Partially strong transparency conditions and a singular localization method in geometric optics

TL;DR

This paper develops a weaker transparency condition and a singular localization method to analyze the stability of WKB solutions in geometric optics, enabling long-term approximation of Klein-Gordon equations by Schrödinger equations.

Contribution

Introduces a compatible condition weaker than strong transparency and a singular localization technique for stability analysis in geometric optics.

Findings

Proves stability of WKB solutions over long time intervals.

Demonstrates long-time approximation of Klein-Gordon by Schrödinger equations.

Provides a new analytical framework near resonances.

Abstract

This paper focuses on the stability analysis of WKB approximate solutions in geometric optics with the absence of strong transparency conditions. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. As an application, we show the long time approximation of Klein-Gordon equations by Schr\"odinger equations in the non-relativistic limit regime.