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

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.