Eigenvectors of the 1-dimensional critical random Schrödinger operator - Rifkind, Ben; Virág, Bálint (Geometric and functional analysis, 2018)
The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator H = Δ + V on ℓ 2 ( Z ) . Here Δ 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 ψ ) ℓ = ψ ℓ - 1 , n + ψ ℓ + 1 , n + v ℓ , n ψ ℓ , ψ 0 = ψ n + 1 = 0 , with v k are independent random variables with mean 0 and variance σ 2 / n . We show that the scaling limit of the shape of a uniformly chosen eigenvector of H n is exp - | t - U | 4 + Z t - U 2 , where U is uniform on [0,1] and Z is an independent two sided Brownian motion started from 0.