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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 13 (8 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.
Optimal Wegner estimates for random Schrödinger operators on metric graphs
, 2008
"... We consider Schrödinger operators with a random potential of alloy type on infinite metric graphs which obey certain uniformity conditions. For single site potentials of fixed sign we prove that the random Schrödinger operator restricted to a finite volume subgraph obeys a Wegner estimate which is ..."
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Cited by 5 (2 self)
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We consider Schrödinger operators with a random potential of alloy type on infinite metric graphs which obey certain uniformity conditions. For single site potentials of fixed sign we prove that the random Schrödinger operator restricted to a finite volume subgraph obeys a Wegner estimate which is linear in the volume and reproduces the modulus of continuity of the single site distribution. This improves and unifies earlier results for alloy type models on metric graphs. We discuss applications of Wegner estimates to bounds on the modulus of continuity for the integrated density of states of ergodic Schrödinger operators, as well as to the proof of Anderson localisation via the multiscale analysis.
A linear Wegner estimate for alloy type Schrödinger operators on metric graphs
, 2006
"... We study spectra of alloytype random Schrödinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloytype m ..."
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Cited by 4 (3 self)
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We study spectra of alloytype random Schrödinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloytype model needs to have fixed sign, but the considered metric graph does not need to have a periodic structure. The second result we obtain is an exhaustion construction of the integrated density of states for ergodic random Schrödinger operators on metric graphs with a Z νstructure. For certain models the two above results together imply the Lipschitz continuity of the integrated density of states.
The modulus of continuity of Wegner estimates for random Schrdinger operators on metric graphs
, 2007
"... We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of ..."
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Cited by 2 (1 self)
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We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.
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.