1210.4224 (Shinichi Kotani et al.)

Level statistics of one-dimensional Schrödinger operators with random
decaying potential    [PDF]

Shinichi Kotani, Fumihiko Nakano

We study the level statistics of one-dimensional Schr\"odinger operator with random potential decaying like $x^{-\alpha}$ at infinity. We consider the point process $\xi_L$ consisting of the rescaled eigenvalues and show that : (i)(ac spectrum case) for $\alpha > \frac 12$, $\xi_L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii)(critical case) for $\alpha = \frac 12$, $\xi_L$ converges to the limit of the circular $\beta$-ensemble.

View original: http://arxiv.org/abs/1210.4224