Barrier-top resonances for non globally analytic potentials

TL;DR

This paper derives the semiclassical asymptotics of barrier-top resonances for Schrödinger operators with smooth, non-globally analytic potentials, extending previous results to a broader class of potentials.

Contribution

It introduces a new propagation of singularities theorem at hyperbolic fixed points for non-globally analytic potentials, refining prior results and providing a novel proof approach.

Findings

Semiclassical asymptotics for barrier-top resonances derived

Propagation of singularities theorem established at hyperbolic fixed points

Extension of resonance results to non-globally analytic potentials

Abstract

We give the semiclassical asymptotic of barrier-top resonances for Schr\"{o}dinger operators on ${\mathbb R}^{n}$, $n \geq 1$, whose potential is $C^{\infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.