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

Boya Fan, Ruipeng Shen

TL;DR

The paper develops a rigorous scattering theory for the linear wave equation with repulsive, inverse-power potentials $V(x)=|x|^{-\beta}$ by reducing to one-dimensional half-line problems and leveraging inward/outward energy theory. It constructs modified wave operators via a phase-shifted Fourier multiplier $\mathbf U(t)$ and a spectral-analytic operator $\mathbf W$, proving unitary bijections between energy spaces for decay $\beta>1/3$, and shows that faster decay ($\beta>1$ or $q\in L^1$) recovers the unmodified operator and free-like asymptotics. Extending to $d\ge 3$ via spherical harmonic decomposition, the authors obtain a parallel modified-operator framework for radial potentials $V(x)=q(|x|)$ and establish dispersion rates characterized by the time-accumulated potential $Q_1(t)=\int_1^t q(s)ds$. The results reveal when and how the asymptotics of finite-energy solutions align with those of the classical wave equation, and quantify energy localization in moving shells for various decay regimes. Overall, the work provides a detailed, phase-corrected scattering description for wave equations with repulsive decays and extends 1D techniques to higher dimensions with explicit phase corrections and energy-flux analyses.

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 $|x|^{-β}$ 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 $β\in (1,2)$.

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

TL;DR

The paper develops a rigorous scattering theory for the linear wave equation with repulsive, inverse-power potentials by reducing to one-dimensional half-line problems and leveraging inward/outward energy theory. It constructs modified wave operators via a phase-shifted Fourier multiplier and a spectral-analytic operator , proving unitary bijections between energy spaces for decay , and shows that faster decay ( or ) recovers the unmodified operator and free-like asymptotics. Extending to via spherical harmonic decomposition, the authors obtain a parallel modified-operator framework for radial potentials and establish dispersion rates characterized by the time-accumulated potential . The results reveal when and how the asymptotics of finite-energy solutions align with those of the classical wave equation, and quantify energy localization in moving shells for various decay regimes. Overall, the work provides a detailed, phase-corrected scattering description for wave equations with repulsive decays and extends 1D techniques to higher dimensions with explicit phase corrections and energy-flux analyses.

Abstract

In this work we consider the wave equation with a repulsive potential, either on the half line or the Euclidean space with . We combine the operator theory and the inward/outward energy theory to deduce a modified wave operator for repulsive potentials decaying like with . In particular the regular wave operator without modification exists if . This implies that the asymptotic behaviour of finite-energy solutions to the wave equation is similar to that of the solutions to the classic wave equation if .
Paper Structure (44 sections, 20 theorems, 403 equations)

This paper contains 44 sections, 20 theorems, 403 equations.

Table of Contents

  1. Introduction
  2. Topic and background
  3. Schrödinger equations
  4. Wave equations
  5. Main idea
  6. The main ingredients
  7. Assumptions and notations
  8. Self-adjoint operators
  9. Fourier transforms
  10. Linear propagation operators
  11. Sobolev spaces
  12. The notation $\lesssim$
  13. Main results
  14. The Structure of this work
  15. Inward/outward energy theory
  16. ...and 29 more sections

Key Result

Theorem 1.6

Assume that $q(x)$ is a type I, II or III repulsive potential with a decay rate $\beta>1/3$. Let $\mathbf{A} = -{\rm d}^2/{\rm d} x^2 + q(x)$ be the self-adjoint operator with zero boundary condition and $\vec{\mathbf{S}}_q$ be its wave propagation operator. We define with phase shift function Then the modified wave operator defined by the strong limit in $\dot{H}^1({\mathbb R}^+)\times L^2({\ma

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6: Wave operator on the half-line
  • Theorem 1.7: Wave operator in the high-dimensional case
  • Remark 1.8
  • Corollary 1.9: dispersion rate in higher dimensions
  • Remark 1.10
  • ...and 45 more