Diophantine tori and nonselfadjoint inverse spectral problems

TL;DR

This paper investigates how the eigenvalues of certain non-selfadjoint quantum operators encode detailed information about classical invariant tori, establishing uniqueness of the quantum normal form near Diophantine tori.

Contribution

It proves the uniqueness of the quantum Birkhoff normal form near a single Diophantine torus based on spectral data in a semiclassical setting.

Findings

Eigenvalues determine the quantum Birkhoff normal form near a Diophantine torus.

The normalization procedure and symmetries of the normal form are characterized.

The work extends inverse spectral theory to non-selfadjoint operators with classical invariant structures.

Abstract

We study a semiclassical inverse spectral problem based on a spectral asymptotics result of arXiv:math/0502032, which applies to small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. The eigenvalues in a suitable complex window have an expansion in terms of a quantum Birkhoff normal form for the operator near several Lagrangian tori which are invariant under the classical dynamics and satisfy a Diophantine condition. In this work we prove that the normal form near a single Diophantine torus is uniquely determined by the associated eigenvalues. We also discuss the normalization procedure and symmetries of the quantum Birkhoff normal form near a Diophantine torus.