We study existence and absence of $\ell^2$-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 $(3.12^2)$ tiling) have $\ell^2$-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 $(4^4)$, $(3^6)$, $(6^3)$, $(3.6)^2$, and $(3.12^2)$ tilings. Our method of proof can be applied to other $\mathbb Z^d$-periodic graphs as well.

中文翻译：

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阿基米德瓦片上的特征函数和状态积分密度

我们通过正凸多边形研究欧几里得平面的 11 个阿基米德平铺上组合拉普拉斯算子的 $\ell^2$-特征函数的存在和不存在。我们证明了这些平铺中的两个（即 $(3.6)^2$“kagome”平铺和 $(3.12^2)$ 平铺）具有 $\ell^2$-eigenfunctions。这些本征函数是无限退化的，由明确描述的本征函数构成，这些本征函数在底层图的有限数量的顶点上得到支持（分别是平铺中的六边形和 12 边形）。此外，我们根据 Floquet 矩阵的特征值提供了拉普拉斯算子在阿基米德拼贴上的综合态密度 (IDS) 的显式表达式，并推导出了拉普拉斯算子在 $(4^4)$、$ (3^6)$、$(6^3)$、$(3.6)^2$ 和 $(3.12^2)$ 平铺。