Eigenvalue statistics in quantum ideal gases

1998/09/04 by Bruno Eckhardt, B. Eckhardt, Eckhardt, B.
Physics and Astronomy · #Advanced Chemical Physics Studies #Chaotic Dynamics (nlin.CD) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Quantum chaos and dynamical systems #chao-dyn #nlin.CD

paper · pdf · doi:10.48550/arxiv.chao-dyn/9809005

16 pages incl. figures, ignore warnings

arxiv created 1998/09/04 · openalex publication_date 1998/09/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The eigenvalue statistics of quantum ideal gases with single particle energies e_n=n^α 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 α>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 α/(α-1). In particular, for α=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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