Width of Shape Resonances for Non Globally Analytic Potentials

TL;DR

This paper derives the asymptotic behavior of shape resonances for semiclassical Schrödinger operators with non-globally analytic potentials, extending previous analytic results using advanced microlocal analysis techniques.

Contribution

It generalizes the asymptotic expansion of shape resonances to smooth potentials lacking global analyticity, employing almost analytic extensions and refined microlocal methods.

Findings

Asymptotic expansion of the imaginary part of shape resonances derived

Extension of results from analytic to smooth potentials

Development of microlocal techniques for non-analytic settings

Abstract

We consider the semiclassical Schroedinger operator with a "well in an island" potential, on which we assume smoothness only, except near infinity. We give the asymptotic expansion of the imaginary part of the shape resonance at the bottom of the well. This is a generalization of a result by Helffer and Sjoestrand in the globally analytic case. We use an almost analytic extension in order to continue the WKB solution coming from the well beyond the caustic set, and, for the justification of the accuracy of this approximation, we develop some refined microlocal arguments in h-dependent small regions.