Eigenfunctions and the Integrated Density of States on Archimedean Tilings

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 on 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.


Introduction and statement of results
The goal of this paper is to provide concrete formulas for the Integrated Density of States (IDS) on Archimedean tilings, viewed as combinatorial graphs, and to study existence or absence of 2 -eigenfunctions for the associated Laplacians.
A plane tiling by regular convex polygons is a countable family of regular convex polygons covering the plane without gaps or overlaps. It is called edge-to-edge if the corners and sides of the polygons coincide with the vertices and edges of the tiling (see [GS89]). The type of a vertex of an edge-to-edge plane tiling by regular polygons describes the order of the polygons arranged cyclically around the vertex, for example the vertices in the honeycomb tiling are all of the type (6.6.6) =: (6 3 ).
Definition 1.1. An Archimedean tiling is an edge-to-edge tiling of the plane by regular convex polygons such that all vertices are of the same type.
Remark 1.2. For periodic graphs with co-finite Z d action, the (distributional) derivative of the IDS, the density of states, is a spectral measure in the sense that is carries all information on the spectrum: The points of increase of the IDS, i.e. the support of the density of states, are the spectrum of ∆ [MY02, p.119], see also [LPV07,Prop. 5.2] for a proof of this statement in a more general context. The set of discontinuities of the IDS constitues the pure point spectrum and the singular continuous spectrum is empty [Kuc16, Theorem 6.10]. Thus, the remaining points of increase are the absolutely continuous spectrum. In particular, we have a complete description of the spectral types on all 11 Archimedean lattices.
The method of proof is based on Floquet theory and can be applied to more general graphs with cofinite Z d -action and not only to Archimedean tilings. Examples include periodic finite hopping range operators on the nearest neighbour graph on Z d or on non-planar, Z 2 -periodic graphs.
2. General results on the IDS and the lattice Z d 2.1. Floquet theory and the IDS. Even though the goal of this article will be to study the 11 (planar) graphs based on Archimedean tesselations, the results of this subsection do not require planarity of the graph. More precisely, let G = (V, E) be an infinite graph with vertex set V and edge set E. We assume that the vertex degree |v| is finite for every v ∈ V.
We also assume that there is a cofinite Z d -action on G, given by The graph Laplacian ∆, a self-adjoint bounded operator on 2 (V), was defined in (1). The (abstract) Integrated Density of States (IDS) where χ (−∞,E] (∆) denotes the spectral projector onto the interval (−∞, E]. Intuitively, the IDS counts the number of states of ∆ below the energy level E per unit volume [LPPV09]. This is also reflected by Formula (2) below. The IDS is non-decreasing and right continuous. In order to apply Floquet theory, we also define the d-dimensional torus T d = R d / (2πZ) d and for every θ ∈ T d the |Q|-dimensional Hilbert space Furthermore, we define on 2 (V) θ the θ-pseudoperiodic Laplacian ∆ θ as that is, ∆ θ acts in the same way as ∆ but on the different vector space 2 (V) θ . Since this is a |Q|-dimensional vector space due to quasiperiodicity, the operator ∆ θ can be viewed as a hermitian |Q|×|Q|-matrix. In Sections 3 and 4 we will give concrete examples of this matrix for the case of the 11 Archimedean lattice graphs. The map T d θ → σ(∆ θ ) is also called dispersion relation.
The following theorem provides an integral expression for the IDS on Z d -periodic graphs, see also [Kuc16,Theorem 6.18].
For the convenience of the reader we now give a proof of Theorem 2.1 using Fourier theory on 2 (Z d ). We Applying the Fourier transform on every component, we obtainf From Fourier theory it follows that f →f is an isometry with the norms We writef θ (v) :=f v (θ) and extendf θ (v) quasiperiodically to V viã We have isometrically identified the spaces Lemma 2.2. For all v ∈ V, all f ∈ 2 (V), and all θ ∈ T d , we have Now, we can identify operators: and ∆f = g, if and only if for Recall that ∆ θ and ∆ are formally defined via the same expressions, but they operate on different spaces: ∆ operates on 2 -functions on G while ∆ θ operates on θ-quasiperiodic functions. From (4), we conclude we are in a position to calculate the IDS. We have The operator, χ (−∞,E] (∆ θ ) is an orthogonal projection onto the finite-dimensional span of eigenfunctions of ∆ θ on 2 (Q) with eigenvalues smaller or equal than E (i.e. a matrix). Hence, the trace Tr χ (−∞,E] (∆ θ ) is the number of eigenvalues of ∆ θ less or equal than E. This finishes the proof of Theorem 2.1.
The next results are useful to show absence of finitely supported eigenfunctions for particular graphs.
Theorem 2.3. The following are equivalent: Proof of Theorem 2.3. The equivalence of items (i), (ii) and (iii) is proved in [Kuc91], see also [LV09, Corollary 2.3] for a proof in a more general setting. It remains to show the equivalence of (i) and (iv). We fix E ∈ R and calculate, using the dominated convergence theorem, This is non-zero if and only if the characteristic polynomial Let us note that the analytic nature of the band functions has been used in similar arguments before, see e.g. [Kuc16, Corollary 6.19].
2.2. The lattice Z d . As a first application of (2), we calculate the IDS of ∆ on the lattice Z d . An elementary cell Q consists of a single point. In the 2-dimensional case, we can view Z 2 as a tiling by unit squares (i.e. as the (4 4 ) tiling) with Z 2 generated by translation vectors ω 1 = (1, 0), and ω 2 = (0, 1), cf. Figure 2. The (1 × 1)-matrix corresponding to ∆ θ has the entry (and hence the only eigenvalue) cos(θ j ).
In dimensions d = 1, 2, the following expressions for the IDS follow directly from (6). In the case d = 2, we derive the expression by applying the substitution t = cos θ 1 .
In dimension d = 2, we have

Concrete integral expressions for the IDS of some Archimedean tilings
In this section, we present concrete integral expressions of the IDS of the Archimedean tilings with vertex types (3 6 ), (6 3 ), (3.6) 2 , and (3.12 2 ). We will denote the corresponding IDS by N (3 6 ) , etc.
We will see that only the last two tilings admit finitely supported eigenfunctions.
3.1. IDS of the (3 6 ) tiling (triangular lattice). A fundamental domain consists of a single point with translation vectors ω 1 = (1, 0), ω 2 = (cos(π/3), sin(π/3)), cf. Figure 3. The corresponding matrix ∆ θ has the only entry and hence the only eigenvalue Therefore, will be relevant later on, we shall discuss it in more detail here. By periodicity, we can consider T 2 as (−π, π) 2 (the boundary is a measure zero set and does not play any role). Using the change of variables u : has the following properties: 2} and F ≤ −1 in the complemetary set (Tri) − ∪ (Tri) + which consists of two rectangular triangles. iv) We have Remark 3.2. It is known that additional symmetries (e.g. a rotational symmetries) of the underlying graph are reflected in symmetries of the dispersion relation. More precisely, in an appropriate basis, the function F is symmetric under rotations by π/3 around its maximum and symmetric under rotations by 2π/3 around its minima. This corresponds to symmetries of the underlying graph, see [BC18, Lemma 2.1] for details. Proof. It is straightforward to check i) to iii) and using symmetry and monotonicity considerations To calculate the area within (Tri) + we consider the upper half (i.e. v ≥ 0) of (Tri) + (i.e. u ≥ π/ √ 2). Therein, cos(u/ √ 2) < 0, whence F (u, v) ≤ L is equivalent to and we found that the area is the area under a graph. Since cos(v/ √ 2) ≤ 1, we conclude that (8) can only be fulfilled if u is in the interval between the two solutions of Together with (8), we find where in the last step, we used the transformation u = √ 2 arccos(t). As for the area in the hexagon H, by an analogous argument, Combining Lemma 3.1 and (7), we find: if 3/2 < E.
In particular, N (3 6 ) is continuous and there are no 2 -eigenfunctions.
For each hexagon H there exists (up to scalar multiples) exactly one eigenfunction with support on H. Every 2 -eigenfunction is a linear combination of these special finitely supported eigenfunctions.
3.4. IDS of the (3.12 2 ) tiling. The (3.12 2 ) tiling is the second Archimedean tiling after the (3.6) 2 (Kagome) tiling which has compactly supported eigenfunctions. It also has the interesting feature that the spectrum consists of the two intervals [0, 2/3] and [1, 5/3], i.e. it has a proper band structure which might make nanomaterials based on this tiling an interesting candidate for applications. A fundamental domain consists of six points, Q = {a, b, c, d, e, f }, cf. Figure 7. We have
Using some elementary algebra and Theorem 2.1, we find Proposition 3.6.
where F is the function explicitly given in Lemma 3.1. For each 12-gon D, there exist (up to scalar multiples) exactly two linear independent eigenfunctions with support on D. Every 2 -eigenfunction is a linear combination of one type of these special finitely supported eigenfunctions.
Remark 3.7. The eigenfunctions on the (6.3) 2 and the (3.12 2 ) are (finite or infinite) linear combinations of eigenfunctions supported on a single hexagon or 12-gon, respectively, see Figure 8 for an illustration. One observes that both these tesselations share the feature that they contain an 2n-gon which is either completely surrounded by triangles or where triangles are adjacent to every second edge. Since the (3.6) 2 tiling and the (3.12 2 ) tiling are the only ones with this property, this might give an intuitive explaination why exactly these two tilings have finitely supported eigenfunctions. However, if one considers periodic graphs which are not based on a tesselation by regular polygons the situation might be different. Figure 9 gives an example of a (non-archimedean) tesselation with finitely supported eigenfunctions. Since we do not always have explicit expressions of the eigenvalues of the operators ∆ θ or the volumes of their sublevels sets are too difficult to handle, we will not provide explicit integral expressions for these IDS', but we are still able to exclude the existence of 2 -eigenfunctions. In fact, for each tiling, we will find the θ-dependent matrix ∆ θ , make two choices θ, θ ∈ T 2 , and see that the sets of eigenvalues of ∆ θ and ∆ θ are disjoint.
It is straightforward to verify that these sets are disjoint and by Theorem 2.3 and Corollary 2.4 we find Proposition 4.6. The (3 4 .6)-tiling has no 2 (V)-eigenfunctions.