Partially strong transparency conditions and a singular localization   method in geometric optics
  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. 
Yong Lu, Zhifei Zhang
math.AP