JournalsjstVol. 4, No. 2pp. 391–413

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

  • Alexander Elgart

    Virginia Tech, Blacksburg, USA
  • Abel Klein

    University of California, Irvine, USA
Ground state energy of  trimmed discrete Schrödinger operators and  localization for trimmed Anderson models cover
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Abstract

We consider discrete Schrödinger operators of the form H=Δ+VH=-\Delta +V on 2(Zd)\ell^2(\Z^d), where Δ\Delta is the discrete Laplacian and VV is a bounded potential. Given ΓZd\Gamma \subset \Z^d, the Γ\Gamma-trimming of HH is the restriction of HH to 2(ZdΓ)\ell^2(\Z^d\setminus\Gamma), denoted by HΓH_\Gamma. We investigate the dependence of the ground state energy EΓ(H)=infσ(HΓ)E_\Gamma(H)=\inf \sigma (H_\Gamma) on Γ\Gamma. We show that for relatively dense proper subsets Γ\Gamma of Zd\Z^d we always have EΓ(H)>E(H)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 Γ\Gamma-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set Γ\Gamma.

Cite this article

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. 391–413

DOI 10.4171/JST/74