On Spatial Conditioning of the Spectrum of Discrete Random Schr\"odinger Operators

TL;DR

This paper proves that the eigenvalues of a broad class of random Schrödinger operators on graphs are number rigid, meaning the count in any region is determined by eigenvalues outside that region, even with complex dependencies.

Contribution

It establishes number rigidity for eigenvalues of non-self-adjoint and long-range dependent random Schrödinger operators on general graphs, extending previous results to more general settings.

Findings

Eigenvalue counts are determined by configurations outside regions.

Number rigidity holds even with non-symmetric operators and long-range noise.

The variance of the trace of the semigroup is controlled via Feynman-Kac.

Abstract

Consider a random Schr\"odinger-type operator of the form $H:=-H_X+V+\xi$ acting on a general graph $\mathscr G=(\mathscr V,\mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $\mathscr G$, $V$ 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 $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); 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 $\mathrm e^{-t H}$ using the Feynman-Kac formula.