95-435 Khuat-duy D.

A semi-classical 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{br-pa-ur}, 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 Weyl-type 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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