Integrated density of states and Wegner estimates for random Schrödinger Operators

Details Integrated density of states and Wegner estimates for random Schrödinger Operators Integrated density of states and Wegner estimates for random Schrödinger Operators

2003-07-29

arxiv.org/abs/math-ph/0307062v2

Spectral theory of Schroedinger operators, 97--183, Contemp. Math., 340, Amer. Math. Soc., Providence, RI, 2004

Physics

Mathematical Physics

87 pages; corrected and extended version

Scientific paper

We survey recent results on spectral properties of random Schr\"odinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a self-averaging IDS which is general enough to be applicable to random Schr\"odinger and Laplace-Beltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schr\"odinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.

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