Width of shape resonances for non globally analytic potentials. (English) Zbl 1210.81037
Summary: We consider the semiclassical Schrödinger 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 the result by B. Helffer and J. Sjöstrand [Mém. Soc. Math. Fr., Nouv. Sér. 24/25, 228 p. (1986; Zbl 0631.35075)] 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.
MSC:
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 35P25 | Scattering theory for PDEs |
| 35B34 | Resonance in context of PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Citations:
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