Some new estimates on the spectral shift function associated with random Schrödinger operators

Jean-Michel Combes (1, 2) , Peter Hislop (3) , Frédéric Klopp (4)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 Department of Mathematics
4 LAGA - Laboratoire Analyse, Géométrie et Applications

We prove some new pointwise-in-energy bounds on the expectations of various spectral shift functions associated with random Schrödinger operators in the continuum having Anderson-type random potentials in both finite-volume and infinite-volume. These estimates are a consequence of our new Wegner estimate for finite-volume random Schrödinger operators. For lattice models, we also obtain a representation of the infinite-volume density of states in terms of a spectral shift function. For continuum models, the corresponding measure is absolutely continuous with respect to the density of states and agrees with it in certain cases. We present a variant of a new spectral averaging result and use it to prove a pointwise upper bound on the SSF for finite-rank perturbations.

Mots clés: random Schrödinger operators; spectral shift function; integrated density of states; Wegner estimate

Domaines: Physique mathématique [math-ph]

Jean-Michel Combes, Peter Hislop, Frédéric Klopp. Some new estimates on the spectral shift function associated with random Schrödinger operators. 2006. ⟨hal-00067801v2⟩