Time scales of ergodic classical dynamics in quantal spectra

The 400 first energy levels of the quantal Sinai billiard has been calculated for different small central disk radii. The spectral rigidity over sequences of L levels, Δ${\mathrm{¯}}_3$(L), agrees with that of the Gaussian orthogonal ensemble of matrices (GOE), for L${\mathit{L}}_{\mathit{d}}$. However, for larger intervals Δ${\mathrm{¯}}_3$ exceeds that of the GOE, and increases linearly. The value of the breakpoint ${\mathit{L}}_{\mathit{d}}$ grows with the radius, probably reflecting the increasing chaos in the classical Sinai billiard. It is suggested that in the semiclassical limit such a behavior is generic for systems with a K-mixing classical analog.