Journal of Spectral Theory
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Published online: 2014-07-13
Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson modelsAlexander Elgart and Abel Klein (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