The Lp-Theory of the Spectral Shift Function,¶the Wegner Estimate, and the Integrated Density¶of States for Some Random Operators

Abstract:

We develop the L ^p-theory of the spectral shift function, for p≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal ℐ _p , 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 Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local Hölder 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.