Decay estimates for one-dimensional wave equations with inverse power potentials

Abstract

We study the one-dimensional wave equation with an inverse power potential that equals c o n s t . x − m const.x^{-m} for large | x | |x| , where m m is any positive integer greater than or equal to 3. We show that the solution decays pointwise like t − m t^{-m} for large t t , which is consistent with existing mathematical and physical literature under slightly different assumptions. Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being c o n s t . x − α const.x^{-\alpha } , where α > 2 \alpha >2 is a real number, as well as potentials of the form c o n s t . x − m + O ( x − m − δ 1 ) const.x^{-m}+O( x^{-m-\delta _1}) with δ 1 > 3 \delta _1>3 .