Discontinuities of the Integrated Density of States for Laplacians Associated with Penrose and Ammann-Beenker Tilings

TL;DR

This paper investigates the spectral properties of Laplacians on aperiodic tilings related to quasicrystals, revealing how locally-supported eigenfunctions cause discontinuities in the integrated density of states.

Contribution

It demonstrates the existence of locally-supported eigenfunctions in Laplacians on Penrose and Ammann-Beenker tilings and quantifies their impact on the spectral density discontinuities.

Findings

Locally-supported eigenfunctions cause jump discontinuities in the integrated density of states.

Lower bounds on the multiplicities of these eigenfunctions are established.

Results suggest further questions about spectral properties of aperiodic tilings.

Abstract

Aperiodic substitution tilings provide popular models for quasicrystals, materials exhibiting aperiodic order. We study the graph Laplacian associated with four tilings from the mutual local derivability class of the Penrose tiling, as well as the Ammann--Beenker tiling. In each case we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in the integrated density of states for these models. By bounding the multiplicities of these locally-supported modes, in several cases we provide concrete lower bounds on this jump. These results suggest a host of questions about spectral properties of the Laplacian on aperiodic tilings, which we collect at the end of the paper.