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

TL;DR

This paper develops a modified wave operator theory for linear wave equations with repulsive potentials decaying at a certain rate, showing that solutions behave like free waves under specific decay conditions.

Contribution

It introduces a new modified wave operator framework for wave equations with decaying repulsive potentials, extending classical scattering theory to these cases.

Findings

Modified wave operators exist for potentials decaying faster than |x|^{-1/3}

Regular wave operators exist if the potential decays faster than |x|^{-1}

Finite-energy solutions behave asymptotically like free solutions for decay rates between 1 and 2

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 $d\geq 3$. We combine the operator theory and the inward/outward energy theory to deduce a modified wave operator for repulsive potentials decaying like $|x|^{-\beta}$ with $\beta>1/3$. In particular the regular wave operator without modification exists if $\beta>1$. This implies that the asymptotic behaviour of finite-energy solutions to the wave equation $u_{tt} - \Delta u + |x|^{-\beta} u =0$ is similar to that of the solutions to the classic wave equation if $\beta \in (1,2)$.