Ground state energy of trimmed discrete Schr\"odinger operators and localization for trimmed Anderson models

TL;DR

This paper studies how trimming parts of a discrete Schrödinger operator affects its ground state energy and uses this to prove localization results for trimmed Anderson models with random potentials supported on specific subsets.

Contribution

It establishes that trimming dense subsets raises the ground state energy and applies this to prove localization for trimmed Anderson models.

Findings

Trimming dense subsets increases the ground state energy.

Wegner estimates are derived for trimmed Anderson models.

Localization at the spectrum's bottom is proven for these models.

Abstract

We consider discrete Schr\"odinger operators of the form $H=-\Delta +V$ on $\ell^2(\Z^d)$, where $\Delta$ is the discrete Laplacian and $V$ is a bounded potential. Given $\Gamma \subset \Z^d$, the $\Gamma$-trimming of $H$ is the restriction of $H$ to $\ell^2(\Z^d\setminus\Gamma)$, denoted by $H_\Gamma$. We investigate the dependence of the ground state energy $E_\Gamma(H)=\inf \sigma (H_\Gamma)$ on $\Gamma$. We show that for relatively dense proper subsets $\Gamma$ of $\Z^d$ we always have $E_\Gamma(H)>E_\emptyset(H)$. We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for $\Gamma$-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set $\Gamma$