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

2025/01/12 by Beceanu, Marius
#35B40 #35L05 #35Q41 #53B20 #58J37 #58J45 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.06957

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 S³, the three-dimensional sphere, and \mathbb H³, the three-dimensional hyperbolic space. Most of the estimates hold for the perturbed Hamiltonian H=H₀+V, where H₀ is the shifted Laplacian H₀=-Δ+κ₀, κ₀ 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.

Related

• Dispersive Estimates for Manifolds with one Trapped Orbit
• Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space
• Pointwise dispersive estimates for Schr "odinger operators on product\n cones
• Decay estimates for Schrödinger's equation with magnetic potentials in three dimensions
• Sharp resolvent and time decay estimates for dispersive equations on asymptotically Euclidean backgrounds