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|^−α for $|x|\rightarrow \infty$ where 2 < α ≤ 4. We prove that the solution decays pointwise like t^−α as $t\rightarrow \infty$ 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, see [5, 6].