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

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 $\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }$ with $\beta >1\mathrm{/}3$. 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 $u_{tt}-\mathrm{\Delta }u+\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }u=0$ is similar to that of the solutions to the classic wave equation if $\beta \in (1,2)$.