Barrier-top resonances for non globally analytic potentials

We give the semiclassical asymptotic of barrier-top resonances for Schr"{o}dinger operators on ${\mathbb{R}}^n$, $n\ge 1$, whose potential is $C^{\mathrm{\infty }}$ 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.