Level statistics of one-dimensional Schr\"odinger operators with random decaying potential

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 $\alpha >\frac{1}{2}$, ${\xi }_L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii)(critical case) for $\alpha =\frac{1}{2}$, ${\xi }_L$ converges to the limit of the circular $\beta$-ensemble.