The Trace Formula

Abstract

This chapter is devoted to the trace formula connecting the spectrum of a finite compact metric graph with the set of closed paths on it. In other words this formula establishes a relation between spectral and geometric/topologic properties of metric graphs.

This chapter is devoted to the trace formula connecting the spectrum of a finite compact metric graph with the set of closed paths on it. In other words this formula establishes a relation between spectral and geometric/topologic properties of metric graphs.

Such a formula was first proved for the Laplacian $\Delta$ defined on a Riemannian manifold $X$ [134, 159, 175, 249] and is known now as Chazarain-Duistermaat-Guillemin-Melrose trace formula

$$\displaystyle \begin{aligned} {} \sum_{\lambda_j \in \mathrm{Spec}\, \Delta} \cos \lambda_j^{1/2} t = \sum_{\gamma} \frac{\ell(\mathrm{prim} (\gamma))}{\vert I- P_\gamma \vert^{1/2}} \delta (t- \ell (\gamma)) + R, \; \; t > 0. \end{aligned} $$

(8.1)

The sum on the left hand side is taken over all eigenvalues of the Laplacian $\Delta$, the sum on the right hand side—over all closed geodesics on the manifold $X.$$\ell (\gamma )$ denotes the length of the geodesic $\gamma$ and $\mathrm {prim} (\gamma )$—the primitive geodesic. $P_\gamma$ is the Poincaré map around $\gamma .$ The remainder term $R$ is a certain (non-specified) function in $L_{1,\mathrm {loc}}$, which means that this formula holds modulo $L_{1,\mathrm {loc}}$-function. Formula (8.1) can be seen as a generalisation of the classical Poisson summation formula in Fourier analysis (see (10.13) below) as well as Selberg’s trace formula.

We are going to prove a direct analogue of formula (8.1) for the case of metric graphs. For simplicity we consider first the standard Laplacian, which is uniquely determined by the metric graph $\Gamma .$ In contrast to (8.1) the formula we are going to prove is exact and does not contain any reminder term. This formula first appeared in a paper by J.-P. Roth [451, 452], but we follow scattering matrix approach suggested in [252, 320, 321] and developed further in [346]. This formula will be used to prove that the spectrum of a quantum graph determines its Euler characteristic. Despite the fact that the original formula is proven for standard Laplacians, it can be generalised to the case of standard Schrödinger operators.

8.1 The Characteristic Equation: Multiplicity of Positive Eigenvalues

Consider the standard Laplace operator on a finite compact metric graph $\Gamma .$ One can easily see that this operator is nonnegative, since its quadratic form is given by

$$\displaystyle \begin{aligned} {} \langle u, L^{\mathrm{st}} u \rangle_{L_2(\Gamma)} = \sum_{n=1}^N \int_{E_n} \vert u'(x) \vert^2 dx, \end{aligned} $$

(8.2)

(see (3.55)). (The operators $A_{S^m}$ appearing in (3.55) are all equal to zero.)

To determine positive eigenvalues we are going to use the characteristic equation on the spectrum derived using the scattering approach in Sect. 5.2. The eigenvalue $\lambda = 0$ needs special attention and will be discussed later on. Let us repeat the derivation of formula (5.47) adjusting formulas to the case of the Laplace operator with standard vertex conditions. Let $\psi$ be an eigenfunction corresponding to the eigenvalue $\lambda = k^2.$ On every edge $[x_{2n-1}, x_{2n}]$ it is a solution to the equation $- \psi '' = \lambda \psi$ and can therefore be written using one of the following two representations:

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \psi (x) & = & \displaystyle a_{2n-1} e^{ i k(x-x_{2n-1})} + a_{2n} e^{-i k (x-x_{2n})} \\ & = & \displaystyle b_{2n-1} e^{ - i k (x-x_{2n-1})} + b_{2n} e^{i k (x-x_{2n})} . \end{array} \end{aligned} $$

(8.3)

The amplitudes $a_j$ of edge-incoming waves are related to the amplitudes $b_j$ of edge-outgoing waves via the edge scattering matrix $S_{\mathbf {e}}^n (k)$

(8.4)

Putting together all amplitudes of incoming and outgoing waves for all edges the last relation can be written as

$$\displaystyle \begin{aligned} {} \vec{B} = \mathbf S_{\mathbf{e}} (k) \vec{A}, \end{aligned} $$

(8.5)

where the matrix $\mathbf S_{\mathbf {e}} (k)$ is formed by $2 \times 2$ block diagonal matrices $S_{\mathbf {e}}^n (k)$ if the amplitudes forming $\vec {A}$ and $\vec {B}$ are indexed according to the endpoints $x_j$.

The second relation between $\vec {A}$ and $\vec {B}$ comes from the vertex conditions

$$\displaystyle \begin{aligned} {} \vec{A} = \mathbf S \vec{B}, \end{aligned} $$

(8.6)

where the matrix $\mathbf S$ has block-diagonal form if the amplitudes $\vec {A}$ and $\vec {B}$ are ordered following the vertices. The matrices on the diagonal are vertex scattering matrices $S^{\mathrm {st}}_d$ given by (3.41) with $d$ equal to $d_m$—the degree of the corresponding vertex. It is important for our derivations that the vertex scattering matrices corresponding to standard vertex conditions do not depend on the spectral parameter $k.$

Putting together (8.5) and (8.6) we arrive at

$$\displaystyle \begin{aligned} {} \mathbb S(k) \vec{A} = \vec{A}, \quad \mathbb S(k) := \mathbf S \mathbf S_{\mathbf{e}} (k) . \end{aligned} $$

(8.7)

Taking the determinant we get Eq. (5.47)

$$\displaystyle \begin{aligned} \det (\mathbb S(k) - \mathbf I) = 0,\end{aligned}$$

determining positive eigenvalues of $L^{\mathrm {st}} (\Gamma ).$ In other words Eq. (5.47) describes all eigenvalues of $L^{\mathrm {st}} (\Gamma )$ with one possible exception $\lambda = 0,$ since the operator is nonnegative.

The $2N \times 2N$ matrix $\mathbb S(k)$ introduced above describes how the waves are penetrating through the collection of edges and vertices forming the graph. We called it the graph scattering matrix, although it is more correct to understand it as the evolution map in a discrete dynamical system associated with the metric graph. We shall use this point of view proving the trace formula.

Let us introduce the function

$$\displaystyle \begin{aligned} {} p(k) := \det (\mathbb S(k) - \mathbf I) \equiv \det \left( \mathbf S \mathbf S_{\mathbf{e}} (k) - \mathbf I \right), \end{aligned} $$

(8.8)

coinciding with the secular trigonometric polynomial $p_\Gamma (k)$ introduced in 6.1, as we agreed to treat these functions projectively:

$$\displaystyle \begin{aligned} p(k) = \det \mathbf S \; p_\Gamma (k) \Rightarrow p(k) = p_\Gamma (k).\end{aligned}$$

Putting together the vertex and edge scattering matrices will help us in the proof.

The zeroes $k_j$ of the trigonometric polynomial $p$ correspond to the eigenvalues $k_j^2$ of the standard Laplacian on $\Gamma$. The zeroes are situated symmetrically with respect to the origin

$$\displaystyle \begin{aligned} p(k_j) = 0 \Rightarrow p(-k_j) = 0 .\end{aligned}$$

We are now interested in the order of the zeroes. One should expect that the orders of zeroes coincide with the multiplicities of the corresponding eigenvalues of $L^{\mathrm {st}} (\Gamma ),$ but this fact needs to be proven. Let $k_j$ be any zero of $p$, then the function $(k^2-k_j^2) p(k)$ is also a characteristic function for the spectrum, but obviously the order of the zero at $k_j$ is different. The orders of zeroes and multiplicities of the eigenvalues coincide only due to the special form of $p$ constructed using the suggested recipe and holds true for nonzero $k$ only. As will be proven later, the order of $k = 0$ may differ from the multiplicity of $\lambda = 0.$

Theorem 8.1

Let$p$be the characteristic function for$L^{\mathrm {st}} (\Gamma )$determined by (8.8) and let$k_j \neq 0$be one of its zeroes. Then the order of the zero of$p(k)$at$k_j$coincides with the multiplicity of the eigenvalue$\lambda _j = k_j^2$of$L^{\mathrm {st}} (\Gamma ).$

Proof

Let us denote by $e^{i \theta _n (k) }, \, n= 1,2, \dots , 2N,$ and $\vec {A}_n(k)$ the eigenvalues and the eigenvectors of the unitary matrix $\mathbb S(k) = \mathbf S \mathbf S_{\mathbf {e}} (k)$

$$\displaystyle \begin{aligned} {} \mathbb S(k) \vec{A}_n (k) = e^{i \theta_n (k)} \vec{A}_n (k). \end{aligned} $$

(8.9)

The determinant (8.8) can be easily calculated in terms of the phases $\theta _n (k)$

$$\displaystyle \begin{aligned} {} p(k) = \prod_{n=1}^{2N} \left( e^{i \theta_n (k)} -1 \right). \end{aligned} $$

(8.10)

For $k_j \neq 0$ there is a one-to-one correspondence (8.3) connecting the amplitudes $\vec {A}$ and the eigenfunctions $\psi$ on $\Gamma .$ Therefore a real number $\lambda _j = k_j^2$ is an eigenvalue of a certain multiplicity $m(\lambda _j)$ if and only if the dimension of the kernel $\mathrm {Ker}\, (\mathbb S(k) - I)$ is equal to $m(\lambda _j)$, in other words, if and only if among $2N$ phases $\theta _n (k_j)$ there are precisely $m(\lambda _j)$ phases equal to $0\; (\mathrm {mod} \; 2 \pi ).$ Hence the function $p$ has a zero at $k_j$ of order at least $m(\lambda _j)$, since precisely $m(\lambda _j)$ terms in (8.10) vanish. On the other hand it may happen that some of the items have zeroes of higher orders.

To prove that the order is precisely equal to $m(\lambda _j)$ it is enough to show that $\theta _n^{\prime } (k_j)$ is different from zero for all $n$ such that $\theta _n (k_j) = 0\; (\mathrm {mod} \; 2 \pi ).$ For such $n$ we have

$$\displaystyle \begin{aligned} {} \mathbb S(k_j) \vec{A}_n (k_j) = \vec{A}_n (k_j). \end{aligned} $$

(8.11)

The matrix $\mathbb S(k) = \mathbf S \mathbf S_{\mathbf {e}} (k)$ possesses the following analytic expansion

$$\displaystyle \begin{aligned} \mathbb S(k) = \displaystyle \mathbb S(k_j) + \mathbb S(k_j) i \mathbf D (k-k_j) + \dots,\end{aligned}$$

where we used the fact that the vertex scattering matrix $\mathbf S$ is independent of the energy and the edge scattering matrix is given by $2 \times 2$ blocks in the basis associated with the edges. The matrix $\mathbf D$ used here is defined as

$$\displaystyle \begin{aligned} {} \mathbf D = \mathrm{diag} \, \left\{ \ell_1, \ell_1, \ell_2, \ell_2, \dots, \ell_N, \ell_N \right\}, \end{aligned} $$

(8.12)

in the edge basis. Since the entries of $\mathbb S(k)$ are analytic functions in $k$, the eigenvalue branches $e^{i \theta _n (k)}$ and the corresponding eigenvectors $\vec {A}_n (k)$ can be chosen analytic

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle e^{i \theta_n(k)} & = & \displaystyle 1 + i \theta_n^{\prime} (k_j) (k-k_j) + \dots, \\ \displaystyle \vec{A}_n (k) & = & \displaystyle \vec{A}_n (k_j) + \vec{A}_n^{\prime} (k_j) (k-k_j) + \dots, \end{array} \quad \mbox{as }k \rightarrow k_j, \end{aligned}$$

where we used the fact that $\theta _n (k_j) = 0.$ Substituting analytic expansions for $\mathbb S(k), \vec {A}_n (k),$ and $\theta _n(k)$ into the eigenfunction Eq. (8.9) we get

$$\displaystyle \begin{aligned} \begin{array}{cl} & \displaystyle \left( \mathbb S(k_j) + \mathbb S(k_j) i \mathbf D (k-k_j) + \dots \right) \left(\vec{A}_n (k_j) + \vec{A}_n^{\prime} (k_j) (k-k_j) + \dots \right) \\ = & \displaystyle \left( 1 + i \theta_n^{\prime} (k_j) (k-k_j) + \dots \right) \left(\vec{A}_n (k_j) + \vec{A}_n^{\prime} (k_j) (k-k_j) + \dots \right). \end{array}\end{aligned}$$

Comparing coefficients to first order in $k - k_j$ we obtain

$$\displaystyle \begin{aligned} \mathbb S(k_j) i \mathbf D \vec{A}_n (k_j) + \mathbb S (k_j) \vec{A}_n^{\prime} (k_j) = i \theta_n^{\prime} (k_j) \vec{A}_n (k_j) + \vec{A}_n^{\prime} (k_j)\end{aligned}$$

$$\displaystyle \begin{aligned} \Leftrightarrow \left( \mathbb S (k_j) - I \right) \vec{A}_n^{\prime} (k_j) = - \mathbb S(k_j) i \mathbf D \vec{A}_n (k_j) + i \theta_n^{\prime} (k_j) \vec{A}_n (k_j).\end{aligned}$$

It remains to take into account that (8.11) implies in particular that $\vec {A}_n (k_j)$ is an eigenvector of the adjoint matrix as well: $\mathbb S^*(k_j) \vec {A}_n (k_j) = \vec {A}_n (k_j).$ This can be seen by acting with $S^*(k_j)$ on both sides of (8.11) and using that $\mathbb S (k_j)$ is unitary. Hence the left hand side in the last displayed formula is orthogonal to $\vec {A}_n (k_j)$. Likewise, on the right assuming $\theta _n^{\prime } (k_j)= 0$, we would have

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0 & =&\displaystyle \langle \vec{A}_n (k_j), \mathbb S(k_j) i \mathbf D \vec{A}_n (k_j) \rangle = \langle \mathbb S(k_j)^* \vec{A}_n (k_j) , i \mathbf D \vec{A}_n (k_j) \rangle\\ & =&\displaystyle i \langle \vec{A}_n (k_j), \mathbf D \vec{A}_n (k_j) \rangle,\end{array} \end{aligned} $$

but the matrix $\mathbf D$ is positive definite. Hence $\theta _n^{\prime } (k_j)$ is different from zero. □

An alternative proof can be found in [463] and [81].

It follows that the analytic function $p$ can be used to determine the spectrum of $L^{\mathrm {st}} (\Gamma )$ including multiplicities of the eigenvalues on $(0, \infty )$ by just calculating all its zeroes and the corresponding orders. The key point in the proof is formula (8.3) describing the one-to-one correspondence between the eigenvectors of $\mathbb S (k_j)$ associated with the eigenvalue 1 and the eigenfunctions of the Laplacian on $\Gamma .$ The eigenvalue $\lambda = 0$ requires more attention as will be seen below.

8.2 Algebraic and Spectral Multiplicities of the Eigenvalue Zero

We have shown that equation $p(k) = 0$ determines the spectrum of $L^{\mathrm {st}} (\Gamma )$ with correct multiplicities for all nonzero values of $k$, but the multiplicity of the zero eigenvalue indicated by (5.47), i.e. the order of the zero, may be different from the correct one. The proof of Theorem 8.1 implies that the order of zeroes of $p$ coincides with the dimension of the kernel $\mathrm {Ker}\, (\mathbf S \mathbf S_{\mathbf {e}} (k) - \mathbf I ) .$ For all $k \neq 0$ the dimension of the kernel coincides with the number of linearly independent eigenfunctions of the Laplacian due to one-to-one correspondence between $\vec {A}$ and $\psi$ on $\Gamma$ (see (8.3)). This correspondence is not valid anymore if $k = 0,$ since the exponentials $e^{\pm i k(x-x_j)} \vert _{k=0} = 1$ coincide.

Therefore, let us introduce two (maybe different) characteristics:¹

• $m_s (0)$—the spectral multiplicity—the dimension of the eigensubspace of $L^{\mathrm {st}} (\Gamma )$ corresponding to the eigenvalue $\lambda = 0,$

• $m_a (0)$—the algebraic multiplicity—the order of zero of $p(k)$ at $k=0$, equal to the dimension of $\mathrm {Ker}\,(\mathbb S(k) - \mathbf I ) = \mathrm {Ker}\,(\mathbf S \mathbf S_{\mathbf {e}} (k) - \mathbf I )$.

It turns out that these multiplicities may be different and the difference depends on the topology of the graph, more precisely on the number of independent cycles. The following theorem connects these multiplicities with the Euler characteristic of $\Gamma$ given by (2.7).

Theorem 8.2

Let $\Gamma$ be a finite compact metric graph with $\beta _0$ connected components and Euler characteristic $\chi = 1- \beta _1,$ and let $L^{\mathrm {st}} (\Gamma )$ be the corresponding standard Laplace operator. Then $\lambda = 0$ is an eigenvalue with spectral multiplicity $m_s (0) = \beta _0$ and algebraic multiplicity $m_a(0) = 2 \beta _0 - \chi = 2 \beta _0 + \beta _1 -1.$

Proof

Spectral Multiplicity

(Easy, quick repetition of Lemma 4.10.) Every eigenfunction corresponding to $\lambda = 0$ minimises the quadratic form (8.2) and therefore is a constant function on every edge. Continuity of the function at all vertices implies that the function is equal to a constant on every connected component of $\Gamma .$ Hence spectral multiplicity of $\lambda = 0$ coincides with the number $\beta _0$ of connected components in $\Gamma .$

Algebraic Multiplicity

To derive Eq. (5.47) we used the representation (8.3) for the eigenfunction. If $k \neq 0$, then the coefficients $a_j$ or $b_j$ are uniquely determined by $\psi (x, \lambda ),$ but it is not the case if $k = 0$: the function $\psi$ determines only the sum $a_{2n-1} + a_{2n} = b_{2n} + b_{2n-1}$. Therefore there is no one-to-one correspondence between $\psi$ and the vectors $\vec {A}, \vec {B} .$²

Assume first that the graph $\Gamma$ is connected. To determine the algebraic multiplicity we have to calculate the dimension of the space of solutions to the following linear system

$$\displaystyle \begin{aligned} \mathbf S \mathbf S_{\mathbf{e}} (0) \vec{A} = \vec{A}.\end{aligned}$$

One may use standard methods of linear algebra like it was done in [346, 408]. We shall instead use the original equations in order to illuminate the relation between the algebraic multiplicity and the fundamental group on $\Gamma .$ The vectors $\vec {A}$ and $\vec {B}$ are related by

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ccc} \vec{B} & = & \mathbf S_{\mathbf{e}}(0) \vec{A}, \\[3mm] \vec{A} & = & \mathbf S \vec{B}. \end{array} \right.\end{aligned}$$

Taking into account that all matrices $S_{\mathbf {e}}^n (0)$ are all equal to $\left ( \begin {array}{cc} 0 & 1 \\ 1 & 0 \end {array} \right )$, the first relation can be written as

$$\displaystyle \begin{aligned} {} a_{2n-1} = b_{2n} \; \; \mbox{and} \; \; a_{2n} = b_{2n-1}, \; \; n =1,2, \dots, N . \end{aligned} $$

(8.13)

The second relation can equivalently be written as (3.37)

$$\displaystyle \begin{aligned} {} \left\{ \begin{array}{l} a_{i} + b_{i} = a_{j} + b_{j}, \; \; x_{i}, x_{j} \in V^m, \\[3mm] \displaystyle \sum_{x_j \in V^m} (a_j - b_j) = 0, \end{array} \right. m = 1,2, \dots, M. \end{aligned} $$

(8.14)

Here we in some sense return back and use standard vertex conditions as they were written originally (2.27) instead of using the vertex scattering matrix.

Excluding the coefficients $b_i$ we get the following linear system with $2N$ unknowns $a_j$

$$\displaystyle \begin{aligned} {} \left\{ \begin{array}{ll} a_{2i-1} + a_{2i} = a_{2j-1} + a_{2j}, & i,j = 1,2,\ldots,N; \\[3mm] \displaystyle \sum_{i: x_i \in V^m} (a_i - a_{i-(-1)^i}) = 0, & m =1,2,\ldots, M. \end{array} \right. \end{aligned} $$

(8.15)

The first set of equations shows that the function $\psi$ corresponding to $\lambda = 0$ is equal to a constant (as one expects taking into account the spectral multiplicity). The reason why the spectral and algebraic multiplicities may be different is that this constant function $\psi (x,0) = c$ may be represented by different vectors $\vec {A}.$

With every edge $E_n$ we associate the flux $f_n$³ defined as follows

$$\displaystyle \begin{aligned} f_n = a_{2n-1} - a_{2n}. \end{aligned} $$

(8.16)

Note that the flux so defined depends on the orientation of the edge $E_n$, i.e. it changes sign if one changes the orientation of the edge. The second set of Eq. (8.15) implies that the total flux through every vertex is zero

$$\displaystyle \begin{aligned} {} \sum_{E_n \; \mbox{starts at }V^m} f_n = \sum_{E_n \; \mbox{ends at }V^m} f_n, \; \; m =1,2,\ldots,M. \end{aligned} $$

(8.17)

Let us prove that the dimension of the space of solutions to this system of equations is equal to the number $g = \beta _1$ of generators for the fundamental group.⁴

Assume that $\Gamma$ is a tree ($N = M-1 \Leftrightarrow \beta _1 = 0$), then the only possible flux is zero. First we note that the flux on all pendant edges is zero. To see this it is enough to look at the relation (8.17) in the case of vertex of degree one—only one sum is present and it contains just one term. Then it is clear that the flux is zero on all edges connected by one of the endpoints to pendant edges. Continuing in this way we conclude that the flux is zero on the whole tree.

Assume now, that $\Gamma$ is an arbitrary connected graph. Then by removing certain $\beta _1 = N-(M-1)$ edges it may be transformed to a certain tree $\mathbf T$ connecting all vertices. Let us denote the removed edges by $E_1, E_2, \ldots , E_{N-M+1}$ so that

$$\displaystyle \begin{aligned} T = \Gamma \setminus \cup_{n=1}^{N-M+1} E_n.\end{aligned}$$

Every removed edge $E_n$ determines one nontrivial class of closed paths on $\mathbf T \cup E_n.$ Consider the shortest paths from this class. There exist precisely two such paths having opposite orientations. To each path we associate basic flux $\mathcal F^n$ supported by it

$$\displaystyle \begin{aligned} \mathcal F^n (E_k) = \left\{ \begin{array}{ll} \pm 1, & \mbox{if} \; E_k \; \mbox{belongs to the path}, \\ 0, & \mbox{if} \; E_k \; \mbox{does not belong to the path}, \end{array} \right.\end{aligned}$$

where the sign in the last formula depends on whether the path runs along $E_k$ in the positive ($+$) or negative ($-$) direction. Without loss of generality we assume that $\mathcal F^n (E_n) = 1.$ This condition fixes the orientation of the shortest path. In what follows we consider just one shortest path associated with $E_n.$ Every constructed flux satisfies the system of Eq. (8.17).

Consider any flux $\mathcal F$ on $\Gamma$ satisfying the conservation law (8.17). We claim that it can be written as a linear combination of the basic fluxes $\mathcal F^n.$ Really the flux

$$\displaystyle \begin{aligned} \mathcal F - \sum_{n=1}^{\beta_1 = N-M+1} \mathcal F (E_n) \mathcal F^n\end{aligned}$$

is supported by the spanning tree $\mathbf T$, it satisfies (8.17) on $\mathbf T$ and therefore it is equal to zero.

Summing up we conclude that for connected graphs the algebraic multiplicity of the zero eigenvalue is given by

$$\displaystyle \begin{aligned} m_a (0) = 1 + \beta_1 = 1+ N-(M-1) = 2 - \chi.\end{aligned}$$

Since the Euler characteristic $\chi$ is additive for not connected graphs, it is straightforward to see that formula $m_a(0) = 2 \beta _0 - \chi$ holds in the general case.

□

This theorem implies that two graph Laplacians can be isospectral only if the underlying graphs have the same number of connected components. It can clearly be seen from the proof that the spectral and algebraic multiplicities for connected graphs are equal only if the fundamental group is trivial $\beta _1 = 0$, i.e. if the graph is a tree.

8.3 The Trace Formula for Standard Laplacians

We prove now the trace formula relating the spectrum of the standard Laplacian to the set of oriented closed paths on the graph. We consider only those paths $\gamma$ which do not turn back in the interior of any edge, but which may turn back at the vertices.

Definition 8.3

Let $\{ y_j \}_{j=1}^{2 d}$ be a finite sequence of edge endpoints on a finite compact metric graph $\Gamma$

$$\displaystyle \begin{aligned} y_1, y_2, y_3, \dots, y_{2 d}, \quad y_j \in \mathbf V = \left\{ x_i \right\}_{i=1}^{2N},\end{aligned}$$

such that

• $y_{2j-1}$ and $y_{2j}$ are endpoints of a certain edge, $j =1,2, \dots , d$,

• $y_{2j}$ and $y_{2j+1}$ belong to a certain vertex, $j =1,2, \dots , d$,

where we used natural cyclic identification $y_{2d+1} = y_1$. Then the oriented closed path$\gamma = ( y_1, y_2, y_3, \dots , y_{2 d})$ is a union of edges

$$\displaystyle \begin{aligned} \gamma = [y_1, y_2] \cup [y_3, y_4] \cup \dots [y_{2d-1}, y_{2d}]\end{aligned}$$

with endpoints $y_{2j}$ and $y_{2j+1}$ identified and inherited orientation. The paths obtained from each other by cyclically permuting the endpoints $y_j (\gamma )$ are identified.

Each pair $(y_{2j-1}, y_{2j} )$ determines not only the edge the path traverses but also the direction of the path on it. The pairs $(y_{2j}, y_{2j+1})$ determine the vertices and their order on the path. Every closed path can be equivalently defined by the sequence of edges indicating path’s direction on each edge.

Topologically every closed path $\gamma$ is a cycle which can be continuously embedded in $\Gamma$ locally preserving the distances. Certain edges may appear in $\gamma$ multiple times. Consider the graph $\Gamma ^\gamma$ obtained from $\Gamma$ by substituting each edge $E_n$ with as many parallel edges identical to $E_n$ as it appears in $\gamma$. If $\gamma$ does not pass along a certain edge, then this edge is missing in $\Gamma ^\gamma$. Therefore the path $\gamma$ can be obtained by cutting $\Gamma ^\gamma$ through the vertices. In other words $\gamma$ can be seen as an Eulerian path on $\Gamma ^\gamma$, i.e. a closed path visiting each edge precisely once.

If the graph has no loops and parallel edges, then every oriented closed path is uniquely determined by the sequence of edges this path goes along. In this case the order of the edges determines the direction in which the path crosses every edge. Alternatively every oriented path is determined by the sequence of vertices in this case.

The discrete length$d = d(\gamma )$ counts how many times the path $\gamma$ comes across an edge, so that contribution from every edge in $\Gamma$ is equal to its multiplicity in $\gamma$ (independently of the direction). The discrete length should not be mixed up with the geometric length$\ell = \ell (\gamma )$ obtained by summing the lengths of the edges respecting their multiplicities in $\gamma$

$$\displaystyle \begin{aligned} \ell (\gamma) = \sum_{j=1}^{d(\gamma)} \big(y_{2j}-y_{2j-1} \big). \end{aligned} $$

(8.18)

The paths having opposite orientations are distinguished, then the path going along the same edges as $\gamma$ in the opposite direction and order can be seen as its inverse.

For any edge the path $\gamma$ going back and forth along the edge coincides with $\gamma ^{-1}$. Moreover, for any even $d = 2 j , \, j \in \mathbb N$, there exists a unique oriented path supported only by the edge and having discrete length $d$. It is a multiple of the primitive path going once back and forth.

For a loop the two paths going in opposite directions are distinguished. For example among the paths supported by the loop there are 3 paths of discrete length $d=2$:

• going twice in the positive direction;

• going twice in the negative direction;

• going once in each direction.

Note that the latter path coincides with its inverse and is primitive.

By the primitive path of $\gamma$, $\mathrm {prim}\,(\gamma )$, we denote the shortest closed path, such that the path $\gamma$ can be obtained by repeating the path $\mathrm {prim}\,(\gamma )$ several times. For example every path supported by an edge $E_0$ is a multiple of the primitive path going back and forth $E_0$ just once.

The set $\mathcal P$ of all closed paths is infinite, but countable.

Assume that the set of edges is fixed, then the flower graph has the largest set of closed paths since any sequence of edges with arbitrary directions is allowed. Otherwise topology of the graph provides certain restrictions on the sequence.

We are ready to formulate the main result of this chapter.

Theorem 8.4 (Trace Formula)

Let $\Gamma$ be a finite compact metric graph with Euler characteristic $\chi$ and the total length $\mathcal L$ , and let $L^{\mathrm {st}} (\Gamma )$ be the corresponding standard Laplacian. Then the spectral measure

$$\displaystyle \begin{aligned} {} \mu := 2 m_s(0) \delta + \sum_{k_n\neq 0} \left( \delta_{k_n} + \delta_{-k_n} \right) \end{aligned} $$

(8.19)

is a tempered positive distribution, such that not only the Fourier transform $\hat {\mu }$ but also $| \hat {\mu }|$ is tempered.

The following two exact trace formulae establish the relation between the spectrum $\{k_n^2\}$ of $L^{\mathrm {st}} (\Gamma )$ and the set $\mathcal P$ of closed paths on the metric graph $\Gamma$

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \mu (k) & = & \displaystyle 2 m_s(0) \delta(k) + \sum_{k_n\neq 0} \left( \delta_{k_n} (k) + \delta_{-k_n} (k)\right) \\[3mm] & = & \displaystyle \chi \delta (k)+ \frac{\mathcal{L}}{\pi}+\frac{1}{\pi}\sum_{\gamma \in \mathcal{P}} \ell(\mathrm{prim}\,(\gamma)) \mathbf S_{\mathbf{v}} (\gamma) \cos k\ell(\gamma), \end{array} \end{aligned} $$

(8.20)

and

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \hat{\mu} (l) & \equiv & \displaystyle 2 m_s(0) + \sum_{k_n \neq 0} 2 \cos k_n l \\[3mm] & = & \displaystyle \chi + 2 \mathcal L \delta (l) + \sum_{\gamma \in \mathcal{P}} \ell(\mathrm{prim}\,(\gamma)) \mathbf S_{\mathbf{v}} (\gamma) \Big( \delta_{\ell(\gamma)} (l)+ \delta_{-\ell(\gamma)}(l) \Big), \end{array} \end{aligned} $$

(8.21)

where

• $m_s (0)$ is the (spectral) multiplicity of the eigenvalue zero; ⁵

• $\mathcal P$is the set of closed oriented paths on$\Gamma$;

• $\ell (\gamma )$is the length of the closed path$\gamma$;

• $\mathrm {prim}\,(\gamma )$is the primitive path for$\gamma$;

• $\mathbf S_{\mathbf {v}} (\gamma )$ is the product of all vertex scattering coefficients along the path $\gamma .$

Proof

We divide the proof of theorem into two parts. The first part concerns general properties of the spectral measure establishing that $\mu$ is not only a tempered distribution, but also $| \hat {\mu }|$ is tempered. This will be important in Sect. 10.2, where we show that spectral measures for metric graphs provide explicit examples of crystalline measures and Fourier quasicrystals. In the second part we prove trace formula connecting the spectral measure associated with the standard Laplacian to the set of periodic orbits on the metric graph.

Part I. General Properties of the Spectral Measure

Step 1.:

Measure$\mu$as a tempered distribution. Consider the spectral measure given by (8.19) where the sum is taken over all non-zero eigenvalues $\lambda _n > 0, \; k_n^2 = \lambda _n, \; k_n > 0$ respecting multiplicities. The formula determines a tempered distribution since the eigenvalues accumulate towards $\infty$ satisfying Weyl’s asymptotics (4.25). All non-zero points (including correct multiplicities) are given by zeros of the analytic secular function $p$ determined by (8.8). The distribution is positive as a sum of delta distributions with non-negative integer amplitudes. The Fourier transform $\hat {\mu }$ is also a tempered distribution.

Step 2.:

Spectral measure and logarithmic derivative of the secular function. The distribution $\mu$ can be obtained by integrating the logarithmic derivative of $p(k)$ (introduced in 8.8) around the zeroes and using Sokhotski-Plemelj formula (see e.g. formula (3.2.11) in [271] )⁶

$$\displaystyle \begin{aligned} \delta_0 = \frac{1}{2 \pi i} \left( \frac{1}{x-i0} - \frac{1}{x+i 0} \right).\end{aligned}$$

Since the zeroes are situated on the real axis, the sum of delta functions with the supports at the zeroes is equal to the jump understood in the sense of distributions

$$\displaystyle \begin{aligned} \frac{1}{2 \pi i} \left( \frac{d}{dk} \log p(k-i0) - \frac{d}{dk} \log p(k+i0) \right).\end{aligned}$$

More precisely we have

$$\displaystyle \begin{aligned} {} \mu (k) &= (2m_s(0)-m_a(0)) \delta (k)\\ &\quad + \frac{1}{2\pi i} \lim_{\epsilon \searrow 0} \left( \frac{d}{dk} \log p(k-i \epsilon) - \frac{d}{dk} \log p(k+i \epsilon )\right). \end{aligned} $$

(8.22)

We used here the fact that $p(k)$ has zero of order $m_a(0)$ at the origin.

Step 3.:

Trigonometric series for the spectral measure$\mu$. Following [350] we shall use that $p(k)$ is a trigonometric polynomial

$$\displaystyle \begin{aligned} p(k) = P(e^{ik {\boldsymbol\ell}}), \quad e^{ik {\boldsymbol\ell}} = (e^{ik \ell_1}, e^{ik \ell_2}, \dots, e^{ik \ell_N}) ,\end{aligned}$$

coming from the secular polynomial $P({\mathbf z})$. The secular polynomial, which is non-zero inside the polydisk

$$\displaystyle \begin{aligned} \mathbb D^N = \mathbb D \times \mathbb D \times \dots \times \mathbb D, \quad \mathbb D = \Big\{ z \in \mathbb C: |z| < 1 \Big\},\end{aligned}$$

can be chosen to satisfy the normalisation condition⁷

$$\displaystyle \begin{aligned} P(\mathbf 0 ) = 1.\end{aligned}$$

Then $\log P({\mathbf z})$ is uniquely defined by putting $\log P(\mathbf 0) = 0$ and using continuous variation along the line $\big \{ s {\mathbf z} \big \}, 0 \leq s \leq 1$. It is an analytic function inside $\mathbb D^N$

$$\displaystyle \begin{aligned} {} \log P({\mathbf z}) = \sum_{{\mathbf n} \in \mathbb{Z}_{+}^{N}} c_{{\mathbf n}} {\mathbf z}^{{\mathbf n}}, \quad {\mathbf z}^{{\mathbf n}} = z_{1}^{n_1} z_{2}^{n_2} \dots z_{N}^{n_{N}}. \end{aligned} $$

(8.23)

Our goal is to prove that the Taylor coefficients $c_{\mathbf {n}}$ are uniformly bounded implying that the above series is convergent in the distributional sense. This follows from the fact that the logarithm of any analytic function is locally integrable over any totally real submanifold in $\mathbb C^n$—a general fact from the theory of functions of several complex variables. Polynomials are analytic functions and the unit torus is a totally real submanifold. For multivariate polynomials the integral we aim to estimate is related to Mahler’s measures connected with the heights of the polynomials (see for example Section 3.2 of [183]).⁸ Instead of looking at the geometric properties of the intersections between algebraic varieties and the unit torus $\mathbb T^N$ we shall, following [350], use explicit formulas for the Taylor coefficients together with the normalisation condition $P(\mathbf 0) = 1$.

The Taylor coefficients can be calculated taking the spherical means

$$\displaystyle \begin{aligned} {} c_{{\mathbf{n}}} = \frac{1}{(2\pi)^N r^{|{\mathbf n}|}} \int_{\mathbf T^N} \log P(r e^{i {\boldsymbol\theta}}) e^{- i {\mathbf n} {\boldsymbol\theta}} d {\boldsymbol\theta} , \end{aligned} $$

(8.24)

where we used notation $|{\mathbf n}| = n_1 + n_2 + \dots + n_N$ and integration is over $\mathbf T^N$ corresponding to the distinguished boundary of the polydisk $\mathbb T^N = \mathbb T \times \mathbb T \times \dots \times \mathbb T$. Hence to prove uniform boundedness it is enough to show that $\log P(r e^{i {\boldsymbol \theta }})$ is absolutely integrable over $\mathbf T^N$ uniformly for all $0 \leq r \leq 1$. Note that absolute integrability of $\log P(re^{i {\boldsymbol \theta }})$ for all $r < 1$ (not uniform) is not enough as $r^{|\mathbf n|}$ in the denominator tends to zero as $|\mathbf n | \rightarrow \infty$.

The real and imaginary parts of the logarithm

$$\displaystyle \begin{aligned} \log P({\mathbf z}) = \ln | P({\mathbf z}) | + i \arg P({\mathbf z})\end{aligned}$$

can be estimated separately.

To estimate the imaginary part, i.e.$\arg P({\mathbf z})$, we consider the function $P(s {\mathbf z})$ which is a polynomial in $s$ of degree at most $2 N$. The constant term in the polynomial is zero and each other term contributes at most $\pi$ to the argument, hence we have

$$\displaystyle \begin{aligned} | \arg P({\mathbf z})) | \leq 2N \pi, \quad {\mathbf z} \in \mathbb D^N, \end{aligned} $$

(8.25)

implying

$$\displaystyle \begin{aligned} {} \int_{\mathbf T^N} | \arg P(r e^{i {\boldsymbol\theta}})) | d {\boldsymbol\theta} \leq r^N (2\pi)^{N+1} N \leq (2\pi)^{N+1} N. \end{aligned} $$

(8.26)

The real part, i.e.$\ln | P({\mathbf z}) |$, is singular on the distinguished boundary $\mathbb T^N$ at the zeroes of $P({\mathbf z})$, but its mean value is zero

$$\displaystyle \begin{aligned} \int_{\mathbf T^N} \ln | P(r e^{i {\boldsymbol\theta}}) | d {\boldsymbol\theta} = \mbox{Re} \, \Big( \int_{\mathbf T^N} \ \log P(r e^{i {\boldsymbol\theta}}) d {\boldsymbol\theta} \Big)= \mbox{Re} \, \Big((2 \pi)^N \log (P(\mathbf 0 )) \Big) = 0\end{aligned}$$

and it is uniformly bounded from above

$$\displaystyle \begin{aligned} \ln |P({\mathbf z}) | \leq \ln K, \quad \mbox{where} \; K = \sup_{{\mathbf z} \in \mathbb D^N} | P({\mathbf z})|. \end{aligned} $$

(8.27)

Hence it is absolutely integrable

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \int_{\mathbf T^N} \Big| \ln | P(r e^{i {\boldsymbol\theta}}) | \Big| d {\boldsymbol\theta} & = & \displaystyle \int_{\mathbf T^N} \Big| \ln | P(r e^{i {\boldsymbol\theta}}) | - \ln K + \ln K\Big| d {\boldsymbol\theta} \\[3mm] & \leq & \displaystyle - \int_{\mathbf T^N} \Big( \ln | P(r e^{i {\boldsymbol\theta}}) | + \ln K ) d {\boldsymbol\theta} + (2\pi)^N \ln K \\[3mm] & = & \displaystyle 2 (2\pi)^N \ln K. \end{array} \end{aligned} $$

(8.28)

Summing (8.26) and (8.28) we obtain the r-independent estimate

$$\displaystyle \begin{aligned} \int_{\mathbf T^N} \Big| \log P(r e^{i{\boldsymbol\theta}}) \Big| d {\boldsymbol\theta} \leq 2 (2 \pi)^N \left( \pi N + \ln K \right) , \end{aligned} $$

(8.29)

implying that Taylor coefficients in (8.24) are uniformly bounded:

$$\displaystyle \begin{aligned} \big| c_{{\mathbf n}} \big| \leq 2 \left( \pi N + \ln K \right) =: C_1. \end{aligned} $$

(8.30)

Here and in what follows $C_j$ denote different positive constants.

We use Taylor’s expansion (8.23) to get

$$\displaystyle \begin{aligned} {} \log p(k+i0) &= \log P(e^{i(k+i0) {\boldsymbol\ell}}) = \sum_{\mathbf n \in \mathbb Z_+^N} c_{\mathbf n} e^{i (k+i0) (\mathbf n \cdot {\boldsymbol\ell})} \\ &= \sum_{m=0}^\infty \sum_{ \begin{array}{c} \scriptscriptstyle \mathbf n \in \mathbb Z_+^N \\ \scriptscriptstyle |\mathbf n | = m \end{array} } c_{\mathbf n} e^{i (k+i0) (\mathbf n \cdot {\boldsymbol\ell})} . \end{aligned} $$

(8.31)

Every test function $\varphi$ from the Schwartz class $\mathcal S$ satisfies the estimate

$$\displaystyle \begin{aligned} {} \Big| \int_{\mathbb R} e^{i k d} \varphi (k) dk \Big| \leq \frac{C_2 }{d^{N+1}}, \quad C_2 = C_2 (\varphi), \end{aligned} $$

(8.32)

therefore we have

implying that the series for $\log p(k)$ is absolutely convergent. The series can also be differentiated termwise.

Taking into account (8.22) we conclude that the spectral measure is given by the series

(8.33)

understood in the sense of distributions. The series converge in the distributional sense as each of the two infinite series is a distributional derivative the series for $\log p(k)$. One can see this directly refining the estimates used above, as it will be done below for $|\hat {\mu }|$.

Step 4.:

$| \hat {\mu } |$is tempered. The Fourier transform of the spectral measure is

(8.34)

and we already know that it is a tempered distribution. Then $| \hat {\mu }|$ is given by

(8.35)

We use the following estimate (similar to (8.32)) valid for any test function from the Schwartz class

$$\displaystyle \begin{aligned} | \varphi(d) | \leq \frac{C_4}{d^{N+2}}.\end{aligned}$$

Then we have

hence the series for $|\hat {\mu }| [\varphi ]$ is absolutely convergent for any test function from $\mathcal S$. It follows that $|\hat {\mu }|$ is a tempered distribution.

Part II. Trace Formula

Step 5.:

Spectral measure via the trace of the scattering matrices. Note that we do not have an explicit formula for the coefficients $c_{\mathbf n}$, hence our next goal will be to get such formula using periodic orbits on $\Gamma$. We shall repeat essentially the same calculations using instead of the secular polynomials formula (8.8) expressing the secular function via edge and vertex scattering matrices. Let us remind that $\mathbb S (k)$ is a product of the edge and vertex scattering matrices $\mathbb S (k) = \mathbf S \mathbf S_{\mathbf {e}} (k)$ and $\mathbf S$ is energy independent. Moreover we use the fact that

$$\displaystyle \begin{aligned} {} \parallel \mathbb S^{\pm 1} (k \pm i \epsilon) \parallel < 1, \quad \epsilon > 0 , \end{aligned} $$

(8.36)

since $\mathbf S$ is unitary and $\mathbf S_{\mathbf {e}}$ satisfies the same inequality. The spectral measure is given by

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle \mu(k) & = & \displaystyle (2m_s(0)-m_a(0)) \delta (k) \\ && \displaystyle + \frac{1}{2\pi i} \lim_{\epsilon \searrow 0} \Big( \frac{d}{dk} \log \det \big( \mathbb S (k-i\epsilon)-I \big) - \frac{d}{dk} \log \det \big(( \mathbb S (k+i\epsilon)-I \big) \Big) \\ && \\ & = & \displaystyle \chi \delta (k) + \frac{1}{2\pi i} \lim_{\epsilon \searrow 0} \Big( \mathrm{Tr}\, \frac{d}{dk} \log \big( \mathbb S (k-i\epsilon)-I\big) \\ && \displaystyle -\mathrm{Tr}\, \frac{d}{dk} \log \big( \mathbb S (k+i\epsilon)-I \big) \Big) \\ && \\ & = & \displaystyle \chi \delta (k) + \frac{1}{2\pi i} \lim_{\epsilon \searrow 0} \Big( \mathrm{Tr}\, \frac{d}{dk} \Big( - \sum_{m=1}^\infty \frac{1}{m} \mathbb S^{-m} (k-i\epsilon) \\ && \displaystyle + \log \mathbb S(k-i\epsilon) + \sum_{m=1}^\infty \frac{1}{m} \mathbb S^m (k+i\epsilon) \Big) \Big) \\ && \\ & = & \displaystyle \chi \delta (k) + \frac{1}{2\pi i} \left( \mathrm{Tr}\, \sum_{m=-\infty}^{+\infty} \mathbb S^m(k) \mathbb S' (k) \right), \end{array} \end{aligned}$$

where we used

• formula $\log \det A = \mathrm {Tr}\, \log A$ is valid modulo $2 \pi i$ for any diagonalisable matrix $A$;

• the fact that $\mathbb S(k\pm i \epsilon )$ are diagonalisable being close to a unitary matrix;

• the series expansion for $\log (1+\alpha )$;

• the fact that under $\mathrm {Tr}$ the matrices may be permuted cyclically.

Taking into account that $\mathbb S'(k) = \mathbf S \mathbf S^{\prime }_{\mathbf {e}} (k) = \mathbf S \mathbf S_{\mathbf {e}} (k) i \mathbf D = \mathbb S (k) i \mathbf D,$ where $\mathbf D$ is the diagonal matrix given by (8.12) we see that the distribution $\mu$ is given by the sum of the series

$$\displaystyle \begin{aligned} \mu(k) = \chi \delta (k) + \frac{1}{2\pi} \left( \mathrm{Tr}\, \sum_{m=-\infty}^{+\infty} \mathbb S^m(k) \mathbf D \right).\end{aligned}$$

Our next goal is to calculate the trace having a geometric picture in mind. We are going to calculate the traces corresponding to each power m separately in direct correspondence with formula (8.31), where the Taylor series was summed putting together terms having the same degree m. It is reasonable to start with small powers.

Step 6.:

Oriented closed paths of discrete lengths$1,2,3$. The trace of a matrix $B$ can be calculated by choosing an arbitrary orthonormal basis $\vec {e}_j$ and calculating the sum $\sum _{j} \langle \vec {e}_j, B \vec {e}_j \rangle .$ In our case it is natural to choose the standard basis of edge incoming waves, so that all except one coordinates of $\vec {e}_j$ are equal to zero and the j-th coordinate is just $1.$ Then every vector $\vec {e}_j$ is naturally associated with one of the edge endpoints. We start by calculating contribution from the first few terms corresponding to $m= 0,1,2$.

• m=0 It is trivial to calculate the contribution from the zero term in the series

  $$\displaystyle \begin{aligned} \mathrm{Tr}\, \mathbb S^0(k) \mathbf D = \mathrm{Tr}\, \mathbf D= 2 \mathcal L .\end{aligned}$$

• m=1 We calculate the contribution $\langle \vec {e}_1, \mathbb S(k) \mathbf D \vec {e}_1 \rangle .$ Let us assume w.l.o.g. that the edge $[x_1, x_2]$ connects together vertices $V^1$ and $V^2$. We get $\mathbb S(k) \mathbf D \vec {e}_1 = \mathbf S \mathbf S_{\mathbf {e}} (k) \mathbf D \vec {e}_1$ by applying the three matrices one after the other:

  

  (8.37)

  (see the first three pictures in Fig. 8.1). We denote here by $\mathbf S_{ij}$ the entry of the matrix $\mathbf S$ corresponding to the transition from the endpoint $x_j$ to the endpoint $x_i$. The result $\langle \vec {e}_1, \mathbb S(k) \mathbf D \vec {e}_1 \rangle$ is non-zero only if both endpoints $x_1$ and $x_2$ belong to the same vertex, in other words if the edge $[x_1, x_2]$ forms a loop (see Fig. 8.2). The contribution is then equal to

  $$\displaystyle \begin{aligned} \langle \vec{e}_1, \mathbb S(k) \mathbf D \vec{e}_1 \rangle = \ell_1 e^{i k \ell_1} \mathbf S_{12}.\end{aligned}$$

  Fig. 8.1

  

  Edge $[x_1, x_2]$ does not form a loop. The numbers indicate positions of the endpoints $x_j$

  Fig. 8.2

  

  Edge $[x_1, x_2]$ forms a loop. Emergence of non-zero contributions

  The remaining first order contributions $\langle \vec {e}_j, \mathbb S(k) \mathbf D \vec {e}_j \rangle , \, j = 1,2, \dots , 2N,$ are calculated in the same way. Using the discrete length $d (\gamma )$ we have

  $$\displaystyle \begin{aligned} \mathrm{Tr} \; \mathbb S(k) \mathbf D = \sum_{ \begin{array}{c} \scriptscriptstyle \gamma\in \mathcal P \\ \scriptscriptstyle d (\gamma) = 1 \end{array} } \ell (\mathrm{prim}\, (\gamma)) \mathbf S_{\mathbf{v}}(\gamma) e^{ik \ell(\gamma)} , \end{aligned} $$

  (8.38)

  where the product of scattering coefficients $\mathbf S_{\mathbf {v}} (\gamma )$ coincides with the single scattering coefficient on the path and every path coincides with its primitive $\ell (\gamma ) = \ell (\mathrm {prim}\, (\gamma ))$. The summation is over all loops in $\Gamma$. If the graph has no loops, then the total contribution from the first term is zero.

  Note that each loop contributes twice since we distinguish paths going in the opposite direction. For example contributions from the loop presented in Fig. 8.2 are

  $$\displaystyle \begin{aligned} \ell_1 e^{i k \ell_1} \mathbf S_{12} \quad \mathrm{and} \quad \ell_1 e^{i k \ell_1} \mathbf S_{21} .\end{aligned}$$

• m=2 As before we start by calculating the contribution $\langle \vec {e}_1, \mathbb S^2(k) \mathbf D \vec {e}_1 \rangle$. Assume first that the edge $E_1$ does not form a loop, then to calculate the contribution we may continue the procedure presented in Fig. 8.1. We need to determine

  $$\displaystyle \begin{aligned} {} \mathbf S_{\mathbf{e}} \Big( \ell_1 e^{ik \ell_1} \sum_{x_j \in V^2} S_{j2} \vec{e}_j \Big) = \ell_1 e^{ik \ell_1} \sum_{x_j \in V^2} S_{j2} \mathbf S_{\mathbf{ e}} \vec{e}_j . \end{aligned} $$

  (8.39)

  Each vector $\mathbf S_{\mathbf {e}} \vec {e}_j$ is just the vector associated with the opposite endpoint of the edge $x_j$ belongs to, multiplied by the exponential. For example if $x_3 \in V^2$ (as in Fig. 8.1), then

  $$\displaystyle \begin{aligned} \mathbf S_{\mathbf{e}} \vec{e}_3 = e^{ik \ell_2} \vec{e}_4.\end{aligned}$$

  Denoting by $V^3$ the vertex $x_4$ belongs to we obtain

  $$\displaystyle \begin{aligned} \mathbf S \mathbf S_{\mathbf{e}} \vec{e}_3 = \sum_{x_i \in V^3} \mathbf S_{i4} e^{ik \ell_2} \vec{e}_i.\end{aligned}$$

  The scalar product with $\vec {e}_1$ is non-zero just in two cases:

  • If $\vec {e}_j$ in (8.39) coincides with $\vec {e}_2$ corresponding to the reflection at $V^2$. Then we have

    $$\displaystyle \begin{aligned} \mathbb S(k) \vec{e}_2 = \sum_{x_i \in V^1} \mathbf S_{i1} e^{ik \ell_1} \vec{e}_i \Rightarrow \langle \vec{e}_1, \ell_1 e^{ik \ell_1} \mathbb S(k) \vec{e}_2 \rangle = \ell_1 e^{ 2 ik \ell_1} \mathbf S_{11} \mathbf S_{22}.\end{aligned}$$

    This term is always present.

  • If $\vec {e}_j$ in (8.39) is different from $\vec {e}_2$ but the corresponding edge is parallel to $[x_1, x_2]$ like the edge $[x_5,x_6]$ in Fig. 8.1. Using notations from the figure we get

    $$\displaystyle \begin{aligned} \mathbb S(k) \vec{e}_5 = \sum_{x_i \in V^1} \mathbf S_{i6} e^{ik \ell_3} \vec{e}_i \Rightarrow \langle \vec{e}_1, \ell_1 e^{ik \ell_1} \mathbb S(k) \vec{e}_5 \rangle = \ell_1 e^{ ik (\ell_1+ \ell_2)} \mathbf S_{16} \mathbf S_{52}.\end{aligned}$$

    This term is present only if there are two parallel edges, i.e. there is a closed path of discrete length 2 supported by two edges.

  Assume now that $E_1$ forms a loop, then modifying formula (8.37) we get

  $$\displaystyle \begin{aligned} \mathbf S \mathbf S_{\mathbf{e}} \mathbf D \vec{e}_1 = \ell_1 e^{ik \ell_1} \sum_{x_j \in V^1} \mathbf S_{j2} \vec{e}_j .\end{aligned}$$

  The contribution

  $$\displaystyle \begin{aligned} \langle \vec{e}_1, \mathbf S \mathbf S_{\mathbf{e}} \mathbf S \mathbf S_{\mathbf{e}} \mathbf D \vec{e}_1 \rangle\end{aligned}$$

  is non-zero in just three cases:

  • the endpoint $x_j$ coincides with $x_2$ with the contribution

    $$\displaystyle \begin{aligned} \ell_1 e^{2 i k \ell_1} \mathbf S_{11} \mathbf S_{22} ;\end{aligned}$$

  • the endpoint $x_j$ coincides with $x_1$ with the contribution

    $$\displaystyle \begin{aligned} \ell_1 e^{2 i k \ell_1} \mathbf S_{12} \mathbf S_{12} ;\end{aligned}$$

  • the endpoint $x_j$ belongs to an edge different from $E_1$, say $E_2$, forming a loop with the contribution

    $$\displaystyle \begin{aligned} \ell_1 e^{i k (\ell_1+ \ell_2)} \mathbf S_{13} \mathbf S_{32} .\end{aligned}$$

  Summing over all $j= 1,2, \dots , 2N$ we see that every oriented path of discrete length 2 contributes to the trace of $\mathbb S(k)^2 \mathbf D$. The paths going through two vertices or two different edges contribute twice.

  Every edge $E_n$ not forming a loop determines the unique path $\gamma _1$ going back and forth along it (see Fig. 8.3a). The contribution from this path comes from the following two scalar products $\langle e_{2n-1}, \mathbb S^2(k) \mathbf D \vec {e}_{2n-1} \rangle$ and $\langle e_{2n}, \mathbb S^2(k) \mathbf D \vec {e}_{2n} \rangle$ and is equal to

  $$\displaystyle \begin{aligned} &\ell_n e^{ i k 2\ell_n} \mathbf S_{2n-1\; 2n-1} \mathbf S_{2n \; 2n} + \ell_n e^{ i k 2\ell_n} \mathbf S_{2n\; 2n} \mathbf S_{2n-1\;2n-1} \\ &= \ell (\mathrm{prim}\,(\gamma_1)) e^{ik \ell (\gamma_1)} \mathbf S_{\mathbf{v}} (\gamma_1) .\end{aligned} $$

  Fig. 8.3

  

  Different paths of discrete length 2

  The corresponding primitive path coincides with $\gamma _1$, hence $\ell (\mathrm {prim}\,(\gamma _1)) = \ell (\gamma _1) = 2 \ell _n$. The product of scattering coefficients is $\mathbf S_{\mathbf {v}} (\gamma _1) = \mathbf S_{2n-1\; 2n-1} \mathbf S_{2n\; 2n}.$

  Consider now the case, where there is an edge parallel to $E_n$. We denote the edge by $E_{n+1}$ and assume $E_n$ and $E_{n+1}$ are oriented in the opposite directions (see Fig. 8.3b). Then there are two more discrete length 2 paths

  $$\displaystyle \begin{aligned} \gamma_2 = (x_{2n-1}, x_{2n}, x_{2n+1}, x_{2n+2} ) \quad \mbox{and} \quad \gamma_3 = (x_{2n+2}, x_{2n+1}, x_{2n}, x_{2n-1} )\end{aligned}$$

  (marked by cyan and magenta colours respectively) contributing via the scalar products

  $$\displaystyle \begin{aligned} \langle e_{2n-1}, \mathbb S^2(k) \mathbf D \vec{e}_{2n-1} \rangle , \quad \langle e_{2n+1}, \mathbb S^2(k) \mathbf D \vec{e}_{2n+1} \rangle\end{aligned}$$

  and

  $$\displaystyle \begin{aligned} \langle e_{2n}, \mathbb S^2(k) \mathbf D \vec{e}_{2n} \rangle , \quad \langle e_{2n+2}, \mathbb S^2(k) \mathbf D \vec{e}_{2n+2} \rangle ,\end{aligned}$$

  respectively. The corresponding contributions are

  $$\displaystyle \begin{aligned} \begin{array}{c} \displaystyle \ell_n e^{ i k (\ell_n+\ell_{n+1})} \mathbf S_{2n-1\; 2n+2} \mathbf S_{2n+1 \; 2n} + \ell_{n+1} e^{ i k (\ell_n+\ell_{n+1})} \mathbf S_{2n+1\; 2n} \mathbf S_{2n-1\;2n+2} \\[2mm] \displaystyle = \ell (\mathrm{prim}\,(\gamma_2)) e^{ik \ell (\gamma_2)} \mathbf S_{\mathbf{v}} (\gamma_4) , \end{array}\end{aligned}$$

  and

  $$\displaystyle \begin{aligned} \begin{array}{c} \displaystyle \ell_n e^{ i k (\ell_n+\ell_{n+1})} \mathbf S_{2n\; 2n+1} \mathbf S_{2n+2 \; 2n-1} + \ell_{n+1} e^{ i k (\ell_n+\ell_{n+1})} \mathbf S_{2n+2\; 2n-1} \mathbf S_{2n\;2n+1} \\[2mm] \displaystyle = \ell (\mathrm{prim}\,(\gamma_3)) e^{ik \ell (\gamma_3)} \mathbf S_{\mathbf{v}} (\gamma_5) , \end{array}\end{aligned}$$

  where $\ell (\mathrm {prim}\,(\gamma _2)) 2 \ell (\mathrm {prim}\,(\gamma _3)) = \ell (\gamma _2) = \ell (\gamma _3) = \ell _n + \ell _{n+1}$ and

  $$\displaystyle \begin{aligned} \mathbf S_{\mathbf{v}} (\gamma_2) = \mathbf S_{2n+1\; 2n} \mathbf S_{2n-1\;2n+2} , \quad \mbox{and} \quad \mathbf S_{\mathbf{v}} (\gamma_3) = \mathbf S_{2n+2\; 2n-1} \mathbf S_{2n\;2n+1} .\end{aligned}$$

  Assume now that the edge $E_n$ forms a loop (see Fig. 8.3b), then we have the path $\gamma _1$ going once back and forth with the same contribution as above

  $$\displaystyle \begin{aligned} \ell (\mathrm{prim}\,(\gamma_1)) e^{ik \ell (\gamma_1)} \mathbf S_{\mathbf{v}} (\gamma_1) .\end{aligned}$$

  In addition we have two more oriented periodic paths

  $$\displaystyle \begin{aligned} \gamma_4 = (x_{2n-1}, x_{2n}, x_{2n-1}, x_{2n}) \quad \mbox{and} \quad \gamma_5 = (x_{2n}, x_{2n-1}, x_{2n}, x_{2n-1} )\end{aligned}$$

  going around the first loop twice in different directions marked by blue and orange colors respectively (see Fig. 8.3c). Each of the paths contributes to just one of the two scalar products with

  $$\displaystyle \begin{aligned} \begin{array}{ll} \displaystyle \gamma_4:& \displaystyle \langle \vec{e}_{2n-1}, \mathbb S^2(k) \mathbf D \vec{e}_{2n-1} \rangle \\ &\quad = \ell_n e^{2 ik \ell_n} \mathbf S_{2n-1\; 2n} \mathbf S_{2n-1\; 2n} = \ell (\mathrm{prim}\,(\gamma_4)) e^{ik \ell (\gamma_4)} \mathbf S_{\mathbf{v}} (\gamma_4), \\[2mm] \displaystyle \gamma_5:& \langle \vec{e}_{2n}, \mathbb S^2(k) \mathbf D \vec{e}_{2n} \rangle \\ &\quad = \displaystyle \ell_n e^{2 ik \ell_n} \mathbf S_{2n\; 2n-1} \mathbf S_{2n\; 2n-1} = \ell (\mathrm{prim}\,(\gamma_5)) e^{ik \ell (\gamma_5)} \mathbf S_{\mathbf{v}} (\gamma_5) , \end{array} \end{aligned}$$

  with $\ell (\mathrm {prim}\,(\gamma _4)) = \ell (\mathrm {prim}\,(\gamma _5))= \ell _n; \ell (\gamma _2) = \ell (\gamma _3) = 2 \ell _n$ and

  $$\displaystyle \begin{aligned} \mathbf S_{\mathbf{v}} (\gamma_4) = \mathbf S_{2n-1\; 2n} \mathbf S_{2n-1\; 2n}, \quad \mathbf S_{\mathbf{v}} (\gamma_5) = \mathbf S_{2n\; 2n-1} \mathbf S_{2n\; 2n-1}.\end{aligned}$$

  We also have analogues of the paths $\gamma _2$ and $\gamma _3$ going first along one of the loops and returning back along the other one. There are four such paths since the loops can be passed in different directions (see Fig. 8.3d). We denote these paths by $\gamma _6, \gamma _7, \gamma _8,$ and $\gamma _9$.

  The result can be written as a sum over all periodic orbits with discrete length 2

  $$\displaystyle \begin{aligned} \sum_{\begin{array}{c} \scriptscriptstyle \gamma\in \mathcal P \\ \scriptscriptstyle d (\gamma) = 2 \end{array} } \ell (\mathrm{prim}\, (\gamma)) \mathbf S_{\mathbf{v}}(\gamma) e^{ik \ell(\gamma)} . \end{aligned} $$

  (8.40)

Step 7.:

Arbitrary oriented closed paths. We are ready now to look at the contributions from higher powers of $\mathbb S(k)$. Our analysis shows that a term $\langle \vec {e}_i, \mathbb S^m(k) \mathbf D \vec {e}_i \rangle$ gives a nonzero contribution only if there is a closed path $\gamma$ on $\Gamma$ with the discrete length $d(\gamma ) = m$ passing through the endpoint $x_i$.

Let us calculate the total contribution from any path $\gamma$ of discrete length $d(\gamma ) = m$. Assume that the path is a multiple of the primitive path

$$\displaystyle \begin{aligned} \mathrm{prim}\, (\gamma) = ( x_{i_1}, x_{i_2}, x_{i_3}, \dots, x_{i_{2Q}}), \quad \gamma = R \; \mathrm{prim}\, (\gamma) ,\end{aligned}$$

where the primitive path is formed by $Q$ edges and this path should be repeated $R$ times to get $\gamma$ so that $m = QR$. This path contributes to the scalar products

$$\displaystyle \begin{aligned} \langle \vec{e}_{i_{2q-1}}, \mathbf S^m \mathbf D \vec{e}_{i_{2q-1}} \rangle, \quad q = 1,2, \dots, Q.\end{aligned}$$

If every endpoint $x_{i_{2q-1}}$ appears just once in $\mathrm {prim}\, (\gamma )$, then the contribution from each point is equal to the length of the edge to which $x_{i_{2q-1}}$ belongs multiplied by $e^{i k \ell (\gamma )}$ and the product $\mathbf S_{\mathbf {v}} (\gamma )$ of all vertex scattering coefficients along $\gamma$. If a certain endpoint appears several times, then the above contribution is multiplied by the number of times $x_{i_{2q-1}}$ appears in $\mathrm {prim}\, (\gamma )$. Summing up contributions from different vertices on the primitive path results in multiplication of $S_{\mathbf {v}} (\gamma ) e^{ik \ell (\gamma )}$ by the length of the primitive path

$$\displaystyle \begin{aligned} \ell(\mathrm{prim}(\gamma)) \mathbf S_{\mathbf{v}} (\gamma) e^{ik \ell(\gamma)} .\end{aligned}$$

Summation over all paths of discrete length m leads to

(8.41)

Formula (8.20) is obtained by summing over all $m$ and taking into account the fact that the contributions from $m$ and $-m$ are complex conjugates of each other.

The second trace formula (8.21) is obtained via Fourier transform.

□

Formula (8.20) can be modified using summation over primitive orbits, provided the graph has more than one edge (is different from $\Gamma _{(1.1)}$ and $\Gamma _{(1.2)}$).

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle \mu (k) & = & \displaystyle 2 m_s(0) \delta(k) + \sum_{k_n\neq 0} \left( \delta_{k_n} (k) + \delta_{-k_n} (k)\right) \\[3mm] & = & \displaystyle \chi \delta (k)+ \frac{\mathcal{L}}{\pi}+\frac{1}{2 \pi}\sum_{\gamma \in \mathcal{P}_{\mathrm{prim}}} \ell(\gamma) \frac{ 2 \;\mathbf S_{\mathbf{v}} (\gamma) (\cos k\ell(\gamma) - \mathbf S_{\mathbf{v}} (\gamma))}{1- 2 \cos k\ell(\gamma) \mathbf S_{\mathbf{v}} (\gamma) + \mathbf S^2_{\mathbf{v}}(\gamma) }, \end{array} \end{aligned} $$

(8.42)

where $\mathcal P_{\mathrm {prim}}$ denotes the set of primitive oriented paths, i.e. those oriented paths that coincide with their primitives. To prove the formula we note that every primitive path $\gamma$ determines a sequence of non-primitive paths: $2 \gamma , 3 \gamma , \dots , n \gamma , \dots$ Taking into account that

$$\displaystyle \begin{aligned} e^{i k \ell(n\gamma)} = e^{i n k \ell(\gamma)} = \left(e^{i k \ell(\gamma)} \right)^n, \quad \mathbf S_{\mathbf{v}} (n \gamma) = \left( \mathbf S_{\mathbf{v}} (\gamma) \right)^n, \end{aligned} $$

(8.43)

we see that contributions from the multiples of $\gamma$ form a convergent geometric progression since $| \mathbf S_{\mathbf {v}} (\gamma ) | < 1$⁹

$$\displaystyle \begin{aligned} \sum_{n=1}^\infty \ell (\mathrm{prim}\;(n \gamma))e^{i k \ell(n \gamma)} \mathbf S_{\mathbf{v}} (n \gamma) &= \sum_{n=1}^\infty \ell (\gamma) \left( e^{i k \ell(\gamma)} \mathbf S_{\mathbf{v}} (\gamma) \right)^n\\ &= \ell(\gamma) \frac{e^{i k \ell(\gamma)} \mathbf S_{\mathbf{v}} (\gamma)}{1- e^{i k \ell(\gamma)} \mathbf S_{\mathbf{v}} (\gamma)}. \end{aligned} $$

Adding the conjugated contribution we get

$$\displaystyle \begin{aligned} \begin{array}{c} \displaystyle \sum_{n=1}^\infty \ell (\mathrm{prim}\;(n \gamma))e^{i k\ell(n \gamma)} \mathbf S_{\mathbf{v}} (n \gamma) + \sum_{n=1}^\infty \ell (\mathrm{prim}\;(n \gamma))e^{-i k\ell(n \gamma)} \mathbf S_{\mathbf{v}} (n \gamma) \\[3mm] \displaystyle = \ell(\gamma) \frac{2 \; \mathbf S_{\mathbf{v}} (\gamma) (\cos k\ell(\gamma) - \mathbf S_{\mathbf{v}} (\gamma)) }{1 - 2 \cos k \ell(\gamma) \mathbf S_{\mathbf{ v}} (\gamma) + \mathbf S^2_{\mathbf{v}} (\gamma)}. \end{array} \end{aligned}$$

The two exceptional graphs $\Gamma _{(1.1)}$ and $\Gamma _{(1.2)}$ lead to classical Poisson summation formula as shown in the examples below.

Example 8.5

Consider the segment graph $\Gamma _{(1.1)}$ having length $\ell$. The spectrum of the standard Laplacian is $\lambda _n = \Big ( \frac {\pi }{\ell }\Big )^2 n^2, \; n= 0,1,2, \dots$, hence $m_s(0) = 1$. The set of periodic orbits is very simple: every orbit is obtained by crossing the interval back and forth n times, so that $\ell (\gamma ) = 2 \ell n$ and $l (\mathrm {prim}\,(\gamma )) = 2 \ell$.

Substitution into the trace formula (8.20) gives:

$$\displaystyle \begin{aligned} 2 \delta(k) + \sum_{n=1}^\infty \Big( \delta_{\frac{\pi}{\ell} n } (k) + \delta_{-\frac{\pi}{\ell} n } (k) \Big) = \delta (k) + \frac{\ell}{\pi} + \frac{1}{\pi} \sum_{n=1}^\infty 2 \ell \cos k 2 \ell n\end{aligned}$$

$$\displaystyle \begin{aligned} \Rightarrow \sum_{n\in \mathbb Z} \delta_{\frac{\pi}{\ell} n }(k) = \frac{\ell}{\pi} \sum_{n \in \mathbb Z} e^{ik 2 \ell n}.\end{aligned}$$

The formula takes the simplest form if one choses $\ell = \pi$

$$\displaystyle \begin{aligned} {} \sum_{n \in \mathbb Z} \delta_n (k) = \sum_{n \in \mathbb Z} e^{i 2 \pi n k}, \end{aligned} $$

(8.44)

which is nothing else than the classical Poisson summation formula. This formula is going to play a very important role in Chap. 10 where crystalline measures are constructed (see in particular formula (10.13)).

Example 8.6

Consider now the cycle graph $\Gamma _{(1.2)}$ of length $\ell$. The spectrum of the standard Laplacian is $\lambda _n = \Big ( \frac {2\pi }{\ell }\Big )^2 n^2, \; n= 0,1,1,2,2,3 \dots$ with $m_s(0) = 1$ and all other eigenvalues double degenerate. The set of periodic orbits is much more complicated: each time one may go along the cycle either clockwise or counter clockwise, but reflection coefficients are zero so that $\mathbf S_{\mathbf {v}} (\gamma ) = 0$ for all orbits that contain reflections at the vertex. Hence, it is enough to consider just the orbits going $m$ times clockwise or counter clockwise, so that $\ell (\gamma ) = \ell n$ and $l (\mathrm {prim}\,(\gamma )) = \ell$.

Substitution into the trace formula (8.20) gives:

$$\displaystyle \begin{aligned} 2 \delta(k) + 2 \sum_{n=1}^\infty \Big( \delta_{\frac{2\pi}{\ell} n }(k) + \delta_{-\frac{2 \pi}{\ell} n }(k) \Big) = 0 \cdot \delta (k) + \frac{\ell}{\pi} + 2 \frac{1}{\pi} \sum_{n=1}^\infty \ell \cos k \ell n ,\end{aligned}$$

where the second sum is taken twice because single sum counts only periodic orbits going in one of the directions. It follows that

$$\displaystyle \begin{aligned} 2 \sum_{n\in \mathbb Z} \delta_{\frac{2 \pi}{\ell} n }(k) = \frac{\ell}{\pi} \sum_{n \in \mathbb Z} e^{ik \ell n}.\end{aligned}$$

One gets formula (8.44) by choosing $\ell = 2 \pi$.

The derived trace formula will be important when applied to inverse spectral problems for metric graphs. Let us mention here that this formula is also interesting from a pure mathematical point of view, since the distribution $\mu (k) - \chi \delta (k)$ possesses the remarkable property: both the distribution itself and its Fourier transform are distributions supported by discrete sets, since both are given by $\delta$-functions supported by $\{ \pm k_n \}$ and $\pm \ell ( \mathcal P)$ respectively. One may think about this formula as a generalisation of the classical Poisson’s summation formula (8.44). We have seen that Poisson’s formula is a special case of (8.20) when $\Gamma = \Gamma _{(1.1)}$ or $\Gamma _{(1.2)}$. If $\Gamma$ is formed by edges with integer lengths, then formulas (8.20) and (8.21) can be obtained by just combining a finite number of classical Poisson formulas, since the spectrum is (more or less) periodic in this case (see Sect. 24.3). For graphs with non compatible edge lengths the spectrum is not periodic and derived formula cannot be obtained as a finite combination of Poisson formulas. One may think about quantum graphs as quasicrystals (see Sect. 10.2).

Problem 34

Consider the two isospectral equilateral graphs presented in Fig. 2.11 (assuming standard vertex conditions). Describe the corresponding sets of periodic orbits and verify trace formula (8.20) by showing directly that the series on the right hand side of the formula are identical.

8.4 Trace Formula for Laplacians with Scaling-Invariant Vertex Conditions

The trace formula can easily be generalised to non-standard conditions at the vertices as well as non-zero potential on the edges [93, 94, 455]. The proof goes almost without modifications for Laplacians with scaling-invariant vertex conditions.

Two points should be taken into account:

• 1. The spectral and algebraic multiplicities of the eigenvalue zero are not given by Theorem 8.2 anymore and depend on the vertex conditions.

• 2. The scattering coefficients at the vertices are not real anymore and therefore contributions from the opposite powers of $\mathbb S(k)$ cannot be combined together as $\mathbf S_{\mathbf {v}} (\gamma ) \cos k \ell (\gamma )$ as is done in formula (8.20).

• 3. The Laplacian with scaling-invariant conditions is non-negative as its quadratic form is given by the Dirichlet integral without any contribution from the vertices.

To clarify the first point consider the following elementary example: Let $\Gamma$ be a compact connected graph with some pendant edges. Let $L^{\mathrm {st}, D}$ be the Laplace operator corresponding to Dirichlet boundary condition at the pendant vertices and standard vertex conditions at all other vertices. Then the operator $L^{\mathrm {st}, D}$ does not have zero as an eigenvalue. The algebraic multiplicity of the eigenvalue zero could be different from zero. Calculations of the algebraic multiplicity can be carried out using essentially the same arguments as before and lead to $m_a (0) = 1 - \chi$.

Calculating contribution from the negative powers of $\mathbb S(k)$ one should take into account that the vertex scattering matrices are unitary and Hermitian. It follows that if $\mathbf S_{\mathbf {v}} (\gamma )$ is the product of scattering coefficients along a path $\gamma$, then the product of inverse scattering coefficients along the same path is given by $\mathbf S_{\mathbf {v}}^* (\gamma ) = \overline {\mathbf S_{\mathbf {v}} (\gamma )}.$

Theorem 8.7

Let$\Gamma$be a finite compact metricgraph with the total length$\mathcal L$and let$L^S (\Gamma )$be the Laplace operator in$L_2 (\Gamma )$determined by properly connecting scaling-invariant vertex conditions at the vertices described by unitary Hermitian matrices$S^m, \; m=1,2, \dots , M$. Then the spectral measure (8.19) is a tempered positive distribution, such that not only the Fourier transform$\hat {\mu }$is tempered but also$| \hat {\mu }|$is tempered.

The following two exact trace formulae establish the relation between the spectrum $\{k_n^2\}$ of $L^S (\Gamma )$ and the set $\mathcal P$ of closed paths on the metric graph $\Gamma$

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \mu (k) & = & \displaystyle 2 m_s(0) \delta(k) + \sum_{k_n\neq 0} \left( \delta_{k_n)}(k)+ \delta_{-k_n}(k) \right) \\ & = & \displaystyle (2 m_s(0) - m_a (0)) \delta (k)+ \frac{\mathcal{L}}{\pi} \\ && \displaystyle +\frac{1}{2\pi}\sum_{\gamma \in \mathcal{P}} \ell(\mathrm{prim}\,(\gamma)) \big( \mathbf S_{\mathbf{v}} (\gamma)e^{ik\ell(\gamma)}+ \mathbf S_{\mathbf{v}}^* (\gamma)e^{-ik\ell(\gamma)} \big), \end{array} \end{aligned} $$

(8.45)

and

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \hat{\mu} (l) & = & \displaystyle 2 m_s(0) + \sum_{k_n \neq 0} 2 \cos k_n l \\ & = & \displaystyle 2 m_s(0) - m_a (0) + 2 \mathcal L \delta (l) \\ & & \displaystyle + \sum_{\gamma \in \mathcal{P}} \ell(\mathrm{prim}\,(\gamma)) \Big(S_{\mathbf{v}} (\gamma) \delta_{\ell(\gamma)}(l) + \mathbf S_{\mathbf{v}}^* (\gamma)\delta_{-\ell(\gamma)}(l) \Big), \end{array} \end{aligned} $$

(8.46)

where

• $m_s (0)$ and $m_a (0)$ are spectral and algebraic multiplicities of the eigenvalue zero;

• $\mathcal P$ is the set of all closed oriented paths on $\Gamma ;$

• $\ell (\gamma )$is the metric length of the closed path$\gamma$;

• $\mathrm {prim}\,(\gamma )$is the primitive path for$\gamma$;

• $\mathbf S_{\mathbf {v}} (\gamma )$is the product of all vertex scattering coefficients along the oriented path$\gamma$.

One may generalise the derived trace formula by including not scaling-invariant vertex conditions [93, 94] or potentials on the edges [455].

Isospectral Laplacians on Two Edges

Consider the metric graph $\Gamma _2$ formed by two edges of length $\pi /2$ connected at one common vertex $V^2$ (see Fig. 8.4). The remaining two vertices are $V^1$ and $V^3$.

Fig. 8.4

Composed graph $\Gamma _2$

The vertex $V^2$ has degree two and we assume there the most general scaling invariant vertex conditions given by any Hermitian unitary $2 \times 2$ matrix $S_2$. Every such matrix has the form:

$$\displaystyle \begin{aligned} S_2 (a, \theta) = \left( \begin{array}{cc} a & \sqrt{1-a^2} e^{i \theta} \\ \sqrt{1-a^2} e^{-i \theta} & - a \end{array} \right), \quad a \in [-1,1], \theta \in [0, 2 \pi). \end{aligned}$$

We assume Neumann conditions at $V^1$ and $V^3$ and denote the corresponding Laplacian by $L(a,\theta )$.

It turns out that the operators $L(a, \theta )$ are isospectral, in particular all operators from the family are isospectral to the Neumann Laplacian on the single interval of length $\pi$—the operator $L(0,0)$. Trace formula (8.45) will help us to understand the reason for isospectrality.

In the proof of Theorem 8.7 it is shown that the spectral measure $\mu (k)$ may be calculated via the formula

(8.47)

where $\mathbb S (k)= \mathbf S \mathbf S_{\mathbf {e}}(k)$. The matrices $\mathbf S, \mathbf S_{\mathbf {e}} (k)$,$\mathbb S (k)$ and $\mathbf D$ are

$$\displaystyle \begin{aligned} \begin{array}{l} \scriptstyle \mathbf S = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & a & \sqrt{1-a^2} e^{i \theta} & 0 \\ 0 & \sqrt{1-a^2} e^{-i \theta} & - a & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \mathbf S_{\mathbf{e}} = \left( \begin{array}{cccc} 0 & e^{i \pi/2 k} & 0 & 0 \\ e^{i \pi/2 k} & 0 & 0 & 0 \\ 0 & 0 & 0 & e^{i \pi/2 k} \\ 0 & 0 & e^{i \pi/2 k} & 0 \end{array} \right) \\ \mathbb S (k) = \left( \begin{array}{cccc} 0 & e^{i \pi/2 k} & 0 & 0 \\ a e^{i \pi/2 k} & 0 & 0 &\sqrt{1-a^2} e^{i\theta} e^{i \pi/2 k} \\ \sqrt{1-a^2} e^{-i \theta} e^{i \pi/2 k} & 0 & 0 & - a e^{i \pi/2 k} \\ 0 & 0 & e^{i \pi/2 k} & 0 \end{array} \right), \quad \mathbf D = \frac{\pi}{2} \mathbf I, \end{array} \end{aligned}$$

in particular implying . Elementary calculations imply

$$\displaystyle \begin{aligned} \mathbb S^2 (k) =e^{i\pi k}\left( \begin{array}{cccc}a&0&0&\sqrt{1-a^2} e^{i\theta} \\ 0&a&\sqrt{1-a^2} e^{i\theta} &0 \\ 0&\sqrt{1-a^2} e^{-i\theta}&-a&0 \\ \sqrt{1-a^2} e^{-i\theta} &0&0&-a \end{array}\right), \end{aligned}$$

and , whereas

$$\displaystyle \begin{aligned} \mathbb S^4 (k) =e^{2i\pi k}\left(\begin{array}{cccc}1&0&0&0 \\0&1&0&0 \\0&0&1&0 \\ 0&0&0&1 \end{array}\right), \end{aligned}$$

and .

We are ready to calculate the sum (8.47) over all closed paths. Only paths of discrete length $4 n, \, n \in \mathbb N$ determine a non-zero contribution $2 \pi e^{2 i \pi k n}$, which is independent of $a$ and $\theta$. Hence the operators $L(a,\theta )$ are isospectral.

We shall return to this graph in Example 14.14.