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Eigenfunctions and the Integrated Density of States on Archimedean Tilings

Peyerimhoff, Norbert; Taeufer, Matthias

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Authors

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

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