Superintegrability of the Caged Anisotropic Oscillator

TL;DR

This paper introduces the superintegrable Caged Anisotropic Harmonic Oscillator, demonstrating its classical closed trajectories and quantum degeneracies, with novel polynomial integrals of motion related to rational frequency ratios.

Contribution

It presents a new superintegrable system with explicit polynomial integrals and analyzes its classical and quantum properties, expanding understanding of anisotropic oscillators with barriers.

Findings

All bound trajectories are closed and periodic.

The quantum eigenstates show SU(3) degeneracy patterns.

The system has five integrals of motion, including polynomial ones.

Abstract

We study "the Caged Anisotropic Harmonic Oscillator", which is a new example of a superintegrable, or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n), but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or cage. In 3 degrees, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m-1) and 2(l+n-1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting multiple copies of the fundamental pattern of the symmetry group SU(3).