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 where 2 < α ≤ 4. We prove that the solution decays pointwise like t−α as provided that there are no resonances at zero energy and no bound states. As an application, we consider the 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].