Dispersive estimates for Schrödinger's and wave equations on Riemannian manifolds

Abstract

This paper proves $L^p$ decay estimates for Schrödinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional curvature. We also prove estimates on ${\mathbb{S}}^3$, the three-dimensional sphere, and ${\mathbb{H}}^3$, the three-dimensional hyperbolic space. Most of the estimates hold for the perturbed Hamiltonian $H=H_0+V$, where $H_0$ is the shifted Laplacian $H_0=-\Delta +{\kappa }_0$, ${\kappa }_0$ is the constant (or asymptotic) sectional curvature, and $V$ is a small scalar potential. The results are based on direct estimates of the wave propagator. All results hold in three space dimensions. The metric is required to have four derivatives.

29 pages