Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d

Abstract

We give an overview and extension of recent results on ergodic random Schrödinger operators for models on ${\mathbb{Z}}^d$. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed Shubin-Pastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.

22 pages; minor extensions and updates

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Cited by in corpus (5)

• Continuity of the integrated density of states on random length metric graphs
• Localization on quantum graphs with random edge lengths
• Anderson Localization for radial tree-like random quantum graphs
• $L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity
• Uniform existence of the integrated density of states on metric Cayley graphs