Semiclassical universality of parametric spectral correlations

We consider quantum systems with a chaotic classical limit that depends on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor K(l, x) which depends on a scaled parameter difference x. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the smallλexpansion of the form factor agrees with random matrix theory for systems with and without time reversal symmetry