The Wegner Estimate and the Integrated Density of States for some Random Operators

Proceedings Mathematical Sciences

P D Hislop¹  J M Combes³  Shu Nakamura²  Frédéric Klopp⁴ 

[1] Département de Mathématiques, Université de Toulon et du Var, 0 La Garde, France$$;L.A.G.A., Institut Galilée, Université Paris-Nord, F-0 Villetaneuse, France$$;Centre de Physique Théorique, CNRS Luminy Case 0, F- Marseille, Cedex , France$$;Mathamatics Department, University of Kentucky, Lexington KY 00–00, USA$$

关键词: Schrödinger operators;    localization;    random potentials.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $mathcal{I}_p$, 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.

【 授权许可】

Unknown