Eigenfunctions and the integrated density of states on Archimedean tilings

  • Norbert Peyerimhoff

    Durham University, UK
  • Matthias Täufer

    FernUniversität Hagen, Germany
Eigenfunctions and the integrated density of states on Archimedean tilings cover
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Abstract

We study existence and absence of -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 “kagome” tiling and the tiling) have -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 , , , , and tilings. Our method of proof can be applied to other -periodic graphs as well.

Cite this article

Norbert Peyerimhoff, Matthias Täufer, Eigenfunctions and the integrated density of states on Archimedean tilings. J. Spectr. Theory 11 (2021), no. 2, pp. 461–488

DOI 10.4171/JST/347