# On spatial conditioning of the spectrum of discrete Random Schrödinger operators

### Pierre Yves Gaudreau Lamarre

University of Chicago, USA### Promit Ghosal

Massachusetts Institute of Technology, Cambridge, USA### Yuchen Liao

University of Warwick, Coventry, UK

## Abstract

Consider a random Schrödinger-type operator of the form $H:=−H_{X}+V+ξ$ acting on a general graph $G=(V,E)$, where $H_{X}$ is the generator of a Markov process $X$ on $G$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $ξ$ 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 $B⊂C$ 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 $ξ$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup $e_{−tH}$ using the Feynman–Kac formula.

## Cite this article

Pierre Yves Gaudreau Lamarre, Promit Ghosal, Yuchen Liao, On spatial conditioning of the spectrum of discrete Random Schrödinger operators. J. Spectr. Theory 12 (2022), no. 3, pp. 1109–1153

DOI 10.4171/JST/415