Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

We consider the Anderson model on the finite grid $G=\mathbb{Z}\mathrm{/}{L}_1\mathbb{Z}\times \cdots \times \mathbb{Z}\mathrm{/}{L}_d\mathbb{Z}$, defined by the random Hamiltonian $H_t=\mathrm{\Delta }+tV$, where $\mathrm{\Delta }$ is the discrete Laplacian and $V=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\{{\omega }_x{\}}_{x\in G})$ 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.