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

Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy

TL;DR

The paper develops a spacetime microlocal framework to study the non-relativistic limit as $c\to\infty$ for the Klein--Gordon equation with time-dependent coefficients on asymptotically flat spacetimes, revealing two phase-space regimes: 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 Schrödinger equation. It introduces a two-regime microlocal strategy (parabolic and natural calculi) and a two-scale decomposition of solutions into positive/negative energy components, yielding an approximation $u \approx e^{-\,i c^2 t} v_- + e^{i c^2 t} v_+$ with $v_\pm$ solving Schrödinger equations with effective potentials; the approximation error is $O(1/c)$ and may be $O(1/c^2)$ under stronger structure. The main results prove uniform-in-$c$ invertibility for the Klein--Gordon operator between tailored Sobolev-type spaces and provide rigorous asymptotics for forced problems and the Cauchy problem, showing the Schrödinger-based approximant captures the leading-order KG behavior. The approach uses new calculi—natural calculus and resolved parabolic calculus—and normal operators $N(D_\pm)$ to connect KG and Schrödinger dynamics, with concrete applications to scalar electrodynamics in Minkowski space and Kerr-like spacetimes, and it outlines open questions and directions toward nonlinear extensions.

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.

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

TL;DR

The paper develops a spacetime microlocal framework to study the non-relativistic limit as for the Klein--Gordon equation with time-dependent coefficients on asymptotically flat spacetimes, revealing two phase-space regimes: 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 Schrödinger equation. It introduces a two-regime microlocal strategy (parabolic and natural calculi) and a two-scale decomposition of solutions into positive/negative energy components, yielding an approximation with solving Schrödinger equations with effective potentials; the approximation error is and may be under stronger structure. The main results prove uniform-in- invertibility for the Klein--Gordon operator between tailored Sobolev-type spaces and provide rigorous asymptotics for forced problems and the Cauchy problem, showing the Schrödinger-based approximant captures the leading-order KG behavior. The approach uses new calculi—natural calculus and resolved parabolic calculus—and normal operators to connect KG and Schrödinger dynamics, with concrete applications to scalar electrodynamics in Minkowski space and Kerr-like spacetimes, and it outlines open questions and directions toward nonlinear extensions.

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.

Paper Structure

This paper contains 49 sections, 356 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Natural units
  4. The ultra-static setting
  5. Reduction to the half-Klein--Gordon equation
  6. Non-relativistic limit for the half-Klein--Gordon equation
  7. The key approximation
  8. Challenges in the non- ultra-static or time-dependent settings
  9. The lack of half-Klein--Gordon operators
  10. Partitioning into positive/negative energy solutions
  11. The lack of local energy estimates
  12. Main theorem and summary of methods
  13. Microlocal methods
  14. Main results
  15. Analytic and geometric assumptions
  16. ...and 34 more sections

Figures (17)

  • Figure 1: \ref{['ex:FEM']}. The real parts of the functions $u,v,u-v$ are shown for three different values of $c$. Evidently, $u\approx v$ if $c\gg 1$. See \ref{['fig:FEMIm']} for the imaginary parts.
  • Figure 2: Continuation of \ref{['ex:FEM']}, showing the $L^2$-density of the Schrödinger solutions $v_\pm$. The solutions are concentrated near the paths $\gamma_\pm$ followed by a classical particle moving in the force-field $F=\mp \nabla V$ generated by the potential $V$ defined by \ref{['eq:V_ex']}; see \ref{['fig:paths']}.
  • Figure 3: Continuation of \ref{['ex:FEM']}, showing (left) a plot of the charge density $\rho= c^{-2}\Im(u^*\partial_t u)$ of $u$, for $c=6$, and (middle) the expected non-relativistic limit, $|v_+|^2-|v_-|^2$. This is just the difference of the two plots in \ref{['fig:FEM2']}. The difference (right) between $\rho$ and $|v_+|^2-|v_-|^2$ is highly oscillatory, but it is $O(1/c^2)$ as $c\to\infty$.
  • Figure 4: Continuation of \ref{['ex:FEM']}, now showing the non-relativistic limit of the current density (still at $c=6$). In this example, $d=1$, so we can view $\mathbf{j}$ as a scalar. It is not true that $\mathbf{j}$ agrees with the non-relativistic prediction $\mathbf{j}_{\mathrm{cl}}=\Im(v_+^*\partial_x v_+) + \Im(v_-^*\partial_x v_-)$ to leading order. Instead, the highly oscillatory, $\Omega(1)$ Zitterbewegung term needs to be subtracted from $\mathbf{j}$ before it converges in the $c\to\infty$ limit to $\mathbf{j}_{\mathrm{cl}}$.
  • Figure 5: The region $U=\operatorname{supp}(1-\chi_T(t/\langle r \rangle))$ in $\mathbb{M}$ where we are modifying the coefficients $V,A,W$ of the PDE in \ref{['thm:simplest']} in order to deduce that theorem from \ref{['thm:Cauchy']}. This does not intersect the region $\{|t|<T\}$ (dark gray).
  • ...and 12 more figures

Theorems & Definitions (16)

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  • ...and 6 more