits of the resolvents
$$R_.(\lambda \pm i0) = \lim_{\epsilon \to 0, \: \epsilon > 0} R_.(\lambda \pm i\epsilon) = \lim_{\epsilon \to 0, \: \epsilon > 0} (L _. -\lambda \mp i \epsilon)^{-1}$$
in the spaces of bounded operators ${\cal L} ({\cal H}^{0,s}, {\cal H}^{0,-s}),\: s > 1/2,$ (see \cite{BrPe} and the definitions of ${\cal H}^{0,s}$ in Section 2) and similarly the resolvents $\tR_.(\lambda \pm i0)$ are determined as bounded operators in ${\cal L}(L^2_s(\RR^n), L^2_{-s}(\RR^n))$, $L^2_s $ being the weight space $L^2(\RR^n,\langle x \rangle^s dx)$. Given two real functions $f,\: g \in C^{\infty}_0(\RR^n)$, we write $f \prec g$, if supp$f$ is contained in the interior of the region, where $g=1$.\\ 

Under the assumptions (2.1)-(2.10) Sj\"{o}strand obtained \cite{Sj1} a trace formula for $\trb (f(L_2) - f(L_1))$ related to the resonances of $L_j,\: j =1,2$,  plus a remainder ${\cal O}_f(h^{-n^{\sharp}})$ depending on $f$. Following this way it seems difficult to obtain directly from the results in \cite{Sj1} a representation for the derivative $\xi'(\lambda)$. On the other hand, for long-range perturbations $P_j = P_j(h)$ acting in a dense set of $L^2(\RR^n)$, Robert \cite{R3} established a representation formula for $\xi'(\lambda)$ related to $P_j, \: j = 1,2$. The representation in \cite{R3} involves some remainder terms related to the constructions of long time parametrices for the propagators $U_j(t) = e^{ith^{-1}P_j}$ and some precise estimates of the difference $U_2(t) - U_1(t).$ This construction is very technical and in the case of "black-box" scattering the situation becomes more complicated.\\

Under suitable assumptions the derivative $\xi'(\lambda)$ coincides modulo some factor with the derivative of the {\em scattering phase}
$$\sigma(\lambda, h) = \frac{1}{2 \pi i}\log \det S(\lambda, h),\: \lambda \in J,$$
where $S(\lambda, h)$ is the scattering operator related to $P_j,\: j =1,2.$ The existence of the scattering operator $S(\lambda, h)$ and that of $\det S(\lambda, h)$ in the general setup of "black box" scattering is far from transparent. On the other hand, in the case of "black box" compactly supported perturbations the asymptotic behaviour of the scattering phase $\sigma(\lambda, h)$ is important for the analysis of the semi-classical Breit-Wigner approximation (see \cite{PZ}).\\

 In this paper we treat the problem of the representation of the derivative $\xi'(\lambda)$ in great generality covering the case of "black box" semi-classical scattering (\cite{Sj1}, \cite{Sj2}, \cite{TZ}). Our arguments are not based neither on the existence of the scattering phase $\sigma(\lambda, h)$ nor on the existence of the scattering operator $S(\lambda, h)$. Under the general "black box" scattering assumptions it seems quite difficult to construct a parametrix for the propagators $U_j(t) = e^{ith^{-1}P_j}$ and for this reason we follow a stationary argument as in \cite{Sj1}, \cite{Sj2} combined with the recent results in \cite{BrPe} concerning the semi-classical resolvent estimates for trapping perturbations. Moreover, we obtain a more precise information for the remainder terms in the formula for $\xi'(\lambda)$ and we  prove a Weyl type asymptotic for $\xi(\lambda)$ generalizing the results of \cite{R3} and \cite{Ch}.\\

 Our  main result is the following.

\begin{thm}\label{thm:repssf} Let $J = ]\mu_0, \mu_1[ \subset \subset \RR^{+}$ be fixed and let  $L_j,\: j =1,2$ satisfy the assumptions $(2.1)-(2.10)$. Let $\rho_0 > R_0$ be as in Proposition \ref{prop1} and let $\chi \in C_0^\infty(\{x: |x| \leq \rho_1\})$ be  equal to $1$ for $|x| \leq 2\rho_0, \: \rho_1 > 2\rho_0$. Then for any $\la \in J$, the operators $\chi E_.'(\lambda) \chi $ and $\chi \tE_.'(\lambda)\chi $ are trace class ones and  we have
\begin{equation}\label{eq:1.1}
\begin{array}{ccl}
\ssf'(\la) = \left[\tr\Big( \chi E_j'(\la) \chi\Big) \right]_{j=1}^2 + \tr T^+(\la+i0) -\tr T^-(\la-i0) \\
+ \tilde{\xi}'(\lambda) - \left[\tr\Big( \chi \tE_j'(\la) \chi\Big) \right]_{j=1}^2 + {\cal O}(e^{-\frac{C}{h}}), 
\end{array}
\end{equation}
where $C > 0,\:\tilde{\xi}(\lambda)$ is the spectral shift function related to $\tL_1$ and $\tL_2$, while  $T^\pm(z)$ are trace class operators for $\pm \Im z > 0$ with
$$\| T^\pm (z) \|_\tr \leq {\cal O}(h^\infty) \; 
 \Big( 1 + \| \tchi R_1(z) \tchi  \|_{{\cal H} \to {\cal H}}
 + \| \tchi R_2(z) \tchi  \|_{{\cal H} \to {\cal H}} \Big), \quad \tchi \succ \chi $$ 
 uniformly with respect to 
$z  \in {\cal B}_{\pm} = \{z \in \C:$ $ (\Re z,\pm \Im z) \in J\times ]0, 1]\}$ and $h \in ]0,h_0]$.  Moreover, the application ${\cal B}_{\pm} \ni z \mapsto \tr(T^\pm(z))$ is holomorphic on ${\cal B}_{\pm}$.
\end{thm}

{\bf Remark.} Notice that the fact that $J$ is a non-trapping and non-critical interval for $\tL_j,\: j =1,2, $ implies the representation
\begin{equation} \label{eq:1.2}
\tilde{\xi}'(\lambda) - \left[\tr\Big( \chi \tE_j'(\la) \chi\Big) \right]_{j=1}^2 = h^{-n}\sum_{k=0}^N d_k(\lambda) h^k + {\cal O}_{\chi}(h^{-n + N + 1}), \: \forall N \in \N,
\end{equation}
where $d_k(\lambda)$ are $C^{\infty}$ functions on $J$ (see \cite{R3}).\\

Our representation yields an estimate for the derivative of $\xi(\lambda)$ by the norms of the cut-off resolvents.

\begin{cor}
Under the assumptions of Theorem 1 we have the estimate
$$|\xi'(\lambda)| \leq C\Bigl(h^{-n} + \| \sum_{\pm}\tchi R_1(\lambda \pm i0) \tchi  \|_{{\cal H} \to {\cal H}}
 + \|\sum_{\pm} \tchi R_2(\lambda \pm i0)) \tchi  \|_{{\cal H} \to {\cal H}} \Bigr),\: \lambda \in J,\: \:  h \in ]0, h_0].$$
\end{cor}

The above estimate shows that the blow up of the derivative $\xi'(\lambda)$ is related to the estimates of the cut-off resolvents which depend on the distribution of the resonances (see \cite{Bu}, \cite{TZ}, \cite{BZ}, \cite{Ste}). Denote by ${\rm Res}\: L_j$ the set of resonances of $L_j,\: j =1,2$ (see \cite{Sj1}, \cite{Sj2} for a precise definition). The representation (\ref{eq:1.1}) combined with the result of Stefanov \cite{Ste} imply the following.

\begin{cor}\label{cor2} 
Assume that $L_j, \: j =1,2$ satisfy the assumptions $(2.1) - (2.10)$. Let $f \in C_0^\infty(J)$ and let $\chi(x)$ be as in Theorem 1.
Suppose that for some $\epsilon > 0, \: C > 0, \: q \geq 1$ we have
\begin{equation} \label{eq:1.3}
{\rm{dist}} \{ {\rm{Res} \;} L_j(h), \: J_1 \} \geq \epsilon \exp(-C h^{-q}), \: \: j =1, 2,  \: h \in ]0, h_0],
\end{equation}
$ J_1 \subset \RR^{+}$ being an open interval containing ${\overline J}$.
Then for  $|t| \geq \delta > 0$ and $h \in ]0,h_0]$ we have
$$\tr_{{\rm bb}} \Bigl(e^{-ith^{-1}L_2} f(L_2) - e^{-ith^{-1}L_1} f(L_1)\Bigr)  = \left[\tr \Big(\chi e^{-ith^{-1}L_j}f(L_j) \chi \Big) \right]_{j=1}^2 + {\cal O}( h^{\infty}).$$
\end{cor}

{\bf Remark.} Applying the recent result of Burq and Zworski \cite{BZ} for $t > h^{-L}$, we can express the term 
$$\left[\tr \Big(\chi e^{-ith^{-1}L_j}f(L_j) \chi \Big) \right]_{j=1}^2$$
 by the resonances of $L_j$ lying in a rectangle close to the real axis. On the other hand, the assumption (\ref{eq:1.3}) is a technical one and in  many cases this assumption is  satisfied (see \cite{Bu}). As it was mentioned in \cite{BrPe}, (\ref{eq:1.3}) implies the estimate
\begin{equation} \label{eq:1.4}
 \|\chi R_j(\lambda \pm i\tau) \chi \|_{{\cal H} \to {\cal H}} \leq C\exp(Ch^{-p}), j =1,2, \: p \geq 1 
\end{equation}
uniformly with respect to $\lambda \in J, \:\; \tau \in ]0,1]$ and $h \in ]0, h_0].$\\

The idea of the proof of Theorem 1 is to use an  approximation of $\ssf(\lambda)$ by a sequence of locally integrable functions. Let $\chi_0   \in C_0^\infty (\RR^n)$ be equal to 1 near $\overline{B(0,R_0)}$. For $\rho > 1$, introduce $\chi_\rho(x) = \chi_0(x / \rho)$ and define the temperate distribution $\ssfr$ by the equality
\begin{equation}\label{eq:3.2}
 \langle \ssfr , f' \rangle_{{\cal S}',{\cal S}} = 
 -\trb \Big(\chi_\rho \Big(f(L_2) - f(L_1)\Big) \chi_\rho \Big)=
- \left[\tr\Big( \chi_\rho f(L_j) \chi_\rho \Big) \right]_{j=1}^2 .
\end{equation}
The multiplication operator defined by $\chi_\rho$ converges strongly as $\rho \longrightarrow \infty$ to the identity in $L^2(\RR^n)$ as well as in ${\cal H}_.$. Consequently, we have in the sense of distributions (see \cite{GoKr})
\begin{equation}\label{eq:3.3}
\lim_{\rho \rightarrow + \infty} \ssfr' = \ssf'.
\end{equation}
 In Section 3 we show that
$$\ssfr'(\lambda)=  \left[\tr\Big( \chi_\rho  E_j'(\la) \chi_\rho \Big) \right]_{j=1}^2$$
and the main difficulty is to investigate the limit $\rho \longrightarrow \infty.$ The operators $E_j'(\la)$ are not trace class so we decompose $\xi'_{\rho}(\lambda)$ in a sum of three terms and we examine the corresponding limits separately. The terms $\tr (T^{\pm}(z))$ are obtained as the limits
$$ (\lambda - z_0)^m\lim_{\rho \rightarrow + \infty} \left[ \tr \Big( \chi_\rho  (L_j-z_0)^{-m} R_j (z) [\tL_j,\chi]\tR_j(z)(1-\chi^2_2) \chi_\rho \Big) \right]_{j=1}^2,\: \Re z = \lambda,$$
where $z_0 \in \C,\: \Im z_0 > 0$ is fixed,  $m > n/2$  and $\chi_2 \in C_0^{\infty}(\RR^n)$ is such that $\chi_2 = 1$ on the support of $\chi.$ The analysis of this limit is based on the localization in the incoming and outgoing regions and the ideas of Robert \cite{R3} to combine the semi-classical estimates in these regions with the cyclicity of the trace. Moreover, we use the existence of non-trapping perturbations $\tilde{L}_.$ such that $L_. (1 - \chi) = \tilde{L}_.(1 - \chi)$, where the support of $\chi$ is related to the existence of a compact set $K(J) \subset {\Bbb R}^n$ containing the bounded trajectories of the Hamiltonian fields related to $L_.$ and to the energy level $\lambda \in J$ (see \cite{BrPe} and section 2).\\

The representation (1.1) is used in Section 6 for the proof of a Weyl type asymptotic for $\xi(\lambda).$ The analysis of the integrals $\int_{\mu_0}^{\mu} \tr \Bigl(T^{\pm}(\lambda \pm i0)\Bigr) d\lambda$ follows closely the argument in \cite{R3}. On the other hand, we deal with the general setup of "black box" semi-classical scattering and in this situation the investigation of $\int_{\mu_0}^{\mu} \tr \Bigl(\chi E'_j(\lambda) \chi \Bigr) d\lambda$ leads to some difficulties related to the analysis of $U_j(t)$ for small $t.$ To overcome these difficulties we use the estimates for the trace in \cite{Sj1} combined with a suitable Duhamel formula based on the results in \cite{BrPe}.\\

The paper is organized as follows. In Section 2 we present some known results for non-trapping and trapping perturbations. In Section 3 we prove Theorem 1 and Corollary 1 and 2. Section 4 is devoted to trace class estimates, while the analysis of the operators $T^{\pm}(z)$ is given in Section 5. In Section 6 we study the  Weyl type asymptotics for the spectral shift function.

\section{Preliminaries}


We start by the abstract ``black box'' scattering assumptions introduced in  \cite{SZ}, \cite{Sj1} and \cite{Sj2}. The operator $\pL$ is defined in a domain ${\cal D}_. \subset {\cal H}_.$ of a complex Hilbert space ${\cal H}_.$ with an orthogonal decomposition      
$${\cal H}_. = {\cal H}_{R_0,.} \oplus L^2({\RR}^n \setminus 
B(0,R_0)),\:B(0,R_0) = \{x \in {\RR}^n: |x| \leq R_0 \},\:\:n \geq 2. $$
Introduce the spaces
$${\cal H}_.^{0,s} = {\cal H}_{R_0,.} \oplus  L^2(\RR^n\setminus B(0,R_0), \langle x {\rangle}^s dx), \:\langle x \rangle = (1 + |x|^2)^{1/2},$$
and denote by $\| . \|_{s,s'}$ the norm in  ${\cal L}({\cal H}_.^{0,s},{\cal H}_.^{0,s'})$.
We suppose that ${\cal D}_.$ satisfies
\begin{equation} \label{eq:2.1} 
{\Bbbone }_{{\RR}^n \setminus B(0,R_0)}{\cal D}_. = 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 ${\cal D}_.$ with the norm $\|(\pL+i)u\|_{{\cal H}_.}.$ Then we require that ${\Bbbone }_{{\RR}^n \setminus B(0,R_0)}: {\cal D}_. \longrightarrow H^2({\RR}^n \setminus B(0,R_0))$ 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)}(\pL+i)^{-1} \hbox{is compact}
\end{equation} 
and
\begin{equation}\label{eq:2.3}
 (\pL u)\vert_{{\RR}^n \setminus  \overline{B(0,R_0)}} = Q_.\Bigl( u\vert_{{\RR}^n \setminus \overline{B(0,R_0)}}\Bigr),
\end{equation}
where $Q_.$ is a formally self-adjoint differential operator
\begin{equation}\label{eq:2.4}
 Q_. u = \sum_{| \nu | \leq 2} a_{.,\nu} (x;h) (hD_x)^\nu u, 
\end{equation}
with $ a_{.,\nu} (x;h)=  a_{.,\nu} (x)$ independent of $h$ for $| \nu | = 2$
 and $a_{.,\nu} \in C_b^\infty(\RR^n)$ 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_{.,0}(x,\xi) =\sum_{| \nu | = 2} a_{.,\nu} (x) \xi^\nu \geq C  {\langle \xi \rangle}^2,\:\:\langle \xi \rangle = (1 + |\xi|^2)^{1/2}.
\end{equation}

 There exists  $\gamma > 0$ such that for every $(\alpha,\beta) \in
 {\N}^n \times \N^n$,  we have
\begin{equation} \label{eq:2.6}
 | {\partial}_{x}^{\alpha}{\partial}_{\xi}^{\beta} 
 \Big(\sum_{| \nu | \leq 2} a_{.,\nu} (x;h) \xi^\nu - |\xi|^2 \Big) |
 \leq C_{\alpha,\beta}  {\langle x \rangle}^{-\gamma- \mid \alpha \mid}{\langle 
\xi \rangle}^2 
\end{equation}
uniformly with respect to $h$.

 There exists  $\overline{n} > n$ such that for every $\alpha \in
 {\N}^n$, $| \nu | \leq 2$  we have
\begin{equation}\label{eq:2.7}
 | {\partial}_{x}^{\alpha} \Big( a_{1,\nu} (x;h) - a_{2,\nu} (x;h) \Big) |
 \leq C_{\alpha}  {\langle x \rangle}^{-\overline{n}- \mid \alpha \mid} \label{eq:1.10*}
\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 $\theta_0 \in ]0,\pi[,\:\epsilon > 0$ and $R_1 > R_0$ so that the coefficients $a_{.,\nu}(x;h)$ of $Q_.$ can be extended holomorphically in $x$ to
\begin{equation} \label{2.8}
\{r\omega;\:\omega \in {\C}^n,\: {\rm dist}\:(\omega, S^{n-1}) < \epsilon, \: r \in {\C},\: |r| > R_1,\:{\rm arg} \: r \in [-\epsilon, \theta_0 + \epsilon)\} \label{eq:1.11}
\end{equation}
and (2.6), (2.7) extend to this larger set.

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

 Denote by $N(\pL^{\#}, [-\lambda^2, \lambda^2])$ the number of eigenvalues of $\pL^{\#}$ in the interval $[-\lambda^2, \lambda^2]$. Then we assume that
\begin{equation}\label{eq:2.9}
N(\pL^{\#}, [-\lambda^2, \lambda^2]) = {\cal O}(\Bigl(\frac{\lambda}{h}\Bigr)^{n_.^{\#}}),\: n_.^{\#} \geq n,\: \lambda \geq 1. \label{eq:1.12}
\end{equation}
Finally,  we suppose that
\begin{equation}\label{2.10}
\sigma_{pp}(\pL(h)) \cap [\mu_0, \mu_1] = \emptyset,\: h \in ]0,h_0]. \label{eq:1.13}
\end{equation}

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 $Q.$ Below we collect some known results for non-trapping perturbations concerning resolvent estimates, microlocal estimates and asymptotic behavior of the spectral shift function for non-trapping perturbations. 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 a classical Hamiltonian $l(x,\xi)$  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
$$ \quad |x(t, x_0,\xi_0)| > R,\:\:\forall | t | > T(R). $$
 We say that $J \subset \subset \RR^+$ is non-trapping for  $\tL(h)$ if every $\la \in J$ is a  non-trapping energy level for the principal symbol of $\tL(h)$.
Following \cite{BrPe}, for a fixed $J \subset \subset \RR^+$, we
can  construct two self-adjoint operators $\tL_1$, $\tL_2$ with domains in $L^2(\RR^n)$ so  that $\tL_.$ is a compactly supported perturbation of $L_.$ and $J$ is non trapping for $\tL_.$.
 In the following, $R_.(z) = (L_.(h) - z )^{-1}$ and  $\tR_.(z) = (\tL_.(h) - z )^{-1}$ denote the resolvent of the operators  ${L}(h)$  and  $\tilde{L}(h),$ respectively.

\begin{prop}\label{prop1} (\cite{BrPe})
Let $L(h)$ satisfy the "black box" assumptions $(\ref{eq:2.1}) - (\ref{eq:2.6})$. Then there exist $\rho_0> R_0$ and  a self-adjoint differential  operator $\tL = \tL(h)$ on $L^2(\RR^n)$, satisfying $(\ref{eq:2.1}) - (\ref{eq:2.6})$, such that each $\lambda \in J$ is a non-trapping and non-critical energy level for  $\tL(h)$ and 
$$ L(h) \psi  =  \tL(h) \psi$$ 
for any $\psi \in C^\infty({\RR}^n)$ supported in $\{x \in \RR^n: \; |x|> \rho_0 \}$.
\end{prop}

 Denote by $\| . \|_{s,s'}$ the norm of ${\cal L}(L^{2,s}, L^{2,s'})$, where $L^{2,s}$ is the weight  space $L^2(\RR^n, \langle x \rangle^s dx)$. Following the Mourre theory (see  \cite{M}, \cite{PSS}, \cite{JMP}, \cite{HN}, \cite{R2}, \cite{R3}, \cite{GM1}, \cite{BrPe}), we have the following.

\begin{lem}\label{lem1} 
Let  $s > 1/2.$ Then for $\la \in J$, the limits 
$$(\tL-\la \mp i0)^{-1}:= \lim_{\epsilon \to 0, \: \: \pm \epsilon > 0}  (\tL-\lambda -i\epsilon)^{-1}$$
exist in   ${\cal L}(L^{2,s}, L^{2,-s})$, and for any differential operator
$P = P(h) = \sum_{|\alpha| \leq p}a_{\alpha}(x)(hD_x)^{\alpha}$,  $p\leq 2$,
  there exist $C > 0$ and $h_0 > 0$ such that
\begin{equation}\label{eq:trap.es}
 \|P (\tL-z)^{-1}  \|_{s,-s} \leq C h^{-1}
\end{equation}
uniformly with respect to 
$z  \in {\cal B}_{\pm} = \{z \in \C:\: (\Re z,\pm \Im z) \in J\times ]0, 1]\}$
 and $h \in ]0,h_0]$.
\end{lem}

Moreover, exploiting the constructions of long time approximations in section 4 of \cite{R3} for long-range perturbations of the Laplacian, we can establish  some microlocal resolvent estimates (see the proof of Lemma 2.3 of \cite{RT2}).
For this purpose we introduce the outgoing and incoming regions of the phase space
\begin{equation}\label{defsortant}
\Gamma^\pm (R,d,\sigma_\pm) = \{ (x,\xi) \in \RR^{2n}\; : \; |x| > R, \; d^{-1} < |\xi | < d, \; \frac{\langle x.\xi \rangle}{|x||\xi|} \;  ^>_< \; \sigma_\pm  \},
\end{equation}
where $d>1$, $-1 < \sigma_\pm < 1$ and $R \gg R_0$ is large enough. Consider a symbol  $ \omega_\pm(x,\xi) \in C^\infty (\RR^{2n})$ such that
 supp $\omega_\pm \subset \Gamma^\pm (R,d,\sigma_\pm)$ and 
\begin{equation}\label{defomegapm}
| \partial_x^\alpha \partial_\xi^\beta  \omega_\pm (x,\xi) | \leq
 C_{\alpha,\beta,L} \langle x \rangle^{-|\alpha |}
 \langle \xi \rangle^{- L },\: \forall \alpha, \: \forall \beta 
\end{equation}
for any $L \gg 1$. Then we have the following.

\begin{lem}\label{lem2} 
For $(\la,\tau) \in J \times ]0,1]$
 and for any differential operator
$P = P(h) = \sum_{|\alpha| \leq p}a_{\alpha}(x)(hD_x)^{\alpha}$,  $p\leq 2$,
 the following assertions hold:

i) For $s > 1/2$, $ \delta >1$ there exist $C > 0$ and $h_0 > 0$ such that for
 $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mlntrap.es1}
 \|P(\tL-\la\mp i\tau)^{-1} \omega_\pm(x,hD_x) \|_{-s+\delta,-s} \leq C h^{-1}.
\end{equation}

ii) If $\sigma_+ > \sigma_-$,  then for any $s \gg 1 $ and any $N \in \N$  there exist $C_N > 0$ and $h_0 > 0$ such that for
 $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mlntrap.es2}
 \| \omega_\mp(x,hD_x)P (\tL-\la\mp i\tau)^{-1} \omega_\pm(x,hD_x) \|_{-s,s} \leq 
C_N h^{N}.
\end{equation}


iii) If $\chi_1 \in C_0^\infty(B(0,R)),\: \chi_1 = 1$ on $\overline{B(0,R_0)}$,  then for any $s \gg 1 $ and $N\in \N$  there exist $C_N > 0$ and $h_0 > 0$ such that for
 $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mlntrap.es3}
 \| \chi_1 P  (\tL-\la\mp i\tau)^{-1} \omega_\pm(x,hD_x) \|_{-s,0} \leq 
C_N h^{N}.
\end{equation}
\end{lem}

Now we will recall some results in the {\em trapping} case.

\begin{prop}\label{prop2}
Under the assumptions of Proposition $1$ and $(2.8)$ and $(2.10)$
 we have

i) For every $s > \frac12$ the limits
$$ R(\lambda \pm i 0) =   \lim_{\epsilon \rightarrow 0, \;\epsilon >0}   R(\lambda \pm i \epsilon), \quad \lambda \in J$$
exist in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$ and 
$J \ni \lambda \mapsto R(\lambda \pm i 0) \in {\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$ is continuous on $J$. Moreover, there exist $C > 0$ and $ h_0 > 0$  such that for $\chi \in C_0^\infty(\{x: |x| \leq \rho_1)$ with $\chi(x) = 1$ for $|x| \leq 2 \rho_0$, we have
$$\| R(z) \|_{s,-s} \leq C h^{-2} \Big( 1 + \| \chi R(z) \chi  \|_{{\cal H} \to {\cal H}} \Big)$$
uniformly with respect to 
$z  \in {\cal B}_{\pm}$ and $h \in ]0,h_0]$.


ii) For any $h \in ]0,h_0]$ the operator $L(h)$ has purely absolutely continuous spectrum in $J$.
\end{prop}

\begin{pf}
The assertion i) was established in \cite{BrPe}. To prove $ii)$, it is sufficient to show that 
$$\liminf_{\epsilon \rightarrow 0, \;\epsilon >0} \; \sup_{\mu_0 \leq \lambda \leq \mu_1} |( R(\lambda \pm i \epsilon) \varphi, \varphi)_{{\cal H}} | \leq C(\varphi),$$
for any $\varphi$ in a dense subset of ${\cal H}$ (see Proposition 1.4 of \cite{CFKS}). The set of compactly supported functions is dense in ${\cal H}$, and, moreover, for every function $\chi \in C^{\infty}_0(\RR^n)$ equal to 1 on $\overline{B(0,R_0)}$ we have
$$\chi R(\lambda+ i0) \chi = \chi (L_{\theta} - \lambda)^{-1}\chi,$$
$L_{\theta}$ being the complex scaling operator related to  $L$ (see \cite{SZ}). The continuity of the cut-off resolvent $\chi R(\lambda + i0) \chi$ implies immediately the above estimate for compactly supported $\varphi$ equal to 1 on $B(0,R_0).$

\end{pf}

According to Stone formula (see \cite{RS1}) and to Proposition \ref{prop2}, i), the spectral projector

$$E(\lambda):={\bf 1}_{[\mu_0,\lambda]}(L) = s-\lim_{\epsilon \rightarrow 0, \;\epsilon >0} \frac1{2i \pi} \int_{\mu_0}^\lambda
 \Big( R(\mu+i\epsilon) -  R(\mu-i\epsilon) \Big) d\mu,\:\: \lambda \in J$$
is differentiable in  ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$, $s>1/2$,  with derivative
\begin{equation}\label{eq:2.17}
E_.'(\lambda) = \frac{1}{2i \pi} \Big( R_.(\la+i0) - R_.(\la-i0) \Big).
\end{equation}
Moreover, for any function $f \in C^1_{b}(\RR)$ we have
\begin{equation}\label{eq:2.18}
f(L.) E_.'(\lambda) = f(\lambda) E_.'(\lambda), \quad \lambda \in J,
\end{equation}
provided $f(L_.)$ is a bounded operator in ${\cal L}({\cal H}^{0,-s},{\cal H}^{0,-s})$. To prove this notice that for $\forall \varphi, \: \forall \psi \in {\cal H}$ by the spectral theorem we get
$$\Bigl([f(L) - f(\lambda)](R(\lambda +i\epsilon) - R(\lambda -i\epsilon))\varphi, \psi\Bigr)_{{\cal H}} = $$
$$\int_{-\infty}^{\infty} (f(\mu) - f(\lambda)) \Bigl[\frac{1}{\mu - \lambda -i\epsilon} - \frac{1}{\mu - \lambda +i\epsilon}\Bigr] d(E_{\mu}\varphi, \psi) \longrightarrow_{\epsilon \to 0} 0.$$ Since the limit
$$\lim_{\epsilon \to 0} [f(L) - f(\lambda)]\Bigl[R(\lambda +i\epsilon) - R(\lambda -i\epsilon)\Bigr] = M$$
exists in  ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$, we must have
$$\lim_{\epsilon \to 0}\Bigl([f(L) - f(\lambda)](R(\lambda +i\epsilon) - R(\lambda -i\epsilon))\psi_s\varphi, \psi_s\psi\Bigr)_{{\cal H}} = (M \psi_s \varphi, \psi_s \psi)_{{\cal H}} = 0$$
for every function $\psi_s$ equal to 1 on $\overline{B(0, R_0)}$ and equal to $<x>^{-s}$ for  $|x|$ large enough. This implies easily that $M = 0$ in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$. 

\begin{prop}\label{prop3} (\cite{BrPe})
 Let $L(h)$ be as in Proposition \ref{prop2} and let  $\psi \in C^\infty({\RR}^n)$ be  supported away from $B(0,\rho_1)$ with $\psi(x) = 1$ for $|x| \gg 1$. Let $\chi \in C_0^\infty(\RR^n)$ be as in Proposition 2. Then for any $(\la,\tau) \in J \times ]0,1]$ and $h \in ]0,h_0] $ the following assertions hold:

i) For any $s > 1/2$, $\delta >1$ and $N\in \N$  there exist $C_N > 0$ and $h_0 > 0$ such that for
 $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mltrap.es1}
 \|\psi R(\la\pm i \tau)\psi \omega_\pm(x,hD_x) \|_{-s+\delta,-s} \leq 
C_N h^{-1} ( 1 + h^N \| \chi  R(\la\pm i \tau) \chi \|_{{\cal H} \to {\cal H}}).
\end{equation}

ii) If $\sigma_+ > \sigma_-$, then for any $s \gg 1 $ and $N\in \N$,  there exist $C_N > 0$ and $h_0 > 0$ such that for  $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mltrap.es2}
 \| \omega_\mp(x,hD_x) \psi R(\la\pm i \tau)\psi \omega_\pm(x,hD_x) \|_{-s,s} \leq 
C_N h^{N} ( 1 + \| \chi  R(\la\pm i \tau) \chi \|_{{\cal H} \to {\cal H}}).
\end{equation}

iii) If $\varphi \in C_0^\infty(B(0,2\rho_0)),\: \varphi = 1$ on $\overline{B(0,R_0)}$,  then for any $s \gg 1 $ and $N\in \N$  there exist $C_N > 0$ and $h_0 > 0$ such that for
 $h \in ]0,h_0]$ we have
\begin{equation}\label{eq:mltrap.es3}
 \| \varphi R(\la\pm i \tau)\psi \omega_\pm(x,hD_x) \|_{-s,0} \leq 
C_N h^{N} ( 1 + \| \chi  R(\la\pm i \tau) \chi \|_{{\cal H} \to {\cal H}}).
\end{equation}
\end{prop}


In the following section we need some results about the SSF for the pair $(\tL_2,\tL_1)$.
The operators $\tL_1$, $\tL_2$ are differential operators satisfying the assumptions (\ref{eq:2.5}), (\ref{eq:2.6}) and (\ref{eq:2.7}). Then for any $f \in {\cal S}(\RR)$, the operator $f(\tL_2) - f(\tL_1)$ is trace class (see \cite{DS}, \cite{R1},  \cite{R3}). Following the theory of Birman-Krein (\cite{WoBa}, \cite{Ya}), the spectral shift function $\tssf \in {\cal S}'(\RR)$ related to $\tL_1$, $\tL_2$, is well defined by the equality
\begin{equation}\label{eq:2.23}
 \langle \tssf , f' \rangle_{{\cal S}',{\cal S}} = -\tr(f(\tL_2) - f(\tL_1)).
\end{equation}
Moreover, $\tssf'(\lambda)$  is a $C^\infty$ function on $J$ and $\tssf'(\lambda)$ coincides with the derivative of the scattering phase and with the average time delay for the pair $(\tL_1,\tL_2)$ (see \cite{R3}). Next, $J$ being non-trapping and non-critical interval for $\tL_1$ and $\tL_2$, Theorem 1.3 of \cite{R3} yields the following.

\begin{prop}\label{prop4}
There exist $h_0>0$ and $C^\infty$ coefficients $c_k(\lambda)$, $k\geq 0$, such that 
\begin{equation}
\tssf'(\lambda) = h^{-n} \sum_{k=0}^{N} c_k(\lambda) h^k + {\cal O}(h^{-n+N+1}), \forall N \in \N
\end{equation}
uniformly with respect to $\lambda \in J$, $h \in ]0,h_0]$.
\end{prop}


\section{Representation formula for the derivative of the spectral shift function}

In this section we will prove Theorem 1 admitting Propositions 5 and 7 below. 
In the next three sections $z_0$ denotes a fixed complex number  such that  $\Im z_0 > 0.$ We start by the following.

\begin{prop}\label{prop5}

Let $L$ satisfy $(2.1) - (2.6)$ and $(2.9).$ Let  $\chi_1 \in C_0^\infty(B(0,2\rho_0))$ be equal to $1$ near $\overline{B(0,\rho_0)}$  and let $\varphi_s \in C^\infty(\RR^n)$ be equal to a constant in a neighborhood of  $\overline{B(0,R_0)}$ so  that $\varphi_s(x) = \langle x \rangle^s$, $s \in \RR$, for $|x| \geq R > R_0$. Then for $m>n^\#/2$ and $f \in {\cal S}(\RR)$,   the operators   
$$\chi_1 (L_.-z_0)^{-m}\varphi_s, \; \; \chi_1 f(L_.) \varphi_s, \; \; \varphi_s \Big( (1-\chi_1) (L_.-z_0)^{-m} (1-\chi_1)-
 (1-\chi_1) (\tL_.-z_0)^{-m} (1-\chi_1) \Big) \varphi_s$$
 can be extended to trace class ones with  trace norm  bounded by ${\cal O } (h^{- n_.^{\#}})$. Moreover, the following assertions hold:

i) for $\chi_2 \in C_0^\infty(\RR^n)$  equal to $1$ on a sufficiently large region, we have
\begin{equation}\label{eq:3.4}
\| \chi_1 (L_.-z_0)^{-m}(1-\chi_2) \varphi_s \|_\tr
+ \| \chi_1 f(L_.)(1-\chi_2) \varphi_s \|_\tr  =
 {\cal O } (e^{-C/h}).
\end{equation}

ii) there exists $\rho_1> \rho_0$ such that for $\chi_2$ equal to $1$ near $\overline{B(0,\rho_1)}$, we have
\begin{equation}\label{eq:3.5}
\|  \varphi_s \Big( (1-\chi_2) (L_.-z_0)^{-m} (1-\chi_2)-
 (1-\chi_2) (\tL.-z_0)^{-m} (1-\chi_2) \Big) \varphi_s \|_{tr} =
 {\cal O } (e^{-C/h}).
\end{equation}
\end{prop}

This proposition will be proved in Section 4 exploiting the functional calculus developed by Sj\"ostrand \cite{Sj1, Sj2}.


\noindent

\begin{prop}\label{prop6}
Let $L$ and $\chi_1$ be as in Proposition 5  and let $\chi_2 \in C_0^\infty(\RR^n)$, $\chi_2 \succ \chi_1$ be  equal to $1$ on  $\{x \in \RR^n: \; |x|\leq \rho_2 \},\: \rho_2>\rho_1$ sufficiently large. Then we have
$$\lim_{\rho \rightarrow + \infty} 
 \left[\tr \Big( \chi_\rho  (L_j-z_0)^{-m} (1-\chi_1) \tE_j'(\la) (1-\chi^2_2)\chi_\rho \Big)\right]_{j=1}^2 $$
$$ = (\lambda -z_0)^{-m} \tssf'(\lambda) 
-(\lambda -z_0)^{-m} \left[ \tr \Big(\chi_2 \tE_j'(\la) \chi_2\Big)\right]_{j=1}^2 + {\cal O}(e^{-C/h})$$
uniformly with respect to $(\la,h) \in J \times ]0,h_0]$.
\end{prop}
\begin{pf}
We write 
$$\chi_\rho  (L_j-z_0)^{-m} (1-\chi_1) \tE_j'(\la) (1-\chi^2_2)\chi_\rho  = 
  \chi_\rho  (\tL_j-z_0)^{-m}  \tE_j'(\la)  (1-\chi^2_2)\chi_\rho
+\chi_\rho {\cal T}_j \tE_j'(\la)  (1-\chi^2_2)\chi_\rho $$
with 
${\cal T}_j := \Big(  (L_j-z_0)^{-m} (1-\chi_1) -  (\tL_j-z_0)^{-m} (1-\chi_1)\Big)
-(\tL_j-z_0)^{-m} \chi_1$ and, according to (3.1) and (3.2), we get
\begin{equation}\label{eq:3.5b}
\|  \varphi_s  (1-\chi^2_2){\cal T}_j  \varphi_s \|_{tr} =
 {\cal O } (e^{-C/h})
\end{equation}
for any $s \in \RR$. Then,  combining the formula (\ref{eq:2.17}) for $\tE_j'$
 and the resolvent estimate (\ref{eq:trap.es}) for $s>\frac12$, we get
$$\| \varphi_s^{-1}  \tE_j'(\la)  \varphi_s^{-1}\| = {\cal O}(h^{-1}).$$
Thus, from the cyclicity of the trace and the estimate (\ref{eq:3.5b}) we deduce

$$\tr \Big( \chi_\rho  (L_j-z_0)^{-m} (1-\chi_1) \tE_j'(\la) (1-\chi^2_2)\chi_\rho \Big) $$
$$ = \tr \Big( \chi_\rho  (\tL_j-z_0)^{-m} \tE_j'(\la) \chi_\rho \Big)
- \tr \Big( \chi_\rho \chi_2 (\tL_j-z_0)^{-m} \tE_j'(\la) \chi_2 \chi_\rho \Big)
+ {\cal O } (e^{-C/h}).
$$

To conclude, we observe that the relations (\ref{eq:2.18}) and  (\ref{eq:3.3}) hold for $\tE_j'$ and  $\tssf'$.
\end{pf}

\noindent

\begin{prop}\label{prop7}
Let  $L$ satisfy $(2.1)-(2.6)$ and $(2.8) - (2.9).$ Let $\chi_1 \prec \chi_2$  be as in Proposition \ref{prop6}. For $\pm \Im z > 0$ we have
$$\lim_{\rho \rightarrow + \infty} 
 (\lambda - z_0)^m\tr \Big( \chi_\rho  (L_.-z_0)^{-m} R_. (z) [\tL_.,\chi_1]\tR_.(z)(1-\chi^2_2) \chi_\rho \Big)=  \tr \Big( T_.^\pm (z) \Big),\: \Re z = \lambda,$$
where  $T_.^\pm(z)$ are trace class operators. Then for $\chi \in C^{\infty}_0({\Bbb R}^n), \:\: \chi_2 \prec \chi$ the following estimates hold
$$\| T_.^\pm (z) \|_\tr = {\cal O}(h^\infty) \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big), \quad \pm \Im z > 0$$
uniformly with respect to $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$ and, moreover, ${\cal B}_{\pm} \ni z \mapsto \tr\Bigl(T_.^\pm(z)\Bigr)$ are holomorphic in ${\cal B}_{\pm}.$
\end{prop}
Proposition \ref{prop7} follows from Proposition \ref{prop8} taking $P=[\tL_.,\chi_1]$. Proposition \ref{prop8} will be proved in Section 5 by localizations in the outgoing and incoming regions of the phase space combined with the resolvent estimates of Section 2.

\noindent

{\it Proof of Theorem \ref{thm:repssf}.}
Applying (\ref{eq:2.18}) and (\ref{eq:2.17}) for 
$f(\lambda) =(\lambda-z_0)^{-m}$, $z_0 \in \C\setminus\RR$ fixed, we have
$$ E_.'(\la)=\frac{(\lambda-z_0)^{m}}{2 i \pi}(L_.-z_0)^{-m}\Big( R(\lambda +i 0) - R(\lambda -i 0) \Big).$$
Then  combining Proposition \ref{prop5}
and Proposition \ref{prop2}, we deduce that for any $\chi \in C_0^\infty(\RR^n)$  equal to 1 near $\overline{B(0,R_0)}$, the operator
 $ \chi  E_.'(\la) \chi$ is trace class one for $\lambda \in J$  with trace norm  uniformly bounded with respect to $\la$. The same argument works for $\tE'$.

The proof of the semi-classical representation formula (\ref{eq:1.1}) follows tree steps.\\
\underline{Step 1}. We will show that  $\ssfr'$ coincides on $J$ with the locally integrable function ${\cal E}_\rho(\lambda) := 
{\cal E}_{\rho,2}(\lambda)-{\cal E}_{\rho,1}(\lambda)$, where 
${\cal E}_{\rho,j}(\lambda) = \tr\Big( \chi_\rho  E_j'(\la) \chi_\rho \Big),\: j = 1,2.$ Let $f \in C^{\infty}_0(\RR)$, supp$f \subset J$. According to (\ref{eq:2.18}), we have
$$\int_\RR {\cal E}_{\rho,j}(\lambda) f(\lambda) d \lambda = 
\int_J \tr\Big( \chi_\rho f(L_j)  E_j'(\la) \chi_\rho \Big)d \lambda.$$
However, for any $s>\frac12$,  $\chi_\rho f(L_j) \varphi_s$ is a trace class operator (see Proposition \ref{prop5}) and ${\varphi_s}^{-1} E_j'(\la) \chi_\rho \in {\cal L}({\cal H})$. Then $ \chi_\rho f(L_j)  E_j'(\la) \chi_\rho$ is the derivative of $ \chi_\rho f(L_j)  E_j(\la) \chi_\rho$ in the space of the trace class operators, and 
$$\int_\RR {\cal E}_{\rho,j}(\lambda) f(\lambda) d \lambda =
\tr\Big( \chi_\rho f(L_j){\bf  1}_J(L_j) \chi_\rho \Big)=
\tr\Big( \chi_\rho f(L_j)   \chi_\rho \Big).$$

Consequently,  $\ssfr'$ becomes a locally integrable function and
\begin{equation}\label{eq:3.7}
\ssfr'(\lambda)=  \left[\tr\Big( \chi_\rho  E_j'(\la) \chi_\rho \Big) \right]_{j=1}^2.
\end{equation}\\
\underline{Step 2}.
Choose $\chi_1 \prec \chi_2$ as in Proposition \ref{prop6}. Then 
 $\tL (1-\chi_2) = L (1-\chi_2)$ and  we have
\begin{equation}\label{eq:3.8}
 R_.(z) (1- \chi_2^2)=  (1-\chi_1)  \tR_. (z)  (1-\chi_2^2) +  R_. (z) [\tL_.,\chi_1]\tR_.(z)(1-\chi_2^2)
\end{equation}
in ${\cal L}({\cal H}_.^{0,s}, {\cal H}_.^{0,-s})$, $ s>1/2$, for $z=\la\pm i 0$ and in ${\cal L}({\cal H}_.)$ for $\Im z \neq 0$.
 Applying formula (\ref{eq:2.17}) once more, we get
$$E_.'(\la)  = E_.'(\la) \chi_2^2 + 
(\lambda -z_0)^m (L_.-z_0)^{-m} (1-\chi_1) \tE_.'(\la) (1-\chi^2_2) $$
\begin{equation}\label{eq:3.9}
 + \frac{(\lambda -z_0)^m}{2 i \pi}  \left(G_.(\lambda + i0)- G_.(\lambda - i0) \right),
\end{equation}
where 
$$ G_.(z)=  (L_.-z_0)^{-m} R_. (z) [\tL_.,\chi_1]\tR_.(z)(1-\chi^2_2).$$
\\
\underline{Step 3}. Combining the relations (\ref{eq:3.3}), (\ref{eq:3.7}), (\ref{eq:3.9}) with Propositions \ref{prop6}, \ref{prop7}, we obtain
$$
\begin{array}{ccl}
\ssf'(\la) = \tssf'(\la) +\left[\tr\Big( \chi_2 E_j'(\la) \chi_2\Big) \right]_{j=1}^2
 - \left[\tr\Big( \chi_2\tE_j'(\la) \chi_2\Big) \right]_{j=1}^2 \\
 + \tr T^+(\la+i0) -\tr T^-(\la-i0) + (\lambda - z_0)^m {\cal O}(e^{-C/h}).
\end{array}
$$
Finally,  $J$ being non-trapping for $\tL_1$ and $\tL_2$, the functions
$\tssf'(\la)$ and   $\tr\Big( \chi_2\tE_.'(\la) \chi_2\Big)$ have  complete asymptotics described by Proposition \ref{prop4} and we deduce  Theorem 1.
\hfill{$\Box$}\\

Corollary 1 follows directly from (\ref{eq:1.2}), the estimate for $\|T^{\pm}(z)\|_{\tr}$ and the representation (\ref{eq:1.1}).\\

{\it Proof of Corollary 2.} According to Theorem 1 and (\ref{eq:1.2}), for $|t| \geq \delta > 0$  we have
\[ \tr_{{\rm bb}} \Bigl(e^{-ith^{-1}L_2} f(L_2) - e^{-ith^{-1}L_1} f(L_1)\Bigr) = \int_J e^{-ith^{-1}\lambda} f(\lambda) \xi'(\lambda) d\lambda \, \]
\[ = \Bigl [ \tr \Bigl(\chi e^{-ith^{-1}L_j} f(L_j)\chi \Bigr) \Bigr]_{j=1}^2 + {\cal F}_h \Big(f(\bullet)  \tr \Bigl(T^+(\bullet+i0)\Bigr)\Big)-{\cal F}_h\Big(f(\bullet) \tr \Bigl(T^-(\bullet - i0)\Bigr)\Big)+ {\cal O}( h^{\infty}) \,, \] where ${\cal F}_h$ denotes the Fourier transform
\[ \Bigl( {\cal F}_h \psi\Bigr)(t) = \int e^{-ith^{-1}\lambda}\psi(\lambda) d\lambda \,. \]
Clearly, the estimate of $\|T^{\pm}(z)\|_{\tr}$ obtained in Theorem 1 yields
\[ \Bigl |{\cal F}_h \Bigl(f(\bullet) \tr \Bigl(T^+(\bullet+i0)\Bigr)\Bigr)\Bigr | \leq {\cal O}(h^{\infty}) \Bigl( \int_J [\|\tilde{\chi} R_1(\lambda) \tilde{\chi} \|_{{\cal H} \to {\cal H}} +  \|\tilde{\chi} R_2(\lambda) \tilde{\chi} \|_{{\cal H} \to {\cal H}}] d\lambda \Bigr) \,. \] 
Applying Proposition 3 in \cite{Ste}, we conclude that the assumption (\ref{eq:1.3}) implies
\[ \int_J \|\chi R_j(\lambda) \chi \| d\lambda \leq C h^{-M}, \:\: j = 1, 2 \, \]
with some $M > 0$ depending on $n^{\sharp}$ and the constants in (\ref{eq:1.3}). Thus the Fourier transform of $f(\lambda) \tr T^{\pm}(\lambda \pm i0)$ give ${\cal O}(h^{\infty})$ terms and the proof is complete.
\hfill{$\Box$}
 


\section{Trace class estimates}

 This section is devoted to the proof of Proposition \ref{prop5}. The main idea is to generalize the trace class estimates of \cite{Sj1} for operators acting in  weight spaces. Let $\Gamma \subset \RR^n$ be a sufficiently widely spaced lattice and let $0 \leq \psi_\nu \in C_0^\infty(\RR^n)$, $\nu \in \Gamma$, be a translation invariant partition of unity $\{\psi_\nu(x)\}_{\nu \in \Gamma},\:\psi_{\nu}(x) = \psi_0(x-\nu)$ with $\psi_0=1$ near  $\overline{B(0,R_0)}$. Denote by  $\mu_j(K)$ the j-th characteristic value of the compact operator $K$ and fix $z_0 \in \C\setminus\RR$. According to Lemma 3.1 in  \cite{Sj1},  we obtain easily the following results.

\begin{lem}\label{lem3}
Let $L$ satisfy $(\ref{eq:2.1})-(\ref{eq:2.6})$ and $(\ref{eq:2.9})$  and let $\tL$ be defined as in  Proposition \ref{prop1}.

i) For $\nu$, $\mu \in \Gamma$ there exists $C>0$ such that
$$\mu_j \Big( \psi_\mu (L-z_0)^{-1} \psi_\nu \Big)   \leq
\frac{C e^{-\frac{1}{Ch}(|\nu - \mu| -C)_+}}{1 + h^2 j^{\frac{2}{n^\#}}}.$$

ii) For $\nu$, $\mu \in  \Gamma\setminus \{0 \}$ there exists $C>0$ such that
$$\mu_j \Big( \psi_\mu (L-z_0)^{-1} \psi_\nu-
                \psi_\mu (\tL-z_0)^{-1} \psi_\nu \Big)      \leq
\frac{C e^{-\frac{1}{Ch}(|\nu |+ |\mu| -C)_+}}{1 + h^2 j^{\frac{2}{n^\#}}}.$$

\end{lem}


\begin{lem}\label{lem4}
Let $m > n^\#/2 $ and let $\varphi_s \in C^\infty(\RR^n)$ be a  constant
in a neighborhood of $\overline{B(0,R_0)}$ so that $\varphi_s(x) = \langle x \rangle^s$, $s \in \RR$, for $|x| \geq R > R_0$.

i) For $\nu$, $\mu \in \Gamma$ the operator
 $\psi_\mu (L-z_0)^{-m} \psi_\nu \varphi_s$ is trace class and  there exists $C>0$ such that
$$\parallel  \psi_\mu (L-z_0)^{-m} \psi_\nu \varphi_s \parallel_\tr     \leq
C h^{-n^\#} \langle \nu \rangle^s  e^{-\frac{1}{Ch}(|\nu - \mu| -C)_+}.$$

ii) For $\nu$, $\mu \in \Gamma\setminus \{0 \}$ the operator
$\varphi_s \Big( \psi_\mu (L-z_0)^{-m} \psi_\nu-
          \psi_\mu (\tL-z_0)^{-m} \psi_\nu \Big)  \varphi_s$ 
 is trace class and there exists $C>0$ such that
$$\parallel \varphi_s \Big( \psi_\mu (L-z_0)^{-m} \psi_\nu-
          \psi_\mu (\tL-z_0)^{-m} \psi_\nu \Big)  \varphi_s  \parallel_\tr   \leq
C h^{-n^\#} \langle \mu \rangle^s  \langle \nu \rangle^s 
e^{-\frac{1}{Ch}(|\nu| + |\mu| -C)_+}.$$ 

\end{lem}

{\it Proof of Proposition \ref{prop5}.}
For $\chi_1 \in C_0^\infty(\RR^n)$, there exists $N_0$ such that 
$\chi_1 = \sum_{|\mu| \leq N_0}\chi_1 \psi_\mu$. Applying Lemma \ref{lem4}, and writing 
$$\chi_1 (L-z_0)^{-m} \psi_\nu \varphi_s =
 \sum_{|\mu| \leq N_0}\chi_1 \psi_\mu (L-z_0)^{-m} \psi_\nu \varphi_s,$$
we conclude that the operator $\chi_1 (L-z_0)^{-m}  \varphi_s \psi_\nu$ is trace class with 
$$\parallel \chi_1 (L-z_0)^{-m} \psi_\nu \varphi_s\parallel_\tr \leq
C h^{- n^\#} \langle \nu \rangle^s e^{-\frac1{Ch}(|\nu | -C)_+}.$$
On the other hand, the series $\sum_{\nu} \langle \nu \rangle^s e^{-\frac1{Ch}(|\nu | -C)_+}$ converges and for $N$ sufficiently large we have 
\begin{equation}\label{eq:4.1}
\sum_{|\nu| \geq N} \langle \nu \rangle^s e^{-\frac1{Ch}(|\nu | -C)_+}
= {\cal O}(e^{-\frac1{Ch}}).
\end{equation}
Consequently, $\chi_1 (L_.-z_0)^{-m}\varphi_s$  can be extended to a trace class operator  with  trace norm bounded by ${\cal O } (h^{- n_.^{\#}})$.
Moreover,
for $\chi_2 \in C_0^\infty(\RR^n)$ equal to $1$ on a sufficiently large region, there exists $N$ such that
$$\chi_1 (L-z_0)^{-m}\varphi_s (1-\chi_2) = 
\sum_{|\nu| \geq N}\chi_1 (L-z_0)^{-m} \psi_\nu \varphi_s (1-\chi_2) .$$
Thus (\ref{eq:3.4}) follows from the estimate (\ref{eq:4.1}). In the same way, for $\Im z \neq 0$, $\chi_1 (L_.-z_0)^{-m}(L_.-z)^{-1}\varphi_s$   can be extended to a trace class operator  with  trace norm bounded by ${\cal O } (h^{- n_.^{\#}}/ | \Im z|)$ and, as in \cite{Sj1}, using the operator version of the Cauchy-Green-Riemann-Stokes formula, we obtain that  the operator  
$ \chi_1 f(L_.) \varphi_s$
 can be extended to trace class one with  trace norm  bounded by ${\cal O } (h^{- n_.^{\#}})$.
 Moreover 
if $\chi_2 \in C_0^\infty(\RR^n)$ is equal to $1$ on a sufficiently large region,
 for $\Im z \neq 0$, the trace norm of $\chi_1 (L_.-z_0)^{-m}(L_.-z)^{-1}\varphi_s (1-\chi_2)$ is  bounded by ${\cal O } (e^{-\frac1{Ch}}/ | \Im z|)$ and, using once more the operator version of the Cauchy-Green-Riemann-Stokes formula, we obtain that trace norm  of  the operator  
$ \chi_1 f(L_.) \varphi_s (1-\chi_2)$
is  bounded by ${\cal O } (e^{-\frac1{Ch}})$.

Now, for $\chi_1$ equal to $1$ near $\overline{B(0,R_0)}$ and such that $\chi_1 \psi_0=\psi_0$ we write
$$(1-\chi_1) (L-z_0)^{-m} (1-\chi_1) =
 \sum_{\nu, \mu \in \Gamma \setminus \{0\}}
       (1-\chi_1) \psi_\mu (L-z_0)^{-m} \psi_\nu(1-\chi_1).$$
Taking into account Lemma \ref{lem4}, the convergence of the series 
$ \sum_{\nu, \mu}\langle \mu \rangle^s  \langle \nu \rangle^s 
e^{-\frac{1}{Ch}(|\nu| + |\mu| -C)_+}$ implies that for $N$  sufficiently large we have 
\begin{equation}\label{eq:4.2}
\sum_{|\mu|,|\nu| \geq N} \langle \mu \rangle^s  \langle \nu \rangle^s 
e^{-\frac{1}{Ch}(|\nu| + |\mu| -C)_+}
= {\cal O}(e^{-\frac1{Ch}}).
\end{equation}
Thus we deduce that the operator
$ \varphi_s \Big( (1-\chi_1) (L-z_0)^{-m} (1-\chi_1)-
 (1-\chi_1) (\tL-z_0)^{-m} (1-\chi_1) \Big) \varphi_s$
 can be extended to a trace class one with  trace norm bounded by ${\cal O } (h^{- n_.^{\#}})$. Moreover,
for $\chi_1$ equal to 1 on a sufficiently large region, there exists $N$ such that
$$ (1-\chi_1) (L-z_0)^{-m} (1-\chi_1) = 
\sum_{|\nu|, |\mu| \geq N}(1-\chi_1) \psi_\mu (L-z_0)^{-m} \psi_\nu(1-\chi_1).$$
Finally, (\ref{eq:3.5}) follows from the estimate (\ref{eq:4.2}) and this completes the proof of Proposition \ref{prop5}.
\hfill{$\Box$}



\section{Limits of traces}

The purpose of this section is to prove Proposition \ref{prop8} which implies Proposition \ref{prop7}.

\begin{prop}\label{prop8}
Let $P$ be a $p$-order $h$-admissible differential operator 
$P(h) = \sum_{|\alpha| \leq p}a_{\alpha}(x)(hD_x)^{\alpha}$, $p\leq 2$, with coefficients $(a_\alpha(x))_{|\alpha| \leq p}$ compactly supported in
$ \{ x \in \RR^{n}:\:R_0 < |x| < 2 \rho_0 \}$. For any $\psi \in C^\infty_b(\RR^n)$, supported away from the supports of $a_\alpha$, there exist trace class operators $T_.^\pm(z)$, defined for $(\Re z,\pm \Im z) \in J \times ]0,1]$, such that the following assertions hold:
$$\lim_{\rho \rightarrow + \infty} 
 \tr \Big( \chi_\rho  (L_.-z_0)^{-m} R_. (z) P \tR_.(z) \psi \chi_\rho \Big)=  \tr \Big( T_.^\pm (z) \Big),$$

$$\| T_.^\pm (z) \|_\tr = {\cal O}(h^N) \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big), \quad \pm \Im z > 0,\: \forall N \in \N, $$
uniformly with respect to $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$, where $ \chi \in C_0^\infty(\RR^n)$ is  equal to $1$ for $|x| \leq 2 \rho_0.$ Moreover, $z \mapsto \tr\Bigl(T_.^\pm(z)\Bigr)$ is holomorphic for $(\Re z,\pm \Im z) \in J \times ]0,1]$.
\end{prop}

In this section, given $s \in \RR$, we denote by  $\varphi_s$ the multiplication operator  by  a $C^\infty(\RR^n)$ function which is  constant  near $\overline{B(0,R_0)}$ and equal to $ \langle x \rangle^s$ for $|x| \geq R > R_0$. Notice that for $s' \in \RR$, the operator $\varphi_s$  is invertible in ${\cal L}({\cal H}^{0,s+s'}, {\cal H}^{0,s'})$. Given two bounded operators $A,\: B$, introduce the operators
\begin{equation}\label{defGrho}
G_\rho(z;B)= \chi_\rho  (L_.-z_0)^{-m} R_. (z) P \tR_.(z) B \chi_\rho,
\end{equation}
\begin{equation}\label{defG}
G(z;A,B)= A  (L_.-z_0)^{-m} R_. (z) P \tR_.(z) B
\end{equation}
which are clearly holomorphic for $\pm \Im z >0$. The proof of Proposition \ref{prop8} is based on a partition of the phase space into outgoing and incoming regions and  on the following lemma.


\begin{lem}\label{lem5}

i) Let $B \in {\cal L}({\cal H})$ be such that $P\tR(z) B \in {\cal L}({\cal H}^{0,-s}, {\cal H})$ with $s > \frac12$ for any $(\Re z,\pm \Im z) \in J \times ]0,1]$. Then  $G_\rho(z;B)$ is trace class and we have
$$\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;B) =  \tr  G(z; {\varphi_s}^{-1}, B\varphi_s) ,$$
$$\|  G(z; {\varphi_s}^{-1}, B\varphi_s) \|_\tr \leq C h^{-2-n_.^{\#}} 
\| P\tR(z) B   \|_{-s,0} \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big)$$
uniformly with respect to $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. Moreover, the trace $\tr  G(z; {\varphi_s}^{-1}, B\varphi_s)$ is holomorphic for  $(\Re z,\pm \Im z) \in J \times ]0,1]$.

ii) Let $s > \frac12$, $\delta \geq 0$. Assume that for $(\Re z,\pm \Im z) \in J \times ]0,1]$ we have  $B \in {\cal L}({\cal H}^{0,s})$ and
$ BR(z) \in {\cal L}({\cal H}^{0,s+\delta}, {\cal H}^{0,s})$ with
 $ B\chi_\rho^2 R(z)$ uniformly bounded with respect to $\rho$ in 
${\cal L}({\cal H}^{0,s+\delta}, {\cal H}^{0,s})$. Then  $G_\rho(z;B)$ is trace class and we have
$$\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;B) =  \tr  G(z;\varphi_s B, {\varphi_s}^{-1}) ,$$
$$\| G(z;\varphi_s B, {\varphi_s}^{-1}) \|_\tr \leq C h^{-1-n_.^{\#}} 
\|  B R(z)\chi  \|_{0,s} +
 {\cal O } (e^{-\frac{C}{h}})\|  B R(z)  \|_{s+\delta,s}$$
uniformly with respect to $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$, for any $\chi \in C_0^\infty(\RR^n)$ equal to $1$ on $\{x:\: |x| \leq 2\rho_0\}$ and the trace $\tr  G(z;\varphi_s B, {\varphi_s}^{-1})$ is holomorphic for  $(\Re z,\pm \Im z) \in J \times ]0,1].$
\end{lem}
\begin{pf}
$i)$ The coefficients of $P$ being compactly supported, there exists $\chi_1 \in C_0^\infty(\RR^n)$ such that $P=\chi_1 P$ and we can write $G_\rho (z;B)$ in the following form
$$G_\rho(z;B)= \chi_\rho  R_. (z)\varphi_s^{-1}\varphi_s (L_.-z_0)^{-m}\chi_1 P \tR_.(z) B \chi_\rho.$$
Then $G_\rho(z;B)$ is trace class because $\chi_\rho  R_. (z)\varphi_s^{-1}$ is bounded according to Proposition \ref{prop2}, while Proposition \ref{prop5} shows that the operator  $\varphi_s (L_.-z_0)^{-m}\chi_1$ is trace class one. Clearly,  $P \tR_.(z) B \chi_\rho \in {\cal L}({\cal H}^{0,-s}, {\cal H}) \subset {\cal L}( {\cal H})$.
Moreover, by the cyclicity of the trace, we have
\begin{equation}
\tr G_\rho(z;B) =\tr \Big( \chi_\rho \varphi_s^{-1} R_. (z) (L_.-z_0)^{-m} P \tR_.(z) B \varphi_s \chi_\rho \Big) = \tr \Big( \chi_\rho G(z; \varphi_s^{-1},  B \varphi_s) \chi_\rho \Big).
\end{equation}
On the other hand, for $s>\frac12$, $ G(z; \varphi_s^{-1},  B \varphi_s)$ is obviously trace class with trace norm satisfying
$$\| G(z; \varphi_s^{-1},  B \varphi_s) \|_{\tr} \leq \|  R_. (z) \|_{s,-s} \;
\| \varphi_s(L_.-z_0)^{-m}\chi_1 \|_{\tr} \; \| P \tR_. (z) B \|_{-s,0}.$$
Consequently, $\tr G_\rho(z;B)$ converges to $\tr G(z; \varphi_s^{-1},  B \varphi_s)$ and applying Propositions \ref{prop5} and \ref{prop2}, we obtain the claimed estimate. Finally, for fixed $h$ and $\pm \Im z > 0$ the resolvent $R(z),\: \tR(z)$ are analytic and we deduce easily the analyticity of the trace $\tr G(z; \varphi_s^{-1})$ for  $(\Re z,\pm \Im z) \in J \times ]0,1]$. 
 
ii) We observe that $G_\rho(z,B)$ is trace class by writing
$$G_\rho(z;B)=\Big( \chi_\rho  R_. (z)\varphi_{s+\delta}^{-1}\Big)
              \Big(\varphi_{s+\delta} (L_.-z_0)^{-m}\chi_1\Big)
              \Big( P \tR_.(z)\varphi_s^{-1}\Big)
              \Big(\varphi_s B\chi_\rho\Big).$$
The  operator  $\varphi_{s+\delta} (L_.-z_0)^{-m}\chi_1$ is trace class according to  Proposition \ref{prop5}, while the other operators are bounded as it is easy to see applying  Proposition \ref{prop2}, Lemma \ref{lem1} and our assumption.
The trace being cyclic, we deduce
$$\tr G_\rho(z;B) = \tr \Big(  \varphi_s B  \chi_\rho^2 R_. (z) (L_.-z_0)^{-m} P \tR_.(z) \varphi_s^{-1}  \Big).$$
Next, our assumption implies that the operator in the R.H.S. of the above relation is uniformly bounded with respect to $\rho$ in  ${\cal L}( {\cal H})$ and it converges weakly to $G(z;  \varphi_s B , \varphi_s^{-1})$ as $\rho \longrightarrow +\infty$. Then $\tr G_\rho(z;B)$ converges to $\tr  G(z;  \varphi_s B , \varphi_s^{-1})$ (see Chapter 6, \cite{GoKr}). Finally, for any $\chi \in C_0^\infty(\RR^n)$ we get
\begin{equation}
G(z;\varphi_s B , \varphi_s^{-1})  = 
\varphi_s B  R_. (z)\chi (L_.-z_0)^{-m}P \tR_.(z)  \varphi_s^{-1}+\varphi_s B  R_. (z)(1-\chi) (L_.-z_0)^{-m}P \tR_.(z)  \varphi_s^{-1}.
\end{equation}
Then, taking $\chi_1 \in C_0^\infty(\RR^n)$ such that $\chi_1 P =P$, and $\chi$ such that $\chi_1 \prec \chi$, $\chi$ equal to $1$  on a sufficiently large region, from Lemma \ref{lem1} and Proposition \ref{prop5} we deduce
$$\| G(z;\varphi_s B , \varphi_s^{-1}) \|_{\tr} \leq
 C \|B  R_. (z)\chi \|_{0,s} h^{-n_.^\#-1} +
 {\cal O}(e^{-C/h}) \|B  R_. (z) \|_{s+\delta,s}.$$
As above we can establish the analyticity of the trace  $\tr  G(z;\varphi_s B, {\varphi_s}^{-1})$ and this  completes the proof.
\end{pf}

{\it Proof of Proposition \ref{prop8}.}
Let $f \in C_0^\infty(\RR^+)$ be equal to $1$ on $\overline{J}$. We have
\begin{equation}\label{eq:4.7}
 \chi_\rho  (L_.-z_0)^{-m} R_. (z) P \tR_.(z) \psi \chi_\rho =
G_\rho(z;f(\tL) \psi) +G_\rho(z;(1-f(\tL)) \psi). 
\end{equation}
Exploiting the ellipticity of $\tL$ and the functional calculus (\cite{DS}, \cite{R1}), we observe that for $\Re z \in J$ the operator $P \tR_.(z)(1-f(\tL))$ becomes a $h$-admissible pseudo-differential operator with compactly supported symbol. Since $a_\alpha \psi = 0$ for any $\alpha$, it follows easily that for any $s \geq 0$ the operator $P \tR_.(z)(1-f(\tL))\psi$ can be extended to a bounded one in ${\cal L}({\cal H}^{0,-s}, {\cal H}),\:s > 1/2$, with norm estimated  by ${\cal O}(h^\infty)$, uniformly with respect to  $(\Re z, \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. Consequently, applying  Lemma \ref{lem5}, $i) $, we deduce
\begin{equation}\label{eq:4.8}
\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;(1-f(\tL)) \psi) = 
 \tr  G(z; {\varphi_s}^{-1}, (1-f(\tL)) \psi\varphi_s),
\end{equation}
$$\|  G(z; {\varphi_s}^{-1}, (1-f(\tL)) \psi\varphi_s) \|_\tr = 
 {\cal O}(h^\infty) \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big)$$
uniformly with respect to  $(\Re z, \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. Next, applying Lemma 5, i), once more we deduce the analyticity of the trace $\tr  G(z; {\varphi_s}^{-1}, (1-f(\tL)) \psi\varphi_s)$ for  $(\Re z,\pm \Im z) \in J \times ]0,1]$.

The analysis of the term $G_\rho(z;f(\tL) \psi)$ is more complicated. First, it is convenient to make a localization in the outgoing and  incoming regions of the phase space
\begin{equation}
\Gamma^\pm (R,d,\sigma_\pm) = \{ (x,\xi) \in \RR^{2n}:\: |x| > R, \; d^{-1} < |\xi | < d, \; \frac{\langle x.\xi \rangle}{|x||\xi|} \;  ^>_< \; \sigma_\pm  \}.
\end{equation}
Let  $R > \rho_1 >2 \rho_0$ and $-1 < \sigma_+ < \sigma_- < 1$. For any function $\psi \in C^{\infty}_{b}(\RR^n)$ such that $\psi = 0$ on a neighborhood of $\overline{B(0,R)}$, there exist $\omega_{\pm}(x,\xi) \in C^\infty(\RR^{2n})$ supported in $ \{ (x,\xi) \in \RR^{2n}: |x| > R, \; \frac{\langle x.\xi \rangle}{|x||\xi|} \;  ^>_< \; \sigma_\pm  \}$ such that
$$\psi(x) =  \omega_{+}(x,h D_x) + \omega_-(x,hD_x).$$
Here for a symbol $a(x,\xi;h)$ we denote by $a(x,h D_x;h)$ the operator given by the Weyl quantization
$$a(x,h D_x;h)\varphi(x)=\frac{1}{(2 \pi)^n} \int \exp(i\langle x - y, \xi \rangle )  
a\Bigl( \frac{x+y}{2}, h\xi;h \Bigr) \varphi(y)dy d\xi,\:\: \varphi \in {\cal 
S}({\bf R}^n).$$
From the  functional calculus (see Appendix) we know that for any $N>0$,
$$f(\tL) = a_N(x,h D_x;h) + {\tilde r}_N(x,h D_x;h),$$
where the symbol $a_N(x,\xi;h)$ is localized in  $ \{ (x,\xi) \in \RR^{2n}:\: \tl_0(x,\xi) \in $ supp$f \}$,  $\tl_0$ being the principal symbol of $\tL$ and 
 the symbol ${\tilde r}_{N}(x,\xi;h)$ satisfies the following estimates: for any $\beta_1$, $\beta_2$ in $\N^n$ and $L\in \N$ there exists $C_{\beta_1,\beta_2, L,N}$ such that
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} {\tilde r}_{N}(h,x,\xi) |
\leq C_{\beta_1,\beta_2, L, N} h^N  \langle x  \rangle^{-N} \langle \xi  \rangle^{-L},$$
uniformly with respect to $(x,\xi) \in \RR^{2n}$, $h \in ]0,h_0]$.
Moreover, under the assumptions (\ref{eq:2.5}), (\ref{eq:2.6}), we can choose $R > 1$ and $d>1$ so that $ \{ (x,\xi) \in \RR^{2n}:\: |x| \geq R,\: \tl_0(x,\xi) \in $ supp$f \}$ is contained in $ \{ (x,\xi) \in \RR^{2n}: d^{-1} < |\xi | < d \}$. Then applying the theorem for composition of pseudodifferential operators (\cite{DS}, \cite{R1}), we conclude that there exist  sequences
$\omega_{\pm,N;h}(x,\xi) \in C^\infty(\RR^{2n})$ and 
$r_{N}(x,\xi;h) \in C^\infty(\RR^{2n})$ satisfying the following conditions:

\begin{itemize}

\item  supp$\omega_{\pm,N} \; \subset \Gamma^\pm (R,d,\sigma_\pm),$

\item for any $\beta_1$, $\beta_2$ in $\N^n$ and $L\in \N$ there exists $C_{\beta_1,\beta_2, L}$ such that
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} \omega_{\pm,N}(x,\xi;h) |
\leq C_{\beta_1,\beta_2, L} \langle x  \rangle^{-|\beta_1 |} \langle \xi  \rangle^{-L},$$
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} r_{N}(x,\xi;h) |
\leq C_{\beta_1,\beta_2, L, N} h^N  \langle x  \rangle^{-N} \langle \xi  \rangle^{-L},$$
uniformly with respect to $(x,\xi) \in \RR^{2n}$, $h \in ]0,h_0]$,

\item $f(\tL) \psi = \omega_{+,N}(x,h D_x;h) \tpsi  + \omega_{-,N}(x,h D_x;h) \tpsi
+ r_{N}(x,h D_x;h) \tpsi$  for any $\tpsi \succ \psi$.
\end{itemize}
Thus we have
\begin{equation}\label{eq:4.10}
G_\rho(z;f(\tL) \psi)=  G_\rho(z;\omega_{+,N}\tpsi) + 
G_\rho(z;\omega_{-,N}\tpsi) + G_\rho(z;r_{N}\tpsi).
\end{equation}

For $s > \frac12$ and  $N>2s$, by construction, $R_N=r_{N}(x,h D_x;h)$ can be extended to an operator in ${\cal L}({\cal H}^{0,-s}, {\cal H}^{0,s})$ with norm equal to  ${\cal O}(h^N)$.
Next, the coefficients of $P$ are compactly supported and Lemma \ref{lem1} implies immediately that $P \tR(z) R_N \in {\cal L}({\cal H}^{0,-s}, {\cal H})$ with norm equal to  ${\cal O}(h^{N-1})$ uniformly with respect to  $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. From Lemma \ref{lem5}, $i)$ we deduce
\begin{equation}\label{eq:4.11}
\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;r_N \tpsi) = 
 \tr  G(z; {\varphi_s}^{-1}, r_N \tpsi\varphi_s),
\end{equation}
$$\|  G(z; {\varphi_s}^{-1}, r_N \tpsi\varphi_s) \|_\tr = 
 {\cal O}(h^{N-3-n^\#}) \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big).$$
uniformly with respect to  $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$.  

The coefficients $(a_\alpha(x))_{|\alpha| \leq p}$ being  compactly supported in $ \{ x \in \RR^{n}: R_0 < |x| < 2\rho_0\}$, an application of Lemma \ref{lem2}, $iii)$ shows that $P\tR(z) \omega_{\pm,N}(x,h D_x)\tpsi \in {\cal L}({\cal H}^{0,-s}, {\cal H})$ for $\pm \Im z \geq 0$ with norm  ${\cal O}(h^\infty)$ uniformly with respect to  $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. Then using Lemma \ref{lem5},  $i)$, once more, we obtain for $\pm \Im z \geq 0$
\begin{equation}\label{eq:4.12}
\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;\omega_{\pm,N} \tpsi) = 
 \tr  G(z; {\varphi_s}^{-1}, \omega_{\pm,N} \tpsi\varphi_s),
\end{equation}
$$\|  G(z; {\varphi_s}^{-1}, \omega_{\pm,N} \tpsi\varphi_s) \|_\tr = 
 {\cal O}(h^\infty) \; 
 \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big)$$
uniformly with respect to  $(\Re z,\pm \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. As above we conclude that traces 
 $$\tr  G(z; {\varphi_s}^{-1}, \omega_{\pm,N} \tpsi\varphi_s), \: \tr  G(z; {\varphi_s}^{-1}, r_N \tpsi\varphi_s)$$
 are analytic for $\Re z \in J,\: \pm \Im z > 0.$

To deal with the terms  $G_\rho(z;\omega_{\pm,N} \tpsi),\: \pm \Im z \leq 0$, we will apply  Lemma \ref{lem5}, $ii)$. First, $\omega_{\pm,N}$ being a pseudodifferential operator, we conclude that
 $ \omega_{\pm,N} \tpsi \in {\cal L}({\cal H}^{0,s})$ for any $s \in \RR$. Secondly, using Proposition \ref{prop3}  and the relation
$$\omega_{\pm,N}(x, hD_x) \tpsi R_.(z) = 
\Big( R_.({\overline z}) \tpsi {\overline \omega}_{\pm,N}(x, hD_x) \Big)^*,$$
 we conclude that $\omega_{\pm,N} \tpsi R_.(z) \in  {\cal L}({\cal H}^{0,s+\delta}, {\cal H}^s)$ for $\delta > 1$ and $s>\frac12$ with norm equal to  $ {\cal O}(h^{-1}) + {\cal O}(h^\infty) \;  \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}}$. The same estimate holds uniformly with respect to $\rho$ replacing $\omega_{\pm,N}$ by $\omega_{\pm,N} \chi_\rho^2$. In fact, the symbol of the pseudo-differential operator 
$\omega_{\pm,N} \chi_\rho^2$ is uniformly bounded in the same class of symbol as $\omega_{\pm,N}$ and, moreover, it has smaller support. Consequently, for $\pm \Im z \leq 0$ an application of  Lemma \ref{lem5},  $ii)$ implies
\begin{equation}\label{eq:4.13}
\lim_{\rho \rightarrow + \infty} 
 \tr G_\rho(z;\omega_{\pm,N} \tpsi) = 
 \tr  G(z; {\varphi_s} \omega_{\pm,N} \tpsi,\varphi_s^{-1}),
\end{equation}
$$\| G(z; {\varphi_s} \omega_{\pm,N} \tpsi,\varphi_s^{-1})  \|_\tr = 
 {\cal O}(h^{-1-n_.^\#})\;  \| \omega_{\pm,N} \tpsi R_.(z) \chi \|_{0,s} +
 {\cal O}(h^\infty) \; \| \omega_{\pm,N} \tpsi R_.(z) \|_{s+\delta,s}$$
uniformly with respect to  $(\Re z,\mp \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. 
Taking into account $iii)$ and $i)$ of Proposition  \ref{prop3}, we obtain
$$\| G(z; {\varphi_s} \omega_{\pm,N} \tpsi,\varphi_s^{-1}) \|_\tr = 
 {\cal O}(h^\infty) \;  \Big( 1 + \| \chi R_.(z) \chi  \|_{{\cal H} \to {\cal H}} \Big)$$
uniformly with respect to  $(\Re z,\mp \Im z,h) \in J \times ]0,1] \times  ]0,h_0]$. Combining the relations
(\ref{eq:4.7}), (\ref{eq:4.8}), (\ref{eq:4.10}), (\ref{eq:4.11}), (\ref{eq:4.12}) and (\ref{eq:4.13}) and applying Lemma 5 for the analyticity of traces, we obtain the assertion of Proposition 8 with 
\begin{equation}\label{defT}
\begin{array}{ccl}
T^\pm(z) & =& \left[ \tr \Big(\varphi_s^{-1}  R_j(z)(L_j-z_0)^{-m} P \tR_j(z) (1-f(\tL_j)) \psi \varphi_s \Big) \right.  \\
&  & +\tr \Big(\varphi_s^{-1}  R_j(z)(L_j-z_0)^{-m} P \tR_j(z) r_N(x,hD_x;h) {\tilde {\psi}}  \varphi_s \Big)\\
&  & +\tr \Big(\varphi_s^{-1}  R_j(z)(L_j-z_0)^{-m} P \tR_j(z) w_{\pm,N}(x,hD_x;h) {\tilde {\psi}}  \varphi_s \Big)\\
&  & \left. +\tr \Big(\varphi_s w_{\mp,N}(x,hD_x;h){\tilde {\psi}} R_j(z)(L_j-z_0)^{-m} P \tR_j(z)    \varphi_s^{-1} \Big)  \right]_{j=1}^2,\\
\end{array}
\end{equation}
for $\pm 
\Im z\geq 0$.

\hfill{$\Box$}

\section{Weyl asymptotics for the spectral shift function}

In this section we obtain a Weyl type asymptotic for the spectral shift function.
We generalize the results of Christiansen \cite{Ch} and Robert \cite{R3} covering the "black box" long-range perturbations of the Laplacian.

 As in Section 2, introduce  $\chi \in C_0^\infty (\{x: |x| \leq \rho_1\})$  equal to 1 for $|x| \leq 2 \rho_0 < \rho_1$.  The constant $ \rho_0 $ depends on the behaviour of the Hamiltonian trajectories of $L_1$ and $L_2$. In this section the operators $L_1^{\sharp}$ and $L_2^{\sharp}$ are defined on the torus $T_{\tilde{R}} = ({\RR}/\tilde{R}{\Z})^n$ with $ \tilde{R} > 2R >  2 \rho_1$.

\begin{thm}\label{weyl} 
Assume that $L_j, \: j =1,2$ satisfy the assumptions $(2.1) - (2.10)$ and suppose that for
 some $\epsilon > 0, \: C > 0, \: q \geq 1$ the assumption $(\ref{eq:1.3})$ fulfilled.
 Moreover, assume that $\mu_0$ and $\mu$ are  non-critical energy levels for $Q_j,\: j=1, 2$.
Then for $h \in ]0, h_0]$ we have the following asymptotic
\begin{equation}\label{asy}
\xi(\mu)-\xi(\mu_0) = \left[ N(L_j^{\sharp}, [\mu_0, \mu]) \right]_{j=1}^2 
-  \left[  \tr\Bigl( {\bf 1}_{[\mu_0,\mu]}(Q_j^{\sharp}) (1 -  \chi^2)\Bigr) \right]_{j=1}^2  +  w_\chi(\mu)h^{-n} + {\cal O}(h^{-n^\sharp + 1})
\end{equation}
with $w_\chi (\mu) \in C^1(J).$
\end{thm}

{\bf Remarks.}
\begin{itemize}
\item{ The smooth function  $w_\chi(\mu)$  is given by the integral  $\int_{\mu_0}^\mu d_0(\lambda) d\lambda$ of the main term $d_0(\lambda)$ in the asymptotic (\ref{eq:1.2}).}
\item{ Recall that  $Q^{\#}$ is the differential operator
$$Q^{\#} = \sum_{|\nu| \leq 2} a_{\nu}^{\#}(x;h)(hD)^{\nu}$$ 
on the torus $T_{\tilde{R}} = ({\RR}/\tilde{R}{\Z})^n$ with $ \tilde{R} > 2R >  2 \rho_1$ and $a_{\nu}^{\#}(x;h) = a_{\nu}(x;h)$ for $|x| < R$, $R> \rho_1 > 2  \rho_0$. Then if  $\mu_0$ and $\mu$ are non critical  for $Q$, it is clear that without loss of generality we can extend $ a_{\nu}^{\#}$ on $T_{\tilde{R}}$ so that $\mu_0$ and $\mu$ remain  non-critical for $Q^{\#}$. Then using the well known method based on  the construction of a parametrix for small $t$ for the unitary group $e^{ith^{-1}Q^{\#}}$, we deduce
$$\tr\Bigl( {\bf 1}_{[\mu_0,\mu]}(Q_j^{\sharp}) (1 -  \chi^2)\Bigr) = \Omega^\sharp_j(\mu) h^{-n} +  {\cal O}(h^{-n+1})$$
with a function $\Omega^\sharp_j(\mu) \in C^1(J).$}

\item{   If $L_2$ is a compactly supported perturbation of $L_1$, then for $R$ large enough we have $Q_1(1 - \chi^2) = Q_2(1 - \chi^2)$  and we can construct $Q^\sharp_j$ and $\tL_j$  so that $Q_1^{\sharp}(1- \chi^2) = Q_2^{\sharp}(1 - \chi^2)$ and $\tL_2=\tL_1$. Consequently, the asymptotic (\ref{asy}) yields
$$\xi(\mu)-\xi(\mu_0) = \left[ N(L_j^{\sharp}, [\mu_0, \mu]) \right]_{j=1}^2 
 + {\cal O}(h^{-n^\sharp + 1}).$$}
\end{itemize}


\begin{pf}  We will examine the asymptotic of the integral
\begin{equation}\label{eq:6.1}
\int_{\mu_0}^{\mu} \xi'(\lambda) d\lambda = \int_{\mu_0}^{\mu} \Bigl[\tr(\chi E_j'(\lambda) \chi)\Bigr]_{j=1}^2 d\lambda + \int_{\mu_0}^{\mu}\tr\Bigl( T^{+}(\lambda +i0)\Bigr)d\lambda
\end{equation}
$$ - \int_{\mu_0}^{\mu}\tr\Bigl(T^{-}(\lambda -i0)\Bigr)d\lambda + w_\chi(\mu) h^{-n} + {\cal O}(h^{-n  +1}),$$
where $w_\chi(\mu)$ was defined in the first remark above.
First,  for fixed $h \in (0,h_0]$ notice that for any $N \in \N$ there exists $\epsilon = \epsilon(N, h)>0$ such that
$$ \left| \int_{\mu_0}^{\mu} \tr\Bigl(T^{\pm}(\lambda \pm i0)\Bigr)d\lambda - \int_{\mu_0}^{\mu} \tr\Bigl(T^{\pm}(\lambda \pm i\epsilon)\Bigr)d\lambda \right| \leq h^N$$
uniformly with respect to $\mu \in \overline{J}$. This follows from the limit
$$\lim_{\epsilon \to 0, \epsilon > 0} \tr\Bigl(T^{\pm}(\lambda \pm i\epsilon)\Bigr) =
\tr\Bigl(T^{\pm}(\lambda \pm i0)\Bigl)
$$
which for fixed $h \in ]0, h_0]$ is uniform with respect to $\lambda \in \overline{J}$.\\

For simplicity we will treat only the term involving $T^{+}$. Consider the rectangle 
$$\Pi_{\epsilon} = \{z \in {\C}: \: \mu_0 \leq \Re z \leq \mu,\: 0 <\epsilon \leq \Im z \leq 1 \}.$$
 According to Proposition 8, the analyticity of $T^{+}(z)$ for $\Im z > 0$ and Stokes formula, we have
$$0 = \int_{\Pi_{\epsilon}} \partial_{\overline{z}} \Bigl(\tr(T^{+}(z))\Bigr)dz = \int_{\partial \Pi_{\epsilon}} \tr T^{+}(z) dz.$$
The integral over the segment $\{ z = \mu +it:\: \epsilon \leq  t \leq 1 \}$ can be estimated by using the representation (see \cite{R3} for a similar argument)
$$\int_{\epsilon}^1 = \int_{\epsilon}^{C\exp(-Ch^{-p})} + \int_{C\exp(-Ch^{-p})}^1.$$
The first integral is estimated taking into account the bound
$$|\tr(T^{+}(z))| \leq C_N h^N \exp(Ch^{-p}),\: \forall N \in \N,$$
where we have used (\ref{eq:1.3}) and the estimates (\ref{eq:1.4}).
For the second integral we exploit the trivial estimate 
$$\|\chi R(z) \chi\|_{{\cal H} \to {\cal H}} \leq \frac{C_1}{\Im z},\: \Im z > 0.$$
 Thus, using Theorem 1 and Lemma 3 of \cite{BrPe}, we conclude that
$$ \int_{\epsilon}^1 \tr\Bigl(T^{+}(\mu + it)\Bigr) dt = {\cal O}(h^{\infty}), \:\int_{\epsilon}^1 \tr\Bigl(T^{+}(\mu_0 + it)\Bigr) dt = {\cal O}(h^{\infty}),\: h \in ]0, h_0],$$
where ${\cal O}(h^{\infty})$ means that we have an estimate by $C_N h^N,\: \forall N \in \N$, uniformly with respect to $\epsilon > 0.$
The analysis of the integral over the segment $\{z = \lambda +i,\: \mu_0 \leq \lambda \leq \mu \}$ is trivial and we conclude that the contribution of $\tr T^{\pm}(\lambda \pm i0)$ to the integral over $(\mu_0, \mu)$ may be estimated by ${\cal O}(h^{\infty}).$ Similarly, exploiting the analyticity of $\tr(T^{-}(z))$ for $\Im z < 0$ we obtain the same result for the integral involving $\tr(T^{-}(\mu - i0))$. \\

 To deal with the first term in the right hand side of (\ref{eq:6.1}), observe that
$$\int_{\mu_0}^\lambda \tr\Bigl(\chi E'_j(t)\chi\Bigr) dt =
  \tr\Bigl(\chi {\bf 1}_{[\mu_0,\lambda]}(L_j) \chi\Bigr).$$

Let $I_{\mu_0} \ni \mu_0$ and $I_{\lambda} \ni \lambda$ be two small intervals which are non-critical for $L_1$ and $L_2$. We introduce $\varphi_1$, $\varphi_2$, $\varphi_3$ so that  $\varphi_1 \in C_0^\infty(I_{\mu_0}; \RR^+)$,  $\varphi_2 \in C_0^\infty(]\mu_0,\lambda[;\RR^+)$, $\varphi_3 \in C_0^\infty(I_{\lambda}; \RR^+)$ and  $\varphi_1 + \varphi_2 + \varphi_3 =1$ on $[\mu_0,\lambda]$. Then the characteristic function ${\bf 1}_{[\mu_0,\lambda]}$ becomes a sum of the smooth function $\varphi_2 $ and two functions ${\bf 1}_{[\mu_0,\lambda]}\varphi_1$, ${\bf 1}_{[\mu_0,\lambda]}\varphi_3$ localized on non-critical intervals.
Thus 
\begin{equation}\label{decssfX}
 \tr\Bigl(\chi {\bf 1}_{[\mu_0,\lambda]}(L_j) \chi\Bigr)= \sum_{k=1}^3
\ssf_{\varphi_k, \chi}(L_j,\lambda),
\end{equation}
where 
\begin{equation}\label{defssfchi}
\ssf_{\varphi, \chi}(L,\lambda)= \tr\Bigl(\chi {\bf 1}_{[\mu_0,\lambda]}(L) \varphi(L) \chi\Bigr).
\end{equation}
 The analysis of  $\ssf_{\varphi_2, \chi}(L_j,\lambda)$ is simple since by definition $ \ssf_{\varphi_2, \chi}(L,\lambda)= \tr\Bigl(\chi  \varphi_2(L) \chi\Bigr)$ is independent on $\lambda$. Then using the  functional calculus for smooth functions and the estimates for the trace \cite{Sj1},  we have
\begin{equation}
\ssf_{\varphi_2, \chi}(L_j,\lambda)  = \tr\Bigl(\chi\varphi_2(L_j^\sharp) \chi\Bigr)+{\cal O}(h^{\infty}) = \tr\Bigl(\varphi_2(L_j^\sharp) \Bigr)- \tr\Bigl(\varphi_2(Q_j^\sharp)(1- \chi^2 )\Bigr)+{\cal O}(h^{\infty}).\label{asymssf22}
\end{equation}
To treat $\ssf_{\varphi_k, \chi}(L_j,\lambda)$ for $k=1,3$, we will exploit a Tauberian argument and for this purpose we will prove the following.

\begin{prop}\label{prop6.1}
Let $\ssf_{\varphi,\chi}(\lambda) = \tr\Bigl(\chi {\bf 1}_{[\mu_0,\lambda]}(L) \varphi(L) \chi\Bigr)$  with $\varphi \in C_0^\infty(I; \RR^+)$,  $I \subset \subset \RR^+$ and let
\[ N^{\sharp}_{\varphi}(\lambda) = \tr\Bigl( {\bf 1}_{[\mu_0,\lambda]}(L^{\sharp}) \varphi(L^{\sharp})\Bigr), \: \: M_{\varphi, \chi}^{\sharp}(\lambda) = \tr\Bigl( {\bf 1}_{[\mu_0,\lambda]}(Q^{\sharp}) \varphi(Q^{\sharp})(1-\chi^2)\Bigr) \,. \]
 Then there exists $h_0>0$ such that 
\begin{equation}
\ssf_{\varphi,\chi}(\lambda) = N^{\sharp}(\lambda) - M_{\varphi, \chi}^{\sharp}(\lambda)  + {\cal O}(h^{-n^\sharp+1}), \; \forall h \in ]0,h_0].
\end{equation}
\end{prop}


\noindent
{\em End of the proof of Theorem 2.} We apply Proposition 9 with $\varphi= \varphi_1,\varphi_3$ and the relations (\ref{decssfX}), (\ref{asymssf22}) and conclude that
$$\tr\Bigl(\chi{\bf 1}_{[\mu_0, \lambda]}(L_j)\chi \Bigr) = \tr\Bigl({\bf 1}_{[\mu_0, \lambda]}(L_j^\sharp) \Bigr) -
\tr\Bigl({\bf 1}_{[\mu_0, \lambda]}(Q_j^\sharp)(1-\chi^2) \Bigr) +
 {\cal O}(h^{-n^\sharp + 1}), \: j =1, 2.$$ 
 Combining this with the asymptotic expansion of the integrals of other terms in (\ref{eq:6.1}), we complete the proof.
\end{pf}

\noindent
{\em Proof of Proposition 9.}
Consider a function $\theta \in C_0^\infty(]-\epsilon_1,\epsilon_1[)$, $\theta(0)=1$, $\theta(-t)=\theta(t)$ such that the Fourier transform of $\theta$ satisfies ${\hat \theta}(\lambda) \geq 0$ on $\RR$ and assume that there exist $\epsilon_0>0$, $\delta_0 >0$ so that ${\hat \theta}(\la) \geq \delta_0 > 0$ for $\mid \la \mid \leq \epsilon_0$. Next introduce 
$$\Bigl({\cal F}_h^{-1} \theta\Bigr)(\lambda) = (2\pi h )^{-1} \int e^{it \lambda/h}\theta (t) dt = (2\pi h )^{-1}{\hat \theta}(- h ^{-1}\lambda).$$

Obviously, the function $\xi_{\varphi,\chi}(\lambda)$ is increasing, $\xi_{\varphi,\chi}(\lambda)$ vanishes for $\lambda \leq \inf (\supp \varphi)$ and it coincides with a  constant for  $\lambda \geq \sup (\supp \varphi)$. We claim that  $\xi_{\varphi,\chi}(\lambda)$ satisfies the following conditions
\begin{equation} \label{eq:6.5}
 \ssf_{\varphi,\chi}(\lambda) =  {\cal O}(h^{-n^\sharp}),
\end{equation}
\begin{equation} \label{eq:6.6}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_{\varphi,\chi})(\lambda) = {\cal O}(h^{-n^\sharp}). 
\end{equation}
 
The estimate (\ref{eq:6.5}) is a consequence of the functional calculus (see \cite{Sj1}) and the fact that ${\bf 1}_{[\mu_0,\lambda]}\varphi(\mu)$ is bounded from above by a smooth function $\Psi(\mu) $ equal to $1$ on ${\rm supp}\:{\bf 1}_{[\mu_0,\lambda]}\varphi(\mu)$. For the second estimate will established the following.


\begin{lem}\label{lem6}
There exists $\epsilon_1 > 0$ such that for $\theta \in C_0^\infty(]-\epsilon_1,\epsilon_1[)$ we have
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_{\varphi,\chi})(\lambda)=\frac{d}{d \lambda}({\cal F}_h^{-1} \theta *N_{\varphi}^{\sharp})(\lambda) -\frac{d}{d \lambda}({\cal F}_h^{-1} \theta *M_{\varphi,\chi}^{\sharp})(\lambda)+  {\cal O}(h^{\infty})$$
uniformly with respect to $\lambda \in \overline{I}.$
\end{lem}

\begin{pf}
For simplicity of the notations, we will omit the subscript $\varphi$ and write $\ssf_\chi, \:\: N^{\sharp}, \:\: M^{\sharp}_{\chi}$ for  $\xi_{\varphi, \chi}, \:\: N_\varphi^{\sharp}, \:\: M_{\varphi, \chi}^{\sharp}$.
 The definition of  $\ssf_\chi$ yields
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda) = {\cal F}_h^{-1}\Bigl(\tr(\chi \theta(t) e^{-ith^{-1} L}\varphi(L) \chi)\Bigr).$$
For $\chi_1 \succ \chi$ we have the Duhamel formula
$$\chi_1 e^{-ith^{-1}L^{\sharp}}\varphi(L^{\sharp})\chi = e^{-ith^{-1}L}\varphi(L)\chi + ih^{-1}\int_0^t e^{-i(t-s)h^{-1}L}[L^{\sharp}, \chi_1 ] e^{-ish^{-1}L^{\sharp}} \varphi(L^{\sharp})\chi ds + B(t)$$
with $\|B(t)\|_{\tr} = {\cal O}(h^{\infty}).$ Here we have used the fact (see \cite{Sj2}, Section 3) that
\begin{equation}\label{eq:sj}
\chi_1 \varphi(L^{\sharp}) \chi = \varphi(L) \chi + B(0),
\end{equation}
$B(0)$ being a trace negligible operator. Let $\tilde{\varphi} \in C^{\infty}_0(I ; \RR^{+})$ be a function equal to 1 on supp $\varphi.$ Then modulo a trace negligible operator we obtain

$$\chi e^{-ith^{-1}L}\varphi(L)\chi = \chi e^{-ith^{-1}L^{\sharp}}\varphi(L^{\sharp})\chi $$
$$ - ih^{-1}\int_0^t \chi e^{-i(t-s)h^{-1}L} \tilde{\varphi}(L) [L^{\sharp}, \chi_1 ] e^{-ish^{-1}L^{\sharp}} \varphi(L^{\sharp})\chi ds,$$
since $\|\chi \tilde{\varphi}(L^{\sharp})(1- \chi_1)\|_{\tr} = {\cal O}(e^{-C/h}).$
Taking $\chi_1 \succ \chi_2 \succ \chi_3 \succ \chi$, we observe that
$$(1 - \chi_2) [L^{\sharp}, \chi_1] = [L^{\sharp}, \chi_1],$$
and applying the Duhamel formula once more,  we get
$$\chi e^{-ith^{-1}L}\tilde{\varphi}(L) (1-\chi_2) = \chi \tilde{\varphi}(L)(1-\chi_3) e^{-ith^{-1}Q}(1- \chi_2)$$
$$ + ih^{-1}\int_0^t \chi e^{-i(t-s)h^{-1}L}\tilde{\varphi}(L) [Q, \chi_3 ] e^{-ish^{-1}Q}(1- \chi_2) ds,$$
since $L(1-\chi_3)= Q (1-\chi_3)$.
Next the semi-classical Egorov theorem (see \cite{DS}, \cite{R1}) implies for sufficiently small  $s$ 
$$ \|\tilde{\chi_3} e^{-ish^{-1}Q}(1-\chi_2)\|_{L^2 \to L^2} = {\cal O}(h^{\infty}),$$
where $\tilde{\chi_3}$ is a cut-off function such that $\chi_2 \succ \tilde{\chi_3} \succ \chi_3$. Moreover,
$$\|\chi \tilde{\varphi}(L)(1 - \chi_3)\|_{L^2 \to L^2} = {\cal O}(h^{\infty}),$$
and we conclude that
$$\| \tilde{\varphi}(L) [Q, \chi_3 ] \|_{L^2 \to L^2} 
 = {\cal O}(1).$$

 Finally, using the estimate  $\|[L^{\sharp}, \chi_1]\varphi(L^{\sharp})\|_{\tr} = {\cal O}(h^{-n^\sharp})$,   we have
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda)=
\tr_{L^2(T_{\tilde{R}})}\Bigl(\chi {\cal F}_h^{-1} \theta(\lambda - L^{\sharp}) \varphi(L^{\sharp}) \chi\Bigr) + {\cal O}(h^{\infty}).$$
By exploiting the cyclicity of the trace this implies
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda)=\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N^{\sharp})(\lambda)-
\tr_{L^2(T_{\tilde{R}})}\Bigl((1-\chi^2) {\cal F}_h^{-1} \theta(\lambda - L^{\sharp}) \varphi(L^{\sharp}) \psi \Bigr) + {\cal O}(h^{\infty})$$
for any function $\psi \in C^\infty(T_{\tilde{R}}),\:\psi \succ (1-\chi^2)\vert_{T_{\tilde{R}}}$, $\psi =0$ on $\overline{B(0,R_0)}$.

To treat the second term we will use the equality
$$\psi e^{-ith^{-1}L^{\sharp}} \varphi(L^{\sharp})(1  - \chi^2)\vert_{T_{\tilde{R}}} = \psi e^{-ith^{-1}Q^{\sharp}} \varphi(Q^{\sharp})(1  - \chi^2)\vert_{T_{\tilde{R}}} $$
$$ + ih^{-1}\int_0^t \psi e^{-i(t-s)h^{-1}Q^{\sharp}} [L^{\sharp}, \psi_1] e^{-ish^{-1}L^{\sharp}} \varphi(L^{\sharp})(1  - \chi^2)\vert_{T_{\tilde{R}}} ds + B_1,$$
where $ \psi_1 \succ \psi \succ (1 - \chi_2)\vert_T,\:\: \psi_1 \in C^{\infty}(T_{\tilde{R}})$ on $\overline{B(0, R_0)}, \:\: \|B_1\|_{\tr} = {\cal O}(h^{\infty}).$
The trace of the second term in the right hand side is ${\cal O}(h^{\infty})$ for $t-s$ small since by Egorov theorem we have
$$\| \psi e^{-i(t-s)h^{-1}Q^{\sharp}} \tilde \psi_1\|_{L^2 \to L^2} = {\cal O}(h^{\infty}),$$
where $\tilde \psi_1 = 0$ on supp $\psi_1.$

 Consequently, we deduce  
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \ssf_\chi)(\lambda)=\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N^{\sharp})(\lambda)-
\tr_{L^2(T_{\tilde{R}})}\Bigl((1-\chi^2) {\cal F}_h^{-1} \theta(\lambda - Q^{\sharp}) \varphi(Q^{\sharp}) \psi \Bigr) + {\cal O}(h^{\infty}).$$
and  this completes the proof of Lemma 6.
\end{pf}

\noindent
{\em End of the proof of Proposition 9.}
According to the assumption (\ref{eq:2.9}), we have
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N^{\sharp})(\lambda) =
(2 \pi h)^{-1}\int e^{it \lambda h^{-1}} \theta (t) \tr \Big(  e^{- it  h^{-1} L^\sharp} \varphi(L^\sharp) \Big) dt = {\cal O}(h^{-n^\sharp}).$$
Recall that $Q^{\sharp}$ is a differential operator on the $n$-dimensional manifolds $T_{\tilde{R}}$, so by a standard argument we obtain
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * M_\chi^{\sharp})(\lambda) = {\cal O}(h^{-n}).$$
Then, applying a Tauberian theorem  (see Theorem IV. 13 in \cite{R1}) for $N^{\sharp}$,  $M_\chi^{\sharp}$ and exploiting Lemma 6 for $\ssf_\chi$,  we deduce
$$N^{\sharp}(\lambda)= ({\cal F}_h^{-1} \theta * N^{\sharp})(\lambda) + 
{\cal O}(h^{- n^\sharp +1}),$$
$$M_\chi^{\sharp}(\lambda)= ({\cal F}_h^{-1} \theta * M_\chi^{\sharp})(\lambda) + 
{\cal O}(h^{- n +1}),$$
$$\ssf_\chi(\lambda)= ({\cal F}_h^{-1} \theta *\ssf_\chi)(\lambda) + 
{\cal O}(h^{- n^\sharp +1}).$$

Finally, applying Lemma 6 once more and integrating,  we get
$$\ssf_\chi(\lambda) =  N^{\sharp}(\lambda) - M_\chi^{\sharp}(\lambda)  + {\cal O}(h^{- n^\sharp +1})$$
and the proof  of Proposition 9 is complete.
\hfill{$\Box$}

In the case of compactly supported perturbations of the Laplacian (or if $\gamma > n$ in the assumption (\ref{eq:2.6})) Theorem 2 holds without the assumption (\ref{eq:1.3}). More precisely, we have the following. 

\begin{thm}\label{weylsr} 
Assume that $L_1 = L$ satisfy the assumptions $(2.1) - (2.10)$ with $L_2=L_0=-h^2 \Delta$. Assume that $\mu_0$ and $\mu$ are  non-critical energy levels for $Q_1$.
Then we have the following asymptotic
\begin{equation}\label{asysr}
\xi(\mu)-\xi(\mu_0) = N(L^{\sharp}, [\mu_0, \mu]) -  \tr\Bigl( {\bf 1}_{[\mu_0,\mu]}(Q^{\sharp}) (1 -  \chi^2)\Bigr)- \tr\Bigl( {\bf 1}_{[\mu_0,\mu]}(L_0^{\sharp})  \chi^2 \Bigr)
\end{equation}
\[  + w_\chi (\mu)h^{-n} + {\cal O}(h^{-n^\sharp + 1}), \: h \in ]0, h_0] \, \]
with a function $w_\chi(\mu) \in C^1(J)$.
\end{thm}

{\bf Remark.} If the perturbation is compactly supported we can construct $ L^{\sharp}$ so that $ Q^{\sharp} = L_0^\sharp = -h^2 \Delta \vert_{T_{\tilde{R}}}$ and $\tL= L_0$. Then the asymptotic (\ref{asysr}) becomes
$$\xi(\mu)-\xi(\mu_0) = N(L^{\sharp}, [\mu_0, \mu]) -  c_n \hbox{vol}(T_{\tilde{R}})(\mu^{\frac{n}{2}}-{\mu_0}^{\frac{n}{2}}) h^{-n} + {\cal O}(h^{-n^\sharp + 1}), \: h \in ]0, h_0],$$
where $c_n = (2 \pi)^{-n}$ Vol($B(0,1)$). 

\begin{pf} 
The analysis of this case is simpler because we can express $\xi(\lambda)$ by a  sum  of monotone functions.  This decomposition was established by the first author \cite{Br} in the case ${\cal H} = L^2(\RR^n)$, while the "black box" scattering has been studied by Christiansen \cite{Ch} for compactly supported perturbations in high energy r\'egime. For $\mu \in J$ we have
\begin{equation}\label{eq:}
 \xi(\mu) - \xi(\mu_0) =
\Bigl[\sigma_1(L_j,\mu_0,\mu)\Bigr]_{j=0}^1 +  \Bigl[\sigma_{W_j}(L_j,\mu_0,\mu)\Bigr]_{j=0}^1,
\end{equation}
where 
$$\sigma_1(L_j,\mu_0,\mu)= \tr(\chi {\bf 1}_{[\mu_0,\mu]}(L_j)  \chi),$$
$\chi$ being as in Theorem 1, and 
$$\sigma_{W_j}(L_j,\mu_0,\mu) = \tr( W_j(h) {\bf 1}_{[\mu_0,\mu]}(L_j)), \:\: j =0,1.$$
Here 
$$W_1(h) = h^{-1}  [\chi^2,L] {\cal A}_h L^{-1}\varphi (L) +
(1- \chi^2)(L-L_0 - h^{-1} [{\cal A}_h,L-L_0]) L^{-1}\varphi (L)$$
$$W_0(h)  = h^{-1} [\chi^2,L_0] {\cal A}_h L_0^{-1}\varphi(L_0), $$
${\cal A}_h :=- \frac{1}{4}(x.h\nabla + h\nabla .x) $ and $\varphi \in C_0^\infty(\RR)$ is equal to $1$ on $J$.
This representation is due to Robert \cite{R2}, \cite{R4} and it is based on the relation $[{\cal A}_h,L_0]= h L_0$ and the cyclicity of the trace.

Now, consider the terms $\sigma_1(L,\mu_0,\mu)$ and $\sigma_{W_1}(L,\mu_0,\mu)$ and remark that a similar representation holds for $L$  replaced  by $L_0$. As in Theorem 2, applying  Proposition 9, we have
$$\sigma_1(L,\mu_0,\mu) = N(L^\sharp,[\mu_0,\mu])  - \tr \Bigl({\bf 1}\vert_{[\mu_0, \mu]}(Q^{\sharp})(1-\chi^2)\Bigr) +{\cal O}(h^{-n^{\sharp} + 1}).$$
To treat $\sigma_{W_1}$, notice that $W_1$ is localized outside $\overline{B(0, R_0)}.$ Then in the representation of $W_1$ we can replace $L$ by $Q_1$ modulo ${\cal O}(h^{\infty})$ and  $W_1$ becomes a  classical $h$-admissible pseudodifferential operator of order
$0$ with  symbol in the space $S(\langle x \rangle^{-\delta})$, $\delta > n$, localized in the non-trapping region $\{x :  |x|\geq 2 \rho_0\}$. Here we used the notation
\begin{equation}\label{symbols}
S(\langle x \rangle^{-\delta}) =
\{s \in C^\infty(\RR^{2n}) :\: \mid \partial_x^\alpha
\partial^\beta_\xi s(x,\xi) \mid \leq C_{\alpha, \beta}\langle x \rangle^{-\delta-\mid \alpha
\mid} \langle \xi \rangle^{ - \mid
\beta \mid}, \:\: \forall \alpha, \:\: \forall \beta \}.
\end{equation}
Following  Lemma 4.2 of \cite{Br}, we decompose $\sigma_{W_1}$ as follows
$$\sigma_{W_1}(L,\mu_0,\mu)= \tr( W_+(h) {\bf 1}_{[\mu_0,\mu]}(L)) - \tr( W_-(h) {\bf 1}_{[\mu_0,\mu]}(L)) + {\cal O}(h^{\infty}),$$
where $ W_\pm (h)$ are  classical, non-negative, $h$-admissible
pseudodifferential operators of order
$0$ with  symbols in $S(\langle x \rangle^{-\delta})$, $\delta > n$, supported  in the non-trapping region. 

By a Tauberian argument for the monotone functions $\sigma_\pm(\mu):= \tr( W_\pm (h) {\bf 1}_{[\mu_0,\mu]}(L))$, we deduce
$$\sigma_\pm(\mu) 
 = ({\cal F}_h^{-1} \theta * \sigma_\pm )(\mu ) + {\cal O}(h^{-n+1}).$$
On the other hand, we have 
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \sigma_\pm )(\lambda )= 
 {\cal F}_h^{-1}\Bigl(\tr( \theta(t) W_\pm  e^{-ith^{-1} L})\Bigr).$$

Next, as in the proof of Theorem 2, for sufficiently small $t$ we apply Egorov theorem and conclude that
$$\tr(  W_\pm  e^{-ith^{-1} L})=\tr(  W_\pm  e^{-ith^{-1} Q }) + {\cal O}(h^{\infty}).$$
Consequently, exploiting the construction of a parametrix for the propagator $e^{ith^{-1}Q}$ for small $t$  (see Proposition 4.3 in \cite{Br}), we get 
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \sigma_\pm )(\mu )= h^{-n} \beta_0(\mu,W_\pm) + {\cal O}(h^{-n+1})$$
with $\beta_0(.,W_\pm) \in C^\infty(J)$.
Finally, 
$$\sigma_{W_1}(L,\mu_0,\mu)= h^{-n} w_{\chi}(\mu) +  {\cal O}(h^{-n+1})$$
with $w_{\chi} \in  C^\infty(J)$ and the proof is complete.

\end{pf}

\section{Appendix}

 Define the class of symbols $S^{p,l}_\nu$ as the set of smooth functions
$b(x,\xi) \in C^\infty(\RR^n \times \RR^n)$ such that for any $\beta_1, \beta_2$ in $\N^n$ we have
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2}  b (x,\xi) |
\leq C_{\beta_1,\beta_2} \langle \xi  \rangle^{p}  \langle x  \rangle^{l-\nu |\beta_1|}.$$

Let $b \in S^{2,0}_\nu$ be an elliptic symbol in the sense that $| b(x,\xi)| \geq
C \langle \xi  \rangle^{2}$. Denote by $B = b(x,hD_x)$ the operator which is the  Weyl quantization of $b(x,\xi)$, that is
$$b(x,h D_x)\varphi(x)=\frac{1}{(2 \pi)^n} \int \exp(i\langle x - y, \xi \rangle )  
b\Bigl( \frac{x+y}{2}, h\xi \Bigr) \varphi(y)dy d\xi,\:\: \varphi \in {\cal 
S}({\bf R}^n).$$

Choosing $m(x,\xi) = \langle \xi  \rangle^{p}  \langle x  \rangle^{l}$, for $\nu = 0$ the class $S^{p,l}_0$ coincides with the class $S^0(m)$ introduced in \cite{DS}. Then,
according to \cite{DS}, we know that given $f \in C_0^\infty(\RR)$ and $b \in  S^{2,0}_0$, the operator $f(B)$ becomes a pseudodifferential operator with symbol $a \in  S^{-k,0}_0$ for any $k \in\N$. 

In the proof of Proposition \ref{prop8}, we needed the following more precise estimate concerning the remainder of the symbol of the operator $f\Big( b(x,hD_x) \Big)$ with $b \in S_{\nu}^{2,0},\: \nu >0.$

\begin{lem}
Let $b \in S^{2,0}_\nu$ be elliptic. Then for any $N \in \N$ we have
\begin{equation} \label{est}
f\Big( b(x,hD_x) \Big) =  a_N(x,hD_x;h) + h^N r_N(x,hD_x;h),
\end{equation}
where for any $k \in\N$ we have  $a_N \in S^{-k,0}_\nu$ and $r_N \in S^{-k,-\nu N}_0$ uniformly with respect to $h \in (0,h_0]$.
\end{lem}
\begin{pf}
Following the calculus in \cite{DS} and applying the Helffer-Sj\"ostrand's formula, it is sufficient to prove the representation (\ref{est}) for the resolvent $(B-z)^{-1}$, $\Im z \neq 0$, $|z | \leq Const$ with an estimate with respect to $z$. On the other hand, the parametrix construction exposed in \cite{R1}, \cite{Se} (see also \cite{DR}) yields
 $$(B-z)^{-1} =  a_N(x,hD_x;h) + h^N (B-z)^{-1} \delta_N(x,hD_x;h),$$
where for any $\beta_1, \beta_2$ in $\N^n$ we have the estimates
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} a_N(x,\xi;h) |
\leq C_{\beta_1,\beta_2} \sum_{j=0}^{N-1} h^j   \langle x  \rangle^{-\nu j -\nu |\beta_1|} |\Im z |^{- (|\beta_1|+ |\beta_2|) -2j -1}$$
$$\qquad\qquad \qquad\leq C_{\beta_1,\beta_2,N} \hbox{Max}\Big(1, \frac{\sqrt h}{|\Im z |} \Big)^{(N-1)/2}  |\Im z |^{- (|\beta_1|+ |\beta_2|) -1}
 \langle x  \rangle^{ -\nu |\beta_1|}.$$
Moreover, there exist $\rho \geq 3/2$ and $N_0$ such that for any $\beta_1, \beta_2$ in $\N^n$we get
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} \delta_N(x,\xi;h) |
\leq C_{\beta_1,\beta_2,N}   |\Im z |^{- (|\beta_1|+ |\beta_2|) -\rho N -N_0}
 \langle x  \rangle^{-\nu N -\nu |\beta_1|}.$$
Next, exploiting the functional calculus of  \cite{DS}, we know that 
$(B-z)^{-1}$ is the Weyl quantization of a symbol $q(x,\xi;h)$ satisfying
$$| \partial_x^{\beta_1}  \partial_{\xi}^{\beta_2} q(x,\xi;h) |
\leq C_{\beta_1,\beta_2} \hbox{Max}\Big(1, \frac{\sqrt h}{|\Im z |} \Big)^{2n+1}  |\Im z |^{- (|\beta_1|+ |\beta_2|) -1}.$$
We complete the proof by using the composition of pseudodifferential operators.
\end{pf}


{\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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