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

San Vũ Ngoc

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 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.

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 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.

Paper Structure

This paper contains 19 sections, 10 theorems, 51 equations.

Table of Contents

  1. EBK and WKB
  2. From WKB to Microlocal Analysis
  3. The lagrangian manifold viewpoint
  4. Semiclassical operators and microlocal solutions
  5. The Bohr-Sommerfeld cocycle
  6. Quasi-modes and eigenvalues
  7. Step 1. Discrete spectrum.
  8. Step 2. Quasimodes.
  9. Step 3. Microsupport.
  10. Step 4. Multiplicities.
  11. Caustics?
  12. Applications
  13. Inverse problems.
  14. Beyond EBK
  15. Smaller errors.
  16. ...and 4 more sections

Key Result

Proposition 5.1

Near any regular point $m$ of $H$, the space of microlocal solutions to $(P-E)\psi = 0$ is one-dimensional, more precisely: In both items the remainder ${\mathcal{O}}(\hbar^\infty)$ is locally uniform in $E$.

Theorems & Definitions (12)

  • Proposition 5.1: san-focus
  • Definition 5.3
  • Proposition 5.4
  • Proposition 5.5
  • Proposition 5.6
  • Theorem 6.1: Bohr-Sommerfeld eigenvalues
  • Theorem 6.3: Bohr-Sommerfeld eigenfunctions
  • Definition 7.1
  • Corollary 8.1: Density of the spectrum
  • Corollary 8.2: $\hbar$-behaviour of individual eigenvalues
  • ...and 2 more