Dispersive estimates for the Schrödinger equation in a model convex domain and applications

Abstract

We consider an anisotropic model case for a strictly convex domain $\Omega \subset {\mathbb{R}}^d$ of dimension $d\ge 2$ with smooth boundary $\partial \Omega \ne \varnothing$ and we describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semiclassical flow: a loss of $(\frac{h}{t}{)}^{1/4}$ occurs with respect to the boundaryless case due to repeated swallowtail-type singularities, and is proven optimal. Corresponding Strichartz estimates allow us to solve the cubic nonlinear Schrödinger equation on such a three-dimensional model convex domain, hence matching known results on generic compact boundaryless manifolds.