Wednesday, January 23, 2013

1301.5268 (Alexander Elgart et al.)

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

Alexander Elgart, Abel Klein
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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