Eigenvectors of the 1-dimensional critical random Schrödinger operator

Abstract

The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator ${H = \Delta + V}$ on ${\ell^2(\mathbb{Z})}$. Here ${\Delta}$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to ${\mathbb{Z}_n}$ and consider the critical model,

$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$

with v_k are independent random variables with mean 0 and variance ${\sigma^2/n}$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of H_n is

$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$

where U is uniform on [0,1] and ${\mathcal{Z}}$ is an independent two sided Brownian motion started from 0.