Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Abstract

We study semiclassical resonances in a box Ω(h) of height h N , N 1. We show that the semiclassical wave front set of the resonant states (including the "generalized eigenfunctions") is contained in the set T of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # (h) with discrete spectrum the number of resonances in Ω(h) is bounded by the number of eigenvalues of P # (h) in an interval a bit larger than the projection of Ω(h) on the real line. As an application, we prove a Weyl type estimate of the number of resonances in Ω(h) in terms of the measure of T . We prove a similar estimate in case of classical scattering by a metric and obstacle.