Eigenvectors of the critical 1-dimensional random Schroedinger operator

TL;DR

This paper investigates the shape of eigenvectors of the critical 1D random Schrödinger operator, revealing their convergence to a stochastic process involving Brownian motion, thus deepening understanding of Anderson localization at criticality.

Contribution

It characterizes the limiting distribution of eigenvector shapes for the critical 1D random Schrödinger operator, a novel insight into eigenvector behavior at the localization transition.

Findings

Eigenvectors converge in distribution to a process involving Brownian motion.

The shape of eigenvectors is described by an explicit stochastic process.

Provides new understanding of eigenvector localization at criticality.

Abstract

The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schroedinger operator H = Delta+V. 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 Z_n and consider the critical model H_n. We show that the shape of a uniformly chosen eigenvector of H_n converges in law to exp (-|t|/4 + Z_t/sqrt(2)), where Z is two-sided Brownian motion.