01-278 MARTINEZ Andre'

Resonance Free Domains for Non-Analytic Potentials (48K, LATeX 2e) Jul 18, 01

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Abstract. We study the resonances of the semiclassical Schr\"odinger operator $P=-h^2\Delta + V$ near a non-trapping energy level $\lambda_0$ in the case when the potential $V$ is not necessarily analytic on all of $\R^n$ but only outside some compact set. Then we prove that for some $\delta >0$ and for any $C>0$, $P$ admits no resonance in the domain $\Omega =[\lambda_0 -\delta ,\lambda_0 +\delta ]-i[0,C h\log (h^{-1})]$ if $V$ is $C^\infty$, and $\Omega =[\lambda_0 -\delta , \lambda_0 +\delta ]-i[0,\delta h^{1-\frac1{s}}]$ if $V$ is Gevrey with index $s$. Here $\delta >0$ does not depend on $h$ and the results are uniform with respect to $h>0$ small enough.

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