Modified wave operators and scattering for linear wave equations with a repulsive potential
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910953012985856 |
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| author | Fan, Boya Shen, Ruipeng |
| author_facet | Fan, Boya Shen, Ruipeng |
| contents | 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 $β\in (1,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modified wave operators and scattering for linear wave equations with a repulsive potential Fan, Boya Shen, Ruipeng Analysis of PDEs Mathematical Physics 35L05(Primary), 47B92(Secondary) 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 $β\in (1,2)$. |
| title | Modified wave operators and scattering for linear wave equations with a repulsive potential |
| topic | Analysis of PDEs Mathematical Physics 35L05(Primary), 47B92(Secondary) |
| url | https://arxiv.org/abs/2505.12838 |