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Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

Oluyinka Lindblad, Ezra Guerrero

TL;DR

This work analyzes when the finite-grid Anderson model H_t = Δ + tV has simple eigenvalues and non-vanishing eigenvectors. It develops a perturbative framework (Kato–Rellich) to classify potentials as good or bad, showing that continuous μ yields P(V good) = 1, while discrete μ can yield a positive probability of bad potentials. In 1D with prime length, it provides exact probability formulas for bad potentials and demonstrates asymptotic vanishing of badness as system size grows for p in (0,1). The results connect algebraic independence and symmetry considerations to spectral properties, informing localization phenomena and nodal structure of eigenvectors.

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 $H_t=Δ+tV$, where $Δ$ is the discrete Laplacian and $V=\mathrm{diag}(\{ω_{x}\}_{x\in G})$ is a random onsite potential with $ω_x\simμ$ i.i.d. We ask the natural question of when $H_t$ has simple eigenvalues and non-vanishing eigenvectors. We prove that, when $μ$ is a continuous probability distribution, $H_t$ has this property for all but finitely many $t$ values with probability $1$. However, when $μ$ 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.

Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

TL;DR

This work analyzes when the finite-grid Anderson model H_t = Δ + tV has simple eigenvalues and non-vanishing eigenvectors. It develops a perturbative framework (Kato–Rellich) to classify potentials as good or bad, showing that continuous μ yields P(V good) = 1, while discrete μ can yield a positive probability of bad potentials. In 1D with prime length, it provides exact probability formulas for bad potentials and demonstrates asymptotic vanishing of badness as system size grows for p in (0,1). The results connect algebraic independence and symmetry considerations to spectral properties, informing localization phenomena and nodal structure of eigenvectors.

Abstract

We consider the Anderson model on the finite grid , defined by the random Hamiltonian , where is the discrete Laplacian and is a random onsite potential with i.i.d. We ask the natural question of when has simple eigenvalues and non-vanishing eigenvectors. We prove that, when is a continuous probability distribution, has this property for all but finitely many values with probability . However, when 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 and is prime.

Paper Structure

This paper contains 9 sections, 15 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Conventions
  4. Continuous Probability Distributions
  5. General Perturbations
  6. Proof of Theorem \ref{['theorem_continuous-prob']}
  7. Symmetry Results
  8. Discrete Probability Distributions
  9. Acknowledgments

Key Result

Theorem 1.2

Let $G$ be any connected graph. If $\mu$ is absolutely continuous with respect to the Lebesgue measure, then $\mathbb P(V\text{ is good})=1$. In particular, for the Anderson model, $\mathbb P(V\text{ is good})=1$ for any $d$ and any side lengths $L_1,\ldots,L_d$.

Figures (1)

  • Figure 1: The localization-delocalization phase transition: $\log(\|\varphi_k\|_{4}^4)$ for the $k$-th eigenvector of $H_t$, as a function of $k$ on the x-axis and $t$ on the y-axis (left) and a random onsite potential (right), with $\mu\sim\operatorname{Uniform}[-1,1]$.

Theorems & Definitions (31)

  • Definition 1.1: Good/Bad Potentials
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 2.1: Kato-Rellich
  • Definition 2.2: Good/Bad Pairs
  • Proposition 2.3
  • ...and 21 more