Discontinuities of the Integrated Density of States for Laplacians Associated with Penrose and Ammann–Beenker 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.

KEYWORDS:

• Laplacians on aperiodic tilings
• eigenvalue computations
• locally-supported eigenfunctions
• quasicrystals

Acknowledgments

The authors thank Michael Baake, Semyon Dyatlov, and Anton Gorodetski for many helpful conversations and the American Institute of Mathematics for hospitality and support through the SQuaRE program during a remote meeting in January 2021 and a January 2022 visit, during which part of this work was completed.

Notes

1 Illustration following https://tilings.math.uni-bielefeld.de/substitution/penrose-rhomb/

2 Illustration following https://tilings.math.uni-bielefeld.de/substitution/penrose-kite–dart/

3 Illustration following https://tilings.math.uni-bielefeld.de/substitution/ammann-beenker/

Additional information

Funding

D.D. was supported in part by NSF grants DMS–1700131 and DMS–2054752, and Simons Fellowship 669836. M.E. was supported in part by NSF grant DMS-1720257. J.F. was supported in part by NSF grant DMS–2213196 and Simons Foundation Collaboration grant #711663.