The scaling limit of the critical one-dimensional random Schrodinger operator (1107.3058v1)

Abstract: We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)l ={\psi}{l-1}+{\psi}{l +1}+v_l {\psi}_l, {\psi}_0={\psi}{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}. 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.