Eigenvalue statistics in quantum ideal gases

TL;DR

This paper investigates the eigenvalue statistics of quantum ideal gases with energies defined by a power law, revealing number theoretic degeneracies and deviations from Poissonian distributions, with implications for classical periodic orbits.

Contribution

It introduces a recursion relation for the partition function and links eigenvalue degeneracies to number theory, providing new insights into spectral statistics of quantum gases.

Findings

Number theoretic degeneracies occur for integer >1

Deviations from Poissonian spacing distribution are observed

Classical length spectra relate to sums of powers and sums of cubes

Abstract

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.