Eigenfunctions and the Integrated Density of States on Archimedean Tilings

Peyerimhoff, Norbert; Taeufer, Matthias

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