# Spectral monodromy of small non-selfadjoint quantum perturbations of completely integrable Hamiltonians

## TL;DR

This paper introduces a spectral monodromy invariant derived from the spectrum of small non-selfadjoint perturbations of integrable quantum systems, linking spectral properties to classical topological invariants.

## Contribution

It defines a new spectral monodromy invariant that connects quantum spectral data with classical integrable system topology.

## Key findings

- Spectral monodromy obstructs global lattice structure of the spectrum.
- Spectral monodromy recovers classical monodromy of the integrable system.
- The invariant provides a bridge between quantum spectrum and classical topology.

## Abstract

We define a monodromy, directly from the spectrum of small non-selfadjoint perturbations of a selfadjoint semiclassical operator with two degrees of freedom, which is classically integrable. It is a combinatorial invariant that obstructs globally the existence of lattice structure of the spectrum, in the semiclassical limit. Moreover this spectral monodromy allows to recover a topological invariant (the classical monodromy) of the corresponding integrable system.