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arXiv · 2512.00278

Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

Oluyinka Lindblad·Ezra Guerrero
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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.

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Oluyinka Lindblad, Ezra Guerrero. 2025-11-29. Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model. https://arxiv.org/abs/2512.00278

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