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

TL;DR

This paper establishes $L^p$ decay estimates for Schr"odinger and wave equations on three-dimensional Riemannian manifolds, including perturbations of constant curvature spaces, using direct wave propagator estimates.

Contribution

It provides new $L^p$ decay estimates for Schr"odinger and wave equations on Riemannian manifolds with small metric perturbations, extending known results to more general geometries.

Findings

Estimates hold for small metric perturbations with four derivatives.

Results include the sphere $S^3$ and hyperbolic space $H^3$.

Most estimates are valid for the perturbed Hamiltonian $H=H_0+V$.

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.