Power spectra and autocovariances of level spacings beyond the Dyson conjecture.

Introduced in the early days of random matrix theory, the autocovariances δI_{k}^{j}=cov(s_{j},s_{j+k}) of level spacings {s_{j}} accommodate detailed information on the correlations between individual eigenlevels. It was first conjectured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay δI_{k}^{j}≈-1/βπ^{2}k^{2}, where β is the symmetry index. In this Letter, we establish an exact link between the autocovariances of level spacings and their power spectrum, and show that, for β=2, the latter admits a representation in terms of a fifth Painlevé transcendent.

Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems.

We present a nonperturbative analysis 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 the main result of this study-a universal, parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent.