Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915490244329472 |
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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 more 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 Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $Ψ_{\natural}$ (a variant of the semiclassical scattering calculus), $Ψ_{\natural\mathrm{res}}$, and $Ψ_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates Hassell, Andrew Jia, Qiuye Sussman, Ethan Vasy, Andras Analysis of PDEs Mathematical Physics Primary 35L05, 35L15. Secondary 35B25, 35Q40, 58J47, 58J50 This is the more 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 Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $Ψ_{\natural}$ (a variant of the semiclassical scattering calculus), $Ψ_{\natural\mathrm{res}}$, and $Ψ_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box. |
| title | Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates |
| topic | Analysis of PDEs Mathematical Physics Primary 35L05, 35L15. Secondary 35B25, 35Q40, 58J47, 58J50 |
| url | https://arxiv.org/abs/2509.09518 |