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Autocorrelation Functions and Form Factors of Statistically Independent Superposition of Spectral Sequences

First I derive the power spectrum (of the cumulative or integrated level density) for the regular (Poissonian) energy spectrum which is 1/t2, and for the fully chaotic one, which is 1/t, as was observed numerically by Relano et al. [2002], and recently discussed independently by Faleiro et al. [2004]. The statement refers to small values of t ≤ 1, i.e. for times smaller than Heisenberg time. For t ≫ 1 it is always 1/t2. Then I analyze the autocorrelation function for spectrum which is a superposition of statistically independent spectral sequences and derive the exact additivity formula for the autocorrelation function, and consequently for the form factor of the density of energy levels and of the cumulative (integrated) density of energy levels (the form factor is the Fourier transform of the corresponding autocorrelation function). Therefore this theory provides prediction for the deep semiclassical regime of sufficiently small effective Planck constant ℏeff. However, in not sufficiently deep semiclassical regime (not sufficiently small ℏeff) we see a deviation from this behavior, as recently demonstrated numerically by Gomez et al. [2004] in a billiard system [Robnik, 1983] where a power law behavior 1/tα is found, and the exponent α goes continuously from 2 to 1, as the system's dynamics goes from complete integrability (Poisson) to full chaoticity (GOE and GUE), respectively.