Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

Leonhardt, Karsten; Peyerimhoff, Norbert; Tautenhahn, Manfred; Veselic, Ivan

Abstract

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.

Citation

Leonhardt, K., Peyerimhoff, N., Tautenhahn, M., & Veselic, I. (2015). Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials. Reviews in Mathematical Physics, 27(04), https://doi.org/10.1142/s0129055x15500075

• Journal Article Type: Article
• Acceptance Date: Apr 3, 2015
• Online Publication Date: May 11, 2015
• Publication Date: May 11, 2015
• Deposit Date: Jan 6, 2016
• Publicly Available Date: Apr 14, 2016
• Journal: Reviews in Mathematical Physics
• Print ISSN: 0129-055X
• Electronic ISSN: 1793-6659
• Publisher: World Scientific Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 27
• Issue: 04
• DOI: https://doi.org/10.1142/s0129055x15500075
• Public URL: https://durham-repository.worktribe.com/output/1392698
• Related Public URLs: http://arxiv.org/pdf/1309.0109v2.pdf