Spectral Statistics and Dynamical Localization: Sharp Transition in a Generalized Sinai Billiard

We consider a Sinai billiard where the usual hard disk scatterer is replaced by a repulsive potential with $V\left(r\right)\sim \lambda r^{-\alpha }$ close to the origin. Using periodic orbit theory and numerical evidence we show that its spectral statistics tends to Poisson statistics for large energies when $\alpha <2\mathrm{}$ and to Wigner-Dyson statistics when $\alpha >2\mathrm{}$, while for $\alpha =2\mathrm{}$ it is independent of energy, but depends on $\lambda$. We apply the approach of Altshuler and Levitov [Phys. Rep. 288, 487 (1997)] to show that the transition in the spectral statistics is accompanied by a dynamical localization-delocalization transition. This behavior is reminiscent of a metal-insulator transition in disordered electronic systems.