  The semiclassical theory for billiards with mixed boundary conditions is
developed and explicit expressions for the smooth and the oscillatory parts of
the spectral density are derived. The parametric dependence of the spectrum on
the boundary condition is shown to be a very useful diagnostic tool in the
semiclassical analysis of the spectrum of billiards. It is also used to check
in detail some recently proposed parametric spectral statistics. The methods
are illustrated in the analysis of the spectrum of the Sinai billiard and its
parametric dependence on the boundary condition on the dispersing arc.
