Commensurability effects on the spectra of integrable systems

Abstract

The eigenvalue spectra of two simple integrable systems-a three-dimensional anisotropic harmonic oscillator (AHO) and a particle in a cuboid-are studied as the aspect ratios are changed from rational to irrational values. For rational aspect ratios (commensurable case) there are large number-theoretic degeneracies which grow with energy. These degeneracies disappear when the aspect ratios are irrational (incommensurable case). However, for a given set of rational aspect ratios, commensurable behavior ensues only at large energies. The number of distinct eigenvalues per unit energy interval, which is indicative of the degree of degeneracies, is seen to approach a constant asymptotically for rational aspect ratios. The asymptotic constant is determined for all sets of aspect ratios for the AHO and is estimated semi-analytically for several sets of aspect ratios for the cuboid. The crossover from incommensurable to commensurable behaviour is studied by following a sequence of rational aspect ratios which approaches an irrational limit. A scaling form the crossover is suggested and explored numerically. A clear indication of scaling is seen for the AHO, while the evidence is suggestive but not definitive for the cuboid. The distribution of energy level spacings in the commensurable regime is determined for the AHO and a sequence of cuboids.