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arXiv · 2412.02070

On the wave equation with Coulomb potential

Liang Li·Shenghao Luo·Ruipeng Shen
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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.

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Liang Li, Shenghao Luo, Ruipeng Shen. 2024-12-03. On the wave equation with Coulomb potential. https://arxiv.org/abs/2412.02070

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