Eigenfunctions and the Integrated Density of States on Archimedean Tilings

Peyerimhoff, Norbert; Taeufer, Matthias

doi:10.4171/jst/347

Eigenfunctions and the Integrated Density of States on Archimedean Tilings

Peyerimhoff, Norbert; Taeufer, Matthias

Authors

Professor Norbert Peyerimhoff norbert.peyerimhoff@durham.ac.uk
Professor

Matthias Taeufer

Abstract

We study existence and absence of ` 2 -eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the .3:6/2 “kagome” tiling and the .3:122 / tiling) have ` 2 -eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely the hexagons and 12-gons in the tilings, respectively). Furthermore, we provide an explicit expression for the Integrated Density of States (IDS) of the Laplacian on Archimedean tilings in terms of eigenvalues of Floquet matrices and deduce integral formulas for the IDS of the Laplacian on the .44 /, .36 /, .63 /, .3:6/2 , and .3:122 / tilings. Our method of proof can be applied to other Z d -periodic graphs as well.

Citation

Peyerimhoff, N., & Taeufer, M. (2021). Eigenfunctions and the Integrated Density of States on Archimedean Tilings. Journal of Spectral Theory, 11(2), 461-488. https://doi.org/10.4171/jst/347

Journal Article Type	Article
Online Publication Date	Mar 17, 2021
Publication Date	2021
Deposit Date	Feb 26, 2020
Publicly Available Date	Oct 14, 2021
Journal	Journal of Spectral Theory
Print ISSN	1664-039X
Publisher	EMS Press
Peer Reviewed	Peer Reviewed
Volume	11
Issue	2
Pages	461-488
DOI	https://doi.org/10.4171/jst/347
Publisher URL	https:/doi.org/10.4171/JST/347