Ehrenfest-time-dependent excitation gap in a chaotic Andreev billiard
  A semiclassical theory is developed for the appearance of an excitation gap in a ballistic chaotic cavity connected by a point contact to a superconductor. Diffraction at the point contact is a singular perturbation in the limit $\hbar\to 0$, which opens up a gap $E_{\rm gap}$ in the excitation spectrum. The time scale $\hbar/E_{\rm gap}\propto\alpha^{-1}\ln\hbar$ (with $\alpha$ the Lyapunov exponent) is the Ehrenfest time, the characteristic time scale of quantum chaos. 
I. Adagideli and C. W. J. Beenakker (Instituut-Lorentz, Universiteit   Leiden)