Content-Type: multipart/mixed; boundary="-------------0109211022449" This is a multi-part message in MIME format. ---------------0109211022449 Content-Type: text/plain; name="01-332.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-332.keywords" schrodinger operator, semi classical limit, spectral shift function, resonances, trace formula, Breit-Wigner approximation, Weyl asymptotic ---------------0109211022449 Content-Type: application/x-tex; name="mero.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mero.tex" \documentstyle[11pt]{amsart} \setlength{\topmargin}{0cm} \setlength{\textheight}{21cm} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{6.5in} \setlength{\parindent}{.25in} \pagestyle{headings} \def\Bbbone{{\mathchoice {1\mskip-4mu \text{l}} {1\mskip-4mu \text{l}} { 1\mskip-4.5mu \text{l}} { 1\mskip-5mu \text{l}}}} \def\squarebox#1{\hbox to #1{\hfill\vbox to #1{\vfill}}} \newcommand{\stopthm}{\hfill\hfill\vbox{\hrule\hbox{\vrule\squarebox {.667em}\vrule}\hrule}\smallskip} \pagestyle{headings} \newcommand{\1}{{\bold 1}} \newcommand{\F}{{\cal F}} \newcommand{\CC}{{\cal C}} \newcommand{\CI}{{\cal C}^\infty } \newcommand{\Oo}{{\cal O}} \newcommand{\K}{{\cal K}} \newcommand{\D}{{\cal D}} \newcommand{\G}{{\cal G}} \newcommand{\Hh}{{\cal H}} \newcommand{\pic}{{\mbox{Pic}}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\RR}{{\Bbb R}} \newcommand{\HH}{{\Bbb H}} \newcommand{\U}{{\cal U}} \newcommand{\A}{{\Bbb A}} \newcommand{\C}{{\Bbb C}} \newcommand{\N}{{\Bbb N}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\vol}{\operatorname{vol}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\half}{\frac{1}{2}} \newcommand{\itt}{\operatorname{it}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\itA}{\operatorname{it}} \newcommand{\im}{\operatorname{Im}} \newcommand{\point}{\operatorname{point}} \newcommand{\comp}{\operatorname{comp}} \newcommand{\loc}{\operatorname{loc}} \newcommand{\Diff}{\operatorname{Diff}} \newcommand{\trb}{\operatorname{tr_{bb}}} \newcommand{\tr}{{\operatorname{tr}}} \newcommand{\rarrow}{\operatornamewithlimits{\longrightarrow }} \newsymbol\circlearrowleft 1309 \newsymbol\restriction 1316 \newcommand{\rest}{\!\!\restriction} \newcommand{\ttt}{|\hspace{-0.25mm}|\hspace{-0.25mm}|} \renewcommand{\Re}{\mathop{\rm Re}\nolimits} \renewcommand{\Im}{\mathop{\rm Im}\nolimits} \theoremstyle{plain} \def\Rm#1{{\rm#1}} \newtheorem{thm}{Theorem} \renewcommand{\thethm}{\arabic{thm}} \newtheorem{cor}{Corollary} \renewcommand{\thecor}{\arabic{cor}} \newtheorem{lem}{Lemma} \renewcommand{\thelem}{\arabic{lem}} %\numberwithin{lem}{section} \newtheorem{prop}{Proposition} \renewcommand{\theprop}{\arabic{prop}} \newtheorem{rem}{Remark} \theoremstyle{definition} \newtheorem{ex}{EXAMPLE}[section] \numberwithin{equation}{section} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\secref}[1]{Section~\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} \newcommand{\exref}[1]{Example~\ref{#1}} \newcommand{\corref}[1]{Corollary~\ref{#1}} \newcommand{\propref}[1]{Proposition~\ref{#1}} \title[Meromorphic continuation] {Meromorphic continuation of the spectral shift function} \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} \def\fh{\frac{1}{h}} %-------------------------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} \newcommand{\tf}{\tilde f} \newcommand{\qjt}{Q_{j, \theta}} \newcommand{\hQ}{\hat{Q}} \newcommand{\hL}{\hat{L}} \newcommand{\ljt}{L_{j, \theta}} \newcommand{\lt}{L_{1, \theta}} \newcommand{\ltt}{L_{2, \theta}} \newcommand{\sh}{\sum_{w \in {\rm{Res}}, \:2h \leq |w - \lambda| \leq C_1}} \newcommand{\ns}{n^{\#}} \newcommand{\brw} {\frac{|\Im w|}{|\mu - w|^2}} %----------------------------pointe-------------------- \newcommand{\pL}{L_.} %----------------------------SSF---------------------- \newcommand{\ssf}{\xi} \newcommand{\ssfr}{\xi_\rho} \newcommand{\DS}[1]{{\displaystyle #1}} \def \rn{{{\RR}^n}} \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 $\|^2u\|_{L^2},\:^2 = 1 + (hD)^2$, 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| 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}^{-\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. 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