Barrier-top resonances for non globally analytic potentials

Abstract

We give the semiclassical asymptotic of barrier-top resonances for Schrödinger operators on ${\mathbb{R}}^n$, $n\ge 1$, whose potential is $C^{\infty }$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained in [8, 29]. 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 [4], and its proof follows another approach.