Arithmetic Structure of the Spectrum and Crystalline Measures

Abstract

We consider applications of the trace formula and spectral theory of metric graphs in Fourier analysis. It turns out that spectral measures associated with metric graphs give explicit examples of crystalline measures.

We consider applications of the trace formula and spectral theory of metric graphs in Fourier analysis. It turns out that spectral measures associated with metric graphs give explicit examples of crystalline measures.

10.1 Arithmetic Structure of the Spectrum

Let us discuss arithmetic structure of the spectra of standard Laplacians on metric graphs. It depends both on the topology of the underlying discrete graph $G$ and on the relations between the edge lengths in the metric graph $\Gamma$.

The trace formula (8.20) is going to play a crucial role in our studies, but we shall write it in a slightly modified way by moving the term $\chi \delta$ from the right hand side to the left hand side

(10.1)

where we assumed that the graph is connected and therefore $m_s (0) = 1$. In the original formula (8.20) the left hand side contains all spectral information while the right hand side collects geometric and topological characteristics of the metric graph. The modified formula (10.1) reflects crystalline¹ structure of the corresponding measures: the left hand side as well the Fourier transform of the right hand side are given by infinite sums of delta functions.

Taking into account the modified variant (10.1) of the trace formula it is natural instead of the discrete eigenvalues $\lambda _j$ to look at their square roots $k_j = \sqrt {\lambda _j} \geq 0$ and in addition to adjust the spectrum at $k= 0$ so that the modified spectrum$\text{Spec} \, (\Gamma )$ of $L^{\mathrm {st}} (\Gamma )$ is

(10.2)

Note that we use this convention for the spectrum only in this chapter, where its arithmetic structure is discussed, and in Chap. 24 devoted to discrete graphs.

We shall discuss two extreme cases:

• the edge lengths are pairwise rationally dependent,

• the edge lengths are rationally independent.

In the first case the length of every edge is an integer multiple of a certain basic length $\ell >0$. Therefore introducing degree two vertices at the distance $\ell$ on every edge makes the metric graph equilateral—all edges have the same length $\ell$. The spectrum for such graphs is directly connected to the spectrum of the corresponding normalised Laplacian matrix $L_N (G)$ (see Sect. 24.1). As the result the spectrum ${\text{Spec}} \, (\Gamma )$ is periodic and hence is given by a finite number of arithmetic sequences.

Let us focus on the second case where the set of edge lengths is rationally independent (see Sect. 9.4). It is clear that each loop in $\Gamma$ leads to the arithmetic sequence $\frac {2 \pi }{\ell _j} n , \; n \in \mathbb Z,$ in the spectrum (here $\ell _j$ is the length of the loop). Each eigenvalue has multiplicity 1. We are going to prove that no other arithmetic sequences occur.

Theorem 10.1

Let$\Gamma$be a compact metric graph on$N$edges with the loops given by the edges$E_1, E_2, \dots , E_\nu$, $\nu \leq N$. Assume that the edge lengths$\ell _j , \; j =1,2, \dots , N,$are rationally independent and$\Gamma$is neither the segment graph$\Gamma _{(1.1)}$, nor the cycle graph$\Gamma _{(1.2)}$, nor the figure eight graph$\Gamma _{(2.4)}$. Then the spectrum of the standard Laplacian on$\Gamma$can be presented as a union of multisets

$$\displaystyle \begin{aligned} {} {\mathrm{Spec}} \,(\Gamma) = L_1 (\Gamma) \cup L_2 (\Gamma) \cup \dots \cup L_\nu (\Gamma) \cup {\mathrm{Spec}}^*(\Gamma), \end{aligned} $$

(10.3)

where$L_j (\Gamma ) = \left \{ \frac {2 \pi }{\ell _j} n, \, n \in \mathbb Z \right \}$are full size arithmetic sequences determined by the loop lengths$\ell _j$, $j =1,2, \dots , \nu ,$and$\mathit{\text{Spec}}^*(\Gamma )$is a discrete set containing no full size arithmetic progression and satisfying:

$$\displaystyle \begin{aligned} {} \# \Big( {\mathrm{Spec}}^*(\Gamma) \cap [-T,T] \Big) = \alpha T + \mathcal O(1), \quad \mathrm{as} \; T \rightarrow \infty , \end{aligned} $$

(10.4)

with

$$\displaystyle \begin{aligned} {} \alpha = \frac{2}{\pi} \left( \mathcal L - \sum_{j=1}^\nu \ell_j \right) = \frac{1}{\pi} \left( \ell_1 + \dots + \ell_\nu + 2 (\ell_{\nu+1} + \dots \ell_N) \right). \end{aligned} $$

(10.5)

Proof

If the graph $\Gamma$ has a loop of length $\ell _j$, then the spectrum $\text{Spec} \, (\Gamma )$ contains the full size arithmetic sequence $\big \{ \frac {2 \pi }{\ell _j} n, \, n \in \mathbb Z \big \}$, hence representation (10.3) is a direct consequence of the fact that each loop gives rise to the eigenfunctions

$$\displaystyle \begin{aligned} \psi (x) = \left\{ \begin{array}{ll} \sin \frac{2 \pi}{\ell_j} n (x-x_{2j-1}), & x \in E_j, \\ 0, & \mbox{otherwise}, \end{array} \right. \quad n \in \mathbb Z.\end{aligned}$$

Formula (10.4) together with (10.5) then follow directly from the Weyl asymptotics (4.25).

It remains to prove that $\text{Spec}^*(\Gamma )$ contains no full size arithmetic sequence. It will be convenient as in Chap. 6 to use simultaneously the complex torus

$$\displaystyle \begin{aligned} \mathbb T^N = \{ {\mathbf z} \in \mathbb C^N: |z_j| = 1, j =1,2, \dots, N \}\end{aligned}$$

and the real torus

$$\displaystyle \begin{aligned} \mathbf T^N = \mathbb R^N\slash 2 \pi \mathbb Z^N.\end{aligned}$$

Consider first the case where $\Gamma$ is not a watermelon graph, then the set $\text{Spec}^*(\Gamma )$ is given as the intersection between the curve

$$\displaystyle \begin{aligned} {} (e^{ik \ell_1}, \dots, e^{ik \ell_N} ) \in \mathbb T^N \end{aligned} $$

(10.6)

and the zero set $\mathbb Z_G^*$ of the reduced secular polynomial $P_G^* ({\mathbf z}) = 0$.

Assume that the reduced spectrum $\text{Spec}^*(\Gamma )$ contains an arithmetic sequence

$$\displaystyle \begin{aligned} a + n b, \quad n \in \mathbb Z,\end{aligned}$$

where $a, b \in \mathbb R$. Consider the corresponding points

$$\displaystyle \begin{aligned} \vec{\psi} (n) = (a + n b) \vec{\ell} = a \vec{\ell} + n b \vec{\ell} \in \mathbf T^N\end{aligned}$$

on the real torus $\mathbf T^N$, where we use the vector $\vec {\ell } = (\ell _1, \ell _2, \dots , \ell _N)$ of edge lengths. Note that $\big \{ b \ell _j \big \}_{ j =1}^{N}$ are rationally independent as $\big \{ \ell _j \big \}_{ j =1}^{N}$ are.

Then there are two possibilities

• if $\big \{ b \ell _j \big \}_{ j =1}^{N}$ are linearly independent modulo $2 \pi$ with respect to $\mathbb Q$, then the points $\vec {\psi } (n)$ densely cover the torus $\mathbf T^N$;

• if $\big \{ b \ell _j \big \}_{ j =1}^{N}$ are linearly dependent modulo $2 \pi$ with respect to $\mathbb Q$, i.e. there exist integers $m_j$ and $m$ such that

  $$\displaystyle \begin{aligned} \sum_{j=1}^N m_j b \ell_j = m 2 \pi ,\end{aligned}$$

  then the points $\vec {\psi } (n)$ densely cover the hypertorus

  $$\displaystyle \begin{aligned} \sum_{j=1}^N m_j \varphi_j = m 2 \pi.\end{aligned}$$

In the first case the zero set of the polynomial $P_G^* ({\mathbf z})$ contains the whole torus $\mathbb T^N$, but this is impossible since $P_G^*$ is not identically zero.

In the second case the polynomial $P_G^*$ vanishes on the hypertorus, which is given on $\mathbf T^N$ by the hyperplanes:

$$\displaystyle \begin{aligned} \frac{m_1}{g} \varphi_1 + \frac{m_2}{g} \varphi_2 + \dots + \frac{m_N}{g} \varphi_N = \frac{m}{g} 2 \pi + \frac{k}{g} 2 \pi, \quad k \in \mathbb Z, \end{aligned}$$

where $g \in \mathbb N$ is the greatest common divisor (GCD) for $\big \{ m_j \big \}_{j=1}^N$. We also assume without loss of generality that GCD for $\{m_1, \dots , m_N, m \}$ is equal to 1.

Consider the polynomial $T(\mathbf z)$ vanishing on one of the hyperplanes

$$\displaystyle \begin{aligned} {} T(\mathbf z) = z_1^{m_1/g} z_2^{m_2/g} \dots z_N^{m_N/g} - e^{i 2 \pi m/g}. \end{aligned} $$

(10.7)

Let $\mathbf I$ be the polynomial ideal generated by $T(\mathbf z)$ with the zero set

$$\displaystyle \begin{aligned} V(\mathbf I) = \big\{ \mathbf z \in \mathbb C^N: F(\mathbf z) = 0, \forall F \in \mathbf I \big\}.\end{aligned}$$

The secular polynomial $P_G^*$ vanishes on $V(\mathbf I)$, then Hilbert’s Nullstellensatz implies that $\big (P_G^* (\mathbf z) \big )^r$ for a certain $r \in \mathbb N$ belongs to the ideal, i.e.

$$\displaystyle \begin{aligned} {} \big(P_G^* (\mathbf z) \big)^r = R(\mathbf z) T(\mathbf z), \end{aligned} $$

(10.8)

for a certain polynomial $R(\mathbf z)$. Since $P_G^*$ is irreducible (Theorem 7.19), the latter equality may hold only if $T(\mathbf z)$ coincides with a certain power of $P_G^* (\mathbf z)$, but $T(\mathbf z)$ given by (10.7) is not a power of any other polynomial. The only possibility that remains is that

$$\displaystyle \begin{aligned} P_G^* (\mathbf z) = T(\mathbf z).\end{aligned}$$

We already know that $P_G^*$ is a first or second degree polynomial in all variables, then (10.7) implies that

$$\displaystyle \begin{aligned} {} P_G^*(\mathbf z) = T(\mathbf z) = \Big( \prod_{j=1}^\nu z_j \Big) \Big( \prod_{j=\nu+1}^N z_j \Big)^2 - e^{i 2 \pi m/g}, \end{aligned} $$

(10.9)

where the first product is over the edges forming loops in $\Gamma$ and the second one—over all other edges. In particular, we have $m_j/g = 1,2$.

To prove that representation (10.9) leads to a contradiction we shall consider contractions of graphs as in Chap. 7 and use the explicit formula (7.6) describing change of the secular polynomials under contraction.

The secular polynomials for all graphs on at most three edges have been listed in Sect. 6.2. Only genuine graphs (i.e. excluding the graphs

$$\displaystyle \begin{aligned} G_{(2.1)}, G_{(2.3)}, G_{(3.1)}, G_{(3.3)}, G_{(3.5)}, G_{(3.6)}, G_{(3.10)}\end{aligned}$$

having degree two vertices) need to be examined. We see that only graphs $G_{(1.1)}$ (segment), $G_{(1.2)}$ (cycle), and $G_{(2.4)}$ (figure eight graph) have secular polynomials compatible with (10.9), but these graphs are excluded by the assumptions of the theorem.

Assume now that $\Gamma$ is a genuine graph on at least 4 edges. Then Lemma 7.7 implies that it can be contracted to a genuine graph on three edges, i.e. to one of the following graphs:

$$\displaystyle \begin{aligned} G_{(3.2)}, G_{(3.4)}, G_{(3.7)} , G_{(3.8)}, G_{(3.9)}, G_{(3.11)}.\end{aligned}$$

The secular polynomials for the graphs $G_{(3.2)}$, $G_{(3.4)}$, $G_{(3.7)}$, $G_{(3.8)}$, and $G_{(3.11)}$ (excluding the watermelon graph $G_{(3.9)}$) are not compatible with (10.9).

It remains to study the case where $\Gamma$ is a watermelon graph. Repeating our argument we arrive at the equation generalising (10.8)

$$\displaystyle \begin{aligned} \Big( P_{\mathbf W_N}^s (\mathbf z) P_{\mathbf W_N}^a (\mathbf z) \Big)^r = R(\mathbf z) T(\mathbf z).\end{aligned}$$

Irreducibility of $P_{\mathbf W_N}^{s} (\mathbf z)$ or $P_{\mathbf W_N}^{a} (\mathbf z)$ and impossibility to write $T(\mathbf z)$ as a power of any other polynomial leads to the conclusion that either

$$\displaystyle \begin{aligned} {} P_{\mathbf W_N}^s (\mathbf z) = T(\mathbf z) \quad \mbox{or} \quad P_{\mathbf W_N}^a (\mathbf z) = T(\mathbf z). \end{aligned} $$

(10.10)

It follows in particular that $T(\mathbf z)$ is first order in all variables. Equality (10.10) does not hold for $G_{(3.9)}$. For watermelon graphs on more than 3 edges contraction to any three edges leads to $G_{(3.11)}$, and factorisation of the corresponding polynomial is not compatible with (10.10). □

The proof of the above theorem shows that the following notations are useful:

• $\mathbb Z^* \subset \mathbb T^N$ and $\mathbf Z^* \subset \mathbf T^N$—the zero sets of the reduced polynomials, to be called reduced zero sets;

• ${\mathrm {Spec}}^*(\Gamma )$—the spectrum determined by the reduced polynomials, to be called reduced spectrum.

The reduced spectrum ${\mathrm {Spec}}^* (\Gamma )$ is obtained by intersecting the line $k \vec {\ell }$ with the reduced zero set $\mathbf Z^*$.

Our original proof of the above theorem [351] was based on Lang’s conjecture from diophantine analysis (proven in [359, 376] and refined in [184, 185]):

Theorem 10.2 (Lang’s Conjecture)

Assume that:

• $V \subset (\mathbb C^*)^N$ is an algebraic subvariety given by the zero set of Laurent polynomiasl; ²

• $\mathbf G$ is a finitely generated subgroup of rank $r$ of the torus $\mathbb T \subset (\mathbb C^*)^N$ considered as a group under coordinate-wise product;

• $\overline {\mathbf G}$ is the division group of $\mathbf G$ , that is

  $$\displaystyle \begin{aligned} \overline{\mathbf G} = \big\{ z \in T: z^m \in \mathbf G \;\mathit{\mbox{for some}} \; m \geq 1 \big\} \subset (\mathbb C^*)^N.\end{aligned}$$

Then there exist finitely many translates of (possibly low dimensional) subtori $T_1, T_2, \dots , T_\mu$ contained in $V$ such that

$$\displaystyle \begin{aligned} \overline{\mathbf G} \cap V = \overline{\mathbf G} \cap (T_1 \cup T_2 \cup \dots \cup T_\mu),\end{aligned}$$

with

$$\displaystyle \begin{aligned} \mu \leq \Big(C (V) \Big)^r,\end{aligned}$$

where $C(V)$ is an effectively computable constant independent of the group.

Full usage of diophantine analysis allows one to prove much more sophisticated properties of the spectrum. In particular the following two statements hold under the assumptions of Theorem 10.1

• The dimension of the reduced spectrum with respect to rationals (more precisely, the rational dimension of the rational linear span of the reduced spectrum) is infinite, despite the fact that the dimension of the length spectrum, that is $\{ \ell (\gamma ) \}_{p \in \mathcal P}$ is always finite [350]

  $$\displaystyle \begin{aligned} \dim_{\mathbb Q} \mathcal L_{\mathbb Q} \Big\{ k_n \Big\}_{k_n \in {\mathrm{Spec}}^*(\Gamma)} = \infty, \quad \dim_{\mathbb Q} \mathcal L_{\mathbb Q} \Big\{ \ell(\gamma) \Big\}_{p \in \mathcal P} = N. \end{aligned} $$

  (10.11)

• The length of possible finite arithmetic progressions—the number of elements in any such progression—can be estimated using the effectively computable constant $C(V)$[351].

10.2 Crystalline Measures

The spectra of standard Laplacians on metric graphs lead to explicit examples of so-called crystalline measures, which were right in the focus of recent studies in Fourier analysis. Before our examples were published [350], it was even conjectured that such positive measures may not exist.

Definition 10.3

A set $S$ is called discrete if every point from the set has a (small) neighbourhood containing no other points from the set.

Following [386, 388] crystalline measures $\mu$ are defined as

Definition 10.4

A tempered distribution $\mu$ is a crystalline measure if $\mu$ and its Fourier transform $\hat {\mu }$ are of the form

$$\displaystyle \begin{aligned} \mu (k) = \sum_{k_n \in K} a_n \delta_{k_n}, \quad \hat{\mu} (s) = \sum_{s_n \in S} b_n \delta_{s_n}, \end{aligned} $$

(10.12)

with $K$ and $S$ discrete subsets of $\mathbb R$.

The set $S$ is in the literature on crystalline measures referred to as the spectrum of the measure. In the case of metric graphs on the contrary the set K is determined by the spectrum ${\mathrm {Spec}} \, (\Gamma )$ (or rather ${\mathrm {Spec}}^*(\Gamma )$) of the Laplacian. In order to avoid possible misunderstanding we shall use the word spectrum only in connection with the spectrum of the operator.

The simplest example of a crystalline measure is given by the Poisson summation formula

$$\displaystyle \begin{aligned} {} \sum_{n \in \mathbb Z} f(n) = \sum_{m \in \mathbb Z} \hat{f} (2 \pi m), \end{aligned} $$

(10.13)

which also can be written as:

$$\displaystyle \begin{aligned} \mu (x) = \sum_{n \in \mathbb Z} \delta_{n} \quad \Rightarrow \quad \hat{\mu} (\gamma) = \sum_{m \in \mathbb Z} \delta_{2 \pi m}. \end{aligned} $$

(10.14)

The measure $\sum _{n \in \mathbb Z} \delta _{t n+c}$, $t>0, c \in \mathbb R,$ is usually called a Dirac comb of period t. Any finite combination of such measures is again a crystalline measure, called generalised Dirac comb. Such measures are considered as trivial crystalline measures and one is interested in constructing non-trivial crystalline measures, i.e. not given by a finite combination of Dirac combs. Periodic generalised Dirac combs are formed by a finite number of elementary Dirac combs having rationally dependent periods. One may always assume that the periods are equal.

The support of a crystalline measure is by definition a discrete set, but we shall need the slightly more subtle notion of uniformly discrete sets. Discreteness of the set (given by Definition 10.3) does not imply that the distance between any two points in the set is bounded from below by a certain strictly positive number.

Definition 10.5

A discrete set $S$ is called uniformly discrete if there is a positive number $d> 0$ such that

$$\displaystyle \begin{aligned} | x_n - x_m | \geq d\end{aligned}$$

holds for any $x_n, x_m \in S, \; n \neq m$.

Note that in the case of multiple eigenvalues the support of the spectral measure (see (10.17) below) may be uniformly discrete, even if the spectrum is not uniformly discrete.

The union of two periodic lattices with rationally independent periods provides an example of a set which is discrete but not uniformly discrete. In what follows uniform discreteness will help us to prove the measures that we shall construct are not generalised Dirac combs.

Lemma 10.6

A measure on $\mathbb R$ given by a finite linear combination of Dirac combs is uniformly discrete if and only if it is periodic.

Proof

Let $\mu$ be a generalised Dirac comb that is given by a finite sum of Dirac combs:

$$\displaystyle \begin{aligned} \mu (x) = \sum_{n=1}^N a_n \Big(\sum_{m\in \mathbb Z} \delta_{t_n m + c_n} \Big). \end{aligned} $$

(10.15)

If all periods $t_n$ are pairwise rationally dependent, i.e.$t_n = q_n t_1, \; q_n \in \mathbb Q$, then the measure is periodic.

Arbitrary generalised Dirac comb can be written as a finite sum of periodic measures with rationally independent periods. To this end let us divide the set $T = \{ t_n \}_{n=1}^N$ of all periods into $N_1 \leq N$ equivalence classes of pairwise rationally dependent periods

$$\displaystyle \begin{aligned} \begin{array}{l} \displaystyle T = \cup_{j=1}^{N_1} T_j ; \quad \; T_i \cap T_j = \emptyset, i \neq j, \\[3mm] \displaystyle t_n, t_m \in T_j \Rightarrow t_n/t_m \in \mathbb Q. \end{array}\end{aligned}$$

In this way the measure $\mu$ is presented as a sum of periodic measures, each being a generalised Dirac comb:

$$\displaystyle \begin{aligned} \mu_j (x) := \sum_{ t_n \in T_j } a_n \Big(\sum_{m\in \mathbb Z} \delta_{t_n m + c_n} \Big), \quad j =1,2, \dots, N_1\end{aligned}$$

$$\displaystyle \begin{aligned} {} \Rightarrow \mu (x ) = \sum_{j=1}^{N_1} \mu_j (x). \end{aligned} $$

(10.16)

The supports of the periodic measures intersect at most at

$$\displaystyle \begin{aligned} \sum_{i <j} |T_i| \; |T_j|\end{aligned}$$

points. This follows from the fact that two sequences $t_i m_i + c_i , \, m_i \in \mathbb Z$ and $t_j m_j + c_j, \, m_j \in \mathbb Z$ have at most one common point, provided $t_i/t_j \notin \mathbb Q$. Assume the opposite:

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ccc} \displaystyle t_i m_i + c_i & = & \displaystyle t_j m_j + c_j \\ \displaystyle t_i m_i^{\prime} + c_i & = & \displaystyle t_j m_j^{\prime} + c_j \end{array} \right. \Rightarrow t_i (m_i^{\prime} - m_i) = t_j (m_j^{\prime}-m_j) , \end{aligned}$$

i.e. $t_i$ and $t_j$ are rationally dependent.

Assume that $\mu$ is not periodic, hence the number of periodic measures in the representation (10.16) is at least two. The measures $\mu _1$ and $\mu _2$ are two periodic measures with rationally independent periods, hence in their supports there exists an infinite sequence of arbitrarily close points:

$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} x_j^1 \in \mathrm{supp}\, \mu_1, \\ x_j^2 \in \mathrm{supp}\, \mu_2 \end{array} \right. \quad |x_j^1-x_j^2| \stackrel{[j \rightarrow \infty]}{\rightarrow} 0 ,\end{aligned}$$

implying that the measure $\mu _1 + \mu _2$ is not uniformly discrete. Only a finite number of points from the sequence are not present in the support of $\mu$ as the supports of the periodic measures intersect at a finite number of points. Hence even the measure $\mu$ is not uniformly discrete in this case.

If the measure $\mu$ is periodic, then it is uniformly discrete: to determine the minimal distance between the atoms it is enough to look at one of the periods. □

This lemma implies that every trivial (given by a generalised Dirac’s comb) uniformly discrete crystalline measure is periodic. This observation will be important in the future.

Consider the standard Laplacian on any finite compact metric graph. Writing the corresponding trace formula (8.20) in the form (10.1) suggests us to introduce the following spectral measure

(10.17)

Remember that the spectrum ${\mathrm {Spec}}$ includes both positive and negative values of $k_n = \pm \sqrt {\lambda _n}, \; \lambda _n \geq 0$. The support of the measure is discrete, since the spectrum of $L^{\mathrm {st}} (\Gamma )$ is discrete. Moreover, the measure is positive as the coefficients in front of delta functions are positive integers ($\beta _1 \geq 0$, multiple eigenvalues are allowed). Taking Fourier transform of the right hand side of the trace formula

$$\displaystyle \begin{aligned} \mu = \frac{\mathcal{L}}{\pi}+\frac{1}{\pi}\sum_{\gamma \in \mathcal{P}} l(\mathrm{prim}\,(\gamma)) S_{\mathrm{v}} (\gamma) \cos kl(\gamma)\end{aligned}$$

leads to the following expression for the Fourier transform of the measure:

$$\displaystyle \begin{aligned} {} \hat{\mu} (l) = 2 \mathcal L \delta + \sum_{\gamma \in \mathcal{P}} l(\mathrm{prim}\,(\gamma)) S_{\mathrm{v}} (\gamma) \Big( \delta_{l(\gamma)} + \delta_{-l(\gamma)} \Big). \end{aligned} $$

(10.18)

The support of this measure coincides with the set of lengths of periodic orbits which are linear combinations with positive integer coefficients of the edge lengths:

$$\displaystyle \begin{aligned} \ell (\gamma) = \sum_{n=1}^N \alpha_n (\gamma) \ell_n, \quad \alpha_n (\gamma) \in \mathbb N, \end{aligned} $$

(10.19)

where $\alpha _n (\gamma ) = 0,1,2, \dots$ counts how many times the orbit $\gamma$ passes through the edge $E_n$. Different periodic orbits may have equal lengths but the number of orbits having a certain length is always finite. On the other hand for any length $\ell (\gamma )$ there is always a non-zero distance to the nearest $\ell (\gamma ')$. This is due to the fact that the coefficients in the representation above are positive integers.³ In other words, the set of lengths is discrete, hence to prove that $\mu$ is a crystalline measure it remains to show that $\mu$ is a tempered distribution, but this follows from the fact that the spectrum of $L^{\mathrm {st}} (\Gamma )$ satisfies Weyl’s asymptotics (4.25) and all non-zero eigenvalues in $\mu (k)$ are counted in accordance to their multiplicities $m_s(\lambda _n)$

(10.20)

Summing up, the spectral measure $\mu (k)$ for any metric graph is crystalline. In order to get an interesting result we need to show that this measure is not a trivial crystalline measure.

If all edge lengths are pairwise rationally dependent, then the spectrum as defined above is periodic in k. We discuss this case in more detail in Sect. 24.3, in particular Theorem 24.6 tells us more about the structure of the spectrum. Periodicity of the spectrum ${\mathrm {Spec}} \, (\Gamma )$ means that the symmetrised spectral measure $\mu$ can be written as a finite sum of Dirac combs with positive integer coefficients and the same periods. One gets a trivial crystalline measure in this case.

In the rest of this section we discuss the opposite case where the edge lengths are rationally linearly independent. We shall also exclude the watermelon graphs from our discussion and will return to them at the end. Theorem 10.1 implies that the spectrum contains arithmetic progressions corresponding to the loops in $\Gamma$. Every such progression can be seen as a Dirac comb and we shall subtract their contributions from the trace formula focusing on the spectrum determined by the set ${\mathrm {Spec}}^*(\Gamma )$ not containing any full size arithmetic progression (see (10.3)). To this end let us introduce the reduced spectral measure

(10.21)

Remember that $\nu$ denotes the numbers of loops in $\Gamma$ and the set ${\mathrm {Spec}}^*(\Gamma )$ is determined by the intersections of the curve $(e^{ik \ell _1}, e^{ik \ell _2}, \dots , e^{ik \ell _N})$ with the zero set of the reduced secular polynomial $P_G^* (\mathbf z)$ and it holds

$$\displaystyle \begin{aligned} {\mathrm{Spec}}^*(\Gamma) = {\mathrm{Spec}} \, (\Gamma) \setminus \Big( \bigcup_{j=1}^\nu L_j(\Gamma) \Big) ,\end{aligned}$$

as multisets. We remind that $L_j (\Gamma )$ denote the arithmetic sequences in the spectrum corresponding to the eigenfunctions supported by the loops. To get an explicit formula for the Fourier transform of $\mu ^*$ we need to subtract from formula (10.18) the terms corresponding to Dirac combs associated with the loops:

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \hat{\mu}^* (l) & = & \displaystyle 2 \mathcal L \delta + \sum_{\gamma \in \mathcal{P}} l(\mathrm{prim}\,(\gamma)) S_{\mathrm{v}} (\gamma) \Big( \delta_{l(\gamma)} + \delta_{-l(\gamma)} \Big) \\[3mm] & & \displaystyle - \sum_{j=1}^\nu \ell_j \sum_{n \in \mathbb Z} \delta_{\ell_j n} \\[3mm] & = & \displaystyle \big(2 \mathcal L - \sum_{j=1}^\nu \ell_j \big) \delta + \sum_{\gamma \in \mathcal{P}} l(\mathrm{prim}\,(\gamma)) S_{\mathrm{v}} (\gamma) \Big( \delta_{l(\gamma)} + \delta_{-l(\gamma)} \Big) \\[3mm] & & \displaystyle - \sum_{j=1}^\nu \ell_j \sum_{n =1}^\infty \Big( \delta_{\ell_j n} + \delta_{-\ell_j n}\Big). \end{array} \end{aligned} $$

(10.22)

Note that all subtracted terms are present in the sum over all periodic orbits and correspond to the orbits obtained by going along one particular loop several times in one particular direction. Only one direction for each loop is present because the eigenvalues determined by the loops have multiplicity one, not two as for the single cycle graph $\Gamma _{(1.2)}$. The periodic orbits involving any two loops are not subtracted.

Our analysis implies that $\mu ^*$ is again a crystalline measure. It remains to show that this measure is not trivial. Before considering the general case, let us study the spectrum of the lasso graph.

10.3 The Lasso Graph and Crystalline Measures

In this section we study the measures associated with the lasso graph $\Gamma _{(2.2)}$. The zero set of the reduced secular polynomial

$$\displaystyle \begin{aligned} P_{(2.2)}^* = 3 z_1 z_2^2 - z_2^2 + z_1 -3\end{aligned}$$

on the real torus $\mathbf T^2$ is presented once more in Fig. 10.1 The polynomial $P^*_{(2.2)}$ is $\mathbb D$-stable, that is it does not have any zeroes inside $\mathbb D^2$, where $\mathbb D$ is the open unit disk $\mathbb D = \{z: |z|<1 \}$. To see this we write the equation $P^*_{(2.2)} (\mathbf z) = 0$ as

$$\displaystyle \begin{aligned} z_2^2 = \frac{z_1-3}{1-3 z_1} .\end{aligned}$$

Fig. 10.1

The zero set $\mathbf Z^*_{G_{(2.2)} }$ on the real torus

The Möbius transformation $z_1 \mapsto \frac {z_1-3}{1-3 z_1}$ maps the unit disk to its complement, hence the equation has no solutions inside $\mathbb D^2$. Moreover we have

$$\displaystyle \begin{aligned} {} \begin{array}{ccc} \displaystyle P_{(2.2)}^* (\frac{1}{z_1}, \frac{1}{z_2}) & = & \displaystyle 3 \frac{1}{z_1} \frac{1}{z_2^2} - \frac{1}{z_2^2} - \frac{1}{z_1} -3 \\[3mm] & = & \displaystyle - z_1^{-1} z_2^{-2} \big( 3 z_1 z_2^2 - z_2^2 + z_1 - 3 \big) \\[3mm] & = & \displaystyle - z_1^{-1} z_2^{-2} P_{(2.2)}^* (z_1, z_2). \end{array} \end{aligned} $$

(10.23)

This relation implies in particular that, if $(z_1, z_2 ), \; z_j \neq 0,$ is a zero of $P_{(2.2)}^*$, then $(1/z_1, 1/z_2)$ is also a zero. Therefore the secular equation $P_{(2.2)}^* (e^{ik \ell _1}, e^{ik \ell _2}) = 0$ cannot have non-real solutions $k_j \notin \mathbb R$. Assume on the contrary that such solution $k_j$ exists. If $\mbox{Im} \, k_j > 0$, then $z_j = e^{ik \ell _j}, \; j =1,2,$ are inside the unit disk $\vert z_j \vert < 1$ and $P_{(2.2)}^* (z_1, z_2) = 0$; this contradicts that $P_{(2.2)}^*$ is stable. If $\mbox{Im} \, k_j < 0$, then $1/z_j = e^{-ik \ell _j}, \; j =1,2,$ are inside the unit disk. The relation (10.23) implies then that

$$\displaystyle \begin{aligned} P_{(2.2)}^* (1/z_1, 1/z_2) = - z_1^{-1} z_2^{-2} P_{(2.2)}^* (z_1, z_2) = 0,\end{aligned}$$

which again contradicts stability of $P_{(2.2)}^*$.

The reduced spectrum of the metric graph is obtained by intersecting the line $(k \ell _1, k \ell _2)$ and the zero set of the function

$$\displaystyle \begin{aligned} L^*_{(2.2)} (\varphi_1, \varphi_2) = 3 \sin ( \frac{\varphi_1}{2} + \varphi_2) + \sin ( \frac{\varphi_1}{2} - \varphi_2) .\end{aligned}$$

The function has a non-zero gradient

$$\displaystyle \begin{aligned} \nabla L^*_{(2.2)} = \Big( \frac{3}{2} \cos ( \frac{\varphi_1}{2} + \varphi_2) + \frac{1}{2} \cos ( \frac{\varphi_1}{2} - \varphi_2), 3 \cos ( \frac{\varphi_1}{2} + \varphi_2) - \cos ( \frac{\varphi_1}{2} - \varphi_2) \Big) ,\end{aligned}$$

implying that the zero set of $L^*_{(2.2)}$ is a smooth curve on the torus $\mathbf T^2$. Moreover, the normal to the curve lies in the first quadrant since the components of the gradient have the same sign

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \frac{\partial L^*_{(2.2)}}{\partial \varphi_1} \frac{\partial L^*_{(2.2)}}{\partial \varphi_2} & = & \displaystyle \frac{1}{2} \Big( 9 \cos^2 (\frac{\varphi_1}{2} + \varphi_2) - \cos^2(\frac{\varphi_1}{2} - \varphi_2) \Big) \\[3mm] & = & \displaystyle \frac{1}{2} \Big( 9 - 9 \sin^2 (\frac{\varphi_1}{2} + \varphi_2) - 1 + \sin^2(\frac{\varphi_1}{2} - \varphi_2) \Big) \\[3mm] & = & \displaystyle 4. \end{array} \end{aligned} $$

(10.24)

where cancelling the two terms on the middle line we used that $L^*_{(2.2)} (\varphi _1, \varphi _2) = 0$. The direction vector for the line $(k \ell _1, k \ell _2)$ belongs to the first quadrant as well, hence the intersection between the line and the zero set is never tangential. One obtains an infinite sequence of simple eigenvalues $\{ k_n\}$ solving the trigonometric equation

$$\displaystyle \begin{aligned} {} L^*_{(2.2)} (k \ell_1 , k \ell_2) = 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big) + \sin k \big(\frac{\ell_1}{2} - \ell_2 \big) = 0. \end{aligned} $$

(10.25)

Plotting the graph of the function one may get an impression how the eigenvalues are placed (see Fig. 10.2).

Fig. 10.2

Graph of the function $L^*_{(2.2)}(k \ell _1 , k \ell _2)$ for $\ell _1 =1, \ell _2 = \sqrt {7}$

To prove that the number of zeroes is infinite, it is enough to take into account that $L^*_{(2.2)} (k \ell _1 , k \ell _2)$ for any $\ell _j >0$ is a continuous function satisfying the two-sided inequality

$$\displaystyle \begin{aligned} 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big) -1 \leq L^*_{(2.2)} (k \ell_1 , k \ell_2) \leq 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big)+1 .\end{aligned}$$

The curved strip

$$\displaystyle \begin{aligned} 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big) -1 \leq y \leq 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big)+1\end{aligned}$$

plotted in Fig. 10.3 crosses the line $y = 0$ infinitely many times because the function $3 \sin k \big ( \frac {\ell _1}{2} + \ell _2 \big )$ does this. The character of the spectrum depends on whether $\ell _1/\ell _2$ is a rational number or not.

• If $\ell _1/\ell _2 \in \mathbb Q$, then the line $(k \ell _1, k \ell _2)$ is periodic on the real torus $\mathbf T^2$ and therefore intersects the zero set at a finite number of points (see left Fig. 10.4).

  Fig. 10.3

  

  The curved strip associated with $L^*_{(2.2)}$ for $\ell _1 =1, \ell _2 = \sqrt {7}$

  Fig. 10.4

  

  Graphical representation of the reduced spectrum of $\Gamma _{(2.2)}$ for rationally dependent (left figure) and rationally independent (right figure) edge lengths

• If $\ell _1/\ell _2 \notin \mathbb Q$, then the line $(k \ell _1, k \ell _2)$ densely covers the real torus $\mathbf T^2$ and therefore the intersection points densely cover the zero set (see right Fig. 10.4).

In the figures above we used the same values of $\ell _1/\ell _2$ as in Fig. 6.3, namely $\frac {\ell _1}{\ell _1} = 3$ and $\frac {\ell _1}{\ell _2} = \frac {\sqrt {5}-1}{2}$.

For arbitrary positive edge lengths there is always a non-zero distance between the subsequent intersections, hence the distance between the subsequent eigenvalues is separated from zero, i.e. the reduced spectrum for $\Gamma _{(2.2)}$ is always uniformly discrete, independently of the actual edge lengths.

It is almost clear that in the case of rationally independent edge lengths the spectrum is not periodic, since $L^*_{(2.2)} (k \ell _1, k\ell _2)$ is not a periodic function of k. To prove this rigorously we may use Theorem 10.1 which states that ${\mathrm {Spec}}^*(\Gamma )$ contains no arithmetic sequences provided the edge lengths are rationally independent, hence the spectrum cannot be periodic.

Let us summarise our findings.

Theorem 10.7

Let$\mu ^* = \delta + \sum _{k_n \neq 0} \delta _{k_n}$where$k_n$are solutions to the trigonometric equation (10.25)

$$\displaystyle \begin{aligned} L^*_{(2.2)} (k \ell_1, k\ell_2) = 3 \sin k \big( \frac{\ell_1}{2} + \ell_2 \big) + \sin k \big(\frac{\ell_1}{2} - \ell_2 \big) = 0\end{aligned}$$

be the reduced spectral measure for the lasso graph$\Gamma _{(2.2)}$with edge lengths$\ell _1$and$\ell _2$.

• (1) If$\ell _1/\ell _2 \in \mathbb Q$, then the measure$\mu ^*$is given by a periodic generalised Dirac comb. In particular:

  • (a) The measure$\mu ^*$is a trivial positive idempotent⁴crystalline measure.

  • (b) The support of the measure$\mu ^*$is given by the union of finitely many arithmetic progressions.

  • (c) The Fourier transform of the measure$\hat {\mu }^*$is again a periodic generalised Dirac comb and is supported by a uniformly discrete set.

  • (d) $|\hat {\mu }^*|$is translation bounded.

• (2) If$\ell _1/\ell _2 \notin \mathbb Q$, then the measure$\mu ^*$is not a generalised Dirac comb. It holds:

  • (a) The measure$\mu ^*$is a non-trivial positive idempotent crystalline measure.

  • (b) The support of the measure$\mu ^*$meets any arithmetic progression in at most a finite number of points.

  • (c) The support of the Fourier transform of the measure$\hat {\mu }^*$is not a uniformly discrete set.

  • (d) $|\hat {\mu }^*|$is not translation bounded.

Proof

The first part of the Theorem is elementary and is presented here just for the record in order to be compared with the second part. Only statement $1(a)$ needs clarification. All zeroes of the function $L^*_{(2.2)} (k \ell _1 , k \ell _2)$ are simple since (10.24) implies that both partial derivatives are non-zero and have the same sign. We did not use in the argument that $\ell _1/\ell _2 \in \mathbb Q$, hence the same proof implies even $2(a)$.

Let us focus on the case $\ell _1/\ell _2 \notin \mathbb Q$. The measure $\mu ^*$ is not a generalised Dirac comb for the following reasons:

• (i) the spectrum of $L^{\mathrm {st}} (\Gamma _{(2.2)})$ is a uniformly discrete set;

• (ii) Lemma 10.6 states that every atomic measure given by a generalised Dirac comb is uniformly discrete only if it is periodic;

• (iii) the measure $\mu ^*$ cannot be periodic as its support does not contain any arithmetic sequence (Theorem 10.1).

The latter statement coincides with $2(b)$.

To prove $2(c)$ let us remember that the Fourier transform of $\mu ^*$ is given by (10.22). The support of the delta functions include lengths $l_0$ of all periodic orbits with non-zero sum of the corresponding scattering coefficients

$$\displaystyle \begin{aligned} \displaystyle \sum_{l(\gamma) = l_0} l(\mathrm{prim}\,(\gamma)) S_{\mathrm{v}} (\gamma) \neq 0.\end{aligned}$$

Since the two edge lengths are rationally independent, at least the orbits supported by each of the two edges alone are present. These periodic orbits have lengths $n_1 \ell _1, \, n_1 \in \mathbb N$ and $2 n_2 \ell _2, \, n_2 \in \mathbb N$. The union of these sets is not uniformly discrete—there are always arbitrarily close lengths for sufficiently large $n_j$. Hence the spectrum is not periodic since otherwise the reduced spectral measure would have been given by a finite sum of Dirac combs with a common period. The Fourier transform of such measures is again given by Dirac combs with equal periods and therefore its support is a uniformly discrete set.

One can also prove $2(c)$ by using the result by Lev-Olevskii [372], which states that in one dimension every crystalline measure with uniformly discrete support of the measure and its Fourier transform is given by a periodic generalised Dirac comb.

We do not have an explicit proof for $2(d)$. It is clear that the number of periodic orbits having length approximately equal to $\ell (\gamma )$ grows as $l(\gamma ) \rightarrow \infty$ and the corresponding scattering coefficients $S_{\mathrm {v}} (\gamma )$ decrease, but it is difficult to compare these quantities even for such simple graphs as $\Gamma _{(2.2)}$. On the other hand translational boundedness of $| \hat {\mu } |$ would contradict Meyer’s Theorem stating that every crystalline measure with $a_\lambda$ from a finite set ($a_\lambda =1$ in our case) and $|\hat {\mu }|$ translation bounded is a generalised Dirac comb:

Theorem (Meyer [385])

If$a_\lambda$takes values in a finite set and$|\hat {\mu }|$is translation bounded, that is,$\displaystyle \sup _{x \in \mathbb R} |\hat {\mu }| (x+ [0,1]) < \infty$, then$\mu$is a generalised Dirac comb.□

Historically, the measure $\mu ^*$ associated with $\Gamma _{(2.2)}$ was the first constructed explicit positive uniformly discrete crystalline measure. All non-trivial examples of crystalline measures known before were less explicit:

• Meyer’s construction [386, 387] following A.P. Guinand [250], where the measure is determined by the non-trivial zeroes of two Dirichlet functions;

• signed measures of N. Lev and A. Olevskii [373] constructed generalising Meyer’s cut-and-project procedure;

• translationally bounded measures by M. Kolountzakis [302] constructed as an infinite sum of Dirac combs.

After those examples were presented it was not clear whether positive uniformly discrete crystalline measures exist, especially in view of Lev-Olevskii theorems stating that every uniformly discrete crystalline measure with uniformly discrete support of the Fourier transform is a generalised Dirac comb in $\mathbb R^1$ [372]. Assuming positivity, the same result holds in $\mathbb R^d$. The above examples of crystalline measures are not explicit, and it is hard to control positivity of the measures or arithmetic properties of the support.

On the other hand those papers contained clearly formulated questions that the measure $\mu ^*$ provides an affirmative answer to:

• (a) Excepting Dirac combs, do there exist nonnegative crystalline measures? [386, page 3158]

• (b) Can one get apositivemeasure in Theorem 1.2—a crystalline measure with the support containing only finitely many elements of any arithmetic progression (part 3 of question 11.2 in [374]).

• (c) Do there exist uniformly discrete setsK? A trivial answer is given by Dirac combs. Are there other examples? [386, page 3158]

• (d) Let$\mu$be a measure on$\mathbb R$, with uniformly discrete supportKand discrete closed spectrum$S$. Does it follow that$S$must be also uniformly discrete? (part 2 of question 11.2 in [374])

After the measure $\mu ^*$ was discovered several alternative explicit constructions of crystalline measures were suggested, in particular using the following mathematical notions: multivariate stable polynomials [350], trigonometric polynomials with real zeroes [414], inner functions in $\mathbb C^N$ [389], linear recurrence relations on lattices and curved model sets [390]. The paper [414] contains in addition characterisation of all idempotent crystalline measures on $\mathbb R^1$ via trigonometric polynomials with real zeroes.

10.4 Graph’s Spectrum as a Delone Set

Laplacians on metric graphs always lead to positive crystalline measures, but measures having uniformly discrete support are of particular interest. In this section we shall focus on how examples of such measures can be obtained. In view of Weyl’s asymptotic 4.15 the support of every such measure is also relatively dense. Discrete sets that are both uniformly discrete and relatively dense are called Delone sets.⁵

The structure of the spectrum depends on whether the edge lengths are rationally dependent or not. If the edge lengths are pairwise rationally dependent, then the support of the reduced spectrum is periodic and therefore is always a Delone set. The corresponding summation formula is a generalised Dirac comb and is not interesting for us.

If the edge lengths are rationally independent, then as we have seen the reduced spectrum is uniformly discrete only if the zero set of the reduced secular polynomial is not singular: the curve $( e^{i k \ell _1}, \dots , e^{i k \ell _N})$ densely covers the unit torus $\mathbb T^N$ leading to close eigenvalues when the curve almost hits the singularities of the reduced zero set (without actually hitting them). Hence looking for measures supported by Delone sets one should either find graphs for which the zero sets of the reduced secular polynomials are not singular, or impose a linear relation on the edge lengths ensuring that the curve $( e^{i k \ell _1}, \dots , e^{i k \ell _N})$ does not come close to the singularities of the reduced zero set.

It is straightforward to describe the reduced spectrum in the case the graph has at most two edges: the reduced spectrum is given by single arithmetic progressions with the only exception of $G_{(2.2)}$

• ${\mathrm {Spec}}^*(\Gamma _{(1.1)}) = {\mathrm {Spec}} \, (\Gamma _{(1.1)})= \{ \frac {\pi }{\ell _1} n, n \in \mathbb Z \}.$

• ${\mathrm {Spec}}^* (\Gamma _{(1.2)}) = \{ \frac { 2\pi }{\ell _1} n, n \in \mathbb Z \}.$

• ${\mathrm {Spec}}^* (\Gamma _{(2.1)}) = {\mathrm {Spec}} \, (\Gamma _{(2.1)})= \{ \frac {\pi }{\ell _1+ \ell _2} n, n \in \mathbb Z \}.$

• ${\mathrm {Spec}}^* (\Gamma _{(2.2)}) = \{ k: 3 \sin k \big ( \frac {\ell _1}{2} +\ell _2 \big ) + \sin k \big ( \frac {\ell _1}{2} - \ell _2 \big ) = 0 \}.$

• ${\mathrm {Spec}}^* (\Gamma _{(2.3)}) = \{ \frac {2\pi }{\ell _1+\ell _2} n, n \in \mathbb Z \}.$

• ${\mathrm {Spec}}^* (\Gamma _{(2.4)}) = \{ \frac {2\pi }{\ell _1+\ell _2} n, n \in \mathbb Z \}.$

The reduced spectrum of $L^{\mathrm {st}} (\Gamma _{(2.2)})$ has already been discussed, therefore let us turn to graphs on three and more edges. The singular subset of the reduced zero set $\mathbf Z_G^*$ may be non-empty, therefore the non-trivial spectrum cannot always be a Delone set. Given a genuine graph on at least three edges, not a dumbbell graph $G_{(3.7)}$, the spectrum is not a Delone set if the edge lengths are rationally independent. Moreover it cannot be made Delone by subtracting a finite number of arithmetic progressions. This follows from Lemma 6.3 stating that all genuine graphs on three edges have singular points in the reduced zero set, provided $G_{(3.7)}$ is excluded.

Having this fact in mind we are going to adopt the opposite strategy and discuss how to choose the edge lengths to get a Delone set, provided the discrete graph $G$ is fixed.

Theorem 10.8

Let $G$ be a discrete graph on at least three edges, not a watermelon graph $G_{(3.9)}$ . Then the edge lengths can be chosen in such a way that the spectrum is a non-periodic Delone set after possibly subtracting a finite number of arithmetic sequences corresponding to the loops.

Proof

To prove the theorem we shall contract $N-3$ edges in the original graph to get a graph on three edges. Not every graph on three edges is suitable—non-genuine graphs should be excluded: these graphs are equivalent to graphs on one or two edges and their spectra are often given by arithmetic sequences. Non-genuine graphs on three edges are:

$$\displaystyle \begin{aligned} {} G_{(3.1)}, G_{(3.3)}, G_{(3.5)}, G_{(3.6)}, \mbox{ and } G_{(3.10)}. \end{aligned} $$

(10.26)

Let us exclude all these graphs. Lemma 7.7 implies that any genuine graph on three or more edges can be contracted to one of the following genuine graphs on three edges:

$$\displaystyle \begin{aligned} G_{(3.2)}, G_{(3.4)}, G_{(3.7)}, G_{(3.8)}, G_{(3.9)}, \mbox{ or } G_{(3.11)}.\end{aligned}$$

One may strengthen Lemma 7.7 by proving that, excluding the watermelon $G_{(3.9)}$, any genuine graph on at least three edges can be contracted to any of the following genuine graphs on three edges:

$$\displaystyle \begin{aligned} {} G_{(3.2)}, G_{(3.4)}, G_{(3.7)}, G_{(3.8)}, \mbox{ or } G_{(3.11)}. \end{aligned} $$

(10.27)

Lemma 7.7 is proven by constructing a sequence of genuine graphs with a decreasing number of edges. Every graph in the sequence is a contraction of the previous graph. This sequence ends up with the watermelon graph $G_{(3.9)}$ only if the previous graph in the sequence is either the watermelon with a loop $\mathbf W_3\mathbf L$, or the watermelon on a stick $\mathbf W_3\mathbf I$ presented in Figs. 7.7 and 7.8. Instead of $G_{(3.9)}$ these graphs can be contracted to $G_{(3.11)}$ and $G_{(3.8)}$ respectively (see Fig. 10.5).

Fig. 10.5

Possible extensions of the watermelon graph $\mathbf W_3= G_{(3.9)}$

In what follows we shall assume that all but except three edges are contracted so that the graph $G$ is equivalent to one of the graphs from the list (10.27). Note that this contraction is not unique and the same original graph can be contracted to different graphs. To get a Delone spectrum we shall subtract arithmetic sequences corresponding to the loops, remember that contraction may lead to new loops not present in the original graph $G$.

To accomplish the proof of the theorem it is enough to show that the edge lengths in the graphs (10.27) can be chosen so that the reduced spectrum is a Delone set. Examining the reduced zero surfaces $\mathbf Z^*$ for $G_{(3.2)}, G_{(3.4)}, G_{(3.7}, G_{(3.8)},$ and $G_{(3.11)}$ (plotted in Figs. 6.10, 6.12, 6.15, 6.16, and 6.19) we see that excluding $\mathbf Z_{(3.7)}^*$ all other surfaces are singular.

Let us discuss the regular case of the dumbbell graph $G_{(3.7)}$ first. The reduced spectrum is determined by the zeroes of the reduced Laurent polynomial $L^*_{(3.7)}$ (see (6.15)). The corresponding reduced zero set $\mathbf Z_{(3.7)}^*$ expanded periodically to $\mathbb R^3$ is formed by smooth two-dimensional sheets separated from each other by a certain non-zero distance.

The normal to the surface is always pointing in the first octant, in other words all coordinates of the gradient have the same sign. This is a general fact following from the monotonicity of the eigenvalues with respect to stretching of the edges: the eigenvalues are non-increasing functions of the edge lengths. This follows from the min-max principle and scaling properties of the Dirichlet integral (giving the quadratic form of the standard Laplacian).⁶ We prove this fact by differentiating $L^*_{(3.7)} (\varphi _1, \varphi _2, \varphi _3)$ directly. The calculations remind us of formula (10.24). We have for example

where we of course used that $L^*_{(3.7)} (\varphi _1, \varphi _2, \varphi _3) = 0$ in the third equality. Almost identical calculations lead to the inequality

$$\displaystyle \begin{aligned} \frac{\partial L^*_{(3.7)}}{\partial \varphi_3} \frac{\partial L^*_{(3.7)}}{\partial \varphi_1} > 0.\end{aligned}$$

The spectrum is obtained by crossing the reduced zero set by the line $k (\ell _1, \ell _2, \ell _3)$ whose direction vector lies in the first octant (as well as the normal to the surface). Hence the distance between any two subsequent eigenvalues is uniformly separated from zero, the reduced spectrum is a Delone set. In order to avoid that the reduced spectrum is given by a generalised Dirac comb, it is enough to assume that $\ell _j$ are rationally independent, no further restriction on the edge lengths is necessary.

Let us turn to the graphs $G_{(3.2)}, G_{(3.4)},$$G_{(3.8)},$$G_{(3.11)}$. If all edge lengths are rationally independent, then it is unavoidable that the reduced spectrum has arbitrarily close points: these points appear when the line $k \vec {\ell }$ comes closer and closer to the singular points. On the other hand Lemma 6.3 lists all singular points ${\boldsymbol \varphi }^j$ for these graphs. Let us choose a hypertorus $\mathcal T$ avoiding all these singular points

$$\displaystyle \begin{aligned} \mathrm{dist}\, \{ \mathcal T, {\boldsymbol\varphi}^j \} \geq d > 0.\end{aligned}$$

The hypertorus for graphs listed in (10.27) can be written as an integer linear relation between the coordinates:⁷

$$\displaystyle \begin{aligned} {} \mathcal T = \{ n_1 \varphi_1 + n_2 \varphi_2 + n_3 \varphi_3 = 0\}, \quad n_j \in \mathbb Z. \end{aligned} $$

(10.29)

The torus can be fixed so that not all integers $n_j$ have the same sign, i.e. the normal does not lie in the first octant. We may always assume that

$$\displaystyle \begin{aligned} {} n_1, n_2 >0, \quad n_3 < 0. \end{aligned} $$

(10.30)

Consider the hyperplane $\varPi \in \mathbb R^3$ given by the same linear equation (10.29)—it avoids all singular points of the reduced zero set on $\mathbb R^3$. The intersection between the zero set of the reduced polynomial and the hyperplane is given by a set of smooth non-singular curves $\mathbf Z_G^* \cap \varPi = \{ \gamma _j \}$. There is a minimal distance between the curves since the picture is periodic and the curves may intersect only at the singular points of $\mathbf Z_G^*$, but these points are avoided.

Let us choose any edge length vector $\vec {l}$ satisfying the same relation (10.29) with positive coordinates that are not pairwise rationally dependent; this is always possible for $\vec {n}$ satisfying (10.30): choose any two rationally independent $\ell _1$ and $\ell _2$ and get positive $\ell _3$ from the relation (10.29). The set $k \vec {\ell }$ densely covers the hypertorus $\mathcal T$ but it is a line on the hyperplane $\varPi$. The intersection points between the line $k \vec {l}$ and the zero set $\mathbf Z_G^*$ on the hyperplane belong to the algebraic curves $\gamma _j \subset \varPi \subset \mathbb R^3.$ The normal to $\mathbf Z_G^*$ lies in the first octant as well as the line’s direction vector $\vec {l}$, hence the two consecutive intersection points belong to different curves $\gamma _j \subset \varPi \subset \mathbb R^3$ and always are separated by a finite distance, hence the reduced spectrum is an uniformly discrete set. □

We illustrate the proof of the above theorem with a few explicit examples.

Example 10.9

This example comes from [63] where the three-star graph $G_{(3.2)}$ is considered. The authors put requirement $\ell _3 = 2 \ell _1$ leading to the equation on the spectrum:

$$\displaystyle \begin{aligned} L_{(3.2)} (k \ell_1. k \ell_2, 2 k \ell_1) = 0\end{aligned}$$

$$\displaystyle \begin{aligned} \Leftrightarrow 3 \sin (3 \ell_1 + \ell_2)k + \sin (-\ell_1 + \ell_2)k + \sin (3 \ell_1 - \ell_2)k + \sin ( \ell_1 + \ell_2)k = 0.\end{aligned}$$

Let us introduce the hypertorus $\mathcal T = \left \{ {\boldsymbol \varphi }: \varphi _3 = 3 \varphi _1 \right \}$. Figure 10.6 shows the zero set and the hypertorus which avoids the singular points $(\pm \pi /2, \pm \pi /2, \pm \pi /2)$. The intersection between the zero set and the hypertorus is given by smooth curves. Projection of these curves to the $(\varphi _1, \varphi _2)$-plane is presented in Fig. 10.7. The curves are clearly separated by a non-zero distance and the spectrum is a Delone set.

Fig. 10.6

Zero set for graph $G_{(3.2)}$ together with the torus $\varphi _3 = 2 \varphi _1$

Fig. 10.7

Intersection between $\mathbf Z_{G_{(3.2)}}$ and the torus $\varphi _3 = 2 \varphi _1$ projected to the $(\varphi _1, \varphi _2)$-plane

Example 10.10

Consider the graph $G_{(3.8)}$ with the reduced Laurent polynomial given by (6.15) (Fig. 10.8). The singular points are $( \pm \pi /2, \pi , \pi ) .$ We choose the hypertorus $\mathcal T$

$$\displaystyle \begin{aligned} \varphi_1 + 2 \varphi_2 - \varphi_3 = 0\end{aligned}$$

avoiding the singular points. The intersection curves projected to the $(\varphi _1, \varphi _2)$-plane are plotted in Fig. 10.9. Different curves are clearly separated by a ***non-zero distance. The reduced spectrum is given by the equation

$$\displaystyle \begin{aligned} L_{(3.8)}^* (k\ell_1, k\ell_2, k (\ell_1 + 2\ell_2)) = 0\end{aligned}$$

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Leftrightarrow 5 \sin \Big(\frac{3}{2} \ell_1 + \frac{3}{2} \ell_2\Big) k + \sin \Big(\frac{1}{2} \ell_1 - \frac{1}{2} \ell_2\Big)k \\ + \sin \Big(\frac{3}{2} \ell_1 + \frac{1}{2} \ell_2\Big)k - 3\sin \Big(\frac{1}{2} \ell_1 - \frac{3}{2} \ell_2\Big)k = 0,\end{array} \end{aligned} $$

where we used that the edge lengths satisfy the relation $\ell _3 = \ell _1 + 2 \ell _2 .$ The reduced eigenvalues form a Delone set, all eigenvalues having multiplicity one. Choosing $\ell _1$ and $\ell _2$ rationally independent the line $(k \ell _1, k \ell _2, k \ell _3)$ densely covers the torus $\mathcal T.$ The reduced spectrum contains no arithmetic sequence.

Fig. 10.8

The reduced zero set for graph $G_{(3.8)}$ together with the torus $\mathcal T$

Fig. 10.9

Projection of the intersection curves to the $(\varphi _1, \varphi _2)$-plane

In both examples obtained Delone sets are simple—all eigenvalues including point $k= 0$ have multiplicity one. Hence the corresponding crystalline measures are not only uniformly discrete but also idempotent. In general to get idempotent measures one has to ensure that the reduced spectrum is simple and that the delta measure at the origin has unit weight. The above examples illustrate two mechanisms how unit weight at the origin can be achieved:

• 1. Let $G$ be a tree, then $\beta _1 = 0$, no reduction of the spectrum is needed and the weight of point zero is one.

• 2. Let $G$ be a graph with all cycles given by loops. Then reducing the spectrum—subtracting precisely $\beta _1$ arithmetic sequences—one gets unit weight at the origin.

Problem 41

Construct your own examples of idempotent uniformly discrete positive crystalline measures using one of the graphs on three edges.

Problem 42

Consider any graph on four edges. Construct an example of idempotent uniformly discrete positive crystalline measure by properly choosing the edge lengths (of course not allowing zero edge lengths, as such graphs should be seen as graphs on three edges).