Propagation of singularities at a non analytic hyperbolic fixed point and an application to the quantization of resonances I

It is well known as Hörmander's theorem that the (semiclassical) wave front set of the solution to a pseudo-differential equation propagates along Hamiltonian flows of the real principal type symbol. We extend this theorem to fixed points of the Hamiltonian vector field. To a hyperbolic fixed point, associated an outgoing and an incoming stable manifolds, and we show that if the semiclassical wave front set is empty on the incoming stable manifold (except at the fixed point), then it is also empty on the outgoing one. We also show how such theorems are applied to a scattering problem of the Schr¥"odinger operator. We give the quantization rule of resonances when the trapped set of the corresponding Hamiltonian system consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. This is a joint work with Jean-Fran¥c cois Bony (Bordeaux), Thierry Ramond (Paris XI) and Maher Zerzeri (Paris XIII)