The Classical and Quantum Monodromy of the Champagne bottle potential

TL;DR

This paper investigates the classical and quantum monodromy of the Champagne bottle potential, revealing how spectral analysis of a perturbed operator can detect geometric modifications in phase space caused by monodromy.

Contribution

It introduces a novel spectral method to detect classical monodromy effects in quantum systems through non-selfadjoint perturbations.

Findings

Spectral signatures reveal classical monodromy effects.

New approach links geometric invariants to spectral properties.

Applicable to semiclassical operators with Champagne bottle Hamiltonian.

Abstract

The Champagne bottle is one of the simplest and typical examples of Liouville integrable systems that exhibit a non-trivial classical monodromy. This geometrical invariant perturbs globally the existence of action-angle coordinates on the phrase space. However, our work shows a new way to detect the geometric modification on the phrase space by looking at the spectrum of a single operator, that is small non-selfadjoint perturbation of a selfadjoint semiclassical operator accepting the Hamiltonian of the Champagne bottle as its principal symbol.