  We study Andreev billiards of box and disk geometries by matching the wave
functions at the interface of the normal and the superconducting region using
the exact solutions of the Bogoliubov-de Gennes equation. The mismatch in the
Fermi wavenumbers and the effective masses of the normal system and the
superconductor, as well as the tunnel barrier at the interface are taken into
account. A Weyl formula (for the smooth part of the counting function of the
energy levels) is derived. The exact quantum mechanical calculations show
equally spaced singularities in the density of states. Based on the
Bohr-Sommerfeld quantization rule a semiclassical theory is proposed to
understand these singularities. For disk geometries two kinds of states can be
distinguished: states either contribute through whispering gallery modes or are
Andreev states strongly coupled to the superconductor. Controlled by two
relevant material parameters, three kinds of energy spectra exist in disk
geometry. The first is dominated by Andreev reflections, the second, by normal
reflections in an annular disk geometry. In the third case the coherence length
is much larger than the radius of the superconducting region, and the spectrum
is identical to that of a full disk geometry.
