# Combinatorial invariant of nearly integrable Hamiltonians

## TL;DR

This paper introduces a spectral monodromy for nearly integrable quantum Hamiltonians, linking spectral properties to the topological structure of classical invariant tori, thus revealing dynamical modifications.

## Contribution

It defines a new spectral monodromy directly from the spectrum of perturbed quantum Hamiltonians, connecting quantum spectral data with classical topological invariants.

## Key findings

- Spectral monodromy detects topological changes in classical dynamics.
- Monodromy aligns with KAM invariant tori monodromy.
- Method applies to small non-selfadjoint perturbations.

## Abstract

We work with small non-selfadjoint perturbations of a selfadjoint quantum Hamiltonian with two degrees of freedom, assuming that the principal symbol of the selfadjoint part is (classically) a nearly integrable system, together with a globally non-degenerate condition. We define a monodromy directly from the spectrum of such an operator, in the semiclassical limit. Moreover, this spectral monodromy allows to detect the topological modification on the dynamics of the nearly integrable system. It can be identified with the monodromy for KAM invariant tori of the nearly integrable system.