Decay estimates for the one-dimensional wave equation with an inverse power potential

Abstract

We study the wave equation on the real line with a potential that falls off like $|x{|}^{-\alpha }$ for $|x|\to \infty$ where $2<\alpha \le 4$. We prove that the solution decays pointwise like $t^{-\alpha }$ as $t\to \infty$ provided that there are no resonances at zero energy and no bound states. As an application we consider the $\ell =0$ Price Law for Schwarzschild black holes. This paper is part of our investigations into decay of linear waves on a Schwarzschild background.

14 pages, added some details in order to match the published version

References in corpus (1)

• Anomalously small wave tails in higher dimensions

Cited by in corpus (5)

• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• On pointwise decay of waves
• Asymptotic Behavior of Massless Dirac Waves in Schwarzschild geometry
• Price's law on Minkowski space in the presence of an inverse square potential