M-Functions: Definitions and Examples

Abstract

M-functions associated with quantum graphs provide an efficient tool not only to describe spectral properties of quantum graphs, but to solve the inverse problems. The goal of this chapter is to give a self-consistent introduction to the theory of the graph’s M-functions.

17.1 The Graph M-Function

M-functions associated with quantum graphs provide an efficient tool not only to describe spectral properties of quantum graphs, but to solve the inverse problems. The goal of this chapter is to give a self-consistent introduction to the theory of the graph’s M-functions.

17.1.1 Motivation and Historical Hints

The classical Titchmarsh-Weyl M-function ${\mathbf {M}}(\lambda )$ (see [501]) connects together the Dirichlet and Neumann data for any solution $\psi$ of the stationary Schrödinger equation on the half-line $[0, \infty )$:

$$\displaystyle \begin{aligned} - \psi''(\lambda, x) + q(x) \psi (\lambda, x) = \lambda \psi (\lambda, x) \Rightarrow {\mathbf{M}}(\lambda) = \frac{\psi^{\prime}_x(\lambda,0)}{\psi ( \lambda,0)} , \quad \mbox{Im} \, \lambda \neq 0.\end{aligned}$$

This function not only accumulates all information about spectral properties of the one-dimensional Schrödinger operator with arbitrary boundary condition at the origin, but can be used to determine the potential $q$. The point $x= 0$ is the unique boundary point and the inverse problem can be seen as recovery of the potential from the boundary observations. Our goal is to generalise this object for the case of quantum graphs.

First of all we need to agree what should be understood as a graph’s boundary. One possibility could be to take all endpoints of the intervals forming the edges. We have already explored this direction in Sect. 5.3, where we derived the characteristic equation using the edge M-functions. In this approach the graph is considered as a collection of intervals and it is hard to see the graph’s topological structure. One may say that this set is too large. Another possibility could be to identify the boundary of $\Gamma$ with all vertices of degree one. The boundary defined in this way appears rather natural and has a clear visual interpretation, especially if the graph under consideration is a tree plotted on a sheet of paper. This definition does not work for all graphs since there are obviously graphs without any degree one vertices. Therefore in what follows we shall speak about the graph’s contact set $\partial \Gamma$—the set of vertices that are used to approach the graph $\Gamma$. Internal points on the edges can be seen as degree two vertices, hence the contact set may contain any finite set of points in $\Gamma$.

We shall also assume that the vertex conditions at the contact vertices are standard. This restriction is not essential, but will make all formulas more transparent.

17.1.2 The Formal Definition

Let $\Gamma$ be a finite compact metric graph formed by $N$ edges joined together at $M$ vertices $V^m.$ The contact set$\partial \Gamma$ is a fixed arbitrary subset of the vertices. Without loss of generality we assume that the vertices are enumerated so that the contact set is formed by the first $M_\partial \geq 1$ vertices: $\partial \Gamma = \{V^j \}_{j= 1}^{M_\partial }.$ Let us denote by $D_\partial$ the total degree of all vertices from $\partial \Gamma$

$$\displaystyle \begin{aligned} {} D_\partial = \sum_{j=1}^{M_\partial} d (V^j). \end{aligned} $$

(17.1)

All vertices in $\Gamma$ are thus divided into

• contact vertices from $\partial \Gamma$,

• internal vertices from ${\mathbf {V}} \setminus \partial \Gamma .$

The magnetic Schrödinger operator $L_{q,a}^S (\Gamma )$ is defined by standard vertex conditions on the contact set $\partial \Gamma$ and arbitrary Hermitian conditions at all internal vertices $V^{{M_\partial }+1}, \dots , V^{M}.$

Consider any nonreal $\lambda : \mbox{Im} \, \lambda \neq 0,$ and any function $\psi (\lambda ,x) \in W_2^2 (\Gamma \setminus {\mathbf {V}})$ which is a solution to the stationary magnetic Schrödinger equation on every edge:

$$\displaystyle \begin{aligned} {} - \left(\frac{d}{dx} - i a(x) \right)^2 \psi (\lambda, x) + q (x) \psi (\lambda, x) = \lambda \psi (\lambda, x). \end{aligned} $$

(17.2)

Every such function is continuous and has continuous first derivative—this is proven repeating the arguments presented in Sect. 4.1 substituting in Eqs. (4.5) and (4.6) the function $f$ with $\lambda \psi$.

It follows that the limiting values of $\psi$ at the vertices are well-defined. We assume that $\psi$ satisfies vertex conditions (3.51) at all internal vertices, but just continuity condition at the contact vertices. Note that no condition on the derivatives is imposed on $\partial \Gamma$.

The vertex conditions at the internal vertices can be written using single $(2 N - D_\partial ) \times (2N- D_\partial )$ irreducible unitary matrix ${\mathbf {S}}^{\mathrm {int}}$ as follows. We first introduce the notations

$$\displaystyle \begin{aligned} {} \vec{\psi}^{\mathrm{int}} = \{ \psi (x_j) \}_{x_j \notin \partial \Gamma}, \; \; \partial \vec{\psi}^{\mathrm{int}} = \{ \partial \psi (x_j) \}_{x_j \notin \partial \Gamma}, \end{aligned} $$

(17.3)

where the $2N -D_\partial$-dimensional vectors $\vec {\psi }^{\mathrm {int}}$ and $\partial \vec {\psi }^{\mathrm {int}}$ collect together all limiting values at the internal vertices. Then putting together the vertex conditions (4.8) at each internal vertex we get the $(2 N - D_\partial ) \times (2N- D_\partial )$ unitary matrix ${\mathbf {S}}^{\mathrm {int}}$, which is block-diagonal if the endpoints are ordered respecting the vertex structure

$$\displaystyle \begin{aligned} {\mathbf{S}}^{\mathrm{int}} = \bigoplus _{V^m \notin \partial \Gamma} S_m, \end{aligned} $$

(17.4)

where $S_m$ are the $d_m \times d_m$ irreducible unitary matrices parameterising vertex conditions at the vertices (4.8). In the rest of this chapter we shall always assume that the vertex conditions

$$\displaystyle \begin{aligned} {} i \left( {\mathbf{S}}^{\mathrm{int}} - {\mathbf{I}} \right) \vec{\psi}^{\mathrm{int}} = \left( {\mathbf{S}}^{\mathrm{int}} + {\mathbf{I}} \right) \partial \vec{\psi}^{\mathrm{int}} \end{aligned} $$

(17.5)

are satisfied.

In addition, we introduce the $M_\partial$-dimensional vectors of the limiting values at contact vertices

$$\displaystyle \begin{aligned} {} \vec{\psi}^\partial = \{ \psi (V^m) \}_{m=1}^{M_\partial}, \; \; \; \partial \vec{\psi}^\partial = \{ \sum_{x_j \in V^m} \partial \psi (x_j) \}_{m=1}^{M_\partial}. \end{aligned} $$

(17.6)

It is important to remember that we assumed that the function $\psi$ is continuous at the contact vertices, hence the values $\psi (V^m)$ are well-defined for $m = 1,2, \dots , M_\partial .$ We use the sums of extended normal derivatives $\sum _{x_j \in V^m} \partial \psi (x_j)$ instead of single values $\partial \psi (x_j)$ at the endpoints for two reasons

• we are aiming to define the M-function as quadratic matrix-valued function;

• the standard conditions at the vertices from $\partial \Gamma$ require additionally that $\sum _{x_j \in V^m} \partial \psi (x_j) = 0$.

Definition 17.1

The graph’s M-function${\mathbf {M}}_\Gamma (\lambda )$ is the $M_\partial \times M_\partial$ matrix-valued function defined by the map:

$$\displaystyle \begin{aligned} {} \displaystyle {\mathbf{M}}_{\Gamma} (\lambda) : \displaystyle \vec{\psi}^\partial \mapsto \displaystyle \partial \vec{\psi}^\partial , \; \; \mbox{Im} \, \lambda \neq 0, \end{aligned} $$

(17.7)

where $\vec {\psi }^\partial$ and $\partial \vec {\psi }^\partial$ are the limiting values for an arbitrary function $\psi (\lambda ,x)$ satisfying the differential equation (17.2), the vertex conditions (17.5) at internal vertices and continuous at contact vertices.

In order to justify this definition we need to show existence and uniqueness of the solutions to:

Dirichlet Problem

For arbitrary vector $\vec {f} \in \mathbb C^{M_\partial }$ find a function $\psi$ solving the differential equation (17.2) satisfying vertex conditions (17.5) at internal vertices, continuous on $\partial \Gamma$ and satisfying the boundary condition

$$\displaystyle \begin{aligned} {} \psi (\lambda, \cdot) \vert_{\partial \Gamma} = \vec{f}. \end{aligned} $$

(17.8)

To show the existence, let us denote by $L^{\mathrm {min}}$ the magnetic Schrödinger operator defined on the functions satisfying vertex conditions (17.5) at all internal vertices and both Dirichlet and Neumann conditions at the contact vertices:

$$\displaystyle \begin{aligned} \vec{\psi}^\partial = 0, \quad \partial \vec{\psi}^\partial = 0.\end{aligned}$$

This operator is clearly symmetric, its adjoint to be denoted by $L^{\mathrm {max}}$ is given by the same differential expression on the domain of functions satisfying vertex conditions (17.5) at the internal vertices and just continuity condition at the contact vertices. Then Eq. (17.2) can be written as

$$\displaystyle \begin{aligned} (L^{\mathrm{max}} - \lambda )\psi (\lambda, x) = 0.\end{aligned}$$

Let $\vec {f}$ be any vector from $\mathbb C^{M_\partial }$, then obviously there exists a function $w \in \mathrm {Dom}\,(L^{\mathrm {max}}),$ such that $w \vert _{\partial \Gamma } = \vec {f}.$ Consider now the function

where $L^{\mathrm {D}}$ denotes the Dirichlet magnetic Schrödinger operator defined on the functions satisfying conditions (17.5) at the internal vertices and Dirichlet conditions at the contact vertices. Here we used that the operator $L^{\mathrm {D}}$ is self-adjoint and its resolvent is defined on the whole Hilbert space. Clearly $L^{\mathrm {max}}$ is an extension of $L^{\mathrm {D}}$ implying that $(L^{\mathrm {max}} - \lambda ) (L^D-\lambda )^{-1}$ is the identity operator. It follows that $\psi$ belongs to the kernel of $L^{\mathrm {max}} - \lambda$

$$\displaystyle \begin{aligned} (L^{\mathrm{max}} - \lambda) \psi = - (L^{\mathrm{max}} - \lambda) w + (L^{\mathrm{max}} - \lambda) w = 0.\end{aligned}$$

Moreover, the restriction of $\psi$ to the contact vertices coincides with $\vec {f}$, since

$$\displaystyle \begin{aligned} (L^D-\lambda)^{-1} (L^{\mathrm{max}} - \lambda) w \in \mathrm{Dom}\,(L^{\mathrm{D}}) \Rightarrow \Big( (L^D-\lambda)^{-1} (L^{\mathrm{max}} - \lambda) w \Big) \vert_{\partial \Gamma} = 0.\end{aligned}$$

Summing up, $u$ is a solution to the Dirichlet problem formulated above.¹

To show uniqueness assume on the contrary, that two solutions to the Dirichlet problem exist, say the functions $\psi _1$ and $\psi _2.$ Then their difference $\psi _2 - \psi _1$ is zero on $\partial \Gamma$ and therefore belongs to the domain of $L^{\mathrm {D}}.$ If $\psi _2 - \psi _1$ is not identically equal to zero, then it is an eigenfunction of $L^{\mathrm {D}}$ corresponding to a non-real $\lambda$, but this is impossible since the operator is self-adjoint. It follows that solution to the Dirichlet problem is unique.

Let $\lambda >0$, then any symmetric matrix with real entries is an M-function for a certain compact metric graph [227] and given $\lambda$.

To understand spectral properties of metric graphs it will be convenient to look at the energy curves—the eigenvalues of the M-function dependent on the real parameter $\lambda$.

17.1.3 Examples

Example 17.2

The M-function for the Laplacian on the single interval $I = [0, 1]$, the contact set $\partial \Gamma$ being one of the endpoints, say $\partial \Gamma = \{ 0\} .$ Two different vertex conditions at the internal vertex are considered:

• (1) Neumann condition at $x = 1$

  $$\displaystyle \begin{aligned} \psi' (1) = 0.\end{aligned}$$

• (2) Dirichlet condition at $x = 1$

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

Case (1)

In the first case the function $\psi (\lambda , x )$ satisfying Neumann condition at $x = 1$ is

$$\displaystyle \begin{aligned} \psi (\lambda, x) = \cos k (x- 1).\end{aligned}$$

The M-function is given by the logarithmic derivative of $\psi$ at $x= 0$

$$\displaystyle \begin{aligned} {\mathbf{M}}_I^N (\lambda) = k \tan k .\end{aligned}$$

See Fig. 17.1, where this function is plotted for real values of the spectral parameter. The function is piece-wise monotone with singularities at $\lambda = \pi ^2 \left ( \frac {1}{2} + n \right )^2 , \; n= 0,1,2, \dots ,$ corresponding to the eigenvalues of the Dirichlet-Neumann Laplacian on $[0, 1]$. The zeroes of ${\mathbf {M}}$ are situated at $\lambda = \pi ^2 n^2, \; n= 0,1,2, \dots ,$ corresponding to the spectrum of the Neumann-Neumann Laplacian on $[0, 1].$

Fig. 17.1

${\mathbf {M}}_I^N (\lambda )$– M-function for the interval, Neumann condition at one of the endpoints

Case (2)

Similarly, for the Dirichlet condition at $x= 1$ we have

$$\displaystyle \begin{aligned} \psi (\lambda, x) = \sin k (x-1).\end{aligned}$$

The logarithmic derivative gives the M-function

$$\displaystyle \begin{aligned} {\mathbf{M}}_I^D (\lambda) = - k \cot k .\end{aligned}$$

The function is plotted in Fig. 17.2. The singularities are located at $\lambda = \pi ^2 n^2, \; n= 0,1,2, \dots ,$ corresponding to the spectrum of the Dirichlet-Dirichlet Laplacian; the zeroes—at $\lambda = \pi ^2 \left (\frac {1}{2} + n \right )^2, \; n= 0,1,2, \dots ,$ corresponding to the spectrum of the Neumann-Dirichlet Laplacian.

Fig. 17.2

${\mathbf {M}}_I^D (\lambda )$– M-function for the interval, Dirichlet condition at one of the endpoints

These examples illustrate that the M-function depends on the conditions at the internal vertices, in this case on the condition at $x= 1.$ The M-function is originally defined for non-real $\lambda$ but to see the spectra of the operators corresponding to different conditions at the contact vertex one has to consider the continuation of ${\mathbf {M}}(\lambda )$ to the real line. This continuation has singularities which explains the reason why only non-real $\lambda$ were used in the definition.

Example 17.3

The M-function for the Laplace operators on the single interval $I_\ell =[0, \ell ]$ and the contact set $\partial \Gamma$ equal to the union of endpoints $\partial \Gamma = \{ 0, \ell \}.$

This function has already been calculated in Sect. 5.3 for the Schrödinger equation. We repeat here explicit calculations in the case of zero potential. Any solution to (17.2) has the form

$$\displaystyle \begin{aligned} \psi = a \cos kx + b \sin kx.\end{aligned}$$

The parameters $a$ and $b$ can be calculated from the values of the function $\psi$ at the contact points $0$ and $\ell$

$$\displaystyle \begin{aligned} \psi (x) = \cos kx \, \psi (0) + \left( - \frac{\cos k \ell}{\sin k \ell} \psi (0) + \frac{1}{\sin k \ell} \psi (\ell) \right) \sin kx.\end{aligned}$$

The normal derivatives are

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ccl} \displaystyle \psi' (0) & = & \displaystyle - k \frac{\cos k \ell}{\sin k \ell} \psi (0) + \frac{k}{\sin k\ell} \psi (\ell), \\ \displaystyle - \psi' (\ell) & = & \displaystyle \frac{k}{\sin k \ell} \psi (0) - \frac{k}{\sin k \ell} \psi (\ell). \end{array} \right.\end{aligned}$$

We can write this system in matrix form as

(17.9)

The corresponding M-function is completely determined by the length of the interval. For almost all real $\lambda$, more precisely for $\frac {\ell ^2}{\pi ^2} \lambda \neq 1 , 4, \dots , n^2, \dots$, this is a Hermitian $2 \times 2$ matrix. Let us plot its eigenvalues (see Fig. 17.3). We see that for every nonsingular $\lambda$ there are precisely two energy values, the energy curves are monotone between the singularities. An interesting feature of this example is that, at the singular points, say $\lambda = 4$, one of the energy curves approaches $\pm \infty$, while the other curve crosses the real line. One may say that the singularities and zeroes of ${\mathbf {M}} (\lambda )$ occur simultaneously.

Fig. 17.3

M-function for the interval with two contact points and $\ell = \pi$. (The energy curves)

Example 17.4

The M-function for the standard Laplacian on the lasso graph $\Gamma _{(2.2)}$ with the loop of length $\ell$ and outgrowth of length $s.$ The contact set formed by the unique vertex of degree one.

This graph is depicted in Fig. 17.4, where parametrisation of the edges is indicated.

Fig. 17.4

The lasso graph $\Gamma _{(2.2)}$

To calculate the M-function it is enough to consider only even functions on the loop (odd functions are equal to zero at the inner vertex and are naturally continued by zero to the outgrowth):

$$\displaystyle \begin{aligned} \psi (x) = \left\{ \begin{array}{ll} \cos kx, & \mbox{on the loop}, \\ a \cos kx + b \sin kx, & \mbox{on the outgrowth}. \end{array} \right.\end{aligned}$$

Standard conditions at the vertex of degree three give

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ccl} \displaystyle \cos \frac{k \ell}{2} & = & \displaystyle a \cos ks + b \sin ks \\[3mm] \displaystyle 2 k \sin \frac{k \ell}{2} & = & \displaystyle - a k \sin ks + b k \cos ks \end{array} \right.\end{aligned}$$

$$\displaystyle \begin{aligned} \Rightarrow \left\{ \begin{array}{ccl} \displaystyle a & = & \displaystyle \cos\frac{k \ell}{2} \cos ks - 2 \sin\frac{k \ell}{2} \sin ks \\[3mm] \displaystyle b & = & \displaystyle 2 \sin\frac{k \ell}{2} \cos ks + \cos\frac{k \ell}{2} \sin ks \end{array} \right.\end{aligned}$$

The M-function is just equal to the ratio $k b/a$

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_{\Gamma_{(2.2)}} (\lambda) = k \frac{ \cos\frac{k \ell}{2} \sin ks + 2 \sin\frac{k \ell}{2} \cos ks}{\cos\frac{k \ell}{2} \cos ks - 2 \sin\frac{k \ell}{2} \sin ks}. \end{aligned} $$

(17.10)

Let us consider the special case $\ell = 2 \pi$

$$\displaystyle \begin{aligned} {\mathbf{M}}_{\Gamma_{(2.2)}}(\lambda) = k \frac{ \cos k \pi \sin ks + 2 \sin k \pi \cos ks}{\cos k \pi \cos ks - 2 \sin k \pi \sin ks}. \end{aligned}$$

The eigenfunctions supported just by the loop correspond to integer values of $k$: $\lambda = n^2, n \in \mathbb N$. These eigenvalues are not seen from the M-function as is indicated by the plot—the function is regular there

$$\displaystyle \begin{aligned} {\mathbf{M}}_{\Gamma_{(2.2)}} (n^2) = n \frac{ \sin ns}{ \cos ns }, \end{aligned}$$

unless $\cos n s = 0.$ For example, for $n= 3$ and $s = \pi /6$ we have $\cos n s = \cos 3 \pi /6 = 0$ and we observe a singularity there. But this singularity has nothing to do with the eigenfunctions supported by the loop: small change of $s$ shifts the singularity to a neighbouring point, while the eigenfunction on the loop and the corresponding eigenvalue remain unchanged (Fig. 17.5).

Fig. 17.5

M-function for the Lasso graph with $\ell = 2\pi , s = \pi /6$

Example 17.5

The M-function for the standard Laplacian on the loop $\Gamma _{(2.3)}$ with two contact points. The lengths of the edges are $\ell _1$ and $\ell _2.$

Consider the graph formed by two intervals $[0, \ell _1]$ and $[0,\ell _2]$ joined pairwise at their endpoints. The contact set is formed by the two vertices (Fig. 17.6).

Fig. 17.6

Cycle with two contact points $\Gamma _{(2.3)}$

The corresponding M-function is equal to the sum of M-functions for the two separate intervals of lengths $\ell _1$ and $\ell _2$ (see (17.9))

$$\displaystyle \begin{aligned} {\mathbf{M}}_{\Gamma_{(2.3)}} (\lambda) = {\mathbf{M}}_{I_{\ell_1}} (\lambda) + {\mathbf{M}}_{I_{\ell_2}} (\lambda)\end{aligned}$$

$$\displaystyle \begin{aligned} = \left( \begin{array}{cc} \displaystyle - k \cot k \ell_2 - k \cot k \ell_2 & \displaystyle \frac{k}{\sin k \ell_1} + \frac{k}{\sin k \ell_2} \\ \displaystyle \frac{k}{\sin k \ell_1} + \frac{k}{\sin k \ell_2}& \displaystyle - k \cot k \ell_1 - k \cot k \ell_2 \end{array} \right). \end{aligned} $$

(17.11)

To understand this formula consider solutions to the Schrödinger equation on the two intervals. Their normal derivatives are related via the corresponding M-functions to the values at the endpoints. Hence the sum of M-functions maps the values at the vertices in $\Gamma _{\ell _1,\ell _2}^{\mathrm {loop}}$ to the sums of normal derivatives.

If the standard Laplacian on the loop has eigenfunctions equal to zero at both contact points, then the corresponding eigenvalues do not cause any singularities of ${\mathbf {M}}(\lambda ).$ This fact is best illustrated by plotting the energy curves for $\ell _1 = \ell _2$ (symmetric case) and $\ell _1 \neq \ell _2$ (non-symmetric case) (see Fig. 17.7). One can see that the number of singularities in the non-symmetric case is doubled compared to the symmetric case. In the non-symmetric case, we have chosen $\ell _1$ close to $\ell _2$, this resulted in the appearance of almost vertical energy curves close to the points $\frac {\ell ^2}{\pi ^2} \lambda = 1, 2^2, \dots$ corresponding to the energies of the invisible eigenfunctions in the symmetric case.

Fig. 17.7

M-function for the graph $\Gamma _{(2.3)}$.(The energy curves)

Problem 75

Calculate the M-function for the watermelon graph formed by three parallel edges of lengths $\ell _1, \ell _2,$ and $\ell _3$ joining together two vertices. Consider two cases

• (1) The contact set is formed by one of the vertices.

• (2) The contact set consists of the two vertices.

Plot the corresponding energy curves for different values of $\ell _j$ including the case where all lengths are equal.

17.2 Explicit Formulas Using Eigenfunctions

The goal of this section is to present explicit formulas for M-functions in terms of the corresponding eigenfunctions. These formulas will be used to study properties of the M-functions, they also may be used to justify the definition itself, since the formulas we obtain will show another one time that the function $\psi$ can be calculated for any vector of boundary values $\vec {f} = \vec {\psi }^\partial .$ The M-function will be given in terms of the eigenfunctions of two differential operators on the same graph $\Gamma :$

$$\displaystyle \begin{aligned} L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{st}} (\Gamma) \; \; \mbox{and} \; \; L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{D}} (\Gamma).\end{aligned}$$

These self-adjoint operators in $L_2 (\Gamma )$ are defined by the same differential expression (2.17), the same vertex conditions (17.5) at the internal vertices and standard respectively Dirichlet conditions at the contact vertices. Sometimes abusing the terminology, we are going to call these operators as standard and Dirichlet operators, also vertex conditions at internal vertices are not assumed to be standard or Dirichlet. Just in this chapter, we are going to use short notations for these operators

$$\displaystyle \begin{aligned} {} L^{\mathrm{st}} := L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{st}} (\Gamma) \; \; \mbox{and} \; \; L^{\mathrm{D}} := L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{D}} (\Gamma) \end{aligned} $$

(17.12)

hoping that this will not lead to any misunderstanding.

One of the usual ways to prove the existence of solutions of boundary value problems is to calculate the resolvent of the corresponding differential operator. Consider the standard operator $L^{\mathrm {st}}$. This is a self-adjoint operator with discrete spectrum and we denote the corresponding eigenvalues and eigenfunctions by $\lambda _n^{\mathrm {st}}$ and $\psi _n^{\mathrm {st}}$, respectively. The eigenfunctions are assumed to form an orthonormal basis, hence for any $f \in L_2 (\Gamma )$ we have the spectral resolution

$$\displaystyle \begin{aligned} f = \sum_{n=1}^\infty \langle \psi_n^{\mathrm{st}}, f \rangle_{L_2 (\Gamma)} \psi_n^{\mathrm{st}}. \end{aligned} $$

(17.13)

This equality can be written using the integral kernel

$$\displaystyle \begin{aligned} k (x,y) = \sum_{n=1}^\infty \psi_n^{\mathrm{st}} (x) \overline{\psi_n^{\mathrm{st}}(y)}\end{aligned}$$

as follows

$$\displaystyle \begin{aligned} f(x) = \int_\Gamma k (x,y) f(y) dy. \end{aligned} $$

(17.14)

If $f \in \mathrm {Dom}\, (L^{\mathrm {st}})$ then

$$\displaystyle \begin{aligned} L^{\mathrm{st}} f = \sum_{n=1}^\infty \lambda_n^{\mathrm{st}} \langle \psi_n^{\mathrm{st}}, f \rangle_{L_2 (\Gamma)} \psi_n^{\mathrm{st}}. \end{aligned} $$

(17.15)

Similar formulas hold for the functions of the operator, in particular, the resolvent is given by

$$\displaystyle \begin{aligned} \left(L^{\mathrm{st}} - \lambda \right)^{-1} f = \sum_{n=1}^\infty \frac{1}{\lambda_n^{\mathrm{st}} - \lambda} \langle \psi_n^{\mathrm{st}}, f \rangle_{L_2 (\Gamma)} \psi_n^{\mathrm{st}}, \end{aligned} $$

(17.16)

or as the bounded integral operator with the Hilbert-Schmidt kernel

$$\displaystyle \begin{aligned} {} r_\lambda (x,y) = \sum_{n=1}^\infty \frac{1}{\lambda_n^{\mathrm{st}} - \lambda} \psi_n^{\mathrm{st}} (x) \overline{\psi_n^{\mathrm{st}} (y)} . \end{aligned} $$

(17.17)

We start with the simplest example of the interval graph $I = [0,1]$ with the contact point $x= 0$ and Neumann condition at $x= 1$ (case ${ (1)}$ in Example 17.2). The corresponding M-function is

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_I (\lambda) = k \frac{\sin k}{\cos k}. \end{aligned} $$

(17.18)

We calculate now the resolvent kernel $r_\lambda (x,y)$ explicitly. It is a solution to the following differential equation

$$\displaystyle \begin{aligned} {} - r_{xx} (x, y) - \lambda r(x,y) = \delta (x-y). \end{aligned} $$

(17.19)

Outside the point $x = y$, the kernel is a solution to the homogeneous equation. Taking into account boundary conditions at $x = 0$ and $x= 1$ we get

$$\displaystyle \begin{aligned} r_\lambda (x,y) = \left\{ \begin{array}{ll} \displaystyle \alpha \cos kx, & \displaystyle x < y, \\[3mm] \displaystyle \beta \cos k(x-1), & \displaystyle x > y. \end{array} \right.\end{aligned}$$

The parameters $\alpha , \beta$ should be chosen so that the function $r_\lambda ( \cdot , y)$ is continuous at $x= y$ and its first derivative has jump $-1$, then differentiating $r_\lambda$ twice one gets the delta-function as required:

$$\displaystyle \begin{aligned} r_\lambda (x,y) = \left\{ \begin{array}{ll} \displaystyle - \frac{\cos k (1-y)}{k \sin k} \cos kx, & \displaystyle x < y, \\[3mm] \displaystyle- \frac{\cos k y}{k \sin k} \cos k(x-1), & \displaystyle x > y. \end{array} \right. \end{aligned} $$

(17.20)

We observe that the following formula holds:

$$\displaystyle \begin{aligned} - \left(r_\lambda (0, 0) \right)^{-1} = k \frac{\sin k }{\cos k} = {\mathbf{M}}_I (\lambda). \end{aligned} $$

(17.21)

The formula connecting $r_\lambda$ and ${\mathbf {M}}_I$ is not just a coincidence, it can be generalised for arbitrary graphs. But let us understand first the reason why this formula holds in our example. We consider the limit $\lim _{\epsilon \rightarrow 0} r_\lambda (x, \epsilon ).$ The function $r_\lambda (x, \epsilon )$ satisfies Neumann condition at $x = 0,$ hence for small $\epsilon > 0$ we have approximately

$$\displaystyle \begin{aligned} \Big( \frac{\partial}{\partial x} r_\lambda \Big) (\epsilon +0, \epsilon) \sim -1.\end{aligned}$$

It follows that $r_\lambda (x,0)$ is a solution to the homogeneous differential equation satisfying the boundary condition

$$\displaystyle \begin{aligned} \Big( \frac{\partial}{\partial x} r_\lambda \Big) (0, 0) = -1.\end{aligned}$$

Then the M-function can be calculated as

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_I (\lambda) = \frac{ \frac{\partial}{\partial x} r_\lambda (0, 0)}{r_\lambda (0,0)} = - \left( r_\lambda (0,0) \right)^{-1}. \end{aligned} $$

(17.22)

This formula can be easily generalised for the case of any finite compact metric graph $\Gamma$ with the contact set $\partial \Gamma$: the corresponding M-function is the matrix valued function with the entries

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) = - \left(\left\{ r_\lambda (V^i, V^j) \right\}_{V^i, V^j \in \partial \Gamma} \right)^{-1}. \end{aligned} $$

(17.23)

This formula holds for Schrödinger operators and arbitrary vertex conditions at internal vertices, but with standard conditions on the contact set $\partial \Gamma .$

The proof follows the same lines as the proof of (17.22) and we are going to assume that the magnetic potential is identically zero $a (x) \equiv 0$. This assumption is not restrictive, since vertex conditions at the internal vertices are arbitrary and elimination of $a$ leads to a special change of those conditions. Consider the resolvent kernel $r_\lambda (V^j, y) , \; V^j \in \partial \Gamma$ as a function of the second argument y, $V^j$ being fixed. For any function $\varphi \in \mathrm {Dom}\,(L^{\mathrm {st}})$ we have

$$\displaystyle \begin{aligned} (L^{\mathrm{st}} - \lambda)^{-1} (L^{\mathrm{st}}-\lambda) \varphi = \varphi,\end{aligned}$$

implying in particular

$$\displaystyle \begin{aligned} {} \int_\Gamma r_\lambda (V^j, y) \Big(- \varphi''(y) + q(y) \varphi(y) - \lambda \varphi(y) \Big) dy = \varphi (V^j). \end{aligned} $$

(17.24)

Taking first $\varphi \in C^\infty _0 (E_n), \, n= 1,2, \dots , N$ we conclude that the resolvent kernel is a weak solution of the differential equation

$$\displaystyle \begin{aligned} - \frac{\partial ^2}{\partial y^2} r_\lambda(V^j, y) + q(y) r_\lambda (V^j, y) = \lambda r_\lambda (V^j, y)\end{aligned}$$

on every edge. Every such solution is continuous and has continuous derivative inside the edges, hence we may integrate by parts in (17.24) taking $\varphi$ smooth on each closed edge:

Consider now test-functions $\varphi$ with the support including the vertex $V^j$ and no other vertex and having all normal derivatives at $x_i \in V^j$ equal to zero. It follows that²

$$\displaystyle \begin{aligned} {} - \sum_{x_i \in V^j} \partial r_\lambda (V^j,x_i) = 1, \end{aligned} $$

(17.25)

where we used that $\varphi$ is continuous at $V^j$ due to standard conditions there.

Relaxing condition that the derivatives are zero and considering all possible test functions we conclude that

$$\displaystyle \begin{aligned} \sum_{x_i \in V^j} r_\lambda (V^j, x_i) \partial \varphi (x_i) = 0\end{aligned}$$

holds whenever $\sum _{x_i \in V^j} \partial \varphi (x_i) = 0,$ implying that $r_\lambda (V^j, y)$ is continuous at $y = V^j.$

Essentially the same calculations imply that the resolvent kernel satisfies standard vertex conditions at all other contact vertices $V^m, m \neq j$. At all internal vertices $r_\lambda (V^j, y)$ satisfies the vertex conditions described by ${\mathbf {S}}^{\mathrm {int}}$—the same conditions as the functions from the domain of $L^{\mathrm {st}} (= L_{q,a}^{{\mathbf {S}}^{\mathrm {int}}, \mathrm {st}}).$

Summing up the resolvent kernel $r_\lambda ( \cdot , V^1)$ is a solution to the differential equation (4.32) satisfying standard vertex conditions outside the boundary, continuous at the boundary vertices and having the following boundary values by (17.25):

$$\displaystyle \begin{aligned} r_\lambda ( V^1, \cdot) \vert_{\partial \Gamma} = \left( \begin{array}{c} r_\lambda ( V^1, V^1) \\ r_\lambda ( V^1, V^2) \\ \vdots \\ r_\lambda ( V^1, V^K) \end{array} \right) , \quad \partial r_\lambda ( V^1, \cdot) \vert_{\partial \Gamma} = \left( \begin{array}{c} -1 \\ 0 \\ \vdots \\ 0 \end{array} \right) .\end{aligned}$$

Similar formulas hold for $r_\lambda ( V^i, \cdot ) \vert _{\partial \Gamma }$ and $\partial r_\lambda ( V^i, \cdot ) \vert _{\partial \Gamma }$, implying that the matrix $\displaystyle - \left \{ r_\lambda (V^i, V^j )\right \}_{i,j=1}^M$ is inverse to ${\mathbf {M}}_\Gamma (\lambda ).$

Theorem 17.6

Let us denote by $\lambda _n^{\mathrm {st}}$ and $\psi _n^{\mathrm {st}}$ the eigenvalues and ortho-normalised eigenfunctions of the Schrödinger operator $L^{\mathrm {st}} = L_{q,a}^{{\mathbf {S}}^{\mathrm {int}},\mathrm {st}}$ on a compact finite metric graph $\Gamma$ with the contact set $\partial \Gamma .$ Let standard conditions be assumed at the contact vertices $V^j \in \partial \Gamma$ and arbitrary Hermitian conditions at the internal vertices $V^j \notin \partial \Gamma .$ Then the M-function for $\Gamma$ is given by.

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) = - \left( \sum_{n=1}^\infty \frac{\displaystyle \langle \psi_n^{\mathrm{st}} \vert_{\partial \Gamma}, \cdot \rangle_{\ell_2 (\partial \Gamma)} \psi_n^{\mathrm{st}} \vert_{\partial \Gamma}}{\lambda_n^{\mathrm{st}} - \lambda} \right)^{-1}. \end{aligned} $$

(17.26)

Proof

We use the explicit expression for the resolvent (17.17) to get:

$$\displaystyle \begin{aligned} r_\lambda (\cdot, y) = \displaystyle \sum_{n=1}^\infty \frac{1}{\lambda_n^{\mathrm{st}} - \lambda} \overline{\psi_n^{\mathrm{st}} (y)} \psi_n^{\mathrm{st}} (\cdot) . \end{aligned}$$

The series is convergent in $L_2 (\Gamma )$. Our goal is to prove that the convergence is pointwise.

We show first that the series

$$\displaystyle \begin{aligned} {} \sum_{n=1}^\infty \frac{\vert \psi_n^{\mathrm{st}} (x) \vert^2}{\lambda_n^{\mathrm{st}} + C } \end{aligned} $$

(17.27)

is (absolutely) convergent for any $x \in (\Gamma \setminus {\mathbf {V}})\cup \partial \Gamma$ and a certain sufficiently large positive $C.$

Consider the case of the Laplacian (with the same vertex conditions). Let us denote the corresponding eigenvalues and normalised eigenfunctions by $\lambda _n^0$ and $\psi _n^0 (x).$ We already established that the Laplacian eigenfunctions are uniformly bounded in the case of standard vertex conditions (11.37), but the proof did not use that the vertex conditions are standard; therefore we have

(17.28)

for any conditions at the internal vertices. The eigenvalues $\lambda _n^0$ satisfy Weyl’s asymptotics (4.25), hence the series (17.27) is absolutely convergent for $C > - \lambda _1^0.$

The delta distribution $\delta _x$, where $x$ is any internal point on an edge, is a bounded functional with respect to the quadratic form of the Laplacian. Then the series (17.27) gives the norm of the delta distribution. To see this consider any test function $\varphi$ from the domain of the Laplacian’s quadratic form and consider the action of the delta distribution on it

$$\displaystyle \begin{aligned} \left\vert \langle \delta_x, \varphi \rangle \right\vert = \left\vert \varphi (x) \right\vert = \left| \sum_n \psi_n^0 (x) \varphi_n^0 \right| = \left| \sum_n (\lambda_n^0+C)^{-1/2} \psi_n^0 (x) (\lambda_n^0 +C)^{1/2} \varphi_n^0 \right| \leq\end{aligned}$$

(17.29)

where $\varphi _n^0 = \langle \psi _n^0, \varphi \rangle _{L_2(\Gamma )}$ are the Fourier coefficient of the function $\varphi$ with respect to the orthonormal system $\{ \psi _n^0 \}_{n=1}^\infty .$ Since the coefficients $( \lambda _n^0 + C) \vert \varphi _n^0 \vert ^2$ can be chosen arbitrarily (of course subject to the convergence of the series), the positive series $\displaystyle \left ( \sum _n \frac {\vert \psi _n^0 (x) \vert ^2}{\lambda _n^0 + C} \right )^{1/2}$ gives the norm of the delta distribution.

The Sobolev-type estimate (11.11)

$$\displaystyle \begin{aligned} \vert \langle q u, u \rangle \vert \leq \epsilon \langle L_0 u, u \rangle + \frac{2}{\epsilon} \| u \|{}^2, \quad q \in L_1 (\Gamma),\end{aligned}$$

with a certain $0 < \epsilon < 1$, implies that the quadratic forms of the Laplacian and of the Schrödinger operator with $L_1$ potential are equivalent:

$$\displaystyle \begin{aligned} (1-\epsilon) \langle L_0 u, u \rangle + (C- \frac{2}{\epsilon}) \| u \|{}^2 \leq \langle L_q u, u \rangle + C \| u \|{}^2 \leq (1+\epsilon) \langle L_0 u, u \rangle + (C+ \frac{2}{\epsilon}) \| u \|{}^2 .\end{aligned}$$

One might need to adjust $C$ to satisfy $C > \frac {2}{\epsilon }.$ Hence the delta function is a bounded functional on the domain of the Schrödinger’s quadratic form. The norm of this functional is calculated as above just changing the upper index from 0 to $\mathrm {st}$:

$$\displaystyle \begin{aligned} {} \left\vert \langle \delta_x, \varphi \rangle \right\vert \leq \left( \sum_n \frac{\vert \psi_n^{\mathrm{st}} (x) \vert^2}{\lambda_n^{\mathrm{st}} + C} \right)^{1/2} \left( \sum_n ( \lambda_n^{\mathrm{st}} + C) \vert \varphi_n \vert^2 \right)^{1/2}. \end{aligned} $$

(17.30)

It follows that the norm is given by (17.27) and the series is absolutely convergent.

We show now that the series $\displaystyle \sum _{n=1}^\infty \frac { \overline { \psi _n^{\mathrm {st}} (y) }}{\lambda _n + C } \psi _n^{\mathrm {st}} (\cdot )$ converges to $r_\lambda (\cdot , y)$ in the norm given by the quadratic form of $L_q$

Equivalence of the Laplace’s and Schrödinger’s quadratic forms means that the series converges to $r_\lambda ( \cdot , y)$ in $W_2^1$-norm implying that the convergence is pointwise.

In particular we have that the positive series $\displaystyle \sum _{n=1}^\infty \frac { | \psi _n^{\mathrm {st}} (x)|{ }^2 }{\lambda _n + C }$ converges pointwise to the function $r_\lambda (x,x)$, which is continuous everywhere on $\Gamma$ outside an arbitrarily small neighbourhood of the internal vertices, where no continuity condition is assumed. Dini’s theorem implies then that the convergence is uniform, including the contact set $\partial \Gamma$, where we have standard conditions.

In particular we have

$$\displaystyle \begin{aligned} r_\lambda (V^i, V^j) = \sum_{n=1}^\infty \frac{\psi_n^{\mathrm{st}} (V^i) \overline{\psi_n^{\mathrm{st}} (V^j)}}{\lambda_n^{\mathrm{st}} - \lambda}, \quad V^i, V^j \in \partial \Gamma,\end{aligned}$$

where we used that the values $\psi _n^{\mathrm {st}} (V^j)$ are well-defined due to standard vertex conditions on $\partial \Gamma$. It remains to take into account formula (17.23). For details see [345]. □

Implications of this explicit formula will be discussed in the following section. Our goal right now will be to obtain a similar formula using the eigenfunctions of the self-adjoint operator $L^{\mathrm {D}} = L_{q,a}^{{\mathbf {S}}^{\mathrm {int}},D}$ defined by Dirichlet conditions at all contact vertices. Formula (17.26) can be obtained using the theory of finite rank singular perturbations [23, 471, 473]. One may consider perturbations of the operator $L_{q,a}^{{\mathbf {S}}^{\mathrm {int}}, \mathrm {st}}$ by the delta-distributions $\delta _{V^j}$ with the support at the contact vertices

$$\displaystyle \begin{aligned} \langle \delta_{V^j}, \varphi \rangle := \varphi (V^j). \end{aligned} $$

(17.31)

Formula (17.30) implies that $\delta _{V^j}$ is a bounded linear functional on the domain of the quadratic form of $L_{q,a}^{{\mathbf {S}}^{\mathrm {int}}, \mathrm {st}}$. The perturbed operator is formally given by

$$\displaystyle \begin{aligned} {} L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{st}} + \sum_{j=1}^{M_\partial} \alpha_j \delta_{V^j} = L_{q,a}^{{\mathbf{S}}^{\mathrm{int}}, \mathrm{st}} + \sum_{j=1}^{M_\partial} \alpha_j \langle \delta_{V^j} \cdot \rangle \;\delta_{V^j}, \end{aligned} $$

(17.32)

where $\alpha _j \in \mathbb R$ are certain coupling parameters, and we use that for any continuous function $\varphi$ it holds

$$\displaystyle \begin{aligned} \delta_{V^j} (x) \varphi (x) = \varphi (V^j) \delta_{V^j} = \langle \delta_{V^j}, \varphi \rangle \; \delta_{V^j}.\end{aligned}$$

Such perturbations are called form-bounded [23, 442] and can be uniquely determined in terms of the quadratic forms: the perturbed operator is given by the same differential expression, the same vertex conditions at internal vertices, but by delta vertex conditions on $\partial \Gamma .$ The central role is played by the Krein’s Q-function, which appears in the formula describing the resolvent of the perturbed operator and therefore encodes the spectral properties. This matrix-valued Herglotz-Nevanlinna function (see Sect. 18.1 below) is given by the bordered resolvent

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle \Big({\mathbf{Q}} (\lambda) \Big)_{ji} &: = & \displaystyle \langle \delta_{V^j}, \left(L^{\mathrm{st}} - \lambda \right)^{-1} \delta_{V^i} \rangle \\[3mm] & = & \displaystyle \left\langle \delta_{V^j}, \sum_{n=1}^\infty \frac{1}{\lambda_n^{\mathrm{st}} - \lambda} \langle \psi_n^{\mathrm{st}}, \delta_{V^i} \rangle \psi_n^{\mathrm{st}} \right\rangle \\[3mm] & = & \displaystyle \sum_{n=1}^\infty \frac{\psi_n^{\mathrm{st}} (V^j) \overline{\psi_n^{\mathrm{st}} (V^i)}}{\lambda_n^{\mathrm{st}} - \lambda} = r_\lambda (V^j, V^i) . \end{array} \end{aligned} $$

(17.33)

We have ${\mathbf {Q}}(\lambda ) = - {\mathbf {M}}^{-1} (\lambda )$ as matrices.

Let us turn now to the perturbations of the Dirichlet operator $L^{\mathrm {D}} =L_{q,a}^{{\mathbf {S}}^{\mathrm {int}}, \mathrm {D}}$. The corresponding eigenvalues and eigenfunctions will be denoted by $\lambda _n^D$ and $\psi _n^D,$${n= 1,2, \dots }$ To perturb operators with Dirichlet conditions the delta-distributions with the support at the contact vertices cannot be used, since they vanish on the functions from the domain of the operators. One has to use more singular distributions like the derivative of the delta-function. Consider for example the distributions $\partial \delta _{V^j}$

$$\displaystyle \begin{aligned} \langle \partial \delta_{V^j} , \varphi \rangle := \sum_{x_i \in V^j} \partial \varphi (x_i) , \; \; j = 1,2, \dots, M_\partial.\end{aligned}$$

These distributions are well-defined on the functions that are continuously differentiable on the edges, in particular on the domain of the Dirichlet operator. But these distributions are not bounded with respect to the quadratic form of the operator, in other words, these distributions are not defined on all functions from the domain of the quadratic form. Such distributions are called form-unbounded [23, 442]. Roughly speaking formal expression generalising (17.32) does not determine the perturbed operator uniquely (even using quadratic form technique)

$$\displaystyle \begin{aligned} {} L^{\mathrm{D}} + \sum_{j=1}^{M_\partial}\alpha_j \langle \partial \delta_{V^j}, \cdot \rangle \partial \delta_{V^j}. \end{aligned} $$

(17.34)

To understand the reason, why such perturbations are not uniquely defined by the formal expression (17.34), let us examine the corresponding bordered resolvent:

$$\displaystyle \begin{aligned} \displaystyle \langle \partial \delta_{V^j}, \frac{1}{L^{\mathrm{D}} - \lambda} \partial \delta_{V^i} \rangle. \end{aligned} $$

(17.35)

The scalar product in this formula cannot be understood even in the sense of distributions, since the element $\left (L^{\mathrm {D}} - \lambda \right )^{-1} \partial \delta _{V^i}$ belongs to the Hilbert space, but not to the domain of the operator. To go around this difficulty one considers the difference between the values of this function at two different points say $\lambda$ and $\lambda '$

$$\displaystyle \begin{aligned} \langle \partial \delta_{V^j}, \frac{1}{L^{\mathrm{D}} - \lambda} \partial \delta_{V^i} \rangle - \langle \partial \delta_{V^j}, \frac{1}{L^{\mathrm{D}} - \lambda'} \partial \delta_{V^i} \rangle = \displaystyle \langle \partial \delta_{V^j}, \frac{\lambda-\lambda'}{\left(L^{\mathrm{D}} - \lambda \right) \left(L^{\mathrm{D}} - \lambda' \right)} \partial \delta_{V^i} \rangle. \end{aligned} $$

(17.36)

The expression on the right hand side is well-defined since

In other words, one needs to regularise the integral determining the Q-function in this case.

Without going further into the theory of singular interactions we formulate the second formula for the M-function, interested readers may consult Chapter 3 of [23] or [345]:

Theorem 17.7

Let us denote by $\lambda ^D_n$ and $\psi ^D_n$ the eigenvalues and ortho-normalised eigenfunctions of the Schrödinger operator $L^{\mathrm {D}} = L_{q,a}^{{\mathbf {S}}^{\mathrm {int}},\mathrm {D}} (\Gamma )$ on a compact finite metric graph $\Gamma$ with the contact set $\partial \Gamma .$ Dirichlet conditions at the contact vertices and arbitrary Hermitian conditions at the internal vertices are assumed, then the M-function for $\Gamma$ satisfies the identity

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) - {\mathbf{M}}_\Gamma (\lambda') = \sum_{n=1}^\infty \frac{\lambda - \lambda'}{(\lambda_n^D - \lambda) (\lambda_n^D - \lambda')} \langle \partial \psi_n^D \vert_{\partial \Gamma}, \cdot \rangle_{\ell_2 (\partial \Gamma)} \partial \psi_n^D \vert_{\partial \Gamma} . \end{aligned} $$

(17.37)

We see that this formula does not allow one to calculate the M-function directly, but just the difference of its values at two regular points. One may say that Dirichlet spectral data allow one to determine M-function up to a constant matrix. The two preceding theorems can be combined together to get the following explicit formula:

(17.38)

We finish this section by providing a couple of clarifying examples.

Example 17.8

M-function for the interval $I = [0,1]$ with Neumann condition at $x=1$ and the boundary point $x=0$.

To illustrate our methods we calculate the M-function ${\mathbf {M}}_I (\lambda )$ using formula (17.26). The spectrum and the eigenfunctions of the Neumann Laplacian are

$$\displaystyle \begin{aligned} \lambda_n^{\mathrm{st}} = (\pi (n-1))^2, n = 1,2, \dots;\end{aligned}$$

$$\displaystyle \begin{aligned} \psi_1^{\mathrm{st}} (x) =1, \; \; \psi_n^{\mathrm{st}} (x) = \sqrt{2} \cos \pi (n-1) x, \quad n =2, 3 \dots.\end{aligned}$$

Substitution into (17.26) gives

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle {\mathbf{M}}_I (\lambda) & = & \displaystyle - \left( - \frac{1}{\lambda} + \sum_{n=2}^\infty \frac{2}{(\pi (n-1))^2 - \lambda} \right)^{-1}\\[3mm] & = & \displaystyle - \left( - \frac{1}{\lambda} - \frac{2}{\pi^2} \sum_{n=2}^\infty \frac{1}{((k/\pi)^2 - n^2} \right)^{-1}\\[3mm] & = & \displaystyle - \left( - \frac{1}{\lambda} - \frac{2}{\pi^2} \left( \cot k - \frac{1}{k} \right) \frac{\pi^2}{2k} \right)^{-1}\\[3mm] & = & \displaystyle k \tan k, \end{array} \end{aligned} $$

(17.39)

where we used formula 1.421.3 from [245]:

$$\displaystyle \begin{aligned} \cot \pi x = \frac{1}{\pi x} + \frac{2x}{\pi} \sum_{n=1}^\infty \frac{1}{x^2 - n^2}.\end{aligned}$$

The result, of course, coincides with formula (17.18).

In particular, we observe that

$$\displaystyle \begin{aligned} {\mathbf{M}}_I (0) = 0.\end{aligned}$$

Hence formula (17.38) takes the form

$$\displaystyle \begin{aligned} {\mathbf{M}}_I (\lambda) = 0 + \sum_{n=1}^\infty \frac{\lambda }{ \lambda_n^D (\lambda_n^D - \lambda)} \langle \partial \psi_n^D \vert_{\partial \Gamma}, \cdot \rangle_{\ell_2 (\partial \Gamma)} \partial \psi_n^D \vert_{\partial \Gamma}.\end{aligned}$$

It remains to substitute the eigenvalues and eigenfunctions of the Dirichlet-Neumann problem:

$$\displaystyle \begin{aligned} \lambda_n^D = \left(\frac{\pi}{2} (2n-1) \right)^2, \quad n = 1,2, \dots;\end{aligned}$$

$$\displaystyle \begin{aligned} \psi_n^{\mathrm{st}} (x) = \sqrt{2} \sin \frac{\pi}{2} (2n-1) x, \quad n = 1, 2 \dots.\end{aligned}$$

We use formula 1.421.1 from [245]

$$\displaystyle \begin{aligned} \tan \frac{\pi}{2} x = \frac{4x}{\pi} \sum_{n=1}^\infty \frac{1}{(2n-1)^2 - x^2}\end{aligned}$$

to get

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle {\mathbf{M}}_I (\lambda) & = & \displaystyle \sum_{n=1}^\infty \frac{\lambda}{ {\left(\frac{\pi}{2} (2n-1) \right)^2} \left( \left( \frac{\pi}{2} (2n-1) \right)^2 - \lambda\right)} 2 { \left(\frac{\pi}{2} (2n-1) \right)^2} \\[3mm] & = & \displaystyle \frac{8\lambda}{\pi^2} \sum_{n=1}^\infty \frac{1}{(2n-1)^2 - (2k/\pi)^2 }\\[3mm] & = & \displaystyle k \tan k, \end{array} \end{aligned} $$

(17.40)

as one should expect.

17.3 Hierarchy of M-Functions for Standard Vertex Conditions

The easiest way to calculate M-functions for graphs is to use the M-functions associated with the single edges. This procedure looks especially simple if standard vertex conditions are assumed at all vertices—only these vertex conditions will be considered in the current section. The functions from the domain of the operator are continuous at the vertices and it is possible to introduce function values at the vertices $\psi (V^m)$ building the M-dimensional vector $\vec {\psi }_{\mathbf {v}} = \{ \psi (V^j) \}_{j=1}^M$.

The first ${M_\partial }$ components of this vector coincide with the vector $\vec {\psi }^{\partial } .$ Similarly the entries of the vector $\partial \vec {\psi }_{\mathbf {v}}$ are equal to the sums of normal derivatives at the vertices. Its first ${M_\partial }$ components coincide with the vector $\partial \vec {\psi }^\partial .$ Denoting by $\vec {\psi }^{\mathrm {int}}$ and $\partial \vec {\psi }^{\mathrm {int}}$ the limiting values at internal vertices we have

$$\displaystyle \begin{aligned} \vec{\psi}_{\mathbf{v}} = \left( \begin{array}{c} \vec{\psi}^\partial \\ \vec{\psi}^{\mathrm{int}} \end{array} \right), \; \; \partial \vec{\psi}_{\mathbf{v}} = \left( \begin{array}{c} \partial \vec{\psi}^\partial \\ \partial \vec{\psi}^{\mathrm{int}_{\mathbf{v}}} \end{array} \right). \end{aligned} $$

(17.41)

This leads to the natural division of the $M$-dimensional space $\mathbb C^M \ni \vec {\psi }_{\mathbf {v}} , \partial \vec {\psi }_{\mathbf {v}}$ into the orthogonal sum of the $M_\partial$ and $M-M_\partial$-dimensional spaces:

The graph’s M-function ${\mathbf {M}}_\Gamma (\lambda )$ is defined as the matrix connecting the limiting values of any function $\psi (\lambda , x)$ solving Eq. (17.2) on the edges and satisfying standard conditions at the internal vertices

$$\displaystyle \begin{aligned} {\mathbf{M}}_\Gamma (\lambda) \; \vec{\psi}^\partial = \partial \vec{\psi}^\partial. \end{aligned} $$

(17.42)

On the other hand, the matrix function ${\mathbf {M}}^{\mathrm {st}} (\lambda )$ introduced in Sect. 5.3.4 describes the relation between the boundary values at all vertices. Denoting by ${\mathbf {M}}^{\mathrm {st}}_{ij} (\lambda ) , i,j = 1,2$ the block components of ${\mathbf {M}}^{\mathrm {st}}$ in the decomposition $\mathbb C^M = \mathbb C^{M_\partial } \oplus \mathbb C^{M-M_\partial }$ we write this relation as

$$\displaystyle \begin{aligned} {} \left\{ \begin{array}{ccl} \displaystyle {\mathbf{M}}^{\mathrm{st}}_{11} (\lambda) \vec{\psi}^\partial + {\mathbf{M}}^{\mathrm{st}}_{12}(\lambda)\vec{\psi}^{\mathrm{int}}& = & \partial \vec{\psi}^\partial \\ \displaystyle {\mathbf{M}}^{\mathrm{st}}_{21}(\lambda) \vec{\psi}^\partial + {\mathbf{M}}^{\mathrm{st}}_{22} (\lambda)\vec{\psi}^{\mathrm{int}}& = & \partial \vec{\psi}^{\mathrm{int}} \end{array} \right. . \end{aligned} $$

(17.43)

Taking into account that standard conditions at the internal vertices imply that

$$\displaystyle \begin{aligned} \partial \vec{\psi}^{\mathrm{int}}= 0\end{aligned}$$

the second equation in (17.43) gives us

$$\displaystyle \begin{aligned} {\mathbf{M}}^{\mathrm{st}}_{21} (\lambda) \vec{\psi}^\partial+ {\mathbf{M}}^{\mathrm{st}}_{22} (\lambda) \vec{\psi}^{\mathrm{int}}= 0.\end{aligned}$$

The block ${\mathbf {M}}^{\mathrm {st}}_{22} (\lambda )$ is invertible for $\mbox{Im} \, \lambda \neq 0$ since otherwise it has a non-trivial kernel implying that the self-adjoint Schrödinger operator on the same metric graph with Dirichlet conditions at $V^1, V^2, \dots , V^{M_\partial }$ and standard conditions at $V^{M_\partial +1}, \dots , V^M$ has a non-real eigenvalue.

Invertibility of $M^{\mathrm {st}}_{22} (\lambda )$ means that the vector $\vec {\psi }^{\mathrm {int}}$ is determined by $\vec {\psi }^\partial$

$$\displaystyle \begin{aligned} \vec{\psi}^{\mathrm{int}} = - \left( {\mathbf{M}}^{\mathrm{st}}_{22} (\lambda) \right)^{-1} {\mathbf{M}}^{\mathrm{st}}_{21} (\lambda) \vec{\psi}^\partial\end{aligned}$$

leading to the explicit Frobenius-Schur formula for the graph’s M-function:

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) = {\mathbf{M}}^{\mathrm{st}}_{11} (\lambda) - {\mathbf{M}}^{\mathrm{st}}_{12} (\lambda) \left( {\mathbf{M}}^{\mathrm{st}}_{22} (\lambda) \right)^{-1} {\mathbf{M}}^{\mathrm{st}}_{21} (\lambda) . \end{aligned} $$

(17.44)

This formula will be useful in the abstract analysis of inverse problems for graphs, but it is often not very practical when one is interested in calculating the spectrum or the graph M-function for a particular graph.

In the proof of (17.44) we did not use that the original M-function ${\mathbf {M}}^{\mathrm {st}} (\lambda )$ is associated with all vertices in $\Gamma$. It is enough to know the M-function associated with a larger contact set $\partial '\Gamma$ containing all vertices from $\partial \Gamma$. We summarise this observation as:

Theorem 17.9

Let $\Gamma$ be a finite compact metric graph with the contact sets $\partial \Gamma$ and $\partial ' \Gamma$ satisfying

$$\displaystyle \begin{aligned} \partial \Gamma \subset \partial' \Gamma. \end{aligned} $$

(17.45)

Then the M-function ${\mathbf {M}}^{\prime }_\Gamma (\lambda )$ associated with the larger contact set $\partial ' \Gamma$ determines the M-function ${\mathbf {M}}_\Gamma (\lambda )$ associated with the smaller contact set $\partial \Gamma$

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) = {\mathbf{M}}^{\prime}_{11} (\lambda) - {\mathbf{M}}^{\prime}_{12} (\lambda) \left( {\mathbf{M}}^{\prime}_{22} (\lambda) \right)^{-1} {\mathbf{M}}^{\prime}_{21} (\lambda), \end{aligned} $$

(17.46)

where${\mathbf {M}}^{\prime }_{ij} (\lambda ), \; i,j =1,2,$are the blocks of${\mathbf {M}}'(\lambda )$in the decomposition$\mathbb C^{M'} = \mathbb C^M \oplus \mathbb C^{M'-M}$with$M' = | \partial '\Gamma |, \, M= |\partial \Gamma |$.

Example 17.10

Let us calculate the M-function for the compact lasso graph $\Gamma _{(2.2)}$ depicted in Fig. 17.8 assuming that the Schrödinger operator is determined by the standard vertex conditions.

Fig. 17.8

The lasso graph $\Gamma _{(2.2)}$

The graph is formed by the edges $[x_1, x_2]$ and $[x_3, x_4]$ joined together at the vertices $V^1 = {x_4}$ and $V^2 = {x_1, x_2, x_3}.$ The first edge forms the loop and the second edge is the outgrowth. Let us denote the corresponding edge M-functions by $M^{1,2} (\lambda )$, each being a $2 \times 2$ matrix function. To build up function ${\mathbf {M}}^{\mathrm {st}}$ we need to write the M-functions associated with the edges in the basis of the vertices:

$$\displaystyle \begin{aligned} {\mathbf{M}}^1 (\lambda) = \left( \begin{array}{cc} \displaystyle 0 & \displaystyle 0 \\ \displaystyle 0 & \displaystyle M^1_{11} + M^1_{22} + M^1_{12} + M^1_{21} \end{array} \right); \quad {\mathbf{M}}^2 (\lambda) = \left( \begin{array}{cc} \displaystyle M^2_{22} & \displaystyle M^2_{21} \\ \displaystyle M^2_{12} & \displaystyle M^2_{11} \end{array} \right).\end{aligned}$$

The scalar M-function associated with the loop is obtained from the M-function for the interval by summing up the entries. Writing it in the basis of the vertices we get the matrix function with all except one entries zero.

Then the $2 \times 2$ M-function ${\mathbf {M}}^{\mathrm {st}}$ is given by

$$\displaystyle \begin{aligned} {\mathbf{M}}^{\mathrm{st}} (\lambda) = {\mathbf{M}}^1 (\lambda) + {\mathbf{M}}^2 (\lambda) = \left( \begin{array}{cc} \displaystyle M^2_{22} & \displaystyle M^2_{21} \\ \displaystyle M^2_{12} & \displaystyle M^1_{11} + M^1_{22} + M^1_{12} + M^1_{21} + M^2_{11} \end{array} \right). \end{aligned} $$

(17.47)

Formula (17.44) gives graph’s M-function with the contact set $V^1$

$$\displaystyle \begin{aligned} {\mathbf{M}}_\Gamma (\lambda) = M^2_{22} - M^2_{21} \left( M^1_{11} + M^1_{22} + M^1_{12} + M^1_{21} + M^2_{11} \right)^{-1} M^2_{12} , \end{aligned} $$

(17.48)

which is a scalar Herglotz-Nevanlinna function. We see immediately that the M-function depends just on the sum $M^1_{11} + M^1_{22} + M^1_{12} + M^1_{21}$, not on the particular form of the entries in $M^1 (\lambda )$. It will be shown later that precisely this feature of lasso’s M-function makes it impossible to solve the inverse problem in the case of standard vertex conditions.

In the case of the Laplace operator (zero magnetic and electric potentials) we may use formulas (5.55) to get

$$\displaystyle \begin{aligned} {\mathbf{M}}^{\mathrm{st}} (\lambda) = \left( \begin{array}{cc} \displaystyle - k \cot k \ell_2 & \displaystyle \frac{k}{\sin k \ell_2} \\ \displaystyle \frac{k}{\sin k \ell_2} & \displaystyle - 2 k \cot k \ell_1 + \frac{2k}{\sin k \ell_1} - k \cot k \ell_2 \end{array} \right) \end{aligned} $$

(17.49)

and

$$\displaystyle \begin{aligned} {} {\mathbf{M}}_\Gamma (\lambda) = - k \cot k \ell_2 - \left( \frac{k}{\sin k \ell_2} \right)^2 \left( - 2 k \cot k \ell_1 + \frac{2k}{\sin k \ell_1} - k \cot k \ell_2 \right)^{-1}. \end{aligned} $$

(17.50)

Problem 76

Show that formulas (17.10) and (17.50) are identical, provided $\ell = \ell _1, s = \ell _2.$

Example 17.11

Calculation of the M-function for the Laplacian on the equilateral star graph.

Let $\mathbb S_d$ be the star graph formed by $d$ edges of length $\ell .$ Consider the Laplace operator on $\mathbb S_d$ defined on the functions satisfying most general vertex conditions (3.21) at the central vertex

$$\displaystyle \begin{aligned} i (S-I) \vec{u} = (S+I) \partial \vec{u}.\end{aligned}$$

To calculate the M-function we need to solve the eigenfunction equation

$$\displaystyle \begin{aligned} - \vec{u}'' (x) = \lambda \vec{u} (x), \, \, \lambda = k^2,\end{aligned}$$

subject to vertex conditions at the origin. Every solution to this differential equation can be written as

$$\displaystyle \begin{aligned} \vec{u} = e^{-ik x} \vec{b} + e^{ikx} \vec{a}.\end{aligned}$$

Taking into account the vertex conditions we arrive at (3.13):

$$\displaystyle \begin{aligned} \vec{a} = S_{\mathbf{v }}(k) \vec{b} = \frac{(k+1) S + (k-1)I}{(k-1) S + (k+1)I} \vec{b}.\end{aligned}$$

It follows that the boundary values of such solution are related via the following matrix

$$\displaystyle \begin{aligned} \begin{array}{l} \displaystyle \vec{u}^\partial = e^{-ikl} \vec{b} + e^{ikl} \vec{a} = \left( e^{-ikl} + e^{ikl} S_{\mathbf{v}}(k) \right) \vec{b}, \\ \displaystyle \partial \vec{u}^\partial = ik e^{-ikl} \vec{b} -ik e^{ikl} \vec{a} = ik \left( e^{-ikl} - e^{ikl} S_{\mathbf{v}}(k) \right) \vec{b}, \end{array}\end{aligned}$$

$$\displaystyle \begin{aligned} {} \Rightarrow {\mathbf{M}} (\lambda) = ik \frac{e^{-ik\ell} - e^{ik\ell} S_{\mathbf{v}} (k) }{e^{-ik\ell} + e^{ik\ell} S_{\mathbf{v}} (k)}. \end{aligned} $$

(17.51)

Problem 77

Consider the case of the star graph with standard vertex conditions. Simplify formulas (17.51) and explain the mechanism behind.

Problem 78

Consider the case of the equilateral star graph with standard vertex conditions. Simplify formulas (17.26) and (17.37) and calculate the corresponding M-function.

Problem 79

Describe the relation between the graph and the edge M-functions in the case of arbitrary Hermitian vertex conditions at inner vertices.