  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 $C(\varphi )$ of level velocities is singular at $\varphi =0$ 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 2nd
moments of level derivatives of appropriate order, and its type is specified to
leading terms for Gaussian, stationary ensembles of GOE, GUE, 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 behaviour 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.
