Europhys. Lett., 76 (6), pp. 1043-1049 (2006)
DOI: 10.1209/epl/i2006-10392-1

Power spectrum of the fluctuation of the spectral staircase function

Boon Leong Lan¹, Jin Aun Ng² and M. S. Santhanam<sup>3

1</sup>  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
published online 24 November 2006

Abstract
The one-sided power spectrum P(f) of the fluctuation N_fluc (E) 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 N_fluc (E) and is a/f² (brown) for Poisson spectrum but c/(1+ df²) (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 N_fluc (E) and is brown for classically integrable systems but Lorentzian for classically chaotic systems. Numerical evidence in support of this conjecture is presented.

PACS
05.45.Mt - Quantum chaos; semiclassical methods.
05.45.Tp - Time series analysis.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.

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