# Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems

## TL;DR

This paper develops a framework for labeling joint spectra of quantum integrable systems to define and compute the quantum rotation number, linking quantum and classical spectral invariants in the semiclassical limit.

## Contribution

It introduces good labellings for asymptotic lattices and provides a constructive method to compute the quantum rotation number from joint spectra.

## Key findings

- Quantum rotation number converges to classical rotation number in the semiclassical limit.
- Constructive algorithm for detecting labellings in two degrees of freedom.
- Application to semitoric systems yields natural formulas.

## Abstract

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.