Level Statistics for Random Schroedinger Operators

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We study the level statistics problem for 1-dimensional Schroedinger operators with random decaying potential. That is, we restrict our Hamiltonian on a bounded interval, consider a point process composed of the rescaled eigenvalues of this truncated Hamiltonian, and then study its infinite volume limit. The result is : (i) rapid decay case : the point process converges to the clock process, and second order asymptotics converges to Gaussian, (ii) critical decay case : the point process converges to that of the circular beta ensemble, as well as that of the Gaussian beta ensemble, thus yielding the coincidence of these two processes as a corollary.