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

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

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