 01327 J. M. Combes, P. D. Hislop, F. Klopp, S. Nakamura
 The Wegner estimate and the integrated density of states for some random operators
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Sep 13, 01

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Abstract. 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 finitevolume 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 H\"older continuity of
the integrated density of states at energies in the unperturbed
spectral gap. The proof depends on the
$L^p$theory of the spectral shift function (SSF), for $p \geq
1$, applicable to pairs of selfadjoint operators whose difference is in
the trace ideal ${\cal I}_p$, for $0 < p \leq 1$.
We present this and other results on the SSF due to other authors.
Under an additional condition of the
singlesite potential, local H\"older continuity is proved at all
energies. Finally, we present extensions of this work to random
potentials with nonsign
definite singlesite potentials.
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