Dispersive estimates for Schr\"{o}dinger's and wave equations on Riemannian manifolds

Abstract

This paper proves $L^p$ decay estimates for Schr\"{o}dinger'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.