Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Z d

We consider ergodic random Schrödinger operators on the metric graph Z d with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the inte...

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Bibliographic Details
Published in:Journal of functional analysis 2007-12, Vol.253 (2), p.515-533
Main Authors: Gruber, Michael J., Lenz, Daniel H., Veselić, Ivan
Format: Article
Language:English
Subjects:
Integrated density of states
Metric graph
Quantum graph
Random Schrödinger operator
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Summary:We consider ergodic random Schrödinger operators on the metric graph Z d with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2007.09.003