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arXiv · 1301.5268

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

Alexander Elgart·Abel Klein
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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 $\Gamma$

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Alexander Elgart, Abel Klein. 2013-01-22. Ground state energy of trimmed discrete Schr\"odinger operators and localization for trimmed Anderson models. https://doi.org/10.4171/jst%2F74

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