Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

TL;DR

This paper investigates the eigenvalue simplicity and eigenvector non-vanishing properties of the Anderson model on finite grids, showing that these properties hold for almost all parameters with continuous disorder but not necessarily with Bernoulli disorder.

Contribution

It establishes conditions under which the Anderson model has simple eigenvalues and non-vanishing eigenvectors, providing exact probabilities in specific Bernoulli cases.

Findings

For continuous distributions, eigenvalues are simple for all but finitely many t with probability 1.

For Bernoulli distributions, the conditions fail with positive probability, and a lower bound is provided.

Exact probability calculations are given for the Bernoulli case when the dimension is 1 and the size is prime.

Abstract

We consider the Anderson model on the finite grid $G = \mathbb Z/L_1\mathbb Z\times\cdots\times\mathbb Z/L_d\mathbb Z$, defined by the random Hamiltonian $H_t=\Delta+tV$, where $\Delta$ is the discrete Laplacian and $V=\mathrm{diag}(\{\omega_{x}\}_{x\in G})$ is a random onsite potential with $\omega_x\sim\mu$ i.i.d. We ask the natural question of when $H_t$ has simple eigenvalues and non-vanishing eigenvectors. We prove that, when $\mu$ is a continuous probability distribution, $H_t$ has this property for all but finitely many $t$ values with probability $1$. However, when $\mu$ is a Bernoulli distribution, the conditions fail with positive probability, for which we give a lower bound. We also calculate the exact probability of these conditions being met in the Bernoulli case when $d = 1$ and $L = L_1$ is prime.