Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: elliptic case

TL;DR

This paper derives Bohr-Sommerfeld quantization conditions for 1D Toeplitz operators near a critical point on a Kähler surface, providing asymptotic eigenvalue expansions and analyzing an example on the torus.

Contribution

It introduces a novel normal form approach to establish Bohr-Sommerfeld conditions around a global minimum for semiclassical Toeplitz operators.

Findings

Derived asymptotic eigenvalue expansions near the critical point.

Extended Bohr-Sommerfeld conditions to the elliptic case.

Analyzed an explicit example on the two-dimensional torus.

Abstract

In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected K\"ahler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.