The Wegner estimate and the integrated density of states for some random operators

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

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22 Scopus citations


The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on the Lp-theory of the spectral shift function (SSF), for p ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal Ip, for 0 < p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local Hölder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.

Original languageEnglish
Pages (from-to)31-53
Number of pages23
JournalProceedings of the Indian Academy of Sciences - Mathematical Sciences
Issue number1
StatePublished - Feb 2002

Bibliographical note

Funding Information:
We thank W Kirsch, A Klein, V Kostrykin, R Schrader, B Simon, K Sinha, P Stollmann, and G Stolz for useful discussions. This research was supported in part by CNRS, NSF grant DMS-9707049 and NATO grant CGR-951351 and, by JSPS grant Kiban B 09440055.


  • Localization
  • Random potentials
  • Schrödinger operators

ASJC Scopus subject areas

  • Mathematics (all)


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