We consider discrete Schrödinger operators of the form on , where is the discrete Laplacian and is a bounded potential. Given , the -trimming of is the restriction of to , denoted by . We investigate the dependence of the ground state energy on . We show that for relatively dense proper subsets of we always have . We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for -trimmed Anderson models, i.e., Anderson models with the random potential supported by the set .
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Alexander Elgart, Abel Klein, Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models. J. Spectr. Theory 4 (2014), no. 2, pp. 391–413DOI 10.4171/JST/74