On spatial conditioning of the spectrum of discrete Random Schrödinger operators

Abstract

Consider a random Schrödinger-type operator of the form $H:=-H_X+V+\xi$ acting on a general graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $H_X$ is the generator of a Markov process $X$ on $\mathcal{G}$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $\xi$ is a random noise with at-most-exponential tails. We prove that the eigenvalue point process of $H$ is number rigid in the sense of Ghosh and Peres (2017); that is, the number of eigenvalues in any bounded domain $B\subset \mathbb{C}$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $\xi$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup ${\mathrm{e}}^{-tH}$ using the Feynman–Kac formula.