Integrated density of states and Wegner estimates for random Schrödinger Operators

Abstract

We survey recent results on spectral properties of random Schrödinger 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 Schrödinger 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 Schrödinger 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.

arXiv comment87 pages; corrected and extended version

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Cited by in corpus (5)

• Spectral Analysis of Percolation Hamiltonians
• Spectral extrema and Lifshitz tails for non monotonous alloy type models
• Lifshitz Tails in Constant Magnetic Fields
• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs
• Groupoids, von Neumann Algebras and the Integrated Density of States