Resonance-free regions for diffractive trapping by conormal potentials

Oran Gannot, Jared Wunsch

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the Schrödinger operator P = h2Δg + V on Rn equipped with a metric g that is Euclidean outside a compact set. The real-valued potential V is assumed to be compactly supported and smooth except at conormal singularities of order −1 − α along a compact hypersurface Y .Forα>2(orevenα>1 if the classical flow is unique), we show that if E0 is a non-trapping energy for the classical flow, then the operator P has no resonances in a region [E0 − δ,E0 + δ] − i[0,ν0hlog(1/h)]. The constant ν0 is explicit in terms of α and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line.

Original languageEnglish (US)
Pages (from-to)1339-1360
Number of pages22
JournalAmerican Journal of Mathematics
Volume143
Issue number5
DOIs
StatePublished - Oct 2021

ASJC Scopus subject areas

  • Mathematics(all)

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