  We study the energy levels of non-interacting electrons confined to move in
two-dimensional billiard regions and having a spin-dependent dynamics due to a
finite Rashba spin splitting. The Green's function for such Rashba billiards is
constructed analytically and used to find the area and perimeter contributions
to the density of states, as well as the smooth counting function. We show
that, in contrast to systems with spin-rotational invariance, Rashba billiards
always possess a negative energy spectrum. A semi-classical analysis is
presented to interpret the singular behavior of the density of states at
certain negative energies. Our detailed analysis of the spin structure of
Rashba billiards reveals a finite out-of-plane spin projection for electron
eigenstates.
