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INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... 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 selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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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 selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami 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.
I.: Existence of the density of states for onedimensional alloytype potentials with small support
 Mathematical Results in Quantum Mechanics
, 2001
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Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Cited by 7 (5 self)
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
LIFSCHITZ TAILS AND LOCALISATION FOR A CLASS OF SCHRÖDINGER OPERATORS WITH RANDOM BREATHERTYPE POTENTIAL
, 2006
"... Abstract. We derive bounds on the integrated density of states of Schrödinger operators with a random, ergodic potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by ..."
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Abstract. We derive bounds on the integrated density of states of Schrödinger operators with a random, ergodic potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models we show that the the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This enables us to prove localisation in certain energy/disorder regimes. 1.
(1)
, 2002
"... Abstract. We study spectral properties of Schrödinger operators with random potentials of alloy type on L 2 (R) and their restrictions to finite intervals. A Wegner estimate for nonnegative single site potentials with small support is proven. It implies the existence and local uniform boundedness o ..."
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Abstract. We study spectral properties of Schrödinger operators with random potentials of alloy type on L 2 (R) and their restrictions to finite intervals. A Wegner estimate for nonnegative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the density of states. Our estimate is valid for all bounded energy intervals. Wegner estimates play a key role in an existence proof of pure point spectrum. 1. Model and results We study spectral properties of families of Schrödinger operators on L 2 (R). The considered operators consist of a nonrandom periodic Schrödinger operator plus a random potential of Anderson or alloy type:
unknown title
, 904
"... Wegnertype bounds for a twoparticle lattice model with a generic quasiperiodic potential ..."
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Wegnertype bounds for a twoparticle lattice model with a generic quasiperiodic potential
Integral Equations and Operator Theory Eigenfunction Expansions for Schrödinger Operators on Metric Graphs
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.