Modified wave operators and scattering for linear wave equations with a repulsive potential

Abstract

In this work we consider the wave equation with a repulsive potential, either on the half line ${\mathbb{R}}^{+}$ or the Euclidean space ${\mathbb{R}}^d$ with $d\ge 3$. We combine the operator theory and the inward/outward energy theory to deduce a modified wave operator for repulsive potentials decaying like $|x{|}^{-{\hat{I}}^2}$ with ${\hat{I}}^2>1/3$. In particular the regular wave operator without modification exists if ${\hat{I}}^2>1$. This implies that the asymptotic behaviour of finite-energy solutions to the wave equation $u_{tt}-\hat{I}”u+|x{|}^{-{\hat{I}}^2}u=0$ is similar to that of the solutions to the classic wave equation if ${\hat{I}}^2\in (1,2)$.

56 pages