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

Abstract

The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schrödinger 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 the microlocal sheaf-theoretic approach of the author (2000). 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.