On the Schrodinger equation outside strictly convex obstacles

TL;DR

This paper establishes sharp Strichartz estimates for the semi-classical Schrödinger equation on manifolds with convex boundaries, leading to results on local existence and scattering for nonlinear Schrödinger equations outside convex obstacles.

Contribution

It provides the first sharp Strichartz estimates for the Schrödinger equation outside strictly convex obstacles, extending the understanding of dispersive PDEs in geometric settings.

Findings

Sharp Strichartz estimates for semi-classical Schrödinger on manifolds with convex boundary

Local existence results for H^1-critical Schrödinger equations in 3D

Scattering results for sub-critical Schrödinger equations in 3D

Abstract

We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3D.