Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model

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.