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

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.