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

2025/05/19 by Fan, Boya, Shen, Ruipeng
#35L05(Primary) #47B92(Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2505.12838

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≥ 3. We combine the operator theory and the inward/outward energy theory to deduce a modified wave operator for repulsive potentials decaying like |x|^-β with β>1/3. In particular the regular wave operator without modification exists if β>1. This implies that the asymptotic behaviour of finite-energy solutions to the wave equation u_tt - Δu + |x|^-β u =0 is similar to that of the solutions to the classic wave equation if β∈ (1,2).

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