  We consider the distribution of eigenvalues for the wave equation in annular
(electromagnetic or acoustic) ray-splitting billiards. These systems are
interesting in that the derivation of the associated smoothed spectral counting
function can be considered as a canonical problem. This is achieved by
extending a formalism developed by Berry and Howls for ordinary (without
ray-splitting) billiards. Our results are confirmed by numerical computations
and permit us to infer a set of rules useful in order to obtain Weyl formulas
for more general ray-splitting billiards.
