 95435 Khuatduy D.
 A semiclassical trace formula for Schr\"odinger operators in the case of a
critical energy level.
(100K, Latex)
Sep 29, 95

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Abstract. Let $\widehat{H}=\frac{\h^2}{2}\Delta + V(x)$ be a Schr\"odinger operator on
$\R^n$, with smooth potential $V(x)\rightarrow +\infty$ as
$x\rightarrow +\infty$. The spectrum of $\widehat{H}$ is discrete,
and one can study the asymptotic
of the smoothed spectral density
\[
\Upsilon(E,\h)=\sum_k\f\left(\frac{E_k(\h)E}{\h}\right),
\]
as
$\h\rightarrow 0$. Here, $\{E_k(\h)\}_{k\in\N}$ is the spectrum of $\widehat{H}$
and $\hf\in C^{\infty}_0(\R)$.
We investigate the case where $E$ is a critical value of the symbol $H$ of
$\widehat{H}$ and, extending the work of Brummelhuis, Paul and Uribe in \cite{brpaur},
we prove the existence of a
full asymptotic expansion for $\Upsilon$ in terms of $\sqrt{\h}$ and $\ln \h$ and
compute the leading coefficient.
We consider some new Weyltype estimates for the counting function :
\mbox{$N_{E_c,\rho}(\h)=\#\{k\in\N \;\;/\;\; E_k(\h)E\leq \rho\h\}$.}
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