Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb{Z}}^d$

Abstract

We consider ergodic random magnetic Schrödinger 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.

17 pages; typos removed, references updated, definition of subgraph densities explained

References in corpus (6)

• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
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• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs
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Cited by in corpus (10)

• Continuity of the integrated density of states on random length metric graphs
• Localization on quantum graphs with random vertex couplings
• Anderson Localization for radial tree-like random quantum graphs
• The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs
• Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d
• Optimal Wegner estimates for random Schroedinger operators on metric graphs
• $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity
• Anderson Localization for a Multi-Particle Quantum Graph
• Uniform existence of the integrated density of states on metric Cayley graphs
• Localization for quantum graphs with a random potential