Arithmetical degeneracies in simple quantum systems

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Abstract
The authors examine the 'accidental' level degeneracies occurring in the quantum mechanical problem of a free particle moving in a polyhedral box, when the problem is integrable. Some remarkable properties of the distribution of degeneracies are studied in several two-, three- and four-dimensional examples and are related to well known problems of number theory. The numerical results of exact enumerations are compared with analytical predictions, or with conjectured expressions in some cases. They consider in particular the asymptotic scaling form of the degeneracy distribution up to some maximal energy E, and the maximal degeneracy occurring for energies less than some given E.