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

TL;DR

This paper establishes decay rates for solutions to the one-dimensional wave equation with inverse power potentials, showing solutions decay like $t^{-eta}$ under certain conditions, with applications to black hole physics.

Contribution

It provides new decay estimates for wave equations with specific inverse power potentials, extending understanding of wave behavior in such settings.

Findings

Solutions decay like $t^{-eta}$ for $2<eta extless 4$ under no resonance conditions.

Decay estimates are applied to Price Law for Schwarzschild black holes.

The results improve previous decay bounds for linear waves with inverse power potentials.

Abstract

We study the wave equation on the real line with a potential that falls off like $|x|^{-\alpha}$ for $|x| \to \infty$ where $2 < \alpha \leq 4$. 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.