Power spectrum of the fluctuation of the spectral staircase function

¹ School of Engineering, Monash University - Selangor, Malaysia
² School of Physics, University of Sydney - Sydney, Australia
³ Max Planck Institute for the Physics of Complex Systems - Dresden, Germany

Received: 9 August 2006
Accepted: 23 October 2006

Abstract

The one-sided power spectrum of the fluctuation and of the spectral staircase function, for respectively the original and unfolded spectrum, from its smooth average part is numerically estimated for Poisson spectrum and spectra of three Gaussian-random matrices: real symmetric, complex Hermitian, and quaternion-real Hermitian. We found that the power spectrum of and is (brown) for Poisson spectrum but (Lorentzian) for all three random matrix spectra. This result and the Berry-Tabor and Bohigas-Giannoni-Schmit conjectures imply the following conjecture: the power spectrum of and is brown for classically integrable systems but Lorentzian for classically chaotic systems. Numerical evidence in support of this conjecture is presented.