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

Alexander Elgart, Abel Klein

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We consider discrete Schrödinger operators of the form H = - Δ + V on l2(ℤd), where Δ is the discrete Laplacian and V is a bounded potential. Given Γ ⊂ ℤd, the Γ-trimming of H is the restriction of H to l2(ℤd \ Γ), denoted by HΓ. We investigate the dependence of the ground state energy Eγ(H) = inf σ(HΓ) on Γ. We show that for relatively dense proper subsets Γ of ℤd we always have Eγ(H) > Eø(H) We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for Γ-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set Γ.

Original languageEnglish
Pages (from-to)391-413
Number of pages23
JournalJournal of Spectral Theory
Volume4
Issue number2
DOIs
StatePublished - 1 Jan 2014
Externally publishedYes

Keywords

  • Anderson models
  • Cheeger's inequality
  • Discrete Schrödinger operators
  • Ground state energy
  • Localization
  • Random Schrödinger operators
  • Trimmed Anderson models
  • Wegner estimates

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