Decay estimates for One-dimensional wave equations with inverse power   potentials
  We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$ where $m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $t^{-m}$ for large $t$, which is consistent with existing mathematical and physical literature under slightly different assumptions (see e.g. Bizon, Chmaj, and Rostworowski, 2007; Donninger and Schlag, 2010; Schlag, 2007). Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being $const.x^{-\alpha}$ where $\alpha>2$ is a real number, as well as potentials of the form $const.x^{-m}+O(x^{-m-\delta_1})$ with $\delta_1>3$. 
O. Costin, M. Huang
math.AP math-ph math.MP