The Scaling Limit of the Critical One-Dimensional Random Schrodinger Operator

We consider two models of one-dimensional discrete random Schrodinger operators (H-n psi)(l) = psi(l-1) + psi(l+1) + upsilon(l)psi(l), in the cases and . Here omega (k) are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.