Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains

Abstract

We prove resolvent estimates for a Schrödinger operator with a short-range potential outside an obstacle with Dirichlet boundary conditions. As a consequence, we deduce integrability of the local energy for the wave equation, and smoothing effect for the Schrödinger equation. Finally, for both equations, we prove that local Strichartz estimates for the free equation outside an obstacle imply global Strichartz estimates with a short-range potential outside the same obstacle. The estimates are all global in time, after projection on the continuous spectrum of the operator.

In this new version, we included Strichartz estimates with loss for the Schrödinger equation