PARAMETRIC SPECTRAL CORRELATIONS OF DISORDERED-SYSTEMS IN THE FOURIERDOMAIN

A Fourier analysis of parametric level dynamics for random matrices pe riodically depending on a phase is developed. We demonstrate both theo retically and numerically that under very general conditions the corre lation C(phi) of level velocities is singular at phi=0 for any symmetr y class; the singularity is revealed by algebraic tails in Fourier tra nsforms, and is milder, the stronger the level repulsion in the chosen ensemble. The singularity is strictly connected with the divergence o f the second moments of level derivatives of appropriate order, and it s type is specified to leading terms for Gaussian, stationary ensemble s of orthogonal (GOE), unitary (GUE), and symplectic (GSE) types, and for the Gaussian ensemble of periodic banded random matrices, in which a breaking of symmetry occurs. In the latter case, we examine the beh avior of correlations in the diffusive regime and in the localized one as well, finding a singularity like that of pure GUE cases. In all th e considered ensembles we study the statistics of the Fourier coeffici ents of eigenvalues, which are Gaussian distributed for low harmonics, but not for high ones, and the distribution of kinetic energies.