# Level statistics of one-dimensional Schrödinger operators with random decaying potential

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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 : iac spectrum case for $\alpha \frac 12$, $\xi L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. iicritical case for $\alpha = \frac 12$, $\xi L$ converges to the limit of the circular $\beta$-ensemble.

Author: Shinichi Kotani; Fumihiko Nakano

Source: https://archive.org/