Density of states in classical and quantum Andreev billiards.

We study the density of states in a small chaotic metal grain connected to a superconductor. Changing the strength of disorder, we obtain the crossover from the gapless regime to a regime with a hard gap. In the dirty limit we reproduce the expression for the density of states derived within random matrix theory [Melsen et.al., Europhys. Lett. 35, 7 (1996)]. We discuss the connection of our results to the problem of the density of states in clean chaotic Andreev billiards. Particularly, we demonstrate that the energy spectrum is different for the classical and quantum cases not only at small energy, but also at higher energies. This discrepancy is due to repeating trajectories, which exist in the classical limit but are destroyed by quantum diffraction in the quantum limit.