The integrated density of states for some random operators with nonsign definite potentials

Peter D. Hislop, Frédéric Klopp

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51 Scopus citations

Abstract

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the Lp-theory of the spectral shift function for p ≥1 (Comm. Math. Phys. 218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys. 167 (1995), 553-569). We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.

Original languageEnglish
Pages (from-to)12-47
Number of pages36
JournalJournal of Functional Analysis
Volume195
Issue number1
DOIs
StatePublished - Oct 20 2002

Bibliographical note

Funding Information:
1To whom correspondence should be addressed. 2Supported in part by NSF Grant DMS-9707049.

Keywords

  • Localization
  • Monotonic variation
  • Schrödinger operators
  • Wegner estimate

ASJC Scopus subject areas

  • Analysis

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