Anyons: Pseudo-integrability, Symmetry reduction and Semiclassical Spectrum

TL;DR

This paper explores the classical and semiclassical properties of anyons with harmonic confinement, highlighting their pseudo-integrability, symmetry reduction, and implications for spectral analysis.

Contribution

It clarifies the concept of pseudo-integrability as symmetry reduction and investigates its effects on the semiclassical spectrum of anyons.

Findings

Pseudo-integrability relates to symmetry group reduction.

Semiclassical trace formula provides evidence of eigenvalues.

Discussion of trajectory ambiguities in classical models.

Abstract

At the classical level anyons with harmonic confinement are known to exhibit two important properties namely partial separability and pseudo-integrability. These stem from the fact that this system is locally identical to isotropic oscillator system but differs in the global topology of the phase space. We clarify the meaning of pseudo-integrability and show that it amounts to a definite reduction of the symmetry group. We elaborate on the role of the fundamental group of the phase space and pseudo-intrgrability in the context of periodic orbit theory and obtain evidence of non-exactly known eigenvalues from the semiclassical trace formula. We also discuss an ambiguity regarding the `half period' trajectories suggested by classical modeling and exhibited by the exactly known propagator for two anyons.