Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator

TL;DR

This paper demonstrates quantum monodromy in the 3D isotropic harmonic oscillator's joint spectrum, revealing fundamental topological obstructions to assigning global quantum numbers due to classical focus-focus singularities.

Contribution

It establishes the presence of quantum monodromy in a well-known quantum system through spectral analysis and links it to classical Hamiltonian monodromy via integrable system singularities.

Findings

Quantum monodromy occurs in the joint spectrum at high energies.

Classical analysis shows a focus-focus singularity causes the monodromy.

Quantum numbers cannot be globally assigned due to monodromy.

Abstract

The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators, one of which is the Hamiltonian. We show that the joint spectrum of the Hamilton operator, the $z$ component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has quantum monodromy for sufficiently large energies. This means that one cannot globally assign quantum numbers to the joint spectrum. The effect can be classically explained by showing that the corresponding Liouville integrable system has a non-degenerate focus-focus point, and hence Hamiltonian monodromy.