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

Abstract

We consider two models of one-dimensional discrete random Schrödinger operators

$$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$$

, ${\psi_0=\psi_{n+1}=0}$ in the cases ${ v_k=\sigma \omega_k/\sqrt{n}}$ and ${ v_k=\sigma \omega_k/ \sqrt{k}}$ . Here ω _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 β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.