Power spectrum of the fluctuation of the spectral staircase function

The one-sided power spectrum $P(f)$ of the fluctuation $N_{fluc} (E)$ and $N_{fluc}(\varepsilon)$ 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 $N_{fluc} (\varepsilon)$ is $a/f^2$ (brown) for Poisson spectrum but $c/(1+ df^2)$ (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 $N_{fluc} (\varepsilon)$ is brown for classically integrable systems but Lorentzian for classically chaotic systems. Numerical evidence in support of this conjecture is presented.