WKB for semiclassical operators: How to fly over caustics (and more)

TL;DR

This paper reviews and unifies the Maslov-WKB semiclassical method using microlocal sheaf theory, providing rigorous proofs of quantization conditions and addressing caustic-related breakdowns in semiclassical analysis.

Contribution

It offers a unified, rigorous microlocal sheaf-theoretic framework for the Maslov-WKB method, improving understanding of semiclassical quantization near caustics.

Findings

Rigorous proof of Bohr-Sommerfeld quantization conditions

Unified microlocal sheaf-theoretic treatment of WKB method

Extension to pseudodifferential and Berezin Toeplitz operators

Abstract

The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schr\"odinger equation lies at the origin of the spectacular development of microlocal and semiclassical analysis. When used naively, the approach appears to break down at caustics, but Maslov showed how a simple generalization could overcome this difficulty. In this paper, after a partial historical review, we take advantage of more recent advances in microlocal analysis to present a unified treatment of this generalized Maslov-WKB method, using a microlocal sheaf-theoretic approach. This framework provides a rigorous proof of the Bohr Sommerfeld Einstein Brillouin Keller quantization conditions for the eigenvalues of general semiclassical operators (pseudodifferential and Berezin Toeplitz) in one degree of freedom. We also review some applications and extensions.