Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models

Abstract

We consider discrete Schrödinger operators of the form $H=-\Delta +V$ on ${\ell }^2({\mathbb{Z}}^d)$, where $\Delta$ is the discrete Laplacian and $V$ is a bounded potential. Given $\Gamma \subset {\mathbb{Z}}^d$, the $\Gamma$-trimming of $H$ is the restriction of $H$ to ${\ell }^2({\mathbb{Z}}^d\setminus \Gamma )$, 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 ${\mathbb{Z}}^d$ we always have $E_{\Gamma }(H)>E_{\varnothing }(H)$. 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 $\Gamma$.