Hamiltonian Monodromy via spectral Lax pairs

TL;DR

This paper introduces a spectral Lax pair method to analyze Hamiltonian Monodromy near focus-focus singularities, providing a new computational approach and applying it to models like the Jaynes-Cummings system and the spherical pendulum.

Contribution

It develops a spectral Lax pair framework to study Hamiltonian Monodromy, enabling straightforward computation of Monodromy matrices near singularities.

Findings

Spectral Lax pairs can be used to analyze Monodromy.

The method simplifies the computation of Monodromy matrices.

Applications to Jaynes-Cummings model and spherical pendulum demonstrate effectiveness.

Abstract

Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by a spectral Lax pair approach. From the Lax pair, we derive a Riemann surface which allows us to compute in a straightforward way the corresponding Monodromy matrix. The general results are applied to the Jaynes-Cummings model and the spherical pendulum.