On the wave equation with Coulomb potential
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913595354251264 |
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| author | Li, Liang Luo, Shenghao Shen, Ruipeng |
| author_facet | Li, Liang Luo, Shenghao Shen, Ruipeng |
| contents | Wave/Schrödinger 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 $-Δ+1/|x|$ 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 Schrödinger 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02070 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the wave equation with Coulomb potential Li, Liang Luo, Shenghao Shen, Ruipeng Analysis of PDEs 35L05 Wave/Schrödinger 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 $-Δ+1/|x|$ 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 Schrödinger 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. |
| title | On the wave equation with Coulomb potential |
| topic | Analysis of PDEs 35L05 |
| url | https://arxiv.org/abs/2412.02070 |