Superintegrability of the caged anisotropic oscillator

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 a cage. In three degrees of freedom, 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 l2m2n2 copies of the fundamental pattern of the symmetry group SU(3).