Superintegrability of the Caged Anisotropic Oscillator

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...

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Bibliographic Details
Published in:arXiv.org 2008-08
Main Authors: Evans, N W, Verrier, P E
Format: Article
Language:English
Subjects:
Anisotropy
Eigenvectors
Harmonic oscillators
Integrals
Oscillators
Planes
Polynomials
Online Access:Get full text
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Summary: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).
ISSN:2331-8422
DOI:10.48550/arxiv.0808.2146