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Spectral Statistics and Dynamical Localization: sharp transition in a generalized Sinai billiard

Ulrich Gerland
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Abstract

We consider a Sinai billiard where the usual hard disk scatterer is replaced by a repulsive potential with $V(r)\simλr^{-α}$ 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 $α<2$ and to Wigner-Dyson statistics when $α>2$, while for $α=2$ it is independent of energy, but depends on $λ$. We apply the approach of Altshuler and Levitov [Phys. Rep. {\bf 288}, 487 (1997)] to show that the transition in the spectral statistics is accompanied by a dynamical localization-delocalization transition. This behaviour is reminiscent of a metal-insulator transition in disordered electronic systems.

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Ulrich Gerland. 1999-07-12. Spectral Statistics and Dynamical Localization: sharp transition in a generalized Sinai billiard. https://doi.org/10.1103/physrevlett.83.1139

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