  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 $K(\tau,x)$ 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 $\tau$ expansion
of the form factor agrees with Random Matrix Theory for systems with and
without time reversal symmetry.
