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An Isaac Newton Institute Workshop

Quantum Graphs, their Spectra and Applications

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

Authors: Michael J. Gruber (TU Chemnitz), Daniel H. Lenz (TU Chemnitz), Ivan Veselic (TU Chemnitz)

Abstract

We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ 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 percolation models.

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