00-168 Martinez A., Yajima K.

On the Fundamental Solution of Semiclassical Schr\"odinger Equations at Resonant Times (50K, LATeX 2e) Apr 6, 00

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Abstract. We consider perturbations of the semiclassical harmonic oscillator of the form ${\ds P=-\frac{h^2}2\Delta + \frac{x^2}2 + h^{\delta}W(x)}$, $x \in {\bf R}^m$, with $W(x)\sim \la x\ra^{2-\mu}$ as $\vert x\vert \rightarrow +\infty$ and $\delta ,\mu \in (0,1)$, and we investigate the fundamental solution $E(t,x,y)$ of the corresponding time-dependent Schr\"odinger equation. We prove that at resonant times $t=n\pi$ ($n\in {\bf Z}$) it admits a semiclassical asymptotics of the form: $E(n\pi ,x,y) \sim h^{-m(1+\nu)/2}a_0e^{iS( x,y)/h}$ with $a_0\not=0$ and $\nu = \delta /(1-\mu)$, under the conditions $x\not= (-1)^ny$ and $\nu <1$.

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