The Lp-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators

J. M. Combes, P. D. Hislop, Shu Nakamura

Research output: Contribution to journalArticlepeer-review

55 Scopus citations

Abstract

We develop the Lp-theory of the spectral shift function, for p ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal Ip, for 0 < p ≤ 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Holder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.

Original languageEnglish
Pages (from-to)113-130
Number of pages18
JournalCommunications in Mathematical Physics
Volume218
Issue number1
DOIs
StatePublished - Apr 2001

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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