Aperiodic order, integrated density of states and the continuous algebras of John von Neumann

TL;DR

This paper explores the connection between aperiodic order, spectral convergence, and von Neumann's continuous algebras, demonstrating uniform spectral convergence for various operators on complex structures.

Contribution

It establishes a novel link between aperiodic order and von Neumann's continuous algebras, proving uniform spectral convergence for several classes of operators.

Findings

Proved uniform spectral convergence for random Schrödinger operators on lattices.

Established spectral convergence for percolation Hamiltonians.

Extended results to pattern-invariant operators on self-similar graphs.

Abstract

Lenz and Stollmann recently proved the existence of the integrated density of states in the sense of uniform convergence of the distributions for certain operators with aperiodic order. The goal of this paper is to establish a relation between aperiodic order, uniform spectral convergence and the continuous algebras invented by John von Neumann. We illustrate the technique by proving the uniform spectral convergence for random Schodinger operators on lattices with finite site probabilities, percolation Hamiltonians and for the pattern-invariant operators of self-similar graphs.