Spectral monodromy of small non-selfadjoint perturbed operators: completely integrable or quasi-integrable case

TL;DR

This paper introduces the spectral monodromy, a new combinatorial invariant derived from the spectrum of small non-selfadjoint operators, analyzing cases of integrable and quasi-integrable classical flows in the semi-classical limit.

Contribution

It constructs the spectral monodromy invariant for non-selfadjoint operators and explores its properties in integrable and quasi-integrable classical flow scenarios.

Findings

Spectral monodromy effectively captures spectral properties in both cases.

The invariant distinguishes between integrable and quasi-integrable dynamics.

Results enhance understanding of spectral behavior under small non-selfadjoint perturbations.

Abstract

We build a combinatorial invariant, called the spectral monodromy from the spectrum of a non-selfadjoint h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. We treat small non-selfadjoint perturbation of selfadjoint h-pseudodifferential operators in two case: in the first, we assume that the classical flow of the unperturbed part is integrable; the second case, more interesting, when this flow is assumed to be quasi-integrable.