Semiclassical inverse spectral theory for singularities of focus-focus type

TL;DR

This paper demonstrates that, under certain conditions, the joint spectrum of a quantum integrable system uniquely determines the classical foliation structure near a focus-focus singularity, linking quantum spectra to classical geometry.

Contribution

It establishes a semiclassical inverse spectral result for focus-focus singularities, assuming Bohr-Sommerfeld rules, applicable to various quantum operators on symplectic manifolds.

Findings

Joint spectrum determines classical foliation near focus-focus singularity

Applicable to h-pseudodifferential and Berezin-Toeplitz operators

Results connect quantum spectra with classical integrable systems

Abstract

We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.