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arXiv · 2511.08724

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics

Andrew Hassell·Qiuye Jia·Ethan Sussman·Andras Vasy
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Abstract

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.

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Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy. 2025-11-11. Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics. https://arxiv.org/abs/2511.08724

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