Spectrum of the Andreev Billiard and Giant Fluctuations of the Ehrenfest Time

The density of states in the semiclassical Andreev billiard is theoretically studied and shown to be determined by the fluctuations of the classical Lyapunov exponent $\lambda$. The rare trajectories with a small value of $\lambda$ give rise to an anomalous increase of the Ehrenfest time ${\tau }_E\approx \mathrm{\mid }\ln \mathrm{\hslash }\mathrm{\mid }\mathrm{/}\lambda$ and, consequently, to the appearance of Andreev levels with small excitation energy. The gap in spectrum is obtained and fluctuations of the value of the gap due to different positions of superconducting lead are considered.