On the Schrödinger equation outside strictly convex obstacles

Abstract

We prove sharp Strichartz estimates for the semiclassical Schrödinger equation on a compact Riemannian manifold with a smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrödinger equation outside a strictly convex obstacle, local existence for the $H^1$-critical (quintic) Schrödinger equation, and scattering for the subcritical Schrödinger equation in three dimensions.