Fluctuations in quantum spectra are known to exhibit a high degree of universality which reflects the nature – regular or chaotic – of the underlying classical dynamics. Following Berry and Tabor (1977), statistics of level spacings in generic quantum systems with completely integrable classical dynamics is expected to mimic statistics of waiting times in a Poisson point process. For generic quantum systems with completely chaotic classical dynamics, Bohigas, Giannoni and Schmit (1984) conjectured that the level spacing distribution coincides with predictions of the Random Matrix Theory.
Recently, an alternative characterization of eigenvalue fluctuations was suggested by Relaño et. al. (2002). Interpreting long eigenlevel sequences as discrete-time random processes, these authors argued that the power spectrum of energy level fluctuations exhibits the 1/w behavior for completely chaotic and 1/w2 behavior for completely regular quantum systems.
In this talk, we present a rigorous theory of the power spectrum of energy level fluctuations in fully chaotic quantum structures. Focusing on systems with broken time-reversal symmetry, we employ a finite-N random matrix theory to derive an exact multidimensional integral representation of the power spectrum. The N→∞ limit of the exact solution furnishes our main result – a universal, parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent. Extensive numerics lends further support to our theory which invalidates a traditional assumption that the power spectrum is merely determined by the spectral form factor of a quantum system.