Uniform existence of the integrated density of states on metric Cayley graphs

TL;DR

This paper proves the uniform convergence of eigenvalue counting functions for ergodic random Schrödinger operators on Cayley graphs of amenable groups, establishing a formula for the integrated density of states and linking spectrum support to eigenfunctions.

Contribution

It demonstrates the uniform convergence of eigenvalue counting functions and provides a Pastur-Shubin formula for the integrated density of states on metric Cayley graphs.

Findings

Eigenvalue counting functions converge uniformly.

Integrated density of states can be expressed by a Pastur-Shubin formula.

Discontinuities in the measure indicate compactly supported eigenfunctions.

Abstract

Given a finitely generated amenable group we consider ergodic random Schr\"odinger operators on a Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states as the limit can be expressed by a Pastur-Shubin formula. The spectrum supports the corresponding measure and discontinuities correspond to the existence of compactly supported eigenfunctions.