Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911259845197824 |
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| author | Hassell, Andrew Jia, Qiuye Sussman, Ethan Vasy, Andras |
| author_facet | Hassell, Andrew Jia, Qiuye Sussman, Ethan Vasy, Andras |
| contents | This is the less technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrödinger equation. As shown in the companion paper, combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. In this paper, we derive asymptotics from those estimates. Our framework differs from those in previous works in that ours is based on spacetime phase-space analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_08724 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics Hassell, Andrew Jia, Qiuye Sussman, Ethan Vasy, Andras Analysis of PDEs Mathematical Physics Primary 35L05, 35L15. Secondary 35B25, 35Q40, 58J47, 58J50 This is the less technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrödinger equation. As shown in the companion paper, combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. In this paper, we derive asymptotics from those estimates. Our framework differs from those in previous works in that ours is based on spacetime phase-space analysis. |
| title | Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics |
| topic | Analysis of PDEs Mathematical Physics Primary 35L05, 35L15. Secondary 35B25, 35Q40, 58J47, 58J50 |
| url | https://arxiv.org/abs/2511.08724 |