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

Citation
Jm. Combes et al., The L-p-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators, COMM MATH P, 218(1), 2001, pp. 113-130
Citations number
33
Categorie Soggetti
Physics
Journal title
COMMUNICATIONS IN MATHEMATICAL PHYSICS
ISSN journal
00103616 → ACNP
Volume
218
Issue
1
Year of publication
2001
Pages
113 - 130
Database
ISI
SICI code
0010-3616(200104)218:1<113:TLOTSS>2.0.ZU;2-A
Abstract
We develop the L-p-theory of the spectral shift function, for p greater tha n or equal to 1, applicable to pairs of self-adjoint operators whose differ ence is in the trace ideal I-p, for 0 < p less than or equal to 1. This res ult is a key ingredient of a new, short proof of the Wegner estimate applic able 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 ga p. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. This new Wegner estimate, together w ith other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.