Eigenvalue statistics in quantum ideal gases

Details Eigenvalue statistics in quantum ideal gases Eigenvalue statistics in quantum ideal gases

1998-09-04

arxiv.org/abs/chao-dyn/9809005v1

Nonlinear Sciences

Chaotic Dynamics

16 pages incl. figures, ignore warnings

Scientific paper

The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer $\alpha>1$ one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power $\alpha/(\alpha-1)$. In particular, for $\alpha=3/2$, the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function.

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