Quantum footprints of Liouville integrable systems

TL;DR

This paper explores how to recover geometric information from the spectra of quantum integrable systems, introducing new concepts like good labellings for asymptotic lattices in complex cases.

Contribution

It introduces the notion of good labellings of asymptotic lattices for quantum integrable systems with multiple degrees of freedom, extending previous results.

Findings

In one degree of freedom, spectral data precisely determine geometric objects.

The concept of good labellings helps in understanding spectral geometry in higher degrees.

The paper advances the theoretical framework for inverse spectral problems in quantum integrable systems.

Abstract

We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labellings of asymptotic lattices.