# Journal of Spectral Theory

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**Volume 4, Issue 2, 2014, pp. 391–413**

**DOI: 10.4171/JST/74**

Published online: 2014-07-13

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

Alexander Elgart^{[1]}and Abel Klein

^{[2]}(1) Virginia Tech, Blacksburg, USA

(2) University of California, Irvine, USA

We consider discrete Schrödinger 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 $\Gamma$.

*Keywords: *Anderson models, trimmed Anderson models, discrete Schrödinger operators, random Schrödinger operators, ground state energy, Cheeger’s inequality, localization,Wegner estimates

Elgart Alexander, Klein Abel: Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models. *J. Spectr. Theory* 4 (2014), 391-413. doi: 10.4171/JST/74