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Black-Box, long-range, resolvent estimate, semi-classique, trapping energy level
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\title[Semiclassical resolvent estimates ]
{Semiclassical resolvent estimates  for trapping perturbations}
\author[V. Bruneau, V. Petkov]{Vincent Bruneau and Vesselin Petkov}

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\begin{document}    
   
\maketitle   
$\:\:\:\:\:$ {\bf Abstract.} {\footnotesize We study the semiclassical estimates of the resolvent
$R(\lambda + i\tau),\:\:\lambda \in J \subset\subset{\RR}^{+},\: \tau \in ]0,1]$ of a self-adjoint operator $L(h)$ in the space of bounded operators ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s}),\:s > 1/2$. In the general case of long-range trapping "black-box" perturbations we prove that the estimate of the cut-off resolvent $\|\chi(x)R(\lambda + i0)\chi(x)\|_{{\cal H} \to {\cal H}} \leq C\exp(Ch^{-p}),\:\chi(x) \in C^{\infty}_0({\RR^n}),\:p \geq 1$ implies the estimate $\|R(\lambda + i\tau)\|_{s,-s} \leq C_1\exp(C_1 h^{-p})$.}
 


 

\section{Introduction}
 
The purpose of this paper is to obtain semiclassical estimates for the resolvent 
$$ R(z)=(L(h)-z)^{-1},\:z \in {\C} \setminus \RR$$
 of a self-adjoint operator $L = L(h)$ depending on $h \in ]0,h_0]$ for
$$z \in {\cal B}_{\pm} = \{z = \lambda + i\tau \in {\C}: \lambda \in J,\:\pm\tau \in ]0,1]\},$$
$J = ]\mu_0, \mu_1[ \subset \subset {\RR}^{+}.$ The operator $L$ is defined in a domain ${\cal D} \subset {\cal H}$ of a complex Hilbert space ${\cal H}$ with an orthogonal decomposition      
$${\cal H} = {\cal H}_{R_0} \oplus L^2({\RR}^n \setminus 
B(0,R_0)),\:B(0,R_0) = \{x \in {\RR}^n: |x| \leq R_0 \},\:\:n \geq 2 \: $$
 and $L$ satisfies the long-range "black box" assumptions (\ref{eq:1.7})-(\ref{eq:1.13}) described below.
Introduce the spaces
$${\cal H}^{0,s} = {\cal H}_{R_0} \oplus  L^2(\RR^n\setminus B(0,R_0), \langle x {\rangle}^s dx), \:\langle x \rangle = (1 + |x|^2)^{1/2}.$$   
 First we show that for $s > 1/2$ and fixed $h > 0$ the limits 
\begin{equation}
\lim_{\epsilon \to 0,\: \epsilon > 0} R(\lambda \pm i\epsilon) =
 R(\lambda \pm i0),\:\:\lambda \in J \label{eq:1.1}
\end{equation}
exist in the space of bounded operators ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$,
where here and below we omit in $R(z)$ the dependence of $h$. Next we give semiclassical resolvent estimates
\begin{equation}
\|R(\lambda \pm i\tau)\|_{s,-s} \leq Cr(h),\: \lambda \in J,\:\:\tau \in ]0,1],\:\: h \in ]0, h_0], \label {eq:1.2}
\end{equation}
where $\|.\|_{s,-s}$ denotes the norm in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$. Such estimates have been obtained by many authors in great generality in the case when every $\lambda \in J$ is a {\em non-trapping energy level} for the principal symbol $l_0(x,\xi)$ of a differential operator $L$ (see  \cite{kn:RT1}, \cite{kn:GM1}, \cite{kn:RT2}, \cite{kn:HN}, \cite{kn:W}, \cite{kn:R2}, \cite{kn:R3},  \cite{kn:VZ} and the survey article \cite{kn:R4} for other references). The proofs of these estimates are  essentially based on Mourre method \cite{kn:M}, \cite{kn:CFKS}, \cite{kn:PSS} and the main point is to construct a conjugate operator $A(h)$ for which the Mourre inequality (\ref{eq:mourreesti}) holds (see Section 5.1). Roughly speaking we call these operators non-trapping perturbations and in this case we have $r(h) = h^{-1}$ in (\ref{eq:1.2}). Moreover, for such perturbations there are no resonances converging sufficiently fast to the real axis as $h \to 0.$\\

The case of {\em trapping energy levels} is more complicated. First, without the non-trapping assumption, in general,  a Mourre type inequality is not true. Secondly, resonances converging exponentially fast to the real axis could exist and, consequently, we must have $r(h) = e^{Ch^{-p}}$ with some $p \geq 1.$  In the special case of a Schr\"{o}dinger operator $-h^2\Delta + V(x)$, with potential $V(x)$ having the form of a "well in an island", the results in \cite{kn:GM2}, \cite{kn:GMR} imply the estimate (\ref{eq:1.2}) with $r(h) = e^{Ch^{-1}}$. Recently, for long-range trapping perturbations in the exterior of an obstacle, Burq \cite{kn:Bu} established the estimate
\begin{equation} 
\label{eq:1.3}
\|\chi(x)R(\lambda +i0)\chi(x)\|_{{\cal H} \to {\cal H}} \leq C\exp(Ch^{-p}),\:\lambda \in J,\:0 < h \leq h_0,
\end{equation}
with $p=1$ and some constants $C > 0,\: h_0 > 0$, provided $\chi(x) \in C^{\infty}_0({\RR}^n)$ is equal to 1 on $\overline{B(0,R_0)}.$ In fact, Burq \cite{kn:Bu} obtained the estimate
\begin{equation}
\|(L_{\theta}(h) - \lambda)^{-1} \|_{L^2({\cal H}_{\theta})} \leq Ce^{C_1 h^{-p}},\:\lambda \in J, \label{eq:1.3*}
\end{equation}
with $p =1,\:L_{\theta}(h)$ being the operator given by complex scaling (see \cite{kn:SZ}, \cite{kn:Sj1}). On the other hand, the estimate (\ref{eq:1.3*}) implies immediately (\ref{eq:1.3}) since we have
$$\chi(x) R(\lambda + i0)\chi(x) = \chi(x)(L_{\theta}(h) - \lambda)^{-1}\chi(x)$$
for suitably chosen $\chi(x)$ (see Lemma 3.5 in \cite{kn:SZ}). Next, denote by Res $L(h)$ the set of resonances of $L(h)$ and assume that there exist  $\epsilon > 0, \:c > 0$ and $q \geq 1$ so that 
$${\rm dist}\: \{ {\rm Res}\:L(h),\: [\mu_0, \mu_1] \} \geq \epsilon \exp(-c h^{-q}),\: 0 < h \leq h_0.$$
Then an application of Lemma 1 in \cite{kn:TZ} with $g(h) = \exp(-\frac{c}{2}h^{-q})$ yields the estimate (\ref{eq:1.3*}) with $p \geq q.$ Thus we have several sufficient conditions leading to the estimate (\ref{eq:1.3}).\\ 


In this work we show that an estimate of the cut-off resolvent $\|\chi(x)R(\lambda + i0)\chi(x)\|_{{\cal H}},\: \lambda \in J,$ implies an estimate of $\|R(\lambda + i\tau)\|_{s, -s}$ with the same order $p$ in the bound $\exp(Ch^{-p}).$
Consequently, we extend the result of \cite{kn:Bu} replacing the cut-off factor $\chi(x)$ by a function $\psi(x) \in C^{\infty}({\RR}^n)$ which is equal to $\langle x \rangle^{-s},\: s > 1/2$ for $|x| \geq R_1 > R_0$. Moreover,  for long-range perturbations in ${\Bbb R}^n$ studied in  \cite{kn:Bu},  we prove the conjecture of Robert \cite{kn:R3} who claimed that (\ref{eq:1.2}) with $r(h) = C\exp(Ch^{-p})$ holds for general trapping perturbations. Under this assumption,  Robert  \cite{kn:R3} obtained a Weyl type asymptotic for the scattering phase related to two $h$-admissible pseudodifferential operators $L_1(h),\:\:L_2(h)$. Our main result is the following.

\begin{thm}\label{theo}
Let $L(h)$ be a self-adjoint operator satisfying the "black box" assumptions
 $(\ref{eq:1.7})- (\ref{eq:1.13})$. Then for fixed $s > 1/2$ and for sufficiently small fixed $h$ the limits  
$$R(\la \pm i0) = \lim_{\Im z \to 0, \: \: \pm \Im z > 0}  R(z), \quad \Re z=\la \in J,$$
exist in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$ uniformly with respect to $\lambda \in J$. Moreover, if  there exist $\delta > 0$ and $p \geq 1$ such that
\begin{equation}
\|\chi(x)R(\lambda \pm i0)\chi(x)\|_{{\cal H} \to {\cal H}} \leq C\exp(Ch^{-p}),\:\lambda \in ]\mu_0 -\delta, \mu_1 + \delta[,\:0 < h \leq h_0,
\label{eq:1.5}
\end{equation}
holds, then for each  $s > \frac12$ there exist $C_s > 0,\:\: h_1 > 0$ such that for
 $(\lambda, \tau)  \in J \times ]0, 1]$ we have
\begin{equation}
\label{eq:1.6}
\| R(\lambda  \pm i \tau) \|_{s,-s} \leq C_s \exp (C_sh^{-p}),\:\: 0 < h  \leq h_1.
\end{equation}
\end{thm}

The idea of our proof is to construct a self-adjoint operator $\tilde{L} = \tilde{L}(h)$ on $L^2({\Bbb R}^n)$ such that each $\lambda \in J$ is a non-trapping energy level for $\tilde{L}$ and $L(h)\psi = \tilde{L}(h) \psi$ for any $\psi \in C^{\infty}_0({\Bbb R}^n)$ supported away from a ball $B(0, R)$. For this purpose we show that the bounded trajectories related to the symbol of $L(h)$ and to the energy levels $\lambda \in J$ are included in a compact set. Next we represent $R(z)$ by a sum of terms involving the resolvent $\tilde{R}(z)$ of $\tilde{L}$
(see Section 3) and an application of the semiclassical estimates for non-trapping energy levels reduces the problem to the estimation of $\|\chi(x)R(z)\chi(x)\|_{{\cal H}}$ (see Proposition 3). In Section 2 our approach is motivated by the proof of Lemma 1.2 in \cite{kn:GMR} concerning the operator  $L(h) = -h^2\Delta + V(x)$, where the potential $V(x)$ involves a "well in an island". Having an "island" around the trapped trajectories, G\'erard, Martinez and Robert have introduced a perturbation  $W_0(x) \in C^{\infty}_0({\RR}^n)$ such that the perturbed  symbol $|\xi |^2 + V(x) + W_0(x)$ becomes a non-trapping Hamiltonian for energy levels $\lambda \in J.$ We prove a similar result in the general case by constructing a perturbation $\tilde l(x,\xi)$ of $l_0(x,\xi)$ (see Proposition \ref{prop2}) so that all $\lambda \in J$ are non-trapping energy levels for $\tilde l(x,\xi)$. The existence of $\tilde l(x, \xi)$ implies immediately the existence of an operator $\tilde{L}(h)$ with the properties mentioned above. Finally, let us mention that the  estimate (\ref{eq:1.6}) combined with the argument of Robert  \cite{kn:R3} lead to a  Weyl asymptotic of the scattering phase for general long-range perturbations in ${\Bbb R}^n$ when we have not a "black box".\\

Now we will recall the abstract "black box" assumptions given in  \cite{kn:SZ}, \cite{kn:Sj1} and \cite{kn:Sj2}. Suppose that ${\cal D}$ satisfies
   \begin{equation}\label{ass1} 
{\Bbbone }_{{\RR}^n \setminus B(0,R_0)}{\cal D} = H^2({\RR}^n \setminus B(0,R_0)), \label{eq:1.7}   
\end{equation} 
uniformly with respect to $h$ in the sense of \cite{kn:Sj1}. More precisely, equip $H^2({\RR}^n \setminus B(0,R_0))$ with the norm $\|<hD>^2u\|_{L^2},\:<hD>^2 = 1 + (hD)^2$ and equip ${\cal D}$ with the norm $\|(L+i)u\|_{{\cal H}}.$ Then we require that ${\Bbbone }_{{\RR}^n \setminus B(0,R_0)}: {\cal D} \longrightarrow H^2({\RR}^n \setminus B(0,R_0))$ is uniformly bounded with respect to $h$ and this map has a uniformly bounded right inverse.

Assume that
%\begin{equation}
%{\rm if}\: u \in H^2({\RR}^n \setminus B(0,R_0))\:\: {\rm vanishes}\:{\rm near}\:\partial B(0,R%_0),\:{\rm then}\: u \in {\cal D}, 
%\end{equation}
\begin{equation}  \label{ass2} 
{\Bbbone}_{B(0,R_0)}(L+i)^{-1} \hbox{is compact}
\end{equation} 
and
\begin{equation}\label{ass3}
 (Lu)\vert_{{\RR}^n \setminus  \overline{B(0,R_0)}} = Q\Bigl( u\vert_{{\RR}^n \setminus \overline{B(0,R_0)}}\Bigr),
\end{equation}
where $Q$ is a formally self-adjoint differential operator
\begin{equation}\label{ass4}
 Q u = \sum_{| \alpha | \leq 2} a_\alpha (x;h) (hD_x)^\alpha u, 
\end{equation}
with $ a_\alpha (x;h)=  a_\alpha (x)$ independent of $h$ for $| \alpha | = 2$
 and $a_\alpha \in C_b^\infty(\RR^n)$ uniformly bounded with respect to $h$.

Next we assume the following properties:

 There exists $C>0$ such that
\begin{equation} \label{ass5}
 l_0(x,\xi) =\sum_{| \alpha | = 2} a_\alpha (x) \xi^\alpha \geq C  {\langle \xi \rangle}^2,\:\:\langle \xi \rangle = (1 + 
|\xi|^2)^{1/2}. \label{eq:1.9}
\end{equation}

 There exists  $\gamma > 0$ such that for every $(\alpha,\beta) \in
 {\N}^n \times \N^n$,  we have
\begin{equation}\label{ass6}
 | {\partial}_{x}^{\alpha}{\partial}_{\xi}^{\beta} 
 \Big(\sum_{| \alpha | \leq 2} a_\alpha (x;h) \xi^\alpha - |\xi|^2 \Big) |
 \leq C_{\alpha,\beta}  {\langle x \rangle}^{-\gamma- \mid \alpha \mid}{\langle 
\xi \rangle}^2 \label{eq:1.10}
\end{equation}
uniformly with respect to $h$.

There exist $\theta_0 \in ]0,\pi[,\:\epsilon > 0$ and $R_1 > R_0$ so that the coefficients $a_{\alpha}(x;h)$ of $Q$ can be extended holomorphically in $x$ to
\begin{equation} \label{ass7}
\{r\omega;\:\omega \in {\C}^n,\: {\rm dist}\:(\omega, S^{n-1}) < \epsilon, \: r \in {\C},\: |r| > R_1,\:{\rm arg} \: r \in [-\epsilon, \theta_0 + \epsilon)\} \label{eq:1.11}
\end{equation}
and (\ref{eq:1.10}) extend to this larger set.

Let $R > R_0,\:T = ({\RR}/\tilde{R}{\Z})^n,\: \tilde{R} > 2R.$ Set
$${\cal H}^{\#} = {\cal H}_{R_0} \oplus L^2(T \setminus B(0, R_0))$$
and consider a self-adjoint operator $L^{\#}: {\cal H}^{\#} \longrightarrow {\cal H}^{\#}$ with domain
$${\cal D}^{\#} = \{u \in {\cal H}^{\#}: \: \chi u \in {\cal D}, \: (1-\chi)u \in H^2 \},$$ 
where $\chi \in C^{\infty}_0(B(0,R); [0,1])$ is equal to 1 near $\overline{B(0,R_0)}.$ Take a differential operator 
$$Q^{\#} = \sum_{|\alpha| \leq 2} a_{\alpha}^{\#}(x;h)(hD)^{\alpha}$$
with $a_{\alpha}^{\#}(x;h) = a_{\alpha}(x;h)$ for $|x| < R$, put
$$L^{\#}u = L(\chi u) + Q^{\#}((1-\chi)u),\: u \in {\cal D}^{\#}$$
and denote by $N(L^{\#}, [-\lambda^2, \lambda^2])$ the number of eigenvalues of $L^{\#}$ in the interval $[-\lambda^2, \lambda^2]$. Then we assume that
\begin{equation}\label{ass8}
N(L^{\#}, [-\lambda^2, \lambda^2]) = {\cal O}(\Bigl(\frac{\lambda}{h}\Bigr)^{n^{\#}}),\: n^{\#} \geq n,\: \lambda \geq 1. \label{eq:1.12}
\end{equation}
Finally, we suppose that
\begin{equation}\label{ass9}
\sigma_{pp}(L(h)) \cap [\mu_0 - \delta, \mu_1+\delta] = \emptyset,\: h \in ]0,h_0], \label{eq:1.13}
\end{equation}
where $\delta > 0$ is the constant in (\ref{eq:1.5}).
A typical example for an operator satisfying the conditions 
(\ref{eq:1.7})-(\ref{eq:1.13}) is an elliptic self-adjoint Schr\"{o}dinger operator
$$P = \sum_{|\alpha| \leq 2} a_{\alpha}(x;h) (hD_x)^{\alpha}$$
with $C^{\infty}$ coefficients in the connex exterior $\Omega = {\RR}^n \setminus {\cal O}$ of a bounded obstacle ${\cal O}$ with $C^{\infty}$ boundary $\partial \Omega$ with Dirichlet or Neumann boundary condition on  $\partial \Omega$ (see \cite{kn:Bu}). In this case ${\cal D} =  H^2 ({\RR}^n \setminus {\cal O}) \cap H_0^1({\RR}^n \setminus {\cal O})$ and $P$ is a positive operator in $L^2({\RR}^n \setminus {\cal O}).$\\

Under the above assumptions the resonances close to the real axis can be defined by the method of complex scaling \cite{kn:SZ}, \cite{kn:Sj1} and they coincide with the poles of the meromorphic continuation of the resolvent $(L(h) - z)^{-1}: {\cal H}_{{\rm comp}} \longrightarrow {\cal D}_{{\rm loc}}$ from $\Im z > 0$ to a conic neighborhood of the positive real axis in the lower half plane. The set of resonances will be denoted by Res $L(h)$.\\

The paper is organized as follows. Section 2 is devoted to the analysis of the non-trapping energy levels and to the construction of the symbol $\tilde l(x,\xi).$ In Section 3 we prove Theorem 1 introducing the operator $\tilde{L}(h)$ and exploiting the semiclassical estimates for non-trapping energy levels. In Section 4 we obtain some microlocal estimates closely related to the bounds of $\|R(z)\|_{s, -s}$. In a further work we will apply the microlocal estimates given in Proposition 5 for the analysis of the semiclassical asymptotic of the scattering phase.\\

{\bf Acknowledgments.} The authors are grateful to Didier Robert for helpful discussions and valuable comments concerning the non-trapping perturbations. We would to thank Maciej Zworski for helpful comments and discussions as well as for his suggestion to study the long-range "black box" perturbations.\\

\section{Classical trajectories and non-trapping energy levels}

Let $l(x,\xi) \in C^\infty(\rn \times \rn :\RR)$ be a classical Hamiltonian,
and let $H_{l}= (\partial_\xi l, - \partial_x l)$
be  the associated Hamiltonian vector field:
$$H_{l}(f)(x,\xi) = \{l,f\}(x,\xi)=(\partial_\xi l \partial_x f-
 \partial_x l \partial_\xi f)(x,\xi),\:\: f  \in C^\infty(\rn \times \rn). $$
 Denote by $\Phi_l^t$ the Hamiltonian flow of $l$ given by
$$\Phi_l^t = \hbox{\rm exp}(tH_{l}) \; : \;
 (x_0, \xi_0) \mapsto \Big( x(t, x_0, \xi_0), \xi(t, x_0, \xi_0)\Big).$$
Therefore for each $\la \in \RR$, the energy surface
$\Sigma_\la=l^{-1}(\la)$ is stable by $\Phi_l^t$. In the exposition below we denote  $J = ]\mu_0, \mu_1[ \subset \subset \RR^{+}$. We will say that $(x,\xi)$ is a non-critical point for $l(x,\xi)$ if  $\nabla_{x,\xi} l(x,\xi) \neq 0$ and that $\lambda$  is a non-critical energy level for $l(x,\xi)$, if any $(x,\xi) \in \Sigma_\la=l^{-1}(\la)$ is a non-critical point for   $l(x,\xi)$.
We start by an examination of the bounded trajectories.

\begin{lem}\label{lem1}
Let $\mu_1 > \mu_0 >  0$ be fixed.
Assume there exist $\mu \in   ]0,\mu_0 [$,
$R>0$ such that 
\begin{equation}\label{eq:2.1}
 | x | \geq R, \;  (x,\xi) \in
l^{-1}(]\mu_0,\mu_1[) \; \Rightarrow \; | \frac14 \{ l, x \cdot \partial_\xi l 
\} (x,\xi) - l(x,\xi) |  \leq \mu.
\end{equation}
Then for each $\la \in J$ and each  $(x_0,\xi_0) \in \Sigma_\la$ with  $|x_0 | > 
R$ the function 
$x(\cdot,x_0,\xi_0)$ is non-bounded.
\end{lem}
\begin{pf}
 By definition, $|x(t)|^2$ satisfies
\begin{equation}\label{eq:2.2}
 \left\{
\begin{array}{ccl}
(|x|^2)'  = \left\{l,x^2 \right\}(x,\xi) = 2 x\cdot \partial_\xi l(x,\xi), 
\\
(x\cdot \partial_\xi l(x,\xi))' = \left\{l,x\cdot \partial_\xi l 
\right\}(x,\xi).
\end{array} \right.
\end{equation}
 For $(x_0,\xi_0) \in \Sigma_\la$ we have $(x(t),\xi(t)) \in
\Sigma_\la$ and this yields
\begin{equation}\label{eq:2.3}
( x\cdot \partial_\xi l(x,\xi))'
 =  4\la +  \left\{l,x\cdot \partial_\xi l \right\}(x,\xi) - 4 l(x,\xi).
\end{equation}
Let $|x_0| > R$ and $(x_0,\xi_0) \in l^{-1}(]\mu_0,\mu_1[)$.
Denote by  $[T_m,T_M]$ the maximal interval
containing $t_0=0$ so that
\begin{equation}\label{eq:2:4}
 |\Big( \{ l, x \cdot \partial_\xi l \}  - 4 l \Big)(x(t),\xi(t))
|  \leq 4 \mu, \:\: t \in [T_m, T_M].
\end{equation}
By the continuity of $x(t)$ and $|\{ l, x \cdot \partial_\xi l \}  -4 l 
|(x(t),\xi(t))$
a such interval exists and it is not reduced to $\{0\}$. From the relation 
(\ref{eq:2.3}) we deduce
$$( x\cdot \partial_\xi l(x,\xi))'(t) \geq 4 (\la - \mu) \geq 4 (\mu_0 -    
\mu),\:\:\forall t \in  [T_m,T_M],$$
 that is 
$$ \left\{\begin{array}{cccc}
 (x\cdot \partial_\xi l(x,\xi))(t) \geq  x_0\cdot \partial_\xi l(x_0,\xi_0) + 
4(\mu_0 - \mu) t,\:\:\forall t \in [0,T_M], \\
 ( x\cdot \partial_\xi l(x,\xi))(t) \leq  x_0\cdot \partial_\xi l(x_0,\xi_0) + 
4(\mu_0 - \mu) t,\:\:\forall t \in [T_m,0].\:\: \\
\end{array} \right. $$
Then applying (\ref{eq:2.2}), we get
$$ |x(t)|^2 \geq |x_0|^2 + 2  x_0\cdot \partial_\xi l(x_0,\xi_0)  t + 4 ( 
\mu_0-\mu)
t^2,\:\:\forall t \in  [T_m,T_M]. $$
Consequently, if $\pm  x_0\cdot \partial_\xi l(x_0,\xi_0) \geq 0$, for $\pm t
\geq 0$, we have
\begin{equation}\label{eq:esti2}
|x(t)|^2 \geq |x_0|^2 +  4 ( \mu_0-\mu) t^2
\end{equation}
which implies immediately that $T_M = + \infty$ (if $ x_0\cdot \partial_\xi 
l(x_0,\xi_0) \geq 0$) or 
 $T_m = - \infty$ (if $ x_0\cdot \partial_\xi l(x_0,\xi_0) \leq 0$) and 
(\ref{eq:esti2}) holds
for any $t \geq 0$ (or $t \leq 0$). Thus, as soon as $|x_0| > R$, and
$l(x_0,\xi_0) \in ]\mu_0,\mu_1[$, the function $x(\cdot,x_0,\xi_0)$ is 
non-bounded.

\end{pf}

\begin{prop}\label{prop.1}
Let $\mu_1 > \mu_0 >  0$ and let  $\displaystyle{\lim_{| \xi | \rightarrow 
\infty} l(x,\xi) = +\infty}$. Assume  there exists $\gamma > 0$ such that for every $(\alpha,\beta) \in \N^n \times \N^n$, $|\alpha| \leq 1$, 
$|\beta | \leq 2$ we have
 $$| {\partial}_{x}^{\alpha}{\partial}_{\xi}^{\beta} \Big( l(x,\xi) - | \xi |^2 \Big) | 
 \leq C_{\alpha,\beta}  {\langle x \rangle}^{-\gamma- \mid \alpha \mid}{\langle 
\xi \rangle}^2.$$
Then there exists a compact set $K(J)\subset \RR^{2n}$ such that for each $\la \in J$, the set  $K(J) \cap \Sigma_\la$ contains all bounded trajectories of the Hamiltonian flow 
$\Phi_l^t$ on $\Sigma_\la$ and all critical points of $l(x,\xi)$   in 
$\Sigma_\la.$
\end{prop}

\begin{rem}
In a more restrictive setting, this result is stated and  exploited by the first author $\cite{kn:Br}$ to study the scattering phase for short-range trapping perturbations of the Laplacian.
\end{rem}



\begin{pf}
First, as $\displaystyle{\lim_{| \xi | \rightarrow \infty} l(x,\xi) = +
\infty}$, it is easy to see that there exists $M_{\mu_1} > 0$ such that
\begin{equation}\label{eq:estixi}
(x,\xi) \in \Sigma_\la, \; \forall \la \leq \mu_1 \; \Rightarrow \; 
|\xi| \leq M_{\mu_1}.
\end{equation}
 Now we will prove that the $x$-component of the 
bounded trajectories and the critical points are included in a compact set. This means that 
outside a fixed  compact set the $x$-component  of the Hamiltonian trajectories 
are non-bounded and  $\nabla_{x,\xi} l(x,\xi) \neq 0$.

Using Lemma \ref{lem1} it is sufficient to establish the existence of $R>0$ such that 
(\ref{eq:2.1}) holds with $\mu <\mu_0$. We write
\begin{equation}\label{calcul1}
\begin{array}{ccl}
 \{ l, x \cdot \partial_\xi l \} =  \partial_\xi l \cdot  \partial_x (x \cdot  \partial_\xi l)
- \partial_x l \cdot \partial_\xi (x \cdot \partial_\xi l) \\
  =  | \partial_\xi l |^2 + 
\sum_{j=1}^{n} \partial_{\xi_j} l \; x \cdot \partial_\xi \partial_{x_j} l  -
\partial_x l \cdot \partial_\xi (x \cdot \partial_\xi l). 
\end{array}
\end{equation}
Set $r(x,\xi)= l(x,\xi) - | \xi |^2$ and  observe that
$$ | \partial_\xi l |^2 = 4 | \xi |^2 +
 4 \xi  \cdot \partial_\xi r +| \partial_\xi r |^2,$$
which implies
\begin{equation}\label{eq:2.8}
 \frac14  | \partial_\xi l |^2 - l(x,\xi) = 
 \xi  \cdot \partial_\xi r + \frac14 | \partial_\xi r |^2 -
r(x,\xi).
\end{equation}
Moreover, since $\partial_{x_j} l = \partial_{x_j} r$, we deduce the following 
representation
 
$$\frac14  \{ l, x \cdot \partial_\xi l \} - l(x,\xi)$$
\[  =\xi  \cdot \partial_\xi r + \frac14 | \partial_\xi r |^2 -
r(x,\xi) + \frac14 \sum_{j=1}^{n} \partial_{\xi_j} l \;
 x \cdot \partial_\xi \partial_{x_j} r  -
\partial_x r \cdot \partial_\xi (x \cdot \partial_\xi l). \]
 From the  assumption on $r$ we obtain
$$| \frac14  \{ l, x \cdot \partial_\xi l \} - l(x,\xi) | \leq
C {\langle x \rangle}^{-\gamma}{\langle \xi \rangle}^4. $$
Thus we can find $R=R(\mu_0,\mu_1,\gamma)$ such that  
(\ref{eq:2.1}) holds. Moreover taking $R$ such that the RHS of (\ref{eq:2.8}) is bounded by $\mu<\mu_0$, we obtain $| \partial_\xi \l | \neq 0$ on $\l^{-1}(J) \cap \{ (x,\xi); |x| >R\}$. Consequently,  each $(x,\xi) \in \l^{-1}(J) \cap \{ (x,\xi); |x| >R\}$ is non-critical for $l$ and for each  $\la \in J$ every trajectory
with initial data $\mid x_0 \mid > R$ is non-bounded. 
\end{pf}


Now we pass to the analysis of the non-trapping energy levels in the sense 
given in \cite{kn:RT1}.
Recall that $\la \in \RR$ is a {\em non-trapping energy level} for
$l(x,\xi)$ if for every $R>0$ there exists $T_R > 0$ such that
for $(x_0,\xi_0) \in \Sigma_\la$, $|x_0| <  R$, we have
$$ \quad |x(t, x_0,\xi_0)| > R,\:\:\forall | t | > T_R. $$

\begin{lem}\label{lem2}
Let $\mu_1 > \mu_0 >  0$ and let 
 $\displaystyle{\lim_{| \xi | \rightarrow \infty} l(x,\xi) =
+\infty}$. If  there exists $\mu \in ]0,\mu_0 [$ such that
\begin{equation}\label{hyp-lem2}
  (x,\xi) \in
l^{-1}(]\mu_0,\mu_1[) \; \Rightarrow \; | \frac14 \{ l, x \cdot \partial_\xi l 
\} (x,\xi) - l(x,\xi) |  \leq \mu,
\end{equation}
then each $\la \in J$  is a non-trapping energy level for $l(x,\xi)$.
\end{lem}
\begin{pf}
Our aim is to prove that $x(\cdot,x_0,\xi_0)$ is non-bounded in both directions 
$t \rightarrow \pm \infty$ for any $(x_0,\xi_0) \in \Sigma_\la$. Following the 
proof of Lemma \ref{lem1}, we have
\begin{equation}
%\label{esti1}
| \Big(\{ l, x \cdot \partial_\xi l \} -4 l \Big)(x(t),\xi(t)) |  \leq 
4\mu,\:\:\forall t \in \RR, \quad
\end{equation}
(there is no restriction on $t$ by the assumption on $l$). Then  we deduce 
$$ |x(t)|^2 \geq |x_0|^2 + 2  x_0\cdot \partial_\xi l(x_0,\xi_0)t
+ 4 (\mu_0-\mu) t^2,\:\:\forall t \in {\RR}.$$
On the other hand the set $\{\xi_0:\:(x_0,\xi_0) \in
\Sigma_\la \;, \; \la \leq \mu_1\}$  is bounded. Consequently, for $R>0$ and $ |x_0 | < R$, we 
conclude that $ x_0\cdot \partial_\xi l(x_0,\xi_0)$ is
bounded, and there exists $C_R >0$ such that
$$ |x(t)|^2 \geq  4  |t| 
\Big(  ( \mu_0-\mu) |t| - C_R \Big).$$
Thus, as soon as $| t | >2  C_R + 2 \sqrt{C_R^2 + (\mu_0-\mu)R^2}$, we will 
have
$| x(t) | > R$, hence $\la$ is a non-trapping energy level.
\end{pf}

Introduce a function  $\chi \in C_0^\infty(\{x \in \rn ; \; |x| \leq 2 \})$,
 $0 \leq \chi \leq 1$ such that $\chi(x) = 1$ for $|x| \leq 1$ and set 
$\chi_R(x)=\chi(\frac{x}{R})$. We have the following 
 

\begin{prop}\label{prop2}
Let $\mu_1 > \mu_0 >  0$. 
%and let  $\displaystyle{\lim_{| \xi | \rightarrow \infty} l(x,\xi) =
%+\infty}$.
 Assume there exists $\gamma > 0$
such that for every $(\alpha,\beta) \in \N^n \times \N^n,\:\:|\alpha| \leq 
1,\:\: |\beta| \leq 2$ we have
 $$| {\partial}_{x}^{\alpha}{\partial}_{\xi}^{\beta} 
 \Big( l(x,\xi) - | \xi |^2 \Big) | 
 \leq C_{\alpha,\beta}  {\langle x \rangle}^{-\gamma- \mid \alpha \mid}{\langle 
\xi \rangle}^2.$$
Then there exists $R_c>0$ such that for $R \geq R_c$, each $\la \in J$
 is a non-trapping and non-critical energy level for the Hamiltonian 
$$\tilde{l}(x,\xi) ={l}(x,\xi) -\chi_R(x) \Big( l(x,\xi) - | \xi |^2 \Big).$$
\end{prop}

\begin{pf}
We will take  $R>0$ large enough in order to satisfy the conditions of Lemma \ref{lem2} for  
$$\tilde{l}(x,\xi) =l(x,\xi) -\chi(\frac{x}{R})\Big( l(x,\xi) - | \xi |^2 \Big).$$
First, we have
$$\tilde{l}(x,\xi) =| \xi |^2 +
\Big(1- \chi(\frac{x}{R}) \Big) \Big( l(x,\xi) - | \xi |^2 \Big),$$
and by the assumption on $l$, we obtain
$$\tilde {l}(x,\xi) \geq {\langle \xi \rangle}^2
 \Big(1-C_{0,0}  {\langle x \rangle}^{-\gamma} \Big) -1.$$
 Thus there exists $R_1$ such that for any $R \geq R_1$,
 $\displaystyle{\lim_{| \xi | \rightarrow \infty} \tl(x,\xi) =
+\infty}$ for $|x| \geq R$ and  this holds also for  $|x| \leq R$, where    
$\tl(x,\xi) = |\xi|^2.$
Now, as in the proof of the Proposition \ref{prop.1}, we have
\begin{eqnarray}\label{tcalcul1}
 \{ \tl, x \cdot \partial_\xi \tl \} = | \partial_\xi \tl |^2 + 
\sum_{j=1}^{n} \partial_{\xi_j} \tl \;
 x \cdot \partial_\xi \partial_{x_j} \tl  -
\partial_x \tl \cdot \partial_\xi (x \cdot \partial_\xi \tl). 
\end{eqnarray}
Next notice that for  $r(x,\xi)= \tl(x,\xi) - | \xi |^2$, the derivative of 
$\tl= | \xi |^2 + (1- \chi_R)r$ satisfies
$$ | \partial_\xi \tl |^2 = 4 | \xi |^2 +
 4 (1- \chi_R) \xi  \cdot \partial_\xi r +
 (1- \chi_R)^2| \partial_\xi r |^2$$
and we deduce 
\begin{equation}\label{tterm1}
 \frac14  | \partial_\xi \tl |^2 - \tl(x,\xi) = 
(1- \chi_R) \xi  \cdot \partial_\xi r + \frac14(1- \chi_R)^2 |
\partial_\xi r |^2 - (1- \chi_R)r(x,\xi).
\end{equation}

On the other hand, it is clear that
$$\partial_{x_j} \tl = \partial_{x_j}(1- \chi_R) r =
(1- \chi_R)\partial_{x_j} r - 
\frac{1}{R} \partial_{x_j} \chi(\frac{x}{R})r, $$
and, consequently,
\begin{eqnarray}
\sum_{j=1}^{n} \partial_{\xi_j} \tl \;
 x \cdot \partial_\xi \partial_{x_j} \tl  -
\partial_x \tl \cdot \partial_\xi (x \cdot \partial_\xi \tl) \nonumber \\
 = (1- \chi_R)\left( \sum_{j=1}^{n} \partial_{\xi_j} \tl \;
 x \cdot \partial_\xi \partial_{x_j} r  -
\partial_x r \cdot \partial_\xi (x \cdot \partial_\xi \tl)\right)
\nonumber\\
- \sum_{j=1}^{n} \frac{1}{R} \partial_{x_j} \chi(\frac{x}{R})
\left( \partial_{\xi_j} \tl \;
 x \cdot \partial_\xi  r  -
 r \cdot \partial_{\xi_j} (x \cdot \partial_\xi \tl)\right).\label{tterm2}
\end{eqnarray}

The function $(1- \chi_R)$ (resp. $\partial_{x} \chi_R$) is supported in
$\{ |x | \geq R \}$, (resp. $\{2R \geq  |x | \geq R \}$) and the assumptions on 
$r(x,\xi)$ combined with the relations (\ref{tcalcul1}), 
(\ref{tterm1}), (\ref{tterm2}) lead to 
$$ | \frac14  \{ \tl, x \cdot \partial_\xi \tl \} - \tl (x,\xi) | \leq
C   {\langle R \rangle}^{-\gamma}{\langle \xi \rangle}^4. $$
Finally, $\xi$ being bounded on the the energy level $\Sigma_{\lambda}$, we can find
$R_c=R_c(\mu_0,\mu_1,\gamma) \geq R_1$ such that for any $R \geq R_c$
the function $\tl$ satisfies (\ref{hyp-lem2}). Thus each $\la \in J$
is a non-trapping energy level for $\tl (x,\xi)$. Moreover, taking
$R_c$ such that the right hand side of  (\ref{tterm1}) is bounded by $\mu< \mu_0$, we obtain $| \partial_\xi \tl | \neq 0$ on $\tl^{-1}(J)$ and, consequently,  each $\la \in J$
is a non-critical energy level for $\tl (x,\xi)$. This completes the proof.    
\end{pf}

\begin{rem}
For a more complete analysis of the qualitative aspects of classical trajectories, we refer to the recent work of  A. Knauf $\cite{kn:K}$. For the Hamiltonian $l(x,\xi)=\frac12 | \xi |^2 + V(x)$, he obtained a topological condition for the existence of trapped trajectories.
\end{rem}

\section{Resolvent estimates}

We start with the following proposition reducing the proof of the resolvent estimates to that for the cut-off resolvent.

\begin{prop}\label{prop:bb}
Let $L(h)$ satisfy the "black box" assumptions $(\ref{ass1}) - (\ref{ass6})$ and let $J = ]\mu_0,\:\mu_1[ \subset \subset {\RR}^{+}$. Then there exists a self-adjoint differential  operator $\tL = \tL(h)$ on $L^2(\RR^n)$, satisfying the assumptions $(\ref{ass1}) - (\ref{ass6})$ such that each $\lambda \in J$ is a non-trapping and non-critical energy level for  $\tL(h)$ and 
$$ L(h) \psi  =  \tL(h) \psi$$ 
for any $\psi \in C^\infty({\RR}^n)$ supported away from $B(0,R_0)$.
% and such that $\psi(x)=1$  for  $x$ sufficiently large.
 Moreover, for every $s > \frac12$ there exists $C > 0,\: h_0 > 0$ and $\rho > R_0$ such that for $\chi \in C_0^\infty(\RR^n)$ with $\chi(x) = 1$ for $|x| \leq \rho$, we have
$$\| R(z) \|_{s,-s} \leq C h^{-2} \Big( 1 + \| \chi R(z) \chi  \|_{{\cal H} \to {\cal H}} \Big),$$
uniformly with respect to 
$z  \in {\cal B}_{\pm} = \{z \in \C;$ $ (\Re z,\pm \Im z) \in J\times ]0, 1]\}$ and $h \in (0,h_0]$.
\end{prop}
\begin{pf}
 Choose a function  $\Phi \in C_0^\infty(\{x \in \rn ; \; |x| \leq 2 \}; \RR)$,
 $0 \leq \Phi \leq 1$ such that $\Phi(x) = 1$ for $|x| \leq 1$ and set 
$\chi_R(x)=\Phi(\frac{x}{R})$ for $R > R_0$, where $R_0$ is the constant in the "black box" assumptions. Introduce the self-adjoint operator on $L^2(\RR^n)$
\begin{eqnarray}\label{deftL}
\tL & = & (1 - \chi_R) L \chi_R +  \chi_R L (1- \chi_R) +
  (1 - \chi_R)  L (1- \chi_R) + \chi_R (-h^2 \Delta ) \chi_R \\
& =&  Q + \chi_R (-h^2 \Delta - Q ) \chi_R,
\end{eqnarray}
where $Q$ is the formally self-adjoint differential operator given in the Introduction.

 According to Proposition \ref{prop2}, there exists $R_c >R_0$ such that if 
$R \geq R_c$, every  $\lambda \in J$ is a non-trapping and non-critical energy level in the sense mentioned in  Section 2 for the principal symbol $\tilde{l}_0(x,\xi)$  of $\tL(h)$ having the 
form
$$\tilde{l}_0(x,\xi) ={l}_0(x,\xi) -\chi^2_R(x) \Big( l_0(x,\xi) - | \xi |^2 
\Big)$$
with ${\DS{l_0(x,\xi) = \sum_{| \a | =2} a_\a (x) \xi^\a}}$.
Clearly, for  $\psi=0$  on the support of $\chi_R$, we have  $ \tL \psi = Q \psi = L \psi $.

 Let $\chi_1 \in C_0^\infty(\RR^n)$ be equal to $1$ on the support of $\chi_R$ and let $\chi_2 \in C_0^\infty(\RR^n)$ be equal to $1$ on the support of $\chi_1$. Using $\tL (1-\chi_1) = L (1-\chi_1)$, we have
\begin{equation}\label{reseq}
 R(z) = R(z) \chi_2 + (1-\chi_1)  \tR (z)  (1-\chi_2) +  R (z) [\tL,\chi_1]\tR(z)(1-\chi_2),
\end{equation}
where 
 $$ \tR(z) = (\tL(h) - z )^{-1},\:\: z \in 
{\C},\:\Im z \neq 0.$$
 Thus we obtain
\begin{equation}\label{reseq1}
\|  R(z) \|_{s,-s} \leq C \Big( \| R(z) \chi_2 \|_{s,-s} + \| \tR(z) \|_{s,-s}
 + \| R(z) \chi_2 \|_{s,-s} \|  [\tL,\chi_1]\tR(z) \|_{s,s} \Big),
\end{equation}
where here and below we denote by $C > 0$ different constants independent on $h$ and $z$.
Next, $[\tL,\chi_1]$ is a first order $h$-admissible differential operator with compactly supported coefficients, so there exists $C > 0$ such that 
$$ \|  [\tL,\chi_1]\tR(z) \|_{s,s} \leq C  \|  [\tL,\chi_1]\tR(z) \|_{s,-s}.$$

Moreover, the interval $J$ contains non-trapping energy levels for $\tL(h)$, and we can estimate $\|\tR(z)\|_{s,-s}$ and $\| [\tL,\chi_1]\tR(z) \|_{s,-s}$ by Lemma \ref{lem4} and Corollary \ref{corlem4} (see Section 5.1).  Consequently, there exist $C>0$ and $h_0>0$ such that for any $h \in (0,h_0]$ we have
\begin{equation}\label{reseq2}
\| R(z) \|_{s,-s} \leq C h^{-1}
\Big( 1 + \|  R(z) \chi_2 \|_{s,-s} \Big) .
\end{equation}

On the other hand, using the equality
 $$ \|   R(z) \chi_2 \|_{s,-s} = \| \chi_2 R({\overline z})   \|_{s,-s}$$
and introducing $\chi_2$ in the estimates  (\ref{reseq1}), (\ref{reseq2}), we obtain
\begin{equation}\label{reseq3}
\|  R(z) \chi_2 \|_{s,-s} \leq C  h^{-1}\Big( 1 + \| \chi_2 R(z) \chi_2 \|_{{\cal H} \to {\cal H}} \Big).
\end{equation}
Finally, (\ref{reseq2}) and (\ref{reseq3}) yield
$$
\| R(z) \|_{s,-s} \leq C h^{-2}\Big( 1 + \| \chi_2 R(z) \chi_2 \|_{{\cal H} \to {\cal H}} \Big).
$$
\end{pf}
\begin{prop}\label{prop:lapbb}
Assume that $L(h)$ satisfies the assumptions $(\ref{eq:1.7})-(\ref{eq:1.13})$. Then  for small $h_0 > 0$ and fixed $s > 1/2,\: h \in (0, h_0]$ the limits
$$R(\la \pm i0) = \lim_{\Im z \to 0, \: \: \pm \Im z > 0}  R(z), \quad \Re z=\la \in J,$$
exist in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$ uniformly with respect to $\lambda \in J.$
\end{prop}
\begin{pf}
Take a cut-off function $\chi \in C_0^{\infty}({\RR}^n)$ equal to 1 on $B(0,R_0)$. We shall prove that there exists $h_0 > 0$ such that for fixed $h \in (0,h_0]$ and $s > 1/2$,
we have 
$$\| R(z) - R(z') \|_{s,-s} \leq C_{h,s} \Big( | z- z'|^{\delta_1} +  | z- z'| + 
\|\chi \Big(  R(z) - R(z') \Big) \chi \|_{{\cal H} \to {\cal H}}\Big),$$
with $\delta_1 \in (0,1/2)$ uniformly with respect to  $z,z'  \in {\cal B}_{\pm}$.

As in the proof of the Proposition \ref{prop:bb}, we choose $R > R_c$ and introduce $\chi_R$ and an operator $\tL = \tL(h)$ with non-trapping energy levels in $J$. Let $\chi_1 \in C_0^\infty(\RR^n)$ be equal to $1$ on the support of $\chi_R$ and 
let $\chi_2 \in C_0^\infty(\RR^n)$ be equal to $1$ on the support of $\chi_1$.
Set $ R(z,z') = R(z) - R(z')$ and  $ \tR(z,z') = \tR(z)-\tR(z')$. 
According to (\ref{reseq}), we have
$$\begin{array}{ccl}
 R(z,z') =  R(z,z') \chi_2 + (1-\chi_1) \tR(z,z')  (1-\chi_2)\\
 + R(z,z') [\tL,\chi_1]\tR(z)(1-\chi_2) + R(z') [\tL,\chi_1]\tR(z,z') (1-\chi_2).
\end{array}$$
and we deduce
$$\begin{array}{ccl}
\| R(z,z') \|_{s,-s} \leq  C
\|  R(z,z') \chi_2 \|_{s,-s} \Big( 1+  \|  [\tL,\chi_1]\tR(z) \|_{s,s}\Big) \\
 + C \| \tR(z,z') \|_{s,-s}  +
C\|  R(z') \chi_2 \|_{s,-s}\;  \|  [\tL,\chi_1]\tR(z,z') \|_{s,s}.
\end{array}$$

For fixed $h$ Lemma \ref{lem6} in Section 5.2  implies that the cut-off resolvent $\chi(x)R(z)\chi(x)$ is analytic in $\overline{{\cal B}_{+}}$ and the norm $\|\chi_2 R(z) \chi_2\|_{{\cal H} \to {\cal H}}$ will be  uniformly bounded with respect to $z \in {\cal B}_{\pm}$. Applying Proposition \ref{prop:bb} 
for  $s>1/2$, we conclude that  $\|R(z') \chi_2 \|_{s,-s}$ is also uniformly bounded with respect to 
 $z  \in {\cal B}_{\pm}.$ On the other hand, since $J$ is non-trapping for $\tL$, we can apply  Corollary \ref{corlem4}. Thus, as in the proof of Proposition \ref{prop:bb},  for any $s > 1/2$ there exists $C_s$ such that 
$$\| [\tL,\chi_1] \tR(z) \|_{s,s} \leq C_s h^{-1}$$
 uniformly with respect to 
 $z  \in {\cal B}_{\pm}$. Moreover, for $s$ and $h$ fixed, there exists $C_{h,s}$ such that 
$$\| \tR(z,z') \|_{s,-s}+ \|[\tL,\chi_1] \tR(z,z') \|_{s,-s} \leq C_{h,s}\Big(| z - z' |+ | z - z' |^{\delta_1}\Big),$$
 uniformly with respect to 
 $z,z'  \in {\cal B}_{\pm}$. Consequently, there exists $C_{h,s} > 0$,  such that 
$$\begin{array}{ccl}
\| R(z,z') \|_{s,-s} & \leq & C_{h,s}  \Big( |z-z'|^{\delta_1}+ |z-z'|
+  \| R(z,z') \chi_2 \|_{s,-s}\Big).
\end{array}$$
Next,  using the equality $ \|   R(z,z') \chi_2 \|_{s,-s} = 
 \| \chi_2 R({\overline z},{\overline {z'}})  \|_{s,-s}$,  and introducing the cut-off function $\chi_2$ in the above estimates, we obtain
$$\| R(z,z') \chi_2\|_{s,-s}  \leq C_{h,s} \Big( |z-z'|^{\delta_1}+ |z-z'|
+  \|\chi_2  R(z,z') \chi_2 \|_{s,-s}\Big).$$
This implies immediately 
$$\|  R(z,z')\|_{s,-s} \leq C_{h,s}  \Big( |z-z'|^{\delta_1}+ |z-z'|
+  \| \chi_2 R(z,z') \chi_2\|_{s,-s} \Big).$$
From Lemma 6 we deduce the analyticity of $\chi_2R(z,z')\chi_2$ so for fixed $h$ and $s > 1/2$, the limits
$$R(\la \pm i0)= \lim_{\Im z \to 0, \: \: \pm \Im z > 0}  R(z), \quad \Re z=\la \in J$$
exist in ${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s})$.
\end{pf}

For the proof of our principal result we need the following

\begin{lem} \label{lem3} Let $L(h)$ satisfy the assumptions $(\ref{eq:1.7})-(\ref{eq:1.13})$ and assume the estimate $(\ref{eq:1.5})$ fulfilled. Then for every $\chi \in C_0^\infty(\RR^n)$ equal to 1 on $B(0,R_0)$ there exist $C>0$ such that for $(\lambda, \tau) \in J\times ]0,1]$ we have
\begin{equation}
 \|\chi(x) R(\lambda \pm i\tau)\chi(x) \|_{{\cal H} \to {\cal H}} \leq Ce^{Ch^{-p}},\:0 < h \leq h_0. 
\label{eq:3.2}
\end{equation}
\end{lem}
\begin{pf}
 Set $S_{\theta} = \{ z \in {\C}: -2\theta < \arg z \leq \pi/4 \},$ where $0 < \theta \leq \theta_0$ is fixed and $\theta_0 \in ]0, \pi[$ was introduced in the assumption (\ref{eq:1.11}) making  possible a complex scaling of $L(h).$ Choose $\mu/2 > \delta > 0$ so that (\ref{eq:1.5}) holds for $\lambda \in [\mu_0 - \delta, \mu_1 + \delta]$ and consider the compact set
\[ \omd = \{ z \in S_{\theta}: \mu_0 - \delta \leq \Re z \leq \mu_1 + \delta \}\,. \]
Then for every $m \in {\N}$,  Lemma 1 in \cite{kn:TZ} yields
\begin{equation} \label{eq:estre}
 \|\chi(L(h) - z)^{-1}\chi \| \leq C(\omd, m) 
\exp\left(C(\omd,m)h^{-k}\right),\: 0 < h \leq h_0(\omd,m)
\end{equation}
for 
$$z \in \Omega_\delta \setminus \bigcup_{z_j \in {\rm Res}\: L(h) \cap \omd} D(z_j : 
h^m)$$
 with $k \geq  n$ and $D(z_j : h^m) = \{ z \in \C: |z - z_j| \leq h^m \}.$  For the number of the scattering resonances we have the estimate
 (see \cite{kn:SZ}, \cite{kn:Sj1})
\[ \sharp \{ z_j: z_j \in {\rm Res}\: L(h) \cap \omd \} \leq C_1(\omd) 
h^{-n^{\sharp}} \]
with $n^{\sharp} \geq n.$ Next we fix an integer $m$ satisfying  $2m \geq n^{\sharp} + 1$ and take $0 < h_1(\omd,m) \leq h_0(\omd,m)$ small enough to arrange
\[4 C_1(\omd) h^{2m - n^{\sharp}} < \delta \, \]
for $0 < h \leq h_1(\omd, m).$
This implies immediately the existence of $\gamma_1(h),\:\gamma_2(h)$ so that
\[ \mu_0/2 \leq \mu_0 - \delta < \gamma_1(h) < \mu_0 - \frac{\delta}{2},\:\: \mu_1 +\frac{\delta}{2} < \gamma_2(h) < \mu_1 + \delta\,, \]
and
\[ \gamma_j (h) + i \tau \notin \bigcup_{z_j \in {\rm Res}\: L(h) \cap 
\omd} D(z_j : h^m),\:\:j =1,2, \: \tau \geq 0\,.\]
Thus for $k > n$ fixed the estimate (\ref{eq:estre}) holds for $z = \gamma_j(h) +i\tau,\:\:j =1,2,\:\: \tau \geq 0.$  Set $\epsilon = \frac{\delta}{10}$ and introduce the domain
\[ \Pi_{\delta} = \{ z = \lambda + i \tau: \mu_0- \delta \leq \lambda \leq 
\mu_1 + \delta,\:\: 0 \leq \tau \leq h^M \} \subset \omd \,, \]
where $2M > k.$ By a modification of the argument in \cite{kn:TZ}, Lemma 2, we may construct a function $f(z,h)$ holomorphic in $\Pi_{\epsilon}$ with the properties:\\

(i) $|f(z,h)| \leq e$ in $\Pi_{\delta}$,\\

(ii) $|f(z,h)| \geq 1/2$ in $[\mu_0 - \epsilon, \mu_1 + \epsilon] + i[0, h^M],$\\

(iii) for $z \in \Pi_{\delta} \cap \{z: \Re z \leq \mu_0 -4\epsilon,\: {\rm or}\: \Re z \geq \mu_1 + 4\epsilon\}$ we have

$$|f(z,h)| \leq C_{\epsilon}\exp\Bigl(-\frac{\epsilon^2}{2h^{2M}}\Bigr), \: 0 \leq h \leq h(\epsilon).$$

To construct $f(z,h)$, take a function $\psi(x) \in C^{\infty}_0({\Bbb R}^n),\: 0 \leq \psi(x) \leq 1,$ such that
$$\psi(x) = \left \{\aligned  0,\:\: x \notin [\mu_0 - 3\epsilon, \mu_1 + 3\epsilon],\\
1, \:\:  x \in [\mu_0 -2\epsilon, \mu_1 + 2\epsilon] \endaligned \right.$$
and put
$$f(z,h) = (\pi \alpha^2)^{-1/2} \int \exp \Bigl( -\frac{(x-z)^2}{\alpha^2}\Bigr)\psi(x) dx,\:\alpha = h^M.$$
The check of (i) is trivial. For (ii) write $z = u +iv$ and take $u \in [\mu_0 -\epsilon, \mu_1 + \epsilon]$. Then
$$|f(z,h) - 1| \leq (\pi \alpha^2)^{-1/2} \exp\Bigl(\frac{v^2}{\alpha^2}\Bigr) \int \exp\Bigl(- \frac{(x-u)^2}{\alpha^2}\Bigr) |\psi(x) -1 |dx$$
$$\leq \pi^{-1/2}e \int e^{-y^2} |\psi(\alpha y + u) - 1| dy \leq 2\pi^{-1/2}e \int_{{\Bbb R} \setminus B(0, \frac{\epsilon}{\alpha})} e^{-y^2} dy \leq \frac{1}{2},$$
for $0 < h \leq h(\epsilon)$ since $\frac{\epsilon}{\alpha} \longrightarrow \infty$ as $h \to 0.$ Finally, to deal with (iii), assume for example that $u  \leq \mu_0 - 4\epsilon..$ We have
$$|f(z,h)| \leq (\pi \alpha^2)^{-1/2} \exp\Bigl(\frac{v^2}{\alpha^2}\Bigr) \int \exp\Bigl(-\frac{(x-u)^2}{\alpha^2}\Bigr) \psi(x)dx$$
$$\leq A\pi^{-1/2} h^{-M}\exp \Bigl(-\frac{\epsilon^2}{\alpha^2}\Bigr) \leq C_{\epsilon}\exp \Bigl(-\frac{\epsilon^2}{2\alpha^2}\Bigr), \: 0 \leq h \leq h(\epsilon).$$
Our assumptions imply that the cut-off resolvent
$$F(z,h) = \chi(x) R(z) \chi(x)$$
has no poles on the real axis (see Appendix 5.2). Thus the function $G(z,h) = f(z,h)F(z,h)$ is holomorphic in $\Pi_{\epsilon}$. For $\Im z = 0$ we have $\|G(z,h)\| \leq eC e^{\frac{C}{h^p}}$, while for $\Im z = h^M$
we get
$$\|G(z,h)\| \leq \frac{Ce}{h^M} \leq e^{\frac{C_1}{h}}.$$
For $z \in \Pi_{\delta},\: \Re z = \gamma_j(h),\:\:j =1,2$,  combining the estimates for $|f(z,h)|$ and $\|F(z,h)\|,$ we obtain
$$\|G(z,h)\| \leq C_{\epsilon}e^{\frac{C}{h^k}} e^{-\frac{\epsilon^2}{h^{2M}}} \leq C_2,\:\: 0 <h \leq h(\epsilon).$$ 
 As in \cite{kn:TZ}, by applying the maximum principle for $G(z,h)$ in $\Pi_{\epsilon}$, we deduce the estimate (\ref{eq:3.2}) for $\lambda \in J,\:\: 0 < \tau \leq h^M.$ The result for $h^M \leq \tau \leq 1$ follows trivially from the inequality $\|\chi(x) R(\lambda + i\tau) \chi(x) \|_{{\cal H} \to {\cal H}} \leq \frac{C}{\tau}$ and the proof is complete. 
\end{pf}

From Proposition \ref{prop:bb} and Lemma \ref{lem3} we deduce Theorem \ref{theo}.

\section{Microlocal resolvent estimates}

In this section we obtain microlocal resolvent estimates in the trapping case. They are inspired by the  microlocal estimates in Lemma 2.3 of \cite{kn:RT2} corresponding to the non-trapping case. In \cite{kn:RT2} (see also \cite{kn:R3} for more general Hamiltonians), Robert-Tamura constructed an approximation of the propagator $\exp(ith^{-1}L)$ by a Fourier integral operator which is  uniform with respect to the time $t$ and $h$. It is convenient to localize in the outgoing or incoming regions of the phase space
\begin{equation}\label{defsortant}
\Gamma^\pm (R,d,\sigma_\pm) = \{ (x,\xi) \in \RR^{2n}\; : \; |x| > R, \; d^{-1} < |\xi | < d, \; \frac{\langle x.\xi \rangle}{|x||\xi|} \;  ^>_< \; \sigma_\pm  \},
\end{equation}
where $d>1$, $-1 < \sigma_\pm < 1$ and $R \gg 1$ is large enough.

Consider a symbol  $ \omega_\pm(x,\xi) \in C^\infty (\RR^{2n})$ such that
 supp $\omega_\pm \subset \Gamma^\pm (R,d,\sigma_\pm)$ and 
\begin{equation}\label{defomegapm}
| \partial_x^\alpha \partial_\xi^\beta  \omega_\pm (x,\xi) | \leq
 C_{\alpha,\beta,L} \langle x \rangle^{-|\alpha |}
 \langle \xi \rangle^{- L } 
\end{equation}
for any $L \gg 1$. Then we have the following

\begin{prop}\label{prop5} 
 Let $J  \subset \subset \RR^+$ and $\psi \in C_b^\infty({\RR}^n)$ be  supported away from $B(0,R_0)$. Then there exists $\chi \in C_0^\infty({\RR}^n),\: \chi = 1$ on $B(0,R_0)$ such that  
for any $(\la,\tau) \in J \times [0,1]$ and $h \in (0,h_0] $ the following assertions hold:

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

ii) If $\sigma_+ > \sigma_-$, then for any $s \gg 1 $,  there exist $C > 0$ and $h_0 > 0$ such that for
 $h \in (0,h_0]$ we have
\begin{equation}\label{eq:mltrap.es2}
 \| \omega_\mp(x,hD_x) \psi R(\la\pm i \tau)\psi \omega_\pm(x,hD_x) \|_{-s,s} \leq 
C h^{-2} ( 1 + \| \chi  R(\la\pm i \tau) \chi \|_{{\cal H} \to {\cal H}}).
\end{equation}

\end{prop}

\begin {pf}
As in the proof of the Proposition \ref{prop:bb}, there exists a self-adjoint differential operator $\tL = \tL(h)$ on $L^2(\RR^n)$, such that each $\lambda \in J$ is a  non-trapping energy level for $\tL$ and the equality (3.3) holds. Then $i)$ is a consequence of Proposition \ref{prop:bb} and $i)$ of Lemma \ref{lem5}. We use also the resolvent estimate 
(as in Corollary \ref{corlem4}) and the fact that $ \langle x \rangle^{-s} \omega_\pm(x,hD_x)\langle x \rangle^{s}$ is a bounded pseudo-differential operator for any $s \in \RR$ (see \cite{kn:DS},  \cite{kn:R1}).

According to (\ref{reseq}) we have
$$\begin{array}{ccl}
 \omega_\mp(x,hD_x) \psi R(z) \psi \omega_\pm(x,hD_x)
           = \omega_\mp(x,hD_x) \psi R(z) \chi_2
           \Big(1 +[\tL,\chi_1]\tR(z)(1-\chi_2)\Big) \psi \omega_\pm(x,hD_x)\\
 + \omega_\mp(x,hD_x)   (1-\chi_1) \tR(z)  (1-\chi_2) \omega_\pm(x,hD_x).
\end{array}$$
However, $\omega_\mp(x,hD_x) \psi R(z) \chi_2 = 
         \Big(  \chi_2  R(\overline{z}) \psi \omega_\mp(x,hD_x)^*  \Big)^*$, then using (\ref{reseq}), the Proposition \ref{prop:bb} and Lemma \ref{lem5} once more we obtain that the operator
$\omega_\mp(x,hD_x) \psi R(\la\pm i \tau) \chi_2 $ exists in
 ${\cal L}(L^2_{-s},L^2_{s})$ for
 $(\la,\tau) \in J \times [0,1]$ and following estimates hold
$$ \| \omega_\mp(x,hD_x) \psi R(\la\pm i \tau) \chi_2 \|_{-s,s} \leq 
C h^{-1} ( 1 + \| \chi_2  R(\la\pm i \tau) \chi_2 \|_{{\cal H} \to {\cal H}}).$$
Consequently, according to Lemma \ref{lem5},  $\omega_\mp(x,hD_x) \psi R(\la\pm i \tau)\psi \omega_\pm(x,hD_x)$ exists in ${\cal L}(L^2_{-s},L^2_{s})$ for
 $(\la,\tau) \in J \times [0,1]$ and these operators satisfy the claimed estimate (\ref{eq:mltrap.es2}).
\end{pf} 



\section{Appendix  }\label{sec.appendix}

\subsection{Resolvent estimates in the  non-trapping case}

In this section we recall some resolvent  estimates in a non-trapping energy interval for a self-adjoint differential operator $\tL = \tL(h)$ defined in a dense domain in $L^2({\RR}^n).$ We say that $J \subset \subset \RR^+$ is non-trapping for  $\tL(h)$ if every $\la \in J$ is a {\em non-trapping energy level} for the principal symbol of $\tL(h)$  in the sense of Section 2. Denote by $\| . \|_{s,s'}$ the norm of ${\cal L}(L^{2,s}, L^{2,s'})$, where $L^{2,s}$ is the weight  space $L^2(\RR^n, \langle x \rangle^s dx)$.

\begin{lem}\label{lem4} 
Let $\tL = \tL(h)$ be a self-adjoint differential operator having  the form $(\ref{ass4})$ and  satisfying $(\ref{ass5})-(\ref{ass6})$. Let $J  \subset \subset \RR^+$ be  an open non-trapping interval for   $ \tL(h)$ and let $s > 1/2.$ Then

i) for $h\in (0,h_0]$ fixed, there exists
$C_{h,s} > 0$ such that for any $z,z'  \in {\cal B}_{\pm}$ we have
 $$\| (\tL-z)^{-1}-(\tL-z')^{-1} \|_{s,-s} \leq C_{h,s} | z - z' |^{\delta_1}, \quad  \delta_1 = (s-1/2)/(2s-1/2).$$

ii) for $\la \in J$, the limits 
$$(\tL-\la \mp i0)^{-1}:= \lim_{\Im z \to 0, \: \: \pm \Im z > 0}  (\tL-z)^{-1}, \quad \Re z=\la,$$
exist in   ${\cal L}(L^{2,s}, L^{2,-s})$, and  there exists $C > 0$ and $h_0 > 0$ such that for
$z \in {\cal B}_{\pm}$ and $h \in (0,h_0]$ we have
\begin{equation}\label{eq:trap.es}
 \|(\tL-z)^{-1}  \|_{s,-s} \leq C h^{-1}.
\end{equation}
\end{lem}

\begin{pf}
This result can be obtained following the Mourre theory. First, there exists a {\em conjugate operator} $A = A(h)$ to $\tL$ satisfying the following Mourre inequality:
 $\forall \Phi \in C_0^\infty(J)$, $\exists \gamma_0>0$ such that for all $h \in (0,h_0]$, we have
\begin{equation}\label{eq:mourreesti}
\Phi(\tL) i^{-1} [\tL,A ] \Phi(\tL) \geq \gamma_0 h \Phi(\tL)^2.
\end{equation}
For the construction of $A(h)$ we refer to \cite{kn:R3}, where a more general setup has been considered.
The operator $A(h)$ has the form of  a $h$-pseudodifferential operator with symbol
$$F(x,\xi) = K(x,\xi) + \frac{2 x.\xi}{\langle \xi \rangle^2},$$
where $K \in C^\infty(\RR^n \times \RR^n)$ is compactly supported with respect to $x$ and $K$ is uniformly bounded with respect to $(x,\xi)$.

According to Mourre type results,  we deduce  that $\tL$ has no eigenvalues in $J$ for $h$ small enough (see \cite{kn:M}, \cite{kn:CFKS}) and for  $s>1/2$ the following assertions hold (see \cite{kn:PSS}, \cite{kn:JMP}  and \cite{kn:HN},\cite{kn:R2}, \cite{kn:R3} for operators depending on $h$):

i') for $h\in (0,h_0]$ fixed,  there exists
$C_h>0$ such that for $z,z'  \in {\cal B}_{\pm} = \{z \in \C;$ $ (\Re z,\pm \Im z) \in J\times ]0, 1]\}$ we have
 $$\| \langle A \rangle^{-s}\Big( (\tL-z)^{-1}-(\tL-z')^{-1}\Big) \langle A \rangle^{-s} \|_{L^2 \to L^2}
 \leq C_h | z - z' |^{\delta_1}
% , \quad  \delta_1 = (s-1/2)/(2s-1/2)
.$$

ii') there exist $C>0$ and $h_0>0$ such that for $z  \in {\cal B}_{\pm}$ and $h \in (0,h_0]$ we have
\begin{equation}\label{eq:trap.es'}
 \|\langle A \rangle^{-s} (\tL-z)^{-1} \langle A \rangle^{-s}  \|_{L^2 \to L^2} \leq C h^{-1}, \: <A> = (1 + A^2)^{1/2} 
\end{equation}
Let us note that the operators  $ \langle x \rangle^{-s} \langle A \rangle^{s},\: \langle A \rangle^{s} \langle x \rangle^{-s} $, $s \geq 0$,  are uniformly  bounded with respect to $h$ in ${\cal L}(L^2)$. This follows from the functional calculus developed by Dimassi-Sj\"ostrand (see Sections 7 and 8 in \cite{kn:DS}) or by using a complexe interpolation as in Lemma 8.2 of \cite{kn:PSS}. Then, writing 
$$\langle x \rangle^{-s} (\tL-z)^{-1} \langle x \rangle^{-s}=
\langle x \rangle^{-s} \langle A \rangle^{s}\; \langle A \rangle^{-s} (\tL-z)^{-1} \langle A \rangle^{-s} \;  \langle A \rangle^{s} \langle x \rangle^{-s},$$
 we complete the proof.
\end{pf}

\begin{cor}\label{corlem4}  Let $P$ be a $p$-order $h$-admissible differential operator, $p\leq 2$ and let $s > 1/2.$ Then, under the assumptions of the lemma \ref{lem4} the following assertions hold:

i'') for $h\in (0,h_0]$ fixed, there exists
$C_{h,s} > 0$ such that for $z,z'  \in {\cal B}_{\pm}$ we have
 $$\| P \Big( (\tL-z)^{-1}-(\tL-z')^{-1} \Big) \|_{s,-s} \leq C_{h,s}\Big(| z - z' |+ | z - z' |^{\delta_1}\Big)
% \quad  \delta_1 = (s-1/2)/(2s-1/2)
.$$

ii'')  there exists $C > 0$ and $h_0 > 0$ such that for
$z \in {\cal B}_{\pm}$ and $h \in (0,h_0]$ we have
\begin{equation}\label{eq:trap.es}
 \| P (\tL-z)^{-1}  \|_{s,-s} \leq C h^{-1}.
\end{equation}
\end{cor}
\begin{pf} 
The ellipticity of $\tL$ and  the functional calculus developed by Dimassi-Sj\"ostrand (see Sections 7 and 8 in  \cite{kn:DS}) imply that  the operator 
$\langle x \rangle^{-s}P(\tL-i)^{-1}\langle x \rangle^{s}$ is uniformly bounded in ${\cal L}(L^2)$ with respect to $h$, for $s \geq 0$.
Then ii'') is a direct consequence of ii) and the resolvent equation
$$(\tL-z)^{-1} =  (\tL-i)^{-1} + (z-i)(\tL-i)^{-1}(\tL-z)^{-1}.$$

In the same way  i'') is a direct consequence of i) and the relation
$$(\tL-z)^{-1}-(\tL-z')^{-1} = (z-z') (\tL-i)^{-1}(\tL-z)^{-1} +
 (z'-i)(\tL-i)^{-1}\Big((\tL-z)^{-1}-(\tL-z')^{-1}\Big).$$
\end{pf}


Finally, we formulate some microlocal resolvent estimates which generalize Lemma 2.3 of \cite{kn:RT2}.
\begin{lem}\label{lem5} 
Let $\tL = \tL(h)$ be a self-adjoint differential operator having  the form $(\ref{ass4})$ and  satisfying $(\ref{ass5})-(\ref{ass6})$. Let $J  \subset \subset \RR^+$ be  an open non-trapping interval for   $ \tL(h)$ and let
$ \omega_\pm(x,\xi)$ be a symbol supported in an outgoing (incoming) region as in  Section 4. Then for any $(\la,\tau) \in J \times [0,1]$ the following assertions hold:

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

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

\end{lem}

The proof of this lemma is analogous to the proof of Lemma 2.3 of \cite{kn:RT2} exploiting the constructions of long time approximations in section 4 of \cite{kn:R3} for a large class of long range perturbations of the Laplacian. The essential point is the construction of the phase $\varphi_\pm$ (see Proposition 4.1 of \cite{kn:R3}).

\subsection{Boundary values of the cut-off resolvent $\chi R(z) \chi$}

In this section we will show that for fixed $h \in (0, h_0]$ the limits
$$\chi(x) R(\lambda \pm i0) \chi(x) = \lim _{\Im z \to 0, \: \: \pm \Im z > 0} \chi(x) R(z) \chi(x),\: \lambda \in J$$
exist in ${\cal L}({\cal H},\: {\cal H}),$ provided $\chi \in C_0^{\infty}({\RR}^n)$ is equal to 1 on $B(0, R_0)$ and $\chi(x) = 0$ for $|x| \geq R_0 + a_0,\: a_0 > 0.$ To do this it is sufficient to show that there are no resonances $z_j$ in the interval $]\mu_0 -\delta, \mu_1 + \delta[.$

\begin{lem}\label{lem6} Assume $(\ref{eq:1.7})-(\ref{eq:1.13})$  fulfilled. Then for $0 < h \leq h_0$ we have
$${\rm Res} \: 
L(h) \cap ]\mu_0 -\delta, \mu_1 + \delta[ = \emptyset.$$
\end{lem}
\begin{pf}
The proof can be obtained from the results  in \cite{kn:SZ}, \cite{kn:Sj1}. Here we will only give indications of how to extract it from there. We assume $h$ fixed and let $z_0 \in J$ be a resonance for $L(h)$. Denote by ${\cal H}_{R_0 +a_0}$ the space of elements of ${\cal H}$ that vanish outside $B(0, R_0 + a_0).$ Then by using a complex scaling argument and Lemma 3.5 in \cite{kn:SZ}, we conclude that for $v \in {\cal H}_{R_0 + a_0}$
and $z$ in a neighborhood of $z_0$ we have
\begin{equation}
(L - z)^{-1}v = \sum_{j =1}^N (z_0 - z)^{-j}v_{-j} + G(z) \label{eq:4.1}
\end{equation}
with $G(z)$ holomorphic in a neighborhood of $z_0$ and $N$ independent of $v$. 
 Consider the spectral projection
$$\pi_{\theta, z_0} = \frac{1}{2\pi i} \int_{\gamma(z_0)} (z - L_{\theta})^{-1}dz,$$
$L_{\theta}$ being the operator obtained by complex dilation defined in a dense set ${\cal D}_{\theta}$ of the space ${\cal H}_{\theta} = {\cal H} \oplus L^2(\Gamma_{\theta} \setminus B(0, R_0))$ with $0 <\theta \leq \theta_0$ (see \cite{kn:Sj1}), while $\gamma(z_0):\: [0,2\pi] \ni s \longrightarrow z_0 + \epsilon e^{is}$ with $ \epsilon > 0$ small enough. Let $F_{\theta, z_0}$ be the image $\pi_{\theta, z_0}({\cal H}_{\theta})$ and let
$$\pi_{0,z_0} = \frac{1}{2\pi i} \int_{\gamma(z_0)} (z - L)^{-1}dz: \: {\cal H}_{{\rm comp}} \longrightarrow {\cal D}_{{\rm loc}}.$$
 Then exploiting the equality (\ref{reseq}) and the fact that the resolvent $\tilde R(z)$ is holomorphic in a small neighborhood of $z_0$, as in Proposition 3.6 in \cite{kn:SZ}, we conclude  that there exists a bijection between the spaces $F_{\theta, z_0}$ and $ \pi_{0,z_0}({\cal H}_{R_0 + a_0}).$ Since $z_0$ is a resonance, the space $F_{\theta, z_0}$ is not trivial and we can find $v \in {\cal H}_{R_0 + a_0}$ so that (\ref{eq:4.1}) holds with $\sum_{j =1}^N (z_0 - z)^{-j}v_{-j} \neq 0$. For simplicity assume that $v_{-j} = 0$ for $j = 2,..., N$ and let $v_{-1} \neq 0.$ Take a cut-off function $\Phi(x) \in C_0^{\infty}({\RR}^n)$ equal to 1 on $B(0, R_0)$ so that $(\Phi, v_{-1})_{{\cal H}} \neq 0.$
Therefore,
$$\lim_{\epsilon \to 0} (\Phi, i\epsilon(L - z_0 + i\epsilon)^{-1}v_{-1})_{\cal H} = (\Phi, P_{\{z_0\}}v_{-1})_{\cal H} \neq 0,$$
$P_{\{z_0\}}$ being the spectral projector of $L$ related to $z_0$, and we get $z_0 \in \sigma_{pp}(L)$ which contradicts (\ref{eq:1.13}). The case of a multiple pole can be treated in the same way.
\end{pf}  

{\footnotesize 


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