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semiclassical, resonances, Breit-Wigner, critical energy level, Schroedinger operator
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\title[ reference operator and semiclassical resonances]
{Eigenvalues of the reference operator and \\ semiclassical resonances}
\author[V. Bruneau, V. Petkov]{Vincent Bruneau and  Vesselin Petkov}

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\begin{document}
\begin{abstract}
We prove that the estimate of the number of the  eigenvalues in intervals $[\lambda - \delta, \lambda + \delta],\:\:\: 0 < \frac{h}{C} \leq \delta \leq C$, of the reference operator $L^{\#}(h)$ related to a self-adjoint operator $L(h)$ is equivalent to the estimate of the integral over $[\lambda - \delta, \lambda + \delta]$ of the sum of harmonic measures associated to the resonances of $L(h)$ lying in a complex neighborhood $\Omega$ of $\lambda > 0$ and the number of the positive eigenvalues of $L(h)$ in $[\lambda - \delta, \lambda + \delta]$. We apply this result to obtain a Breit-Wigner approximation of the derivative of the spectral shift function near critical energy levels.
\end{abstract}

\maketitle
\section{Introduction}

This paper is devoted to the analysis of the connection between the distribution of the semi-classical resonances $z_j(h)$ of a Schr\"odinger type  operator $L = L(h),\: 0 < h \leq h_0$, and the  behavior of the counting function
$$N(L^{\#}(h), [\alpha, \beta]) = \# \{ \mu \in \RR: \: \mu \in {\rm sp}_{pp} L^{\#}(h),\: \alpha \leq \mu \leq \beta \}$$
 of the so called reference operator $L^{\#}(h)$ related to $L(h)$ (see Section 2). Under the general "black box" assumptions (2.1)-(2.7) we may define the semi-classical resonances $w \in \overline{\C}_{-}$ by complex scaling \cite{SZ}, \cite{Sj3}. Let ${\rm Res}\:L$ be the set of resonances of $L$. Then for every relatively compact open domain $\Omega \subset\subset \{ z \in \C: \Re z > 0 \}$ the estimate
\[ N(L^{\#}(h), [-\lambda, \lambda]) = {\mathcal O}\Bigl(\Bigl(\frac{\lambda}{h^2}\Bigr)^{\ns/2}\Bigr),\:\: \ns \geq n,\: \lambda \geq 1 \, \]
implies the bound
\begin{equation}\label{eq:1.0}
 \# \{ w \in {\rm Res}\: L \cap \Omega \} \leq C(\Omega) h^{-\ns},\: 0 < h \leq h_0 \, 
\end{equation}
(see \cite{SZ}, \cite{Sj3} and, \cite{V} for the classical case). Given an interval $[E_0, E_1],\: 0 < E_0 < E_1$, such that every $\lambda \in [E_0, E_1]$ is a non-critical energy level for the principal symbol of $L$, a more precise result holds and recently J. F. Bony \cite{Bo1} proved (see also \cite{Bo3} for similar results concerning critical energy levels) that the condition
\begin{equation} \label{eq:1.1}
 N(L^{\#}(h), [\lambda - \delta, \lambda + \delta]) \leq C\delta h^{-\ns}
\end{equation}
for all $\lambda \in [E_0, E_1],\:\: 0 <\frac{h}{C_1} \leq \delta \leq C_1$, implies
\begin{equation} \label{eq:1.2}
\# \{ w \in \C:\: w \in {\rm Res}\: L,\: |w - \lambda| \leq \delta\} \leq C \delta  h^{-\ns}
\end{equation}
for $\lambda \in [E_0, E_1]$ and $0 < \frac{h}{B} \leq \delta \leq B$ (see also \cite{PZ3} for the case of compact perturbations). Moreover, under the assumption (\ref{eq:1.1}) we can obtain a Weyl type asymptotics and a Breit-Wigner approximation  of the spectral shift function $\xi(\lambda, h)$ (see \cite{BrPe2} for more details). Finally, there exists a close relation between the behavior of $\xi(\lambda, h)$ and that of $N(\lambda) = N(L^{\#}, ]-\infty, \lambda])$. This relation has been studied by S. Nakamura  \cite{Na} in the case of short range perturbations of the Schr\"odinger operator $L = -h^2\Delta + V(x)$ and by the authors \cite{BrPe1}, \cite{BrPe2} in the setup of "black box" long range scattering.

It is natural to expect that some information on the distribution of the resonances in a complex neighborhood $\Omega$ of $[E_0, E_1]$ will imply (\ref{eq:1.1}) and via the results in \cite{BrPe2} the asymptotics of $\xi(\lambda, h)$. To our best knowledge it seems that there are no such results in the literature. The purpose of this paper is to show that (\ref{eq:1.1}) is equivalent to a similar condition involving the sum of the {\em harmonic measures}
$$\omega_{\C_{-}}(w, J) = \int_{J} \frac{|\Im w|}{\pi |t - w|^2} dt,\:\: J \subset \RR= \partial \C_{-}$$ related to the resonances $w, \: \Im w < 0,$ lying in  a {\em complex} neighborhood of $[E_0, E_1]$. We refer to  \cite{M}, \cite{GMR}, \cite{PZ1}, \cite{PZ3}, \cite{BSj} for the results concerning the Breit-Wigner approximations and the harmonic measures $\omega_{\C_{-}}(w,.)$. More precisely, the condition (\ref{eq:1.1}) is equivalent to the same condition for the function
\[ M_\Omega(\lambda) = \sum_{w \in {\rm Res}\:L,\: w \in \Omega, \atop \Im w \neq 0} \omega_{\C_{-}}(w, ] -\infty, \lambda]) + \#\{ \mu \in ]-\infty, \lambda] \cap \Omega: \: \mu \in {\rm sp}_{pp}\: L \} \, \] 
(Theorem 1) which may be considered as an analogue of the counting function of eigenvalues. Notice that the positive eigenvalues $\mu \in {\rm sp}_{pp}\: L$ coincide with the resonances $w \in \RR^{+}$ and the function $M_\Omega(\lambda)$ is completely determined by the resonances in $\Omega.$ In particular, from Theorem 1 we obtain a new proof of the implication (1.1) $\Rightarrow$ (1.2) established by J. F. Bony \cite{Bo1} (Corollary 2).\\

 We can define the spectral shift function $\xi(\lambda, h)$ for $L(h)$ and $\tilde{L}(h)$, where $\tilde{L}(h)$ is an intermediate operator defined in Proposition 1. On the other hand, for short range perturbations the spectral shift function $\xi(\lambda, h)= \xi(L, \tilde{L})$ can be defined for the pair of operators $L, L_0 = -h^2 \Delta$. The importance of Theorem 1 is that we have the equivalence of three conditions $i)-iii)$ and exploiting $i)$ and $iii)$ we obtain after minor modifications of the arguments of Section 6 in \cite{BrPe2} a Breit-Wigner approximation of the derivative of the spectral shift function $\xi(\lambda, h)$ (Theorem 2). \\

In the case when we have no "black box" and $L(h)$ is a h-pseudodifferential self-adjoint operator in $L^2(\RR^n)$ with principal symbol $l_0(x, \xi)$  we should stress that the assumptions (2.1)-(2.9) do not concern the eventual {\it critical points} of $l_0(x,\xi)$ lying in $\{(x,\xi) \in \RR^n:\: |x| \geq R_0 > 0 \}.$ Thus we can cover the case of critical energy levels choosing an appropriate {\it weight factor} $r(h)$ with $\inf_{h \in ]0, h_0]} r(h) > 0.$ For non-degenerate critical points the results of J. F. Bony \cite{Bo3} imply the assumption $i)$ of Theorem 1 with suitable $r(h)$ and combining this with Theorem 2 we obtain some  applications for non-degenerate critical points (see Section 4). There are only few results concerning a Breit-Wigner approximation of $\frac{\partial \xi}{\partial \lambda}(\lambda, h)$ near critical energy levels (see \cite{GMR}, \cite{FR2}, \cite{FR3}). In this direction Theorem 2 and Corollary 3 present some general results. In Section 5 we compare our results in the one dimensional case with those obtained recently by Fujii\'e and Ramond \cite{FR2}, \cite{FR3}. In the paper we denote by $C$  positive constants, independent on $h$, which may change from line to line.\\

{\bf Acknowledgments.} The authors would like to thank Jean-Fran\c{c}ois Bony for many useful discussions concerning the application of Propositions 1 and 4. We would like to thank also Mouez Dimassi for his remarks and comments concerning the paper.\\

\section{Assumptions and results}


We start by the abstract ``black box'' scattering assumptions introduced
in  \cite{SZ}, \cite{Sj3} and \cite{Sj5}. The operator $L(h) = L, \:\: 0 < h \leq h_0,$ is defined in a domain ${\cal D} \subset
{\cal H}$ of a complex Hilbert space ${\cal H}$ with an orthogonal
decomposition
$${\cal H} = {\cal H}_{R_0} \oplus L^2({\RR}^n \setminus
B(0,R_0)),\:B(0,R_0) = \{x \in {\RR}^n: |x| \leq R_0 \},\:\: R_0 > 0,
\:\:n \geq 1. $$
Below $h > 0$ is a small parameter. We suppose that ${\cal D}$ satisfies
\begin{equation} \label{eq:2.1}
{\Bbbone }_{{\RR}^n \setminus B(0,R_0)}{\cal D} = H^2({\RR}^n \setminus
B(0,R_0)),
\end{equation}
uniformly with respect to $h$ in the sense of \cite{Sj3}. 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}  \label{eq:2.2}
{\Bbbone}_{B(0,R_0)}(L+i)^{-1} \hbox{is compact}
\end{equation}
and
\begin{equation}\label{eq:2.3}
 (L u)\vert_{{\RR}^n \setminus  \overline{B(0,R_0)}} = Q\Bigl(
u\vert_{{\RR}^n \setminus \overline{B(0,R_0)}}\Bigr),
\end{equation}
where $Q$ is a formally self-adjoint differential operator
\begin{equation}\label{eq:2.4}
 Q u = \sum_{| \nu | \leq 2} a_{\nu} (x;h) (hD_x)^\nu u
\end{equation}
with $ a_{\nu} (x;h)=  a_{\nu} (x)$ independent of $h$ for $| \nu | =
2$
 and $a_{\nu} \in C_b^\infty(\RR^n)$ uniformly bounded with respect to
$h$.

We assume also the following properties:

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

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

There exist $\theta_0 \in ]0, \frac{\pi}{2}[,\:\epsilon > 0$ and $R_1 >
R_0$ so that the coefficients $a_{\nu}(x;h)$ of $Q$ can be extended
holomorphically in $x$ to
\begin{equation} \label{2.8}
\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 $R > R_0,\:T_{\tilde{R}} = ({\RR}/\tilde{R}{\Z})^n,\: \tilde{R} > 2R.$ Set
$${\cal H}^{\#} = {\cal H}_{R_0} \oplus L^2(T \setminus B(0, R_0))$$
and consider a differential operator
$$Q^{\#} = \sum_{|\nu| \leq 2} a_{\nu}^{\#}(x;h)(hD)^{\nu}$$
on $T$ with $a_{\nu}^{\#}(x;h) = a_{\nu}(x;h)$ for $|x| \leq R$
satisfying (2.3), (2.4), (2.5) with $\RR^n$ replaced by $T_{\tilde{R}}$. Consider a
self-adjoint operator
$L^{\#}: {\cal H}^{\#} \longrightarrow {\cal H}^{\#}$
 defined by
$$L^{\#}u = L\varphi u +Q^{\#}(1-\varphi)u, \: u \in {\cal
D}^{\#},$$
with domain
$${\cal D}^{\#} = \{u \in {\cal H}^{\#}: \: \varphi  u \in {\cal D},
\: (1-\varphi)u \in H^2 \},$$
where  $\varphi \in C^{\infty}_0(B(0,R); [0,1])$ is equal to 1 near
$\overline{B(0,R_0)}.$ Denote by $N(L^{\#}, [-\lambda, \lambda])$ the number of eigenvalues of
$L^{\#}$ in the interval $[-\lambda, \lambda]$. Then we assume that
\begin{equation}\label{eq:2.9}
N(L^{\#}, [-\lambda, \lambda]) = {\mathcal
O}(\Bigl(\frac{\lambda}{h^2}\Bigr)^{n_{}^{\#}/2}),\: n^{\#} \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(h) \subset [-C, \infty[, \:\: 
\end{equation}
where sp $(L)$ denotes the spectrum of $L.$ \\

Following \cite{Sj3}, \cite{Sj5}, we define the resonances $w \in
\overline{\C}_{-}$ by the complex scaling method as the eigenvalues of the
complex scaling operator $L_{ \theta}$. Denote by
${\rm{Res}}\: L(h),$ the set of resonances.  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 $Q.$
Since $L(h)$ tends to $-h^2 \Delta$, for $\lambda > 0$ fixed, the set of the critical points of the  Hamiltonian $l(x,\xi)$ in $\Sigma_\la$ is compact. Then taking $R_0$ sufficiently large, we can suppose that $\lambda$ is non critical for $Q$ and we can construct $Q^{\#}$ so that  $\lambda$ is non critical for $Q^{\#}$, too.

 We fix $E_1> E_0 > 0$ and introduce an intermediate operator $\tL(h)$ having no resonances in a complex neighborhood of $[E_0,E_1]$ and each $\lambda \in [E_0, E_1]$ is a non critical energy level for $\tilde{L}$ (see Proposition 1). Moreover, the estimate (3.1) makes possible to introduce the spectral shift function $\xi(\lambda, h)$ for the pair $(L(h),\tL(h))$ (see Section 3) and, as  in \cite{BrPe2}, we define 
$$\xi(\lambda, h) = \lim_{\epsilon \to 0,\: \epsilon > 0} \xi(\lambda + \epsilon, h).$$
 Our main result is the following.

\begin{thm}\label{thm1}
Assume that $L$ satisfies the assumptions $(2.1) - (2.9)$ and  suppose  that
each $\lambda \in[E_0,E_1]$ is a  non-critical energy level for $Q$ and $Q^{\#}$.
Then for any real valued function $r(h)$, $h \in ]0,h_0]$ such that 
$\inf_{h \in ]0,h_0]}  r(h)>0$, the following assertions are equivalent:

i) There exist positive constants $B_1, \:C_1$,\: $\epsilon_1$,\: $h_1$  such that for
any $\lambda \in[E_0 - \epsilon_1, E_1 + \epsilon_1]$, $h \in ]0,h_1]$ and $h/B_1\leq \delta \leq
B_1$ we have
$$ \# \{ \mu  \in \RR: \mu  \in {\hbox{sp}}(
L^{\#}(h))\cap[\lambda-\delta,\lambda+\delta] \}  \leq C_1 \delta r(h)
h^{-\ns}.$$

ii) For every complex relatively compact neighborhood $\Omega \subset \{ z \in \C: \: \Re z > 0 \}$ of $[E_0,E_1]$, independent on $h$ there exist positive constants  $B_2,\:C_2$,\: $\epsilon_2,\: \: h_2$, depending on $\Omega$,  such that for
any $\lambda \in[E_0 - \epsilon_2, E_1 + \epsilon_2]$, $h \in ]0,h_2]$ and $h/B_2 \leq \delta \leq
B_2$ we have
$$\sum_{w \in {\rm{Res}}\: L(h)\cap \Omega, \atop \Im w \neq 0}
\omega_{\C_{-}} ( w, [\lambda - \delta, \lambda + \delta]) +  \# \{ \mu  \in \RR: \mu  \in {\mbox{sp}_{pp}}(L(h))\cap[\lambda-\delta,\lambda+\delta] \}
 \leq C_2 \delta r(h)  h^{-\ns}.$$

iii) There exist positive constants $B_3,\:C_3, \: \epsilon_3,\: h_3$ such that for any $\lambda \in [E_0 - \epsilon_3, E_1 + \epsilon_3]$, $ h \in ]0, h_3]$ and $\frac{h}{B_3} \leq \delta \leq B_3$ we have
 $$|\xi(\lambda + \delta,h) - \xi(\lambda - \delta,h)| \leq C_3\delta r(h) h^{-\ns}.$$

\end{thm}

{\bf Remarks.  1.} In the assertion $ii)$ it is sufficient to establish the bound for one complex neighborhood $\Omega$ of $[E_0, E_1]$ with constants depending on $\Omega$. Then for every other complex neighborhood $\Omega_1 \supset \Omega$ the sum of the harmonic measures related to the resonances lying in $\Omega_1 \setminus \Omega$ is easily estimated by ${\mathcal O}(h^{-\ns})$ by using the bound of the function counting the resonances. On the other hand, it is clear that if every $\lambda \in [E_0, E_1]$ is a non-critical energy level for $Q$ the same is true for a small neighborhood of $[E_0, E_1]$.\\

{\bf 2.} The assumption $iii)$ does not depend on the choice of the operator $\tilde{L}$. This follows from the equivalence of $ii)$ and $iii)$, as well as from the observation that if we have two operators $\tilde{L_i},\:i = 1,2,$ with the properties of Proposition 1, then $\xi(L, \tilde{L_1})- \xi(L, \tilde{L_2}) = \xi(\tilde{L_2}, \tilde{L_1})$ and for $\xi(\tilde{L_2}, \tilde{L_1})$ we obtain easily $iii)$ since the operators $\tilde{L_i}, \: i=1,2$ have non-trapping energy levels in $[E_0, E_1].$ In the case of short range perturbations we can take $\tilde{L}(h) = L_0 = -h^2\Delta$ and the estimate (3.1) (see Section 3) holds for the coefficients of $L$ and $L_0$. Thus we can define the spectral shift function related to $L$ and $L_0$.\\

{\bf 3.} If $L$ is h-pseudodifferential operator in $L^2(\RR^n)$, then the assumptions (2.2)-(2.9) don't exclude the existence of critical points $(x,\xi)$ of the principal symbol of $L$ lying in $B(0, R_0)$. Thus Theorem 1 covers the case of critical energy levels and we will present some applications in Sections 4 and 5.\\

The assertion $ii)$ is independent of  the choice of a reference operator $L^{\#}(h)$ so we obtain the following.

\begin{cor}
Let $L_1^{\#}$, $L_2^{\#}$ be two reference operators for $L$ satisfying the conditions $(2.1) -(2.9)$ and suppose that each $\lambda \in[E_0,E_1]$ is a  non-critical energy level for $Q$, $Q_1^{\#}$ and $Q_2^{\#}$. Then $L_1^{\#}$ satisfies $i)$ if and only if $L_2^{\#}$ satisfies $i).$
\end{cor}


 From the implication $i) \Rightarrow ii) $ we deduce an upper bound for the counting function of resonances in small domains. In fact, as in the proof of Lemma 6.1  in \cite{PZ3},  for $0<y<\delta$ and  $|x-\lambda | < \delta$ we have
$$\int_{\lambda-2\delta}^{\lambda+2\delta} \frac{y}{(x-\mu)^2 + y^2} d\mu
\geq \int_{-\delta / y}^{\delta / y} \frac{1}{1 + r^2} dr \geq \pi/2.$$
Thus we deduce 
$$\sum_{w \in {\rm{Res}}\: L(h)\cap \Omega, \atop \Im w \neq 0}
\omega_{\C_{-}} ( w, [\lambda - 2\delta, \lambda +2 \delta]) +  \# \{ \mu  \in \RR: \mu  \in {\hbox{sp}_{pp}}(
L(h))\cap[\lambda-\delta,\lambda+\delta] \}$$
$$ \geq \frac12
\# \{ z \in {\rm{Res}}\: L(h), \:  \Im z \neq 0,\: |z - \lambda| \leq
\delta \} + \# \{ z \in {\rm{Res}}\: L(h) \cap [\lambda - \delta, \lambda+
\delta]\}$$
and we obtain the following.


\begin{cor}
The assumption $i)$ of Theorem $1$ implies  the existence of positive constants  $C,\: B \:$, $b,\: \:h_0$  such that
for any $\lambda \in[E_0- b, E_1+ b]$, $h \in ]0,h_0]$ and $h/B \leq \delta
\leq B$ we have
$$\# \{ z \in \C: z \in {\rm{Res}}\: L(h),\: |z - \lambda| \leq \delta \}
\leq C \delta r(h) h^{-\ns}.$$
\end{cor}

In the non-critical case we can take $r(h) = 1$ and this corollary gives a new proof of a recent result of J. F. Bony \cite{Bo1} (see also \cite{PZ3} for the case of compact perturbations). In the critical case the statement of Corollary 2 implies the results of J. F. Bony \cite{Bo3} for differential operators $L$ and dimension $n \geq 2$ (see Section 4). The results in \cite{Bo3} in the case $n = 1$ for h-pseudodifferential operators $L$ are more precise since the upper bounds $r(h)$ is replaced by $r(\delta)$.\\

We may obtain a Breit-Wigner approximation for the derivative of the spectral shift function $\xi(\lambda, h)$ defined before Theorem 1. In fact, by using the assertions $i)$ and $iii)$ of Theorem 1 and repeating with minor modifications the arguments of Section 6 in \cite{BrPe2}, we obtain the following generalization of Corollary 1 in \cite{BrPe2}.

\begin{thm} Assume that $L$ satisfies the assumptions $(2.1)-(2.9)$ and suppose that $[E_0, E_1]$ is a non-critical energy level for $Q$ and $Q^{\#}.$ Let $r(h),\: h\in ]0,h_0],$ be a real valued function such that $\inf_{h \in ]0, h_0]} r(h) > 0.$ Then if one of the assumptions $i) - iii)$ of Theorem $1$ holds, then for each $E \in ]E_0, E_1[$ there exist constants $C_2 > C_1 > 0, \: h_0' > 0$ so that for $|\lambda - E| \leq C_1h,\: h \in ]0, h_0'],$ we have
\begin{equation}
\frac{\partial \xi}{\partial \lambda} (\lambda, h) =-\frac{1}{\pi}\sum_{|E- w| \leq C_2 h,\: \atop w \in {\rm Res}\: L(h)} \frac{\Im w}{|\lambda - w|^2} + \sum_{|E - w| \leq C_1h,\: \atop w\: \in {\rm sp}_{pp}\:\: L(h) \ } \delta(\lambda - w)  +  \co\Bigl(r(h) h^{-n^{\#}}\Bigr).
\end{equation}
\end{thm} 



\section{Proof of Theorem 1}

The proof of Theorem 1 is based on a representation formula for the spectral shift function (see Theorem 1 in \cite{BrPe2}). 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
$$ \quad |x(t, x_0,\xi_0)| > R,\:\:\forall | t | > T(R). $$

We introduce an intermediate operator exploiting
the following result of J. F. Bony (see also \cite{Sj4}).

\begin{prop}[\cite{Bo2}]\label{prop:bony} Let $L$  satisfy the
assumptions of Section $2$ and  let $0 < E_0 < E_1$. Then there exists a differential
operator 
$$\tL(h) =  \sum_{| \nu | \leq 2} \tilde{a}_{\nu} (x;h) (hD_x)^\nu,$$
satisfying assumptions $(2.4)-(2.7)$ and the following properties:\\

$(a)$  There exists  $\overline{n} > n$ such that we have
\begin{equation}\label{eq:2.7}
 \Bigl| a_{\nu} (x;h) - \tilde{a}_{\nu} (x;h) \Bigl|
 \leq {\mathcal O}(1)  {\langle x \rangle}^{-\overline{n}}
\end{equation}
 for $x \in \Gamma$ introduced in $(2.7)$ ,uniformly with respect to $h$,\\

$(b)$ The operator $\tL$ has no resonances in a complex neighborhood
$\Omega_0$ of $[E_0,E_1]$ and $\Omega_0$  is independent on $h$,\\

$(c)$ There exists an open interval $I_0 \subset ]0, + \infty[$ containing $[E_0,E_1]$,  such that each $\mu \in I_0$ is non-trapping  energy level for $\tL$.
\end{prop}

The property $(a)$ guarantees that for every  $f \in C_0^\infty(\RR)$ the
operator $f(L) - f(\tilde{L})$ is ``trace class near infinity''. More precisely, if we denote $L_2=L$ and $L_1=\tilde{L}$, 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{Sj3}, \cite{Sj5}, by the equality
\[ \trb \Bigl(f(L_2) - f(L_1)\Bigr) = [\tr (\chi f(L_j)\chi + \chi f(L_j)
(1 - \chi) + (1 - \chi)  f(L_j)\chi) ]_{j=1}^2 \, \]
\[  + \tr [(1-\chi)f(L_j)(1- \chi)]_{j=1}^2 \,,\]
where we use the notation $[a_j]_{j=1}^2 = a_2 - a_1.$  The spectral shift function
$\xi(\lambda, h)$ is a distribution in ${\mathcal D}'(\RR)$ such that
\[ <\xi'(\lambda, h), f(\lambda) >_{{\mathcal D}'(\RR), {\mathcal D}(\RR)}
= \trb \Bigl(f(L(h)) -  f(\tilde{L}(h))\Bigr), f(\lambda) \in C_0^{\infty}
(\RR)  \,. \]


 Applying Theorem 1 of \cite {BrPe2} in the domain $\Omega_0$, we deduce that there
exists a
function $g_+(z,h)$, holomorphic in $\Omega_0$, such that for $\mu \in I_0 =
W_0 \cap \RR$, $W_0 \subset \subset \Omega_0$ we have
\begin{equation} \label{eq:5.3}
 \xi'(\mu, h) = \frac{1}{\pi} \Im g_+(\mu, h) + \sum_{w \in {\rm Res}\:L
\cap \Omega_0, \atop \Im w \neq 0} \frac{-\Im w}{\pi |\mu - w|^2} +
\sum_{w  \in {\rm Res}\:L \cap I_0} \delta (\mu - 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 $h \in ]0, h_0].$

 Property (c) shows that $\tilde{L}$ has no critical energy levels $\lambda \in [E_0, E_1]$. In the following, we fix  an open interval $I_0 \subset \RR^+\cap \Omega_0$ containing $[E_0,E_1]$ so that
each $\lambda \in I_0$ is a non-critical energy level for the operators $Q$, $\tL$ and
we introduce open intervals $I_2  \subset \subset I_1 \subset \subset
I_0$ containing $[E_0,E_1]$. We suppose that $|\lambda - z | \geq \eta_0 > 0$ for $\lambda \in I_1,\: z \notin \Omega_0.$

Consider a function $\theta \in C_0^\infty(]-\epsilon_4,\epsilon_4[)$, $\theta(0)=1$,
$\theta(-t)=\theta(t)$ such that the Fourier transform of $\theta$
satisfies ${\hat \theta}(\lambda) \geq 0$ on $\RR$. Assume that there
exist $\epsilon_0>0$, $\delta_0 >0$ so that ${\hat \theta}(\la) \geq
\delta_0 > 0$ for $\mid \la \mid \leq \epsilon_0$ and introduce the function
$$\Bigl({\cal F}_h^{-1} \theta\Bigr)(\lambda) = (2\pi h )^{-1} \int e^{it
\lambda/h}\theta (t) dt = (2\pi h )^{-1}{\hat \theta}(- h ^{-1}\lambda).$$
The next lemma, established in \cite{BrPe2}, yields a connection between the derivatives
of the functions $M_{\varphi,\Omega_0}$ and $N^{\#}_{\varphi}.$

\begin{prop}[\cite{BrPe2}]\label{prop1}
Let $\varphi \in C^\infty_0(I_1; \RR^+)$ and let 
$$N_{\varphi}^{\#}(\mu)= \tr \Big( \varphi(L^\#){ \bf
1}_{]-C^{\#}, \mu]}(L^\#) \Big),$$
$$G_{\varphi}(\mu) = \frac{1}{\pi} \int_{]-\infty,\mu]}\Im g_+(\nu, h) {\varphi}(\nu) d \nu, $$
\begin{equation} \label{defM}
M_{\varphi, \Omega_0}(\mu) =  \sum_{w \in {\rm Res}\:L \cap \Omega_0, \atop \Im
w \neq 0} \int_{]-\infty,\mu]}\frac{-\Im w}{\pi |\nu -
w|^2}{\varphi}(\nu) d \nu + \sum_{w  \in {\rm Res}\:L \cap ]- \infty,
\mu]} \varphi (w).
\end{equation}
 Then there exists
$\omega_{\varphi} \in C_0^0(\RR)$ such that
\begin{equation}\label{eq:5.5}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * M_{\varphi, \Omega_0}
)(\mu)=\frac{d}{d \lambda}({\cal F}_h^{-1} \theta
*N_{\varphi}^{\#})(\mu)-
 G_{\varphi}'(\mu)+  \omega_{\varphi}(\mu) h^{-n} +  {\cal O}( h^{1-n^{\#}}),
\end{equation}
where ${\mathcal O}(h^{1-\ns})$ is uniform with respect to $\mu \in \RR.$
\end{prop}

 For our argument we need a Tauberian theorem  involving the factor $r(h)$. A such theorem can be obtained by modifying the proof of the Tauberian theorem in \cite{PR}, \cite{R1}. For the sake of completeness we present a version of the Tauberian theorem related to a real valued function $r(h)$, $h \in ]0,h_0]$ such that $\inf_{h \in ]0,h_0]}  r(h)>0$.

\begin{thm}
Let $\sigma(\lambda, h)$, $h\in ]0,h_0],$ be  a set of real valued increasing functions. Assume that there exist $a,b,c \in \RR$ and $d \in \N$ independent of $h$ so that\\
$\sigma(\lambda, h) = 0$ for $\lambda \leq a$, $\sigma(\lambda, h) = c$ for $\lambda \geq b$,\\
$\sigma(\lambda, h) = {\mathcal O}(h^{-d})$ uniformly with respect to $\lambda \in \RR$ and $h\in ]0,h_0]$.

Then the following assertions are equivalents:

i) There exists positive constant $C_1$ such that for any $\lambda \in \RR$, $h\in ]0,h_0]$ we have
$$\Bigl|\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * \sigma(., h) )(\lambda)\Bigr|\leq C_1r(h) h^{-d}.$$

ii)  There exists positive constant $C_2$ such that for any $\lambda \in \RR$, $h\in ]0,h_0]$ and $\eta \geq 0$ we have
$$\sigma(\lambda+\eta, h)- \sigma(\lambda-\eta, h) \leq C_2 (\eta +h)r(h) h^{-d}.$$
Moreover, $ii)$ implies

iii) There exists positive constant $C_3$ such that for any $\lambda \in \RR$, $h\in ]0,h_0]$ we have

$$|\sigma( \lambda, h) - ({\cal F}_h^{-1} \theta * \sigma(., h) )(\lambda)|
\leq C_3 r(h) h^{1-d}.$$

\end{thm}

\begin{pf}
 We assume $i)$ and we are going to prove $ii)$. It is clear that we can assume that $\eta$ is bounded. Since $\sigma(\lambda, h)$ is constant outside $[a,b]$, it is sufficient to prove $ii)$ for $\delta$ bounded and  $\lambda$ in a compact set. As in the proof of the Tauberian theorem
(see \cite{R1} or \cite{PR} for more details), the inequality
$\hat{\theta}(\epsilon)\geq \delta_0$ for $|\epsilon | \leq \epsilon_0$, implies
$$\sigma(\mu + \epsilon h, h )-\sigma(\mu -
\epsilon h, h)   \leq \frac{2 \pi h}{\delta_0} \frac{d}{d \lambda}({\cal
F}_h^{-1} \theta *\sigma(., h ))(\mu),\:\: |\epsilon| \leq \epsilon_0, \:\: \forall \mu \in \RR.$$
Exploiting $i)$ for $\eta \leq \epsilon_0 h $, we have
$$\sigma( \mu+\eta, h)- \sigma(\mu-\eta, h) \leq Cr(h)  h^{1-d}.$$
On the other hand, for $\eta \geq \epsilon_0 h $ applying  the above inequality
for $\mu= \lambda + \eta - (2j+1) \epsilon_0 h$ at the right hand side of
$$\sigma( \mu+\eta, h)- \sigma(\mu-\eta, h)
 \leq \sum_{j=0}^{\left[\eta/ \epsilon_0 h \right]}
\Big(\sigma(\lambda + \eta - 2j \epsilon_0 h, h)-
\sigma(\lambda + \eta - 2(j+1) \epsilon_0 h, h) \Big),$$
we obtain
$$\sigma( \mu+\eta, h)- \sigma(\mu-\eta, h) \leq C (\frac{\eta}{\epsilon_0 h} +1)r(h) h^{1-d}$$
and this implies $ii)$.

Now let us assume $ii)$ fulfilled. Then
$$ \frac{d}{d \lambda}({\cal F}_h^{-1} \theta *\sigma( ., h ))(\lambda) = \frac{1}{2 \pi h} \int_\RR {\hat \theta}(\frac{\mu - \lambda}{h}) d \sigma (\mu, h)$$
and this implies
$$\Bigl|\frac{d}{d \lambda}({\cal F}_h^{-1} \theta *\sigma(.,h ))(\lambda)\Bigr| \leq
\frac{C}{2 \pi h}(\sigma( \lambda+h, h)- \sigma(\lambda-h, h)) $$
$$+ \frac{1}{2 \pi h} \sum_{k =1}^{\infty} 
 \int_{kh \leq |\mu - \lambda| < (k + 1)h} {\hat \theta}(\frac{\mu - \lambda}{h}) d \sigma (\mu, h).$$
Combining this with the estimate $|{\hat \theta(\nu)}| \leq C(1 + |\nu|)^{-2}$ and applying $ii)$,  we deduce
$$|\frac{d}{d \lambda}({\cal
F}_h^{-1} \theta *\sigma(., h ))(\lambda)| \leq 
Cr(h)h^{-d} + \frac{C}{2 \pi h}  \sum_{k =1}^{\infty} \frac{1}{k^2}r(h)h^{1-d},$$
which yields $i)$. Here  we have used that 
$$|\sigma( \lambda \pm (k+1)h, h)- \sigma(\lambda \pm kh, h) | \leq Cr(h) h^{1-d}.$$

The proof of $ii) \Rightarrow iii)$ follows from the relation
$$\sigma (\lambda, h) - ({\cal F}_h^{-1} \theta * \sigma(., h) )(\lambda)=
\frac{1}{2\pi} \int_\RR (\sigma( \lambda, h) - \sigma(\lambda + \nu h, h)) {\hat \theta}(\nu) d\nu,$$
where we have used that $\theta(0) = 1.$
\end{pf}
{\bf Remark.} In the applications below we will use the estimate
$$\sigma(\lambda + \eta) - \sigma(\lambda - \eta) \leq C_2\eta r(h)h^{-d}$$
for $\frac{h}{B} \leq \eta \leq B$. Since $\sigma(\lambda, h)$ is increasing, this is equivalent to the assumption $ii)$ of Theorem 3 for $\eta \geq 0.$\\

{\em Proof of Theorem $1$.} We assume $i)$ and we are going to prove $ii)$.
Let $[E_0,E_1] \subset I_2  \subset \subset I_1 \subset \subset I_0$ be as above and let each $\mu \in I_0$ be a non critical energy level for $Q,\:\: Q^{\#}$. Choosing $\varphi \in C_0^\infty(I_1; \RR^{+})$, 
$\varphi =1$ on $I_2$ as above, we will show that for $\frac{h}{B_2}\leq \delta \leq B_2$ we have
\begin{equation} \label{estm}
M_{\varphi, \Omega_0}(\lambda + \delta) -
M_{\varphi, \Omega_0}(\lambda - \delta) = {\cal O}_{\varphi}(\delta) r(h)  h^{-\ns}, \quad \lambda \in [E_0 - \epsilon_2, E_1 + \epsilon_2].
\end{equation}
 According to Theorem 3, to obtain (\ref{estm}) it  is sufficient to show that
\begin{equation} \label{eq:n}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * M_{\varphi,\Omega_0})(\mu)
= {\cal O}(r(h)h^{-n^\#})
\end{equation}
uniformly with respect to $\mu \in \RR$.
Exploiting the assumption $i)$, Theorem 3 and the Remark above, we get the estimate
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N_{\varphi}^{\#})(\mu)
={\cal O}(r(h)h^{-n^\#})$$
uniformly with respect to $\mu \in \RR.$
 This implies (\ref{eq:n}) using the representation of Proposition 2.

Now let $\Omega \subset\subset \{ z \in \C: \: \Re z > 0 \}$ be a relatively compact open neighborhood of $[E_0, E_1]$ in $\{ z \in \C: \: \Re z > 0 \}$ containing $\Omega_0$. Then taking into account (1.1), we obtain
$$\sum_{w \in {\rm Res}\:L \cap (\Omega \setminus \Omega_0),\: \atop \Im w \neq 0} \int_{\mu - \delta}^{\mu + \delta} \frac{-\Im w}{\pi |\nu - w|^2} \varphi(\nu) d\nu \leq \frac{C \delta}{\eta_0^2} h^{-\ns},\: \forall \delta > 0.$$
Consequently, the function
\[ M_{\varphi, \: \Omega}(\mu) =  \sum_{w \in {\rm Res}\:L \cap \Omega, \atop \Im
w \neq 0} \int_{]-\infty,\mu]}\frac{-\Im w}{\pi |\nu -
w|^2}{\varphi}(\nu) d \nu + \sum_{w  \in {\rm Res}\:L \cap ]- \infty,
\mu]} \varphi (w) \, \]
satisfies the estimate
\[ M_{\varphi,\: \Omega} (\lambda + \delta) - M_{\varphi,\: \Omega} (\lambda - \delta) \leq C_{\varphi}\delta r(h)h^{-\ns},\:\: \lambda \in [E_0 - 2\epsilon_2, E_1 + 2\epsilon_2], \: \frac{h}{C_1} \leq \delta \leq C_1 \, \]
with a sufficiently small $\epsilon_2 > 0.$ Moreover, the constant $C_{\varphi} > 0$ depends on $\eta_0,\: \Omega$ and $C_{\varphi}$ is independent on $h$. Since $\varphi$ is equal to $1$ on a neighborhood of $[E_0,E_1]$, we deduce $ii)$.\\


The proof of the implication $ii) \Rightarrow i)$ is very similar. As in the analysis of the function  $M_{\varphi, \Omega_0}(\mu)$, the estimate of $N^\#_{\varphi}(\lambda + \delta) - N^\#_{\varphi}(\lambda - \delta)$ is a consequence of the bound
$$\frac{d}{d \lambda}({\cal F}_h^{-1} \theta * N_{\varphi}^{\#})(\mu) =
 {\cal O}_{\varphi}(r(h)h^{-n^\#}), \quad \mu \in \RR.$$
%For  $\mu \notin I_0$, this derivative is easily estimated by ${\cal O}(h^\infty)$ since $\hat{\theta} \in {\mathcal S}(\RR)$. For $\mu \in I_0$,
According to  the representation given in Proposition \ref{prop1}, we have to prove that
\begin{equation}\label{mphi*}
\frac{d}{d \lambda}({\cal F}_h^{-1} \theta *M_{\varphi, \Omega_0})(\mu) =
 {\cal O}_{\varphi}(r(h)h^{-n^\#}).
\end{equation}

First, using the notations introduced above, notice that
 \[ M_{\varphi,\Omega_0} (\lambda + \delta) - M_{\varphi,\Omega_0}(\lambda - \delta) \leq 
 M_{\varphi, \: \Omega} (\lambda + \delta) - M_{\varphi, \: \Omega}(\lambda - \delta) + C_{\varphi}\delta h^{-\ns} \,. \]
Secondly, it is clear that
$$ M_{\varphi, \: \Omega} (\lambda + \delta) - M_{\varphi, \: \Omega}(\lambda - \delta)  \leq  \sum_{w \in {\rm Res}\: L(h) \cap \Omega,\: \atop \Im w \neq 0}\omega_{\C_{-}} (w, [\lambda - \delta, \lambda + \delta])$$
$$ +  \# \{ \mu  \in \RR: \mu  \in {\mbox{sp}_{pp}}(L(h))\cap[\lambda-\delta,\lambda+\delta] \} \leq C\delta r(h) h^{-\ns},$$
where in the second inequality we have used $ii)$. Combining these estimates, we get
\begin{equation} \label{estvar}
M_{\varphi, \Omega_0}(\lambda + \delta) - M_{\varphi, \Omega_0}(\lambda - \delta) \leq C_{\varphi}\delta r(h) h^{-\ns}, \: \frac{h}{C_2} \leq \delta.
\end{equation}
It is clear that for $0 \leq \delta \leq \frac{h}{C_2}$ the last estimate remains true if we replace $C_{\varphi}$ by $\frac{C_{\varphi}}{C_2}(\delta + h).$ Thus we can apply Theorem 3 for $M_{\varphi, \Omega_0}(\lambda + \delta) - M_{\varphi, \Omega_0}(\lambda - \delta)$ and  this implies  (\ref{mphi*}). By using that $\varphi$ is equal to $1$ on a neighborhood of $[E_0,E_1]$, we   complete the proof of $i)$.
The equivalence $ii) \Leftrightarrow iii)$ is a consequence of (\ref{eq:5.3}) and (\ref{eq:1.4bis}).
\hfill{$\Box$}\\



\section{ Applications of Theorems 1 and 2}

First we will examine the connection between the condition (\ref{eq:1.2}) and Theorems 1 and 2. We have the following.

\begin{prop}
Assume that $L$ satisfies the assumptions $(2.1) - (2.9)$ and suppose that each $\lambda \in [E_0, E_1]$ is a non-critical energy level for $Q$ and $Q^{\#}.$ Assume that for $\lambda \in [E_0 -\epsilon_2, E_1 +\epsilon_2],\: \epsilon_2 > 0$ and $0 < \frac{h}{B} \leq \delta \leq B$ we have
\begin{equation} \label{eq:4.1}
 \# \{ w \in \C:\: w \in {\rm Res}\: L,\: |w - \lambda| \leq \delta\} \leq C \delta  h^{-\ns}.
\end{equation}
Then we can apply Theorem $1$ and Theorem $2$ with $r(h)= \ln (1/h)$.
\end{prop}
\begin{pf} It is sufficient to prove that the assertion $ii)$ of Theorem 1 holds with $r(h)= \ln (1/h)$.
 It is easy to show that (\ref{eq:4.1}) implies 
\begin{equation} \label{eq:4.2}
\sum_{w \in \:({\rm Res}\: L) \: \cap \Omega\:, \atop 0 < |\Im w| \leq A \delta} \omega_{\C_{-}}(w, [\lambda - \delta, \lambda + \delta]) \leq {\mathcal O}(\delta)h^{-\ns},\: \frac{h}{B_2} \leq \delta \leq B_2.
\end{equation}
In fact, taking into account the estimate
\begin{equation} \label{estint}
\int_{\alpha}^{\beta}\frac{|\Im w|}{|\nu - w|^2} d \nu \leq \pi,\:\:-\infty\leq \alpha < \beta \leq \infty,
\end{equation}
 we obtain
\begin{equation} \label{eq:4.3}
 \sum_{w \in {\rm Res}\:\: L,\: \Im w \neq 0, \atop |w - \lambda| \leq 2\delta} \int_{\lambda - \delta}^{\lambda + \delta} \frac{|\Im w|}{\pi |t - w|^2} dt \leq C\delta h^{-\ns}.
\end{equation}
On the other hand, for $|t - \lambda| \leq \delta \leq 1/2$ we get
\[ \sum_{w \in {\rm Res}\:L,\: 0 < |\Im w | \leq A \delta, \atop |w - \lambda| >2\delta}  \frac{|\Im w|}{|t - w|^2} \, \]
\[ \leq \sum_{k = 1}^{C \log \frac{1}{\delta}} \sum_{2^k \delta < |w - \lambda| \leq 2^{k+1}\delta, \atop |\Im w | \leq A \delta}  \frac{|\Im w|}{|t - w|^2} \leq \sum_{k = 1}^{C \log \frac{1}{\delta}} \frac{C \delta (2^{k+1}\delta) h^{-\ns}}{(2^k \delta)^2} \leq Ch^{-\ns} \,, \]
and after an integration over the interval $[\lambda - \delta, \lambda + \delta]$ we obtain immediately (\ref{eq:4.2}) since the case $\delta > 1/2$ is trivial. 

To obtain the assumption $ii)$, we will show that
\begin{equation} \label{eq:4.4}
 \sum_{w \in\: ({\rm Res}\: L)\:\cap \Omega,\: \atop \Im w \neq 0} \omega_{\C_{-}}(w, [\lambda - \delta, \lambda + \delta]) \leq {\mathcal O}(\delta) \max \Bigl(\log \frac{1}{\delta}, 1\Bigr) h^{-\ns},\: \frac{h}{B_2} \leq \delta \leq B_2 .
\end{equation}
To see this, first we apply (\ref{eq:4.2}) with $A = 2$. Next for $|t- \lambda| \leq \delta \leq 1/2$ we have
\[ \sum_{w \in \:({\rm Res}\:L)\: \cap \Omega,\: |\Im w | \geq  2\delta } \frac{1}{|t - w|}  \, \]
\[ \leq \sum_{k =1}^{C \log \frac {1}{\delta}} \sum_{ 2^k \delta \leq |w - \lambda| \leq 2^{k+1}\delta} \frac{1}{|t - w|} \leq \sum_{k=1}^{C\log \frac {1}{\delta}} \frac{C 2^{k+1} \delta h^{-\ns}}{2^k \delta} \leq C \Bigl(\log \frac {1}{\delta}\Bigr) h^{-\ns} \,. \]
So writing
\[ \frac{|\Im w|}{|t - w|^2} = \frac{1}{2i}\Bigl( \frac{1}{t - \overline{w}} - \frac{1}{t - w}\Bigr) \,, \]
we obtain (\ref{eq:4.3}) for $\delta \leq 1/2.$ The analysis of the case $1/2 < \delta \leq B_2$ is trivial.
\end{pf}

{\bf Remark.} There are examples, where the result of Proposition 3 is sharp. In fact consider the case $n = 1$ and let $L(h) = -h^2\Delta + V(x)$. If $V(x)$ has an absolute non-degenerate maximum at only one point $\alpha$, then the analysis in  \cite{Sj2} shows that (4.1) holds, while following the approximation of the resonances given in \cite{BCD}, \cite{Sj1}, \cite{FR1} the assumption $iii)$ of Theorem 1 is satisfied with $r(h) = \lh.$ We will discuss with more details this example in the next Section.\\

Next consider a classical h-pseudodifferential operator $L(h) = L$  on $L^2(\RR^n)$ with symbol 
$$l(x,\xi; h) \sim \sum_{j \geq 0} l_j(x,\xi) h^j,\:\: l_j(x,\xi) \in S_0^{-j}(<\xi>^2),$$
where we use the notations of \cite{DS} for the symbols of h-pseudodifferential operators.
 Assume that there is no "black box" and that  $L(h)$ satisfies the conditions (2.2)-(2.9). Moreover, we suppose that there exist constants $C_1 > 0,\:\; C_0 > 0$ so that
\begin{equation} \label{eq:4.6}
l_0(x, \xi) \geq C_1 |\xi|^2 - C_0,\:\: \forall (x,\xi) \in \RR^n,
\end{equation}
$l_0(x,\xi)$ being the  principal symbol of $L.$ As we have mentioned in Section 2, the symbol $l_0(x, \xi)$ may have critical points. Given a critical point $E_c \in [E_0, E_1]$, we assume that the set $\mc$  of the critical points of $l_0(x, \xi)$  on the surface $\{(x,\xi) \in \RR^n:\: l_0(x,\xi) = E_c\}$ is a  submanifold of $T^{*}(\RR^n)$ such that the Hessian of $l_0(x, \xi)$ is non degenerate on the subspace normal to $\mc.$ This implies that $E_c$ is an isolated critical point of $l_0(x,\xi)$ and the conditions on ${\mathcal C}$ are the same as those in \cite{BPU}, \cite{Bo3}. Moreover, the assumption (2.6) shows that $\mc$ is a finite union of connected compact sets $\mc = \mc_1 \cup ... \cup \mc_N$. Let $(r_j, s_j)$ be the signature of the Hessian of $l_0(x,\xi)$ on the subspace normal to $\mc_j.$  The codimension of $\mc_j$ is equal to $r_j + s_j$. Notice that if $L$ is a differential operator, the ellipticity condition (\ref{eq:4.6}) implies (see \cite{Bo3}) that  $r_j + s_j \geq n +1.$ In order to apply Theorem 1, we need the following result of J. F. Bony.
\begin{prop}[\cite{Bo3}]
Under the above assumptions on the set of the critical points $\mc$ for $E$ in a small neighborhood of $E_c$, $h \in ]0, h_0]$ and $h/B \leq \delta \leq B$ we have
$$\# \{\mu \in \RR: \mu \in {\rm sp}_{pp}\: L^{\#}(h),\:\: |\mu - E| \leq \delta \} \leq C\delta r(\delta) h^{-n},$$
where in the general case $r(\delta) = \Bigl(|E - E_c| + \delta\Bigr)^{-1/2}.$ Moreover, if for all $1 \leq j \leq N$ we have $r_j + s_j \geq 2,$ then $r(\delta) = |\log(\delta + |E - E_c|)|$ and if for all $1 \leq j \leq N$ we have  $\max\{r_j, s_j\} \geq 2$, then $r(\delta) = 1.$
\end{prop}

The proof in \cite{Bo3} is based on the estimation of the trace norm
$$\Bigl \| f\Bigl(\frac{L^{\#} - E}{\delta}\Bigr)\Bigr \|_{\tr}$$
for a cut-off function $f \in C_0^{\infty} (\RR; [0, 1]),\:\: f = 1$ on $[-1, 1]$ following the tools developed in \cite{BPU}. Under the above assumptions the critical points are isolated and by using a finite covering of $[E_0, E_1]$, we obtain a global version of Proposition 4 with constants $B > 0, \: C > 0$ and $h_0 > 0$ which are uniform with respect to $E \in [E_0-\epsilon, E_1 +\epsilon],\:\:\epsilon > 0.$\\

\begin{cor} Under the above assumptions on ${\mathcal C}$, we can apply Theorems $1$ and $2$  for the $h$-pseudodifferential operator $L$ with $r(h)$ given above.
\end{cor}



\section{Breit-Wigner approximation near critical energy levels}

In this section we assume that the critical manifold $\mc$ has the form described in Section 4. Thus the assumption $i)$ of Theorem 1 holds and we are going to discuss the form of the sum of harmonic measures related to the resonances. Throughout this section we assume that
$$L(h) = -h^2\Delta + V(x),$$
where $V(x)$ is real valued on $\RR^n$ and
$$|V(x)| \leq C(1 + |x|) ^{-n-\sigma}, \:\: \sigma > 0.$$
Moreover, we suppose that $V(x)$ is holomorphic in the domain
$$\{ x \in \C^n:\: |\Im x | \leq \tan \theta_0 |\Re x | \} \cup \{x \in \C:\: |\Im x | \leq \delta_0\}$$
for $0 < \theta_0 < \pi/2$ and $\delta_0 > 0.$\\

Denote by $\xi(\lambda, h)$ the spectral shift function related to $L(h)$ and $L_0(h) = -h^2 \Delta.$ Let $l(x, \xi) = |\xi|^2 + V(x)$ be the symbol of $L(h) = L$ and let the set ${\mathcal C}$ of the critical points of $l(x,\xi)$ have the form described in the previous section.\\ 

First we will treat the case $n=1.$ An application of Corollary 3 yields the following.
\begin{prop} Assume $n = 1$ and let the set of the critical points in $l^{-1}([E_0, E_1])$ have the form  ${\mathcal C} = \cup_{i=1}^N \{(\alpha_i, 0)\}$ with $V(\alpha_i) = E_i,\:V'(\alpha_i) = 0,\: V''(\alpha_i) \neq 0, \: i = 1,...,N.$ Then for each $E_i,\: i=1,2,...,N$, $h \in ]0, h_0]$  and for $|\lambda - E_i| \leq C_1h,\: C_2 > C_1,$ we have
$$\frac{\partial \xi}{\partial \lambda}(\lambda, h) = -\frac{1}{\pi}\sum_{|E_i- w|\leq C_2h ,\:\atop w \in {\rm Res}\: L(h)} \frac{\Im w}{|\lambda - w|^2} + \co \Bigl(h^{-1}\lh\Bigr).$$
\end{prop} 
 Our next purpose is to obtain an estimate of the term involving the  harmonic measures. Consider the simplest case when the manifold $\mc$ is given by a single point $\{(\alpha, 0)\},$ where 
$$V(\alpha) = \max_{x \in \RR} V(x) =  E_c,\:\; V'(\alpha) = 0,\: V''(\alpha) = -\frac{1}{\rho} < 0.$$
The resonances in a disk $D(E_c, r)$ for $r > 0$ sufficiently small have the form (see \cite{BCD}, \cite{Sj1}, \cite{FR1})
$$w_k = E_c -i(k + \frac{1}{2}) \frac{1}{\rho} h + {\mathcal O}(h^2),\:\: k \in \N.$$
and for $|\lambda - E_c| \leq C_1 h < \frac{1}{2\rho} h$ we have
$$-\frac{1}{\pi} \sum_{|E_c - w_k| \leq \frac{r}{2}} \frac{\Im w_k}{|\lambda - w_k|^2}= -\frac{1}{\pi}\sum_{\frac{1}{2\rho}h \leq |E_c - w_k| \leq \frac{r}{2}} \frac{\Im w_k}{|\lambda - w_k|^2} = \frac{\rho}{\pi h} \sum_{k=1}^{C/h} \frac{1}{k} + {\mathcal O}(h^{-1})$$
$$ = \frac{\rho}{\pi h} \int_{1}^{C/h} \frac{dx}{x} + {\mathcal O}(h^{-1}) = \frac{\rho}{\pi}h^{-1}\log \frac{1}{h} + {\mathcal O}(h^{-1}).$$
Thus applying Theorem 1 in \cite{BrPe2} in the disk $D(E_c, r/2)$,  we get for $\lambda \in \RR,\:|\lambda - E_c| \leq C_1h$ 
\begin{equation} \label{eq:5.1}
\frac{\partial \xi}{\partial \lambda}(\lambda, h) = \frac{\rho}{\pi} h^{-1} \lh + \co (h^{-1})
\end{equation}
and we obtain the result of Theorem 2.2 in \cite{FR2} concerning the case of an unique non-degenerate maximum point.\\

Next assume that $\mc = (\alpha_1, 0) \cup (\alpha_2, 0),\:\: \alpha_1 < \alpha_2,$ with
$$V(\alpha_i) = \max_{x \in \RR} V(x) = E_c, \: V'(\alpha_i) = 0, \: V''(\alpha_i) = -\frac{1}{\rho_i} < 0, \: i =1,2.$$
Following the results of \cite{FR1}, the resonances in a disk $D(E_c, rh),\: r > 0$, have the form
$$w_k = E_c + \frac{S_0 - (k + 1/2)\pi h +  ih \log 2 }{K \log h} + {\mathcal O}\Bigl(\frac{h}{(\log h)^2}\Bigr),\: k \in \N$$
if $E_c -\Re w_k = {\mathcal O}(h/\log h)$ and
$$z_k = E_c + \frac{S_0 - (k + 1/2)\pi h }{K \log h} + \co (\frac{h}{\log h}),\:\: k \in \N$$
in the exterior of this domain, where $S_0 \in \RR$ and  $K = \frac{1}{2}(\rho_1 + \rho_2).$

First we are going to estimate for $|\lambda - E_c| \leq C_1 \frac{h}{\lh},\:\: C_1 \leq \frac{\pi}{2K}$, the sum
$$-\frac{1}{\pi} \sum_{|E_c - z| \leq rh, \atop z \in {\rm Res}\: L(h)} \frac{\Im z}{|\lambda - z|^2} \leq \#\{z \in {\rm Res} L(h) :\: |E_c - \Re z| < \frac {\pi h}{K \lh} \} \co \Bigl(h^{-1} \lh\Bigr) $$
$$ +\frac{C h}{\lh} \sum_{k=1}^{C \lh} \sum_{\frac{k \pi h}{K \lh} \leq |E_c - \Re z| < \frac{(k+1)\pi h}{K \lh} \atop z \in {\rm Res}\: L(h)} \frac{1}{|\lambda - \Re z|^2}$$
$$\leq Ch^{-1} \lh \sum_{k=1}^{\infty} \frac{1}{k^2} + C_2 h^{-1} \lh  = C_3 h^{-1} \lh .$$
Here we have used the fact that there are only finite number resonances $z$ for which
$$\frac{k \pi h}{K \lh} \leq |E_c - \Re z| < \frac{(k+1)\pi h}{K \lh}, \: k \in \N.$$
By using the lower bound
$$ -\Im z \geq  C_0 \frac{h}{\lh},\: C_0 > 0, \: E_c - \Re z = {\mathcal O}(h / \log h),\: z \in {\rm Res}\: L(h),$$
a similar argument yields
$$-\frac{1}{\pi} \sum_{|E_c - z| \leq rh, \atop z \in {\rm Res}\: L(h)} \frac{\Im z}{|\lambda - z|^2} \geq \ C_4h^{-1} \lh , \: C_4 > 0.$$
 
Secondly, the sum of the Breit-Wigner factors related to the resonances $z$  lying in 
$$\{z \in \C: rh < |E_c - z| \leq r/2 \}$$
 can be estimated as above by $\Bigl(\frac{\rho_1 + \rho_2}{2 \pi}\Bigr) h^{-1} \lh + \co(h^{-1})$. Thus for $\lambda \in \RR,\: |\lambda - E_c| \leq C_1 \frac{h}{\lh}$ we get the following representation
\begin{equation} \label{eq:5.2}
\frac{\partial \xi}{\partial \lambda}(\lambda, h) =  -\frac{1}{\pi} \sum_{|E_c - z| \leq rh, \: \atop z \in {\rm Res}\: L(h)} \frac{\Im z}{|\lambda- z|^2} + \Bigl(\frac{\rho_1 + \rho_2}{2 \pi}\Bigr) h^{-1} \lh + \co (h^{-1})
\end{equation}
and
\begin{equation} \label{eq:resb}
 C_4 h^{-1} \lh \leq  -\frac{1}{\pi} \sum_{|E_c - z| \leq rh, \atop z \in {\rm Res} \: L(h)} \frac{\Im z}{|\lambda- z|^2} \leq  C_3 h^{-1} \lh.
\end{equation}
Consequently, we obtain a result similar to that of Theorem 2.2 in \cite{FR2}. More precisely, Fujii\'e and Ramond \cite{FR2} proved that for $|\lambda - E_c| \leq C_1\frac{h}{ \lh}$ we have
$$\pi \frac{\partial \xi}{\partial \lambda}(\lambda, h) = \Bigl( \frac{\rho_1 + \rho_2}{2}\Bigl(1 +  \frac{\gamma}{(1- \gamma^2)\cos^2(\sigma_i/h) + \gamma^2}\Bigr)\Bigr) h^{-1} \lh + \co(h^{-1}),$$
where the function $\gamma(\lambda, h)$ is holomorphic in $[E_c - \epsilon, E_c + \epsilon] + i[-Ch, Ch],\: \epsilon >0$ and the function $\sigma_i(\lambda, h)$ is holomorphic in a disk $D(E_c, Ch)$ .

Comparing these representations, we get
 $$- \sum_{|E_c - z| \leq rh, \:\atop z \in {\rm Res}\: L(h)} \frac{\Im z}{|\lambda- z|^2} = \frac{\rho_1 + \rho_2}{2}\Bigl(\frac{\gamma}{(1- \gamma^2)\cos^2(\sigma_i/h) + \gamma^2}\Bigr) h^{-1}\lh + \co(h^{-1}).$$
 For $\gamma$ small we have spikes at each zero of $\cos(\sigma_i/h)$, while the spikes in (\ref{eq:5.3}) are related to the real part of the resonances.\\

To treat the case $\lambda \in \RR,\:|\lambda - E_c| \leq rh,$ notice that we have $\co \Bigl( \lh \Bigr)$ resonances in $D(E_c, rh)$ and the lower bound of the imaginary part of the resonances implies
\begin{equation} \label{eq:5.4}
\frac{\partial \xi}{\partial \lambda}(\lambda, h)
 = \co \Bigl(h^{-1} \Bigl(\lh\Bigr)^2 \Bigr) + \Bigl(\frac{\rho_1 + \rho_2}{2 \pi}\Bigr) h^{-1} \lh + \co (h^{-1}).
\end{equation}

Now consider the case $n \geq 2.$ In this situation Corollary 3 yields for $|\lambda - E| \leq C_1 h$ the representation
\begin{equation} \label{eq:rn}
\frac{\partial \xi}{\partial \lambda}(\lambda, h) = -\frac{1}{\pi}\sum_{|E- w|\leq C_2h ,\:\atop w \in {\rm Res}\: L(h)} \frac{\Im w}{|\lambda - w|^2} + \co (h^{-n}).
\end{equation}
Let us discuss the simplest case when the set $K$ of trapping points of $L(h)$ lying in $l^{-1}(E_c)$ is given by a single point $\{( \alpha, 0 \}$ so that
$$ V(\alpha) = E_c,\: \nabla_x V(\alpha) = 0.$$
Assume that the Hessian of $V(x)$ at $\alpha$ is non-degenerate and let $(n-d, d),\:\; d \geq 1,$ be the signature of this Hessian. Then the linearization of the Hamiltonian field $H_l$ at $( \alpha, 0)$ has eigenvalues
$$\pm i \lambda_j, \: 1 \leq j \leq n-d,$$
$$ \pm \lambda_j,\: n-d + 1 \leq j \leq n$$
with $\lambda_j > 0$ (see \cite{Sj1}, \cite{Sj2}). Following the results in \cite{Sj1}, \cite{KK}, the resonances of $L(h)$ in a disk $D(E_c, Ch)$ admit an asymptotic representation and the condition $d \geq 1$ implies easily that 
$$-\Im w \geq c_0 h,\: c_0 > 0,\: \:\forall  w \in D(E_c, Ch) \cap {\rm Res}\: L(h).$$
On the other hand, the result of \cite{Sj2} says that the number of the resonances $w$ lying in $D(E_c, C_2h)$ is at most $C_0 h^{d-n}$ and for $|\lambda - E_c| \leq C_1h,\: C_1 < C_2$ we obtain the estimate
\begin{equation}
-\frac{1}{\pi}\sum_{|E- w|\leq C_2h ,\:\atop w \in {\rm Res}\: L(h)} \frac{\Im w}{|\lambda - w|^2} = \co(h^{d-n-1}).
\end{equation}
In contrast to the case $n =1$ for $n \geq 2$ the sum of the Breit-Wigner factors is bounded by a term having at most the same order as the remainder $\co(h^{-n})$ in (\ref{eq:rn}). In this direction we notice the analysis of the radial case in \cite{FR3} concerning the partial scattering phases $\sigma_l(\lambda, h),\:\: l \in \N,$ for a potential having an absolute maximum ($d = n$). By using the asymptotics of $\frac{\partial \sigma_l}{\partial \lambda}(\lambda, h),\:\: l \in \N$, it seems difficult to obtain a representation for $\frac{\partial \xi}{\partial \lambda}(\lambda, h)$ with remainder $\co(h^{-n}).$ 

{\footnotesize


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\end{thebibliography}}


\bigskip

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

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