The scaling limit of the critical one-dimensional random Schrodinger operator

TL;DR

This paper analyzes the scaling limits of eigenvectors and eigenvalues of one-dimensional random Schrödinger operators at criticality, revealing delocalization, stochastic differential equation limits, and connections to random matrix theory.

Contribution

It introduces two models of critical one-dimensional random Schrödinger operators and characterizes their eigenvalue and eigenvector behavior, including stochastic limits and eigenvalue statistics.

Findings

Eigenvectors are delocalized at criticality.

Transfer matrix evolution converges to a stochastic differential equation.

Eigenvalue point processes match those of beta-ensembles in the second model.

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.