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

TL;DR

This paper develops a microlocal analysis framework to study the non-relativistic limit of the Klein--Gordon equation, providing asymptotic descriptions in different phase space regimes and unifying the analysis with a spacetime phase-space approach.

Contribution

It introduces a novel microlocal framework based on spacetime phase-space analysis for the Klein--Gordon equation's non-relativistic limit, extending previous methods.

Findings

Derived asymptotics from uniform estimates in phase space regimes

Unified analysis of different asymptotic regimes in Klein--Gordon equation

Framework applicable to time-dependent coefficients in wave equations

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 Schr\"odinger 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.