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

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

2009-11-16

arxiv.org/abs/0911.3174v2

Int. Math. Res. Not. 2010, No. 22, 4276-4300 (2010)

Mathematics

Analysis of PDEs

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

Scientific paper

10.1093/imrn/rnq038

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.

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