Parametric spectral correlations of disordered systems in the Fourier domain

A Fourier analysis of parametric level dynamics for random matrices periodically depending on a phase is developed. We demonstrate both theoretically and numerically that under very general conditions the correlation C(y) of level velocities is singular at y=0 for any symmetry class; the singularity is revealed by algebraic tails in Fourier transforms, and is milder, the stronger the level repulsion in the chosen ensemble. The singularity is strictly connected with the divergence of the second moments of level derivatives of appropriate order, and its type is specified to leading terms for Gaussian, stationary ensembles of orthogonal (GOE), unitary (GUE), and symplectic (GSE) types, and for the Gaussian ensem- ble of periodic banded random matrices, in which a breaking of symmetry occurs. In the latter case, we examine the behavior of correlations in the diffusive regime and in the localized one as well, finding a singularity like that of pure GUE cases. In all the considered ensembles we study the statistics of the Fourier coefficients of eigenvalues, which are Gaussian distributed for low harmonics, but not for high ones, and the distribution of kinetic energies.