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Semiclassical universality of parametric spectral correlations

We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor $K(τ,x)$ 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 small $τ$ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry.