On the wave equation with Coulomb potential

TL;DR

This paper investigates the wave equation with a Coulomb potential in dimensions three and higher, analyzing the asymptotic behavior of solutions, their energy distribution, and scattering properties, especially for radial finite-energy solutions.

Contribution

It provides the first detailed analysis of the global behavior and scattering of solutions to the wave equation with Coulomb potential in higher dimensions.

Findings

Describes asymptotic behavior and scattering profiles of linear Coulomb wave solutions.

Establishes global existence and scattering for radial finite-energy solutions.

Develops radial Strichartz estimates for the Coulomb wave equation.

Abstract

Wave/Schr\"{o}dinger equations with potentials naturally originates from both the quantum physics and the study of nonlinear equations. The distractive Coulomb potential is a quantum mechanical description of distractive Coulomb force between two particles with the same charge. The spectrum of the operator $-\Delta +1/|x|$ is well known and there are also a few results on the Strichartz estimates, local and global well-posedness and scattering result about the nonlinear Schr\"{o}dinger equation with a distractive Coulomb potential. In the contrast, much less is known for the global and asymptotic behaviour of solutions to the corresponding wave equations with a Coulomb potential. In this work we consider the wave equation with a distractive Coulomb potential in dimensions $d\geq 3$. We first describe the asymptotic behaviour of the solutions to the linear homogeneous Coulomb wave equation, especially their energy distribution property and scattering profiles, then show that the radial finite-energy solutions to suitable defocusing Coulomb wave equation are defined for all time and scatter in both two time directions, by establishing a family of radial Strichartz estimates and combining them with the decay of the potential energy.