Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence

TL;DR

This paper investigates the properties of discrete Hamiltonians on various geometric structures, establishing a link between jumps in the integrated density of states and eigenspaces with compactly supported eigenfunctions, leading to uniform convergence results.

Contribution

It introduces a unified framework for analyzing IDS jumps and uniform convergence across diverse discrete structures, including quasiperiodic, periodic, and random operators.

Findings

Eigenspaces at fixed energies are spanned by compactly supported eigenfunctions.

Jumps in the IDS correspond to the equivariant dimension of these eigenspaces.

Finite volume approximants of the IDS converge uniformly with respect to the spectral parameter.

Abstract

We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasi-transitive graphs, and operators on percolation graphs.